SUPPLY CHAIN SYSTEMS FOR PERISHABLE PRODUCTS ISSUES, MODELS, AND STRATEGIES. Mutib Algannas

Transcription

1 SUPPLY CHAIN SYSTEMS FOR PERISHABLE PRODUCTS ISSUES, MODELS, AND STRATEGIES Mutib Algannas Master of Engineering Manageent and Technology, Portland State University, 2009 Bachelor of Electrical Engineering, King Fahd University of Petroleu and Minerals, 2003 Subitted to the Departent of Industrial and Manufacturing Engineering and the faculty of the Graduate School of Wichita State University in partial fulfillent of the requireents for the degree of Doctor of Philosophy Deceber 2016

2 Copyright 2016 by Mutib Algannas All Rights Reserved

3 SUPPLY CHAIN SYSTEMS FOR PERISHABLE PRODUCTS ISSUES, MODELS, AND STRATEGIES The following faculty ebers have exained the final copy of this dissertation for for and content, and recoend that it be accepted in partial fulfillent of the requireent for the degree of Doctor of Philosophy with a ajor in Industrial Engineering. Krishna Krishnan, Coittee Chair Ehsan Salari, Coittee Meber Deepak Gupta, Coittee Meber Mehet Barut, Coittee Meber Raazan Asatulu, Coittee Meber Accepted for the College of Engineering Royce Boyden Accepted for the Graduate School Dennis Livesay iii

4 ACKNOWLEDGEMENTS I would first and foreost like to express y deep gratitude to y dissertation chair, Professor Krishna Krishnan. He guided e through y dissertation research with y highest best effort. His wisdo, knowledge and coitent to the highest standards inspired and otivated e. I acknowledge y coittee ebers Dr. Ehsan Salari, Dr. Deepak Gupta, Dr. Mehet Barut, and Dr. Raazan Asatulu for agreeing to serve in y dissertation research. I would like to give y heartfelt appreciation to y parents and y wife who always supported and encouraged e to study and get graduate degree. iv

5 ABSTRACT This dissertation develops a ulti-objective fraework for optial replenishent strategy for perishable product to optiize revenue and transportation costs considering shelf life and delivery planning objectives under varying price scenario. Varying prices restrictions of shelf life tie for product is introduced into this odel to axiize the total profit of supply chain for perishable product. Using the fraeworks developed, optial delivery schedules for perishable products when the product prices vary based on the deteriorated or reaining shelf life of the product are deterined. In this dissertation, ixed integer nonlinear (MINLP) atheatical odels are developed for ulti-objective probles with an objective to axiize revenue while taking into consideration shelf life, and iniization of transportation cost in order to deterine the best schedule for delivery. The objective of the next research is in the study of location-allocation of depots for the developent of supply chains of perishable products in which the prices depend on the tie of delivery. This study developed odels with an objective to axiize revenue while deterining the need for adding new depots in order to iniize the total cost of the supply chain while eeting the due dates for delivery of the perishable product. MINLP odels are developed for axiization of revenue while taking into consideration shelf life, the costs associated with depots, and iniization of transportation cost while eeting due dates for delivery. The last research objective in this dissertation develops a ulti-objective optial replenishent strategy for perishable product to optiize revenue and transportation costs considering shelf life and flexible custoer deands. This research extends the initial objective to include deand profiles for ultiple days for each retailer with an objective to axiize the total profit of the supply chain for perishable product. v

6 TABLE OF CONTENTS Chapter Page 1. INTRODUCTION Research Motivation and Background Research Scope and Objectives Dissertation Overview References LITERATURE REVIEW Introduction Tie Window-Based Delivery Classical Vehicle Routing Proble Vehicle Routing Proble with Tie Windows Perishable Product Factor within the Supply Chain Syste How Perishable Products are Changeable with Varied Arrival Tie Supply Chain Manageent Applied on Perishable Products Transportation of Perishable Product in Supply Chain Syste Transportation Planning Task Logistics and Cost of Transportation of Single Perishable Product in Supply Chain Conclusion References OPTIMAL REPLENISHMENT STRATEGY FOR PERISHABLE PRODUCTS Introduction Model to Maxiize Total Profit Proble Definition Notation and Forulation MTP Model MTP-R Model MTP-T Model Maxiize Total Profit (MTP) Experiental Evaluation Experiental Design and Tests Case Study One Case Study Two: Revenue and Transportation Model with Unliited Vehicle Capacity Additional Case Study for Validation: Revenue and Transportation Model with Liited Vehicle Capacity...44 vi

7 TABLE OF CONTENTS (continued) Chapter Page Additional Case Study for Validation: Revenue and Transportation Model with Liited Vehicle Capacity Conclusion References OPTIMAL LOCATION ALLOCATION AND DELIVERY STRATEGY FOR PERISHABLE PRODUCTS Introduction Model to Maxiize Total Profit of Location Allocation Proble Definition Notation and Forulation MTP-L Model MTP-RE Model MTP-AT Model Maxiize Total Profit (MTP-L) Experiental Evaluation Experiental Design and Tests Case Study One Case Study Two Conclusion References OPTIMAL REPLENISHMENT STRATEGY FOR MULTIPLE DAYS OF DEMAND OF PERISHABLE PRODUCTS Introduction Model to Maxiize Total Profit Proble Definition Notation and Forulation MTP-MD Model MTP-RD Model MTP-TD Model Maxiize Total Profit (MTP-MD) Experiental Evaluation Experiental Design and Tests Case Study One Case Study Two: Revenue and Transportation Model with Unliited Vehicle Capacity Conclusion References vii

8 TABLE OF CONTENTS (continued) Chapter Page 6. CONCLUSIONS AND FUTURE WORK Conclusions Future Work REFERENCES viii

9 LIST OF TABLES Table Page 3.1. D τt, Deand for Product with Deteriorated Shelf Life τ at Retailer Inventory Capacity O of Retailer Initial Inventory with Deteriorated Shelf Life τ at Retailer Vehicle Capacity in Units of Products B k and Dispatch Cost Q k Tie (days) for Travel fro Depot or Retailer to Retailer (T ) Distance (iles) of Travel fro Depot or Retailer to Retailer C S τt, Quantity of Product with Deteriorated Shelf Life τ Delivered at Retailer using CPLEX x tt Nuber of Units to Ship fro Depot to Retailer during Tie Period t using CPLEX S τt, Quantity of Product with Deteriorated Shelf Life τ Delivered at Retailer using BARON x tt Nuber of Units to Ship fro Depot to Retailer during Tie Period t using BARON D τt, Deand for Product with Deteriorated Shelf Life τ at Retailer Inventory Capacity O of Retailer Initial Inventory with Deteriorated Shelf Life τ at Retailer Vehicle Capacity in Units of Products B k and Dispatch Cost Q k Tie (days) for Travel fro Depot or Retailer to Retailer (T ) Distance (iles) of Travel fro Depot or Retailer to Retailer C S τt, Quantity of Product with Deteriorated Shelf Life τ Delivered at Retailer using CPLEX x tt Nuber of Units to Ship fro Depot to Retailer during Tie Period t using CPLEX...41 ix

10 LIST OF TABLES (continued) Table Page Vehicle Capacity in Units of Products B k and Dispatch Cost Q k S τt, Quantity of Product with Deteriorated Shelf Life τ Delivered at Retailer using BARON x tt Nuber of Units to Ship fro Depot to Retailer during Tie Period t using BARON Vehicle Capacity in Units of Products B k and Dispatch Cost Q k S τt, Quantity of Product with Deteriorated Shelf Life τ Delivered at Retailer using CPLEX x tt Nuber of Units to Ship fro Depot to Retailer during Tie Period t using CPLEX S τt, Quantity of Product with Deteriorated Shelf Life τ Delivered at Retailer using BARON x tt Nuber of Units to Ship fro Depot to Retailer during Tie Period t using BARON D τt Deand for Product with Deteriorated Shelf Life τ at Retailer Inventory Capacity O of Retailer Initial Inventory with Deteriorated Shelf Life τ at Retailer Vehicle Capacity in Units of Products B k and Dispatch Cost Q k Fixed Cost for Opening Depot per Period Tie Tie (days) for Travel fro Depot w or Retailer to Retailer (T w )...75 w 4.7. Distance (iles) of Travel fro Depot w or Retailer to Retailer C S τt, Quantity of Product with Deteriorated Shelf Life τ Delivered at Retailer...76 w 4.9. x tt Nuber of Units to Ship fro Depot w to Retailer during Tie Period t Vehicle Capacity in Units of Products B k and Dispatch Cost Q k...81 x

11 LIST OF TABLES (continued) Table Page S τt, Quantity of Product with Deteriorated Shelf Life τ Delivered at Retailer...82 w x tt Nuber of Units to Ship fro Depot w to Retailer during Tie Period t D τt Deand for Product with Deteriorated Shelf Life τ at Retailer Inventory Capacity O of Retailer Initial Inventory with Deteriorated Shelf Life τ at Retailer Vehicle Capacity in Units of Products B k and Dispatch Cost Q k Tie (days) for Travel fro Depot or Retailer to Retailer (T ) Distance (iles) of Travel fro Depot or Retailer to Retailer C S τt, Quantity of Product with Deteriorated Shelf Life τ Delivered at Retailer x tt Nuber of Units to Ship fro Depot to Retailer during Tie Period t Vehicle Capacity in Units of Products B k and Dispatch Cost Q k S τt, Quantity of Product with Deteriorated Shelf Life τ Delivered at Retailer x tt Nuber of Units to Ship fro Depot to Retailer during Tie Period t xi

12 LIST OF FIGURES Figure Page 3.1. Space-Tie Network MTP Model Flow Chart Case 1 Results of Space Tie Network using CPLEX Space-Tie Network Proble with Three Retailers and One Depot using CPLEX Case 1 Results of Space Tie Network using BARON Space-Tie Network Proble with Three Retailers and One Depot using BARON Case 2 Results of Space Tie Network using CPLEX Space-Tie Network Proble with Eight Retailers and One Depot using CPLEX Case 3 Results of Space Tie Network using BARON Space-Tie Network Proble with Eight Retailers and One Depot using BARON Case 4 Results of Space Tie Network using CPLEX Space-Tie Network Proble with Eight Retailers and One Depot using CPLEX Space-Tie Network Proble with Eight Retailers and One Depot using BARON Space-Tie Network MTP-L Model Flow Chart Case 1 Results of Space Tie Network Space-Tie Network Proble with Three Retailers and Two Depots Case One Sensitivity Analysis of fixed depot cost h w Case One Sensitivity Analysis of travel C w ij and fixed depot costs h w Case 2 Results of Space Tie Network Space-Tie Network Proble with Eight Retailers and Two Depots...84 xii

13 LIST OF FIGURES (continued) Figure Page 4.9. Case Two Sensitivity Analysis of fixed depot cost h w Case Two Sensitivity Analysis of travel C w ij and fixed depot costs h w Space-Tie Network MTP-MD Model Flow Chart Space-Tie Network Proble with Eight Retailers and One Depots Space-Tie Network Proble with Eight Retailers and One Depots xiii

14 LIST OF ABBREVIATIONS BARON CPLEX CPU GAMS MINLP MTP MTP-L MTP-MD MTP-R MTP-RE MTP-RD MTP-T MTP-AT MTP-TD Branch-And-Reduce Optiization Navigator C Prograing Language Siplex Method Central Processing Unit The General Algebraic Modeling Syste Mixed-Integer Nonlinear Prograing Maxiize Total Profit Maxiize Total Profit of Location Allocation Maxiize Total Profit of Multiple Deand Days Maxiize Total Profit Revenue Maxiize Total Profit Revenue of Location Allocation Maxiize Total Profit Revenue of Multiple Deand Days Miniize Cost of Transportation Miniize Cost of Location Allocation and Transportation Miniize Cost of Transportation of Multiple Deand Days RMC SCS SVRPTW TSP VRP VRPTW VRPTWTD Ready-Mixed Concrete Supply Chain Syste Stochastic Vehicle Routing Proble along with Tie Window Traveling Sales Proble Vehicle Routing Proble Vehicle Routing Proble with Tie Windows VRP with Tie Windows and Tie-Dependent xiv

15 LIST OF SYMBOLS N(t) A(t) K T, fmtp R Set of Nodes Set of Arcs Set of Trucks Set of Tie-Space Set of Retailers Objective Function of Maxiizing Revenue fmtp T Objective Function of Miniizing Cost of Transportation xv

16 CHAPTER 1 INTRODUCTION The topic of supply chain anageent has recently received considerable attention. The supply chain anageent of perishable products pose further challenges due to the liited shelf life of products and the tie constraints in product distribution. The ain goal of the supply chain syste (SCS) is to fulfill the deand for products or services to retailers or custoers at the requested tie at iniu cost. It is iportant to design an optiu network coprised of depots, retailers, and custoers to achieve this goal. A supply chain network is coprised of different decision categories, including inventory, location allocation, production, and distribution. This dissertation addresses two decision probles: distribution (routing) and location allocation of perishable product. The distribution proble, or vehicle routing proble (VRP), concerns designing optial delivery or transportation routes fro one or several depots to a set of retailers or custoers. Basically, the VRP is a for of the traveling salesan proble (TSP). Clearly, the VRP helps to deterine the best vehicle routes beginning at the depot, oving to a set of retailers or custoers, and then returning to the sae depot. The goal of the VRP is to iniize the total transportation cost. On the other hand, the location allocation proble involves a set of depots and a set of retailers or custoers. The goal of this proble is to locate a set of depots and allocate a set of retailers or custoers in order to iniize the cost of distribution. 1.1 Research Motivation and Background Due to the increasing need for distribution of perishable products, such as beverages, dairy, fresh fruits, fresh flowers, and live seafood, fro depots to retailers, the supply chain has becoe 1

17 ore coplicated. Potentially, the depot, distribution syste, and retailers involved in the supply chain could suffer substantial losses. It is clear that coordination of the supply chain can play an iportant role in the distribution of perishable products (Cai, Chen, Xiao, & Xu, 2010). Many researchers have highlighted the iportance of the association between perishable products and transportation tie. (Shukla & Jharkharia, 2013) noted that 20 to 60 percent of the total food production has been lost or wasted in the food supply chain. Yared Lea and Gatew (2014) identified the need to axiize the avalibility of food products to society. Perishable products ay cause pollution effects to the environent if it is not used within their shelf lives. This has a large ipact on the world econoy and in turn affects all organizations involved in the hierarchy of the supply chain, including the final custoer (Mena, Adenso-Diaz, & Yurt, 2011). Studies to anage perishable products in the supply chain syste have been reported in literature (Deniz, 2007). Many variables are considered in iproving the shipping of perishable products, including transportation cost, inventory cost, and eeting due dates. Heelayr, Doerner, Hartl, & Savelsbergh (2009) studied product delivery schedules and deterined the best routing in order to distribute blood supplies across a region. Bräysy & Gendreau (2005) developed a etaheuristic algorith to solve the VRP with tie windows, with the intention of identifying the lowest-cost routes fro a depot to the retailers or custoers. In the supply chain of perishable products, ost studies have focused on anaging or shipping perishable products fro one or several depots to a set of retailers or custoers, focusing on inventory anageent integrated with the routing proble. However, in ost perishable products scenario, the delivery tie and the available shelf life can deterine the price and deand for the product. For exaple, consider the delivery of newspapers. The deand of the newspaper can be influenced by the tie of delivery. If the newspaper is delivered to a retailer late in the day, the deand ay not be significant. In addition, 2

18 the retailer ay deand a lower price since there is a higher risk of not selling the newspaper. Siilar situations can be seen in the delivery of food ites especially in supply chain systes where refrigerated or cliate-controlled delivery systes are not used. 1.2 Research Scope and Objectives The objective of this research is to develop the odel for axiizing the total profit of supply chain syste for perishable products while considering the vehicle route, shelf life, varying deand and price based on shelf-life of product. In order to accoplish this objective, a nuber of questions are addressed: (1) How does one design an SCS that satisfies syste requireents and axiizes revenue while considering the varying prices of the product s shelf life? (2) What is the optial delivered quantity fro the depot to retailers? (3) How does one design an SCS that iniizes the transportation syste of the supply chain? (4) What is the optial delivery tie fro the depot to the retailers? (5) What is the ipact of liiting the vehicle capacity? (6) What is the optial nuber of depots considering the shelf life of the product? (7) How does one design an SCS that iniizes the transportation syste and nuber of open depots in the supply chain? 1.3 Dissertation Proposal Outline The reainder of this dissertation is organized as follows: Chapter 2 reviews the current state-ofthe-art in the design optiization and analysis ethods related to design of supply chain systes for perishable products, along with product shelf life, and transportation of perishable goods. Chapter 3 presents the odels for axiizing profit in a supply chain syste for perishable products, and deterining the delivery schedules when taking into consideration deand and price variability when date of delivery influences the shelf-life of the products. Chapter 4 presents a proposed design for the optial nuber and allocation of depots to iniize the cost of shipping perishable products. Chapter 5 discusses the optiization odel for deterining delivery 3

19 schedules for a single depot ultiple retailer proble when the deand for the retailers are known for each day in the tie planning horizon. Finally, Chapter 6 suarizes the dissertation and discusses future work. 1.4 References Bräysy, O., & Gendreau, M. (2005). Vehicle routing proble with tie windows, Part I: Route construction and local search algoriths. Transportation science, 39(1), Cai, X., Chen, J., Xiao, Y., & Xu, X. (2010). Optiization and coordination of fresh product supply chains with freshness keeping effort. Production and Operations Manageent, 19(3), Deniz, B Essays on perishable inventory anageent. Doctoral dissertation, Tepper School of Business, Carnegie Mellon University, Pittsburgh. Heelayr, V., Doerner, K. F., Hartl, R. F., & Savelsbergh, M. W. (2009). Delivery strategies for blood products supplies. OR spectru, 31(4), Mena, C., Adenso-Diaz, B., & Yurt, O. (2011). The causes of food waste in the supplier retailer interface: evidences fro the UK and Spain. Resources, Conservation and Recycling, 55(6), Shukla, M., & Jharkharia, S. (2013). Agri-fresh produce supply chain anageent: a state-ofthe-art literature review. International Journal of Operations & Production Manageent, 33(2), Yared Lea, D. K., & Gatew, G. (2014). Loss in Perishable Food Supply Chain: An Optiization Approach Literature Review. International Journal of Scientific & Engineering Research, 5(5),

