Dual sourcing : with arbitrary stochastic demand and stochastic lead times

Schimpel, Ulrich

2010

KIT Scientific Publishing

ompanies with high-performing supply chains enjoy essential competitive ad- vantages. However, supply chain management faces an environment of rising risk that endangers these competitive advantages. One of the reasons is to outsource parts of their business. This bears the risk of significantly increased lead times and lead time variability. It is the impact of lead time variability on inventory management that is the central aspect of this book. It describes a mathematical model for dual sourcing with two reorder points, shows the deviation between stochastic and deterministic calculations in a sensitivity analysis, and investigates different relaxations of a traditional dual-sourcing policy.

Business .

Management

English

978-3-86644-528-4 (print)

Ulrich Schimpel
Dual sourcing
with arbitrary stochastic demand and stochastic lead times

Dual sourcing
with arbitrary stochastic demand and stochastic lead times
by
Ulrich Schimpel

Dissertation, Karlsruher Institut für Technologie
Fakultät für Wirtschaftswissenschaften,
Tag der mündlichen Prüfung: 27. April 2010
Referent: Prof. Dr. Christof Weinhardt
Korreferent: Prof. Dr.-Ing. Kai Furmans

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KIT Scientific Publishing 2010
Print on Demand
ISBN 978-3-86644-528-4

Acknowledgements
My great thanks go to all the people that have supported me directly or indirectly
throughout the last years. In particular, I like to mention and thank:
Christof Weinhardt, who gave me the opportunity to join his great team, continuously supported my thesis, and opened my horizon to the strengths of decentralized
mechanisms. His words proved to be truly correct that the most critical factor in the
whole process of obtaining a Dr. degree is to succeed in making a dot at the end.
Richard Bödi, my mentor, colleague, and good friend. His expertise in mathematics, practical inventory optimization, and the combination of both is admirable.
Our discussions on the academic and practical challenges of supply chain management and inventory optimization were usually long and very insightful. We had good
laughs and great working together not only in the customer projects that were a permanent and substantial reference throughout the whole process of writing.
Kai Furmans and his team for the detailed feedback on my thesis and the very
good discussions on the challenges of operational supply chain management.
The colleagues at IBM: Jean-Philippe Pellet, Eric Cope, Ulf Nielsen, Gianluca Antonini, André Elisseeff, Abderrahim Labbi, Eleni Pratsini, Gautier Stauffer, Peter Haas,
Annie Chen, Dieter Sommer, Samuel Müller, and Michael Wahler for many fruitful
and humorous discussions not only on the topics of Statistics, Operations Research,
and Algorithms. Peter Korevaar for many discussions on customer projects. I have
learnt a lot from his very pragmatic and effective way of solving problems.

The colleagues at the university of Karlsruhe: My numerous advisors – apparently,
I stayed pretty long, or (ab-) used them above the normal level, or both – Clemens van
Dinther, Henner Gimpel, Stefan Seifert, and Dirk Neumann. They have spent considerable time in giving directions and valuable advice way beyond clarifying, rephrasing, and restructuring many of my thoughts. The whole team at IM and FZI, especially Carsten Block, Stefan Luckner, Marc Adams, and Ilka Weber for the discussions
about common challenges that occur when writing a Ph.D. thesis, the good exchange
of hints regarding technical problems and solutions, and a great time not only limited
to Karlsruhe. Susanne Heidenreich for the support in many administrative tasks.
Last but definitely not least, my family and friends: Karin for her naturalness,
openness, great understanding, and many smiles even in rough times which took their
small and large concessions. My parents and parents-in-law for their unperturbed
support and the great freedom for all my activities and ideas. Susanne, Jutta, and all
my close friends for the great time together and their faithfulness – no matter what
happens.

Contents
I

Introduction and literature

1

1

Introduction

3

1.1

Research questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2

Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2

Literature review
2.1

2.2

11

Replenishment with deterministic lead times . . . . . . . . . . . . . . . . 13
2.1.1

Continuously and daily review policies . . . . . . . . . . . . . . . 14

2.1.2

Periodic review policies . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.3

Optimality of base-stock policies . . . . . . . . . . . . . . . . . . . 21

Replenishment with stochastic lead times . . . . . . . . . . . . . . . . . . 22
2.2.1

Order expediting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.2

Order splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3

Related replenishment policies . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

II

Modeling and practical application

37

3

A model for stochastic dual sourcing (SDMR)

39

3.1

Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
I

II

C ONTENTS

3.2

3.3

3.4

3.5

3.6
4

3.1.1

Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1.2

Time window for triggering second order . . . . . . . . . . . . . . 45

3.1.3

Categorization of replenishment cycles . . . . . . . . . . . . . . . 47

3.1.4

Notation conventions . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.1.5

Definition of an appropriate probability space . . . . . . . . . . . 53

3.1.6

Measure of applicability . . . . . . . . . . . . . . . . . . . . . . . . 64

Probabilities of the reorder cycle scenarios . . . . . . . . . . . . . . . . . . 68
3.2.1

General probabilities of one-order and two-order cycles . . . . . 68

3.2.2

Detailed probabilities of two-order cycles . . . . . . . . . . . . . . 70

Expected shortage of the reorder cycle scenarios . . . . . . . . . . . . . . 71
3.3.1

One-order cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3.2

Two-order cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Expected average cycle stock . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.4.1

One-order cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.4.2

Two-order cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Calculation of derived model parameters . . . . . . . . . . . . . . . . . . 116
3.5.1

Order quantities and cycle demand . . . . . . . . . . . . . . . . . 117

3.5.2

Service levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.5.3

Cycle time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.5.4

Average stock per time unit . . . . . . . . . . . . . . . . . . . . . . 120

3.5.5

Number of replenishment cycles and orders . . . . . . . . . . . . 120

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Considerations for a practical application of the SDMR model
4.1

127

Discretization of the formal model . . . . . . . . . . . . . . . . . . . . . . 128
4.1.1

Discretization of continuous distributions . . . . . . . . . . . . . . 130

4.1.2

Bounds of integration and summation . . . . . . . . . . . . . . . . 131

4.1.3

Start of the replenishment cycle . . . . . . . . . . . . . . . . . . . . 132

C ONTENTS

4.2

4.3

III
5

6

III

4.1.4

Time and demand discretization . . . . . . . . . . . . . . . . . . . 134

4.1.5

Time window for second order . . . . . . . . . . . . . . . . . . . . 135

4.1.6

Compliance with model assumptions . . . . . . . . . . . . . . . . 137

Determining convolution of distributions . . . . . . . . . . . . . . . . . . 139
4.2.1

Analytical determination . . . . . . . . . . . . . . . . . . . . . . . 140

4.2.2

Limits of an analytical determination . . . . . . . . . . . . . . . . 141

4.2.3

Numerical determination . . . . . . . . . . . . . . . . . . . . . . . 142

4.2.4

Algorithm for a numerical determination . . . . . . . . . . . . . . 144

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Evaluation, results, and conclusions
Overview of the evaluation approach

149
151

5.1

Comparison of replenishment policies . . . . . . . . . . . . . . . . . . . . 152

5.2

Employed cost structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5.3

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Deterministic versus stochastic dual sourcing
6.1

6.2

161

Measures of divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.1.1

Preliminary observations . . . . . . . . . . . . . . . . . . . . . . . 164

6.1.2

Ratio of costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.1.3

Ratio of service levels . . . . . . . . . . . . . . . . . . . . . . . . . 169

Sensitivity analysis on replenishment-relevant parameters . . . . . . . . 171
6.2.1

Daily demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.2.2

Lead time of the first supply mode . . . . . . . . . . . . . . . . . . 174

6.2.3

Lead time of the second supply mode . . . . . . . . . . . . . . . . 180

6.2.4

Order quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

6.2.5

Reorder points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

IV

C ONTENTS
6.2.6
6.3

6.4
7

Case study – a spare parts warehouse . . . . . . . . . . . . . . . . . . . . 198
6.3.1

Setup for comparing the case study and the sensitivity analysis . 200

6.3.2

Capital costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

6.3.3

Normal order costs . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

6.3.4

Rush order costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

6.3.5

Back order costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

6.3.6

Service level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

6.3.7

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

Comparison of stochastic replenishment policies

233

7.1

Overview of stochastic replenishment scenarios . . . . . . . . . . . . . . 234

7.2

Total savings of different replenishment scenarios . . . . . . . . . . . . . 235

7.3

Savings induced by relaxed replenishment restrictions . . . . . . . . . . 237

7.4

7.5
8

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

7.3.1

Negative reorder point . . . . . . . . . . . . . . . . . . . . . . . . . 238

7.3.2

Using emergency orders as normal orders . . . . . . . . . . . . . 241

7.3.3

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

Savings induced by dual sourcing . . . . . . . . . . . . . . . . . . . . . . 246
7.4.1

Small difference between the lead times . . . . . . . . . . . . . . . 247

7.4.2

Increasing difference between the lead times . . . . . . . . . . . . 249

7.4.3

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

Summary and outlook

257

8.1

Contributions to the research area . . . . . . . . . . . . . . . . . . . . . . 257

8.2

Critical review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

8.3

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

C ONTENTS

IV

Appendices

A Explanations regarding the used probability space

V

267
269

A.1 Example of a discrete random variable and its σ-algebra . . . . . . . . . 269
A.2 Expected value for a continuous random variable . . . . . . . . . . . . . 271
A.3 Sum of expected values using indicator functions . . . . . . . . . . . . . 272
A.4 Expected value using an indicator function and adding a scalar . . . . . 275
A.5 Expected value using an indicator function and multiplying a scalar . . 275
A.6 Conditional expectations for multiple random variables . . . . . . . . . . 276
A.7 Time window for triggering a second order . . . . . . . . . . . . . . . . . 283
A.8 Conditional expected time until stock depletion . . . . . . . . . . . . . . 284
B Shortage formulas

287

C Stock formulas

291

C.1 One-order cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
C.2 Two-order cycles where the first order arrives first . . . . . . . . . . . . . 293
C.3 Two-order cycles where the second order arrives first . . . . . . . . . . . 296
C.4 Two-order cycles where both orders arrive simultaneously . . . . . . . . 300
C.4.1 Case 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
C.4.2 Case 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
D Formulas for dual sourcing with lost sales

305

D.1 Adjusted probability space for the reorder point scenarios . . . . . . . . 306
D.2 Definition of additional random variables and event sets . . . . . . . . . 308
D.3 Probabilities of the reorder cycle scenarios . . . . . . . . . . . . . . . . . . 311
D.3.1 All reorder points are non-negative . . . . . . . . . . . . . . . . . 312
D.3.2 At least one reorder point is negative . . . . . . . . . . . . . . . . 313
D.4 Expected number of direct shipments . . . . . . . . . . . . . . . . . . . . 314

VI

C ONTENTS
D.4.1 All reorder points are non-negative . . . . . . . . . . . . . . . . . 315
D.4.2 At least one reorder point is negative . . . . . . . . . . . . . . . . 319
D.5 Expected average cycle stock . . . . . . . . . . . . . . . . . . . . . . . . . 319
D.5.1 Both reorder points are non-negative . . . . . . . . . . . . . . . . 320
D.5.2 At least one reorder point is negative . . . . . . . . . . . . . . . . 326

Bibliography

329

List of Figures
1.1

Structure of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1

Exemplified cycle of a (R1 , R2 , Q1 , Q2 ) replenishment policy with stochas-

8

tic lead times and demand . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2

Time line with events during a two-order replenishment cycle . . . . . . 47

3.3

Different cases for cycles with one or two orders . . . . . . . . . . . . . . 51

3.4

Stochastic variables (red) in a one- and two-order replenishment cycle . 67

3.5

Relationship between the shortages before and after the first delivery . . 74

3.6

The two possible scenarios for a one-order cycle . . . . . . . . . . . . . . 76

3.7

Scenarios for a two-order cycle where the first order arrives . . . . . . . 86

3.8

Scenarios for a two-order cycle where the second order arrives first . . . 94

3.9

Scenarios for a two-order cycle with simultaneous arrivals . . . . . . . . 96

3.10 Exemplified scenarios of the stock on hand during a replenishment cycle 100
3.11 The two possible scenarios for a one-order cycle . . . . . . . . . . . . . . 102
3.12 The two cases for a two-order cycle where the first order arrives first . . 108
3.13 The two cases for a two-order cycle where the second order arrives first 112
3.14 The two scenarios for a two-order cycle with simultaneous arrivals . . . 114
4.1

Algorithm to compute the convolution of two distributions . . . . . . . . 145

5.1

Obtaining a deterministic model by reducing the histogram to one entry 153

5.2

Representing single-sourcing by using a very small reorder point R2 . . 154
VII

VIII

L IST O F F IGURES

5.3

Comparison of two scenarios . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.4

Employed cost structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.1

Cost ratios τ min , τ max , and τ̄ for a changing demand fluctuation σD . . . 173

6.2

Cost ratios τ min , τ max , and τ̄ for a changing average lead time µ L1 . . . . 176

6.3

Cost ratios τ min , τ max , and τ̄ for a changing lead time fluctuation σL1 . . 179

6.4

Cost ratios τ min , τ max , and τ̄ for a changing average lead time µ L2 . . . . 182

6.5

Cost ratios τ min , τ max , and τ̄ for a changing lead time fluctuation σL2 . . 185

6.6

Cost ratios τ min , τ max , and τ̄ for a changing order quantity Q1 . . . . . . 188

6.7

Cost ratios τ min , τ max , and τ̄ for a changing order quantity Q2 . . . . . . 190

6.8

Cost ratios τ min , τ max , and τ̄ for a changing reorder point R1 . . . . . . . 193

6.9

Cost ratios τ min , τ max , and τ̄ for a changing reorder point R2 . . . . . . . 195

6.10 Range of capital cost ratios τCC for the sensitivity analysis and the warehouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
6.11 Range of order cost ratios τOCN for the sensitivity analysis and the warehouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
6.12 Range of order cost ratios τOCR for the sensitivity analysis and the warehouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
6.13 Range of order cost ratios τOCB for the sensitivity analysis and the warehouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
6.14 Range of β service level ratios ρ for the sensitivity analysis and the
warehouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
7.1

Total warehousing costs for the 4 replenishment scenarios. . . . . . . . . 236

List of Tables
1

List and description of utilized parameters . . . . . . . . . . . . . . . . . XVI

2

List and description of used acronyms and abbreviations . . . . . . . . . XVIII

3.1

Permutations of possible events in a replenishment cycle . . . . . . . . . 49

3.2

Valid permutations of possible events in a replenishment cycle

3.3

Legend of the triple notation

3.4

Investigated cases in dual sourcing . . . . . . . . . . . . . . . . . . . . . . 53

3.5

Definition of stochastic variables in the probability space Ω

3.6

Conditions of a shortage in Case 1 to 8

3.7

Possible settings for choosing the reorder points R1 and R2 . . . . . . . . 75

3.8

Notation of the different cases of expected shortage in a two-order cycle

3.9

Summary of all functions to determine the expected shortage . . . . . . 99

. . . . . 50

. . . . . . . . . . . . . . . . . . . . . . . . . 52

. . . . . . . 56

. . . . . . . . . . . . . . . . . . . 72

3.10 Conditions of stock on hand in Case 1 to 8

87

. . . . . . . . . . . . . . . . . 124

3.11 Summary of all functions to determine the expected stock . . . . . . . . 125
3.12 Summary of major equations derived from the SDMR model . . . . . . . 126
4.1

Exemplified conditions for two-order cycles in a business scenario

5.1

Summary of used cost parameters . . . . . . . . . . . . . . . . . . . . . . 157

6.1

Exemplified cost structure of two SKUs . . . . . . . . . . . . . . . . . . . 163

6.2

Results for the deterministic and the stochastic scenario . . . . . . . . . . 165
IX

. . . 137

X

L IST O F TABLES
6.3

Initial input parameter setting

. . . . . . . . . . . . . . . . . . . . . . . . 171

6.4

Ratios τ and ρ for a changing demand fluctuation σD . . . . . . . . . . . 172

6.5

Ratios τ and ρ for a changing average lead time µ L1 . . . . . . . . . . . . 175

6.6

Ratios τ and ρ for a changing lead time fluctuation σL1

6.7

Ratios τ and ρ for a changing average lead time µ L2 . . . . . . . . . . . . 181

6.8

Ratios τ and ρ for a changing lead time fluctuation σL2

6.9

Ratios τ and ρ for a changing order quantity Q1 . . . . . . . . . . . . . . 186

. . . . . . . . . . 178
. . . . . . . . . . 183

6.10 Ratios τ and ρ for a changing order quantity Q2 . . . . . . . . . . . . . . 189
6.11 Ratios τ and ρ for a changing reorder point R1 . . . . . . . . . . . . . . . 192
6.12 Ratios τ and ρ for a changing reorder point R2 . . . . . . . . . . . . . . . 194
6.13 Excerpt of Table 6.4 on page 172

. . . . . . . . . . . . . . . . . . . . . . . 202

6.14 Deviation of the capital costs between the DET and the STOCH scenario 205
6.15 Statistics for the capital costs of the DET and the STOCH scenario . . . . 206
6.16 Selection of SKUs with highest deviation for OCR between DET and
STOCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
6.17 Statistics on the ratio of back order costs between DET and the STOCH

217

6.18 Partial results of the regression analysis on all input parameters for τOCB 221
6.19 SKUs with the 10 highest deviations for OCB between DET and STOCH 222
6.20 Statistics on the ratio of (1 − β) between STOCH and DET . . . . . . . . 227
7.1

Overview of replenishment scenarios . . . . . . . . . . . . . . . . . . . . 235

7.2

List of SKUs and the probability p(2, ., .) of triggering a second order in
RS2,relax

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

7.3

Contribution of the relaxed restriction R1 < 0 to the total savings . . . . 238

7.4

Wilcoxon Rank Sum Test on price and demand of SKUs where R1 < 0 . 240

7.5

Contribution of the relaxed restriction R1 < 0 to the total savings . . . . 242

7.6

Wilcoxon Rank Sum Test on price and demand of SKUs with exchanged
supply modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

L IST O F TABLES

XI

7.7

Contribution of the relaxed restrictions to the total savings . . . . . . . . 245

7.8

Total cost savings by the second supply channel with relaxed restrictions

7.9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

Comparison of single and dual sourcing for an increasing lead time µl1 . 251

7.10 Cost development for RS1,trad and RS2,trad for an increasing lead time
µ l1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

8.1

Findings regarding different stochastic replenishment policies . . . . . . 260

8.2

Contributions to the area of dual sourcing

. . . . . . . . . . . . . . . . . 261

D.1 Possibility of different events to happen for the three reorder point scenarios

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

D.2 Probabilities of direct shipments and one- and two-order cycles . . . . . 311
D.3 Conditions for direct shipments – expressions in blue differ from the
backlog policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

XII

L IST O F TABLES

List of parameters
parameter
R1
R2

description
reorder point for first supply mode can be positive or negative,
R1 > R2
reorder point for second supply mode can be positive or negative,
R1 > R2

Q1

quantity delivered by the first supply mode, Q1 > 0

Q2

quantity delivered by the second supply mode, Q2 > 0

L1

lead time distribution of the first supply mode

l1

density function of L1

lt1

length of the lead time for first supply mode in written text

y

length of the lead time for first supply mode in formulas

L2

lead time distribution of the second supply mode

l2

density function of L2

lt2

length of the lead time for second supply mode in written text

z

length of the lead time for the second supply mode in formulas

D (t)

distribution of cumulated non-negative demand within t periods

F ( x, t)

cumulative probability of a demand less or equal to x in t time units

f ( x, t)

density function of F
continued on next page
XIII

XIV

L IST O F PARAMETERS

parameter

description

f disc ( x, t)

density function for continuous demand x within discrete time t

x, x 0 , x 00

amount of demand in formulas

dt
dmax
λ

amount of demand during time t in written text
maximum demand during tc allowed by Assumption 8 on page 43
demand rate (sometimes conditional to a certain case)

G (t)

probability to reach the reorder point R2 within t time units

g(t)

density function of G

T, t

length of a time interval

t0

time when the order of the first supply mode is triggered

t A1

time when the order of the first supply mode arrives

t A2

time when the order of the second supply mode arrives

tc

cycle time, i.e. time between t0 and reaching R1 again after all
deliveries

tg

inter-order time, i.e. time between triggering both orders

tw

length of the time window to trigger a second order

t̄c

expected cycle time, t̄c = E[tc ]

R

the set of real numbers

N

the set of integer numbers

h

some function depending on the context

H

hypothesis used with subscripts for specification

(Ω, κ , P)

probability space, see definition starting on page 55

Ω

sample space

ω

element in the sample space, ω ∈ Ω

κ
P, p

σ-algebra on Ω

probability measures used with subscript or argument, see page 52
continued on next page

L IST O F PARAMETERS
parameter
pE
X, Y

XV
description

probability that demand exceeds dmax , violation of Assumption 8
some random variables

Xs

random variable in context of the stock level

M

set of elements, usually related to sample space Ω

∅

the empty set

1M

indicator function on some set M

E[ X ]

expected value of a random variable X

µD

average value of the distribution D

σD

standard deviation of the distribution D

Φ

the Gaussian distribution

φ

density function of Φ

η ( x, R)
i, j, k
N, n, m

function used in context of the expected time when the stock depletes
(integer) indices
some integer values

SH

shortage

CC

capital costs, see TC

OCB

order costs for unmet demand, see TC

OCN

order costs for first supplier, see TC

OCR

order costs for second supplier, see TC

TC

total cost including CC, OCN , OCR , and OCB

r

interest rate, used in context of CC

ρ

ratio of β service levels between DET and STOCH, ρ =

T

set of ratios τi , used with subscript for specification

τ

ratio of costs between DET and STOCH, τ =

1−STOCH β
1−DET β

DET value
STOCH value

continued on next page

XVI

L IST O F PARAMETERS

parameter
τ min
τ max
τ̄

description
minimum ratio in a given set T

maximum ratio in a given set T

average value of all ratios τi ∈ T

notation conventions, see page 52
dµ

used in context of multiple integrations

a

+

equivalent to max(0, a)

a

−

equivalent to min(0, a)

(2, A1 , =)
p

triple that refers to a certain case
probability measure used with subscript or argument
Table 1: List and description of utilized parameters

List of Acronyms and Abbreviations

XVII

XVIII

A CRONYMS & A BBREVIATIONS

expression

description
β service level to customers

β
CC

capital costs, see TC

CPFR

collaborative planning, forecasting, and replenishment

DET

scenario where demand is stochastic and all lead times are
deterministic, see STOCH

disc

discrete, usually used as a superscript of a function, hdisc
and refers to the discrete version of the continuous function h

DS

replenishment policy with direct shipments

ERP

enterprise resource planning

fix

fixed cost, see var

KPI

key performance indicator

OCB

order costs for unmet demand, see TC

OCN

order costs for first supplier, see TC

OCR

order costs for second supplier, see TC

relax

relaxed replenishment restrictions, see trad

RQ

research question

RS

replenishment scenario

SCM
SDMR model
SH
SKU
STOCH

supply chain management
stochastic dual-mode replenishment model
shortage
stock-keeping unit
scenario where demand and all lead times are stochastic, see DET

TC

total cost including CC, OCN , OCR , and OCB

trad

traditional replenishment restrictions, see relax

var

variable cost, see fix

VMI

vendor-managed inventory

WH

warehouse

Table 2: List and description of used acronyms and abbreviations

Part I
Introduction and literature

1

Chapter 1
Introduction
Companies with high-performing supply chains (SCs) enjoy essential competitive advantages. They are able to offer a more punctual and more complete service to the
customer at lower logistics costs than their competition in many cases, see [CRW09].
However, supply chain management (SCM) faces an environment of rising risk
that endangers these competitive advantages. In a study by McKinsey 33% of 273
supply chain executives state that their supply chain risk has increased significantly
over the past 5 years [PN08]. The reasons are a higher complexity of products and
services, rising energy prices, and the decision to outsource and offshore parts of their
business. The latter has a great impact on the risk of their inbound logistics and there
is a strong need to ”improve the effectiveness of the inbound supply chain, because
lead times and variability have increased significantly.” [Kla09, p. 5].
It is the impact of lead time variability on inventory management, a prominent
area of SCM, that will be the central aspect of this work. We investigate how a fast
and expensive second supplier can mitigate the effects of the stochastic lead time
of our slower and cheaper primary supplier. Moreover, we analyze the deviation
from a deterministic approximation in our scenario where the demand and both lead
times are stochastic. Thereby, we focus on the two most common key performance
3

4

C HAPTER 1. I NTRODUCTION

indicators (KPIs) used in SCM, namely the total costs and the customer service level
[Pay09]. This aligns well with the most important objectives of SCM in days of a critical economic situation, namely cost reduction and customer service improvement, see
[PN08], [Kla09], [PK09].

1.1

Research questions

Let us consider a company which has two suppliers with stochastic lead times and
which faces stochastic customer demand. For each supplier i ∈ {1, 2} an individ-

ual reorder point Ri and order quantity Qi should be determined in such a way that

minimal total costs occur. Such ( R1 , R2 , Q1 , Q2 ) replenishment policies are called dual
sourcing. In this context, the fundamental Research Question (RQ) that guides us
through all chapters is:
RQ 0: Is it beneficial to use dual sourcing in a practical situation where the demand
and both lead times are stochastic? Moreover, is it feasible to use a deterministic approximation in such a situation?
We will use the data of an existing warehouse with 2, 751 stock-keeping units (SKUs)
for the evaluation of our work.
Our present work extends the existing literature on SCM by a ( R1 , R2 , Q1 , Q2 ) replenishment policy, where two suppliers – or more generally two supply modes – with
stochastic lead times are available. A major part of this work is to elaborate a mathematical model for such a ( R1 , R2 , Q1 , Q2 ) policy that allows for arbitrary distributions
regarding the demand and the lead times of both suppliers. We will call it the SDMR
model which stands for Stochastic Dual-Mode Replenishment model. In particular,
we derive the following questions from RQ 0:

Introduction – Motivation and research questions

5

RQ 1: How can we define and model a ( R1 , R2 , Q1 , Q2 ) replenishment policy with
stochastic demand and lead times? Currently, there does not exist any model in
the literature that describes a ( R1 , R2 , Q1 , Q2 ) policy when the demand and both
lead times are arbitrary stochastic variables. Therefore, we first want to set up
an appropriate framework in which we can express the probabilities of issuing
orders with one or two suppliers and the probability of a particular sequence of
the order arrivals. Second, we want to be able to calculate KPIs such as the average stock level, the customer service level, the expected number of orders for
each supplier, and their related costs. This is the core of the SDMR model. One
important feature of the SDMR model is its independence of the type of distributions for the demand and both lead times so that it can be used in potentially
every industry or business.
RQ 2: How can the SDMR model be applied in practice? Here, we cover the questions regarding the discretization of the SDMR model and how to incorporate
practical rules like the time window in which it still makes sense to trigger a second order even if the first order is likely to arrive soon. It is important to answer
this question in order to keep up with our objective to provide a model which
can be used operationally.
RQ 3: How much and in which situations does a scenario with deterministic lead
times deviate from the stochastic scenario of the ( R1 , R2 , Q1 , Q2 ) replenishment policy? More simplistically, one could rephrase the question by: Why
should we use a complex stochastic model instead of an existing, simpler model
which assumes deterministic lead times? Here, we want to compare the calculated values for the given KPIs between the deterministic scenario (DET) and the
stochastic scenario (STOCH). Moreover, we want to investigate and understand
in more detail the different effects that lead to this deviation.

6

C HAPTER 1. I NTRODUCTION
Of course, one expects the deviation between DET and STOCH to depend on
several parameters of the SKUs like the fluctuation of both lead times or the
probability to use the second supplier. In fact, all four replenishment parameters
R1 , R2 , Q1 , and Q2 turn out to have an influence. After the formulation of the
SDMR model it will become apparent that it is much more difficult to answer
Question 3 than one might initially have expected.
In order to iteratively gain a better understanding of the impact that different
parameters have on the deviation between DET and STOCH we take a twostep approach. First, we conduct a sensitivity analysis on each input parameter
individually for an exemplary SKU. This allows us to study the gradual change
of a single input parameter and its effect on the deviation between DET and
STOCH. Second, we analyze the deviation in a real warehouse with 2, 751 SKUs
and compare these results with our findings from the sensitivity analysis.
This two-step approach will, on the one hand, provide us with greater insights
into the mechanisms and relations between the input parameters and the KPIs
and their impact on the deviation between DET and STOCH. On the other hand,
it will show the relevance of our findings for different SKUs and groups of SKUs
of a real warehouse.

RQ 4: How much total costs can be saved by moving from single sourcing with
traditional restrictions to dual sourcing with relaxed restrictions when lead
times are stochastic? There exist common restrictions regarding the replenishment of SKUs in practice as well as in literature. The two most prominent are
non-negative reorder points and a fixed sequence of orders if there is the option
of several suppliers. We want to see by how much the total costs can be reduced
in our real warehouse if we relax these restrictions. Therefore, we have to optimize the different scenarios regarding their total costs and compare their results.

Introduction – Structure

7

In addition, we want to understand the mechanisms that lead to these savings.
Here, we are only interested in comparing scenarios with stochastic lead times.
A large part of the evaluation is based on a warehouse with 2,751 SKUs, the only one
for which we possess real data, especially regarding both suppliers. This warehouse
contains automotive spare parts. Usually, spare parts have a low demand in common. This characteristic is rather unfortunate for dual sourcing as we will experience
throughout the evaluation chapter. Many times a second supplier does simply not pay
off. Nevertheless, we have decided to include all 2,751 SKUs in all our observations
for several reasons. First, they nicely reveal the limits of a dual sourcing. Second, the
low-demand spare parts do not influence the various effects of those SKUs where dual
sourcing is favorable. Third, despite the low demand of those SKUs their wide range
of prices yield interesting results especially related to the cost-optimization when we
compare the different stochastic scenarios in context of our question RQ 4.
Last but not least we have to be aware that each SKU and its particular parametrization throughout all our sensitivity analyses are only examples and equally important
from a phenomenological perspective. This is important to keep in mind as different
warehouses are likely to have a completely different assortment of SKUs. What is an
”exceptional” SKU in one warehouse might constitute the majority of SKUs in another
warehouse. Therefore, we are very much interested also in ”exceptional” SKUs and
will elaborate on them at many places throughout our evaluations.

1.2

Structure

Our work is organized in four parts with 8 chapters, see Figure 1.1. Part I contains a
review of the relevant literature in Chapter 2 which influences most of the subsequent
chapters as indicated by its graphical location. Each of the four research questions
will be addressed in a dedicated chapter of Part II or Part III. Finally, we conclude

8

C HAPTER 1. I NTRODUCTION

with a summary and outlook in Chapter 8 at the end of Part III. The appendix with
additional information is found in Part IV. Part II and Part III represent our major
work and will be described in more detail in the following paragraph.

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Figure 1.1: Structure of this work

Part II contains all chapters that address the SDMR model. Chapter 3 covers question
RQ 1. We derive a mathematical model for the ( R1 , R2 , Q1 , Q2 ) replenishment policy
with stochastic demand and stochastic lead times. The basis of the SDMR model consists of a suitable probability space and its random variables. On this basis we give a
set of formulas for calculating the expected shortage, the expected physical stock, and
other expected values that are necessary to calculate the total costs of our warehouse.
Chapter 4 covers question RQ 2 which is related to the practical application of the
SDMR model. The two main topics are the discretization of the model that we have
established in Chapter 3 and the calculation of convolutions which play a major role
in the SDMR model.

!

Introduction – Structure

9

Part III is dedicated to evaluating the SDMR model in different scenarios. We give
an overview of the evaluation approach in Chapter 5. Chapter 6 is devoted to the comparison between our model with stochastic lead times (STOCH) and an approximation
(DET) with deterministic lead times for both suppliers. This corresponds to question
RQ 3. First, we conduct a sensitivity analysis on individual input parameters. Second,
we quantify the deviation between DET and STOCH in case of our real warehouse.
Chapter 7 exclusively looks at scenarios with stochastic lead times. It compares the
savings potential if our warehouse operation moves from a single-source replenishment with traditional restrictions to a dual-source replenishment with relaxed restrictions. This covers question RQ 4. Chapter 8 completes our work. First, it summarizes
our findings and contributions in the area of inventory management. Second, we critically review our work. Finally, we give an outlook for future research and possible
extensions related to our work.

Chapter 2
Literature review
In this section we will position our work within the existing literature for SCM. Most
likely, the first contact with the SCM-related literature will be overwhelming as the
number and topics of publications is large. The structure of the paper ”Perspectives
in supply chain risk management” by Tang, see [Tan06], gives a good overview of
the different areas in SCM because risk is basically omnipresent in a supply chain.
He distinguishes four main SCM areas: supply management, demand management,
product management, and information management. Each of these areas contains
several activities and responsibilities which are often interlinked.
Supply management usually contains all aspects from design the SC network over
the supplier relation management to the daily replenishment of articles. Demand
management is widely used in SCM to align demand with existing supply especially
in times of excess inventory or stock-out situations. Product management usually
involves decisions about the push-pull boundary within the SC. In other words, the
question is where products are built on order and where they are built to be stocked in
a warehouse. Information management is often focused on a better customer service
by enabling faster reactions and on mitigating the effects of fluctuation like in the case
of the overstressed phenomenon called bullwhip effect. Here the integration across
11

12

C HAPTER 2. L ITERATURE R EVIEW

functional or organizational entities usually plays a major role. Well-known examples in this area are Vendor-Managed Inventories (VMI) and Collaborative Planning,
Forecasting, and Replenishment (CPFR).
Our work is located in the area of supply management. More precisely, it contributes to the topic of replenishment policies for a single-item warehouse with dual
sourcing, meaning that two suppliers or, more generally, two supply modes are available. The literature on multiple suppliers is still very ample which accounts for the
various situations and their complexity in which companies seek to leverage the benefits of several supply modes. Minner gives a very good review of that topic [Min03].
He partitions the literature of multiple sourcing into strategic aspects, single-echelon
models, multi-echelon models, and reverse logistics models. The majority of publications is in the area of single-echelon models which look at an isolated warehouse
and its suppliers. Our work describes such a single-echelon, single-item replenishment policy but with substantially weaker assumptions than in the existing literature,
especially regarding the distribution of the demand and both supplier lead times.
In contrast, multi-echelon policies take several inventories and their hierarchical
structure into account. It is apparent that the increased complexity of multi-echelon
situations usually leads to very restrictive assumptions, like Poisson customer demand or constant lead times, see [NLC01], [LP00]. We believe, that this is also the
main reason why publications on the topic of global sourcing focus mainly on strategic aspects. Many times this leaves a large gap between global decisions and local
implementations as we have experienced at many companies. However, the alignment of global and local objectives is repeatedly reported to be a key success factor for
SCM in practice, see [CRW09], [Kla09].

The company of our interest uses dual sourcing and can completely backlog unsatisfied customer demand in a single-echelon, single-item environment. The demand

Literature – Replenishment with deterministic lead times

13

and both lead times are arbitrary, known random variables. The company employs a

( R1 , R2 , Q1 , Q2 ) replenishment policy which triggers an order of size Q1 and Q2 if the
stock level drops to or below the reorder point R1 and R2 , respectively. Mainly due to
its complexity, dual sourcing has been studied in the context of random demand and
random lead times only in a few papers.
Our model somewhat follows the logic of the ( R1 , R2 , Q1 , Q2 ) replenishment policy with deterministic lead times of Moinzadeh and Nahmias [MN88]. If we relate
our work to the segmentation used by Minner, see [Min03], it is closer to the stream of
publications with deterministic lead times and emergency deliveries in cases of low
inventory than it is to models with stochastic lead times that leverage a shorter effective lead time – time until the first order arrives – by splitting orders among several
suppliers. Therefore, we first review policies with deterministic lead times in a singleechelon, multi-supplier context. Second, we look at policies with stochastic lead times
but in less detail. Third, we mention a few policies in other areas that are interesting
for our work.

2.1

Replenishment with deterministic lead times

In the literature most models for dual sourcing that assume deterministic lead times
and stochastic demand are already fairly complex. Optimal policies have strong restrictions like immediate deliveries or lead times that differ by exactly one day. Many
times fixed order costs or the arrival of emergency orders in previous periods are assumed to be negligible. This section will help positioning a model for dual sourcing
with stochastic demand and stochastic lead times.

14

C HAPTER 2. L ITERATURE R EVIEW

2.1.1

Continuously and daily review policies

The first studies on a replenishment policy with two suppliers date back to Barakin
[Bar61]. He determines a single-period policy with two critical stock levels s1 > s2
and negligible fixed ordering costs. If the initial stock x is below s1 , an order of Q1 ( x )
units is placed which arrives after one single period to cover potential backlogged
demand. In addition, an emergency order of a predetermined amount Q2 is immediately delivered if x < s2 . This policy has been extended to n > 1 periods by several
authors. Daniel allows for changing values of Q1 and Q2 within the n periods [Dan62].
Thereby, the maximal stock level x̄i = xi + Q1,i + Q2,i is not necessarily identical for
all i ∈ {1, ..., n} periods. Bulinskaya introduces a constant stock level ȳ that has to be
reached by an immediate emergency order if the current stock level falls below ȳ, see

[Bul64]. Still, the maximal stock level x̄i can vary from period to period. This stands
in contrast to the optimal replenishment policy of Scarf in the absence of emergency
orders [Sca60].
Scarf describes the fundamental results of the well-known (s, S) base-stock policy
which replenishes up to S when the stock level drops to s, see [Sca60]. He requires a
known demand distribution which can change over time. Further, he assumes linear
ordering, holding, and backlog costs and allows for fixed order costs. This standard

(s, S) policy has been extended to dual-source (normal and emergency) replenishment
scenarios. For example, Neuts gives an optimal dual-source (S1 , S2 ) policy with the
two base-stock levels S1 > S2 , see [Neu64]. If the stock x of the current period is below
S2 , an emergency order brings the stock immediately back to S2 . In addition, S1 − S2

units are ordered via the regular channel which arrive with a delay of one period. If
the stock x is between S2 and S1 , a regular order of a size Q( x ) with 0 < Q( x ) ≤
S1 − S2 units is placed. Otherwise, no order is triggered.

Fukuda and Veinott show that Q( x ) = S1 − x is optimal for the latter policy and

extend it to fixed costs which results in a very similar optimal (s, S1 , S2 ) policy [Fuk64],

Literature – Continuously and daily review policies

15

[Vei66]. The difference is that orders are only triggered if the current stock x is below
the critical level s with S2 < s ≤ S1 . Fukuda extends this approach to cases in which
arbitrary lead times k ≥ 0 and k + 1 are allowed for emergency and normal orders, re-

spectively [Fuk64]. Fukuda also gives an optimal base-stock policy (S1 , S2 , S3 ) with 3
suppliers where orders can only be placed every other period [Fuk64]. The lead times
are k ≥ 0, k + 1, and k + 2 periods, respectively, and there are only variable ordering

costs allowed. Wright confirms the optimality of the dual-source (S1 , S2 ) policy and
extends it to 2 articles with a limited, joint capacity [Wri69]. Whittemore and Saunders
describe conditions sufficient for ordering nothing by the normal or by the emergency
channel, assuming arbitrary deterministic delivery times [WS77].

All these optimal polices issue a normal and possibly an emergency order simultaneously. Triggering several orders at the same time is called order splitting and is
usually related to stochastic lead times. Thus, the mentioned policies can be regarded
as conditional order splitting policies with deterministic lead times, where the number
of suppliers is not fixed but depends on the current stock level. They are not suitable
in our case because both lead times are only allowed to differ by one period. Consequently, the delayed triggering of an emergency order – one of our major interests – is
useless as it never arrives before the normal order.
A more promising policy for our needs is described by Moinzadeh and Nahmias
[MN88]. They extend the ( R, Q) policy, in which a fixed quantity Q is ordered whenever the current stock drops to the reorder point R, to an approximately optimal

( R1 , R2 , Q1 , Q2 ) policy. This approach assumes arbitrary constant lead times, stochastic demand, and R1 > R2 and allows for fixed ordering costs unlike most papers we
have mentioned so far. Whenever the on-hand inventory drops to R1 , a normal order
of Q1 units is triggered. If the on-hand stock even decreases to R2 at a later point in
time, an emergency order of Q2 units is triggered as long as it arrives before the out-

16

C HAPTER 2. L ITERATURE R EVIEW

standing normal order. Their approach considers only the on-hand inventory without
regarding quantities of open orders. Therefore, they have to restrict the number of
open orders to one per supply mode. The evaluation by extensive simulation shows
that most savings are achieved when backlogging demand is expensive. High fixed
ordering costs lead to larger order quantities and diminish the benefit of this policy.
Johansen and Thorstenson extend this policy by replacing the order quantity Q2
with an order-up-to stock level S2 and by using a backorder cost rate instead of a
one-time cost [JT98]. The lead times are L and 1/L for normal orders and emergency
orders, respectively. The customer demand is restricted to Poisson distributions. They
conclude that an extension to arbitrary demand distributions and random lead times
are interesting topics for future research.
Axsäter studies a model with compound Poisson demand which is similar to the
latter [Axs07]. However, several outstanding orders are allowed and the emergency
lead time does not need to be negligible. A ( R, Q) policy is used for the regular order
and its values are assumed to be given. The decision rule is based on two parameters
∆ E and ∆ N . He defines ∆ N as the deviation between the long-run average costs C̄ and
the expected future costs given that the ( R, Q) policy is continuously used without
emergency orders. Analogously, ∆ E is the deviation between C̄ and the expected costs
CE if an emergency order is immediately placed. Thereby, the order quantity minimizes CE in the current situation. The decision places an emergency order if ∆ E < ∆ N .
Axsäter finds that his policy yields comparable results as the approach from Johansen
and Thorstenson, see [JT98], but can be applied more widely and easily. The assumption of given R and Q is disadvantageous particularly if the optimal values for the
plain ( R, Q) policy differ significantly from the policy with emergency orders.

Veeraraghavan and Scheller-Wolf describe an interesting heuristic called ”dual-index
base-stock policy” for arbitrary constant lead times and arbitrary demand distribu-

Literature – Continuously and daily review policies

17

tions [VSW08]. For each period they derive the cost-minimal values for the two orderup-to levels by considering all outstanding orders that arrive during the lead time of
the possibly new emergency order and normal order, respectively. Their solution approach is much faster than dynamic programming and extendable to other scenarios
like limited capacities. Its state-dependency makes it rather complicated to implement, though. In numerous examples where, among others, lead times and demand
variability are changed they find their results to be close to the optimal dynamic programming solution. The maximum deviation is less than 5% for the service level and
less than 8% for the total costs. However, the examples are of rather small complexity
due to time restrictions imposed by the dynamic programming approach.
An interesting variant of a two-supplier replenishment policy is described by Lee
et al. and includes the presence of an electronic market as the emergency channel
[LLB06]. The company is able to sell excessive stock to the electronic market and to
purchase items with immediate delivery from the electronic market. They give an optimal (S0 , S1 , S2 ) policy with S0 < S1 < S2 in cases where the regular lead time is one
day. Each day the stock level is investigated. If it falls below S0 it is immediately refilled to S0 via the electronic market. Whenever the stock is above S1 it is reduced to S1
by selling to the electronic market. After resetting the stock to S0 and S1 , respectively,
a regular order is triggered to increase the stock level to S2 . For lead times longer than
one day they propose 3 heuristic policies and compare them.

All policies and their models that have been mentioned so far are either continuous
policies or they investigate the stock level at each time unit. This is exactly the situation we find at the company of our interest. In contrast, periodic replenishment
policies consider long review cycles like weeks or even months. Both types of policies
are closely related. We will give an overview for this rather large topic in the literature.

18

C HAPTER 2. L ITERATURE R EVIEW

2.1.2

Periodic review policies

In an early paper, Gross and Soriano give a simple decision rule on how many items
to order via the emergency channel and the regular channel, respectively, when the
stock is investigated periodically [GS72]. Their rule is based on a (s, S) policy but they
do not specify how to obtain the optimal parameters. Rosenshine and Obee describe
a ( T1 , R2 , Q1 , S2 , S̄) model where a standing order of size Q1 arrives every T1 periods
and at most one immediate emergency delivery can be triggered in T1 to bring the
stock level back to S2 , see [RO76]. The stock level to trigger an emergency order is
R2 and the maximal warehouse capacity is S̄. Both are assumed to be given. They
optimize Q1 and S2 given that excessive stock above S̄ is sold. Chiang examines the
same scenario but optimizes S2 and S̄, see [Chi07].
Chiang and Gutierrez introduce a periodic replenishment policy ( T, R2 , S) with an
order-up-to stock level S and a critical stock level R2 , see [CG96]. They find that after
each review period T either a normal order or an emergency order is used to bring the
stock up to S if the current stock is above R2 or below R2 , respectively. They state that
this policy is especially attractive when the unit shortage cost is large, the demand
variance is high, or the fixed costs for the emergency order is small. This policy is not
an optimal policy. Chiang extends this model by allowing the variable costs of both
supply modes to be different and by considering the simultaneous triggering of a normal order and an emergency order at the review times [Chi03]. The policy is similar
to the earlier work of Fukuda and Veinott, see [Fuk64] and [Vei66], but considers long
review cycles instead of daily cycles. Chiang finds that this base-stock policy for an
emergency order is only optimal when fixed costs are neglected.
In another paper, Chiang and Gutierrez extend their earlier work, see [CG96], to
review cycles of many periods where emergency orders can also be issued throughout
a cycle and the fixed costs of placing an emergency order is negligible [CG98]. This
policy is similar to the continuous ( R1 , R2 , Q1 , Q2 ) policy of Moinzadeh and Nahmias,

Literature – Periodic review policies

19

see [MN88], except for the fact that Chiang and Gutierrez allow normal orders to be
placed only at the review times [CG98]. Later, Chiang describes a simple algorithm
to find the cost-minimal solution for the latter policy but only in the case where both
lead times differ by one period [Chi01].

Tagaras and Vlachos describe an approximate model for a base-stock policy ( T, ST , S1 )
where normal orders bring the inventory position back to its maximum value ST every
T periods [TV01a]. In addition, one emergency order can be placed within the T periods so that it arrives exactly at T − 1, the day before the periodic delivery and where

the stockout probability is the highest. The emergency order is only triggered if the
on-hand stock is below S1 and then brings it back to S1 . They assume that the probability of unmet demand is negligible in periods before T − 2. Moreover, emergency

orders in the previous review cycle are said to be negligible, as well. Two heuristic algorithms are given that yield near-optimal results. In 32 examples they find that using
emergency orders leads to 15.83% of cost savings in average and exceeds 30% in some
cases. The benefit of their model is high if demand variability, penalty for stockouts,
or the difference in both lead times are large.
In a follow-up paper, Vlachos and Tagaras compare the latter policy, called ”lateordering”, with an ”early-ordering” which places possible emergency orders so that
they arrive exactly at T − 2, see [VT01]. An additional assumption for this model is a

capacity restriction for the emergency order. They find that the capacity constraint has
a significant influence on the total costs for the early-ordering and the late-ordering
policy, especially if T is large. In contrast, the cost difference between both policies is
rather small about 3% at maximum in all 72 examples they consider. Thereby, earlyordering outperforms late-ordering in 38 of 72 examples and the reverse holds for the
remaining 34 examples.
Teunter and Vlachos, see [TV01b], investigate a periodic ( T, ST , S1 , ..., Sk ) policy in

20

C HAPTER 2. L ITERATURE R EVIEW

a similar context to Tagaras and Vlachos, [TV01a] and [VT01], where k emergency orders with immediate delivery can be triggered in the review cycle of T periods. The
replenishment of the emergency orders follows a base-stock policy in the k days before
T. The objective is to find optimal values for ST , S1 , ..., Sk for a given k. Their observation is that the additional cost savings by introducing new emergency orders are
highest for the first emergency order and decrease strongly after that. Moreover, they
find that total cost savings are higher for fast-moving articles than for slow-moving
articles. The average savings for one emergency order are 7.6% and 3.0% for fast- and
slow-moving articles, respectively. In some cases the cost reductions are more than
10%.

Kiesmüller et al. describe a periodic replenishment policy ( T, S0 , S2 ) with random demand, a constant production time L0 , and two constant transport times L1 > L2 , see
[KdKF05]. Every T time units an order of Q( T ) units is placed to bring the stock X ( T )
back to S0 . After the production time L0 the transport of all Q( T ) units can be split.
There are 0 ≤ Q2 ≤ Q( T ) units delivered via the fast mode up to the amount where

the current stock X ( T + L0 ) reaches S2 . The remaining Q( T ) − Q2 units are delivered

after L1 time units. They describe restrictions on the search area for the optimal values
of S0 and S2 . Their numerical study shows that the usage of the fast supply mode
depends strongly on the inventory holding costs, the value of the product, the lead
time difference L1 − L2 , and the ratio of transportation costs between the slow and the

fast supply mode. They find that using a slow and cheap supplier can yield high cost
savings.
Matta and Guerrero analyze an interesting replenishment policy with long review
periods that purely focuses on the availability of articles [MG90]. It is primarily used
for naval vessels, uses 3 reorder points, and all orders arrive at the end of the review
period, for example by a supply ship. By simulation they find that the best availability

Literature – Optimality of base-stock policies

21

is achieved when the 3 reorder points are distributed equidistantly between 0 and the
maximum capacity onboard.
An important characteristic of all mentioned policies is their optimality. We have seen
optimal and non-optimal multi-supplier replenishment policies. Are there indications
whether we can expect to find an optimal policy for our setting with arbitrary stochastic lead times?

2.1.3

Optimality of base-stock policies

The base-stock policy is widely applied also for dual-source scenarios. It is known to
be optimal in the single-supplier case, see [Sca60]. Moreover, we have seen optimal
base-stock policies with two suppliers in rather restrictive models where the two lead
times are 0 and 1 days, for example. Zhang states to have found an optimal base-stock
policy for three suppliers with lead times of k, k + 1, and k + 2 periods [Zha96]. However, Feng et al. proof that this is not true in general, show counter examples, and give
more insights into the problem [FSYZ06a], [FSYZ06b]. They find that the optimality
of the base-stock policy closely related to the structure of the cost function. More precisely, base-stock policies are optimal when the cost function is separable. This is the
case for one supplier or two suppliers whose lead times differ by one period. Whenever the lead times differ by more than two periods there are more than two suppliers,
or the cost function is complex for other reasons then the cost function is usually not
separable and a base-stock policy is not optimal. In the context of our work lead times
are random and we have to expect a complex cost function especially due to the various possibilities when and in which sequence orders arrive.
This concludes our review on replenishment policies with deterministic lead times
which are most similar to the situation we want to investigate. In this section we have

22

C HAPTER 2. L ITERATURE R EVIEW

seen common assumptions and the usual environment of these policies. Moreover,
the empirical cost savings have ranged between 3% and 30% for the various policies.
This is a good benchmark for our model. Most importantly, the findings of Feng et
al. indicate that we can not expect to find an optimal (base-stock) policy in our case
for which lead times are random.

2.2

Replenishment with stochastic lead times

In multi-supplier scenarios with stochastic lead times many assumptions must be revised. Usually, this renders deterministic models obsolete. For example, the common
assumption that orders arrive in a given sequence does not hold anymore. This phenomenon is called order crossovers and becomes more and more common in contemporary supply chains, see [Rie06]. Intuitively, order crossovers complicate calculations
considerably. This might be one of the reasons why a major part of the literature covers only special cases like order expediting or order splitting. These special cases will
be addressed in separate subsections. First however, we have a look at the few publications about more general multi-supplier replenishment policies where at least one
lead time is stochastic.

One of the first papers on multiple suppliers with random lead times has been published by Verrijdt et al. in the area of reverse logistics where failed parts return to the
warehouse and can be repaired via a normal repair channel and an emergency repair
channel [VAdK98]. Both repair times L1 and L2 are exponentially distributed. The
demand in form of failed parts arrives according to a Poisson process. This is a key
assumption as it allows to use the Poisson Arrivals See Time Average (PASTA) property, see [Wol82], and simple expressions can be given for the expected number of
items in stock and the expected waiting time of backlogged demand. Thus, Verrijdt

Literature – Replenishment with stochastic lead times

23

et al. are able to formulate exact expressions for the costs and other measures. They
employ a Markov process to obtain the required probabilities of the different states
of their model. Here, the assumption of exponentially distributed lead times allows
for an exact formulation of all parameters. They give numerical results and find that
substantial cost savings of more than 50% can be achieved by using an emergency
supply. However, the difference of the expected service level, i.e. the fill rate, between
their stochastic model and a model with deterministic lead times are negligible. They
conclude that the assumption on the type of the lead time distribution is therefore not
restrictive. These findings are very interesting. We will execute similar analyses for
our model and compare our results to these findings.
This model is interesting for us, since it satisfies our primary requirement of
stochastic lead times and stochastic demand. Moreover, its formulation is astonishingly simple. However, there are several drawbacks. The model does not include
fixed ordering costs and is limited to Poisson demand and exponential lead times.
Our intention is to find a model which allows for more complicated cost structures
and which is independent of the lead time distribution and the demand distribution.
Verrijdt et al. give two limiting cases, namely L1 = L2 and L2 = 0 for which their
model is independent of the lead time distribution. These are interesting cases. However, they still do not cover many scenarios which we frequently find at companies.

Mohebbi and Posner describe an exact formulation of the ( R1 , R2 , Q1 , Q2 ) policy
where orders of size Q1 and Q2 are placed at the reorder points R1 and R2 , respectively [MP99]. While their replenishment parameters are identical to Moinzadeh and
Nahmias, see [MN88], they assume exponential distributed lead times and a Poisson
process for the customer demand. Moreover, unmet customer demand is not backlogged but lost. Similar to the paper by Verrijdt et al. the PASTA property, see [Wol82],
plays a key element when formulating the equations for the probability of a certain in-

24

C HAPTER 2. L ITERATURE R EVIEW

ventory level and deriving the total costs. Under the assumption of exponential lead
times the model becomes a renewal process with a stationary distribution of the inventory level if time goes to infinity. They employ a so called system-point method of
level crossings to find this stationary distribution. Level crossing divides the whole
state space into subsets where no, one normal, one emergency, or two orders are outstanding. By equating the rates in which the renewal process enters and leaves each of
these subsets one finds the desired stationary distribution of the inventory level after
long calculations. For details about system-point methods and level crossing we refer
the interested reader to the papers of Brill and Posner [BP77], [BP81].
The cost functions of their model include fixed and variable costs for ordering unlike many other papers. Despite the fact that all their equations are exact, the joint
optimization of R1 , R2 , Q1 , and Q2 can not be done analytically. In fact, Mohebbi
and Posner use a computationally intensive four-dimensional numerical search and
report the existence of local minima. This makes it extremely difficult to assure that
the global minimum has been found. In several sensitivity analyses they find that cost
savings are up to 20% if emergency orders are used. In some cases it is not beneficial
to use emergency orders. They state that the ratio between holding and shortage costs
has a strong influence on the total costs. Moreover, the average lead time and the variable costs for emergency orders have more impact on the total costs than the change
of the average demand and the fixed costs of the emergency order, for example.
This model by Mohebbi and Posner gives very interesting insights for us. Especially, it shows the complexity of such a system. We should be prepared to employ
numerical approaches for the optimization which are able to overcome non-global
minima. The savings resulting from emergency orders are significant. Unfortunately,
no comparison is given to a model where lead times are simplistically assumed to be
deterministic. The disadvantages of this model are its strong dependence on a Poisson process for the demand and the exponentially distributed lead times. The authors

Literature – Replenishment with stochastic lead times

25

state that their model could be extended to generalized hyperexponential lead time
distributions by utilizing results of Botta et al. [BHM87]. However, this still does not
solve the restriction on the demand distribution. Moreover, it remains to be proven
that generalized exponential distributions really cover the variety of lead time distributions that occur in certain industries and whether it is practically feasible to do so.
Another major difference to our situation is the assumption of loosing unmet demand.
The scenario we are interested in must additionally keep track of the backlogged demand. We expect this to further complicate their model.

Kouvelis and Li describe a ( Q1 , Q2 , T, T1 ) policy with constant demand, a random
normal lead time L1 and a deterministic emergency lead time L2 , see [KL08]. They
only consider one completely independent period of T time units. The order of Q1
units with a random lead time L1 is placed T1 time units before the interval T starts.
In addition, an emergency order with deterministic lead time L2 = 0 can be placed
upon knowledge of the exact arrival time of the Q1 units. They derive formulas for
the cost-minimal values for T1 and the ratio between Q1 and Q2 . They extend this
model, among others, to settings where the second lead time L2 > 0 is increased and
to cases where the knowledge about the actual value of L1 is postponed into the interval T. The formulation of this model is rather simple mainly due to the assumptions of
a completely independent interval T and only one stochastic variable L1 . The model
describes a quite different scenario than the continuous and stochastic one we are interested in. Therefore, it is only of limited use for us.

A practical approach for a ( R1 , R2 , Q1 , Q2 ) policy is given by Korevaar et al. which
is successfully employed at a German automotive manufacturer [KSB07]. They use a
simulation approach to find the β service level for a given pair ( R, Q) for normal and
emergency order separately. This simulation is able to cover detailed operational con-

26

C HAPTER 2. L ITERATURE R EVIEW

straints like lot sizes and capacities. The value for Q is determined beforehand by the
standard economic order quantity, for example. They yield a ( R, β) curve separately
for normal and emergency orders by iteratively increasing the appropriate reorder
point R. The joint service level is calculated by a simple approximation. The total
costs which incorporate a very detailed cost structure and penalties are derived from
the simulation results and the joint service level. They use a threshold accepting algorithm for the optimization. The result has to satisfy various constraints including
a total budget and a system-wide average service level from which single articles can
deviate to a limited extent. They report that total cost savings of 30% are soon to be
reached for the entire warehouse.
This approach satisfies our request of flexibility regarding arbitrary distributions
and a detailed cost structure with variable and fixed costs. However, it lacks a solid
analytical foundation that allows to understand the dynamics of this model in greater
detail.

This closes our review on general multi-source policies with stochastic lead times. We
can state that these kinds of problems are highly complex and current models do not
cover arbitrary distributions. The cost savings have shown to be significant in all papers. A common approach is to utilize sensitivity analysis to gain more insights about
the behavior of the policy in different situations. We keep in mind that Verrijdt et
al. have found negligible differences in the β service level between their stochastic repair model and an approximation with deterministic lead times. We will see whether
this statement also holds in our case.
Even if we are able to give exact formulas for the total costs a numerical optimization algorithm will most likely be necessary. We also have to expect several local
minima when jointly optimizing R1 , R2 , Q1 , and Q2 .

Literature – Order expediting

27

We conclude this section by briefly summarizing the literature on the special cases of
order expediting and order splitting with stochastic lead times.

2.2.1

Order expediting

Order expediting refers to a replenishment policy where the delivery of an outstanding order can be accelerated by some additional fee. This option is usually chosen
in cases of low inventory. Even if both lead times are constant but the time of expediting is random, the effective lead time until the arrival of the order becomes random.
Allen and D’Esopo describe a ( R1 , R2 , Q) policy with random demand and two constant lead times L1 > L2 , see [AD68]. An order of Q units is placed at time t0 when the
stock decreases to R1 . If the stock drops further to R2 within t0 and t0 + L1 − L2 , the Q

units are expedited and arrive L2 time units later. They give the formula for the total
expected costs and find approximately cost-minimal values of R1 , R2 , and Q with an
iterative procedure.
Chiang extends this policy by a time parameter T ≤ L1 − L2 until which it still

makes economical sense to expedite the order given the additional costs [Chi02]. This
window of T time units is an interesting aspect for our scenario where both lead times
are random. In his paper he also describes a heuristic policy when the lead time L1
consists of a constant production time L0 and a random transportation time L10 where
L1 = L0 + L10 , see [Chi02]. After the production at L0 the buyer expedites the order
if the stock is R2 or below. Their examples show that the higher the service level,
the more beneficial these policies are. This policy is similar to the periodic ( T, S0 , S2 )
policy of Kiesmüller et al., see Section 2.1.2, which assumes constant lead times and
allows to expedite part of the total order quantity [KdKF05].
Durán et al. describe a continuous ( R1 , R2 , Q) policy when the production time L0
is constant and there exist two constant transportation times L1 > L2 , see [DGZ04].

28

C HAPTER 2. L ITERATURE R EVIEW

This policy places an order if the stock s drops to R1 and expedites it if s ≤ R2 at time

L0 . They give an algorithm to find the cost-minimal values for R1 , R2 , and Q. The

numerical examples with exponential demand are interesting. Their first observation
are the existence of opposing mechanisms as L1 increases. Initially, the difference
R1 − R2 decreases which makes expediting more probable and more attractive. Later,
R1 − R2 decreases again because R1 grows continuously which makes expediting less

necessary. A second observation is that never expediting or always expediting can be
optimal. We expect similar phenomena for our model.

Dohi et al. describe a ( T, Q) policy with two constant lead times L1 > L2 , see [DKO95].
A normal order is scheduled to be placed at time T which would arrive at T + L1 .
However, if the stock preliminarily depletes at t0 < T, then the order is immediately
placed and arrives at t0 + L2 . They derive sufficient conditions for the existence of
some T that minimizes the expected costs. In numerical studies they find that the
optimal Q is much more influenced by the expedited lead time L2 than by L1 .
Dohi, Shibuya, and Osaki extend this analysis to a similar policy ( T 0 , Q) in two papers [DSO97], [SDO98]. This policy only investigates the stock at time T 0 and bases the
expediting decision on the current stock. The objective is to find cost-minimal values
for T 0 and Q. For both models, ( T, Q) and ( T 0 , Q), they give a closed representation
for the expected costs under the assumption of a Poisson distribution for the demand
and a Gamma distribution for both lead times L1 and L2 . In numerical examples they
find that the uncertainty especially of the regular lead time L1 influences the optimal
policy significantly. This is a very interesting observation for our work.
Dohi et al. introduce an optimal ( R, Q, T ) policy called order-limit policy, see
[DOO99]. They assume a fixed and known demand rate d and two lead times L1
and L2 . A normal order of random lead time L1 and quantity Q is placed when the
stock drops to R at t0 . The order is expedited if it does not arrive until T, some point

Literature – Order splitting

29

of time after the depletion of the stock at t0 + Q/d. They derive conditions for the
existence of a cost-minimal value for T. However, the joint optimization of R, Q, and
T is very complex. In their examples, they use a numerical optimization approach and
assume a Weibull distribution for the lead time L1 . They find that the additional use
of T is superior to the regular ( R, Q) policy.
The reviewed literature on expediting orders with two random lead times has shown
several interesting points for our work. First, stochastic lead times seem to significantly influence the cost-minimal policy. Second, the joint optimization of scenarios
with random lead time is very complex. Third, we can expect to gain insights on
various mechanisms that influence the optimal solution and on interesting cases like
always or never using the faster delivery option.

2.2.2

Order splitting

Order splitting defines a replenishment policy where the company uses one reorder
point and places several orders each with a fraction of the total order quantity Q at
one or more suppliers. The amount of ordered units can vary from order to order but
sum up to Q.
Sculli and Wu were among the first to show that splitting an order between two suppliers with independently normal distributed lead times reduces the reorder level and
the buffer stock when compared with replenishment with only one supplier [SW81].
Since then, many extensions and specializations have followed. Ramasesh et al. give
a detailed comparison between sole sourcing and order splitting in the case of a regular (s, Q) replenishment policy with either uniformly or exponentially distributed
lead times [ROHP91]. They find that order splitting provides savings in holding and
backordering costs, which increase if the demand volatility increases or the lead time

30

C HAPTER 2. L ITERATURE R EVIEW

distributions are skewed and have a long tail.
Furthermore, they divide the order-splitting approaches into two groups [ROHP91].
First, macro studies analyze the effect of order splitting on the whole replenishment
process, including the relation to the supplier. For example, a competition among
the suppliers (producers) can lead to lower prices and better quality. Second, micro
studies focus on the inventory perspective and savings induced by lower ordering,
holding, and shortage costs.
Despite the attention given by the academic world to the concept of reducing lead
time risk by order splitting, it has received considerable criticism by Thomas and Tyworth [TT06]. Regarding the micro focus, they argue that savings in holding and
shortage costs are more than compensated for by increased ordering costs in reality.
They see the gap between literature and reality mainly in the neglected transportation
economies of scale and underestimated transporting costs. In a more macroscopic
view, they question whether the savings from a reduced average cycle stock in one
inventory are still valid or significant for the whole supply chain. Many approaches
neglect the in-transit inventory, which can lead to additional costs.
Thomas and Tyworth suggest that future research should focus on other models of
dual sourcing like the cost performance differences in modes of transportation [TT06].
With respect to our scenario, we can support many statements of Thomas and Tyworth, since the cost structure and practical replenishment constraints such as transportation economies of scale do not allow for an order-splitting approach.

2.3

Related replenishment policies

Time window for second supply mode. Time windows for placing orders in a
multi-supplier environment are used for different purposes. Dohi et al. use a periodic replenishment policy and identify a critical time t0 before the next regular review

Literature – Multi-echelon inventory policies with multiple suppliers

31

time where a depletion of the stock immediately triggers an expedited order [DKO95].
In another paper an outstanding order with random lead time is expedited only if it
has not arrived by a certain time T, see [DOO99]. Chiang describes a policy with constant lead times L1 and L2 where an order is not necessarily expedited up to L1 − L2

time units before the arrival of the next order [Chi02]. In fact, expediting is only allowed up to a time T < L1 − L2 where its benefit by avoiding unmet demand exceeds
the additional costs for expediting.

Moinzadeh and Schmidt describe a (S − 1, S, Ŝ) policy with two deterministic sup-

ply modes [MS91]. Upon each unitary customer demand an order is placed to bring
the stock position back to S. They allow for several open orders and include the arrival time T of the next order into the decision about using the faster supply mode.
Actually, the fast supply mode is only used if the stock on hand is below Ŝ and if the
emergency order arrives before T. Moinzadeh and Aggarwal extend this model to a
multi-echelon environment with one warehouse and M retailers [MA97].

The concept of a time window in which the usage of a faster delivery still makes sense
is very important for us. In our scenario both lead times are random and a company
might want to control the interval in which the placement of an emergency order still
seems to be reasonable.

Multi-echelon inventory policies with multiple suppliers.

Multi-echelon supply

chains obtain a lot of attention in practice as well as in the literature. Some work
even exists on inventory management with multiple suppliers. The complexity of
multi-echelon supply chains is inherently high. We are interested in the common assumptions and solution approaches that allow to merge the complexity of multiple
suppliers and multi-echelon inventory models.
One of the first papers on this topic is from Muckstadt and Thomas [MT80]. A cen-

32

C HAPTER 2. L ITERATURE R EVIEW

tral warehouse supplies several local retailers each of which faces a Poisson demand
process from its customers. The described situation is in the area of spare parts where
average demand is very low. Thus, it is plausible that each retailer uses a (S − 1, S)

base-stock policy. In case of stock-outs at the retailers emergency orders can be triggered. They compare different heuristic multi-echelon policies and find that they yield
significant savings compared to single-echelon policies.
Moinzadeh and Aggarwal describe a multi-echelon model with one central warehouse and several regional retailers [MA97] which is based on the (S − 1, S, Ŝ) pol-

icy of Moinzadeh and Schmidt mentioned before [MS91]. The retailers face demand
in form of a Poisson process and the two transportation modes between the warehouse and the retailers have constant lead times. Moinzadeh and Aggarwal give an
expression for the expected total cost rate and explain an algorithm to find the optimal
parameters.
Ng et al. state that it is very difficult to find an optimal policy for a multi-echelon inventory system [NLC01]. Therefore, they concentrate on giving a cost-effective heuristic. In their policy they consider two central warehouses W1 and W2 each of which has

one regional retailer w1 and w2 attached to it. All lead times are constant and both
retailers are identical in terms of the same ( R, Q) replenishment policy and identical Poisson customer demand. In this context they compare a policy in which each
retailer wi only orders from its assigned warehouse Wi with a policy which allows ordering from the non-assigned warehouse in case of a supply shortage, as well. The
latter policy yields significantly more savings in their sensitivity analysis on the order
quantities and the lead times.

From these publications we see that strong assumptions are applied for combined
multi-echelon and multi-supplier scenarios. Furthermore, the optimization can only
be achieved numerically. This is a good indication for the complexity of the problem.

Literature – Summary

33

On the one hand, our single-echelon scenario is less complex than these multi-echelon
scenarios. On the other hand, we employ much less restrictive assumptions especially
about the distribution of the demand and the lead times. Thus, we expect the necessity to determine cost-minimal parameters by a heuristic and to perform extensive
sensitivity analysis to gain more insights about the dynamics of our model.

2.4

Summary

The literature on multi-source models and especially on dual-source models is vast.
The models range from continuous and periodic policies with deterministic lead times
to order expediting and order splitting with stochastic lead times. The modeling and
calculation techniques range from dynamic programming over Markov processes to
simulation. All of them share the conclusion that emergency orders can substantially
reduce the total costs in these various situations.

A central finding by Feng et al. is that (s, S) base stock policies are generally not optimal when the lead times differ by more than one period [FSYZ06a], [FSYZ06b]. It
is very hard if not impossible to find an optimal policy for mostly all practical situations even if lead times are deterministic. Nevertheless, one can find optimal or
cost-minimal parameter values for a given policy, of course.
The majority of multi-supplier replenishment policies with stochastic lead times
are special cases called order expediting and order splitting. While these models are
barely useful to us some of the results are still interesting. In papers about order expediting the authors find that the variability of the lead times influences the cost-minimal
solution significantly. This supports the importance of our work. Moreover, the joint
optimization of the replenishment parameters seems to be rather complex. This is
what we should keep in mind if we start to develop an optimization approach for

34

C HAPTER 2. L ITERATURE R EVIEW

our model. Order splitting has received a considerable attention in literature. Here,
Thomas and Tyworth question the practical applicability of these approaches as the
splitting of one large order into small pieces neglects the common reality of large
transportation economies of scale and considerable fixed ordering costs [TT06].
In fact, only very few papers address the situation of our interest where orders of
quantity Q1 and Q2 are placed upon reaching the reorder points R1 and R2 , respectively. The first paper on this ( R1 , R2 , Q1 , Q2 ) policy is by Moinzadeh and Nahmias.
They do not specify the demand distribution but only allow for deterministic lead
times.

There exist two papers assuming exponentially distributed lead times and a Poisson
process for the demand which provide interesting insights, see [VAdK98] and [MP99].
Mohebbi and Posner are able to give an exact formulation of the total costs. However,
they have to use a computational-intensive numerical search algorithm for the joint
optimization of R1 , R2 , Q1 , and Q2 . They encounter several local minima during their
optimization. This fact is important to keep in mind as it complicates the optimization
considerably and excludes gradient descending algorithms, for example. Verrijdt et
al. state that there is only a negligible difference in the β service level if the scenario
with stochastic lead times is approximated by a model with deterministic lead times.
Basically, this means that the distribution of the lead time does not matter at all. We
doubt this statement and give counter examples in our sensitivity analysis.
The formulation in both papers is rather complex, especially compared to the rest
of the mentioned literature, even though they can simplify their formulas due to the
PASTA property, see [Wol82], which plays a key role in their models. Unfortunately,
this PASTA property is restricted to the demand Poisson process and cannot be used
in our scenario.

Literature – Summary

35

The naturally more complicated approaches for multi-echelon replenishment policies
do not yield useful mechanisms for our situation either. To our knowledge the only
publication that models the daily demand and both lead times as arbitrary stochastic
variables is the practice-oriented paper by Korevaar et al. [KSB07]. While this paper uses a heuristic approach based on simulation, the intention of our work is to
investigate the problem more formally, to identify and quantify possible cost reductions of dual sourcing. Thereby, we follow the approach of Moinzadeh and Nahmias
which seems most promising to us because they do not restrict the demand distribution [MN88]. In addition, we will use the concept of a time window for the second
supplier which has been used in some related replenishment policies.

Part II
Modeling and practical application

37

Chapter 3
A model for stochastic dual sourcing
(SDMR)
In the course of this chapter we define a formal model called ”Stochastic Dual-Mode
Replenishment” model (SDMR) which represents a stochastic replenishment policy
where two different suppliers, or more generally two supply channels, are available
and the warehouse manager has to decide when and how much to order from each
supply channel. This addresses our research question RQ 1 on page 5. The SDMR
model is inspired by the work of Moinzadeh and Nahmias [MN88]. Many aspects
and our practical experience relating to the definition of the following stochastic dualsourcing replenishment policy result from customer projects which have been published to some extent, see [KSB07].
The structure of this chapter is as follows. First, we describe the SDMR model,
which consists of eight cases, and elaborate a proper probability space for it. Then
we develop the formulas for important parameters in inventory management in the
subsequent sections. Section 3.2, 3.3, and 3.4 give formulas for the probability, the
expected shortage, and the expected average stock on hand, respectively, for each of
the eight cases. These represent the main part and the core formulas of the SDMR
39

40

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

model. In Section 3.5 we develop the basics for calculating the total inventory costs.
Finally, we conclude the chapter with a summary.

3.1

Model description

Consider the common situation at a warehouse where the stock is monitored continuously and one reorder point R1 has been thoughtfully chosen to trigger replenishment
orders. In the beginning there are several items on stock. The stock level is diminished
due to stochastic customer demand until R1 is reached. Then, an order of predetermined Q1 units is placed at the supplier which arrives after some stochastic period of
time. Upon arrival of every order the stock level is raised and the process starts anew.
Note, several orders can be on the way, also referred to as outstanding orders, but we
will neglect this aspect here. Now, we will extend this simple scenario by introducing
a second reorder point R2 and a second order quantity Q2 . Moinzadeh and Nahmias
are the first of the few people who have investigated this kind of replenishment policy
so far [MN88]. We abolish two of their strong restrictions. First, we allow for stochastic lead times for both orders, normal and emergency. Second, reorder points are not
limited to positive numbers but can be negative, as well. Figure 3.1 exemplifies our
model which we describe in detail below.

