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Three Essays on Inventory Management THREE ESSAYS ON INVENTORY MANAGEMENT By JIANG ZHANG Submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Thesis Advisor: Dr. Matthew J. Sobel Department of Operations CASE WESTERN RESERVE UNIVERSITY August 2004 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the dissertation of ______________________________________________________ candidate for the Ph.D. degree *. (signed)_______________________________________________ (chair of the committee) ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ (date) _______________________ *We also certify that written approval has been obtained for any proprietary material contained therein. I grant to Case Western Reserve University the right to use this work, irrespective of any copyright, for the University’s own purposes without cost to the University or to its students, agents and employees. I further agree that the University may reproduce and provide single copies of the work, in any format other than in or from microforms, to the public for the cost of reproduction. JIANG ZHANG To my mother: Huishu Jiang my father: Shuqing Zhang my wife: Dr. Yan Cao Contents ListofTables.................................................... viii ListofFigures................................................... ix Acknowledgements............................................... x Abstract......................................................... xii 1 InventoryReplenishmentwitha FinancialCriterion........... 1 1.1Introduction.............................. 1 1.2ModelFormulation.......................... 7 1.3DynamicProgrammingAnalysis................... 11 1.4 Optimality of (sn, Sn)ReplenishmentPolicies........... 14 1.5 Infinite Horizon Convergence . 18 1.6ModelswithSmoothingCosts.................... 19 1.7ConcludingRemarks......................... 23 v 2 Fill Rate of General Review Supply Systems. 25 2.1Introduction.............................. 25 2.2GeneralPeriodicReviewSystem................... 29 2.3 Uncapacitated Single-stage Systems . 32 2.4 Gamma and Normal Demand in Single-stage Systems . 35 2.4.1 GammaDemandDistribution................ 35 2.4.2 NormalDemandDistribution................ 36 2.4.3 Fill Rate Approximation for Normal Demand Distribution 39 2.5Multi-StageGeneralReviewSystems................ 40 2.5.1 FillRateinTwo-StageSystems............... 42 2.5.2 Fill Rate in Two-Stage Systems with General Leadtime . 48 2.5.3 NumericalExample...................... 51 2.6FillRateinaThree-StageSystem.................. 51 2.7Conclusion............................... 56 3 Interchangeability of Fill Rate Constraints and Backorder Costs inInventoryModels.............................................. 60 3.1Introduction.............................. 60 3.2ModelandProblemFormulations.................. 66 3.3ContinuousDemand......................... 72 3.4DiscreteDemand........................... 76 3.5 Interchangeability . 79 vi 3.6Examples............................... 82 3.6.1 StrictlyPositiveDemandDensity.............. 83 3.6.2 NonStrictlyPositiveDemandDensity........... 84 3.6.3 DiscreteDemand....................... 86 3.7GeneralizationsandSummary.................... 89 3.8Appendix............................... 91 Bibliography .................................................... 94 vii List of Tables 2.1 Fill Rate and its Approximation for Normal Demand ......... 57 2.2 Fill Rate of Two-stage Systems for Normal Demand (a) ....... 58 2.3 Fill Rate of Two-stage Systems for Normal Demand (b) ....... 59 3.4 Distribution Function and Expected Number of Backorders ...... 87 3.5 Values of G−1(·)andB−1(·) ...................... 88 3.6 S-optimal Base-Stock Levels and Fill Rates at which they are F-optimal 88 3.7 F-optimal Base-Stock Levels and Unit Stockout Costs at which they are S-optimal ................................ 89 viii List of Figures 2.1 The standard N-stage serial inventory system . 30 2.2 The Fill Rate Integral for a system with Normal Demand . 38 3.3 Dependence of S-Optimal Base-Stock Level on Stockout Cost: Non- negativeDensity........................... 84 3.4 Locus of {(b, f)} with the Same Optimal Base-Stock Level: Non- negativeDensity........................... 