20 CHAPTER 2 LITERATURE REVIEW 2.1 Introduction During the last decade, the supply chain has becoe an iportant phenoenon because of cost restrictions that can lead to the achieveent of an organization s success in a product s field. However, shipping perishable products to different retailors is ore coplicated. The supply chain ust iprove the delivery of perishable products fro suppliers to retailers. In order to design a odel for designing a supply chain syste for perishable food, the literature was reviewed relative to the ain key factors of the practical proble: a tie window-based delivery syste, replenishent strategy for a single perishable product, and transportation of the perishable product. Taking a wider look at perishability in producing and distributing products, such as daily agazines, perishability is based on the certainty of the product, which requires that the cobination of supply chain processes be accurate. (Cagliano, 2015) Other instances of perishability can be found in cheical industries, agricultural food businesses, processed food anufacturing, and blood banks. In the progressively copetitive arket, transporting perishable products focuses ore on opportunities to attain consuer satisfaction and gain efficiency. This requires a higher diension in the supply chain of perishable products, where products can becoe daaged during transportation, causing industries to incur additional costs (Deep, Nagar, Pant, & Bansal, 2012). Furtherore, products are highly associated with consuer satisfaction; hence, better anageent of perishable goods can be a strong copetitive advantage. At the strategic level, perishability can also play a significant role in forcing good relations and further integration between supply chain networks of organizations. For instance, any food supplies depend on trucks or vehicles to 5

21 transport their products. A better relation between transportation channels could lead to additional benefits. At the other end of the supply chain, anyone can iagine? A vendor-anaged inventory associating a supplier of perishable unprocessed products and a copany that processed these products. This could possibly result in less spoilage and, thus, fewer expenses (Cagliano, 2015). Section 2.2 of this chapter addresses tie window-based delivery/transportation. Section 2.3 deals with aspects of perishable product within the supply chain syste. Section 2.4 discusses transportation of perishable products within a supply chain syste. And, Section 2.5 focuses on the cost of transportation and the logistics of a single perishable product within the supply chain Tie Window-Based Delivery A tie window is an identified period involving stops along the delivery route (for exaple, 8 a and 5 p). The transportation anager can plan the route to ensure that every stop is toured only in its tie window. Advanced routing services will also allow routes to be optiized using certain criteria. These criteria include vehicle attributes (such as size and height of the vehicle), physical condition of the road (such as speed liit or nuber of lanes), and traffic status (such as real-tie, predicted, or historical) (Qi, Shen, & Dou, 2013). Tie windows are particularly helpful in circustances where deadlines are crucial, such as delivery, repair copanies, and utilities. To ake certain that a delivery is effective after logistics and delivery schedules have been set, the salesperson or soeone involved in the transportation s contribution is required. Take an instance in which you will receive an approxiate delivery date in the shopping cart (Q. Zhang, Ma, Luo, & Luo, 2012). You should take into consideration that this does not guarantee the delivery tie of the order. Specific areas can be prolonged because of the nuber of deliveries to the delivery zip codes. A syste is set up to calculate this approxiated date, taking into account 6

22 the average tie to process an order, including packaging of the products, transit tie to the delivery supplier, and transit day to the delivery address (Eanouilidis, Taisch, & Kiritsis, 2013). When the delivery supplier receives your order or request and is prepared to plan a delivery date, they will try to telephone you through the phone nuber you provided in the order (X. Zhang, Qiu, & Yi, 2012). At that tie, they will infor you of what days of the week they can deliver to your location. Deliveries will be ade between the hours you selected on the delivery date you chose. The delivery copany will contact you before the scheduled delivery and will offer you an estiated delivery tie window. To ensure that perishable products are not delayed, it is iportant to ensure that these products are delivered in keeping with the tie window provided (Deep et al., 2012) Classical Vehicle Routing Proble To establish ideal routes, such as start and end of delivery destinations, for the least aount of cost, the classical vehicle routing proble is used. This integer prograing and cobinational optiization proble queries: What are the nuerous optial routes for a fleet of transport vehicles to the network in order to deliver to a particular group of clients? The VRP is derived fro the traveling salesan proble (TPS), which is known as the proble of locating a cobination of routes originating fro various depots to serve a group of retailers having coon deands. Each client ought to be visited just once, and all transport vehicles should be returned to the depot where they originated. Clientele deand on a particular route should not surpass the capacity of the vehicle (Yuan, 2012). 7

23 2.2.2 Vehicle Routing Proble with Tie Windows The vehicle routing proble with tie windows (VRPTW) is an extension of the renowned VRP. This proble is usually sipler than real-life liitations probles. Though any real-life liitations are not included, the research odel characteristically odels the fundaental properties and thus gives the key results eployed in the analysis and adoption of the syste in real-life issues. One of the well-known routing probles is siilarly the siplest one, which is the TSP (Gao & Ryan, 2014). Many etropolises ust be toured by salespersons that ust return to the typical etropolis fro where they started, and the route ust be designed to reduce the distance traveled. Nuerous heuristics and etaheuristics have been proposed for solving the VRPTW proble under different constraints. However, there have been actions on solution ethods to fluctuation in deands as a factor of tie. Deep et al. (2012) reviews the forulation and exact (coplete/optial solution) algorith of the VRPTW, and categorizes the forulations into four ain ethods: spanning free tree forulation, path forulation, arc forulation, and arc-node forulation. In the arc forulation, each arc of a basic directed graph is related to a binary variable (Gao & Ryan, 2014). In the graph, these binary variables are related to nodes in the arc-node forulation of the VRPTW proble. The spanning free tree forulation is a ethod to obtain lower constraints for the VRPTW proble (Cagliano, 2015). The path forulation is also a ethod to obtain lower boundaries for the VRPTW proble. In the current cobinational (integer and ixed-integer) forulations of the VRPTW proble, the sequence of visits for vehicles ay not be deterinate, unless the arcs are linked to the correct order (i.e., TSP). Hence, the analysis of the order of visits in every route is obtained once the solution is received and cannot be utilized as input when the costs incurred rely on factors ipacted by a sequence of node visits within the 8

24 route, such as deand (Liu & Ma, 2008). It is believed that the node-based forulation approach overcoes this insufficiency, and whenever the order of visits causes any additional cost, then the arc-node-based proble forulation is an ideal ethod. Other research has eployed a tiedependent paraeter into the VRPTW proble. However, aong the four forulation ethods, only one sort of tie-dependent deand paraeter ay be used (i.e., indexed-tie forulation) (Qi et al., 2013). As a result, the objective of the VRPTW is to reduce the nuber of vehicles needed, the average traveling tie, and the average travel distance covered by the fleet of trucks. Note that in the arc-node forulation, the set arcs usually represent linkages between the depot and the retailers (Bai, 2010). 2.3 Perishable Product Factor within Supply Chain Syste Perishable products are those likely to becoe unsafe to consue or use if refrigerated iproperly or transported poorly. Exaples of products that should be transported with care or refrigerated suitably include dairy products, eat, vegetables, eggs, glasses, and any others. Proper refrigeration delays bacterial growth and proper transportation prevents perishable products fro spoiling (Deep et al., 2012). Most researchers have developed a ixed-integer Nonlinear prograing (MINLP) syste with a certain heuristic algorith (i.e., dispatching rules such as shortest transporting tie or first-coe first-served) to deterine the best ethod of transportation for perishable ites fro a single producer (depot) to a single vendor (retailer). As a result, the researchers provide positive results to reduce the total delivery cost (i.e., copletion tie) and inventory cost between the depot and the retailer. Yuan (2012) studied the scheduling and transporting of perishable products, and provided a nonlinear odel as a production planning and vehicle routing proble together with tie windows for perishable ites. In further research, Gao and Ryan (2014) pointed out that the 9

25 ain objective of studying perishable products in the supply chain is to increase the total turnover fro the depot (supplier) by choosing the right aount at the right tie going to the right location in order to transport the products to the right consuer Changeability of Perishable Ites with Varied Arrival Tie Liu and Ma (2008) classified the deteriorating properties of inventory into three categories: deterioration (such as flowers, vegetables, and fruits), physical depletion (such as alcohol and gasoline), and obsolescence (such as radioactive ites, and with the depreciation of value in inventory, such as uraniu). Ma et al. (2012) pointed out the associated utility loss and differentiate two types of products those that experience functionality degrading over tie (for exaple, ilk, fruits, and vegetables) and those without functionality degrading over tie but custoer-assued utility degrading over tie (such as fashion attire, newspapers, and hightechnology ites with a teporary life cycle). Wang (2012) differentiated two categorizations of perishability: fixed lifetie and rando lifetie. In the fixed lifetie category, the ite s lifespan is specified earlier, and thus the effect of depreciating factors is considered when fixing it. The product s value declines in its lifetie until the ite perishes totally and is of no use to the custoer, for exaple, blood and yogurt (X. Zhang et al., 2012). In a rando lifetie, based on the work of Bourlakis Vlachos, and Zeipekis (2011), there is no given lifetie for products such as gasoline. Consequently, the lifetie of the product (i.e., flowers, vegetables, gasoline, and fruits) can be presented as a odel that is variable in proportion to a specified probability distribution. Indeed, with these products, it is difficult to describe a dependency between perishability and age because the products ay be stored indefinitely, although the products undergo natural wear and tear while being delayed in inventory, thus degrading their status (Vijayakuar, 2014). 10

26 Another idea strongly associated with deterioration and perishability is shelf life, which refers to the period following the production of a product in which it is in the state of an acceptable quality. This is the duration of tie that a specified product can stay in a arketable condition on the trader s shelf or counter. Note that shelf life does not essentially consider the physical condition of a product because any products only deteriorate occasionally once their shelf life expires. However, the product s saleable life can be considered (Bourlakis, Vlachos, & Zeipekis, 2011) Supply Chain Manageent Applied to Perishable Products The challenge for industries in their anageent of the supply chain of perishable products worsens considerably after a period of tie at a level that is very reliant on the environent. Huidity and teperature are basic factors in this process. Another latent issue regarding the production of perishables and delivery odels all over the globe is that they should be odel-dependent rather than traditional production ethods. Perishable products in a supply chain are ore vulnerable to environental change, econoic shock, or even anageent istakes supported by a deficiency in knowledge. Future perishables will have to integrate four key features to anage and satisfy consuer deand: resilience, copetitiveness, sustainability, and capability (Liu & Ma, 2008). Due to losses related to the delay in transporting perishable ites, custoers have becoe ore indful of the nutritional content and origin of their foodstuffs. As a result, iproveent in the attention to freshness, traceability, and high-quality fors of food is expected. Therefore, anufacturers have expanded the variety of products in order to eet the desires of custoers. This leads to ore coplicated decisions and increased transportation costs (Deep et al., 2012). Another fundaental challenge for industries is that the population is aging continuously and in an unstructured anner. This will eventually affect food availability and security where it 11

27 becoes a vital concern to incorporate public records into the supply chain anageent. For exaple, fresh products (i.e., vegetables, flowers, and live seafood) becoe a challenging operation at risk rates. Eanouilidis et al. (2013) points out that the transporter can incur considerable losses resulting fro poor logistics before and during delivery. 2.4 Transportation of Perishable Products in Supply Chain Syste Bourlakis et al. (2011) expanded the VRPTW by taking into account certain perishability properties during dispatching and arrived at a stochastic vehicle routing proble along with the tie window (SVRPTW) to reduce total costs. In their proposed odels, they prove that SVRPTW is ore effective than nuerous traditional VRPTWs. Since they encopass both energy and inventory costs, the result is a considerable cost iniization in the delivery process. In line with previous studies (as cited in Peng, 2011), Vijayakuar (2014) presented a deonstrated odel blend of the VRP with a tie window and tie-dependent travel tie (VRPTWTD). He also used a heuristic ethod to address the delivery proble. Here, he identified quality as a easure to reduce the total transportation costs by considering travel tie, vehicle nuber, and quality loss Transportation Planning Task Transportation/distribution planning tasks take into consideration solutions, for exaple, fleet diensioning or delivery frequencies. A copany producing perishable products for a retailer can desire to increase the frequencies of supplies to attain a better consuer service concerning product freshness, which, as a result, can ipact fleet diensioning (Peng, 2011). Siilarly, during the id-ter planning prospect, the distribution of perishable products between producer and transportation centers can be influenced by the varied shelf-lives of products. On an additional operational degree, a transport strategy that deterines the quantities of distribution between 12

28 classes of the outbound vehicle routings and the supply chain is a planning issue that can siilarly be influenced by perishability (Yuan, 2012). Replenishent probles are siilarly incorporated because soe of the inventory anageent researchers fail to differentiate whether a procureent section orders ites fro outside the supplier or whether warehouse delivery orders involved processed ites fro the production plant in a for of in-house way (Liu & Ma, 2008). The VRPTW odel takes into account the effect of perishability on general distribution costs. Also, the randoness of the perishable food shipent process is considered and presented in a stochastic VRPTW odel to deterine routes, vehicle, loading, departure, and delivery frequencies at the delivery centers. The objective function considers inventory costs as a result of the deterioration of perishable products incurred during the cooling of distribution vehicles (Wang, 2012). The delivery of ready-ixed concrete (RMC), which is extreely perishable, has received significant attention. However, in this scenario, the practice indicates that the transportation process is so intensive in relation to the tie that the perishability course is handled by enforcing very narrow, tough tie-windows and by iposing a stringently continuous supply of concrete (Eanouilidis et al., 2013). 2.5 Logistics and Cost of Transportation of Single Perishable Product in Supply Chain How to transport an industry s single products fro one position to another is one of the ost significant decisions that the decision-aker can ake based on his/her preferences. In fact, transporting products cheaper and ore rapidly will ake consuers happier. Here, logistics plays a role. If you need to axiize your return, then it is vital to reain abreast of present trends in the transport sector. For exaple, several corporations are starting to shift their shipping centers and supply chains to regions where cost is effective (Cagliano, 2015). 13

29 Bai (2010) indicates that shipping of perishable products ay be a coplex process in the supply chain. However, oving single, perishable products quickly in order to guarantee freshness can be costly, and onitoring conditions such as huidity and teperatures increases those costs. Therefore, systeatic logistics needs to be carried out on single products before delivery coences. 2.6 Conclusion This chapter provided a brief literature review on a supply chain of perishable product and presented solution ethods for solving ulti-objective optiization proble. Interest here focused on shelf life, vehicle routing proble, logistic and cost of transportation, and perishable product with varied arrival tie. To conclude, the effective logistics of production and transportation is becoing significant in the area of perishable products. The classification of perishable ites is soewhat variable. Thus, anufacturers and distributors need to thoroughly research suitable ways of handling various perishable products, in order to avoid incurring logistical and overdue costs as they ai to satisfy consuer deand. Producers should eploy a unified fraework and a new transversal approach while classifying perishability. Tie window delivery ethods have proven to be responsive in addressing the delivery of perishable products fro distribution centers to consuers. 2.7 References Bai, X. (2010). Research on coordination of perishable products supply chain with the cobined option and quantity discount contract. Proceedings of the ICLEM 2010 slogistics for Sustained Econoic Developent: Infrastructure, Inforation, Integration, pp Bourlakis, M., Vlachos, I. P., & Zeipekis, V. (2011). Intelligent agrifood chains and networks: John Wiley & Sons. Bräysy, O., & Gendreau, M. (2005). Vehicle routing proble with tie windows, Part II: Metaheuristics. Transportation Science, 39(1),

30 Cagliano, A. C. (2015). Networks against tie. Supply chain analytics for perishable products. Production Planning & Control (ahead-of-print), 1-2. Deep, K., Nagar, A., Pant, M., & Bansal, J. C. (2012). Advance in Intelligent Systes and Coputing, Proceedings of the International Conference on Soft Coputing for Proble Solving (SocProS 2011) Deceber 20-22, 2011: Volue 2 (Vol. 236, pp ): Springer Science & Business Media. Eanouilidis, C., Taisch, M., & Kiritsis, D. (2013). Advances in Production Manageent Systes. Copetitive Manufacturing for Innovative Products and Services: IFIP WG 5.7 International Conference, APMS 2012, Rhodes, Greece, Septeber 24-26, 2012, Revised Selected Papers (Vol. 398): Springer. Gao, N., & Ryan, S. M. (2014). Robust design of a closed-loop supply chain network for uncertain carbon regulations and rando product flows. EURO Journal on Transportation and Logistics, 3(1), Liu, B.-l., & Ma, W.-h. (2008). Application of quantity flexibility contract in perishable products supply chain coordination. Paper presented at the Control and Decision Conference, CCDC Chinese. Peng, Q. (2011). ICTE 2011 Proceedings of the Third International Conference on Transportation Engineering, July 23-25, 2011, Chengdu, China. Reston, VA: Aerican Society of Civil Engineers. Qi, E., Shen, J., & Dou, R. (2013). International Asia Conference on Industrial Engineering and Manageent Innovation (IEMI2012) Proceedings: Core Areas of Industrial Engineering: Springer Science & Business Media. Vijayakuar, A. (2014). Title of paper. International Conference of OR for Developent (ICORD 2002). Heidelberg: Springer. Wang, X. (2012). Optial pricing with dynaic tracking in the perishable food supply chain. In Decision-Making for Supply Chain Integration (pp ). Springer. Xu, X. (2010). Optial decisions in a tie-sensitive supply chain with perishable products. Yuan, Q. (2012). Title of paper, Proceedings of the First International Conference on Transportation Engineering, July 22-24, 2007, Southwest Jiaotong University, Chengdu, China. Reston, Va.: [Aerican Society of Civil Engineers]. Zhang, Q., Ma, C., Luo, Y., & Luo, Z. (2012). Green supply chain perishable products pricing: Considering price-driven substitution. Paper presented at the ICLEM 2012: Logistics for Sustained Econoic Developent Technology and Manageent for Efficiency. Where and when????? Zhang, X., Qiu, Z., & Yi, P. (2012). ICLEM 2012: Logistics for Sustained Econoic Developent Technology and Manageent for Efficiency. 15

31 16

32 CHAPTER 3 OPTIMAL REPLENISHMENT STRATEGY FOR PERISHABLE PRODUCTS 3.1 Introduction As stated in Chapter 2, Literature Review, it is iportant to enhance supply chain strategy logistics involving perishable products effectively and efficiently, especially in their distribution. Perishable products are those that decay and are likely not to be safe for their intended use after the shelf-life period. This ay render the value of products to be zero or negative (if disposal costs are involved). If the supply chain is not effective in the delivery of these products, then the price and hence the profits of the products ay decrease. Exaples of such products include coposite aterials, ost food ites, flowers, etc. Most perishable products have a predeterined shelf life, which designates the tie period for which the product is safe for its intended use. Over the past decade, supply chains have becoe a vital part of the anufacturing industry. Many researchers have highlighted the iportance of the association between perishable products and transportation tie. Shukla and Jharkharia (2013)) stated that 20 to 60 percent of total food production has been lost or wasted in the food supply chain. Yared Lea and Gatew (2014) identified the need to axiize the avalibility of food products for society. Perishable products ay cause pollution effects to the environent if not used within their shelf life. This has a large ipact on the world econoy and in turn affects all organizations involved in the hierarchy of the supply chain, including the final custoer (Mena et al., 2011). Shipping perishable products with a liited shelf life to retailers akes the supply chain network design ore coplex. Product perishability forces additional constraints on diverse supply chain processes such as production planning, procureent, inventory anageent, and the supply chain network. For exaple, in the yogurt industry, perishability is a factor at every link in 17