There exist two order modes with the order quantities Q1 and Q2 and the lead times
lt1 and lt2 , respectively. These lead times follow some given stochastic distributions L1
and L2 , respectively, with the according probability density functions l1 (y) and l2 (z).
Moreover, the demand x during any time interval of length t is distributed according
to the demand distribution D (t) with the cumulative distribution function F ( x, t) and
the probability density function f ( x, t). Given two arbitrary reorder points R1 and
R2 for the two order modes with R1 , R2 ∈ R and R1 > R2 the inter-order time t g is

The SDMR model – Model description

41

"#$%&''&()*+,-#.+)/&'0(#/&102)34/ /&'0(#/&105

/&'09

8,#+)&1-,)$!!

#!
;,';+,;)
='1.&/
%!

#7
1.&,;%';+,;)&1-,)!"

%7
8,#+)&1-,)$!7

!6

!"

!( 7

!( !

&1-,

&1-,)$1.+'$)!' &')&;1<<,;)';+,;)7
0:08,)&1-,)!&

Figure 3.1: Exemplified cycle of a (R1 , R2 , Q1 , Q2 ) replenishment policy with stochastic
lead times and demand

a random variable with its associated distribution G (t) and the probability density
function g(t). The point of time where the two orders are triggered are denoted by t0
and t g , respectively. The notation t0 indicates that it is the starting point of the reorder
process and, thus, the first point in time of our interest.
Note, the demand distribution D (t) represents the probability that a demand x occurs in a fixed period of time t and so F ( x → ∞, t) = 1 holds for each time interval

t with R 3 t > 0. The demand distribution might be completely different for two

periods t1 6= t2 so D specifies a whole set of distributions and not just a single distri-

bution. Thus, arithmetic operations on the probability function with different periods
t1 6= t2 are not necessarily easy to interpret and have to be handled with care. For ex-

ample, the result of integrating the expression f ( x, t) regarding t and for a fixed value

!

42

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

x is not limited to the interval [0, 1] but might be any positive number. Moreover,
F ( x, t → ∞) = 1 does usually not hold.
The objective of the model is, first, to adequately calculate the expected number of
shortage and the average stock during a replenishment cycle. Second, we will derive
common measures in SCM like costs and service level. Finally, we will utilize this
model to optimize the four replenishment parameters R1 , R2 , Q1 , and Q2 .

3.1.1

Assumptions

Usually, it is assumed that regular orders, mode 1, are slower and less expensive than
emergency orders, mode 2. This is true in most cases. However, one should question the restriction which in practice is commonly imposed on the ordering process
as a direct consequence of this assumption: the slower order has to be triggered first
and then one can possibly place an emergency order if the lower reorder point R2 is
reached. We do not make this restriction because we believe there might exist (theoretical) cases in which the interplay of costs, lead times and demand could lead to a
scenario where it is cheaper to not use the regular orders at all, for example.
Assumption 1. The association of the two available delivery modes with the two possible identifiers 1 and 2 is insignificant and exchangeable.
Assumption 2. The reorder points R1 and R2 can take positive or negative real values
and they always follow the relation R1 > R2 .
Assumption 3. The order quantities Q1 and Q2 are positive real values.
Assumption 4. All distributions, for the demand and both lead times, are continuous
and have a finite expected value.

The SDMR model – Assumptions

43

Assumption 5. The demand per time unit is identically and independently distributed
(iid) and non-negative. The demand dt during a period t > 0 is non-negative and
dt = 0 for t = 0.
Assumption 6. Both lead times, lt1 and lt2 , are positive.
Assumption 7. At any point in time there is only one order outstanding per supply
mode.
Assumption 8. The order quantities must be big enough to bring the stock level above
the first reorder point R1 . Consequently, the maximum demand is limited to the order
quantity Q1 and Q1 + Q2 during a one-order cycle and two-order cycle, respectively.
According to Assumption 5 the demand per time unit – like one minute, hour,
or day – is identically and independently distributed. This can be justified as long
as there exist no strong trend or seasonality in the demand for the observed period
of time and single customers have no significant influence on the demand. For simplicity, we refer to the demand distribution D (1) per time unit also as daily demand
distribution.
Example 3.1.1. A company has a large (potential) customer base with similar needs
for one specific product. Only a small fraction, e.g. < 1%, of these customers are
willing to buy some units of the product every day. Obviously, the satisfied demand
of the current day does not significantly influence the number of customers and their
demand on the next day.
Even if the customer base is not very homogeneous and single customers request
a multiple of the average number of units this is usually well covered by the daily
demand distribution in practice. This is especially true for a rolling replenishment,
e.g. a (daily) repetition of the calculations, where the daily demand distribution is
permanently adjusted according to the latest customer demands. In this way even a

44

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

seasonality or trend in the demand can successfully be covered. In the spare parts
business of a world-wide operating automotive manufacturer or in the case of large
retailers Assumption 5 will usually hold. For the supplier of a single power plant,
however, this assumption is most likely not valid. We focus on applications with
a large customer base and only consider time frames that do not involve too large
changes in the demand distribution. Thus, we can defend this assumption for our
purposes.
Assumption 8 forces the stock to be equal to or above the first reorder point, R1,
after the last order in a replenishment cycle has arrived. It assures that the replenishment policy can be applied repetitively without causing a constantly decreasing stock
level. This might be the case otherwise because we do not allow for several outstanding orders per supply mode as it is stated in Assumption 7.
Obviously, Assumption 8 is violated with a positive probability if the demand distribution does not have a finite upper bound, for example. However, we do not want
to restrict the type of distributions used in our model. Thus, in order to make statements about the applicability of our model it is desirable and essential to have a way
to measure whether Assumption 8 is violated and, if yes, how much.
Another practical restriction results from the fact that many companies do not want to
issue a second (emergency) order if the outstanding order is just about to arrive. This
restriction is, for example, applied in the heuristic approach described by Korevaar et
al. [KSB07] and we will use it in our stochastic model, as well. Note, this rule is easy
to apply in case of deterministic lead times [MN88]. There, the second (fast) order is
only triggered if it arrives before the first order. In a setting with stochastic lead times
a simple and obvious rule does not exist.
Assumption 9. The second order is only allowed to be triggered if the stock level drops
to or below the second reorder point, R2 , within a given time window tw .

The SDMR model – Assumptions

45

Companies are free to choose and adapt the rule or policy that determines the
length of the time window tw in a way that it proofs most appropriate in respect of
their business needs. The only requirement for our model is that they are able to specify the value of tw . Further details on the time window are given in the following.
For the formulation of the formal model we make the following assumption which
will be relaxed in Chapter 4 about the practical application of this model.
Assumption 10. The time is considered to be continuous and the demand is a continuous function over time.
In other words the stock will gradually deplete over time and we can give the exact
point in time when a certain stock level is reached. This is reflected in the integration
notation of our formulas. Of course, this assumption does usually not hold in practice.
Thus, in Chapter 4 we will show how this assumption can be relaxed while we can still
utilize the given formulation.

3.1.2

Time window for triggering second order

In a scenario with stochastic lead times the decision up to which point of time a second
order should be triggered is not straight-forward because the exact arrival of the first
order is unknown. One possibility is to determine a time window of length tw for
triggering a second order in such a way that the expected arrival of the second order
never exceeds the expected arrival of the first order. Equation (3.1.1) expresses this
condition where E[ x ] denotes the expected value of a stochastic variable x.
tw = E[lt1 ] − E[lt2 ]

(3.1.1)

However, there is no restriction on how to determine tw . Another possibility is to
set the time window tw in such a way that the second order arrives at least in q percent

46

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

of all cases before the first order. In this case the equation
1−q =

⇒ 1−q =

Z tw
0

Z tw
0

l1 (y)dy +
l1 (y)dy +

Z ∞Z ∞
tw
Z ∞
tw

y−tw

l2 (z)dz l1 (y)dy

[1 − L2 (y − tw )] l1 (y)dy

(3.1.2)

has to be solved for tw . Note, a solution of this equation is only possible if 1 − q

is greater or equal to the right-hand side of Equation (3.1.2) with tw = 0, see Appendix A.7. In other words the equation can only be solved if the condition holds that
the first order arrives after the second order in at least q percent of all cases when both
orders are triggered simultaneously, i.e. the inter-order time t g equals zero.
Even though we favor the second case the value of tw can be determined and
adapted in any arbitrary way to best represent the employed replenishment policy
at a specific company. Thus, this concept is very flexible to cover the needs in practical
inventory management. Once the value of tw is calculated it serves as a constant to
the rest of the replenishment model. Note that we implicitly assume that the lead time
distributions L1 and L2 remain unchanged. Whenever these distributions change the
value of tw has to be adjusted accordingly.
In the current time more and more sophisticated IT systems support the processes
throughout the whole supply chain. Consequently, it is getting much easier to improve and adapt statements about the expected arrival time in the course of the whole
delivery process. Obviously, one can use this information to adjust tw accordingly.
Upon hitting the second reorder point one can dynamically decide how likely a new
emergency order will arrive before the next regular order and whether it is feasible to
place a fast order or not.
However, one still has to distinguish between planning and operational activities.
While the operational people can use the latest information to optimize their current
behavior the planning people often have to rely on historical data, rules and experi-

The SDMR model – Assumptions

47

ence to make statements and decisions that reach far into the future. On the one hand,
the planners might want to use one of the two approaches shown above, see equations (3.1.1) and (3.1.2), to determine a preliminary value for tw which enables them to
make further decisions on how to set the reorder points and order quantities, for example. On the other hand the purchasing staff can possibly use the latest information
about an outstanding order from the supplier and decide that it is not feasible to place
another order regardless of the original value for tw given by the planners.

3.1.3

Categorization of replenishment cycles

After the time window tw for possibly triggering a second order has been determined
further calculations of different scenarios, e.g. a cycle consists of one or two orders,
and their probability of occurrence can be made. The number of order cycle scenarios
and their necessary conditions are not trivial to formulate. Thus, in the following we
give a systematic view on one- and two-order cycles and their interplay of demand

"#$%&'#(%&)*&%+%(,-

and lead times.

When we look at the time line of possible events in a replenishment cycle, see Figure 3.2, we can give a number of straight-forward observations.
,/#00%/&
)/1%/&2

,/#00%/&
)/1%/&3

4//#+4'&
)/1%/&3

%(1&)*&,#$%&
5#(1)5&,)&
,/#00%/&)/1%/&3

4//#+4'&
)/1%/&2

%(1&)*&
)/1%/&676'%

!.

!"

!&3

!#

!&2

!$

,#$%

Figure 3.2: Time line with events during a two-order replenishment cycle
9(1&)*&,#$%&

"/#00%/&
"/#00%/& 8//#+4'&
First,
all our replenishment
cycles5#(1)5&,)&
start at t0 , when 8//#+4'&
the first order is9(1&)*&
triggered be)/1%/&2

)/1%/&3

)/1%/&3

!%

!"

!&(

,/#00%/&)/1%/&3

)/1%/&2

)/1%/&676'%

!&'

!$

cause the stock level has fallen to (or below) the first reorder point R1 . This is the very

!#

,#$%

48

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

basic event for our observations. For this reason we assume it happens at an arbitrary
point of time and do not further regard it.
Second, every replenishment cycle is terminated after tc time units when the stock
drops to (or below) R1 again once all orders have arrived that had been triggered since
t0 . This event will always be the last one in a cycle and is completely determined by
the demand and the delivered quantities within this cycle. Of course, the end of one
replenishment cycle is the start of the succeeding replenishment cycle.
In between t0 and tc one can think of n = 4! = 24 permutations of events, see Table 3.1. Without deeper knowledge one has to consider all 24 possible cases when
developing formulas, for example, for the average stock level during a replenishment
cycle. Luckily, some permutations violate against logical constraints and others do not
comply with our replenishment policy. The requirements for valid permutations are:
1. an order must arrive after it has been triggered (t A2 > t g ) and
2. the second order can only be triggered within the valid time window tw and
before the arrival of the first order, so t g ≤ min(t A1 − e; tw ) where e → 0 is an

arbitrarily small positive number.

Eliminating all invalid elements, indicated by a grey background in Table 3.1, from
the 24 cases one yields many identical, redundant rows. Now, the single sequence
of events, (tw , t A1 ) completely represents the valid part of the nine rows 1 − 6 and

19 − 21. Similarly, the rows 13 − 18 and 22 − 24 can be reduced to the valid sequence

(t A1 , tw ). Both sequences, (tw , t A1 ) and (t A1 , tw ), only consist of the arrival of the first

order t A1 and the time window tw . In other words, they describe the two possible
alternatives for a one-order cycle.
The remainder of Table 3.1, rows 7 − 12, consists of valid constellations for two-order

The SDMR model – Assumptions

49

event sequence

event sequence

no.

1st

2nd

3rd

4th

no.

1st

2nd

3rd

4th

1

tw

tg

t A1

t A2

13

t A1

tw

tg

t A2

2

tw

tg

t A2

t A1

14

t A1

tw

t A2

tg

3

tw

t A1

tg

t A2

15

t A1

tg

tw

t A2

4

tw

t A1

t A2

tg

16

t A1

tg

t A2

tw

5

tw

t A2

tg

t A1

17

t A1

t A2

tg

tw

6

tw

t A2

t A1

tg

18

t A1

t A2

tw

tg

7

tg

tw

t A1

t A2

19

t A2

tw

tg

t A1

8

tg

tw

t A2

t A1

20

t A2

tw

t A1

tg

9

tg

t A1

tw

t A2

21

t A2

tg

tw

t A1

10

tg

t A1

t A2

tw

22

t A2

tg

t A1

tw

11

tg

t A2

tw

t A1

23

t A2

t A1

tw

tg

12

tg

t A2

t A1

tw

24

t A2

t A1

tg

tw

legend:

invalid event

valid event

Table 3.1: Permutations of possible events in a replenishment cycle
cycles. At first glance, the six rows can not be further compacted. However, if we
look closer there are still two redundant lines. Fixing t g as the first event, which is a
necessary and sufficient condition for a two-order cycle, the arrival sequence of the
two orders influences the average stock and amount of shortage whenever the two
order quantities are not identical. So the relative position between t A1 and t A2 has to
be preserved. On the other hand, the maximal time t g to trigger a second order is
decreased below tw if t A1 < tw . So we have to preserve the relative positions between
t A1 and tw , as well. However, the relative positions of tw and t A2 are not important to
us as it does neither influence the stock level nor the amount of shortage. Considering

50

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

this fact the event sequences (t g , tw , t A2 , t A1 ) and (t g , t A2 , tw , t A1 ) are equivalent just as
the sequences (t g , t A1 , tw , t A2 ) and (t g , t A1 , t A2 , tw ) are.
This reduces Table 3.1 to six lines, which represent all possible different and valid
scenarios for a replenishment cycle. They correspond to the lines 1 − 6 in Table 3.2,
which are prefixed with the associated row numbers from Table 3.1.

events
no.

1st

2nd

3rd

4th

1

tw

t A1

13

t A1

tw

7

tg

tw

t A1

t A2

8

tg

tw

t A2

t A1

10

tg

t A1

t A2

tw

12

tg

t A2

t A1

tw

tg

tw

t A1 = t A2

tg

t A1 = t A2

description

one-order cycles

tw

two-order cycles

two-order cycles –
special cases

Table 3.2: Valid permutations of possible events in a replenishment cycle

Up to now we neglected the fact, that events might occur simultaneously. In general
this is not a problem, as we can specify rules whether a second order will still be
triggered in special cases like t g = t A1 or t g = tw . As we will see, most of these rules
can easily be incorporated into the domain bounds of our formulas, e.g. for the stock
level, without changing the structure of the formula itself. Only the simultaneous
arrival of both orders, t A1 = t A2 , changes the structure of the formula significantly.
Thus, we include those scenarios separately in our investigations, see lines 7 − 8 in

Table 3.2. Note, simultaneous arrivals are only of interest for a discrete time line and

The SDMR model – Assumptions

51

have a probability of zero if time is measured continuously.
The six scenarios of a replenishment cycle shown in Table 3.2 can be divided into
cycles with one and two orders. Obviously, a one-order cycle will occur if the demand
D (tw ) during the time window tw satisfies D (tw ) < R1 − R2 . However, this condition

is not sufficient for a one-order cycle in accordance with Table 3.2. Special care has to
be taken here because a second order will not be triggered after an early arrival of the
first order, lt1 < tw , even before tw time units have elapsed. The complete conditions
for a one- and two-order cycle are given in Figure 3.3. Furthermore, two-order cycles
"#$$%&%'()*+,%,)- ,(.*/+,(#*)0%+1)(#2%,)+'1)1%2+'1
can be distinguished by the sequence of their two orders, namely t A1 < t A2 , t A1 > t A2 ,
and t A1 = t A2 .

1%(%&2#'%)!"
#$ !4.&1%&)*5*0%)%

#$&34.&1%&)*5*0%)%

!).&1%&,

3).&1%&

&%67#&%2%'(,8)
9 1%2+'1)! '3)- '! #')(!3 +'1)(!3 ")!"
9 1%2+'1)!)'3)- '! #')!" +'1)(!3 :)!"

&%67#&%2%'(,8
9 1%2+'1);)'3)- '! #')(!3 +'1)(!3 ")!"
9 1%2+'1);)'3)- '! #')!" +'1)(!3 :)!"

Figure 3.3: Different cases for cycles with one or two orders

In order to fully understand the process of dual sourcing and to derive commonly
valid statements and formulas, our first goal is to investigate the eight cases of Table 3.2 separately. We will develop formulas for the expected shortage and the expected cycle for each of these cases and for positive and negative reorder points. Afterwards we will unify the individual formulas to generally valid formulas where it is
possible.

!

52

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

3.1.4

Notation conventions

In order to have a more convenient notation of the different cases that are investigated
in the following sections, we introduce the triple notation (# of orders, lead time relation,
arrival sequence). The attributes of each of these elements are encoded as shown in
+

Table 3.3. Moreover, we use the expressions x and max(0, x ) interchangeably, as well
−

as x and min(0, x ).
# of orders:

1:
2:

time window:
arrivals:

A1 :

one-order cycle, so D (min(lt1 , tw )) < R1 − R2
two-order cycle D (min(lt1 , tw )) ≥ R1 − R2
order one arrives before tw , so lt1 ≤ tw

tw :

order one arrives after tw , so lt1 > tw

A1 :

order one arrives before order two, so lt1 < lt2 + t g

A2 :

order two arrives before order one, so lt2 + t g < lt1

=:

both orders arrive at the same time, so lt1 = lt2 + t g

Table 3.3: Legend of the triple notation

Example 3.1.2. The triple (1, A1 , .) specifies all one-order cycles where lt1 ≤ tw

whereas the notation (2, ., =) represents all two-order cycles with simultaneous arrivals.
Now, we can refer to the eight cases by the notations shown in Table 3.4. Keep in
mind that a triple, e.g. (1, ., .) for one-order cycles, describes a certain subset of the set
containing all possible scenarios. However, it just represents a specific set and is not a
set itself.
Example 3.1.3. The identical expressions p(Case 2), p(1, tw , .), and p(1,tw ,.) stand for the
probability of a one-order cycle where the order arrives after tw .

The SDMR model – Definition of an appropriate probability space
name

triple

# of orders

time window

arrival sequence

Case 1

(1, A1 , .)

1

none

Case 2

(1, tw , .)

1

A1 ≤ t w
A1 > t w

none

Case 3

(2, A1 , A1 )

2

A1 < A2

Case 4

(2, A1 , A2 )

2

A1 ≤ t w

Case 5

(2, A1 , =)

2

A1 = A2

Case 6

(2, tw , A1 )

2

A1 ≤ t w

Case 7

(2, tw , A2 )

Case 8

(2, tw , =)

A1 ≤ t w

A1 > A2

A1 > t w

A1 < A2

2

A1 > t w

A1 > A2

2

A1 > t w

A1 = A2

53

Table 3.4: Investigated cases in dual sourcing
Moreover, we introduce the following equivalent notations for multiple integration
Z T Z X Z X0
0

a

b

f ( x 0 − x ) g( x ) h(t) dx 0 dx dt ≡

0
ZT ZX ZX
t

0

a

x

b

x0

f ( x 0 − x ) g( x ) h(t) dµ

which we think improves the readability of the next chapters significantly.

3.1.5

Definition of an appropriate probability space

One of our milestones is to calculate the expected shortage and the expected average stock on hand during a replenishment cycle. It would be convenient to simply
combine the individually calculated values for the different cases, Case 1 to Case 8.
However, the expected value of a random variable X, defined as
E[ X ] =

Z ∞

−∞

x f ( x ) dx,

(3.1.3)

always integrates over the whole sample space. In order to look just at a partial set
of the whole event space there exists the conditional expected value of X under the

54

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

condition of some random variable Y is defined in the discrete case as
E [Y | X = x ] =

n
1
· ∑ yn h( x, yi )
f ( x ) i =1

(3.1.4)

where f ( x ) is the probability of the event x and h is the joint distribution of Y and X.
In the continuous case things become more complicated especially when there are
more than two random variables. Thus, we have to expand our considerations to
some more exhaustive probability theory for a moment. After that we are able to sum
up the conditional values for disjoint cases.
Example 3.1.4. Given two sets A and B with special properties, which are subject to
further investigation, the expected stock E[stock] can be obtained by
(3.1.5)

E[stock] = E[stock 1 A ] + E[stock 1 B ]

where stock is a random variable X and 1 A is the indicator function of A that yields
1 for all elements in A and 0 otherwise. The special properties include that both sets
A, B ⊂ Ω are disjoint but cover the whole event space, so A ∩ B = ∅ and A ∪ B = Ω

and that they have to be measurable regarding a σ-algebra κ on Ω. Thus, both sets
A and B can be expressed by means of a second random variable Y. Additionally,
this example can be extended to n pairwise disjoint sets Ai where

Sn

i =1

Ai = Ω for

i ∈ {1, ..., n} which can be specified using one single random variable Y.
E[stock] =

n

∑ E[stock 1 Ai ]

(3.1.6)

i =1

We start by giving some definitions concerning probability spaces, see e.g. [Fel71],
before we create an adequate probability space and some random variables. Finally,
we will proof the statements of Example 3.1.4.

The SDMR model – Definition of an appropriate probability space
3.1.5.1

55

Definition of the probability space

Definition 3.1.1 (Probability space). A probability space is a triple (Ω, κ , P) of a sample space Ω, a σ-algebra κ of sets in it, and a probability measure P on κ .
Definition 3.1.2 (σ-algebra, Borel sets). A σ-algebra is a family κ of subsets of a given
set Ω with the following properties:
1. If a set A is in κ so is its complement A = Ω\ A.
2. If { An } is any countable collection of sets in κ , then also their union
intersection

T

An belong to κ .

S

An and

Given any family z of sets in Ω, the smallest σ-algebra containing all sets in z is called
the σ-algebra generated by z.
In particular, the sets generated by the intervals of R n are called Borel sets of R n .
Definition 3.1.3 (Probability measure). A probability measure P on a σ-algebra κ of
sets in Ω is a function assigning a value P( A) ≥ 0 to each set A ∈ κ such that
P(Ω) = 1 and that P({∪ An }) =
non-overlapping sets An ∈ κ .

1

∑ P( Ak ) holds for every countable collection of

n =i

Let us now define an appropriate sample space Ω. From Figure 3.4 on page 67
one can see that the demand d, the two lead times lt1 and lt2 , and the inter-order
time t g are the stochastic variables in our model. Furthermore, the demand is split
up into demand during the lead times dlt1 and dlt2 and demand dtw during the time
window tw for triggering a second order. Note that the sequence of arriving orders in
the two-order case does not influence the number of stochastic variables. The demand
during the total cycle length, tc , is Q1 and Q1 + Q2 in a one-order and a two-order
cycle, respectively. Moreover, the demand dtg during the inter-order time is R1 − R2

in a two-order cycle as we assume that demand does not appear bulky according to

56

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

Assumption 10 on page 45. The variables t g , lt2 , and dlt2 do not exist in a one-order
cycle and we set them to −1. It might also happen that an order cycle is finished before
tw and so the demand dtw cannot be specified. We may set dtw = −1 without any harm

whenever lt1 < tw because we only need tw for cases where the second order is not
allowed to be triggered although R2 is met. This situation can only occur if lt1 ≥ tw .

Considering all those random variables and given the endogenous and fixed pa-

rameters, R1 , R2 , tw , Q1 , and Q2 we define the sample space as
Ω R1 ,R2 ,tw ,Q1 ,Q2 := Ω = {(lt1 , lt2 , t g , tc , dlt1 , dlt2 , dtw )}

(3.1.7)

with the 7-tuple consisting of the parameters as described in Table 3.5. Note, the sample space changes if only one of the endogenous parameters is changed. Furthermore,
definition
lt1

=

lt2

=

tg

=

tc

=

dlt1

=

dlt2

=

d tw

=

range

description

R >0

lead time of order triggered at R1

R >0 ∪ {−1}

lead time of order triggered at R2

R >0 ∪ {−1}

inter-order time

tc

R >0

total cycle time

dlt1


dlt
2

R ≥0

demand during the lead time lt1

R ≥0 ∪ {−1}

demand during the lead time lt2

R ≥0 ∪ {−1}

demand during time window tw

lt1


lt2


 −1


t g

 −1


 −1


dt
w

 −1

if 2-order cycle
else
if 2-order cycle
else

if 2-order cycle
else
if tw ≤ lt1
else

Table 3.5: Definition of stochastic variables in the probability space Ω

The SDMR model – Definition of an appropriate probability space

57

let κ be the σ-algebra generated by the Borel sets of Ω. Given an arbitrary probability
measure P we can define the probability space (Ω, κ , P).
3.1.5.2

Definition of the random variables

Usually we want to assign a probability to a set of events that fulfill certain criteria.
For this purpose random variables are usually used which assign a real value to all elements in Ω which can be conveniently used as selection criteria. Formally, a random
variable is defined as follows, see [Fel71].
Definition 3.1.4 (Random variable, distribution function). A random variable X is a
real function which is measurable with respect to the underlying σ-algebra κ . The
function FX defined by FX (t) = P({ X ≤ t}) is called the distribution function of X.
Thus, a random variable X is a mapping X : Ω → R1 in such a way that X ( A) =

{ X ( a) ∈ R1 | a ∈ A} = B for an arbitrary set A ⊆ Ω. If a set A is determined by

A = {ω | ω ∈ Ω and a < X (ω ) ≤ b} it is then mapped to a subinterval B with

X ( A) = B ⊆ ( a, b]. Moreover, the probability P( A) = FX (b) − FX ( a) is assigned to A.

The term measurable in Definition 3.1.4 refers to the fact that each set of events

A = {ω | ω ∈ Ω and X (ω ) ≤ t}, specified by a random variable X and an arbitrary

value t ∈ R, has to be part of the σ-algebra κ again.

Definition 3.1.5 (Measurability). Let κ be an arbitrary σ-algebra of sets in Ω. A realvalued function g on Ω is called κ -measurable if for each t ∈ R the set of all points x

with g( x ) ≤ t belongs to κ .

It can be shown that all κ -measurable functions form a closed class or family (under pointwise limits) where the smallest of these closed classes is called Baire class.
The interested reader is referred to chapter 4 of Feller [Fel71]. Moreover, in R n the
class of Baire functions is identical with the class of functions measurable with respect
to the σ-algebra κ of Borel sets.

58

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

Definition 3.1.6 (Baire class, Baire functions). The smallest closed class of functions
containing all continuous functions is called the Baire class and its members are called
Baire functions.
One can also reverse the construction of a σ-algebra by using an existing random variable X. Take an arbitrary Borel set B ∈ R1 one can specify the set A ⊆ Ω which

elements are mapped to a value within B by the random variable X. The set A is
called preimage of X ( B) and denoted by
A = X −1 ( B) = {ω | ω ∈ Ω and X (ω ) ∈ B}.

(3.1.8)

Moreover, we also use the notations
A1 = X −1 (t) = {ω | ω ∈ Ω and X (ω ) = t}

A2 = X −1 ( x < t) = {ω | ω ∈ Ω and X (ω ) < t}

(3.1.9)
(3.1.10)

to specify the sets A1 , A2 ∈ Ω with a parameter t ∈ R1 . It can be shown that the col-

lection of sets derived from all Borel sets form a σ-algebra κ1 which may be identical
to κ , but it is usually smaller. This κ1 is also called ”the σ-algebra generated by the
random variable X”, see [Fel71].
After this more elaborate definitions concerning probability spaces we can now define
the following random variables each with the mapping X : Ω → R where ω ∈ Ω.
Xlt1 := ω 7→ lt1 (3.1.11)
Xlt2 := ω 7→ lt2 (3.1.12)
Xtg := ω 7→ t g (3.1.13)
Xdlt

1

Xd tw : = ω 7 → d t w
Xs,A1 := ω 7→ R1 − dlt1

(3.1.15)
(3.1.16)

Xs+Q1 ,A1 := ω 7→ R1 + Q1 − dlt1 (3.1.17)

:= ω 7→ dlt1 (3.1.14)

Note, the first five random variables are projections of ω ∈ Ω to one of its coordinates.

The SDMR model – Definition of an appropriate probability space

59

All random variables starting with Xs calculate a stock level with certain conditions
denoted in the subscript. Thereby, the random variable Xs+Q1 ,A1 is equivalent to the
expression Xs,A1 + Q1 .
Example 3.1.5. The stock level in a one-order cycle just before the arrival of the order
can be expressed by R1 − dlt1 where dlt1 is the demand during the lead time lt1 . This
corresponds to the definition of Xs,A1 .

Of course, several more random variables could be defined here but we will introduce them when needed in later sections. It is important to consider that the random
variables are defined on the whole probability space which might not always be meaningful. Thus, we will have to adapt the set on which the random variables operate in
certain cases.
Example 3.1.6. The random variable Xs,A1 does not consider a potential second order
that has already delivered Q2 units. Thus, one has to be careful using these random
variables and, for example, restrict their domain on a subset A ⊂ Ω.
3.1.5.3

Definition of event sets

By means of the specified random variables Xtg , Xlt1 , and Xs,A1 we can now define the
set of
one-order cycles

Mt g , < 0 = [ X t g ] − 1 ( x < 0 ) ,

(3.1.18)

cycles where the 1st order arrives before tw

Mlt1 ,≤tw = [ Xlt1 ]−1 ( x ≤ tw ), and

(3.1.19)

cycles with shortage before A1 (neglecting Q2 )

Ms,A1 ,<0 = [ Xs,A1 ]−1 ( x < 0)

(3.1.20)

each of which is a set of a real-valued random variable, which is based on κ , and so the
sets itself are an element of κ , as well. Moreover, the set Mlt1 ,≤tw is identical to the set
described by the triple (., A1 , .), see the notation convention in Section 3.1.4, because

there is only a constraint on the arrival of the first order, namely A1 = lt1 < tw .

60

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

Similarly, Mtg ,<0 can be referred to by (1, ., .). Using the probability measure P : κ →

R ≥0 as defined in the probability space (Ω, κ , P) one can assign a probability value to
each of the elements in the sigma algebra κ .

Example 3.1.7. In the continuous case the probability of a shortage per cycle is given
by
Z

P( Ms,A1 ,<0 ) =

f Xs,A ( x ) dx =
1

Z0

x

f Xs,A ( x ) dx = FXs,A (0)
1

1

(3.1.21)

−∞

Xs,A1 ( Ms,A1 ,<0 )

for Case 1 where f Xs,A is the density function of the distribution FXs,A which is speci1

1

fied by the random variable Xs,A1 .
3.1.5.4

Expected values using indicator functions

Attention has to be paid to express the expected value which can only be given in
connection with at least one random variable X, denoted by E[ X ]. Obviously, for a
set M ∈ Ω the undefined expression E[ M] also does not make sense as the real value

attached to each element of the sample space by the random variable X is missing.

In the simple case where the random variable X only maps to a countable number of
different values a1 , a2 , ..., an , so |range( X )| = n, the expected lead time lt1 is given by
E[ Xlt1 ] =

n

∑ ai · P ( Ai )

(3.1.22)

i =1

with Ai = X −1 ( ai ). Note, Ai ∈ κ always holds for i ∈ {1, 2, ..., n} because the random

variable X has to be measurable concerning the σ-algebra κ , see Appendix A.1 for an
example.
In the continuous case we have to use Borel sets for the sample space Ω. Fur-

thermore, the distribution function F, as defined in Definition 3.1.4, may be used.

The SDMR model – Definition of an appropriate probability space

61

According to Feller one can now specify an arbitrary random variable X and specifically choose two simple (discrete) random variables, X and X to give an upper and
 
lower bound for the expected value of X [Fel71]. Diminishing E X − E [ X ] to an
infinitesimal small value we yield the common formula for the expected value of X
E[ X ] =

Z∞

t

−∞

t · f X (t) dt.

(3.1.23)

For more information see Appendix A.2. Note, these findings are not restricted to
one-dimensional sample spaces but also valid for random variables that are based on
a σ-algebra generated by the Borel sets in R n .
We want to express the expected value of a random variable which is restricted to
a subset of Ω. One possibility is to use conditional expectations. Usually, E[Y | X ] is

defined for two arbitrary random variables X and Y. Due to several theoretical problems, for example different domains of X and Y, the formal definition of conditional
expectation is quite complex.
Definition 3.1.7 (Conditional expectation by Doob). Let (Ω, κ , P) be a probability
space, and κ1 a σ-algebra of sets in κ (that is, κ1 ⊂ κ ). Let Y be a random variable with expectation. A random variable U is called conditional expectation of Y
with respect to κ1 if it is κ1 -measurable and equation
E [Y 1 A ] = E [ U 1 A



(3.1.24)

holds for all sets A ∈ κ1 where 1 A is the indicator function for each A. In this case we

write U = E(Y | κ1 ). In the particular case that κ1 is the σ-algebra generated by the
random variables X1 , ..., Xn the variable U reduces to a Baire function of X1 , ..., Xn and
will be denoted by E(Y | X1 , ..., Xn ).
For more details the interested reader is referred to Appendix A.6 or to the elaborate explanations by Feller [Fel71]. Fortunately, we do not need the full capabilities of

62

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

conditional expectations. Given two random variables X and Y which are measurable
on the σ-algebra κ , and the joint probability f ( x, y) of X and Y it is more convenient
to introduce the notation
E[ X 1 A ] =

Z∞ Z∞
y

1 A (y) x f ( x, y) dµ =

x

−∞ −∞

Z

y

Z∞

x

x f ( x, y) dµ

(3.1.25)

Y ( A) −∞

where A ∈ κ and 1 A is the indicator function of A defined as


1 if Y −1 (y) ∈ A
1 A (y) :=

0 else.

(3.1.26)

Note, this can easily be extended to several random variables Y1 , ..., Yn which are all
measurable on κ and their associated sets A1 , ..., An ∈ κ .
Z∞

E[ X 1 A1 ... 1 An ] =

y1

−∞

Z

=

y1

...

Z∞

Z∞

yn

1 A1 (y1 ) ... 1 An (yn ) x f ( x, y1 , ..., yn ) dµ

x

−∞ −∞

...

Z

yn

Z∞

x

x f ( x, y1 , ..., yn ) dµ

(3.1.27)

Y1 ( A1 ) Yn ( An ) −∞

Moreover, this concept can also be applied to only one random variable X, its density
function f , and a set A ∈ κ in the way
E[ X 1 A ] =

Z∞

x

1 A ( x ) x f ( x ) dx

−∞

=

Z

x

x f ( x ) dx.

(3.1.28)

X ( A)

Indicator functions prove to be very flexible in this context. Recall the initial problem that random variables are defined on the whole probability space Ω but their
application might only make sense on a particular subset of Ω. In fact, we can utilize
indicator functions as an easy and convenient mechanism to apply random variables
just to such a subset A ⊂ Ω.

The SDMR model – Definition of an appropriate probability space

63

Example 3.1.8. The expected stock of one-order cycles just before the arrival of its sole
order and where lt1 < tw is given by

E[ Xs,A1 1 Mlt

1 ,≤ t w

1 Mt g , <0 ] =

=

Z∞

x2

Z∞

x1

−∞ −∞
Ztw Z0
x2

x1

1 Mlt

1 ,≤ t w

1 Mtg ,<0 x1 f ( x1 , x2 ) dµ

x1 f ( x1 , x2 ) dµ

(3.1.29)

−∞ −∞

with f ( x1 , x2 ) = f Xs,A ( x ) · f Xlt (y) due to the assumption of statistical independence
1

1

between the demand and the lead time.

Let us now come back to the statement of Example 3.1.4 that

E[ X ] = E[ X 1 A ] + E[ X 1 B ]

(3.1.30)

holds for a random variable X under certain properties for sets A, B ∈ Ω.

Proof. Let X1 , X2 : Ω → R1 be arbitrary random variables on the sample space Ω with
a σ-algebra κ , two individual density functions f X1 , f X2 , and a joint density function

f . Moreover, the sets A1 and A2 are specified by the random variable X2 according
to A1 = X2 −1 ( B1 ) and A2 = X2 −1 ( B2 ) where B1 , B2 ⊂ R1 . Whenever X2 ( A1 ) ∪

64

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

X2 ( A2 ) = X2 (Ω) = B1 ∪ B2 and A1 ∩ A2 = ∅ it then holds
E [ X1 1 A 1 ] + E [ X1 1 A 2 ]

=

Z∞

x2

Z∞

x1

1 A1 x1 f ( x1 , x2 ) dµ +

−∞ −∞

=

Z∞

x1

−∞

=

=

Z∞

x1



x1 f ( x1 , x2 ) dx2 +

X2 ( A 1 )

Z∞

x2

x2

Z∞

x1

1 A2 x1 f ( x1 , x2 ) dµ

−∞ −∞

Z




Z∞

Z

X2 ( A 2 )




x1 f ( x1 , x2 ) dx2  dx1

x1 f ( x1 , x2 ) dµ

−∞ −∞
Z∞
x1

x1 f X1 ( x1 ) dx

−∞

= E [ X1 ] .

(3.1.31)

This shows that the expected value of a random variable can be computed by sum of
two expected values if the restricting sets A1 and A2 are thoughtfully chosen.
Note, f ( x1 , x2 ) = f X1 ( x1 ) · f X2 ( x2 ) in case of statistical independence of X1 and X2

like in our case of lead time and demand. Moreover, this proof can be extended to
several disjoint sets A1 , ..., An , see the appendices A.3, A.4, and A.5.
Now we have the proper theory to describe not only the probability of the eight scenarios Case 1 to Case 8 but also the two major types of inventory costs, namely holding
and shortage costs. These formulas are developed in the following sections.

3.1.6

Measure of applicability

During all the calculations we have to keep in mind that the stock has to be refilled
at least up to the reorder point R1 the moment the last order in a cycle arrives, see
Assumption 8. In this way we assure a certain steady state situation of the stock

The SDMR model – Definition of an appropriate probability space

65

level where there is at most one outstanding order per supply mode, i.e. normal or
emergency delivery. Therefore, the occurring demand in a cycle has to be limited to a
maximum of
dmax
(1,.,.) = Q1

(3.1.32)

dmax
(2,.,.) = Q1 + Q2

(3.1.33)

in a one-order cycle and to

in a two-order cycle.
These limits are independent of the amount of shortage and whether the reorder
points are positive or negative. Due to the fact that both order quantities are subject
to change, for example in an optimization process, we do not consider the maximum
demand while developing the formulas for the expected shortage and the average cycle stock on hand. Of course, these limits will be incorporated by means of a simple
minimum operation, e.g. min(dmax , demand), later in the implementation of these
formulas.
However, for practical reasons it is important to have an indicator of how applicable our model is. This can be achieved by calculating the probability p E of all cases
where the demand exceeds its maximum limits. In contrast to our eight cases we can
reduce the number of cases where the demand exceeds its limits to four.
First, in Case 1 the demand must be between 0 and R1 − R2 until the delivery time.

Otherwise, we will trigger a second order. Thus, it is only possible to exceed the
maximum demand if R1 − R2 > dmax
. The associated probability is given by
(1,.,.)
p E,(1,A1 ,.) =

1 ,R1 − R2 )
Ztw R1Z− R2max(QZ
y

0

x

0

which yields zero if R1 − R2 ≤ Q1 .

x0

Q1

f ( x 0 , y) f ( x, y) l1 (y) dµ

(3.1.34)

66

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)
Second, in Case 2 we will not trigger a second order after tw even if R2 is met. Thus,

we have to consider possibly unlimited demand after tw .
p E,(1,tw ,.) =

Z∞ R1Z− R2 Z∞
y

x

tw

0

x0

f ( x 0 , y) f ( x, tw ) l1 (y) dµ

(3.1.35)

Q1

Third, for all two-order cycles where the first order arrives before tw the demand
is also possibly unlimited because we will only trigger one order per supply mode in
a replenishment cycle. Moreover, now the maximum demand Q1 + Q2 must not be
exceeded until the second delivery time, see Assumption 8.
p E,(2,A1 ,.) =

Ztw Zy Z∞
y

0

t

0

z

Z∞

x

Z∞

x0

f ( x 0 , max(y, z + t)) f ( x, y) l2 (z) g(t), l1 (y) dµ

(3.1.36)

0 R1 − R2 Q1 + Q2

Last, for all cases (2, tw , .) the probability of exceeding the maximum demand is
p E,(2,tw ,.) =

Z∞ Ztw Z∞
y

tw

t

0

z

Z∞

x

Z∞

x0

f ( x 0 , max(y, z + t)) f ( x, tw ) l2 (z) g(t), l1 (y) dµ

(3.1.37)

0 R1 − R2 Q1 + Q2

The total probability p E is given by the sum
p E = p E,(1,A1 ,.) + p E,(1,A1 ,.) + p E,(1,A1 ,.) + p E,(1,A1 ,.)

(3.1.38)

due to the fact that the subcases are disjoint.
A high probability p E indicates that the given setting of replenishment parameters,
Q1 , Q2 , R1 , and R2 , will most likely lead to an unusual replenishment cycle, i.e. not
covered by our model. Assumption 8 assures that the stock level remains in a certain steady state with having at maximum one outstanding order per supply mode
simultaneously. In other words p E is not only an indication for the applicability of the
model regarding the given replenishment parameters but also an indication for how
likely additional orders will be necessary to remain a steady average stock level across
several replenishment cycles.

The SDMR model – Definition of an appropriate probability space

67

Saw-tooth stochastic parameter 1-order (demand stochastic, LTs stochastic)
stock

lt1

R1
dtw

Q1

tg= –1
lt2= –1
dlt 2= –1

R2

t0

dt c =Q1
dlt1

tA

tw

tc

time

1

"#$%&''&()*&'+(#*&,+)-#.#/0&0.)1%'.20.)320/#42)*&'+(#*&,+5)67* *&'+(#*&,+8
(a) One-order cycle

*&'+;

!"!

#:

&! '#: ( #1
<

%"

$:

#

#1

%!"

"$

%!""

!

&! '$:
+
*+$1

$1

!""
!9

!%

!"

"#

!) 1

!) :

&,/0

(b) Two-order cycle

Figure 3.4: Stochastic variables (red) in a one- and two-order replenishment cycle

1

68

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

3.2

Probabilities of the reorder cycle scenarios

In this chapter we will give the probability of occurrence for each of the eight cases
mentioned in Table 3.4.

3.2.1

General probabilities of one-order and two-order cycles

In the following expressions for the probabilities p(one-order cycle) and p(two-order cycle)
to trigger one and two orders, respectively, are developed.
Only one order will be triggered in the two cases represented by Case 1 and Case 2
of Table 3.4. In Case 1, or (1, A1 , .), the lead time lt1 is smaller than tw and the second
reorder point is not reached before lt1 . In Case 2, or (1, tw , .), the lead time lt1 and the
second reorder point is not reached till tw .
p(Case 1) = p(1, A1 , .) =

Ztw R1Z− R2
x

y

0
R
−
Z∞ 1Z R2

f ( x, y) l1 (y) dµ

(3.2.1)

f ( x, tw ) l1 (y) dµ

(3.2.2)

0

p(Case 2) =

p(1, tw , .) =

y

tw

x

0

Both equations (3.2.1) and (3.2.2) look very alike at the first glance. However, the big
difference is until when the second reorder point is not allowed to be met in order
to still remain a one-order scenario. In Case 2 the demand during the whole time
window tw is regarded. In Case 1 the time frame for the demand to reach R2 is limited to the arrival time of the first order, expressed by f ( x, y). As a consequence the
probability p(1, ., .) for a one-order cycle to occur is given by
p(1, ., .) = p(1, A1 , .) + p(1, tw , .) =

Z∞ R1Z− R2
y

0

x

f ( x, min(y, tw )) l1 (y) dµ.

(3.2.3)

0

Similarly, we can give the probabilities p(2, A1 , .) and p(2, tw , .) for two-order cycles
if we neglect the sequence of arrivals for a moment. However, now the second reorder

The SDMR model – Probabilities of the reorder cycle scenarios

69

point R2 always has to be met. Note, reaching a stock level of R2 or below in t time
units is equivalent to a demand of x ≥ R1 − R2 during t time units.
Ztw

p(2, A1 , .) =

Z∞

y

x

0 R1 − R2
∞
Z∞
Z

p(2, tw , .) =

x

y

f ( x, y) l1 (y) dµ

(3.2.4)

f ( x, tw ) l1 (y) dµ

(3.2.5)

t w R1 − R2

Adding both equations (3.2.4) and (3.2.5) leads to the probability p(2, ., .) of a twoorder cycle.
p(2, ., .) = p(2, A1 , .) + p(2, tw , .) =

Z∞

y

Z∞

x

f ( x, min(y, tw )) l1 (y) dµ

(3.2.6)

0 R1 − R2

Of course, the total probability of one- and two-order cycles has to be equal to one.
This coincides with the sum of p(1, ., .) and p(2, ., .).
p(., ., .)

= p(1, ., .) + p(2, ., .)
=

=

Ztw Z∞
y

0
tZw

y

0

x

f ( x, y) l1 (y) dµ +

Z∞ Z∞
y

tw

0

1 · l1 (y) dy +

Z∞

y

tw

x

f ( x, tw ) l1 (y) dµ

0

1 · l1 (y) dy

=1

(3.2.7)

Note, a demand distribution D (t) gives the probability that a demand x ≥ 0 will occur
within a given period t. Thus, D (t) is a complete distribution for each 0 < t ∈ R and
Z∞

x

f ( x, t) dx = 1

(3.2.8)

0

always holds an arbitrary t > 0. Consequently, D specifies a whole set of distributions and not just a single distribution. Moreover, the demand distribution might be
different for two periods t1 6= t2 .

70

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

3.2.2

Detailed probabilities of two-order cycles

In the latter section we did not consider the arrival sequence in the two-order cycle.
Now we will further break down the two-order equations (3.2.4) and (3.2.5) according
to the different arrival patterns. In other words, we develop the probabilities for Case 3
till Case 8 shown in Table 3.4. Thereby, we utilize the inter-order-time distribution G
given by
G(T ) =

Z∞

ZT

t

x

f ( x, t) dµ

(3.2.9)

0 R1 − R2

and its density function g(t) and replace the formulation by means of the demand
density function f ( x, t) used before in the equations (3.2.4) and (3.2.5).
p(Case 3) =

p(2, A1 , A1 ) =

Ztw Zy Z∞

l2 (z) g(t) l1 (y) dµ

(3.2.10)

Ztw Zy yZ−t

pcont (Case 5) =

p(2, A1 , A2 ) =

l2 (z) g(t) l1 (y) dµ

(3.2.11)

pcont (2, A1 , =) =

p(2, tw , A1 ) =

p(2, tw , A2 ) =

pcont (2, tw , =) =

t

y

z

0 y−t

l2 (y − t) g(t) l1 (y) dµ

= 0 (3.2.12)

Z∞ Ztw Z∞

l2 (z) g(t) l1 (y) dµ

(3.2.13)

Z∞ Ztw yZ−t

l2 (z) g(t) l1 (y) dµ

(3.2.14)

y

t

z

0 y−t

y

tw

pcont (Case 8) =

z

0 0 0
tZw Zy yZ−t

tw

p(Case 7) =

t

y

0

p(Case 6) =

z

0 y−t

0

p(Case 4) =

t

y

t

z

0
Z∞ Ztw yZ−t
y

tw

0

t

z

0 y−t

l2 (y − t) g(t) l1 (y) dµ

= 0 (3.2.15)

In the latter equations the first integration sign over the lead time y = lt1 addresses
the constraint that the first order arrives before or after the end of the time window tw ,
so t A1 < tw or t A1 > tw , respectively. The integration over the inter-order time t = t g

The SDMR model – Expected shortage of the reorder cycle scenarios

71

assures that the second order is triggered in time, so t g < min(y, tw ). Note that y is
always identical to t A1 because we start the replenishment cycle with triggering the
first order at t0 = 0. The third integration over the lead time z = lt2 takes care of the
arrival sequence of order one and two, where t A2 = lt1 + t g .
Further, note that the probability of a simultaneous arrival of both orders is zero in
the continuous case, so pcont (Case 5) = 0 and pcont (Case 8) = 0, and can be neglected. However, in the discrete case (e.g. days) the probability pdisc (Case 5) and
pdisc (Case 8) is not necessarily zero and has to be considered in the calculations.
p

disc

(Case 5) =

tw

y

∑∑

y =2 t =1

pdisc (Case 8) =

∞

l2 ( y − t ) g ( t ) l1 ( y ) ≥ 0

tw

∑ ∑

y = t w t =1

l2 ( y − t ) g ( t ) l1 ( y ) ≥ 0

(3.2.16)
(3.2.17)

Adding all these probabilities must yield the total probability that a two-order cycle
occurs.
p(2, ., .) = p(Case 3) + p(Case 4) + p(Case 5) + p(Case 6) + p(Case 7) + p(Case 8)
(3.2.18)
These detailed probabilities for each of the eight cases answer the first part of our
research question RQ 1 on page 5. Next, we address the expected shortage in a reorder
cycle.

3.3

Expected shortage of the reorder cycle scenarios

The objective of investigating the amount of shortage is to determine the expected
costs that arise from not satisfying customer demand. This is a cut-off function


−bv · y if y < 0
CB ( y ) =
(3.3.1)

0
else

72

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

with y, the amount of units on stock and bv , the variable costs per unit of shortage
(backorder). This cut-off mechanism can be conveniently described using random
variables and the indicator functions of appropriate subsets of Ω.
h
Example 3.3.1. The shortage costs in Case 1 is CB,(1,A1 ,.) = −bv E Xs,A1 1(1,A1 ,.) 1 Ms,A

1 ,<0

i

In the following we will first give a general and systematic overview of the random
variables, sets, and indicator functions that are needed to calculate the shortage for all
scenarios, Cases 1 – 8. Then, we will go into details for the single cases.
case
Case 1

(1, A1 , .)

Case 2

(1, tw , .)

shortage

Case 3

(2, A1 , A1 )

Case 6

(2, tw , A1 )

dlt1 − R1

before unique arrival, A1 , if R1 − dlt1 < 0

dlt1 − R1

before 1st arrival, A1 , if R1 − dlt1 < 0

dlt2 − R2 − Q1
R1 − dlt1 − R2 + dlt2

Case 4

(2, A1 , A2 )

Case 7

(2, tw , A2 )

Case 5

(2, A1 , =)

Case 8

(2, tw , =)

shortage condition

dlt2 − R2

dlt1 − R1 − Q2
R2 − dlt2 − R1 + dlt1

dlt1 − R1

(dlt2 − R2 )

before 2nd arrival, A2 , if R1 − dlt1 ≥ − Q1
and R2 + Q1 < dlt2

before 2nd arrival, A2 , if R1 − dlt1 < − Q1
before 1st arrival, A2 , if R2 − dlt2 < 0

before 2nd arrival, A1 , if R2 − dlt2 ≥ − Q2
and R1 + Q2 < dlt1

before 2nd arrival, A1 , if R2 − dlt2 < − Q2

before simultaneous arrivals, A1 = A2 , if
R1 − dlt1 < 0 (or equiv. R2 − dlt2 < 0)

Table 3.6: Conditions of a shortage in Case 1 to 8
In all eight scenarios, Case 1 to 8, there exist different probabilities and formulas for
the expected shortage. Table 3.6 gives an overview over the different conditions for a
shortage and the middle column illustrates that there exist three types of sources for a
shortage, namely the demand, the order quantities, and the reorder points. Of course,

.

The SDMR model – Expected shortage of the reorder cycle scenarios

73

there will be some shortage if the demand is unexpectedly high but a negative reorder
point or too small order quantities can lead to shortage, as well. This corresponds to
our intuition.
All four cases in the first and the last row of Table 3.6 have in common that all
orders, one or two, arrive at one single point in time and, thus, only a single condition
suffices to capture all stockout situations. Note, for Case 5 and 8 the two conditions are
equivalent just as their formulas for the shortage. The other four cases, displayed in
the two middle rows of Table 3.6, are cycles with two separate delivery times. Before
each of them shortage might occur as illustrated in Figure 3.5a.
Example 3.3.2. In Figure 3.5a the replenishment cycle starts at t0 with a positive stock
level of R1 units where a normal order is triggered. The stock declines and hits the
second reorder point, R2 , within the allowed time window tw after t g time units and
an emergency order is triggered. Before the first order arrives, either the normal or
the emergency order, the stock depletes. At the time of the first arrival the stock out
has accumulated to SH A units which can be completely satisfied by the first delivery
containing Q A units. Until the arrival of the second order the stock depletes again but
the stock out which amounts to SHB units can be completely covered by the delivered
Q B units. Moreover, the second delivery brings the stock level above the first reorder
point R1 as required by Assumption 8. In total there are two separate periods where
stock out occurs which accumulates in total to SH A + SHB units.
Unfortunately, the amount of the second shortage is influenced by the first one as
shown in the two drawings of Figure 3.5. Depending on the size of the shortage before the first arrival there are two cases which require different treatments. Namely,
the shortage is small enough to be covered by the first delivery or not. This is caused
by our policy to serve backlogged demand as soon as possible, analogous to the approach of Moinzadeh and Nahmias [MN88] which, in our experience, is also a common practice in the spare parts industry, for example.

74

C HAPTER 3. A M ODEL F OR "#$%&''&()*+,-,.#/*0%'.1,.*232/,*4('.&#5,*67
S TOCHASTIC D UAL S OURCING
(SDMR)
9":;*64&'2(#4&<2;

"#$%&''&()*+,-,.#/*0%'.1,.*232/,*4('.&#5,*6"7!89:*64&'2(#4&;2:

8

4&'2=

4&'2>

#9

#8

!"

#0
!"
<

=
#0

$%

!<

!=

!"

&<?,
$%

)*(

$(
!"

)*%

)*(

$(

&;>,

)*%

!'

!'

!&

!&

(a) First shortage < order quantity

(b) First shortage > order quantity

Figure 3.5: Relationship between the shortages before and after the first delivery
The condition for a shortage before the first arrival in a two-order cycle is similar to
the condition for some shortage in a one-order cycle: a shortage occurs if the demand
during the lead time of the earliest delivery is higher than the associated reorder point.
This translates into the conditions R1 − dlt1 <! 0 and R2 − dlt2 < 0, mentioned in

Table 3.6 for all cases (., ., A1 ) and all cases (., ., A2 ) where the first and the second
order, respectively, arrives first.

Whenever not all potentially backlogged demand can be satisfied by the first delivery, the shortage between the first and second arrival is identical to the demand
during this time, see Figure 3.5b. Let us denote the first arriving order by i and the
second one by j. Then, the demand d[t A −t A ] between the two arrivals is
j

i

d[t A −t A ] = Ri − dlti − R j + dlt j
j

i

(3.3.2)

which coincides with the shortage stated in Table 3.6 for the condition Ri − dlti < − Qi .

Otherwise, if the stock level is refilled to or above zero, the demand between the

arrivals can initially be satisfied. The stock level after the first delivery i is equal to
Ri − dlti + Qi . Subtracting the positive stock from d[t A −t A ] yields a shortage SH of
j

i

SH = d[t A −t A ] − ( Ri − dlti + Qi )
j

i

= dlt j − R j − Qi

(3.3.3)

!

The SDMR model – Expected shortage of the reorder cycle scenarios

75

which is again identical to the expressions given in Table 3.6 for the condition Ri −
dlti ≥ − Qi after appropriately substituting 1 and 2 for i and j.

From Table 3.6 one can now derive appropriate random variables and associated indicator functions to determine the expected shortage in the different cases. In favor of
a better understanding this is done directly in the sections for the individual cases.
For each of the eight basic scenarios, Case 1 through 8, the calculation of the shortage strongly depends on the setting of the reorder points R1 and R2 . Because we allow
for positive and negative reorder points there are three possible settings which are displayed in Table 3.7. We indicate the different conditions for the reorder points in the
R1 > R2 ≥ 0

both reorder points are positive

0 > R1 > R2

both reorder points are negative

R1 ≥ 0 > R2

only second reorder point is negative

Table 3.7: Possible settings for choosing the reorder points R1 and R2
subscript of the function. Note, only one of the three possibilities is effective at a time
since the reorder points R1 and R2 are fixed and known in advance.
Example 3.3.3. If we look at Case 1 and impose the condition R1 ≥ 0 > R2 we write

p R1 ≥0> R2 (1, A1 , .) for its probability. Obviously, the probability of Case 1 does not

change with different settings for the reorder points, so p R1 ≥0> R2 (1, A1 , .) = p(1, A1 , .).

Note, the probability of a shortage, for example in Case 1, depends on the choice
of R1 and R2 but it can never exceed the probability of its underlying case (Case 1).
h
Example 3.3.4. The expression ER1 ≥0> R2 − Xs,A1 1(1,A1 ,.) 1 Ms,A

1 ,<0

i

represents the ex-

pected shortage of Case 1 in a scenario where the first reorder point is positive and

76

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

the second is negative. In contrast to Example 3.3.3 the shortage does strongly depend on positive or negative reorder points.
Combining the random variables with the indicator functions for the defined subsets
representing the shortage and the eight cases, Case 1 to 8, we are in a position to express for each of the cases the formulas for the three scenarios concerning the reorder
point settings. Afterwards we will try to condense the formulas to the total shortage
and a combined expression covering all reorder point settings.

3.3.1

One-order cycles

The two possible cases of one-order cycles are graphically displayed in Figure 3.6. We
will consider Case 1 and 2 jointly in this section to show the parallelism in developing
the formulas"#$%&''&()*+#,-*.*,('/&#0-*1,&'2(#,&324
for both cases. In accordance with
Table 3.4 one can summarize the
"#$%&''&()*+#,-*.*,('/&#0-*1,&'2(#,&324
,&'27

,&'27

#!.

".

#!6

"6

$.

$6
".

"6

!5

!&

!' .

&38-

!5

!' 6

!&

&38-

!%

!%

(a) Case 1

(b) Case 2

Figure 3.6: The two possible scenarios for a one-order cycle
conditions for both one-order scenarios as follows:
Case 1:

The lead time lt1 does not exceed the time window tw and the demand
during lt1 is too small to trigger a second order, so lt1 ≤ tw and D (lt1 ) <
R1 − R2 .

!

!

The SDMR model – Expected shortage of the reorder cycle scenarios
Case 2:

77

The lead time lt1 is longer than tw and the second reorder point is not
reached during tw , so lt1 > tw and D (tw ) < R1 − R2 .

Table 3.6 shows that there is one single condition for both cases which leads to a stockout situation, namely when R1 − dlt1 < 0 just before the arrival A1 . If we consider the

set of elements in Ω which refer to (1, A1 , .) and the set Ms,A1 ,<0 , as specified in Equation (3.1.20), then the associated indicator functions are given by 1(1,A1 ,.) and 1 Ms,A

1 ,<0

,

respectively. Together with the random variable Xs,A1 the expected shortage for Case 1
can now be expressed by E[− Xs,A1 1(1,A1 ,.) 1 Ms,A

1 ,<0

]. Similarly, one can give the ex-

pected shortage for Case 2. Both results can be summed up to E[− Xs,A1 1(1,.,.) 1 Ms,A

1 ,<0

],

the expected shortage for both one-order scenarios.
E[− Xs,A1 1(1,.,.) 1 Ms,A

1 ,<0

]=

E[− Xs,A1 1(1,A1 ,.) 1 Ms,A

1 ,<0

] + E[− Xs,A1 1(1,tw ,.) 1 Ms,A

1 ,<0

]

(3.3.4)

The three possible settings of reorder points, see Table 3.7, can be illustrated by
moving the x-axis of Figure 3.6a and 3.6b vertically. Next, we will separately elaborate
the formulas for these three possible settings for the reorder points before stating a
unified equation.
3.3.1.1

Positive reorder points

We know that the two reorder points follow the inequality R1 > R2 , see Assumption 2.
Further we temporarily assume that R2 ≥ 0. Now, it directly follows from D (lt1 ) <

R1 − R2 in the description of Case 1 that the demand during the lead time satisfies the

inequality D (lt1 ) < R1 . Having R1 units on stock in t0 and a demand D (lt1 ) < R1
during the lead time there will be no stockout until the first order arrives.




p R1 > R2 ≥0 1(1,A1 ,.) and 1 Ms,A ,<0
= 0
1
h
i
ER1 > R2 ≥0 − Xs,A1 1(1,A1 ,.) 1 Ms,A ,<0 = 0
1

(3.3.5)
(3.3.6)

78

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

A stockout situation is therefore restricted to Case 2. The expected shortage is determined by all cases where the lead time follows the inequality y = lt1 > tw and the
two conditions x = D (tw ) < R1 − R2 and x 0 = D (lt1 ) > R1 hold for the demand. This

leads to the formula for the amount of shortage where the inequalities are directly
transferred to the bounds of the integrations.
h

ER1 > R2 ≥0 − Xs,A1 1(1,tw ,.) 1 Ms,A
Z∞ R1Z− R2 Z∞
y

tw

x

0

x0

R1 − x

1 ,<0

i

=

( x 0 − R1 + x ) f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dµ

(3.3.7)

The associated probability is given by


p R1 > R2 ≥0 1(1,tw ,.) and 1 Ms,A
Z∞ R1Z− R2 Z∞
y

tw

x

0

x0

1 ,<0

R1 − x



=

f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dµ

(3.3.8)

which is always equal to the expression of the expected shortage without the scaling
factor, here ( x 0 − R1 + x ), just after the most inner integration sign. Due to the close

and simple relation between the expression for the expected value and its probability
we refrain from explicitly writing the formula for the probability unless it serves a
better understanding.

3.3.1.2

One positive and one negative reorder point

Now, let us revoke the constraint that both reorder points have to be positive and
replace R1 > R2 ≥ 0 by R1 ≥ 0 > R2 in a first step. Again, the two possibilities

Case 1 and Case 2 that can cause an one-order cycle remain the same. In contrast to
h
i
non-negative reorder points the expected shortage ER1 ≥0> R2 − Xs,A1 1(1,A1 ,.) 1 Ms,A ,<0
1

in Case 1 is positive whenever the demand during the lead time exceeds R1 . In order

The SDMR model – Expected shortage of the reorder cycle scenarios

79

to not become a two-order cycle the demand must remain below R1 − R2 .
h

ER1 ≥0> R2 − Xs,A1 1(1,A1 ,.) 1 Ms,A

1 ,<0

i

=

Ztw R1Z− R2
y

0

x

R1

( x − R1 ) f ( x, y) l1 (y) dµ

(3.3.9)

In Case 2 additional shortage can occur before the second reorder point is reached.
h
i
ER1 ≥0> R2 − Xs,A1 1(1,tw ,.) 1 Ms,A ,<0 =
1
R
Z∞

y

tw



Z 1 Z∞
x

x0

0 R1 − x

R1Z− R2 Z∞
x

0

R1

x0

( x 0 − R1 + x ) f ( x 0 , y − tw ) f ( x, tw ) dµ +


( x 0 − R1 + x ) f ( x 0 , y − tw ) f ( x, tw ) dµ l1 (y) dy

(3.3.10)

The two summands in Equation (3.3.10) above are very similar and can be unified
if the lower bound R1 − x of the most inner integration is truncated to non-negative
+

values. By replacing R1 − x with [ R1 − x ] we can write the simplified expression
h
ER1 ≥0> R2 − Xs,A1 1(1,tw ,.) 1 Ms,A
Z∞ R1Z− R2
y

tw

x

0

Z∞

x0

[ R1 − x ] +

1 ,<0

i

=

( x 0 − R1 + x ) f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dµ.

(3.3.11)

+

Note, even though we changed the lower integration bound to [ R1 − x ] in Equa-

tion (3.3.11) this is not reflected in the first factor inside the integration. The reason
lies in the fact that R1 − x < 0 coincides with the shortage (possibly between 0 and

− R2 ) till tw is reached and has to be added to the shortage x 0 beyond tw till the end of

the lead time lt1 . Equation (3.3.11) is identical to Equation (3.3.7) for the case of positive reorder points except for restricting the expression R1 − x to non-negative values
+

[ R1 − x ] in the lower bound of the most inner integration.

80

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

In total one would expect higher shortages in the case of a negative second reorder
point because the demand can now be higher without triggering a second order. This
is also reflected in the equations (3.3.9) and (3.3.11).
First, if one compares the formulas for shortage in Case 1 then Equation (3.3.9)
yields a positive value which is greater than zero, the result of its counterpart: Equation (3.3.6).
Second, a closer look at Equation (3.3.11) reveals that the interval [0, R1 − R2 ] is

larger for a negative R2 than for R2 ≥ 0. The same is the case in Equation (3.3.7).

Given R1 ≥ 0 the reduction of R2 increases the result of the integration ceteris paribus.
Moreover, a negative or reduced second reorder point R2 also increases the total
probability p(1, ., .) that only one order is issued during a cycle, see Equation (3.2.3).

3.3.1.3

Negative reorder points

If both reorder points are negative the amount of shortage has already accumulated
to R1 even before the first order is triggered at t0 . Thus, we have to add this initial
shortage to Case 1 and 2. According to the calculation rules, see Appendix A.4, the
additional term to be added to the expected shortage in Case 1 is given by

= t0
SH(t1,A

1 ,.)

= − R1 · p(1, A1 , .)
= − R1 ·

Ztw R1Z− R2
y

0

x

0

f ( x, y) l1 (y) dµ.

(3.3.12)

The SDMR model – Expected shortage of the reorder cycle scenarios

81

In total the expected amount of shortage for Case 1 is determined by
i
h
E0> R1 > R2 − Xs,A1 1(1,A1 ,.) 1 Ms,A ,<0
1

= (− R1 ) ·

Ztw R1Z− R2
y

0

Ztw R1Z− R2

=

y

x

0

0

x

f ( x, y) l1 (y) dµ +

0

Ztw R1Z− R2
y

0

x

x f ( x, y) l1 (y) dµ

0

( x − R1 ) f ( x, y) l1 (y) dµ.

(3.3.13)

Similar to Equation (3.3.13) the expected shortage for Case 2 is given by
i
h
E0> R1 > R2 − Xs,A1 1(1,tw ,.) 1 Ms,A ,<0 =
1

Z∞ R1Z− R2 Z∞
y

x

tw

0

0

x0

( x 0 − R1 + x ) f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dµ.

(3.3.14)

Note, the expression x 0 − R1 + x in Equation (3.3.14) consists of three positive terms.

First, R1 < 0 reflects the fact that now there exists no positive reorder point and that
we already start with a shortage of − R1 in t0 . Second and third, all demand x and

x 0 after t0 and until the arrival of the order immediately increases the amount of total

shortage.
In sum the expected shortage is given by
h
i
E0> R1 > R2 − Xs,A1 1(1,.,.) 1 Ms,A ,<0
1

=

Ztw R1Z− R2
y

0

+

x

( x − R1 ) f ( x, y) l1 (y) dµ

0
R
Z∞ 1Z− R2 Z∞
y

tw

x

0

0

x0

( x 0 − R1 + x ) f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dµ

(3.3.15)

There exists always some shortage in this scenario and, thus, the stockout probability
p(SHR1 1 <0 ) is equal to the maximum possible value, namely the probability p(1, ., .)
that a one-order cycle occurs. We stated before that the expected value and the probability are closely related. Neglecting the first factors ( x − R1 ) and ( x 0 − R1 + x ) in

82

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

both terms of Equation (3.3.15) leads to the probability that a shortage occurs either in
Case 1 or Case 2.

p0> R1 > R2 1(1,.,.) and 1 Ms,A

1 ,<0



=

Ztw R1Z− R2

=

Ztw R1Z− R2

f ( x, y) l1 (y) dx dy +

Z∞ R1Z− R2

f ( x, min(y, tw )) l1 (y) dx dy

y

0

y

x

x

0

x

0

0

Z∞ R1Z− R2
y

tw

0

y

y

tw

0

0

=

f ( x, y) l1 (y) dx dy +

x

Z∞ R1Z− R2 Z∞

x

0

x0

f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dx 0 dx dy

1 · f ( x, tw ) l1 (y) dx dy

0

= p(1, ., .).

(3.3.16)

This result corresponds to the probability of a one-order cycle given in Equation (3.2.3).

3.3.1.4

Combined formulas for all reorder point scenarios

Most of the formulas for the expected stock in the three scenarios R1 > R2 ≥ 0,
R1 ≥ 0 > R2 , and 0 > R1 > R2 are very similar. In fact, it is possible to combine the

three formulas for the expected shortage and write a compact expression.
h
E − Xs,A1 1(1,.,.) 1 Ms,A

=

Ztw R1Z− R2
y

0

+

R1

x
+

tw

x

0

+

[ x − R1 ] f ( x, y) l1 (y) dµ

Z∞ R1Z− R2
y

1 ,<0

i

Z∞

x0

[ R1 − x ] +

( x 0 − R1 + x ) f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dµ.

(3.3.17)

The SDMR model – Expected shortage of the reorder cycle scenarios

83

The associated probability that a stockout situation occurs in a one-order cycle is

p 1(1,.,.) and 1 Ms,A

=

Ztw R1Z− R2
y

0

R1

x
+

1 ,<0



=

f ( x, y) l1 (y) dµ +

Z∞ R1Z− R2
y

tw

x

0

Z∞

x0

[ R1 − x ] +

f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dµ (3.3.18)

Note that the basic assumption R1 > R2 holds throughout these calculations. As a
consequence, the first term for the expected shortage in Equation (3.3.17) can only be
positive if the second reorder point R2 is negative. Otherwise, the first term is equal
to zero.
+

+

Moreover, the expressions [ R1 ] and [ R1 − x ] for the lower bound of the most

inner integral take care of the fact that there might be some stock left at t0 to cover
some customer demand between t0 and the arrival of the order before running into a
stockout situation. However, when R1 is negative or the demand x reaches R1 every
additional demand immediately increases the amount of shortage.
It is important to mention that we do not specify which supply mode, normal or emergency, is assigned to the identifiers 1 and 2, see Assumption 1 on page 42. Consequently, the reorder point of the normal supply mode might be represented by R1 and
R2 , respectively. The concrete assignment of the supply mode to the identifiers depends on the given values for both reorder points as the constraint R1 > R2 always
has to hold. Due to this exchangeability of the identifiers the equations (3.3.17) and
(3.3.18) for the expected amount of shortage and the associated stockout probability,
respectively, are valid for all combinations of positive and negative reorder points.
The only case excluded here are identical reorder points R1 = R2 which represents an
order splitting approach. The interested reader is referred to the literature about order
splitting, see Chapter 2.2.2 on page 29.

84

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

So far, we have presented the SDMR model as a probability space in Section 3.1 and
eight different cases have been identified, see Table 3.4 on page 53. In the first two
cases only one order is triggered. The present section has developed formulas for
the expected shortage in these two cases. In the following Section 3.3.2 analogous
formulas are developed for all six two-order cases.

3.3.2

Two-order cycles

Similar to the one-order case one can develop the formulas for the shortage in a twoorder cycle. The difference mainly lies in the incorporation of a second reorder point
and a second stochastic lead time. Table 3.6 on page 72 shows that the pair of cases

(3, 6), (4, 7), and (5, 8) can be jointly analyzed. Therefore, we explain only the first of
the two cases per pair and refer the interested reader to Appendix B for the remaining
cases. In order to avoid tedious repetitions we will not develop the formulas for all
reorder point settings explicitly either but only give the formulas for positive reorder
points and the unified formulas.
We can directly derive the necessary random variables and indicator functions for
their associated sets. In addition and analogous to the random variable Xs,A1 = R1 −

dlt1 which has been defined in Equation (3.1.16) on page 58 we define the random

variable

Xs,A2 = ω 7→ R2 − dlt2

(3.3.19)

which represents the stock level at the point of time t A2 when the emergency order
arrives. Using the random variables Xs,A2 and Xs,A1 we further define the following

The SDMR model – Expected shortage of the reorder cycle scenarios

85

subsets of Ω
Ms,A2 ,<0 = [ Xs,A2 ]−1 ( x < 0)

(3.3.20)

Ms,A1 ,<−Q1 = [ Xs,A1 ]−1 ( x < − Q1 )

(3.3.21)

Ms,A1 ,<−Q2 = [ Xs,A1 ]−1 ( x < − Q2 )
Ms,A2 ,<−Q1 = [ Xs,A2 ]−1 ( x < − Q1 )
Ms,A2 ,<−Q2 = [ Xs,A2 ]−1 ( x < − Q2 )

(3.3.22)
(3.3.23)
(3.3.24)

which contain all elements that suffice a certain condition on the stock level at the
points of time t A2 or t A1 , respectively, e.g. a negative stock level at t A2 in Equation (3.3.20).
All two-order cycles trigger the second order after a stochastic positive time, the
inter-order time t g , see Figure 3.1 on page 41. The inter-order time follows the distribution G with its density function g and can be calculated by means of the two reorder
points, R1 and R2 , and the demand density function f .
G (t) =

Z∞

f ( x, t) dx

(3.3.25)

R1 − R2

Note, in the following we do not consider the maximum limits for the demand
dmax
and dmax
which enforce Assumption 8 on page 43 and which are given explicitly
(1,.,.)
(2,.,.)
in equations (3.1.32) and (3.1.33) on page 65. These limits depend strongly on the
choice of Q1 and Q2 as mentioned before and will be considered by the measure of
applicability, see Chapter 3.1.6.
3.3.2.1

First order arrives first

The two possible cases of two-order cycles where the first order arrives first are Case 3
and Case 6. Both cases are intimately related to each other as one can see from Figure 3.7. Given the fact that a second order is triggered and the first order arrives before

86

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

the second it only matters whether the first order arrives before or after tw , the end of
the time window to trigger an emergency order. This is indicated by the blue arrow in
Figure 3.7. It is the only difference between Case 3 and Case 6 and due to this strong
#$%&'((')*+,$-.+/+0+1+-)(2'$3.+4-'(5)$-'657
similarity we can develop
their formulas jointly.
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Figure 3.7: Scenarios for a two-order cycle where the first order arrives
Note, in the continuous case the event t A1 = tw is a null set and has a probability
of zero. In a discrete scenario, however, Case 6 includes the event t A1 = tw except for
a replenishment policy that does not allow to trigger an emergency order during the
time unit, e.g. day, where the normal order arrives.
According to Table 3.6 at the beginning of this chapter on page 72 there are three
different conditions how shortage can occur. Thereby, we distinguish between the
shortage that occurs before and after the first arrival, A1 . However, this is only an
!"
artificial separation in favor of a more systematic investigation. We denote the two

different cases in the superscript of the specific parameters. This is exemplarily summarized for the expected shortage in Table 3.8.

