ESSAYS ON INVENTORY MANAGEMENT, CAPACITY MANAGEMENT, AND RESOURCE-SHARING SYSTEMS by Yang Bo APPROVED BY SUPERVISORY COMMITTEE: Milind Dawande, Co-Chair Ganesh Janakiraman, Co-Chair Alp Muharremoglu Anyan Qi Copyright c 2017 Yang Bo All rights reserved To my dear family and friends. ESSAYS ON INVENTORY MANAGEMENT, CAPACITY MANAGEMENT, AND RESOURCE-SHARING SYSTEMS by YANG BO, BS DISSERTATION Presented to the Faculty of The University of Texas at Dallas in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY IN MANAGEMENT SCIENCE THE UNIVERSITY OF TEXAS AT DALLAS August 2017 ACKNOWLEDGMENTS I am deeply thankful to my advisors, Drs. Milind Dawande and Ganesh Janakiraman, for patiently guiding me in my research endeavors, and for providing me with constructive advice and feedback. My frequent beneficial interactions with them have sharpened my research ability and have enabled me to become an independent researcher. I owe much to my parents and my wife; their unconditional love and support continues to give me courage and make me stronger. I would also like to express my gratitude to the following professors in my department, from whom I have benefited a lot: Alain Bensoussan, Metin Cakanyildirim, Dorothee Honhon, Elena Katok, Alp Muharremoglu, Shun-Chen Niu, Ozalp Ozer, Anyan Qi, Suresh P. Sethi, Kathryn Stecke, Serdar Simek, Shouqiang Wang, John J. Wiorkowski, and Shengqi Ye. Last but not the least, I would like to thank my fellow PhD students: Ying Cao, Bahriye Cesaret, Jiayu Chen, Shaokuan Chen, Wei Chen, Ilhan Emre Ertan, Zhichao Feng, Blair Flicker, Shivam Gupta, Harish Guda, Bharadwaj Kadiyala, Ismail Kirci, Chungseung Lee, Jingyun Li, Ting Luo, Sandun Perera, Xi Shan, Sina Shokoohyar, Yulia Vorotyntseva, Xiao Zhang, and many others. Many thanks for making my stay at UTD joyful. April 2017 v ESSAYS ON INVENTORY MANAGEMENT, CAPACITY MANAGEMENT, AND RESOURCE-SHARING SYSTEMS Yang Bo, PhD The University of Texas at Dallas, 2017 Supervising Professors: Milind Dawande, Co-Chair Ganesh Janakiraman, Co-Chair This dissertation consists of three essays, each focusing on one of the following three impor- tant topics in operations management: inventory management, capacity management, and management of resource-sharing systems. These topics are each summarized below. In the first essay, we investigate the following integrality question for inventory control on distribution systems: Given integral demands, does an integral optimal policy exist? We show that integrality holds under deterministic demand, but fails to hold under stochastic demand. In distribution systems with stochastic demand, we identify three factors that influence the gap between integral and real optimal policies: shipping cost variation across time, holding cost difference across stages, and economies of scale. We then obtain a tight worst-case bound for the gap, which captures the impact of all these factors. In the second essay, we investigate the computational complexity of determining the capacity of a process, a fundamental concept in Operations Management. We show that it is hard to calculate process capacity exactly and, furthermore, also hard to efficiently approximate it to within a reasonable factor; e.g., within any constant factor. These results are based on a novel characterization, which we establish, of process capacity that relates it to the vi fractional chromatic number of an associated graph. We also show that capacity can be efficiently computed for processes for which the collaboration graph is a perfect graph. The third essay addresses an important problem in resource-sharing systems. We study the minimum-scrip rule in such a system: whenever a service request arises, among those who volunteer and are able to provide service, the one with the least number of scrips (also known as coupons) is selected to provide service. Under mild assumptions, we show that everybody in the system being always willing to provide service is a Nash Equilibrium under the minimum-scrip rule. This suggests that the minimum scrip rule can lead to a high level of social welfare. vii TABLE OF CONTENTS ACKNOWLEDGMENTS . v ABSTRACT . vi LIST OF FIGURES . xi CHAPTER 1 INTRODUCTION . 1 CHAPTER 2 ON INTEGRAL POLICIES IN DETERMINISTIC AND STOCHASTIC DISTRIBUTION SYSTEMS . 4 2.1 Introduction . 5 2.2 Preliminaries . 8 2.2.1 Network Representation and Notation . 8 2.2.2 Economies of Scale Structure . 10 2.3 Distribution Systems with Deterministic Demand . 10 2.4 Distribution Systems with Stochastic Demand . 13 2.4.1 Failure of Integrality When Assumption S Does Not Hold . 14 2.4.2 Failure of Integrality When Assumption H Does Not Hold . 16 2.4.3 Bounding the Integrality Gap . 17 2.5 Managerial Implications of Results . 22 2.5.1 Unpredictable Demand: Variation in Shipping Costs Across Time . 22 2.5.2 Unpredictable Demand: Holding Cost Variation Across Stages . 