UNIVERSITY OF CALGARY. An Inventory Model For Multi-Echelon Supply Chains With Price- and Inventory-Level- Dependent Demand. Zaman Forootan A THESIS

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1 UNIVERSITY OF CALGARY An Inventory Model For Multi-Echelon Supply Chains With Price- and Inventory-Level- Dependent Demand by Zaman Forootan A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTRATE OF PHILOSOPHY IN OPERATIONS MANAGEMENT GRADUATE PROGRAM IN MANAGEMENT CALGARY, ALBERTA APRIL, 2015 Zaman Forootan 2015

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3 Abstract Given the nature of highly competitive markets, most supply chains try to optimize their processes by reducing costs and increasing customer satisfactions. Hence, managers attempt to closely follow changes in customers buying behaviour relative to product price and availability. Such behaviour directly affects the inventory management of a supply chain since the demand patterns change. This research targets decision making for pricing and replenishing a product, i.e. price, reorder point, and order quantity, to benefit the overall supply chain. An inventory model for a multi-echelon supply chain with price- and inventory level- sensitive demand is developed in this research in a progressive manner. First, the inventory system of a retailer facing a price and inventory level dependant demand function is studied. The results of the analytical model in this section answer the questions related to the order quantity, and the magnitude and timing of price changes in order to maximize the profit for the retailer. Second, the inventory system for a supply chain consisting of a retailer and a manufacturer with a price-dependent demand is analyzed both analytically and using discrete-event simulation-optimization. The results of these analyses address the benefits of cooperation in pricing and replenishment decisions. This section also answers the questions related to the retailers order quantity, the production quantity of the manufacturer, and the price level. Finally, an inventory system of a supply chain including a retailer, a distribution centre and a manufacturer is analyzed. Simulation-optimization is used to find the optimal strategy for the joint pricing and replenishment problem. The results of this last section provide answers as to the timing and quantity of the orders, the pricing, and how the ordering policy is affected. This research also explores the differences in pricing and replenishment policies for different demand functions and answers the question of whether it is worthwhile, considering the computational complications, to contemplate price- and stock- iii

4 dependency of demands while coordinating supply chains. In this research an empirical research method is also used to evaluate the assumptions of price- and inventory level- dependency of demand for different products available in a grocery store. The results of this thesis support the proposition that collaborations in inventory management decision-makings benefit the whole supply chain, even if all the firms do not participate in collaboration. It also supports investigating the price- and inventory level- dependency of demand in order to maximize the profits for the supply chain. iv

5 Acknowledgements I would like to express my special appreciation and thanks to my advisors Professor Dr. Jaydeep Balakrishnan and Dr. Silvanus Theodore Enns, you have been tremendous mentors for me. I would like to thank you for encouraging my research and for allowing me to grow as a research scientist. I would also like to thank my committee members, Professor Diane Bischak, Professor Giovani C. da Silveira, Professor Paul Tu, and Professor Timothy Urban for serving as my committee members and putting in the effort to provide valuable suggestions that improved the thesis. I also want to thank you for letting my defense be enjoyable and memorable. I would like to thank the staff at Haskayne School of Business, especially Lesley Diamrzo, our beloved Graduate Program Officer. All of you have been there to support me when I needed help and support during my Ph.D. program. A special thanks to my family. Words cannot express how grateful I am to my mother, father, my mother-in law, and father-in law for all of the sacrifices that you ve made on my behalf. Your love, support, and prayer were what sustained me thus far. I would also like to thank all of my friends, especially Nancy Southin, who supported me in writing, and incented me to strive towards my goal. Last but not the least, I would like express appreciation to my beloved wife Mozhde who was always my support in the moments when there was no one to answer my queries. v

6 Dedication To my son Mauhan, whose laughter brings the most precious moments to my life. To my wife Mozhde, who holds the most precious place in my life. vi

7 Table of Contents Abstract... iii Acknowledgements...v Dedication... vi Table of Contents... vii List of Tables... xi List of Figures and Illustrations... xiii List of Symbols, Abbreviations and Nomenclature... xvii Epigraph... xviii CHAPTER ONE: INTRODUCTION...1 CHAPTER TWO: BACKGROUND Overview Joint Economic Lot-sizing Problem (JELS) Inventory management with price sensitive demands Inventory management with stock- sensitive demands Inventory management with price and stock sensitive demands Simulation-based optimization Gaps in the literature and potential research areas Summary...25 CHAPTER THREE: OBJECTIVES Overview Price- and stock-dependent demand assumption Joint pricing and replenishment policies Using simulation Summary...30 CHAPTER FOUR: METHODOLOGY Methods and proposed approaches Analytical approach Discrete-event simulation Simulation- optimization approach Design of experiment and statistical analysis Summary...40 CHAPTER FIVE: EMPIRICAL ANALYSIS ON SALES DATA FOR A RETAILER Overview Introduction to database Data preparation Invalid data Missing data Price sensitivity of demand Inventory level sensitivity of demand Other possible forms of dependency of demand in price and stock level...54 vii

8 5.6.1 Price inelasticity and positively price dependent demand Effect of scarcity on demand Correlation between price-dependence and inventory level dependence Three models for general price- and stock-dependent demand Summary...59 CHAPTER SIX: SINGLE STAGE SUPPLY CHAIN MODEL Overview Single replenishment inventory model with price- and stock-dependent demand Summary of proposed optimal solution for linear demand function The model with exponential demand function The model with power demand function Improved model for re-pricing cost Numerical examples Optimal solutions comparisons Optimal number of price settings Sensitivity analysis Pricing policies Summary...84 CHAPTER SEVEN: TWO STAGE SUPPLY CHAIN WITH PRICE SENSITIVE DEMAND Overview Problem definition and notation Model formulation General model for the lot sizing problem Models for the independent policy The model for joint policy Numerical comparisons of solution methods Results Conclusion A numerical example Profit allocations Summary CHAPTER EIGHT: MULTI-ECHELON SUPPLY CHAIN WITH PRICE AND STOCK- LEVEL DEPENDENT DEMAND Overview EOQ for multi-stage supply chain with deterministic level demand Independent policy (R-D-M) Partial joint policy (R-DM) Partial joint policy (RD-M) Joint policy (RDM) Summary of optimal solutions: Changes in profit based on coordination policy Revenue Costs for the retailer viii

9 Costs for the DC Costs for the manufacturer Total supply chain costs Profits for firms in different coordination policies R-D-M R-DM Comparing R-DM to R-D-M RD-M Comparing RD-M to R-D-M RDM Comparing RDM to R-D-M Comparing RDM to R-DM Comparing RDM to RD-M Conclusion A numerical example: Profit allocation Proportion approach Partial joint policies: Joint policy: Shapley value approach Results Simulation Customer demand Retailer Distribution Centre (DC) Manufacturing plant Performance measures Simulation parameters Validating the model Conceptual model validation Computerized model verification Operational validation Optimization Optimization formulation Assumptions Pricing and replenishment scenarios Design of experiments and statistical analysis Experimental model parameters Results Comparing different replenishment policies under each demand function Comparing different demand function assumptions A more practical solution Profit allocation Sensitivity analysis CHAPTER NINE: SUMMARY AND CONCLUSIONS Overview ix

10 9.2 Significance of this research Limitations References APPENDIX A: A.1. Model for joint policy with equal-shipment assumption A.2. Solution for case 1 (x1 > 0, x2 > 0) A.3. Solution for case 2 (x1 > 0, x2 = 0) A.4. Solution for case 3 (x1 > 0, x2 < 0) A.5. Solution algorithm for the joint policy A.6. Price and profits for the firms APPENDIX B: B.1. Arena model modules B.2. SIMAN code for Arena model B.3. Arena experimental File x

11 List of Tables Table 1. Different commercial optimization packages (adapted from (Fu, 2002 Grewal, 2012 Law & Kelton, 2007)) Table 2. List of product categories in Dominick s database Table 3. Price and demand correlation and exponential function for different products (demand per week) Table 4. Stock level and demand correlation (Spearman s non-linear correlation) Table 5. Comparison of demand functions behaviour to marginal inventory changes Table 6. Comparison of Demand functions behaviour to extreme cases Table 7. Comparing linear and exponential models for different number of price changes Table 8. Comparing two pricing policies Table 9. Effect of retailer s holding cost on deviations from optimal profit Table 10. Effect of manufacturer s setup cost on deviations from optimal profit Table 11. Effect of manufacturer s production rate on deviations from optimal profit Table 12. Effect of price dependency of demand on deviations from optimal profit Table 13. Numerical example results Table 14. Effects of parameter b using the Independent vs. Joint policy Table 15. Profit allocation using different allocation approaches Table 16. Optimal values for order/production quantities Table 17. The relation of optimal order quantity for the DC in different coordination Table 18. Summary of optimal order quantities Table 19. Optimal solution for a supply chain with level demand Table 20. Profit allocation in the multi-echelon supply chain for the level demand analytical example Table 21. Probability distributions for stochastic variables in the model Table 22. Arena report for the simple inventory model xi

12 Table 23. Experimental design for multi-echelon supply chain performance Table 24. Summary results of the simulation-based optimization Table 25. Results of comparing different demand functions for a supply chain with price and stock-dependent demand Table 26. Profits for P&SDD scenario with positive individual firm s profit constraint Table 27. Profit allocation using Shapley value method in price and inventory level dependent demand scenario Table 28. Values of the function, first and second derivatives of f at critical points xii

13 List of Figures and Illustrations Figure 1. Number of publications considering JELS... 9 Figure 2. Optimization simulation process Figure 3. The research framework Figure 4. Test for normality of residuals using P-P curve for Paper Towel UPCs Figure 5. Spearman s non-linear Correlation test for product Paper Towel Figure 6. Comparing goodness-of-fit for linear vs. exponential price-dependent demand function Figure 7. Curve estimation analysis for linear and exponential function of demand as a function of price Figure 8. Saturation factor Figure 9. Inventory function Figure 10. Demand as linear and exponential functions of price and stock level Figure 11. Inventory level and demand function for two price changes Figure 12. Optimal profit and the optimal number of price changes for the linear model Figure 13. The effect of changes in re-pricing cost on optimal number of re-pricings Figure 14. The effect of change in holding cost on optimal number of price changes Figure 15. Two pricing policies, one with allowed price increase, one without price increase both for n= Figure 16. Comparing pricing policies for n= Figure 17. Inventory levels for retailer, manufacturer and supply chain, m=5 and n=7 using geometric then equal shipment policy Figure 18. Profit behaviour for the numerical example Figure 19. Effect of parameters b and c on profit improvement using the joint policy Figure 20. Cost curves for firms in supply chain. (Top:, Bottom: ) xiii

14 Figure 21. Cost curves for the supply chain. (Top:, Bottom: ) Figure 22. Comparison between the profits for different policies Figure 23. Process flow of the supply chain simulation Figure 24. System reaches its steady state after about 1500 days Figure 25. Model development process, Adopted from Sargent (2013) Figure 26. Inventory levels for different firms in supply chain, used to validate the model Figure 27. OptQuest while optimizing is in process Figure 28. Different demand function assumptions for a price and inventory level dependent demand Figure 29. Test of homogeneity of variances Figure 30. One-way ANOVA for the effect of coordination policy for a supply chain with level demand Figure 31. One-way ANOVA for the effect of coordination policy for a supply chain with price-dependent demand Figure 32. One-way ANOVA for the effect of coordination policy for a supply chain with stock-dependent demand Figure 33. One-way ANOVA for the effect of coordination policy for a supply chain with price and stock-dependent demand Figure 34. Comparing different policies Figure 35. Changes in profits for firms in different replenishment policies Figure 36. Effect of stock dependency of demand on firms inventory levels Figure 37. Process Analyzer window showing different scenarios Figure 38. Effect of different demand functions for a supply chain with price and stockdependent demand Figure 39. Average profit and 95% CI for different demand functions Figure 40. Profit sensitivity to production rate xiv

15 Figure 41. Profit sensitivity to retailer s order cost Figure 42. Profit sensitivity to DC s order cost Figure 43. Profit sensitivity to manufacturer s setup cost Figure 44. Profit sensitivity to retailer s holding cost Figure 45. Profit sensitivity to DC s holding cost Figure 46. Profit sensitivity to manufacturer s holding cost Figure 47. Profit sensitivity to parameter a in demand function. (Left: relations not considered, Right: relations considered) Figure 48. Profit sensitivity to parameter b in demand function. (Left: relations not considered, Right: relations considered) Figure 49. Profit sensitivity to parameter alpha in demand function. (Left: relations not considered, Right: relations considered) Figure 50. Profit sensitivity to parameter beta in demand function. (Left: relations not considered, Right: relations considered) Figure 51. Different cases for function f Figure 52. Profit function and its first derivative (Case 3, Scenario A) Figure 53. Profit function first derivative and related profit function (Case 3, Scenario B) Figure 54. Retailer profit as a function of c under an Independent policy Figure 55. Modules for updating demand based on price and inventory level at the retaile Figure 56. Retailer s system, where customers arrive and buy items Figure 57. Retailer s inventory system Figure 58. DC s Information system Figure 59. DC s inventory system Figure 60. Manufacturer s information system Figure 61. Manufacturer s inventory system xv

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17 List of Symbols, Abbreviations and Nomenclature Symbols and abbreviations are introduced in each Chapter separately. xvii

18 Epigraph For every complex problem there is a simple solution that is wrong. G.B. Shaw xviii

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20 Chapter One: Introduction Most firms try to optimize their supply chain processes by reducing costs and increasing customer satisfaction to increase their market share and profit. One effective method of optimizing these processes is by managing inventory and replenishment systems effectively. Inventory management plays an important role in every manufacturing, retail and other type of firms. Firms keep inventories for different reasons such as protection against market uncertainties, customer service, and taking advantage of economic order quantities (Davis et al., 2007 Reid & Sanders, 2007). In order to manage an inventory system there are several factors involved, such as demand rates, holding costs, order (setup) costs, lost sales due to stock-outs, and demand functions. Putting it in a more practical way, as stated by Silver et al. (1998), managers should resolve the following three key issues that take into account all the above factors (Silver et al., 1998, p. 235): How often the inventory status should be determined? When a replenishment order should be placed? How large the replenishment order should be? One of the most important factors that influence inventory system decisions is demand and its patterns considering customers behaviour. Lau and Lau (2003) demonstrates that changes, even very small, in the demand pattern can affect optimal inventory control decisions. Since demand patterns based on customers buying behaviour may change due to factors such as product price and availability, managers should closely follow these factors (You & Hsieh, 2007). 1

21 Traditional inventory models assume that demand is either constant or time dependent but exogenous to the inventory management decisions, including product price and the inventory level (Goyal, 1977a Ouyang et al., 2009). Nowadays though, many researchers and practitioners consider demand to be an endogenous factor within the inventory management system. The concept of correlations between demand and selling price was first discussed in the classical economic order quantity (EOQ) model by Whitin (1955). Although for years after Whitin (1955) this topic was not investigated any further, recently studies on inventory systems with price sensitive demand and product pricing strategies have received much attention (Forootan et al., 2014 Hong et al., 2011 Petruzzi & Dada, 1999 Sajadieh & Akbari Jokar, 2009 Shib, 2011). The models that consider the correlation between price and demand will be discussed in more detail in later sections. There has been some research that has observed that besides sensitivity to price, demand is sensitive to stock level, especially in a retail store. These studies suggest that demand is positively correlated with the quantity of items available in stores (Balakrishnan et al., 2004 Krishnan & Winter, 2010 Mandal & Maiti, 1999, 2000 Silver & Peterson, 1985 Van Donselaar et al., 2010 You & Hsieh, 2007 Zhou & Yang, 2005). High inventories may increase demand for different reasons, such as more visibility, signalling popularity of product, and confidence of higher service levels and future availability (Balakrishnan et al., 2004). Managers can improve their profitability by considering the above-mentioned factors in inventory systems in their firms. Following the success of lean manufacturing in Japanese industry, researchers and practitioners turned their focus on some best practices in management, particularly cooperation with other firms in their supply chains. Especially with emerging global 2

22 markets and more online sales, the competition has turned from firm-to-firm competition to supply chain-to-supply chain competition (Hong et al., 2011). It has been noted by several researchers that competitive advantage can be reached through optimizing the whole supply chain. Therefore the Joint Economic Lot-sizing problem (JELS or JELP) has been the topic of many research studies, including of this thesis. This research targets decision-making insights which will benefit the overall supply chain. Related work falls primarily into the areas of economic lot-sizing and revenue management under price and inventory level sensitive demand with deterministic and stochastic assumptions. The goal of the research is to find the optimal pricing and replenishment policy for a multiechelon supply chain with price and inventory level dependent demand. In order to reach this goal, a progressive approach is taken. In this journey, first the idea of dependency of demand on price and inventory level is investigated by empirical analysis of data that is made available by the Chicago Booth business school from a chain grocery store (Dominick s Finer Food). Then, the pricing and replenishment policy for a retailer that faces a price and inventory level dependent demand for a seasonal product is discussed. Next, entering the realm of supply chain and joint lot sizing, inventory management for a two-stage supply chain that faces a pricedependent demand is modeled analytically. Equipped with insights from the first three milestones, joint pricing and replenishment policies are investigated for a three-echelon supply chain with price and inventory level dependent demand. This final milestone is reached by means of discrete-event simulation and optimization tools. In the empirical analysis section the positive correlation between and price and demand is proved for some products. It is also shown that some products are price inelastic. The data 3

23 doesn t include direct information about the inventory level at stores. However, the number of different SKUs for specific products is assumed as a proxy for inventory level. It is shown that there is a positive correlation between the number of SKUs at the store and the demand. In the chapter on the retailer EOQ model, the optimal order quantity and optimal set of selling prices for a seasonal item is modeled assuming that the firm will set the price n times during the time span of L and demand is dependent on price and stock level. In this model the number of price changes is one of the decision variables to be optimized. Further the same model is investigated with different patterns for demand dependence on price and stock level i.e. linear, exponential, and power functions. It is shown that the exponential function is the most appropriate dependence function. However, sometimes for the sake of simplicity one can use a linear function to represent the dependence of demand on price and stock level, as explained in a model by You and Hsieh (2007). In the extended model it is assumed that the re-pricing cost has two portions. One portion is fixed, such as the advertisement cost for the new price, and the other is dependent on the inventory level, such as changing the price tags on the items. The sensitivity of the optimal solution on re-pricing costs is discussed. In the next chapter the supply chain consists of a retailer and a manufacturer that face deterministic price-dependent demand. Generic profit functions for firms and the supply chain are developed. Using these models different pricing and replenishment policies, such as equal lot-size and geometrically increasing lot-sizes, are optimized and compared in independent and joint lot-sizing scenarios. It is shown that policies with unequal shipments do not produce meaningfully better results than the policy with equal shipments. Hence, firms can confidently optimize their process without going through calculations requiring sophisticated methods. The 4

24 results in this chapter also demonstrate higher total profit of joint policy than if each member optimizes its profit independently. Finally, in the chapter for the multi-echelon supply chain a simulation model is developed along with an optimization process. These are used to study the effect of different individual or joint pricing policies on scenarios with different demand functions. The simulation model developed in this chapter is a flexible model that provides the capability of studying the effects of different decision parameters in the system, as well as assuming different stochastic variables affecting the performance of the supply chain, such as transportations and setup times. It is shown that cooperation in supply chains, even partial cooperation that does not involve all the firms in the supply chain, benefits the overall supply chain. It is also discussed that it is beneficial for the supply chain if firms invest in understanding the characteristics of the demand and consumers reactions to available inventory and price at the retail store. The sensitivity of the optimal solution to different parameters in the model, including the estimated coefficients for the price and inventory level dependent demand function, is also discussed. The rest of this thesis is organized as follows. First, the background literature on major topics related to this thesis is reviewed. This includes joint economic lot-sizing models, inventory models with price and stock-dependent demands, and simulation- optimization models. This background leads us to the objectives of the thesis stated in Chapter Three. Chapter Four introduces methods used in this thesis, most of which are based on operations research methodologies and mathematical modelling. Chapters Five to Eight present detailed analysis for four main topics of this thesis. These topics are empirical analysis, the single stage retailer model, the two-stage supply chain model, and multi-echelon supply chain model. Each model is 5

25 presented and the results are discussed. Finally in Chapter Nine the overall summary of the thesis along with its findings and limitations are presented. This thesis also has two appendices. 6

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27 Chapter Two: Background 2.1 Overview As mentioned in the introduction, since markets are competitive, firms try to gain market share by meeting a specific customer service level, while reducing costs. Moreover, as firms compete in global markets the firm s survival may depend on success in supply chain to supply chain competition (Hong et al., 2011). In order to ensure higher service levels as measured by a higher rate of immediately satisfied demands, inventory, which reduces the impact of unforeseen events, should be kept in the supply chain. Therefore, topics in inventory management such as pricing policies, replenishment, and supply chain coordination have received enormous attention from both practitioners and researchers. Consequently, there is a vast body of literature addressing these topics, and it is impossible to review it all. In this section, the general historical trend of research is addressed by citing some of the well-known, most cited, or most related papers to this research. The inventory management literature is categorized in three major areas important for this thesis: joint replenishment and pricing policies, price-sensitive-demand models, and stock-sensitive-demand models. At the end of this section, the gaps in these areas are discussed. These lead to the objectives of this thesis. 8

28 2.2 Joint Economic Lot-sizing Problem (JELS) An early analysis of the JELS was done by Goyal (1977b, 1977c). Following Goyal a stream of research emerged on this topic. Glock (2012) identified 155 papers published on JELS until the end of Figure 1, captured from his review paper, shows the exponentially increasing trend of publications on this topic. This shows the increasing importance of supply chain coordination in the current business environment. Figure 1. Number of publications considering JELS Joint economic lot-sizing models are useful as planning tools for supply chains in which the firms have long-term trustful relationships which is a common scenario in automotive industries. In this case firms, knowing that the gains obtained by cooperation will be distributed among the members of the supply chain, the participant firms will cooperate in optimizing the supply chain replenishment strategies (Glock, 2012). 9

29 Only the most important and relevant papers to this thesis are discussed in the following paragraphs. For a complete list of JELS papers and more details on them refer to Glock (2012) and its online supplement. Goyal (1977b, 1977c) dealt with a single vendor-product-buyer situation under deterministic demand. The objective was to minimize relevant vendor and buyer costs, including both holding and order/setup costs. It was demonstrated that an integrated model that simultaneously considered both vendor and buyer costs resulted in lower total costs than if the buyer and then the vendor independently minimized their costs. Interestingly, this early work already raised the issue of how costs (or profits) should be allocated under integrated model assumptions. Goyal (1977b) mentioned since no vendor cost was assumed under joint optimization, in practice naturally, a mechanism for treating both the vendor and buyer fairly needs to be addressed. In his 1977 paper, Goyal assumed the vendor lot size could be a multiple of the buyer order size but that the production rate was infinite (Goyal, 1977c). Banerjee (1986) relaxed the assumption of an infinite production rate but assumed vendor production lot sizes were equal to buyer order sizes. Joint optimization was proved to yield lower costs than independent optimization. Goyal (1988) then demonstrated that, under finite production rates, relaxing Banerjee s assumption of equal vendor and buyer lot sizes could further reduce total costs. Goyal s model allowed the vendor lot size to be a multiple of the buyer order size but was based on the implied assumption that the vendor s production lot had to be completed prior to shipping any buyer orders. Another extension by Goyal and Szendrovits (1986) was to assume the vendor s production process involved multiple, serial manufacturing stages. In this case partial production lots were allowed to be transferred to the next stage. In other words, lot sizes from stage to stage did not 10