The SDMR model – Expected shortage of the reorder cycle scenarios
time

87

reorder point settings

condition

R1 > R2 ≥ 0

R1 ≥ 0 > R2

0 > R1 > R2

ER≥1A>1R2 ≥0

ER≥1A≥10> R2

E0≥>AR11 > R2

ER<1A>1R2 ≥0

t < t A1
t ≥ t A1

ER<1A≥10> R2

E0<>AR11 > R2

Table 3.8: Notation of the different cases of expected shortage in a two-order cycle
Referring to Table 3.6 on page 72 again, the three different parts which contribute
to the shortage in Case 3 and Case 6 can be expressed by random variables and sets
of our probability space. For the first scenario of Case 3 the translation of the shortage
and its condition is
dlt1 − R1 → − Xs,A1
R1 − dlt1 < 0 → Ms,A1 ,<0 .
In addition, we only consider Case 3, (2, A1 , A1 ), and the shortage before the first
arrival, A1 . In total this results in the expression
E

< A1

h

− Xs,A1 1(2,A1 ,A1 ) 1 Ms,A

1 ,<0

i

(3.3.26)

where the condition for the reorder point setting still has to be added in the subscript
of E< A1 . Similarly, the second shortage scenario of Case 3 is translated by
dlt2 − R2 − Q1 → − Xs,A2 − Q1
R1 − dlt1 ≥ − Q1 → Ms,A1 ,<−Q1
R2 + Q1 < dlt2 → Ms,A2 ,<−Q1
into
E

> A1

h

−( Xs,A2 + Q1 ) 1(2,A1 ,A1 ) 1 Ms,A

1 ,<− Q1

1 Ms,A

2 ,<− Q1

i

.

(3.3.27)

88

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

The last scenario can be translated by
R1 − dlt1 − R2 + dlt2 → Xs,A1 − Xs,A2
R1 − dlt1 < − Q1 → Ms,A1 ,<−Q1
into

h
E> A1 ( Xs,A1 − Xs,A2 ) 1(2,A1 ,A1 ) 1 Ms,A

1 ,<− Q1

i

.

(3.3.28)

By exchanging the indicator function 1(2,A1 ,A1 ) with 1(2,tw ,A1 ) the formulas are valid
for Case 6. In the following we will translate the given expressions for the shortage in
Case 3 and 6 into mathematical formulas and thereby consider the different possible
settings of the reorder points.
For Case 3, (2, A1 , A1 ), we know that the first order arrives

Positive reorder points.

before tw , so 0 ≤ lt1 ≤ tw . This implies that the second reorder point, R2 , has to be

reached during the lead time lt1 , so t g < lt1 . Moreover the second order must arrive
after the first one which translates into t g + lt2 > lt1 .
We are interested in the amount of shortage. First, we look at the shortage before
the first arrival as specified by the expression (3.3.26). Then, demand f ( x, y) between
triggering and receiving the first order at time t = 0 and y = lt1 , respectively, must be
greater than the reorder point R1 . This condition is equivalent to a demand f ( x, y −

t) > R2 between triggering the second order at time t = t g and the arrival of the first

order after y = lt1 time units. Due to simplifications we use the latter condition in the
formula for the shortage given by
h

ER<1A>1R2 ≥0 − Xs,A1
Ztw Zy Z∞ Z∞
y

0

t

z

x

0 y − t R2

1(2,A1 ,A1 ) 1 Ms,A

1 ,<0

i

=

( x − R2 ) f ( x, y − t) l2 (z) g(t) l1 (y) dµ.

(3.3.29)

We can directly derive the formula for Case 6, where lt1 > tw , from Case 3. The only

The SDMR model – Expected shortage of the reorder cycle scenarios

89

things changing are the new condition tw < lt1 < ∞ for the lead time lt1 and an interorder time t g that is now limited by the time window tw instead of lt1 , so 0 ≤ t g ≤ tw .
h

ER<1A>1R2 ≥0

− Xs,A1 1(2,tw ,A1 ) 1 Ms,A

Z∞ Ztw Z∞ Z∞
y

tw

t

z

x

0 y − t R2

1 ,<0

i

=

( x − R2 ) f ( x, y − t) l2 (z) g(t) l1 (y) dµ.

(3.3.30)

Now we come to the shortage between the two arrivals. If the first delivery
cannot cover all backlogged demand, as assumed in the expression (3.3.28), then
the additional shortage is identical to the demand x 0 between the two deliveries, so
0 < x 0 < ∞. Of course, we have to consider that the demand x before the first arrival
is too high to be covered by the delivered order quantity Q1 . If we use the second
reorder point R2 as a reference the inequality R2 + Q1 < x < ∞ must hold.
h
ER≥1A>1R2 ≥0 ( Xs,A1 − Xs,A2 ) 1(2,A1 ,A1 ) 1 Ms,A
Ztw Zy Z∞
y

t

z

Z∞

x

Z∞

0 y − t R2 + Q1 0

0

x0

1 ,<− Q1

i

=

x 0 f ( x 0 , z + t − y) f ( x, y − t) l2 (z) g(t) l1 (y) dµ (3.3.31)

Again, we can easily write down the related formula for Case 6 by considering the
new conditions on the lead time tw < lt1 < ∞ and the inter-order time 0 ≤ t g ≤ tw .
h
ER≥1A>1R2 ≥0 ( Xs,A1 − Xs,A2 ) 1(2,tw ,A1 ) 1 Ms,A
Z∞ Ztw Z∞
y

tw

t

z

Z∞

x

Z∞

0 y − t R2 + Q1 0

x0

1 ,<− Q1

i

=

x 0 f ( x 0 , z + t − y) f ( x, y − t) l2 (z) g(t) l1 (y) dµ (3.3.32)

Last but not least we look at the scenario where the first delivery can satisfy all
demand that has been backlogged so far. This is represented by the expression (3.3.27).
In addition, there might be a positive stock to cover future demand after the first order
arrives. The condition for the demand x between triggering the second order and the
arrival of the first order is 0 ≤ x ≤ R2 + Q1 . The demand x 0 after the first arrival must

90

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

now be at least as large as the potentially positive stock, so R2 + Q1 − x ≤ x 0 < ∞.
h
ER≥1A>1R2 ≥0 −( Xs,A2 + Q1 ) 1(2,A1 ,A1 ) 1 Ms,A
Ztw Zy Z∞ R2Z+Q1
y

t

z

0 y−t

0

x

0

Z∞

x0

R2 + Q1 − x

1 ,<− Q1

1 Ms,A

2 ,<− Q1

i

=

( x 0 − R2 − Q1 + x ) f ( x 0 , z + t − y)·
(3.3.33)

f ( x, y − t) l2 (z) g(t) l1 (y) dµ
The equivalent formula for Case 6 is given by
ER≥1A>1R2 ≥0

h

−( Xs,A2 + Q1 ) 1(2,tw ,A1 ) 1 Ms,A

Z∞ Ztw Z∞ R2Z+Q1
y

tw

t

z

0 y−t

x

0

Z∞

x0

R2 + Q1 − x

1 ,<− Q1

2 ,<− Q1

=

( x 0 − R2 − Q1 + x ) f ( x 0 , z + t − y)·

f ( x, y − t) l2 (z) g(t) l1 (y) dµ.
Arbitrary reorder points.

1 Ms,A

i

(3.3.34)

In this section we will relax all restrictions on how to set

the reorder points. We begin by only setting R2 < 0, leave R1 ≥ 0, and give formulas

that are also valid for the last section where both reorder points are non-negative.
Then, we will set both reorder points to a negative value. Interestingly, the formulas
are identical because we always take the second reorder point as a reference for the
demand and shortage before the first arrival. In this way we will obtain formulas that
are valid for all possible values for the reorder points.
Looking at Equation (3.3.29) for the shortage before the first arrival we now already
have a shortage of − R2 when the second order is triggered. In fact, the first term

within the integration, ( x − R2 ), is not limited to a positive value of R2 and adds the

already cumulated shortage to the regular demand until the first order arrives. Thus,
there is no need for change at this point. The rest of the terms within the integration
are independent of R2 . Only the lower bound of the most inner integration sign will

become negative. Restricting this lower bound to non-negative values, by writing

The SDMR model – Expected shortage of the reorder cycle scenarios

91

+

[ R2 ] , we obtain the equation
h
E< A1 − Xs,A1 1(2,A1 ,A1 ) 1 Ms,A
Ztw Zy Z∞
y

t

Z∞

z

x
+

0 y − t [ R2 ]

0

1 ,<0

i

=

( x − R2 ) f ( x, y − t) l2 (z) g(t) l1 (y) dµ

(3.3.35)

which is valid for both cases R1 > R2 ≥ 0 and R1 ≥ 0 > R2 . Note, the negative R2

is always met here because we look at a two-order cycle. Thus, there will always be a
shortage of at least − R2 . The identical restriction of the lower bound to non-negative

values gives an equivalent formula for Case 6 which can be found in the Appendix B.
In the case where the first order cannot cover all backlogged demand R2 occurs
only once in the formula for the shortage given in Equation (3.3.31). We can restrict
the lower bound R2 + Q1 to positive values for the integration of the demand x until
the first arrival because a negative demand does not make sense. This yields
E

≥ A1

h

( Xs,A1 − Xs,A2 ) 1(2,A1 ,A1 ) 1 Ms,A

Ztw Zy Z∞
y

0

t

z

Z∞

Z∞

x

0 y − t [ R2 + Q ] + 0
1

x0

1 ,<− Q1

i

=

x 0 f ( x 0 , z + t − y) f ( x, y − t) l2 (z) g(t) l1 (y) dµ

(3.3.36)

which is again valid for non-negative reorder points, as well. Note, whenever R2 +
Q1 ≤ 0 the latter equation reduces to
h

ER≥1A>1R2 ≤−Q1 ( Xs,A1
Ztw Zy Z∞ Z∞
0
y

0

t

z

0 y−t 0

x0

− Xs,A2 ) 1(2,A1 ,A1 ) 1 Ms,A

1 ,<− Q1

i

=

x f ( x 0 , z + t − y) l2 (z) g(t) l1 (y) dµ

which yields the expected demand between the two arrivals t A1 and t A2 . This result
is easy to verify. Whenever the normal order arrives first and R2 ≤ Q1 holds then
the first delivery, consisting of Q1 units, cannot bring the stock level above zero at

t A1 . Consequently, all demand between t A1 and t A2 remains unsatisfied and, thus, the

92

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

expected shortage between t A1 and t A2 corresponds to the expected demand during
that time. The equivalent formula for Case 6 is derived analogously and can be found
in the Appendix B.
Equation (3.3.33) for the scenario where the stock level after the first delivery is
positive consists of three places where R2 is used. Starting with the bounds R2 + Q1
and R2 + Q1 − x of the integration of the demand until the first order arrives and

the demand between the two deliveries, respectively, it is clear that negative values
do not make sense and we restrict their bounds to non-negative values. The term

( x 0 − R2 − Q2 + x ) within the integration represents the shortage that occurs given

the four parameters. Here again, the term is not restricted to positive values of R2 but
increases the total sum if R2 < 0. This corresponds to the fact that the demand after

the first arrival now needs to be less high to cause a stockout situation and is actually
equivalent to an increased demand before the first arrival. Thus, we do not need to
change this term.
h
E≥ A1 −( Xs,A2 + Q1 ) 1(2,A1 ,A1 ) 1 Ms,A
Ztw Zy Z∞ [ R2 +ZQ1 ]
y

0

t

z

0 y−t

+

Z∞

x0

x

0

[ R2 + Q1 − x ] +

1 ,<− Q1

1 Ms,A

2 ,<− Q1

i

=

( x 0 − R2 − Q1 + x ) f ( x 0 , z + t − y ) ·

f ( x, y − t) l2 (z) g(t) l1 (y) dµ

(3.3.37)

Again, this equation also holds for the case R1 > R2 ≥ 0. Note, if R2 + Q1 ≤ 0 then the
whole expression is 0 because the integration of the demand x before the first arrival
reduces to

Z0
0

x

f ( x, y − t) dx = 0.

In other words, the second reorder point is so negative that there is no possibility
for the first order to refill the stock level above zero. Of course, then the shortages
between the two arrivals are solely determined by the formula for the previous case

The SDMR model – Expected shortage of the reorder cycle scenarios

93

where the first delivery cannot cover all backlogged demand. The equivalent formula
for Case 6 is listed in Appendix B.
The sum of the latter three formulas yield the total shortage one expects in Case 3.
See Appendix B for Case 6.
So far we have not considered cases where both reorder points are negative. Interestingly, the formulas do not change because the reorder point R1 does not appear in
any of the equations given so far for two-order cycles. So these formulas hold for all
possible settings of the reorder points R1 and R2 .
This can be explained by the fact that R2 always has to be reached in Case 3 and
Case 6. Then, there exists shortage prior to reaching R2 if and only if R2 < 0 which
is completely independent from how the demand has evolved before R2 . Further, we
are interested in the amount of shortage but do not consider how long backlogged
demand remains unsatisfied. Consequently, it suffices for the calculation of the expected shortages to look at R2 and at the development of the demand after R2 has
been reached. This changes, of course, when we want to calculate the average cycle
stock which will be addressed in Chapter 3.4.

3.3.2.2

Second order arrives first

The two possible cases for a two-order cycle where the second order arrives first are
Case 4 and Case 7. These are displayed in Figure 3.8. Here again, both cases are closely
related and differ only in the condition whether the first order arrives before or after
the end of the time window tw to trigger an emergency order.
For Case 4, analogous to Case 3, we can give expressions for the three different
sources of shortage by means of random variables, subsets of Ω and their indicator
function. We skip explicitly elaborating them here because they are shown in the
coming equations and are straight forward to verify my means of Case 3.

94

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)
"#$%&''&()*+#,-*.*/*0*,('1&#2-*3,&'4(#,&546
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Figure 3.8: Scenarios for a two-order cycle where the second order arrives first

For Case 4 and 7 there exist again three different conditions for the shortage as
specified in Table 3.6, and three possible settings for the reorder points. In the following we directly give the universal formulas that apply to all three possible setting of
the reorder points, R1 > R2 ≥ 0, R1 ≥ 0 > R2 , and 0 > R1 > R2 .
!!

The Case 4 is specified by (2, A1 , A2 ). First, this means that the lead time lt1 of
the first order is shorter than the time window tw , so 0 < y = lt1 < tw . Second,

this implies that the second order must be triggered before the first order arrives, so
0 < t = t g < lt1 . Third, the second order arrives before the first one which translates
into 0 < z = lt2 < y − t. In addition the demand dlt2 during the lead time of the

second order must exceed R2 in order to cause shortage, so R2 < x = dlt2 < ∞.
Consequently, the formula for the shortage before the first order arrives, which is the

The SDMR model – Expected shortage of the reorder cycle scenarios

95

order associated with R2 in this case, yields
h
E< A2 − Xs,A2 1(2,A1 ,A2 ) 1 Ms,A
Ztw Zy yZ−t Z∞
y

0

t

z

0

x
+

0 [ R2 ]

2 ,<0

i

=

( x − R2 ) f ( x, z) l2 (z) g(t) l1 (y) dµ.

(3.3.38)

Note, this formula is already independent of the value of the two reorder points. For
the case where the first delivery cannot satisfy all backlogged demand at the time
of its arrival the demand dlt2 during the lead time of the second order must exceed
R2 + Q2 . Thus, R2 + Q2 < x < ∞ and any positive demand x 0 between the two
arrivals immediately leads to some shortage. Obviously, only non-negative bounds of
the integration of the demand x make sense.
h
E≥ A2 ( Xs,A2 − Xs,A1 ) 1(2,A1 ,A2 ) 1 Ms,A
Ztw Zy yZ−t
y

0

t

0

z

Z∞

Z∞

x

0 [ R + Q ]+ 0
2
2

x0

2 ,<− Q2

i

=

x 0 f ( x 0 , y − z − t) f ( x, z) l2 (z) g(t) l1 (y) dµ (3.3.39)

For the last scenario, where the order quantity Q2 is large enough to satisfy all backlogged demand at the time of its arrival
h
E≥ A2 −( Xs,A1 + Q2 ) 1(2,A1 ,A2 ) 1 Ms,A
Ztw Zy yZ−t [ R2 +ZQ2 ]
y

0

t

0

z

0

x

0

+

Z∞

x0

[ R2 + Q2 − x ] +

2 ,<− Q2

1 Ms,A

1 ,<− Q2

i

=

( x 0 − R2 − Q2 + x ) f ( x 0 , y − z − t ) ·

f ( x, z) l2 (z) g(t) l1 (y) dµ

(3.3.40)

determines the shortage between the first and the second arrival.
Without any problems the equivalent formulas for Case 7 can be given, see Appendix B.

96

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

3.3.2.3

Both orders arrive simultaneously

In a setting where at least one of the two lead times, lt1 or lt2 , are continuous variables
the event of a simultaneous arrival is a null set with a probability of zero. However,
due to the practical implementation of this model with discretized lead times we have
to consider these scenarios. They are related to the one-order cases regarding their
single arrival time, A1 = A2 . Several aspects have to be considered when discretizing
our formulation. These are addressed in Chapter 4.1 on page 128. Note, the demand
can still be a continuous variable even though the lead times are not.
There are two cases, Case 5 and Case 8, with a simultaneous arrival of both orders
which are illustrated in Figure 3.9. Case 5 is specified by (2, A1 , =) which implies
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!&

!"

!(!) !("
5$-.+1

'6:.
5$-.+/

!'

Figure 3.9: Scenarios for a two-order cycle with simultaneous arrivals

several properties. These are reflected in the integration bounds of the formula for the

!"

The SDMR model – Expected shortage of the reorder cycle scenarios

97

expected shortage
h
E − Xs,A1 1(2,A1 ,=) 1 Ms,A
t w y −1 y − t

∑∑ ∑

y =2 t =1 z = y − t

Z∞

1 ,<0

i

=

( x − R2 ) f disc ( x, z) l2disc (z) gdisc (t) l1disc (y) dx.

x
+

[ R2 ]

(3.3.41)

First, the lead time lt1 of the first order is shorter than the time window tw , so 0 <
y = lt1 < tw . Consequently, the second order must be triggered before the first order
arrives, so 0 < t = t g < lt1 . Third, the first order arrives together with the second one
which translates into z = lt2 = y − t. In addition the demand dlt2 till the delivery of
the second order must exceed R2 in order to cause shortage, so R2 < x = dlt2 < ∞.

Note, the density functions f disc , l1disc , l2disc , and gdisc are the discrete counterparts
of f , l1 , l2 , and g, respectively. In the case of the demand distribution f disc ( x, z) this
represents the probability that a continuous amount of demand x ∈ R occurs within
0 < z ∈ N time units.

Further note, a second order can only be triggered if y = lt1 > 1 holds for the

lead time lt1 of the first order. This can be deducted from the fact that we prohibit simultaneous ordering, called order splitting, and that only strictly positive lead times,
z = lt2 > 0, are allowed. Consequently, the second order can be triggered one time
unit after t0 the earliest, so t g ≥ 1, and it cannot arrive before t0 + 2 time units. This is
reflected in Equation (3.3.41) where the lower bound for the lead time is y = lt1 = 2

and the upper bound for the inter-order time t = t g is y − 1.
For Case 8 one yields the closely related formula
h
E − Xs,A1 1(2,tw ,=) 1 Ms,A
∞

tw

y−t

∑ ∑ ∑

y = t w +1 t =1 z = y − t

Z∞

1 ,<0

x
+

[ R2 ]

i

=

( x − R2 ) f disc ( x, z) l2disc (z) gdisc (t) l1disc (y) dx (3.3.42)

98

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

where the lead time of the first order is greater than tw and the demand during tw must
be large enough to trigger the second order. Note, the second order is still triggered at
time tw .
This concludes our elaboration of formulas for the shortage in all different cases. This
is a key result for calculating the customer service level as requested in our research
question RQ 1 on page 5. An overview of all final equations is given in Table 3.9. In
favor of a compact presentation we only list the equation number and page where the
complete equation can be found.
The total amount of expected shortage during a replenishment cycle is denoted by
E[SH ]. It is the sum of all shortage formulas for Case 1 to Case 8 listed in Table 3.9 and
can be expressed by
E[SH ] =

8

∑ E[SH Case i].

(3.3.43)

i =1

In the next chapter we develop the formulas for the second large cost driver, the average stock on hand throughout the replenishment cycle.

The SDMR model – Expected shortage of the reorder cycle scenarios

scenario
Case 1

+
Case 2

Case 3

Case 4

Case 5

Case 6

Case 7

Case 8

function name
h
i
E − Xs,A1 1(1,.,.) 1 Ms,A1 ,<0
h
i
E< A1 − Xs,A1 1(2,A1 ,A1 ) 1 Ms,A1 ,<0
h
i
E≥ A1 ( Xs,A1 − Xs,A2 ) 1(2,A1 ,A1 ) 1 Ms,A1 ,<−Q1
h
i
E≥ A1 −( Xs,A2 + Q1 ) 1(2,A1 ,A1 ) 1 Ms,A ,<−Q 1 Ms,A2 ,<−Q1
1
1
h
i
E< A2 − Xs,A2 1(2,A1 ,A2 ) 1 Ms,A2 ,<0
h
i
E≥ A2 ( Xs,A2 − Xs,A1 ) 1(2,A1 ,A2 ) 1 Ms,A2 ,<−Q2
h
i
E≥ A2 −( Xs,A1 + Q2 ) 1(2,A1 ,A2 ) 1 Ms,A ,<−Q 1 Ms,A1 ,<−Q2
2
2
h
i
E − Xs,A1 1(2,A1 ,=) 1 Ms,A1 ,<0
h
i
E< A1 − Xs,A1 1(2,tw ,A1 ) 1 Ms,A1 ,<0
h
i
E≥ A1 ( Xs,A1 − Xs,A2 ) 1(2,tw ,A1 ) 1 Ms,A1 ,<−Q1
h
i
≥
A
1
E
−( Xs,A2 + Q1 ) 1(2,tw ,A1 ) 1 Ms,A ,<−Q 1 Ms,A2 ,<−Q1
1
1
h
i
<
A
2
E
− Xs,A2 1(2,tw ,A2 ) 1 Ms,A2 ,<0
h
i
E≥ A2 ( Xs,A2 − Xs,A1 ) 1(2,tw ,A2 ) 1 Ms,A2 ,<−Q2
i
h
E≥ A2 −( Xs,A1 + Q2 ) 1(2,tw ,A2 ) 1 Ms,A ,<−Q 1 Ms,A1 ,<−Q2
2
2
h
i
E − Xs,A1 1(2,tw ,=) 1 Ms,tw ,<0

99

reference

Eq. (3.3.17),

p. 82

Eq. (3.3.35),

p. 91

Eq. (3.3.36),

p. 91

Eq. (3.3.37),

p. 92

Eq. (3.3.38),

p. 95

Eq. (3.3.39),

p. 95

Eq. (3.3.40),

p. 95

Eq. (3.3.41),

p. 97

Eq. (B.0.1),

p. 287

Eq. (B.0.2),

p. 287

Eq. (B.0.3),

p. 288

Eq. (B.0.4),

p. 288

Eq. (B.0.5),

p. 289

Eq. (B.0.6),

p. 289

Eq. (3.3.42),

p. 97

Table 3.9: Summary of all functions to determine the expected shortage

100

3.4

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

Expected average cycle stock

In the following we want to determine the average stock on hand during a replenishment cycle. In contrast to the calculation of the shortage which causes timeindependent costs the holding costs are specified per time unit. Thus, we have to
consider the development of the stock on hand over time and calculate the area under the demand curve, as indicated by the colored area in Figure 3.10. However, we
#$%&'((')*+!&(,-.,+/0/1.2+3'(/4+5.6.,$1+73'(/)$3'8/9

#$%&'((')*+,&(-./-+0102/3+4'(05+6/7/-$2+84'(0)$4'90:

3'(/4

4'(05

"!

#!

#,
";

!:

!% !

!$
!#

(a) One-order cycle

'8<.

!;

!%

!"

!& ,

!& !

'9</

!$

(b) Two-order cycle

Figure 3.10: Exemplified scenarios of the stock on hand during a replenishment cycle

cannot reproduce the exact development of the demand over time from our probability space Ω. While an adaptation of Ω is possible it implies considerable effort. It is
much more promising to divide the replenishment!" cycle into several intervals and use
an interpolation between the start and the end point. This approach yields the exact
value for the expected stock for our assumptions, see Appendix A.8. Additionally, it
can be used as an approximation when our assumptions do not hold.
In many cases we need to know the expected point of time when the stock level
depletes. This is especially essential in combination with negative reorder points and
shortages. Here, the approach of interpolation can be applied, as well. Therefore, we

!"

The SDMR model – Expected average cycle stock

101

introduce the function
η ( x, R) =



1


R
x

if x = 0 or x ≤ R

(3.4.1)

else

which will be used in a multiplication with a time t in the following sections. The expression η ( x, R) · t indicates the fraction of t at which the stock is expected to deplete.

Note, in a two-order cycle the stock might deplete twice and we have to consider both
times separately.
Analogous to the calculations of the shortage we distinguish between different
cases, Case 1 to 8. Moreover, we split the whole replenishment cycle into several
periods for which we calculate the average stock on hand separately, see Table 3.10.
The notation for the subinterval is used in the superscript of the function or parameter.
Example 3.4.1. The expression E< A1 [stock] refers to the average stock on hand before
the arrival of the first order. One should be aware that E[ Xs,A1 1 Ms,A

1 ,>0

] is not identical

to E< A1 [stock]. The main difference lies in the fact that the first expression is just a
snapshot of the stock on hand before t A1 while the second expression represents the
average stock on hand over the whole time until the arrival of the first order. We are
interested in the average stock over time and, thus, can only indirectly utilize random
variables like Xs,A1 .

3.4.1

One-order cycles

In an one-order cycle there exist two possible cases, Case 1 and 2, as shown in Figure 3.11. Note, conditional demand rates λ(1,A1 ,.) and λ(1,tw ,.) apply in the first period
of the reorder cycle as all case-dependent conditions, like not triggering a second order, must hold. Throughout the rest of the time the unconditional average demand
rate µ D applies.

102

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)
!"#$%&&%'()*"+,)-)+%&./)0,1,0)2 3,%"40,3)5+%&.'"+%4.6
+%&./

!"#$%&&%'()*"+,)-)+%&./)0,1,0)2 3,%"40,3)5+%&.'"+%4.6
+%&./

!%

"!-

!%

")

&-

#!8

")

"8

"5-A(-AB6

89)"! :
-

'-

)"!

!58=!&=>6

-

(!

C

:;(! <
C

&;

$8

")

:;(#! 2 (! <

"-

!$
!7
,=>,.%,3)+%&./(

C

8

!&

89!#:

!( -

,=,<>04?4,3)@,"0)+%&./(

!7

%4<,

:;!%<

,?@,.%,3)+%&./(

(a) Case 1

!' 8

%49,

,?,9@04A4,3)B,"0)+%&./(

(b) Case 2

Figure 3.11: The two possible scenarios for a one-order cycle

Just as we considered different reorder point settings for determining the amount
of shortage in a reorder cycle we have to consider these settings for the average stock,
as well, namely, R1 > R2 ≥ 0, R1 ≥ 0 > R2 , and 0 > R1 > R2 .

3.4.1.1

The order arrives within the window to trigger a second order

Positive reorder points.

Let us assume for a moment that the lead time lt1 and the

demand dlt1 during the lead time is fixed. Then, for Case 1 the average stock on hand
is

h
i
ER<1A>1R2 ≥0 stock 1(1,A1 ,.) =
h
i
≥ A1
ER1 > R2 ≥0 stock 1(1,A1 ,.) =

R1 lt1 −

dlt1 lt1
2

Q1 − dlt1
(tc − t A1 ) R1 + Q1 − dlt1 −
2



The SDMR model – Expected average cycle stock

103

before and after the arrival till the end of the cycle, respectively. Now we have to
consider that lead time y = lt1 and demand x = dlt1 are random variables.
ER<1A>1R2 ≥0

h

i

stock 1(1,A1 ,.) =

h

i

Ztw R1Z− R2
y

0

x

0

ER≥1A>1R2 ≥0 stock 1(1,A1 ,.) =
Ztw R1Z− R2
y

0

x

0



Q −x
( t c − y ) R1 + 1
2



x
y( R1 − ) f ( x, y) l1 (y) dµ
2

(3.4.2)

f ( x, y) l1 (y) dµ

(3.4.3)

Note, in this case the stock cannot be negative, because R2 is never reached during the
cycle time – otherwise, a second order would have been triggered – and R2 is nonnegative by definition here. Moreover, the total demand during the order cycle time
tc is exactly Q1 , the time when R1 is hit again, also see Assumption 8.
The remaining cycle time tc − t A1 is a random variable which depends on the stock

level right after the arrival of the order and the demand rate. Here, we use the expected time E[tc − t A1 ] in which the stock level is reduced to R1 again. The exact value
is given by means of the demand distribution D ( x, t) which yields




E tc − t A1 = E D | x = Q1 − dlt1

(3.4.4)

in our current case with positive reorder points. Whenever the latter expression is
hard to calculate, a simple approximation is to divide the amount of stock that exceeds
R1 by the average demand rate µ D which leads to
E [ t c − t A1 ] =
Negative second reorder point.

Q1 − dlt1
.
µD

(3.4.5)

If exclusively the second reorder point is negative

we have to consider the possibility that the stock on hand drops to zero before the
arrival of the order at time y = lt1 . This expected time of depletion is a fraction of
lt1 which can be expressed by the function η ( x, R1 ) as defined in the beginning of this

104

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

chapter. Now, we can give the average stock on hand by
h

ER<1A≥10> R2

stock 1(1,A1 ,.) =

Ztw R1Z− R2
y

x

0

i

0

η ( x, R1 ) y



min( R1 , x )
R1 −
2



f ( x, y) l1 (y) dµ.

(3.4.6)

If the stock during the complete lead time lt1 is positive then we know that dlt1 < R1
holds for the demand and η ( x, R1 ) y = y. For a demand R1 ≤ dlt1 ≤ R1 − R2 it holds
that η ( x, R1 ) y =

R1
x

y is the expected time of depletion.

Note, the formula for the stock on hand after the arrival of the order remains the
same as in the case R1 > R2 ≥ 0 because the stock level has to reach R1 > 0 when Q1
units are delivered.

Negative reorder points.

Whenever both reorder points are negative there is no

stock on hand until the arrival of the first order.
E0<>AR11 > R2

h

i

stock 1(1,A1 ,.) = 0

(3.4.7)

If the delivered quantity is large enough to bring the stock level above zero then we
have a positive average stock on hand after the arrival. The expected on-hand inventory is given by
h
i
E0≥>AR11 > R2 stock 1(1,A1 ,.) =
Ztw R1Z− R2
y

0

x

0

+

+

( R1 + Q1 − x ) ( R1 + Q1 − x )
·
f ( x, y) l1 (y) dµ.
µD
2

(3.4.8)

The SDMR model – Expected average cycle stock

105

The formulas for universal reorder point settings are de-

Arbitrary reorder points.

rived from the equations above.
E

< A1

h

stock 1(1,A1 ,.) =
+

Ztw R1 Z− R2
y

+

y

E

x
y( R1 − ) f ( x, y) l1 (y) dµ +
2

0

Ztw R1 Z− R2
≥ A1

+

x

0

0

h

i

R1

x
+

−

R
η ( x, R1 ) y 1
2

+

f ( x, y) l1 (y) dµ

(3.4.9)

i

stock 1(1,A1 ,.) =
 −
+ "
#+
−
Ztw R1Z− R2 R1 + Q1 − x
Q1 − x
R1
+
· R1 −
f ( x, y) l1 (y) dµ (3.4.10)
x
y
µD
2
2
0

0

Note that the second summand of Equation (3.4.9) vanishes whenever R1 is negative
or R2 is positive and that the whole equation is equal to zero if R1 and R2 are negative.
3.4.1.2

The order arrives after the window to trigger a second order

In contrast to Case 1 there are now the three different sections ( < tw , < A1 , and ≥ A1 ),
see Table 3.10.

Positive reorder points.

For the beginning we assume both reorder points to be pos-

itive. Then, the expected stock on hand before tw is
w
ER<1t>
R2 ≥0

h

i

stock 1(1,tw ,.) =

Z∞ R1Z− R2
y

tw

x

0


x
f ( x, tw ) l1 (y) dµ.
t w R1 −
2

(3.4.11)

We have to consider possible negative stock between tw and the arrival of the order
because lacking a second order does not necessarily mean that the stock always has to
be above R2 . It can also happen that a second order is not placed due to exceeding the

106

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

time window tw .
ER<1A>1R2 ≥0


h

i

stock 1(1,tw ,.) =

R1 − x −

min( x 0 , R
2

Z∞ R1Z− R2 Z∞
y

x

tw

1 − x)



0

0

x0

η ( x 0 , R1 − x ) (y − tw )·

f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dµ.

(3.4.12)

Note that one assumption is that the stock is above R1 after the last order of a cycle
has arrived. Thus, strictly spoken we have to limit the upper bound of the integration
of x 0 to Q1 − x. We will consider this fact in the implementation but do not regard it

here. We can split up the formula and yield
ER<1A>1R2 ≥0

h

stock 1(1,tw ,.) =

1 −x
Z∞ R1Z− R2 RZ
y

x

tw

0

0

x0

Z∞ R1Z− R2 Z∞
y

x0

x

tw

i

0

R1 − x

0





η ( x 0 , R1 ) ( y − t w )

R1 − x
·
2

x0
( y − t w ) R1 − x −
2

f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dµ +

f ( x , y − tw ) f ( x, tw ) l1 (y) dµ.

(3.4.13)

Considering that R1 > 0 and neglecting the upper bound Q1 − x for x 0 , the average
stock on hand between the arrival of the order and the end of the cycle is given by
ER≥1A>1R2 ≥0

h

stock 1(1,tw ,.) =

Z∞ R1Z− R2 Z∞
y

tw

x

0

i

0

x0


+ 
Q1 − x − x 0
( Q1 − x − x 0 )
0
· R1 + Q1 − x − x −
·
µD
2

f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dµ.

(3.4.14)

Here exists no logical limit like the lead time for the remaining cycle time and we
refrain from using a minimum operator.

The SDMR model – Expected average cycle stock

107

Arbitrary reorder points. The formulas for R1 ≥ 0 > R2 and 0 > R1 > R2 can be

found in the Appendix C.1. We directly give the formulas that hold for every setting
of R1 and R2 . Note that we do not consider the maximum limit of demand here again.
h

+

i

E<tw stock 1(1,tw ,.) =
+

Z∞ R1 Z− R2
y

tw

R1

x
+

Z∞ R1 Z− R2
y

tw

−

η ( x, R1

+

+

x

0

R
) tw 1
2


x
t w R1 −
f ( x, tw ) l1 (y) dµ +
2

+

f ( x, tw ) l1 (y) dµ

(3.4.15)

Note, Equation (3.4.15) is zero if both R1 and R2 are negative. Moreover, the second
summand is only positive, if R1 > 0 and R2 < 0.
h
i
E< A1 stock 1(1,tw ,.) =
+

+

1 −x
Z∞ R1 Z− R2 RZ
x

y

tw

0

0

+

Z∞ R1 Z− R2
y

x

tw

0

0

+

x0

Z∞

x0

R1 − x

(y − tw ) ·



x0
R1 − x −
2

η ( x 0 , R1 − x ) ( y − t w )



f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dµ +

R1 − x
·
2

f ( x , y − tw ) f ( x, tw ) l1 (y) dµ

(3.4.16)

Note that Equation (3.4.16) is only zero if both reorder points, R1 and R2 , are smaller
or equal to zero. Whenever only R2 is negative then the upper bound of x is R1 . In
other words, the stock on hand after tw is only positive if the demand during tw can
be buffered by the initially existing stock R1 .
h
i
E≥ A1 stock 1(1,tw ,.) =
 −
+ "
#
0
−
Z∞ R1Z− R2 Z∞
R1 + Q1 − x − x
R1
Q1 − x − x 0
· R1 −
+
·
y
x
x0
µD
2
2
tw

0

0

0

f ( x , y − tw ) f ( x, tw ) l1 (y) dµ

(3.4.17)

108

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

According to Assumption 8 the stock must exceed R1 after the last arrival. Thus, it
is not allowed to have x + x 0 > Q1 and Equation (3.4.17) can only be zero if R1 ≤ 0.

More precisely, R1 has to be so negative that R1 ≤ x + x 0 − Q1 which refers to the fact
that the stock is not refilled above zero when Q1 units are delivered.

3.4.2

Two-order cycles

3.4.2.1

First order arrives first

According to Table 3.10 there are 4 different periods during a reorder cycle in which
a positive stock might occur for Case 3 and Case 6. These can also been seen in Figure 3.12. Let us observe Case 3.
!"#$%&&%'()*"+,)-).)/)+%&01)2,3,2)4+%&0'"+%506
+%&01

$!8

#8

"-

!49>' >' 6
8

8

;<(! " =

%9

%8

#9

"-

;<($!8 )!**" =
!"
!7
,?@,0%,B)+%&01(
,?,:@25A5,B)C,"2)+%&01(

;<($!9 +*!**)
=
" $!8

"$!9
!"

;<!&=

!' 8

0"+,)/

!' 9

%5:,

0"+,)!,

Figure 3.12: The two cases for a two-order cycle where the first order arrives first

The SDMR model – Expected average cycle stock

109

If both reorder points are positive then the stock before trig-

Positive reorder points.

gering the second order is given by
<t g
ER1 > R2 ≥0 [stock 1(2,A1 ,A1 ) ]

=

Ztw Zy Z∞
y

t

z

0 y−t

0

t·



R − R2
R1 − 1
2



l2 (z) g(t) l1 (y) dµ. (3.4.18)

The demand during the two periods denoted by < A1 and < A2 is theoretically unlimited and only restricted by the Assumption 8 about the maximal demand which we
postpone to the implementation again. However, there are other logical restrictions to
consider.
First, there is no stock on hand if the demand exceeds the remaining stock which is
equal to R2 and R2 + Q1 − x for the period < A1 and < A2 , respectively. Second, the
maximum time of a positive stock on hand is lt1 − t g and t g + lt2 − lt1 for the periods

< A1 and < A2 , respectively. This leads to the two formulas
ER<1A>1R2 ≥0 [stock 1(2,A1 ,A1 ) ] =
Ztw Zy Z∞ Z∞
t

y

0

z

x

0 y−t 0

η ( x, R2 ) (y − t)



min( R2 , x )
R2 −
2



·

f ( x, y − t) l2 (z) g(t) l1 (y) dµ

(3.4.19)

and
ER<1A>2R2 ≥0 [stock 1(2,A1 ,A1 ) ] =
Ztw Zy Z∞ Z∞ Z∞
y

0

t

z

0 y−t 0

x

0

x0



+
η x 0 , ( R2 + Q1 − x )
(t + z − y) ·



+
0
min
(
R
+
Q
−
x
)
,
x
2
1
 ( R2 + Q1 − x ) + −
·
2


f ( x 0 , t + z − y) f ( x, y − t) l2 (z) g(t) l1 (y) dµ.

(3.4.20)

110

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

The average stock between the last arrival and the end of the cycle is given by
ER≥1A>2R2 ≥0 [stock 1(2,A1 ,A1 ) ] =
Ztw Zy Z∞ Z∞ Z∞
y

0

0

x

z

t

0 y−t 0

0

+

E [tc − t − z] ·

x0

( R2 + Q1 + Q2 − x 0 − x + R1 )
·
2

f ( x , t + z − y) f ( x, y − t) l2 (z) g(t) l1 (y) dµ

(3.4.21)

where the expected remaining cycle time E [tc − t − z] can be expressed by the demand distribution D with

h

0

E [ t c − t − z ] = E D | x = ( R2 + Q1 + Q2 − x − x − R1 )

+

i

(3.4.22)

or simply by dividing the stock level exceeding R1 at time t A2 by the mean demand
rate µ D . The latter leads to
+

( R2 + Q1 + Q2 − x − x 0 − R1 )
.
E[tc − t − z] ≈
µD

(3.4.23)

Merging these formulas for the average stock during the

Arbitrary reorder points.

four periods with the formulas for the cases R1 ≥ 0 > R2 and 0 > R1 > R2 , which

can be found in the Appendix C.2, yields the following formulas which are valid for
all possible settings of R1 and R2 .
E<tg [stock 1(2,A1 ,A1 ) ] =
Ztw Zy Z∞
y

t

z

0 y−t

0

R1

t·

+

+

R1 − R2

+

−

·

R1 + R2
2

+

!

l2 (z) g(t) l1 (y) dµ

(3.4.24)

Note, the latter equation is equal to zero if R1 , R2 ≤ 0. The average stock on hand
between t g and t A1 is given by

E< A1 [stock 1(2,A1 ,A1 ) ] =
Ztw Zy Z∞ Z∞
y

0

t

z

0 y−t 0

x



+
+
η ( x, R2 ) (y − t)  R2 −

f ( x, y − t) l2 (z) g(t) l1 (y) dµ



+

min R2 , x
2



·

(3.4.25)

The SDMR model – Expected average cycle stock

111

which is only positive if R2 > 0. The average stock on hand between both arrivals is
E< A2 [stock 1(2,A1 ,A1 ) ] =
Ztw Zy Z∞ Z∞ Z∞
y

t

z

x

0 y−t 0

0



0

x0



+
η x 0 , ( R2 + Q1 − x )
(t + z − y) ·

 ( R2 + Q1 − x ) + −



min ( R2 + Q1 − x )

+

, x0

2

f ( x 0 , t + z − y) f ( x, y − t) l2 (z) g(t) l1 (y) dµ



·

(3.4.26)

which might be positive independent of the setting of R1 and R2 . The average stock
on hand between the last arrival and the end of the cycle can also be positive for all
reorder point settings and is given by
E≥ A2 [stock 1(2,A1 ,A1 ) ] =

+
+
Ztw Zy Z∞ Z∞ Z∞
R2 + Q1 + Q2 − x − x 0 − R1
y

0



t

z

0 y−t 0

x

0

x0

R2 + Q1 + Q2 − x − x 0 + R1

µD

+

·

+

·
2
f ( x 0 , t + z − y) f ( x, y − t) l2 (z) g(t) l1 (y) dµ.

(3.4.27)

Note, Case 6 differs from Case 3 only regarding the fact that lt1 > tw instead of lt1 ≤

tw . Accordingly, the formulas for Case 6 are obtained by replacing the bounds 0 and
tw of the first integration by tw and ∞. Moreover, the inter-order time has to be within

the allowed window, so 0 ≤ t g ≤ tw is used as bound for the integration of t g . The
complete set of formulas can be found in the Appendix C.2.
3.4.2.2

Second order arrives first

The Cases 4 and 7 are intimately related to the Cases 3 and 6, respectively, as only the
order of arrivals is interchanged. According to Table 3.10 there are 4 different periods

112

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

during a reorder cycle in which a positive stock might occur for Case 4 and Case 7.
These can also been!"#$%&&%'()*"+,)-).)/)+'&0%"1,)2+%&3'"+%435
seen in Figure 3.13. Let us look at Case 4.
+%&39

$!7

",

#7
!28>!''>(
+
85

%7

;<?%" =

%8

",

;<)$!7 * !''*
=
" $! 8

#8
;<?@% 8 =

",

!"

$!8

!6

;<!&=

!"

!( 7

!( 8

,AB,3%,?)+%&39(
,A,:B@4C4,?)0,"@)+%&39(

3"+,)/

%4:,
3"+,)-

!+

Figure 3.13: The two cases for a two-order cycle where the second order arrives first

Arbitrary reorder points. We give the formulas that are valid for all possible settings
of both reorder points, R1 and R2 . The detailed formulas for the different cases are
listed in the Appendix C.3. These formulas can be joined and yield
E<tg [stock 1(2,A1 ,A2 ) ] =
Ztw Zy yZ−t
y

0

t

0

z

0

t·

R1
+

+

+

R1 − R2

−

·

R1 + R2
2

+

!

l2 (z) g(t) l1 (y) dµ

(3.4.28)

for the stock before triggering the second order. This equation is equal to zero whenever the first reorder point is negative, i.e. 0 > R1 > R2 . Whenever only R2 is negative
the stock on hand depletes before the second order is triggered according to the proportion of positive stock R1 and the complete delta R1 − R2 .

The SDMR model – Expected average cycle stock

113

The stock between triggering and receiving the second order is only positive if
R2 > 0. It can be expressed by
E< A2 [stock 1(2,A1 ,A2 ) ] =
Ztw Zy yZ−t Z∞
y

0

t

z

0

x

0



η x, R2

0


+

f ( x, z) l2 (z) g(t) l1 (y) dµ

 + 
min
R2 , x
+
·
z  R2 −
2


(3.4.29)

which is equal to zero whenever R2 ≤ 0. Depending on the stock level just after the

first arrival, given by Q2 + R2 − x, the stock on hand will always, partly, or never
remain above zero between both arrivals. This leads to
E< A1 [stock 1(2,A1 ,A2 ) ] =
Ztw Zy yZ−t Z∞ Z∞
z

t

y

0

0

0



x

0

0

x0



+
η x 0 , ( R2 + Q2 − x )
(y − t − z) ·

 ( R2 + Q2 − x ) + −



min ( R2 + Q2 − x )
2

f ( x 0 , y − t − z) f ( x, z) l2 (z) g(t) l1 (y) dµ.

+

, x0



·

(3.4.30)

For the last period in the reorder cycle the stock on hand might always be zero, i.e. if
the first reorder point is set to a very large negative value.
E≥ A1 [stock 1(2,A1 ,A2 ) ] =
Ztw Zy yZ−t Z∞ Z∞
y

0



t

0

z

0

x

0

0

x0

+ +

( R2 + Q2 + Q1 − x − x 0 − R1 )
·
µD

R2 + Q2 + Q1 − x − x 0 + R1

+

+

·
2
f ( x 0 , y − t − z) f ( x, z) l2 (z) g(t) l1 (y) dµ.

(3.4.31)

Case 7 is closely related to Case 4. The only thing changing is that the lead time of the
first order exceeds tw . Thus, the equivalent formulas for Case 7 can be again derived

114

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

from Case 4. First, the bounds of the first integration have to be changed from 0 and
tw to tw and ∞, respectively. Second, the inter-order time is not limited by the lead
time lt1 anymore but by the time window tw which has to be considered in the upper
bound of t g . The complete list of formulas for Case 7 are found in the Appendix C.3.

3.4.2.3

Both order arrive simultaneously

Both Cases 5 and 8 differ from other two-order cycles by the fact that both orders arrive
at the same time. While the probability of such a coincident is zero in a continuous
environment it has to be considered in a discrete environment like the implementation
on a computer. According to Table 3.10 there are 3 different periods within the reorder
cycle in which a positive stock might occur for Cases 5 and 8. These can also been seen
in Figure 3.14. Here, we look at Case 5.
!"#$%&&%'()*"+,)-).)/)+%&01)2+%&0'3)4,5"647)4,%,853)9,"4)%:5,+;
+%&01

$!=

"-

#=
!2>7+=7E;

?@'! A

%=

#>
"-

?@'$! = ( !))" A

$!>

!"
!<

%>

?@!&A

,BC,0%,4)+%&01(
,B,5C9:D:,4)8,"9)+%&01(

!"

!+ =, !+ >
0"+,)/

%:5,
0"+,)-

!*

Figure 3.14: The two scenarios for a two-order cycle with simultaneous arrivals

The SDMR model – Expected average cycle stock

115

Arbitrary reorder points In the following, we give the formulas that are valid for all
possible settings of both reorder points, R1 and R2 . The details for the different cases
are listed in the Appendix C.4.1. These formulas can be joined and yield
E<tg [stock 1(2,A1 ,=) ] =
y

tw

∑∑

y =2 t =1

t·

R1

+

+

+

R1 − R2

−

R1 + R2
2

·

+

!

l2disc (y − t) gdisc (t) l1disc (y) (3.4.32)

for the average stock on hand until the second order is triggered. Whenever both
reorder points are negative, there is no stock on hand until the arrival and the latter
equation yields zero. This coincides with the equivalent formula for Case 3 and Case 4.
The average stock on hand can deplete between triggering the second order and the
simultaneous arrival if the demand during the lead time lt2 = y − t exceeds a positive
reorder point R2 .

E< A1 [stock 1(2,A1 ,=) ] =
tw

y

∑∑

y =2 t =1

f

disc

Z∞

x

0



η x, R2

+



+

(y − t)

R2

+

min( R2 , x )
−
2

!

·

( x, y − t) l2disc (y − t) gdisc (t) l1disc (y) dx

(3.4.33)

This equation reduces to zero if R2 ≤ 0. Depending on the setting of the reorder point
R1 and the order quantities the stock on hand might not be positive at all between t A1

and tc even though it exceeds R1 . This is reflected by the formula for the average stock
on hand between t A1 and tc .
E≥ A1 [stock 1(2,A1 ,=) ] =

+
+
tw y Z∞
R2 + Q2 + Q1 − x − R1

∑∑

y =2 t =1

f

disc

x

0

µD

·



( x, y − t) l2disc (y − t) gdisc (t) l1disc (y) dx

R2 + Q1 + Q2 − x + R1
2

+

+

·

(3.4.34)

116

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

Case 8 is closely related to Case 5. Again, the only thing changing is the lead time of
the first order which exceeds tw now. Thus, the equivalent formulas for Case 8 can
be easily derived from Case 5. First, the bounds of the first integration have to be
changed from 0 and tw to tw and ∞, respectively. Second, the inter-order time is not
limited by the lead time lt1 anymore but by the time window tw which has to be considered in the upper bound of t g . The complete list of formulas for Case 8 are found
in the Appendix C.4.2.
This concludes the development of the formulas for the expected stock on hand as
required by our question RQ 1 on page 5. A summary of all final formulas is given in
Table 3.11. The sum of all formulas shown in this table represents the total expected
stock during a replenishment cycle E[stock]. This is expressed by
E[stock] =

8

∑

E[stock Case i ].

(3.4.35)

i =1

So far we have developed all the formulas necessary to calculate the major two cost
drivers in inventory management for a given ( R1 , R2 , Q1 , Q2 ) replenishment policy:
the expected shortage and the expected stock on hand. However, there are still important parameters missing like the customer service level, the expected replenishment
cycle time or the expected number of orders during a given planning horizon. The
calculation of these parameters is topic of the next chapter.

3.5

Calculation of derived model parameters

In the latter sections we have developed rather complex formulas for the probabilities,
the shortage, and the average stock within the different cases. Of course, these are not
the only parameters of interest regarding a replenishment cycle. People in the area of
inventory management will demand for other parameters like the expected cycle time

The SDMR model – Derived model parameters

117

or the expected order quantity rather quickly. Moreover, these derived parameters are
also part of the question RQ 1 on page 5.
We found that all parameters of interest can be expressed within our specified
probability space. Moreover, most of them can be calculated rather easily. In the
following we will give the formulas for important parameters which, for example, are
required for determining the expected total costs for a given planning horizon.

3.5.1

Order quantities and cycle demand

One basic parameter in inventory management is the expected order quantity E[ Q]
per cycle. Of course, the order quantity is Q1 in the one-order cases and Q1 + Q2 in
the two-order cases. We can use the probabilities p(1, ., .) and p(2., ., .), that have been
specified in Chapter 3.2, for one- and two-order cycles, respectively. The expected
order quantity per cycle is then given by
E[ Q] = E[ Q | one-order cycle] p(1, ., .) + E[ Q | two-order cycle] p(2, ., .)

= Q1 p(1, ., .) + ( Q1 + Q2 ) p(2, ., .)
= Q1 + Q2 p(2, ., .).

(3.5.1)

Equation (3.5.1) specifies how many units are expected to be delivered via the two
supply channels during one replenishment cycle.
Now, we are turning our focus to the expected demand during one replenishment
cycle. Let us consider the scenario where all unsatisfied demand is back-logged and
there are only the two supply channels available that are related to Q1 and Q2 , respectively. In this case, the total expected demand per replenishment cycle E[ D (t¯c )] is
given by
E[ D (t¯c )] = E[ Q]

= Q1 + Q2 p(2, ., .)

(3.5.2)

118

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

which is identical to the expected order quantity. Note that t¯c = E[tc ] is the expected
cycle time. Consequently, D (t¯c ) is the distribution of the joint demand which occurs
during the expected length of a replenishment cycle. The values of t¯c and D (t¯c ) are
irrelevant at this point and we only use it for a correct notation of the parameters. In
a subsequent section we will specify the formula for E[tc ] as it is of significant interest
in inventory management.
In contrast to backlogging unsatisfied demand, one could also think of a scenario
where unsatisfied customer demand is covered via a third supply channel, e.g. a direct
shipment. This additional supply channel increases the amount of supplied units and
the expected cycle time t¯c 0 . Consequently, the expected demand during t¯c 0 is increased
by exactly the amount of unsatisfied, directly shipped demand E[SH ], see page 98.
E[ D (t¯c 0 )] = E[ D (t¯c )] + E[SH ]

= Q1 + Q2 p(2, ., .) + E[SH ]

(3.5.3)

The latter scenario was rather easy to model. However, one might think of scenarios where unsatisfied customer demand is lost. In this case, the formulas for E[SH ]
and E[S] from Chapter 3.3 and 3.4, respectively, have to be adopted, as well. However,
one can still utilize the logic and the majority of definitions which are provided by the
model framework in Section 3.1.
The calculation of the expected cycle demand and order quantity per cycle is the
basis for the calculation of various other parameters.

3.5.2

Service levels

There exist several different types of service levels. Here we focus on the two main
representatives, namely the α and the β service level.
The beta service level β, also known as the fill rate, is defined as the ”fraction of

The SDMR model – Derived model parameters

119

demand that can be satisfied immediately from stock on hand”, [Axs06, p. 94]. In our
model, the expected value E[ β] can be calculated by
E[ β] = 1 −

E[SH ]
E[ D (t¯c )]

(3.5.4)

which represents a service level of 100% minus the expected fraction of unsatisfied
demand.
The α service level is defined as the probability that all demand during one replenishment cycle can be satisfied [Tem06]. This measure does not consider the amount
of occurring shortage but defines the shortage within one single replenishment cycle
as a binary ”yes-no” variable. Therefore, α can also be interpreted as the fraction of
replenishment cycles with no shortage. This is expressed by
E[α] = 1 − p(SH )

(3.5.5)

where p(SH ) is the probability that a shortage occurs within one single replenishment
cycle. Note, the formulas of p(SH ) and E[SH ] are closely related to each other. Simply
spoken, the formula for E[SH ] is identical to the demand-weighted formula of p(SH ).
An example has been given in Section 3.3.1.1 on page 77.
Other definitions of service levels are common in many industries. Usually, they are
based on the probability or the amount of shortage- and ordering-related parameters
which can be expressed by means of our formal SDMR model.

3.5.3

Cycle time

So far we know how much demand, shortage, and stock we have to expect during a
replenishment cycle but we have no idea how long this replenishment cycle will be.
The expected duration of a replenishment cycle E[tc ] is identical to the time when we
expect to trigger the next normal order. This again is equivalent to the time it takes

120

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

until the delivered order quantities are depleted by customer demand. Knowing the
average demand rate µ per time unit yields
E[ Q]
µ
Q1 + Q2 p(2, ., .)
=
µ

E[tc ] =

(3.5.6)

for the expected cycle time E[tc ].

3.5.4

Average stock per time unit

A direct result from knowing the expected cycle time is the possibility to calculate
the expected stock on hand per time unit E[stock per time unit]. So far, we have only
specified E[stock], the expected stock on hand during a complete replenishment cycle.
We can derive the equation
E[stock]
E[tc ]
µ E[stock]
=
E[ Q]

E[stock per time unit] =

(3.5.7)

for the average stock on hand per time unit in a straight-forward way.

3.5.5

Number of replenishment cycles and orders

Given a certain period of time or planning horizon T, it is of great interest how often
one will have to replenish a certain article when the replenishment parameters R1 , R2 ,
Q1 , and Q2 are applied. This is especially true for calculating and possibly minimizing
the total inventory costs during time T. Usually, T is one year, one quarter, or one
month.
The number of expected replenishment cycles can only be determined if we have
an estimation, e.g. a forecast, of the total demand occurring within T. Then, the expected number of replenishment cycles E[# cycles] within the time horizon T is given

The SDMR model – Derived model parameters
by
E[# cycles] =

121

FC
E[ Q]

(3.5.8)

where FC is the demand forecast for T.
The expected number of replenishment cycles E[# cycles] during T is closely linked
to the number of triggered first and second orders:
E[# first orders] = E[# cycles]
E[# second orders] = E[# cycles] p(2, ., .)

(3.5.9)
(3.5.10)

Naturally, the expected number of first order coincides with the expected number of
replenishment cycles in T as we will always trigger at least the first order within one
replenishment cycle. The number of second orders coincides with the number of twoorder cycles.
There are several ways to treat the unmet customer demand. In the simplest case, one
calculates the opportunity costs by multiplying the unmet demand, (1 − β) · FC, with

the observed opportunity cost factor.

In the backlog case, where a customer is willing to wait for the delivery, one can use
the same calculations and replace the opportunity cost factor with other cost factors,
e.g. covering price reductions or accelerated processes. Additionally, one can include
fixed costs in the calculations. We are using a backlog approach: Unmet customer
demand is processed via a privileged shipment once the new supply has arrived. The
expected number of these privileged shipments E[# privileged shipments] is given by
dividing the expected amount of shortage by the average size of a customer demand.
The average demand size is given by the total demand quantity divided by the number of customer requests and its value differs usually strongly from the average demand per day µ.
In yet another business scenario the unsatisfied customer demand might directly

122

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

be shipped to the customer from the supplier via a third supply channel without going through the warehouse. We have the flexibility to cover this scenario, as well.
While the calculations for E[# privileged shipments] apply here without any changes,
as well, one has to reduce the expected number of first and second orders so they do
not include the demand covered by the third, direct supply channel.

3.6

Summary

The research question RQ 1 on page 5 has been: How can we define and model a

( R1 , R2 , Q1 , Q2 ) replenishment policy with stochastic demand and lead times? In fact,
we have elaborated such a model in this chapter, where orders can be sent to two suppliers by utilizing two different reorder points R1 and R2 and two order quantities Q1
and Q2 . This has been achieved by defining a proper probability space for the SDMR
model in Chapter 3.1 which is completely independent from the type of distributions.
Based on this probability space we have specified the formulas for various essential
variables in inventory management in the Chapters 3.2 to 3.5. A summary of these
variables is given in Table 3.12. Usually, these parameters are sufficient to quantify
the total costs and the customer service level for a given set of replenishment parameters R1 , R2 , Q1 , and Q2 .
Summing up, we are able to give the probabilities for one-order and two-order cycles,
the average stock level, the customer service level, the number of orders for each supplier, and the related costs once we know the exact values of the cost structure and
of the replenishment parameters. Thus, this chapter has satisfactorily answered the
research question RQ 1 on page 5 in all its details.

In specific cases, one might need to extend the number of parameters or change their

The SDMR model – Summary

123

definition according to the concrete (business) scenario at hand. The effort for this
can vary significantly from case to case. However, we have experienced that adaptations are rather simple whenever they do not affect the underlying logic of the formulated probability space and its rather general and flexible definitions. For example,
the SDMR model can easily be adapted to a replenishment model where unsatisfied
demand is lost, see Appendix D.
So far, we have defined the SDMR model from a rather theoretical perspective. Of
course, the intention of the SDMR model is its practical application. Many times this
includes not only a description of a specific situation at the warehouse but also a process to find values for the replenishment parameters that improve or optimize the total
costs. Due to the complexity of the SDMR model we do not see a way to analytically
find the optimal solution. Therefore, we need to use numerical approaches for implementing the SDMR model and for finding an optimal solution. These practical issues
are addressed in the next chapter.

124

C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

case
Case 1

Case 2

notation

(1, A1 , .)

(1, tw , .)

interval within the replenishment cycle

< A1

E[stock] before 1st arrival

≥ A1

E[stock] after 1st arrival

< tw

E[stock] before tw

< A1

E[stock] between tw and 1st arrival

≥ A1

E[stock] after 1st arrival

< tg

E[stock] before triggering 2nd order

Case 3

(2, A1 , A1 )

< A1

E[stock] between triggering 2nd order and 1st arrival

Case 6

(2, tw , A1 )

< A2

E[stock] between the two arrivals

≥ A2

E[stock] after the 2nd arrival

< tg

E[stock] before triggering 2nd order

Case 4

(2, A1 , A2 )

< A2

E[stock] between triggering and receiving 2nd order

Case 7

(2, tw , A2 )

< A1

E[stock] between the two arrivals

≥ A1

E[stock] after the 2nd

< tg

E[stock] before triggering 2nd order

< A1

E[stock] before the simultaneous arrival

≥ A1

E[stock] after the simultaneous arrival

Case 5

(2, A1 , =)

Case 8

(2, tw , =)

E[stock]: average on-hand stock during a specific period, e.g. before 1st arrival
Table 3.10: Conditions of stock on hand in Case 1 to 8

The SDMR model – Summary
scenario
Case 1

Case 2

Case 3

Case 6

Case 4

Case 7

Case 5

Case 8

function name
h
i
E< A1 stock 1(1,A1 ,.)
h
i
≥
A
1
E
stock 1(1,A1 ,.)
h
i
E<tw stock 1(1,tw ,.)
h
i
E< A1 stock 1(1,tw ,.)
h
i
E≥ A1 stock 1(1,tw ,.)

125
reference
Eq. (3.4.9),

p. 105

Eq. (3.4.10),

p. 105

Eq. (3.4.15),

p. 107

Eq. (3.4.16),

p. 107

Eq. (3.4.17),

p. 107

E<tg [stock 1(2,A1 ,A1 ) ]

Eq. (3.4.24),

p. 110

E< A1 [stock 1(2,A1 ,A1 ) ]

Eq. (3.4.25),

p. 110

E< A2 [stock 1(2,A1 ,A1 ) ]

Eq. (3.4.26),

p. 111

E≥ A2 [stock 1(2,A1 ,A1 ) ]

Eq. (3.4.27),

p. 111

E<tg [stock 1(2,tw ,A1 ) ]

Eq. (C.2.7),

p. 295

E< A1 [stock 1(2,tw ,A1 ) ]

Eq. (C.2.8),

p. 295

E< A2 [stock 1(2,tw ,A1 ) ]

Eq. (C.2.9),

p. 295

E≥ A2 [stock 1(2,tw ,A1 ) ]

Eq. (C.2.10),

p. 295

E<tg [stock 1(2,A1 ,A2 ) ]

Eq. (3.4.28),

p. 112

E< A2 [stock 1(2,A1 ,A2 ) ]

Eq. (3.4.29),

p. 113

E< A1 [stock 1(2,A1 ,A2 ) ]

Eq. (3.4.30),

p. 113

E≥ A1 [stock 1(2,A1 ,A2 ) ]

Eq. (3.4.31),

p. 113

E<tg [stock 1(2,tw ,A2 ) ]

Eq. (C.3.11),

p. 299

E< A2 [stock 1(2,tw ,A2 ) ]

Eq. (C.3.12),

p. 299

E< A1 [stock 1(2,tw ,A2 ) ]

Eq. (C.3.13),

p. 299

E≥ A1 [stock 1(2,tw ,A2 ) ]

Eq. (C.3.14),

p. 299

E<tg [stock 1(2,A1 ,=) ]

Eq. (3.4.32),

p. 115

E< A1 [stock 1(2,A1 ,=) ]

Eq. (3.4.33),

p. 115

E≥ A1 [stock 1(2,A1 ,=) ]

Eq. (3.4.34),

p. 115

E<tg [stock 1(2,tw ,=) ]

Eq. (C.4.10),

p. 302

E< A1 [stock 1(2,tw ,=) ]

Eq. (C.4.11),

p. 302

E≥ A1 [stock 1(2,tw ,=) ]

Eq. (C.4.12),

p. 303

Table 3.11: Summary of all functions to determine the expected stock

Eq. (3.5.1),
Eq. (3.5.2),

E[SH ]
E[stock]
E[stock per time unit]
E[ Q]
E[ D (t¯c )]
E[α]
E[ β]
E[tc ]
E[# cycles]
E[# first orders]
E[# second orders]

expected shortage per cycle

expected stock on hand per cycle

expected stock on hand per time unit

expected order quantity

expected cycle demand

expected service level α

expected service level β

expected cycle time

expected number of replenishment cycles

expected number of 1st orders

expected number of 2nd orders

p. 98

p. 69

p. 68

p. 121

p. 121

p. 120

p. 119

p. 119

p. 117

p. 117

p. 120

Eq. (3.5.10), p. 121

Eq. (3.5.9),

Eq. (3.5.8),

Eq. (3.5.6),

Eq. (3.5.4),

Eq. (3.5.5),

Eq. (3.5.7),

Eq. (3.4.35), p. 116

Eq. (3.3.43),

Eq. (3.2.6),

p(2, ., .)

probability of a two-order cycle

Eq. (3.2.3),

reference

p(1, ., .)

function name

probability of a one-order cycle

variable name

126
C HAPTER 3. A M ODEL F OR S TOCHASTIC D UAL S OURCING (SDMR)

Table 3.12: Summary of major equations derived from the SDMR model

Chapter 4
Considerations for a practical
application of the SDMR model
In this chapter we address question RQ 2 on page 5, how to feasibly apply the SDMR
model in practice. Several challenges arise when we transfer the SDMR model from
theory to practice. These are mainly related to the usage of empirical data and to the
complexity of the calculations within the SDMR model. Consequently, this chapter is
divided into two parts. First, we will look at how the SDMR model can be discretized.
The discretization is usually a direct implication of using empirical data. Second, we
will show how the computational complexity of the SDMR model can be reduced.
Thereby, we focus on its major impact factor which is the efficient determination of
the joint demand distribution D (t) due to its time-consuming calculation by means of
convolutions of the daily demand distribution D (1).
Throughout this chapter we augment single aspects with examples from our practical experience that illustrate the wide range of restrictions and requirements which
are frequently found in inventory management.

We want to briefly mention another important aspect of implementing the SDMR
127

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C HAPTER 4. P RACTICAL A PPLICATION O F T HE SDMR M ODEL

model for a practical application. It is the integration into the existing IT landscape
at a company. Its success depends strongly on the individual situation of a company because contemporary inventory management involves a wide range of IT, from
databases to complex enterprise-resource-planning (ERP) systems [Zip00]. Without
going into further details we remark that the SDMR model has been integrated into
the inventory management tool ”IBM Dynamic Inventory Optimization Solution”1 .
Thereby, the SDMR model can utilize existing interfaces which assure a successful
integration with various IT systems.

4.1

Discretization of the formal model

In this chapter we address the topic of discretizing the SDMR model which has been
formally introduced in Chapter 3. One will usually agree that a discretization has to
be primarily considered for all parts of a model that represent a discrete parameter.
The decision whether a parameter is continuous or discrete is rather intuitive in
most cases. For the demand quantity, it translates into the divisibility of the considered
product. For example, gas and liquids are in general infinitely divisible while cars
and computers are not. Screws and paper take an intermediate role as they are often
considered to be infinitely divisible in practice [Zip00]. Similarly, the time can be seen
as a continuous or a discrete parameter. Examples are a continuous-time production
process with a certain demand rate versus a periodic replenishment policy even if the
orders are placed every 24 hours [Tem06], [Zip00].
Knowing that a parameter is discrete still does not necessarily answer the question
whether it should also be modeled in a discrete way for a practical application. The
following examples will highlight some scenarios in the context of the SDMR model.
Example 4.1.1. Company A has collected the discrete daily demands over the last two
1

Solution of IBM Corporation

Practical application – Discretization of SDMR model

129

years. In favor of a simple model, A only extracts the average µ D and the standard
deviation σD of the daily demands from the historic data. The daily demand distribution is then approximated by the Gaussian distribution Φ(µ D , σD ). The lead time of
a supplier is approximated by a Gaussian distribution Φ(µ L , σL ), as well. The values
for µ L and σL are not measured but manually set for each supplier once a year.
In this case we theoretically do not have to discretize the SDMR model. All parameters and convolutions can be derived analytically. However, in practice the replenishment parameters are usually rounded to integer values at some point because
fractional order quantities or reorder points are rather hard to interpret in the context of discrete demands and lead times. Consequently, one could also consider to
discretize the input parameters of the SDMR model.
Example 4.1.2. Company B collects the daily demands and the lead times of the suppliers and uses histograms for calculating all parameters that are relevant to operate
the inventory.
This is the most common scenario we have encountered in practical inventory
management and, here, the SDMR model has to be able to use discrete input histograms. Note, this procedure is superior to Example 4.1.1 because it incorporates
more information about the distribution of the demand and the lead time. However,
very rare events that are represented by the tails of distributions will insufficiently be
covered by a limited time series of historical data.
Example 4.1.3. Company C collects the daily demands and lead times. Instead of using them directly in their calculations they try to find the best fitting continuous distribution for each histogram. These continuous distributions are then used for further
calculations.
In this example, we theoretically do not have to discretize the SDMR model. However, the convolution of arbitrary continuous distributions cannot always be given

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C HAPTER 4. P RACTICAL A PPLICATION O F T HE SDMR M ODEL

analytically. Then, numerical techniques have to be utilized and a discretization of the
SDMR model is necessary. Moreover, even if an analytical solution can be given, the
individual result values might be rounded to integer values for a similar reason as we
gave in context of Example 4.1.1.
Finding the best fitting continuous distribution from histograms like described in
Example 4.1.3 is an elaborate and quite accurate way to capture the stochastic nature of
parameters like the daily demand. However, the required calculations are extremely
time-intensive and barely feasible on a per-SKU basis especially for large warehouses.
One feasible approach here is to utilize clustering techniques as described by Bödi and
Schimpel [BS05].
In summary, the examples 4.1.1 till 4.1.3 show that the discretization of the SDMR
model plays a crucial role for a practical application. The following sections address
important issues that arise when the SDMR model is discretized. The sections 4.1.1
and 4.1.2 look at the transformation from a continuous to a discrete model on a general
level. The remaining sections cover more SDMR-specific aspects.

4.1.1

Discretization of continuous distributions

Whenever continuous distributions are available but the business scenario requires a
discretized SDMR model, we have to find a way to discretize these input distributions.
The two possibilities to discretize a continuous distribution are intervals of fixed
size and variable size. The choice is easy for the SDMR model because we have to
calculate the distribution D (t) of the cumulated demand over t days by means of
numerical convolution techniques. These require intervals of identical size, see Section 4.2.3.
A practical requirement is to preserve important characteristics of the continuous
distribution after the discretization. This is especially true for the lower moments of

Practical application – Bounds of integration and summation

131

a distribution. Consequently, the average demand value of the original, possibly analytical, distribution D should match the average value of its discretized counterpart
D 0 with a sufficient precision. Our experience shows that this can be achieved very
well by creating the density function of D 0 according to the rule
p D0 (k ) = FD ((k + 0.5)c D ) − FD ((k − 0.5)c D ),

(4.1.1)

where FD is the cumulative distribution function of D and p D0 is the probability function of D 0 . The distribution D 0 consists of n + 1 intervals with 1 ≤ k ≤ n + 1 and a

fixed interval size of c D . In our case we only allow for non-negative demand. Thus,
the probability of the initial interval 0 is given by
p D0 (0) = FD (0.5c D ).

(4.1.2)

The magnitude of deviation between the expected value of D and D 0 depends
strongly on the size and the number of the buckets. Given a desired accuracy of the
model one can choose an appropriate quantile interval of the distribution and divide
it into several hundreds of buckets. Of course, a trade-off between accuracy and calculation time has to be found, which is strongly influenced by the business purpose
or the investigated scenario.

4.1.2

Bounds of integration and summation

In the continuous case no thought has to be spent on how to treat the bounds of an
integral. These bounds represent a so called null set and two arbitrary distributions
that only differ in a null set are equivalent [Fel71]. Thus, for an integrable function h
the equation
Zb
a

h( x ) dx = lim

e →0

bZ+e
a

h( x ) dx = lim

e →0

Zb

a+e

h( x ) dx = lim

e →0

bZ+e
a+e

h( x ) dx

(4.1.3)

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always holds. This simplifies the formulation of equations. Certainly, the equation
Za

f ( x ) dx +

Z∞

f ( x ) dx = 1

(4.1.4)

a

−∞

always holds for every density function f ( x ) defined in R and an arbitrary a ∈ R.

However, the formulation of equations becomes less obvious in the discrete case

where these bounds play a crucial role. Using the integer-discretized version f d of
the density function f in Equation (4.1.4) and given an arbitrary integer value b, the
equation
b

∑

x →−∞

f d (x) +

∞

∑

x =b

f d (x) = 1

(4.1.5)

does not hold except for the lucky coincidence where f d (b) = 0. In every other case
the probability f d (b) is counted twice.
Thus, one has to take special care to not exclude arguments or include them twice.
Especially, if the bounds separate two cases which require different formulas, the decision about the exact bounds has to be based on a thoughtful and sometimes non-trivial
semantic interpretation of the formula.

4.1.3

Start of the replenishment cycle

In theory, we are free to choose when the time line starts even if we apply discrete time
intervals. Thus, we can decide to individually start the replenishment cycle for each
SKU exactly when the first order is triggered, denoted by t0 . In practice, however, this
is not applicable as one is bound to certain times like a daily recurring departure of
the supply truck at 6:00h in the morning. Of course, the reorder point R1 is usually
not hit exactly at that time. This means we have to treat the fraction of a day until the
next truck departs separately regarding the lead time and the stock depletion. In the
following we show how we can relax Assumption 10 on page 45 in order to capture
time intervals and the bulkiness of the demand.