33 the supply chain, fro raw aterials to the final product. Thus, the supply chain ust enhance the transportation of perishable products fro suppliers to retailers by considering their shelf lives. Perishable products can be classified as products with either a rando shelf life or a fixed shelf life. Products with a rando shelf life, such as fruits and vegetables, experience continuous deterioration and have expiration dates that depend on specific deterioration rates. Products with a fixed shelf life, such as ilk and juice, are given an expiration date at the tie of production and are considered to have the sae quality until that expiration date passes (Madduri, 2009). Madduri (2009) developed a odel for inventory order policies for perishable products with a fixed shelf life. Here, fixed shelf life refers to the period of tie beginning with the anufacturing of the product until the expiration date, during which tie the product reains in good quality (Valero, Carrasco, & García-Gieno, 2012). In other words, the fixed shelf life is the aount of tie for a given perishable product to expire (Ministry for Priary Industries, 2014). Due to the perishable nature of products, especially those with a fixed shelf life, effective delivery planning can offer ore savings in logistic operations. Siilarly, the optiization of vehicle routes aids in the distribution of perishable goods and is incorporated into the research regarding distribution optiization. The ain challenge in the decision support syste of shipping perishable products is the developent of the vehicle schedule. This paper focuses on iniizing the transportation cost while taking into consideration the aount of product being shipped (i.e., vehicle capacity) and the cost of transportation (i.e., vehicle dispatch cost and route cost). A considerable aount of literature ay be found on the vehicle routing proble, which has been extended to include the tie window constraints. For exaple, Deep et al. (2012) reviewed the forulation and exact (coplete/optial solution) algorith of the VRPTW and categorized the forulation into four 18

34 ain ethods: arc forulation, spanning free tree forulation, path forulation, and arc-node forulation. In the arc forulation, each arc of a basic directed graph is related to a binary variable (Gao & Ryan, 2014). In this graph, these binary variables are related to nodes in the arc-node forulation of the VRPTW proble. The spanning free tree forulation is a ethod for obtaining lower constraints for the VRPTW proble (Cagliano, 2015). The path forulation is also a ethod to obtain lower boundaries for the VRPTW proble. In the current cobinational (integer and ixed-integer) forulations of the VRPTW proble, the sequence of visits for vehicles ay not be deterinate, unless the arcs are linked in the correct order (i.e., traveling salesan proble). Hence, an analysis of the order of visits in every route is obtained once the solution is received and cannot be utilized as input when the costs incurred rely on factors ipacted by a sequence of node visits within the route, such as deand (Liu & Ma, 2008). It is believed that the node-based forulation approach overcoes this insufficiency, and whenever the order of visits causes any additional cost, then the arc-node-based proble forulation is an ideal ethod. Other research has eployed a tie-dependent paraeter into the VRPTW proble. However, aong the four forulation ethods, only one sort of tie-dependent deand paraeter ay be used (Qi et al., 2013). As a result, the objective of the VRPTW is to reduce the nuber of vehicles needed, the average traveling tie, and the average travel distance covered by the fleet of trucks. Note that in the arc-node forulation, the set arcs usually represent linkages between the depot and the retailers (Bai, 2010). This paper focuses on finding the optial delivery strategy for a single perishable product for which the deand and price as a function of shelf life are known. It is assued that the product is fresh and has all of its shelf life when it leaves the depot. A tie continuu for the shelf life of a product is assigned, starting fro tie zero when it leaves the depot to the designated shelf-life 19

35 duration. The product is fresh at tie zero and starts losing its price as tie passes. To odel this proble, the objective function is decoposed into two coponents: one that axiizes the total revenue during the planning horizon (i.e., revenue fro the sale of the product) and the other that iniizes the total transportation cost. The proble is odeled as an MINLP atheatical proble and solved using the CPLEX and BARON solvers. Solving the odel provides a solution to axiize total profit (MTP) (total revenue inus total transportation cost). Finally, a detailed experiental test proble is provided in order to gain insight into the dynaics of the odel. The key contributions of this paper are as follows: (1) developent of a odel to axiize MTP during the planning horizon for perishable products, (2) utilization of ixed-integer linear prograing to forulate the odel, (3) provision of a detailed experiental test proble to obtain insight into the odel dynaics by varying different paraeters, (4) utilization of the CPLEX solver to solve the proposed odel sequentially, and (5) utilization of the BARON solver to solve the proposed odel siultaneously, where both solvers provide values for the decision variables of the supply chain proble. 3.2 Model to Maxiize Total Profit The MTP odel axiizes the total profit and utilizes the tie-indexed variable forulation in which the planning horizon is discretized into N intervals of one unit length each. This objective functions consists of two different coponents: to axiize revenue (MTP-R) and to iniize the cost of transportation (MTP-T). The MTP odel is developed as a ixed-integer atheatical prograing odel. In its ipleentation, after solving each objective (MTP-R and MTP-T), the MTP-R objective is subtracted fro the MTP-T objective in order to obtain the profit during the planning horizon. 20

36 3.2.1 Proble Definition In this paper, the ain goal of the MTP odel is to find the optial delivery strategy for a single perishable product for which the deand and price as a function of shelf life are known. Several retailers are served by one ain depot, and they are connected via a supply chain network. A fleet of vehicles is used to ship the product fro the depot to a set of retailers. Each retailer has a deterinistic inventory capacity and initial inventory, the deand for the product each day is known, and the deand depends on the reaining shelf life of the product. It is assued that the product has its axiu shelf life at tie zero, which is when it leaves the depot. However, it is assued that the product starts deteriorating as soon as it is loaded on the truck. The shelf life of the product is less than or equal to the order tie (i.e.,τ t). It is assued that the depot has enough quantity of the product to eet the deand. The length of the planning horizon is assued to be greater than the length of the shelf life (i.e., T > τ ). In addition, the product has a unit price that varies with the shelf life over the planning horizon. Also, it is assued that the inventoryholding cost and loading and unloading costs are negligible. The objective is to optiize (axiize) profit by axiizing revenue (i.e., axiize units of product with shelf life τ delivered at retailer during tie period t) and by iniizing transportation cost Notation and Forulation Notations relative to the MTP-R and MTP-T odels are defined as follows: Sets N(t): set of nodes in space-tie network A(t): set of arcs in space-tie network K: set of trucks T: set of space-tie 21

37 Indices k K: truck index k = 1,, K t, t T: tie period index t, t = 1,, T τ T: deteriorated shelf life, which is the nuber of days that the product has deteriorated τ = 0,, T, N(t): index for retailer, 0,, M, where = 0 and = 0 represents depot l N(t): set of retailers () satisfied at tie period (t) i, j N(t): nodes () to ( ) Paraeters B k : capacity of truck k D τt : deand for product with deteriorated shelf life τ at retailer during tie period t P τ : price of product with deteriorated shelf life τ I τ0 : initial inventory with deteriorated shelf life τ at retailer O : inventory capacity of retailer Q k : fixed cost associated with dispatching truck k C ij : cost of travel fro node i to node j (i =, j = ) T : tie for travel fro retailer to, G : upper bound on x vector values Decision Variables x tt : nuber of units to ship fro depot at tie period t to be delivered to retailer at tie period t ; aggregate vector of all decision variables x = {x tt : = 1,, M, t = 1,, T, t = t,, T} represents product delivery plan for planning horizon 22

38 e τt {0,1}: binary variable; e τt = 1 if deand for product with shelf life τ during tie period t at retailer is satisfied; otherwise, e τt = 0 I τt : inventory level with deteriorated shelf life τ at retailer at end of tie period t S τt : product with deteriorated shelf life τ delivered by retailer during tie period t Note that to define other decision variables, first, the truck index ust be augented, and the pair (k, t) ust be used to represent truck k that leaves depot at tie period t. u kt {0,1}: binary variable; u kt = 1 if truck (k) is used at tie t; otherwise, u kt = 0 kt y (i,j) kt {0,1}: binary variable; y (i,j) = 1 if truck (k) is used at tie t with travels arc kt (i, j) A(t); otherwise, y (i,j) Auxiliary Variables = 0 v l kt {0,1}: binary variable; v l kt = 1 if truck (k) visits node l N(t) at tie t; otherwise, v l kt = 0 z kt : arrival tie of truck (k, t) back to depot η tt {0,1}: binary variable indicating whether retailer ε M receives any shipent on day t, which was sent on day t. In other words, η tt = { 1 x tt > 0 0 otherwise MTP Model The objective of the MTP odel is to axiize total profit during the planning horizon, which is the difference between the objective function of the MTP-R odel and the objective function of the MTP-T odel: 23

39 Max profit f f MTPR MTPT T T M T K T K kt kt t ( i, j) ( i, j) k Max (( P s ) ( C y Q u )) t1 0 1 t1 k1 ( i, j) A0 ( t) t1 k MTP-R Model follows: The MTP-R odel axiizes the total revenue obtained fro the product delivered as Max P s t (3.1) t1 0 1 st.. S I x 1..., M, t 1,..., T, 0 (3.2) t t t, t t t 1, t1 t, t S I I x 1..., M, t 1,..., T, 1,..., t 1 (3.3) t t I 1, t1 S I 1..., M, t 1,..., T, 1,..., T (3.4) T 0 t e T T M T I t O 0 St etd τt xtt' ax k k tt ' t t e 1 1,..., M, t 1,..., T (3.5) 1..., M, t 1,..., T (3.6) 1..., M, 0,..., T, t 1,..., T (3.7) B 1..., M, t 1,..., T, t' 1,..., T (3.8) x, I, S , M, t 1,..., T, T, t' 1,..., T, 1,..., T (3.9) t {0,1} 1..., M, t 1,..., T, 1,..., T (3.10) The first objective (3.1) axiizes total revenue. The inventory constraint sets (3.2), (3.3), and (3.4) ensure a balance in the inventory level with deteriorated shelf life τ at each retailer at the end of tie period t. Constraint set (3.5) ensures that each retailer selects only one deand for the product with deteriorated shelf life τ during tie period t. Constraint set (3.6) ensures the inventory level with deteriorated shelf life τ at retailer at end of tie period t does not exceed the inventory capacity at each retailer. Constraint set (3.7) states that the selected units of product sold per tie period t ust be less than or equal to deand. Constraint set (3.8) ensures liiting the units of products shipped to any retailer per tie period t in order to not exceed the 24

40 axiu truck capacity. Finally, constraint sets (3.9) and (3.10) are integrality and nonnegativity, respectively. After the MTP-R odel is solved, the objective value and delivery plan to arrive at each retailer (x tt, i.e., nuber of units to ship fro depot at tie period t and to arrive at retailer at tie period t ) are obtained in order to solve the MTP-T odel MTP-T Model The MTP-T odel attepts to iniize the total transportation cost incurred in order to carry out a given replenishent plan (x tt ). Evaluation of this odel requires solving a nested optiization proble that deterines the optial routes and fleets used. In particular, this can be odeled siilar to the VRPTW constraints. The VRP is concerned with finding the optial nuber and type of trucks, associated routes, and starting and end ties for trips in order to satisfy a retailer s deand within specified tie windows while iniizing the cost of transportation. An adaptation of the VRP forulation suited for this proble is discussed next. The supply chain network odel includes a set of nodes (N) consisting of retailers and a depot, and a set of arcs (A) representing routes. In order to characterize the routes, Τ space-tie networks, each associated with a tie period t = 1,, T in the planning horizon, are constructed, as shown in Figure 3.1. As can be seen, the horizontal and vertical nodes (N(t), A(t)) represent space (retailer index) and tie (tie period index), respectively. The first and last nodes (0, t) represent the depot. An arc appears between every two nodes, only if the difference between the tie periods is equivalent to the travel tie between the corresponding retailers (or depot and retailer). The final arc of the route connects it back to the depot to coplete the route. Finally, the arc costs represent the distance between the corresponding nodes (or depot and retailer). 25

41 Figure 3.1: Space-tie network (N(t), A(t)) The space-tie network shown in Figure 3.1 is denoted by (N(t), A(t))(t = 1,, T), where N(t) and A(t) are the sets of nodes and arcs, respectively. In particular, for network (N(t), A(t)), the set of nodes represents pairs of retailers and tie periods as N(t) = { (, t ): = 1,..., M, t = t,..., T} A node corresponding to the depot is added as N 0 (t) = N(t) {(0, t)} Additionally, an arc occurs between every two nodes ( 1, t 1 ) and ( 2, t 2 ), if a vehicle can traverse the distance fro retailer 1 to 2 in exactly (t 2 t 1 ) tie periods. More specifically, the set of arcs is defined as A(t) = {(( 1, t 1 ), ( 2, t 2 )): 1 = 0,..., M, 2 = 1,..., M, 1 2, t 1 = t,..., = t 1 + T } There is also an arc between all nodes in N(t) and the depot to close the loop (see Figure 3.1) as A 0 (t) = A(t) {(i, (0, t)): i N(t) } For notational convenience, R s (( 1, t 1 ), ( 2, t 2 )) = 1 R e (( 1, t 1 ), ( 2, t 2 )) = 2 26

42 where retailer is associated with the start and end of the arc (( 1, t 1 ), ( 2, t 2 )), respectively. Finally, with a slight change of notation, T C represents the tie and cost of traversing arc (i, j) A 0 (t), respectively, which can be obtained fro the travel tie and cost atrices, respectively: T (i,j) = T Rs (i,j),r e(i,j) The MTP-T atheatical odel follows: C (i,j) = C Rs (i,j),r e(i,j) st.. T K T K kt ( i, j) ( i, j) t1 k 1 ( i, j) A ( t ) t1 k 1 Min C y Q u kt kt ( i, ) (, j) ( i, ) A( t) (, j) A( t) kt kt ((0, t), ) (,(0, t)) N ( t ) N ( t ) ( i, j) A( t) ( i, j) A( t) tt ' (, t ') N ( t ) 0 k kt (3.11) y y N( t), k 1,..., K, t 1,..., T (3.12) y y 1 k 1,..., K, t 1,..., T (3.13) kt ( i, j) y k 1,..., K, t 1,..., T (3.14) kt ( i, j) ( i, j) y z k 1,..., K, t 1,..., T (3.15) tt ' kt j kt x G 1,..., M, k 1,..., K, t 1,..., T (3.16) K kt ( t, ') k 1 kt kt kt tt' (, t ') k x B u k 1,..., K, t 1,..., T (3.17) tt' (, t ') N( t), t 1,..., T (3.18) kt kt kt ( i, j) l tt ' kt ' ( t z ) t ' (1 u ) k 1,..., K, t 1,..., T, t ' t 1,..., T (3.19) u, y,, {0,1} 1,..., M, k 1,..., K, t 1,..., T, t ' t 1,..., T, N( t), ( i, j) At ( ) (3.20) kt z 0 k 1,..., K, t 1,..., T (3.21) The first objective (3.11) iniizes the total transportation cost, which consists of variable and fixed costs. The balance constraint set (3.12) ensures that if the vehicle visits a node (retailer) other than the depot on the network, it ust leave that node (retailer). Constraint set (3.13) ensures 27

43 that each route originates and terinates at the node corresponding to the depot(0, t). Constraint sets (3.14), (3.15), and (3.16) ensure that the auxiliary variables are connected to the ain decision variables, where G is an upper bound on the x-vector values. Constraint set (3.17) ensures that the shipped quantity is less than the capacity of the vehicle. Constraint set (3.18) ensures that each node in the space-tie network is visited, if the corresponding retailer is included in the delivery plan. Constraint set (3.19) ensures that a vehicle can only be used again after it returns to the depot. This constraint can be cast as the following logical constraints: t t if z kt > t, then u kt = 0. Finally, constraint sets (3.20) and (3.21) are integrality and non-negativity, respectively Maxiize Total Profit (MTP) The objective of the MTP-odel is Max profit f f MTPR MTPT T T M T K T K kt kt t ( i, j) ( i, j) k t1 0 1 t1 k1 ( i, j) A0 ( t) t1 k1 Max (( P s ) ( C y Q u )) Figure 3.2 shows the MTP odel flow chart. 28

44 MTP SEQUENTIAL MODEL Max profit = f MTP R f MTP T s. t. Ax b x 0 and integer MTP-R Model Paraeters Max f MTP R s. t. Ax b x 0 and integer f MTP R Value x tt MTP-T Model Paraeters Min f MTP T s. t. Ax b x 0 and integer f MTP T Value kt y (i,j) Figure 3.2: MTP odel flow chart 3.3 Experiental Evaluation The General Algebraic Modeling Syste (GAMS) is used to odel the MTP proble, and the proble is solved sequentially using the solver CPLEX 12.5 (ILOG, 2012) and siultaneously using the solver BARON The coputational experients are perfored on an Intel i GHz processor with 4 GB of eory and a Windows 7 operating syste. 29

45 3.4 Experiental Design and Tests Case Study One: A case study is used to provide an illustration for solving the MTP odel sequentially using CPLEX. It helps to obtain the solution of the shipping quantity as the first part of the odel, and then use the solution as the nonlinear constraint (3.17) to obtain the objective value. This case study involves one depot, three retailers, and two vehicles. The length of the tie horizon T =3, and the axiu shelf life is 2. Table 3.1 suarizes the deand for each retailer while ordering a product with different deteriorated shelf lives τ each day. Note that a retailer can place an order each day (t) with deteriorated shelf life of 0 to τ = t 1. For exaple, Table 3.1 shows that retailer 2 places an order on day 2. Hence, retailer 2 can order the product with a deteriorated shelf life of τ = 0 or 1 (i.e., D 2 02 = 50 or D 2 12 = 40). Table 3.1: D τt, Deand for Product with Deteriorated Shelf Life τ at Retailer Tie Horizon Deteriorated Shelf Life τ Price with Deteriorated Shelf Life P τ Deand Deand Deand Table 3.2 illustrates the inventory capacity O for each retailer, whereas Table 3.3 shows the initial inventory I τ0 with deteriorated shelf life τ at retailer. The vehicle capacity in ters of the nuber of units of products is provided in Table

