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Supply chain consolidation and cooperation in the agriculture industry

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Supply chain consolidation and cooperation in the agriculture industry

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Content Supply Chain Consolidation and Cooperation in the Agriculture Industry by Christine Nguyen A dissertation submitted in partial fulllment of the requirements for the degree Doctor of Philosopy (Industrial and Systems Engineering) University of Southern California August 2014 Committee: Maged Dessouky - Chair Alejandro Toriello James E. Moore II Peter Gordon Table of Contents List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Chapter 1 Introduction 1 1.1 Direct Shipping Consolidation Model . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Inventory Consolidation Model . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Chapter 2 Literature Review 10 2.1 Freight Consolidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Inventory Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.1 Economic Lot-Sizing Problem . . . . . . . . . . . . . . . . . . . . . . 16 2.2.2 Joint Replenishment Problem . . . . . . . . . . . . . . . . . . . . . . 19 2.2.3 One Warehouse Multi-Retailer . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Supply Chain Cooperation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 Perishable Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5 Research Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Chapter 3 Deterministic Direct Shipping Consolidation Model 34 3.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Cost Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3.1 Transition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3.2 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4 -based Consolidation Heuristic . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.5.1 Case Study: California Cut Flower Industry . . . . . . . . . . . . . . 46 3.5.2 Comparison of Solution Approaches . . . . . . . . . . . . . . . . . . . 57 Chapter 4 Stochastic Direct Shipment Consolidation Model 62 4.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Stochastic Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3 Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3.1 -based Consolidation Heuristic . . . . . . . . . . . . . . . . . . . . . 67 4.3.2 Other Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 i 4.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.4.1 Comparison of Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.4.2 Sensitivity to Consolidation . . . . . . . . . . . . . . . . . . . . . . . 74 4.4.3 Cost Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Chapter 5 Inventory Consolidation Model 78 5.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2 Dynamic Programming Model . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2.1 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.3 Comparison Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Chapter 6 Conclusions 94 6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 ii List of Figures Figure 3.1 A Transportation Network with a Consolidation Center and Direct Shipping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Figure 3.2 The Shipping Cost Function . . . . . . . . . . . . . . . . . . . . . . . 38 Figure 3.3 Probability Plot for 2008 data from Farm A . . . . . . . . . . . . . . 48 Figure 3.4 Probability Plot for 2010 data from Farm A . . . . . . . . . . . . . . 49 Figure 3.5 Annual Transportation Costs for Current Practices and Consolidation 56 Figure 3.6 Volume sent by Less-Than-Truckload and Full truckload . . . . . . . 56 Figure 4.1 Demand distributions for two suppliers of the California cut ower industry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Figure 4.2 Ratio of consolidating versus operating independently . . . . . . . . . 74 Figure 4.3 Average Number of Suppliers per Full Truckload Shipped . . . . . . . 75 iii List of Tables Table 3.1 Two-Sample t-test Summary . . . . . . . . . . . . . . . . . . . . . . . 49 Table 3.2 Detailed Results for Scenario 1 . . . . . . . . . . . . . . . . . . . . . . 53 Table 3.3 Detailed Results for Scenario 2 . . . . . . . . . . . . . . . . . . . . . . 53 Table 3.4 Detailed Results for Scenario 3 . . . . . . . . . . . . . . . . . . . . . . 54 Table 3.5 Detailed Results for Scenario 4 . . . . . . . . . . . . . . . . . . . . . . 54 Table 3.6 Detailed Results for Scenario 5 . . . . . . . . . . . . . . . . . . . . . . 55 Table 3.7 Detailed Results for Scenario 6 . . . . . . . . . . . . . . . . . . . . . . 55 Table 3.8 General Theta Results Using CA Cut Flower Deterministic Demand . 59 Table 3.9 Average Runtime per Destination for General Theta (in seconds) . . . 60 Table 4.1 Numerical Results of Algorithms and Heuristics . . . . . . . . . . . . . 71 Table 4.2 Average Runtime for the Dynamic Programming Model (in seconds) . 73 Table 4.3 Rolling Horizon and -based Heuristic Comparison . . . . . . . . . . . 73 Table 4.4 Grand Coalition Average Daily Costs per Supplier . . . . . . . . . . . 76 Table 4.5 Ratio Comparison Against the Grand Coalition . . . . . . . . . . . . . 77 Table 5.1 Case 1: All Inventory Costs are $0.01 . . . . . . . . . . . . . . . . . . 87 Table 5.2 Case 1: CPLEX Details . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Table 5.3 Case 1: Comparison Heuristic 1 details . . . . . . . . . . . . . . . . . 88 Table 5.4 Case 2: All Inventory Costs are $0.50 . . . . . . . . . . . . . . . . . . 88 Table 5.5 Case 2: CPLEX Details . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Table 5.6 Case 2: Comparison Heuristic 1 Details . . . . . . . . . . . . . . . . . 89 Table 5.7 Case 3: All Inventory Costs are $1.00 . . . . . . . . . . . . . . . . . . 89 Table 5.8 Case 3: CPLEX Details . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Table 5.9 Case 3: Comparison Heuristic 1 Details . . . . . . . . . . . . . . . . . 90 Table 5.10 Case 4: Inventory Costs . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Table 5.11 Case 4: Dierent Inventory Costs . . . . . . . . . . . . . . . . . . . . . 91 Table 5.12 Case 4: CPLEX Details . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Table 5.13 Case 4: Comparison Heuristic 1 Details . . . . . . . . . . . . . . . . . 92 iv Acknowledgements The journey to receive a Ph.D. was lled with people who supported and encouraged me. I am very thankful to every person I met. To my advisors: Maged Dessouky and Alejandro Toriello. I am privileged and fortunate to have two advisors who helped me navigate the PhD program. You taught me to trust my instincts and to understand my abilities. Your vast knowledge of the area, enthusiasm, and guidance were integral in training me to be an eective researcher. Our conversations brought me great joy and laughter, especially in the most dicult times, and I can never thank you enough. To my committee members: James Moore II and Peter Gordon. You have been with me since I began my Ph.D. and saw my potential immediately. Thank you for your dedication to your students, and your unwavering support. To my seniors: Joongkoo Cho and Chen Wang. You are both successful scholars, and I aspire to live up to your example. Thank you for your unwavering faith in my abilities, and your continuous encouragement. To my friends who helped me adjust to life in Los Angeles and at USC: Caitlin Hawkins, Yihan Xie, Pai Liu, Rahul Schro, Fei Li, Lijuan Xu, Wang Li, and Jae Kim. You explored the city with me, and danced through the night with me. To my oce mates: Michael Poremba, Weihong Hu, Xiaoqing Wang, Huayu Xu, Lunce Fu, Han Zou, Yihuan Shao, Wentao Zhang, and Liang Liu. There was never a boring day with you. v To the most important people in my life: my family. In the most dicult times, you reminded me that sometimes I need to trip and fall in order to achieve the motivation to climb higher. You encouraged me to enjoy every moment in my journey, all the good and especially the bad ones, because there is always something to learn. To Steven, for always knowing how to cheer me up and make me laugh. To David, Mengting, and Jordan, for reminding me to explore other frontiers. To my best friend, who always saw my potential and kept me grounded. Thank you for being by my side, and always encouraging me to be true to myself. vi Abstract Freight transportation supports the economic activity of the United States. Trucking is the most frequently used method of transportation in the agriculture industry. The majority of suppliers in the agriculture industry are small farmers with little demand. We investigate a supply chain system with small farmers. Two models are considered in this research for the shipping of perishable goods. Both models consider a transportation network with direct shipping modes to and from a single consolidation center. We represent perishability as a hard constraint on the total time products are allowed to stay at the consolidation center. In the rst model, we assume inventory costs are negligible, and only consider transportation costs. We develop a consolidation heuristic that is ecient and easy-to-implement in prac- tice. The consolidation strategy exploits economies of scale from three shipping methods: full truckload rates, less-than-truckload rates, and courier rates. A dynamic programming model is developed to calculate an optimal solution. Both solution approaches solve the deterministic and stochastic cases of the direct shipping consolidation model with multiple suppliers and seasonal demand. The second model adds inventory costs, a soft constraint for perishability, to the rst model. A dynamic programming algorithm and a heuristic that calculates the trade-o between shipping and holding inventory are developed. Numerical results show that the heuristic performs well for high inventory costs. However, if the inven- tory costs are signicantly large, the best solution is to ship product everyday and hold no inventory. vii Chapter 1 Introduction The