Practical application – Start of replenishment cycle

133

Let us look at time intervals of one day. Given a specific stock level we can use
the daily demand distribution to calculate the probability that the stock level drops
to R1 during one of the subsequent days. As we might not have detailed information
about the demand distribution within one day we can simply determine t0 by interpolating between the starting and the ending stock per day. The result is a probability
distribution of the possible cycle start times t0 which can be directly translated into the
fraction of a day of additional lead time y0 and the demand x 0 during this additional
lead time, y0 .

Now, there exist several options. The first one is to base the calculations on the
original reorder point R1 and to shift the regular daily demand distribution D ( x ) for
the first and only the first day by x 0 . Moreover, the regular lead time distribution L1
is shifted by y0 . So f 1 ( x + x 0 ) = f ( x ) holds for the demand density function of the
first day and l11 (y + y0 ) = l1 (y) is used as the lead time of the first supplier. Having
done so, the formal model can be used as before for all 1-order-cycle cases. For the
2-order-cycle cases one has to repeat the same procedure with the reorder point R2 .

Another option is to reduce the reorder point R1 by x 0 and to continue calculating
based on our original model. In order to obtain the correct result, for example, for the
expected stock on hand per cycle we have to add the additional demand x 0 during the
time y0 , respectively. In a 2-order cycle the same approach has to be repeated for the
interval where the stock level drops below the second reorder point, R2 .

These approximative approaches can be easily combined with our formal model. This
is a very important step for relaxing Assumption 10 which allows the successful application of the SDMR model in many practical situations.

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4.1.4

C HAPTER 4. P RACTICAL A PPLICATION O F T HE SDMR M ODEL

Time and demand discretization

The time line of the replenishment cycle is divided into identical, arbitrarily long intervals, starting at t0 . Both lead times and the inter-order time can now be expressed by
an integer number of intervals. All occurring demand distributions D (t) are related to
a positive number t ∈ {1, 2, 3, ...} of time intervals, as well. The demand during a time

of zero intervals is zero just as the probability of an instantaneous lead time is usually
zero in practice, as well. The detailed occurrence of demand within a single interval
cannot be measured and is unknown. Besides the time component also the quantity of
the demand can be discretized. In many industries the demand is an integer number
of units like one kilogram or a single item. Note, one is completely free to choose the
size of the time interval and the demand unit.
Example 4.1.4. Company A does not trigger an order immediately when the stock
level reaches the reorder point but only twice per day, once in the morning and once
in the afternoon. In this case it is completely sufficient to set the time interval to half a
day.
Example 4.1.5. In company B orders are triggered instantaneously after reaching the
reorder point but the supplier delivers only once a day in the morning. Here, a time
interval of one day will be appropriate for the SDMR model.
Example 4.1.6. Company C runs a fully-automated replenishment system. Orders are
sent immediately to the supplier upon reaching the reorder point. The supplier is able
to guarantee a delivery time accuracy of one hour. One could use an interval of one
hour to model their operative business scenario here.
Example 4.1.7. Company D is in the same situation as company C. However, D aggregates the demand and lead times on a daily basis for its data warehouse. Hence, one
cannot derive the lead time and demand distribution on an hourly basis from the historical data. Then, one should use an interval length of one day for the SDMR model.

Practical application – Time window for second order

135

Otherwise one might obtain wrong results induced by the lack of information when
exactly during the day the demand occurs and orders arrive.
Of course, the computation time can increase significantly when small intervals are
used to discretize the time and the demand. Thus, a tradeoff between accuracy of
the model and the time performance of the system usually has to be found. Note,
that the accuracy of the model is also limited by the quality and the aggregation of the
input data from the company. In many practical projects we have encountered that the
properties of the input data dominate the technical restrictions especially regarding
the accuracy of the results.

4.1.5

Time window for second order

The time interval to trigger a second order is restricted by two parameters. First, the
maximum time window tw for placing a second order. Second, the arrival time t A1 of
the first order because we do not trigger a second order after the arrival of the first
order. In most cases this is quite obvious. However, the decision whether a second
order can be triggered exactly in interval t A1 or tw is more difficult.
First, consider the case t A1 > tw , where the first order arrives after tw . One can argue that tw is the maximum time between triggering the first and second order which
includes the possibility to issue an order at tw . However, the precise value of tw might
be a non-integer result of some given formula and one has to decide whether to include the last day or not based on the specific business case.
Second, let the first order arrive no later than tw , so t A1 ≤ tw . Now, the question

arises whether one can still place a second order on the day where the first order
arrives. This depends heavily on what is known about the arrival of the orders. On
the one hand, if we know the arrival interval t A1 of the first order ex ante, then it does
not make sense to place a second order at t A1 because it cannot arrive before the first

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one. On the other hand, if the exact arrival interval is only known ex post we will
usually place a second order at t A1 . Sometimes we have additional knowledge about
the delivery time within an interval. We can leverage this knowledge in order to set
up a SDMR model that represents the business practice more appropriately.
In the following, different scenarios are exemplified that influence the decision up
to when a second order is allowed to be triggered in the SDMR model.
Example 4.1.8. Company A has a sophisticated IT system for managing its inventory
and the relations to its suppliers. The possible time frame of the delivery is constantly
updated by the supplier and the logistics company. Consequently, the time frame of
delivery is constantly narrowed down over time. The delivery period t A1 is fixed one
period in advance. In addition, the time window tw is constantly updated in the way
that the probability of the first order arriving before the second is less than 50%.
Company A has a minimal lead time of one period for the second order. Therefore,
A stops placing the second order one day before t A1 . In other words, a second order is
only triggered in the SDMR model if the inter-order time t g satisfies the inequality t g ≤

min(tw , t A1 − 2). Sometimes it happens that the replenishment orders have already
been placed at day t A1 − 2 before the day of delivery is fixed. Then, the accurate way
of modeling the inequality for t g is t g ≤ min(tw , t A1 − 1).

Example 4.1.9. Company B is supplied no more than once per day and does not exactly
know the day of delivery t A1 . However, the supply always arrives in the morning
before the replenishment orders are sent to the suppliers.
In this case, B will not trigger a second order on day t A1 and the SDMR model has
to use the restriction t g ≤ min(tw , t A1 − 1) for all two-order cycles.
Example 4.1.10. Company C places its order already in the early morning. However,
C neither knows the exact day of delivery nor has a fixed delivery time throughout
the day. Therefore, it is quite common for C to place a second order even if the first
order arrives at the same day.

Practical application – Compliance with assumptions

137

In this case, the constraint t g ≤ min(tw , t A1 ) is used in the SDMR model. The same
holds for all scenarios where, throughout the daily schedule, C places its replenishment orders before the potential supply arrives.
The examples show different conditions up to when a second order can be triggered.
The challenge is to employ the restrictions within the SDMR model in such a way that
they meet the practical situation best. A summary of the conditions from the latter
examples is given in Table 4.1.
delivery interval
notification or

before placing orders

delivery time

after placing orders

known ex post

known ex ante

t g ≤ min(tw , t A1 − 1)

t g ≤ min(tw , t A1 − 2)

t g ≤ min(tw , t A1 )

t g ≤ min(tw , t A1 − 1)

Table 4.1: Exemplified conditions for two-order cycles in a business scenario

4.1.6

Compliance with model assumptions

Last but not least the discretization and implementation of the model has to comply
with the assumptions of the formal model described in Chapter 3.1.1, see pages 42 to
44. Most of them are easy to assure.
Initially, both supply modes and their specific parameter settings, e.g. lead time distributions, order quantities, and price structures, are arbitrarily associated with different
identifiers, namely 1 and 2. This complies with Assumption 1.
In order to satisfy Assumption 2 we simply exchange their identifiers whenever
the relation R1 > R2 is violated between both reorder points. Additionally, whenever
R1 = R2 holds our model is not applicable and we either change R1 or R2 , or we
have to exclude the SKU from further calculations. This exchange of identifiers and

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C HAPTER 4. P RACTICAL A PPLICATION O F T HE SDMR M ODEL

the enforcement of the restriction R1 6= R2 will be extensively used in an optimiza-

tion process where both reorder points and order quantities are continuously adapted
while searching for a cost-minimal solution, for example.
We have to enforce positive lead times lti > 0, order quantities qi > 0, and a nonnegative daily demand d ≥ 0 for i ∈ {1, 2}, see Assumptions 3, 5, and 6. Whenever

this is not possible for a specific SKU we exclude it from our investigations. Note,
by using empirical data one usually complies with these restrictions already. In other
cases we restrict the values of a parameter to its valid range. For example, in the
case of a Gaussian-distributed daily demand distribution D ( x, 1) = Φ(µ, σ) this can
be achieved by using the truncated Gaussian distribution given by φ( x |0 ≤ x < ∞)

where φ is the probability density function of Φ(µ, σ).

Convolving D ( x, 1), the daily demand distribution, n times leads to an aggregated
demand distribution D ( x, n) over n ∈ {1, 2, 3, ...} days which is identically and independently distributed as requested by Assumption 5.

Both Assumptions 7 and 9 relate to rules when orders are allowed to be triggered.
These rules are inherently assured by the implementation of the model and its formulas.
Last but no least, Assumption 8 limits the maximum demand that is allowed to occur during one replenishment cycle. The demand during a cycle is potentially infinite
for unbounded distributions like Gaussian distributions. Therefore, we only consider
SKUs where the probability p E , that the demand exceeds its maximum limit, is below a certain threshold, e.g. p E < 10−10 . The probability p E has been introduced as
a measure of applicability in Chapter 3.1.6 on page 64 and is influenced by the four
parameters R1 , R2 , Q1 , and Q2 . Whenever these parameters cannot be changed we exclude all SKUs with a too high value for p E . However, whenever these parameters are
not fixed, like in an optimization process, we adjust the parameters in such a way that
the requirement for p E is met.

Practical application – Determining convolutions

139

In summary, we are able to discretize the SDMR model and still assure all the assumptions that have been made for the continuous SDMR model in Chapter 3.1.1.
This concludes our section about how to discretize the SDMR model and answers the
first part of question RQ 2 on page 5.

4.2

Determining convolution of distributions

We mentioned in the beginning of Chapter 4 that the computational complexity of
the SDMR model is a key success factor for a practical application. In our case, the
complexity in time dominates the complexity in space of the SDMR model. This can
be explained by the fact that all SKUs in a warehouse are processed sequentially on
one or several computers. Thus, only the time of the calculations scales linearly with
the number of SKUs while the space of the calculations remains constant.
Example 4.2.1. Company C owns a warehouse with 10, 000 SKUs and introduces a
new replenishment policy for reducing the total costs of the warehouse. An appropriate theoretic model is developed for this new policy. This enables C to investigate the
behavior of the policy.
Now, C wants to find the set of parameters that reduces – and preferably minimizes
– its total inventory costs. The replenishment model is too complex to be optimized
analytically. Therefore, a numeric optimization algorithm is used. On average, the
optimization algorithm needs 50 iterations to find a cost-minimal solution. In sum,
this involves 500, 000 recalculations of the theoretic model with different values of the
replenishment parameters.
One requirement of C is to adapt the cost-minimal replenishment parameters every Sunday based on the data of the past week. This way, the replenishment of C
incorporates only information that is younger than one week. One full day consists of
86, 400 seconds. Without parallelization this restricts the average calculation time for

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C HAPTER 4. P RACTICAL A PPLICATION O F T HE SDMR M ODEL

each of the 500, 000 iterations to 0.1728 seconds.
A time restriction as mentioned in Example 4.2.1 can be found in many practical
inventory management projects. Consequently, we have to identify time-consuming
calculations that are frequently used in the SDMR model. The determination of convolutions of distributions is such a candidate especially if the calculation cannot be
expressed analytically. Therefore, the following sections will investigate when it is
possible to use fast analytical methods to retrieve convolutions and what can be done
to reduce the computational complexity of determining the convolutions when a numerical calculation is necessary. This will be exemplified by using the convolutions
of the daily demand distribution D (1). Of course, it can be transferred to arbitrary
distributions, as well.

4.2.1

Analytical determination

In most cases companies observe the customer demand on a daily basis over a certain period of time and derive its distribution, called daily demand distribution D (1).
Throughout the SDMR model we need to know how the cumulated demand over
i ∈ {1, 2, 3, ...} days is distributed, represented by the notation D (i ). Consequently, D

is a family of distributions.

The question is how to obtain D. We assume that the daily demand distribution
D (1) does not change throughout the replenishment cycle. Therefore, the daily demand distributions are identically and independently distributed, see Assumption 5
on page 42. Then the joint demand distribution for i days can be simply calculated by
the i-fold convolution of the daily demand distribution, D (1), which is denoted by
D (i ) = D (1) ∗ ... ∗ D (1) .
|
{z
}

(4.2.1)

i-times

In the simplest case one observes (or assumes) that the daily demand follows a distribution for which the n-fold convolution can be given analytically by a closed form

Practical application – Limits of analytically determining convolutions

141

expression. In this case, one can directly calculate the joint demand distribution D (n)
for n days. In a second step, one discretizes D (n) adequately. This approach is very
fast and easy.
Example 4.2.2. Let the daily demand D (1) be exponentially distributed with a rate of
λ and an expected value of E[ D (1)] =

1
λ.

Then the joint demand distribution for n

days is given by the Gamma distribution D (n) = Γ(n, λ).
Example 4.2.3. Let the demand follow a Gaussian distribution Φ with mean µ and
a standard deviation of σ. Then the demand occurring within n days is distributed
according to D (n) = Φ(nµ, nσ2 ).
However, problems arise if the demand distribution deviates from such convenient
distributions. For example, the Gaussian distribution usually has to be truncated to
non-negative values as negative demands do not occur. This is especially significant
for small values of µ or large values of σ.

4.2.2

Limits of an analytical determination

Sometimes the practically observed (or assumed) demand distribution D (1) can be
simply obtained by truncating a distribution F0 . Then, there might still exist a compact
expression for D (1).
Example 4.2.4. Let the daily demand D (1) be a truncated version of the Gaussian distribution F0 = Φ(µ, σ), where the probability of all negative values is added to the
probability of zero demand. The probability density function of F0 is denoted by f 0 .
This leads to a Gaussian distribution with an initial peak for the value zero:

D ( x, 1) =



0


 F0 ( x )

if x < 0
else

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Note, distributions with a high peak for zero demand per day and a second peak for
a positive demand are very common in the spare parts business. The average µtrunc of
D (1) is given by
µtrunc
where α is specified by




f 0 (α)
= (1 − F0 (0)) µ + σ
,
1 − F0 (α)
α=

0−µ
µ
=− .
σ
σ

While one can use this compact representation of the daily demand distribution it
is usually not possible to derive an explicit representation of its n-fold convolution
anymore. This is where numerical approaches come into play.
Besides the technical details that are necessary for a practical application of the
SDMR model, we have to consider some common business requirements and restrictions of applied inventory management. We briefly highlight the aspects that we consider to be most relevant.

4.2.3

Numerical determination

Most data in practical inventory management origins from empirical observations.
Due to the wide range of industries we cannot rely on certain assumptions that are
often to be found in the theory of inventory management. For example, academic
publications frequently assume that parameters like the lead time or the demand are
constant, known, or governed by special distributions, like a Gaussian or an exponential distribution. This is also reflected in the paper by Minner where he gives an
overview of research in multi-supplier policies [Min03]. In many industries, however,
such simplifying assumptions do not hold for various, sometimes rather simple reasons.
Example 4.2.5. In the spare parts industry the average yearly demand is often as little
as one unit per month. In addition, the coefficient of variation can be quite high as

Practical application – Determining convolutions numerically

143

there are months with two or three units of demand but in the majority of the months
there occurs no demand. Consequently, the error caused by modeling the demand
with a non-truncated Gaussian distribution would be high in this example.
Example 4.2.6. A company has private and business customers. The private customers
tend to buy small quantities while the business customers usually make bulk orders.
When both customers are equally served from the same warehouse the appropriate
approach would be to use a bimodal distribution. The usage of a more convenient or
simpler distribution can lead to severe miscalculations here.
The big advantage of a numerical approach for convolving distributions is that
most of them are able to convolute arbitrary discrete distributions. However, they
require a discretizion of the daily demand distribution prior to its convolution. Of
course, these approaches are far more complex and time-consuming than an analytical approach mentioned in Section 4.2.1. Still they have been successfully applied by
the SDMR model for a practical case.
The general definition of the convolution f ∗ g of two real functions f and g is given
by

f ∗ g : x 7→

Z∞

−∞

f (τ ) g( x − τ )dτ.

(4.2.2)

In the case of a discrete SDMR model, the functions f and g are defined for n and m different input intervals, respectively. The minimum number of operations to calculate
f ∗ g in the traditional way, by using all possible combinations of input parameters, is

(n − 1)m which leads to an almost quadratic computational complexity O( N 2 − N ) in

the case where m = n. Obviously, this complexity class is not very convenient especially for large N. Fortunately, the result can be calculated much faster by means of the
Fourier transform and the family of so called Fast Fourier Transform (FFT) algorithms
which usually have a complexity of O( Nlog2 N ) [PTVF07].

144

C HAPTER 4. P RACTICAL A PPLICATION O F T HE SDMR M ODEL

However, one has to be careful about the concrete usage of these FFT algorithms.
Despite of its general definition, the context of a convolution strongly affects its detailed computation and usage, for example when convolving distributions. Recall that
we want to calculate the distribution of the cumulated demand over k days. Naturally,
this means that the demand distribution will shift towards higher demand values. For
example, the average demand during k days amounts to k times the average daily demand. Consequently, by increasing k, the range of cumulated demand during these k
days with a positive probability will grow, as well.
In contrast, the handling of convolutions, for example in image processing, usually truncates the values after a certain point. This makes sense as the array a =

( a1 , a2 , ..., an ) usually represents an image and the array b is a signal for filtering or
processing the image. All values that exceed the size of the image will simply be cut
off. In other words, the size n of the array a which represents the image will always
remain the same even though the value for each element in the array might change.
In other cases, for example electrical engineering, each of the two arrays a and b
represents a periodic wave function. Here the values exceeding the range of a affect
the beginning of the succeeding wave function which is identical to the beginning of
the current wave and, thus, is identical to the beginning of the array a itself. Certainly,
these exceeding values have to be added to the first elements of a. While the size of a
remains constant, just as in the case of image processing above, the exceeding values
are not cut off. Consequently, the results of both convolutions will differ.

4.2.4

Algorithm for a numerical determination

In the following an efficient algorithm for calculating the convolution of two distribution densities f and g will be explained in more detail, see Figure 4.1. It represents a
major requirement to make the SDMR model feasible for a practical application and it
describes how convolutions of probability distributions – in contrast to wave function

Practical application – Numerical algorithm for determining convolutions

145

or images – can be obtained. A good source for further information on the numerical
computation of convolutions is the book ”Numerical Recipes: The Art of Scientific
Computing” by Press et al., [PTVF07].
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314

Figure 4.1: Algorithm to compute the convolution of two distributions

Start.

Let f be the n-array of real numbers representing the histogram of the one-day

demand probability density function, D (1). The objective is to calculate the histogram
of the distributions D (2), D (3), ..., D (m) representing the cumulative demand during
2, 3, ..., m days.
Step 1. Copy the one-day histogram f to the base array g.
Step 2. Convert the arrays f and g with real numbers to the two arrays f C and gC
containing the complex numbers representing f and g, respectively, by simply setting

146

C HAPTER 4. P RACTICAL A PPLICATION O F T HE SDMR M ODEL

the imaginary value to zero. A common representation of such an array with complex
numbers is to alternate the real and imaginary value in the array. Consequently, an
array with n real values is replaced by its complex counterpart containing 2n entries.
Step 3 and 4. We have to pad both arrays gC and f C with additional entries of value
0. This is essential in order to avoid an overlap of results being calculated at the end
of the array with values at the beginning of the array.
Note, this origins from the assumption that the used histogram is just one instance
of a repetitive pattern, like a wave function in electrical engineering. In this case,
the calculations at the end of gC which exceed its original array size will impact the
beginning of the successive array which is identical to the beginning of the current
array gC .
In our case, however, the average demand increases the more daily demand distributions we convolve. This implies that our histogram will increase its size towards
higher demand values and the values at the tail of the distribution must not influence
the beginning of our density function.
Step 5.

The array f C which consists of N complex numbers after padding is trans-

formed into its Fourier transform f F by the calculation rule
f F,k =

N −1

∑

j =0

kj

f C,j e−i2π N

(4.2.3)

where f C,j is the jth complex number in the array f C . The same procedure is applied
to the array gC , as well. Here the choice and usage of an efficient transformation
algorithm like the FFT is crucial [PTVF07].
Step 6. The Fourier-transformed complex values of g F and f F are pairwise multiplied with each other according to the multiplication rule for complex numbers

( a j + b j i )(c j + d j i ) = ( a j c j − b j d j ) + (b j c j + a j d j )i

(4.2.4)

Practical application – Numerical algorithm for determining convolutions

147

where a j and c j are the real part and b j and d j are the imaginary part of the jth complex
number in array f and g, respectively,. The result is stored in the array h F of complex
and Fourier-transformed values.
Step 7. The array h F of Fourier-transformed complex numbers is now transformed
back into the original domain by an inverse Fourier transform.
hC,k =

1
N

N −1

∑

j =0

kj

h F,j ei2π N

(4.2.5)

Step 8. Usually a cleaning process is necessary for array hC in order to correct small
numerical calculation errors. For example, insignificant small values are set to zero
because they can induce large errors once hC is used for further convolutions. Moreover, we can delete elements that were needed for the sole purpose of padding but do
not contain any values for the convoluted result.
Step 9.

All imaginary parts of the complex numbers are zero in our case if the con-

volution process has been successful. So we can simply reduce the array hC to a realvalued array h by deleting the imaginary part.
Step 10, 11, and 12.

We store the result h of convoluting f and g. Whenever we do

not need to calculate the joint demand distribution of more days the algorithm stops
here. Otherwise we can simply convolve hC and f C by using them as input to step 3
of the algorithm.
For example, hC is the result of convolving two one-day demand distributions
D (1), represented by the arrays f and g. The result is the joint distribution D (2) for
two days which can be convolved with f C , the array of complex numbers representing
D (1). This yields D (3), the joint demand distribution for three days.

148

C HAPTER 4. P RACTICAL A PPLICATION O F T HE SDMR M ODEL

This concludes the description of a FFT algorithm that is able to determine the convolution of arbitrary distributions in a time-efficient way. In practice we encounter very
different types of distribution for the demand, for example. Many of these are hard to
express analytically by a closed-form expression, not to mention their convolutions.
Therefore, we suggest that an implementation of the SDMR model should employ an
FFT algorithm as described above.

4.3

Summary

The discretization of the SDMR model is a key requirement to make it applicable in
practice. In this chapter we showed two important aspects related to the discretization
of the SDMR model. First, we highlighted how to discretize different parameters of
the SDMR model according to specific business scenarios. These discretized versions
of the SDMR model still comply with the assumptions from Chapter 3.1.1. Second, we
investigated how to convolve distributions which is a time-intensive and heavily used
calculation within the SDMR model. In Chapter 4.2.4 we described a FFT algorithm
that is able to calculate the convolution of arbitrary distributions in a time-efficient
manner. The elaboration throughout this chapter describes how to apply the SDMR
model in a practical scenario which answers our research question RQ 2 on page 5.
The discrete SDMR model is able to represent different discrete business scenarios.
While it is designed to deal with arbitrary distributions, the time-intensive calculation
of convolutions is often practically infeasible. Fortunately, the usage of a fast and
flexible FFT algorithm enables us to represent all possible distributions of demand
that might occur in different industries, for example. These are essential properties for
a practical application of the SDMR model. In the next chapter we are able to leverage
these properties by evaluating the SDMR model with data from a real warehouse.

Part III
Evaluation, results, and conclusions

149

Chapter 5
Overview of the evaluation approach
So far we have introduced a formal model for a stochastic dual-source replenishment
policy, called the SDMR model, and we have addressed some of the issues that are essential for an application in practice. Now, we want to evaluate whether the usage of
the rather complex SDMR model can be justified in practice and whether it is beneficial
to replenish via two suppliers. Throughout our comparisons we focus mainly on the
two key performance indicators (KPIs) total costs and service level. We have encountered these two KPIs to be predominant throughout most of our projects. This choice
is supported by one of the key findings in the Gartner Research report ”Top KPIs for
Supply Chain Management” by Payne: ”Although all supply chains have numerous
departmental or operational KPIs, most supply chains have only two enterprise-level
KPIs: total delivered cost and customer service, or variants of these.” [Pay09, p. 1].
We will focus on the two research questions RQ 3 and RQ 4 as stated in the Introduction. The remainder of this chapter is organized around these two questions.
First, Section 5.1 elaborates why we can answer both questions by exclusively using
the SDMR model, given that we use the appropriate input data and parameters. Second, Section 5.2 describes the cost structure that applies for all our evaluations. Finally,
we summarize the key points in Section 5.3.
151

152

5.1

C HAPTER 5. O VERVIEW O F T HE E VALUATION A PPROACH

Comparison of replenishment policies

Our questions RQ 3 and RQ 4 that have been posed in the Introduction on page 5 are:
RQ 3: How much and in which situations does a scenario with deterministic lead
times deviate from the stochastic scenario of the ( R1 , R2 , Q1 , Q2 ) replenishment
policy?
RQ 4: How much total costs can be saved by moving from single sourcing with traditional restrictions to dual sourcing with relaxed restrictions when lead times are
stochastic?
In the process of answering RQ 3 we want to iteratively gain a better understanding
about the impact that different parameters have on the deviation between a scenario
with deterministic lead times (DET) and the one with stochastic lead times (STOCH).
We take a two-step approach. First, we conduct a sensitivity analysis on each input
parameter individually of an exemplary SKU. This allows us to study the gradual
change of a single input parameter and its effect on the deviation between DET and
STOCH. Second, we analyze the deviation in a real warehouse with 2,751 SKUs and
compare these results with our findings from the sensitivity analysis. This two-step
approach will, on the one hand, provide us with greater insights into the mechanisms
and relations between the input parameters and the KPIs and their impact on the
deviation between DET and STOCH. On the other hand, it will show the relevance of
our findings in the example of a real warehouse.
If we want to exclude effects that result from a different modeling of the DET scenario and the STOCH scenario we have to use the same model. Fortunately, the SDMR
model is able to represent STOCH as well as DET. The reason is that the SDMR model
runs on a computer program and utilizes histograms for all its distributions. Therefore, every deterministic dual-source policy can be represented by a stochastic dual-

Evaluation approach – Comparison of replenishment policies

153

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source policy where all histograms consist only of one single positive value, namely
100%, see Figure 5.1.
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9:

3

4

5

6

7

9:

3

4

5

6

7

Figure 5.1: Obtaining a deterministic model by reducing the histogram to one entry

For the question RQ 4 we investigate common restrictions regarding the replenishment of SKUs that exist in literature as well as in practice. We want to see by how
much the total costs can be reduced in our real warehouse if we relax these restrictions.
Therefore, we have to optimize the different scenarios regarding their total costs and
compare their results. In addition, we want to understand the mechanisms that lead
to these savings. Here, we are only interested in comparing scenarios with stochastic
lead times.
In order to answer question RQ 4 we have to model single-souring and dualsouring replenishment policies with and without relaxed replenishment restrictions.
Luckily, it is easy to represent the different replenishment restrictions with the SDMR
model by limiting the values of the input parameters Q1 , Q2 , R1 , and R2 . Similarly,
single sourcing can be represented by the SDMR model by setting the second reorder
point R2 to a very negative value, such that it will never be reached, see Figure 5.2.
An intermediate conclusion is that we can perform our comparisons solely by means
of the SDMR model. Only the input to the SDMR model is different for each of the

!

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154

C HAPTER 5. O VERVIEW O F T HE E VALUATION A PPROACH
/4+56

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Figure 5.2: Representing single-sourcing by using a very small reorder point R2

scenarios used in the comparisons for our questions RQ 3 and RQ 4. This is illustrated
in the three-step approach (Input, Calculation, Comparison) of Figure 5.3.
We use the term comparison for the whole process of comparing two scenarios.
Each scenario consists of a specific set of input data and parameters called setting, a
replenishment policy, a calculation model, and a corresponding result. Consequently,
different combinations of settings and replenishment policies lead to different scenarios. Moreover, a comparison always has to be seen in context of its specific settings
and models. This accounts for the common observation that the advantages and disadvantages of applying a replenishment policy cannot be determined up front but
depends very much on the specific situation (setting) at a warehouse.
In accordance with Figure 5.3 the input of each scenario is defined in a first step. This
input consists of external data like the demand distribution, the chosen replenishment
policy, and internal parameter settings like R1 , R2 , Q1 , and Q2 . Most comparisons that
we perform in context of the questions RQ 3 and RQ 4 involve the stochastic dualsource scenario which will be used as a reference scenario. The opponent replenishment scenario is called counterpart scenario. The input of the reference scenario may

!"

Evaluation approach – Comparison of replenishment policies
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Figure 5.3: Comparison of two scenarios

be influenced by the input of the counterpart scenario which is indicated by the arrow
between the two data boxes in Figure 5.3. For example, the optimized parameters of
the counterpart scenario might be used for both scenarios.
In the second step, the complete input information including the replenishment
policy is fed into the SDMR model which calculates the appropriate results. The third
step compares the output of the reference scenario and the counterpart scenario regarding total costs, for example.

In summary, we can use the SDMR model for each of the comparisons that are required for answering our research questions RQ 3 and RQ 4. Moreover, most comparisons involve the stochastic dual-source replenishment policy which is used as a
so-called reference scenario. Now, we will briefly present the cost structure that will
be applied in our comparisons.

156

5.2

C HAPTER 5. O VERVIEW O F T HE E VALUATION A PPROACH

Employed cost structure

Most evaluations in the subsequent chapters are based on the total inventory costs.
!"#$%#$&'($'&)
Figure 5.4 illustrates the detailed cost structure that we apply. Note, the formulas
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Figure 5.4: Employed cost structure

that are necessary to calculate all parts of this cost structure have been introduced
throughout Chapter 3 and are summarized in Table 3.12 on page 126. We use these
formulas to define the total costs TC as
TC = CC + OCN + OCR + OCB .

(5.2.1)

Details about the components of TC are given in Table 5.1. The total costs TC contain
the inventory holding costs CC and the ordering costs OCN , OCR , and OCB . Thereby,
the ordering costs include all transport costs.
The inventory holding costs for one SKU only consist of the capital costs CC
which are induced by the interest rate r, the unit price, and the average positive stock
E[stock per time unit] according to
CC = E[stock per time unit] · price · r,

(5.2.2)

Evaluation approach – Employed cost structure
cost type
name

cost factors
symbol

variable

fixed

total costs

TC

capital costs

CC

r

cost 1st orders

OCN

c N,var

c N,fix

cost 2nd orders

OCR

c R,var

c R,fix

cost back orders

OCB

c B,var

c B,fix

157

comment
TC = CC + OCN + OCR + OCB
e.g. interest rate

Table 5.1: Summary of used cost parameters
see Table 3.12. Of course, one can also add volume-dependent storage costs to CC, for
example.
The order costs are the sum of fixed and variable costs for each of the three order modes: regular, rush, and back order. The regular order costs and the rush order
costs refer to the two different supply modes which are associated to R1 and R2 , respectively. The back order costs represent the costs in case of a stock-out situation.
These might have completely different implications in certain business scenarios. In
our case, a stock out is backlogged and the customer will be served by a privileged
process, called back order, which causes additional fixed and variable costs. However,
these back orders can be interpreted as the well-known opportunity costs for not serving a customer or a compensation payment to the customer. There exist many ways
of calculating the order costs which heavily depend on the usage of these costs in an
operational or more strategic context. For the remaining chapters we define the order
costs as follows.
OCN = E[# first orders] · ( Q1 · c N,var + c N,fix )

(5.2.3)

OCR = E[# second orders] · ( Q2 · c R,var + c R,fix )
E[SH ]
( Q B · c B,var + c B,fix )
OCB =
QB

(5.2.4)
(5.2.5)

158

C HAPTER 5. O VERVIEW O F T HE E VALUATION A PPROACH

Again, all equations for the expected values can be found in Table 3.12 on page 126.
The parameter Q B in Equation (5.2.5) represents the average demand quantity of one
single customer which can easily be determined by looking at the sales data, for example. Consequently, the fraction of E[SH ] and Q B yields the expected number of
privileged shipments to customers whose demand had to be backlogged.
Please notice that the variable and the fixed costs are both calculated on basis of the expected, potentially fractional number of orders. Here we are interested in the multipleperiod average value for the costs. The costs per planning period, like month, quarter,
or year, might be different. Depending on the purpose of the calculation one will have
to use different formulas for calculating the costs. For example, when the focus lies on
how much ordering expenses will occur in a particular period then the formulas


0
OCN = E[# first orders] ( Q1 · c N,var + c N,fix )


0
OCR = E[# second orders] ( Q2 · c R,var + c R,fix )


E[SH ]
0
OCB =
( Q B · c B,var + c B,fix )
QB
are used by many companies in an operational context because they have to pay for
the complete order immediately. Moreover, the orders are often bound to certain batch
sizes or lot sizes independent of whether only a fraction of the ordered units can be
used for a subsequent planning period or not.
Note, the fixed cost factors still play a substantial role when determining or optimizing the replenishment parameters Q1 , Q2 , R1 , and R2 , because they significantly
influence the expected number of orders and the expected shortage. However, once
the replenishment parameters have been determined, the fixed costs can be interpreted as an increase in the variable costs c N,var according to
c0N,var = c N,var +

c N,fix
Q1

Evaluation approach – Summary

159

in case of the normal orders, for example.
This concludes the description of the cost structure that is used throughout all evaluations in the subsequent chapters.

5.3

Summary

In the course of this chapter, we have argued why the SDMR model can be used to
represent all scenarios of the comparisons that are necessary to answer our questions
RQ 3 and RQ 4. This eliminates the danger to observe differences between the scenarios that originate from a different modeling. Most of the comparisons in the following
chapters involve the stochastic dual-source scenario, for which the SDMR model has
been developed in Part II. Therefore, this scenario is called the reference scenario.
The setup and process of a comparison is illustrated in more detail by Figure 5.3 on
page 155.
All comparisons consider the total inventory costs of the chosen replenishment
policies. The applied cost structure includes fixed and variable inventory holding
costs, ordering costs, and stock-out costs. The latter are also called back order costs in
our case. The cost structure is illustrated in Figure 5.4 on page 156.
The two questions RQ 3 and RQ 4 are addressed in the next chapters.

Chapter 6
Deterministic versus stochastic dual
sourcing
This chapter addresses our research question RQ 3 on page 5, why we should use a
complex stochastic model instead of a simpler model that assumes deterministic lead
times. Therefore, we compare the calculated values for selected KPIs between the
deterministic scenario (DET) and the stochastic scenario (STOCH).
We take a two-step approach to iteratively gain a better understanding about the
impact that different parameters have on the deviation between DET and STOCH.
First, for an exemplary SKU we conduct a sensitivity analysis on each input parameter
individually. This allows us to study the gradual change of a single input parameter
and its effect on the deviation between DET and STOCH. Second, we analyze the
deviation in a real warehouse with 2,751 SKUs and compare these results with our
findings from the sensitivity analysis. This two-step approach will, on the one hand,
provide us with greater insights into the mechanisms and relations between the input
parameters and the KPIs and their impact on the deviation between DET and STOCH.
On the other hand, it will show the relevance of our findings for different SKUs in
the example of a real warehouse. These analyses are also motivated by our common
161

162

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

observation that strongly simplified models are used to represent a complex process
in practice. For the preparation of the sensitivity analysis and its execution we will
use the following example.
Example 6.0.1. Company A observes that the demand for some SKU is represented by
the truncated Normal distribution F = Φ( x; 1; 1| x ≥ 0) with a mean and a standard
deviation of one. Both lead times are distributed according to a truncated Gaussian
distribution, as well. More precisely, the first lead time distribution is specified by
L1 = Φ(y; 5; 0.5|y > 0) and the second one is given by L2 = Φ(z; 1; 0.5|z > 0).
The company A uses an order quantity of fifty units for both supply modes, so
Q1 = Q2 = 50 due to high fixed order costs. For now, the detailed cost structure does
not play a role. The reorder point R1 for the first supplier is set to R1 = 10 and a rush
order is triggered when the last item has been sold, so R2 = 0. To make calculations
easier they employ a deterministic approximation of their stochastic replenishment
process.
The results of such a simplifying deterministic approximation are quite often used
as a basis for concrete business decisions. However, most of the times the question remains unanswered how accurate and feasible this simplified approach is. An answer
to this question would be of great interest for companies. Either they can continue to
use an approximation due to its proximity to the cost-minimal solution or they can
obtain significant benefits by moving to a more complex and more accurate approach
that considers stochastic lead times.
In Section 6.1 we define some measures that allow us to appropriately compare different KPIs, like service level and total costs, between the deterministic and the stochastic
scenario. These measures are then applied in the sensitivity analysis of Section 6.2. Finally, we optimize the parameters for a real warehouse with stochastic demand and
sochastic lead times. These optimal parameters are used for DET and STOCH and we

Deterministic vs. stochastic – Measures of divergence

163

compare the value of their KPIs.
Note, all parameters related to the deterministic model are indicated by the subscript DET while the stochastic counterpart uses the subscript STOCH. Chapter 5.1 on
page 152 has shown that the SDMR model is able to represent both the DET scenario
and the STOCH scenario. Thus, we can exclude effects that result from a different
modeling.

6.1

Measures of divergence

The objective here is to define measures that allow a comparison between the KPIs
of the DET scenario and the STOCH scenario. We start by assigning two different
sets of costs to Example 6.0.1 as illustrated in Table 6.1. In practical words, we look
at the two SKUs A and B that share the same stochastic behavior and only differ in
their monetary aspects. Due to the huge variety and span of costs and prices it would
significantly simplify the comparison between the deterministic and the stochastic
approach if we found measures that are independent of all money-related parameters.

interest

unit

supply mode 1

supply mode 2

back orders

rate

price

var.

fixed

var.

fixed

var.

fixed

SKU A

12.0%

75.00

0.30

4.50

0.50

8.00

0.10

5.00

SKU B

10.0%

2.00

0.80

1.50

1.10

1.80

12.00

150.00

name

legend:

prices in currency units

var. = variable

Table 6.1: Exemplified cost structure of two SKUs

We can use the formal model described in Part II to calculate the KPIs for the deterministic (DET) and the completely stochastic (STOCH) scenario. Intuitively, one

164

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

expects that the cost-independent values, i.e. the service level, are identical for SKU
A and SKU B because the underlying stochastic relationship between the demand
and the lead times is identical. Following the same arguments one will expect the
cost-dependent values to be different between SKU A and SKU B at least in absolute
terms. However, one could suspect that some relative measure between the costs of
the deterministic scenario and the costs of the stochastic scenario remains constant
independent of SKU A and SKU B. We will elaborate on these aspects now. Note,
the deterministic scenario considers the stochastic demand but only takes the average
lead time for each supplier into account.

6.1.1

Preliminary observations

The result of both scenarios, DET and STOCH, is shown in Table 6.2. One can see that
the β service level in the DET scenario exceeds β of STOCH by 0.05 percentage points.
The reason lies in the fact that the DET scenario yields a higher probability for triggering a 2nd order. This leads to a higher average stock on hand and higher capital costs
CC for the DET scenario. Consequently, in the DET scenario more customer demand
can be satisfied. The 2-order probability is as little as 0.1367% and 0.1336% for DET
and STOCH, respectively. Still, they lead to a visible difference in the service level
which is considerable regarding the fact that they are close to 100%. The low probability of a second order and the high service level lead to little costs for the second
supply mode OCR and the stock outs OCB .
Referring to the service level and the 2-order probability one can see that they
are identical for SKU A and SKU B. This complies with our intuition that costindependent parameters remain constant between SKUs with the same stochastic
characteristics. In contrast, the total costs differ significantly between SKU A and B
and even the ratio of TC, is not identical. The deterministic calculations do not even
systematically over- or underestimate the total costs but they seem to depend on the

16.219

95.01
5.8284

5.9365
101.85

101.85

ratio

abs. = absolute value

ratio =

DET abs.
STOCH abs.

in %

OCR = order costs rush mode

99.93

15.409

262.28

267.14

abs.

OCN = order costs normal mode

0.1336

STOCH

99.98

416.53

100.95

ratio

CC

TC = total costs

0.1367

DET

99.93

420.49

abs.

TC

SL β = beta service level in %

0.1336

STOCH

99.98

SL β

prices in currency units

0.1367

order)

p(2-

DET

model

legend:

SKU B

SKU A

name

100.00

100.00

ratio

0.0167

0.0171

0.3452

0.3530

abs.

ratio

102.28

102.28

OCR

1.2527

0.3347

1.2434

0.3322

abs.

ratio

26.72

26.72

OCB

OCB = opportunity costs (back order)

CC = capital costs

p(2-order) = probability to place a 2nd order in %

9.1211

9.1209

152.67

152.66

abs.

OCN

Deterministic vs. stochastic – Preliminary observations
165

Table 6.2: Results for the deterministic and the stochastic scenario

166

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

specific cost structure of SKU A and SKU B instead. So we cannot give a general statement or rule about the deviation of the costs between the scenario DET and STOCH
even if the underlying stochastic characteristics of two SKUs are identical. At least
this is true in the case of the aggregated parameter TC.
However, a look at the individual cost types of the total costs in Table 6.2 yields a
different picture. While the absolute values for the capital cost CC and the three types
of order costs OCN , OCR , and OCB are different between SKU A and B, their ratios
remain constant. This observation is easy to verify once we recall the formulas for
calculating the various costs in Chapter 5.2, see Equation (5.2.2) till (5.2.5) starting at
page 156. For example, the formula for the 1st order costs is
OCN = E[# first orders] · ( Q1 · c N,var + c N,fix )
and we know that all replenishment parameters, including Q1 , and all cost factors
like c N,var and c N,var are identical for DET and STOCH while their expected number
of orders usually differs. Then the ratio between DET and STOCH can be reduced to
EDET [# first orders] · ( Q1 · c N,var + c N,fix )
OCN,DET
=
OCN,STOCH
ESTOCH [# first orders] · ( Q1 · c N,var + c N,fix )
EDET [# first orders]
=
.
ESTOCH [# first orders]

(6.1.1)

Equation (6.1.1) shows that all cost-relevant parameters can be eliminated. This
relation between OCN,DET and OCN,STOCH holds even for cases where the number of
expected orders has to be rounded up or down to the next integer value. Moreover,
equivalent equations can be formulated for CC, OCR , and OCB . This insight allows us
to easily compare DET and STOCH regarding the relative cost difference for an arbitrary SKU without bothering about specific cost factors – as long as the replenishment
parameters and cost factors are identical for DET and STOCH. Of course, a sensitivity
analysis becomes much more signifant now, as well, because it holds for all possible
values within our general cost structure introduced in Chapter 5.2 on page 156.

Deterministic vs. stochastic – Ratio of costs

167

Let us come back to the ratio of TC between DET and STOCH. Here a simplification
as in Equation (6.1.1) is not possible because the ratio is defined by
CCDET + OCN,DET + OCR,DET + OCB,DET
TCDET
=
TCSTOCH
CCSTOCH + OCN,STOCH + OCR,STOCH + OCB,STOCH

(6.1.2)

where different cost factors and order quantities apply for CC, OCN , OCR , and OCB .
Thus, the ratio of TC can strongly vary for different cost factors.
Example 6.1.1. We can state from the four rightmost ratios in Table 6.2 that TCDET can
be as low as 26.72% of TCSTOCH or as high as 102.28% of TCSTOCH if the back order
costs or the rush order costs, respectively, are the only cost types that play a role in
our business scenario. Consequently, in a situation where the stock out costs outweigh the other cost factors it would not be a good idea to use the DET scenario as
an approximation. However, the absolute value of TCDET is a quite good approximation of TCSTOCH for SKU A. A company that focuses on the service level has to decide
whether the difference between 99.98% and 99.93% is significant for their business or
not.
Now, we have good insights that help us to elaborate on some measures of divergence
between DET and STOCH.

6.1.2

Ratio of costs

Let us define τ as the cost ratio between DET and STOCH given by
τ=

costs determined by DET
.
costs determined by STOCH

(6.1.3)

In the subscript we indicate which costs τ refers to so that τCC , τOCN , τOCR , and τOCB
represent the ratio between DET and STOCH regarding the costs CC, OCN , OCR , and
OCB , respectively. Further, we can now define an arbitrary set T of ratios τi with

168

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

i ∈ {0, 1, 2, 3, ...}. To get a fast and simple picture of the extreme cases we can use the

minimum and the maximum cost ratio τ min and τ max , respectively.
τ min = min(T )

(6.1.4)

τ max = max(T )

(6.1.5)

We define the special set TSKU which contains the 4 cost types of TC for a certain SKU.

TSKU = {τCC , τOCN , τOCR , τOCB }.

(6.1.6)

The set TSKU is used extensively in our sensitivity analysis and will be used as the

default set of ratios. Thus, whenever we do not explicitly mention a certain set T then
τ min , τ max , and τ̄ refer to TSKU .

In the situation shown in Table 6.2 we yield τ min = 0.2672 and τ max = 1.0228 and

the lower bound τ min is associated with the costs for stock out for SKUs A and B.
Despite the low value of τ min its effect on the deviation of the total costs is alleviated
by the very small probability of a stock out situation. Of course, one could multiply the
cost ratios with some weight of relevance like the probability of occurrence. However,
this is still no guarantee for a more precise picture of the situation as extremely high
or low absolute cost values can still overrule the weights of relevance. In favor of
simplicity we do not consider this kind of measures.
Second, we introduce the mean value τ̄ of all elements in the set T by
τ̄ =

1
|T |

∑

τ

(6.1.7)

τ ∈T

which indicates how much the costs between DET and the STOCH differ on average.
For the set of basic cost types TSKU we yield
τ̄ =

1
(τCC + τOCN + τOCR + τOCB )
4

which amounts to τ̄ = 0.8271 in our example of Table 6.2. Be aware that τ̄ is only
reasonable if the individual absolute costs are of a similar magnitude or all cost ratios

Deterministic vs. stochastic – Ratio of service levels

169

are about the same value. If this does not hold one has to be careful with interpreting
τ̄. Referring to our example, the statement that the DET scenario underestimates the
total costs of STOCH on average by 1 − 0.8271 = 17.29% can be problematic and it
does not hold for any of our two SKUs A and B. Yet τ̄ represents all basic cost types

and not solely the two extreme cases. This might become even more attractive when
additional basic cost types are considered. Consequently, we use τ̄ occasionally but it
has to be seen rather in context of τ min and τ max than as a stand-alone measure.

6.1.3

Ratio of service levels

The service level is an essential KPI in many industries like retail. Therefore, we will
include a measure ρ for the divergence of the service levels, as well. Note, we only
consider the beta service level β here. However, one can derive other service levels
from the formal model in Part II, as well. Companies that use β as a KPI are usually
interested in a high customer satisfaction and target a β close to 100%. A deviation
between two service levels of 98% and 99% is not considered to be just one percentage
point but, in fact, it is often seen as an improvement of 50% or as a decline of 100% in
practice. Clearly, these companies regard β = 100% as their reference point. Therefore,
we use a ratio based on 1 − β for defining ρ.
ρ=

1 − STOCH β
1 − DET β

(6.1.8)

In case of SKU A this leads to ρ = 3.5 which indicates that the DET scenario calculates
a service level which is 3.5 times, or 350%, better or higher than STOCH β. Note, this
definition of ρ also supports a monetary perspective because the sensitivity of the total
costs usually increases steeply when β approaches 100%.
Example 6.1.2. It is easy to show why the total costs are usually very sensitive to small
changes in the beta service level β whenever β is close to 100%. First, a substantial

170

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

part of the total inventory costs are storage costs CC. Mostly, the value of CC is directly proportional to the number of items on stock. Second, a so-called safety stock
is needed to buffer the variability of the demand and the lead times. Therefore, the
safety stock and β positively correlate with the quantile of the demand distribution for
given lead time distributions. Third, many times the measured demand distributions
show high marginal quantiles when β approaches 100%. This translates directly into
a very sensitive and high safety stock and correlates one-to-one to the amount and
sensitivity of the storage costs CC.
In our example of SKU A, see Table 6.2, this indicates that the necessary costs for
reaching a service level of 99.98% can be significantly higher than the 420.49e calculated
by the DET scenario whenever the lead times are not deterministic as DET assumes.
This finishes the definition of our measures of divergence. We want to point out that
the measures τ min , τ max , τ̄, and ρ are sufficient for our purposes. Nevertheless, one
might want to adapt or add some measures for a specific problem at hand.
The defined measures put us in a position where we can evaluate the difference in β
on the one hand and on the other hand give minimal and maximal bounds for the ratio
of the total costs between the DET and the STOCH scenario. All measures are valid for
every possible cost structure. Moreover, the absolute deviation of TC between DET
and STOCH for a given SKU can be calculated by simply inserting its specific cost
factors. The only things we need are the values for the replenishment parameters R1 ,
R2 , Q1 , and Q2 and some knowledge or approximation of the stochastic characteristics
of the demand and the lead times. Of course, these can be quite different for individual
SKUs in an inventory. However, products from the same supplier often have quite
similar lead time distributions, for example. Naturally, one of the next questions is
how the deviation between the DET and the STOCH scenario reacts on changes in
the demand, the lead times and the replenishment parameters. These questions are
addressed in the next chapter.

Deterministic vs. stochastic – Sensitivity analysis on daily demand

6.2

171

Sensitivity analysis on replenishment-relevant parameters

In this chapter we perform a sensitivity analysis on the differences between the DET
scenario and the STOCH scenario for each of the input parameters: the demand distribution D, the lead time distributions L1 and L2 , the order quantities Q1 and Q2 , and
the reorder points R1 and R2 . Thereby, we utilize the measures defined in Section 6.1.
The basic settings for all parameters is copied from Example 6.0.1, page 162, but
we change the first reorder point from R1 = 10 to R1 = 5. Note, we conduct a partial sensitivity analysis where the values of all but the investigated parameter remain
constant. An overview of the used initial parameter setting is given in Table 6.3. The
Gaussian distribution is denoted by Φ.
parameter

setting

parameter

setting

Q1

50

D

Q2

50

L1

Φ( x; 1.0; 1.0| x ≥ 0)

R1

5

L2

R2

0

Φ(y; 5.0; 2.5|y > 0)
Φ(z; 1.0; 0.5|z > 0)

Table 6.3: Initial input parameter setting
Note, for some parameter settings the probability to violate Assumption 8, page 43,
is above the threshold of 10−10 . The results of these settings are excluded from our
analysis and their entries in tables are set to N.A..

6.2.1

Daily demand

The initial daily demand distribution is given by Φ( x; 1.0; 1.0| x ≥ 0). Now, we in-

crease the demand fluctuation σD from 0.0 to 2.0 in steps of 0.2. The results for the cost

172

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

ratios and ρ are shown in Table 6.4 and Figure 6.1.
ratio

standard deviation σD of the daily demand distribution D (1) = Φ( x; 1.0; σD | x ≥ 0)
0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

CC

102.0

102.4

107.3

110.0

111.0

111.4

111.6

111.7

111.7

111.7

111.7

OCN

100.0

99.8

96.7

94.5

93.3

92.7

92.3

92.1

92.0

91.9

91.9

OCR

100.0

152.3

147.0

140.3

134.7

130.4

127.1

124.5

122.4

120.8

119.3

OCB

0.0

13.8

25.4

33.5

39.3

43.6

46.9

49.5

51.5

53.1

54.5

ρ

NAN

7.23

3.93

2.98

2.54

2.29

2.13

2.02

1.94

1.88

1.83

legend:

ratio =

DET
STOCH

· 100

ρ=

1−STOCH β
1−DET β

NAN = not a number

Table 6.4: Ratios τ and ρ for a changing demand fluctuation σD
Let us investigate the results of Table 6.4 in more detail. The first column shows the
ratios for a scenario with deterministic demand as the variation is zero. The costs for
normal and rush orders are identical for the DET and the STOCH scenario. Actually,
there are no rush orders in both scenarios. This is easy to verify. Keep in mind that
the demand per day is exactly 1.0 units. Then it takes exactly 5 days till the demand
cumulates to the delta between the two reorder points, R1 = 5 and R2 = 0. Looking
up the detailed parameters for the scenario reveals that the time window to trigger a
second order tw is set to four days. Consequently, R2 is never reached in time and all
demand has to be covered by the first orders. This also explains why the cost ratio
OCN is 100%.
The DET scenario only uses average lead times. In our example, this implies that
the first order arrives exactly after 5 days, just when the stock level drops to zero. Consequently, there are never any stock outs and their related costs and the service level
β is 100%. In the STOCH scenario the lead time distribution is L1 = Φ(y; 5; 2.5|y > 0)

and so there is a positive probability that the order will arrive after more than 5 days.
This translates directly into a service level of less than 100% and positive stock out

Deterministic vs. stochastic – Sensitivity analysis on daily demand

173

costs. It also explains why OCB ratio = 0 and why ρ is not a number due to a division
by zero.
Starting with a division by zero, the values of ρ decrease with an increasing demand fluctuation. Still, the DET scenario consequently overestimates the service level
lead time fluctuation σL1 lead time fluctuation σL2

β. In the case where σD = 0.2 the values DET β = 99.65% and STOCH β = 97.45%
average lead time µ L1 average lead time µ L2

differ significantly. Even for σD = 2.0 the values are 98.70% and 97.61% for DET and
STOCH, respectively. Companies for which a precise value of β is important should
order quantity Q1 order quantity Q2

be careful when using the DET scenario
2 as an approximation for the STOCH scenario
2 2

2

in %

20%
20%20%

0%

DET
deviation
DET
DET
STOCH
deviation
deviationSTOCH
STOCH
demand
fluctuation
demand
fluctuation
demand
fluctuation
cost
measures
cost
measures
cost measures
average
lead
time
average
lead
time
average
lead
time

lead
time
fluctuation
lead
time
fluctuation
lead
time
fluctuation
order
quantity
order
quantity
order
quantity

τ min τ max τ̄

2

cost measures

CC OCB OCR OCN

value

80%80%
60%
40%
60%60%
40%
20% 40%40%

RS1,trad RS2,trad RS1,relax RS2,relax

100%
100%

80%
60%

1−STOCH
1−DET

120%
120%

100%
80%

deviation

140%
140%
120%
100%

DET
STOCH

120%
160%
140%
160%

value

DET
deviation
value
STOCH

140%

deviation

demand
fluctuation σD
value

160%

reorder point R1 reorder point R2

for SKUs with the current parameter settings.

reorder
point
reorder
point
reorder
point
DET
DETDET in %
deviation
deviation
inin%%
deviation
STOCH
STOCH
STOCH

1−STOCH
1−1STOCH
−STOCH
deviation
deviation
deviation
1−DET
1−1DET
−DET

DET
deviation
RSRSRS1,trad
RSRSRS2,trad
RS
RS
RS RS1,relax
RS RS2,relax

1,trad
2,relax
STOCH
1,trad
2,trad
1,relax
2,relax
0.0
0.2
0.4
0.6
0.8
1.02,trad
1.2 1,relax
1.4
1.6
1.8
0%0% 0%
0.0 0.0 0.0 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 0.6 0.8 0.8 0.8 1.0 1.0 1.0 1.2 1.2 1.2 1.4 1.4 1.4 1.6 1.6 1.6 1.8 1.8 1.8 2.0 2.0 2.0
demand
fluctuation
σ
standard
deviation
D
CC
OC
OC
OC
CCstandard
OC
OC
BR R OC
RNN N
B deviation
B deviation
standard
standard
deviation

2.0

CC OC OC OC

min
max
max
min
max
average
µ
avg_t
t_min
t_max
τ min
τ
τ̄avg_t
τlead
τtime
τ̄L1 average lead time µ L2
t_min
t_max
τt_min
τt_max
τ̄avg_t
t_min
t_max
avg_t

Figure 6.1: Cost ratios τ min , τ

leadσ time
fluctuation
lead
σL2
Lσ
1Q
σ1LLσ11Lµ1σLµ
RQ
σσ2 2Q
Q212time
R112RR
Q
Rfluctuation
221 R2
Dσµ
Lµ
Dσ
121Q
D
L21Lσ
L
2µLL2L
1 Lµ
order quantity Q1 order quantity Q2
max

, and τ̄ for a changing demand fluctuation σD

reorder point R1 reorder point R2
deviation

DET
STOCH

deviation

1−STOCH

in %

1−DET τ min and τ max deviate significantly
A look at Figure 6.1 reveals that the cost ratios

RS1,trad RS2,trad RS1,relax RS2,relax

from the desired 100%. It is interesting to see that all values of τ min are associated
CC OCB OCR OCN

with OCB . This means that DET consequently
underestimates the number of stock
min max
τ

τ

τ̄

outs which is, of course, directly linked to the overestimation of the service level.
Moreover, all but the first value of τ max are associated with OCR . DET systematically

174

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

overestimates the usage of the second order channel which is partly responsible for
the overestimation of the service level and the underestimation of the stock out costs.
Even if the distance between τ max and τ min decreases with increasing demand fluctuation the deviation is always more than 20%. Especially, companies with high costs
for stock outs or for the second order channel are affected by these deviations. If the
storing costs are the main cost type one has to expect deviations between 2% and 11%
which is acceptable in many cases.
Note, we will not include the average daily demand into our analysis here because
one can always normalize it to one unit per day. By adapting the reorder points and
the order quantities in subsequent chapters this will have the same effect as changing
the average daily demand.

6.2.2

Lead time of the first supply mode

In this section we will look at two things separately: the effect of a changing average
lead time and the effect of a changing lead time fluctuation.
When the average lead time µ L1 of the normal supply mode increases, our intuition
is that the second order will successively gain more importance. For the DET scenario
this means that in addition to neglecting the variability of one variable, namely L1 , the
effect of a second stochastic variable, L2 , might lead to higher deviations between DET
and STOCH. The strength of these effects could reach a peak where both supply modes
are equally strong. Whenever µ L1 is increased further on, some parameters should be
dominated by the second supply mode. Then the deviation between DET and STOCH
might be dominated only by the stochastic variable L2 and, thus, decreases.
Regarding the standard deviation σL1 of L1 things should be more straight forward.
One will most likely expect that the DET scenario will yield an increasingly worse
approximation of the STOCH scenario the more the standard deviation increases.

Deterministic vs. stochastic – Sensitivity analysis on lead time L1
6.2.2.1

175

Changes in the average lead time

Let us now look at the effect of different average lead times for the first supply mode.
We increase the value from 1 to 10 in steps of one. The result is shown in Table 6.5 and
Figure 6.2.

ratio

1

average µ L1 of the lead time distribution L1 = Φ(y; µ L1 ; 2.5|y > 0)
2

3

4

5

6

7

8

9

10

CC

104.9

103.7

103.9

108.1

111.4

112.3

111.8

110.5

108.6

106.6

OCN

100.0

100.0

99.4

96.3

92.7

91.3

91.9

93.6

95.5

97.1

OCR

0.0

152.4

151.6

143.0

130.4

120.3

113.4

108.6

105.3

103.2

OCB

0.00

1.87

13.0

29.0

43.6

55.0

63.0

68.4

70.4

70.2

ρ

NAN

53.72

7.710

3.451

2.292

1.820

1.589

1.461

1.420

1.424

legend:

ratio =

DET
STOCH

· 100

ρ=

1−STOCH β
1−DET β

NAN = not a number

Table 6.5: Ratios τ and ρ for a changing average lead time µ L1
For 1 ≤ µ L1 ≤ 2 the probability p(2, ., .) that a second order is placed is very low.

Consequently, more than 99.99% of the demand is satisfied by the normal orders and
the cost ratios OCN are rounded to 100.0%. If the lead time of the first order is exactly 1
day, as assumed by DET, then there is no time for triggering a second order. However,
p(2, ., .) is small but positive in the STOCH scenario for µ L1 = 1 because it considers
the fluctuation of both lead times. That is the reason why OCR = 0.0% for µ L1 = 1. If
µ = 2 then DET calculates p(2, ., .) > 0, as well, which yields a ratio of OCR of 152.4%.
Another effect caused by the deterministic lead times of the DET scenario is an
overestimation of the service level. For µ L1 = 1 and µ L1 = 2 the DET scenario yields a
β of 100.00% and 99.98%, respectively, while the associated values for the STOCH scenario are only 99.51% and 99.08%, respectively. This is already a significant deviation
for some companies with highly available SKUs.

2

20%
0%

20%
20%20%

DET
deviation
DET
DET
STOCH
deviation
deviationSTOCH
STOCH
demand
fluctuation
demand
fluctuation
demand
fluctuation
cost
measures
cost
measures
cost
measures
average
lead
time
average
lead
time
average
lead
time

lead
time
fluctuation
lead
time
fluctuation
lead
time
fluctuation
order
quantity
order
quantity
order
quantity

τ min τ max τ̄

2

cost measures

CC OCB OCR OCN

RS1,trad RS2,trad RS1,relax RS2,relax

1−STOCH
1−DET

in %

deviation

120%
120%
100%
80% 100%
100%
80%
60% 80%80%
60%
40% 60%60%
40%
40%40%

DET
STOCH

100%

deviation

140%
140%
120%

2

2 2
C HAPTER 6. D ETERMINISTIC
V ERSUS S TOCHASTIC D UAL S OURCING

reorder point R1 reorder point R2

160%
140%
160%

120%

value

DET
deviation
valueSTOCH

140%

order quantity Q1 order quantity Q2

160%

lead time fluctuation σL1 lead time fluctuation σL2

demand
fluctuation σD
value

180%

average
lead time µ L1 average lead time µ L2
value

176

reorder
point
reorder
point
reorder
point
DET
DETDET in %
deviation
deviation
inin%%
deviation
STOCH
STOCH
STOCH

1−STOCH
1−1STOCH
−STOCH
deviation

deviation
deviation
DET 1−1DET
1−DET
−DET
deviation
STOCH

RSRSRS1,trad
RSRSRS2,trad
RS
RS
RS RS1,relax
RS RS2,relax

1,trad
2,trad
2,relax
1,trad
2,tradσD1,relax
1,relax
2,relax
demand
fluctuation
0%0%
0%
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
0.0 0.0 0.0 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 0.6 0.8 0.8 0.8 1.0 1.0 1.0 1.2 1.2 1.2 1.4 1.4 1.4 1.6 1.6 1.6 1.8 1.8 1.8 2.0 2.0 2.0
CC
OC
OC
OC
CC
OC
OC
BR time
average
lead
B deviation
Btime
R
N N
average
lead
µOC
lead time µ L2
LR1Naverage
standard
standard
deviation
standard
deviation

CC OC OC OC

10.0

min
max
min
max
min
t_min
t_max
τ
ττ
ττ̄avg_t
τ̄
t_min
t_max
τt_min
τ max
τ̄avg_t
lead
time
fluctuation
σL1 lead time fluctuation σL2
t_min
t_max
avg_t
t_max
avg_t

order
order
Q
σDσquantity
µDσLµ
σ1LLσ11Lµ1σQ
Q
σ2 2Q
Q2RQ
R112RR
σQ
Q
R2221 R2
Lµ
1quantity
D
L211Lσ
L
2µLL2L
1 Lµ
21 21
reorder
point
R2
Figure 6.2: Cost ratios τ min , τ max
, andpoint
τ̄ forRa1 reorder
changing
average
lead time µ L1
deviation

DET
STOCH

deviation

1−STOCH
1−DET

in %

RS1,trad RS2,trad RS1,relax RS2,relax
The measure ρ for the service level
falls rapidly towards the desired value of 1.0
CC OCB OCR OCN

for 2 ≤ µ L1 ≤ 9 but then starts to slightly
rise again. This is a common observation
min max
τ

τ

τ̄

here. All but the OCR ratio in Table 6.5 develop in one direction up to a certain value
of µ L1 before they start changing in the opposite direction again. The reasons for this
kind of behavior are manifold.

First, a major reason is the consequent over- or underestimation of the values for a
given parameter by the DET scenario. Due to the fact that the probability of a 2-order
cycle cannot exceed 100% the value of the STOCH scenario can catch up over time and
the delta decreases. This effect is nice to observe in the development of the OCR ratio.
Second, let us look at the increase of ρ for µ L1 = 10. The explanation lies in the
gradually later arrival of the first order. Without the second order the service level
would decrease quickly. Fortunately, the 2-order probability p(2., ., .) increases as µ L1
approaches 10 days. However, p(2, ., .) is saturating and reaches its maximum effect
to cover excessive demand. Consequently, ρ decreases and the ratio of OCB 22increases
2

again for µ L1 = 10.

2

Deterministic vs. stochastic – Sensitivity analysis on lead time L1

177

Third, there are several effects in case of the holding costs. Initially, µ L1 = 1 holds
and the DET scenario assumes that the first order arrives in the minimum possible
time. Consequently, DET neglects the cases where the stock decreases further on until
a late arrival of the first order in the STOCH scenario. This results in a high ratio for
CC. For 2 ≤ µ L1 ≤ 3 things are more balanced as the first order in the STOCH scenario

can arrive also before µ days. All effects of a second order are restricted due to its little
probability p(2, ., .). Consequently, the ratio of CC decreases in comparison to µ L1 = 1.

For µ L1 ≥ 3 the second orders are gaining more influence. Due to the deterministic

lead times in the DET scenario the probability that both orders arrive at the same day
or consecutive days is increasing up to µ L1 = 6 which is exactly one day after R2 is
expected to be reached. Keep in mind that R1 = 5, R2 = 0, and the average demand is
1.0 per day. This implies that the stock level is raised by both order quantities Q1 and
Q2 within a short time which leads to high storage costs under the premise that the
lead times are deterministic. When we take into account fluctuations, the increase of
the stock level is dampened. Therefore, the ratio of CC is increasing for 4 ≤ µ L1 ≤ 6.
For µ > 6 the two peaks of stock level, induced by both order quantities, are drifting

further apart because the expected placement and arrival of the second order remains
constant but µ keeps increasing. Again, this effect is smoothed by the fluctuation of
the STOCH scenario and the ratio of CC is slowly decreasing for 7 ≤ µ L1 ≤ 10.
Figure 6.2 shows that the bounds are not monotonically changing with increasing
lead time. Moreover, the associated ratios for the upper bound are changing. The
value of τ max is the ratio of CC for µ L1 = 1 and µ L1 ≥ 8. For 2 ≤ µ L1 ≤ 7 the value of
τ max is the ratio of OCR . The lower bound is constantly related to the ratio of OCB .

In summary the deviation between DET and STOCH is over 10% for most cost ratios.
Only the ratios for the normal ordering are consistently below 10%. Again, the underestimation of stock outs is striking here. Therefore, it is quite difficult to predict how

178

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

well the DET scenario approximates the STOCH scenario when the lead time of the
first order changes.

6.2.2.2

Changes in the lead time fluctuation

Let us now have a closer look at the impact of varying the fluctuation of the default
lead time distribution L1 = Φ(y; 5.0; 2.5|y > 1). The result is shown in Table 6.6 and

Figure 6.3. Note, the last column in Table 6.6 is not filled because the probability for
violating Assumption 8 on page 43 is 1.14 · 10−10 and exceeds our maximum tolerance
of 10−10 .
ratio

standard deviation σL1 of the lead time distribution L1 = Φ(y; 5.0; σL1 |y > 0)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

CC

100.4

102.0

105.3

108.0

110.0

111.4

112.6

113.6

114.5

115.4

N.A.

OCN

100.0

99.6

97.3

95.3

93.8

92.7

92.0

91.3

90.9

90.5

N.A.

OCR

100.0

101.1

108.9

117.3

124.5

130.4

135.2

139.1

142.4

145.1

N.A.

OCB

89.1

62.2

58.0

53.0

48.1

43.6

39.6

36.2

33.1

30.5

N.A.

ρ

1.12

1.61

1.72

1.89

2.08

2.29

2.52

2.77

3.02

3.28

N.A.

legend:

ratio =

DET
STOCH

· 100

ρ=

1−STOCH β
1−DET β

N.A. = not available

Table 6.6: Ratios τ and ρ for a changing lead time fluctuation σL1
In the deterministic case where σL1 = 0 the first order always arrives exactly after
5 days for DET and STOCH. Moreover, both scenarios consider the identical demand
distribution D. Consequently, the conditions for placing a second order are identical
which leads to identical cost ratios OCN and OCR . In the DET scenario, the absolute
values of all parameters, like costs and the service level, remain constant for the different settings of σL1 . This comes to no surprise as we are only changing a parameter
that is not considered by DET.

order quantity Q1 order quantity Q2

lead time fluctuation σL1 lead time fluctuation σL2

value
average
lead time µ L1 average lead time µ L2

2

20%
20%20%

0%

2

in %

DET
deviation
DET
DET
STOCH
deviation
deviationSTOCH
STOCH
demand
fluctuation
demand
fluctuation
demand
fluctuation
cost
measures
cost
measures
cost
measures
average
lead
time
average
lead
time
average
lead
time

lead
time
fluctuation
lead
time
fluctuation
lead
time
fluctuation
order
quantity
order
quantity
order
quantity

τ min τ max τ̄

80%80%
40%60%60%60%
40%
20% 40%40%

179

cost measures

CC OCB OCR OCN

100%
100%

60%80%

RS1,trad RS2,trad RS1,relax RS2,relax

120%
120%
80%
100%

1−STOCH
1−DET

140%
140%
120%

deviation

160%
140%
160%

100%

DET
STOCH

120%

value

DET
deviation
valueSTOCH

140%

2 2

deviation

value
demand
fluctuation σD

160%

reorder point R1 reorder point R2

Deterministic vs. stochastic – Sensitivity
2 analysis on lead time L1

reorder
point
reorder
point
reorder
point
DET
DETDET in %
deviation
deviation
inin%%
deviation
STOCH
STOCH
STOCH

deviation

DET

1−STOCH
STOCH
1−1STOCH
−STOCH
deviation
deviation
deviation
1−DET
1−1DET
−DET

demand fluctuation σD

RSRSRS1,trad
RSRSRS2,trad
RS
RS
RS RS1,relax
RS RS2,relax

1,trad
2,relax
1,trad 2,trad
2,trad 1,relax
1,relax
2,relax
0%0% 0%
average
lead
L1 average
0.00.0 0.0 0.2
0.50.2 0.4 0.4
1.0 0.6 0.6 1.5
2.0 time µ
2.5
3.0 lead time
3.5 µ L2 4.0
0.0
0.2
0.4
0.6 0.8 0.8 0.8 1.0 1.0 1.0 1.2 1.2 1.2 1.4 1.4 1.4 1.6 1.6 1.6 1.8 1.8 1.8 2.0 2.0 2.0
CC
OC
OC
OC
CC
OC
OC
OC
BR R
N time fluctuation σ
B deviation
B deviation
N
standard
deviation
standard
lead
time
fluctuation
σRLN
lead
standard
deviation
standard
L2
1

CC OC OC OC

4.5

minminmin
max
maxmax

t_max
avg_t
t_min
t_max
τ t_min
ττ
ττ̄avg_t
τ̄avg_tquantity Q2
τquantity
τ t_max
t_min
t_max
avg_t
order
Qτ̄
1 order
t_min

reorder
point
σDσµDσ
σ1LLσ1R1Lµ
σ2 2Q
Q
Q2R
R112RR
2Q
11σreorder
σpoint
Q
R221 R2
Lµ
Lµ
121Q
D
L21Lσ
L
12R
2µLL2L
1 Lµ
DET

%
Figure 6.3: Cost ratios τ min , τ maxdeviation
, and τ̄ STOCH
for a in
changing
lead time fluctuation σL1
deviation

1−STOCH
1−DET

RS1,trad RS2,trad RS1,relax RS2,relax
CC OCB OCR OCN

The service level is systematically
overestimated by DET also when σL1 = 0. The
τ min τ max τ̄
explanation is that DET assumes a deterministic lead time for the second order of 1
day. This is the minimum possible lead time and in the stochastic case of STOCH the
second orders can only arrive later than DET assumes. Therefore, additional stock out
can occur in the case of STOCH which leads to a ratio of CC above 100% and ρ > 1.0.
Whenever the fluctuation of L1 increases this means that less second orders are
triggered because the time window tw remains constant but the probability that an
early arrival of the first order prohibits the placement of a second order increases.
Consequently, the number of second orders is consequently and increasingly overestimated when σL1 increases. An overestimation of the probability p(2, ., .), to place a
second order, by DET can be directly translated in its overestimation of the service
level and its underestimation of the stock out costs OCB . This is represented in an
increasing value for ρ and a decreasing ratio of OCB .
22
The holding costs are always overestimated by DET as well. Similar explanations

2

2

180

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

as for the lead time average in the previous Section 6.2.2.1 are valid here too. It depends very much on the time between the two peaks of stock that are induced by the
arrival of both orders. In the case where σL1 = 0 the first order arrives exactly after
5 days. Whenever the second order is triggered shortly after the first one, then DET
assumes that the order will arrive the very next day. This is the maximum distance
between the arrival of both orders for the case that the second order arrives first. Consequently, DET should underestimate CC. However, the second order is most likely
triggered shortly before the first order arrives – after exactly 5 days. Then DET calculates that both orders arrive shortly after each other while in the STOCH scenario
the second order arrives sometimes much later. The latter effect prevails here even for
σL1 = 0 and amplifies the more we increase σL1 .
Looking at Figure 6.3 our intuition is confirmed that with an increasing fluctuation
of the lead time distribution the approximation of the STOCH scenario by DET, which
uses only the average value, is getting increasingly worse.
In the last two sections we could see that it is not always easy to predict the direction
and the magnitude of the deviation between stochastic dual sourcing and its deterministic approximation. However, the more variability is involved the more unprecise the
approximation is. This coincides with our intuition. Moreover, the poor approximation of the service level and of the costs related to the stock outs is striking.