46 Table 3.2: Inventory Capacity O of Retailer Retailer M Capacity Table 3.3: Initial Inventory with Deteriorated Shelf Life τ at Retailer Retailer M Deteriorated Shelf Life τ I τ Table 3.4: Vehicle Capacity in Units of Products B k and Dispatch Cost Q k Vehicle K Capacity of Vehicle B k Cost Q k k k In this ipleentation, three space-tie networks are constructed, corresponding to tie period t = 1,..., 3 in the planning horizon. In each space-tie network, an arc can occur between two nodes, only if the difference between tie periods is equal to the travel tie between nodes. Table 3.5 presents the tie (T ) required for travel fro retailer to retailer (, = 0,, M, where 0 iplies the depot). Table 3.6 shows the distance (iles) of travel fro retailer to (, = 0,, M, where 0 iplies the depot). Note that, the cost per ile of travel fro depot or retailer to retailer is five cents per ile. Table 3.5: Tie (Days) for Travel fro Depot or Retailer to Retailer (T ) Depot

47 Table 3.6: Distance (iles) of Travel fro Depot or Retailer to Retailer C Depot This odel obtains the solution for a product delivery plan, including the tie of departure fro each node (depot or retailer) and the tie of arrival at each node. The delivery plan also provides the nuber of units delivered to each retailer in the route ( S τt ). Table 3.7 represents solutions for the nuber of products with deteriorated shelf life τ delivered at the retailer during tie period t. It can be seen that each retailer receives the ordered quantity of product. The price of the product is deterined by the deteriorated shelf life τ (Table 3.1). For exaple, retailer 2 receives a quantity of 50 units with deteriorated shelf life of τ = 0 at tie t = 2 (i.e., S 2 02 = 50). The odel provides the total revenue for this case study as $1,420. Table 3.7: S τt, Quantity of Product with Deteriorated Shelf Life τ Delivered at Retailer during Tie t using CPLEX Retailer M Deteriorated Shelf Life τ Table 3.8 represents the product-delivery plan for the planning horizon. For exaple, the depot ships 50 units at tie 2, and it arrives at retailer 2 on the sae day (i.e., x 2 22 = 50). As a result, retailer 2 has a deand of 50 units to be delivered on day 2 with zero deteriorated shelf life. It can be seen that the order for retailer 2 is satisfied on the sae day t = 2 and eets the deteriorated shelf life value of zero (i.e., t t = 0). Tie Delivered t

48 Table 3.8: x tt Nuber of Units to Ship fro Depot to Retailer during Tie Period t using CPLEX After obtaining all x tt, the delivery plan for the planning horizon, the MTP-T odel is solved to obtain the best route in order to iniize total transportation cost which includes variable and fixed costs. Retailer ( ) Tie Depart t Tie Arrive t For this case study, the routes selected after optiization is shown in Figure 3.3. As can 1 3 be seen, in network (a), vehicle k 1 is used to ship x 11 and x 13 via arcs y 11 (0,1), y 11 (1,3), and y 11 (3,0), with a total travel tie of four days. The vehicle has enough capacity to cobine the two orders. 33

49 NETWORK kt y (i,j) x tt C ($) T (days) t = 1 t 11 = 1 y (0,1) Fixed Cost = 40 x 1 11 = 50 C 0,1 = 100 T 0,1 = 1 t = 2 t 11 = 3 y (1,3) x 3 13 = 70 C 1,3 = 50 T 1,3 = 2 t = 4 11 y (3,0) C 3,0 = 100 T 3,0 = 1 Total Shipping =120 Total Cost = 290 Total Travel Tie = 4 (a) t = 2 Fixed Cost = 30 t 22 = 2 y (0,2) x 1 12 = 50 C 0,2 = 100 T 0,2 = 1 t = 3 22 y (2,0) C 2,0 = 100 T 2,0 = 1 (b) Total Shipping =50 Total Cost = 230 Total Travel Tie = 2 Figure 3.3: Case 1 results of space-tie network using CPLEX Figure 3.4 shows the space tie network solution. Retailers 1 and 3 are assigned to one route and use vehicle k 1, and retailer 2 is assigned to another route and uses vehicle k 2. In this case, the transportation cost achieved is $520, and the total profit (i.e., revenue inus transportation cost) is $900. Objective Value: $ Vehicle k 1 Route: y (0,1) 22 Vehicle k 2 Route: y (0,2) 11 y (1,3) 22 y (2,0) 11 y (3,0) 34

50 Figure 3.4: Space-tie network proble with three retailers and one depot using CPLEX The objective function value ($900) obtained by sequentially solving using GAMS/CPLEX shows an absolute gap of $45 and a relative gap of 8.03%. Therefore, in order to check if the odel is able to achieve a better solution, the sae case study is solved siultaneously using BARON This coercial ixed-integer nonlinear prograing (MINLP) solver ipleents a branch-and-reduce algorith (Tawaralani & Sahinidis, 2002). By using BARON, the odel solution resulted in an optiu solution for the product delivery plan, which included the tie of departure fro each node (depot or retailer) and the tie of arrival at each node. The delivery plan also provides the nuber of units delivered to each retailer in the route ( S τt ). Table 3.9 represents the new solution and shows the nuber of products along with their tie of delivery for each retailer. It can be seen that each retailer receives the ordered quantity of product. In this case, retailer 2 receives a quantity of 40 units instead of 50 in the sequential case with deteriorated shelf life τ = 1 at tie t = 2 (i.e., S 2 12 = 40). When solving the sae odel using BARON the total revenue obtained is reduced to $1,

51 Table 3.9: S τt, Quantity of Product with Deteriorated Shelf Life τ Delivered at Retailer during Tie t using BARON Retailer M Deteriorated Shelf Life τ Table 3.10 represents the product-delivery plan for the planning horizon. For exaple, the depot ships 50 units at tie 1, and it arrives at retailer 2 on the next day (i.e., x 2 12 = 40). As a result, retailer 2 has a deand for 40 units to be delivered on day 2 with one deteriorated shelf life. It can be seen that the order for retailer 2 is satisfied on the sae day t = 2, and eets the deteriorated shelf life value of zero (i.e., t t = 1). Tie Delivered t Table 3.10: x tt Nuber of Units to Ship fro Depot to Retailer during Tie Period t using BARON Retailer ( ) After obtaining all x tt, the delivery plan for the planning horizon, the MTP-T odel is solved to obtain the best route in order to iniize total transportation cost. By using the BARON solver for this case study, Figure 3.5 illustrates the routes selected 1 after optiization. As can be seen, in network (c), vehicle k 2 is used to ship x 11, x , and x 13 via arcs y 21 (0,1), y 21 (1,2), y 21 (2,3), and y 21 (3,0), with a total travel tie of four days. The vehicle has enough capacity to cobine the three orders. Tie Depart t Tie Arrive t

52 t = 1 NETWORK kt y (i,j) t 21 = 1 y (0,1) x tt C ($) T (days) Fixed Cost = 30 x 1 11 = 50 C 0,1 = 100 T 0,1 = 1 t 21 = 2 y (1,2) x 2 12 = 40 C 1,2 = 25 T 1,2 = 1 t 21 = 3 y (2,3) t 11 = 4 y (3,0) x 3 13 = 70 C 2,3 = 25 T 1,3 = 1 C 3,0 = 100 T 3,0 = 1 Total Shipping =160 Total Cost = 280 Total Travel Tie = 4 (c) Figure 3.5: Case 1 results of space-tie network using BARON Figure 3.6 shows the space tie network solution. Retailers 1, 2, and 3 are assigned to one route and use vehicle k 2. In this case, the transportation cost when solving using BARON is reduced to $280, and hence the total profit (i.e., revenue inus transportation cost) increased to $960. Objective Value: $ Vehicle k 2 Route: y (0,1) 21 y (1,2) 21 y (2,3) 21 y (3,0) Figure 3.6: Space-tie network proble with three retailers and one depot using BARON 37

53 Results of the siultaneous solution ($960.00) by using GAMS/BARON has an absolute gap of E-7 and a relative gap of 0.00 with central processing unit (CPU) coputing tie of 0.30 seconds Case Study Two: Revenue and Transportation Model with Unliited Vehicle Capacity A larger case study is used to illustrate the MTP odel sequentially using the solver CPLEX. In this case study, there is one depot and eight retailers and there are two vehicles with unliited capacity. The length of the tie horizon T = 6, and the axiu shelf life is 5. Table 3.11 suarizes the deand for each retailer while ordering a product with a different deteriorated shelf life τ each day. Table 3.11: D τt Deand for Product with Deteriorated Shelf Life τ at Retailer Note that a retailer can place an order on each day (t) with deteriorated shelf lives of 0 to τ = t 1. Also, the travel tie between the depot and the retailer ust not exceed the product s (deand) shelf life. For exaple, in Table 3.11, retailer 2 places an order on day 4, and hence 38

54 it can order the product with τ = 0 to 3. Note that retailer 5 can order a product with deteriorated shelf life of zero because the travel tie between depot and retailer 2 is one day (i.e., D 2 04 = 71, D 2 14 = 86, D 2 24 = 95, D 2 34 = 123). The price of the product (P τ ) decreases as the deteriorated shelf life of the product increases. For exaple the price of the product with a shelf life τ = 2 is $9, while the price of the product when the shelf life τ = 4 is $6. Table 3.12 illustrates the inventory capacity O for each retailer. Table 3.13 shows the initial inventory I τ0 with deteriorated shelf life τ at retailer. The vehicle capacity in ters of the nuber of units of products is provided in Table Table 3.12: Inventory Capacity O of Retailer Retailer M Capacity Table 3.13: Initial Inventory with Deteriorated Shelf Life τ at Retailer Retailer M Deteriorated Shelf Life τ I τ

55 Table 3.14: Vehicle Capacity in Units of Products B k and Dispatch Cost Q k Vehicle K Capacity of Vehicle B k Fixed Cost Q k k k Table 3.15 presents the tie (T ) required in day for travel fro retailer to (, = 0,, M, where 0 iplies the depot). Table 3.16 shows the distance (iles) of travel fro retailer to (, = 0,, M, where 0 iplies the depot). Note that, the cost per ile of travel fro depot or retailer to retailer is five cents per ile. Table 3.15: Tie (Days) for Travel fro Depot or Retailer to Retailer (T ) Depot After case study two is solved, the odel obtains the solutions for the product delivery plan x tt and nuber of units of product delivered S τt. Table 3.17 shows solutions for the units of products with deteriorated shelf life τ delivered during tie period t. It can be seen that each retailer receives the ordered quantity of product. The price of the product is deterined by the deteriorated shelf life τ (Table 3.11). For instance, retailer 5 receives a quantity of 120 units of product with deteriorated shelf life of τ = 1 delivered at tie t = 2 (i.e., S 5 12 = 120). Based on 40

56 the quantity deanded and specific price (deterined by shelf life), the total revenue obtained for this case is $8,768. Table 3.16: Distance (iles) of Travel fro Depot or Retailer to Retailer C Depot Table 3.17: S τt Quantity of Product with Deteriorated Shelf Life τ Delivered at Retailer during Tie t using CPLEX Retailer M Deteriorated Shelf Life τ Tie Delivered t Table 3.18 represents the product delivery plan for the planning horizon. For exaple, the depot ships 120 units at tie 1 and arrives at retailer 5 on day 2 (i.e., x 5 12 = 120). One result of the odel is that retailer 5 has a deand for 120 units to be delivered on day t = 2 with deteriorated shelf life of τ = 1; however, it is clear that retailer 5 is satisfied by receiving the 41

57 order on the sae day t = 2, which eets the deteriorated shelf life value of τ = 1 (i.e., t t = 2 1 = 1). By solving this optiization odel, to deterine all decision variables in the MTP-R odel, the transportation objective is iniized by obtaining the right x tt delivery plan for the planning horizon for each retailer. Table 3.18: x tt Nuber of Units to Ship fro Depot to Retailer during Tie Period t using CPLEX Retailer M Tie Depart t Tie Arrive t After obtaining the delivery plan for the planning horizon, the MTP-T odel is solved using CPLEX. Figure 3.7 represents the best routes to ship all aounts of the product (x tt ) fro the depot to retailers. As can be seen, network (d) shows the best route using vehicle k 1. For this proble, decision variables x 7 11 = 131, x 5 12 = 120, x 4 13 = 104, x 2 14 = 123, x 1 15 = 142, and x 3 16 = 135 were obtained. The product is shipped via arcs y 11 (0,7), y 11 (7,5), y 11 (5,4), y 11 (4,2), y 11 (2,1), y 11 (1,3), and y 11 (3,0), respectively, with a total travel tie of seven days fro depot to retailers and back to depot. Network (e) shows the best route for vehicle k 2 to deliver x 8 11 = 121, and x 6 12 = 122. Vehicle k 2 delivers the product via arcs y 21 (0,8), y 21 (8,6), and y 21 (6,0), with a total travel tie of four days. As shown in network (e) in Figure 3.7, the vehicle carries ultiple orders and copletes the route. 42

58 NETWORK kt y (i,j) x tt C ($) T (days) t = 1 t = 1 11 y (0,7) t = 2 11 y (7,5) t = 3 11 y (5,4) t 11 = 4 y (4,2) t = 5 11 y (2,1) t = 6 11 y (1,3) t = 7 11 y (3,0) Fixed Cost = 360 x 7 11 = 131 C 0,7 = 75 T 0,7 = 1 x 5 12 = 120 C 7,5 = 50 T 7,5 = 1 x 4 13 = 104 C 5,4 = 50 T 5,4 = 1 x 2 14 = 123 C 4,2 = 50 T 4,2 = 1 x 1 15 = 142 C 2,1 = 25 T 2,1 = 1 x 3 16 = 135 C 1,3 = 50 T 1,3 = 1 C 3,0 = 75 T 3,0 = 1 (d) Total Shipping =755 Total Cost = 735 Total Travel Tie = 7 t = 1 Fixed Cost = 360 t 21 = 1 y (0,8) t 21 = 2 y (8,6) t 21 = 4 y (6,0) x 8 11 = 121 C 0,8 = 75 T 0,8 = 1 x 6 12 = 122 C 8,6 = 50 T 8,6 = 1 C 6,0 = 125 T 6,0 = 2 (e) Total Shipping =243 Total Cost = 610 Total Travel Tie = 4 Figure 3.7: Case 2 results of space-tie network using CPLEX Figure 3.8 shows the space tie network solution. Retailers 1, 2, 3, 4, 5, and 7 are assigned to one route and use vehicle k 1, and retailers 6 and 8 are assigned to another route and use vehicle k 2. As a result, the odel obtains a transportation cost of $1,345. Therefore, the total profit (i.e., revenue inus transportation cost) in this case is $7,423. As a result, it can be concluded that the 43

59 MTP odel provides the revenue and transportation cost using the MTP-R and MTP-T odels, respectively. Objective Value: $7, Vehicle k 1 Route: y (0,7) 11 y (7,5) 11 y (5,4) 11 y (4,2) 11 y (2,1) 11 y (1,3) 11 y (3,0) 21 Vehicle k 2 Route: y (0,8) 21 y (8,6) 21 y (6,0) Figure 3.8: Space-tie network proble with eight retailers and one depot using CPLEX The objective function value ($7,423) obtained by sequentially solving using GAMS/CPLEX shows an absolute gap of and a relative gap of 9.08%. Therefore, in order to check if the odel is able to achieve a better solution, the sae case study is solved siultaneously using BARON. By using BARON, the odel solution resulted in an optiu solution for the product delivery plan, which included the tie of departure fro each node (depot or retailer) and the tie of arrival at each node. The solution obtained includes the product delivery plan x tt and units of product delivered S τt with the total revenue, which in this case is $8,768. The odel resulted in a reduced transportation cost of $1,320. Hence, the total profit (i.e., revenue inus transportation cost) increased in this case to $7,

60 Objective Value: $7, Vehicle k 1 Route: y (0,7) 11 y (7,5) 11 y (5,4) 11 y (4,2) 11 y (2,1) 11 y (1,3) 11 y (3,0) 21 Vehicle k 2 Route: y (0,8) 21 y (8,6) 21 y (6,7) 21 y (7,0) The results of a siultaneous solution ($7,448) by using GAMS/BARON shows an absolute gap of E-6 and a relative gap of 0.00 with CPU tie of seconds. As a result, it can be concluded that the MTP odel provides the optial revenue and transportation cost using GAMS/BARON by using the MTP-R and MTP-T odels, siultaneously Additional Case Studies for Validation: Revenue and Transportation Model with Liited Vehicle Capacity This section describes a case study in which the vehicle capacity is liited. This ay result in infeasible condition when using GAMS/CPLEX software. Hence, using BARON ay help to identify a feasible solution. If the vehicle capacity is liited copared to the deand of the retailers, then the odel becoes infeasible. Hence, the odel need to be solved siultaneously. This case study is siilar to the one presented in section However, in this case study, the vehicles have liited capacity. In the odified case study, the vehicle capacity is liited to 380 units, as shown in Table Table 3.19: Vehicle Capacity in Units of Products B k and Dispatch Cost Q k Vehicle K Capacity of Vehicle B k Cost Q k ($) k k The requests for orders fro each retailer are shown in Table Therefore, in order to check if the odel is able to obtain a solution, the sae case study is solved using BARON. The resulting solution with liited vehicle capacity is given in Tables 3.20 and

61 After solving case study three, the odel obtains the optiu and feasible solutions for product delivery plan x tt and also the nuber of units of product delivered S τt. Table 3.20 shows solutions for the nuber of products with deteriorated shelf life τ delivered at the retailer during tie period t. Copared to the solutions in the previous case, the BARON solver deterined that for optial profit, it will do a partial-order fill for retailer 1 and supply only 33 units instead of 142 as shown in Tables 3.11, 3.20, and So, the odel lowers the quantity of the products delivered to retailer 1 instead of delivering the original quantity of 142. In addition, when the vehicle capacity is liited, BARON solves the odel by taking into consideration the availability of the vehicle and ensures that profit is axiized. Siilarly, the solvers recoended the best deand fro retailer 3 in ters of profit axiization which is 56 in tie period 6. In this case study, the total revenue is $8,111 copared to the total revenue for the previous case, which was $8,768. Table 3.20: S τt Quantity of Product with Deteriorated Shelf Life τ Delivered at Retailer during Tie t using BARON Retailer M Deteriorated Shelf Life τ Tie Delivered t Table 3.21 represents the product delivery plan for the planning horizon. By optiizing all decision variables in the MTP-R odel, the revenue objective is axiized to obtain the delivery quantities x tt and the delivery plan for the planning horizon for each retailer. 46

