INVENTORY CONTROL IN A TWO-LOCATION TRANSSHIPMENT MODEL WEI WEI (B.Eng UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF INDUSTRIAL AND SYSTEMS ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2007 Acknowledgements I would like to express my sincere gratitude to my supervisors, Dr Wikrom Jaruphongsa and Dr Ng Kien Ming for their utmost support and professional guidance throughout my whole research work I greatly acknowledge the support from Department of Industrial and Systems Engineering (ISE) for providing the scholarship and the facilities I wish to thank Mr Victor Cheo Peng Yim, Ms Ow Lai Chun and Ms Celine Neo Siew Hoon for their care and assistance during these two years I would also like to thank Dr Mabel Chou and Dr Melvyn Sim in Department of Decision Science I benefit a lot from the seminars organized by Decision Science Department I would like to take this opportunity to express my appreciation for my parents I thank them for their eternal encouragement and support Without them, I may never reach this level I will thank my seniors Jiang Hong, Zhu Zhecheng and Sun Hainan Jiang Hong helps me clarify many problems associated with statistics Zhu Zhecheng provides guidance when I encounter obstacles in programming and Latex usage Sun Hainan gives me many useful advices to help me setup my research topic I will thank my colleagues Han Dongling, Yao Zhishuang, Yuan Le, Qu Huizhong, Mehdi Souidae, Wang Xiaoying, Wang Yuan, Pan Jie, Liu Rujing, Liu Shudong for discussions in operations research and their related research areas Especially, i Acknowledgements ii I would thank Wong Wai Peng for her recommendation in module selection in the second semester and Aldy Gunawan for his assistant in the registration and preparation for the Asia-Pacific Systems Engineering Conference 2007 I would thank Masruroh Nur Aini, Mutingi Michael, Dr Ng Szu Hui and Shen Yan for their help in my teaching assistant job My thanks also go to my friends in the ISE Department: Yin Jun, Wu Yanping, Zhang Haiyun, Liu Xiao for the hilarious life brought by them during these two years Contents Introduction 1.1 Background 1.2 Motivation 1.3 Purpose of the Study 1.4 Organization Literature 2.1 2.2 Transshipment Problem 2.1.1 Proactive Transshipment 2.1.2 Arbitrary Transshipment Time 2.1.3 Reactive Transshipment 2.1.4 Transshipment after Observing Demand 10 Supply Chain Coordination 11 2.2.1 One Supplier, One Retailer 12 2.2.2 One Supplier, Multiple Retailer 14 Centralized Transshipment Model 16 3.1 Model Description 3.2 Transshipment Amount 18 3.2.1 16 Linear Cost Function 20 iii CONTENTS iv 3.2.2 Container-based Transportation Cost 22 3.3 Time of Transshipment 25 3.3.1 Definition 25 3.3.2 Proactive Transshipment vs JIT Transshipment 26 3.3.3 Postponed Transshipment 28 3.4 Initial Order Quantity 33 3.5 Extensions 37 3.6 3.5.1 Lead Time in Transshipment 38 3.5.2 Opportunity for Higher Margin 40 3.5.3 Multiple Location 41 Conclusion 42 Numerical Experiment 4.1 4.2 4.3 44 Newsboy Heuristic and Effect of Transshipment 44 4.1.1 Effect of Margin 45 4.1.2 Effect of Demand Rate 48 Simulation 50 4.2.1 Statistical Analysis 50 4.2.2 Effect of Margin 53 4.2.3 Effect of Demand Rate 55 Conclusion 59 Decentralized Transshipment Model 61 5.1 Definition 61 5.2 Double Marginalization 62 5.3 Rule of the Game 63 5.4 Transshipment Amount 64 CONTENTS v 5.5 Numerical Result 65 5.6 Conclusion and Discussion 68 Conclusion and Future Work 70 6.1 Main Contributions 70 6.2 Future Work 71 6.2.1 Further Properties 72 6.2.2 Multiple Transshipment Opportunity 73 6.2.3 Pricing and Demand Shifting 74 6.2.4 Contract Design 75 Summary Transshipment is widely adopted in multiple location inventory management We study a two-location logistics system Each location holds a certain amount of perishable products before the selling season Leftover inventory after the selling season would have no value The demand at each location is assumed to be an independent random variable At most one transshipment is allowed within the selling season We first study a centralized controlled system We prove that the expected revenue function is concave with respect to the transshipment amount at any point of time The optimal decision process is developed to determine the initial order quantity for each location, the time to conduct transshipment and the transshipment amount In our numerical experiments, we examine the performance of the newsboy heuristic for the initial order quantity decision We also conduct sensitivity analysis for certain parameters in our model The model is further investigated from the supply chain coordination aspect We study a decentralized system with two decision makers We find that the system-wide revenue