30 need to be equal. A heuristic was developed to minimize costs. Results determined that unequal lot sizes were advantageous. Alternate types of lot-sizing models were reviewed by Goyal and Gupta (1989). They categorized integrated models into four types. First were the JELS models, such as those of Goyal (1977c). These did not consider a vendor price (buyer cost) in joint optimization. Second were models concurrently reflecting vendor and buyer lot sizes with the vendor price as a decision variable. Third were models that sequentially considered vendor and buyer lot sizes and vendor discounts. Discounts may induce buyers to place larger orders than their economic order quantity (EOQ), thus reducing vendor setup costs. Finally were the models where demand is related to price. In such models price is a decision variable and, since revenue is no longer constant, the objective is profit maximization. This review paper also pointed out that the literature usually assumes costs across the supply chain are known when joint optimization is pursued. This may be quite unrealistic in practice. Lu (1995) considered a vendor-buyer problem with multiple buyers. A heuristic was developed to minimize costs by taking advantages of production coordination in meeting buyer orders. Shipments of orders to individual buyers were allowed prior to completion of larger production lot sizes. These shipments were equal in size. Goyal (1995) recognized that it may not be optimal to have equal buyer order sizes shipped during a production run. He realized that smaller lot sizes should initially be shipped to meet demand as product becomes available from a new production run. This reduces inventory-holding costs. As manufactured inventory builds up at the vendor, lot sizes should be increased to reduce buyer order costs. Goyal suggested that lot sizes should increase geometrically and showed that costs could be reduced relative to using equal lot sizes. Hill (1999) proved that geometrically increasing lot sizes are optimal initially. 11

31 However, once there is sufficient inventory built up at the vendor, order shipments to the buyer should be of equal size. The size of these equal shipments should be based on the trade-off between the buyer order cost and inventory holding costs, assumed to be higher for the buyer than for the vendor. Here, using geometrically increasing shipments, initially minimum system inventory at the beginning of a production cycle is guaranteed. Using equal-sized shipments later guarantees that the buyer s inventory is not increased more than necessary. This paper also recognized that the structure of order shipments had similarities to the heuristic approach developed by Goyal and Szendrovits (1986) for multiple stage, serial production lines. Khouja (2003) examined a three-stage divergent model in which a supplier ships to multiple manufacturers, who then each ship to multiple buyers. Cycle times, or lot sizes, were allowed to be integer powers of two of the next downstream stage. However, all lot sizes shipped from a given stage were assumed equal. As well, orders could only be shipped after each production run was complete. This study concluded that integer power of two cycle times reduced costs. Hill and Omar (2006) relaxed the assumption made in Hill (1999) that the buyer s inventory holding costs should be more than those at the vendor and showed that the problem structure is identical to that of the Hill (1999) and solved it optimally. Ben-Daya and Al Nassar (2008) relaxed the assumption of order shipments only after completion of production lots, and were able to further reduce costs. Ben-Daya, Darwish, and Ertogral (2008) reviewed the JELS problem along with several extensions and proposed a unified framework for the problem. They also conducted a comparative empirical study of different proposed solution policies and assessed their deviation from the optimal solution. In recent years research in joint pricing has been very broad and covers topics such as coordinating the supply chain channel, game theoretic approaches toward 12

32 pricing, and progressive payment and trade credit schemes (Ouyang et al., 2009 Soni et al., 2010 Teng et al., 2011b). Recently, Glock (2012) reviewed and categorized papers on the joint economic lot-sizing problem with respect to the coordination of different echelons in supply chains. Lee (2005) considered three-stages with one supplier, one manufacturer and one buyer. An integrated model was developed that simultaneously optimized decision variables across the supply chain that consisted of a raw material supplier, a manufacturer, and a buyer (retailer). He suggested this model was a combination of previous two-stage models, which he referred to as the Integrated Procurement-Production (IPP) and Integrated Vendor-Buyer (IVB) models. In the IVB model a buyer (retailer) and a vendor (manufacturer) collaborate in optimizing the inventory system for the supply chain by deciding the quantities of ordering lot size and production batch size. In the IPP model, on the other hand, a raw material supplier and a manufacturer collaborate in order to minimize the cost of raw material ordering, holding, manufacturing setup cost, and finished goods holding cost by determining the raw material procurement lot size and the manufacturing batch size. In classic joint-pricing policy papers, it is assumed that the base price of the product is given. However, choosing a correct price is an important step to be done by sellers, especially when the price of the product affects the demand level. In the following section, literature on inventory management that assumes price sensitive demands is discussed. 2.3 Inventory management with price sensitive demands The body of literature dealing with price decision-making is less substantial. When revenue is not fixed, due to price-sensitive demand, price becomes a decision variable and the objective is to maximize profit. Whitin (1955) introduced demand as a linear function of price in a single- 13

33 stage problem with EOQ assumptions. He combined the inventory theories from operations management (business persons) and economics (economists) and addressed the missing considerations in each area, i.e. price dependency in the business field and lot size and uncertainty in economics. In conclusion, the analysis links inventory control policy and price policy in different models and derives the combined policy, which yields the highest profits (Whitin, 1955). Similarly Kunreuther and Richard (1971) solved the EOQ model for a retailer under linear price-sensitive demand. They considered two situations of simultaneous and sequential pricing and noted that the more inelastic the demand curve for a particular product, the more desirable it is to use a simultaneous rather than a sequential solution procedure (Kunreuth.H & Richard, 1971). Abad (1988a, 1988b) offered mathematical techniques to calculate the optimal prices and order quantities for a linear price sensitive demand function and a constant elasticity demand function when the supplier offers incremental or all-unit quantity discounts. He also noted that for either form of price dependency, the procedure for calculating the optimal solution is the same, i.e., by using ordinary differential equation (ODE) (Abad, 1988a, 1988b). Burwell et al. (1991, 1997) extended Abad s model by permitting shortages and considering freight discounts, correspondingly. Porteus (1985) used similar assumptions in a study on setup time reduction and came up with an explicit solution. The initiative for this research was to explain how reductions in setup cost contribute in moving toward just-in-time (JIT) inventory management. One of the interpretations taken from the results is that ceteris paribus, there is a threshold for the sales level such that if and only if the sales rate is above that level, investment is made in reducing setups cost and when such investment is made, the optimal lot size is independent of the sales rate (Porteus, 1985). Viswanathan and Wang (2003) examined 14

34 optimal discount pricing to maximize profit in a vendor-buyer problem under price-sensitive demand. In this study the vendor and buyer sought to maximize their profits independently. Using a similar linear price-sensitivity assumption, Sajadieh and Jokar (2009) demonstrated that in a supply chain with price-dependent demand, the sales price is higher for the end customer if the vendor and buyer do not coordinate, due to the effect of double marginalization. They also showed that cooperation gains are higher when the demand sensitivity to price is higher. Lau and Lau (2003) explored the effects of price-demand relationships in one, two and three stage supply chains. They considered different demand curves including linear and exponential curves and the results showed that the form of the demand curve influences optimal solutions. Yano and Gilbert (2004) reviewed the literature on the integration of pricing in joint supplier/manufacturer model (IPP). Ray et al. (2005) examined a single-stage model with setup costs and price-sensitive demand. Results showed that under high demand elasticity, order sizes do not necessarily increase with higher setup costs. Dye (2007) established a deterministic price-inventory model for deteriorating items with a time-dependent backlogging rate. In investigating a market selection problem, Bakal et al. (2008) considered an EOQ problem with price-sensitive deterministic demand. They also examined a Newsboy problem with stochastic, price-sensitive demand (Bakal et al., 2008). The studies on price-sensitive demand can be divided into two major groups, considering their formulation of price-demand relations: additive demand type and multiplicative demand type (Petruzzi & Dada, 1999). The former type is a linear function such as, where p is the price, the intercept a is the demand if price were zero and b is the slope of the demand curve i.e. the demand increase rate if price increases by one unit. You and Hsieh (2007) 15

35 used the linear model in a single-stage replenishment problem, which will be discussed in detail later. Sajadieh and Jokar (2009) used this approach to optimize shipment and pricing policies in a two-stage supply chain. The second group, the multiplicative demand type, uses a power equation such as. Most of the above studies examined price-inventory models at retailers only. However, there are some studies that integrated the price sensitive demands in joint pricing and replenishment problems. Esmaeili (2009) solved the joint pricing and replenishment problem for the general price-dependent function. Esmaeili, Aryanezhad, and Zeephongsekul (2009) and Esmaeili and Zeephongsekul (2010) also solved the joint pricing policy and lot sizing problem with price sensitive demand using game theoretic approaches. In their review paper, Chan et al. (2004) classified the literature assuming of price sensitive demand according to a number of characteristics of the problem or assumptions made by the researchers such as the time horizon, constraints, restocking methods, and demand type, function, and input parameters (including inventory-level). Some researchers have observed that in addition to price sensitive demands, higher inventory levels in the store may stimulate demand. These studies are summarized in the following section. 2.4 Inventory management with stock- sensitive demands High inventories on hand may encourage demand for several reasons. For example, larger stacks of a product can foster greater visibility, thus promoting latent demand. Holding more inventories might also be an indication of a popular product. Larger inventories also provide consumers with confidence of higher service levels and future availability. Having many units of a product on hand also allows a retailer to split up the product across multiple locations on the 16

36 floor and potentially to capture additional demand (Balakrishnan et al., 2004). Researchers, particularly in marketing, have long recognized the demand-stimulating role of retail inventories (Balakrishnan et al., 2004 Krishnan & Winter, 2010 Levin et al., 1972 Mandal & Maiti, 1999, 2000 Silver & Peterson, 1985 Van Donselaar et al., 2010 Zhou & Yang, 2005). As Abbott and Palekar (2008) stated Retailers have long recognized the relationship between display space and product sales and the display space or the facings of a product makes it more visible and the visibility, in turn, creates additional demand. In an early attempt at showing stock dependency of demand, Wolfe (1968) offered empirical evidence that demonstrated sales for sport clothes and women s dresses are proportional to the amount of stock displayed. Among the first attempts to quantify the stock dependency of demand, G.L. Urban (1969) introduced shelf-space allocation models in the marketing literature. Later, Baker and T.L. Urban (1988a, 1988b) established EOQ models for inventory-leveldependent demand patterns. They assumed a power form inventory-level-dependent demand. Gerchak and Wang (1994) developed periodic-review inventory models with stock-dependent demand. Bhunia and Maiti (1998), using deterministic variables and the linear inventory and time dependent demand model, studied two inventory models for deteriorating items, one with shortage and the other without shortage. Their models show the optimal average cost and its dependence on the demand and the deterioration parameters. In a similar research study Chung et al. (2000) extended a paper published in 1995 by Padmanabhan et al. adding backlogging variables for an inventory model with a linear stock-dependent demand. Wang and Gerchak (2001) stated retailers can often affect sales volume of a product by increasing the shelf space allocated to it. Therefore, they studied supply chain coordination when demand is shelf-space 17

37 dependent. T.L. Urban (2005a, 2005b) in two comprehensive reviews studied over 60 papers and demonstrated that the stock dependence for demand is modelled in two general forms among researchers: the initial-level inventory dependent demand and instantaneous-inventory level dependent demand. He then showed that in certain situations these two types are equivalent. He also proposed an approach for conducting sensitivity analysis on demand parameters. Goyal and Chang (2009) also determined an operating policy for an inventory system in which the demand is dependent on the display stock level. Sajadieh et al. (2010) developed an inventory model for a two-stage supply chain with stock-dependent demand where the manufacturer produces an integer number of the buyer s lot size. Their analysis confirms the higher profitability for the system with joint lot-sizing. They also showed that the effect of double marginalization provides a link between the non-coordinated and the coordinated case. Recently, researchers have incorporated other aspects into inventory models with stock-dependent demands, such as progressive payment schemes and game theoretic approaches (Min et al., 2010 Shib Sankar, 2011 Teng et al., 2011b). Similar to price dependence, in the extant literature, stock dependence takes two forms: linear form and power form dependence (Mandal & Maiti, 1999). The linear form is such that (Bhunia & Maiti, 1998 Chung et al., 2000 Teng & Chang, 2005 You & Hsieh, 2007). The power form uses the equation (Balkhi & Benkherouf, 2004 Chung, 2003 Mandal & Maiti, 1999). It is expected that the demand function be an increasing function as inventory level increases. However, it even when the inventory level is zero, there is likely be some demand. The power form mentioned above lacks this characteristic, therefore Larson and DeMarais (1990) proposed to add a constant to the power function ( 18

38 ) in order to enhance the behaviour of the model. Another approach proposed is to assume demand is sensitive to inventory level down to some level, beyond which it is constant (Datta & Pal, 1990). Another important characteristic for a demand function mentioned by Urban (2005a, 2005b), is its capability to reflect diminishing returns i.e. as stock level increases, the demand should increase at a decreasing rate. The linear form of demand function does not have this characteristic. In his paper, Urban (2005b) identifies different future research areas with roots in supply chains with inventory dependent demand. Model generalization by relaxing some of the assumptions, considering stochasticity, and empirical studies are suggested. As mentioned earlier there has not been any empirical studies on inventory dependent demand models since (Wolfe, 1968). Further supply chain coordination focusing on retail supply chains, particularly those with products that have inventory-level dependent demand is suggested as a future research avenue. Although, there are research studies that address supply chain coordination for products with price and/or stock sensitive demand, none has answered the question To what extent should inventory be pushed forward in the distribution channels to respond to ILDD [inventory level dependent demand] patterns? (T.L. Urban, 2005b) 2.5 Inventory management with price and stock sensitive demands Although researchers have acknowledged the importance of dependency of demand on price and stock levels, few have developed models that incorporate both of these aspects into the analysis. Urban and Baker (1997) generalized the EOQ model considering demand as a multivariate function of price, time, and inventory level. Datta and Paul (2001) analyzed a finite period inventory system, in which both price and displayed stock influence the demand. They 19

39 assumed price and order quantity as decision variables and maximized the profit (Datta & Paul, 2001). Teng and Chang (2005) imposed a ceiling for the number of items on display for an economic production quantity (EPQ) system with price- and stock-sensitive demand and proposed necessary conditions for optimality in profit maximization. You and Hsieh (2007) assumed a linear function for stock sensitivity and developed a continuous-review inventory model in order to determine the EOQ and price setting strategy for a retailer selling a seasonal single item in a finite time horizon. Recently, Dye and Hsieh (2011) and Shib Sankar (2011) solved the EOQ problem for inventory models with price- and stock-dependent demand with different sophisticated assumptions. Most of the above papers assumed demand to be a deterministic function of price and stock level and proposed analytical solutions for optimizing the profit. Moreover, all of the price- and stock-dependent demand models were developed for a single retailer inventory system. It must be noted that incorporating stochastic variables and/or multi-stage supply chain assumption into an analytical model makes it significantly more complex such that it is almost impossible to solve. However, an effective tool in order to both evaluate the behaviour and incorporate stochasticity in variables is the use of discrete-event simulation. To the best of our knowledge, there is no research in the extant literature that has used simulation methods for price-dependent demand, stock-dependent demand, or price- and stock-dependent demand problems. 2.6 Simulation-based optimization As illustrated in the following chapters increasing the number of parameters and assumptions considered in an inventory model adds to the complexity of the problem exponentially and it becomes difficult or impossible to optimize the system analytically. In such situations 20

40 researchers have actively developed meta-heuristics search techniques to search for optimal solutions (Grewal, 2012). Using these optimization techniques along with simulation is an emerging method in operations research. Most commercial discrete-event simulation packages have integrated some sort of optimization toolkit (Fu, 2002). Simulation-based optimization is the process of optimizing a set of performance measures based on outputs from a stochastic simulation. Usually a generic optimization package is linked with a generic discrete-event simulation engine. The optimization module uses the outputs from each previous simulation iteration to determine the set of inputs for the next simulation iteration. The simulation module then uses the inputs to provide new performance outputs. This process continues until an acceptable set of performance outputs are obtained or a termination rule is met. Figure 2 illustrates this process (Rogers, 2002). Figure 2. Optimization simulation process Many of the commercial simulation-optimization packages use meta-heuristics for searching the problem space globally rather than locally (Fu, 2002). Meta-heuristic methods are global 21

41 search techniques that search for optimal solutions without knowing the mathematics of the complex system (Grewal, 2012). Genetic Algorithm, Tabu Search, Neural Network, Ant Colony, Simulated Annealing, and Scatter Search are among the most common meta-heuristics. Table 1 shows a list of available optimization packages and the associated meta-heuristics used in them (adapted from (Fu, 2002 Grewal, 2012 Law & Kelton, 2007)). Simulation-based optimization is becoming a widespread method to solve optimization problems in operations research. Simulation-based optimization involving simultaneous perturbation stochastic approximation (SPSA) is used by Schwartz et al. (2006) for identifying optimal decision parameters for inventory management in supply chains with supply and demand uncertainty. The results of their study show that safety stock levels can be significantly reduced and financial benefits can be improved while sustaining acceptable operating performance in the supply chain (Schwartz et al., 2006). Table 1. Different commercial optimization packages (adapted from (Fu, 2002 Grewal, 2012 Law & Kelton, 2007)) Optimization package supported Simulation package Evolutionary approaches (GA, Ant Colony) AutoStat AutoMod, AutoSched X Neural Network search methods Scatter Search Simulated Anealing Tabu Search Optimiz Simul8 X Optimizer WITNESS X OptQuest Arena, Crystal Ball, Quest, Micro Saint, Flexsim ED, SIMPROCESS, X X X Simrunner2 Promodel, MedModel, Service Model X 22

42 Daniel and Rajendran (2005) also used simulation along with a Genetic Algorithm (GA) as the optimization module, to optimize the inventory levels at different firms in a supply chain and minimize the total supply chain cost. They simulated different supply chain settings (deterministic and stochastic replenishment lead times) to analyze the performance of the supply chain for base-stock levels generated by the optimization package. This method appears to perform very well in generating solutions that are not significantly different from the optimal solutions found by complete enumeration of solution space. The authors also claim that the simulation along with GA needs significantly less computing effort than complete enumeration (Daniel & Rajendran, 2005). In an attempt to extend the simulation-based optimization framework, Wan et al. (2005) proposed a method of iterative construction of an alternative model based on accumulated simulation results to identify the causal relationship between the decision variables and supply chain performance. The decision variables can then be optimized using the alternative model instead of single simulation runs to save on the overall computational effort (Wan et al., 2005). For more research studies using simulation-based optimization refer to Ingalls (1998) Joines et al. (2002) Truong and Azadivar (2003). OptQuest is a well-known commercial package for global optimization. For reasons such as its superior performance on a wide variety of problems and as a matter of convenience and availability, it has been used as a benchmark for empirical testing (Grewal, 2012). OptQuest uses a composite search algorithm combining tabu search, neural network, and scatter search (See Table 1) to intelligently search the input parameter space (Law & Kelton, 2007). Jafferali et 23

43 al. (2005) compared the performance of OptQuest and SimRunner and concluded that OptQuest provided results that are better than or equal to those of SimRunner. Kleijnen and Wan (2007) used OptQuest and Arena to simulate an inventory system with random lead times and constrained service levels and found very good estimates of the true optimal inputs. Grewal et al. (2010) and Grewal et al. (2015) used simulation-optimization to compare Kanban and reorder point replenishment strategies and find the best parameters. Although the simulation-based optimization method is becoming a common method in optimization problems in stochastic supply chains, there appears to be no evidence that this method has been used to compare different pricing and lot-sizing policies. 2.7 Gaps in the literature and potential research areas There are gaps and shortcomings in the literature, as mentioned above in different sections. These gaps provide opportunities for further research in the inventory management area. In this section a summary of the observed gaps relevant to the following sections of this thesis are stated. a. There are only a few papers that integrate price-sensitive demands and stock-sensitive demands into one model. However, numerous scholars acknowledge the importance of both of these factors and have modelled inventory systems with one of them. b. Most of the papers that considered either price dependency or stock dependency for demand have modelled their inventory systems for a single retailer. There are only a few papers that solved joint pricing and replenishment for multi-echelon supply chains. As Urban (2005a) mentions, it is important to understand with inventory-level- 24

44 dependent demand, to what extent should inventory be pushed forward in the distribution channels to respond to inventory-level-dependent demand functions? c. As can best be determined, there has been no research that uses simulation to model the price- and stock-dependent demands in inventory systems with assumptions such as stochastic variables, multi-echelon supply chains, and replenishment and joint pricing policies. d. Most of the papers have formulated the price-dependent or stock-dependent demands as linear or power functions. In the empirical research of Chapter 5, it is shown that for some products an exponential function is a better form to model dependency of demand on price and inventory level. e. Urban (2005a) claimed that the sensitivity of stock-dependent demand models to the stock dependency factors are considerably overestimated in the literature. Therefore, the relevance of price- and stock-dependency in the inventory models, especially for supply chains, is an important question. f. Related to the previous point, in reality price- and stock-dependency relationships are likely to be unknown, which is another source of error in these kinds of models. Therefore, performance sensitivity to errors in estimating the dependency relationships is another issue 2.8 Summary Considering the above gaps, in this manuscript an inventory model for a multi-echelon supply chain with price- and stock-sensitive demand is modelled using a general exponential demand function. A simulation-optimization method is used in order to find the optimal strategy 25

45 for joint pricing and replenishment problem. Details of the research such as objectives, methods, and its significance are discussed in the following sections. 26

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47 Chapter Three: Objectives 3.1 Overview In this section, following the relevant issues discussed in the literature, objectives of this research will be defined. Some of the objectives are followed by propositions, which will be tested in this research. 3.2 Price- and stock-dependent demand assumption Some authors performed sensitivity analysis to evaluate the effect of stock sensitivity on profit for a single retailer. However, overall, there is not enough support to claim that incorporating price- and stock-dependency factors in inventory models have significant performance advantages in comparison with models that assume demand functions independent from the price and inventory level. Therefore one of the objectives in this thesis is to evaluate the importance of these factors by comparing supply chain models with different demand functions. Proposition 1a: Considering price- and stock-dependency of demand significantly affects the performance of the inventory models. There are few studies that have addressed dependency of demand on price and stock levels together in one model. Therefore another objective in this research is to investigate the 28

48 simultaneous influence of price and inventory level dependency of demand on optimality of inventory and pricing. Proposition 1b: Considering both price and stock dependency of demand results in more effective inventory systems compared to models that incorporate only one of the factors. Moreover, an exponential function for price- and stock-dependent demand is assumed as a more realistic demand function at least in some cases. 3.3 Joint pricing and replenishment policies It has also been observed that higher benefits are achieved when firms in different stages of supply chain share information and coordinate their inventory systems by setting the price and lot sizes jointly (Frohlich & Westbrook, 2001). It has also been noted that if a firm cooperates with both its buyer and supplier, higher performance would result compared to situations with asymmetric information sharing (Frohlich & Westbrook, 2001). Therefore another objective of this research is to develop a model for joint pricing and replenishment problems in a multiechelon supply chain in order to test the following propositions. Proposition 2a: Joint pricing and replenishment policies result in higher profits for the firms in the supply chain than policies with independent pricing and replenishment strategies. Proposition 2b: In a multi-echelon supply chain a pricing and replenishment policy that considers all of the stages of the supply chain results in more effective inventory systems than 29