23 2.5.3 Unpredictable Demand: Economies of Scale . 24 2.5.4 Predictable Demand . 24 2.6 Proofs of Results . 24 2.6.1 Proof of Theorem 2.3.1 . 24 2.6.2 Proof of Proposition 2.4.1 . 26 2.6.3 Proof of Proposition 2.4.2 . 29 2.6.4 Proof of Theorem 2.4.1 . 31 CHAPTER 3 ON FINDING PROCESS CAPACITY . 40 3.1 Introduction . 40 3.1.1 A Brief Overview of the Related Literature . 43 viii 3.2 Preliminary Results . 44 3.2.1 Formal Definition of Process Capacity . 45 3.2.2 Sufficiency of Cyclic Policies and Redundance of Precedence and Non- Preemption Constraints . 47 3.2.3 Static Planning Problem for Collaboration . 50 3.3 Characterizing Process Rates for Subclasses of Cyclic Policies . 52 3.3.1 The Maximum Process Rate Over 1-Unit Cyclic Policies . 52 3.3.2 The Maximum Process Rate Over k-Unit Cyclic Policies . 54 3.3.3 The Maximum Process Rate Over Cyclic Policies (Process Capacity) 55 3.4 Hardness and Inapproximability Results . 57 3.4.1 Inapproximability of Determining Process Capacity . 57 3.5 Good News . 58 3.6 Extension: Relaxation of Assumption (3.2.14) . 61 3.7 Proofs of Some Results . 63 3.7.1 Proof of Lemma 3.2.1 . 63 3.7.2 Proof of Lemma 3.3.2 . 65 3.7.3 Proof of Theorem 3.4.1 . 66 3.7.4 Proof of Theorem 3.5.3 . 66 3.7.5 Proof of Lemma 3.6.2 . 71 CHAPTER 4 ANALYSIS OF SCRIP SYSTEMS . 76 4.1 Introduction . 76 4.2 Model . 78 4.3 Analysis and Results . 79 4.3.1 The Main Result . 80 4.3.2 Intermediate Results . 82 4.3.3 Proof of Theorem 4.3.1 . 83 4.4 Proofs of the Intermediate Results { Lemmas 4.3.2 { 4.3.4 . 84 4.4.1 Proof of Lemma 4.3.2 . 84 4.4.2 Proof of Lemma 4.3.3 . 86 ix 4.4.3 Proof of Lemma 4.3.4 . 87 APPENDIX PROOFS OF LEMMA 3.2.2 . 95 REFERENCES . 104 BIOGRAPHICAL SKETCH . 108 CURRICULUM VITAE x LIST OF FIGURES 2.1 (a) A Distribution System and (b) An Assembly System. 9 2.2 The Distribution Network in the Proofs of Propositions 2.4.1 and 2.4.2. 15 A.1 Policy π1 in Example 1 . 97 A.2 Policyπ ^ in Example 1 . 98 A.3 Policy π0 in Example 2 . 101 A.4 Policyπ ˇ in Example 2 . 102 xi CHAPTER 1 INTRODUCTION This dissertation includes three essays. The first two essays focus on inventory and capacity management, and the third essay addresses an important problem in resource-sharing sys- tems. We now briefly describe the problems analyzed in these essays and summarize our contributions. In the first essay, we study the \integrality" question for dynamic optimization models of inventory control: Given integral initial inventory levels, capacity constraints, and demand realizations, does there exist an integral optimal policy? One practical implication of this question lies in whether or not full-truckload (FTL) shipping is optimal if customer demand is in integral number of truckloads. In this essay, we investigate the integrality question in single-product, multi-echelon distribution systems and show that integrality holds under deterministic demand, but fails to hold under stochastic demand. In distribution systems with stochastic demand, Less-Than-Truckload (LTL) shipping can be significantly cheaper than the cost of the optimal FTL shipping policy, even in the presence of economies of scale. For instance, this occurs in settings where shipping costs are expected to increase in the future and/or inventories are more expensive to hold upstream than downstream. In such situations, our results highlight the importance of strategically positioning inventory: LTL shipments can offer a more balanced allocation of inventory across the distribution network, leading to benefits that can exceed the savings from FTL shipments due to economies of scale. However, when the cost parameters are fairly constant across time and inventory holding costs are not significantly higher upstream than downstream, then the difference between the costs of optimal FTL and optimal LTL shipping is provably marginal. The second essay investigates the analytical tractability of process capacity. The concept of the capacity of a process and the associated managerial insights on the investment and management of capacity are of fundamental importance in Operations Management. Most 1 OM textbooks use the following simple approximation to determine process capacity: the capacity of each resource is first calculated by examining that resource in isolation; process capacity is then defined as the smallest (bottleneck) among the capacities of the resources. In a recent paper, (Gurvich and Van Mieghem, 2015) show that this \bottleneck formula" can be significantly inaccurate, and obtain a necessary and sufficient condition under which it correctly determines process capacity. We provide further clarity on the intractability of process capacity by showing that it is hard to calculate process capacity exactly and, furthermore, also hard to efficiently approximate it to within a reasonable factor; e.g., within any constant factor.