shrinks along with the transshipment amount in a decentralized system The transshipment deal may also fail if either party finds it to be nonprofitable for him to conduct the transshipment vi List of Tables 3.1 Parameter setup for postponed transshipment 29 4.1 Parameter setup for effect of acquisition cost 45 4.2 Parameter setup for effect of demand rate 48 4.3 Parameter setup for ANOVA 52 4.4 Parameter setup for effect of acquisition cost c (simulation) 54 4.5 Parameter setup for effect of demand rate λ (simulation) 55 4.6 Parameter setup for effect of high acquisition cost c (simulation) 58 4.7 Effect of high acquisition cost cn (simulation) 58 4.8 Parameter setup for effect of low acquisition cost c (simulation) 59 4.9 Effect of low acquisition cost c (simulation) 59 5.1 Parameter setup for the performance of decentralized system 66 5.2 Parameter setup for failed deal 5.3 Retailers’ revenue 68 vii 67 List of Figures 3.1 Lateral transshipment 17 3.2 Revenue with trucking cost: yT 3.3 Revenue with trucking cost: RT and CT 3.4 Delayed transshipment 30 3.5 Immediate transshipment 30 3.6 Expected profit with regard to initial order quantity 37 4.1 Difference between order quantities with change of c 46 4.2 Difference between performances with change of c 47 4.3 Difference between order quantities with change of λ 48 4.4 Difference between performances with change of λ 49 4.5 Box plot of ANOVA 52 4.6 Difference between order quantities with change of c (simulation) 54 4.7 Difference between profit with change of c (simulation) 55 4.8 Difference between order quantities with change of λ (large c, sim- 23 24 ulation) 56 4.9 Difference between profit with change of λ (large c, simulation) 56 4.10 Difference between order quantities with change of λ (small c, simulation) 57 4.11 Difference between profit with change of λ (small c, simulation) 57 viii LIST OF FIGURES ix 5.1 Centralized vs decentralized: transshipment amount 66 5.2 Centralized vs decentralized: expected revenue 67 CHAPTER DECENTRALIZED TRANSSHIPMENT MODEL 67 Comparison of expected revenue in centralized and decentralized systems 95 centralized system decentralized system 90 85 yT 80 75 70 65 60 55 0.2 0.4 0.6 0.8 1.2 1.4 c T Figure 5.2: Centralized vs decentralized: expected revenue We also find another obstacle hindering lateral transshipment One player in the decentralized system, either retailer or retailer 2, would find that it is no longer profitable for him to implement transshipment, i.e., the benefit does not cover the opportunity cost Hence, one of the players would not accept the transshipment proposal due to his concern of his own revenue, even though the overall performance of the entire system can be improved In this case, the decentralized system would deteriorate to the no transshipment case with a constant expected revenue function, which is lower than the summation of the expected revenue from two retailers In this experiment, we set the value of parameters in Table 5.2 and the result is shown in Table 5.3 Parameter Value r1 r2 K 10 cT 0.6 λ1 10 λ2 10 t Table 5.2: Parameter setup for failed deal CHAPTER DECENTRALIZED TRANSSHIPMENT MODEL Player Retailer Retailer System-wide 68 Expected revenue Expected revenue with transshipment without transshipment 49.88 50 10.85 10 60.73 60 Table 5.3: Retailers’ revenue In the above example, even though transshipment leads to a higher systemwide expected revenue, $60.73 > $60, the deal still cannot be made Retailer would not trade with retailer simply because transshipment makes him worse off ($49.88 < $50) The system-wide benefit from transshipment is $0.73 The unfair distribution eliminates retailer 1’s incentive to conduct transshipment With the same setting of parameters, the optimal transshipment amount in a centralized system is 37, leading to a system-wide profit of $63.15 We also investigate an extreme case with no transshipment cost at all, i.e., K = and cT = Given that the value of other parameters remain the same, the expected revenue of the centralized system is $97.94 with a transshipment amount of 45 The expected revenue of the decentralized system is $90.69 with 31 units transshipped The experiment result is consistent with the analytic result That is, the optimal transshipment amount in the decentralized system is different from that in the centralized system In our case, the transshipment amount in the decentralized system is much less than the system-wide optimal transshipment amount 5.6 Conclusion and Discussion In this chapter, we study a decentralized two-location lateral supply chain Through numerical experiment, we detect two reasons explaining the worse performance of CHAPTER DECENTRALIZED TRANSSHIPMENT MODEL 69 a