49 partial joint pricing and replenishment policies (one that involves only a part of the supply chain). 3.4 Using simulation The use of simulation models, in cases where uncertainty in variables result in complicated and unsolvable analytical models, is common in operations research. The simulation model for a multi-echelon supply chain scenario can also evaluate the impact of additional influencing variables, such as lead times and back orders, on performance. The simulation approach can also be used to validate and modify results of analytical models. The use of a simulation model is important in this research and complements analytical models. Therefore, another objective of this research is to develop a flexible parametric discrete-event simulation in order to test independent and joint pricing policies in different environments. Simulation is done using Arena and statistical analysis is performed on the outputs using SPSS and Excel. 3.5 Summary In order to meet the above-mentioned objectives, four major phases in the research were planned. Phase one is an empirical study on data from a grocery chain store. In this phase the assumptions of price- and stock-dependent demand are studied. The data is then used to estimate the coefficients of the proposed exponential demand function. Phase two is to develop an analytical model for a single stage (one retailer) supply chain with a deterministic exponential price- and stock-dependent demand function and discuss the optimality of pricing and order quantity determination. 30

50 In the third phase, the analytical model is extended to a two-stage supply chain (one retailer and one manufacturer) scenario, with price- sensitive demand. Optimal profits of the firms for independent and joint pricing and replenishment policies are compared. This supply chain is simulated in Arena and the simulation results are compared to those of the analytical model in order to validate the simulation outputs 1. The problem of profit allocation is a relevant issue in the joint pricing policy. In depth consideration of this issue is beyond the scope of this research. However, two approaches for profit allocation are discussed in this phase and next phase. In the last phase the supply chain model is further extended to a multi-echelon supply chain (one retailer, one distribution centre (DC), and one manufacturer). Different combinations of pricing and replenishment policies besides the sensitivity of the optimal strategy to different variables are discussed. In the following diagram the overall framework for this research is presented. 1 Verification and validation of the simulation models are discussed in chapter 8 31

51 Figure 3. The research framework 32

52 Figure 3 shows the overall framework of the research. In this research a comprehensive model for a multi-echelon supply chain is developed in a progressive step-by-step fashion. The purpose of the model in each stage is to investigate supply chain coordination and pricing and replenishment policies under price- and stock-dependent demand assumption. The influence of the price and stock dependency on optimal policies along with sensitivity analysis is discussed in each phase. In the next chapter methods and approaches used in this research are discussed in detail. 33

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54 Chapter Four: Methodology 4.1 Methods and proposed approaches In the previous chapter an overview of this manuscript was provided as a framework, shown in Figure 3. There are four major methods used in this research. First, an analytical method is used in the second phase of this research. The exponential function proposed for demand is the core to the analytical method and later will be used in the simulation as well. Mathematical formulations used to model a retailer s ordering and pricing policies in phase two provide insights for further extensions of the model in the following phases. Second is the discrete-event simulation method used to model the supply chains and to evaluate the influence of different factors in a more realistic situation, considering variable uncertainties. Arena is the software chosen for this method. Third is the iterative simulation-optimization method that will be used to optimize the performance measures for each policy. Finally, design of experiments and statistical analysis (i.e. factorial design and ANOVA) are used for the empirical study of phase one and testing propositions in later phases. SPSS is used for statistical analysis. In the following sections these methods will be discussed in more detail. 4.2 Analytical approach Analytical modeling has been used for decades in order to model systems in operations research. In inventory management, similar to other operations fields, the classical papers proposed simple models for the problem in hand. With the emergence of computer applications, 35

55 analytical models have incorporated numerical methods for finding optimal solutions for more complex problems. As mentioned earlier, all of the works on optimal pricing policies with priceand stock-dependent demands are analytical-numerical models. Therefore, for the first phase of this research a similar approach is used to solve the order quantity and pricing problem with the assumption of exponential price- and stock-dependent demand functions. This analytical solution will be used as a benchmark to verify the results of the proposed simulation-optimization model for the same problem. However, as the problem gets more complex in later phases, use of analytical modeling is not possible and only the simulation-optimization model will be extended to address complexities. 4.3 Discrete-event simulation Discrete-event simulation is a powerful computer-based tool used by many researchers and practitioners to analyze management science problems. It deals with mimicking a system over time by assuming that the state of the system changes instantaneously at discrete points in time. This method is suitable for modeling supply chains as it can analyze complex systems with uncertainty in variables. Simulation modeling is one of the techniques used by researchers to model the behaviour of complex systems. There are many advantages that make simulation a tool useful for research, one of which is its ability to approximate real life situations closely. For the specific problem in this paper, the benefits of simulation can be summarized as followed: a. There are simplifying assumptions in theoretical models that can be avoided in the simulation process, which results in more accurate outcomes. 36

56 b. The supply chain and the variables in this research are highly uncertain in real life. These kinds of stochastic variables cannot be dealt with analytically. Simulation, on the other hand, is a suitable tool for stochastic experimental research. c. Changing the number of stages and using different parameters and variables to perform what if analyses can expand the usefulness of the simulation model of a supply chain. d. Simulation allows modeling of non-stationary behaviour as well as dynamic adjustments of decision variables during the modeling run. For example, the demand rate can be adjusted according to price and stock levels. Arena is used to develop simulation models for the supply chain scenarios considered in this research. Arena is built based on the earlier SIMAN simulation language and was first released in It is a flexible discrete-event simulation package that allows researchers to create models that represent any real system. Some of the specific reasons for using Arena in this research are: a. It is a very popular package therefore the software and expert technical support is readily available. b. It uses graphical modules that make understanding and debugging the model easier and more time efficient. c. It has a built-in process analyzer that makes running structured experimental designs easy and time efficient. d. It interfaces with an optimizing package (OptQuest ) that makes the simulationoptimization process feasible. 37

57 e. It can interface with other software, such as MS Excel, that makes data import and export for further statistical analysis easy. Simulation is used as part of phase three in order to compare the simulation results with analytical results. It is also used extensively in phase four, which compares pricing and replenishment policies in the multi-echelon supply chain model. 4.4 Simulation- optimization approach As mentioned in the previous section, simulation is a powerful tool to model complex problems. However, it is not sufficient for optimizing decision variables. Therefore, simulationbased optimization is utilized to find the optimal strategies experimentally. In simulation-based optimization packages simulation and optimization tools are linked so that outputs from stochastic simulations are linked to optimization search techniques. The use of simulation-based optimization in this research can be justified for the following reasons: a. Different optimal policies for pricing and replenishment in a supply chain are compared. Analytical optimization is not feasible due to the complexity of the problem and the stochastic variables involved. However, simulation-optimization is practical. b. There are ongoing adjustments of optimization inputs from simulation outputs and vice versa adjusting these variables in two separate packages for simulation and optimization is not efficient. However, integrated simulation-optimization packages are designed for this purpose. c. There are several stochastic decision variables interacting in the models of the research and this can be handled easily in a simulation-optimization. 38

58 Some optimization methods that can be used in optimizing simulation results have shortcomings, such as being trapped in local optimums, lack of learning ability, and the absence of intelligent search guides. Meta-heuristic methods have been developed in order to overcome these flaws. OptQuest, an optimization package integrated into Arena, uses three metaheuristic methods in its search for the optimum. It uses tabu search, scatter search, and neural networks in its optimization algorithm. In short, the main reasons that OptQuest is suitable for this research are: a. It is able to deal with stochastic problems. b. It is able to deal with nonlinear formulas in the objective function and boundary conditions. c. It is not easily trapped in local optimums. d. It is integrated in Arena and has been used widely by researchers and practitioners. 4.5 Design of experiment and statistical analysis In order to be able to test the propositions, statistical analyses are required in phase two and three. Analysis of variances (ANOVA) alone or as part of regression analysis is applied as the method of choice for statistical analysis on the performance measure. These analyses are performed in Chapters Five to Eight for empirical input parameter analysis and on the results of the simulation-optimization process. The performance measure considered in this research is profit. However, service level will also be calculated and reported. To perform statistical analyses, experiments are designed using a full factorial design method in Chapter Eight. Statistical analysis for this research is performed in SPSS and Microsoft Excel. 39

59 4.6 Summary In this chapter, methodologies used in this dissertation have been introduced. However, an in-depth explanation of each method is not provided. In the following sections, while using these methodologies, comprehensive explanations will be provided. This research is synthesized from four major studies. First is an empirical analysis on data from a chain grocery store in order to investigate the idea of price and inventory level dependency of demand. Next, a retailer who faces a price and inventory level dependent demand for one of its products is modelled and the optimal pricing and order quantity is discussed. Third is a model for a two-stage supply chain including a retailer and a manufacturer with deterministic price-dependent demand. A comprehensive analytical model is developed for this supply chain and different pricing and replenishment strategies are compared. Finally, a simulation-optimization model is developed for a multi-echelon supply chain consisting of a retailer, a distribution centre, and a manufacturer with stochastic price and stock level dependent demand. Different aspects of a supply chain, such as transportation times, setup time at the manufacturer, and different coordination and information sharing policies, are considered. The optimal pricing and replenishment policies for different possible demand functions are discussed. The sensitivity of the model to different parameters is analyzed at the end of this section. 40

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61 Chapter Five: Empirical Analysis on Sales Data for a Retailer 5.1 Overview Two of the major assumptions in this thesis are the price sensitivity and stock sensitivity of demand. Studying price and stock-dependent demand is one of the major contributions of this research to the joint economic lot-sizing and pricing research area. Dependency of demand on price is a known phenomenon, especially in economics. However, the dependency of demand on the available stock level in stores, although acknowledged conceptually by some research studies, is not well-studied empirically. In similar research, only a few studies provide empirical evidence that demand is actually dependent on stock level. Wolfe (1968) offered empirical evidence that demonstrated sales for sport clothes and women s dresses are proportional to the amount of stock displayed. Balakrishnan et. al. (2004) interviewed a manager of a retail store and claimed that the demand is dependent on the available inventory. In this chapter first using the available data from a retail store, evidence for price- and stockdependent demand is tested. One function is selected as best for further consideration. Moreover, fitting a curve with the proposed formula for price- and stock-dependent demand, to the actual sales data allows the coefficients for this formula to be captured. Next, three different price and inventory level dependency functions that are used 42

62 widely in similar research are compared conceptually and according to their behaviour to extreme price and inventory levels. In the following pages first an introduction to the database is provided. Later some statistical analyses are performed to test the hypothesis that demand is dependent on price and stock level for at least some products. Finally, the demand functions are discussed. 5.2 Introduction to database The University Of Chicago Booth School Of Business and Dominick's Finer Foods (DFF) entered into a partnership for store-level research into shelf management and pricing from 1989 to Randomized experiments were conducted for more than 25 different categories throughout all stores in this 130-store chain. As a by-product of this research cooperation, approximately nine years of store-level data on the sales of more than 3,500 Universal Product Codes (UPCs) is available in this database. This data is unique for the breadth of its coverage and for the information available on retail margins. The Dominick's database covers store-level scanner data collected at Dominick's Finer Foods over a period of more than seven years. The database contains two types of files: categoryspecific files and general files. The general files contain information pertaining to all the categories in the database. The data contains information on: Customer Count Files These files contain information on store traffic and coupon usage, by store. Store-Specific Demographics 43

63 Demographics file contains detailed information based on the US Government census. Universal Product Code (UPC) Files These files contain one record for each UPC in a product category. They contain information about product name, size, commodity code, etc. Movement Files These files contain weekly sales data for each UPC in each store for over 5 years. The variables included in these files comprise: price, unit sold, profit margin, deal code, etc. The files are sorted by UPC, store and week. Dominick Store locations Map of Dominick stores There are 29 different product categories in this database, which are listed Table 2. These products and the associated sale numbers and prices are analyzed to evaluate the price dependency of demand. When the Dominick's data was collected, DFF priced products by 16 zones. Within each zone, there was supposed to be a uniform regular price (promoted prices are the same, chain-wide). However, these 16 zones amounted to four price tiers: Cub-Fighter, low, medium, and high. This will help to evaluate the price dependency of demand by considering different UPCs in each product category in similar price tier zones as well as studying the demand between different price tier zones. In the data there is no direct measure for the number of products on hand (stock level) at the time of each sale. Therefore, the number of different UPCs for any product category is assumed as a proxy for the stock level in the study. For example, if there are 50 different UPCs for a 44

64 product, such as soap, in one store and 70 different UPCs for soap in another store, it is assumed that the second store holds more inventory of soap and the inventory levels are proportional to number of UPCs. Table 2. List of product categories in Dominick s database Product Category Analgesics Crackers Paper Towels Bath Soap Dish Detergent Refrigerated Juices Beer Fabric Softener Soap Bottled Juice Front-End-Candies Soft Drinks Canned Soup Frozen Dinners Shampoos Canned Tuna Frozen Entrees Snack Crackers Cereals Frozen Juices Toothbrushes Cigarettes Grooming Products Toothpastes Cheese Laundry Detergents Toilet Papers Cookies Oatmeal 5.3 Data preparation In order to be able to analyze a set of data using different methods there are some assumptions that must be met. In other words if the assumptions required for a specific method are not satisfied by the data, that method could not be used. There are two important assumptions that almost always must be tested: Normality and absence of correlated errors. These assumptions are tested in this thesis for all of the methods that are used. To use a database for statistical analysis the first step is always dealing with missing/invalid data. The issues of missing/invalid data for Dominick s database issues are investigated next. 45

65 5.3.1 Invalid data The owners of the database have defined a variable in each data-set called OK that flags data points that are suspected of being erroneous. There is no further explanation provided indicating the type or source of the error. Therefore, any data point that was flagged by this variable is removed from the analysis Missing data All the data points missing values for one or more of the necessary variables for the analysis of this thesis are removed. There are situations where the demand for a UPC in a branch is not recorded. In those cases it is assumed that the demand for that UPC in that branch at that date was zero. It is necessary to keep those data in the analysis since one of the major factors tested is the dependency of demand on price. Removing data with zero demand may skew the demand upward. 5.4 Price sensitivity of demand In this chapter each of the 29 different products is analyzed for price dependency of demand. It is expected that demand for some products is less sensitive to price (price inelastic) than for other products (price elastic), such as most consumer products. It is not expected demand would increase as the price increases for any of the products in Table 2. In economics such products are called Veblen or Giffen goods, each of which has specific requirements that are not met by any of the 29 products listed. In order to make sure that the data is appropriate for correlation tests, tests for normality and endogeneity are performed for each product category. Figure 4 shows a sample normality curve for Paper Towel. 46

66 Figure 4. Test for normality of residuals using P-P curve for Paper Towel UPCs It is explained below that the negative correlation between price and demand is better explained with an exponential function rather than a linear function. Therefore, Pearson s correlation test (specific for linear correlations) is not used. Instead the Spearman s correlation test is used, which is adequate for nonlinear correlations. The correlation results tables for the product Paper Towel are shown in Figure 5. In the correlation test the null hypothesis is that there is no correlation between price and demand. There are two factors that are of interest. First, the correlation coefficient factor ( ) which indicates the extent of the correlation. Here it is expected to see a negative correlation between price and demand. The second factor is the p-value for testing the hypothesis of no 47

67 correlation against the alternative that there is a nonzero correlation. The smaller the p-value, the more statistically significant the correlation is. Figure 5. Spearman s non-linear Correlation test for product Paper Towel. Table 3 summarizes the correlation between demand and price for different products. All of these correlation factors are statistically significant at the p-value 0.01 level. Except for Beer, Paper Towel, and Toilet Paper all the correlations are negative, meaning that as the price increases, demand reduces. The correlation between price and demand for Beer (Positive) and Grooming Products (Negative) are not very strong, meaning that they are to some extent price inelastic. For the case of Beer, the price inelastic behaviour could be due to the loyalty of consumers to their favourite brand and the fact that they usually buy the same brand no matter how much the price changes. In general the correlation factors are not very high (in the range). This fact could be due to comparison between different UPCs of a similar product. Therefore, the same analysis is performed for each individual UPC to evaluate the correlation between price and demand. Although there is higher correlation observed in this study, the study suffers from the small sample size problem. 48

68 Table 3. Price and demand correlation and exponential function for different products (demand per week) Product Category Correlation factor a b Analgesics Bath Soap Beer NA NA Bottled Juice Canned Soup Canned Tuna Cereals Cigarettes Cheese Cookies Crackers Dish Detergent Fabric Softener Front-End-Candies Frozen Dinners Frozen Entrees Frozen Juices Grooming Products Laundry Detergents Oatmeal Paper Towels NA NA Refrigerated Juices NA NA NA Soap Soft Drinks Shampoos Snack Crackers Toothbrushes Toothpastes Toilet Papers NA NA 49

69 Linear, Power, and exponential curves are fitted to the data using the Curve Estimation option in SPSS. For each of the functions ANOVA is performed to evaluate the significance of the coefficients of the fitted curve. This process is similar to regression analysis with the difference being that different curve types can be used. In all cases the goodness-of-fit for the exponential function was better than that of the linear and power functions, suggesting the proposition that an exponential function is better than a linear and power functions in modeling the price dependency of demand. The coefficients for the exponential function, a and b in, are also shown in Table 3. A comparison of the fitted linear function versus fitted exponential function is also shown in Figure 6. It is shown that the R2 of fit for the exponential function is higher than for the linear function, both when outliers are included in the data and when they are removed. This graph is for a single UPC of product Soap. Figure 7 exhibits a sample of the curve estimation in SPSS for Soap. Figure 6. Comparing goodness-of-fit for linear vs. exponential price-dependent demand function. 50

70 Figure 7. Curve estimation analysis for linear and exponential function of demand as a function of price 51

71 5.5 Inventory level sensitivity of demand The Dominick s data doesn t provide the inventory level at the stores. However, the number of different UPCs for each product type is available. In this section it is assumed that the number of different UPCs can be considered as a proxy for the stock level at the store, while controlling for store size and demographic aspects of customers in the areas that stores are located using the price tier zone for each store. That is, comparing two stores with similar characteristics, the one that has more UPCs for a product holds more inventory of that product and that store is expected to have more demands for that product. Therefore, the proposition would be the higher the number of UPCs for a product at each store, the higher the demand for that product would be. Table 4 summarizes the correlation between demand and stock level (number of UPCs) for different products. All of these correlation factors are statistically significant at the p-value 0.01 level. As expected the correlation between stock level and demand is highly positive, indicating that as the number items available in the store increases the demand for that product also increases. It must be noted that this statement doesn t indicate the direction of causality. However, we can consider a reinforcing loop. In other words as the number of UPCs for a product category increases, the demand for that product category increases and the store manager realizes the increased revenue for that product category and therefore orders more of that product, which in turn increases the demand further. It must be noted that the counter argument for this assumption is plausible. In other words it can be argued that higher number of UPCs may not be correlated with higher inventory levels. An example would be a case that there are several UPCs available but the number of items for each UPC is a small number and the total inventory is not necessary high. In that case perhaps 52

72 higher demand could be the cause of the higher variety available. This situation, which is an interesting topic for marketing research, is not part of the scope of this thesis. As mentioned earlier, this database does not provide the inventory levels for each product. Therefore it is not possible to capture the coefficient factors for stock dependency part of the demand function (i.e. and ). However, for the demand function used for the multi-echelon supply chain model, a one to one relation between the number of UPCs and inventory level is assumed and an exponential function is fitted to the data. This method helps yield a model that is not totally arbitrary. Table 4. Stock level and demand correlation (Spearman s non-linear correlation) Product Category Correlation factor Product Category Correlation factor Analgesics Frozen Entrees Bath Soap Frozen Juices Beer Grooming Products Bottled Juice Laundry Detergents Canned Soup Oatmeal Canned Tuna Paper Towels Cereals Refrigerated Juices NA Cigarettes Soap Cheese Soft Drinks Cookies Shampoos Crackers Snack Crackers Dish Detergent Toothbrushes Fabric Softener Toothpastes Front-End-Candies Toilet Papers Frozen Dinners

73 5.6 Other possible forms of dependency of demand in price and stock level It is assumed in this research that there is a negative correlation between price and demand. That is demand decreases as price increases and demand increases as price decreases. It is also assumed that there is a positive correlation between inventory levels observed by customers and demand. That is higher inventory levels available at the shelves stimulate the demand. There are other products and/or other situations that the effect of price, inventory level, or a combination of the two on demand can be different from what is assumed in this model. These situations are not in the scope of this thesis. However, with some minor changes to the proposed demand function, most of these situations can be studied. Here are a few examples of these situations Price inelasticity and positively price dependent demand There are some products for which demand is not dependent on their price. An observed example in the DFF data is Beer. There are also some products that demand increases as the price of the product increases. An observed example in DFF data is Toilet Paper. Studying the inventory system for these types of products is not in the scope of this thesis Effect of scarcity on demand There are situations in which the scarcity of a product stimulates demand. That is lower inventory levels observed by customers increases the demand. For the products in DFF data this may happen in case of a natural disaster. In these situations a low inventory level may signals a scarce product that in turn may increases its demand. Again, studying the inventory system for these types of products is not in the scope of this thesis. Also in a normal market situation, a scarcity may imply high demand, which may indicate that the product is valued. This may in turn increased demand from buyers not previously interested. 54

74 5.6.3 Correlation between price-dependence and inventory level dependence It is assumed in this research that dependency of demand on price and inventory levels are not correlated. In other words there is no interaction between price and inventory level effects on demand. There are examples of counter arguments. An example of this correlated effect could be the situation of a natural disaster, where higher price and lower inventory levels signal shortage of a product which in-turn increases the demand. Inventory systems with products that has the correlated price and inventory level dependent demands are not studied in this research either. 5.7 Three models for general price- and stock-dependent demand As mentioned earlier, two major functions have been used for showing the dependence of demand on both price and stock level. These two are linear (additive) and power (multiplicative) functions. In this thesis a third function, the exponential model, is used for showing both dependencies. The linear form is such that (Bhunia & Maiti, 1998 Chung et al., 2000 Teng & Chang, 2005 You & Hsieh, 2007) and the power form is such that (Balakrishnan et al., 2004 Balkhi & Benkherouf, 2004 Chung, 2003). In these formulas is the price and I is the inventory level. In both of these models as inventory level goes to infinity, the demand goes to infinity. This is an unrealistic assumption. Therefore a new exponential model is developed so that when the stock level goes to infinity the demand tends toward a stable demand level that depends on the price. This formula takes an exponential form such that: (1) 55

75 in which. The constant is named the saturation factor and affects the way that the demand function increases with the inventory level and reaches its final value. As shown in Figure 8, the larger the, the faster the demand merges to its maximum level. This behaviour is consistent for all relevant values for and. Figure 8. Saturation factor Table 5 shows the comparison of the three formulas based on their behaviour to marginal inventory level changes. 56

76 Table 5. Comparison of demand functions behaviour to marginal inventory changes All of the functions are increasing, as their derivatives are positive ( ). In other words all these functions behave such that as the inventory level increases, demand will increase. The third column in Table 5, ( ) presents the behaviour of the demand function with the rate of change in inventory level. A negative value for this column demonstrates the diminishing return characteristic of the function. The linear function does not have this characteristic, as its second derivative is not negative. In reality it is expected that the rate of demand growth decreases as inventory level increases. 57

77 Table 6. Comparison of Demand functions behaviour to extreme cases Table 6 shows the response to extreme stock levels and prices. For the proposed exponential function, as the stock level goes toward zero the demand goes toward the demand rate that is specified by the price. On the other hand if the stock level goes toward infinity the demand will not exceed a specific level, which is defined by price and. An unrealistic behaviour of the power function is the fact that as the price goes toward zero, the demand would be infinity. In other words, the power model suggests that the demand for a free item is unlimited, which is not true in reality. Also, it can be seen in Tables 2 and 3 that in the linear model for prices larger than, demand is negative and in the extreme case if price goes to infinity, demand is. While it might be argued that these extreme conditions are unlikely to occur, these are additional reasons (other than the fit of actual data described earlier) to believe that an exponential function is the best choice to represent the dependence of demand on price and stock level. However, the coefficients for the exponential function need to be obtained by experimental investigation. In the next section, studying a chain grocery store database, these coefficients are explored for different products. 58