86 ix Acknowledgements I would like to express my sincere gratitude to my mentor, Professor Matthew J. Sobel, who encouraged and guided me through various phases of my doctoral studies with patience. I would also like to thank him for his incredible effort and willingness to help me at any time and any where. I would specially like to thank my dissertation committee members, Profes- sors Lisa Maillart, Peter Ritchken, and Yunzeng Wang for their generous insight, comments, and support on this work. In addition, my thanks are also owed to Pro- fessors Apostolos Burnetas, Hamilton Emmons, Kamlesh Mathur, Daniel Solow, and George Vairaktarakis for their help throughout my doctoral studies. I would like to express my appreciation to the Department of Operations, Case Western Reserve University, for their generous financial support. Special thanks to department’s staff, Elaine Iannicelli, Sue Rischar, and Emily Anderson for their help throughout my study in the department. I had a fabulous time at Case which would not have been possible without the company of friends like Junze Lin, Zhiqiang Sun, Yuanjie He, Huichen Chiang, x Wei Wei, Xiang Fang, Qiaohai Hu, Will Millhiser, Ant Printezis, Halim Hans, and Kang-hua Li who have always helped and cheered me up in every possible way. Finally, I would like to thank my family for their unconditional love, support and encouragement. My special thanks are for my wife Yan who is always there for me through everything. xi Three Essays on Inventory Management Abstract by JIANG ZHANG This dissertation consists of three essays that are related to inventory man- agement. The first essay models a single-product equity-owned firm which orders prod- ucts from an outside supplier, borrows short-term capital for solvency, and issues dividends to its shareholders while facing financial risks and demand uncertainty. The firm maximizes the expected present value of the time stream of dividends. If there is a setup cost in this model, we show that an (s, S) replenishment policy is optimal by jointly optimizing the firm’s operational and financial decisions. The analysis is not a straightforward copy of Scarf’s argument. The second part of this essay studies the same model with a smoothing cost (instead of a setup cost) and shows that the optimal policy has the same form as the traditional smoothing xii cost model. Although operational decisions and financial decisions interact with each other in these models, the optimal inventory policies have standard forms. The second essay obtains fill rate formulas for general review inventory models with base-stock-level policies. Ordering decisions in a general review model are made every R (R ≥ 1) periods but demand arises every period. We provide exact fill rate formulas for single-stage model with a general demand distribution. A simple fill rate expression is derived for the model with normally distributed demand. For multi-stage models, we first discuss a general review procedure at each stage and then provide exact fill rate formulas for two-stage and three-stage models. There are parallel streams of literature which analyze identical models except that one stream has backorder costs and the other has fill rate constraints. The third essay clarifies redundancy in the two streams of dynamic inventory models with linear purchase costs. We show that optimal policies for either kind of model can be inferred from the other. That is, inventory fill rate constraints and backorder costs are interchangeable in dynamic newsvendor models. xiii Chapter 1 Inventory Replenishment with a Financial Criterion 1.1 Introduction Nearly all the literature on optimal inventory management uses criteria of cost minimization or profit maximization. An inventory managers’ goal for example, is modeled as minimizing cost or maximizing profit while satisfying customers’ demands. If inventory decisions do not affect the revenue stream, these two crite- ria result in the same optimal replenishment policy. Most of this literature treats firms’ inventory decisions and financial decisions separately. This dichotomy is perhaps due to the perception that inventory managers in a large firm cannot in- fluence the firm’s financial policy and that financial officers are usually detached 1 2 from the inventory decisions. This separate consideration of financial and opera- tional decisions simplifies management and has its foundation in corporate finance. The pathbreaking papers, Modigliani and Miller (1958) and Modigliani and Miller (1963) (hereafter referred to as M-M), show that the firm’s capital structure and its financial decisions should be independent of the firm’s investment and opera- tional decisions if capital markets are perfect. However, when market imperfections such as taxes and transaction fees are in- troduced, the results characterized from these separate treatments may no longer hold. “Treating real and financial decisions of the firm as independent may not be justified.”(Dammon and Senbet 1988). Other sources of market imperfections include asymmetric information between supplier and retailer, asymmetric infor- mation between shareholders and managers, and differential
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