decentralized system The first reason is that the sender sets the price based on his own benefit, which forces the receiver to order less stock from the sender The second reason is that the transshipment cannot be made when either party cannot benefit from the deal The potential lateral trading deal may also affect each party’s decision in the initial order We only use symmetric parameter in the previous numerical example Yet in real business, different parties have different comparative advantages Some companies have the cutting edge warehouse management system, some are good at marketing and sales, certain companies have the expertise in logistics and own a well connected transportation network in their business territories Difference in core competition power leads to different focus in decision making Our study not only highlights the tipping point for companies who are concerned about inventory management, but also demonstrates the importance of collaboration Lateral trading based on proper transshipment contract can create win-win situation for both parties involved in the deal Logistics companies also have a broader market if lateral transshipment among different parties becomes popular Chapter Conclusion and Future Work We summarize the main contribution in this study Based on the existing work, we discuss several potential research areas for future work 6.1 Main Contributions Chapter discusses the basic two location logistics model and its extensions We investigate the transshipment problem from the operational level and determine the optimal transshipment time and amount precisely We also provide the optimal ordering policy to help the managerial team make decisions before the selling season In addition, we discuss several extensions, including lead time, opportunity for higher margin and multiple location We further investigate our model through numerical study in Chapter We conduct sensitivity analysis on two parameters: product margin and demand rate We compare the performance between newsboy heuristics and optimal order policy We find that newsboy heuristics can provide a decent solution We also demonstrate that transshipment can improve the performance of the system significantly 70 CHAPTER CONCLUSION AND FUTURE WORK 71 In Chapter 5, we highlight the supply chain coordination issue in lateral supply chain The previous research work in supply chain coordination mainly focus on vertical supply chain which is composed of a supplier and a retailer (or several retailers) A number of supply chain contracts which can be adopted in vertical supply chain to achieve coordination no longer work in the lateral chain In a lateral supply chain, double marginalization still exists and diverts the system-wide performance from optimality In addition, conflicting incentives from different parties may hinder collaborative deal The potential of improvement, which can be achieved through strategic alliance, is significant and can benefit both parties if the contract is well-designed 6.2 Future Work We list several directions for future work based on this study The first one is to investigate the unimodality of the revenue function with respect to delayed transshipment time If we can prove the unimodality, we can save a lot of time in the searching process while deciding whether we should conduct transshipment or not We can also relax the assumption of allowing at most one transshipment in the selling season It could be profitable to conduct several transshipment, especially when we are facing a long selling season The problem can be modeled as a continuous Markov chain decision process Another extension is to incorporate pricing as a decision variable and investigate the demand shifting phenomenon in the market We also provide a brief review in this research area In Chapter 5, we have shown the potential benefit in lateral collaboration or integration in a decentralized supplier chain However, we did not provide any CHAPTER CONCLUSION AND FUTURE WORK 72 coordinating mechanism in this study Future research can be done to design coordinating contract for a lateral supply chain 6.2.1 Further Properties In Chapter 3, we have obtained the optimal decision making process for the basic two-location transshipment model However, if we incorporate the lead time into the model, since we have not proved any unimodality of the revenue function with respect to delayed time ∆t, we need to exhaust all the possible values of ∆t ≤ t to ensure that further delay would lead to suboptimality The same problem exists when it comes to the initial order decision For the basic model, we need to calculate every pair of Q1 , Q2 in a reasonable searching scale to ensure we get the optimal