6.2.3

Lead time of the second supply mode

This chapter shows the effect of changes in the lead time distribution for the second
order, L2 . We will first alter the average lead time and then its standard deviation.
Intuitively, one would expect that the number and the effect of the second supply
mode decreases the more its average lead time µ L2 increases. This is simply the result of a decreasing time window tw for placing a second order based on the lower

Deterministic vs. stochastic – Sensitivity analysis on lead time L2

181

probability that the second order will arrive before the first one.
The effect of an increase in the standard deviation σL2 of L2 should not effect the
probability of a second order but decrease the service level until the regular arrival
of the first order compensates for the possibly very late arrival of the second order.
Moreover, we expect that DET yields constant results because it completely neglects
the fluctuation.
6.2.3.1

Changes in the average lead time

In our analysis, the average lead time µ L2 of the second order is subsequently increased from 1 day to 10 days. Table 6.7 only lists the results up to day 6, though,
because the time window tw for placing the second order decreases to zero time units
already for µ L2 = 5. Consequently, no second orders are placed for µ L2 ≥ 5 and the
results are identical. This corresponds with Table 6.7 and our expectations.

ratio

average µ L2 of the lead time distribution L2 = Φ(z; µ L2 ; 0.5|y > 0)

1

2

3

4

5

6

CC

111.4

105.4

102.5

102.2

102.2

102.2

OCN

92.7

97.7

99.8

100.0

100.0

100.0

OCR

130.4

122.0

112.8

105.8

100.0

100.0

OCB

43.6

52.9

53.6

53.4

53.4

53.4

ρ

2.29

1.89

1.86

1.87

1.87

1.87

legend:

ratio =

DET
STOCH

· 100

ρ=

1−STOCH β
1−DET β

Table 6.7: Ratios τ and ρ for a changing average lead time µ L2
Note, the cost ratio for OCN is 100% already for µ L2 = 4 because the 2-order probability p(2, ., .) is so small that over 99.995% of the demand must be satisfied by the
first supply mode.

182

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

Once again, the number of rush orders is overestimated by the DET scenario here
again because DET assumes that the first order arrives exactly after the average lead
time µ L1 . Therefore, DET does not account for the decrease of the 2-order probability,
p(2, ., .), which is induced by early-arriving first orders. In these cases, second orders
can only be placed within the first min(l1 , tw ) days. Consequently, a second order is
not necessarily triggered if the stock level reaches R2 within min(µ L1 , tw ) days. An
overestimation of the number of second orders implies higher stocks and a higher
service level, as well. Note, this effect only exists as long as p(2, ., .) > 0 and ceases for
µ L2 ≥ 5.

Then the question arises why CC and ρ are still overestimated for µ L2 ≥ 5. This is
lead time fluctuation σL1 lead time fluctuation σL2

linked to the fact that DET again neglects early arrivals of the first order. One part of
value
average
lead time µ L1 average lead time µ L2

an early supply can cover additional demand that would be left unsatisfied otherwise.
This also manifests in the low ratio of stock out costs, OCB . The other part of the

2

0%

20%
20%20%

deviation

in %

DET
deviation
DET
DET
STOCH
deviation
deviationSTOCH
STOCH
demand
fluctuation
demand
fluctuation
demand
fluctuation
cost
measures
cost
measures
cost
measures
average
lead
time
average
lead
time
average
lead
time

lead
time
fluctuation
lead
time
fluctuation
lead
time
fluctuation
order
quantity
order
quantity
order
quantity

τ min τ max τ̄

100%
100%
80%
80%80%
40%60% 2
60%60%
20%40%40%40%

cost measures

CC OCB OCR OCN

60%

RS1,trad RS2,trad RS1,relax RS2,relax

120%
120%
100%

1−STOCH
1−DET

80%

deviation

140%
140%
120%

value

DET
deviation
valueSTOCH

160%
160%
100%140%

DET
STOCH

120%

2 2

deviation

value
demand
fluctuation σD

140%

reorder point R1 reorder point R2

costs.

order quantity Q1 order quantity Q2

supply increases the stock earlier than assumed
by DET and causes additional holding
2

reorder
point
reorder
point
reorder
point
DET
DETDET in %
deviation
deviation
inin%%
deviation
STOCH
STOCH
STOCH

1−STOCH
1−1STOCH
−STOCH
deviation
deviation
deviation
1−DET
1−1DET
−DET

DET
STOCH

RSRSRS1,trad
RSRSRS2,trad
RS
RS
RS RS1,relax
RS RS2,relax

1,trad
2,relax
1,trad 2,trad
2,trad 1,relax
1,relax
2,relax
demand fluctuation
σD
0%0% 0% 1.0
2.0
3.0
4.0
5.0
6.0
0.0 0.0 0.0 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 0.6 0.8 0.8 0.8 1.0 1.0 1.0 1.2 1.2 1.2 1.4 1.4 1.4 1.6 1.6 1.6 1.8 1.8 1.8 2.0 2.0 2.0

CC
OC
CC
OC
OC
OC
CC
OC
OC
average
BRtime
RNOC
BOC
Blead
R
N N
average lead time µ L1 average
lead
time
µOC
standard
deviation
standard
deviation
L
standard
deviation
2

min
max
min
max
max
t_min
t_max
t_min
t_max
τ
τ
τ̄avg_t
τtime
τfluctuation
τ̄
τ min
τt_max
τ̄avg_t
t_min
t_max
avg_t
t_min
avg_t σL2
lead time fluctuation σL1 lead

σDσµDσLµ
σ1Q
σ112Lµ1σLµ
σ2 2Q
Q212RQ
R112RR
σQ
Q
R221 R2
order quantity Q1 order quantity
LL
121Q
D
L21Lσ
L
2µLL2L
1 Lµ
pointτRmin
R2for a changing average lead time µ L
1 reorder
Figure 6.4: reorder
Cost ratios
, τ max ,point
and τ̄
2
deviation

DET
STOCH

deviation

1−STOCH
1−DET

in %

RS1,trad RS2,trad RS1,relax RS2,relax
CC OCB OCR OCN
τ min τ max τ̄

Deterministic vs. stochastic – Sensitivity analysis on lead time L2

183

In Figure 6.4 the upper bound τ max is initially dominated by the deviation in the
costs OCR for the second supply mode. This effect diminishes with a decreasing probability p(2, ., .) until no second orders are placed anymore and the rather small overestimation of the holding costs CC take over for µ L2 ≥ 5. The lower bound τ min is

identical to the ratio of OCB .

According to our intuition the increase of the average lead time µ L2 improves the
approximation of STOCH by DET. However, we can see that just by neglecting the
variability of L1 one obtains a very poor approximation of the back order costs and
the service level. In the case where µ L2 ≥ 5 the service level is 98.01% and 96.27% for

DET and STOCH, respectively. This deviation is significant for most businesses.
6.2.3.2

Changes in the lead time fluctuation

In the following, the standard deviation σL2 of the lead time distribution L2 is gradually increased from 0.0 – a deterministic lead time – to 2.0. It might come to a surprise
ratio

standard deviation σL2 of the lead time distribution L2 = Φ(y; 1.0; σL2 |y > 0)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

CC

111.7

111.4

111.4

111.5

111.5

111.5

111.5

111.6

111.6

111.6

111.7

OCN

92.7

92.7

92.7

92.7

92.7

92.7

92.7

92.7

92.7

92.7

92.7

OCR

130.4

130.4

130.4

130.4

130.4

130.4

130.4

130.4

130.4

130.4

130.4

OCB

48.6

43.8

43.7

43.5

43.2

42.8

42.5

42.2

41.9

41.7

41.5

ρ

2.06

2.29

2.29

2.30

2.32

2.34

2.35

2.37

2.38

2.40

2.41

legend:

ratio =

DET
STOCH

· 100

ρ=

1−STOCH β
1−DET β

Table 6.8: Ratios τ and ρ for a changing lead time fluctuation σL2
that the value for most ratios in Table 6.8 and both bounds in Figure 6.5 are almost
constant over the different values for σL2 .

184

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

Once we know that the time window tw is equal to the difference between both average lead times in our case, so tw = µ L1 − µ L2 , we can see why the ratios of OCN and

OCR remain constant. The probability p(2, ., .) that a second order is placed depends

only on the time window tw , the probability that a first order arrives before tw , and the
distribution of the demand. All three parameters remain constant here unlike in the
latter Section 6.2.3.1 where a changing average lead time µ L2 affected the value of tw .
Consequently, the over- and underestimation of OCN and OCR , respectively, are constant in Table 6.8. Note, just like in Section 6.2.2.2 the absolute values of all parameters
remain constant as the only changing parameter is not considered by DET.
The high values for ρ might be surprising, as well. Especially for the deterministic
case where σL2 = 0 one could expect that the service levels DET β = 99.15% and
of STOCH β = 98.24% are closer together. Once again, it turns out that neglecting
the fluctuation has a strong influence on the quality of the results even if only one
lead time is stochastic. Here, much less second orders than DET calculates are placed
due to early arrivals of first orders. This leads to a significant overestimation of the
service level and a correlated underestimation of the stock out costs which is reflected
in the values of ρ and the ratio of OCB , respectively. Of course, the approximation is
continuously getting worse if we increase the variability of the second lead time, see
Table 6.8.
In general, DET overestimates the holding costs CC due to more second orders.
Given the constant value for CC of DET, the convex curve of the ratios for CC is solely
determined by the change of CC in the STOCH scenario. When σL2 = 0 holds, the
second order arrives as early as possible with the given setting because there are no
lead times below 1 day. Consequently, the stock peak induced by the second order
is the furthest ahead of the stock peak caused by the first order which leads to less
holding costs for STOCH. With increasing fluctuation, 1 ≤ σL2 ≤ 2, the expected

delivery of the second order coincides more and more with the arrival of the first

order quantity Q1 order quantity Q2

lead time fluctuation σL1 lead time fluctuation σL2

average lead time µ L1 average lead time µ L2

2

DET
deviation
DET
DET
STOCH
deviation
deviationSTOCH
STOCH
demand
fluctuation
demand
fluctuation
demand
fluctuation
cost
measures
cost
measures
cost
measures
average
lead
time
average
lead
time
average
lead
time

CC OCB OCR OCN

lead
time
fluctuation
lead
time
fluctuation
lead
time
fluctuation
order
quantity
order
quantity
order
quantity

τ min τ max τ̄

in %

demand fluctuation σD

20%
20%20%

0%

RS1,trad RS2,trad RS1,relax RS2,relax

value

80%
80%80%
40%
60%
60%60%
DET
deviation STOCH
40%
20%
40%40%

1−STOCH
1−DET

120%
120%
100%

60%100%
2 100%

deviation

140%
140%

120%
80%

DET
STOCH

100%
160%
140%
160%

value

DET
deviation
value
STOCH

120%

185

cost measures

deviation

value
demand
fluctuation σD

140%

reorder point R1 reorder point R2

2
Deterministic vs. stochastic – Sensitivity
analysis on lead time L2
2 2

reorder
point
reorder
point
reorder
point
DET
DETDET in %
deviation
deviation
inin%%
deviation
STOCH
STOCH
STOCH

1−STOCH
1−1STOCH
−STOCH
deviation
deviation
deviation
1−DET
1−1DET
−DET

RSRSRS1,trad
RSRSRS2,trad
RS
RS
RS RS1,relax
RS RS2,relax

1,trad
2,trad
2,relax
1,trad µ
2,trad 1,relax
1,relax
2,relax
average
lead time
lead
0%0% 0%
0.0
0.2
0.4 µ L1 average
0.6
0.8 time
1.0 L2 1.2
1.4
1.6
1.8
0.0 0.0 0.0 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 0.6 0.8 0.8 0.8 1.0 1.0 1.0 1.2 1.2 1.2 1.4 1.4 1.4 1.6 1.6 1.6 1.8 1.8 1.8 2.0 2.0 2.0
OC
OC
OC
CC
OC
OC
OC
standard
BR R
B deviation
Bdeviation
lead time fluctuation σL1 leadCC
time
fluctuation
σRLN2 N N
standard
standard
deviation
standard
deviation

2.0

CC OC OC OC

minminmin
max
maxmax

τ t_min
ττ
ττ̄avg_t
τ̄
t_min
t_max
τ
τt_max
τ̄avg_t
t_min
avg_t
t_min
t_max
avg_t
order quantity Q1 order quantity
Q
2 t_max
σDRσ2µDσLµ
σ1LLσ11Lµ1σLµ
σ2 2Q
Q212RQ
R112RR
reorder point R1 reorder point
σQ
Q
R221 R2
121Q
D
L21Lσ
L
2µLL2L
1 Lµ
DET min
deviation
Figure 6.5:
Cost ratios
, τ max , and τ̄ for a changing lead time fluctuation σL2
STOCHτ in %

deviation

1−STOCH
1−DET

RS1,trad RS2,trad RS1,relax RS2,relax

order. This increases
CC
and
CC OCB OC
R OC
N the ratio CC between DET and STOCH declines. After
min

max

τ
τ delivery
τ̄
that the expected
of the second order gradually moves further behind the

arrival of the first order. This leads to a smaller value of STOCH CC and to a higher
ratio.

6.2.4

Order quantities

In contrast to the sections before, the order quantity is a parameter that can usually
be varied by the owner of an inventory. Here again, we anticipate that DET overestimates the service level, the costs for second orders, and the holding costs while it
underestimates the stock out costs and costs for first orders. Surely, one expects that
the absolute value of the β service level increases with higher order quantities because
the majority of the demand has already been covered before a new order is placed and
we run into the increased danger of facing stock outs.
22

2

2

186

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

6.2.4.1

Order quantity of the first supply mode

From Table 6.9 we can see that an order quantity Q1 = 30 is not sufficient to satisfy all
the demand until the next replenishment cycle. The probability for violating Assumption 8, page 43, is 1.68 · 10−9 which exceeds our tolerance of 10−10 . Consequently, we
do not consider this case.

order quantity Q1

ratio

30

40

50

60

70

80

90

100

250

500

CC

N.A

113.3

111.4

110.0

108.8

107.8

107.0

106.3

102.5

101.3

OCN

N.A

91.4

92.7

93.6

94.4

94.9

95.4

95.8

98.2

99.1

OCR

N.A

128.6

130.4

131.7

132.8

133.6

134.2

134.8

138.1

139.4

OCB

N.A

43.0

43.6

44.1

44.4

44.7

44.9

45.1

46.2

46.6

ρ

N.A

2.32

2.29

2.27

2.25

2.24

2.23

2.22

2.16

2.15

ratio =

DET
STOCH

legend:

· 100

ρ=

1−STOCH β
1−DET β

N.A. = not available

Table 6.9: Ratios τ and ρ for a changing order quantity Q1
In general, we see the usual picture that the values for CC, OCR , and β are indeed
overestimated by DET which leads to an underestimation of the remaining parameters in Table 6.9. This is reflected in the associated ratios. The absolute value of the
holding costs rises with an increasing order quantity Q1 . While the absolute difference between DET CC and STOCH CC remains fairly constant, their ratio gradually
decreases towards 100% when Q1 approaches 500 units. One could try to use a similar
argumentation to explain the decreasing ratios of the service level and the stock out
costs OCB . Then, one refers the decrease ρ to a saturation of β in the DET case while
the value of β from the STOCH scenario is gradually catching up. This is certainly
true to some extend. However, the question remains for OCB why its ratio does not
converge to 100%.

Deterministic vs. stochastic – Sensitivity analysis on order quantity Q1

187

This brings us to the interesting observation that the ratio for OCN increases towards 100% and the ratio of OCR gradually increases to values close to 140% at the
same time. Several interdependent causes lead to this result. First, the overall demand per year is constant for different values of Q1 . Second, the 2-order probability
p(2, ., .) remains 24.10% and 33.91% for STOCH and DET, respectively, independent
of Q1 . This can be explained by the fact that neither the underlying distribution nor
the time window tw is changed. Third, even though p(2, ., .) remains constant more
demand is covered by the first orders because Q1 increases. This leads to higher absolute costs OCN and to a lower absolute value for OCR . Consequently, the systematic
overestimation of the percentage of second orders, p(2, ., .), has gradually less impact
on the total costs of the first orders OCN for an increasing value of Q1 . This implies
that the ratio of OCN converges to 100% for Q1 → ∞.
Analogous to the latter argumentation, the total costs for the second orders, OCR ,
decrease when Q1 increases. Consequently, the constant overestimation of p(2, ., .)
gains more impact on the total costs of the second orders OCR .

Let us come back to the question why the ratio of OCB does not converge to 100%. We
stated that the overestimation of p(2, ., .) looses importance for the total cost and the
number of first orders. In the same way the importance of the second orders vanishes
in general and the whole scenario gradually changes towards a situation where only
one (stochastic) lead time exists. This also makes sense if we exchange the increase
of Q1 in relation to Q2 by a decrease of Q2 . Once Q2 equals 0 only one supply mode
effectively remains.
In all cases this would imply that the values of ρ and of the ratio of OCB origin
from the poor approximation by DET even if only one lead time is stochastic. Most
interestingly, the values of ρ and the ratio of OCB for Q1 = 500 from Table 6.9 are very
similar to the values of Table 6.5 where σL2 = 0. This result fosters our argumentation.

2

20%
0%

cost measures
DET
deviation
DET
DET
STOCH
deviation
deviationSTOCH
STOCH
demand
fluctuation
demand
fluctuation
demand
fluctuation
cost
measures
cost
measures
cost
measures
average
lead
time
average
lead
time
average
lead
time

20%20%
20%

lead
time
fluctuation
lead
time
fluctuation
lead
time
fluctuation
order
quantity
order
quantity
order
quantity

τ min τ max τ̄

2

CC OCB OCR OCN

RS1,trad RS2,trad RS1,relax RS2,relax

1−STOCH
1−DET

in %

deviation

40%

DET
STOCH

60%

100%
100%
80%
80%
80%
60%
60%
60%
40%
40%
40%

deviation

120%
120%
80%100%

2
C HAPTER 6. D ETERMINISTIC
V ERSUS S TOCHASTIC D UAL S OURCING
2 2

reorder point R1 reorder point R2

160%
140%
160%

100%120%
140%
140%

value

DET
deviation
value
STOCH

120%

order quantity Q1 order quantity Q2

140%

lead time fluctuation σL1 lead time fluctuation σL2

demand
fluctuation σD
value

160%

value
average
lead time µ L1 average lead time µ L2

188

reorder
point
reorder
point
reorder
point

deviation

DET

STOCH
DET
DETDET in %
deviation
deviation
inin%%
deviation
STOCH
STOCH
STOCH

demand fluctuation
σD
1−STOCH
1−1STOCH
−STOCH
deviation
deviation
deviation
1−DET

1−1DET
−DET
average lead time
µ L1 average lead time µ L2

RSRSRS1,trad
RSRSRS2,trad
RS
RS
RS RS1,relax
RS RS2,relax

1,trad
2,relax
1,trad 2,trad
2,trad 1,relax
2,relax
lead
σ1,relax
time
fluctuation
σL2 500
0% 0%40
0%
L1 lead 100
50
60
70 time fluctuation
80
90
250
0.0 0.00.0 0.2 0.20.2 0.4 0.40.4 0.6 0.60.6 0.8 0.80.8 1.0 1.01.0 1.2 1.21.2 1.4 1.41.4 1.6 1.61.6 1.8 1.81.8 2.0 2.02.0
CC
OC
OC
OC
order
1OC
CC
OC
OC
BRQ
RNN N
B quantity
B deviation
R
order
quantity
quantity Q2
standard
deviation
standard
deviation
1 order
standard

CC OC OC OC

min
max
max
max
t_min
t_max
avg_t
τ min
ττ
ττ̄avg_t
τ̄avg_t
t_min
t_max
τ min
τ
τ̄R
t_min
t_max
reorder
point
reorder point R2
t_min
t_max
1avg_t

Figure 6.6: Cost ratios τ min , τ

deviation
σDσµDσLµ
σ1LSTOCH
µσµ σin
Q
σ1DET
Q212RQ
R112RR
µσ%
σQ121QQ
R221 R2
D
L
1 Lµ
22 L
1L1 LL21L2 LL2L
1−STOCH
deviation 1−DET
max

, and τ̄ for a changing order quantity Q1

RS1,trad RS2,trad RS1,relax RS2,relax

CC OCB OCR OCN
τ min τ max τ̄

Of course, the scenario in Table 6.8 includes the effects of second orders but they cannot have a big influence on the two ratios because the second orders are identically
modeled by DET and STOCH when σL2 = 0.

6.2.4.2

Order quantity of the second supply mode

In the latter section we argued that an increasing value of Q1 diminishes the influence
of the second order which should lead to a similar effect as reducing the order quantity
Q2 to zero. This is exactly what we are doing in the following. The order quantity Q2
is increased from 0 to 500 units. The results are shown in Table 6.10 and Figure 6.7.
From a direct comparison between the column for Q1 = 500 in Table 6.9 and the
result for Q2 = 0 of the Table 6.10 we see that the ratio for OCB is 46.6% versus 43.3%,
respectively, and ρ is 2.15 versus 2.31, respectively. These values are neither identical nor very alike. So our argumentation of Section 6.2.4.1 does not seem2 to hold.
2

2
However, we can say that the rough magnitude is similar, at least. The question
is
2

Deterministic vs. stochastic – Sensitivity analysis on order quantity Q2

189

order quantity Q2

ratio

0

15

30

45

60

75

90

150

250

500

CC

103.1

106.2

108.9

110.9

112.4

113.3

113.9

114.0

112.2

108.5

OCN

100.0

97.3

95.1

93.2

91.6

90.3

89.0

85.4

81.8

77.7

OCR

100.0

136.9

133.8

131.2

128.9

127.0

125.3

120.2

115.1

109.3

OCB

43.3

45.8

44.8

43.9

43.1

42.5

41.9

40.2

38.5

36.6

ρ

2.31

2.18

2.23

2.28

2.32

2.35

2.39

2.49

2.60

2.74

ratio =

DET
STOCH

legend:

· 100

ρ=

1−STOCH β
1−DET β

Table 6.10: Ratios τ and ρ for a changing order quantity Q2

whether we find a good explanation for the deviation which is able to rehabilitate our
argumentation.
If we look at the service levels β for Q1 = 500 in the latter section they are 99.99%
for DET and 99.76% for STOCH. The values for Q2 = 0 in this section are 98.59%
for DET and 96.75% for STOCH. We mentioned in the latter section that β is reaching
a certain level of saturation for Q1 = 500 while the value of β is catching up over
time in the STOCH case. This is not valid for Q2 = 0 in this section. The service
level is far from saturation and the usual strong overestimation of β by DET is still
effective. Consequently, the underestimation of the ratio of OCB is larger here, as well.
To complete the argumentation for Q1 = 500 in Section 6.2.4.1 it requires to include
the impact of entering a level saturation for β on the overestimation of ρ by DET.
Putting the special case aside where Q2 = 0 the remaining entries for ρ and the
ratio of OCB follow a monotonous line as the value of Q2 increases. We know that
DET tends to overestimate the number and the costs of the second orders. Here, one
can see that this effect is amplified with an increasing importance of the second orders
which is induced by higher values of Q2 . This behavior coincides with our intuition.

2

20%20%
0% 20%

RS1,trad RS2,trad RS1,relax RS2,relax

DET
STOCH

lead
time
fluctuation
lead
time
fluctuation
lead
time
fluctuation
order
quantity
order
quantity
order
quantity

τ min τ max τ̄

deviation

CC OCB OCR OCN

1−STOCH
1−DET

deviation

in %

DET
deviation
DET
DET
STOCH
deviation
deviationSTOCH
STOCH
demand
fluctuation
demand
fluctuation
demand
fluctuation
cost
measures
cost
measures
cost
measures
average
lead
time
average
lead
time
average
lead
time

DET
STOCH

100%
60%100%
80%
80%
80%
60%
40%
60%
60%
40%
20%
40%
40%

2

cost measures

deviation

120%
80%
120%
100%

2
C HAPTER 6. D ETERMINISTIC
V ERSUS S TOCHASTIC D UAL S OURCING
2 2

reorder point R1 reorder point R2

100%

140%
140%
120%

order quantity Q1 order quantity Q2

120%

140%
160%
160%

value

DET
deviation
value value
STOCH

140%

lead time fluctuation σL1 lead time fluctuation σL2

demand
fluctuation σD
value

160%

average lead time µ L1 average lead time µ L2

190

reorder
point
reorder
point
reorder
point
DET
DETDET in %
deviation
deviation
inin%%
deviation
STOCH
STOCH
STOCH

demand fluctuation σD

1−STOCH
1−1STOCH
−STOCH
deviation
deviation
deviation

1−DET
1−1DET
−time
DET
average lead time µ L1 average lead
µ L2

RSRSRS1,trad
RSRSRS2,trad
RS
RS
RS RS1,relax
RS RS2,relax

2,trad
1,relax
2,relax
1,trad
2,trad
1,relax σL150
2,relax 250
lead
fluctuation
time
0% 0%0% 0
15 time 30
45 σ1,trad
60
75 fluctuation
90
500
L1 lead
2
0.0 0.00.0 0.2 0.20.2 0.4 0.40.4 0.6 0.60.6 0.8 0.80.8 1.0 1.01.0 1.2 1.21.2 1.4 1.41.4 1.6 1.61.6 1.8 1.81.8 2.0 2.02.0

order
quantity
2OC
CC
OC
CC
OC
OC
OC
CC
OC
OC
BROC
BOC
B deviation
R
order quantity Q1 order
quantity
Q
standard
deviation
2 RNN N
standard
deviation
standard
min
max
max
max
t_min
t_max
t_min
t_max
t_min
τ min
ττ
ττ̄avg_t
τ̄avg_t
τ min
τ t_max
reorder point R1 reorder
point
Rτ̄2avg_t
t_min
t_max
avg_t

deviation

DET
STOCH

in %
σDσµDσLµ
σ1LLσ11Lµ1σLµ
σ2 2Q
Q212RQ
R112RR
σQ
Q
R221 R2
121Q
D
L21Lσ
L
2µLL2L
1 Lµ

1−STOCH

deviation
−DET
Figure 6.7: Cost
ratios τ1min
, τ max , and τ̄ for a changing order quantity Q2
RS1,trad RS2,trad RS1,relax RS2,relax

CC OCB OCR OCN
τ min τ max τ̄

The argumentation of a shifting importance between the first orders and the second
orders from Section 6.2.4.1 can be applied for the ratios of OCN and OCR in Table 6.7,
as well. With an increasing value of Q2 , the second orders cover more demand and the
ratio of OCR develops towards 100%. Simultaneously, the reducing importance of first
orders makes the general underestimation of the OCN ratio by DET more apparent
and its value falls to 77.7%. Only the ratios of OCN and OCR for Q2 = 0 do not fit into
the series of values. An explanation is easy to give as the total demand is exclusively
covered by the first supply mode when Q2 = 0. This equally holds for DET and
STOCH and yields exactly 100.0% for the ratios of OCN and OCR .
We expect rising holding costs if Q2 increases which is confirmed by the absolute
values of CC in our two scenarios DET and STOCH. The ratio of CC is increasing for
Q2 ≤ 150 due to the increasing influence of the second orders and their overestimation
22

2

2

Deterministic vs. stochastic – Sensitivity analysis on the reorder point R1

191

by DET. Yet, it is surprising to see that the value of the ratios decreases for Q2 > 150.
The reason are potentially unlimited holding costs – unlike the service level. While the
effect of overestimating the probability of second orders, p(2, ., .) remains constant, the
absolute difference of CC between DET and STOCH keeps increasing. Therefore, it is
only due to the higher absolute values of CC that its ratio decreases from 114.0% to
108.5%.

6.2.5

Reorder points

A company is usually completely free to choose the reorder points. This liberty does
not even exists for the order quantities in many cases as suppliers dictate lot sizes.
We will make full usage of this liberty to set Q1 and Q2 . In particular, this means
that we will explore the possibility and effect of setting negative reorder points in
compliance with Assumption 2, page 42. This can be useful in cases where the supplier
of the secondary orders is expensive and dictates large quantities, for example. Then
one could wait until the backlog of demand is big enough and a second order pays
off. In addition, we will allow that the second orders are triggered before the first
ones, so R1 < R2 . Obviously, this violates Assumption 2. However, we can simply
relabel the first and the second supply mode and apply the formal model without any
further changes and without any restrictions. Note, only the case R1 = R2 is explicitly
excluded here which would lead to a order-splitting policy.
6.2.5.1

Reorder point of the first supply mode

In the first two columns of Table 6.11 we set R1 to a negative value. This requires us
to relabel supply mode 1 and two by exchanging their names internally before we can
apply the formal model again. However, we will still use the original name of the
supply modes in the following text.
Setting R1 < 0 implies that the second order is triggered first. Therefore, we also

192

ratio

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING
reorder point R1
-2

-1

1

2

3

4

5

6

7

8

9

CC

102.3

102.3

110.5

111.9

112.6

112.5

111.4

108.8

105.6

103.4

102.4

OCN

100.0

100.0

94.7

92.1

90.5

90.7

92.7

95.5

97.9

99.2

99.8

OCR

100.0

100.0

106.0

110.3

115.8

122.6

130.4

138.4

144.9

148.9

151.0

OCB

65.7

65.7

68.6

68.4

63.6

54.6

43.6

32.5

22.3

13.9

7.7

ρ

1.52

1.52

1.46

1.46

1.57

1.83

2.29

3.08

4.49

7.21

12.98

legend:

ratio =

DET
STOCH

· 100

ρ=

1−STOCH β
1−DET β

Table 6.11: Ratios τ and ρ for a changing reorder point R1
have to adapt the time window tw to 0 days because µ L1 > µ L2 holds for the average
lead times in our case and there is no chance that orders triggered at R1 will arrive
first. Consequently, all demand is covered by the second supply mode. There occur
no costs OCN and the absolute ordering costs OCN and OCR are identical for DET
and STOCH. The service level and the holding costs, CC, are overestimated by DET
because all lead times over µ L2 = 1.0 days are neglected. The same argumentation
leads to an explanation of the underestimation of OCB by DET for R1 < 0.
Let us switch to the results for R1 > 0. The service-level-related values, ρ and the
ratio of OCB , in Table 6.11 show a completely monotonous behavior. However, behind
the scenes some interesting effects occur. The service level β starts at 98.91% and
98.41% for DET and STOCH when R1 = 1, respectively. Then β falls until R1 = 2 and
R1 = 5 for DET and STOCH, respectively. After that β starts rising again. Intuitively,
one might expect that β should monotonously increase due to a higher implicit safety
stock induced by a rising reorder point R1 . This effect is valid without exception.
However, the increasing gap between R1 and R2 lowers the probability that a second
order is triggered. This has a reducing effect on β. Of course, both effects change their
strength of influence in direct relation to the importance of the first and the second

order quantity Q1 order quantity Q2

lead time fluctuation σL1 lead time fluctuation σL2

value
average
lead time µ L1 average lead time µ L2

2

2

0%

20%
20%20%

DET
deviation
DET
DET
STOCH
deviation
deviationSTOCH
STOCH
demand
fluctuation
demand
fluctuation
demand
fluctuation
cost
measures
cost
measures
cost
measures
average
lead
time
average
lead
time
average
lead
time

lead
time
fluctuation
lead
time
fluctuation
lead
time
fluctuation
order
quantity
order
quantity
order
quantity
DET
deviation
STOCH
reorder
point
reorder
point
reorder
point
demand
fluctuation
σ

τ min τ max τ̄

2

193

cost measures

CC OCB OCR OCN

in %

RS1,trad RS2,trad RS1,relax RS2,relax

100%
100%
80%
80%80%
60%
40% 60%60%
40%
20% 40%40%

60%

1−STOCH
1−DET

120%
120%

80%100%

deviation

140%
140%
100%120%

DET
STOCH

120%140%
160%
160%

value

DET
deviation
valueSTOCH

140%

deviation

value
demand
fluctuation σD

160%

reorder point R1 reorder point R2

2 2
Deterministic vs. stochastic – Sensitivity
analysis on the reorder point R1

D

DET
DETDET in %
deviation
deviation
in
%%
deviation
STOCH
STOCH
STOCH
average
lead
time
µin
L1 average lead time µ L2
1−STOCH
1−1STOCH
−STOCH
deviation
deviation
deviation
1−DET
lead
time fluctuation
σL1 lead time fluctuation σL2
1−1DET
−DET
RS
RS
RSquantity
RS2,relax
RSRS
RSRS
RS
RS
RS
RS2,relax
1,trad
1,relax
1,trad
2,trad
1,relax
1,trad
2,trad
1,relax
2,relax
order
quantity
Q2,trad
order
Q

15
0%0%-2
0%
-1
1
2
3
4
6
7 2
8
0.0 0.0 0.0 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 0.6 0.8 0.8 0.8 1.0 1.0 1.0 1.2 1.2 1.2 1.4 1.4 1.4 1.6 1.6 1.6 1.8 1.8 1.8 2.0 2.0 2.0

9

CC
OC
OC
CC
OC
OC
OC
CC
OC
OC
OC
reorder
BRR
RNOC
B
B point
R
N N point R2
reorder
point
standard
deviation
standard
deviation
11 reorder
standard
deviation
min
max
min
max
DET
t_min
t_max
t_min
t_max
τ
τ̄avg_t
ττ
τSTOCH
τ̄
τ min
τ max
τ̄avg_t
t_min
t_max
avg_t
t_min
t_max
avg_t
deviation
in %

STOCH
σDσµDσLµ
σ1LLσ111L−1µ1σ
µ21Lσ
σ2 2Q
Q212RQ
R112RR
σQ
Q
R221 R2
deviation
121Q
D
L
2µLL2L
1 Lµ
−LLDET

RS,1,trad
Figure 6.8: Cost ratios τ min , τ max
andRS
τ̄ 2,trad
for aRSchanging
reorder point R1
1,relax RS2,relax
CC OCB OCR OCN
τ min τ max τ̄

supply mode, respectively. It is the combination of these effects that leads to an initial
decrease and later increase of β. Analogously, the absolute costs CC and OCB initially
rise before they fall again. Only the absolute costs for both supply modes, OCN and
OCR , monotonously increase and decrease, respectively. This is easy to verify upon
the presence of a gradually decreasing probability that a second order is triggered.
One observes that the ratios of CC and OCN tend towards 100% with an increasing
R1 . The main reason lies in the decreasing weight of the second orders which manifests in a probability p(2, ., .) of 0.64% and of 0.42% for DET and STOCH, respectively.
Exactly these small values for p(2, ., .), the inherent little coverage of total demand,
and the general overestimation of second orders by DET lead to gradually decreasing
ratios for OCB . Directly linked to the decreasing values of the OCB ratios are the increasing values of ρ. This effect is amplified by the circumstance that DET calculates
values for β which are close to 100%, namely 99.94% for R1 = 9. In contrast, β is only
99.22% for STOCH in this case.

22

2

2

194

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

Finally, we should remark that in our example here the absolute holding costs CC
are significantly lower for Q1 < 0 than for Q1 > 0. Depending on the remaining
cost structure it could be beneficial to replenish an SKU with the given characteristics
mainly via the second order channel. While this idea is counter-intuitive at first sight
and there is no guarantee that it holds for the cost-optimal solution, it is still an option
worth considering for an optimization process.

6.2.5.2

Reorder point of the second supply mode

Here we will explore a wide range of values for R2 , as well. This includes negative
values and values greater than R1 . One might expect similar results and effects as in
the latter section but on a higher niveau of service level as R1 = 5 remains fixed here
in contrast to the constant condition R2 = 0 in the latter section.
Indeed, the service levels are in general higher here than in the latter section and
reach 100.0% already for R2 ≥ 4. However, Table 6.12 differs quite a lot from Table 6.11

at first sight. The same hold for the two Figures 6.8 and 6.9. At a second sight, we

ratio

reorder point R2
-10

-8

-6

-4

-2

0

2

4

6

8

10

CC

102.2

102.2

102.2

102.6

105.8

111.4

112.2

110.1

101.9

101.8

101.7

OCN

100.0

100.0

100.0

99.8

97.9

92.7

90.5

94.7

100.0

100.0

100.0

OCR

100.0

153.2

152.3

151.0

144.9

130.4

115.8

106.0

100.0

100.0

100.0

OCB

53.4

53.4

53.4

53.4

52.4

43.7

18.1

1.2

0.0

0.0

100.0

ρ

1.87

1.87

1.87

1.87

1.91

2.29

5.52

79.8

NAN

1.00

1.00

legend:

ratio =

DET
STOCH

· 100

ρ=

1−STOCH β
1−DET β

NAN = not a number

Table 6.12: Ratios τ and ρ for a changing reorder point R2
find identical values for the ratios of OCN and OCR in both tables but in a reverse

Deterministic vs. stochastic – Sensitivity analysis on the reorder point R2

195

order. For example, the ratios of OCR are 144.9% and 151.0% for R1 = 7 and R1 = 9,
respectively, in Table 6.11 where R2 = 0 applies. Identical values appear in Table 6.12
for R2 = −2 and R2 = −4, respectively, where R1 = 5 remains constant. This makes

perfect sense, once we consider the fact that in both examples the delta between R1

and R2 is identical, namely 7 and 9 units. Consequently, here the underestimation and
overestimation by DET regarding the ratio of OCN and OCR , respectively, is solely
related to the probability p(2, ., .) of placing a second order. The reason for this is
lead time fluctuation σL1 lead time fluctuation σL2

apparent as OCN , OCR , and their ratios only depend on the number of triggered orders
average lead time µ L1 average lead time µ L2

and the individual order quantity. All these parameters do not change with different
order quantity Q1 order quantity Q2

reorder points as long as their delta remains the same. There is no need to further

2

20%
0%

20%
20%20%

DET
deviation
DET
DET
STOCH
deviation
deviationSTOCH
STOCH
demand
fluctuation
demand
fluctuation
demand
fluctuation
cost
measures
cost
measures
cost
measures
average
lead
time
average
lead
time
average
lead
time

DET
STOCH

lead
time
fluctuation
lead
time
fluctuation
lead
time
fluctuation
order
quantity
order
quantity
order
quantity

τ min τ max τ̄

in %

deviation

2 cost
2
measures

CC OCB OCR OCN

40%

100%
100%
80%
80%80%
60%
60%60%
40%
40%40%

RS1,trad RS2,trad RS1,relax RS2,relax

60%

120%
120%
100%

2

1−STOCH
1−DET

80%

140%
140%
120%

deviation

100%

140%
160%
160%

DET
STOCH

120%

deviation

DET
deviation
valueSTOCH

140%

value

160%

value
value

demand fluctuation σD

180%

reorder point R1 reorder point R2

explain the ratios of OCN and OCR as they
are identical to the ones in Section 6.2.5.1.
2

reorder
point
reorder
point
reorder
point

demand fluctuation σD

DET
DETDET in %
deviation
deviation
inin%%
deviation
STOCH

STOCH
STOCH
average lead time µ L1 average
lead time µ L2

1−STOCH
1−1STOCH
−STOCH
deviation
deviation
deviation

1fluctuation
−DET
−1DET
−DET
lead time fluctuation σL1 lead 1time
σL2

RSRSRS1,trad
RSRSRS2,trad
RS
RS
RS RS1,relax
RS RS2,relax

1,trad
2,trad
2,relax
1,trad
2,trad
1,relax
2,relax
order
Q1 order
quantity
Q22 1,relax
0%
-100% 0% -8
-6 quantity
-4
-2
0
4
6
8
0.0 0.0 0.0 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 0.6 0.8 0.8 0.8 1.0 1.0 1.0 1.2 1.2 1.2 1.4 1.4 1.4 1.6 1.6 1.6 1.8 1.8 1.8 2.0 2.0 2.0

CC
OC
OC
CC
OC
OC
OC
CC
OC
OC
OCOC
reorder
BRR
B
B point
R
reorder point R1 reorder
point
standard
deviation
standard
deviation
2 1 RNN N
standard
deviation
deviation

min
max
max
DET
t_min
t_max
t_min
t_max
τ min
ττ
ττ̄avg_t
τ̄avg_t
τ min
τ max
τ̄avg_t
t_min
t_max
avg_t
%
t_max
STOCH in t_min

deviation

1−STOCH
σ1LLσ11Lµ1σLµ
σ2 2Q
Q212RQ
R112RR
σQ
Q
R221
Lµ
Dσ
121Q
D
L21Lσ
L
1−DET σDσµ
2µLL2L
1 Lµ

10

R2

RS1,trad RS
RS1,relax RS2,relax
Figure 6.9: Cost ratios
τ min2,trad
, τ max
, and τ̄ for a changing reorder point R2
CC OCB OCR OCN
τ min τ max τ̄

We do not find identical values for the other ratios because they all depend on
the stock level and on when orders arrive. These values change for different reorder
points and whenever we swap the fast and the slow supply mode.

196

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

The development of the ratios for CC is not identical but still similar in Table 6.11
and Table 6.12. The deviation is small if one of the two supply modes dominates,
so for R2 ≤ −4 and R2 ≥ 6. In that case, the deviation originates from neglecting the
variability of one supply mode. Whenever both supply modes are used the ratio of CC

rises which indicates that the fluctuation of both lead times do not compensate each
other but amplify themselves. Keep in mind that for R2 > 5 all orders are exclusively
placed via the second channel as tw = 0 and R1 < R2 .
As long as only the first supply mode is used, the values for ρ are 1.87, due to the
average lead times used by DET. In the subsequent columns, ρ rises up to 79.8% as
more second supply modes are placed. For DET the service level β rises faster here as
the fluctuation of the second lead time is neglected. Finally, for R2 = 6 DET calculates
β = 100% while this value is reached by STOCH only for R2 ≥ 8. Consequently, ρ is

not a number due to a division by zero for R2 = 6 and is 1.00 for larger values of R2 .

A similar argumentation as for ρ also holds for the ratio of OCR . Initially, ratio
indicates the overestimation of the service level by DET induced by not considering
the lead time fluctuation. This value rises as the two stochastic variables interact more
heavily. For R2 = 6 DET already calculates β = 100% unlike STOCH and the ratio of
OCB is zero. For R2 = 10 both service levels are 100% and the ratio is 100.0%. One
would expect the same for R2 = 8. However, the service level β of STOCH is close
enough to 100.0% to get ρ = 1.00 in that case. However, β is still not exactly 100.0%
and there exist still small costs for stock out. This leads to a ratio of OCB of 0.0%.

6.2.6

Summary

In the sensitivity analysis one out of 9 input parameter was altered at a time. The
resulting changes of the deviation between the DET scenario and the STOCH scenario
were investigated in the context of various cost types and the β service level. This
represents our first approach to answer research question RQ 3.

Deterministic vs. stochastic – Summary of the sensitivity analysis

197

Our observation was that the deviation between DET and STOCH varies greatly. Especially, for the service level and the stock out costs the deviations were large and
easily exceeded ±40%. In extreme cases they even reached 100%. The deviation de-

creased for a reduced or even eliminated variability of some parameters. However, in
reality one will usually face significant lead time fluctuation. The same is true for the
demand, especially in cases where the inventory faces consumer demand.

Another observation is that the deviation between DET and STOCH decreased if
one of the two supply modes predominated. Consequently, we cannot derive from
our observations that the variability of several parameters cancel out each other and a
deterministic model is fortunate for approximating the stochastic scenario.

In our examples, DET never underestimated the number of rush orders because
normal orders were expected to arrive late and there was a good chance to place a
second order. However, this does not need to be true for all parameter settings. Such
a case can simply be constructed by a negatively skewed distribution for the first lead
time. Then the average lead time for the first order is lower than its mode. Consequently, regular orders are assumed to arrive early which reduces the probability to
place a second order.

Due to the manifold interrelations between the individual stochastic variables it is
hard to generally predict the implications of a deterministic approximation for the
stochastic scenario. However, from the frequent and large deviations between DET
and STOCH in the sensitivity analysis we conclude that STOCH should be clearly
favored to calculate the KPIs when demand and both lead times are stochastic. This is
especially true if a high accuracy of the values for the KPIs is required.

198

6.3

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

Case study – a spare parts warehouse

The results of the sensitivity analysis in the previous chapter give a good impression
about how large the span of divergence between the results of the deterministic DET
scenario and the stochastic STOCH scenario can get. However, these results are rather
theoretical and one can well imagine that the ratios in reality are different because
the random variables are not Gaussian distributed. Moreover, it does not answer the
question of how much the results of DET and STOCH differ when several parameters
are changed simultaneously.
The joint change of the 9 parameters yields 29 = 512 combinations if there exist
only 2 different values for each parameter. Most likely one is interested in 5 to 10
different values per parameter which easily exceeds 1,000,000 possible combinations.
This extensive approach is definitely not feasible. Even if we restrict the scope of
parameters we still face the problem of limited knowledge whether the combination
of different values are relevant or not. For example, how close are they to the costminimal values for these parameters? Actually, one will have to repeat the sensitivity
analysis for each warehouse by considering its specific situation. This way one could
give a valid statement about the differences between DET and STOCH for a specific
warehouse.
Fortunately, we have the data of a warehouse with 2,751 SKUs at hand. In the following, we will evaluate our findings from the sensitivity analysis in the context of this
empiric situation. Now, how should we compare the results of the sensitivity analysis
with the real data? An individual comparison for each of the SKUs is just as unfeasible as another simple sensitivity analysis on a few selected representative SKUs. A
more promising approach is to consider the 2,751 SKUs to represent the members of a
sensitivity analysis where all input parameters have been changed simultaneously.
The rational behind this approach lies in the differences among the 2,751 SKUs.
The demand distribution, the lead time distributions, and the costs influence the cost-

Deterministic vs. stochastic – Case study

199

minimal values of the replenishment parameters R1 , R2 , Q1 , and Q2 . This leads to
many different settings for which we can compare the DET scenario and the STOCH
scenario. We apply the ratio τ between DET and STOCH to each of the costs CC,
OCN , OCR , and OCB . For each of these costs we can also derive the minimal, maximal, and average ratio over all SKUs. These can be compared to the values that we
have obtained in the sensitivity analysis of Chapter 6.2. The comparison requires the
following four steps.
Step 1. The replenishment parameters (R1 , R2 , Q1 , and Q2 ) are optimized for the deterministic DET scenario in order to minimize the total costs for each individual
SKU.
Step 2. We calculate the value of all individual costs types twice, once for the deterministic DET case and once for the stochastic STOCH case, always using the
optimal replenishment parameters from step 1. Recall, that the cost types are
normal, rush, and back order costs (OCN , OCR , OCB ) as well as the capital costs
(CC).
Step 3. The empiric ratios τ min , τ max , and τ̄ from the warehouse case study are calculated for each cost type.
Step 4. The ratios of Step 3 are compared to the ratios that have been observed in the
sensitivity analysis, see Section 6.2.

Note that we obtain the optimal replenishment parameters in Step 1 by means of a
Threshold-Accepting Algorithm (TAA) described by Dueck and Scheuer in [DS90].
This optimization algorithm was developed by the IBM Science Center in Heidelberg
in the late 1980s. It is very similar to the well-known simulated annealing algorithm
as described by Kirkpatrick et al. [KGV83] and Eglese [Egl90], but its acceptance rules

200

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

are different, and the TAA leads to more stable results in a number of experiments as
reported by Dueck et al. [DSW91].
We have adapted this general-purpose optimization algorithm to our needs and
it has proven to be very stable and to return close-to-optimal results in all our cases.
However, our investigations are by no means limited to the usage of this particular
optimization algorithm. Other non-linear optimization techniques could have been
used to determine the cost-minimal replenishment parameters, as well. That’s why
we do not go into details about the concrete implementation and parametrization of
the applied threshold-accepting algorithm.
Recall, the DET scenario assumes deterministic lead times lt1 and lt2 while STOCH
considers stochastic lead times distributions L1 and L2 . This is the sole difference
between DET and STOCH.

6.3.1

Setup for comparing the case study and the sensitivity analysis

Case study.

In our case study we calculate the ratios τCC , τOCN , τOCR , and τOCB

for the costs CC, OCN , OCR , and OCB , respectively, just as in the sensitivity analysis, see Chapter 6.1.2. For each of these costs we can also derive the minimal,
maximal, and average ratio over all SKUs, namely τAmin , τAmax , and τ̄A , where A ∈

{CC, OCN , OCR , OCB }. However, we point out here that these values are not directly

comparable to τ min , τ max , and τ̄ of the sensitivity analysis. The sensitivity analysis
determines these values for a particular input parameter like R1 or Q2 . In contrast, the
case study changes all of these input parameters simultaneously and, thus, τ min , τ max ,
and τ̄ have to be defined per cost parameter like CC or OCN . Luckily it only takes
limited effort to make them comparable and this change to a cost perspective will
give us additional and different insights than we have from the sensitivity analysis in
Chapter 6.2.

Deterministic vs. stochastic – Setup of the comparison

201

Recall from Chapter 6.1.2 that τ min , τ max , and τ̄ are defined regarding a set T =

{τ1 , ..., τn } of n arbitrary ratios. In the sensitivity analysis we used the set
TSKU = {τCC , τOCN , τOCR , τOCB }

in each individual section about the input parameters D, L1 , L2 , R1 , R2 , Q1 , and Q2 .
min was omitted for simplicity and better readThe subscript SKU in expressions like τSKU

ability. Now, we define the appropriate sets for our case study and its warehouse of
2,751 SKUs by

TWH,A = {τA,1 , τA,2 , ..., τA,2751 }

(6.3.1)

where A ∈ {CC, OCN , OCR , OCB } represents the currently investigated type of costs.

For example, the average ratio τ̄ of capital costs between DET and STOCH in our
warehouse can be expressed by
τ̄WH,CC =

1 2751
τCC,i .
2751 i∑
=1

(6.3.2)

Sensitivity analysis. Now, we define similar sets for the sensitivity analysis that allow us to compare its measures like τ̄ to the respective measures of the case study.
Recall that only one input parameter Bi ∈ B = {σD , µ L1 , σL1 , µ L2 , σL2 , Q1 , Q2 , R1 , R2 }

has been changed in the sensitivity analysis at a time. Then, for this input parameter Bi there exist k i ratios τA,Bi ,j for each cost type A ∈ {CC, OCN , OCR , OCB } with
j ∈ {1, 2, ..., k i }.

Example 6.3.1. In Table 6.13 the standard deviation σD of the Gaussian demand distribution D is set to 11 different values. Thus, k i = 11 for the capital costs CC and
τCC,σD ,2 = 102.4. Note that only entries with a number are considered and all other
entries like ”not defined” are ignored.
We can define the set of all ratios that belong to a certain cost type A within the sensitivity analysis (SA) for a specific input parameter Bi by

TSA,A,Bi = {τA,Bi ,1 , τA,Bi ,2 , ..., τA,Bi ,ki }

(6.3.3)

202

ratio
CC
legend:

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING
standard deviation σD of the daily demand distribution D (1) = Φ( x; 1.0; σD | x ≥ 0)
0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

102.0

102.4

107.3

110.0

111.0

111.4

111.6

111.7

111.7

111.7

111.7

ratio =

DET
STOCH

· 100

Table 6.13: Excerpt of Table 6.4 on page 172
and for all input parameters Bi ∈ B by

TSA,A =

[

Bi ∈ B

TSA,A,Bi =

[

Bi ∈ B

{τA,Bi ,1 , τA,Bi ,2 , ..., τA,Bi ,ki }.

(6.3.4)

The average ratios τ̄SA,CC,σD and τ̄SA,CC between DET and STOCH are given by
τ̄SA,CC,σD =

1

|TSA,CC,σD | τ ∈T ∑
i

τ̄SA,CC =

i

(6.3.5)

SA,CC,σD

1

|TSA,CC | τ ∈T∑

τi

τi

(6.3.6)

SA,CC

for the cost type CC within our sensitivity analysis. This is all we need to compare the
ratios of the sensitivity analysis with the ones of our case study.

Comparison. A link between the sensitivity analysis and the case study can now be
established via these definitions. For example, the set TSA,A of Equation (6.3.4) is the

pendent to TWH,A in Equation (6.3.1) and the expression τ̄SA,CC of Equation (6.3.6) is
the counterpart to τ̄WH,CC in Equation (6.3.2).

Our comparisons will be focused on charts for a specific cost type like Figure 6.10
for CC. Each of the first 9 columns illustrate the range of the ratios in the set TSA,CC,Bi

for one input parameter Bi that we have observed in the sensitivity analysis in Chapter 6.2. The last column shows the range of the ratios in the set TWH,CC of our case
study.

Deterministic vs. stochastic – Case study regarding capital costs CC

203

Example 6.3.2. The minimal, maximal, and average value in the first column σD of Figmin
max
ure 6.10 are identical to the capital cost ratios τSA,CC,µ
= 102.0%, τSA,CC,µ
= 111.7%,
D
D

and τ̄SA,CC,µD = 109.3% derived from Table 6.13. This table is an excerpt from Table 6.4
on page 172 of our sensitivity analysis. The values for the eight middle columns in
Figure 6.10 correspond to the first row of the corresponding tables of the sensitivity
analysis in Chapter 6.2, namely Table 6.5, Table 6.6, ..., and Table 6.12, respectively.
Note, the definition of the various sets and the various ratios for the β service level
can be derived analogously to the ones for the cost types in this section.
The following sections compare the range of ratios between the sensitivity analysis
and the warehouse for each cost type and the β service level. All sections share a common structure. First, we visually compare the ranges of ratios for the current cost type
between the sensitivity analysis and the warehouse and discuss our findings. Second,
we go into detail for the most interesting observations. Third, we derive analytical
expressions for the deviation between DET and STOCH for several KPIs. Finally, we
summarize or key observations.

6.3.2

Capital costs

This section compares the capital costs CC between the sensitivity analysis and the
warehouse. Figure 6.10 reveals in its first 9 columns that the sensitivity analysis gives
a too pessimistic picture for the situation at our warehouse, see last column. The
ratios of CC between DET and STOCH in the warehouse are smaller than those of
the sensitivity analysis for each of the 9 input parameters. More precisely, we obtain
min
max
τWH,CC
= 99.33%, τWH,CC
= 106.30%, and τ̄ = 100.40% for the warehouse while the
min
max = 114.00%,
lowest values in the sensitivity analysis are τSA,CC
= 101.70% and τSA,CC

respectively.
The most interesting observation can be made regarding the average ratio τ̄. The
value τ̄SA,CC = 104.3% indicates an average overestimation of CC by at least 4.3%

2

2

tc

(0.0.5)

i

(0.0.6)

T

DET DET DET DET
DET DET
DET
deviation
deviation
deviation
deviation
deviation
deviation
deviation
deviation
deviation
T
STOCH
STOCH (0.0.7)
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH

Ri

demand
demand
σD σD(0.0.8)
σD
demand
fluctuation
σDfluctuation
demand
fluctuation
demand
σDfluctuation
σDfluctuation
demand
demand
fluctuation
demand
σDfluctuation
demand
σDfluctuation
Rσ
i Dfluctuation

Qi

2

2

2

2

2

DET

2

2

DET

average
lead
average
time
lead
µLL21 lead
average
time
average
time
µ L2 time
(0.0.9)
average
lead
time
µ L1 lead
average
time
µ
average
lead
time
average
µ L1 average
lead
time
µ L1lead
time
average
µ
time
µµLL21 lead
average
time
µQLi1 lead
average
time
µ L2 lead
average
lead
time
average
µ L1lead
lead
average
average
time
lead
µLL21 lead
time
average
time
µµLL21 lead
average
time
µ L2 lead
time
µ L2 µ L2tw

105

Capital costs

lQ
(0.0.11)
quantity
quantity
Q
Q
quantity
Q2 Q2 Q2
1DET
quantity
Q
order
quantity
Q
orderorder
quantity
order
Q
quantity
Q
order
quantity
Q12 order
quantity
Q
order
quantity
Q
order
quantity
Q1 order
order
quantity
quantity
quantity
Q12 order
Q
quantity
Q
quantity
12 order
deviation
2order
2 quantity
1 order
1 order
1 order
1DET
12 order
DET
STOCH

deviation
deviationSTOCH
STOCH
fpoint
(reorder
x,
T ) reorder
(0.0.12)
demand
fluctuation
reorder
point
reorder
reorder
point
reorder
R2 point
reorder
point
R1 reorder
point
R
reorder
point
reorder
R1 reorder
point
R
reorder
R
RR2 1point
reorder
point
R
RR
reorder
point
reorder
R1point
reorder
point
reorder
R
point
RR2 1point
reorder
R2 point
R2
2 1point
2
2 1point
1
1
demand
fluctuation
demand fluctuation
cost
measures
cost
measures
cost measures
DET
average
DET DET DET average
DET
DET
x%
=
−inlead
dt
w
lt
1 lead
deviation
deviation
in %DET
in %
lead
time
time
deviation
deviation
deviation
%DET
indeviation
% indeviation
inddeviation
%
%
deviation
inaverage
intime
%DET
in %(0.0.13)

f ( x, T )

R2 x = dlt1 − dtw

STOCH
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH
lead
time
fluctuation
1
−
STOCH
1−STOCH
µ
(0.0.14)
1−STOCH
1−STOCH
1−
STOCH
1time
−STOCH
1−STOCH
1fluctuation
−STOCH
1−STOCH
lead
lead
time
fluctuation
D
140%
140% deviation
deviation
deviation
120%
deviation
deviation
deviation
1−DET
1−DET
1−deviation
DET
1−DET
1−DET
1−deviation
DET
1−deviation
DET
1−DET
1−DET
140%
160%
160%

µD

order
quantity
order
quantity
order
quantity

120%
120%
100%

τ min τ max τ̄

110

2
Clead
HAPTER
6.
D ETERMINISTIC
Vlead
ERSUS
Sfluctuation
TOCHASTIC
UAL
Sfluctuation
OURCING
2lead
time
fluctuation
σ σlead
time
σ σlead
σL2 σL2l1 σL2
twlead
(0.0.10)
time
fluctuation
σlead
lead
time
fluctuation
lead
timelead
fluctuation
lead
time
σ2Llead
fluctuation
time
fluctuation
σlead
time
σlead
fluctuation
lead
time
fluctuation
σLtime
lead
time
fluctuation
σLtime
time
fluctuation
σLlead
fluctuation
time
fluctuation
σLfluctuation
time
σD
σlead
fluctuation
time
fluctuation
σL2fluctuation
Ltime
Ltime
2
1
1
1
1
1
1 L2 L1 L2 L1 L2 L1 L2

CCvalue
OCB OCR OCN

1−STOCH
1−DET

115

deviation

RS
RS
RS
RS
RS
RS
RS2,relax
RS2,relax
RS RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS2,relax
RS2,relax
1,trad
2,trad
1,trad
1,relax
2,trad
1,relax
2,relax
2,trad
1,trad
1,relax
2,trad
2,relax
1,relax
2,relax
1,trad
2,trad
1,relax
1,trad
2,trad
1,trad
1,relax
2,trad
1,trad
2,relax
1,relax
1,relax
100%
100%1,trad 1,trad
R12,relax
−1,relax
R2,trad
∞2,trad
∞
2

�
reorder
point � �
reorder
point
reorder
point
E
SH
·
1
=
[
]
CC
OC
CC
OC
OC
OC ( x − R1 + dtw ) f ( x, lt1 − tw ) f (dtw , tw ) l
Fall
Fall
A
CC OC
OC
OCBRN
CC OC
OC
CC
OCBRNOC
OC
OC
CC
OC
OC
OC
CC OC
OC
CC
OCBRN
OC
CC
OC
OC
OC
OC
BRDET
NOC
ROC
N
BA
BRN
RNOC
dtw RN x N
DETDET
ltB
deviation
in
%
1
deviation
inin%
deviation
%
STOCH
STOCH
STOCH
( x − R1 + dtw ) f ( x, lt1 − tw ) f (dtw , twtw) l1 (0lt1 ) Rdµ
(0.0.15)
]=
1 − dtw
max
min
maxττ̄
min
maxττ̄
maxτττ̄
maxττ̄min
1max
−STOCH
1−1STOCH
τ−ττ̄STOCH
ττ̄max τ̄
τtw τ0τ R1ττ
τ̄ τττ̄min
τ min
τmin
deviation
deviation
deviation
− dtw
1
−
DET
1−1DET
−DET

80%
80%80%
∞ R1�− R2
∞
60% �
B � BR
60%60%
95Fall A 40% min
dmin
lt
xmax
tmax
min
w
40%40%1

100

E [SHFall A · 1

90

value

DET
deviation
in %
ratioSTOCH
in %

120

RS1,trad
RS2,trad RS1,relax RS2,relax
value

204

reorder point R1 reorder point R2

order quantity Q1 order quantity Q2

lead time fluctuation σL1 lead time fluctuation σL2

average lead time µ L1 average lead time µ L2

demand fluctuation σD

2

20%
20%20%

RSRSRS1,trad
RSRSRS2,trad
RS
RS
RS RS1,relax
RS RS2,relax

1,trad
2,relax
1,trad 2,trad
2,trad 1,relax
1,relax
2,relax
mu_D
sigma_LT1
mu_LT2
Q1
Q2
µ µσσ µσL
µσσQ
Rµ
R
σQσQ
µσL
σQ
QµσRµ
Q
RσQ
σ0%0%µ0%σσmu_LT1
µσL
R
µ
µ
σQR
µσL
σQ
R
σ
Q
µRσQRσµ
RQRσQRQRR1QR
σQsigma_LT2
µσRµσQσ
σR
µ
Q

R2 2.0
2D
22L210.8
21L
22L111.4
2 22L0.8
211L1L1.2
L10.2
L
L0.6
L
D L0.0
L10.0
L
D0.4
2L0.6
21.4
1L10.4
11 D
1L1.0
D2L0.2
L22L0.6
D1.0
L112D
1L110.8
112L
1L21.4
11L1.6
1 1.8
2 11.8
22R
D
L2111.0
L22L1.2
L222 1.6
2L10.4
22
1L0.2
1 D0.0
11
12
22
12 L
1.2
1.6
1.812.0 2.0
CC
CC
OC
OC
OC
CC
OC
OC
OC
dimensions
BROC
RNOC
BOC
B deviation
R
N N
standard
deviation
standard
deviation
standard

R2

TSA,CC,.

T

TWH,CC
(0.0.16)
min
max
min
max
t_min
t_max
t_min
t_max
τ
ττ
ττ̄avg_t
τ̄
τ min
τ max
τ̄avg_t
t_min
t_max
avg_t
T
SA,OCN ,.
TWH,OCN min max avg
(0.0.17)
σDσµDσLµ
σ1LLσ11Lµ1σLµ
σ2 2Q
Q212RQ
R112RR
σQ
Q
R221 TR2
121Q
D
L21Lσ
L
2µLL2L
1 Lµ
SA,OC ,.
TWH,OCR
(0.0.18)R

TSA,OCB ,.

TWH,OCB

TSA,CC,.
TSA,OCN ,.

WH

TWH,CC
TWH,OCN
TWH,OCR

SA,OCR ,.
Figure 6.10: Range of capital
cost ratios τCC for the sensitivity analysis and the ware-

house

TSA,β,.

TWH,β

TSA,ρ,.

TWH,ρ

TSA,OCB ,.
(0.0.19)

T

TWH,OCB

SA,β,.
(0.0.20)

TWH,β

TSA,ρ,.
(0.0.21)

TWH,ρ

in the sensitivity analysis where1 the DET scenario is used in a setting which
corre1
sponds to the STOCH scenario. The empiric average deviation, however, is only 0.4%
as τ̄WH,CC = 100.4% shows. In fact, τ̄WH,CC is as low as the minimal value in our sensitivity analysis, see column σL1 . The small deviation of τ̄WH,CC from 100% leads to the
conclusion that the simple DET scenario can be used as a quite good approximation
for calculating the capital costs of most SKUs in our warehouse even though the lead
times are stochastic. This is quite surprising.
Let us have a closer look at the ratios τWH,CC,i for the 2,751 SKUs. Table 6.14 shows
that DET calculates CC for 2,376 SKUs (86.37%) with a maximal deviation of 1% from
the value given by STOCH. Moreover, CC of both scenarios match exactly in all 173
cases where no capital cost occur. The DET model deviates from the STOCH value
for CC only in 202 cases (7.34% of all SKUs) by more than 1%. While all these facts
22

2
speak for using the DET model when it comes to calculating CC for our warehouse
2
2
2 2
2
2
2 2
one can observe the unfortunate trend that τWH,CC increases with increasing values

2

Deterministic vs. stochastic – Case study regarding capital costs CC
range of τCC

# of

% of

CC for DET

SKUs

SKUs

avg

min

max

1 − 10−2

1 − 10−3

1

0.04

1 + 10−4

0

0.00

1 + 10−4

1 + 10−3

138

5.02

0.127

1 + 10−3

1 + 10−2

2,237

81.32

4.949

1 + 10−2

1 + 10−1

202

7.34

66.279

173

6.29

0

2,751

100.00

1 − 10−3

N.D.
sum:
legend:

CC for STOCH
min

20.008

max

N.D. = not defined

avg

20.142

—

CC = capital costs

205

—
8.918 · 10−3
2.114 · 10−2

5.867

0.524

0.127

36.311

4.925

1,567.8

64.510

0

τCC =

CC of DET
CC of STOCH

avg = average

Table 6.14: Deviation of the capital costs between the DET and the STOCH scenario

for CC. This is true for the minimal, the maximal and the average value for the three
groups of SKUs where 1 + 10−4 < τCC ≤ 1 + 10−1 .
Let us have a brief look at a representative of each of the SKU groups listed in Table 6.14 in order to gain a better understanding.
There exists only one SKU where DET slightly underestimates the capital costs of
STOCH. This SKU is rather expensive (632.00e ) which tells us that the small difference of capital costs – 20.008e for DET vs. 20.142e for STOCH – results from a very
small deviation in calculating the average stock level between DET and STOCH. Taking a closer look, the difference can be exclusively explained by the fact that DET uses
the average lead time µ L1 instead of the complete lead time distribution L1 because the
optimal solution for DET abandons the second supply mode and DET and STOCH are
identical in all other aspects. The reason for not using the second supply mode simply

206

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

lies in the fact that it is equally fast as the first one but more expensive.
All 138 SKUs where DET overestimates CC of STOCH by values between 0.01%
and 0.1% have very low absolute capital costs in common. These articles are very
cheap with prices between 0.03e and 2.94e and an average price of 0.48e . The average stock level for these articles lies between 1.49 and 31.05 units and an average of
3.47 units. Obviously, these 138 SKUs with a sum of 17.53e of capital costs play only
a minor role in this warehouse regarding the total capital costs of above 20, 000.00e .
The marginal deviation from the STOCH results as well as the minimal absolute influence of these 138 SKUs on the total capital costs are strong arguments for using the
DET scenario instead of the calculation-intensive STOCH scenario here.
For the majority of SKUs, 81.32% or 2,237 SKUs, the capital costs are overestimated
by DET between 0.1% and 1.0%. The price span increases with a minimum of 0.06e , a
maximum of 684.58e , and an average price of 36.64e over all 2,237 SKUs. The average
stock level is between 0.02 units and 36.31 units with an average of 4.92 units.
The largest deviation of DET from STOCH occurs for 202 SKUs with a price span
between 2.13e and 3, 432.00e and an average of 211.54e . The average stock level per
SKU lies between 0.05 and 285.67 units with an average of 8.44 units over all 202 SKUs.
Last but not least, τWH,CC,i is not defined for 173 SKUs. The reason lies in the fact
these SKUs are very expensive with only very little demand, see Table 6.15. Whenever
price in e

CC in

# of

% of

e

SKUs

SKUs

min

max

avg

0.00

173

6.29

93.81

6,261.00

> 0.00

2,578

93.71

0.03

3,432.00

sum:

2,751

100.00

legend:

CC = capital costs

yearly demand in units
min

max

avg

514.37

1

24

1.51

48.64

1

2,860

16.22

avg = average

Table 6.15: Statistics for the capital costs of the DET and the STOCH scenario

Deterministic vs. stochastic – Case study regarding capital costs CC

207

the demand is very low for an expensive SKU one will try to reduce its average stock
level as much as possible. If the costs associated with a stock out situation are low
compared to the inventory holding costs it is beneficial to order this expensive SKU
after the customer has requested it. In other words, the reorder point is below 0 for
this SKU in the SDMR model. Of course, this leads to a service level of 0% and the
calculated capital costs are identical for DET and STOCH. In these cases, one will
prefer DET to STOCH.
Interestingly, a negative reorder point is the optimal solution for 173 SKUs in our
warehouse. For many companies there exist other reasons like customer satisfaction
or corporate image which make them impose a minimum service level, usually far
above 50%, that has to be met. In our case we do not consider such a restriction.
Another good possibility to reduce the expected capital costs for such expensive and
slow-moving SKUs has been introduced by Schultz, see [Sch89], which delays the
placement of the order for a certain time after the inventory level has dropped to zero.

Key observations. The large deviation of capital costs between DET and STOCH
from the sensitivity analysis do not occur in our warehouse of 2,751 SKUs. The average the ratio τ̄WH,CC between DET and STOCH is very close to 100% in our warehouse.
Here, DET can be used as a good approximation of CCSTOCH . The most expensive
SKUs are not stocked in our warehouse because the capital costs outweigh the costs
for shortage and there exist no other restrictions to store these SKUs. For all these
cases DET and STOCH yield CC = 0 and DET can be used to calculate CC. Nevertheless, there exists a trend that the deviation of between DET and STOCH increase
with increasing capital costs. Eventually, one will have to check whether DET is still
a good approximation for STOCH when the price of a certain SKU is very high and
there exist restrictions that enforce this SKU to be stored.

2

deviation
deviation
deviation
deviation
deviation
deviation
1−DET
1−DET
1−deviation
DET
1−DET
1−DET
1−deviation
DET
1−deviation
DET
1−DET
1−DET
CC OCB OCR OCN

RS1,tradvalue
RS2,trad RS1,relax RS2,relax

1−STOCH
deviation
value 1−DET

140%
140%
120%

order quantity

R −R

120%
120%
order
quantity
order
quantity �∞ 1� 2 �∞
100%
∞ R1�− R2
∞
�
�
2,trad
1,trad
1,relax
2,trad
2,relax
1,relax
2,trad
1,relax
2,relax
2,trad
1,trad
1,relax
2,trad
2,relax
1,relax
2,relax
1,trad
2,trad
1,relax
2,relax
1,trad
2,trad
1,trad
1,relax
2,trad
1,trad
1,relax
2,trad
2,relax
1,relax
100%
tw 2,relax 1
1 2,relax
Fall
A1,trad
Fall
A2,relax
100%1,trad 1,trad
dtw
90
lt1
x
80%
reorder
point
reorder
point
t
w
t
w
1
1
1
1
Fall A 80%80%
w
w
tw
0
R1 − d t w
dtw
lt1
x
RN
N
R
NDETDET
60% t
BRN BRN BRN BRN BRDET
BRN BRN
85
0 B R1 −dB
w
tw
deviation
in
%
deviationSTOCH
in
%
60%60%
STOCH
STOCH
40% min min
max−STOCH
min
max
max min
max min
max min
max min
max min
max min
1max
−STOCH
80
40%40%
deviation 1−1STOCH

E [SHFall A · 1

75

RS
RS
RS
RS
RSRS
RS
RS RS RS
RSRS
RS
RS
RS
RS
RSRS
RS
RS
RS RS
RS
RS
RS
RS
RS
RSRS
RS RS
ERS
· 1RS
(RS
x − RS
R + dRS) f ( x, lt − tw ) f (dtw , tw ) l1
[SH
]=
reorder point

]=

τ min τ max τ̄

95

value

DET
deviation
in %
ratioSTOCH
in %

reorder point R1 reorder point R2

order quantity Q1 order quantity Q2

lead time fluctuation σL1 lead time fluctuation σL2

average lead time µ L1 average lead time µ L2

demand fluctuation σD

{(lt1 , lt2 , t g , tc , dlt1 , dlt2 , dtw )} = Ω =: Ω R1 ,R2 ,tw ,Q1 ,Q2
{(lt1 , lt2 , t g , tc , dlt1 , dlt2 , dtw )} = Ω =: Ω R1 ,R2 ,tw ,Q1 ,Q2
(0.0.1)
lti
lti
(0.0.2)
dT
dT
(0.0.3)
tg
208
C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING
tg
(0.0.4)
tc
t
(0.0.5)
c
6.3.3 Normal order costs
i
i
(0.0.6)
2
2
2 2
2
2 2
2 2
T
In this section we compare the normal order costs OCN between the DET scenario
T
(0.0.7)
Ri
DET
and the STOCH scenario.
In DET
Figure
weDET
seeDET
that
the ratios
are very close to 100%
DET
DET
DET6.11
DET DET
deviation
deviation
deviation
deviation
deviation
deviation
deviation
deviation
deviation
Ri STOCH
STOCH
STOCH (0.0.8)
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH
Qi
for all SKUs in our warehouse, see the column ”WH”. More precisely, the values
demand
demand
σD σD(0.0.9)
σD
demand
fluctuation
σDfluctuation
demand
fluctuation
demand
σDfluctuation
σDfluctuation
demand
demand
fluctuation
demand
σDfluctuation
demand
σDfluctuation
Qσi Dfluctuation
min
max
tw
for range between τWH,OC
= 99.61% and τWH,OC
= 100.00% with an average of
N
average
lead
average
time
lead
µLL21 lead
average
time
average
time
µ L2 time
(0.0.10)
average
lead
time
µ L1 lead
average
time
µ
average
lead
time
average
µ L1 average
lead
time
µ L1lead
time
average
µ
time
µµLL21 lead
average
time
µtLNw1 lead
average
time
µ L2 lead
average
lead
time
average
µ L1lead
lead
average
average
time
lead
µLL21 lead
time
average
time
µµLL21 lead
average
time
µ L2 lead
time
µ L2 µ L2
l1
τ̄WH,OCN = 99.99%. This is a totally different
picture from what we have observed in
2
2
2
lead
time
lead
fluctuation
time
fluctuation
σ
lead
time
σ
lead
fluctuation
time
fluctuation
σ
l
(0.0.11)
time
fluctuation
σ
lead
time
fluctuation
σ
lead lead
timelead
fluctuation
lead
time
σ
fluctuation
lead
time
fluctuation
σ
lead
time
σ
fluctuation
σ
lead
time
fluctuation
σ
lead
time
fluctuation
σ
time fluctuation
σLlead
fluctuation
lead
fluctuation
σL1 Llead
time
σ 1σLlead
fluctuation
timeLfluctuation
σL2 L2 σL2 σ
1time
Ltime
Llead
Ltime
Lfluctuation
2 L1 L2 Lmin
2
2 L1 L2
1
1
1
1
1
f ( x, T )
our sensitivity analysis, see columns 1-9, where the minimal value is τSA,OC
=
77.70%
N
f
(
x,
T
)
(0.0.12)
DETorder
order
quantity
order
quantity
Q
order
Q
quantity
order
quantity
Q
Q
quantity
Q
order
quantity
Q
orderorder
quantity
order
Q
quantity
Q
order
quantity
Q
order
quantity
Q
order
quantity
Q
order
quantity
order
Q
order
quantity
order
quantity
Q
quantity
Q
Q
quantity
order
Q
quantity
Q
2
2
1
1
2 DET
2
2
2
2
1
1
1
1DET
1 deviation
12
12
STOCH
deviation
x = dlt1 − dtw
and the lowest average value is τ̄SA,OCdeviation
= 90.14%.
STOCH
STOCH
N ,Q2
xdemand
=point
dRlt1point
−
dtwpoint
(0.0.13)
fluctuation
reorder
point
reorder
point
reorder
R2 point
reorder
point
R1 reorder
Rreorder
reorder
point
reorder
R1 reorder
point
ROrder
reorder
RCosts
RR2 1point
reorder
point
reorder
RR
reorder
point
reorder
RNormal
reorder
reorder
R
point
RR2 1point
reorder
R2 point
R2 R2
2 1 reorder
2fluctuation
2 1point
1point
1fluctuation
1point
demand
demand
µD
cost
measures
cost
measures
cost
measures
DET
DET (0.0.14)
average
lead
time
DET DET DET average
DET
DET
µ
DET
DET
D
105
deviation
in STOCH
% in % in %
lead
time
time
deviation
deviation
deviation
indeviation
%
indeviation
%
%lead
indeviation
%
deviation
inaverage
% indeviation
in %
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH
160%
140%
160%
lead
time
fluctuation
100
1−STOCH
1−STOCH
1−STOCH
1−STOCH
1−
STOCH
1time
−STOCH
1−STOCH
1fluctuation
−STOCH
1−STOCH
lead
lead
time
fluctuation
( x − R + d ) f ( x, lt − t ) f (d , t ) l (lt ) dµ (0.0.15)

CC
OC
OC
CC
OCinOC
CC OC
OC
OC OC
CC OC
OC
CC
OC OC
OC
OC
CC
OC
OC OC
CC OC
OC
CC
OC OC
OC
CC
OCOC
OC
OC
deviation
% OC
τ

20%
20%20%

ττ

τττ̄

τττ̄

τττ̄

τττ̄deviation
τ̄ 1−DET
ττ̄
τττ̄ deviation
ττ̄ 1−ττDET
1−DET

τ̄

TSA,CC,.
RSRSRS1,trad
RSRSRS2,trad
RS
RS
RS RS1,relax
RS RS2,relax

TWH,CC

1,trad
1,relax
2,relax
1,trad 2,trad
1,relax
2,relax(0.0.16)
mu_D
sigma_LT1
sigma_LT2
Q12,trad
Q2
R1
R2
WH
0%0% 0% mu_LT1
TSA,CC,. mu_LT2TWH,CC
2D
22L210.8
21L
22L111.4
2 22L0.8
211L1L1.2
L10.2
L
L0.6
L
D L0.0
L10.0
L
D0.4
2L0.6
21.4
1L10.4
11 D
1L1.0
D2L0.2
L22L0.6
D1.0
L112D
1L110.8
112L
1L21.4
11L1.6
1 1.8
2 11.8
221.812.0 2.0
2 2.0
D
L2111.0
L22L1.2
L222 1.6
2L10.4
22
1L0.2
1 D0.0
11
12
22
12 L
1.2
1.6
dimensions
TSA,OC
TWH,OC
,.
CC OC OC OC

µσσQ
Rµ
σQσQ
µσL
σQ
QµσRµ
Q
RσQ
σ µ σσ µ µσσ µσL
µσL
Rµ
µ
σQR
µσL
σQ
R
σ
Qµ
µRσQRσµ
σQR
µσRµσQσ
σR
QRQRσQRQR QR R R

CC
OC
OC
OC
CC
OC
OC
OC
N
N
BR R
RNN N
B deviation
B deviation
standard
standard
deviation
standard
TWH,OC
(0.0.17)
N
min
max
min
max
max
min
avgτ̄
t_min
t_max
t_min
t_max
TSA,OCR ,.
TWH,OCR
τ
ττ
ττ̄avg_t
τ min
τmax
τ̄avg_t
t_min
t_max
avg_t
TSA,OCR ,.
TWH,OCR
(0.0.18)
σDσµDσLµ
σ1LLσ11Lµ1σLµ
σ2 2Q
Q212RQ
R112RR
σQ
Q
R
R2 ,.
121Q
D
L21Lσ
L
T221SA,OC
TWH,OCB
2µLL2L
1 Lµ
B
TSA,OCB ,.
TWH,OCB
(0.0.19)
Figure 6.11: Range of order cost ratios τOCN for the sensitivity analysis
wareTSA,β,.and Tthe
WH,β
TSA,β,.
TWH,β
(0.0.20)
house
TSA,ρ,.
TWH,ρ
TSA,ρ,.
TWH,ρ
(0.0.21)
1
1

TSA,OCN ,.