78 5.8 Summary It was shown in Chapter 2 that there is a huge gap in empirical literature supporting the price and stock-level dependencies as a whole and the exponential function specifically. Therefore, in this chapter, using data from a chain grocery store t available from The University of Chicago Booth School of Business and Dominick s Finer Foods (DFF) the price and inventory level dependency of demand are studied. There is a list of 29 different product categories and more than 3500 UPCs, sold in more than 135 stores. The correlation analysis demonstrates for most products there is negative nonlinear correlation between price and demand. Furthermore, using curve-fitting methods the correlations are formulated in the form of, proposed in this thesis. The regression methods used show first that the proposed model fits to data better than linear and power functions and second that the fitted model is statistically significant for these products. Since, there is no data available for the inventory levels at stores, it is assumed that the number of UPCs for a product in each store, controlling for other factors such as price tier zone and store specific demographics, is an indicator of that product inventory held in that store. The correlation analysis shows that there are strong and significant positive correlations between the number of UPCs at the store and demand. These results are used to justify that demand for some products depends on available inventory level, besides factors such as price. In this chapter three different models for dependence of demand on price and stock level used in literature are compared. Besides the evidence from the data showing the superiority of exponential function, it is argued that the models with linear and power dependence functions have some characteristics that are difficult to justify for real life situations. Therefore, the 59

79 proposed exponential function is believed to be a better model to interpret customers behavior toward price and stock level. In the next chapters, the price- and inventory- dependent demand functions are used in analyzing pricing and replenishment policies at the firm and supply chain levels. 60

80 61

81 Chapter Six: Single Stage Supply Chain Model 6.1 Overview In this chapter the model introduced by You and Hsieh (2007) for a single-stage singleperiod replenishment problem with the assumption of price and stock-dependent demand is investigated further. In the original model the optimal order quantity and optimal set of selling prices for a seasonal item is modeled assuming that the firm will set the price n times during the time span of L and demand is linearly dependent on price and stock level (You & Hsieh, 2007). Different from You and Hsieh (2007), in the extended model introduced here, the number of price changes is not assumed as a given number but is one of the decision variables to be optimized. Further the same model is investigated with different patterns for demand dependence on price and stock level, i.e. exponential and power functions. It is shown in the previous sections that the exponential function is likely the most realistic dependence function. However, sometimes for the sake of simplicity one can use a linear function to represent the dependence of demand on price and stock level, as explained in You and Hsieh s model (2007). Further in the original model the cost for re-pricing the items was assumed to be independent of the remaining quantity of the product left to sell. In the extended model it is assumed that re-pricing cost has two portions. One is fixed, such as that of the cost of advertising the new lower price, and the other portion is dependent on the on-hand inventory level, such as changing the price tags on the items. 62

82 6.2 Single replenishment inventory model with price- and stock-dependent demand You and Hsieh (2007) noted that firms stimulate demand for seasonal products by strategies such as reducing price. Nevertheless, cases with multiple price changes when the demand is price- and stock-dependent seldom appear in the literature (You & Hsieh, 2007). Therefore they studied a single replenishment inventory model to determine the optimal order quantity and optimal set of selling prices for a seasonal item assuming that the firm will adjust the price n times during the time span of L (You & Hsieh, 2007). The general assumptions and notations for this problem are as follows. Assume that a retailer purchases Q units of a product in each order period and sells them all in time horizon L. In order to maximize its benefit, the retailer considers changing (reducing) its price n times during this time horizon. It is assumed that the last item always sells on the last day of the time horizon, though reducing the price during the time horizon would result in selling more items upfront and consequently reducing the holding cost. The trade-off for numerous price changes is the re-pricing cost, which will be discussed later. The retailer faces a price and stock sensitive demand, which can take the following forms (You and Hsieh (2007) only considered the linear form): (2) 63

83 The retailer divides the time horizon into n equal time segments and is able to reset the price at the beginning of each time segment. In You and Hsieh (2007) n is assumed to be a given number. The length of each segment is (for all j=1 m). Index j shows the j th time segment of the horizon and denotes the price during segment j. Ij(t) represents the retailer s inventory level at time t ( ) of the segment j. The retailer s holding cost per item per unit time is assumed to be hr and manufacturer s price is c. It also is assumed that changing the price incurs costs such as changing product price tags, changing price lists, and advertisements. The price change/adjustment cost is assumed to be a constant, K (You & Hsieh, 2007). The objective is to maximize the retailer s profit by concurrently optimizing the order quantity and the selling price. Since all demand at the retailer store is provided by available inventory, the rate of decrease in inventory level during the time is equal to the demand. Therefore: (3) This ordinary differential equation (ODE) must be solved to find the inventory level at each point in time. You and Hsieh (2007) used the linear model for price- and stock- sensitive demand and proposed a solution method for the optimal order quantity and selling prices. However, they couldn t find an analytical solution for the general form of n prices. Therefore they discussed the situations with no price change (n=1), single price change (n=2), and two price changes (n=3). Later, Mo et al. (2009) solved the general form of the problem for any given n in You and Hsieh (2007) along with the optimal prices and the optimal order quantity that maximizes the profit. 64

84 The question that is not fully explored in these two papers (Mo et al., 2009 You & Hsieh, 2007), is: What is the optimal number of price changes (n) that maximizes the profit? In the following sections the models developed by You and Hsieh (2007) and Mo et al. (2009) are investigated in order to find the optimal number of price changes in the single replenishment inventory model. Moreover, the same approach is used for two other models of price- and stockdependencies (i.e. power and exponential) and the results are compared and discussed Summary of proposed optimal solution for linear demand function Solving the ODE in (3), You and Hsieh (2007) found the inventory level function, assuming linear price and stock sensitive demand as: (4) where. The sales revenue is then obtained by: (5) where qj is the inventory level at the beginning of the j th period. The holding cost for each period is found by integrating the inventory level function and the sum of the holding costs for all the n periods. The total holding cost is: (6) 65

85 Considering the equations for revenue (5) and holding cost (6) and the fact that the maximum profit is achieved when the inventory level at the end of time span is zero, You and Hsieh (2007) showed that the total profit function over the time horizon is: (7) where, (8) (9) and (10) Mo et al. (2009) determined the unique optimal value of selling prices such that the profit is maximized as follows: (11) where (12) (13) 66

86 Mo et al. (2009) also mentioned that an optimal value for n existed and proposed a recursive algorithm in order to find the optimal n. However they did not discuss the results and implications of this optimal value of n The model with exponential demand function The exponential demand function is shown in (2). It is used to solve the ODE in (3) knowing the initial condition that at time zero of the first time segment the inventory level is equal to the quantity ordered by retailer, I1(0)=Q. The solution steps of this ODE are not presented. However, the final result is: (14) where and ln is the natural logarithm function. This equation holds for all segments j=1 n. Now we have the inventory level as a function of time, the order quantity Q, and the set of prices ( ). Using this inventory level function, one can calculate the revenue and holding cost. Incremental revenue for any time segment is the number of products sold in that segment. If one assumes inventory function curve on a diagram where y- axis is the inventory level and x-axis is time (Figure 9), holding cost in each time segment is equal to the area under the inventory function curve multiplied by the unit holding cost. The following sections show these calculations. 67

87 Figure 9. Inventory function Sales revenue Let qj be defined as the inventory level at the beginning of the time segment j (You & Hsieh, 2007). In other words Ij(0)=qj. Therefore, during time segment j the number of products that are sold is equal to: sold in segment j is (You & Hsieh, 2007). Since the fixed price for each of the items, the total sale revenue is: 68

88 (15) This formulation shows the sum of the revenues made for all time segments. Inventory holding cost Holding cost for an item in inventory is calculated by the total time that the item remains in the warehouse multiplied by the holding cost per item per year. To add the holding time for all of the items in the inventory for a specific time segment, one can calculate the area under the inventory level curve by time, Ij(t) (see Figure 9). This area for each time segment j, calculated by the integral in (16), multiplied by the holding cost per item per year gives the holding cost for time segment j. The sum of holding costs for all time segments gives the total holding cost: (16) However, calculating this integral is not analytically feasible. Therefore numerical methods such as Simpson s method or Trapezoidal method need to be utilized. Using the Trapezoidal method, the estimated value of the integral in (16) is: 69

89 (17) Equation (17) is the estimated area under the inventory curve in time segment j. Order quantity As mentioned in the problem statement, the retailer s goal is to sell all of the ordered items in the time horizon L. In other words the objective is to have In(T)=0. Calculating In(T) in (14) and equating it to zero, one can find Qn, the order quantity from the producer that results in In(T)=0: (18) Profit function Let be the total profit for one time horizon. Thus F is the revenue minus all the costs, which are buying, holding, re-pricing, and order costs. Then: (19) Substituting Q=Qn from (18) into (15) and (17), one can recalculate R(n) and H(n) respectively. The maximum profit for the retailer can be calculated by finding the derivative of function F with respect to, which results in solving a system of nonlinear equations. Later in this chapter this system of nonlinear equations is solved in a numerical example. 70

90 6.2.3 The model with power demand function The use of the power function for presenting the dependence of demand on price and stock level is observed in some papers, including Mandal & Maiti (1999), Chung (2003), Balkhi & Benkherouf (2004), and Balakrishnan et al. (2004). The power demand function is: (20) This function is used to solve the ODE of Equation (2) using initial condition I1(0)=Q. The final result is: (21) where. This equation holds for all segments j=1 n. Having the inventory level as a function of time, the order quantity Q, and the set of prices ( ), one can calculate the revenue and holding cost as follows. Revenue As explained earlier, the number of units sold during time segment j is equal to: (You & Hsieh, 2007). Since the fixed price for each of the items sold in segment j is, the total sale revenue is: (22) These formulations show the sum of the revenues earned for all time segments. 71

91 Inventory holding cost Using a similar formulation to that presented in (16), the holding cost for time segment j can be calculated. The sum of holding costs for all time segments gives the total holding cost: (23) Order quantity Similar to other models, Qn, the order quantity from the producer that results in In(T)=0 can be calculated as: (24) and consequently: (25) (26) Profit function The profit function introduced in (7) and (19) can be used here as well. The final formula using (24) - (26) is: 72

92 (27) Similar to the exponential model, the maximum profit for the retailer can be calculated by finding the derivative of function F with respect to, which results in solving a system of nonlinear equations. However, solving for maximum profit using the power function is not within the scope of this thesis. In this section three different formulas for dependence of demand on price and stock level were discussed and the replenishment and pricing problem was formulated for each of them. It must be noted that the latter two models, i.e. exponential and power models, require solving a system of nonlinear equations that appears to be analytically impractical. Thus numerical computation is necessary to solve these problems. 6.3 Improved model for re-pricing cost It is assumed that the retailer sets the price n times to optimize the total benefit. Each time the price is changed there is a cost associated, which is referred to as the re-pricing cost in You and Hsieh (2007). As mentioned, it was assumed that this cost is fixed regardless of the number of items remaining in stock to be re-priced. The argument in this research is that this is not a realistic assumption and the re-pricing cost should comprise a portion dependent on the number of items currently in inventory. Therefore, the following formula is proposed to replace the constant re-price cost (K). 73

93 (28) where Kf is a fixed cost such as advertisement costs, and is the inventory dependent costs such as the cost of changing price tags. It is expected that the dependent portion will be smaller than the constant portion, especially with the improvements in technology and computer software used for inventory management. Re-pricing is done at the beginning of each time segment j, when the inventory level is Ij(0). Therefore, the formula in (28) will be: (29) where is per item cost of changing price. The total cost of re-pricing through the time horizon L is then: (30) Ij(0) in this formula can be calculated by substituting t=0 in (4), (14), and (21) for linear, exponential, and power models respectively. The profit function then changes to: (31) These new formulas can easily be reduced to the fixed cost model by You and Hsieh (2007) by substituting. In the next section the two models for exponential and linear demand functions are compared vis-a-vis the optimal solution for n=1, 2, and 3. 74

94 6.4 Numerical examples Optimal solutions comparisons In real life the demand function should be a forecast based on historical data. As explained earlier, because of its characteristics, it is believed that exponential function is the closest function to the real behaviour of customers demand at least for some products when it is dependent on price and stock level. However, it can be assumed that managers, for sake of simplicity, could use linear regression to show dependence of demand on price and stock level. This would result in the function presented in You and Hsieh (2007). On the other hand, they can fit an exponential function, similar to the one proposed in this thesis, to the same data. Since this presumed data is not available to be able to compare the results of the two models, the curve that is used for exponential function should follow a path close to that of the linear function. In other words, to be able to compare the optimal solutions with that of the linear function in You and Hsieh (2007), coefficients for the exponential model should be defined in a way that the resulting curve is close to that of the proposed linear function. Using the power function is not discussed here. The reason being that there is no power function that can follow a path close to that of the linear function. This is because the boundary conditions for a power function are different from those of the linear and exponential functions at zero-inventory or zero-price (See Table 6). Therefore at this point only the results from the exponential function are compared to the results presented in You and Hsieh (2007). The linear function in You and Hsieh (2007) is ( ). Using numerical analysis and solving a system of equations one can find the exponential function that 75

95 closely follows the linear curve as ( ). Figure 10 shows these functions. The R 2 for the fitted function is 94.37%. All other variables are assumed to be similar and as follows: L = 120, hr = 0.005, c = 20, Kf = 500 and = 0. Figure 10. Demand as linear and exponential functions of price and stock level. Optimal solutions for no price change (n=1) Here it is assumed that there is no price change. The retailer decides on the price once at the beginning of the time span and doesn t change it. According to You and Hsieh (2007), the optimal order quantity is Q = and optimal price is for a total profit of F = 14, The exponential model proposed in this paper, on the other hand, resulted in an optimal order quantity Q = and optimal the price is for a total profit of F = 25, This model allows for lower order quantities and higher price, which eventually results in lower holding cost and higher profit. 76

96 Optimal solutions for one price change (n=2) For the case of one price change, i.e. at the middle of the time horizon, You and Hsieh (2007) found the optimal solution to be Q = and optimal prices as, for a total optimal profit of F = 16, The exponential model resulted in an optimal Q = and optimal prices as, for a total optimal profit of F = 25, Optimal solutions for two price changes (n=3) For the case of three price settings, i.e. at the beginning and then, one third, and two thirds into the time horizon, You and Hsieh (2007) found the optimal Q = and optimal prices as,, and total optimal profit of F = 17, The exponential model resulted in an optimal Q = and optimal prices as,, and total optimal profit of F = 25, Figure 11 shows how the demand and inventory levels change in this case for both linear and exponential models. Figure 11. Inventory level and demand function for two price changes 77

97 These results, summarized in Table 7, show that for similar demand functions, the exponential model results in higher profits. Therefore, whenever possible, the managers are encouraged to invest more time and resource to investigate whether their product s demand follows an exponential function or not. If it follows the exponential function, it is beneficial to consider it exponential and use this method to determine the optimal price and order quantities. However, the question remains as to what the optimal number of price changes for the product is. In the following section this question will be discussed. Table 7. Comparing linear and exponential models for different number of price changes n=1 n=2 n=3 Demand Model Linear Exponential Optimal Order Quantity Optimal Price set δ1 $26.79 $39.29 Optimal Profit $14, $25, Optimal Order Quantity Optimal Price set δ1 $31.27 $41.19 δ2 $22.31 $36.20 Optimal Profit $16, $25, Optimal Order Quantity δ1 $33.00 $41.82 Optimal Price set δ2 $26.77 $38.73 δ3 $20.58 $35.09 Optimal Profit $17, $25, Optimal number of price settings As n appears at the upper limit of the summations in formulas of profit function components (see equations (15) to (18) for example), one should use the Euler-Maclaurin formula to transform the summations to integral form and then solve the optimization problem for the upper limit of the integral. The Euler-Maclaurin formula is as follows: 78

98 (32) and its derivative with respect to n is: (33) This approach is not computationally efficient and presents several difficulties. Another approach to find optimal value of n is to use numerical methods. The simplest approach would be to solve the optimization problem for different values of n and find the optimal value that maximizes the profit. This approach is used in the following analysis. Figure 12 shows the optimal profit for different values of n, i.e. n=1 to n=10, considering the linear method. Figure 12. Optimal profit and the optimal number of price changes for the linear model For the given set of variables, n = 3 is the optimal number of price changes for the linear model. The major trade-off in this case is between the re-pricing cost and the holding cost. In other words, as the number of price changes increase, more units of the given product will be sold faster and that reduces the holding cost. On the other hand by increasing the number of price 79

99 changes the total re-pricing cost will increase. Later, the effect of different variables on the optimal number of price changes will be discussed. A similar curve is observed for the exponential function where the optimal number of price changes is n = Sensitivity analysis The sensitivity of the optimal number of price changes to the fixed re-pricing cost (Kf), per unit re-pricing cost ( ), holding cost (hr), and purchase cost (c) is evaluated in this section. Fixed re-pricing cost (Kf) As seen in Figure 13 if the re-pricing cost is zero, the profit is non-decreasing. That means if there is no cost to change the price, the more often the retailers change the price (reduce the price), the more they can manipulate the demand to increase their profits. However, the change in profit diminishes by increasing the number of price changes. Mathematically speaking, the profit curve is convex and after several price changes, the increase in profit per price change is almost zero. Figure 13. The effect of changes in re-pricing cost on optimal number of re-pricings 80

100 If the re-pricing cost is not zero, then there exist an optimum number of re-pricings that maximize the profit. The number of re-pricings is usually a small number, such as 2 or 3 times. Moreover, it can be seen from Figure 10 that increasing the re-pricing cost decreases the profit in general. Per item re-pricing cost ( ) The effect of per item re-pricing cost is very similar to that of the fixed portion of re-pricing, but with less effect. This re-pricing cost in reality should be small compared to the fixed portion. However, in Figure 13 the values selected for are fairly large to demonstrate its effect better. Holding Cost (hr) Looking at Figure 14 one can see that increasing the per-item per-unit-time holding cost would reduce the profit. However, it doesn t change the optimal number of price changes Pricing policies Figure 14. The effect of change in holding cost on optimal number of price changes Another interesting and counter intuitive result should be noted. In some cases, such as for higher holding costs (hr= 0.05) and higher stock-level dependence of demand ( ), the 81

101 optimal solution is to increase the price a few times at the beginning of the time horizon rather than decreasing it. At the beginning of the time horizon there is more inventory available, which stimulates higher demand. Therefore the retailer can increase the price and obtain higher revenue. The incremental revenue offsets the lower demand of higher prices in this case. Later in the time horizon, as there is not enough inventory available for this phenomenon to occur, the retailer needs to reduce the price to increase the demand and profit. If the optimization problem is constrained to allow only prices that are not higher than the price in previous step, then a suboptimal solution is the result. The demand and inventory level for a case with three price changes are demonstrated in Figure 15. The inventory levels for these solutions follow a very similar curve. The initial price for the solution that allows for price increase is slightly lower than the one without the price increase. This would result in higher demand at the beginning. At one-third into the time horizon, where there is still high demand due to higher inventory, the retailer can increase the price that would reduce the demand. If no price increase is allowed, the solution at this point is to keep the price constant. The last one-third of the time horizon (after day 80), where the demand drops due to lower inventory levels, the retailer can stimulate the demand by reducing the price. Both solutions produced similar prices at this point and the curves for the demands overlap. 82

102 Figure 15. Two pricing policies, one with allowed price increase, one without price increase both for n=3. Table 8 shows different costs, revenue, profit, and optimal decision variables (price and order quantity) for solutions with and without price increase. The profit for the suboptimal solution is lower than the optimal profit, but the deviation is minimal. Therefore, it is logical for retailers to keep the price fixed for two thirds of the time horizon and reduce the price in last one third. That would result in less price change inconvenience and perhaps reduced loss of customer loyalty due to price increases. Table 8. Comparing two pricing policies 83

103 A similar situation is shown in Figure 16 below, where n=9. That means the retailer has 9 opportunities to set the price. Following the no-price-increase policy, the retailer can hold the price constant for the first 5 decision intervals and start reducing the price from the 6 th, when demand is dropping as a result of lower inventory levels. 6.5 Summary Figure 16. Comparing pricing policies for n=9. A pricing and ordering problem is considered where a retailer is selling one product, the demand of which is dependent on price and stock level, during a specified time horizon. The solutions for three demand function models are provided and linear and exponential models are compared using a numerical example. It is shown that if the demand dependence on price and stock level is follows a function that is close to both linear and exponential assumptions substantial profit will be sacrificed if the linear function is used. Thus it is advantageous to use 84

104 the exponential function. Empirical research, such as the one in Chapter Five, can be used to investigate the coefficients of the proposed exponential function. It is shown in this chapter that there is an optimal number of times that the retailer should change the price. On the one hand, changing the price several times allows demand to be manipulated so holding costs are reduced. On the other hand, there is more re-pricing cost. A more comprehensive form of re-pricing cost is introduced that includes both a fixed cost and a variable cost dependent on the number of items remaining in inventory. The optimal number of price changes is sensitive to parameters of the re-pricing cost. However, it is almost always in the range of two to four price changes for the scenarios considered. Finally, it is shown that in some cases the optimal solution is to increase the price at the beginning of the time horizon to capture more revenue by sacrificing some demand. However, retailers could opt to keep the price constant at points where the price increase is suggested by the solution. That would reduce the profit slightly but retailers can benefit from less inconvenience and avoid potential loss of goodwill due to increased prices. This chapter provides analytical motivations to use price and stock level dependent models where appropriate. Especially, the proposed exponential model is believed to mirror customers behaviour toward price and stock level dependence and improve the retailer s inventory system performance. 85

105 86

106 Chapter Seven: Two Stage Supply Chain with Price Sensitive Demand 7.1 Overview In Chapter Five a new exponential formula for price and stock-dependent demand was introduced. This model was used to find the optimal number of price changes, selling prices and order quantity during a time horizon for a retailer. The analytical solutions and solving procedure, although unique to the retailer with a defined time horizon, provide insights for the more general ordering and pricing policies in a supply chain. In this chapter the simultaneous joint lot-sizing and pricing problem for a supply chain is examined. Initially, the problem addressed is for a two-echelon supply chain consisting of a retailer and a manufacturer with price-dependent demand. Production, ordering and pricing decisions across a two-stage supply chain are optimized simultaneously. The model developed involves the solution of an embedded JELS problem. However, it also involves determining pricing under price-sensitive demand such that profits will be maximized for the supply chain as a whole. This model is important in that it allows issues of coordination and joint optimization among different members in a price-sensitive supply chain to be tested and explored. In the next section, the problem scenario, notation and assumptions are introduced. The section after that examines models that allow supply chain members to independently optimize their profits, as well as models that allow joint optimization. Section 7.4 empirically compares the solution of the proposed algorithm with the generic models presented by Ben-Daya et al. 87