solution Both searches cost considerable time Consider the continuous review policy with unlimited transshipment opportunity Since it is hard to foresee whether we should conduct transshipment in the future, a discrete time model can be deployed to simulate the continuous model when the time interval is extremely small Another potential research area is to find the threshold state (I1 , 0, t) for given r1 , r2 , cT , λ1 , λ2 The numerical experiment indicates that yT might be unimodal with respect to ∆t, the time of delay It can be written as follows: If yT (0) > yT (δt), where δt is a small time, or equivalently, ∂yT (∆t) |∆t=0 ∂∆t < 0, we can get yT (0) > yT (∆t), ∀∆t, which means transshipment should be implemented immediately Although numerical experiment indicates such a unimodal property, it is difficult to prove The optimal transshipment amount qT∗ does not have a closed form while the demands are Poisson distributed, which implies that the objective function cannot be expressed by ∆t explicitly CHAPTER CONCLUSION AND FUTURE WORK 6.2.2 73 Multiple Transshipment Opportunity We only allow at most one transshipment during the selling season in current study When the selling season is long, after the transshipment, the inventory levels of two locations may become imbalanced again In this case, it may be profitable to conduct transshipment again The system state is denoted by [I1 , I2 , t], i.e., the on hand inventory and the time to go before the end of selling season When the system state is [I1 , 0, t] and we incur a new demand at location 2, we need to evaluate whether we should conduct transshipment to fulfill this demand or forgo it y(I1 , 0, t) = max{yN T (I1 , 0, t), yT (I1 − qT , qT − 1, t)} yN T (I1 , 0, t) denotes the expected revenue if we forgo the demand at location and not conduct transshipment yT (I1 − qT , qT − 1, t) denotes the expected revenue if we conduct transshipment to fulfill this demand yN T (I1 , 0, t) = E[y(I1 − D1 (t˜2 ), 0, t − t˜2 ) + r1 D1 (t˜2 )] = E [E [y(I1 − D1 (t˜2 ), 0, t − t˜2 ) + r1 D1 (t˜2 )|t˜2 ]] D1 (t˜2 ) t˜2 t˜2 is the arrival time of the next demand at location D1 denotes the demand at location during t˜2 The calculation of yT (I1 − qT , qT − 1, t) is much more complicated We need to introduce the time of one location incurs stock out during the time t Referring to Section 3.4, we can analyze three different cases and compute the conditional expected revenue for each case CHAPTER CONCLUSION AND FUTURE WORK 6.2.3 74 Pricing and Demand Shifting Pricing strategy is also widely adopted in retail industry to maximize retailer’s profit One way to deal with imbalanced inventory is to mark up the price in the store whose inventory level is relatively low and mark down the price in the store which holds more than enough stock Since the demand and price are negatively correlated, the decision maker can use pricing to make demand meet inventory On the other hand, transshipment is more about making inventory meet demand Numerous researches have been done in pricing for the newsvendor Gallego and van Ryzin [21] studied a newsvendor model with a finite time horizon SimchiLevi et al [43] synthesized the previous work and gave a systematic review on the subject of the integration of inventory and pricing However, the above research only focus on the pricing strategy for the one retailer case Compared with transshipment, pricing does not cost any logistics fees However, marking up or down may cost employee’s working hours as well as noneligible advertising fees in certain situations In addition, if the stores are operating under the same brand name, keeping prices in different stores in the same region inline with each other helps to gain recognition from customers Many franchisors would set the retail price by themselves for their franchisees to guarantee the same price of the same product for customers Demand shifting is commonly observed in the market A customer who is willing to buy a certain product may visit stores in a larger region if he cannot get it from any nearby stores A supermarket may incur significant demand loss when demand shifting comes in Customers usually come to supermarkets for one-stop shopping They not tend to visit several supermarkets in a day If they find that one supermarket incurs stock out on products they commonly consume, they may turn to another supermarket which always keeps enough stock The loss of CHAPTER CONCLUSION AND FUTURE WORK 75 demand from certain products may be amplified to the loss of customers, who are supposed to be regular visitors to the store Demand shifting is not only related to service level but also sensitive with price 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