Inventory Rationing (eBook)

Karin Möllering published her work with Kölner Wissenschaftsverlag until 2018.

Contents 8
List of Tables 14
List of Figures 16
List of Abbreviations 18
List of Symbols 20
1 Introduction 23
1.1 Motivation 24
1.2 Research Objectives 25
1.3 Outline 26
Part I Foundations of Stochastic Inventory Control 29
2 Basic Concepts of Inventory Management 30
2.1 Inventory Management and Control 31
2.2 Motivations for Holding Inventory 32
2.3 Classification of Inventories 34
2.3.1 The Strategic Perspective 34
2.3.2 The Operational Perspective 35
2.4 Inventory-related Costs 37
2.4.1 Order Cost 37
2.4.2 Holding Cost 38
2.4.3 Penalty Cost 39
2.5 Service Level 40
3 Stochastic Inventory Control 42
3.1 Characteristics of Inventory Models 43
3.1.1 Management Decisions 43
3.1.2 System-Inherent Characteristics 44
3.2 Types of Inventory Control Policies 46
3.3 Periodic Inventory Control 48
3.3.1 Single-Period Inventory Control 49
3.3.2 Multi-Period Inventory Control 50
3.3.3 Service-Constraint Multi-Period Inventory Control 53
Part II Essential Stochastic Processes 54
4 Markov Chains 55
4.1 The Markov Property 56
4.2 Homogeneous Markov Chains 57
4.3 Limit Distribution and Invariant Distribution 58
4.4 Communication Classes 60
4.5 Class Properties 61
4.5.1 Aperiodicity 61
4.5.2 Recurrency and Transiency 61
4.5.3 Class Criteria 63
4.6 Uniqueness of Limit Distributions 64
4.7 Multi-Dimensional Markov Chains 65
4.8 Applications of Markov Chains 66
5 Numerical Solution of Infinite Markov Chains 67
5.1 System Reduction Approaches 68
5.1.1 Reduced System Approaches 68
5.1.2 Geometric Tail Distributions 69
5.2 State Space Reduction Approaches 70
5.2.1 Augmentations 70
5.2.2 Pointwise Convergence 72
5.3 Algorithms 74
5.3.1 Sheskin’s Partitioning Algorithm 74
5.3.2 Power Iteration 76
5.4 Criteria for Choosing a Specific Algorithm 76
6 Comparing Stochastic Processes 78
6.1 Stochastic Ordering for Distribution Functions 78
6.2 The Sample Path Method 80
Part III Stochastic Inventory Control with Customer Segmentation 82
7 Introduction to Inventory Rationing 83
7.1 Examples for Different Customer Classes 84
7.2 Related Areas 85
7.3 Rationing Rules 87
7.4 Backorders and Backorder Clearing Mechanisms 89
7.5 Literature Review 91
7.5.1 Characterizations of the Optimal Policy 91
7.5.2 Evaluations and Optimizations of Critical Level Policies 94
7.5.3 Other Studies Involving Critical Level Rationing 98
7.6 Classification of Our Work 99
8 A Markov-Chain Based Modeling Approach 101
8.1 Modeling Framework 102
8.2 Recursive Expressions for Backorders 102
8.3 Markov Chain 104
8.4 Structural Results 105
8.5 Proofs 108
9 Prioritization by Penalty Costs 115
9.1 Optimization Approach 116
9.1.1 Convexity of Objective Function 116
9.1.2 Transition Matrix 118
9.1.3 Cost Function 119
9.1.4 Numerical Optimization Algorithm 120
9.2 Numerical Results 120
9.2.1 Critical Level Policy versus Benchmark Policies 121
9.2.2 Combination Heuristic 124
9.3 Conclusion 124
9.4 Proofs 126
10 Prioritization by Service Levels 129
10.1 Analytical Insights 130
10.1.1 Service Level Constraints 130
10.1.2 Potentially Optimal Parameter Constellations 133
10.1.3 Structural Results for the Optimal Solution 135
10.1.4 Cost Function 137
10.1.5 Model Alteration: ?-Service Levels 138
10.2 Numerical Results 139
10.2.1 Optimization Algorithm 140
10.2.2 Critical Level Policy versus Benchmark Policies 141
10.2.3 Heuristic Approach 144
10.3 Conclusion 145
10.4 Proofs 147
11 Dynamic Rationing Policies 153
11.1 The Next Period Optimization Policy 154
11.1.1 The Policy 154
11.1.2 Structural results 155
11.2 The Linear Critical Level Policy 156
11.2.1 The Policy 156
11.2.2 Structural results 157
11.3 Efficiency of the Dynamic Rationing Policies 160
11.3.1 Next Period Optimization and Constant Rationing Policy 161
11.3.2 Linear Critical Level Policy and Constant Rationing Policy 165
11.3.3 Next Period Optimization and Linear Critical Level Policy 168
11.4 Conclusion 170
11.5 Proofs 171
12 Conclusion and Critical Review 174
12.1 Contributions 174
12.2 Critical Review 176
12.3 Future Research 177
Bibliography 179