107 (2008) to evaluate its deviation from optimal solutions in the context of price sensitive demand. Section 7.5 provides a numerical example, along with analysis. Section 7.6 compares two possible methods for allocating profits between firms when jointly optimizing total profit. Finally, conclusions and further research directions are summarized in Section Problem definition and notation This research assumes a deterministic, two-stage supply chain with one manufacturer (vendor) and one retailer (buyer). The supply chain produces one product, produced by the manufacturer, purchased by the retailer and then sold to customers. Following the unified framework presented in Ben-Daya et al. (2008) the retailer places n orders of size qi (i=1 n), where n is assumed to be an integer, to the manufacturer when the next customer arrives and the inventory is zero. An ordering cost, Ar, is incurred by the retailer for each of the n orders. Inventory holding costs, hr, are also incurred so there is a trade-off between incurring ordering and holding costs. The manufacturer produces units of the product at a rate of P. There is a setup cost, Am, incurred which may make it beneficial to produce enough units in one run (Q) to meet multiple retail orders ( ). Manufacturer s inventory holding costs, hm, are also incurred so there is a trade-off between incurring setup and holding costs. The customer demand is a function of the price, with demand increasing as the price falls. This means both the manufacturer and retailer profits are dependent on this price. The objective is therefore to maximize profits, with as an additional decision variable to those of Ben-Daya et al. (2008). In some cases profit can be maximized individually by the retailer and then by the 88

108 manufacturer, given the value of Q specified by the retailer. However, of more interest is the maximization of profit across the whole supply chain by considering the manufacturer and retailer simultaneously. Notations used are as follows: D: customer demand rate, in items per unit time P: production rate of the manufacturer, in items per unit time Q: manufacturer s production lot-size equal to retailer s order quantity Am: manufacturer s setup cost for a production run Ar: retailer s ordering cost for any order a: constant cost parameter in the customer demand function b: exponent cost parameter in the customer demand function c: manufacturer s selling price cp: manufacturer s cost per item (including raw material and process) : final price to the end customer (retailer s selling price) hm: inventory holding cost per item per unit time for the manufacturer hr: inventory holding cost per item per unit time for the retailer n: number of shipments to retailer from one production run, where n is always an integer m: number of shipments that follow the geometric policy Im: average manufacturer s inventory Ir: average retailer s inventory Is: average system inventory, Is=Im+Ir 89

109 qi: retailer s order quantity TP: average total profit per unit time. Subscripts r, m, and j are used for retailer, manufacturer and joint profits. : geometric size factor for shipments Additional model assumptions and clarifications may be summarized as below: a. The manufacturer has a finite production rate, P, which is always greater than the demand rate, D. b. The retailer orders a lot of size qi at the point a new customer arrives but the inventory has been depleted. Receipt of this shipment from the manufacturer is instantaneous. c. The shipment size, qi, for the first m shipments follow the geometric policy with the ratio of. The remaining n-m shipments stay equal to the last shipment size computed using the geometric policy. d. The manufacturer runs a production batch of size Q following each setup. e. The manufacturer s price to the retailer is c. The final price to the customer is. This includes both the manufacturer and retailer mark-up. There is no re-pricing occurring in this chapter. f. The retailer faces negative exponential customer demand, where the rate, D, is a function of the final price,. g. The retailer incurs a setup cost, Ar, every time an order of size qi is placed. The manufacturer incurs a setup cost of Am every time a production batch of size Q is started. 90

110 h. The manufacturer and retailer incur a holding cost equal to hm and hr per item per unit time, respectively. It is assumed. 7.3 Model formulation The general model for the lot sizing problem is presented for the price sensitive condition using the unified model presented in Ben-Daya et al. (2008). In the general model the firms cannot optimize their profit independently because of common decision variables in the profit functions. However, in the special case of the equal lot size problem the profit functions can be decoupled and firms can maximize their profit independently. This approach will be referred to as the Independent policy. Other models of an integrated supply chain in which information is shared will be referred to as the Joint policy General model for the lot sizing problem The retailer faces exponential demand that is a function of the price and takes the form. The retailer s profit is equal to the demand times the profit margin minus the retailer s order and holding costs. The retailer s profit margin is equal to the total profit margin,, minus the manufacturers profit margin, c. The manufacturer s profit is equal to the manufacturer s price, c, times the demand minus the setup and inventory holding costs. When both parties cooperate under a joint policy it is the combined total profit that is of interest. Since the profit for the manufacturer, Dc, is a cost for the retailer, these terms cancel out. Therefore, total profit for the retailer, manufacturer and joint supply chain, respectively are: (34) 91

111 (35) (36) where (37) (38) (39) (40) (41) (42) In the optimal solution by Hill (1999) is equal to P/D and qe in Equation (40) is the remaining items of the production batch equally divided among the rest of shipments: (43) 92

112 In the model presented by Ben-Daya et al. (2008), could be a decision variable or could be equal to P/D and qe in Equation (40) is equal to the last shipment size calculated by the geometric policy: (44) Using Ben-Daya et al. (2008), substituting (39), (40), and (44) into (37) and (38), and simplifying results in: (45) and (46) where (47) Substituting (45) and (46) into (34)- (36) results in: (48) (49) (50) 93

113 7.3.2 Models for the independent policy For cases where or these equations reduce to the equal lot-size problem, where is the shipment size for all n shipments and the production size is Q=nq. In this situation the retailer and manufacturer can optimize their profit independently. The retailer wishes to maximize its profit function, TPr, through the optimal choice of the final price,, and order quantity, q, since c is assumed given. Thus, the retailer s decision variables are. Therefore the objective function for the retailer is as follows. (51) The optimal replenishment order size is then the economic order quantity. This is obtained by setting. (52) Substituting the optimal order size, q *, into objective function (51) and simplifying, results in a retailer profit function, TPr, that depends only on the final price,. Alternatively, since price determines the demand,, profit can also be stated in terms of demand, D. (53) The second derivative of the profit function has one root, equal to. The profit function is concave if demand is greater than, and convex otherwise. Maximum profit will occur in the concave region. The first derivative has two roots, which will be designated s1 and s2. The root s2 is the largest and indicates maximum retailer profit since it is in the concave 94

114 region. Therefore the optimal retailer s decision variables are,, and. The manufacturer now needs to be considered. It is assumed a setup cost is incurred and therefore it may be optimal to produce n retail shipments of size q* in one production run, where n is an integer. Therefore, the manufacturer s decision variable is n. The time interval between setups will then be nq/d, while the length of a production run will be nq/p. Figure 17 provides further clarification. The average inventory at the manufacturer is given by the following. (54) The manufacturer s objective function, TPm, is given by the following. (55) Since q and are already specified by the retailer, the manufacturer only has n as a decision variable. Taking the derivative with respect to n and solving shows that TPm is concave. Therefore the optimal value of n is given as follows. (56) 95

115 Figure 17. Inventory levels for retailer, manufacturer and supply chain, m=5 and n=7 using geometric then equal shipment policy. Since n must be an integer variable, its optimal value is either the truncated portion of n * or n * +1. The optimal integer order shipments from one production run will be designated as. The maximum total profit under the independent policy can now be determined based on the retailer s order quantity,, the final price,, and the number of shipments filled from one production run,. (57) 96

116 7.3.3 The model for joint policy As mentioned earlier, when both parties cooperate under a joint policy it is the combined total profit that is of interest. Following the general model presented in Section 7.3, the objective function for the supply chain is to maximize the joint total profit presented in (50). The second derivative of the profit function with respect to q1 shows that the total profit is concave. Hence, following the economic order quantity method, the optimal joint policy is reached when and the optimal order quantity is as follows: (58) The total profit objective function,, can now be written as a function of ( ). (59) Note that as mentioned earlier, the geometric size factor,, in these equations could be a decision factor larger than one or equal to the production-to-demand ratio,. In the first case, as a decision variable is independent from demand function, D. Hence, for a given value of n, TP * j can be rewritten as follows: (60) in which: 97

117 (61) In (61) x 1 is always non-negative and depending on different values for, m, and n, x 2 can take on different values as follows: If ( If If If ) then: then then then If then. Condition (a) reduces the lot-sizing problem to the equal lot-size policy, which is explored in detail in Forootan et al. (2014) and Appendix A. Condition (b) represents all other policies and will be discussed later. 7.4 Numerical comparisons of solution methods The following policies are compared as shown in Table 10. a. The equal-shipment policy ( or ), denoted by E. b. The geometric policy ( and ), where is a given factor, denoted by GC. c. The geometric policy ( and ), where is a decision factor, denoted by GV. 98

118 d. The geometric-then equal policy ( and ), where is a given factor, denoted by GEC. e. The geometric-then equal policy ( and ), where is a decision factor, denoted by GEV Results Equations (37)-(50) were programmed in MATLAB. The optimal solution for each policy was then found using the optimization toolbox. The values of all parameters were the same as those used in Ben-Daya et al. (2008). Besides the price sensitive demand assumption, another contribution in this section is assuming m as a decision variable in the GEC and GEV policies, where Ben-Daya et al. (2008) assumed m=2 for these cases. Comparisons were made based on sensitivity analyses with respect to key parameters i.e. the retailer s holding cost (hr), manufacturer s setup cost (Am), price sensitivity factor (b), and production rate (P). The following parameters were used in the numerical comparisons. Am=400, Ar= 25, hm=4, hr=5, P=3200, a=1000, b=0.3 cp=0. The results are discussed below. The sensitivity of the results to variations in some of these parameters is also presented. For the inventory system with price insensitive demand (b=0 in the exponential model), results and conclusions are similar to those of Ben-Daya et al. (2008). Slight differences in numerical values could not be explained without having the details for the Ben-Daya et al. (2008) optimization models, but were judged to be trivial. Conjecture is that the differences are due to round-off-error within different optimization packages. 99

119 For retailer s holding cost (hr). As shown in Table 9 the lowest average deviation from optimal is for the geometric-thenequal policy with as a decision variable (GEV), followed by the geometric policy with as a decision variable (GV). However, the differences from the optimum solution for different methods are very close to each other and one cannot be selected as the best method. For the profit resultant from each model is very close to zero. Therefore the deviation from optimal cannot be calculated. Table 9. Effect of retailer s holding cost on deviations from optimal profit Deviation from optimal Profit (%) hr E GC GEC GV GEV Mean St. dev For manufacturer s setup cost (Am). The lowest average deviation from optimal in this case is for the geometric policy with as a decision variable (GV). The second lowest deviation from optimal is associated with the equalshipment policy (E). Here again, the average deviation from the optimal solution is similar for different methods and the best method cannot be selected. For the calculated profits are very close to zero and the deviations from optimal cannot be evaluated. 100

120 Table 10. Effect of manufacturer s setup cost on deviations from optimal profit Deviation from optimal Profit (%) Am E GC GEC GV GEV Mean St. dev For production rate (P) Table 11. Effect of manufacturer s production rate on deviations from optimal profit Deviation from optimal Profit (%) P E GC GEC GV GEV Mean St. dev All the methods produce similar average deviations from optimal except for geometric policy with as a constant (GC). It had a deviation from optimal higher than the other methods. The 101

121 lowest average deviations from optimal here are for policies with as a decision variable (GV and GEV). For price dependency factor (b). Table 12. Effect of price dependency of demand on deviations from optimal profit Deviation from optimal Profit (%) b E GC GEC GV GEV Mean St. dev As seen in Table 12, as the dependency on price increases, the deviation from the optimal solution increases as well. On the other hand, for very small dependency factors, i.e., the deviation from Hill s optimal increases as well Conclusion Based on the results above it can be seen that different methods produce results with similar deviations from Hill s optimal solutions. Second, the average deviation from optimal for the equal-shipment policy is among the lowest average deviations resulted from different methods. Therefore, one can see that more sophisticated policies with non-equal shipment sizes produce final solutions that are very similar to those of the much simpler equal-shipment policy. Moreover, when implementing the policy it is much easier for both retailer and manufacturer to 102

122 have shipments with equal sizes. In conclusion, this empirical evidence suggests that firms can implement the equal-shipment policy as their method of choice for the price sensitive products, knowing that other methods are less convenient and yet do not produce meaningfully higher benefits. In Appendix A the method introduced for equal-shipment policy is summarized in an algorithm for programming in spreadsheets. 7.5 A numerical example An example is considered with the following input variables: P = 3200/year, Am = $200/setup, Ar = $10/order, hm = $2/item/year, hr = $5/item/year, a = 1500, b = 0.3 and c = $5/item. The manufacturing cost of goods and process for each item (cp) is assumed zero in this example. Therefore the profits reported in the following section for manufacturer and for the joint policy are adjusted profits. The optimal solutions for both the Independent and Joint policies are given in Table 13. Table 13. Numerical example results Decision Variables Performance Measures Policy Price ( ) Q n D TPr TPm TP * Independent Joint The following profit improvement index was used to evaluate the change in profit, where TPj and TPi =TPr+m are the total profits from the Joint and Independent policies respectively. 103

123 (62) Table 13 shows that total profit improved by 88.56% when the Joint policy was used. This is largely the result of demand increasing by over 400%, due to a lower price being charged to the customers. It is interesting to note that the optimal final price,, for the joint policy is actually lower than the manufacturer s price, c, under the Independent policy. Obviously, this is an extreme case in which the manufacturer has set c too high with respect to optimizing profits across the overall system. If cp is not assumed zero, then it can be a lower limit for manufacturer s selling price to the retailer ( ) that also dictates a lower limit for the retailer s final price to customers ( ). The effect of the manufacturer s price, c, on total profit is explored in Figure 18. The profit under the Joint policy, TPj, is independent of c. The total profit under the Independent policy, TPr+m, increases as c decreases, becoming asymptotic to TPj as c approaches 0. However the benefits go mainly to the retailer. The retailer profit, TPr, increases at the expense of the manufacturer s profit, TPm, as c approaches 0. The manufacturer s profit reaches a maximum around c = 4. Under the Independent policy the manufacturer sets c initially and then the retailer optimizes its selling price and order quantity using this information. However, it must be recognized that the manufacturer cannot determine its optimal c value prior to knowing the retailers ordering behaviour and therefore it cannot exploit the profit maximization information in Figure 18 a priori. 104

124 Figure 18. Profit behaviour for the numerical example The effects of the demand price-sensitivity parameter, b, also warrant analysis. Suppose different values are selected such that b [0.05, 0.1, 0.15, 0.20, 0.25, 0.30]. Results are shown for both the Independent and Joint policies in Table 14. It can be observed that as the price sensitivity increases, the percent profit improvement, PI, increases rapidly. Also, if demand is not very sensitive to price, as is the case when b is low, optimal final prices,, are much higher. Note that at b = 0 demand is independent of price and revenue is usually considered fixed, making the problem one of cost minimization. Table 14 also indicates that the optimal final prices are higher when the supply chain members try to maximize their own profits independently. As a result the demand is lower. However, it must be recognized that there is an interaction effect involving parameter c, the 105

125 manufacturer s price. Therefore, the effect of c and its interaction effect with b also needs to be explored. Table 14. Effects of parameter b using the Independent vs. Joint policy b Independent Policy Joint Policy δi Q i ni TPr TPm TPm+n δj Qj nj TPj PI % % % % % % % Figure 19 shows a plot of the profit improvement, PI, as a function of the price-sensitivity parameter, b, and the manufacturer s price, c. The relative improvement due to using the Joint policy increases with both b and c. This graph also indicates that as c moves toward 0 the difference between the Joint and Independent policies moves toward 0. In other words, setting c equal to 0 for the Independent policy becomes equivalent to using the Joint policy with respect to the overall system profit. However, there is a critical difference. If c is set to 0 under the Independent policy, the manufacturer is forgoing all profit. In fact it will be operating at a loss due to setup and holding costs. Therefore, this is not a decision that would ever be made under an Independent policy. Only if the manufacturer were assured profits under some type of profit sharing agreement, such as would be implemented under a Joint policy, would it be willing to ignore its selling price, c. 106

126 Figure 19. Effect of parameters b and c on profit improvement using the joint policy This issue is further explored using the relationships shown in the Appendix A. If c increases under the Independent policy, it is proved that the optimal final price,, also increases and the maximum retailer profit decreases. It is obvious that when members make decisions independently the manufacturer will set c to a higher value and therefore the total potential profit for the supply chain will decrease. 7.6 Profit allocations As mentioned earlier, the joint policy results in higher profits for the supply chain as a whole. The question left is how to fairly divide the benefits attained by joint policy among different firms, in this case retailer and manufacturer. A fair allocation of the achieved benefits, gain sharing, has to be ensured to entire cooperation. To make sure gain sharing mechanism is fair, the marginal contributions of each party to the total gain must be quantified. Two approaches are 107

127 compared. First, the profits are allocated in the same proportions as when the firms optimized their profits individually. Second, solution procedures from cooperative game theory are employed. Cooperative game theory models the negotiation process within a group of cooperating agents (in this case supply chain firms) and allocates the generated profits (Reyes, 2006). Different fairness properties are represented by well-known allocation rules such as the Shapley value (Shapley, 1952). The Shapley value has the following properties: 1) Individual fairness: each player definitely gets at least the amount he or she would have got if he or she had not collaborated, 2) Efficiency: the Shapley value distributes the total value of a grand cooperation among the players, 3) Symmetry: if the marginal contributions of different players i and j to any cooperation S are the same, they receive the same share of total value of grand cooperation, 4) Dummy: if a player does not cooperate with any other players, it should receive zero, 5) Strong monotonicity: if the marginal contributions of all the players increase, the payoffs also increase. Myerson (2000) showed that there exists a unique function satisfying the above properties, which is called the Shapley value. Since the above properties are consistent with the assumptions of cooperation, Shapley values are considered a fair division of collaboration benefits among firms. The general formula for the share of firm i ( ) in any cooperation among N firms is obtained by the average added contribution of firm i to all cooperation S that don t already contain i: (63) where V(S) is the value (or total profit) obtained by any set S of the firms. 108

128 Using the first approach, i.e. keeping the profit proportions same as before, the share for each of the two firms is obtained by: (64) (65) where, TPr,i and TPm,i are the total profits of the retailer and the manufacturer in the Individual policy and TPr,j and TPm,j are the total profits of these firms in the Joint policy. The Shapley value for the profits, on the other hand, is obtained by: (66) (67) Using the numerical example above (shown in Table 13), profit shares for the retailer and the manufacturer based on each approach are summarized in Table 15. Table 15. Profit allocation using different allocation approaches Approach Proportion Shapley Value Independent Joint Gain from cooperation Retailer Manufacturer Retailer Manufacturer Retailer Manufacturer There are two major observations in Table 15. First, both firms gain more profit when they collaborate in lot sizing and pricing. Second, using the Shapley value approach the gain from collaboration for both firms are equal. The intuition behind this, based on Shapley fairness definition, is that, since without any of the firms the collaboration is meaningless. Therefore, 109

129 their contribution in cooperation is equal and hence they both should gain an equal extra profit from jointly making pricing and lot sizing decisions. 7.7 Summary In this chapter an integrated ordering, pricing and production model for a two-stage supply chain is developed. The contribution is that it adds optimal pricing under non-linear demand sensitivity to the optimal joint economic lot-sizing relationships used in previous integrated manufacturer retailer models. The pricing is assumed to affect end-customer demand exponentially and members in the supply chain simultaneously optimize decision variables so profits for the overall system are maximized. This research demonstrates that such joint optimization in a supply chain model is beneficial under price-sensitive demand. Since pricesensitive demand is commonly observed in practice, the results have practical implications. The key insights may be summarized in the following discussion. First, as shown by the comparison of different policies, the policies with unequal-shipments (e.g. geometric then equal shipment policy) do not produce meaningfully better results than the policy with equal-shipment. Evidence suggests that firms can confidently optimize their process without having to deal with overly sophisticated methods. If it is accepted that equal-shipment policy is best based on both performance and feasibility, several additional insights can be identified as followed. If the members of a supply chain are working independently the upstream manufacturer will not likely be aware of demand price-sensitivity at the retail level. It will likely base its selling price on the cost of goods manufactured plus a profit margin. The retailer will then set its customer price based on its cost and knowledge of market demand. The manufacturer will later 110

130 see a stream of orders from the retailer and attempt to optimize production by adjusting the number of retailer order sizes in a production lot size to maximize profits. This scenario is unlikely to maximize profits across the complete supply chain, as shown by the Independent policy results presented. Alternately, if members of the supply chain jointly maximize profits the retailer s price will be lower. Demand will then be disproportionately higher, given the assumption of exponential sensitivity. This research shows that the net effect will be higher total profit than if each member optimizes its profit independently. Therefore it should be possible to allocate profit to each supply chain member in such a way that all are better off. The issue of profit allocation is discussed by the effect of the internal pricing in the supply chain, which is the selling price of the manufacturer to the retailer. Under the Independent policy the manufacturer s profit margin, c, will dictate its profit, along with the volume of orders received from the retailer. However, as this profit margin is increased, the potential profit of the overall supply chain decreases. This may or may not mean the manufacturer s profit also decreases. However, under the Joint policy there still needs to be profit allocated to the manufacturer, even though a manufacturer s selling price is not incorporated in the model. Since it is reasonable to do this on a volume basis, a value based on the profit margin, c, is still useful for profit allocation. If profits are allocated according to some proportion, such as equal split profits, it is easy to calculate the appropriate manufacturer s selling price. The relative benefits of the Joint policy over the Independent policy are dictated by the demand price sensitivity. As the price sensitivity parameter b decreases the benefit of joint optimization will diminish. In this case the manufacturer s profit margin, c, will also have little 111

131 impact on total profits under the Independent policy. However, it will have a major effect on the allocation of profit between the manufacturer and the retailer. There are two profit allocation schemes that do not require the calculations of the internal price c. The first approach is allocating the profit gains by the Joint policy based on the proportion of the profits in Independent policy. Another way is to use a collaborative game approach and the Shapley value. Using this method, the extra benefits from cooperation are divided equally between the two firms. The intuition behind this approach can be explained as the equal necessity of both firms being present to form cooperation. In other words without the presence of either firms, the other one is not able to gain the extra profit of the Joint policy. Therefore, they should share the extra profit equally. 112

132 113

133 Chapter Eight: Multi-echelon Supply Chain with Price and Stock-level Dependent Demand 8.1 Overview In the previous chapters the analytical model for the supply chain consisting of one retailer and one manufacturer was presented. The demand for that supply chain was assumed to be price sensitive. For supply chains with more than two firms and the more complicated assumptions of price and inventory dependent demands, analytical solutions are very hard to achieve. Therefore these supply chains are analyzed using simulation methods. First, a simple deterministic model of a supply chain with one retailer, one Distribution Centre (DC) and one manufacturer is analyzed using the Economic Order Quantity (EOQ) approach. Demand is assumed to be a deterministic and level. Using this model, the research propositions are discussed. Later the details of the simulation process for this multi-stage supply chain and inventory model is explained. Following this, the model is validated and verified by testing its behaviour. This model is then solved using simulation-optimization software and the results are discussed. Finally, sensitivity analysis is done on select variables of the model. 8.2 EOQ for multi-stage supply chain with deterministic level demand The EOQ model is fundamental to inventory analysis and appears in almost all of the operations management textbooks. The basic model is a cost minimization model. However, it can be used in profit maximization by introducing a revenue term. The profit for a firm with deterministic constant demand can be defined as: 114

134 Profit = (sales revenue) (purchase cost) (holding cost) (order/ setup cost) Using similar notations to the previous sections, the profit function can be derived: TPi: Total profit for firm i D: Customer demand rate, in items per unit time P: Production rate of the manufacturer, in items per unit time Q: Retailer s order quantity Am: Manufacturer s setup cost for a production run Ad: Distributing centre ordering cost for any order Ar: Retailer s ordering cost for any order cp: Manufacturer s production cost per unit cm: Manufacturer s price to the DC cd: DC s price to the retailer : Final price to the end customer hm: Inventory holding cost per item per unit time for the manufacturer hd:inventory holding cost per item per unit time for the DC hr:inventory holding cost per item per unit time for the retailer m: DC s lot-size multiplier. DC orders m*q on each order n: Manufacturer s lot-size multiplier. Manufacturer produces n*m*q units on each production run. The profits for each of the three firms are as follows: (68) 115