extensive network of highways, railways, and waterways supports the economic activ- ity of the United States by providing access between businesses, households, and markets. Trucking is the most frequently used transportation mode to ship commodities in the United States. Including for-hire and private use, trucks haul approximately 70% of commodities (by total value) and dominate shipment distances of less than 500 miles [148]. In 2011, 10% of the U.S. GDP was spent on transportation services [149]. The agricultural sector is the largest user of the freight transportation services in the United States [47,147]. In particular, the agriculture industry depends on trucking more than rail, barge and ocean vessel. Ac- cording to a joint study by the U.S. Departments of Agriculture and Transportation [147], trucks are the only form of agriculture transportation in at least 80% of U.S. cities and communities. The agriculture industry greatly depends on trucking at all stages of its supply chain. Trucking is usually the rst link and last link of the supply chain and sometimes all the transportation in between. Trucking connects farmers with distant markets and other modes of transportation. Factors such as the lack of rail service and restrictions on access and routing to competing railroads cause farmers in rural areas to depend on trucking services 1 instead of rail. Rail and barge rate increases tend to divert shipments to trucks. The operating exibility of trucks matches the exibility farmers need during their planting and harvest seasons. Trucks deliver supplies essential to the planting season, and pick-up produce during the harvest season. However, the planting and harvest seasons are highly variable because of factors such as the weather, eld condition and crop maturity. The variability and massive needs for transportation at concentrated periods of time makes it dicult for farmers to hire and contract temporary drivers. Therefore, a large percentage of the nal price of agricultural products is spent on transportation services [147]. Farmers prefer trucks to ship their perishable products because trucks are best in providing refrigeration, eciency, special handling and operational exibility. However, the transportation costs of perishable products are substantially greater than shipping bulk commodities [124]. Small farms make up about 90% of all farms in the United States, and in many sectors of the agriculture industry, they are responsible for the majority of production [86]. They depend on third-party carriers to deliver essential supplies, such as fertilizer, in the planting season and to load their vehicles of fresh product during the harvest season to be shipped domestically and internationally. The shipping rates are generally in full truckload (FTL) and less-than-truckload (LTL) rates. Growers who partially ll a truck are charged less-than- truckload rates while those with enough shipping volume to send a near full or full truck are charged a more advantageous full truckload rate. In a few sectors, growers ship very small volumes using a courier service such as Federal Express (FedEx), United Postal Service (UPS) or Ontrac. Small and medium-sized growers do not generate enough volume for the majority of their shipments to be at the FTL rate. While advancements in technology such as plant genetics, application of fertilizer, farming equipment, and pest control methods have improved the overall productivity of farms, it is increasingly dicult for small and medium-sized farms without access to these advancements to compete successfully [47]. Third-party carriers consolidate their shipments when there are multiple partial truckload 2 shipments, but these savings are rarely passed on to growers. Many agriculture sectors with small and medium-sized farms are realizing the importance to consolidate shipments and reduce transportation costs [44]. The dierence in transportation costs place small growers at a competitive disadvantage with the larger growers because they do not produce enough volume to ship at more benecial rates. States like California and Oregon greatly depend on trucks to transport agricultural products to the rest of nation. For example, California produces over 99% of the production of 14 commodities for the United States, such as almonds, peaches, grapes, dates and pomegranates [124]. An example of this phenomenon exists in the California cut ower industry. The growers in the California cut ower industry are at a competitive disadvantage with imported cut owers from South America, who currently dominate the U.S. cut ower industry. The 1991 Andean Trade Preference Act (ATPA) suspended import duties on owers entering the United States from South American countries like Colombia and Ecuador. The APTA was due to expire in 2011, but a trade deal with Colombia was passed by Congress in October 2011 that will further facilitate their dominance in the market. The South American growers compete eectively with U.S. growers because they share a cross-docking and distribution facility established in Miami and have cheaper labor costs in South America. Miami is an important point-of-entry for imports, and the facility provides a single consolidation and pick- up location for South American growers. Being able to consolidate allows South American growers to negotiate favorable trucking rates due to the magnitude of their volume. The competitive advantage of lower transportation and labor costs is re ected in less expensive owers from South America. Columbia alone controls 70% of the U.S. ower markets [10] while California's share of the national ower market has reduced from 64% to 20% in the past two decades, according to the California Cut Flower Commission. California grown owers are at a disadvantage. California growers operate independently of each other and no consolidation takes place. The majority of California growers generate volume that ll up 3 partial trucks, and they pay LTL rates even though multiple growers from the same region are shipping to the same region or state. Many rms believe they cannot continue to compete individually due to the increase in competition, globalization and demanding customers [103]. The advantages of collaboration between rms includes the opportunity to share risks, reduce costs, minimize unsatised customer demand and increase competitive advantage [28]. Consolidation policies can re- duce system-wide transportation costs but since this problem considers multiple suppliers, allocating the cost can be dicult. The cost allocation problem focuses on nding a way to allocate the costs among those participating in a cooperation such that every player benets from working together. Incentives, such as subsidies or side payments, might be necessary to encourage enough participation to experience savings through consolidation. The notion of stability is extremely important while designing cost allocation rules. A stable cost allo- cation policy is an allocation such that no player or subset of players can reduce their costs by leaving the collaboration. Currently, many suppliers in the agriculture industry operate independently and suer from the large transportation costs as well as increased competition from domestic and foreign businesses. The goal of this work is to establish an ecient consolidation system for perishable products and suppliers that generate low volumes. We aim to take advantage of economies of scale by consolidating more product to ship at the FTL rate. We focus on two major questions: 1) how much should be consolidated, and 2) when should consolidation take place? Since our focus is on long-haul transportation costs, the transportation cost from sup- pliers to the consolidation center are assumed to be signicantly small in comparison. This assumption is reasonable when the system possesses the following characteristics: 1. suppliers harvest the product just before they are shipped to the consolidation center, 4 2. the suppliers are located near the consolidation center, 3. each supplier ships their product directly to the consolidation center, 4. perishability is represented as a xed limit on the total time products are allowed to stay at the consolidation center, and 5. the unit cost of the products are small compared to the shipping cost. The perishable nature of the products is represented by a hard time constraint. We dene as the maximum number of time periods an item is allowed to stay at the consolidation center. Two models with the above characteristics are formulated. Both models are inventory control problems where the decision maker must decide how to manage the inventory. The unique aspect of this problem is the consolidation of perishable products. The rst model is a direct shipping consolidation model that considers only long-haul transportation costs. The second model disregards the 5th characteristic and includes transportation and inventory costs. 1.1 Direct Shipping Consolidation Model A set of suppliers ship their product to a consolidation center directly, using the least cost method. Under the assumptions stated in the previous section, only transportation costs are considered. Perishable products that are heading to the same destination can be consol- idated. Since there are no inventory costs, there is no cost penalty for a product to stay in inventory. The goal is to decide when and how much to consolidate. If the decision is to not consolidate with today's outgoing shipment, then the inventory stays at the consolidation center. Otherwise, we must decide how much to ship today. We dene to be a xed time limit on the number of periods a product can be held in inventory. 