The explanation for 1,954 SKUs, or 71.03%, in the warehouse is simple. In these
cases the optimization returns a solution that does not use the second supply channel
so the probability, p(2, ., .), to trigger a second order is 0. Consequently, the total demand has to be covered exclusively via the normal orders independent of stochastic
lead times and possible stock out situations. Remember, all backlogged demand still
has to be delivered to the warehouse via the normal orders but additional costs occur

Deterministic vs. stochastic – Case study regarding ordering costs OCN

209

due to a privileged process between the warehouse and the waiting customer. Consequently, OCN has to be identical between DET and STOCH for these 1,954 SKUs and
τWH,OCN ,i = 100.00%.
One might ask why so many articles are not using the second supply mode. Here,
we have to consider that the warehouse is from the spare parts and service industry
which usually faces low demands. The implications for the inventory management
caused by spare parts might reach quite far and can be nicely shown in the following
example.

Example 6.3.3. The minimum number of normal orders per year is as low as 0.14 in
our warehouse. In other words, there exists a SKU that is only ordered every seven
years in average. Why is a SKU ordered only every seven years in the optimal case in
reality?
If we exclude the option of a mistake, then the reason must be that the sum of
(fixed) ordering costs is much higher than the yearly holding costs. In our case the
fixed ordering costs are 4.40e per order. Following our argumentation, the yearly
holding cost must be much smaller than 4.40e . Indeed, the demand is stochastic with
a mean of 1 unit per year and the price is 0.49e . Consequently, the annual holding
costs, 12% of the inventory value, are below 4.40e if we apply the optimal order quantity of Q1 = 7 units. The result including the expected replenishment cycle of seven
years makes sense from a computational point of view.
However, for economic reasons companies usually employ additional constraints
for such extreme cases. For example, some companies restrict the maximal order
quantity to the total demand of one year. The reason is that people are usually reluctant to make predictions that reach too far into the future.

Now, let us have a look at the remaining 797 SKUs where p(2, ., .) > 0 is observed.

210

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

Here, we can make the observation that
τOCN =

OCN,DET
Q + Q2 · pSTOCH (2, ., .)
= 1
OCN,STOCH
Q1 + Q2 · pDET (2, ., .)

(6.3.7)

holds for all 797 SKUs. Obviously, the ratio in Equation (6.3.7) is completely independent from all costs factors. Moreover, the nominator and the denominator coincide
with the expression for the expected number of ordered units per cycle E[ Q] which
has been introduced in Chapter 3 and which is part of the summarizing Table 3.12 on
page 126.
Let us explain how Equation (6.3.7) can be derived and why its parameters make
sense. From the sensitivity analysis we know that the ratio between ordering costs is
independent from the actual cost. According to the Equation (6.1.1) mentioned during
our initial observations on page 166 it holds that
OCN,DET
E
[# first orders]
= DET
= τOCN .
OCN,STOCH
ESTOCH [# first orders]

(6.3.8)

We recall how the expected number of normal orders, E[# first orders], is calculated in
Chapter 3.5 starting on page 116 and we yield
E[ Q] = Q1 + Q2 p(2, ., .)
FC
E[# cycles] =
E[ Q]
E[# first orders] = E[# cycles] =
τOCN =

FC
Q1 + Q2 · p(2, ., .)

FC
Q1 + Q2 · pDET (2,.,.)
FC
Q1 + Q2 · pSTOCH (2,.,.)

=

Q1 + Q2 · pSTOCH (2, ., .)
.
Q1 + Q2 · pDET (2, ., .)

(6.3.9)

In Equation (6.3.9) the forecast FC is identical for DET and STOCH and cancels out
while the probability of a second order is pDET (2, ., .) and pSTOCH (2, ., .), respectively.
Thus, we yield for τOCN exactly the Equation (6.3.7). This proofs that the ratio of costs
for normal orders, τOCN , between DET and STOCH is equal to the reciprocal ratio
of their expected order quantity EDET [ Q] and ESTOCH [ Q], respectively, given that the

Deterministic vs. stochastic – Case study regarding ordering costs OCR

211

costs are calculated as in our warehouse. Consequently, the little deviation of τOCN
from 100% in our warehouse can be explained by large values for Q1 , small values for
Q2 , minor difference of p(2, ., .) between DET and STOCH, or a combination of these
effects.
Key observations. All values of τOCN are very close to 100% and DET is a very good
approximation of OCN,STOCH in our warehouse. This observation stands in great contrast to the findings of our sensitivity analysis.
The costs of normal orders, OCN , for the 71.03%-majority of the SKUs in our warehouse are identical between DET and STOCH because the second supply channel is
not used. This is rather common for a spare parts warehouse like ours. Therefore,
all units have to be ordered via the only used supply channel and the order costs are
identical for DET and STOCH. For these SKUs, DET is absolutely sufficient to calculate
OCN . The cost structure of our warehouse allows to rewrite τOCN as
τOCN =

Q1 + Q2 · pSTOCH (2, ., .)
Q1 + Q2 · pDET (2, ., .)

which shows that the reason for τOCN ≈ 100% can be a high value for Q1 , a small value

for Q2 , a small deviation between pSTOCH (2, ., .) and pDET (2, ., .), or a combination of
these effects.

6.3.4

Rush order costs

In this section we investigate the rush order costs OCR and their ratios between DET
and STOCH in the sensitivity analysis and our warehouse. In the warehouse, OCR = 0
for 1,954 SKUs (71.03%) of all 2,751 SKUs because the optimization algorithm returns
a solution where only the normal supply mode is used, as mentioned before. For
these 1,954 SKUs both scenarios, DET and STOCH, yield identical values, namely
OCR,DET = OCR,STOCH = 0. The division by zero is not defined, so we cannot calculate

2

2

2

2

2

i

(0.0.6)

T

(0.0.7)

Ri

(0.0.8)

i
T
Ri
Qi

DET

DET

DET DET DET DET
DET DET
DET
deviation
deviation
deviation
deviation
deviation
deviation
deviation
deviation
deviation
STOCH
STOCH
tw STOCH
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH

tw

(0.0.10)

l1

2

demand
demand
σ
τOCR . However,
we
have
chosen
τ σas
a σmeasure
toDfluctuation
indicate
theσrelative
difference
beD σD (0.0.11)
D
demand
fluctuation
demand
fluctuation
demand
σ
demand
σD
demand
fluctuation
demand
σDfluctuation
demand
σDfluctuation
Dfluctuation
Dfluctuation
l fluctuation
1

140

deviation
2 DET
2 (0.0.14)
2
2
2
1 Rush1 Order
1
1DET
12
Costs
STOCH
µ12
deviation
deviation
STOCH
STOCHD
fluctuation
reorder
point
reorder
point
reorder
R2 point
reorder
point
R1 reorder
Rreorder
reorder
point
reorder
R1 reorder
point
Rdemand
reorder
Rreorder
RR2 1point
reorder
point
R
reorder
RR
reorder
point
reorder
R1point
reorder
point
R
reorder
point
RR2 1point
reorder
R2 point
R2 R2
2 1point
2fluctuation
2 1point
1point
1point
demand
demand
fluctuation
1

cost
measures
cost
measures
cost
measures

DET
average
time
DET DET DET average
DET
DETlead
DET
DET%DET in %
deviation
deviation
time
time
R1�
− Rin
deviation
deviation
deviation
indeviation
%
indeviation
%
indeviation
%lead
inlead
%
deviation
inaverage
%
%
2 STOCH
�∞in
�∞in %
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH

CCvalue
OCB OCR OCN

160

1−STOCH
1−DET

DET
deviation
in %
ratio inSTOCH
%

180

RS1,trad
RS2,trad RS1,relax RS2,relax
value

f ( x, T )
tween the results
of
DET
and
STOCH.
are
identical
case.
average
lead
average
time
lead
µLL21in
average
time
lead
average
time
µ L2 time
average
lead
time
µThese
average
time
µ
average
lead
time
average
µ L1 average
lead
time
lead
µ L1lead
time
µ
time
µµ
average
time
average
lead
time
µ LTherefore,
average
lead
time
average
µ L1costs
lead
average
average
time
lead
time
average
time
µµthis
average
time
lead
µ L2 lead
time
µ L2 µ L2
LL(0.0.12)
L1 lead
LL21 lead
21
2
faverage
(µx,L1Tlead
)µLL21 lead
2
x = dfluctuation
− dtw
we define τOClead
to be
100%
whenever
both
scenarios,
DET
and
STOCH,
2lead
time
fluctuation
σ σlead
time
σ σlead
fluctuation
lead
time
fluctuation
σlead
lead
time
fluctuation
time
fluctuation
lead
time
σ2Llead
fluctuation
time
fluctuation
σlead
lead
time
σlead
fluctuation
lead
time
fluctuation
σ time
lead
time
fluctuation
σrush
lead
time
fluctuation
σLlead
fluctuation
lead
time
fluctuation
fluctuation
σLfluctuation
time
σ σcalculate
lead
fluctuation
time
fluctuation
σL2 lt1 σL2 σ
R
Ltime
Ltime
Ltime
L
L2 σ
2
2 L1 L2
1x =
1
1 dt L1
1d
1 L2 L1 L2 L1 (0.0.13)
−
w
lt1
µD
ordering costsorder
of 0.00e
.quantity
With
this
definition
of
τOC
and
majority
of
quantity
order
quantity
Q
order
Q
quantity
quantity
Q271.03%
RDET
order
Q
order
quantity
Q
quantity
order
Q
quantity
Q
order
quantity
Q aorder
order
quantity
Q
order
quantity
Q
order
quantity
order
Q order
order
quantity
order
quantity
Q
quantity
order
Q1with
Q
quantity
Q
quantity
Q Q2
1 order

reorder point R1 reorder point R2

order quantity Q1 order quantity Q2

lead time fluctuation σL1 lead time fluctuation σL2

2

2

(0.0.5)

(0.0.9)
C HAPTER 6. D ETERMINISTIC V ERSUS QSiTOCHASTIC D UAL
S OURCING

212

140%
160%
160%

�∞ R1�− R2

�∞

E [SH

·lead
1 time
] =fluctuation

( x − R + d ) f ( x, lt1 − tw ) f (dtw , tw )

60

140%
140%
120%

τ min τ max τ̄

80

value

1−STOCH
1−STOCH 1
1−STOCH
1−STOCH
1−
STOCH
1time
−Fall
STOCH
tw
1−STOCH
1Afluctuation
−STOCH
1−
STOCH
lead
Fall
A
lead
time
fluctuation
dtw
lt1
x
deviation
deviation
deviation
deviation
deviation
deviation
deviation
deviation
1(−
DET
1(−
DET
1−
DET
1−(DET
1
−
DET
1
−
DET
1−ddeviation
DET
1
−
DET
1
−
DET
E [SHFall A · 1Fall A ] =
x
−
R
+
)
f
(
x,
lt
−
t
)
f
d
,
t
)
l
lt
)
dµ
(0.0.15)
t
w
t
w
1
1
1
1
w
tw w 0
R1 − d t w
dtw
order
quantity
lt1
x
120%
120%
order
quantity
order
quantity
100
100%
tw 1,trad
0RS1,trad
R
−
d
RS
RS
RS
RS
RS
RS
RS2,relax
RS2,relax
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS2,relax
RS2,relax
t
1
w
1,trad
2,trad
1,trad
1,relax
2,trad
1,relax
2,trad
1,relax
2,relax
2,trad
1,trad
1,relax
2,trad
2,relax
1,relax
2,relax
1,trad
2,trad
1,relax
2,relax
1,trad
2,trad
1,trad
1,relax
2,trad
1,trad
2,relax
1,relax
2,trad
1,relax
100%
100%
120

deviation

average lead time µ L1 average lead time µ L2

demand fluctuation σD

2

2

tc

80%
80%
80%
60%
60%
60%
40% min
40%
40%

reorder
point
reorder
point
reorder
point

CC
OC
CC
OC
OCBin
CCB OC
OC
OCBRN
CC OC
OC
CC
OCBRNOC
OC
OC
CC
OC
OC
OCBRDET
CCBROC
OC
CC
OCBRN
OC
CC
OC
OC
OC
OC
NOC
N
BROC
BRN
RNOC
DETDET
deviation
% RNOCN
deviation
inin%%
deviationSTOCH
STOCH
STOCH

TSA,CC,.
TWH,CC
(0.0.16)
20%20%
TSA,OCN ,.
TWH,OCN
20%
0
RS
RS
RS1,relax
RS2,relax(0.0.17)
RSRS
RSRS
RS
RS
RS
RSR1
1,trad
2,trad
1,trad
2,trad
1,relax
2,relax
1,trad
2,trad
1,relax
2,relax
T
T
mu_D
sigma_LT1
mu_LT2
sigma_LT2
Q1
Q2
R2
WH
0% 0%0% mu_LT1
WH,OC
SA,OC
,.
N
σ2Lµ
µσ12LD
R
R
σQ
µ
σ0.6
Q
Q
R
RσQ
σD µ Lσ0.0
σLµ10.0
σL
Q
µσL
R11Lµ
R
R
σ10.2
σ0.4
σL
σ
µ
R
σ
R
σ
Q
Q
µ
Rσ111.4
R1.822R
σ1Q
σ1121.2
QR
σ11LQ
R
Q
R2 2.0
2D
2Q
2σ
22µ
22L
2σL
2σ2Q
2σ
Lµ
L
L
2L
2µ
2σ
21.4
22R
111.0
1Lµ
1Q
1NLµ
D
Lσ2Q
L0.8
L
D
Lµ
L1.2
L
1Lµ
11Q
11L1L
1µ
1Q
2R11.8
Lµ
D
LR
L2Q
2L10.4
2σ
2L
2L
22
21.4
1L0.2
1D
10.8
1 D0.0
10.4
1D
1L
22
12 L
0.6
1.0
1.6
0.2
0.6
0.8
1.0
1.6
1.2
1.6
1.8,.12.0 2.0
T
T
WH,OC
SA,OC
dimensions
R
R
CC
CC
OC
OC
OC
CC
OC
OC
OC
BROC
RNOC
BOC
B deviation
R
N N
standard
deviation
standard
deviation
standard
TSA,OCR ,.
TWH,OC
(0.0.18)
R
min
max
max
min
max
t_min
t_max
t_min
t_max
avg_t
τ min
ττ
ττ̄avg_t
τ̄
τmin
τ
τ̄avg
TSA,OCB ,.
TWH,OCB
t_min
t_max
avg_t
max
TSA,OCB ,.
TWH,OCB
(0.0.19)
σDσµDσLµ
σ1LLσ11Lµ1σLµ
σ2 2Q
Q212RQ
R112RR
σQ
Q
R221T R2
121Q
D
L21Lσ
L
2µLL2L
1 Lµ
TWH,β
SA,β,.
T
T
(0.0.20)
SA,β,.ratiosWH,β
Figure 6.12: Range of order cost
τOCR for the sensitivity analysis and the wareTSA,ρ,.
TWH,ρ
T
T
(0.0.21)
WH,ρ
SA,ρ,.
house
1
1
40
20

τ

max
min
max ττ̄min
maxττ̄
min
maxττ̄
maxτττ̄
maxττ̄min
1max
−STOCH
1−1STOCH
τ−ττ̄STOCH
ττ̄max
ττmin
τ maxτττ̄min
τ min
τmin

TSA,CC,.

deviation
TWH,CC
deviation
deviation
1−DET
1−1DET
−DET

τ̄

of all SKUs not using the second supply mode, it does not come as a surprise that
τ̄WH,OCR = 100.46% is so close to 100%.
After excluding all 1,954 SKUs which do not use the second order channel we
0
0 min
0 max
obtain the ratios τ̄WH,OC
= 101.84%, τWH,OC
= 0.00%, and τWH,OC
= 107.78%.
R
R
R
0
While τ̄WH,OC
testifies a small deviation between DET and STOCH even for the 797
R
min
two-order cases in our warehouse the value of τWH,OC
= 0.00% is very interesting.
R

For 2 SKUs the DET scenario calculates a two-order probability of pDET (2, ., .) =
0 by assuming deterministic lead time values, µ L1 and µ L2 , while STOCH yields
pSTOCH (2, ., .) > 0 when considering the lead time distributions L1 and L2 . Conse-

22

2

2

2

2

2

2

2

2

2

2

Deterministic vs. stochastic – Case study regarding ordering costs OCR

213

quently, OCR = 0 for DET while OCR for STOCH are 1.0 · 10−6 e and 4.0 · 10−6 e ,

respectively. Coincidentally, these values are very low and favor the usage of DET
instead of STOCH. However, the low values for STOCH are simply a matter of the
lead time distributions and the given costs – they can be arbitrarily high if the price
for the SKU or the costs for rush order increases significantly, for example. It is nice to
min
see from Figure 6.12 that the minimal ratio τWH,OC
= 0 matches with the results and
R

explanation of our sensitivity analysis regarding the average lead time µ L1 in Chapter 6.2.2.1 starting on page 175.
For all remaining 795 SKUs we see that DET overestimates OCR compared to
STOCH. Further, we know that
τOCR =

OCR,DET
E
[# second orders]
= DET
OCR,STOCH
ESTOCH [# second orders]

(6.3.10)

must hold in analogy to the observations for the normal order costs in Chapter 6.3.3
and Equation (6.1.1) on page 166. If we replace the expressions in Equation (6.3.10)
with the appropriate formulas summarized in Table 3.12 on page 126 we yield
EDET [# second orders]
ESTOCH [# second orders]
EDET [# first orders] · pDET (2, ., .)
=
ESTOCH [# first orders] · pSTOCH (2, ., .)
( Q1 + Q2 · pSTOCH (2, ., .)) · pDET (2, ., .)
=
( Q1 + Q2 · pDET (2, ., .)) · pSTOCH (2, ., .)
ESTOCH [ Q] · pDET (2, ., .)
=
EDET [ Q] · pSTOCH (2, ., .)

τOCR =

(6.3.11)

which is equivalent to the ratio of the expected order quantities between STOCH and
DET weighted with the reciprocal ratio of their probability for a 2-order cycle. Again,
the cost factors do not play a role for the ratio τOCR but it is mainly influenced by
the deviation of p(2, ., .) between DET and STOCH. Equation (6.3.11) mathematically
confirms our previous observations when pSTOCH (2, ., .) > 0 and pDET (2, ., .) = 0.

214

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

Table 6.16 lists the highest values for τWH,OCR ,i . We see from the first row that the
relative measure τOCR can be large in cases where the absolute costs for STOCH and
DET are negligible for most companies. In our spare parts warehouse this can be
rank

τOCR in %

OCR,DET

OCR,STOCH

pDET (2, ., .)

pSTOCH (2, ., .)

in e

in e

in %

in %

1.80 · 10−4

1.67 · 10−4

3.70 · 10−3

3.40 · 10−3

Q1

Q2

2

2

5.02

31

4

1

107.78

2

104.85

3

104.65

5.45

5.20

3.73

3.56

16

13

10

103.94

0.25

0.24

0.88

0.85

1

1

100

102.57

12.11

11.80

14.02

13.65

9

4

4.68

4.46

5.26

Table 6.16: Selection of SKUs with highest deviation for OCR between DET and
STOCH
explained with a low probability to trigger a second order, p(2, ., .). Moreover, DET
and STOCH calculate similar probabilities for p(2, ., .) due to the rather short lead
times of mostly 9 days in average and a standard deviation of 2 days. Therefore, we
have to constitute that the DET scenario is probably sufficient for approximating the
rush order costs for the 2,751 SKUs for most companies.
Key observations. In general, DET overestimates OCR for many SKUs in our warehouse when the second supply channel is used. However, the overestimation is not
as strong as we have observed in the sensitivity analysis and the average value of
τWH,OCR is close to 100%. DET could still be used as an acceptable approximation of
OCR in most cases but not as unconfined as for CC and OCN .
For 71.03% of all SKUs rush orders do not pay off because demand of these spare
parts is too small. Here DET and STOCH calculate identical order costs OCR . Neither
DET nor STOCH is needed to calculate this trivial result for OCR . We can give an exact

Deterministic vs. stochastic – Case study regarding ordering costs OCB

215

expression for τOCR due to the cost structure of our warehouse again
τOCR =

( Q1 + Q2 · pSTOCH (2, ., .)) · pDET (2, ., .)
( Q1 + Q2 · pDET (2, ., .)) · pSTOCH (2, ., .)

which shows that a deviation in p(2, ., .) between DET and STOCH influences τOCR
much more than this has been the case for τOCN , for example.
There exist two SKUs where τOCR = 0 because DET yields pDET (2, ., .) = 0 while
STOCH calculates a small positive value for pSTOCH (2, ., .) under consideration of the
stochastic lead times. This result coincides with the observation we have made during the sensitivity analysis. All remaining SKUs are overestimated by DET in terms
of OCR up to 7.78%. However, the absolute difference between the rush order costs
are rather small in our warehouse. Of course, this depends on the individual cost
structure of the SKU and might lead to problems at other warehouses.

6.3.5

Back order costs

In this section we compare the ratio τOCB of back order costs OCB between the sensitivity analysis and our warehouse. In case of the back order costs the ratio τ is much
less affected by the situation whether a SKU is partially replenished via the rush orders or not. Even if only normal orders are used to satisfy the customer demand, τOCB
can still be high because DET neglects the variability of the lead times which might
substantially influence the amount of occurring shortage. If we apply Equation (5.2.5)
from page 157 the value of τOCB can be determined by
τOCB =

EDET [SH ] · Q B
E
[SH ]
= DET
.
Q B · ESTOCH [SH ]
ESTOCH [SH ]

(6.3.12)

We can rewrite Equation (6.3.12) by means of the β service level for DET and STOCH
and the forecast FC.

(1 − β DET ) · FC
1 − β DET
EDET [SH ]
=
=
ESTOCH [SH ]
(1 − β STOCH ) · FC
1 − β STOCH

(6.3.13)

lt1

lt2

(0.0.1)

lti

(0.0.2)

dT

(0.0.3)

tg

(0.0.4)

lti
dT
tg

(0.0.6)

i

i
T

cause they both rely on the expected shortage, E[SH ], where the expression

Ri

2

2

2

T

(0.0.7)

Ri

(0.0.8)

8

(0.0.9) (6.3.14)

2 ∑ 2E[SH
2 CaseQi ]i
2 E[SH
2 ]=
i =1

(0.0.10)

tw

D

D

D

D
f ( x, TD)

D

D

(0.0.12)

Qi
tw
l1
f ( x, T )

2

DET DET DET
DET DET
DET
DET
DET the
DET
is the weighted
sum
of
non-trivial
terms
for
shortages
in each of our 8 cases,
deviation
deviation
deviation
deviation
deviation
deviation
deviation
deviation
deviation
STOCH
STOCH
l1 STOCH
(0.0.11)
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH
E[SH Case i ],demand
see Chapter
3fluctuation
in Table
3.9σon
page
demand
fluctuation
demand
σD σ σD
demand
fluctuation
demand
σ fluctuation
σ99.
demand
fluctuation
σ fluctuation
demand
fluctuation
demand
σ fluctuation
demand
σ fluctuation

x=d

− dtw

120

RS1,trad
RS2,trad RS1,relax RS2,relax
value

average
lead
average
time
lead
µLL21 lead
average
time
lead
average
time
µ L2 time
average
lead
time
µ L1 lead
average
time
µ
average
lead
time
average
µ L1 average
lead
time
µdL1lead
time
average
µ
time
µµLL(0.0.13)
average
time
µ L1 lead
average
time
µ L2 lead
average
lead
time
average
µ Lx1lead
lead
average
average
time
lead
µLL lead
time
average
time
µµLL21 lead
average
time
µ L2 lead
time
µlt1L2 µ L2
21
=
w 21
lt1 − dt
2
µ
Figure 6.13 shows that the deviations
between
DET
and
STOCH
regarding
OCBtime
are fluctuation
2lead
time
fluctuation
σ σlead
time
σ σlead
fluctuation
time
fluctuation
σlead
lead
time
fluctuation
lead lead
timelead
fluctuation
lead
time
σ2Llead
fluctuation
time
fluctuation
σlead
lead
time
σlead
fluctuation
lead
time
fluctuation
σ time
lead
time
fluctuation
σLfluctuation
time
fluctuation
σLlead
fluctuation
lead
time
fluctuation
fluctuation
σLfluctuation
time
σ σLlead
fluctuation
time
σL2 σL2 σDL2 σL
Ltime
Ltime
2
2 L1 L2
1
1
1 µ L1
1
1 L2 L1 L2 L1 (0.0.14)
min D
max
quite widespread. The values range between τWH,OC
=order
9.30% and τWH,OC
=
DETorder
B
order
quantity
order
quantity
Q
Q
quantity
order
quantity
Q
quantity
Q
order
quantity
Q
orderorder
quantity
order
Q
quantity
Q
order
quantity
Q
quantity
Q
quantity
Q
order
quantity
Q
quantity
order
quantity
Q
quantity
Q
Q
quantity
order
Q
quantity
2 QB2 Q2
Back
Order
1
1
deviation
2
2 order
2
2
2
2
1 order
1 order
1Costs
1DET
1 order
1
1
DET
STOCH

reorder point R1 reorder point R2

order quantity Q1 order quantity Q2

tc

Both equations (6.3.12) and (6.3.13) are equivalent and cannot easily be reduced be-

2
lead time fluctuation σL1 lead time fluctuation σL2

R1 ,R2 ,tw ,Q1 ,Q2

w

(0.0.5)
C HAPTER 6. D ETERMINISTIC V ERSUS tSc TOCHASTIC D UAL
S OURCING

216

deviation
deviationSTOCH
STOCH

fluctuation
Rreorder
reorder
point
point
reorder
R2 point
2R
reorder
point
R1 reorder
Rreorder
reorder
point
reorder
R1 reorder
point
Rdemand
reorder
Rreorder
R
reorder
point
R
reorder
RR
reorder
point
reorder
R1point
reorder
point
R
point
RRR21�−
R2 point
R2 R2
1point
�∞reorder
�∞reorder
2 1point
2fluctuation
2 1point
2 1point
1point
1point
demand
demand
fluctuation
�∞ R1�− R2

�∞

cost
measures
cost
measures
cost
E [SH
· measures
1
]=

( x − R + dt ) f ( x, lt1 − tw ) f (dtw , tw )

40

order
quantity
order
quantity
order
quantity

120%
120%
100%

τ min τ max τ̄

deviation

60

value

1−STOCH
1−DET

100

CCvalue
OCB OCR OCN

DET
A average
Fall ADET
lead
w
DET DET DET Fall
DET
DET
din
lttime
x
t
deviation
deviation
%DET
in %1
average
lead
time
deviation
deviation
deviation
in
%
in
%
%lead
%
deviation
deviation
in
%DET
in
in
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH
E [SHFall A · 1Fall A ]140%
=160%
(
x STOCH
−
Rdeviation
+
ddeviation
)
f average
(
x,
ltin
−
tin
)
ftime
(
d1tw%
,0tww) RlSTOCH
(
ltdt1 )%dµ (0.0.15)
t
w
1
1
1
w
160%
t
−
w
d
1
lt1
x
w
tw
time
fluctuation
1−STOCH
1−STOCH
1−STOCH
STOCH
1−
STOCH
1lead
−STOCH
1−STOCH
1fluctuation
−STOCH
1−STOCH
lead
time
lead
time
fluctuation
tw
0deviation
140%
R1deviation
−1d−
140%
80
deviation
deviation
tw deviation
120%
deviation
deviation
deviation
deviation
1−DET
1−DET
1−DET
1−DET
1−DET
1−DET
1−DET
1−DET
1−DET
DET
deviation
in %
STOCH
ratio in
%

average lead time µ L1 average lead time µ L2

demand fluctuation σD

1

{(lt1 , lt2 , t g , tc , dlt1 , dlt2 , dtw )} = Ω =: Ω R1 ,R2 ,tw ,Q1 ,Q2

RS
RS
RS
RS
RS
RS
RS2,relax
RS2,relax
RS RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS2,relax
RS2,relax
1,trad
2,trad
1,trad
1,relax
2,trad
1,relax
2,trad
1,relax
2,relax
2,trad
1,trad
1,relax
2,trad
2,relax
1,relax
2,relax
1,trad
2,trad
1,relax
2,relax
1,trad
2,trad
1,trad
1,relax
2,trad
1,trad
2,relax
1,relax
2,trad
1,relax
100%
100%1,trad 1,trad

80%
80%80%
60%
60%60%
40% min
40%40%

reorder
point
reorder
point
reorder
point

TSA,CC,.

TWH,CC

CC
OC
CC
OC
OCBin
CCB OC
OC
OCBRN
CC OC
OC
CC
OCBRNOC
OC
OC
CC
OC
OC
OCBRDET
CCBRTOC
OC
CC
OC
OC
CC
OC
OC
OC
OC
T
NOC
BRN
N
BROC
BRN
RNOC
WH,CC
SA,CC,.
DETDET
deviation
% RNOCN (0.0.16)
deviation
inin%%
deviationSTOCH
STOCH
STOCH

TSA,OCN ,.
TWH,OCN
(0.0.17)
20%
TSA,OCR ,.
TWH,OCR
20%20%
0
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
1,trad
2,trad
1,relax
2,relax
1,trad
2,trad
1,relax
2,relax
1,trad
2,trad
1,relax
2,relax
TSA,OC
T
(0.0.18)
WH,OC
,.µ
mu_D
sigma_LT1
mu_LT2
R1R Q
R2
R
σ2Lµ
µσ
R
R
σQ
σ0.6
Q
Q
R
RσQ
σ0%D0%µ0%
σmu_LT1
σL
Q
µσL
R11Lµ
R
R
σ10.2
σ0.4
σL
σ
µ
Q
R
σ
R
σ
Q
Q
µ
Rσ11Q2
σ1sigma_LT2
σ11Q1
QR
σ11LQ
R
Q
R2 2.0T WH
2D
2Q
2σ
22µ
22L
2σL
2σ2Q
2σ
Lµ
L
L
Lσ
Lµ10.0
L
D
2L
2µ
2σ
21.4
22R
111.0
1Lµ
1Q
1RLµ
D
Lσ2Q
L0.8
L
D
Lµ
L1.2
L
1Lµ
11Q
11L1L
1µ
1R
2T11.8
22R
D
Lµ
LR
L2Q
2L10.4
2σ
2L
2L
22
2D
21.4
1L0.2
1D
10.8
1 D0.0
10.4
12
10.8
1L
22
12 L
0.0
0.6
1.0
1.2
1.6
2.0 2.0
0.2
0.6
1.0
1.4
1.6
1.8
1.2
1.6
1.81
WH,OCB
SA,OC
,.
B
dimensions
CC
OC
OC
OC
CC
OC
OC
OC
CC
OC
OC
OC
BR R
RNN N
B deviation
B deviation
standard
deviation
standard
standard
TSA,OCB ,.
TWH,OC
(0.0.19)
B
min
max
min
max
min
max
TSA,β,.
TWH,β
t_min
t_max
t_min
t_max
avg
τ
ττ
ττ̄avg_t
τ̄
τmin
τmax
τ̄avg_t
t_min
t_max
avg_t
TSA,β,.
TWH,β
(0.0.20)
σDσµDσLµ
σ1LLσ11Lµ1σLµ
σ2 2Q
Q212RQ
R112RR
σQ
Q
R221TR2
121Q
D
L21Lσ
L
2µLL2L
1 Lµ
TWH,ρ
SA,ρ,.
T
T
(0.0.21)
WH,ρ
SA,ρ,.
Figure 6.13: Range of order cost ratios τOCB for the sensitivity analysis and the ware1
house
1
20

τ

max
min
max ττ̄min
maxττ̄
min
maxττ̄
maxτττ̄
maxττ̄min
1max
−STOCH
1−1STOCH
τ−ττ̄STOCH
ττ̄max
ττmin
τ maxτττ̄min
τ min
τmin

TSA,OCN ,.

deviation
TWH,OC
deviation
deviation
N
1−DET
1−1DET
−DET

τ̄

100.00% with an average of τ̄WH,OCB = 91.56%. This is an average underestimation
for OCB of 8.44% by DET. This average deviation is much higher than in the previous
cases for CC, OCN , and OCR . Still, it is much less than the average values τ̄SA,OCB ,Bi
from our sensitivity analysis. According to Equation (6.3.12) the value of τOCB can be
solely explained by the difference in calculating the expected shortages, E[SH ], be-

Deterministic vs. stochastic – Case study regarding ordering costs OCB

217

tween DET and STOCH, which is caused by neglecting and considering, respectively,
the stochastic lead times and their rather complex interplay with the stochastic demand and the replenishment parameters R1 , R2 , Q1 , and Q2 .
Now, we take a closer look at the values of τOCB for the 2,751 SKUs. Table 6.17 reveals
that DET never overestimates the costs related to the back orders because τWH,OCB ,i ≤

1.0 for all 2,640 SKUs where τOCB is defined. According to the definition of τOCB it is
range of τOCB

SKUs

OCB,DET

min

max

#

%

1.00

1.00

173

6.29

10.39

0.99

< 1.00

151

5.49

0.90

< 0.99

1,586

0.00

< 0.90

0.00

0.00
N.D.

sum:
legend:

min

price

demand
avg

max

avg

avg

5.51

77.10

10.39

514.37

1.51

14.91

2.95

301.68

14.97

204.76

3.84

57.65

2.30

500.60

2.42

44.64

5.07

730

26.54

2.52

7.57 · 10−3

109.79

3.44

29.53

43.78

0

0.00

111

4.03

19.17

11.12

0.82∗

8.84∗

2,751

avg

OCB,STOCH

6.90 · 10−4

0.00

—
0.00

100.00

OCB : back order costs
τOCB =

OCB of DET
OCB of STOCH

N.D.: not defined
∗:

avg: average

excl. 1 high-demand & expensive SKU

Table 6.17: Statistics on the ratio of back order costs between DET and the STOCH
not defined whenever the expected shortage is zero in the STOCH scenario. This is
true for 111 SKUs in our warehouse out of which 110 SKUs are so cheap – the average
price is 0.82e – that the capital costs for storing are much lower than possible back
order costs. Moreover, the average demand is only 8.84 units per year. Therefore, the
optimization algorithm sets the replenishment parameters in a way that the inventory
level is high enough to eliminate all shortages and back order costs.

218

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

Only one out of the 111 SKUs does not follow this pattern. It is expensive with
2, 037.00e and has a rather high average demand of 262 units per year. Here, the
optimal replenishment exclusively uses the rush orders with the lead time distribution
L2 where µ L2 = 3 days and σL2 = 1 day instead of the normal lead time distribution
L1 with µ L1 = 9 and σL1 = 3. Due to the short lead time one can reduce the expected
shortage to zero. The additional fixed costs of 2.1% for using the rush orders instead of
the normal orders are much lower than the additional back order costs or capital costs
that are expected due to the cumulated stochastic demand during the much longer
lead time of the normal orders.
For all these 111 SKUs the reorder points and safety stock, respectively, are high
enough that both scenarios, DET and STOCH, calculate E[SH ] = 0. Trivially, DET and
STOCH yield the same back order costs OCB = 0 and τOCB is actually not defined.
However, due to the fact that τOCB has been introduced as a measure of divergence
and OCB,DET = OCB,STOCH we set τOCB = 1 for these 111 SKUs. Here, it does not
make sense to use DET or STOCH to calculate the back order costs. One should be
careful, though, because E[SH ] is not an input parameter and its value has to be calculated first. We can not assume that EDET [SH ] = 0 always implies ESTOCH [SH ] = 0
because the latter considers the stochastic lead times. In the worst case EDET [SH ] = 0
and ESTOCH [SH ] > 0 which leads to τOCB = 0, a case that has not occurred in our
warehouse.
Another interesting set of SKUs from Table 6.17 is characterized by positive and
identical back order costs for DET and STOCH, so OCB > 0 and τOCB = 1. One might
think that the reason for these 173 SKUs is a certain constellation of lead time distributions. A look at the capital costs gives a different and simple explanation. All 173 SKUs
appear in Table 6.14 on page 205 in the row where τCC is not defined. Consequently,
these 173 SKUs are too expensive to be held on stock, as explained in Chapter 6.3.2,
which is obvious if we look at their average price of 514.37e per unit. Their reorder

Deterministic vs. stochastic – Case study regarding ordering costs OCB

219

points are negative, their β service level is 0%, and all customer demand has to be
satisfied via back orders. Thus, DET and STOCH yield the same amount of back order
costs and τOCB = 1, independent of the lead time distributions. Of course, one will
prefer the fast DET scenario to the complex STOCH scenario for calculating the back
order costs in this situation. The high average value for OCB results from exclusively
using back orders.

It is very important to mention that the non-existing deviation of back order costs
between DET and STOCH, so τOCB = 100%, is a result of the optimized replenishment
parameters and not related to the stochastic nature of the lead times. We see that
very expensive and slow-moving items are not stocked at all while very cheap items
are replenished in a way to cause no shortage because the costs associated with a
shortage outweigh the stock holding or capital costs. In both cases either all demand
or no demand has to be covered by the back orders independent of the lead times.
Thus, trivially both scenarios DET and STOCH yield the same back order costs.
While our current analysis on ratio τOCB does not change for different costs parameters and different prices – given identical replenishment parameters – the result
of the optimization algorithm for the replenishment parameters heavily depends on
the cost structure. For example, more and more shortage will be accepted even for
cheap SKUs when the stock holding cost increase. Similarly, expensive SKUs will be
stocked to a greater extent when the back order cost factor increases. Consequently,
with a changed cost structure the optimal solution for the replenishment parameter
changes which changes the expected number of shortages and back orders in return.
For those changed replenishment parameters DET and STOCH might calculate different back order costs which brings us to the investigation of the majority of SKUs
where τOCB 6= 100%.

220

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

For 151 SKUs, or 5.49% of all SKUs, in Table 6.17 the maximal deviation is only 1%.
Here again, the rather high unit price of 204.76e in average causes the optimization
algorithm to reduce the average stock by setting the replenishment parameters accordingly. This manifests in the low β service level of 40.7% for STOCH. Exactly this high
probability of unsatisfied customer demand and its associated back order costs OCB
lead to a little relative deviation of back order costs between DET and STOCH and to
a value of τOCB that is close to 100% and should be negligible for most companies and
industries. Here, it should be sufficient to use DET for calculating OCB again.

The back order costs of 1,586 SKUs (57.65%) in our warehouse are calculated by DET
with a deviation between 1.0% and 10.0% regarding OCB,STOCH . Compared to the
latter group with a maximal deviation of 1%, the average price of the SKUs has now
dropped to 44.64e and their average demand has increased to 5.07 units per year. The
SKUs are not that expensive anymore and the optimized replenishment parameters
induce a stock level that covers 96.0% of the customer demand in average for STOCH
compared to β = 40.7% before. A deviation in back order costs of up to 10% can be
important for companies. It would be very helpful to have an approximation for τOCB
based on the given input parameters.
Therefore, we conduct a simple regression analysis on τWH,OCB ,i , see Table 6.18. It is
based on the input parameters but can not even explain two thirds of the variation as
R2 = 57.24% and R2 adjusted = 56.83%. The parameters for this regression analysis
are: the price, the demand distribution with µ D , σD ,
with µ L1 , σL1 ,

µ L1
σL1 ,

µ L2 , σL2 ,

µ L2
σL2 ,

µD
σD ,

the lead time distributions

the order quantities Q1 , Q2 , Q B , and the reorder-point-

related expressions R1 and R1 − R2 . Note that we use the coefficient of variation for the

distributions of the demand D and the two lead times L1 and L2 which is defined as
µ

the ratio between the average and the standard deviation, σ . The regression coefficient
of most parameters is statistically not significant from zero. The parameters that are

Deterministic vs. stochastic – Case study regarding ordering costs OCB

221

statistically most significant are listed in Table 6.18.
parameter

value

regressor

coefficient

R2

0.5724

intercept

a0 =

R2 adjusted

0.5683

price

a1 =

observations

1,586

Q1

a11 =

Q2

a12 =

QB

a13 =

R1

a14 =

R1 − R2

a15 =

9.701 · 10−1

P value
0

1.613 · 10−5

9.994 · 10−15

−2.100 · 10−2

5.546 · 10−81

−9.173 · 10−4
−3.972 · 10−3
−5.128 · 10−3
2.258 · 10−2

8.305 · 10−18
1.388 · 10−18
3.873 · 10−38
3.404 · 10−83

Table 6.18: Partial results of the regression analysis on all input parameters for τOCB
We know that τOCB is identical to the ratio of expected shortage, E[SH ], between
DET and STOCH. From the formulas in Chapter 3 about the SDMR model we know
that E[SH ] is strongly influenced by the distributions D, L1 , and L2 and the reorder
points. Interestingly, all parameters that reflect the variability of the demand or the
lead times do not have a coefficient that is significantly different from zero in our regression analysis. Apparently, the effect of the stochastic variables on τOCB cannot
be adequately captured by a simple linear combination of the average, the standard
deviation, and the coefficient of variation in our warehouse. The intercept is by far
the most significant parameter in Table 6.18 which sets the baseline for all τOCB in the
warehouse to 97.01%. This is slightly above the average value of τOCB for these 1,586
SKUs with 94.82%.
The last group to be discussed in Table 6.17 consists of the 730 SKUs where DET
underestimates OCB by more than 10%. A regression analysis with the same input
parameters as described before yields R2 = 53.58% and R2 adjusted = 52.61% and can
only explain half of the variation of τOCB . However, completely different parameters

222

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

now have a coefficient that is significantly different from zero, namely all parameters
that are related to the distributions D, L1 , and L2 .
An overview of the SKUs with the lowest value for τWH,OCB ,i is given in Table 6.19.
In favor of a better comparability we give the average and the standard deviation for
each distribution independent of their type. Interestingly, all but two rows illustrate
τOCB

OCR in e

demand distribution

lead time distributions

pS (2, ., .)

in %

DET

STOCH

type

µD

σD

µ L1

σL1

µ L2

σL2

9.3

1.03

11.03

Beta

2.62

3.76

21

7

21

7

0.00

18.4

10.05

54.55

Tri

5.12

7.36

9

2

3

1

0.00

22.9

0.79

3.47

Exp

1.57

2.26

9

2

3

1

0.00

27.1

1.70

6.28

Exp

1.47

2.11

9

2

3

1

0.00

28.7

1.30

4.55

Beta

1.63

2.33

9

2

3

1

0.00

30.0

1.45

4.84

Exp

1.16

1.34

9

2

3

1

0.00

31.6

15.59

49.29

Tri

4.34

6.24

3

1

9

2

0.00

32.8

0.09

0.29

Beta

2.15

3.09

9

2

3

1

0.00

36.4

6.92

19.01

Exp

3.22

4.62

9

2

3

1

0.07

37.4

9.33

24.92

Beta

1.73

2.48

9

2

3

1

5.02

legend:

Exp: (shifted) Exponential

Tri: Triangular

in %

pS (2, ., .): pSTOCH (2, ., .)

all lead time distributions are truncated Gaussian distributions

Table 6.19: SKUs with the 10 highest deviations for OCB between DET and STOCH
SKUs where pSTOCH (2, ., .) = 0 and so they do not use a second supply channel but
the standard single-source replenishment is applied instead. Nevertheless, the importance to include the stochastic lead time in the calculations even for single sourcing
is apparent. In addition, the last two rows in Table 6.19 show large deviations in the
back order costs between DET and STOCH for SKUs with dual sourcing.

Deterministic vs. stochastic – Case study regarding ordering costs OCB

223

From these results we can conclude that the incorporation of the stochastic lead
times are essential for determining the costs for back orders or shortages. Thereby, we
do not claim that the accuracy of the complex SDMR model and the related, rather
slow time performance is necessary for all SKUs. However, it is hard to predict how
strong the deviation between DET and STOCH is except for the extreme cases where
all or no demand is satisfied via the back orders. A simple linear model of input parameters as represented by a regression analysis will not be enough, as we have seen.
More advanced approximations seem to be necessary. From the formulas for E[SH ]
we strongly suspect that they have to depend on the type and the parameters of the
distributions, on the reorder points, and on the order quantities. The question remains
how good those other approximations can predict τOCB while they are still faster than
the SDMR model.
We expect that similar observations and statements can be given for the analysis of
the β service level in the next section as β is closely linked to the expected amount of
shortages.
Key observations. Large deviations occur for the back order costs OCB between DET
and STOCH which coincides with the results from the sensitivity analysis. The average value of τOCB is 91.56% and DET never overestimates OCB,STOCH . Given the cost
structure of our warehouse τOCB is identical to
τOCB =

EDET [SH ]
.
ESTOCH [SH ]

Expensive and low-demand SKUs are not stocked at all by the optimization algorithm. Trivially, all demand leads to a shortage and both scenarios, DET and STOCH,
yield the same value for OCB . The variability of the lead times does not have any effect on OCB and one can even calculate its value manually. The value of τWH,OCB ,i is
close to 100% for all SKUs where the majority of the demand is covered via back or-

224

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

ders. Here, DET is a good approximation for OCB,STOCH . Very cheap SKUs are stored
to a great amount by the optimization algorithm because the back order costs per unit
outweigh the capital costs per unit by far. This leads to no shortages and a β service
level of 100%. In such cases OCB = 0 holds and neither DET nor STOCH is needed.
For all non-trivial cases we observe a trend that increasing demand leads to larger
deviations of OCB between DET and STOCH. Our attempt to find a simple indication
for τWH,OCB ,i by means of a regression analysis based on the input parameters could
only explain 56.83% of the variation. This holds also for the rather simple case where
only one supplier is used. One could elaborate more complicated approximation algorithms. From our knowledge about the formulas for E[SH ] we suspect that a good
approximation has to consider the reorder points, the order quantities and the three
distributions D, L1 , and L2 at least to a great extent. Such a solution would be very
similar to STOCH.
The large deviations of back order costs between DET and STOCH in our warehouse reveal the advantage and justification for the SDMR model in combination with
the complex STOCH.

6.3.6

Service level

The intention of this section is to analyze which impact the usage of DET instead of
STOCH has in an environment of stochastic demand and lead times on companies
that are focused on a high service level instead of ordering costs. Especially the retail
industry is much more focused on service level than on ordering costs as customers
tend to do their entire shopping at a different shop if they do not find a particular SKU.
Moreover, the retailers are delivered frequently by large logistics companies anyhow.
Thus, the fixed transportation costs are spread among many different SKUs and do
not play a primary role for a single SKU. The utilization of the existing transportation,
accurate forecasts – both of which we are not covering in this work – and a high service

Deterministic vs. stochastic – Case study regarding service level β

225

level are very important performance indicators for the supply chain in their business.
We expect that the results of the last chapter on back order costs are similar to the
results we obtain on the β service level as E[SH ] and β are intimately related. In fact,
it holds that
1 − β STOCH
1 − β DET


E
[SH ]
1 − 1 − STOCH
FC


=
E
[SH ]
1 − 1 − DETFC

ρ =

=

1
ESTOCH [SH ]
=
EDET [SH ]
τOCB

(6.3.15)

for a given demand forecast FC. Equation (6.3.15) is identical to the reciprocal value
of τOCB , defined in Equation (6.3.12) on page 215, but only for the cost structure that
has been defined for our warehouse. Consequently, Figure 6.14 is the reciprocal illustration of Figure 6.13 and highlights where the shortages of STOCH, ESTOCH [SH ],
are a high multiple value of EDET [SH ]. We see from Figure 6.14 that the shortages in
the stochastic case of our warehouse are up to 10 times higher than predicted by DET.
This value for ρ lies in the middle of the experienced values of the sensitivity analysis.
We have covered τOCB in the last chapter. Therefore, we will not exhaustively discuss the deviation of β between DET and STOCH and only illustrate the values of ρ in
Table 6.20 because ρ gives us insight on the service level without multiplying it with
the back order cost factors. Table 6.20 shows in the first and the last row that DET and
STOCH calculate the same β for all 284 SKUs of the two extreme cases where no or all
demand is covered immediately. The reasons for this have been explained in detail in
the previous Chapter 6.3.5. It can be nicely observed that ρ is small for small β STOCH
service levels. The deviation from a small targeted β STOCH is uncritical in most cases
because these low values are usually only applied to SKUs where the customer can
wait. Therefore, it should be sufficient to use DET as an approximation of β for SKUs
with a low target service level.

2

2

2

2

2
DET

f ( x, T )

DET

DET DET DET DET
DET DET
DET
deviation
deviation
deviation
deviation
deviation
deviation
deviation
deviation
fSTOCH
( x,deviation
TSTOCH
)
STOCH(0.0.12)
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH

demand
fluctuation
demand
σD σ(0.0.13)
x =σD
dfluctuation
−
demand
fluctuation
σDfluctuation
demand
fluctuation
demand
σDfluctuation
σDdt
demand
demand
fluctuation
demand
demand
σDfluctuation
w σDfluctuation
lt
D σD
1

x = dlt1 − dtw
µD

2

2lead
time
fluctuation
σ σlead
time
σ σSlead
fluctuation
σL2 σL2 σL2
Clead
HAPTER
6.
D ETERMINISTIC
V
ERSUS
TOCHASTIC
OURCING
time
fluctuation
σlead
lead
time
fluctuation
lead
timelead
fluctuation
lead
time
σ2Llead
fluctuation
time
fluctuation
σlead
lead
time
σlead
fluctuation
lead
time
fluctuation
σLStime
lead
time
fluctuation
σLtime
time
fluctuation
σLlead
fluctuation
lead
time
fluctuation
fluctuation
σLfluctuation
time
σDσUAL
lead
fluctuation
time
fluctuation
σL2fluctuation
Ltime
Ltime
2
1
1
1
1
1
1 L2 L1 L2 L1 L2 L1 L2

226
RS1,trad
RS2,trad RS1,relax RS2,relax
value

reorder point R1 reorder point R2

average
lead
average
time
lead
µLL21 lead
average
time
average
time
µ L2 time
µ
average
lead
time
µ L1 lead
average
time
µ
average
lead
time
average
µ L1 average
lead
time
µ L1lead
time
average
µ
time
µµLL21 lead
average
time
µDL1 lead
average
time
µ L2 lead
average
lead
time
average
µ L1lead
lead
average
average
time
lead
µLL21 lead
time
average
time
µµLL2(0.0.14)
lead
average
time
µ L2 lead
time
µ L2 µ L2
1
Service level Ratio

�∞ R1�− R2

�∞

1

σD µ L1 σL1 µ L2 σL2 Q1 Q2 R1 R2

0

R1 − d t w

cost
measures
cost
measures
cost
measures

DET
average
DET DET DET average
DET
DETlead
DET
DET
deviation
deviation
in STOCH
%DET
time
time
deviation
deviation
deviation
indeviation
%
indeviation
%
indeviation
%lead
inlead
%
deviation
inaverage
%
intime
%
in % in %
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH
STOCH

140%
160%
160%

lead time fluctuation

1−STOCH
1−STOCH
1−STOCH
1−STOCH
1−
STOCH
1time
−STOCH
1−STOCH
1fluctuation
−STOCH
1−STOCH
lead
TSA,CC,.
lead
time
fluctuation
deviation
deviation
deviation
deviation
deviation
deviation
1−DET
1−DET
1−deviation
DET
1−DET
1−DET
1−deviation
DET
1−deviation
DET
1−DET
1−DET

TWH,CC
T
(0.0.16)
WH,CC order quantity
120%
120%
order
quantity
order
quantity
100%
TSA,OCN ,. RSTWH,OCN
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS
RS2,relax
RS2,relax2,relax
1,trad
2,trad
1,trad
1,relax
2,trad
2,relax
1,relax
2,trad
1,relax
2,relax
2,trad
1,trad
1,relax
2,trad
2,relax
1,relax
2,relax
1,trad
2,trad
1,relax
2,relax
1,trad
2,trad
1,trad
1,relax
2,trad
1,trad
2,relax
1,relax
2,trad
1,relax
100%
100%1,trad 1,trad
T
T
(0.0.17)
reorder
point
WH,OC
SA,OCN ,.
80%
N
reorder
point
reorder
point
TSA,OCR ,.
TWH,OCR
80%80%
CC
OC
CC
OC
OCBin
CCB OC
OC
OCBTRN
OC
CC
OCBRNOC
OC
OC
CC
OC
OC
OCBRDET
CC
OC
CC
OCBRN
OC
CC
OC
OC
OC
OC
RNOC
N
60% CC OC
BROC
NOC
N
BROC
BRN
RNOC
DETDET
deviation
%
T
(0.0.18)
deviation
in
%
WH,OC
SA,OC
,.
deviation
in
%
60%60%
STOCH
R
R
STOCH
STOCH
TSA,OCB ,.
TWH,OCB
40% min min
min
max
min
maxττ̄min
maxττ̄
min
maxττ̄
maxτττ̄
min
maxττ̄min
1max
−STOCH
1−1STOCH
−
STOCH
40%
τ
τ̄
ττ̄max τ̄
τ40% ττ maxτττ̄min
τ
τ
τ
τ
deviation
deviation
deviation
TSA,OCB ,.
TWH,OC
(0.0.19)
1
−
DET
1−1DET
−DET
B
20%
20%20%
T
TWH,β
RS
RS
RS1,relax
RS2,relaxSA,β,.
RS
RS
RS
RS
RS
RS
RS
RS2,relax
1,trad
2,trad
1,trad
2,trad
1,relax
1,trad
2,trad
1,relax
2,relax
T
T
(0.0.20)
0%0% 0% mu_LT1
mu_D
sigma_LT1
mu_LT2
sigma_LT2
Q1
Q2
R1
R2
WH
SA,β,.
WH,β
σ2Lµ
µσ12LD
R
R
σQ
µ0.6
σ0.6
Q
Q
R
RσQ
σD µ Lσ0.0
σLµ10.0
σL
Q
µσL
R11Lµ
R
R
σ10.2
σ0.4
σL
σ
µ
R
σ
R
σ
Q
Q
µ
Rσ111.4
R22R
σ1Q
σ112D
QR
σ11LQ
R
Q
R2.0
2D
2Q
2σ
22µ
22L
2σL
2σ2Q
2σ
Lµ
L
L
2L
2µ
2σ
21.4
22R
111.0
1Lµ
1Q
1Lµ
D
Lσ2Q
L0.8
L
D
Lµ
L1.2
L
1Lµ
11Q
11L1L
1µ
2R1Q
2 2.0 T
D
Lµ
LR
L2Q
12.0
2L10.4
2σ
2L
2L
22
21.4
1L0.2
1D
10.8
1 D0.0
10.4
10.8
1L
22
12 L
1.0
1.2
1.6
0.2
0.6
1.0
1.6
1.2
1.611.8 1.8T
1.8
WH,ρ
SA,ρ,.
CC
OC
OC
OC
CC
OC
OC
OC
CC
OC
OC
OC
dimensions
B
R
N
B deviation
RR
NN
B deviation
deviation
standard
standard
TSA,ρ,.
TWH,ρstandard
(0.0.21)
min
max
min
max
min
max
t_min
t_max
t_min
t_max
avg_t
τ
ττ
ττ̄avg_t
τ̄
τ min
τmax
τ̄avg
t_min
t_max
avg_t
1
1
σDσµDσLµ
σ1LLσ11Lµ1σLµ
σ2 2Q
Q212RQ
R112RR
σQ
Q
R221 R2
121Q
D
L21Lσ
L
2µLL2L
1 Lµ

τ min τ max τ̄

140%
140%
120%

value

10

tw

CCvalue
OCB OCR OCN

DET
STOCH in %
1−STOCH β
deviation
ratio in
1−%DET β

DETorder
order
quantity
Q
Q
quantity
order
quantity
Qd2 )Qf (2x, ltQ2− t ) f (d , t ) l
quantity
Q
order
quantity
Q
order
order
Q
quantity
Q
order
quantity
Q12 order
quantity
Q
order
quantity
Q
order
quantity
Q[1SH
order
quantity
quantity
Q
quantity
Q12 order
Q
quantity
Q
quantity
12( x
Rquantity
deviation
2 quantity
2order
2R1 +
2
1 order
1 order
1 order
1DET
�∞ R1�−order
�∞
DET
E
−
] 1=
STOCH
tw
w
tw w 1
1
Fall A · 1Fall
A
deviation
deviation
dtw
lt1
x
STOCH
STOCH
E [SHFall A · 1100
(
x
−
R
+
d
)
f
(
x,
lt
−
t
)
f
(
d
,
t
)
l
(
lt
)
dµ
(0.0.15)
t
w
t
w
1
1
1
1
Fall A ] =
w
w
t
0
R
−
d
dtw point
twreorder
fluctuation
lt1
x
reorder
point
reorder
point
R2 point
reorder
point
R1 reorder
Rreorder
reorder
reorder
R1 reorder
point
Rdemand
reorder
Rreorder
RR2 1point
reorder
point
R
reorder
R1R
reorder
point
reorder
R1point
reorder
point
R
reorder
point
RR2 1wpoint
reorder
R2 point
R2 R2
2 1point
2fluctuation
2 1point
1point
1point
demand
demand
fluctuation

deviation

order quantity Q1 order quantity Q2

lead time fluctuation σL1 lead time fluctuation σL2

l1

(0.0.11)

l1

2

2

2

TSA,CC,.

Figure 6.14: Range of β service level ratios ρ for the sensitivity analysis and the warehouse

The value of ρ increases with higher β service levels. Of course, the delta between
β DET and 100% decreases which favors high values of ρ even if the absolute difference
in β between STOCH and DET remains constant. However, many retail companies
have target β service levels of 99% and above especially for their fast-moving SKUs.
Interestingly, those fast-moving SKUs also exist in our spare parts warehouse and
Table 6.20 reflects that the SKUs with the highest value for ρ are the ones with the
highest demand.
For those fast-moving SKUs even a 0.5% of difference for β has a great impact on
their business. This is the reason why we have introduced the definition of ρ as the
ratio of gap to 100% between STOCH and DET.
A β service level of 100% can be achieved with a little amount of stocked units for
slow-moving items like in the spare parts industry. In the retail business, however, it
22
is almost impossible to reach a service level of 100% for most of the fast-moving
SKUs
2
2
2
2 2
2
2
2
without running into serious space problems. Therefore, the number of SKUs where

2

2

Deterministic vs. stochastic – Case study regarding service level β
range of ρ

SKUs

β DET

min

max

#

%

1.00

1.00

173

6.29

0.00

> 1.00

1.01

151

5.49

40.64

6.34

70.74

> 1.01

1.10

1,542

56.05

96.18

24.11

> 1.10

2.00

751

27.30

99.05

> 2.00

10.0

22

0.80

99.81

> 10.0

∞

1

0.04

99.95

111

4.03

100.00

N.D.

avg

β STOCH

sum:

2,751

legend:

β: β service level in %

min

max

avg

0.00

227

price

demand

avg

avg

514.37

1.51

40.46

204.76

3.83

99.94

95.98

44.98

4.71

84.47

99.99

98.87

29.91

29.05

98.42

99.99

99.48

23.78

468.95

99.46

4.50

607.00

100.00

19.17

11.12

0.82∗

8.84∗

100.00

ρ=

1− β STOCH
1− β DET

N.D.: not defined
∗:

avg: average

excl. 1 high-demand & expensive SKU

Table 6.20: Statistics on the ratio of (1 − β) between STOCH and DET
β = 100% for DET and STOCH will be rather limited for fast-moving SKUs in the
retail environment.
Key observations.

The deviation of the β service level between DET and STOCH is

large but not as large as we have observed in the sensitivity analysis. There exists an
intimate relation between the ρ and τOCB for the given cost structure in our warehouse.
ρ=

ESTOCH [SH ]
1
=
EDET [SH ]
τOCB

The value of β coincides between DET and STOCH for all SKUs where β = 0% and
β = 100% in our warehouse. Here, DET can be used without large concerns instead
of STOCH.
For all other SKUs we observe the trend that a small targeted β service level is

228

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

better approximated by DET than a β close to 100%. This is especially problematic
as the retail industry usually targets at very high β service levels for important fastmoving SKUs where even small deviations from β have a great impact on the business.
In this case DET cannot be used as an appropriate approximation of β STOCH .

6.3.7

Summary

The objective of this case study was to see whether our conclusion from the sensitivity
analysis, to prefer the STOCH scenario to the DET scenario whenever the lead times
are stochastic, also holds in reality. This represents our second approach to answer
question RQ 3 on page 5.
We investigated whether the large deviation of the KPIs between DET and STOCH
from the sensitivity analysis can also be observed at our spare parts warehouse that
contains 2,751 SKUs. Therefore, the replenishment parameters R1 , R2 , Q1 , and Q2
had been optimized beforehand for all SKUs regarding the minimal total costs TC =
CC + OCN + OCR + OCB . Based on the optimal replenishment parameters the case
study compared the values of the ratios τSA and ρSA from the sensitivity analysis with
the values of τWH and ρWH from the warehouse with 2,751 SKUs. In general we found
that the deviations of KPIs between DET and STOCH are much smaller than in the
sensitivity analysis. In order to avoid tedious repetitions we move the more detailed
summary to the next section.

6.4

Summary

The purpose of this chapter was to find an answer to the research question RQ 3 about
the differences between a stochastic scenario STOCH and a simpler approximative
scenario DET that assumes deterministic lead times. More precisely, we gained an
understanding where one can use the DET scenario instead of the STOCH scenario to

Deterministic vs. stochastic – Summary

229

calculate various KPIs like costs and service level, and where DET is not sufficient.
Our comparisons between DET and STOCH concentrated on the five KPIs: the capital costs CC for holding inventory, the order costs OCN , OCR , and OCB for normal,
rush, and back orders, respectively, and the β service level. In order to obtain meaningful comparisons we introduced the two relative measures τ and ρ for the deviation
of the KPIs between DET and STOCH. Fortunately, τ and ρ are independent of the
cost factors for ordering and holding stock. Thus, we were able to derive statements
which apply for arbitrary values of the cost factors by only considering the interplay
between the single replenishment input parameters and the stochastic variables.
DET assumes stochastic demand but deterministic lead times for both suppliers
while STOCH considers all demand and lead times to be stochastic. Fortunately, the
SDMR model was able to represent the DET scenario as well as the STOCH scenario
so that we can eliminate effects within our comparisons that origin from a different
modeling approach. Our investigations were two-fold.

Sensitivity analysis. We conducted a sensitivity analysis on how partial changes of
the input parameters affect the deviation of our KPIs between DET and STOCH. The
deviation was measured by means of the ratios τ and ρ. The sensitivity analysis was
of a more theoretic nature and used certain assumptions like Gaussian-distributed
demand and lead times. The input parameters are σD for the demand, µ L1 , σL1 , µ L2 ,
and σL2 related to the lead times, and our four replenishment parameters Q1 , Q2 , R1 ,
and R2 .
Within the sensitivity analysis we found that the range of deviation between DET
and STOCH is very high. DET underestimated some of our KPIs up to 100% while
it overestimated other KPIs usually by 40% and above for at least one value of each
input parameter. Sometimes these large deviations related to extreme values for single
input parameters. In any case, we found that the reasons for a deviation between

230

C HAPTER 6. D ETERMINISTIC V ERSUS S TOCHASTIC D UAL S OURCING

DET and STOCH are very diverse and one has to be aware of the rather complex
interplay between the various input parameters, especially if the deviations should be
sufficiently explained. The conclusion from the sensitivity analysis is to clearly favor
the complex STOCH scenario for calculating our KPIs in a setting with stochastic lead
times because the magnitude of deviation of DET is hardly predictable.

Case study. In the case study we compared the findings of our sensitivity analysis
with the results of a spare parts warehouse containing 2,751 SKUs. In general, we
found that the deviations of KPIs between DET and STOCH are much smaller than in
the sensitivity analysis. For some KPIs like the normal order costs OCN there existed
almost no deviation while DET underestimates especially KPIs that are related to the
β service level up to a factor of 10. The latter proves that the conclusion of Verrijdt
et al. in their context of an emergency repair SC does not hold in general [VAdK98].
They found that the expected β service level of a model which considers exponentially
distributed lead times deviates only negligibly from an approximation with deterministic lead times. From this result they concluded that the assumption on the variability
and the type of the lead time distribution is not restrictive in their case. Clearly, our
results show this result cannot be transferred to the context of a ( R1 , R2 , Q1 , Q2 ) policy
when the demand and the lead time distribution can be arbitrary.
The case study has further shown that τ and ρ are not only influenced by the distribution of the demand and the lead times but also by the result of the optimization
of the replenishment parameters Q1 , Q2 , R1 , and R2 . This leads to many special cases
and even trivial results like single costs of 0.00e . For several KPIs we were able to analytically express the deviation between DET and STOCH. All expressions contain at
least one term that is complex and calculation-intensive to determine like the service
level or the probability of a two-order cycle in the STOCH scenario. Thus, there is no
general and easy way to predict the deviation between DET and STOCH.

Deterministic vs. stochastic – Summary

231

After all, the case study relativizes our conclusion from the sensitivity analysis to
clearly prefer the STOCH scenario to the DET scenario. However, if reliable values are
essential we still advise to use the STOCH scenario to calculate the KPIs.
This brings us to the conclusion that the approximation with deterministic lead times
is sufficient for calculating a part of the KPIs for many SKUs even when both lead
times are stochastic. However, great discrepancies can occur especially regarding the
shortage costs and the β service level. These discrepancies might even grow for longer
lead times and greater variability. Due to the fact that there are many special cases and
it is hard to quantify the deviation between DET and STOCH a priori we suggest to
use the STOCH scenario at least once for all SKUs. Then, one can identify the SKUs
and KPIs for which the deviation is unacceptable from a business perspective of the
company. One should continue to use the SDMR model with the STOCH scenario for
those critical SKUs and KPIs in the future, as well, while the SDMR model and the
DET scenario can be applied to the other SKUs and KPIs.
Summa summarum, our observations in this chapter show that the answer to question
RQ 3 is very complex and strongly depends on the specific situation.

Chapter 7
Comparison of stochastic
replenishment policies
In this chapter we are interested in the savings of total costs that can be achieved by
moving from a single-sourcing replenishment policy with traditional restrictions to a
dual-sourcing policy with relaxed restrictions. Moreover, we also want to understand
the mechanisms that lead to these savings. This corresponds to our research question
RQ 4 on page 6. In particular, we compare four stochastic replenishment policies in
the context of our warehouse with 2,751 SKUs. The approximative DET scenario with
deterministic lead times is not considered here.
First, Chapter 7.1 gives an overview of the different combinations of replenishment
policies and restrictions. Chapter 7.2 quantifies the savings of total costs in our warehouse for these combinations. Then, we analyze the savings exclusively related to the
relaxed restrictions in Chapter 7.3. In Chapter 7.4 we investigate the savings related to
dual sourcing in our warehouse with short and long lead times for the first supplier.
Finally, Chapter 7.5 summarizes our findings. Note, in this chapter we use the same
optimization approach, a Threshold Accepting Algorithm, as we have mentioned in
our case study in Chapter 6.3.
233

234

7.1

C HAPTER 7. C OMPARISON O F S TOCHASTIC R EPLENISHMENT P OLICIES

Overview of stochastic replenishment scenarios

Only a small fraction of companies use two suppliers simultaneously in our experience. However, most of them apply certain restrictions for their replenishment more
or less deliberately. The most prominent restrictions are non-negative reorder points
and a fixed sequence of triggering the normal order before the rush order. We question
these restrictions and investigate whether more relaxed restrictions inhibit a saving
potential for the total costs. Therefore, we first define the traditional restrictions and
the relaxed restrictions.

Definition 7.1.1 (Traditional restrictions). The normal order is always associated with
the reorder point R1 and the rush order is assigned to the reorder point R2 , if used
at all. The reorder points R1 and possibly R2 are non-negative and a normal order
has to be triggered before a rush order. Consequently, it must hold that R1 ≥ 0 and
R1 > R2 ≥ 0 for single sourcing and dual sourcing, respectively.

Definition 7.1.2 (Relaxed restrictions). The assignment of normal orders and rush orders to the reorder point R1 and R2 , respectively, does not matter. The reorder points
can be negative and R1 > R2 holds.

We have on the one hand the option between the commonly applied, traditional
restrictions versus the relaxed restrictions. On the other hand we have the choice between single-source and dual-source replenishment. The combination of the number
of suppliers and the restrictions leads to the four options represented in Table 7.1.
Next, we look at the savings of the total costs when we move from RS1,trad to one
of the other replenishment scenarios. All of them can be represented by the SDMR
model.