62 Table 3.21: x tt Nuber of Units to Ship fro Depot to Retailer during Tie Period t using BARON Retailer M Tie Depart t Along with the delivery quantities x tt and the delivery plan for the planning horizon, the MTP-T odel also obtains the best route in order to iniize the total transportation cost. For this case study, Figure 3.9 shows the space-tie network solution. Tie Arrive t

63 NETWORK kt y (i,j) x tt C ($) T (days) t = 1 t = 1 11 y (0,7) t = 2 11 y (7,5) Fixed Cost = 265 x 7 11 = 131 C 0,7 = 75 T 0,7 = 1 x 5 12 = 120 C 7,5 = 50 T 7,5 = 1 t = 3 t = 4 11 y (5,2) t = 6 11 y (2,0) x 2 14 = 123 C 5,2 = 75 T 5,2 = 2 C 2,0 = 75 T 2,0 = 1 t = 6 (f) Total Shipping = 374 Total Cost = 540 Fixed Cost = 265 Total Travel Tie = 5 t 16 = 6 y (0,3) t 16 = 8 y (3,0) x 3 66 = 56 C 0,3 = 75 T 0,3 = 1 C 3,0 = 75 T 3,0 = 1 t = 1 (g) Total Shipping = 56 Total Cost = 415 Fixed Cost = 265 Total Travel Tie = 2 t 21 = 1 y (0,8) t 21 = 2 y (8,6) t 21 = 3 y (6,4) x 8 11 = 121 C 0,8 = 75 T 0,8 = 1 x 6 12 = 122 C 8,6 = 50 T 8,6 = 1 x 4 13 = 104 C 6,4 = 75 T 6,4 = 1 t = 4 t 21 = 5 y (4,1) t 21 = 7 y (1,0) x 1 15 = 33 C 4,1 = 75 T 4,1 = 2 C 1,0 = 75 T 1,0 = 1 (h) Total Shipping = 380 Total Cost = 615 Total Travel Tie = 6 Figure 3.9: Case 3 results on space-tie network by using BARON 48

64 As can be seen in Figure 3.10, retailers 2, 5, and 7 are assigned to one route by using vehicle k 1 ; retailer 3 is assigned to another route by using vehicle k 1 ; and retailers 1, 4, 6, and 8 are assigned to a different route by using vehicle k 2. As a result, the odel is able to achieve the iniu transportation cost, which is $1,570. Therefore, the total profit (i.e., revenue inus transportation cost) in this case is $6,541. As a result, it can be concluded that the MTP odel provides the optial revenue and transportation cost using the MTP-R and MTP-T odels, siultaneously. Therefore, the decision aker will be able to deterine the correct aount of product that can be shipped to the retailers and the correct vehicles to be dispatched, including the best routes that the vehicles can take to satisfy the objective of axiizing the total profit. Details of the solution are shown in Figure This type of solution is helpful for supply chain copanies that serve ultiple supply chain syste with liited vehicle capacity. Objective Value: $6, Vehicle k 1 Route: y (0,7) 11 y (7,5) 11 y (5,2) 11 y (2,0) 16 Vehicle k 1 Route: y (0,3) 16 y (3,0) 21 Vehicle k 2 Route: y (0,8) 21 y (8,6) 11 y (6,4) 21 y (4,1 ) 21 y (1,0) Figure 3.10: Space-tie network proble with eight retailers and one depot using BARON 49

65 Results of the siultaneous solution ($6,541) by using GAMS/BARON shows the absolute gap of E-6 and the relative gap of 0.00 with CPU tie of seconds. As a result, it can be concluded that the MTP odel provides the optial revenue and transportation cost using GAMS/BARON by using the MTP-R and MTP-T odels, siultaneously Additional Case Studies for Validation: Revenue and Transportation Model with Liited Vehicle Capacity This section describes a case study in which the vehicle capacity is liited. As explained in the previous case, this ay cause infeasibility when using CPLEX to solve the odel sequentially. Thus, other possible solutions ay have to be explored. In order to avoid an infeasible solution by using the sae solver, the nuber of vehicles ay need to be increased. If the vehicle capacity is liited copared to the deand for retailers, then the odel becoes infeasible. Hence, the nuber of vehicles ust be increased. This case study is siilar to the one presented in section However, in this case, the vehicles have liited capacity. In the odified case study, the vehicle capacity is liited to 550 units, as shown in Table Thus, with two vehicles, the solution is infeasible. Table 3.22: Vehicle Capacity in Units of Products B k and Dispatch Cost Q k Vehicle K Capacity of Vehicle B k Cost Q k ($) k k The requests for orders fro each retailer is shown in Table Therefore, in order to check if the odel is able to achieve a solution, the sae case study is used to illustrate the MTP odel sequentially using solver CPLEX to obtain the solution. Because of the liited capacity of vehicles, the proble becoes infeasible. Therefore, the nuber of vehicles is increased to three. 50

66 Vehicle k 3 with siilar capacity and cost was added to the fleet. The resulting solution with the increased nuber of vehicles is provided in Tables 3.23 and After case study four is solved, the odel obtains the solutions for the product delivery plan x tt and units of product delivered S τt. Table 3.23 shows solutions for the nuber of products with deteriorated shelf life τ delivered at the retailer during tie period t. In this case study, the total revenue is $8,768. Table 3.23: S τt Quantity of Product with Deteriorated Shelf Life τ Delivered at Retailer during Tie t using CPLEX Retailer M Table 3.24 represents the product delivery plan for the planning horizon. By optiizing (i.e., operation research) all decision variables in the MTP-R odel, the revenue objective is axiized and the nuber of units of products delivered and the delivery plan for the planning horizon for each retailer is obtained. Deteriorated Shelf Life τ Tie Delivered t

67 Table 3.24: x tt Nuber of Units to Ship fro Depot to Retailer during Tie Period t using CPLEX Retailer M Tie Depart t After obtaining the x tt delivery plan for the planning horizon, the MTP-T odel is run to obtain the best route in order to iniize the total transportation cost. For this case study, solved using CPLEX, Figure 3.11 shows the space-tie network solution. Tie Arrive t

68 NETWORK kt y (i,j) x tt C ($) T (days) t = 1 Fixed Cost = 265 t = 1 t = 2 11 y (0,6) x 6 12 = 122 C 0,6 = 125 T 0,6 = 2 t = 3 t 11 = 4 y (6,2) t = 5 11 y (2,1) t = 6 11 y (1,3) x 2 14 = 123 C 6,2 = 75 T 6,2 = 2 x 1 15 = 142 C 2,1 = 25 T 2,1 = 1 x 3 16 = 135 C 1,3 = 50 T 1,3 = 1 t = 7 t = 1 (h) 11 y (3,0) Total Shipping = 522 C 3,0 = 75 T 3,0 = 1 Total Cost = 615 Fixed Cost=265 Total Travel Tie= 7 t 21 = 1 y (0,7) t 21 = 2 y (7,5) t 21 = 3 y (5,4) t 21 = 5 y (4,0) x 7 11 = 131 C 0,7 = 75 T 0,7 = 1 x 5 12 = 120 C 7,5 = 50 T 7,5 = 1 x 4 13 = 104 C 5,4 = 50 T 5,4 = 1 C 4,0 = 100 T 4,0 = 2 t = 1 (i) Total Shipping =355 Total Cost = 540 Fixed Cost = 265 Total Travel Tie= 5 t 31 = 1 y (0,8) x 8 11 = 121 C 0,8 = 75 T 0,8 = 1 31 y (8,0) C 8,0 = 75 T 8,0 = 1 t = 2 (j) Total Shipping =121 Total Cost = 415 Total Travel Tie= 2 Figure 3.11: Case 4 results on space-tie network by using CPLEX 53

69 As can be seen in Figure 3.12, retailers 1, 2, 3, and 6 are assigned to one route by using vehicle k 1 ; retailers 4, 5, and 7 are assigned to another route by using vehicle k 2 ; and retailer 8 is assigned to a different route by using vehicle k 3. As a result, the odel achieves a transportation cost of $1,570. Therefore, the total profit (i.e., revenue inus transportation cost) in this case is $7,198. Details of the solution are shown in Figure Objective Value: $7, Vehicle k 1 Route: y (0,6) 11 y (6,2) 11 y (2,1) 11 y (1,3) 11 y (3,0) 21 Vehicle k 2 Route: y (0,7) 21 y (7,5) 21 y (5,4) 21 y (4,0) 31 Vehicle k 3 Route: y (0,8) 31 y (8,0) Figure 3.12: Space-tie network proble with eight retailers and one depot by using CPLEX The objective function value ($7,189) obtained by sequentially solving using CPLEX shows an absolute gap of and a relative gap of 9.238%. Therefore, in order to check if the odel can achieve a better solution, the sae case study is solved siultaneously using BARON, After solving case study four, the odel obtains the optiu and feasible solutions for product delivery plan x tt and also the nuber of units of product delivered S τt. Table 3.25 shows solutions for the nuber of products with deteriorated shelf life τ delivered at the retailer during tie period t. Copared to the solutions in this case by using CPLEX, the BARON solver 54

70 deterined that for optial profit, BARON solves the odel by taking into consideration the availability of the vehicle when the vehicle capacity is liited and ensures that profit is axiized. Siilarly, the solvers recoended the best deand fro retailer 3 in ters of profit axiization which is 56 in tie period 6. In this case study, the total revenue is $8,765 copared to the total revenue for this case by using CPLEX, which was $8,768. Table 3.25: S τt Quantity of Product with Deteriorated Shelf Life τ Delivered at Retailer during Tie t using BARON Retailer M Deteriorated Shelf Life τ Tie Delivered t Table 3.26 represents the product delivery plan for the planning horizon. By optiizing all decision variables in the MTP-R odel, the revenue objective is axiized to obtain the delivery quantities x tt and the delivery plan for the planning horizon for each retailer. 55

71 Table 3.26: x tt Nuber of Units to Ship fro Depot to Retailer during Tie Period t using BARON Retailer M Tie Depart t Along with the delivery quantities x tt and the delivery plan for the planning horizon, the MTP-T odel also obtains the best route in order to iniize the total transportation cost. As can be seen in Figure 3.13, retailers 4, 5, and 7 are assigned to one route by using vehicle k 1 ; retailer 3 is assigned to another route by using vehicle k 1 ; and retailers 1, 2, 6, and 8 are assigned to a different route by using vehicle k 2. As a result, the odel is able to achieve the iniu transportation cost, which is $1,520. Therefore, the total profit (i.e., revenue inus transportation cost) in this case is $7,245. As a result, it can be concluded that the MTP odel provides the optial revenue and transportation cost using the MTP-R and MTP-T odels, siultaneously. Therefore, the decision aker will be able to deterine the correct aount of product that can be shipped to the retailers and the correct vehicles to be dispatched, including the best routes that the vehicles can take to satisfy the objective of axiizing the total profit. Details of the solution are shown in Figure This type of solution is helpful for supply chain copanies that serve ultiple supply chain syste with liited vehicle capacity. Tie Arrive t

72 Objective Value: $7, Vehicle k 1 Route: y (0,7) 11 y (7,5) 11 y (5,4) 11 y (4,0) 16 Vehicle k 1 Route: y (0,3) 16 y (3,0) 21 Vehicle k 2 Route: y (0,8) 21 y (8,6) 11 y (6,2) 21 y (2,1 ) 21 y (1,0) Figure 3.13: Space-tie network proble with eight retailers and one depot using BARON Wherein the odel solution resulted in an optiu solution for the product delivery plan, which included the tie of departure fro each node (depot or retailer) and the tie of arrival at each node. The solution obtained includes the product delivery plan x tt and the nuber of units of product delivered S τt. When solving using BARON, a total revenue of $8,765 is obtained. The odel resulted in a decreased transportation cost of $1,520 as copared to solving using CPLEX. Hence, the total profit (i.e., revenue inus transportation cost) in this case is increased to $7,245. Results of the siultaneous solution ($7,245) by using GAMS/BARON shows an absolute gap of E-6 and a relative gap of 0.00 with CPU tie of seconds. As a result, it can be concluded that the MTP odel provides the optial revenue and transportation cost using GAMS/BARON by using the MTP-R and MTP-T odels, siultaneously. 57

73 3.5 Conclusion In this chapter, a novel approach for an optial replenishent strategy for a single perishable product and its associated atheatical odel is presented. This odel is decoposed into two objective coponents: axiizing total revenue based on the shelf life of the product, and iniizing the total transportation cost. In this paper, a ixed-integer linear prograing odel is proposed to forulate the odel sequentially, and is solved using CPLEX. Also, it is proposed to forulate the odel siultaneously, and is solved using BARON. Case studies are provided to obtain insight into the odel behavior and dynaics. In sequential case studies, the MTP-R odel was able to axiize the total revenue objective by obtaining the optial delivery plan for the planning horizon for each retailer, whereas the MTP-T odel was able to iniize the total transportation cost by identifying the optial route to deliver the product to the retailers. Finally, the MTP odel was able to provide the decision aker with a coprehensive view of the delivery plan while axiizing total profit for the proposed proble. In the siultaneous case studies, the MTP odel was able to provide the optial solution by axiizing the total revenue and iniizing the total transportation cost siultaneously. For future studies, this research could be extended to the proble involving a dynaic perishable product in which the retailer can have ultiple orders within the planning horizon with varying shelf lives. The odel could also be extended to investigate cases in which ultiple products and varying shelf lives are considered. The current odel does not take into account depot location. Thus, the odel could be extended to a ulti-product, ulti-retailer, and depotlocation study. 58

74 3.6 References Bai, X. (2010). Research on coordination of perishable products supply chain with the cobined option and quantity discount contract. Proceedings of the ICLEM 2010 slogistics for Sustained Econoic Developent: Infrastructure, Inforation, Integration, pp Cagliano, A. C. (2015). Networks against tie. Supply chain analytics for perishable products. Production Planning & Control, vol.26 n. 6, pp Deep, K., Nagar, A., Pant, M., & Bansal, J. C. (2012). Advance in Intelligent Systes and Coputing, Proceedings of the International Conference on Soft Coputing for Proble Solving (SocProS 2011) Deceber 20-22, 2011: Volue 2 (Vol. 236, pp ): Springer Science & Business Media. Gao, N., & Ryan, S. M. (2014). Robust design of a closed-loop supply chain network for uncertain carbon regulations and rando product flows. EURO Journal on Transportation and Logistics, 3(1), ILOG, I. (2012). Inc. CPLEX 12.5 User Manual: Soers, NY, USA. Liu, B.-l., & Ma, W.-h. (2008). Application of quantity flexibility contract in perishable products supply chain coordination. Paper presented at the Control and Decision Conference, 2008 Madduri, V. S. R. (2009). Inventory policies for perishable products with fixed shelf lives. The Pennsylvania State University. Mena, C., Adenso-Diaz, B., & Yurt, O. (2011). The causes of food waste in the supplier retailer interface: evidences fro the UK and Spain. Resources, Conservation and Recycling, 55(6), Qi, E., Shen, J., & Dou, R. (2013). International Asia Conference on Industrial Engineering and Manageent Innovation (IEMI2012) Proceedings: Core Areas of Industrial Engineering: Springer Science & Business Media. Shukla, M., & Jharkharia, S. (2013). Agri-fresh produce supply chain anageent: A state-ofthe-art literature review. International Journal of Operations & Production Manageent, 33(2), Valero, A., Carrasco, E., & García-Gieno, R. M. (2012). Principles and ethodologies for the deterination of shelf life in foods. In Trends in Vital Food and Control Engineering, Yared Lea, D. K., & Gatew, G. (2014). Loss in perishable food supply chain: An optiization approach literature review. International Journal of Scientific & Engineering Research, 5(5),

75 CHAPTER 4 OPTIMAL LOCATION ALLOCATION AND DELIVERY STRATEGY FOR PERISHABLE PRODUCTS 4.1 Introduction As discussed in the Chapter 3, the distribution of perishable products provides an additional challenge in the supply chain due to their fixed shelf life. To further clarify, the aount of perishable product delivered fro a depot to the retailers is often liited by its shelf life. Therefore, the nuber of depots and their locations ay increase the cost and difficulty in vehicle routing in order to satisfy deand. Due to the challenges involved in the distribution of perishable products, the location of depots and routing of vehicles are very iportant issues. The transportation syste plays a significant role in the distribution of perishable products with respect to tie of delivery, eeting deands, and ensuring a low cost. It is iportant for supply chain systes of perishable products to have an optial nuber and locations of depots so that retailer deand is satisfied and total transportation cost is iniized. Many researchers have stressed the iportance of linkage between the supply chain of perishable products, location allocation, and distribution routes. In a perishable supply chain, a suitable plan involving depot locations and transportation routing tie ust be designed (Khalili-Daghani, Abtahi, & Ghasei, 2015). Aluur and Kara (2007) ention that there is a ajor relation between the logistics cost and the transportation cost. In fact, the deand for depots and the nuber of depot centers for perishable products have increased year by year (Tsao, 2013; Aori, Meyr, Aleder, & Alada-Lobo, 2013). Therefore, having a plan for the supply chain of perishable products is all the ore iportant. Perishable products are classified as having either a rando shelf life or a fixed shelf life. Products with a rando shelf life, such as fruits and vegetables, have expiration dates that are tied 60

76 to specific deterioration rates. Products with a fixed shelf life, such as ilk and juices, are given an expiration date at the tie of production and are considered to have the sae quality fro that tie until the expiration date (Madduri, 2009). Shelf life plays an iportant role in the odeling of the location allocation proble of any food supply chain for a product (Sahai & Nabiar, 2013). This paper considers the proble of optiizing the distribution, nuber of depots, transportation cost, and revenue for products with a fixed shelf life. Here, the fixed shelf life refers to the period of tie starting fro anufacturing the product (on the sae day of leaving the depot) until the expiration date, during which tie the product reains in good quality (Valero, Carrasco, & García-Gieno, 2012). In other words, the fixed shelf life is the aount of tie for a given perishable product to expire (Ministry for Priary Industries, 2014, July 22). Due to the perishable nature of certain products, effective delivery planning and the optial nuber and location of depots can offer greater savings in logistics operations. Siilarly, the optiization of vehicle routes aids in the distribution of perishable goods and is incorporated into the research regarding distribution optiization. In the last several years, facility location has becoe the priary focus of uch research and literature. Also, any researchers and organizations have focused on the challenge of a facility location that increases the value of the supply chain. In reviewing the facility location literature, it is evident that there are nuerous definitions for both the facility location proble and the fixed-cost facility location proble. The facility location proble is an optiization proble that iniizes cost by deterining the optial nuber and location of depots to serve a set of retailers or custoers (Wu, Zhang, & Zhang, 2006). This cost ay include the fixed cost of opening the depot and the transportation cost fro the depot to retailers. The fixed-cost facility location proble is a traditional location proble and fors the basis of any of the 61