5 We formulate the objective function to minimize the total long-haul transportation cost to ship product from the consolidation center to a break-bulk destination. Due to the second assumption from the previous section, we do not need to consider the transportation cost from the suppliers to the consolidation center. The direct shipping consolidation problem is formulated as a mixed-integer programming (MIP) model that considers the FTL, LTL and courier shipping methods. We propose a dynamic programming model to solve the optimization problem. A practical and ecient heuristic for implementation should not require demand to be known in advance. The heuristic should exploit the characteristics of the dierent shipping methods and ship as many full truckloads as possible. In the deterministic case, numerical experiments are conducted to compare the heuristic, the dynamic programming algorithm and a commercial solver for varying values of . Then, using data from the California cut ower industry, we conduct a study that measures the dierence in annual costs of the consolidation model against the California cut ower growers' current transportation practice to determine whether a consolidation system would be benecial in practice. In the stochastic case, we develop a stochastic dynamic programming model, and modify the heuristic to consider stochastic demand. Additional solution approaches such as a rolling horizon heuristic and an Every + 1 Policy are considered for comparison. Numerical experiments test the four solution approaches. We compare their performance with each other, and perform a sensitivity analysis on whether consolidation is benecial as demand changes. The question of whether suppliers would cooperate with each other for consolidation benets remains. We explore the cost allocation problem for the stochastic case. We propose a proportional cost allocation policy based on the volumes that are included in an outgoing shipment. Numerical experiments are conducted to determine whether this policy is viable or not. 6 1.2 Inventory Consolidation Model The Inventory Consolidation model expands the Direct Shipping Consolidation Model by including inventory costs. We assume that inventory costs are large enough to impact the consolidation decisions and cannot be ignored. By allowing dierent holding costs per sup- plier, we consider perishable products with dierent decay rates and measurements. For example, we can consider dierent products such as melons and potatoes, each with a dif- ferent rate of deterioration. We formulate a mixed-integer programming model with an objective function to minimize transportation and inventory costs. A dynamic programming model is formulated to solve the problem, and we propose a heuristic that looks at the trade-o between shipping product today versus tomorrow. The dynamic programming model is unable to handle large problems because the number of states now includes the number of suppliers. In the previous model, the state variable represents aggregate inventory for all the suppliers. However, this model's state variable also includes the inventory originating from each supplier. The complexity increases dramatically with respect to , the time horizon, the discretization level, and the number of suppliers. The increased complexity and runtime for the dynamic program is a motivator to develop an ecient heuristic that considers both transportation and inventory costs. The heuristic should consider the trade-o of holding inventory to ship in the future or consolidating with the shipment that must leave immediately. Numerical experiments are conducted to demon- strate the eectiveness of the solution approaches and a commercial solver. We perform a sensitivity analysis for varying inventory costs when all suppliers have the same cost. Then we analyze situations where inventory costs are dierent for all suppliers. 7 1.3 Contribution In this research, we study the eects of consolidation for an agricultural supply chain. The freight consolidation problem considers multiple suppliers, a single consolidation center and perishable products. The problem is studied in two models: with transportation costs only and combined transportation and inventory costs. We concentrate on a model that considers only transportation costs for highly perishable items with a low unit cost compared to the shipping cost. For perishable products with a higher unit cost, inventory costs cannot be ignored in the analysis. A large time horizon must be considered to capture the peak and nonpeak times of the year that correspond with plant and harvest seasons. We focus on developing a solution approach that determines the amount of product to consolidate and when the consolidation should take place for a large-scale optimization problem. The contributions of this thesis include: analyze a direct shipping consolidation model for perishable items with a maximum inventory time constraint, deterministic and stochastic demands, multiple suppliers and a single consolidation point, formulate a mixed integer programming model that minimizes long-haul transportation costs, develop an ecient heuristic and formulate a dynamic programming model to minimize long-haul transportation costs, propose a cost allocation policy and numerically study whether it encourages suppliers to consolidate jointly, formulate a mixed integer programming model that minimizes long-haul transportation costs and inventory costs, 8 develop an ecient heuristic that considers inventory and transportation costs, and conduct numerical experiments to compare the proposed solution approaches. The next chapter contains a review of the literature on consolidation practices, common inventory control problems, supply chain cooperation solution approaches and perishability. We study the freight consolidation problem with only transportation costs in the deter- ministic case rst (Chapter 3). Chapter 4 introduces the same problem but with stochastic demand. Then we include inventory costs in the freight consolidation problem and investigate dierent solution approaches (Chapter 5). The conclusions of these studies are summarized in the nal chapter. 9 Chapter 2 Literature Review This chapter reviews well-known problems in the literature that are relevant to the pro- posed models in this thesis. First, we review freight consolidation problems and solution approaches. Next, we study the most common inventory control problems: economic lot- sizing problem, joint replenishment problem and one-warehouse multi-retailer problem. We review solution approaches for supply chain cooperations, and nally, we summarize the approaches to representing perishability in the literature. 2.1 Freight Consolidation Freight consolidation is the process of combining the transportation of multiple products; larger load sizes take advantage of lower transportation costs. This process can occur in in- ventory, in vehicles, and in terminals [75]. A survey of 53 U.S. rms re ects the importance of freight consolidation practices in terms of cost reductions. The majority of the cost reduc- tion comes from taking advantage of economies of scale [92]. There are three classications of basic consolidation strategies: spatial, product and temporal [115]. We focus primarily on a product consolidation strategy. More recent literature use the terms shipment consolidation 10 and freight consolidation. The freight consolidation literature focuses on two questions: how much to consolidate and when to consolidate. Early literature concentrates on developing shipment-release policies for freight consoli- dation problems. The quantity policy calculates an optimal accumulation quantity for the system. The system will hold volume until it reaches the optimal accumulation quantity, after which it will schedule an immediate shipment. Gupta and Bagchi [73] utilize stochastic clearing theory to develop a quantity policy for a inbound freight consolidation under the just-in-time procurement system. Time policies determine the maximum holding time of orders before a shipment is re- leased. The time policy is the most frequently implemented policy in practice [91]. Mark- lund [111] develops two heuristics for determining near optimal shipment intervals for a divergent supply chain of a central warehouse and multiple retailers. The time-based policy is used at the warehouse along with real-time point-of-sale information. The most popular policy in recent literature is the time-and-quantity policy, a hybrid of the time policy and quantity policy. The decision maker holds all orders for a destination until the earliest of: 1) a maximum holding time or 2) a minimum accumulation volume or weight. Bookbinder and Higginson [25] apply the stochastic clearing theory. They formulate the probability density function for the total target weight accumulated in time t, which is based on a Poisson arrival process for the orders and a Gamma distribution to generate the weight of each order. The authors develop a nomograph, which consists of 4 graphs that relate the optimal consolidated weight, vehicle utilization, and the expected total cost per consolidation cycle given a set of order-arrival parameters. The nomograph provides a method to perform sensitivity analysis and guidance for management, rather than a method to be used in real-time. Mutlu et al [119] develop an analytical model for a time-and-quantity consolidation policy by using renewal theory. Aside from shipment-release policies, the literature contains simulation-based models 11 that study and compare the three shipment-release policies in dierent environments. In general, the quantity-based policy can outperform the time-based policy [32]. However, a good consolidation policy should be time-based when considering the eects on service characteristics such as the transit time of an order [145]. Jackson [91] develops a computer- based simulation model to study a four-factor, full factorial design. The simulation measures system performance based on the average cost per order, the average order cycle length and the variance of the order cycle. The author conduct multiple numerical experiments by varying the scheduled consolidation cycle and the number of consolidation points in the network in order to study the performance of the system. Zhou et al [169] study the dierence between a strategic alliance collaboration and a