135 (69) (70) Total profit for the supply chain is the sum of the profits of the individual firms. In this case the selling price for the manufacturer and DC becomes internal costs from one firm to the other and do not contribute toward the total supply chain profit: (71) Independent policy (R-D-M) The EOQ for the independent inventory management policy can be established. That is, each firm maximizes its profit based on the orders received from the downstream firm. The EOQ model for the retailer is the classical EOQ, and can be derived as follows: (72) which results in: (73) Using this order quantity as an input, the DC can now maximize its profit using a similar EOQ approach, which is: (74) which results in: 116

136 (75) Similarly, knowing Q and m, the manufacturer can maximize its profit by solving: (76) which results in: (77) Partial joint policy (R-DM) In this case the retailer would optimize its own profit and then the DC and manufacturer would optimize their total profit together. The retailer part remains the same, but the joint total profit for the DC and manufacturer is: (78) They can optimize their total profit by solving the following system of differential equations simultaneously: (79) The optimal values for m and n along with all other solutions are presented in Table 16 below Partial joint policy (RD-M) In this case the retailer and the DC optimize their profit jointly and then the manufacturer optimizes its total profit. The joint total profit for the retailer and the DC is: 117

137 (80) The system of differential equations for them is: (81) See Table 16 below for optimal values for Q, m, and n for this policy Joint policy (RDM) If all three firms coordinate the inventory system together and solve for the optimal profit of the whole supply chain, they should solve the following system: (82) See Table 16 below for optimal values Summary of optimal solutions: As shown in Table 16 the order quantity for the retailer (Q * ) is the same for policies where it acts as an individual (R-D-M and R-DM) and when coordinating with other firms, either with the DC only or with both, its order quantity increases. The manufacturer has a similar situation. If it acts as an individual (R-D-M and RD-M), its lot size (n * ) is larger than when coordinating with other firms. 118

138 Table 16. Optimal values for order/production quantities Q * m * n * R-D-M R-DM RD-M RDM For the DC the situation is more complicated. Table 17 shows the relation among different values of m* for different policies. To read the table one must examine whether m* for the policy of each row is smaller, larger or equal to m* of each column. For example m* for the R-DM policy is always larger than m* for the R-D-M policy. Table 17. The relation of optimal order quantity for the DC in different coordination m* R-D-M R-DM RD-M RDM R-D-M = R-DM > = RD-M < < = RDM Conditional < > = The situation for R-D-M and RDM is conditioned based on the following critical relation between the DC s holding cost and the others holding costs. 119

139 (83) If the left hand side is larger than the right hand side, then m* for the R-D-M policy is larger than that of the RDM policy and its opposite if the relation is opposite. According to Table 16, the actual number of orders/productions for firms can be summarized as shown in Table 18. The optimal profit for different scenarios for firms in the supply chain can then be calculated as explained in the following sections. Table 18. Summary of optimal order quantities Retailer DC Manufacturer 120

140 8.2.6 Changes in profit based on coordination policy Revenue The revenue for all the three firms is independent from the collaboration policy. Revenue is only dependent on the price and the exogenous constant demand rate assumed in this section. Therefore the changes in profits for the firms based on the cooperation policy are only dependent on different costs for each firm Costs for the retailer When the retailer cooperates with the DC, the retailer saves on the order cost but its holding cost is increases, as its order quantity is higher. However, the amount of saving on order cost cannot cover all the extra holding cost. Therefore the retailer s total cost is higher when cooperating with the retailer (RD-M and RDM) compared to when acting independently (R-D-M and R-DM). (84) Costs for the DC The order cost for the DC depends on whether the DC collaborates with the manufacturer or not. If the DC collaborates with the manufacturer (R-DM and RDM) its order cost is lower than when does not collaborate with the manufacturer (R-D-M and RD-M). (85) The holding cost for the DC is dependent on its collaboration with both the retailer and the manufacturer. Comparing the individual policy (R-D-M) to one-way partial collaborations (RD- M) and (R-DM), if the DC collaborates with its upstream firm (R-DM) its holding cost increases. If the DC collaborates with its downstream firm (RD-M) its holding cost decreases. (86) 121

141 Comparing the partial collaborations with each other and to the whole supply chain collaboration (RDM) result in similar situations. When collaborating with its upstream firm the DC s holding cost is higher than when it is not collaborate with its upstream firm. When collaborating with its downstream firm the DC s holding cost is lower than when not collaborating with its downstream firm. (87) A comparison of the holding costs for the DC in whole supply chain cooperation (RDM) and individual policy (R-D-M) is dependent on the relationship among the holding costs for all firms. In general the amount that the DC saves on the order cost is lower than the increase in holding cost. Therefore the total cost for the DC increases as its holding cost increases and decreases as its holding cost decreases. Hence, the total cost for the DC increases when collaborating with its upstream firm and decreases when collaborating with its downstream firm. (88) Costs for the manufacturer The setup cost for the manufacturer doesn t change with changing policies as it is always producing the constant n*m*q* items. The manufacturer s holding cost however is lower in policies that it cooperates with the DC (R-DM and RDM) than in policies that it acts independently (R-D-M and RD-M). Therefore the total manufacturer s cost is dependent on whether it cooperates with the DC or not. (89) Figure 20 shows the total cost curves as function of order/ production quantity for the firms in supply chain for two conditions considering the relations among the holding costs for firms. 122

142 Based on what is shown in Figure 20 and Equation (89) when the manufacturer collaborates with the DC its holding cost, and therefore its total cost, reduces. Hence, the manufacturer (upstream firm) always benefits from collaboration. A similar situation is true for the DC. That is whenever the DC collaborates with the retailer its total cost is lower. However, if collaborating with the manufacturer, the amount of its gain reduces. That is why the cost for the DC is lowest when collaborating only with the retailer (RD- M) and is highest when collaborating only with the manufacturer (R-DM). For the retailer the total cost curve us always the same for all policies. However, its order quantity is at its optimum (EOQ) when acting individually, and increases when collaborating with the DC, which results in higher total cost. All of the benefits and losses mentioned above are before allocating the profits among the firms. When firms collaborate the total supply chain cost always reduces (See Figure 21). 123

143 Figure 20. Cost curves for firms in supply chain. (Top:, Bottom: ) 124

144 Figure 21. Cost curves for the supply chain. (Top:, Bottom: ) 125

145 Therefore, all firms can benefit from collaboration by allocating the profits fairly. In the next section the total profits for the firms are discussed in detail for different policies and the effects of relationship between the holding costs of the firms is studied Total supply chain costs Based on the above discussions it can be concluded that the total supply chain cost is the highest for the independent policy (R-D-M) and is the lowest for the whole supply chain coordination (RDM). The costs for the partial joint policies are higher than the whole supply chain joint policy and lower than the independent policy. In the following sections the total profits for the firms in different policies are calculated using the optimal solution of Table 18. These firm profits are compared for different policies and similar conclusions are reached by comparing the profits directly Profits for firms in different coordination policies R-D-M (90) (91) (92) R-DM (93) 126

146 (94) (95) Comparing R-DM to R-D-M Using these formulas for firms profits and holding the independent pricing policy (R-D-M) as the benchmark, one can calculate changes in the firms profits compared to the benchmark. (96) (97) (98) Since, it can be proved that: (99) This means the total profit for the DC is always lower for the R-DM scenario than for the R- D-M scenario and the manufacturer s profit is always higher for R-DM than for R-D-M. It can 127

147 be proved that the total profit for the whole supply chain is always higher for the R-DM scenario than for the R-D-M scenario RD-M (100) (101) (102) Comparing RD-M to R-D-M (103) (104) (105) Again, since, it can be proved that: (106) This means the total profit for DC is always higher for the RD-M scenario than for the R-D- M scenario and the retailer s profit is always lower for RD-M than for R-D-M. It can also be 128

148 proved that the total profit for the whole supply chain is always higher for the RD-M scenario than for the R-D-M scenario RDM (107) (108) (109) Comparing RDM to R-D-M (110) (111) (112) Once more, since, it can be proved that: 129

149 (113) This means the total profit for the retailer is always lower for the RDM scenario than for the R-D-M scenario and the manufacturer s profit is always higher for RDM than for R-D-M. For the retailer the change in total profit is dependent on the relationship among the values for holding costs of the three firms. However, it can be proved that the total profit for the whole supply chain is always higher for the RDM scenario than for the R-D-M scenario Comparing RDM to R-DM (114) (115) (116) Once more, since, it can be proved that: (117) This means the total profit for the retailer is always lower for the RDM scenario than for the R-DM scenario and the DC s profit is always higher for RDM than for R-DM. However, it can 130

150 be proved that the total profit for the whole supply chain is always higher for the RDM scenario than for the R-DM scenario Comparing RDM to RD-M (118) (119) (120) Again, since, it can be proved that: (121) This means the total profit for the DC is always lower for the RDM scenario than for the RD- M scenario and the manufacturer s profit is always higher for RDM than for RD-M. It can be proved that the total profit for the whole supply chain is always higher for the RDM scenario than for the RD-M scenario Conclusion Based on the EOQ and joint replenishment models explained above, the following conclusions can be made: 131

151 a. It is always beneficial for the whole supply chain if the firms coordinate inventory system decision-making. b. Even for partial coordination, total profit for the supply chain is always higher when compared to individual replenishment policy. c. When all firms cooperate fully in decision-making for pricing and replenishment the profits are always higher than when they cooperate partially. d. In any two-firm coordination scenario, the upstream firm benefits from coordination while downstream firm is disadvantaged. However, the net amount of gain for the upstream firm is always higher than the loss for the downstream firm. Therefore the net benefit of the contract is positive. e. In the 3-firm coordination policy, the retailer always suffers and manufacturer always benefits from coordination. The middle firm, the DC, may suffer or benefit, depending on the ratios of the inventory holding costs of the firms. Nonetheless, the overall net benefit to the supply chain is positive when coordinating compared to individual policies. f. In any coordination scenario, firms should negotiate in order to share the extra profit gained from coordination. In other words, upstream firms could compensate for the downstream firms losses. The profit allocation is discussed in section of this chapter A numerical example: Consider a three-stage supply chain with the following parameters: D= 745 units per year P= 1000 units per day 132

152 Am= $100 per production Ad= $20 per order Ar= $10 per order cp= $0.1 per item cm= $1 per item cd= $1.5 per item = $2.82 per item hm= $ per item per day hd= $ per item per day hr= $0.02 per item per day The decision variables are: Q: Retailer s order quantity m: DC s lot-size multiplier. The DC orders units on each order. n: Manufacturer s lot-size multiplier. Each production run is units. Using the formulas presented, the optimal solution for each of the four policies can be found. Table 19 shows the summary of the optimal solutions. Table 19. Optimal solution for a supply chain with level demand Retailer DC Manufacturer Profit Q m n Retailer DC Manufacturer Supply Chain R-D-M $ $ $ $1, R-DM $ $ $ $1, RD-M $ $ $ $1, RDM $ $ $ $1,

153 It can be seen that the profit for the supply chain improves if the firms coordinate in the decision making process and the maximum profit is achieved when all three firms coordinate together. Figure 22 illustrates the changes in the amount of profit for different policies compared to the Independent pricing policy. As mentioned before, when firms coordinate with an upstream partner their profit drops and when they coordinate with a downstream partner their profit improves. However, the net profit for the supply chain is always better for the coordination scenario. In the next section two approaches for profit allocations are discussed. Figure 22. Comparison between the profits for different policies 134

154 Profit allocation Using the same two approaches explained in Chapter 7, the profits can be allocated for different policies. The notation for total firm profit in the following formulas is TPfirm,policy. The independent policy is denoted as i Proportion approach When profits of the joint policies are allocated proportional to the profits gained by firms when optimizing their inventory system independently, profit shares are as follow. Partial joint policies: (122) (123) (124) (125) Joint policy: (126) (127) (128) 135

155 Shapley value approach The Shapley value approach gives the following values for profit shares: Partial joint policies: (129) (130) (131) (132) Joint Policy: (133) (134) (135) Results These sharing approaches are used for the numerical example discussed in Table 19 for a level demand situation. Table 20 shows the results. First, it can be observed that, in any sharing approach the firms involved in cooperation gain extra profit from collaboration. Second, in partial joint policies the extra profit is divided equally between the participant firms when using the Shapley value approach. Finally, in the total joint policy where all firms are involved, the DC gains more of the extra profit. The intuition for this fair sharing can be explained by the DC s 136

156 role in shaping the cooperation. Without the DC, the collaboration could not be formed, as it is the link between the three firms. Therefore, its contribution in the collaboration and hence its share is more than (equal to the sum of) the other two firms. Table 20. Profit allocation in the multi-echelon supply chain for the level demand analytical example. Policy Profit Profit Share Gain by collaboration Sharing Approach Retailer Distributer Manufacturer Retailer Distributer Manufacturer R-D-M Individual $ $ $ No Collaboration Proportion $ $ $ $- $9.54 $15.68 R-DM Shapley Value $ $ $ $- $12.61 $12.61 Proportion $ $ $ $5.89 $1.34 $- RD-M Shapley Value $ $ $ $3.62 $3.62 $- Proportion $ $ $ $20.27 $4.61 $7.58 RDM Shapley Value $ $ $ $3.62 $16.23 $12.61 In the rest of this chapter the supply chain is simulated using Arena. The inventory system is optimized for different situations of price and stock-level dependencies of demand using OptQuest. 8.3 Simulation Arena discrete-event-simulation software was used to model the supply chain. Figure 23 illustrates the process flow for the supply chain. The dotted lines show the flow of information and solid lines show the flow of the material. As an example, there is one information line that goes from the retailer to the DC, indicating the purchase orders for the products. After the purchase order is processed at the DC, there is a material flow line from the DC to the retailer, indicating the shipment of the products from the DC to the retailer. 137

157 Supply Chain consisted of one retailer, one D.C., and one Manufacturer D.C Manufacturer Is there enough products in inventory to fill DC s demand? Record the Back order and wait till the inventory is filled Check the inventory position against the reorder point (R_DC) Record Back order Take the products from inventory and ship to D.C Order to Manufacturer (quantity = m*q) Start the production (Produce n*m*q) Record the sale and holding cost Wait for production setup time and till at least m*q is produced Wait for products to arrive (transportation + manufacturer preparation) Record setup cost and time Record the order cost and buying cost Is there enough products in inventory to fill retailer s demand? Take the products from inventory and ship to retailer Record the sale and holding cost Check the inventory position against the reorder point (R) Order to DC (quantity = Q) Wait for products to arrive (transportation + DC preparation) Record the order cost and buying cost Retailer Customer arrives Check the inventory at retailer Take from inventory Record the sale, holding cost and update demand rate Customer leaves the retailer Record the lost sale Figure 23. Process flow of the supply chain simulation 138

158 The process starts by the customer arriving at the retailer. If there is enough inventory at the retailer, the customer s demand is satisfied. The sale is recorded and the customer leaves the system. If, on the other hand, there is not enough inventory the customer leaves the system and the sale is lost. The inventory level at the retailer is constantly monitored and as soon as the inventory level is equal to reorder point an order is placed to the DC. The retailer receives the order after a transportation delay, provided that sufficient inventory to fill the order is available at the DC. The order and buying costs are recorded at this stage for retailer. When the order arrives at the DC, the inventory is checked against the order quantity. If there is enough inventory available the order is shipped and sale is recorded. If enough inventory is not available, a backorder is recorded and the order will be sent at a later time. The inventory level at the DC is also constantly monitored against the DC reorder point. When the inventory is less than or equal to the reorder point an order is placed to the manufacturer. The DC waits for the manufacturing process for m Q items to be completed and then the products are shipped (with a transportation time) to the DC. The ordering and item purchase costs are recorded at this stage for DC. When the order arrives at the manufacturer, the order is shipped and the sale is recorded if there is enough inventory available. If enough inventory is not available, a backorder is recorded and a production request is sent to the production plant to produce another batch. The production process starts, if there is no production already in process, and the setup time, setup cost and production time are calculated. As soon as enough items are produced to fill the DC demand (m Q), the items are shipped, the sale is recorded, and the rest of the production batch is stored in the manufacturer inventory. 139

159 In order to capture the behaviour of the system the inventories for all the firms in the supply chain are stored in Hold modules in Arena, rather than just calculating the inventory levels by mathematical formulations. This method enables capturing the holding cost and flow time for each entity in the system. In other words as soon as an item (entity) enters the inventory (Hold module) a timer records its entrance time. When the item leaves the inventory to be sold to the other parties in the system, the timer is stopped and total duration of holding and holding cost are calculated. That means inventory levels are calculated as a time-persistent statistic and holding cost is calculated by integrating the inventory level over time. This method would also support studying special products with limited shelf life. In the following paragraphs more detail is provided for the variables and processes in the simulation. The time unit for the simulation is days and it is assumed that the firms work 365 days each year. Costs and profits are reported annually. The underlying SIMAN code for the model is provided in Appendix B Customer demand It is assumed that each customer needs only one item and that customer arrival is a Poisson process with inter-arrival time following a negative exponential distribution. However the mean customer inter-arrival time is dictated by the price- and stock- sensitive function that follows the same formula as mentioned before. (136) In this formula changing and results in different levels of stock dependency and changing b results in different levels of price dependency. If either or are zero, the demand function would be price sensitive only. If either or are zero and b is zero as well, demand 140

160 would be a level demand equal to a. These conditions are used for testing the first two propositions of Chapter 3 and performing sensitivity analyses Retailer Items are sold at the retailer. As mentioned earlier, the retailer s inventory is replenished from the DC inventory through a reorder point policy. Inventory position will be reviewed whenever the inventory changes and an order will be initiated if the inventory position is equal to the defined reorder point. Therefore the model is a continuous review, order-point, orderquantity (R, Q) system. Inventory position is calculated by the following formula (Silver et al., 1998): (137) A stochastic transportation-time is assumed between the order initiation and inventory replenishment. The transportation time from the DC to the retailer is assumed to follow a Triangular distribution. The most probable time for transporters to deliver the product to retailer is one day, the lowest possible time is half a day and the latest that the retailer receives the product is in two days. This distribution is defined by a triangular distribution in Arena as TRIA (0.5,1,2) Distribution Centre (DC) The inventory system in the DC, similar to that of the retailer, follows a continuous review, reorder-point, order-quantity (R, Q) system. The reorder point is based on the inventory position, which is defined similar to that of the retailer. The order quantity for the DC is a multiple of the retailer order quantity (mq), where m is a decision variable for the DC. The transportation time from the manufacturer to the DC is also assumed to follow a Triangular distribution. The most probable time for transporters to deliver the product to DC is 5 days, the lowest possible time is 2 141

161 days and the latest that DC receives the products is in 10 days. This distribution is defined in Arena as TRIA(2,5,10) Manufacturing plant Once the manufacturer receives the order, demand will be filled immediately if inventory is available. Otherwise, the order will be backordered and production process will be initiated. The manufacturer produces a multiple of the DC order quantity in each production run (nmq), where the multiplication factor, n, is a decision variable to be optimized. However, as soon as enough items are produced to fill the ordered quantity (mq), items will be shipped to DC and the remaining items go into the manufacturer s inventory item by item. These items are shipped to the DC in batches of mq units, upon order. There is a constant production rate assumed for the manufacturer, which is larger than the demand rate ( ). This is a common assumption in inventory literature (Silver et al., 1998). It is an essential assumption in capacity constrained systems if all demand is to be satisfied, otherwise the system is unstable. A setup is required before each production run which incorporates setup cost and setup time. Setup cost is assumed to be constant for each run and the setup time is assumed to follow a gamma distribution. These are commonly used assumptions in the literature (Grewal et al., 2015 Silver et al., 1998). The mean and coefficient of variance parameters are used for the Gamma distribution in Arena. A Gamma(, ) distribution results in non-negative values and is defined by the following probability density function (Law & Kelton, 2007): (138) 142

162 where, the Gamma function is. In this is shape function and is scale function. The mean and variance for the gamma distribution are: (139) In this research the Gamma distribution defined in Arena as GAMMA(0.5,1) days. That is, the mean and variance for the gamma distribution are 0.5 and 0.25 respectively. There is a single process that products go through in the manufacturer s plant. If the resource is busy, orders in queue will be processed in first-in-first-out (FIFO) order Performance measures Total profit of the supply chain and customers service level are used in this research as performance measures. Profit is used by all of the researchers assuming price- and/or stockdependent demand assumptions (Urban, 2005a). Service level is commonly used in the general supply chain literature (see literature review in Grewal (2012)). However, using service level is not common in the literature with the price- and stock-dependent demand assumptions. One of the reasons is that the stock dependency of demand may make it desirable to order large quantities, resulting in stock remaining at the end of the cycle, due to the potential profits resulting from the increased demand (Urban, 1992). The unsold stock results in no lost sales and 100% service levels. This is especially true for the linear demand functions, which do not consider diminishing returns. IN this situation the service level is not a differentiator in performance. A few papers address this issue by imposing a ceiling for the inventory level (Teng et al., 2011a). However, in this research the exponential stock-dependent demand curve flattens 143

163 out very fast for larger saturation factors ( ), meaning that, after a specific point, demand will not increase due to higher stock levels. There are some other papers in the field that assumed penalty costs for backorders and lost sales in order to consider the influence of service level in the optimization objective function (Dye & Hsieh, 2011). In this research this later approach is used to address the service level in the profit function. However, for more clarity, the service level will be calculated using fill-rate formula, Equation (142), as well. a) Total profit The total profit is defined in this research as the sum of the retailer s, DC s, and manufacturer s profits. The profit for each of the firms is defined as: (140) The lost sales penalty cost is assumed to be equal to the opportunity cost of lost profit on the sale of the item: (141) b) Service level The customer service level is calculated by the following formula, known as the fill-rate (Silver et al., 1998). (142) 144

164 8.3.6 Simulation parameters The main parameters related to the simulation process used in this supply chain model are the warm-up period, the run length, and number of replications. As shown in Figure 24 the simulation reaches an acceptable steady-state around 1500 days. To be on the safe side, the warm-up period is set to six year (2190 days) and the simulation is run for an additional 20 years (7300 days) in Arena. In total the simulation run length is 9490 days. Figure 24. System reaches its steady state after about 1500 days The number of simulation replications in the optimization process in OptQuest was set to be between 3-6 replications for each search point. The search number of simulation-runs to find the optimal solution was set to automatic. However, the usual number of simulations searched was in the range of simulations. The termination tolerance was set to After termination, the top 5 solutions were each run for another 10 replications in Arena, without using OptQuest, in order to make sure the best solution is chosen. Common random numbers 145

165 were used to reduce the within-group variance. Table 21 shows the formulas used for each of the stochastic values in the model. Table 21. Probability distributions for stochastic variables in the model. Parameter Distribution Stream Number Arrival Rate Expo(Demand_rate) 1 Transportation time to retailer TRIA(0.5,1,2) 11 Manufacturer s setup time GAMMA(0.5,1) 12 Transportation time to DC TRIA(2,5,10) Validating the model Before using the simulation model for the optimization process, it should be verified and validated. Model verification is defined as ensuring that the computer program of the computerized model and its implementation are correct. Model validation is defined as substantiation that a model within its domain of applicability possesses a satisfactory range of accuracy consistent with the intended application of the model (Sargent, 2013). The simulation model development process and the required verifications and validations are illustrated in Figure 25. It must be noted that the process of developing a valid simulation model is an iterative process of the following steps. After developing a conceptual model it must be validated. Next the computerized model is developed and verified. At the end the operational validity is performed on the computerized model. The model changes during each of the verification and validation steps until a valid simulation model is prepared (Sargent, 2013). Data validity, being very difficult, time consuming, and costly, is often not considered as part of the model validation (Sargent, 2013). However, in this research data validity is done by an empirical analysis performed on the Dominick s chain store data. Values for the parameters of the simulation model are estimated using the statistical results from the empirical analysis. 146