77 location odels that have been used in supply chain design (Daskin, Snyder, & Berger, 2005). Beasley (1990) defines the facility location proble as a cobination of a set of facility locations and a set of retailers or custoers, that is, each retailer or custoer needs to be served by the depot, and the depot utilized to serve each retailer or custoer ust be identified. The ain challenge in the decision support syste of shipping perishable products is the creation of a schedule for the truck or vehicle that will be transporting the products. However, this paper focuses attention on iniizing the transportation cost by identifying the best vehicle scheduling solution along with the aount of products being shipped (i.e., truck dispatch cost and route cost). Generally, the vehicle scheduling proble can be classified as a traditional proble that has been studied for any years. Many ethods have been developed for solving the VRP with tie windows under different constraints. For exaple, Deep (2012) reviewed the forulation and exact (coplete/optial solution) algorith of the VRPTW and categorized the forulation into four ain ethods: spanning free tree forulation, path forulation, arc forulation, and arc-node forulation. In the arc forulation, each arc of a basic directed graph is related to a binary variable (Gao & Ryan, 2014). In this graph, these binary variables are related to nodes in the arc-node forulation of the VRPTW proble. The spanning free tree forulation is a ethod for obtaining lower constraints for the VRPTW proble (Cagliano, 2015). This paper involves a study of finding the optial nuber of depots and delivery strategies for a single perishable product for which the deand and price as a function of shelf life are known. It is assued that the product has a deteriorated shelf life of zero once it leaves the depot (i.e., the product will be fresh and will start to lose its quality once it leaves the depot). A tie continuu is assigned for the deteriorated shelf life of a product, starting fro tie zero when it leaves the 62

78 depot to the designated shelf life duration. The product is fresh at tie zero and starts losing its quality and undergoes price reduction as tie passes. To odel this proble, the objective function is decoposed into two coponents: axiizing total revenue during the planning horizon (i.e., revenue due to sale of the product) and iniizing costs of opening depots and transportation costs (i.e., cost incurred fro opening depots and transportation activities). The proble is odeled as MINLP proble and solved using the BARON solver. The solution obtained axiizes the total profit, which is equal to the total revenue inus the total cost of opening depots and total transportation cost. Finally, a detailed experiental test proble is provided in order to gain insight into the dynaics of the odel. The key contributions of this paper are as follows: (1) developent of a odel to axiize total profit during the planning horizon that enables the decision aker to obtain the right tie, right location (best route and retailer), right quantity for the given supply chain proble, and right nuber of depots; and (2) developent of a detailed experiental test proble to obtain insight into the odel dynaics with varying different paraeters. 4.2 Model to Maxiize Total Profit of Location Allocation In this paper, creating a odel that axiizes the total profit of the location allocation of the perishable product (MTP-L) involves utilizing the tie-indexed variable forulation in which the planning horizon is discretized into N intervals of one unit length each. This odel consists of two different objective functions: (a) to axiize total profit revenue of the location allocation (MTP-RE) and (b) to iniize the cost of location allocation (opening a depot) and transportation (MTP-AT). Note that each objective function ust satisfy the constraints of the odel. Also, the MTP-L odel is developed as an MINLP odel. In the ipleentation, after solving each 63

79 objective, the MTP-AT objective value is subtracted fro the MTP-RE objective value in order to obtain the profit for the planning horizon Proble Definition This study deterines the nuber of depots to be opened and their locations for shipping a single perishable product for which the deand and price as a function of shelf life are known. With respect to the location of depots, one of the ost iportant factors is the tie of delivery because the perishable product decays. Thus, the objective here is to deterine the nuber of depots and their locations in order to iniize the total cost of the supply chain while eeting the due dates for delivery of the perishable product and considering varying prices of shelf life in order to axiize profits. In this proble, several retailers are served by a set of depots, and they are connected via a supply chain network. A fleet of trucks is used to ship the product fro a set of depots to a set of retailers. A truck can visit a nuber of retailers and return back to the sae depot. The inventory capacity, initial inventory, and deand for the product for given shelf lives for each delivery day is known for each retailer. It is assued that the product is fresh when it leaves the depot. Also, the length of the planning horizon is greater than the length of the shelf life (i.e., T > τ ). In addition, the product has a unit price with a varying shelf life over the planning horizon. It is assued that the fixed cost of opening a depot is aortized over n tie periods with 0% interest rate. For exaple, if the fixed cost of opening a depot is $10,400 with a pay-back period of two years, then the fixed cost is aortized over a tie period of two years, and hence the onthly fixed cost is $433, and the weekly cost is $100. Also, it is assued that the depot has enough product to ship, and the product has a deteriorated shelf life of zero when it leaves the depot. As a result, by opening the optial nuber of depots, the supply chain network can axiize profit by axiizing the total revenue while iniizing the costs of opening depots 64

80 and transportation. In addition, it is assued that the inventory-holding cost and loading-andunloading costs are negligible. Also, it is assued that the vehicle capacity is the sae for all vehicles at all depots Notation and Forulation The notations relative to odels MTP-RE and MTP-AT are as follows: Sets N(t): set of nodes in space-tie network A(t): set of arcs in space-tie network K: set of trucks T: set of space-tie Indices k K: truck index k = 1,, K t, t T: tie period index t, t = 1,, T τ T: deteriorated shelf life, which is the nuber of days that the product has deteriorated τ = 0,, T, N(t): index for retailer, 0,, M, where = 0 and = 0 represent depot l N(t): set of retailers () satisfied at tie period (t) i, j N(t): nodes () to ( ) w N(t): set of candidate depots w = 1,, W Paraeters B k : capacity of truck k D τt : deand for product with deteriorated shelf life τ at retailer during tie period t P τ : price of product with deteriorated shelf life τ 65

81 I τ0 : initial inventory with deteriorated shelf life τ at retailer O : inventory capacity of retailer Q k : fixed cost associated with dispatching truck k C ij w : cost of travel fro node i to node j (i =, j = ) for each depot w T w : tie for travel fro retailer to for each depot w, G : upper bound on x vector values h w : fixed cost per period tie of opening depot w N(t) Decision Variables x w tt : nuber of units to ship fro depot w at tie period t and to arrive at retailer at tie period t ; aggregate vector of all decision variables x = {x w tt : = 1,, M, t = 1,, T, t = t,, T} represents product delivery plan for planning horizon e τt {0,1}: binary variable indicating if deand for product with shelf life τ during tie period t at retailer is satisfied I τt : inventory level with deteriorated shelf life τ at retailer at end of tie period t S τt : product with deteriorated shelf life τ delivered by retailer during tie period t Note that to define other decision variables, first, the truck index ust be augented and the pair (k, t) ust be used to represent truck k that leaves depot at tie period t. u ktw {0,1}: binary variable indicating if truck (k, t, w) is used (1 if used) y ktw (i,j) {0,1}: binary variable indicating if truck (k, t, w) travels arc (i, j) A(t) (1 if used) g w {0,1}: binary variable indicating if depot is selected (1 if selected) Auxiliary Variables v l ktw {0,1}: binary variable indicating if truck (k, t, w) visits node l N(t) (1 if used) 66

82 z ktw : arrival tie of truck (k, t) back to depot w η w tt {0,1}: binary variable indicating whether retailer ε M receives any shipent fro depot w on day t, which was sent on day t. In other words, MTP-L Model η w tt = { 1 x w tt > 0 0 otherwise The objective of the MTP-L odel is to axiize total profit during the planning horizon, that is, the objective function MTP-RE inus the objective function MTP-AT: Max profit f f MTPRE MTPAT T T M W T K W T K W w ktw ktw t w w ( i, j) ( i, j) k t1 0 1 w1 t1 k1 w1 ( i, j) A0 ( t) t1 k 1 w1 Max(( P s ) ( h g C y Q u )) MTP-RE Model follows: The MTP-RE odel axiizes total revenue obtained fro the product delivered as Max st.. T T M P s t W w t t t, t w1 S I x 1..., M, t 1,..., T, 0 (4.2) W w t t 1, t1 t, t w1 t I t I 1, t1 S I I x 1..., M, t 1,..., T, 1,..., t 1 (4.3) S T 0 t , M, t 1,..., T, 1,..., T ( 4.4) T I t 0 t t w tt' w tt' t e , M, t 1,..., T (4.5) O 1..., M, t 1,..., T (4.6) τt S e D 1..., M, 0,..., T, t 1,..., T (4.7) x ax B 1..., M, t 1,..., T, t' 1,..., T, w 1,..., W (4. 8) k t t k x, I, S , M, t 1,..., T, T, t' 1,..., T, 1,..., T, w 1,..., W (4.9) t e {0,1} 1..., M, t 1,..., T, 1,..., T ( 4.10) (4.1) 67

83 The first objective (4.1) axiizes total revenue. The inventory constraint sets (4.2), (4.3), and (4.4) ensure a balance in the inventory level with shelf life τ at each retailer at the end of tie period t. Constraint set (4.5) ensures that each retailer selects only one deand for the product with shelf life τ during tie period t. Constraint set (4.6) ensures that the inventory level with deteriorated shelf life τ at retailer at end of tie period t does not exceed the inventory capacity at each retailer. Constraint set (4.7) states that the selected units of product delivered per tie period t ust be less than or equal to deand. Constraint set (4.8) ensures liiting the units of product shipped to any retailer per tie period t in order to not exceed the axiu truck capacity. Finally, constraint sets (4.9) and (4.10) ensure integrality and non-negativity, respectively MTP-AT Model The MTP-AT odel attepts to iniize the total transportation cost incurred in order to carry out a given replenishent plan (x w tt ). Evaluation of this odel requires solving a nested optiization proble that deterines the optial routes and fleets used. In particular, this can be odeled siilar to that of the VRPTW constraints. The VRP is concerned with finding the optial nuber and type of trucks, associated routes, and starting and ending ties for trips in order to satisfy a retailer s deand within the specified tie windows while iniizing the cost of transportation. An adaptation of the VRP forulation odified to take into consideration the varying price odel is discussed next. The supply chain network odel includes a set of nodes (N) consisting of retailers and a depot, and a set of arcs (A) representing routes. In order to characterize the routes, Τ space-tie networks, each associated with a tie period t = 1,, T in the planning horizon, are constructed, as shown in Figure 4.1. As can be seen, the horizontal and vertical nodes (N(t), A(t)) represent 68

84 space (retailer index) and tie (tie period index), respectively. The first and last node of any route is (w, t), which represents the depot. An arc appears between every two nodes, only if the difference between the tie periods is equivalent to the travel tie between the corresponding retailers (or depot and retailer). The final arc of the route connects it back to the depot to coplete the route. Finally, the arc costs represents the distance between the corresponding nodes (or depot and retailer). Figure 4.1: Space-tie network (N(t), A(t)) The space-tie network shown in Figure 4.1 is denoted by (N(t), A(t))(t = 1,, T), where N(t) and A(t) are the sets of nodes and arcs, respectively. In particular, for network (N(t), A(t)), the set of nodes represents pairs of retailers and tie periods as N(t) = { (, t ): = 1,..., M, t = t,..., T} A node corresponding to the depot is added as N 0 (t) = N(t) {(w, t)} Additionally, an arc occurs between every two nodes ( 1, t 1 ) and ( 2, t 2 ), if a vehicle can traverse the distance fro retailer 1 to 2 in exactly (t 2 t 1 ) tie periods. More specifically, the set of arcs is defined as 69

85 A(t) = {(( 1, t 1 ), ( 2, t 2 )): 1 = 0,..., M, 2 = 1,..., M, 1 2, t 1 = t,..., = t 1 + T w } There is also an arc between all nodes in N(t) and the depot to close the loop (see Figure 4.1) as For notational convenience, A 0 (t) = A(t) {(i, (w, t)): i N(t) } R s (( 1, t 1 ), ( 2, t 2 )) = 1 R e (( 1, t 1 ), ( 2, t 2 )) = 2 where retailer is associated with the start and end of the arc ( 1, t 1 ), ( 2, t 2 ), respectively. w w Finally, T C represents the tie and cost of traversing arc (i, j) A 0 (t), respectively, which can be obtained fro the travel tie and cost atrices, respectively, as T w w (i,j) = T Rs (i,j),r e(i,j) C w w (i,j) = C Rs (i,j),r e(i,j) After the MTP-RE odel is solved, the objective value and replenishent plan to arrive at w each retailer x tt (i.e., nuber of units to ship fro depot at tie period t and to arrive at retailer at tie period t ) are obtained. The MTP-AT atheatical odel follows: 70

86 st.. W T K W T K W w kt w kt w w w ( i, j) ( i, j) k Min h g C y Q u w1 t1 k 1 w1 ( i, j) A0 ( t) t1 k 1 w1 kt w kt w ( i, ) (, j) ( i, ) A( t) (, j) A( t) y y N( t), k 1,..., K, t 1,..., T, w 1,.., W (4.12) kt w kt w y(( w, t), ) y(,( w, t) ) N() t N() t kt w ( i, j) ( i, j) A( t) ( i, j) A( t) w tt ' w kt w ( i, j) ( i, j) w tt ' kt w j (4.11) g k 1,..., K, t 1,..., T, w 1,.., W (4.13) w y k 1,..., K, t 1,..., T, w 1,.., W (4.14) kt w y z k 1,..., K, t 1,..., T, w 1,.., W (4.15) x G 1,..., M, k 1,..., K, t 1,..., T, w 1,..., W (4.16) (, t ') N ( t ) K k t w ( t, ) k 1 ktw w kt w kt w tt' (, t ') k x B u k 1,..., K, t 1,..., T, w 1,.., W (4.17) w tt' (, t ') N( t), t 1,..., T, w1,..., W (4.18) kt ' w ( t z ) t ' (1 u ) k 1,..., K, t 1,..., T, t ' t 1,..., T, w 1,.., W (4.19) kt w kt w kt w w ( i, j) l tt' u, y,,, g {0,1} 1,..., w 1,.., W, M, k 1,..., K, t 1,..., T, t ' t 1,..., T, N ( t),( i, j) A( t) (4.20) kt w w z 0 k 1,..., K, t 1,..., T, w 1,.., W (4.21) The first objective (4.11) iniizes the total transportation cost, which consists of variable and fixed costs. The balance constraint set (4.12) ensures that if the vehicle visits a node (retailer) other than the depot in the network, then the vehicle ust leave that node (retailer). Constraint set (4.13) ensures that each route originates and terinates at the node corresponding to the depot (w, t). Constraint sets (4.14), (4.15), and (4.16) ensure that the auxiliary variables are connected to the ain decision variables, where G is an upper bound on the x-vector values. Constraint set (4.17) ensures that the shipped quantity does not exceed the capacity of the truck. Constraint set (4.18) ensures that each node in the space-tie network is visited, if the corresponding retailer is included in the delivery plan. Constraint set (4.19) ensures that a vehicle can be assigned to a new route only after the vehicle has returned to the depot. This constraint can be enforced using the following logical constraints: t t if z ktw > t, then u ktw = 0. Finally, constraint sets (4.20) and (4.21) are integrality and non-negativity, respectively. 71

87 MTP-L Model The objective of the MTP-L odel is to axiize total profit during the planning horizon, that is, the objective function MTP-RE inus the objective function MTP-AT: Max profit f f MTPRE MTPAT T T M W T K W T K W w ktw ktw t w w ( i, j) ( i, j) k t1 0 1 w1 t1 k1 w1 ( i, j) A0 ( t) t1 k 1 w1 Max (( P s ) ( h g C y Q u )) Figure 4.2 illustrates the MTP-L odel flow chart. MTP-L SIMULTANEOUS MODEL Max profit = f MTP RE f MTP AT s. t. Ax b x 0 MTP-RE Model Paraeters Max f MTP RE s. t. Ax b x 0 x tt MTP-AT Model Paraeters Min f MTP AT s. t. Ax b x 0 f MTP AT Value kt y (i,j) f MTP RE Value Figure 4.2: MTP-L odel flow chart 72

88 4.3 Experiental Evaluation The General Algebraic Modeling Syste (GAMS) is used to odel the MTP-L proble, and the proble is solved siultaneously using the solver BARON The coputational experients are perfored on an Intel i GHz processor with 4 GB of eory and a Windows 7 operating syste. 4.4 Experiental Design and Tests Case Study One: Revenue and Transportation Model with Unliited Vehicle Capacity In this case study, there are two depots, eight retailers, and two vehicles. It is assued that the vehicles have unliited capacity. The length of the tie horizon is T = 6, and the axiu shelf life is 5. Table 4.1 suarizes the deand for each retailer while ordering a product with a different deteriorated shelf life τ each day. Note that a retailer can place orders on each day (t) with deteriorated shelf lives of 0 to τ = t 1. Also, the travel tie between the depot and the retailer ust not exceed the product s (deand) shelf life. For exaple, in Table 4.1, retailer 2 places an order on day 4 and hence it can order the product with τ = 0 to 3. Note that retailer 5 can order a product with a deteriorated shelf life of zero because the travel tie between depot w 1 and retailer 2 is one day (i.e., D 2 04 = 71, D 2 14 = 86, D 2 24 = 95, D 2 34 = 123). The price of the product (P τ ) decreases as the deteriorated shelf life of the product increases. For exaple, the price of the product with a shelf life τ = 2 is $9, while the price of the product when the shelf life τ = 4 is $6. 73

89 Table 4.1: D τt Deand for Product with Deteriorated Shelf Life τ at Retailer Table 4.2 illustrates the inventory capacity O for each retailer. Table 4.3 shows the initial inventory I τ0 with deteriorated shelf life τ at retailer. The vehicle capacity in ters of the nuber of units of products is provided in Table 4.4. The cost of opening a depot is provided in Table 4.5. Table 4.2: Inventory Capacity O of Retailer Retailer M Capacity

90 Table 4.3: Initial Inventory with Deteriorated Shelf Life τ at Retailer Retailer M Deteriorated Shelf Life τ I τ Table 4.4: Vehicle Capacity in Units of Products B k and Dispatch Cost Q k Vehicle K Capacity of Vehicle B k Fixed Cost Q k k k Table 4.5: Cost of Opening Depot per Period Tie Depot W Opening Depot Cost h w w 1 0 w Table 4.6 presents the tie (T w ) required in day for travel fro the retailer to (, = 0,, M where w iplies the depot). Table 4.7 shows the distance of travel fro retailer to retailer (, = 0,, M, where 0 iplies the depot). Note that, the cost per ile of travel fro depot or retailer to retailer is five cents per ile. 75