full collaboration for a freight consolidation model with multiple agents that use a hybrid policy. If the policy triggers a less-than-full truckload shipment release, then the collaborating agents have the opportunity to transfer shipments to utilize the unlled capacity. They conclude that collaborative methods are benecial but dependent on the arrival pattern, shipment deadline distribution and truck choice. Higginson and Bookbinder [84] use a discrete-event system simulation model to compare the quantity, time and hybrid shipment-release policies. The authors use an unshifted Gamma distribution for order weight and a Poisson arrival process. Using sample parameters, they compare the mean cost per cwt (hundredweight) and mean delay per order for dierent holding times and order arrival rates. Closs and Cook [39] develop a dynamic simulation model that replicates complex operations in a multi-stage distribution network for use in real-world applications. It uses a consolidation strategy that ships orders with a high priority and holds orders with a lower priority to be consolidated with future shipments. More recently, a series of papers emphasizes vendor-managed inventory applications and argue that consolidation policy parameters and inventory policy parameters should be stud- ied simultaneously. These papers are considered to be integrated inventory and consolidation 12 problems while the problems mentioned up to this point are pure consolidation problems. C etinkaya and Lee [29] calculate the optimal replenishment quantity for the inventory pol- icy and the optimal dispatch frequency for the consolidation policy. The authors use a renewal-theoretical model and Poisson demands. Axsater [13] propose an exact optimiza- tion algorithm for the previous problem. C etinkaya et al [30] revisit the problem in C etinkaya and Lee [29] and consider the case where the supplier applies a quantity consolidation pol- icy. The authors develop approximations to solve the compound renewal process. Mutlu and C etinkaya [120] consider joint inventory replenishment related costs in the derivation of a time-based policy and a quantity-based policy. Freight consolidation is considered in more complex problems. For example, Ulk u [146] considers environmental impacts and total costs in the objective function of a discrete-time based consolidation model where shipments are released only at specic moments of the day. Using a discrete-time Markov decision process, Higginson and Bookbinder [85] study a sequential shipment consolidation model where the shipper is required to reconsider the dispatch decision at the arrival of every order. Howard and Marklund [87] study the cost benets of using state-dependent myopic allocation policies instead of a simple rst-come- rst-serve rule in allocating shipments to retailers. While in some circumstances the myopic allocation policy is considerably better, rst-come-rst-serve remains an attractive policy in practice. Song et al [140] consider a third-party logistics environment where consolidation and pick-up decisions need to be made. The authors show that the nonlinear optimization problem can be solved by a dual-based algorithm. The solutions are on average 3.24% within optimality. Several studies analyze the design of consolidation systems and policies. Cooper [42] de- signs a simulation to study consolidation strategies and warehousing in the design of physical distributions systems. The author applies multivariate analysis of variance to determine the lowest cost systems and performs a regression analysis. The study shows that consolidation 13 should be considered for high-valued products. In Cooper [43], dierent combinations of centralized and decentralized distribution systems are studied. The author examines the trade-o between cost and delivery time and discusses the various in uencing factors in de- ciding between establishing a centralized or decentralized distribution system. Uster and Agrahari [150] study a freight design problem that decides the locations and capacities of consolidation and deconsolidation centers, shipment routes from origins to destinations, and capacities on line-haul links. A Benders decomposition-based approach solves the optimiza- tion problem. Mesa-Arango and Ukkusuri [114] study combinatorial bids, a mechanism used for freight consolidation in the shipper's perspective. The shipper makes bids to multiple customers in hopes that it will ll his capacity. The problem integrates consolidation benets into a multi-commodity one-to-one pick-up-and-delivery vehicle routing problem, which is solved using a branch-and-price algorithm. The solution determines which customers the shipper should place a bid to best use his capacity and the corresponding pick-up schedule. Ha et al [74] perform an analysis of how consolidation strategies and their impact on the number of consolidation points and special delivery requirements aect logistics system performance. Transportation costs, inventory costs, total costs, mean delivery time, and delivery time variance are used as measurements for the comparison of two types of networks: direct LTL shipments and line-haul transportation of consolidated shipments. They assume that the rm maintains a network of warehouses and inventories to satisfy multiple customers in a region. The results show that the special delivery requirement (the number of special orders that must be sent immediately) greatly impacts the system costs because it directly aects the volume that can be consolidated. The number of consolidation points impacts the system very little. Popken [127] considers a multi-attribute, multi-commodity problem with consolidation at multiple transshipment terminals. The proposed model is dicult to solve globally so the developed algorithm focuses on nding local optima and uses a heuristic search to improve it. Pooley and Stenger [126] study the impact of 5 factors on shipment 14 consolidation in a logistics environment: network design, mean order size, internal cycle time constraints, LTL carrier discount, and geographic distribution of customer demand. The solution approach includes a shipment consolidation heuristic, a dynamic simulation model, and a mixed integer mathematical programming network design model. The mean order size, internal cycle time, and LTL carrier discount have the most signicant impact on unit costs for most businesses and echelons. Practical applications and case studies provide further insight on the behavior of real world scenarios to consolidation policy implementation. Marcucci and Danielis [110] per- form a stated-preference study to investigate a rms' willingness to use and pay for an urban freight consolidation center (UFCC). Based on the sample data, UFCC service cost, delivery time, annual cost, and parking distance from the shop have a major in uence on choosing between UFCC and private transport. Lee et al [106] apply a quantity-based pol- icy to a third party warehouse. Tyan et al [144] consider a system with shipments made between a distribution center and an airport. They focus on maximizing aircraft utiliza- tion under various consolidation policies for a global third party logistics rm. The authors develop a mathematical model that determines dierent shipment allocations to ights for each day. Bookbinder and Barkhouse [24] describe the necessary requirements and modules for a logistics information system to be eective for a consolidation center. Consolidation practices have been successfully implemented by Mobil Oil Cooperation and Kellogg Com- pany. Baush et al [18] develop a tool to aid in dispatching orders for heavy products in bulk and packages from lube plants to customers. They solve set partition integer programs to determine potential schedules for the dispatchers to review before releasing. Mobil has been using the proposed system for 3 years and reduced transportation costs by approximately $1 million dollars. Kellogg Company's planning system reduces costs, plans production and makes distribution decisions. The system estimates savings of $35 million per year from consolidating [26]. 15 2.2 Inventory Control Problems The class of inventory control problems aims to construct a replenishment schedule (or a production schedule) to satisfy demand. The problems reviewed in this section are the economic lot-sizing model, the joint replenishment model and the one-warehouse multiple- retailer problem. We will also include signicant research of other similar models. 2.2.1 Economic Lot-Sizing Problem The classic economic lot-sizing problem (ELSP) is heavily studied in the inventory control and production planning community. The economic lot-sizing problem searches for an order schedule or production schedule based on calculating an optimal lot size for a single item, over a discrete xed time horizon, and with dynamic demand. When the problem is considering an ordering environment, the replenishment decision is made while minimizing total costs: a xed order cost for each period an order is made, a unit cost for each item purchased, and a holding cost for each time period an item is held in inventory. In the production environment, the costs include the setup cost for production and a unit cost for each item produced. In the literature, the economic lot-sizing problem is also referred to as thedynamic lot-sizing problem. Wagner and Whitin [155] propose the dynamic economic lot-sizing model and develop a dynamic programming algorithm as a viable solution approach. The ELSP continues to be of interest because it serves as the core problem to many applications. Due to developments in production and inventory management in the past several decades, the majority of the literature contains extensions to the basic economic lot-sizing problem. Jans and Degraeve [95] provide an overview of various extensions of the deterministic lot-sizing problem. Quadt and Kuhn [129] review the capacitated lot-sizing problem with backorders, linked lot sizes, sequence-dependent setups, and parallel machines. Robinson et al [132] summarize exact and heuristic solution approaches for 1988-2009. 