Stochastic replenishment – Total saving
replenishment

# of

scenarios

suppliers

235

restrictions

RS1,trad

1

traditional

RS1,relax

1

relaxed

description
traditional single-source replenishment
supplier selection with the option of a negative
reorder point

RS2,trad

2

traditional

dual-source replenishment with traditional setting of the replenishment parameters

RS2,relax

2

relaxed

dual-source replenishment with the option of
negative reorder points

Table 7.1: Overview of replenishment scenarios

7.2

Total savings of different replenishment scenarios

This chapter quantifies the possible reduction of total costs TC in our warehouse by
applying the different replenishment scenarios of Table 7.1. This addresses the first
part of question RQ 4.
We assume that our warehouse currently uses only one supplier with traditional
replenishment restrictions for each SKU. The manager wants to evaluate the cost reductions of the other scenarios. Thus, the status quo is RS1,trad and we compare
RS1,relax , RS2,trad , and RS2,relax against it. In Figure 7.1 we see that the total warehousing costs of the initial scenario RS1,trad is normalized to 100%. We refer to these
total costs as TCRS1,trad . The total costs of our warehouse can be reduced from 100% to
94.3% if the replenishment is changed from RS1,trad to a two-supplier replenishment
with relaxed restrictions, RS2,relax . These 5.7% of savings in total costs is a significant
amount of money given the fact that the total costs of large warehouses easily reach
several million euros per year.
The largest saving amount of our warehouse origins from the relaxed replenish-

2
2

2

110%

100%

L1

L2

1

order
quantity
Q1σorder
quantity
leadlead
time
fluctuation
time
average
time
lead
time fluctuation
µ LQ
demand
fluctuation
σD µ L1 average
L1 lead
2 2
110%

32.5%

31.8%

2

100%

2

2

24.5%

24.2%

2

32.5%

Total Costs

7.8%
0.0%

40%

STOCH

1−STOCH β
DET
CC
OC
OCRSOC
11.0% deviation
11.4%
RS
deviation
N RS2,relax
CC
%B2,trad
7.4%
1,trad
1,relax
1in
−RS
DET
β R
STOCH
demand fluctuation
demand fluctuation
σD
σD
demand
fluctuation
σD
demand
fluctuation
σD
min τ
1CC
−
STOCH
β max τ̄
11.0%deviation
11.4%
τ
7.4%
7.8%
CC
OC
OCN RS2,relax
RS1,trad
RS
RS
Bβ OC
R 1,relax
OC_B
2,trad
0.3%
1−DET
0.0%
average
average
time µ
lead
time
µaverage
average
time
lead
time
µaverage
average
lead
time
µ L1µ
average
lead
timelead
µ L2 time OC_B
L1 average
Llead
Ltime
L2 0.3%
lead
µ L 0.3%
µmin
0.0% lead
0.3%
2
1
L max
0.0%
τ2,trad
τ
τ̄
CCRS
OC
OC
OC
RS1,trad
RS
RS2,relax
B
R
N
OC_R
σDOC_R
µ L σ1,relax
L µ L σL Q
1 Q2 R1 R2
lead time fluctuation
lead time lead
fluctuation
σL1 time
lead lead
time
σL1time
fluctuation
leadfluctuation
time
fluctuation
σL2 time
σL2time fluctuation
fluctuation
σL1 lead
fluctuation
σL2 OC_N
σL lead
σL
min
max
τ B OC
τ R OC_N
τ̄ N
CC OC
OC
σD µ L σL µ L σL Q1 Q2 R1 R2

70%
60%

1

59.7%

2

2

order quantity
orderQquantity
Q
Qquantity
orderquantity
quantity
Qquantity
quantity
order
Q21 order Q
quantity
2 1 order Q
1 order
1 order
2
minQ2max
59.7%

2

1

1

40%

1

1

2

2

τ σ τµ στ̄ µ σ Q Q R R
D L1 L1 L2 L2 1 2 1 2

58.9%

58.8%

reorder point
reorder
R1 reorder
point
Rpoint
R2 Rpoint
R2Rpoint
reorder
point
R2 point R2
reorder
point
1 reorder
1 reorder
1 reorder

σ µ L1 σL1 µ L2 σL2 Q1 Q2 R1 R2

20%

D
DET
DET
DET
DET
59.7%
58.9%
deviation deviation
in deviation
%
58.8%
deviation
STOCH in %
STOCH
STOCH in %
STOCH in %

10%

20%

2

1

50%

30%

59.7%

DET

deviation
in %fluctuation
reorder
point
Rlead
point
orderfluctuation
quantity
Qσ1Lorder
quantity
Q2R2 σL2
lead time
time
1 reorder
1STOCH

STOCH β
DET 1−
RS
RS
RS2
RS2,relax
deviation
deviation
%
2,trad
24.2%reorder
point
RSTOCH
reorder
point
1 1,trad
1in
−DET
β R1,relax

31.8%

DET
DET
DET 24.5%
DET
deviation deviation
deviation
deviation
STOCH
STOCH
STOCH

80%

reorder
reorder
order
quantity
Qσ1LRorder
quantity
QR22
leadlead
time
fluctuation
lead
timepoint
fluctuation
σL2
average
time
µ L1 point
average
1 1 lead time µ L2
DET1−STOCH β
deviation
deviation
%
point
RSTOCH
point
orderreorder
quantity
Q1 order
quantity
1 reorder
1−in
DET
β Q2R2

90%

50%

30%

D

STOCH

70%
60%

DET lead time µ average lead time µ
average
deviation
demand
fluctuation σD L1
L
STOCH

C HAPTER 7. C OMPARISON O F S TOCHASTIC R EPLENISHMENT
P OLICIES
DET
lead
time
fluctuation
σL lead
time
fluctuat
demand
fluctuation
σ µ average
average
lead
time
lead
time
µ
deviation

90%
80%

DET
STOCH

DET
demand
fluctuation σD
deviation
STOCH

2

236

deviation

1−STOCH β 1−STOCH β 1−STOCH β 1−STOCH β
1−DET β

deviation deviation
1−DET β deviation
1−DET β deviation
1−DET β

0%

RS
RS
RS
RS2,relax
RS1,trad RSRS
RS
RS
RS
RS
RS
RS
RS
RS2,relax
RS1,trad
RS2,trad
RS1,relax
1,trad
2,tradRS
1,relaxRS2,relax
2,trad
1,trad
1,relax
2,trad
2,relax
1,relax
1,trad
2,trad
1,relax
2,relax

10%

CCR OC
CC OCB OC
CCR OCBN OC
OCBNOC
CCROC
OCBNOCR OCN
min

max

min

max min

max

min

max

ττ̄ for
τ the
τ̄ 4 replenishment scenarios.
τ
τ warehousing
τ̄
τ
τ
ττ̄ τcosts
Figure 7.1: Total

0%

σ1 Qµ2LRσ1LRQ
R1 R2
σD µ L1 σL1 µσLD
σµ 2 1Qσ1LQ
µσR2 σ
R
µ
σL11Q
µσ2LDR
σµRQ
LL212Q
1RS1,relax
2 1 Q2RS2,relax
2 LL
1 2LD
2 1LL2 2 L
RS2,trad

RS1,trad

1

1

2

2

ment restrictions. The difference between the single sourcing and dual sourcing plays
only a minor role. This result corresponds to our findings in Chapter 6.3 where we
have explained that only a very small percentage of the 2,751 SKUs actually uses the
second supply channel due to the sparse demand which is common in the spare parts
business. Table 7.2 shows that 69.47% of all SKUs have a cost-minimal replenishment
solution with only one supplier even though a second supplier is allowed in scenario

2

RS2,relax . Only 12 SKUs or 0.44% use the second supply channel with a probability of

2
2

p(2, ., .) ≤

0.00

0.01

0.05

0.10

0.15

0.20

0.25

0.30

# of SKUs

1,911

2,605

2,720

2,739

2,746

2,748

2,749

2,751

% of SKUs

69.47

94.69

98.87

99.56

99.82

99.89

99.93

100.00

Table 7.2: List of SKUs and the probability p(2, ., .) of triggering a second order in
RS2,relax

2

2

2

2

more than 10%. Despite the low usage of the second supply channel in RS2,trad , the
additional spending for a second supply mode of roughly 0.3% of TCRS1,trad reduces

Stochastic replenishment – Total saving by relaxed restrictions

237

the total costs by 0.8% of TCRS1,trad . For the relaxed scenarios we yield net reduced
costs of about 0.5% by spending 0.3% of the total costs on the second supply channel.
The fraction of costs for the normal orders, those orders assigned to the reorder
point R1 , remains almost unchanged close to 60% while the reduction of the capital
costs CC contributes the biggest part to the total cost savings for RS2,trad and RS2,relax .
It is interesting to see that the reduced capital costs CC lead to increased costs OCB for
unsatisfied customer demand.
Summing up, we can give first answers to our research question RQ 4. First, we have
found that the total costs of our warehouse can be reduced by 5.7% if a dual-source
replenishment policy with relaxed restrictions is used instead of single sourcing with
traditional restrictions. Second, dual sourcing is not used for most SKUs in our warehouse. Consequently, their contribution to the overall savings of total costs TC is low.
Third, the main part of the savings in TC originates from the relaxed restrictions. This
savings effect can be observed in Figure 7.1 for single sourcing and dual sourcing alike
and will be the topic of the next chapter.

7.3

Savings induced by relaxed replenishment restrictions

This chapter investigates in more detail the savings of total costs that are caused by
the relaxed replenishment restrictions. We are interested in the mechanisms that lead
to these savings. This is related to the second part of question RQ 4. We know from
Figure 7.1 that the saving amount is almost identical for RS1,relax and RS2,relax . For
the sake of simplicity we only consider the single-supplier scenario RS1,relax here and
compare it to the initial scenario RS1,trad . Both scenarios can be represented by the
SDMR model.

238

C HAPTER 7. C OMPARISON O F S TOCHASTIC R EPLENISHMENT P OLICIES

According to the Definition 7.1.1 and Definition 7.1.2 the difference between the
traditional and the relaxed replenishment restrictions can only be a negative reorder
point, so R1 < 0, or the replacement of normal order with the emergency order in the
single-supplier scenario. We look at both cases in the following which happen to be
disjoint sets of SKUs in our warehouse.

7.3.1

Negative reorder point

In total there are 327 SKUs in our warehouse which use a negative reorder point,
R1 < 0, in the scenario RS1,relax . We divide the 2,751 SKUs into the two sets
MR1 <0 : set of SKUs where R1 < 0
MR1 ≥0 : set of SKUs where R1 ≥ 0.
Table 7.3 shows that MR1 <0 accounts for 95.84% of all savings in total costs that are
gained by using the relaxed replenishment restrictions compared to the traditional
scenario RS1,trad . These savings are achieved by a strong reduction of the capital costs
restriction

SKUs

% of total savings

price (e )

#

%

sum

CC

OCN

OCB

avg

M R1 <0

327

11.9

95.84

152.39

23.62

-80.17

M R1 ≥0

2,424

88.1

4.16

0.18

-0.24

total

2,751

100.0

100.00

152.57

23.38

legend:

demand (units)

stddev

avg

387.9

634.0

2.9

4.0

4.23

0.36

80.0

17.0

80.6

-75.95

77.9

257.7

15.3

75.8

CC: stock holding costs

OCN : normal order costs

OCB : back order costs

stddev: standard deviation

stddev

avg: average

Table 7.3: Contribution of the relaxed restriction R1 < 0 to the total savings
CC. The first row of Table 7.3 shows that the reduction of CC by less stock is even
larger (152.39%) than the total savings in the end. The reduction of CC is mitigated,

Stochastic replenishment – Total saving by a negative reorder point

239

though, by the increased costs for non-satisfied customer demand (-80.17%). Summing up the savings and additional costs for CC, OCN , and OCB we yield the 95.84%
of total savings for MR1 <0 .
The average price and the average yearly demand differ largely between MR1 <0
and MR1 ≥0 . We want to quantify these observations statistically and state the follow-

ing hypotheses:
p

H0 : The price distribution of MR1 <0 does not dominate the one of MR1 ≥0
p

H1 : The price distribution of MR1 <0 dominates the one of MR1 ≥0
H0d : The demand distribution of MR1 <0 dominates the one of MR1 ≥0
H1d : The demand distribution of MR1 <0 does not dominate the one of MR1 ≥0 .
p

The null hypotheses H0 and H0d are tested against the α = 0.01 level of significance.
We use the single-sided Wilcoxon rank-sum test as described by Hartung et al. and
Hollander et al. [HEK05], [HW99]. The number of elements exceeds 20 for MR1 <0 and
MR1 ≥0 and the Standard Gaussian Distribution can be used as an approximation. The

critical value of rejection is given by the (1 − α) quantile, Φ(0.99; 1; 0) = 2.33. The

results are shown in Table 7.4.

We find that the empirical value T = 27.09 for the price is greater than the 99%
quantile of the Standard Gaussian distribution Φ(0.99; 0; 1) = 2.33. For the demand
T = −10.84 is smaller than Φ(0.99; 0; 1). The rule of rejection, see Table 7.4, is satisfied
p

for both cases and we have to reject both null hypotheses H0 and H0d at the significance
level of α = 0.01.

240

C HAPTER 7. C OMPARISON O F S TOCHASTIC R EPLENISHMENT P OLICIES

Wilcoxon rank-sum test for
Wn1 ,n2 :
T:

demand (units)

rank sum of MR1 <0

815,244.00

303,793.00

standardized value, Gaussian distributed

27.09

-10.84

standard Gaussian distribution

2.33

2.33

rule of rejection

T > Φ (.)

T < Φ (.)

null hypotheses

H0 rejected

H0d rejected

n 1 = | M R1 <0 |

n 2 = | M R1 ≥0 |

Φ(0.99; 0; 1)

H0 :
legend:

price (e )

MR1 <0 : set of SKUs where R1 < 0
T=

Wn1 ,n2 −0.5n1 (n1 +n2 +1)

p

Wn1 ,n2 : rank sum of MR1 <0

√1

12 n1 n2 ( n1 + n2 +1)

Table 7.4: Wilcoxon Rank Sum Test on price and demand of SKUs where R1 < 0
'

Consequently, we can state that the
SKUs that are cost-optimally replenished

Info-Box 7.1: SKU A

General information

with a negative reorder point R1 have a

description:

engine

higher price and a lower demand than

demand:

(units / year)

price:

(e )

the remaining SKUs with R1 ≥ 0. More

precisely, the price distribution of the
SKUs in MR1 <0 dominates the price distribution of the SKUs in MR1 ≥0 accord-

ing to the first order stochastic dominance. The demand distribution derived
from MR1 ≥0 dominates the demand distribution derived from MR1 <0 .

$

Comparison

1.00
5019.00

RS1,trad

RS1,relax

R1 ; Q1 :

0;1

-1 ; 1

β in %:

97.15

0.00

TC in e :

304.63

24.70

CC in e :

292.41

0.00

OCN in e :

11.85

11.85

OCB in e :

0.37

12.85

&

%

We will show that this result is plausible in an economic view, as well. First, the
major cost for these expensive, low-demand SKUs are the capital costs. A primary
objective is to lower the average stock and, thus, decrease the capital costs. Second,
even with a reorder point R1 = 0 the stock will be above 0 most of the time because

Stochastic replenishment – Total saving by switching orders

241

the probability of demand during the lead time is close to zero for those low-demand
SKUs. This also causes the β service level to be close to 100%. Consequently, R1 has to
be below 0 to effectively reduce the average stock level and the holding costs. Third,
the total amount of unmet customer demand and all related costs are rather small
even if R1 < 0 due to the low total demand of these SKUs. Consequently, the total
costs can be reduced for expensive and low-demand SKUs by setting R1 < 0. This
makes the result of Table 7.4 plausible from an economic view. An example within our
warehouse is SKU A in Info-Box 7.1 where the total costs can be reduced by 91.89%
from 304.63e to 24.70e . Alternatively to setting R1 < 0 one could set R1 = 0 but delay
the reordering for some time as described by Schulz [Sch89].

7.3.2

Using emergency orders as normal orders

We look now at the SKUs for which the optimal result is achieved by switching the
supply modes. We divide all SKUs into the two sets
M R1
R2 :

set of SKUs where the two supply modes have been exchanged

M R1 6
R2 :

set of SKUs where the supply modes are not exchanged.

Table 7.5 shows that MR1
R2 accounts for the remaining 4.16% of all savings that can
be gained by using the relaxed replenishment restrictions RS1,relax . The first row of Table 7.5 shows that the exchange of supply modes applies only to 6 SKUs. The largest
fraction of savings for MR1
R2 is achieved by reducing the costs OCB for unmet customer demand. However, this saving effect is mitigated by additional costs OCN for
normal orders which occur due to the shorter lead time.
The average price and the average yearly demand differ largely between MR1
R2
and MR1 6
R2 . Similar to the previous chapter we want to quantify these observations

242

C HAPTER 7. C OMPARISON O F S TOCHASTIC R EPLENISHMENT P OLICIES

restriction

SKUs
#

% of total savings
%

sum

CC

OCN

price (e )
OCB

avg

demand (units)

stddev

avg

stddev

M R1
R2

6

0.2

4.16

0.18

-0.24

4.23

266.1

226.3

172.3

130.3

M R1 6
R2

2,745

99.8

95.84

152.39

23.62

-80.17

77.5

257.6

15.0

75.3

total

2,751

100.0

100.00

152.57

23.38

-75.95

77.9

257.6

15.3

75.8

legend:

CC: stock holding costs

OCN : normal order costs

OCB : back order costs

stddev: standard deviation

avg: average

Table 7.5: Contribution of the relaxed restriction R1 < 0 to the total savings
statistically and state the following hypotheses:
p

H0 : The price distribution of MR1
R2 does not dominate the one of MR1 6
R2
p

H1 : The price distribution of MR1
R2 dominates the one of MR1 6
R2

H0d : The demand distribution of MR1
R2 does not dominate the one of MR1 6
R2
H1d : The demand distribution of MR1
R2 dominates the one of MR1 6
R2 .
p

We want to test the null hypotheses H0 and H0d against the α = 0.01 level by
means of the single-sided Wilcoxon rank-sum test, see [HW99] and [HEK05]. However, we find that the cumulated price distributions of MR1
R2 and MR1 6
R2 cross.

Thus, the Wilcoxon rank-sum test is not applicable. For the demand, the number of
elements of MR1 6
R2 exceeds 20 and the Standard Gaussian Distribution can be used

as an approximation. The critical value of rejection is given by the 1 − α quantile,

Φ(0.99; 1; 0) = 2.33. Table 7.6 shows that T > 2.33 for the demand which satisfies the

rule of rejection.
Consequently, we have to reject the null hypothesis H0d at the 0.01-level of signifip

cance but we cannot make a statement about the null hypothesis H0 . This means that
the SKUs that are cost-optimally replenished with interchanged supply modes have a
higher demand than the remaining SKUs. Technically spoken, the demand distribu-

Stochastic replenishment – Total saving by switching orders
Wilcoxon rank-sum test for

price (e )

demand (units)

rank sum of MR1
R2

N.A.

15,817.50

standardized value, Gaussian distributed

N.A.

3.89

standard Gaussian distribution

2.33

2.33

rule of rejection

N.A.

T > Φ (.)

null hypotheses

N.A.

H0d rejected

Wn1 ,n2 :
T:

243

Φ(0.99; 0; 1)

H0 :
legend:

MR1
R2 : set of SKUs with exchanged supply modes
n 1 = | M R1
R2 |

n 2 = | M R1 6
R2 |

Wn1 ,n2 : rank sum of MR1
R2

T=

Wn1 ,n2 −0.5n1 (n1 +n2 +1)

√1

12 n1 n2 ( n1 + n2 +1)

N.A.: not applicable

Table 7.6: Wilcoxon Rank Sum Test on price and demand of SKUs with exchanged
supply modes
tion of the SKUs in MR1
R2 dominate the corresponding distribution of the SKUs in
MR1
R2 according to the first order stochastic dominance.

244

C HAPTER 7. C OMPARISON O F S TOCHASTIC R EPLENISHMENT P OLICIES

This result can be explained by the
fact that shorter lead times allow for a

'

$

Info-Box 7.2: SKU B

General information

faster reaction after an unexpectedly high

description:

turbo charger

demand. This is crucial to avoid stock

demand:

(units / year)

125.00

outs especially when the average demand

price:

(e )

534.00

is high. Thus, the stock out costs can

c N,fix ; c N,var :

(e )

4.40 ; 0.63

c R,fix ; c R,var :

(e )

4.40 ; 0.63

be considerably reduced for SKUs with a
high demand. Of course, the additional
costs for the faster supply must not exceed the saved stock out costs. An example is given by SKU B in Info-Box 7.2
where the average demand is 125 per
year and the fast supplier delivers at the
same ordering costs c R,fix and c R,var than
the slow supplier. The total costs can be

Comparison

RS1,trad

RS1,relax

9;2

3;1

R1 ; Q1 :

0 ; 11

1;7

β in %:

66.17

90.19

TC in e :

552.84

441.87

CC in e :

169.11

210.62

OCN in e :

128.75

157.32

OCB in e :

254.98

73.93

µ L1/2 ; σL1/2 :

&

%

reduced by 20.07% while the β service level increases. This is a highly beneficial combination in practice.

7.3.3

Summary

In this section we have focused on cost savings in our warehouse that are induced by
applying single sourcing with relaxed restrictions instead of traditional restrictions.
Moreover, the underlying mechanisms have been investigated. This addresses the
second part of our research question RQ 4.
The two components of the relaxed restrictions are the possibility of negative reorder points and the exchange between the first and the second supply mode, see
Definition 7.1.2 on page 234. Consequently, all SKUs were divided into the set MR1 <0
with a negative reorder point R1 < 0 and the set MR1 ≥0 of the remaining SKUs. Sim-

Stochastic replenishment – Summary of total saving by relaxed restrictions

245

ilarly, we divided all SKUs in a set MR1
R2 where both supply modes are exchanged
and the set MR1 6
R2 . The sets MR1 <0 and MR1
R2 are disjoint which allows us to sumrestric-

SKUs

tion

% of total savings

price (e )

#

%

sum

CC

OCN

OCB

avg

327

11.9

95.84

152.39

23.62

-80.17

387.9

634.0

2.9

4.0

6

0.2

4.16

0.18

-0.24

4.23

266.1

226.3

172.3

130.3

Mrest

2,418

87.9

0.00

0.00

0.00

0.00

35.55

78.5

16.58

80.1

total

2,751

100.0

100.00

152.57

23.38

-75.95

77.9

257.7

15.3

75.8

M R1 <0
M R1
R2

legend:

stddev

demand (units)
avg

stddev

CC: stock holding costs

OCN : normal order costs

OCB : back order costs

stddev: standard deviation

Mrest : Ω \ { MR1 <0 ∪ MR1
R2 }

avg: average

Table 7.7: Contribution of the relaxed restrictions to the total savings
marize our findings in one single Table 7.7.
We showed for our warehouse that the SKUs in MR1 <0 , which were cost-optimally
replenished with a negative reorder point, had a price distribution that stochastically
dominates the distribution of MR1 ≥0 . The demand distribution of MR1 <0 was stochastically dominated by the distribution of MR1 ≥0 . In one example of a very expensive

SKU the total costs were reduced by over 90% by using a negative reorder point. In
total, the SKUs with a negative reorder point contributed 95.84% of all savings. This
means that in our warehouse most of the savings originated from reducing the capital costs by lower inventories. The explanation for this are high prices, large holding
costs, or little demand. These characteristics are common for spare parts and, thus,
the large fraction of total savings induced by a negative reorder point does not come
as a surprise.
The remaining 4.16% of savings came from exchanging the first and the second

246

C HAPTER 7. C OMPARISON O F S TOCHASTIC R EPLENISHMENT P OLICIES

supply mode. Here, we were able to show that the SKUs in MR1
R2 had a price distribution that stochastically dominated the equivalent distribution of MR1 6
R2 . The

savings originated mainly from reduced costs for unmet demand. This is plausible
because substituting the faster supply mode for the slow supply mode allows for a
faster reaction. Especially, in times of unexpectedly high demand this leads to a faster
restocking and less stock out. In one example the total costs were reduced by over
20% while the β service level increased by over 20 percentage points from 66.17% to
90.19%. This combined effect is highly beneficial for companies.

We recall that the relaxed restrictions are one of two options that contribute to the
overall savings. In order to fully answer our research question RQ 4 we investigate
the second option, the usage of a second supplier, in the next section.

7.4

Savings induced by dual sourcing

In this section we have a closer look at the cost savings that can be realized by using
a second supply channel. From Chapter 7.2 we know that these savings are marginal
from a global perspective. The reason is the high percentage of spare parts in our
warehouse for which dual sourcing does not pay off. Therefore, we now go into more
detail on examples where dual sourcing is applied in the cost-minimal situation. This
will help giving a more profound anwer to the second part of question RQ 4.
We will find that the lead times for the first and second supplier are relatively small
in our warehouse. Our suspicion is that the savings will increase with longer lead
times. Therefore, we will investigate the current situation at our warehouse in Section 7.4.1. An example of how longer lead times can affect the total savings achieved
by dual sourcing is given in Section 7.4.2. We summarize our findings in Section 7.4.3.

Stochastic replenishment – Total saving when lead time difference is small

7.4.1

247

Small difference between the lead times

In the following we explain how the cost savings can be achieved if we move from
a single-supplier to a two-supplier replenishment policy with relaxed restrictions
RS1,relax and RS2,relax .

'

$

Let us have a look at the two exam-

Info-Box 7.3: SKU C

ples ”fog lamp right” (SKU C) from Info-

General information
description:

fog lamp right

demand:

(units / year)

price:

(e )

Box 7.3 and the ”mirror glass heated”
1,007.00

(SKU D) from Info-Box 7.4. The demand

27.14

is in average greater than one item per

Lead time information

day for both SKUs which distinguishes

µ l1 :

9

σl1 :

2

µ l2 :

3

σl2 :

1

&

and µl2 = 3 days, respectively.
Especially the value for µl1 is small
compared to the lead times of overseas

them from the low-demand spare part
we have discussed in earlier chapters.

%

The average lead times are µl1 = 9 days

'

$

Info-Box 7.4: SKU D

General information

suppliers which can easily exceed 30 days.

description:

mirror glass heated

Still, we will see that the second supplier

demand:

(units / year)

307.00

is able to reduce the total costs for these

price:

(e )

139.75

short lead times.
Table 7.8 gives a comparison between
the two supply scenarios RS1,relax and
RS2,relax for each of the two SKUs. In the

Lead time information
µ l1 :

9

σl1 :

2

µ l2 :

3

σl2 :

1

&

%

first row we find that the savings in total costs TC is 8.24% for SKU C and 7.42% for
SKU D. Usually, cost reductions of this magnitude are already quite attractive for
companies especially if they are coupled with an increase of the β service level. We
can find this combination of cost reduction and service level increase for SKU C. In

248

C HAPTER 7. C OMPARISON O F S TOCHASTIC R EPLENISHMENT P OLICIES

fact, all costs (CC, OCN , and OCB ) are reduced for SKU C with the introduction of the
second supplier. For SKU D the second supplier reduces the capital costs CC to such
an extent that the increases of OCN , OCR , and OCB are more than compensated.
SKU C: fog lamp right

TC
β in %
R1

RS1,relax

RS2,relax

∆ in %

RS1,relax

RS2,relax

∆ in %

544.59

499.69

-8.24

456.01

422.16

-7.42

98.12

100.00

1.92

98.48

96.77

-1.73

54

52

14

11

22

R2
Q1

SKU D: mirror glass heated

61

Q2
p(2, ., .) in %

2

50

26

21

14

10

24.01

14.46

CC

143.99

128.12

-11.02

301.10

226.18

-24.88

OCN

341.89

334.77

-2.08

132.96

135.54

1.94

36.80

N.D.

13.93

N.D.

0.00

-100.0

46.52

111.86

OCR
OCB
legend:

58.71
∆:
β:

RS2,relax
RS1,relax

−1

β service level

21.96
p(2, ., .):
N.D.

2-order probability
not defined

Table 7.8: Total cost savings by the second supply channel with relaxed restrictions
From Table 7.8 we further see that the reorder point R1 and the order quantity Q1
are reduced for both SKUs by moving to a replenishment with two suppliers. This
reduction of R1 and Q1 decreases the average stock and leads to less stock holding
costs CC. A large cost reduction for SKU C is achieved by increasing the β service
level to 100% which saves 58.71e of back order costs OCB .
In this section we have seen that the introduction of a second supplier can lead

Stochastic replenishment – Total saving when lead time difference increases

249

to reduced total costs even when the difference between the lead times of the two
suppliers are rather small. Moreover, the introduction of a second supplier can have
the positive side effect to increase the β service level. In our warehouse, the TC savings
are 8.24% and 7.42% for 2 SKUs with lead times of µl1 = 9 days and µl2 = 3 days. We
expect the usage and the savings effect of the second supplier to increase even further
for a long lead time µl1 .

7.4.2

Increasing difference between the lead times

Many companies have suppliers in other continents and need to consider long lead
times for their supply chain calculations. The lead times of overseas shipments can
easily exceed several weeks or months. Naturally, we expect that the total costs will
increase with increasing lead times. In addition, we expect that the saving in total
costs increases ceteris paribus for long lead times, as well.
In our spare parts warehouse the maximal lead time is µl1 = 10 days for ar-

'

General information

ticles where dual sourcing is possible.

description:

brake repair kit

This is very short. Thus, we take the ex-

demand:

(units / year)

ample of a ”break repair kit” (SKU E) as

price:

(e )

described in the Info-Box 7.5 with its initial lead times and increase µl1 = 9 iteratively by 2 days until it reaches 27 days.
The initial savings by RS2,trad compared
to RS1,trad are 15.76e or 2.05% and we
expect these values to increase for longer

$

Info-Box 7.5: SKU E

851.00
84.00

Lead time information
µ l1 :

9

σl1 :

2

µ l2 :

3

σl2 :

1

RS1,trad

RS2,trad

768.31

752.55

95.34

96.40

Comparison
TC:
β in %:
&

%

lead times.
For each value of µl1 we optimize the replenishment parameters regarding total
costs for both scenarios RS1,trad and RS2,trad . Then we determine the difference of to-

250

C HAPTER 7. C OMPARISON O F S TOCHASTIC R EPLENISHMENT P OLICIES

tal costs between the two scenarios and observe how this difference develops with
increasing values for µl1 . We deliberately use the traditional replenishment restrictions here to avoid a switch of the first and the second supplier as we are interested
in the saving effect of a second supplier when such a switch is not possible, see Definition 7.1.1 on page 234. Many times such a switch is not possible in reality at least in
short term due to capacity restrictions or supply contracts, for example.
The results of RS1,trad and RS2,trad and their differences are given in Table 7.9 for
increasing values of µl1 . The standard deviation σl1 of the lead time increases together
with µl1 in such a way that the coefficient of variation remains constant at 29 . First, we
clearly see that the absolute value of TC increases monotonously for both scenarios.
This is not surprising as we do not allow for overlapping reorder cycles and longer
lead times require a higher average stock to limit the amount of stock outs. Second,
the relative savings ∆ TC between RS1,trad and RS2,trad monotonously increase from
about 2% for µl1 = 9 days up to more than 20% for µl1 = 27 days. These observations
coincide with our expectations but let us take a closer look at the mechanisms behind
the increasing benefits of using a second supplier for large values of µl1 .
On the one hand, we find in Table 7.9 that the probability to use the second supplier increases almost monotonously from 8.96% to 99.97% with a large jump between
µl1 = 9 and µl1 = 11. On the other hand, the quantity ordered via the first supplier,
Q1 , is always lower in the dual-source scenario RS2,trad . Both developments have the
effect to raise the influence of the second supplier in terms of amount of customer
demand being satisfied by the individual supplier. Due to this rising influence of the
second supplier and its constant lead time of µl2 = 3 the warehouse is able to remain
reactive to excessive demand even for high values of µl1 – unlike in the RS1,trad scenario. This allows the warehouse manager to keep the average stock level and its costs
at a low level without decreasing the β service level. The development of the warehousing costs CC for RS1,trad and RS2,trad can be observed in Table 7.10 in more detail.

Stochastic replenishment – Total saving when lead time difference increases
∆ TC

∆β

RS1,trad

µ l1

σl1

9

2.00

2.05

1.06

768.3

95.34

11

2.44

6.14

2.43

803.9

13

2.89

11.59

4.44

15

3.33

12.07

17

3.78

19

in %

251

RS2,trad
p(2, ., .)

Q1

Q2

96.40

8.69

51

14

754.6

95.08

98.24

19

45

65

758.3

94.40

99.31

19

45

95.97

67

770.5

96.07

99.80

19

57

906.4

93.38

78

770.8

94.11

99.91

33

45

3.72

952.9

91.95

91

779.9

95.67

99.93

33

57

19.09

0.07

980.1

94.41

91

793.0

94.48

99.94

33

57

5.11

20.85

8.02

1,025.6

87.83

104

811.8

95.84

99.97

33

69

25

5.56

20.95

1.24

1,044.6

93.41

105

825.8

94.65

99.98

33

69

27

6.00

21.67

3.03

1,073.3

91.26

114

840.7

94.29

99.97

47

69

TC

β

52

752.5

92.65

53

857.7

89.96

0.11

876.2

14.96

0.73

4.22

18.16

21

4.67

23

legend:

TC

β

Q1

µ l1 :

average lead time 1

TC:

total costs

∆ TC :

β service level

∆β:

2-order probability

Qi :

β:
p(2, ., .):

σl1 :

standard deviation lead time 1
TCRS

1,trad

− TCRS2,trad

TCRS

1,trad

β RS2,trad − β RS1,trad

order quantity supplier i

Table 7.9: Comparison of single and dual sourcing for an increasing lead time µl1

Consequently, one main reason for the increasing gap in total costs between RS1,trad
and RS2,trad is the moderate increase of the capital costs due to a more extensive usage
of the second supplier with its constant lead time µl2 = 3.

Another potential source of cost savings is related to the service level. Interestingly,
we find a non-monotonous change of the β service level for each of the scenarios in
Table 7.9. One might have expected a declining value of β due to increased lead times.
In fact, this non-monotonic behavior can be explained by the pure cost-optimization

252

C HAPTER 7. C OMPARISON O F S TOCHASTIC R EPLENISHMENT P OLICIES

µ l1

σl1

9

∆ TC

RS1,trad

in %

TC

2.00

2.05

11

2.44

13

RS2,trad

CC

OCN

OCB

TC

CC

OCN

OCR

OCB

768.3

369.7

299.2

99.5

752.5

370.7

293.5

11.5

76.9

6.14

803.9

349.1

297.9

157.0

754.6

305.3

126.2

218.1

105.1

2.89

11.59

857.7

357.8

285.3

214.7

758.3

294.5

125.2

218.8

119.8

15

3.33

12.07

876.2

506.4

283.6

86.2

770.5

360.7

105.1

220.8

83.9

17

3.78

14.96

906.4

488.9

276.0

141.5

770.8

321.5

143.5

179.8

126.0

19

4.22

18.16

952.9

511.5

269.4

172.0

779.9

376.4

124.4

186.5

92.6

21

4.67

19.09

980.1

591.3

269.4

119.4

793.0

364.2

124.4

186.5

118.0

23

5.11

20.85

1,025.6

501.0

264.5

260.2

811.8

421.6

109.7

191.6

88.8

25

5.56

20.95

1,044.6

639.6

264.1

140.9

825.8

410.0

109.7

191.6

114.4

27

6.00

21.67

1,073.3

625.1

261.4

186.8

840.7

426.0

124.2

168.5

122.0

legend:

σl1 :

standard deviation lead time 1

µ l1 :

average lead time 1

TC:

total costs

∆ TC :

TCRS1,trad − TCRS2,trad
TCRS1,trad

CC:

capital costs

OCN :

normal order costs

rush order costs

OCB :

back order costs

OCR :

Table 7.10: Cost development for RS1,trad and RS2,trad for an increasing lead time µl1

of the replenishment parameters of both suppliers where the actual value of β is just
a side effect of balancing the different cost components. Coincidentally, in a pairwise
comparison the scenario RS2,trad achieves better β service levels than RS1,trad for every
value of µl1 . This positive effect of the second supplier on the β service level should
not be underestimated. Many companies have to put a lot of effort and intelligence
into keeping client service levels at an acceptable level. This is especially true for long
lead times. However, the dual-source replenishment RS2,trad cannot always guarantee
a better service level than RS1,trad for the mentioned reason that β is just a side effect
of the cost-optimization.

Stochastic replenishment – Summary of total savings by dual sourcing

253

We have seen a case where the introduction of a second supplier decreased the
total costs but also led to a worse β service level. This refers to SKU D, see Table 7.8 on
page 248. Despite the existence of cases where β decreases by shifting to dual sourcing,
RS2,trad makes it possible to leverage the much shorter lead time of a second supplier.
This allows a faster reaction in the replenishment and, thus, can mitigate the negative
effect of an exceptional high demand on the β service level. In our warehouse RS2,trad
saves a lot of back order costs OCB especially where µl1 = 23, see Table 7.10. The same
holds for all other values of µl1 even if the amount of savings are not that high.
Last but not least Table 7.10 reveals that the ordering costs OCN + OCR of RS2,trad
are higher than the ordering costs OCN of the single-source replenishment RS1,trad .
This observation coincides with the common experience that the usage of a second,
more expensive supplier leads to additional costs. Of course, these additional replenishment costs make only sense if they lead to other savings of at least the same amount.
In summary, we can state that there do not only exist SKUs in our warehouse where
the availability of a second supplier in the RS2,trad scenario reduces the total costs, but
more importantly, that there are SKUs for which the savings in total costs increase with
a longer lead time µl1 of the first supplier. In our case of SKU E the savings increased
from 2% for µl1 = 9 days to 21% for µl1 = 27 days.

7.4.3

Summary

In the last two sections we have investigated the cost saving effect by using dual sourcing instead of single sourcing. First, we saw that the introduction of a second supplier
can lead to reduced total costs and even an increased β service level.
Second, we saw exemplarily that the reduction of total costs by introducing a second supplier increases for longer lead times of the first supplier. These findings show
that dual sourcing can be highly beneficial when the demand and both lead times are
stochastic. The detailed summary is moved to the next section.

254

7.5

C HAPTER 7. C OMPARISON O F S TOCHASTIC R EPLENISHMENT P OLICIES

Summary

In this chapter we have compared four combinations of stochastic replenishment policies and their execution with different restrictions for our warehouse of 2,751 SKUs. In
particular, we were interested in how much total costs can be saved in a stochastic environment if we move from a single-source replenishment with traditional restrictions
to a dual-source replenishment with relaxed restrictions. Moreover, we investigated
the mechanisms that lead to these savings. This relates to question RQ 4. It was possible to represent and calculate all four scenarios with the SDMR model. This enabled
us to eliminate effects on the results by different modeling approaches and proved the
flexibility of the SDMR model once more.
After a proper definition of the four scenarios in Chapter 7.1 we quantified the
savings of total costs in our warehouse in Chapter 7.2. Our findings were three-fold.
First, total costs can be reduced by 5.7% if a dual-source replenishment policy with
relaxed restrictions is used instead of a single-source replenishment policy with traditional restrictions. Second, dual sourcing is not used for most SKUs in our warehouse
due to the low demand of most spare parts. Consequently, their contribution to the
overall savings of total costs is low. Third, the main part of the savings in total costs
originates from the relaxed restrictions.
Regarding the first part of question RQ 4 we conclude at this point that moderate
savings of 5.7% are induced by moving from traditional single sourcing to dual sourcing with relaxed restrictions in the example of our warehouse. Most of the savings are
contributed by the relaxed replenishment restrictions and not by dual sourcing.
Relaxed restrictions.

The savings and the mechanisms related to the relaxed restric-

tions were analyzed in Chapter 7.3 in more detail. The two components of the relaxed
restrictions were first, the possibility of negative reorder points and second, the ex-

Stochastic replenishment – Summary

255

change between the first and the second supply mode. On the one hand, the effect of
negative reorder points contributed 95.84% of all savings. The reason are lower capital
costs CC due to the reduced amount of inventory. Interestingly, we were able to show
that all SKUs which were cost-optimally replenished with a negative reorder point
had a higher price and a lower demand than the rest of all SKUs in our warehouse.
In one example of a very expensive SKU the total costs were reduced by over 90%
by this effect. On the other hand, the remaining 4.16% of total savings resulted from
exchanging the first and the second supply mode. The savings were mainly caused
by lower costs for unmet customer demand. This is the result of a shorter lead time
which enables to replenish faster in case of unexpectedly high demand. Here, we were
able to show that the SKUs in the set MR1
R2 , which are cost-optimally replenished
by only using the faster second supplier, had a demand distribution that dominated
the corresponding distribution of the remaining SKUs in MR1 6
R2 according to the first

order stochastic dominance. In the case of one SKU with high demand the total costs
were reduced by over 20% while the β service level increased from about 66% to over
90%. This effect is highly beneficial in practice.
In respect to question RQ 4 and the relaxed replenishment restrictions we can identify two major mechanisms that lead to cost savings in our warehouse. First, the negative reorder points cause a reduction of inventory and its related capital costs CC.
Second, the substitution of the faster supplier for the slower and possibly cheaper
supplier allows for a faster replenishment which leads to lower stock out costs especially in times of exceptionally high demand.

Dual sourcing. In Chapter 7.4 we investigated the savings and the mechanisms related to dual sourcing in our warehouse when the lead time µl1 of the first supplier is
short or long. Our findings were three-fold. First, the introduction of a second supplier lead to reduced total costs by over 7% for two examples in our warehouse even

256

C HAPTER 7. C OMPARISON O F S TOCHASTIC R EPLENISHMENT P OLICIES

when the supply lead times were as short as µl1 = 9 days and µl2 = 3 days. Second,
the introduction of a second supplier increased the β service level for two of our three
examples in addition to the reduced total costs even for small lead times. Third, we
saw exemplarily for one SKU that the reduction of total costs by introducing a second
supplier increased from about 2% to over 20% when the lead time of the first supplier
rose from 9 days to 27 days.
Our conclusion for question RQ 4 is a large benefit of dual sourcing for several
SKUs in our warehouse. The savings can exceed 20% especially when the lead time of
the first supplier is long. The main reason for the savings are reduced stock out costs
due to the possibility of a faster replenishment.
Wrapping up, this chapter allows to give the following answers and insights regarding
our research question RQ 4. First, it shows that there is only a relatively small total
saving potential for dual sourcing with relaxed restrictions in a warehouse with little
demand (spare parts) and small lead times of 10 days.
Second, for single SKUs the usage of relaxed restrictions and dual sourcing can reduce the total costs significantly. We saw examples where the total costs were reduced
by more than 20% and even more than 90% for one SKU. In several cases the usage of
the second supplier did not only decrease the total costs but also increase the customer
service level. This combined effect is very interesting and advantageous in practice.
We also saw that the savings increased with longer lead times for the first supplier.

Chapter 8
Summary and outlook
The central research question throughout this work has been whether it is beneficial
to use dual sourcing in a stochastic environment, and if yes, whether it is feasible to
use a deterministic approximation, see RQ 0 on page 4.
In order to answer this question we need to bring together the answers of our
questions RQ 1 to RQ 4. These four questions have been addressed throughout the
previous chapters and will be recapped in the following together with our contributions. We will also critically review our approach and its limitations and give a short
outlook to possible future work in this area.

8.1

Contributions to the research area

The contribution to the literature on dual sourcing is mainly related to our question RQ 11 . We have elaborated the SDMR model in Chapter 3 which represents a

( R1 , R2 , Q1 , Q2 ) replenishment policy, where the demand and both lead times have
arbitrary distributions. To our knowledge it is the first publication on a dual-sourcing
1

RQ 1: How can we define and model a ( R1 , R2 , Q1 , Q2 ) replenishment policy with stochastic demand
and lead times?

257

258

C HAPTER 8. S UMMARY A ND O UTLOOK

( R1 , R2 , Q1 , Q2 ) replenishment policy that allows for arbitrary distributions for the
daily demand and the stochastic lead times. Only a few other publications consider
stochastic demand and stochastic lead times. All of them make strong assumptions
on the type of the distribution like Poisson demand and exponential lead times, see
[VAdK98], [MP99].
We have given exact formulas for the expected shortage, approximated expressions
for expected stock, and a series of formulas for various essential variables in inventory
management like different service levels, number of orders for each supply mode, and
total costs. The advantage of the SDMR model is its flexibility. For example, it takes
moderate effort to change the SDMR model from the scenario where unmet customer
demand is backlogged, see Chapter 3, to a scenario where backlogged demand is lost,
see Appendix D.
Our contribution to the practical application of the SDMR model relates to RQ 22 and
can mainly be found in Chapter 4. A key requirement for a practical application is the
discretization of the SDMR model. We have highlighted two important aspects. First,
we have shown how to discretize different parameters of the SDMR model according
to specific business scenarios. Second, we have investigated how to determine the
convolution of distributions, usually a time-intensive and heavily used calculation,
more time-efficiently by means of a FFT algorithm.
The investigations regarding question RQ 33 contribute to the insights on the deviations between a scenario where the demand and both lead times are stochastic and an
approximation where the lead times are assumed to be deterministic. Our investiga2

RQ 2: How can the SDMR model be applied in practice?

3

RQ 3: How much and in which situations does a scenario with deterministic lead times deviate from
the stochastic scenario of the ( R1 , R2 , Q1 , Q2 ) replenishment policy?

Summary and outlook – Contributions

259

tions have been two-fold. First, in a sensitivity analysis we have found that the range
of deviation between DET and STOCH is very high. DET underestimates some of our
KPIs up to 100% while it overestimates other KPIs usually by 40% and above for at
least one value of each input parameter. The reasons for a deviation between DET
and STOCH are very diverse and one has to be aware of the rather complex interplay
between the various input parameters. The conclusion from the sensitivity analysis is
to clearly favor the complex STOCH scenario for calculating our KPIs in a setting with
stochastic lead times because the magnitude of deviation of DET is hardly predictable.
Second, we have compared the results of the sensitivity analysis with a case study
in form of a spare parts warehouse. In general, we have found that the deviations
between DET and STOCH are much smaller than in the sensitivity analysis. For some
KPIs there exists almost no deviation. Larger deviations occur mostly regarding the
β service level. We have observed that the deviations are not only influenced by the
stochastic variables but also by the optimization of the replenishment parameters like
R1 or Q2 . For several KPIs we could analytically express and explain the deviation
between DET and STOCH. However, all expressions contain at least one term that is
complex and calculation-intensive to determine like the service level or the probability
of a two-order cycle in the STOCH scenario. Thus, there is no general and easy way
to predict the deviation between DET and STOCH.
This brings us to the conclusion that the approximation with deterministic lead
times might be sufficient for calculating several KPIs for many SKUs even when both
lead times are stochastic. However, great discrepancies can occur regarding the shortage costs and the β service level. Unfortunately, it is hard to predict the magnitude
of the deviation due to complex interplays between the various parameters. Summa
summarum, an answer to research question RQ 3 is anything but trivial and strongly
depends on the specific situation.

260

C HAPTER 8. S UMMARY A ND O UTLOOK

Our analysis in context of question RQ 44 yield several contributions regarding the
cost savings and the responsible mechanisms in the context of stochastic dual sourcing. First, we show that relaxed replenishment restrictions can be very beneficial. On
the one hand, the substitution of the fast supplier for the slow and usually cheaper
supplier has proven to reduce the total costs by more than 20% while increasing the β
service level at the same time. This effect is very beneficial and applies especially for
SKUs with a high demand. On the other hand, negative reorder points can save a significant amount of inventory holding costs especially for expensive and low-demand
SKUs. In one example the savings exceeded 90% of the total costs. Instead of a negative reorder point R1 < 0 one can set R1 = 0 and delay the replenishment by some
time as described by Schulz [Sch89].
SKU attributes

replenishment option to consider

section

low demand, high price

negative reorder point

7.3.1

p. 238

high demand

substituting fast supplier for slow supplier

7.3.1

p. 238

long lead time for slow supplier

dual sourcing

7.4

p. 246

Table 8.1: Findings regarding different stochastic replenishment policies
Second, dual sourcing can have large benefits over single sourcing in a scenario
with stochastic demand and stochastic lead times. Thereby, the savings depend
strongly on the characteristics of a SKU and the applied replenishment restrictions.
While dual sourcing is not applied for many spare parts due to their low demand, the
relaxed replenishment restriction of a negative reorder point is very beneficial even
for single sourcing. There exist SKUs where dual sourcing can reduce the total costs
by more than 20% especially for long lead times of the first supplier. In addition, dual
sourcing increased the β service level in several cases. This combination of reduced
4

RQ 4: How much total costs can be saved by moving from single sourcing with traditional restrictions
to dual sourcing with relaxed restrictions when lead times are stochastic?

Summary and outlook – Contributions

261

costs and an increased service level is very attractive for most companies. We summarize the findings in Table 8.1.
Finally, we bring all these results together in order to answer our research question
RQ 0. We have seen from the answers to RQ 1 and RQ 2 that we are able to formulate
and practically implement a model for stochastic ( R1 , R2 , Q1 , Q2 ) dual sourcing. In the
context of RQ 4, dual sourcing has been able to save more than 20% of total costs and
even increase the β service level in some cases compared to single sourcing. Moreover,
the substitution of the fast supplier for the slow supplier also yielded savings of over
20% while increasing β. However, for SKUs with a low demand the option of a second
supply mode is usually not exercised. These results are also summarized in Table 8.2.
area

contributions

model for ( R1 , R2 , Q1 , Q2 )

independent of distribution type

replenishment policy with

exact formulas for expected shortage

stochastic demand and lead
times (RQ 1)
practical application (RQ 2)
comparison of KPIs between

approximated formulas for expected stock
formulas for important KPIs (e.g. service levels, costs)
adaptation of the model to various business scenarios
feasible calculation of convolutions
sensitivity analysis (cost-independent)

stochastic model and

shows strong deviation

approximation with

case study (spare parts warehouse)

deterministic lead times (RQ 3)

shows small deviation

different saving potentials

negative reorder point

in a stochastic setting
(case study, RQ 4)

substituting fast supplier for slow supplier
dual sourcing for longer lead times

Table 8.2: Contributions to the area of dual sourcing

262

C HAPTER 8. S UMMARY A ND O UTLOOK

Consequently, the answer to the first part of RQ 0 is: Yes, dual sourcing is beneficial in several cases where the demand and all lead times are stochastic. The savings
usually increase with an increasing lead time of the primary supplier.
The answer to RQ 3 has shown that the results of the stochastic scenario can substantially deviate from an approximation where both lead times are deterministic. The
answer to the second part of RQ 0 is: An approximation with deterministic lead
times is usually not feasible especially for the β service level and the shortage costs.
While it might be feasible for certain SKUs and their KPIs it is hard to predict the
magnitude of deviation beforehand.

8.2

Critical review

Despite our ambition to make the assumptions for our SDMR model as little restrictive
as possible, there still exist a few assumptions in Section 3.1.1 on page 42 that can be
restrictive in practice.

Ad Assumption 5. The assumption states that the daily demand is identically and
independently distributed over time. This might not be true, of course. Currently, we
are using the demand distribution D (t) which is the convolution of t identical daily
demand distributions D (1). However, it is mostly a technical effort to replace D (t)
by a convolution D 0 ( a, a + t − 1), where the t single-day demand distributions for the

days a, a + 1, ..., a + t − 1 are different. Of course, one has to take care about the specific
t-day time interval at hand as D 0 ( a, a + t − 1) 6= D 0 ( a + 1, a + t) in general. Besides

this, all equations of the SDMR model can be used as before. Theoretically spoken, one
could analogously incorporate correlated demand over time into the SDMR model, as
well. However, the quickly growing problem space will most likely hinder a practical
application.

Summary and outlook – Critical review
Ad Assumption 7.

263

The most restrictive assumption is that at most one order can be

outstanding per supply mode at any point in time. Especially when lead times are
long, there usually exist several outstanding orders in practice. An appropriate extension of the SDMR model to one or more outstanding orders causes substantial effort.
All equations for the shortage and the stock level have to be extended to the different possible arrival times of these outstanding orders. This requires at least one more
integration within most of the formulas. Nevertheless, this extension is very interesting and crucial for an application in industries with long lead times like overseas
shipments of several months.
Ad Assumption 8.

We do not allow for a cumulated demand that exceeds the supply

within a replenishment cycle. This assumption should be met in practice at least in the
long run. Otherwise one is bound to run out of stock sooner or later. In combination
with several outstanding orders, which we excluded in this work, this assumption
will generally not hold as many orders arrive before the one arrives that has just been
placed.
Ad Assumption 10. For the formulation of the SDMR model we considered time to
be continuous and demand to be non-bulky so that a supply order is immediately
processed at the point in time when the stock level reaches R1 or R2 . Usually this
is not true in practice. Especially, when the supply process is restricted by certain
departure times of supplying vehicles one has to wait a certain fraction of the time
interval after reaching R1 until the next departure is scheduled. Chapter 4 shows how
this assumption can be relaxed by approximative approaches where the demand distribution, the lead time distribution, or the reorder points are adjusted accordingly.
These approximative approaches allow for a practical application of the SDMR model
in many practical approaches.

264

C HAPTER 8. S UMMARY A ND O UTLOOK

Besides the criticism of the assumptions we want to mention a few points one should
keep in mind about the SDMR model. First, other replenishment policies than the

( R1 , R2 , Q1 , Q2 ) policy with backlogged demand might be required in certain situations. In the case of lost sales we have shown that the flexibility of the SDMR model
allows for certain extensions and adaptation without too much effort, see Appendix D
on page 305. We expect a similar effort to adapt the SDMR model to a ( R1 , R2 , S1 , S2 )
base-stock policy, for example. For other dual-source replenishment policies one has
to decide case by case whether and how they can be incorporated in the SDMR model.
Second, once we move from pure planning to daily operations, we have to be
aware of a repetitive application of the SDMR model. Currently, we are not considering the possibility that the situation and information tomorrow might deviate from
our predictions today. This is especially critical when an order from the past is still
outstanding. This situation relates somewhat to our comments on Assumption 7 before. Moreover, the demand and the lead time distributions might change the next
day and, thus, also the optimal values for the replenishment parameters R1 , R2 , Q1 ,
and Q2 will change. In practice this situation can happen frequently and the question
is how to apply the SDMR model repetitively day after day. Of course, we could simply ignore prior days as a first approximation and apply the SDMR model each day
anew. If there is an outstanding normal order we will still calculate R1 , Q1 , R2 , and
Q2 but only the latter two parameters will be applied for the second supply mode. In
addition, one could substitute the remaining lead time distribution of the outstanding normal order for the normal lead time distribution L1 . While this approach for a
repetitive usage of the SDMR model could be feasible in practice, it is anything but
satisfying from a theoretical point of view.
Third, the optimization of the SDMR model for our warehouse with 2, 751 SKUs
takes about 12 hours. Large warehouses easily contain more than 100, 000 SKUs. Thus,
the current implementation of the SDMR model is not feasible in such cases.

Summary and outlook – Outlook

8.3

265

Outlook

One can imagine many extensions and adaptations of the current SDMR model. We
only want to mention a few. Probably the most apparent extensions are more policies
like the base-stock policy, more practical constraints like warehouse capacities, and
the introduction of a third supplier. However, in our opinion the most interesting extension of the SDMR model is the consideration of several outstanding orders. This
allows for applying the SDMR model in scenarios with long lead times. Further, it
enables us to use the SDMR model not only for planning but also for daily operations
where an order has been placed just the day before, for example. Closely related to the
daily operational usage of the SDMR model is the consideration of current purchase
prices on the market and the decision whether it is beneficial to purchase today even
though the inventory is still sufficiently high. This is an exciting but non-trivial extension.
In practice, inventory management faces a multitude of further restrictions and considerations every day. These can significantly influence the timing and magnitude of
a replenishment order. Common examples are minimum order quantities, lot sizes,
and certain time limits of the supplier. More complex topics include rules for ordering
a bundle of different SKUs from the same supplier, price discounts, or adjustments to
better utilize the capacity of the supplying trucks. These topics are already quite hard
to optimize individually, not to mention a combination with dual sourcing.
Future improvements for the current implementation of the SDMR model should
target at a better time performance especially regarding the employed optimization
algorithm. Only this way an application in large warehouses is practically feasible.

Part IV
Appendices

267

Appendix A
Explanations regarding the used
probability space
A.1

Example of a discrete random variable and its σalgebra

In the simple case where the random variable X only maps to a countable number of
different values a1 , a2 , ..., an , so |range( X )| = n, the expected lead time lt1 is given by
E[ Xlt1 ] =

n

∑ ai · P ( Ai )

(A.1.1)

i =1

with Ai = X −1 ( ai ). In this case Ai ∈ κ always holds for i ∈ {1, 2, ..., n} because the

random variable X has to be measurable concerning the σ-algebra κ . This means that
the set of elements in Ω specified by A j = X −1 ( x ≤ t) has to be an element of κ for

all possible t ∈ R. Moreover, also its complement A j = Ω\ A j has to be a member

of κ due to the definition of a σ-algebra, see Definition 3.1.2. Then, by intersection of

appropriate sets in κ one can extract exactly the set Ai = X −1 ( ai ) which, by the very
definition of a σ-algebra, has to be an element of κ , as well.
269

270

A PPENDIX A. E XPLANATIONS R EGARDING T HE U SED P ROBABILITY S PACE

Example A.1.1. Ω = {(0, 1], (1, 2], (2, 3], (3, 4]} defines a sample space of intervals in

R1 . One possible σ-algebra can be specified by


κ = Ω, ∅, {(0, 1], (1, 2]}, {(2, 3], (3, 4]},


{(0, 1], (1, 2], (2, 3]}, {(3, 4]}, {(0, 1], (1, 2], (3, 4]}, {(2, 3]} .

Moreover, a real function X on Ω is given.

X:Ω→R=




(0, 1] 7→ −10






(1, 2] 7→ −10



(2, 3] 7→ 20





(3, 4] 7→ 40

For every arbitrary t ∈ R the set A = X −1 ( x ≤ t) has to be an element of the σ-algebra

κ , due to the definition of measurability, see Definition 3.1.5. In our case the results
are

t < −10 :

A−10 = ∅

t < 20 :

A20 = {(0, 1], (1, 2]}

t < 40 :

A40 = {(0, 1], (1, 2], (2, 3]}

t ≥ 40 :

A≥40 = {(3, 4]}.

All sets – A−10 , A20 , A40 , and A≥40 – are members of κ and it follows that X is mea-

surable on κ . Moreover, X is a real function X : Ω → R. Thus, X is a random variable
according to Definition 3.1.4.

Appendix A – Expected value of a continuous random variable

A.2

271

Expected value for a continuous random variable

In the simple case where the random variable X only maps to a countable number of
different values a1 , a2 , ..., an , so |range( X )| = n, the expected lead time lt1 is given by
E[ Xlt1 ] =

n

∑ ai · P ( Ai )

(A.2.1)

i =1

with Ai = X −1 ( ai ). Note, that Ai ∈ κ always holds for i ∈ {1, 2, ..., n} because the ran-

dom variable X has to be measurable concerning the σ-algebra κ , see Appendix A.1
for an example.
In the continuous case we have to use Borel sets for the sample space Ω. Furthermore, the distribution function F, as defined in Definition 3.1.4, may be used. According to Feller one can now specify an arbitrary random variable X, an arbitrary ε > 0,
and two simple random variables X and X with | range( X ) | = | range( X ) | = n,
X = X + ε, and X ≤ X ≤ X. Whenever E[ X ] and E[ X ] exist then
E[ X ] ≤ E[ X ] ≤ E[ X ]

(A.2.2)

must hold for any reasonable definition of E[ X ], see [Fel71]. With this in mind one can
define the expected value E[ X ] of an arbitrary random variable X. Thereby, the two
notations

∞

E[ X ] = lim ∑ k · ε · P ((k − 1) e < X ≤ k ε)
and

ε →0 − ∞

E[ X ] =

Z∞

t

−∞

t · FX {dt}

(A.2.3)

(A.2.4)

are equivalent. Note, that the expression FX {dt} actually refers to an interval function
FX { I } that assigns a probability value to the interval I = (t − ε, t] where I = ε = dt.

Consequently, if ε → 0 the expression FX {dt} represents the density function f X (t)

associated with the distribution function FX {( − ∞, t]} = FX (t). So we can rewrite the

272

A PPENDIX A. E XPLANATIONS R EGARDING T HE U SED P ROBABILITY S PACE

formula for the expected value of X as

E[ X ] =

Z∞

t

−∞

t · f X (t) dt.

(A.2.5)

Note, these findings are not restricted to one-dimensional sample spaces but also valid
for random variables that are based on a σ-algebra generated by the Borel sets in R n .

A.3

Sum of expected values using indicator functions

In this section we give proofs for several conditions in which the expected value of
one random variable used in combination with several indicator functions and their
underlying sets is simply the expected value of the random variable in combination
with one indicator function based on the union of all individual sets.

Proof. Let X1 , X2 : Ω → R1 be arbitrary random variables on the sample space Ω with
a σ-algebra κ , two individual density functions f X1 , f X2 , and a joint density function

f . Moreover, the sets A1 and A2 are specified by the random variable X2 according
to A1 = X2 −1 ( B1 ) and A2 = X2 −1 ( B2 ) where B1 , B2 ⊂ R1 . Whenever X2 ( A1 ) ∪

Appendix A – Sum of expectations with indicator functions

273

X2 ( A2 ) = X2 (Ω) = B1 ∪ B2 and A1 ∩ A2 = ∅ it then holds
E [ X1 1 A 1 ] + E [ X1 1 A 2 ]
Z∞

=

Z∞

x1

x2

1 A1 x1 f ( x1 , x2 ) dµ +

−∞ −∞



Z∞

=

−∞

Z∞

=

=

x1

x1 f ( x1 , x2 ) dx2 +

X2 ( A 1 )

Z∞

x2

x2

Z∞

x1

1 A2 x1 f ( x1 , x2 ) dµ

−∞ −∞

Z




x1

Z∞

Z

X2 ( A 2 )




x1 f ( x1 , x2 ) dx2  dx1

x1 f ( x1 , x2 ) dµ

−∞ −∞
Z∞

x1 f X1 ( x1 ) dx

x1

−∞

= E [ X1 ] .

(A.3.1)

This shows that the expected value of a random variable can be simply computed by
sum of two expected values if the restricting sets A1 and A2 are thoughtfully chosen.

Note, that f ( x1 , x2 ) = f X1 ( x1 ) · f X2 ( x2 ) in case of statistical independence of X1

and X2 like in our case of lead time and demand. Moreover, this proof can easily be
extended to several disjoint sets A1 , ..., An .
Proof. Using the same assumptions as in the proof before the equations
E[ X1 1 A1 ] + ... + E[ X1 1 An ]


=

Z∞

x1

−∞




X2 ( A 1 )

= E [ X1 ]
hold whenever

[

Z

i ∈{1,...,n}

x1 f ( x1 , x2 ) dx2 + ... +

Z

X2 ( A n )




x1 f ( x1 , x2 ) dx2  dx1

(A.3.2)

Ai = Ω and Ai ∩ A j = ∅ for all i, j ∈ {1, ..., n} and i 6= j.

274

A PPENDIX A. E XPLANATIONS R EGARDING T HE U SED P ROBABILITY S PACE

This concept can be simply applied to arbitrary disjoint sets that not necessarily
have to sum up to Ω.
Proof. For two κ -measurable sets A1 and A2 with A1 ∩ A2 = ∅ and two random variables X1 and X2 with X2 ( A1 ) ⊆ R1 and X2 ( A2 ) ⊆ R1 it holds
E [ X 1 A1 ∪ A2 ]

Z∞

=

Z

x1

−∞

Z∞

=

x1

−∞

x1 f ( x1 , x2 ) dx2 dx1

X2 ( A 1 ∪ A 2 )





Z

x1 f ( x1 , x2 ) dx2 +

X2 ( A 1 )

Z

X2 ( A 2 )




x1 f ( x1 , x2 ) dx2  dx1

(A.3.3)

= E [ X 1 A1 ] + E [ X 1 A2 ]

Thus, even for two arbitrary disjoint sets A1 and A2 one can simply sum up the two
expected values of a random variable X1 by applying indicator functions and yield
the joint expected value restricted by the indicator function of A1 ∪ A2 .
The simplicity of using indicator functions is not limited to multiple sets but can
also be extended to multiple random variables X2 , ..., Xm .
E[ X1 1 A1,2 ... 1 A1,m ] + ... + E[ X1 1 An,2 ... 1 An,m ]


=

Z∞

x1

−∞




X2 ( An,2 )
∞
Z Z∞
x1

x2

−∞ −∞

= E [ X1 ]
holds whenever

[

i ∈{1,...,n}

...

...

Z

X2 ( A1,2 )

Z

=

Z

...

Xm ( A1,m )

Xm ( An,m )
∞
Z
xm

Z

x1 f ( x1 , x2 , ..., xm ) dµ + ...+



x1 f ( x1 , ..., xm ) dµ dx1

x1 f ( x1 , x2 , ..., xm ) dµ dx1

−∞

(A.3.4)

Ai,j = Ω for each j ∈ {2, ..., m}, all Ai,j are pairwise disjoint,

and f ( x1 , ..., xm ) is the joint distribution of the random variables X1 , ..., Xm .

Appendix A – Expectation value with indicator function and multiplying a scalar 275

A.4

Expected value using an indicator function and adding
a scalar

If we want to calculate the sum of a random variable X and a scalar c in a certain range
using the indicator function of a set A one has to consider the probability of set A. The
following proof can easily be adjusted for the case of substraction E[ X − c], as well.
Proof. Given the arbitrary random variables X : Ω → R1 and Y : Ω → R1 and an
interval B ⊂ R1 then there always exists a set A ⊂ Ω with A = Y −1 ( B). Further, the

indicator function 1 A yields 1 whenever for an arbitrary y ∈ R1 it holds Y −1 (y) ∈ A.

Otherwise 1 A = 0. Let c be an arbitrary real value and let f be the probability density
function associated with X. Then it holds that
E[( X + c) 1 A ] =

Z∞

1 A ( x + c) f ( x ) dx

Z

( x + c) f ( x ) dx

−∞

=

Y ( A)

=

Z

B

c f ( x ) dx +

Z

x f ( x ) dx

B

= P( A) c + E[ x 1 A ]

(A.4.1)

If A = Ω then P( A) = 1, the indicator function 1 A = 1 for all y ∈ Y (Ω), and we yield

the commonly known expression E[ X + c] = c + E[ X ].

A.5

Expected value using an indicator function and multiplying a scalar

If we want to calculate the expected value of a random variable X which is multiplied
with a real number we can use the well-known equation E[ X · c] = c · E[ X ] even in the

276

A PPENDIX A. E XPLANATIONS R EGARDING T HE U SED P ROBABILITY S PACE

case of applying an indicator function. The following proof also holds for the division
E[ Xc ] as one can easily verify.
Proof. Given the arbitrary random variables X : Ω → R1 and Y : Ω → R1 and an
interval B ⊂ R1 then there always exists a set A ⊂ Ω with A = Y −1 ( B). Further, the

indicator function 1 A yields 1 whenever for an arbitrary y ∈ R1 it holds Y −1 (y) ∈ A.

Otherwise 1 A = 0. Let c be an arbitrary real value and let f be the probability density
function associated with X. Then it holds that
E[( X · c) 1 A ] =

=

Z∞

1 A ( x · c) f ( x ) dx

Z

x · c · f ( x ) dx

−∞

Y ( A)

= c·

Z

x f ( x ) dx

B

= c · E [ x 1 A ].

(A.5.1)

If A = Ω then the indicator function 1 A = 1 for all y ∈ Y (Ω), and we yield the

commonly known expression E[ X · c] = c · E[ X ].

A.6

Conditional expectations for multiple random variables

One can create a very flexible mechanism that allows to use an arbitrary countable
number of different random variables, for example for calculating the conditional expectations. This approach is based on the book of Feller to which we refer the interested reader for details [Fel71].
The joint probability P( X, Y ) of any two intervals A, B ∈ R1 is given, where R1 is

Appendix A – Conditional expectations for multiple random variables

277

the range of the random variables X and Y.
P( X, Y ) = P( X ∈ A, Y ∈ B).

(A.6.1)

Similarly to the density distribution f X : R → [0, 1] which is defined on single points
x ∈ R the marginal distribution or density distribution gX : [ a, b] → [0, 1] is defined

on arbitrary intervals I = [ a, b] with a, b ∈ R and a ≤ b. Using Equation (A.6.1) the
function gX can now be expressed by

gX ( A) = P( X ∈ A, X ∈ R1 ) = P( X ∈ A).

(A.6.2)

Given that gX ( A) > 0 one can now write the conditional probability
P (Y ∈ B | X ∈ A ) =

P( X ∈ A, Y ∈ B)
P( X ∈ A, Y ∈ B)
=
gX ( A)
P( X ∈ A)

(A.6.3)

which is very similar to the well known formula for conditional probability on two
sets. But here we do have the explicit notation of two different random variables.
Note, for gX ( A) = 0 the expression P(Y ∈ B | X ∈ A) is undefined. If we reduce

the length of the interval A towards zero, indicated by Aε = ( x, x + ε), we obtain the
marginal conditional probability
P( X ∈ Aε , Y ∈ B)
.
ε →0
gX ( Aε )

q( x, B) = lim

(A.6.4)

Example A.6.1. Assuming that a joint density function h( x, y) exists for the two random variables X and Y, the conditional probability of the event {Y ∈ B} and X = x
is
1
q( x, B) =
f X (x)

Z

h( x, y) dy.

(A.6.5)

B

This leads to the definitions for the joint probability of X and Y and for the conditional
probability distribution.

278

A PPENDIX A. E XPLANATIONS R EGARDING T HE U SED P ROBABILITY S PACE

Definition A.6.1 (Joint probability). Let X and Y be two random variables and let the
set B be fixed. By P(Y ∈ B | X ) (in words, ”a conditional probability of the event

{Y ∈ B} for given X”) is meant a function q( X, B) such that for every set A ∈ R1
Z

P( X ∈ A, Y ∈ B) =

q( x, B) gX (dx )

(A.6.6)

A

where g is the marginal distribution of X (on interval dx).
Definition A.6.2. A conditional probability distribution of Y for given X refers to a
function q of two variables, a point x and a set B, such that
1. for a fixed set B
q( X, B) = P(Y ∈ B | X )

(A.6.7)

is a conditional probability of the event { X ∈ B} for given X.
2. q is for each x a probability distribution.
Now, one can define the conditional expected value. First we assume that the
probability of the random variable X is always positive.
Definition A.6.3. A conditional expectation E[Y | X ] is a function of X assuming at x

the value

E (Y | x ) =

Z∞

y

y q( x, dy)

(A.6.8)

−∞

provided the integral converges (except possibly on a x-set of probability zero).
Note, in this case it holds that
E (Y ) =

Z∞

x

−∞

E(Y | x ) g(dx ) = E( E(Y | X )).

(A.6.9)

However, if the probability of X is not exclusively positive on R1 , e.g. not defined, we
use the Borel set A ∈ R1 and define the random variable 1 A ( X ) which is one for all

Appendix A – Conditional expectations for multiple random variables

279

values X ∈ A and zero otherwise. Now, we can integrate over Equation (A.6.8).
E(Y1 A ( X )) =

Z

A

Z∞

E(Y | x ) g(dx ) =

x

−∞

1 A ( x ) E(Y | x ) g(dx )

(A.6.10)

The random variable 1 A ( X ) is also referred to as indicator function of A. Moreover,
1 A ( X ) influences B in a way that B is the set of all points in Ω which X maps to a
value in A. In other words, B is a σ-algebra of sets in Ω which is called ”the algebra
generated by X” [Fel71], p 163, and which we denote by 1 B . Thus, the expression
U = E(Y | X ) is a function for which
E (Y 1 B ) = E ( U 1 B )

(A.6.11)

holds for every set B in the σ-algebra generated by X.
Based on Feller we give a comprehensive example for the two-dimensional space
R2 [Fel71].
Example A.6.2. We take the plain (R R>0 × R ) with random variables X : R >0 →

R and Y : R → R as sample space and there exists a strictly positive continuous

probability density f ( x, y). The random variable X maps a constant value to all lines

parallel to the y-axis and intersecting with the positive x-axis. If A is an arbitrary set
on the positive x-axis then the set B contains all those lines that run through the points
of A. So the left side of Equation (A.6.11) is equal to the ordinary integral of y f ( x, y)
and the expected value can be calculated by
E (Y 1 B ) =

Z

dx

A

Z∞

y f ( x, y) dy.

y

(A.6.12)

−∞

The right side of Equation (A.6.11) is the ordinary integral of a function U ( x ) f 1 ( x ),
where f 1 ( x ) is the marginal density of X. Thus, in this case one can write
1
U (x) =
f1 (x)

Z∞

y

−∞

y f ( x, y)dy

(A.6.13)

280

A PPENDIX A. E XPLANATIONS R EGARDING T HE U SED P ROBABILITY S PACE

which is identical to Definition A.6.3.
Note that f 1 ( x ) represents the marginal density of X based on its underlying σalgebra κ1 which is identical to the Borel sets of R >0 in this case. So f 1 ( x ) > 0 is not
restricted to values from A and for an arbitrary A ⊂ R >0 it must hold that
Z

f 1 ( x ) dx <

A

Z

f 1 ( x ) dx = 1

(A.6.14)

R >0

due to the strict positivity of f 1 regarding the fact that the denominator cannot be zero.
This approach can easily be extended to several conditional random variables
which leads to the definition of Doob.
Definition A.6.4 (Conditional expectation by Doob). Let (Ω, κ , P) be a probability
space, and κ1 a σ-algebra of sets in κ (that is, κ1 ⊂ κ ). Let Y be a random variable
with expectation. A random variable U is called conditional expectation of Y with
respect to κ1 if it is κ1 -measurable and Equation (A.6.11) holds for all sets B ∈ κ1 . In
this case we write U = E(Y | κ1 ).

In the particular case that κ1 is the σ-algebra generated by the random variables

X1 , ..., Xn the variable U reduces to a Baire function of X1 , ..., Xn and will be denoted
by E(Y | X1 , ..., Xn ).
Note, that Definition 3.1.7 is not limited to random variables but to any set of event
sets κ1 as long as κ1 is a σ-algebra and κ1 ⊆ κ . Of course, one can define a random
variable X generated by the σ-algebra κ1 , as well. Thus, there are two ways of constructing a conditional expected random variable U.
Now we are able to express the probability and the expected values for our random
variables X1 , X2 , and X3 , the three sets M1 , M2 , M3 ∈ κ , and arbitrary intersections

and unions of these sets. For example, the conditional expected amount of shortage in

Appendix A – Conditional expectations for multiple random variables

281

a cycle with shortage is given by

E[ X1 1 M1 ] =

=

=

=

=

Z∞

x

−∞
Z∞
x

−∞
Z∞
x

−∞
Z∞
x

−∞
Z0
x

−∞

1 M1 ( x ) · E[ X1 | x ] · g(dx )


1 M1 ( x ) · 


1 M1 ( x ) ·

1
·
∩ dx )

g (R 1



Z∞

y

−∞



y P(dy ∩ dx ) · g(dx )

g(dx )
· x · g(dx )
g(dx )

1 M1 ( x ) · x · g(dx )
(A.6.15)

x · g(dx )

where g(dx ) is the marginal probability of X1 . The result becomes clear when one
considers the fact that x ∈ M1 denotes the shortage in a cycle and that X1 extracts

the (possibly negative) stock from every triple element (s, lt1 , d0 ) of an arbitrary set
A ∈ κ defined by our probability space (Ω, κ , P). Thus, for a given negative stock x

the probability P(dy ∩ dx ) > 0 only holds if dx = dy and so the expected value of the
stock E[ X1 | x ] must be equal to x.