91 Table 4.6: Tie (days) for Travel fro Depot w or Retailer to Retailer (T w ) Depot w 1 Depot w w Table 4.7: Distance (iles) of Travel fro Depot w or Retailer to Retailer C Depot w 1 Depot w After the case study is solved, the odel obtains solutions for the product delivery plan w x tt and nuber of units of product delivered S τt. Table 4.8 shows an optiu solution for the nuber of units of products with deteriorated shelf life τ delivered during tie period t. It can be seen that each retailer receives the ordered quantity of product. The price of the product is 76

92 deterined by the deteriorated shelf life τ (Table 4.1). For instance, retailer 5 receives a quantity of 120 units of product with deteriorated shelf life τ = 1 at tie delivered t = 2 (i.e., S 5 12 = 120). Based on the quantity deanded and specific price (deterined by shelf life), the odel is able to deterine the possible total revenue, in this case $8,768. Table 4.8: S τt Quantity of Product with Deteriorated Shelf Life τ Delivered at Retailer during Tie t Retailer M Deteriorated Shelf Life τ Table 4.9 shows the product delivery plan for the planning horizon. For exaple, the quantity of 120 units ships fro depot w 2 at tie 1 and arrives at retailer 5 on day 2 (i.e.x = 120). One result of the odel is that retailer 5 has a deand for 120 units to be delivered on day t = 2 with a deteriorated shelf life of τ = 1; however, it is clear that retailer 5 is satisfied by receiving the order on the sae day t = 2, which eets the deteriorated shelf life value of τ = 1 ( i.e., t t = 2 1 = 1). By solving this optiization odel, to deterine all decision variables w in the MTP-RE odel, the transportation objective is iniized by obtaining the right x tt delivery plan for the planning horizon for each retailer. Tie Delivered t

93 w Table 4.9: x tt Nuber of Units to Ship fro Depot w to Retailer during Tie Period t Retailer M Depot W Tie Depart t Tie Arrive t w Along with the delivery quantities x tt and the delivery plan for the planning horizon, the MTP-AT odel also obtains the best route in order to iniize the total transportation cost. For this case study, Figure 4.3 shows the space-tie network solution. As can be seen, network (a) shows the best route using vehicle k 1. For this proble, decision variables x = 121, x = 120, x = 104, x = 123, x = 142, and x = 135 were deterined. The product is shipped fro depot w 2 via arcs y 112 (0,8), y 112 (8,5), y 112 (5,4), y 112 (4,2), y 112 (2,1), y 112 (1,3), and y 112 (3,0), respectively, with a total travel tie of seven days fro depot to retailers and back. Network (b) shows the best route for vehicle k 2 to deliver x = 131, and x = 122. Vehicle k 2 ships the product depot w 2 via arcs y 212 (0,7), y 212 (7,6), and y 212 (6,0), with a total travel tie of three days. As shown in networks (a) and (b) in Figure 4.3, the vehicle carries ultiple orders and copletes the route. Figure 4.4 shows the space tie network solution. Retailers 1, 2, 3, 4, 5, and 8 are assigned to one route and use vehicle k 1, and retailers 6 and 7 are assigned to another route and use vehicle k 2. As a result, the odel is able to achieve the iniu transportation cost of $1,170 with the cost of opening a new depot of $100 as a period cost. Therefore, the total profit (i.e., revenue cost inus 78

94 transportation cost) in this case is $7,498. As a result, it can be concluded that the MTP-L odel provides the optial revenue and transportation cost using the MTP-RE and MTP-AT odels, siultaneously. NETWORK ktw y (i,j) w x tt w w C ($) T (days) t = 1 Fixed Cost = 360 t 112 = 1 y (0,8) t 112 = 2 y (8,5) t 112 = 3 y (5,4) t 112 = 4 y (4,2) t = y (2,1) t 112 = 6 y (1,3) x = 121 C 2 08 = 50 T 2 08 = 1 x = 120 C 2 85 = 50 T 2 85 = 1 x = 104 C 2 54 = 50 T 2 54 = 1 x = 123 C 2 42 = 50 T 2 42 = 1 x = 142 C 2 21 = 25 T 2 21 = 1 x = 135 C 2 13 = 50 T 2 13 = 1 t = 7 (a) 112 y (3,0) C 2 30 = 50 T 2 30 = 1 Total Cost = 685 Total Travel Tie = 7 t = 1 Fixed Cost = 360 t 212 = 1 y (0,7) t 212 = 2 y (7,6) x = 131 C 2 07 = 50 T 2 07 = 1 x = 122 C 2 76 = 25 T 2 76 = 1 t = 3 (b) 212 y (6,0) C 2 60 = 50 T 2 60 = 1 Total Cost = 485 Total Travel Tie = 3 Objective Values: $7,498 Figure 4.3: Case 1 results of space-tie network 112 Vehicle k 1 Route: y (0,8) 112 y (8,5) 112 y (5,4) 112 y (4,2) 112 y (2,1) 112 y (1,3) 112 y (3,0) 212 Vehicle k 2 Route: y (0,7) 212 y (7,6) 212 y (6,0) 79

95 Figure 4.4: Space-tie network proble with eight retailers and two depots Results of the siultaneous solution ($7,498) by using GAMS/BARON shows an absolute gap of 7.49E-6 and a relative gap of 0.00 with CPU tie of seconds. As a result, it can be concluded that the MTP-L odel provides the optial revenue and transportation cost using GAMS/BARON by eploying the MTP-RE and MTP-AT odels siultaneously. Decisions on the location and nuber of depots have a long-ter planning horizon. In addition, there is uncertainty with respect to the fixed depot cost, traveling cost fro the depot and retailers, and vehicle dispatch cost. A sensitivity analysis with respect to fixed costs of a new depot has been copleted, and the optial solution for each value of fixed cost is shown in Figure 4.5. The solid line represents the optial solution with respect to the fixed cost of opening depot w 2. As can be seen, a fixed cost of $150/week represents the point at which the new depot w 2 should not be opened and depot liited to the existing depot w 1. At any price below $150/week, the depot w 2 can be opened and will yield a better profit. Sensitivity analysis with respect to the fixed cost of opening a new depot h w and the cost of travel fro the depot to retailers C w ij is shown in Figure 4.6. In this analysis, both the travel costs and the fixed cost per week for opening depot w 2 are analyzed. In general, the odel is sensitive to changes in these paraeters. Fro the obtained solution, the area below the line represents the region in which depot w 2 can be operated, whereas the area above the line represents the region 80

96 Travel Cost fro Depot to Optial Solution where operating depot w 2 is not feasible. This line represents the points of indifference between the two decisions. As can be seen fro Figure 4.6, as the travel cost is reduced, the feasible region for opening depot w 2 becoes larger. As the travel cost is increased, the fixed cost for opening the depot w 2 ust be reduced in order for it to be feasible to open it. Optial Solution vs Depot Fixed Cost (hw) Depot Fixed Cost (hw) for w2 Optial Solution using w2 Optial Solution using w1 Figure 4.5: Case one sensitivity analysis of fixed depot cost h w Travel Cost vs Depot Fixed Cost (hw) Retailers (Cij^w) Do not Open Depot w2 Open Depot w Depot Fixed Cost (hw) for w2 Feasible Solution Region for w2 Figure 4.6: Case one sensitivity analysis of travel C ij w and fixed depot costs h w 81

97 4.4.2 Case Study Two: Revenue and Transportation Model with Liited Vehicle Capacity This section describes a case study in which the vehicle capacity is liited. If the vehicle capacity is liited copared to the deand for retailers, then the odel ust be tested. This case study is siilar to the one presented in section The difference is that there is liited vehicle capacity. In the odified case study, the vehicle capacity is liited to 300 units, as shown in Table Table 4.10: Vehicle Capacity in Units of Products B k and Dispatch Cost Q k Vehicle K Capacity of Vehicle B k Cost Q k ($) k k As entioned in the previous case, the requests for orders fro each retailer is shown previously in Table 4.1. The resulting solution with the liited capacity of vehicles is given in Tables 4.11 and After solving case study two, the odel obtains the optiu and feasible solutions for w product delivery plan x tt and also the nuber of units of product delivered S τt. Table 4.11 shows solutions for the nuber of products with deteriorated shelf life τ delivered at the retailer during tie period t. In this case, when the vehicle capacity is liited, BARON solves the odel by taking into consideration the availability of the vehicle and ensures that profit is axiized. For exaple, the solvers recoended the best deand fro retailer 3 in ters of profit axiization, which is 73 in tie period 6, as shown in Tables 4.1, 4.11, and In this case study, the total revenue is $8,316. Table 4.12 represents the product delivery plan for the planning horizon. By optiizing all decision variables in the MTP-RE odel, the revenue objective is axiized by obtaining the w correct x tt delivery plan for the planning horizon for each retailer. In this case, as shown in 82

98 Tables 4.12, the BARON solver deterined that for optial profit, it will do a partial-order fill for retailer 4 and supply 55 and 49 units fro depots w 1 and w 2, respectively. Since the vehicle capacity is liited, the odel divides the quantity of the products delivered to retailer 4 and delivers the original quantity of 142 by using different vehicles and supply fro different depots. w Along with the delivery quantities x tt and the delivery plan for the planning horizon, the MTP-AT odel also obtains the best route in order to iniize the total transportation cost. For this case study, Figure 4.7 shows the space-tie network solution. Table 4.11: S τt Quantity of Product with Deteriorated Shelf Life τ Delivered at Retailer during Tie t Retailer M Deteriorated Shelf Life τ Tie Delivered t w Table 4.12: x tt Nuber of Units to Ship fro Depot to Retailer during Tie Period t Retailer M Tie Depart t Depot w Truck k Tie Arrive t

99 NETWORK ktw y (i,j) w x tt w w C ($) T (days) t = 1 Fixed Cost=265 t 112 = 1 y (0,7) t 112 = 2 y (7,5) t 112 = 3 y (5,4) x = 131 C 2 0,7 = 50 T 2 07 = 1 x = 120 C 2 7,5 = 50 T 2 75 = 1 x = 49 C 2 5,4 = 50 T 2 54 = 1 t = y (4,0) C 2 4,0 = 50 T 2 40 = 2 (c) Total Shipping = 300 Total Cost = 465 Total Travel Tie = 5 t = 4 Fixed Cost=265 t 242 = 4 y (0,2) t 242 = 5 y (2,1) t 242 = 6 y (1,3) x = 71 C 2 0,2 = 50 T 2 02 = 1 x = 85 C 2 2,1 = 25 T 2 21 = 1 x = 73 C 2 1,3 = 50 T 2 13 = y (3,0) C 2 3,0 = 50 T 2 30 = 1 t =7 (d) Total Shipping = 219 Total Cost = 440 Total Travel Tie = 4 t = 1 Fixed Cost=265 t 111 = 1 y (0,8) t 111 = 2 y (8,6) t 111 = 3 y (6,4) x = 121 C 1 0,8 = 75 T 1 08 = 1 x = 122 C 1 8,6 = 50 T 1 86 = 1 x = 55 C 1 6,4 = 75 T 1 64 = 1 t = y (4,0) C 1 4,0 = 100 T 1 40 = 2 t = 5 (e) Total Shipping = 298 Total Cost = 565 Total Travel Tie = 5 Figure 4.7: Case 2 results on space-tie network 84

100 As can be seen in Figure 4.8, retailers 4, 5, and 7 are assigned to one route fro depot w 2 by using vehicle k 1 ; retailers 1, 2, and 3 are assigned to another route fro depot w 2 by using vehicle k 2 ; and retailers 4, 6, and 8 are assigned to a different route fro depot w 1 by using vehicle k 1. As a result, the odel is able to achieve the iniu transportation cost of $1,470 with the cost of opening new depot of $100 as a period cost. Therefore, the total profit (i.e., revenue cost inus transportation cost) in this case is $7,169. As a result, it can be concluded that the MTP-L odel provides the optial revenue and transportation cost using the MTP-RE and MTP-AT odels, siultaneously. Details of the solution are shown in Figure 4.7. This type of solution is helpful for supply chain copanies that serve ultiple supply chain syste with existing additional vehicles. Objective Value: $7, Vehicle k 1 Route: y (0,7) 112 y (7,5) 112 y (5,4) 112 y (4,0) 242 Vehicle k 2 Route: y (0,2) 242 y (2,1) 242 y (1,3) 242 y (3,0) 111 Vehicle k 1 Route: y (0,8) 111 y (8,6) 111 y (6,4) 111 y (4,0) Figure 4.8: Space-tie network proble with eight retailers and two depots Results of the siultaneous solution ($7,169) by using GAMS/BARON shows an absolute gap of 7.319E-6 and a relative gap of 0.00 with CPU tie of seconds. As a result, it can be 85

101 Optial Solution concluded that the MTP-L odel provides the optial revenue and transportation cost using GAMS/BARON by using the MTP-RE and MTP-AT odels, siultaneously. In general, profit is relatively sensitive to changes in the paraeters, particularly changes in cost to operate new depot. In this case, a sensitivity analysis with respect to the fixed costs of a new depot was copleted, and the optial solution for each value of fixed cost is shown in Figure 4.9. The solid line represents the optial solution with respect to the fixed cost of opening depot w 2. As can be seen, a fixed cost of $440/week, represents the point at which the new depot w 2 should not be opened and the depot liited to the existing depot w 1. At any price below $440/week, the depot w2 can be opened and will yield better profit Optial Solution vs Depot Operation Cost (hw) Depot Operation Cost (hw) for w2 Optial Solution using w1 and w2 Optial Soultion using w1 Figure 4.9: Case two sensitivity analysis of fixed depot cost h w Sensitivity analysis perfored with respect to the fixed cost of opening a new depot h w and the cost of travel fro the depot to retailers C ij w is shown in Figure In this analysis, both travel costs and the fixed cost per week for opening depot w 2 is analyzed. In general, the odel 86

102 Travel Cost fro Depot to Retailers is sensitive to changes in these paraeters. Fro the obtained solution, the area below the line represents the region in which depot w 2 can be operated, for exaple, at point 150 of the fixed depot cost h w with the point 50 of travel cost fro depot to retailer falling in the area of opening depot w 2, whereas the area above the line represents the region where operating depot w 2 is not feasible. The line represents the points of indifference between the two decisions. As can be seen fro Figure 4.10, as the travel cost is reduced, the feasible region for opening depot w 2 becoes larger. As the travel cost increases, it can be seen that the fixed cost for opening the depot ust be reduced in order for it to be feasible to open w Travel Cost vs Depots Operation Cost Do not Open Depot w Open Depot w Depot Operation Cost for w2 Feasible Solution Region of using w2 Figure 4.10: Case two sensitivity analysis of travel C w ij and fixed depot cost h w 4.5 Conclusion In this chapter, a novel approach for an optial replenishent strategy fro a ultiple depot for a single perishable product and its associated atheatical odel is presented. This odel is decoposed of two objective coponents: axiizing total revenue based on the shelf life of the product, and iniizing the total transportation cost and fixed cost of opening a new 87

103 depot. In this paper, a ixed-integer nonlinear prograing odel is proposed to forulate the odel siultaneously and is solved using BARON. Case studies are provided to obtain insight into the odel behavior and dynaics. In sequential case studies, the MTP-RE odel was able to axiize the total revenue objective by obtaining the optial delivery plan for the planning horizon for each retailer, whereas the MTP-AT odel was able to iniize the total transportation cost by identifying the optial route to deliver the product to the retailers. Finally, the MTP-L odel was able to provide the decision aker with a coprehensive view of the delivery plan while axiizing total profit for the proposed proble. In siultaneous case studies, the MTP-L odel was able to provide the optial solution by axiizing the total revenue and iniizing the total transportation cost siultaneously. For future studies, this research could be extended to the proble involving a dynaic perishable product in which the retailer can have ultiple orders within the planning horizon with varying shelf lives. The odel could also be extended to investigate cases in which ultiple products and varying shelf lives are considered. The current odel does not take into account when the deands for each day is known with existing ultiple depots location. Thus, the odel could be extended to a ulti-product, utli-deand days, ulti-retailer, and depot location study. 4.6 References Aori, P., Meyr, H., Aleder, C., & Alada-Lobo, B. (2013). Managing perishability in production-distribution planning: a discussion and review. Flexible Services and Manufacturing Journal, 25(3), Cagliano, A. C. (2015). Networks against tie. Supply chain analytics for perishable products. Production Planning & Control(ahead-of-print), 1-2. Daskin, M. S., Snyder, L. V., & Berger, R. T. (2005). Facility location in supply chain design Logistics systes: design and optiization (pp ): Springer. 88

104 Gao, N., & Ryan, S. M. (2014). Robust design of a closed-loop supply chain network for uncertain carbon regulations and rando product flows. EURO Journal on Transportation and Logistics, 3(1), ILOG, I. (2012). Inc. CPLEX 12.5 User Manual: Soers, NY, USA. Khalili-Daghani, K., Abtahi, A.-R., & Ghasei, A. (2015). A New Bi-objective Locationrouting Proble for Distribution of Perishable Products: Evolutionary Coputation Approach. Journal of Matheatical Modelling and Algoriths in Operations Research, 14(3), Madduri, V. S. R. (2009). Inventory Policies for Perishable Products with Fixed Shelf Lives. The Pennsylvania State University. Tsao, Y.-C. (2013). Designing a Fresh Food Supply Chain Network: An Application of Nonlinear Prograing. Journal of Applied Matheatics, Valero, A., Carrasco, E., & García-Gieno, R. M. (2012). Principles and Methodologies for the Deterination of Shelf Life in Foods. Trends in Vital Food and Control Engineering, Wu, L.-Y., Zhang, X.-S., & Zhang, J.-L. (2006). Capacitated facility location proble with general setup cost. Coputers & Operations Research, 33(5), Kartikeya Mohan Sahai, & Siddhartha Nabiar. (2013). Network design for transporting perishable edible coodities with the application of queuing theory. Buffalo, New York, University at Buffalo, State University of New York. Available fro 89