16 Many exact algorithms for the ELSP have been developed, and a majority of them are dynamic programming-based. Bai and Xu [17] consider an economic lot-sizing problem where the retailer replenishes his inventory from multiple suppliers, each with a dierent ordering cost structure. The authors design optimal algorithms based on dynamic programming for cases where each supplier is characterized by incremental quantity discount or multiple set-ups cost structure. A stochastic lot-sizing problem is studied by Guan and Miller [71]. The authors dene a production-path property of an optimal solution which allows them to develop a backward dynamic programming recursion. They show that the value function is continuous, piecewise linear, and convex, and develop an ecient dynamic programming algorithm. Guan and Liu [70] study the same basic problem but with additional inventory bounds and order capacities. Uncertainty exists for customer demands, inventory bounds, and costs throughout the nite planning horizon. Hwang et al [90] introduce the concept of a basis path for the multi-level lot-sizing problem with production capacities and concave cost functions. The dynamic programming algorithm uses the basis path, which is characterized by time and stage, instead of the classic decomposition based on time. Instead of a dynamic programming-based algorithm, Guan et al [69] develop a branch-and-cut algorithm to solve a stochastic uncapacitated lot-sizing problem. Certain studies focus on nding properties in reformulations of the ELSP. Ek sio glu [50] develop a primal-dual algorithm that generates tighter lower and upper bounds for the ELSP with multi-mode replenishment. The author formulates it as a network ow problem with multiple setup costs and additional side constraints. Zhou and Guan [170] propose an ex- tended linear programming formulation for a single-item two-stage stochastic lot-sizing prob- lem under cost parameter uncertainty. Through numerical experiments, they show that the new formulation is more ecient and more stable than the two-stage stochastic mixed-integer programming formulation. Constantino and Gouveia [41] apply a discretization reformula- tion technique to the ELSP and achieve an LP relaxation solution that is integer. Wu et 17 al [160] propose a facility location formulation that yields tight lower bounds. Extensions of the economic lot-sizing problem continue to increase in complexity and be- come computationally expensive to solve, making heuristics and metaheuristics more appeal- ing. Jans and Degraeve [94] summarize metaheuristics developed for the economic lot-sizing problem up to 2004. This includes tabu search, simulated annealing, and genetic algorithms. Lee et al [105] study an ELSP that includes transportation costs and does not allow inven- tory shortage. The authors develop a genetic algorithm to solve the problem. Stadtler [141] modies the dynamic single-level uncapacitated lot-sizing problem by considering a shortest route representation. The author proposes an algorithm that uses Wagner-Whitin and a myopic heuristic that looks beyond the planning horizon by using a simple moving aver- age. This algorithm is compared to the part-period heuristic [45], Silver-Meal [137], Gro's heuristic [68], and a fathoming algorithm [171], and the author shows that his rolling horizon algorithm performs well when there is erratic demand. Wu et al [160] propose a Lagrangian relaxation-based heuristic that performs well compared to a commercial solver and other heuristics. The following work focuses on heuristics and metaheuristics for the stochastic economic lot-sizing problem and its extensions. Beraldi et al [22] develop a x-and-relax heuristic that partitions the stochastic lot-sizing problem problem into subproblems and solves it in a reasonable amount of time. Alonso-Ayuso et al [7] develop a x-and-relax coordination framework that selectively explores the nodes of the search tree based on the characteristics of the non-anticipativity constraints. Fisher et al [58] propose an ending-inventory valuation algorithm in a rolling horizon framework that includes a term to oset end-eects. Their algorithm outperforms the Wagner-Whitin algorithm and the Silver-Meal heuristic [137]. Using a combined rolling horizon heuristic and dynamic programming algorithm, Zhang et al [168] solves the stochastic uncapacitated ELSP with incremental quantity discounts. The production environment is considered by Raa and Aghezzaf [130]. The authors propose a dy- 18 namic, probabilistic heuristic approach that solves the newsboy problem in each production period. Absi et al [2] decomposes the multi-item capacitated lot-sizing problem with lost sales and solves it using a metaheuristic that combines Lagrangian relaxation, a non-myopic heuristic based on a probing strategy, and dynamic programming. Xiao et al [161] develop an improved variable neighborhood search for the uncapacitated multi-level lot-sizing problem. The authors demonstrate that the new search performs better than all existing algorithms for this type of problem. Fazle Baku et al [52] propose a heuristic to solve the ELSP with remanufacturing and product recovery. The authors exploit an alternative formulation that allows them to deconstruct each feasible solution into blocks, that are solved separately, before a heuristic determines the optimal blocks to be constructed together to form the nal solution. 2.2.2 Joint Replenishment Problem The joint replenishment problem is a variation of an inventory problem that focuses on nding the optimal ordering quantities for multiple products ordered by the same retailer while minimizing inventory and setup costs. The supplier generally charges a major ordering cost regardless of order size and a minor ordering cost dependent on the number of items. Khouja and Goyal [100] review the literature on the classical, stochastic and dynamic joint replenishment problem from 1989 to 2005. It summarizes heuristics and special approaches such as power-of-two policies and genetic algorithms. Several solutions have been introduced for the deterministic JRP. Silver [137] proposes an iterative heuristic that nds the time between replenishments. Kaspi and Rosenblatt [98] propose an improvement to Silver's heuristic. Kaspi and Rosenblatt [99] develop the RAND algorithm that computes the upper and lower bound for the optimal cycle length. They apply Silver's heuristic within these bounds at strategic locations, which nds the optimal cycle length the majority of the time. Nilsson and Silver [121] propose a non-iterative improvement 19 routine to be performed on the solution from Silver's heuristic. It's ease-of-use makes it more appealing for real world applications, even though it does not perform better than existing optimization approaches and other sophisticated heuristics. Khouja et al [101] modify the RAND algorithm to solve the JRP with continuous unit cost change. The RAND algorithm can also solve the JRP with resource restriction after additional modications [117]. The exact algorithms for the joint replenishment problem with constant demand focuses on nding an optimal cyclic policy. Goyal [67] determines the strict-cyclic policy by obtaining lower and upper bounds for the optimal cycle. Once the bounds are known, they nd the optimal solution through enumeration. Van Eijs [152] argues that the strict-cyclic policy is not the same as the optimal cyclic policy for low values of the major setup cost. The author's algorithm is similar to Goyal's but a dierent lower bound is used to obtain the optimal among all cyclic policies. Viswanathan's [153] algorithm tightens the lower and upper bound so that the computational eort to consider every enumeration is much lower than Goyal's and Van Eijs' algorithms. Fung and Ma [60] analyze the previously mentioned algorithms to develop an improved algorithm that evaluates fewer intervals for an optimal cycle. However, it does not guarantee an optimal solution for the optimal strict cyclic policy. Viswanathan [154] improves Fung and Ma's algorithm by providing a lower bound that will ensure an optimal strict cyclic policy. While the previous algorithms depend on an enumeration procedure, Wildeman et al [159] propose a global-optimization algorithm based on a dynamic Lipschitz constant that generates a solution with little computation time. Frenk et al [59] develop a procedure that solves a convex-programming problem before applying a feasibility procedure and global optimization technique. Jackson et al [93] considers the JRP with a powers-of-two restriction, where each reorder interval is a power-of-two multiple of the base planning period. The authors propose a sorting algorithm, which yields a solution whose average annual cost is within 6% of the long-run 20 minimum average annual cost of the general problem. Federgruen and Zheng [56] propose a two-stage algorithm that nds an optimal power-of-two policy for a JRP with general joint cost structures. The two-stage algorithm includes a decomposition algorithm and a rounding procedure. Heuristics and metaheuristics are attractive solution approaches for more complex exten- sions of the joint replenishment problem. Hariga [76] develop two heuristics that iteratively generates a near-optimal replenishment policy by starting with order frequencies derived from a solution of the relaxed joint replenishment problem. Federgruen and Tzur [55] develop a partitioning heuristic that splits the horizon into JRP subproblems, which are solved using a branch-and-bound method. Praharsi et al [128] develop a heuristic that balances the order cost and the inventory costs for the classical, centralized, and decentralized JRP models. It can also be implemented in the stochastic setting where random variables are generated by Monte Carlo simulation. Zhang et al [167] study a JRP with complete backordering and correlated demand. The proposed heuristic adjusts the replenishment frequencies of minor items to solve the problem. Amaya et al [8] develop a linear programming-based heuristic framework to solve the constrained joint replenishment problem. It denes a range of values for the time between replenishment and solves a linear programming model for each value in that range. Robinson et al [131] show that a simulated annealing metaheuristic performs well for the dynamic joint replenishment problem. Another popular metaheuristic for the JRP is genetic algorithm. Moon et al [118] apply the genetic algorithm to the JRP with resource restriction. Lin and Hsiau [109] dene a new gene based on the basic cycle time for a modied genetic algorithm. Its gene is dened as the ratio of each product delivery cycle time to the basic cycle time. Wang et al [157] propose a dierential evolution algorithm that uses direct and indirect grouping, which allows for interdependence of minor ordering costs. A modied adaptive dierential evolution algorithm is applied to the stochastic joint replenishment and 21 delivery scheduling problem with uncertain costs by Wang et al [156]. 