166 Figure 25. Model development process, Adopted from Sargent (2013) There are several validation and verification techniques available and used the literature. Below a list of different techniques that are used in this research to validate the simulation model are presented and in some cases an example of the validation test is provided. Each technique is presented under the subtitle of the type of validity it was to test Conceptual model validation Conceptual model validity indicates the correctness of the theories and assumptions of the conceptual model. It also evaluates whether the model s mathematical, logical, and causal relationships are reasonable for the purpose of the simulation (Sargent, 2013). 147

167 The main validation techniques used for conceptual validity are: Face Validation For face validity, individuals knowledgeable about the system being simulated are asked whether the model is logical or input/output relations are reasonable. To ensure the face validity, the model in this research was reviewed by the members of the supervisory committee. Another resource for face validity is related literature, which is addressed in the Chapter 2. Specifically inventory system models, such as the reorder point- order quantity (r, Q) model and queuing theory were referred to. Distributions for the arrival process, production process time, setup time, and transportation time were discussed and chosen to ensure the face validity. The overall face validity of the model can be evaluated by following the conceptual process flow presented in Figure 23. Trace An entity in the simulation is traced (followed) throughout the processes in the model to ensure the logic is correct and the necessary accuracy is obtained. Another trace method used in this research is using the break points tool available in Arena. Using this tool, the outputs from critical processes in the simulation were monitored whenever an entity passed through them Computerized model verification Computer programming and implementation of the conceptual model was tested using computerized model verifications (Sargent, 2013). Using a special-purpose simulation language, such as Arena, results in having fewer errors than if a general purpose or high-level generalpurpose language, such as C# or C++, is used (Sargent, 2013). The other advantage of the computerized model verification process is that the inefficiency and/or complexity of the specific 148

168 modelling methods will surface and the model can be improved in order to reduce the computational endeavour and time. The main techniques used for computer model verification are structure walkthrough and trace, which was explained in last section Operational validation Whether the simulation model s outputs follow the necessary logic and have the required accuracy for the intended purpose of the model over its applicable domain is tested using operational validation (Sargent, 2013). This is basically the last and most important validation process, as it tests whether the model is delivering what it is supposed to deliver. The validation techniques used in this stage are: Animation and operational graphs Animation is one of the features available in Arena that is useful for tracing entities in the simulation. Operational graphs are useful to capture the dynamic behaviour of the model to ensure that the model is behaving logically (Sargent, 2013). For example, Figure 26 demonstrates inventory levels for different firms in the supply chain when the demand is assumed constant, reorder points are set to zero, and there is no transportation lead time. This figure is used to verify that the model is valid compared to theoretical models for inventory systems. The comparison verification is explained in the next section. 149

169 Figure 26. Inventory levels for different firms in supply chain, used to validate the model Comparison to other models Results and behaviour of the model is compared to other valid models. As an example, in this research, the results of a simple inventory case are compared to the results of a known analytical model. The simple model for annual profit of an inventory system with constant demand D=365 items per year (365 days), order cost of A=$10 per order, holding cost of h=$0.01 per item per day, selling price of p=$8 per item and buying cost of c=$5 per item that orders Q=10 items whenever the inventory level reaches zero is calculated by (it is a transformed profit model based on the cost formula in (Silver et al., 1998)): (143) 150

170 Using the simulation model the same result is obtained for the retailer, as shown in Table 22 below. The minor differences observed are due to the simulation initial conditions. In the simulation model the inventory level at time zero is equal to the order quantity. The Min and Max values shown in this table are basically the first and last values recorded for each of the variables in any of the replications in Arena and the Average of these two are reported as the average value of the variable for each 20 years. However, what should be considered as the average annual revenue, cost, or profit to compare with the theoretical results is the maximum values for 20 years divided by 20. These values are reported in the last column of the table. Table 22. Arena report for the simple inventory model Parameter Results for 20 years Average for Average Min Max one year Revenue $29,204.0 $8.0 $58,400.0 $2,920 Buying Cost $18,225.0 $- $36,450.0 $1,822.5 Holding Cost $200.7 $0.0 $401.4 $20.1 Order Cost $3,655.0 $10.0 $7,300.0 $365 Profit $7,128.3 $8.0 $14,248.6 $712.4 Data relationship correctness This type of validation requires results or data generated by model to follow correct relationships among different parameters (Sargent, 2013). For example it is expected that the dollar amount for the retailer buying cost match to that of the DC s selling revenue. Table 18 verifies that these numbers are balanced. Internal validity Several replications (runs) of a stochastic model are made to determine the amount of (internal) stochastic variability in the model (Sargent, 2013). If this variability is very large, it 151

171 may signal an invalid or questionable model of the system being investigated. The internal validity of the model is discussed later in the results and conclusions section of this chapter. Parameter variability-sensitivity analysis The extent of effects of individual parameters of the model on the outputs can be tested by changing the input variables within a range and studying the direction and amount of the change in the outputs. This method is a common method used in decision-making. The parameters that the model is very sensitive to should be made sufficiently accurate prior to using the model. Sensitivity analysis of the parameters in this research is discussed in detail in section Optimization Optimization processes are defined in OptQuest based on the simulation model. OptQuest is integrated with Arena and uses search-based heuristics to find optimal solutions. Setting up the problem in OptQuest is done by defining the decision variables (e.g. price set, reorder points, lot sizes, and production size), objective function (maximizing profit) and constraints. The process of defining decision variables, objective function, and constraints in OptQust is by selecting the variables from a list of available variables previously defined in the Arena model. Therefore, when building the model in Arena, all of these variables must be included. The procedure to setup the optimization problem in OptQuest is as follows: 1- Specify the decision variables. These are called controls in OptQuest and are selected from the list of variables defined in the Arena model. 2- Specify lower, suggested, and upper values for these controls. Selecting these values is necessary to define the search space. Wise choices improve the search speed and accuracy by limiting the search space. It is a challenge to select suitable bounds for decision variables. The method that is used in this research 152

172 is to initially select a wide range and then to tighten it based on the initial results. If a solution ended up with a decision variable at or near a bound, that bound was relaxed. 3- Identify the responses, such as profits and service level. These are the outputs of interest and may be used to define the objective function and constraints. 4- Define any constraints besides the search boundary constraints for the decision variables. 5- Set the objective function. Here it is to maximize the profit for each firm in the individual policies and the sum of the profits for the joint policies. These objectives are selected from the variables previously defined for statistical analysis in Arena. 6- Define the options for the optimization procedure, such as number of simulation replications for each set of decision variables or the precision of the search method. These are used as exit flags for the search process. 7- Run the optimum seeking process. As the methods used by OptQuest are heuristics, the final solution may not be optimal but should be very close. Increasing the number of simulations in the research and tightening the tolerances on the stopping rules increases the chance of finding the optimal decision variables. Comparing the top solution with the four next best solutions using multiple replications further improves the chance of identifying the optimal, as well as allowing confidence intervals on objective function values to be constructed. 153

173 Figure 27 shows the OptQuest windows while running a search. The left side of this window is the control panel, the larger central part shows the search process and improvements in the objective function, and the upper part shows the decision variables values for sequential experiments in the search. The optimization process was performed for each of the experimental design settings discussed below. Figure 27. OptQuest while optimizing is in process Optimization formulation The optimization problem must be formulated before conducting the simulation-based optimization in OptQuest. The equation below shows the generic form of the optimization formula: 154

174 (144) subject to: (145) where TP R,TP DC, and TP M are total profits for the retailer, DC, and manufacturer, respectively. The total profits are formulated following Equation (140). is the final price to the customer and pr DC, and pr M are the internal prices in the supply chain selling to the retailer and DC respectively. Q is the order quantity at the retailer and m and n are batch sizes at the DC and manufacturer. Finally, R R, and RDC are the reorder points at the retailer and DC. Using OptQuest, the integer constraint for all the variables was relaxed in order to speed up the search. Then the integers immediately above and below the best value found were used in simulation with Arena to select the best integer solution Assumptions Besides the assumptions and parameters introduced in previous chapters, the following assumptions were made for the simulation and optimization models. a. The supply chain operates 24 hours a day, with 365 days in a year. b. Raw material is always available at the manufacturing plant. c. There are no resource breakdowns or failures. d. Customer demand that cannot be filled immediately is lost. e. Demands that cannot be filled at the DC and manufacturer will be backordered and shipped when stock becomes available. f. The sequence of processing orders and inventory are first in first out (FIFO). 155

175 g. Transportation system capacity is unlimited. h. In independent policies with price sensitive demand, only the retailer decides the final price and the intermediate prices will remain constant. If this assumption were not made, the optimization would be trapped in an endless loop Pricing and replenishment scenarios There were three pricing scenarios considered in the simulation-based optimization process. a) Independent pricing and replenishment policy In this scenario the performance measure for the retailer was maximized using the sales price ( ), reorder point (R), and order quantity (Q) as decision variables. Then, by fixing these variables the objective performance for the DC was maximized by use of the reorder point ( ) and order quantity (the factor m) as decision variables. Finally, the manufacturer performance was optimized by deciding on the production-batch size (the factor n). Note that in this scenario the DC s and manufacturer s price were assumed to be known values. b) Joint-pricing and replenishment policy In this scenario the performance measures for all firms was maximized simultaneously by deciding on the sales price ( ), reorder points (R and ), order quantity (Q and mq), and production-batch size (nmq). c) Partial joint-pricing and replenishment policy Two partial joint policy configurations were assumed in order to test proposition 2b, stated in Chapter 3. a. Independent retailer, joint DC and manufacturer (R-DM in experimental design below). 156

176 b. Joint retailer and DC and independent manufacturer (RD-M in experimental design below) Design of experiments and statistical analysis In order to test the propositions, the experimental design consisted of two factors. As shown in Table 23, the first factor was the coordination policy, run at 4 levels. These policies were the independent policy (R-D-M), independent retailer and joint DC and manufacturer (R-DM), joint retailer and DC and independent manufacturer (RD-M), and completely joint policy (RDM). The second factor was the demand function, also run at 4 levels. These were level demand (LD), stock-dependent demand (PDD), stock-dependent demand (SDD), and price- and stockdependent demand (P&SDD). Therefore, a full factorial design resulted in 16 combinations of settings. These were evaluated using the simulation- optimization procedure previously described. Performance results from the final stage simulation experiment were then analyzed using the analysis of variance (ANOVA) techniques. Table 23. Experimental design for multi-echelon supply chain performance. Factor Levels # of levels Proposition tested Policy Demand function Full factorial combinations: R-D-M, R-DM, RD-M, RDM LD, PDD, SDD, P&SDD 4 Proposition 2a, b 4 Proposition 1a, b 16 In order to study different policies for each demand function, the simulation model was set to one of the demand functions and then the optimization process is performed for each of the policies. As an example, to find the optimal parameters for the RD-M policy in a supply chain 157

177 with a price-dependent demand, the demand function in the simulation was defined in a way that it is not dependent on stock level. That is in, which is. The optimization problem was solved by first maximizing the total profit of the retailer and DC (i.e. Maximize ) by means of the final price ( ), DC s selling price to retailer ( ), order quantity (Q), reorder point for retailer ( ), order quantity at DC (m), and reorder point for DC ( ). After the optimal solution was found for this problem, these parameters were set in the Arena simulation model. Then the next step of optimization was performed for manufacturer using OptQuest i.e.,, to maximize the profit for the manufacturer by means of manufacturer production batch size (n). It must be noted that in this scenario, if the manufacturer gets to optimize its profit by including its selling price to DC ( ) as a decision variable, the optimal solution for the DC and retailer would change and they would have to reoptimize their profit. This would lead to an infinite loop of optimizations. After all optimizations were performed, the best solution was chosen from the top 5 OptQuest solutions by comparing their simulation-only results based on in Arena. The optimal parameters of the best solution were then set in the simulation and the model was run one more time for 20 replications each equal to 20 years of supply chain operation. Then the mean and standard deviations of the average annual profits and service levels for each replication were computed. In order to study the effect of using a specific demand function as the basis of decision making in the supply chain the following method is used. When the optimal solution for each demand function was finalized, the simulation model in Arena was set to have the price- and 158

178 stock-dependent demand, called Actual Demand. Then the optimal parameters of each of the demand functions and pricing policies were used to simulate the system for another 20 replications and the profits were compared. In other words, the actual demand is price- and stock-dependent but the firms may assume the dependency of demand on price, inventory level, both, or none. The firms assumption about demand is called Assumed Demand. The supply chain performances for these situations were then compared (See Figure 28). This supported testing the first and second propositions about the importance of considering price and inventory level dependency of demand in choosing an optimal replenishment and pricing policy. In order to show the effect of price dependency of demand in Figure 28, it is assumed that the retailer discounts the product in the middle of the year. Figure 28. Different demand function assumptions for a price and inventory level dependent demand 159

179 8.4.5 Experimental model parameters The parameters for the demand function were taken from one of the products in the Dominick s database. The product that was used is Soap, which demonstrated an acceptable match with the exponential price and stock-dependent demand formula proposed in this research. Based on the data, the average price for this product was $2.82. The fitted exponential function for price-dependent demand, shown in Table 5 was per week. Fitting the exponential function of inventory level dependency of demand to the actual demand of this product the following formula was obtained for the daily demand The rest of the variables are as follow: P= 1000 units per day Am= $200 per production Ad= $20 per order Ar= $10 per order cp= $0.1 per item cm= $1 per item (when not a decision variable) cd= $1.5 per item (when not a decision variable) = $2.82 per item (when not a decision variable) hm= $ per item per day hd= $ per item per day hr= $0.02 per item per day 160

180 8.5 Results Comparing different replenishment policies under each demand function Table 24 shows the summary of the optimal decision variables along with the profits for firms in the supply chain. As can be seen in the table, for each demand function the lowest profit for the supply chain is that of the individual policy (R-D-M). The partial joint policies (R-DM and RD-M) always resulted in better supply chain profits while the whole supply chain joint policy (RDM) resulted in the best total profit. Between the two partial coordination policies, there is no consistent advantage for one over the other. The service level for all scenarios is higher than 95% and under stock-dependent demand situations it is always 100%. This is due to the fact that there is a penalty cost associated with lost sales in the model. In the case of stock-dependent demand where increasing inventory levels would result in higher demands and higher revenues, it is natural to find higher service levels. In order to check whether the differences between the average profits are statistically significant, one-way ANOVA in SPSS was used. As mentioned earlier, to use ANOVA one should make sure of the homogeneity of variances. Although, these results are from the simulation in Arena and there should be no correlations among the replications, Levene s test was performed. As shown in Figure 29 for each demand function the variances of different policies are not significantly different (high Sig. value), which means the variances are homogenous. Figure 30 - Figure 33 show the ANOVA results for each demand function. 161

181 Table 24. Summary results of the simulation-based optimization 162

182 Figure 29. Test of homogeneity of variances. Figure 30. One-way ANOVA for the effect of coordination policy for a supply chain with level demand. 163

183 Figure 31. One-way ANOVA for the effect of coordination policy for a supply chain with price-dependent demand. Figure 32. One-way ANOVA for the effect of coordination policy for a supply chain with stock-dependent demand 164

184 Figure 33. One-way ANOVA for the effect of coordination policy for a supply chain with price and stockdependent demand. As seen in these tables, there are significant differences between the profits of the different coordination policies. These tables support the propositions 2a and 2b: Proposition 2a: Joint pricing and replenishment policies result in higher total profits for the supply chain than policies with independent pricing and replenishment strategies. Proposition 2b: In a multi-echelon supply chain the pricing and replenishment policy that considers all of the stages of supply chain results in more effective inventory systems than partial joint pricing and replenishment policies. It must be noted that these ANOVA analyses indicate that there is a significant difference among different policies. However, they do not compare policies one to one. To compare each 165

185 pair of policies, one can do ANOVA analysis for each pair. Instead of taking a multiple comparison approach, box and whisker plots with error bars indicating the 95% confidence intervals for the means are used to visually detect mean profit differences. Figure 34 shows the different policies in different demand function situations and their 95% confidence intervals. Figure 34. Comparing different policies. As seen in this figure, there is a significant difference between the joint policy (RDM) and independent policy (R-D-M) in all cases. Except for RD-M in level demand and stock-dependent demand situations, all partial joint policies result in significantly higher profits than those of the individual policies (R-D-M). Except for R-DM in level demand and both partial policies in price-dependent demand, the joint policy (RDM) performs better than the partial joint policies (R-DM and RD-M). Therefore, it can be concluded that in most situations propositions 2a and 2b are true. It is also important to note that the average profit for the RDM was the best in every situation. 166

186 Looking at changes in profits for individual firms, it can be seen that when firms collaborate in pricing and replenishment, upstream firms benefit and downstream firms suffer, regardless of whether partial or complete joint coordination is used. Figure 35 illustrates two examples of profit changes for firms, one with a price-dependent demand function case and the other with price and stock-dependent demand. Figure 35. Changes in profits for firms in different replenishment policies One of the questions that was mentioned in the literature survey is To what extent should inventory be pushed forward in the distribution channels to respond to ILDD [inventory level dependent demand] patterns? (Urban, 2005b) Comparing the situations with ILDD (i.e. SDD and P&SDD) to others that are not inventory dependent (i.e. LD and PDD), it can be seen that dependency of demand on the inventory level mostly affects the retailer and this effect is dampened, to some extent, from propagating upstream in the distribution channel. In fact the inventory levels at the manufacturer in stockdependent scenarios is not any higher than those of other demand functions. In these situations the retailer, by pushing the reorder point higher, tries to increase its demand (and its revenue). Inventory levels at the DC are affected by increased order sizes from retailer, but the dependency of demand on inventory level does not affect the inventory levels at the manufacturer. Figure 36 shows this effect both as the change in average inventory levels of firms in different demand 167

187 function situations and as the ratio of increased inventory levels compared to the level demand situation. Figure 36. Effect of stock dependency of demand on firms inventory levels Comparing different demand function assumptions In order to compare different demand functions, the Process Analyzer in Arena was used. Figure 37 shows a snap shot of this software. Using this software one can define different scenarios by changing input values and then compare the outputs statistically. In this research 16 different scenarios were defined. These were based on the previous design of experiments, with the input values being the optimal parameters for each policy and demand function, shown in Table 24. These optimal policies were used to run the model with price and stock-dependent demand to study the effect of the demand function assumption on the performance measures. Therefore it must be noted that the results shown in Table 25 are not based on re-optimizing the decision variables for different scenarios. They are results of re-running the model for P&SDD scenario in Arena only, using the exact values for decision variables from Table

188 Figure 37. Process Analyzer window showing different scenarios Table 25 shows the results of the process analysis for average profit and service level for the whole simulation time. Service levels were always above 99%. This means that assuming any of the demand functions and optimizing the system will result in a very high service level. As mentioned earlier this is due to the fact that there is a cost associated with losing sales. Figure 38 shows the differences in average profits of supply chains for different demand function assumptions. It shows that if the demand is actually price and stock-dependent (P&SDD), there is a significant benefit in considering both of these factors in the demand function. 169

189 Table 25. Results of comparing different demand functions for a supply chain with price and stock-dependent demand In the case of an actual price and stock-dependent demand, even if firms only consider the dependency of demand on price or on inventory levels, they would benefit considerably more than if they assume that demand is constant (Compare LD to PDD and LD to SDD in Figure 38). The figure above only shows the average profits. In order to differentiate between different scenarios with 95% confidence, Figure 39 is used. It must be noted that since the scale of the difference between profits is very high, it is very difficult to see the 95% confidence interval bars. 170

190 Figure 38. Effect of different demand functions for a supply chain with price and stock-dependent demand. Figure 39. Average profit and 95% CI for different demand functions 171

191 All of the above tables and figures justify propositions 1a and 1b: Proposition 1a: Considering price- and stock-dependency of demand significantly affect the performance of the inventory models. Proposition 1b: Considering both price dependency and stock dependency of demand results in more effective inventory systems compared to models that incorporate only one of the factors. The conclusion from sections and is that if demand is dependent on price or inventory level, then considering the dependency in the model will benefit the supply chain, especially when firms jointly decide in replenishment parameters. If firms find demand is dependent on either price or stock level, they should investigate dependency of demand on the other factor as well, since if demand is dependent on both price and stock level, considering both will benefit the supply chain significantly A more practical solution Although the results shown in Table 24 and Table 25 are the optimal solutions, there are situations in which the profit for one or two of the firms is negative. For example in the case of individual policy (R-D-M) with price and stock depended demand (P&SDD) the retailer maximizes its profit by deciding on price and order quantity in a way that there won t be any room for profitability of the DC. Hence, the DC is not profitable, while the retailer s profit is very high. In such a case the DC would not accept to be part of this supply chain. On the other hand, when the firms collaborate in a joint policy (RDM) the retailer loses money and the manufacturer s profit is very high. This is the most profitable situation for the supply chain as a 172

192 whole. Even if the profits are allocated using any of the two approaches explained before, one of the firms (the DC) will not be profitable. In order to demonstrate a viable solution to this impracticality, a set of constraints were introduced to the optimization process to force positive profits for all firms in the solution for P&SDD scenario. These constraints have the following form and were added to the optimization constraints shown in (145). (146) Using these new constraints the simulation-optimization was redone for all four policies (R- D-M, R-DM, RD-M, and RDM) in the P&SDD scenario. The results shown in Table 26 are the average profits for different policies, with the new constraints. Table 26. Profits for P&SDD scenario with positive individual firm s profit constraint Comparing Table 26 to Table 25, one should note that first, applying the constraints of equation (146) didn t change the combinations of profits for the firms and the total supply chain profit for the RD-M policy. This policy had already resulted in positive profits for all the firms prior to imposing the new constraint. Second, the total supply chain profit for R-D-M and R-DM policies are higher for the constrained solution compared to the unconstrained model. The reason is that these policies need sequential optimization for the retailer first and then for the DC and the manufacturer and the constraints are added to each step. In the unconstrained model, the fact 173

193 that the retailer maximized its profit first meant that the rest of the firms in the problem have less solutions space and the overall supply chain profit was poorer. Where as in the constrained, the fact that the retailer has to leave some space for other firms to gain positive profits increased the solution space for the other steps and therefore the total supply chain profit was higher. As can be seen the profit for the retailer for these policies in the constrained models (Table 26) are lower than those of the unconstrained models (Table 25). Finally, as the RDM policy is a one-step optimization, the total supply chain profit is lower in constrained model than that of the unconstrained model. A managerial insight for this phenomenon is the emphasis on collaboration. That is when the firms act individually and optimize their profit the retailer maximizes its profit without considering possible losses to other firms in the supply chain. The non-negative profit constraint can be assumed as a surrogate for collaboration, either in the form of negotiation or sharing information. In this case, the retailer has to give up part of its profit to the benefit of other firms in the supply chain, which finally results in total higher profits for the whole supply chain. On the other hand when firms are collaborating (RDM), they don t need the non-negative profit constraint, if they trust this collaboration. Part of the collaboration is the profit allocation of the higher total supply chain profits. In other words, they may accept the temporary losses (negative individual profits) knowing that the higher profits of the whole supply chain will be allocated in a way that they all will benefit from collaboration. The profit allocation is discussed in the next section Profit allocation If the two approaches to profit allocation, using equations (129)-(135), are applied to the results shown in Table 26, The extra profits obtained by collaboration compared to individual 174