105 CHAPTER 5 OPTIMAL REPLENISHMENT STRATEGY OF PERISHABLE PRODUCTS WHEN THE DEMANDS FOR EACH DAY DURING THE PLANNING HORIZON IS KNOWN 5.1 Introduction As stated in Chapter 2, Literature Review, it is iportant to enhance supply chain strategy logistics involving perishable products effectively and efficiently, especially in their distribution. Perishable products are those that decay and are likely not to be safe for their intended use after the shelf life period. This ay render the value of products to be zero or negative (if disposal costs are involved). If the supply chain is not effective in the delivery of these products, then the price and hence the profits of the products ay decrease. Exaples of such products include coposite aterials, ost food ites, flowers, etc. Most perishable products have a predeterined shelf life, which designates the tie period for which the product is safe for its intended use. Over the past decade, supply chains have becoe a vital part of the anufacturing industry. Many researchers have highlighted the iportance of the association between perishable products and transportation tie. Shukla and Jharkharia (2013) stated that 20 to 60 percent of total food production has been lost or wasted in the food supply chain. Yared Lea and Gatew (2014) identified the need to axiize the avalibility of food products for society. Perishable products ay cause pollution effects to the environent if not used within their shelf life. This has a large ipact on the world econoy and in turn affects all organizations involved in the hierarchy of the supply chain, including the final custoer (Mena et al., 2011). Shipping perishable products with a liited shelf life to retailers akes the supply chain network design ore coplex. Product perishability forces additional constraints on diverse supply chain processes such as production planning, procureent, inventory anageent, and the 90

106 supply chain network. For exaple, in the yogurt industry, perishability is a factor at every link in the supply chain, fro raw aterials to the final product. Thus, the supply chain ust enhance the transportation of perishable products fro suppliers to retailers by considering their shelf lives. Perishable products can be classified as products with either a rando shelf life or a fixed shelf life. Products with a rando shelf life, such as fruits and vegetables, experience continuous deterioration and have expiration dates that depend on specific deterioration rates. Products with a fixed shelf life, such as ilk and juice, are given an expiration date at the tie of production and are considered to have the sae quality until that expiration date passes (Madduri, 2009). Madduri (2009) developed a odel for inventory order policies for perishable products with a fixed shelf life. Here, fixed shelf life refers to the period of tie beginning with the anufacturing of the product until the expiration date, during which tie the product reains in good quality (Valero, Carrasco, & García-Gieno, 2012). In other words, the fixed shelf life is the aount of tie for a given perishable product to expire (Ministry for Priary Industries, 2014). Due to the perishable nature of products, especially those with a fixed shelf life, effective delivery planning can offer ore savings in logistic operations. Siilarly, the optiization of vehicle routes aids in the distribution of perishable goods and is incorporated into the research regarding distribution optiization. The ain challenge in the decision support syste of shipping perishable products is the developent of the vehicle schedule. This paper focuses on iniizing the transportation cost while taking into consideration the aount of product being shipped (i.e., vehicle capacity) and the cost of transportation (i.e., vehicle dispatch cost and route cost). A considerable aount of literature ay be found on the vehicle routing proble, which has been extended to include the tie window (VRPTW) constraints. For exaple, Deep et al. (2012) reviewed the forulation and 91

107 exact (coplete/optial solution) algorith of the VRPTW and categorized the forulation into four ain ethods: arc forulation, spanning free tree forulation, path forulation, and arcnode forulation. In the arc forulation, each arc of a basic directed graph is related to a binary variable (Gao & Ryan, 2014). In this graph, these binary variables are related to nodes in the arcnode forulation of the VRPTW proble. The spanning free tree forulation is a ethod for obtaining lower constraints for the VRPTW proble (Cagliano, 2015). The path forulation is also a ethod used to obtain lower boundaries for the VRPTW proble. In the current cobinational (integer and ixed-integer) forulations of the VRPTW proble, the sequence of visits for vehicles ay not be deterinate, unless the arcs are linked in the correct order (i.e., traveling salesan proble). Hence, an analysis of the order of visits in every route is obtained once the solution is received and cannot be utilized as input when the costs incurred rely on factors ipacted by a sequence of node visits within the route, such as deand (Liu & Ma, 2008). It is believed that the node-based forulation approach overcoes this insufficiency, and whenever the order of visits causes any additional cost, then the arc-node-based proble forulation is an ideal ethod. Other research has eployed a tie-dependent paraeter into the VRPTW proble. However, aong the four forulation ethods, only one sort of tie-dependent deand paraeter ay be used (Qi et al., 2013). As a result, the objective of the VRPTW is to reduce the nuber of vehicles needed, the average traveling tie, and the average travel distance covered by the fleet of trucks. Note that in the arc-node forulation, the set arcs usually represent linkages between the depot and the retailers (Bai, 2010). This paper focuses on finding the optial delivery strategy for a single perishable product for which the deand and price as a function of shelf life are known. It is assued that the product is fresh and has all of its shelf life when it leaves the depot. A tie continuu for the shelf life of 92

108 a product is assigned, starting fro tie zero when it leaves the depot to the designated shelf life duration. The product is fresh at tie zero and starts losing its price as tie passes. To odel this proble, the objective function is decoposed into two coponents: one that axiizes the total revenue during the planning horizon (i.e., revenue fro the sale of the product) and the other that iniizes the total transportation cost. The proble is odeled as a ixed-integer atheatical prograing proble and solved using the BARON solver. Solving the odel provides a solution to axiize total profit when the deands for each day during the planning horizon is known (MTP-MD) (total revenue inus total transportation cost). Finally, a detailed experiental test proble is provided in order to gain insight into the dynaics of the odel. The key contributions of this paper are as follows: (1) developent of a odel to axiize MTP-MD during the planning horizon for perishable products, (2) utilization of ixed-integer linear prograing to forulate the odel, (3) provision of a detailed experiental test proble to obtain insight into the odel dynaics by varying different paraeters, and (4) utilization of the BARON solver to solve the proposed odel siultaneously, where both solvers provide values for the decision variables of the supply chain proble. 5.2 Model to Maxiize Total Profit The MTP-MD odel axiizes the total profit when the deands for each day during the planning horizon is known days and utilizes the tie-indexed variable forulation in which the planning horizon is discretized into N intervals of one unit length each. This objective functions consists of two different coponents: to axiize total profit revenue of (MTP-RD) and to iniize the cost of transportation (MTP-TD). The MTP-MD odel is developed as an MINLP odel. In its ipleentation, after solving each objective (MTP-RD and MTP-TD), the MTP-RD 93

109 objective is subtracted fro the MTP-TD objective in order to obtain the profit during the planning horizon Proble Definition In this paper, the ain goal of the MTP-MD odel is to find the optial delivery strategy for a single perishable product for which the deand and price as a function of shelf life are known. Several retailers are served by one ain depot, and they are connected via a supply chain network. A fleet of vehicles is used to ship the product fro the depot to a set of retailers. Each retailer has a deterinistic inventory capacity and initial inventory, the deand for the product every day during the tie horizon is known, and the deand depends on the reaining shelf life of the product. It is assued that the product has its axiu shelf life at tie zero, which is when it leaves the depot. However, it is assued that the product starts deteriorating as soon as it is loaded on the truck. The shelf life of the product is less than or equal to the order tie (i.e.,τ t). It is assued that the depot has enough quantity of the product to eet the deand. The length of the planning horizon is assued to be greater than the length of the shelf life (i.e., T > τ ). In addition, the product has a unit price that varies with the shelf life over the planning horizon. Also, it is assued that the inventory-holding cost and loading and unloading costs are negligible. The objective is to optiize (axiize) profit by axiizing revenue (i.e., axiize units of product with shelf life τ delivered at retailer during tie period t) and by iniizing transportation cost Notation and Forulation Notations relative to the MTP-RD and MTP-TD odels are defined as follows: Sets N(t): set of nodes in space-tie network A(t): set of arcs in space-tie network 94

110 K: set of trucks T: set of space-tie Indices k K: truck index k = 1,, K t, t T: tie period index t, t = 1,, T τ T: deteriorated shelf life, which is the nuber of days that the product has deteriorated τ = 0,, T, N(t): index for retailer, 0,, M, where = 0 and = 0 represents depot l N(t): set of retailers () satisfied at tie period (t) i, j N(t): nodes () to ( ) Paraeters B k : capacity of truck k D τt : deand for product with deteriorated shelf life τ at retailer during tie period t P τ : price of product with deteriorated shelf life τ I τ0 : initial inventory with deteriorated shelf life τ at retailer O : inventory capacity of retailer Q k : fixed cost associated with dispatching truck k C ij : cost of travel fro node i to node j (i =, j = ) T : tie for travel fro retailer to, G : upper bound on x vector values 95

111 Decision Variables x tt : nuber of units to ship fro depot at tie period t to be delivered to retailer at tie period t ; aggregate vector of all decision variables x = {x tt : = 1,, M, t = 1,, T, t = t,, T} represents product delivery plan for planning horizon e τt {0,1}: binary variable; e τt = 1 if deand for product with shelf life τ during tie period t at retailer is satisfied; otherwise, e τt = 0 I τt : inventory level with deteriorated shelf life τ at retailer at end of tie period t S τt : product with deteriorated shelf life τ delivered by retailer during tie period t Note that to define other decision variables, first, the truck index ust be augented, and the pair (k, t) ust be used to represent truck k that leaves depot at tie period t. u kt {0,1}: binary variable; u kt = 1 if truck (k) is used at tie t; otherwise, u kt = 0 kt y (i,j) kt {0,1}: binary variable; y (i,j) = 1 if truck (k) is used at tie t with travels arc kt (i, j) A(t); otherwise, y (i,j) Auxiliary Variables = 0 v l kt {0,1}: binary variable; v l kt = 1 if truck (k) visits node l N(t) at tie t; otherwise, v l kt = 0 z kt : arrival tie of truck (k, t) back to depot η tt {0,1}: binary variable indicating whether retailer ε M receives any shipent on day t, which was sent on day t. In other words, η tt = { 1 x tt > 0 0 otherwise 96

112 5.2.3 MTP-MD Model The objective of the MTP-MD odel is to axiize total profit during the planning horizon, which is the difference between the objective function of the MTP-RD odel and the objective function of the MTP-TD odel: Max profit f f MTPRD MTPTD T T M T K T K kt kt t ( i, j) ( i, j) k Max (( P s ) ( C y Q u )) t1 0 1 t1 k1 ( i, j) A0 ( t) t1 k MTP-RD Model follows: The MTP-RD odel axiizes the total revenue obtained fro the product delivered as Max st.. t1 0 1 t t t, t t t 1, t1 t, t t t 1, t1 t1 0 (5.1) S I x 1..., M, t 1,..., T, 0 (5.2) S I I x 1..., M, t 1,..., T, 1,..., t 1 (5.3) S I I 1..., M, t 1,..., T, 1,..., T (5.4) T T T 0 tt ' t T T M e t tt ' t t τt P s t , M (5.5) O 1..., M, t 1,..., T (5.6) S e D 1..., M, 0,..., T, t 1,..., T (5.7) x t I t ax B 1..., M, t 1,..., T, t' 1,..., T (5.8) k k x, I, S , M, t 1,..., T, T, t' 1,..., T, 1,..., T (5.9) e t {0,1} 1..., M, t 1,..., T, 1,..., T (5.10) The first objective (5.1) axiizes total revenue. The inventory constraint sets (5.2), (5.3), and (5.4) ensure a balance in the inventory level with deteriorated shelf life τ at each retailer at 97

113 the end of tie period t. Constraint set (5.5) ensures that each retailer selects only one deand for the product with satisfying on any day during the tie horizon and deteriorated shelf life τ during tie period t. Constraint set (5.6) ensures the inventory level with deteriorated shelf life τ at retailer at end of tie period t does not exceed the inventory capacity at each retailer. Constraint set (5.7) states that the selected units of product sold per tie period t ust be less than or equal to deand. Constraint set (5.8) ensures liiting the units of products shipped to any retailer per tie period t in order to not exceed the axiu truck capacity. Finally, constraint sets (5.9) and (5.10) are integrality and non-negativity, respectively. After the MTP-RD odel is solved, the objective value and delivery plan to arrive at each retailer (x tt, i.e., nuber of units to ship fro depot at tie period t and to arrive at retailer at tie period t ) are obtained in order to solve the MTP-TD odel MTP-TD Model The MTP-TD odel attepts to iniize the total transportation cost incurred in order to carry out a given replenishent plan (x tt ). Evaluation of this odel requires solving a nested optiization proble that deterines the optial routes and fleets used. In particular, this can be odeled siilar to the VRPTW constraints. The VRP is concerned with finding the optial nuber and type of trucks, associated routes, and starting and end ties for trips in order to satisfy a retailer s deand within specified tie windows while iniizing the cost of transportation. An adaptation of the VRP forulation suited for this proble is discussed next. The supply chain network odel includes a set of nodes (N) consisting of retailers and a depot, and a set of arcs (A) representing routes. In order to characterize the routes, Τ space-tie networks, each associated with a tie period t = 1,, T in the planning horizon, are constructed, as shown in Figure 5.1. As can be seen, the horizontal and vertical nodes (N(t), A(t)) represent 98

114 space (retailer index) and tie (tie period index), respectively. The first and last nodes (0, t) represent the depot. An arc appears between every two nodes, only if the difference between the tie periods is equivalent to the travel tie between the corresponding retailers (or depot and retailer). The final arc of the route connects it back to the depot to coplete the route. Finally, the arc costs represent the distance between the corresponding nodes (or depot and retailer). Figure 5.1: Space-tie network (N(t), A(t)) The space-tie network shown in Figure 5.1 is denoted by (N(t), A(t))(t = 1,, T), where N(t) and A(t) are the sets of nodes and arcs, respectively. In particular, for network (N(t), A(t)), the set of nodes represents pairs of retailers and tie periods as N(t) = { (, t ): = 1,..., M, t = t,..., T} A node corresponding to the depot is added as N 0 (t) = N(t) {(0, t)} Additionally, an arc occurs between every two nodes ( 1, t 1 ) and ( 2, t 2 ), if a vehicle can traverse the distance fro retailer 1 to 2 in exactly (t 2 t 1 ) tie periods. More specifically, the set of arcs is defined as A(t) = {(( 1, t 1 ), ( 2, t 2 )): 1 = 0,..., M, 2 = 1,..., M, 1 2, t 1 = t,..., = t 1 + T } 99

115 There is also an arc between all nodes in N(t) and the depot to close the loop (see Figure 5.1) as For notational convenience, A 0 (t) = A(t) {(i, (0, t)): i N(t) } R s (( 1, t 1 ), ( 2, t 2 )) = 1 R e (( 1, t 1 ), ( 2, t 2 )) = 2 where retailer is associated with the start and end of the arc ( 1, t 1 ), ( 2, t 2 ), respectively. Finally, with a slight change of notation, T C represents the tie and cost of traversing arc (i, j) A 0 (t), respectively, which can be obtained fro the travel tie and cost atrices, respectively: T (i,j) = T Rs (i,j),r e(i,j) The MTP-TD atheatical odel follows: C (i,j) = C Rs (i,j),r e(i,j) 100

116 st.. T K T K kt ( i, j) ( i, j) t1 k 1 ( i, j) A ( t ) t1 k 1 Min C y Q u kt kt ( i, ) (, j) ( i, ) A( t) (, j) A( t) kt kt ((0, t), ) (,(0, t)) N ( t ) N ( t ) ( i, j) A( t) ( i, j) A( t) tt ' (, t ') N ( t ) 0 k kt (5.11) y y N( t), k 1,..., K, t 1,..., T (5.12) y y 1 k 1,..., K, t 1,..., T (5.13) kt ( i, j) y k 1,..., K, t 1,..., T (5.14) kt ( i, j) ( i, j) y z k 1,..., K, t 1,..., T (5.15) tt ' kt j kt x G 1,..., M, k 1,..., K, t 1,..., T (5.16) K k 1 kt ( t, ') kt kt kt tt' (, t ') k x B u k 1,..., K, t 1,..., T (5.17) tt' (, t ') N( t), t 1,..., T (5.18) kt kt kt ( i, j) l tt ' kt ' ( t z ) t ' (1 u ) k 1,..., K, t 1,..., T, t ' t 1,..., T (5.19) u, y,, {0,1} 1,..., M, k 1,..., K, t 1,..., T, t ' t 1,..., T, N( t), ( i, j) At ( ) (5.20) kt z 0 k 1,..., K, t 1,..., T (5.21) The first objective (5.11) iniizes the total transportation cost, which consists of variable and fixed costs. The balance constraint set (5.12) ensures that if the vehicle visits a node (retailer) other than the depot on the network, it ust leave that node (retailer). Constraint set (5.13) ensures that each route originates and terinates at the node corresponding to the depot(0, t). Constraint sets (5.14), (5.15), and (5.16) ensure that the auxiliary variables are connected to the ain decision variables, where G is an upper bound on the x-vector values. Constraint set (5.17) ensures that the shipped quantity is less than the capacity of the vehicle. Constraint set (5.18) ensures that each node in the space-tie network is visited, if the corresponding retailer is included in the delivery 101

117 plan. Constraint set (5.19) ensures that a vehicle can only be used again after it returns to the depot. This constraint can be cast as the following logical constraints: t t if z kt > t, then u kt = 0. Finally, constraint sets (5.20) and (5.21) are integrality and non-negativity, respectively Maxiize Total Profit (MTP-MD) The objective of the MTP-MD odel is Max profit f f MTPRD MTPTD T T M T K T K kt kt t ( i, j) ( i, j) k t1 0 1 t1 k1 ( i, j) A0 ( t) t1 k1 Max (( P s ) ( C y Q u )) Figure 5.2 shows the MTP-MD odel flow chart. 102

118 MTP-MD SIMULTANEOUS Max profit = f MTP RD f MTP TD s. t. Ax b x 0 MTP-RD Model Paraeters Max f MTP RD s. t. Ax b x 0 x tt MTP-TD Model Paraeters Min f MTP TD s. t. Ax b x 0 f MTP TD Value kt y (i,j) f MTP RD Value Figure 5.2: MTP-MD odel flow chart 5.3 Experiental Evaluation The General Algebraic Modeling Syste (GAMS) is used to odel the MTP-MD proble, and the proble is solved siultaneously using the solver BARON The coputational experients are perfored on an Intel i GHz processor with 4 GB of eory and a Windows 7 operating syste. 103

119 5.4 Experiental Design and Tests Case Study One: A case study is used to provide an illustration for solving the MTP-MD odel siultaneously using BARON. It helps to obtain the solution of the shipping quantity as the first part of the odel and then use the solution as the nonlinear constraint (5.17) to obtain the objective value. This case study involves one depot, eight retailers, and two vehicles with unliited capacity. The length of the tie horizon T =6, and the axiu shelf life is 5. Table 5.1 suarizes the deand for each retailer while ordering a product with different deteriorated shelf lives τ each day. Note that a retailer can place an order ultiple days (t) with deteriorated shelf life of 0 to τ = t 1. For exaple, Table 5.1 shows that retailer 2 places an order on day 2. Hence, retailer 2 can order the product with a deteriorated shelf life of τ = 0 or 1 (i.e., D 2 02 = 50 or D 2 12 = 40). Table 5.1: D τt Deand for Product with Deteriorated Shelf Life τ at Retailer 104