2.2.3 One Warehouse Multi-Retailer The one warehouse, multi-retailer (OWMR) problem generalizes the single-item lot-sizing problem and the joint replenishment problem. The goal is to determine the optimal lot- sizing policy for a warehouse and to satisfy demand for a set of retailers. Federgruen [53] provides a review of the OWMR problem. Solyali and Sural [139] propose a new integer programming formulation and show that it is stronger than the echelon stock-based and transportation-based formulations in the literature. Several pieces of literature focus on determining the optimal continuous review (R;Q) policy for the OWMR problem. When inventory falls below a specied level, R, a batch of size Q is ordered. Axs ater [11] considers a two-level OWMR problem with identical retailers. Retailers face a stationary and independent Poisson demand and all facilities follow a continuous review policy. The author proposes an exact algorithm to determine the optimal parameters for R and Q while considering holding and shortage costs. Axs ater and Zhang [16] solves a modied version of the problem, where the retailers follow a separate reordering policy from the warehouse. The trigger to place an order at the warehouse, R, is now based on the sum of the retailers' inventory. If the sum of the retailers' inventory drops to a joint reorder point R r , the retailer with the lowest inventory position places an order of size Q r . Assuming the batch sizes for the warehouse and retailers are given, the authors obtain steady-state probabilities for the long-run average costs and can optimally nd the reorder points for the retailers and warehouse. In his 2000 paper [12], Axs ater revisits the problem where all retailers and the warehouse apply dierent continuous review policies. The author derives the probability distributions of the retailer inventory levels in steady state. Chen and Zheng [33] study the OWMR with centralized stock information. The retailers are nonidentical with independent compound Poisson demand processes. The authors propose 22 two approximation methods to calculate the long-run average holding and backorder costs of any feasible continuous review (R;Q) policy. Both approximation procedures determine the steady-state distributions of the inventory levels by disaggregating the backorders at the warehouse. The periodic review of the two-echelon OWMR problem is studied by Axs ater et al [15]. Warehouse replenishments must be multiples of a given lot size, and the objective is to minimize holding and backorder costs. The authors propose a virtual assignment ordering rule for warehouse replenishments and a two-step allocation rule for allocating stock from the warehouse to the retailers. The virtual assignment ordering rule states that the warehouse should order until the additional holding costs are higher than the savings in the retailer costs. The two-step allocation rule determines how much of the available inventory should stay at the warehouse and how much is available to allocate to each retailer. Wang [158] develops a periodic review policy for the two-level OWMR with stochastic demand. The author derives an exact optimal solution and develops a heuristic based on the balance condition because the optimization procedure is computationally intensive. Policies dierent from the popular continuous and periodic review are proposed as well. Axs ater and Marklund [14] derive a new policy that is optimal in the class of position- based policies and relies on complete information at all facilities. Retailers replenish their inventory using a standard (R;Q) policy where the batch quantity is equal for all retailers but the reorder point is retailer-dependent. This new policy orders all the requests from retailers by arrival time. Each request holds two pieces of information: whether it has been satised by a warehouse order request and the corresponding retailer's inventory level. The optimal policy determines when to place a warehouse order to cover the requests from the retailers. The authors use a dynamic programming framework to nd the optimal position- based warehouse ordering policy. Single-cycle policies are studied in Hsiao [88]. The author studies a multi-echelon OWMR problem with deterministic demand. An optimal single-cycle 23 policy is computed by applying recursive tightening methods. The power-of-two ordering policy introduced for the joint replenishment problem is also considered for the OWMR problem. Chu and Shen [38] propose a plane partition method and develop an algorithm to nd a power-of-two policy, which is guaranteed to be no more than 1.26 times the optimal power-of-two policy cost. A few papers incorporate distribution and transportation decisions with the OWMR problem. Monthatipkul and Yenradee [116] study the OWMR in an inventory and distribu- tion control system. They propose a mixed-integer linear programming model that calculates an optimal inventory and distribution plan for the inventories in the supply chain. It is shown to outperform the echelon-stock (R;s;S) control policy where R is the periodic review, s is the reorder point and S is the order-up-to level. Kochel and Thiem [102] propose a particle swarm optimization method for an OWMR problem with transportation considerations. Heuristics and metaheuristics are developed for more complex problems and when com- putation is extensive. Levi et al [108] develop the rst approximation algorithm for the deterministic OWMR problem with ordering costs and holding costs. The algorithm rounds the optimal solution of the LP relaxation to a feasible solution and guarantees to be at most 1.8 times the optimal cost. Cha et al [31] apply a hybrid genetic algorithm to the joint replenishment and delivery scheduling of the OWNR problem. Abdul-Jabar et al [1] develop an easy-to-implement and ecient heuristic to solve the multi-echelon OWMR prob- lem where customer demand arrives to the retailer at a constant rate. They propose an iter- ative approach to calculate integer-ratio policies which focuses on nding a balance between replenishment and holding costs. Cachon [27] takes an entirely dierent approach. The author shows that a Nash equilib- rium, a set of reorder points such that retailer can deviate from and lower its cost, exists and demonstrates how to nd all the reorder points. In addition, the author shows that shifting operation decision control to the supplier leads to optimal supply chain performance. In 24 other words, the supplier decides the reorder points for all the retailers and itself. Yang [163] performs a full factorial simulation experiment to identify environmental fac- tors and policies that impact the performance of a OWMR system. Factors such as the number of stores, order processing time, demand variability, warehouse location, vehicle- scheduling rules, inventory rules and order size were tested. Four performance measures were collected for each simulation run: total store inventory, demand changes observed at the warehouse, mean number of vehicles and total travel distance per day. The number of stores and demand variability have the most in uence on total store inventory, and order size has the most impact on demand changes at the warehouse. The number of stores, order size, and warehouse location have the largest eect on the number of vehicles used and total travel distance per day. 2.3 Supply Chain Cooperation Advantages of cooperation have been studied in several areas of supply chain management including production, inventory and routing. When multiple suppliers or retailers work together by consolidating their shipments, they can take advantage of economies of scale and decrease system-wide costs. The cost allocation problem attempts to nd a fair allocation of the total costs among the group. In this case, a fair and stable cost allocation is an allocation such that every participating member would not benet more if they left the coalition to form another. Fiestras-Janeiro et al [57] provides a review of solution approaches to the cost allocation problem, of which the majority use the concept of the core. If the core is proven to be non-empty, then a stable allocation exists. Several researchers consider an inventory control environment based on the Economic Order Quantity (EOQ). Meca et al [113] study a basic EOQ model where a collection of rms order the same product type, but they have their own demands and storage capabilities. The 25 authors propose an allocation rule such that the ordering cost is divided proportionally to the square of the individual ordering cost. Meca et al [112] introduce a family of cooperative games based on the EOQ problem and propose various cost allocation rules for the games. Economic lot-sizing games involve multiple retailers sharing the cost of joint orders from the same manufacturer [151]. Extensive reviews of solution approaches to the joint economic lot-sizing problem can be found in Ben-Daya et al [21] and