194 policy are shared among the firms fairly. The profits after allocation are always higher or equal to each firm s profit gained by individual policy. Table 27 shows the results of profit allocation. Using proportion approach, firms share the profits proportional to their profits when making decisions individually. However, using Shapley value approach the profits are shared relative to the extra profits added to the whole supply chain when each of the firms joined the cooperation. Table 27. Profit allocation using Shapley value method in price and inventory level dependent demand scenario Sensitivity analysis The conclusions made in the previous pages are all based on the use of certain values for the parameters. The next question to answer is to what extent the conclusions are dependent on different model inputs. Sensitivity analysis is used to answer this question. One method that is used commonly for sensitivity analysis is to change one parameter at a time and study the effect of that parameter on the optimal solution. This method is used for the following parameters. P: Production rate of the manufacturer, in items per unit time Am: Manufacturer s setup cost for a production run Ad: Distributing centre ordering cost for any order Ar: Retailer s ordering cost for any order 175

195 hm: Inventory holding cost per item per unit time for the manufacturer hd: Inventory holding cost per item per unit time for the DC hr: Inventory holding cost per item per unit time for the retailer Sensitivity analysis was performed on the supply chain model with price- and stockdependent demand that uses the joint pricing policy (RDM policy in P&SDD). Decision variables were kept constant at the optimal level. Using Process Analyzer in Arena each of the above mentioned parameters was changed (increased and decreased up to 25%). Changes in profit and service level were recorded and illustrated on different charts. As shown in Figure 40, profit sensitivity to the production rate is very low in general. The DC is most influenced by the production rate. As the production rate increases, items arrive at the DC more quickly, and since the re-order point was kept constant, that would result in higher inventory on average, which reduces the profit. The same phenomenon happens at the manufacturer. Figure 40. Profit sensitivity to production rate. 176

196 As expected the sensitivity of profit to order and setup costs is inverse. As the order cost increases the profit decreases. Supply chain profit is more sensitive to setup cost. This could be due the fact that the setup cost is an order of magnitude larger than the retailer s and DC s order cost. Figure 41. Profit sensitivity to retailer s order cost. Figure 42. Profit sensitivity to DC s order cost. 177

197 Figure 43. Profit sensitivity to manufacturer s setup cost. Holding costs, especially those of the retailer and DC, are major factors in determining the profit and EOQ for the supply chain in this research. Hence, as shown in Figure 44 - Figure 46, profits are very sensitive to holding costs. The retailer s holding cost has, by far, the most influence on profit. The trade-off between holding cost and order cost is another important aspect of EOQ models. In the case of stock-dependent demand, there is also a trade-off between the cost of keeping more inventory and increasing demand. Finally, the trade-off between the cost of keeping more inventory and the penalty cost of losing sales is another important factor in the model. Thus, there are several benefits to lowering holding costs in price and stock-dependent model. As shown in Figure 44, 25% reduction in the retailer s holding cost would increase the total supply chain profit by more than 30%. 178

198 Figure 44. Profit sensitivity to retailer s holding cost. Figure 45. Profit sensitivity to DC s holding cost. 179

199 Figure 46. Profit sensitivity to manufacturer s holding cost. The same sensitivity analysis approach can be used for the parameters of the demand function. However, as discussed by Urban (2005a) the relationships among these parameters must be considered when analyzing the sensitivity. Urban (2005a) argues that as the usual method for capturing these variables is using regression analysis (which is also true for this thesis), changing one parameter would change the other demand function parameters to keep the square errors of the regression formula at their minimum. Therefore, if is used in estimating the stock-dependent demand then while using for sensitivity analysis, the following value should be used for, considering the average demand and inventory level. (147) where and are the average demand and inventory levels. Furthermore, when using for sensitivity analysis, should be set to: 180

200 (148) The same idea is used when changing a and b for the price dependency function : (149) and (150) where is the average selling price to customers. In the figures below the sensitivity of profits to each of these parameters are shown, once without considering the relationships above and once with considering the relationships. It is obvious that if these relationships are not considered, profits are very sensitive to changes in any of these parameters. However, if considered, the profits are quite insensitive to the parameters of demand function. In the following figures the graphs on the left side are sensitivity analyses without considering the relationships between demand function parameters. The graphs on the right side are sensitivity analysis with considering the relationships. Figure 47 shows the two types of sensitivities to parameter a in the demand function. If relation between a and b is not considered, reducing a would reduce the profit dramatically, as a is directly related to average demand. In the case where the relation between a and b is considered, the average demand would not change by changing only a, as the price dependency 181

201 is adjusted to keep the average demand fixed. In this case profits are not sensitive to changes in a. The small variations in profits observed in this case are believed to be due to numerical errors in simulations. Figure 47. Profit sensitivity to parameter a in demand function. (Left: relations not considered, Right: relations considered) The price dependency parameter b has the most influence on the profit, if changes are considered to be not correlated to changes in a. If demand is less dependent on price, the difference between the current optimal price of $5.84 will not contribute to a huge drop in demand and revenue, so the profits increase. If demand is more sensitive to price, the difference between the current price and the average price of $2.82 will result in a significant demand reduction and profit loss. Again, the small variations in profits observed in the case where the relationship between a and b is considered is believed to be due to numerical errors in the simulations. 182

202 Figure 48. Profit sensitivity to parameter b in demand function. (Left: relations not considered, Right: relations considered) The parameters of demand price dependency, and, are other influential factors in the optimal profit, if considered independently. That is, if the demand is not so very dependent on the inventory level, then the high order quantity and reorder point do not contribute to attracting more demand and therefore the current values of Q=R=200 are not optimal. Therefore, reducing the dependency on demand (reducing or ) would reduce the profit dramatically. However, if the interrelation of and is considered, increasing one would reduce the other to keep the average demand fixed. Therefore, changes in one of them will not affect the profit (see Figure 49 and Figure 50). Figure 49. Profit sensitivity to parameter alpha in demand function. (Left: relations not considered, Right: relations considered) 183

203 Figure 50. Profit sensitivity to parameter beta in demand function. (Left: relations not considered, Right: relations considered) 184

204 185

205 Chapter Nine: Summary and Conclusions 9.1 Overview In this thesis, first, using data made available from a chain grocery store, the price and inventory level dependency of demand are studied. The statistical analyses demonstrate that for some products the demand function defined as is better than a linear function. Correlation analyses also show that there are strong and significant positive correlations between the number of UPCs at the store and demand. These results are used to justify that demand for some products depends on available inventory level. Then, three different models for dependence of demand on price and stock level (i.e. Linear, power and exponential) are compared. It is conceptually argued that the exponential function is also a better interpretation of customers behavior toward price and inventory level than the linear model extensively used in literature. Later a pricing and ordering problem is considered where a retailer is selling one product, the demand of which is dependent on price and stock level, during a specified time horizon. It is shown that there is an optimal number of times that the retailer should change the price and those prices and the order quantity are optimized. A comprehensive form of re-pricing cost is introduced that includes both a fixed cost and a variable cost dependent on the number of items remaining in inventory. It is shown that the optimal number of price changes is sensitive to the parameters of the re-pricing cost. Finally, it is shown that in some cases the optimal solution is to increase the price at the beginning of the time horizon. This increase will result in higher revenue despite decreasing demand. 186

206 In the next chapter an integrated ordering, pricing and production model for a two-stage supply chain is developed. This model synthesizes several lot sizing policies in one formula. The contribution is to add optimal pricing for exponentially price-dependent demand to the optimal joint economic lot-sizing relationships used in previous integrated manufacturer retailer models. This research demonstrates that joint optimization in a supply chain model is beneficial under price-sensitive demand. It is also shown that the policies with unequal-shipments (e.g. geometric then equal shipment policy) do not produce meaningfully better results than the policy with equal-shipments. Firms can confidently optimize their process without using these sophisticated methods. This research also shows that higher total profit for the whole supply chain is achieved when firms jointly make decisions on price and lot-sizes than if each member optimizes its profit independently. Therefore it should be possible to allocate profit to each supply chain member in such a way that all are better off. The relative benefits of the Joint policy over the Independent policy are dictated by the demand price sensitivity. As the price sensitivity decreases, the benefit of joint optimization will diminish. Next an EOQ model is discussed for a supply chain with three echelons and a constant deterministic demand. Based on the EOQ and joint replenishment models it is shown that the highest benefits for the whole supply chain results if all three firms share information and optimize the replenishment policy together. Even partial coordination, when two of the three firms coordinate, will result in higher total profit for the supply chain when compared to the independent replenishment policy. It is shown that in any two-firm coordination scenario, the firm on the upstream side of the coordination contract benefits while the downstream firm suffers from the coordination. This is due to two major phenomena happening in the cost structure of the firms. First, whenever 187

207 collaborating, the order quantity for the downstream firm is larger than its EOQ resulting in higher holding cost, and consequently higher total cost for the downstream firm. Second, when collaborating, the upstream firm selects its lot size multiplier (m* for the DC and n* for the manufacturer) in a way that the total number of items holding (m*q* for the DC and n*m*q* for the manufacturer) is constant. They also ship one batch as soon as they receive/produce the required amount, resulted in reducing their holding cost. In all these cases the net amount of gain for the upstream firm is always higher than the loss of the downstream firm. Therefore the net benefit of contracting is positive and profit sharing makes it possible for both firms to be better off. In general, for any coordination scenario, firms should negotiate in order to share the extra profit gained from coordination. Two approaches for profit allocations are compared. In the first approach firms share the profits with the same ratio as when they independently optimize their inventory system. In the second approach, the Shapley value method (a game theoretic process) is used. In this method each firm s share is proportional to its contribution to the collaboration. Using both methods, the profits for both firms are higher than those of when they optimize their profit independently. Next, a simulation model is developed to model a supply chain with three echelons. This model is used to analyze the pricing and replenishment policies for more complex situations with stochastic variables and price and stock-dependent demand. Using the simulation-optimization methods it is shown that, similar to simpler models discussed above, cooperation in decision making for inventory systems benefits the whole supply chain even if it is partial cooperation. It is also shown that if demand is only dependent on price or stock level, considering it in the model benefits the supply chain, particularly when firms coordinate. However, if demand is dependent on both price and stock level, considering the dependency on only one of them will 188

208 not benefit the supply chain as much as when considering both. In this case, firms should consider both price and inventory-level dependency when making decisions in order to improve the supply chain profitability. The effect of price-sensitivity of demand on the decisions and providing higher profits, when considered, is mostly related to revenue management. When collaborating, firms can reduce the effect of double marginalization and hence reduce the final consumer price. The reduced price will affect the demand exponentially. The higher demands will bring in more revenue in turn. The contribution of inventory-level-dependency of demand on profits is due to better allocation of inventory in supply chain. When demand is inventorylevel-dependent, it is more beneficial to keep more inventory at the retail store and visible to consumers to stimulate demand. Hence both order quantity and reorder points are high at the retail store. When collaborating, the retailer will share this information. Therefore the DC is also keeping slightly higher inventories and a higher re-order point. However, the manufacturer is not affected as much and there is not a significantly higher production batch size at the manufacturer. This could be due to very high production rate at the manufacturer. If the production rates were lower, it could become a capacity constraint and the situation might be different. Sensitivity analysis for the complete joint replenishment policy under the price and inventory level demand shows that the most influential factors on the profit are holding costs at the retailer and the DC and the manufacturer s setup cost. As mentioned above in the stock-sensitive situations highest inventory levels are kept at the retailer and then at the DC. Therefore, the holding cost at this firms have the most influence on the profits. Profit allocation methods are used in order to support that idea that when coordinating in decision making for inventory systems in a supply chain, all firm could benefit from cooperation. 189

209 9.2 Significance of this research This research produces the following contributions to the area of inventory management and supply chain coordination: 1. This study is one of the few studies that address demand price and inventory level dependency using real data and empirical analysis. This was one of the shortcomings in the field since it was not clear that the results were practical when real data was not used in building the models. 2. This study provides insights on managing inventories and dynamic pricing under price- and stock-dependent demand by using appropriate re-pricing schemes. The issue of when and how much the price should change, and how that affects the ordering policy is explored at the retailer level. The questions of when and how many items to order are also answered for supply chains consisting of two and three echelons with different demand functions. 3. This research also investigates the importance of price and stock dependency of demand on optimal supply chain coordination policies through sensitivity analysis. It also explores the potential differences in pricing and replenishment policies for different demand functions and answers the question of whether it worth considering demands price- and stock- dependency while coordinating supply chains. This is an important issue relating to required computational power when dealing with sophisticated models for price and stock dependencies. 4. This research determines the extent to which inventory dependent demand affects different stages in the supply chain. 190

210 5. This research makes a contribution in supply chain modelling. A parametric flexible simulation model is developed for multi-echelon supply chains, considering different stochastic factors. This simulation model can provide a benchmark for validating the heuristics proposed in the literature for optimizing inventory systems under price- and/or stock-dependent demand. 6. The issue of profit allocation for joint policies is addressed in this research. This research is one the few studies in this area that uses a game theoretic approach to address collaborative games in supply chains. 9.3 Limitations 1. One of the limitations of this research is that it takes only the dependency of demand on price and stock level into consideration. However, it is clear that demand is also dependent on several other factors such as quality and time (seasonal demands). 2. Although, an empirical study is done in an attempt to capture the price- and inventory level- dependency of demand, justifying dependency of demand on inventory level is still limited to an assumption made about the direct relation between number of UPCs for a product and its inventory level. It is possible to argue that although demand is positively correlated to number of UPCs for a product in stores, it is not a good measure to show that demand is dependent on available inventory. 3. In this research firms profits and service levels are used separately as the measures for performance of the inventory systems. There are other recognized performance 191

211 measures that can be used. For example combining different measures in order to define a multi-performance measure has been used in literature. 4. There are other factors that are involved in the pricing and replenishment policies of inventory systems that are not included in this research. Deterioration, quantity discount, and multiple items involved are among these factors. 5. The profit allocation approaches used in this research can be studied more extensively using other game theoretic approaches. It is especially beneficial if different firms are considered at each stage of the supply chain and the competition is addressed for those firms in the supply chain. 192

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218 APPENDIX A: A.1. Model for joint policy with equal-shipment assumption If, then the EOQ and total profit presented in equations (58) and (59) will reduce to: (A.1) (A.2) If is considered to be a fixed value, maximizing TPj is equal to minimizing the expression under square root, which will be termed TCj. (A.3) Using the assumption that the second derivative of the cost function, TCj, with respect to n is always positive. Consequently TCj is strictly convex with respect to n so the optimal value of n is as follows. (A.4) Based on this expression there is a non-increasing relation between the value of n* and the value of. Since, there are minimum and maximum bounds for n*. These bounds, rounded to integers, are specified as nmin and nmax. 199

219 (A.5) For a given value of n, TPj can then be rewritten as (27) and x1 and x2 in (28) reduce to: (A.6) Taking the first and second derivative of (27) yields the following. (A.7) (A.8) Now it is required to solve the problem for each of the three cases (i, ii, and iii) introduced above for condition (a). The first step is to find the optimal demand, Dj*, given an assumed value of n. Once Dj* is known, the optimal price,, can be determined based on the demand function,. (A.9) Finally, the optimal order quantity,, can be determined by substituting into (A.1). 200

220 A.2. Solution for case 1 (x1 > 0, x2 > 0) Case 1 assumes n = 1. Since the second derivative of the profit function has only one root, identified as Ds. However, the first derivative has a concave parabolic shape with two roots, s1 and s2, and one maximum point at Ds. The problem is considered feasible if the profit is positive. This means it will be feasible if. Equivalently, using (A.6) it will be feasible if. The first and second derivative of the profit function can be used to find the optimal solution for Case 1 as follows: If, the profit is negative. Therefore set: Else and the profit is positive. Therefore evaluate: If s2 < a the optimal solution is: Else s2 > a and the optimal solution is: 201

221 A.3. Solution for case 2 (x1 > 0, x2 = 0) Case 2 assumes n = 2. The second derivative of the profit function has one root, equal to. The first derivative has a parabolic shape with two roots, s1 and s2, and a maximum at. The problem is feasible, resulting in a positive profit, if. Alternately, the profit is positive if. Therefore the optimal solution is given by the following. If, the profit is negative. Therefore set: Else and the profit is positive. Therefore evaluate: If s2 < a the optimal solution is: Else s2 > a and the optimal solution is: 202

222 A.4. Solution for case 3 (x1 > 0, x2 < 0) Case 3 assumes. All positive values of D, which make the expression under the square root in Equation (27) positive, must be considered. This means must be satisfied. Therefore the feasible ranges of D will be limited to the following. (A.10) Solving the problem and finding solution is based on first finding the shape of the second derivative of profit function and then successively considering the first derivative and the profit function. First, the roots of the second derivative of the profit function need to be determined. From (A.8) it is apparent the terms within the brackets must be 0. This leads to the following. (A.11) This expression, slightly rearranged, will now be defined as the f function to simplify the solution procedure. (A.12) Taking the first and second derivative of the profit function respect to D results in the following. (A.13) The second derivative is a parabolic function and has two roots. (A.14) 203

223 The first derivative is a function of power 3 but also has two roots. All critical points of these functions are summarized in Table 28. Table 28. Values of the function, first and second derivatives of f at critical points D f(d) f (D) f (D) The critical values between the upper and lower bounds, x1/4x2 and x1/2x2 can now be plugged into (A.12) to determine the behavior of function f, which is the second derivative of the profit function with respect to demand, D. The function f may take on the following cases, which have also been plotted in Figure 51, Case I: If and Then and must be true. Therefore Case II: If and 204

224 Then and must be true. Therefore Case III: If and Then and must be true. Therefore Figure 51. Different cases for function f occur. Based on Figure 51 it is obvious two different possibilities, defined as Scenario A and B, may Scenario A: The function f doesn t have any root and (Case III in Figure 51). Scenario B. The function f has two roots and (Case I and II in Figure 51). These two scenarios will now be dealt with separately in finding the optimal solution. Function f has no roots (Scenario A) If function f has no roots, the second derivative of the profit function TPj also has no root. 205

225 and Therefore the first derivative of f has just one root. and Consequently the profit function has just one extremum point and is continuously convex in its domain. Figure 52 shows the profit function and its first derivative. Figure 52. Profit function and its first derivative (Case 3, Scenario A) Therefore the optimal solution for Scenario A (Case III in Figure 51) is given by the following. If the optimal solution is: 206

226 Else and the optimal solution is: Function f has two roots (Scenario B) Since the function f has two roots, the second derivative of the profit function TPj has also two roots, t1 and t2. The limiting range of its values continues to be between 0 + and x1/x2, same as for Scenario A. The profit function can take on various shapes related to the first derivative. Since the derivative of TPj would have two extremums and it starts from and reaches, four possible cases for may occur. These are illustrated as Cases i iv in Figure 53. Generally, the first derivative of f has one maximum at its first extremum and a minimum at its second. The first derivative of the profit function may have one (s1), two (s1 and s2) or three roots (s1, s2 and s3). The optimal solutions for Scenario B (Cases I and II in Figure 51) can now be determined. If Case i, iii or iv in Figure 53 applies, there are less than three roots and the following is evaluated. 207

227 Figure 53. Profit function first derivative and related profit function (Case 3, Scenario B) 208

228 If the optimal solution is: Else and the optimal solution is: Else Case ii in Figure 53 applies since there are three roots and the following is evaluated. If the following is evaluated: If the optimal solution is: Else and the optimal solution is: Else If and the following is evaluated: the optimal solution is: 209

229 Else and the optimal solution is: These sophisticated calculations show how to find the analytical solution for the condition (a), equal lot-size policy. Later these calculations will be summarized into an algorithm that can be programmed in a user friendly user interface such as excel and be used easily. However, for condition (b), policies with non-equal lot-sizes, the analytical solution will get even more complicated and impossible to be solved analytically. Therefore, it is a good idea to repeat some of Ben-Daya et al. (2008 #753) comparisons in the context of price sensitive demand to see how much benefit will be gained by solving the more sophisticated solutions. A.5. Solution algorithm for the joint policy The following steps outline an algorithm to solve the model presented in Sections A.1 - A.4. For the lowest feasible n, the optimal demand, D n, to maximize profit, TP(D) is determined. The value of n is then incremented until the n yielding the highest profit is found. This will lead to an optimal set of decision variables, n *,, and q *. Note that the Newton method can be embedded in this algorithm to find the required profit function roots of the first and second derivatives with respect to D. 1. Set as a low amount of profit, where M is a large number. Initialize n *,, and q * equal to zero. 2. Calculate and using (A.115). Set and evaluate the following. If n = 1 or 2, go to Step

230 Else go to Step Use the first derivative of the profit function with respect to D to find the larger root, s2. If then Else Go to Step 5. (Refer to Sections A.2. (n=1) and A.3. (n=2) for details) 4. Since, multiple cases will need to be evaluated. If evaluate the following: If then Else. Go to Step 5. (Refer to Section A.3. for details) Else and determine the roots of the first derivative of the profit function. If there are less than three roots, evaluate the following: If then Else. (Refer to Section A.4. Scenario B with Cases i, iii and iv in Figure 53) Else select the second of the three roots, s2, and evaluate the following: If evaluate the following: 211

231 If then Else. Else If and evaluate the following: then Else. (Refer to Section A.4. Scenario B with Case ii in Figure 53) 5. Calculate the optimal price for the given n, using (A.9). 6. Calculate the optimal order quantity for the given n, q n using (A.1). 7. Calculate the maximum joint profit for the given n,, using the (A.2). 8. If replace previous solution with the new solution: Set,,, and Go to Step 9. Else go to Step If increment n by 1, then If n < 3, go to Step 3 Else go to Step 4. Else, go to Step Stop, since all values in Step 8 are globally optimal. 212

232 A.6. Price and profits for the firms This section examines the relationships between retailer profit, manufacturer s price and final prices. The retailer s profit is given by. Assuming parameter a is set to 1.0, results in the following profit function. (A.15) If then. Substituting and replacing with function f allows the following derivatives to be defined. (A.16) (A.17) (A.18) Since the value of x is always positive, the profit function is concave. The value at which the global maximum retailer profit occurs is then determined from the first derivative. Setting f equal to 0 results in the following. (A.19) (A.20) (A.21) At the retailer profit, TPr, is equal to c, while as approaches infinity the profit approaches

233 (A.22) (A.23) Furthermore, zero profit, f = 0, will occur as and x = 0, which means and. The relationship between retailer profit, TPr, and the total markup,, is plotted in Figure 54. As well, two important conclusions can be stated. First, from (A.20) it is clear that as the manufacturer s markup, c, increases the total markup,, must also be increased in order to maximize profits. Second, from (A.21) it is clear that as c increases the retailer profit, TPr, decreases. Figure 54. Retailer profit as a function of c under an Independent policy 214

234 APPENDIX B: B.1. Arena model modules Figure 55. Modules for updating demand based on price and inventory level at the retailer 215

235 Figure 56. Retailer s system, where customers arrive and buy items. Figure 57. Retailer s inventory system 216

236 Figure 58. DC s Information system Figure 59. DC s inventory system 217

237 Figure 60. Manufacturer s information system Figure 61. Manufacturer s inventory system 218