Note that E[ X1 1 M1 ] is neither identical to the expected stock E[ X1 ] nor identical to the
conditional expected stock E[ X1 | X1 < 0] but the relations

E [ X1 ] =

Z∞

x

−∞

x · g(dx )

= E[ X1 1 M1 ] +

Z

Ω\ M1

x · g(dx )

= E[ X1 1 M1 ] + E[ X1 1 M1 ]

(A.6.16)

282

A PPENDIX A. E XPLANATIONS R EGARDING T HE U SED P ROBABILITY S PACE

and
1
E [ X1 | X1 < 0 ] =
FX1 (0)

=

Z

M1

x · g(dx )

1
· E[ X1 1 M1 ]
FX1 (0)

(A.6.17)

hold between them where g(dx ) is the marginal probability of X1 , namely
g(dx ) = g( Ah ∈ R1 ) = P({ω | ω ∈ Ω, x ≤ X1 (ω ) < x + h, and h → 0}),

(A.6.18)

and FX1 (t) is the distribution function of X1 given by
FX1 (t) = P({ω | ω ∈ Ω and X1 (ω ) ≤ t}).

(A.6.19)

Given a random variable Y and an arbitrary Borel set A on R1 with the associated
set B = {Y ∈ A} of elements in the event space the conditional expected shortage
E[ X1 1 M1 ,B ] is determined by the intersection of M1 and B. Its value is given by

E[ X1 1 M1 ∩ B ] =

=

Z∞

x

−∞

=

Z∞

x

−∞

1 M1 ∩ B ( x ) · E[ X1 | M1 , B] · g(dx )

1 M1 ∩ B ( x ) · x · g(dx )
Z

x

range( M1 ∩ B)

x · P({ω | ω ∈ Ω, x ≤ X1 (ω ) < x + h, and h → 0}). (A.6.20)

Moreover, Case 1 can now be expressed by M2 ∩ M3 and Case 2 can be expressed

by M2 ∩ M3 . Coming back to the formula (3.3.4) which we like to proof and which

calculates the expected shortage in a one-order cycle we can now write the equivalent

Appendix A – Time window for triggering a second order

283

expressions
E[ X1 1 M1 ∩ M3 ] = E[ X1 1SH ∩(1,.,.) ]

=

=

=

Z∞

x

−∞
Z∞
x

−∞
Z∞
x

−∞
∞
Z
x

−∞

1 M1 ∩ M3 ( x ) · E[ X1 | x ] · g(dx )
1 M1 ∩( M2 ∪ M2 )∩ M3 ( x ) · x · P({ω | ω ∈ Ω, x ≤ X1 (ω ) < x + h, and h → 0})
1 M1 ∩ M2 ∩ M3 ( x ) · x · P({ω | ω ∈ Ω, x ≤ X1 (ω ) < x + h, and h → 0}) +

1 M1 ∩ M2 ∩ M3 ( x ) · x · P({ω | ω ∈ Ω, x ≤ X1 (ω ) < x + h, and h → 0})

= E[ X1 1 M1 ∩ Case 1 ] + E[ X1 1 M1 ∩ Case 2 ]
= E[ X1 1 M1 ∩ M2 ∩ M3 ] + E[ X1 1 M1 ∩ M2 ∩ M3 ].

A.7

(A.6.21)

Time window for triggering a second order

We want to show that the expression
f (tw ) =

Z tw
0

l1 (y)dy +

Z ∞Z ∞
tw

y−tw

l2 (z)dz l1 (y)dy

(A.7.1)

is monotonously increasing regarding tw with tw ≥ 0.

First, the integration over a probability density is non-negative and cannot exceed

the unity. Given any arbitrary value for y we can replace the value of the inner integral
of the second summand by a and it holds that 0 ≤ a ≤ 1. Then it holds for any
arbitrary y that

l1 ( y ) ≥ a l1 ( y ) .

(A.7.2)

This result can be transferred to the sum of more than one value, y1 , y2 , ..., yn , and the

284

A PPENDIX A. E XPLANATIONS R EGARDING T HE U SED P ROBABILITY S PACE

inequality
Z x
tw

l1 (y)dy ≥

Z xZ ∞
tw

y−tw

l2 (z)dz l1 (y)dy

(A.7.3)

must be true, as well. Knowing this, it is easy to verify that with an increasing tw the
first summand of Equality (A.7.1) increases at least as much as its second summand
decreases. Consequently, f (tw ) is monotonously increasing and its minimum value is
reached for tw = 0, the minimum value for tw by definition.

A.8

Conditional expected time until stock depletion

Often we are facing the situation where we know the stock level Xs (t) at two times
t0 < t1 . For example, Xs (t0 ) = R1 and Xs (t g ) = R2 . Whenever R1 > 0 > R2 is given
it is an interesting question what the expected depletion time is without regarding
incoming orders. In fact, we can show that
E[time when stock of R1 depletes] =

R1
· ( t g − t0 )
R1 − R2

(A.8.1)

holds in this scenario under some mild assumptions. While this equation seems intuitively right, the proof needs some further considerations1 . In our setting we know
the demand distribution over time rather than the stock distribution. Therefore, we
reformulate the problem in terms of: what is the expected time E[t D ] at which the demand accumulates to R1 given a total demand of R1 − R2 is observed at time t g .
Let the demand distribution D ( x, t) be a continuous variable in amount x and time t.
We refer to D ( x = R, t) as D ( R) for an arbitrary demand R ≥ 0. Then, in our example
we formally want to calculate the value for

1



E D ( R1 ) | D ( R1 − R2 ) = t g .

Many thanks to Eric Cope and Peter Haas for pointing out this elegant proof.

(A.8.2)

Appendix A – Conditional expected time until stock depletion

285

Further, we happen to observe a demand of R after time T ∈ R. We assume the

demand to be independently and identically distributed for each point in time. Moreover, the time distributions for incremental demand are identical, as well. Then we
can consider the evolvement of the demand over time to be a time-homogeneous
Markovian process. Now we split R into two equal amounts R1 + R2 = R. Due to
our assumption about the Markovian process we can state that D ( R1 ) ∼ D ( R2 ) are

two random time variables that are identically and independently distributed. Then
it holds that D ( R1 ) + D ( R2 ) = D ( R) and
E[ D ( R1 ) | D ( R1 ) + D ( R2 )] =

D ( R)
.
2

Now, we bisect the interval ( R1 , R) which leads to a third amount R3 =

(A.8.3)
R1 + R
2

=

3
4

· R.

Note that D (α · R) is identically distributed independent over time for any α ≥ 0. This
leads to

D ( R1 ) D ( R1 )
D ( R1 )
+
= D ( R3 ) +
2
2
2
E[ D ( R3 )] = E[ D ( R1 )] + E[ D ( R3 ) − D ( R1 )]
D ( R ) = D ( R1 ) +

E[ D ( R3 ) | D ( R)] = E [ E[ D ( R3 ) | D ( R1 ), D ( R)] | D ( R)] .

(A.8.4)
(A.8.5)
(A.8.6)

We can derive from the latter two equations
 


D ( R1 )
E[ D ( R3 ) | D ( R)] = E E D ( R) −
| D ( R1 ), D ( R ) | D ( R )
2


D ( R1 )
= E D ( R) −
| D ( R)
2


D ( R)
= E D ( R) −
| D ( R)
4
3
=
· D ( R ).
(A.8.7)
4
Putting all pieces together, we have chosen the three amounts R, R1 , and R3 with R1 =
1
2

· R and R3 = 34 · R. In the beginning we set D ( R) = T and from Equations (A.8.3) and

(A.8.7) we know that E[ D ( R1 ) | D ( R) = T ] =

1
2

· T and E[ D ( R3 ) | D ( R) = T ] =

3
4

· T,

286

A PPENDIX A. E XPLANATIONS R EGARDING T HE U SED P ROBABILITY S PACE

respectively.
We can recursively apply this mechanism of bisection k times. This allows us to
calculate the conditional expected time E[ D ( x ) | D ( R)] for all x = i ·

{0, 1, ..., k + 1} which yields
 
E D

i
·R
k+1





| D ( R) =

R
k +1

i
· D ( R ).
k+1

with i ∈

(A.8.8)

Passing to the limit k → ∞ we are able to verify Equation (A.8.8) for all demands α · R

where 0 ≤ α ∈ R which leads to the desired equation

(A.8.9)

E [ D (α · R) | D ( R)] = α · D ( R)
under the given mild assumptions.
In our example, α =
tion time


E D



R1
R1 − R2

and D ( R) = R1 − R2 , so we yield for the expected deple-

R1
· ( R1 − R2 ) | D ( R1 − R2 ) = t g
R1 − R2



=

R1
· tg .
R1 − R2

Appendix B
Shortage formulas
Case 6.

This case is specified by (2, tw , A1 ). First, this means that time window tw

does not exceed the lead time lt1 of the first order, so tw ≤ y = lt1 < ∞. Second, this
implies that the second order must not be triggered after tw , so 0 < t = t g ≤ tw . Third,

the first order arrives before the second one which translates into 0 < y − t < z = lt2 .
In addition, the demand x between triggering the second order and the arrival of the

first order must exceed R2 in order to cause a shortage, so R2 < x < ∞. Consequently,
the formula for the shortage before the first order arrives yields
h
E< A1 − Xs,A1 1(2,tw ,A1 ) 1 Ms,A
Z∞ Ztw Z∞
y

tw

t

z

Z∞

x
+

0 y − t [ R2 ]

1 ,<0

i

=

( x − R2 ) f ( x, y − t) l2 (z) g(t) l1 (y) dµ

(B.0.1)

which is valid for both cases R2 ≥ 0 and R2 < 0. For the condition that the first

delivery cannot satisfy all backlogged demand the formula is given by
E

≥ A1

h

( Xs,A1 − Xs,A2 ) 1(2,tw ,A1 ) 1 Ms,A

Z∞ Ztw Z∞
y

tw

t

z

Z∞

x

Z∞

0 y − t [ R2 + Q ] + 0
1

x0

1 ,<− Q1

i

=

x 0 f ( x 0 , z + t − y) f ( x, y − t) l2 (z) g(t) l1 (y) dµ. (B.0.2)
287

288

A PPENDIX B. S HORTAGE F ORMULAS

Whenever the first delivery can cover all backlogged demand the formula for Case 6
is
h
E≥ A1 −( Xs,A2 + Q1 ) 1(2,tw ,A1 ) 1 Ms,A
Z∞ Ztw Z∞ [ R2 +ZQ1 ]
y

tw

t

z

+

Z∞

x0

x

0 y−t

0

1 ,<− Q1

[ R2 + Q1 − x ] +

1 Ms,A

2 ,<− Q1

i

=

( x 0 − R2 − Q1 + x ) f ( x 0 , z + t − y ) ·
(B.0.3)

f ( x, y − t) l2 (z) g(t) l1 (y) dµ.
.
Case 7.

This case is specified by (2, tw , A2 ) which is similar to Case 6. First, this

means that time window tw does not exceed the lead time lt1 of the first order, so
tw ≤ y = lt1 < ∞. Second, this implies that the second order must not be triggered

after tw , so 0 < t = t g ≤ tw . Third, the second order arrives before the first one which

translates into 0 < z = lt2 < y − t. In addition, the demand dlt2 during the lead time

of the second order must exceed R2 in order to cause shortage, so R2 < x = dlt2 < ∞.

Consequently, the formula for the shortage before the second order arrives, which is
the order associated with R2 in this case, yields
E

< A2

h

− Xs,A2 1(2,tw ,A2 ) 1 Ms,A

Z∞ Ztw yZ−t Z∞
y

tw

t

0

z

x
+

0 [ R2 ]

2 ,<0

i

=

( x − R2 ) f ( x, z) l2 (z) g(t) l1 (y) dµ.

(B.0.4)

Note, this formula is already independent of the value of the two reorder points. For
the case where the first delivery cannot satisfy all backlogged demand at the time
of its arrival the demand dlt2 during the lead time of the second order must exceed
R2 + Q2 . Thus, R2 + Q2 < x < ∞ and any positive demand x 0 between the two
arrivals immediately leads to some shortage. Note, that only non-negative bounds of

Appendix B – Case 7

289

the integration of the demand x make sense.
h
E≥ A2 ( Xs,A2 − Xs,A1 ) 1(2,tw ,A2 ) 1 Ms,A
Z∞ Ztw yZ−t
y

tw

t

0

Z∞

Z∞

z

x

0 [ R + Q ]+ 0
2
2

x0

2 ,<− Q2

i

=

x 0 f ( x 0 , y − z − t) f ( x, z) l2 (z) g(t) l1 (y) dµ

(B.0.5)

For the last scenario, where the order quantity Q2 is large enough to satisfy all backlogged demand at the time of its arrival
h
E≥ A2 −( Xs,A1 + Q2 ) 1(2,tw ,A2 ) 1 Ms,A
Z∞ Ztw yZ−t [ R2 +ZQ2 ]
y

tw

z

t

0

0

x

0

+

Z∞

x0

[ R2 + Q2 − x ]

2 ,<− Q2

1 Ms,A

1 ,<− Q2

i

=

( x 0 − R2 − Q2 + x ) f ( x 0 , y − z − t ) ·

f ( x, z) l2 (z) g(t) l1 (y) dµ
determines the shortage between the first and the second arrival.

(B.0.6)

Appendix C
Stock formulas

C.1

Case 2.

One-order cycles

Consider the case where R1 ≥ 0 > R2 holds. It is then possible that the stock

depletes within tw without triggering a second order. The average stock on hand
before tw is

h
i
w
ER<1t≥
stock
1
(1,tw ,.) =
0> R2
R1
Z∞ Z
y

tw

x

0


x
t w R1 −
f ( x, tw ) l1 (y) dµ +
2

Z∞ R1Z− R2
y

x

tw

R1

η ( x, R1 ) · tw ·
291

R1
f ( x, tw ) l1 (y) dµ.
2

(C.1.1)

292

A PPENDIX C. S TOCK F ORMULAS

For the time after tw and until the arrival of the order we (partially) have stock only if
the demand x during tw has been less than R1 .
h
i
ER<1A≥10> R2 stock 1(1,tw ,.) =
R1 Z∞
Z∞ Z
y

x

tw

0

0

0

η ( x , R1 − x ) ( y − t w )

x0



min( x 0 , R1 − x )
R1 − x −
2



·

0

f ( x , y − tw ) f ( x, tw ) l1 (y) dµ.

(C.1.2)

Note, that for x > R1 the stock level is zero and can be neglected. We can split up the
formula and yield
h
i
ER<1A≥10> R2 stock 1(1,tw ,.) =
R1 RZ
1 −x
Z∞ Z
y

tw

x

0

0

x0

R1 Z∞
Z∞ Z
y

tw

x0

x

0 R1 − x

(y − tw ) ·



x0
R1 − x −
2

η ( x 0 , R1 − x ) ( y − t w )



f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dµ +

R1 − x
f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dµ.
2

(C.1.3)

Considering that R1 > 0 and the average stock on hand between the arrival of the
order and the end of the cycle can be approximated by
h
i
ER≥1A≥10> R2 stock 1(1,tw ,.) =
Z∞ R1Z− R2 Z∞
y

x

tw

0

0

x0


+ 
( Q1 − x − x 0 )
Q1 − x − x 0
0
· R1 + Q1 − x − x −
·
µD
2

0

f ( x , y − tw ) f ( x, tw ) l1 (y) dµ.
Negative reorder points.

(C.1.4)

The last option is 0 > R1 > R2 . Here the stock on hand is

zero at least until the arrival of the order.
E0<>twR1 > R2

h

h

stock 1(1,tw ,.)

E0<>AR11 > R2 stock 1(1,tw ,.)

i

i

= 0

(C.1.5)

= 0

(C.1.6)

Appendix C – Two-order cycles where the first order arrives first

293

Only if the order quantity is large enough to bring the stock level not only above R1
but even above 0 the average stock on hand is positive. Assuming an average demand
rate of µ D throughout the rest of the cycle time tc − t A1 yields an expected stock of
h
i
E0≥>AR11 > R2 stock 1(1,tw ,.) =


+ 2
Z∞ R1Z− R2 Z∞
( R1 + Q1 − x − x 0 )
y

tw

C.2
Case 3.

x

0

0

2µ D

x0

f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dµ. (C.1.7)

Two-order cycles where the first order arrives first
We assume that R1 ≥ 0 > R2 holds for the reorder points. The stock on hand

must now deplete before triggering the second order at time t = t g . The expected
depletion time is given by
R1
· tg
R1 − R2

under some mild conditions, see proof in Appendix A.8. The expected stock on hand
before t g is then given by

<t g
ER1 ≥0> R2 [stock 1(2,A1 ,A1 ) ]

=

Ztw Zy Z∞
y

0

t

z

0 y−t

R1
R
· t · 1 l2 (z) g(t) l1 (y) dµ.
R1 − R2
2

(C.2.1)

Whenever R2 < 0 holds then the stock on hand obviously remains zero between triggering the second order and the arrival of the first order.
ER<1A≥10> R2 [stock 1(2,A1 ,A1 ) ] = 0

(C.2.2)

294

A PPENDIX C. S TOCK F ORMULAS

Between the arrival of both orders the stock might be above zero for a while but not
longer than t g + lt2 − lt1 time units.
ER<1A≥20> R2 [stock 1(2,A1 ,A1 ) ] =
Ztw Zy Z∞ Z∞ Z∞
y

0

t

z

x

0 y−t 0

0

x0



+
η x 0 , ( R2 + Q1 − x )
(t g + z − y) ·



+
0
min
(
R
+
Q
−
x
)
,
x
2
1
·
 ( R2 + Q1 − x ) + −
2


f ( x 0 , t + z − y) f ( x, y − t) l2 (z) g(t) l1 (y) dµ

(C.2.3)

The average stock between the last arrival and the end of the cycle is identical to the
case R1 > R2 ≥ 0 given in Section 3.4.2.1.
Here, we assume that 0 > R1 > R2 holds. Obviously, the

Negative reorder points.

stock on hand before the first arrival must be zero.
<t g

E0> R1 > R2 [stock 1(2,A1 ,A1 ) ] = 0

(C.2.4)

E0<>AR11 > R2 [stock 1(2,A1 ,A1 ) ] = 0

(C.2.5)

Between the arrival of both orders the stock might be above zero for a while but not
exceeding t g + lt2 − lt1 just as for the condition R1 ≥ 0 > R2 . Thus, the formula is
identical to Equation (C.2.3).

The stock on hand after the last arrival is not necessarily positive at any time even
if it exceeds R1 at time t A2 . Thus, the average stock on hand is given by
E0≥>AR21 > R2 [stock 1(2,A1 ,A1 ) ] =


+ 2
0
Ztw Zy Z∞ Z∞ Z∞
( R2 + Q1 + Q2 − x − x )
y

t

0

0

z

0 y−t 0

x

0

x0

2 µD

f ( x , t + z − y) f ( x, y − t) l2 (z) g(t) l1 (y) dµ.

·
(C.2.6)

Appendix C – Two-order cycles where the first order arrives first
Case 6.

295

Case 6 differs from Case 3 only by the lead time lt1 > tw instead of lt1 ≤ tw .

Thus, we obtain the formulas for Case 6 by simply adjusting the bounds 0 and tw of the
integration of lt1 to tw and ∞. Moreover, the inter-order time is restricted by t g ≤ tw .
E<tg [stock 1(2,tw ,A1 ) ]
Z∞ Ztw Z∞
y

tw

t

z

0 y−t

R1

=
+

+

+

R1 − R2

−

R1 − R2
2

·t·

E< A1 [stock 1(2,tw ,A1 ) ] =
Z∞ Ztw Z∞ Z∞
y

tw

t

x

z



η x, R2

0 y−t 0


+

+



!

+
( y − t )  R2 −

l2 (z) g(t) l1 (y) dµ



+

min R2 , x
2

(C.2.7)



·

f ( x, y − t) l2 (z) g(t) l1 (y) dµ

(C.2.8)

E< A2 [stock 1(2,tw ,A1 ) ] =
Z∞ Ztw Z∞ Z∞ Z∞
y

tw



t

z

x

0 y−t 0

0

x0



+
η x 0 , ( R2 + Q1 − x )
(t + z − y) ·

 ( R2 + Q1 − x ) + −



min ( R2 + Q1 − x )

+

, x0

2

f ( x 0 , t + z − y) f ( x, y − t) l2 (z) g(t) l1 (y) dµ



·

E≥ A2 [stock 1(2,tw ,A1 ) ] =

+
+
Z∞ Ztw Z∞ Z∞ Z∞
R2 + Q1 + Q2 − x − x 0 − R1
y

tw



t

z

0 y−t 0

x

0

x0

R2 + Q1 + Q2 − x −

µD

x0

+ R1

+

(C.2.9)

·

+

·
2
f ( x 0 , t + z − y) f ( x, y − t) l2 (z) g(t) l1 (y) dµ

These formulas hold for all possible settings of R1 and R2 .

(C.2.10)

296

C.3

A PPENDIX C. S TOCK F ORMULAS

Two-order cycles where the second order arrives first

Case 4.

Let us assume positive reorder points R1 > R2 ≥ 0 for a moment. The stock

on hand cannot deplete before the second order is triggered at time t = t g .

<t g
ER1 > R2 ≥0 [stock 1(2,A1 ,A2 ) ]

=

Ztw Zy yZ−t
y

t

0

0

z

0

t·



R − R2
R1 − 1
2



l2 (z) g(t) l1 (y) dµ (C.3.1)

The demand between triggering and receiving the second order is theoretically unlimited but must occur within the lead time z = lt2 .
ER<1A>2R2 ≥0 [stock 1(2,A1 ,A2 ) ] =
Ztw Zy yZ−t Z∞
0

z

t

y

0

0

x

0



min( R2 , x )
η ( x, R2 ) · z · R2 −
2



f ( x, z) l2 (z) g(t) l1 (y) dµ. (C.3.2)

The demand between both arrivals is unlimited as well but timely restricted to a duration of lt1 − t g − lt2 . Depending on the previous demand x the stock level might be

above zero the entire time, just for a while, or never. This leads the formula
ER<1A>1R2 ≥0 [stock 1(2,A1 ,A2 ) ] =
Ztw Zy yZ−t Z∞ Z∞
t

y

0

0

z

0

x

0

0

x0



+
η x 0 , ( R2 + Q2 − x )
(y − t − z) ·



+
0
min
(
R
+
Q
−
x
)
,
x
2
2
 ( R2 + Q2 − x ) + −
·
2


f ( x 0 , y − t − z) f ( x, z) l2 (z) g(t) l1 (y) dµ

(C.3.3)

The stock after the second arrival must be above R1 according to our assumptions.
Here R1 > 0 holds and the average stock between the last arrival and the end of the

Appendix C – Two-order cycles where the second order arrives first

297

cycle is given by
ER≥1A>1R2 ≥0 [stock 1(2,A1 ,A2 ) ] =
Ztw Zy yZ−t Z∞ Z∞
y

0

t

0

z

0

x

0

0

x0

E [ t c − t A1 ] ·

R2 + Q2 + Q1 − x − x 0 + R1
·
2

f ( x 0 , y − t − z) f ( x, z) l2 (z) g(t) l1 (y) dµ.

(C.3.4)

where the remaining cycle time after the second arrival is expected to be
+

( R2 + Q1 + Q2 − x − x 0 − R1 )
E [ t c − t A1 ] =
.
µD

(C.3.5)

Negative second reorder point. We assume that R1 ≥ 0 > R2 holds. The stock on
hand must deplete now before triggering the second order.
<t g
ER1 ≥0> R2 [stock 1(2,A1 ,A2 ) ]

=

Ztw Zy yZ−t
0

z

t

y

0

0

R1
t · l2 (z) g(t) l1 (y) dµ
R1 − R2

(C.3.6)

Whenever R2 < 0 holds then the stock on hand obviously remains zero between triggering and receiving the second order.
ER<1A≥20> R2 [stock 1(2,A1 ,A2 ) ] = 0

(C.3.7)

Between the arrival of both orders the stock might be above zero for a while but never
longer than lt1 − t g − lt2 . In fact, the formula does not differ from ER<1A≥10> R2 [stock 1(2,A1 ,A2 ) ]

given in Equation (C.3.3). The average stock between the last arrival and the end of
the cycle is identical to the case R1 > R2 ≥ 0, as well, see Equation (C.3.4).
Negative reorder points. We assume that 0 > R1 > R2 holds. Obviously, the stock
on hand before the first arrival must be zero.
<t g

E0> R1 > R2 [stock 1(2,A1 ,A2 ) ] = 0

(C.3.8)

E0<>AR21 > R2 [stock 1(2,A1 ,A2 ) ] = 0

(C.3.9)

298

A PPENDIX C. S TOCK F ORMULAS

Between the arrival of both orders the stock might be above zero for a while just as
for the conditions R1 > R2 ≥ 0 and R1 ≥ 0 > R2 . Thus, the formula is identical

to Equation (C.3.3). The stock on hand between the last arrival and tc might not be
positive at any time even if it exceeds R1 at time t A1 . Thus, the average stock on hand
is given by

E0≥>AR11 > R2 [stock 1(2,A1 ,A2 ) ] =


+ 2
Ztw Zy yZ−t Z∞ Z∞
( R2 + Q2 + Q1 − x − x 0 )
y

t

0

0

z

0

x

0

0

x0

0

2µ D

f ( x , y − t − z) f ( x, z) l2 (z) g(t) l1 (y) dµ.

·
(C.3.10)

All the formulas of this section can be joined to universal formulas that apply for every
setting of the reorder points R1 and R2 .

Case 7.

Case 7 is closely related to Case 4. The only thing changing is that the lead

time of the first order exceeds tw . Thus, the equivalent formulas for Case 7 can be
again easily derived from Case 4. First, the bounds of the first integration have to
be changed from 0 and tw to tw and ∞, respectively. Second, the inter-order time is
not limited by the lead time lt1 anymore but by the time window tw which has to be

Appendix C – Two-order cycles where the second order arrives first

299

considered in the upper bound of t g .
E<tg [stock 1(2,tw ,A2 ) ] =
Z∞ Ztw yZ−t
y

tw

t

0

z

0

R1

+

+

+

R1 − R2

−

·t·

E< A2 [stock 1(2,tw ,A2 ) ] =
Z∞ Ztw yZ−t Z∞
y

tw

t

0

z

0

x

0

R1 + R2
2

+

!

l2 (z) g(t) l1 (y) dµ

 + 
min
R2 , x
+
+
·
η ( x, R2 ) · z ·  R2 −
2

(C.3.11)



f ( x, z) l2 (z) g(t) l1 (y) dµ

(C.3.12)

E< A1 [stock 1(2,tw ,A2 ) ] =
Z∞ Ztw yZ−t Z∞ Z∞
y

tw

t

0

z

0

x

0

0

x0



0

η x , ( R2 + Q2 − x )

+



(y − t − z) ·



+
0
min
(
R
+
Q
−
x
)
,
x
2
2
·
 ( R2 + Q2 − x ) + −
2


f ( x 0 , y − t − z) f ( x, z) l2 (z) g(t) l1 (y) dµ

E≥ A1 [stock 1(2,tw ,A2 ) ] =

+
+
0
Z∞ Ztw yZ−t Z∞ Z∞
R2 + Q2 + Q1 − x − x − R1
y

tw



t

0

z

0

x

0

0

x0

R2 + Q2 + Q1 − x − x 0 + R1

µD

+

(C.3.13)

·

+

·
2
f ( x 0 , y − t − z) f ( x, z) l2 (z) g(t) l1 (y) dµ.

(C.3.14)

These formulas hold for all three possible scenarios of the reorder points, R1 > R2 ≥ 0,
R1 ≥ 0 > R2 , and 0 > R1 > R2 .

300

C.4

A PPENDIX C. S TOCK F ORMULAS

Two-order cycles where both orders arrive simultaneously

C.4.1

Case 5

Note that the relation lt1 = t g + lt2 must hold between the two lead times for a simultaneous arrival of both orders. This case occurs with a probability of zero in the
continuous case but with a positive probability in the discrete case. Thus, we only
consider discrete time units, demands, and the probability functions are the discrete
pendants to their original definition in the following.
Positive reorder points. Both reorder points are positive so that R1 > R2 ≥ 0 holds.

The stock on hand cannot deplete before the second order is triggered.
<t g

ER1 > R2 ≥0 [stock 1(2,A1 ,=) ] =


tw y
R1 − R2
l2disc (y − t) gdisc (t) l1disc (y)
∑ ∑ t · R1 − 2
y =2 t =1

(C.4.1)

Note, that the lead time of the first order has to be two time units or longer. Otherwise,
the probability of a two-order cycle is zero because we assume a minimum lead time
of 1 time unit and do not allow for simultaneous ordering, e.g. order splitting. Thus,
we would yield an one-order cycle for lt1 < 2.
ER<1A>1R2 ≥0 [stock 1(2,A1 ,=) ] =
tw

y

∑∑

y =2 t =1

f

disc

Z∞

x

0

η ( x, R2 ) (y − t)



min( R2 , x )
R2 −
2

( x, y − t) l2disc (y − t) gdisc (t) l1disc (y) dx



·
(C.4.2)

The stock after the simultaneous arrival must be above R1 according to our assumptions. Here R1 > 0 holds and the average stock between the arrival and the end of the

Appendix C – Two-order cycles where both order arrive simultaneously

301

cycle is given by
ER≥1A>1R2 ≥0 [stock 1(2,A1 ,=) ] =
tw

y

∑∑

y =2 t =1

f

disc

Z∞

x

0

E[tc − y] ·

R2 + Q2 + Q1 − x + R1
·
2

( x, y − t) l2disc (y − t) gdisc (t) l1disc (y) dx

(C.4.3)

where E[tc − y], the remaining cycle time after the arrivals is

+

( R2 + Q2 + Q1 − x − R1 )
.
E[tc − y] =
µD
Negative second reorder point.

(C.4.4)

We assume that R1 ≥ 0 > R2 holds for the reorder

points. Now the stock on hand must deplete before triggering the second order at
time t = t g .The expected average stock level before triggering the second order is
then given by
<t g

ER1 ≥0> R2 [stock 1(2,A1 ,=) ] =
 
tw y
R1
R1
∑ ∑ R1 − R2 · t · 2 l2disc (y − t) gdisc (t) l1disc (y).
y =2 t =1

(C.4.5)

Whenever R2 < 0 the stock on hand obviously remains zero between triggering the
second order and receiving both orders.
ER<1A≥10> R2 [stock 1(2,A1 ,=) ] = 0

(C.4.6)

The average stock between the simultaneous arrival and the end of the cycle is identical to the case R1 > R2 ≥ 0 given in Equation (C.4.3).
Negative reorder points.

We assume that 0 > R1 > R2 holds for the reorder points.

Obviously, the stock on hand before the simultaneous arrival must be zero.
<t g

E0> R1 > R2 [stock 1(2,A1 ,=) ] = 0

(C.4.7)

E0<>AR11 > R2 [stock 1(2,A1 ,=) ] = 0

(C.4.8)

302

A PPENDIX C. S TOCK F ORMULAS

The stock on hand between t A1 and tc might not be positive at any time even if it
exceeds R1 at time t A1 . Thus, the average stock on hand is given by
E0≥>AR11 > R2 [stock 1(2,A1 ,=) ] =


+ 2
tw y Z∞
( R2 + Q1 + Q2 − x )

∑∑

y =2 t =1

f

disc

2 µD

x

0

·

( x, y − t) l2disc (y − t) gdisc (t) l1disc (y) dx.

(C.4.9)

All the formulas of this section can be joined to universal formulas that apply for every
setting of the reorder points R1 and R2 .

C.4.2

Case 8

Case 8 is closely related to Case 5. The only thing changing is that the lead time of
the first order exceeds tw . Thus, the equivalent formulas for Case 8 can be again easily
derived from Case 5. First, the bounds of the first integration have to be changed from
0 and tw to tw and ∞, respectively. Second, the inter-order time is not limited by the
lead time lt1 anymore but by the time window tw which has to be considered in the
upper bound of t g .
E<tg [stock 1(2,tg ,=) ] =
∞

tw

∑ ∑

y = t w +1 t =1

R1

+

+

+

R1 − R2

−

·t·

R1 + R2
2

+

!

l2disc (y − t) gdisc (t) l1disc (y)

E< A1 [stock 1(2,tg ,=) ] =
∞

tw Z∞

∑ ∑

y = t w +1 t =1

f

disc

x

0



η x, R2

+



+

(y − t)

R2

+

min( R2 , x )
−
2

( x, y − t) l2disc (y − t) gdisc (t) l1disc (y) dx

!

(C.4.10)

·
(C.4.11)

Appendix C – Two-order cycles where both order arrive simultaneously
E≥ A1 [stock 1(2,tg ,=) ] =

+
+
tw Z∞
∞
R2 + Q2 + Q1 − x − R1

∑ ∑

y = t w +1 t =1

f

disc

x

0

µD

( x, y − t) l2disc (y − t) gdisc (t) l1disc (y) dx

·



R2 + Q1 + Q2 − x + R1
2

303

+

+

·

(C.4.12)

These formulas hold for all three possible scenarios of the reorder points, R1 > R2 ≥ 0,
R1 ≥ 0 > R2 , and 0 > R1 > R2 .

Appendix D
Formulas for dual sourcing with lost
sales
In this chapter we develop the formulas for the expected shortage and the average
stock on hand for scenarios where unsatisfied demand cannot be backlogged but is
lost. To quantify the lost sales we can use most of the random variables and indicator
function from Chapter 3 without any changes. However, some of the random variables and the formulas for expected stock and shortages have to be slightly adapted.
Interestingly, one can also interpret the opportunity costs for lost sales as shipping
costs from another warehouse, for example, in a two- or multiple-echelon inventory
system. In this case all unsatisfied demand is not lost but satisfied by a direct shipment from another warehouse, the central warehouse or another warehouse on the
same level within the multi-echelon supply network, for example. Throughout this
chapter we refer to this new policy as direct shipment policy (DS) in contrast to the
backlog policy described in Chapter 3 where unsatisfied demand is backlogged until
the arrival of new supply.
In the following we will first investigate the effect of different reorder point settings. Second, we will look at the implications of satisfying unmet demand by other
305

306

A PPENDIX D. F ORMULAS F OR D UAL S OURCING W ITH L OST S ALES

warehouses and introduce the necessary changes to the random variables. In the last
three section we will develop the formulas to determine the probabilities of the different cases, the expected amount of direct shipments, and the average stock on hand. To
distinguish between the formulas of the two different replenishment policies we add
the initials DS to the superscript of the formulas of the direct shipment policy.

D.1

Adjusted probability space for the reorder point scenarios

One implication of the direct shipment policy is a changing role of negative reorder
points. Obviously, unsatisfied demand will occur before a negative reorder point is
reached. This demand is covered by another warehouse and there will be no pending
unsatisfied demand. Thus, the stock cannot fall below zero and the negative reorder
point is never reached. Consequently, a negative reorder point implies that this supply
channel is not taken into consideration. This is exactly our interpretation when using
the direct shipment policy.
A negative second reorder point, R2 , prohibits the use of the second supply channel and reduces our model to a traditional replenishment policy without the option
of a second supply channel. Furthermore, a negative reorder point R1 implies that
all demand is covered by another warehouse on the same or a different level in the
supply network. These scenarios are illustrated in Table D.1 and complies with the
Assumption 2 on page 42 which states that one is free in assigning supply channels to
R1 and R2 but the relation R1 > R2 must always hold.
Of course, setting R1 < 0 for all articles makes a warehouse obsolete because everything is directly shipped to the customer. However, this option is commonly used
in the spare parts business for single articles that are very expensive to store.

Appendix D – Adjusted probability space

scenario

direct shipment
(shortage)

307

1-order case

2-order case

R1 > R2 ≥ 0

possible

possible

possible

possible

possible

not possible

0 > R1 > R2

possible

not possible

not possible

R1 ≥ 0 > R2

Table D.1: Possibility of different events to happen for the three reorder point scenarios

Example D.1.1. Consider a situation where a very expensive article has very little demand on a regional level – a car engine, for example. Storing one engine at each
warehouse facing the customer will result in significant costs in terms of space and
fixed capital. Then it is most likely beneficial to store this article only on a central level
in the supply network and deliver it upon request directly to the customer, a garage
or factory, for example.
One might expect that the different implications of negative reorder points have a
strong affect on the probability space
Ω R1 ,R2 ,tw ,Q1 ,Q2 := Ω = {(lt1 , lt2 , t g , tc , dlt1 , dlt2 , dtw )}
as it is defined for the backlog policy in Equation (3.1.7) on page 56. However, this is
not the case. We can still use this definition and make some additional statements.
We define the probability space for the direct shipment policy according to the
backlog policy by
DS
ΩDS
= {(lt1 , lt2 , t g , tc , dlt1 , dlt2 , dtw )}
R1 ,R2 ,tw ,Q1 ,Q2 : = Ω

(D.1.1)

For all scenarios where R1 > R2 ≥ 0 there are no differences between both event

spaces and we can state Ω ≡ ΩDS although we will have to change some of the ran-

308

A PPENDIX D. F ORMULAS F OR D UAL S OURCING W ITH L OST S ALES

dom variables in order to model the average stock on hand and the expected number
of direct shipments correctly.
For a scenario where R1 ≥ 0 > R2 the definition of Ω and ΩDS is still identical.

However, there will be no two-order cycles in the direct shipment policy. Consequently, the event sets are differently, Ω 6= ΩDS , and we have to adapt the random
variables.

In the last scenario where 0 > R1 > R2 there will be no replenishment. Consequently, the probability space ΩDS is empty but we can still use the same definition as
for Ω, of course. In addition we can make the statement that ΩDS = ∅ 6= Ω. For a di-

rect shipment policy there exist no cycle times, lead times, and all the order-dependent
variables in this reorder point scenario. The average stock on hand is zero and the expected number of direct shipments per time unit is identical to the average demand
rate per time unit.
We have defined the proper probability space, ΩDS , for the direct shipment policy in
this section. In the following we will adapt the random variables which are necessary
to correctly calculate the average stock on hand and the expected number of direct
shipments.

D.2

Definition of additional random variables and event
sets

In the direct shipment replenishment policy all shortage is covered by other warehouses. However, instead of abandoning the formulas for shortages from the backlog
policy they can still be used after some slight adjustments to quantify the number of
direct shipments in the direct shipment policy.

Appendix D – Definition of additional random variables

309

Whenever the direct shipment policy is applied all arriving supplies can be fully used
to cover new demand. Potential old shortages have already been compensated by
other warehouses. Consequently, we have to introduce some new random variables
to correctly reflect the stock on hand after each delivery. Here, we proceed analogously
to the definition of the random variables for the backlog policy, see Section 3.1.5.2 on
page 57. Again, each random variables represents a mapping from a probability space
ΩDS to the set of real numbers R, formally expressed by X : ΩDS → R with ω ∈ ΩDS .

Random variables which are meaningful only in the context of the direct shipment
policy are denoted by the superscript DS.
Both random variables (D.2.1) and (D.2.2) represent the stock on hand just before
the first and second order arrives, respectively, without considering any other potentially incoming supply.

+
DS
Xs,A
:
=
ω
7
→
R
−
d
1
lt
1
1

+
DS
Xs,A2 := ω 7→ R2 − dlt2

(D.2.1)
(D.2.2)

Moreover, we can still use the random variables Xs,A1 := ω 7→ R1 − dlt1 and
Xs,A2 := ω 7→ R2 − dlt2 of the backlog policy. They represent the possibly negative

stock just before the arrival of both orders without considering any intermediate delivery, see Section 3.1.5.2 on page 57. Consequently, whenever they have a negative
value they quantify the amount of direct shipments from other warehouses for the
direct shipment policy.
The two random variables (D.2.3) and (D.2.4) represent the stock on hand after
the arrival of the first and second order, respectively. Both are intimately related to
the random variables Xs+Q1 ,A1 and Xs+Q2 ,A2 , respectively, of the backlog policy in
Section 3.1.5.2 on page 57. However, using the direct shipment policy the stock after

310

A PPENDIX D. F ORMULAS F OR D UAL S OURCING W ITH L OST S ALES

the arrival of each order cannot fall below Q1 and Q2 , respectively.
+

+ Q1
XsDS
+ Q1 ,A1 : = ω 7 → R1 − dlt1

+
XsDS
+ Q2
+ Q2 ,A2 : = ω 7 → R2 − dlt2

(D.2.3)
(D.2.4)

The stock level, not to be confused with the stock on hand, just before the second
order arrives in a two-order cycle is given by the random variables (D.2.5) and (D.2.6).
Note, both equations consist of the stock on hand just before the first arrival plus
the delivered order quantity minus the demand that occurs between both arrivals.
A positive value represents the stock on hand just before the second arrival while a
negative value represents the number of direct shipments that occur between the first
and the second delivery, i.e. the demand that cannot be satisfied by the warehouse.



+
XsDS
+ Q1 − dlt2 + ( R1 − R2 ) − dlt1
+ Q1 ,A2 : = ω 7 → R1 − dlt1
XsDS
+ Q2 ,A1

= ω 7→ XsDS
+ Q1 ,A1 − Xs,A1 + Xs,A2

+


:= ω 7→ R2 − dlt2 + Q2 − dlt1 − ( R1 − R2 ) − dlt2
= ω 7→ XsDS
+ Q2 ,A2 − Xs,A2 + Xs,A1

(D.2.5)

(D.2.6)

Once these random variables are defined we can use them to define additional event
sets that are required, for example, to calculate the expected stock on hand in the direct
shipment policy.
cycles with shortage before A2 (including Q1 )

DS
−1
MsSD
+ Q1 ,A2 ,<0 = [ Xs+ Q1 ,A2 ] ( x < 0)

(D.2.7)

cycles with shortage before A1 (including Q2 )

DS
−1
MsSD
+ Q2 ,A1 ,<0 = [ Xs+ Q2 ,A1 ] ( x < 0)

(D.2.8)

With all these formal preparations we are almost ready to set up the formulas for
the probabilities, expected number of direct shipments and the average stock on hand.
The only thing missing is to investigate the effect of different reorder point settings on
the calculations.
Now, we are ready to set up the different formulas for the direct shipment policy.

Appendix D – Probabilities of Case 1 till Case 8

D.3

311

Probabilities of the reorder cycle scenarios

For each of the eight cases, Case 1 to Case 8, the probability of occurrence does not
differ much between the backlog policy and the direct shipment policy. In fact, they
are identical for the scenario where R1 > R2 ≥ 0 holds. However, a supply channel

with a negative reorder point is not used for replenishment within the direct shipment
policy. Consequently, the formulas get much simpler. The relation between different
reorder point scenarios and the probabilities for the occurrence of direct shipments
and one-and two-order cases are illustrated in Table D.2. Recall that the notation 1

is used for the indicator function. For example, the expression 1(1,.,.) represents the
indicator function for one-order cycles.
scenario
R1 > R2 ≥ 0
R1 ≥ 0 > R2
0 > R1 > R2

direct shipment (shortage)
p R1 > R2 ≥0 (direct shipment)

≥0

p R1 ≥0> R2 (direct shipment) =


p RDS
1
and
1
M
,A
,
<
0
(
1,.,.
)
s
1
1 ≥0> R2
p0> R1 > R2 (direct shipment)
=1

1-order case


p RDS
1
(
1,.,.
)
1 > R2 ≥0

≥0


2-order case


p RDS
1
(
2,.,.
)
1 > R2 ≥0

≥0


p RDS
1(1,.,.)
1 ≥0> R2



p RDS
1(2,.,.)
1 ≥0> R2



p0DS
> R1 > R2 1(1,.,.)



p0DS
> R1 > R2 1(2,.,.)



=1

=0

=0

=0

Table D.2: Probabilities of direct shipments and one- and two-order cycles
In summary, the Table D.2 shows that one- and two-order cycles and direct shipments
are possible as long as both reorder points are not negative. The probability of direct
shipments for R1 > R2 ≥ 0 encompasses many subcases so it shall be enough here to
state that the probability of direct shipments can be positive.

Whenever only the second reorder point is negative the probability of a two-order
cycle is zero. Then there will only be one-order cycles to cover the demand. The probability of direct shipments can be simply given as the probability of some shortage

312

A PPENDIX D. F ORMULAS F OR D UAL S OURCING W ITH L OST S ALES

before the arrival of the first and only order.
In the case where both reorder points are negative there will be no replenishment.
This results in probabilities of zero for one- and two-order cycles. Furthermore, all
demand has to be covered by direct shipments.

For the development of the corresponding formulas it appears fortunate to divide all
possibilities according to their condition on the reorder points. Thereby one should
keep in mind that there exist two major differences between the direct shipment policy
and the backlog policy:
1) different implications of negative reorder points
2) compensation of shortage by future supply vs. other warehouses.
Let us first look at the probability of the different cases where both reorder points are
positive.

D.3.1

All reorder points are non-negative

For this scenario the condition R1 > R2 ≥ 0 holds and so the implication of negative

reorder points has no effect. More precisely, a second order will be triggered if and
only if R2 is met within the time window tw . Here R2 is positive and all shortages
must occur after the second order possibly has been triggered.
In summary, the decision whether to trigger a second order is neither influenced
by the different treatment of shortages nor by the implications of negative reorder
points. Consequently, the formulas for a direct shipment policy are identical to their
counterparts of the backlog policy, see Section 3.2 on page 68. This conclusion is also

Appendix D – Probabilities of Case 1 till Case 8

313

reflected in the formulas (D.3.1) to (D.3.8).
pDS
R1 > R2 ≥0 (1,

A1 , .) =

p(1, A1 , .) =

pDS
R1 > R2 ≥0 (1, tw , .) =

p(1, tw , .) =

pDS
R1 > R2 ≥0 (2, A1 , A1 ) =

p(2, A1 , A1 ) =

pDS
R1 > R2 ≥0 (2,

disc
pDS,
R1 > R2 ≥0 (2,

A1 , A2 ) =

f ( x, y) l1 (y) dµ

(D.3.1)

f ( x, tw ) l1 (y) dµ

(D.3.2)

l2 (z) g(t) l1 (y) dµ

(D.3.3)

l2 (z) g(t) l1 (y) dµ

(D.3.4)

l2disc (y − t) gdisc (t) l1disc (y)

(D.3.5)

y

x

0
0
R
−
∞
1
Z
Z R2
y

x

tw
0
tZw Zy Z∞
y

t

Ztw Zy yZ−t

p(2, A1 , A2 ) =

A1 ) =

p(2, A1 , =) =
p(2, tw , A1 ) =

y

t2

p(2, tw , A2 ) =

disc
pDS,
R1 > R2 ≥0 (2, tw , =) =

p(2, tw , =) =

y

∑∑

y =2 t =1

t

0

z

0

Z∞ Ztw Z∞

l2 (z) g(t) l1 (y) dµ

(D.3.6)

Z∞ Ztw yZ−t

l2 (z) g(t) l1 (y) dµ

(D.3.7)

t

y

tw

pDS
R1 > R2 ≥0 (2, tw , A2 ) =

z

0 y−t

0

0

A1 , =) =

pDS
R1 > R2 ≥0 (2, tw ,

Ztw R1Z− R2

0 y−t

y

tw

∞

z

z

t

0

0

tw

∑ ∑

y = t w t =1

l2disc (y − t) gdisc (t) l1disc (y)

(D.3.8)

These formulas will change for scenarios where one or more reorder points are negative.

D.3.2

At least one reorder point is negative

Whenever the second reorder point is negative, R2 < 0, the second supply channel
will not be considered. In other words, there will only be one-order cases independent
from the amount of occurring demand. This can be simply expressed by the following

314

A PPENDIX D. F ORMULAS F OR D UAL S OURCING W ITH L OST S ALES

formulas.
A1 , .) =

Ztw Z∞

pDS
R1 ≥0> R2 (1, tw , .) =

Z∞ Z∞

pDS
R1 ≥0> R2 (1,

y

0

x

f ( x, y) l1 (y) dµ =

Ztw

l1 (y) dy

(D.3.9)

f ( x, y) l1 (y) dµ =

Z∞

l1 (y) dy

(D.3.10)

0

0

y

tw

x

tw

0

pDS
R1 ≥0> R2 (2, ., .) =

0

(D.3.11)

The probability of both one-order cases sum up to the unity. This can be easily verified.
In the case where both reorder points are negative, 0 > R1 > R2 , the warehouse will
not replenish this article. All demand will be satisfied by other warehouses. Consequently, the probability for one- and two-order cycles is zero and the probability of
direct shipments, p(DS), is equal to one.
pDS
0> R1 > R2 (1, ., .) = 0

(D.3.12)

pDS
0> R1 > R2 (2, ., .) = 0

(D.3.13)

p(DS) = 1

(D.3.14)

In the next section we will address the expected number of direct shipments.

D.4

Expected number of direct shipments

In this chapter the formulas for the expected number of direct shipments is developed.
Here again, we will first give a general and systematic overview over the different
conditions for a shortage. All expressions that differ from the backlog policy, compare
Table 3.6 on page 72, are indicated by a blue font color.
A look at the Table D.3 reveals that major adaptations have to be made only for
one formula for each Case 3, 4, 6, and 7. All remaining formulas need slight changes

Appendix D – Number of direct shipments
case

direct shipments

Case 1

(1, A1 , .)

Case 2

(1, tw , .)

dlt1 − R1

Case 3

(2, A1 , A1 )

dlt1 − R1

Case 6

(2, tw , A1 )

dlt2 + ( R1 − R2 ) −
dlt1 − Q1 −
+

Case 4

(2, A1 , A2 )

Case 7

(2, tw , A2 )

[ R1 − dlt1 ]
dlt2 − R2

dlt1 − ( R1 − R2 ) −
dlt2 − Q2 −
+

Case 5

(2, A1 , =)

Case 8

(2, tw , =)

[ R2 − dlt2 ]
dlt1 − R1

(dlt2 − R2 )

315
direct shipment condition
before unique arrival, A1 , if R1 > 0 and
R1 − dlt1 < 0

before 1st arrival, A1 , if R1 > 0 and
R1 − dlt1 < 0

before 2nd arrival, A2 , if R1 > R2 ≥ 0 and
dlt2 + ( R1 − R2 ) − dlt1 > Q1 + [ R1 − dlt1 ]

+

before 1st arrival, A2 , if R1 > 0 and
R2 − dlt2 < 0

before 2nd arrival, A1 , if R1 > R2 ≥ 0 and
dlt1 − ( R1 − R2 ) − dlt2 > Q2 + [ R2 − dlt2 ]

+

before simultaneous arrivals, A1 = A2 , if
R1 > R2 ≥ 0 and R1 − dlt1 < 0

(or equiv. R2 − dlt2 < 0)

Table D.3: Conditions for direct shipments – expressions in blue differ from the backlog policy
regarding their condition and can be directly derived from the backlog policy. Note,
all major adaptations occur only in two-order cycles for which both reorder points
have to be non-negative.
In the following we will give the formulas for the direct shipment policy separately
for each of the reorder point scenarios.

D.4.1

All reorder points are non-negative

For Case 1 and 2 the expected number of direct shipments is identical to the shortage
of a warehouse in the backlog policy, i.e. the customer demand that cannot be imme-

316

A PPENDIX D. F ORMULAS F OR D UAL S OURCING W ITH L OST S ALES

diately satisfied by the warehouse itself, see equations (3.3.6) and (3.3.7) on page 77.
h
i
ERDS1 > R2 ≥0 − Xs,A1 1(1,A1 ,.) 1 Ms,A ,<0 = 0
1
i
h
DS
ER1 > R2 ≥0 − Xs,A1 1(1,tw ,.) 1 Ms,A ,<0 =

(D.4.1)

1

Z∞ R1Z− R2 Z∞
y

x0

x

tw

0

R1 − x

( x 0 − R1 + x ) f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dµ

(D.4.2)

Note, the random variable Xs,A1 originates from the backlog policy. The number of
direct shipments before the first arrival is identical to the amount of shortage in the
backlog policy. This applies to all two-order cases. For a comparison with the formulas
of Cases 3 to 8 used in the backlog policy see Table 3.9 on page 99.
h
ER<1A>1R,DS
− Xs,A1 1(2,A1 ,A1 ) 1 Ms,A
2 ≥0
Ztw Zy Z∞ Z∞
t

y

z

x

0 y − t R2

0

y

0

t

z

0

0

ER<1A>2R,DS
2 ≥0

h

x

R2

∑∑ ∑

y =2 t =1 z = y − t

ER<1A>1R,DS
2 ≥0

h

y

tw

t

Z∞

x

R2

1 ,<0

z

x

0 y − t R2

i

1 ,<0

i

(D.4.3)

=
(D.4.4)

=

( x − R2 ) f disc ( x, z) l2disc (z) gdisc (t) l1disc (y) dx

− Xs,A1 1(2,tw ,A1 ) 1 Ms,A

Z∞ Ztw Z∞ Z∞

2 ,<0

i

( x − R2 ) f ( x, z) l2 (z) g(t) l1 (y) dµ

− Xs,A1 1(2,A1 ,=) 1 Ms,A

t w y −1 y − t

=

( x − R2 ) f ( x, y − t) l2 (z) g(t) l1 (y) dµ

h
ER<1A>2R,DS
− Xs,A2 1(2,A1 ,A2 ) 1 Ms,A
2 ≥0
Ztw Zy yZ−t Z∞

1 ,<0

i

(D.4.5)

=

( x − R2 ) f ( x, y − t) l2 (z) g(t) l1 (y) dµ

(D.4.6)

Appendix D – Number of direct shipments
h
− Xs,A2 1(2,tw ,A2 ) 1 Ms,A
ER<1A>2R,DS
2 ≥0
Z∞ Ztw yZ−t Z∞
y

tw

t

0

z

0

x

R2

tw

2 ,<0

i

=
(D.4.7)

( x − R2 ) f ( x, z) l2 (z) g(t) l1 (y) dµ

h
− Xs,A1 1(2,tw ,=) 1 Ms,A
ER<1A>2R,DS
2 ≥0
∞

317

y−t

∑ ∑ ∑

y = t w +1 t =1 z = y − t

Z∞

x

R2

1 ,<0

i

=

( x − R2 ) f disc ( x, z) l2disc (z) gdisc (t) l1disc (y) dx

(D.4.8)

Last but not least we look at the number of direct shipments between the arrival of
both orders for the cases 3, 4, 6, and 7. We will explain the formula using the example
of Case 3. The argumentation can be applied analogously to the remaining cases. In
Case 3 we first want to know the demand d[ A1 ,A2 ] that occurs between both arrivals,
A1 and A2 .
(D.4.9)

d[ A1 ,A2 ] = dlt2 + ( R1 − R2 ) − dlt1

The demand d[ A1 ,A2 ] is calculated by subtracting the demand dlt1 during the lead time
lt1 from the total demand dlt2 + ( R1 − R2 ) that occurs between triggering the first

order and the arrival of the second order, A2 . In Equation (D.4.10) the demand d[ A1 ,A2 ]
is represented by the variable x.

Second, we know that there are at least Q1 units on stock after the first order arrives
at A1 due to the fact that potential unmet demand is covered by other warehouses. So
Q1 units have to subtracted from the demand between A1 and A2 .
Third, any stock on hand just before the first order arrives at A1 will additionally
+

decrease the number of direct shipments. This is expressed by [ R1 − x ] in Equation (D.4.10).

In total the formula for the number of expected direct shipments between A1 and

318

A PPENDIX D. F ORMULAS F OR D UAL S OURCING W ITH L OST S ALES

A2 in Case 3 is given by

≥ A1 ,DS
ER1 > R2 ≥0 − XsDS
+ Q1 ,A2 1(2,A1 ,A1 ) 1 MSD

s+ Q1 ,A2 ,<0

Ztw Zy Z∞ Z∞
z

t

y

x

Z∞

x0

0 y−t 0 [ R − x ]+ + Q
2
1

0



=
+

( x 0 − [ R1 − x ] − Q1 ) f ( x 0 , z + t − y ) ·
(D.4.10)

f ( x, y − t) l2 (z) g(t) l1 (y) dµ
The same approach can be applied to the formula of Case 6


≥ A1 ,DS
DS
=
ER1 > R2 ≥0 − Xs+Q1 ,A2 1(2,tw ,A1 ) 1 MSD
s+ Q1 ,A2 ,<0

Z∞ Ztw Z∞ Z∞
y

tw

t

z

x

Z∞

x0

0 y−t 0 [ R − x ]+ + Q
2
1

+

( x 0 − [ R1 − x ] − Q1 ) f ( x 0 , z + t − y ) ·
(D.4.11)

f ( x, y − t) l2 (z) g(t) l1 (y) dµ

and to the scenarios where the second order arrives first, namely, the formulas for
Case 4 and Case 7. These are given by the equations (D.4.12) and (D.4.13), respectively.


≥ A1 ,DS
DS
=
ER1 > R2 ≥0 − Xs+Q2 ,A1 1(2,A1 ,A2 ) 1 MSD
s+ Q2 ,A1 ,<0

Ztw

y

0

Zy yZ−t Z∞
t

0

z

0

x

Z∞

x0

0 [ R − x ]+ + Q
2
2

+

( x 0 − [ R2 − x ] − Q2 ) f ( x 0 , y − z − t ) ·
(D.4.12)

f ( x, z) l2 (z) g(t) l1 (y) dµ

ER≥1A>1R,DS
2 ≥0



− XsDS
+ Q2 ,A1

Z∞ Ztw yZ−t Z∞
y

tw

t

0

z

0

x

1(2,tw ,A2 ) 1 MSD

s+ Q2 ,A1 ,<0

Z∞

x0

0 [ R − x ]+ + Q
2
2



=
+

( x 0 − [ R2 − x ] − Q2 ) f ( x 0 , y − z − t ) ·

f ( x, z) l2 (z) g(t) l1 (y) dµ

(D.4.13)

Appendix D – Average cycle stock

319

All these formulas hold if both reorder points are positive. Now, we investigate the
cases where this is not true.

D.4.2

At least one reorder point is negative

Whenever only the second reorder point is negative, so R1 ≥ 0 > R2 holds, there will
exclusively occur one-order cycles.
ERDS1 ≥0> R2

h

− Xs,A1 1(1,A1 ,.) 1 Ms,A

h
ERDS1 ≥0> R2 − Xs,A1 1(1,tw ,.) 1 Ms,A

1 ,<0

1 ,<0

i

=

i

=

Ztw Z∞
y

x

0 R1
∞
Z Z∞
y

x

t w R1

( x − R1 ) f ( x, y) l1 (y) dµ

(D.4.14)

( x − R1 ) f ( x, tw ) l1 (y) dµ (D.4.15)

The remaining cases 3 to 8 have a probability of zero and there are no direct shipments.
Thus, both expressions can be unified and yield the expression for the number of
expected shipments.
ERDS1 ≥0> R2

h

− Xs,A1 1 Ms,A

1 ,<0

i

=

Z∞ Z∞
y

0

x

R1

( x − R1 ) f ( x, min(y, tw )) l1 (y) dµ (D.4.16)

In the last reorder point scenario, 0 > R1 > R2 , there will be no replenishment. Consequently, we cannot specify replenishment parameters like the average cycle time
either. In practice, however, this means that all customer demand has to be satisfied
via direct shipments. Then the expected number of direct shipments per day equals
the average customer demand per day.

D.5

Expected average cycle stock

Compared to the backlog policy the stock after each delivery of an order can never
be negative due to the fact that unsatisfied demand is covered by direct shipments

320

A PPENDIX D. F ORMULAS F OR D UAL S OURCING W ITH L OST S ALES

in the direct shipment policy. Consequently, there will be no differences in the stock
formulas for the time before the first delivery.

D.5.1

Both reorder points are non-negative

In Case 1 the stock cannot be negative because the second reorder point, R2 , is positive
and never reached.
ER<1A>1R,DS
2 ≥0

h

i

stock 1(1,A1 ,.) =

h

x

y

0

0

y

x

0

i

ER≥1A>1R,DS
stock 1(1,A1 ,.) =
2 ≥0
Ztw R1Z− R2

Ztw R1Z− R2

Q1 − x
µ



0


x
y R1 −
f ( x, y) l1 (y) dµ
2

Q −x
R1 + 1
2



f ( x, y) l1 (y) dµ

(D.5.1)

(D.5.2)

For Case 2 the average stock on hand is composed of three parts, the stock before tw ,
between tw and t A1 , and the stock after t A1 until the end of the cycle.
w ,DS
ER<1t>
R2 ≥0

h

ER<1A>1R,DS
2 ≥0

h



Z∞ R1Z− R2

i

Z∞ R1Z− R2 Z∞

stock 1(1,tw ,.) =

min( x 0 , R
2

i

y

x

tw

stock 1(1,tw ,.) =

R1 − x −

h

i

1

0

y

tw

− x)

tw

x



0

0



x0

x
R1 −
f ( x, tw ) l1 (y) dµ
2


R −x
(y − tw ) min 1, 1 0
x



(D.5.3)

·

f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dµ

Z∞ R1Z− R2 Z∞

+

( Q1 − x − x 0 )
·
y
x
x0
µ
tw
0
0
"
#
+
( Q1 − x − x 0 )
0 +
( R1 − x + Q1 − x ) −
f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dµ
2

ER≥1A>1R,DS
2 ≥0

stock 1(1,tw ,.) =

(D.5.4)

(D.5.5)

Appendix D – Average cycle stock

321

The formulas for all two-order scenarios are given in the following. For Case 3 the
expected stock on hand is given by the following four formulas.

<t g ,DS
E R1 > R2 ≥0

h

ER<1A>1R,DS
2 ≥0

h

ER<1A>2R,DS
2 ≥0

i

Ztw Zy Z∞

i

Ztw Zy Z∞ Z∞

stock 1(2,A1 ,A1 ) =

y

h

R2 −

y

i

stock 1(2,A1 ,A1 ) =
+

min 1,

( R2 − x ) + Q1
x0

t



R − R2
t R1 − 1
2

z

0 y−t 0

0

min( R2 , x )
2

z

0 y−t

0

stock 1(2,A1 ,A1 ) =


t



l2 (z) g(t) l1(y) dµ (D.5.6)



R
(y − t) min 1, 2
x
x



·
(D.5.7)

f ( x, y − t) l2 (z) g(t) l1 (y) dµ

Ztw Zy Z∞ Z∞ Z∞
y

0



!

t

z

x

0 y−t 0

0

x0

(t + z − y)·

 ( R2 − x ) + + Q1 −

min



x0 , ( R

2

0



z

0 y−t 0

x

0

x0

h
i+

 ( R2 − x ) + + Q1 − x 0 + Q2 −



µ

h

·

+

·

( R2 − x ) + Q1 − x 0

f ( x 0 , t + z − y) f ( x, y − t) l2 (z) g(t) l1 (y)dµ



(D.5.8)

h
i
ER≥1A>2R,DS
stock
1
(2,A1 ,A1 ) =
2 ≥0
h
+
i+
+
( R2 − x ) + Q1 − x 0 + Q2 − R1
Ztw Zy Z∞ Z∞ Z∞
t

− x ) + Q1

2

f ( x 0 , t + z − y) f ( x, y − t) l2 (z) g(t) l1 (y) dµ

y

+

2

i+

+ 
+ Q2 − R1 

·



(D.5.9)

322

A PPENDIX D. F ORMULAS F OR D UAL S OURCING W ITH L OST S ALES

Closely linked to the formulas of Case 3 are the formulas for Case 6.
i
h
<t g ,DS
ER1 > R2 ≥0 stock 1(2,tw ,A1 ) =
Z∞ Ztw Z∞
y

tw

ER<1A>1R,DS
2 ≥0

ER<1A>2R,DS
2 ≥0

h

h

t

z

0 y−t

i

stock 1(2,tw ,A1 ) =


( R2 − x ) + Q1
x0



y

t

z

0 y−t 0





R
(y − t) min 1, 2
x
x

tw

!

t

z

x

0 y−t 0



0

x0

 ( R2 − x ) + + Q1 −

min



x0 , ( R

2

tw



z

0 y−t 0

x

0

x0

h
i+

 ( R2 − x ) + + Q1 − x 0 + Q2 −



·

+

·

( R2 − x ) + Q1 − x 0

f ( x 0 , t + z − y) f ( x, y − t) l2 (z) g(t) l1 (y)dµ

− x ) + Q1



(D.5.12)

µ

h

+

2

h
i
ER≥1A>2R,DS
stock
1
(2,tw ,A1 ) =
2 ≥0
h
+
i+
+
( R2 − x ) + Q1 − x 0 + Q2 − R1
Z∞ Ztw Z∞ Z∞ Z∞
t

(D.5.11)

(t + z − y)·

f ( x 0 , t + z − y) f ( x, y − t) l2 (z) g(t) l1 (y) dµ

y

·

f ( x, y − t) l2 (z) g(t) l1 (y) dµ

Z∞ Ztw Z∞ Z∞ Z∞
y

(D.5.10)

l2 (z) g(t) l1(y) dµ

Z∞ Ztw Z∞ Z∞
tw

stock 1(2,tw ,A1 ) =

min 1,

R − R2
t R1 − 1
2

min( R2 , x )
R2 −
2

i

+



2

i+

+ 
+ Q2 − R1 

·



(D.5.13)

Appendix D – Average cycle stock

323

For Case 4 the stock on hand is composed by four subcases, as well.
<t g ,DS
E R1 > R2 ≥0

h

i

stock 1(2,A1 ,A2 ) =
Ztw Zy yZ−t 
y

0

ER<1A>2R,DS
2 ≥0

h

t

0

ER<1A>1R,DS
2 ≥0

h

z

0

i

stock 1(2,A1 ,A2 ) =
R2 −

i

stock 1(2,A1 ,A2 ) =

R − R2
t R1 − 1
2

Ztw Zy yZ−t Z∞
y

t

0

z

0

0

min( R2 , x )
2

z

t

y

0

0

l2 (z) g(t) l1 (y) dµ



R2
z min 1,
x
x
0
!
+

x

0



0

x0

(D.5.14)

·

f ( x, z) l2 (z) g(t) l1 (y) dµ

Ztw Zy yZ−t Z∞ Z∞
0



(D.5.15)

(y − t − z)·



!
0 , ( R − x )+ + Q
+
min
x
2
2
+
( R2 − x ) + Q2 
·
min 1,
( R2 − x ) + Q2 −
0
x
2

f ( x 0 , y − t − z) f ( x, z) l2 (z) g(t) l1 (y) dµ

(D.5.16)

h
i
ER≥1A>1R,DS
stock
1
(2,A1 ,A2 ) =
2 ≥0
h
+
i+
+
0
( R2 − x ) + Q2 − x
+ Q1 − R1
Ztw Zy yZ−t Z∞ Z∞
y

0

t

0

z

0

x

0

0

x0

h
i+

 ( R2 − x ) + + Q2 − x 0 + Q1 −



µ

h

+

·

( R2 − x ) + Q2 + x 0

f ( x 0 , y − t − z) f ( x, z) l2 (z) g(t) l1 (y) dµ

2

i+

+ 
+ Q1 − R1 

·



(D.5.17)

324

A PPENDIX D. F ORMULAS F OR D UAL S OURCING W ITH L OST S ALES

Once again the formulas of Case 7 are closely related to the formulas of Case 4.
h
i
<t g ,DS
ER1 > R2 ≥0 stock 1(2,tw ,A2 ) =

Z∞ Ztw yZ−t 
y

tw

ER<1A>2R,DS
2 ≥0

h

t

z

0

0

h

stock 1(2,tw ,A2 ) =

i

stock 1(2,tw ,A2 ) =



Z∞ Ztw yZ−t Z∞

i

y

t

tw

R2 −

ER<1A>1R,DS
2 ≥0

R − R2
t R1 − 1
2

z

0

0

min( R2 , x )
2



R2
z min 1,
x
x
0
!
+

tw

z

t

0

0

x

0



0

x0

(D.5.18)

·

f ( x, z) l2 (z) g(t) l1 (y) dµ

Z∞ Ztw yZ−t Z∞ Z∞
y

l2 (z) g(t) l1 (y) dµ

(D.5.19)

(y − t − z)·



!
0 , ( R − x )+ + Q
+
min
x
2
2
+
( R2 − x ) + Q2 
·
min 1,
( R2 − x ) + Q2 −
0
x
2

f ( x 0 , y − t − z) f ( x, z) l2 (z) g(t) l1 (y) dµ
ER≥1A>1R,DS
2 ≥0

h

(D.5.20)

i

stock 1(2,tw ,A2 ) =
h
+
i+
+
0
y
−
t
( R2 − x ) + Q2 − x
+ Q1 − R1
Z∞ Ztw Z Z∞ Z∞
y

t

w

t

0

z

0

x

0

0

x0

h
i+

 ( R2 − x ) + + Q2 − x 0 + Q1 −



µ

h

+

·

( R2 − x ) + Q2 + x 0

f ( x 0 , y − t − z) f ( x, z) l2 (z) g(t) l1 (y) dµ

2

i+

+ 
+ Q1 − R1 

·



(D.5.21)

Appendix D – Average cycle stock

325

Finally, we develop the formulas for the two scenarios where both orders arrive simultaneously. First, for Case 5 these are given by the following three equations.
h
i
<t g ,DS
ER1 > R2 ≥0 stock 1(2,A1 ,=) =


tw y
R1 − R2
l2disc (y − t) gdisc (t) l1disc (y)
∑ ∑ t R1 − 2
y =2 t =1
ER<1A>1R,DS
2 ≥0


h

i

stock 1(2,A1 ,=) =

min( R2 , x )
R2 −
2



h

tw

∑∑

y =2 t =1



Z∞
0

R2
(
y − t) min 1,
x
x



·

f disc ( x, y − t) l2disc (y − t) gdisc (t) l1disc (y) dx

i

ER≥1A>1R,DS
stock 1(2,A1 ,=) =
2 ≥0


y

(D.5.22)

tw

y

∑∑

y =2 t =1

Z∞

h

x

0

h

+

( R2 − x ) Q1 + Q2 − R1
µ

+

( R2 − x ) + Q1 + Q2 − R1
+

(
R
−
x
)
+
Q
+
Q
−
 2
2
1
2

(D.5.23)

i+

·

i+ 

f disc ( x, y − t) l2disc (y − t) gdisc (t) l1disc (y) dx


·

(D.5.24)

For Case 8 the formulas are very similar to Case 5.
<t g ,DS
E R1 > R2 ≥0

h

i

stock 1(2,tw ,=) =


∞ tw
R1 − R2
l2disc (y − t) gdisc (t) l1disc (y)
∑ ∑ t R1 − 2
y = t w t =1


h
i
∞ tw Z∞
R2
< A1 ,DS
ER1 > R2 ≥0 stock 1(2,tw ,=) = ∑ ∑
(y − t) min 1,
·
x
x
y = t w t =1
0


min( R2 , x )
R2 −
f disc ( x, y − t) l2disc (y − t) gdisc (t) l1disc (y) dx
2

(D.5.25)

(D.5.26)

326

A PPENDIX D. F ORMULAS F OR D UAL S OURCING W ITH L OST S ALES
h

i

ER≥1A>1R,DS
stock 1(2,tw ,=) =
2 ≥0


∞

tw Z∞

∑ ∑

y = t w t =1

h

x

0

h

+

( R2 − x ) Q1 + Q2 − R1
µ

+

( R2 − x ) + Q1 + Q2 − R1
+

 ( R2 − x ) + Q1 + Q2 −
2

f disc ( x, y − t) l2disc (y − t) gdisc (t) l1disc (y) dx

i+

·

i+ 


·

(D.5.27)

This completes the set of formulas to calculate the average stock on hand when both
reorder points are non-negative. The remaining scenarios are investigated in the next
section.

D.5.2

At least one reorder point is negative

First, we will assume that only the second reorder point is negative. Thus, the second
order channel will never be used and we will only observe one-order cycles no matter
how much customer demand occurs.
h
i
ER<1A≥10,DS
stock
1
(1,A1 ,.) =
> R2
Ztw Z∞



R1
min( x, R1 )
y min 1,
R1 −
f ( x, y) l1 (y) dµ
y
x
x
2
0 0
h
i
ER≥1A≥10,DS
stock
1
(1,A1 ,.) =
> R2
!
+
+
Ztw Z∞
( Q1 − x )
( Q1 − x )
R1 +
f ( x, y) l1 (y) dµ
y
x
µ
2
0



(D.5.28)

(D.5.29)

0

In Equation (D.5.28) above it is easy to see that the stock on hand will always be positive as long as R1 > 0 holds. For Case 2 one additional equation is needed to deter-

Appendix D – Average cycle stock

327

mine the expected stock on hand.
h
i
w ,DS
stock
1
ER<1t≥
(1,tw ,.) =
0> R2



Z∞ Z∞
R1
min( x, R1 )
tw min 1,
R1 −
f ( x, tw ) l1 (y) dµ
y
x
x
2
tw

0

h

i

ER<1A≥10,DS
> R2 stock 1(1,tw ,.) =


 ( R1 − x ) + −
h



h

Z∞ Z∞ Z∞
y

x

tw

0

0

x0

min x 0 , ( R1 − x )

i

ER≥1A≥10,DS
> R2 stock 1(1,tw ,.) =


(D.5.30)

+

0
 ( R1 − x ) + Q1 − x

+

2

Z∞ Z∞ Z∞
y

x

tw

i+

0

−

f ( x 0 , y − tw f ( x, tw ) l1 (y) dµ

h

0

(y − tw ) min 1,

x0

( R1 − x )
x0

!

+

·


h

 f ( x 0 , y − tw ) f ( x, tw ) l1 (y) dµ
+

( R1 − x ) + Q1 − x 0 − R1
µ

+

( R1 − x ) + Q1 −
2

x0

− R1

i+

(D.5.31)

·

i+ 


·

(D.5.32)

This completes the set of formulas for the stock on hand in the case where R1 ≥ 0 >

R2 . Moreover, this concludes our elaboration of the formulas needed to calculate the

average stock on hand due to the fact that for the last reorder point scenario, 0 > R1 >
R2 , there will be no replenishment process at all and, thus, no stock on hand.

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