Glock [63]. Chen and Zhang [36] formulate the problem as stochastic programs and use the strong duality of the stochastic linear program to nd a cost allocation. Van den Heuvel et al [151] show that the economic lot-sizing game with equal demand for each player and the 2-period ELS game are concave. A primal-dual algorithm developed by Gopaladesikan et al [65] can nd a cost allocation in the core of an economic lot sizing game. Xu and Yang [162] develop a cost-sharing method that possesses properties such as cross-monotonicity, fairness and competitiveness, that make it a viable methodology. Toriello and Uhan [143] develop a dynamic cost allocation for an economic lot-sizing game where each player incurs demand in each period and coalitions can pool orders. The joint replenishment problem focuses on making joint orders, and the cost allocation is interested in how to allocate the joint ordering costs. Anily and Haviv [9] apply optimal power-of-two policies for the rst-order interaction model of the joint replenishment problem, and Zhang [166] applies a strong duality theorem to show the cor is nonempty. Elomri et al [51] develop a procedure to form ecient coalition structures in a non-superadditive game of the JRP. He et al [82] apply general results from polymatroid optimization to the joint replenishment problem, and show that the cooperative game is submodular. Bauso and Timmer [20] design robust allocation rules for a joint replenishment problem, while Dror et al [49] focus on how the joint replenishment costs should be allocated across the individual products such that no subset of products subsidizes another subset. They propose a method to test the sensitivity of a stable cost allocation to cost parameters, and a less excessive 26 computational procedure to calculate the stable cost allocation. A centralized inventory system is studied by Hartman and Dror [78]. A set of stores with common ownership use the centralized inventory system, and each store pays a share of the inventory cost. The authors identify stability, justiability, and polynomial computability as three necessary criteria for cost allocation selection. Hartman et al [80] study a cooperative inventory centralization game with holding and penalty costs. The authors prove that a nonempty core exists for demands with symmetric distributions and for joint multivariate normal demand distributions. Hartman and Dror [79] show that when holding and penalty costs are the same, a cooperative game based on optimal expected costs is subadditive. If the demands are normally distributed, then the core is not empty. When the holding and penalty costs are dierent, the cooperative game could have an empty core. There might be no benets in centralizing the inventory. Hartman and Dror [81] consider single period inventory models with normally distributed, correlated individual demands. The authors propose a step-by-step greedy optimization scheme that nds an optimal centralization solution. Leng and Parlar [107] study a three-level supply chain with demand information sharing. They construct a three-person cooperative game and derive necessary stability conditions. They use the Nash arbitration scheme and the Shapley value to nd a unique allocation scheme. Cost allocation methods are applied to other similar inventory control problems. Bauso et al [19] study a system with one warehouse and multiple retailers where the retailers share reordering costs. They show that the Nash equilibrium reordering policies possess a threshold structure on the number of retailers interested in reordering. The allocation guarantees each player an average allocation amount over time. Dror et al [48] consider a newsvendor inventory centralization game. The authors propose a repeated cost allocation scheme that converges to a least square value or the core of the expected game. Jokar and Sajadieh [97] show that the vendor and buyer benet more from cooperating with each other in an integrated vendor-buyer production-inventory model. 27 Work in supply chain cooperation includes routing games, such as the vehicle routing game and the traveling salesman game. The vehicle routing game allocates the cost of an optimal route conguration among the customers. A constraint generation method can compute the nucleolus if the core is non-empty [66]. Derks and Kuipers [46] study a traveling salesmen game in the context of a repairman who must visit every customer once before returning home. The cost of his trip is allocated to the customers and Derks and Kuipers design an algorithm which computes a core element of a routing game if the core is non-empty. Slikker et al [138] dene the general newsvendor game, where expected joint prots result from coordinating orders, followed by transshipments after demand realization. The authors show that they have nonempty cores. Ozen et al [122] study the cooperation between multiple newsvendors with warehouses. They show that the associated cooperative game between the retailers has a cost allocation. Sanchez-Soriano et al [134] show that a cost allocation exists for transportation games. Yengin [165] characterizes the Shapley value for a xed- route traveling salesman game with appointments by using a property that shows players do not benet from mergers. Fiestras-Janeiro et al [57] study an inventory transportation system with a single item and multiple retailers placing joint orders using the economic order quantity (EOQ) policy. The authors show that a stable allocation exists after identifying several properties. Research in supply chain games are not limited to routing and inventory. Goemans and Skutella [64] study the core of a cooperative game based on a facility location problem. The goal is to nd an allocation such that no coalition of customers has any incentive to build their own facility or ask a competitor for service. Ozener et al [123] study the potential of collaborative opportunities between truck carriers, called the multi-carrier lane-covering problem. The problem is to nd an assignment of shipments to multiple truck carriers that minimizes the total transportation cost of all the carriers or the maximum benet of a collaboration. The authors develop and test multiple exchange mechanisms, with respect 28 to the amount of information sharing and side payments, that allow carriers to realize cost saving opportunities. Schulz and Uhan [136] study supermodular cost cooperative games and look at a least core, which calculates the minimum defection penalty needed to sustain cooperation among the participants. 2.4 Perishable Products The agriculture industry contains perishable products or age-dependent perishability. While the majority of products considered in inventory management and inventory control problems are assumed to be able to stay in stock indenitely to meet future demand, perishable products become unusable after some time. Perishable products in the agriculture industry include food products such as produce, seafood and meat, while barley, grain, and oats are examples of agricultural goods with extremely low deteriorating rates. This research focuses on perishable agricultural products but other industries include deteriorating products or goods such as blood samples, medicine, ready-mix concrete, and electronic components. In this review, we are interested in how perishability is measured and studied in the literature. Ahumada and Villalobos [5] review all applications of planning models for the agri-food supply chain. Chen [37] reviews integrated production and outbound distribution scheduling models for time-sensitive products. The added complexity of perishable products aects production, inventory, and routing decisions. An environment with perishable goods can impact multiple aspects of a supply chain. Thron et al [142] use a discrete-event simulation to evaluate the impacts of supply chain advancements on safety stock levels, replenishment policies, throughput policies and demand due to a collaboration between the manufacturer and retailers. Blackburn and Scudder [23] develop a hybrid model that minimizes the lost value in a supply chain based on a product's marginal value of time. A product's value begins deteriorating exponentially after harvest 29 until the product is refrigerated. The hybrid includes a response model for the period from post-harvest to the beginning of refrigeration. The ecient model considers the beginning of refrigeration to the end of the supply chain, where a product's deterioration is dampened. The results show that the coordination of decisions from post-harvest to the end of the supply chain are not needed to achieve supply chain optimization. The majority of the literature deals with perishable products in two ways: 1) dening usability of the product by its age or 2) applying a rate of deterioration. Most inventory, production and distribution models limit the number of periods that a perishable item can stay in inventory. Aggoun et al [3] tracks the age of the product, which has a limited life in inventory before it is discarded. The authors also measure the products' survival to the next time period as a probability distribution. A markov decision process is used to solve the problem. Federgruen et al [54] classify their products as old or fresh, where fresh products become old after a certain number of time periods. A heuristic algorithm allocates the inventory to dierent delivery points while minimizing the expected shortage, transportation and out-of-date costs. All old units must be distributed before they become outdated and unusable. Geismar et al [62] consider products with a short shelf life. The objective is to determine the required minimum time to complete producing and delivering a product and meet customer demand. They use a two-phase approach that includes a genetic algorithm, a memetic algorithm, and a shortest path algorithm. Ahuja et al [4] studies a multi-period single-sourcing problem where retailers are assigned to facilities while taking into account production, inventory, and perishability constraints. The perishability constraints ensure that product do not stay longer than a predetermined number of periods. The authors propose a greedy heuristic and a very-large-scale-neighborhood-search method improvement step to solve the nonlinear assignment problem. Le [104] proposes a combined column generation and tabu search algorithm to solve an inventory routing problem where out-of-stock never occurs. The author considers a perishable product to be in good condition 30 until it is discarded after a xed number of periods. Jia and Hu [96] use a xed lifetime for the perishable item in a dynamic ordering and pricing problem with one supplier and one retailer. Chen and Sapra [35] consider products with a two-period lifetime for a periodic review inventory model. The value or quality of a perishable item can be measured using a rate of decay or deterioration. Huang and Yao [89] concentrates on nding lot-sizing policies for a single- vendor, multiple buyer system with items that deteriorate continuously. Chen et al [34] model a nonlinear mathematical model that decides production start times and quantities for each commodity, and constructs the delivery routes. The delivery planning is modeled as a vehicle routing problem with time windows. The quality of the product is measured using a rate of deterioration over time. Rong et al [133] integrate food quality degradation and temperature control in a mixed-integer linear programming model. Yang and Wee [164] incorporate a rate of decay for perishable items in a mathematical model to determine a production-inventory policy for a system with a single-vendor and multiple-buyers. Sazvar et al [135] model the deterioration process by applying a nonlinear holding cost in an inventory model with stochastic lead time. Coelho and Laporte [40] incorporate the age and value of a perishable item in the holding cost. The unit holding cost is formulated as a function of the age of the item. Some problems approach perishability dierently. Perry and Stadje [125] study a marko- vian model with random input and an external source of obsolescence. After a lifetime generated by a Poisson distribution, the entire system's product is spoiled and must be discarded. Hariga [77] uses both a xed lifetime and an exponential rate of decay for an inventory problem. Real world applications and case studies with perishable products cover a variety of products and systems. Gaur and Fisher [61] solve a periodic inventory routing problem for a supermarket chain in the Netherlands. Ahumada and Villalobos [6] develop a model to be 31 used for making production and distribution decisions while maximizing revenues for large fresh produce growers in Mexico. Guan and Philpott [72] formulate a production planning problem for a New Zealand dairy company as a multi-stage stochastic quadratic programming model. A decomposition algorithm nds an optimal sales policy. Hemmelmayr et al [83] study a vendor-managed inventory resupply policies of blood products for the Austrian Red Cross. The problem is formulated as an integer programming problem and the authors use a variable neighborhood search. 2.5 Research Gap A supply chain is composed of inventory, production, and scheduling operations between the suppliers, distribution centers, and retailers. The literature review summarizes current methodologies and systems studied in freight consolidation, supply chain cooperation, inven- tory control, and perishability. The freight consolidation literature is extensive in shipment release policies that follow a time or quantity policy. However, to our knowledge, perishable products are not considered when developing these policies. Some problems in inventory control include consolidation in the manner of joint shipments or replenishment, as seen in the joint replenishment problem. However, the joint orders are not motivated by perish- able products or the advantages that would be achieved for small suppliers. Cost allocation problems for supply chain cooperations vary from inventory control to routing problems. Inventory control and freight consolidation problems have been studied extensively, how- ever not many problems, as far as we know, combine inventory with consolidation, perisha- bility, and cooperation over a long planning horizon. The proposed model considers (1) small suppliers generating LTL shipments, (2) consol- idation practices, and (3) perishable agricultural commodities. The challenge is in designing a system that reduces transportation costs for all the suppliers by exploiting economies of 32 scale and FTL rates. The system must also be simple enough for an ecient solution and decentralized control, and allow for fair cost allocation schemes. We propose studying two aspects, and cost allocation. We solve a large-scale mixed-integer programming problem that considers multiple sup- pliers, multiple shipping methods and perishable agricultural products where suppliers can enter into a cooperation. The planning horizon will be suciently large to incorporate the long harvest and planting seasons of agriculture. 33 Chapter 3 Deterministic Direct Shipping Consolidation Model The implementation of consolidation strategies is very important in the agriculture environ- ment where a large number of suppliers have low demands. Larger shipment volumes can take advantage of economies of scale and negotiate more favorable trucking rates. However, small suppliers are unable to achieve better transportation rates because they have low de- mands. Consolidation across the small growers would allow them to ship at better rates. In this chapter, we address the direct shipping model with deterministic demand and seek to nd an ecient heuristic that allows the suppliers to ship at more favorable rates. In this chapter, we rst introduce the problem and model formulation. We discuss the cost function structure and its properties. Next, the dynamic programming model is proposed with an explanation on the set of states and the transition function. We develop an ecient heuristic that aims to consolidate products when a full or near-full truckload can be achieved. Next, we test the performance of the heuristic and dynamic programming model against an optimal solution and lower bound provided by CPLEX. Finally, a case study of the California cut ower industry is introduced. 34 3.1 Model Formulation Figure 3.1: A Transportation Network with a Consolidation Center and Direct Shipping A set of suppliers ship a common product to a consolidation center, where products going for the same break-bulk destination leave in combined shipments. Direct shipment occurs between each supplier and the consolidation center, then from the consolidation center to the destination (see Figure 3.1). This model does not consider routing strategies from the suppliers or to the destinations, therefore products headed towards dierent destinations do not share the same transportation resource. From here on, we will formulate this model as a single break-bulk destination problem. Each destination is solved separately. Inventory capacity is not an issue because perishability keeps inventories low at the consolidation center. The products in this model are assumed to be perishable and have limited inventory time. We let represent the maximum number of time periods the product can stay at the consolidation center. We assume the consolidation center is closely located to the suppliers, so the transportation cost from the suppliers to the consolidation cost are signicantly smaller than the long-haul costs. Therefore, we do not consider the costs of transporting the product to the consolidation center. We assume the product is harvested immediately before shipment from the supplier to the consolidation center, and that inventory is held primarily at the consolidation center. The following is a list of all the parameters for this problem: 35 S: Set of suppliers T : Time horizon d it : Demand at period t from supplier i,8i2S;t = 1:::T c F : Transportation cost for a full truck, ($ per truck) c L : Transportation cost for an LTL unit, ($ per cubic foot) c U : Transportation cost for a courier service, ($ per pound) : Conversion factor (pounds per cubic foot) : Maximum inventory time at the consolidation center F : Maximum capacity for a truck, in cubic feet L : Maximum capacity for an LTL unit, in cubic feet Because the product is perishable, it stays at the consolidation center a relatively short amount of time. We assume that the unit cost of the product is small compared to the shipping cost, therefore this model does not consider inventory costs. The inventory costs are much lower than the transportation cost. In Chapter 4, we relax this assumption and include inventory costs. A mixed integer program mathematical model represents the consolidation model with direct shipping. The decision variables include the number of trucks charged at FTL rates (x tF ) and at LTL rates (x tL ). The volume on the full truck is dened as y istF , and the volume sent by LTL is dened as y istL . Small volumes are sent using courier services and are denoted as y istU . x tF : Number of full trucks leaving the consolidation center at time t,8t = 1:::T x tL : Number of LTL units leaving the consolidation center at time t,8t = 1:::T y istF : Volume sent at the FTL rate with origin supplier i, dispatched on period s, that must leave the consolidation center by period t,8i2 S;s = 1:::T;t = s:::minfs +;Tg y istL : Volume sent at the LTL rate with origin supplier i, dispatched on period s, that must leave the consolidation center by period t,8i2 S;s = 1:::T;t = s:::minfs +;Tg y istU : Volume sent using a courier service with origin supplier i, sent on period s, that must leave the consolidation center by period t,8i2 S;s = 1:::T;t = s:::minfs +;Tg The mixed integer programming formulation is the following: 36 Minimize: T X t=1 (c F x tF +c L x tL ) +c U X i2G T X s=1 minfs+;Tg X t=s y istU (3.1) Subject to: X i2S minfs+;Tg X t=s y istF F x sF ; 8s = 1:::T (3.2a) X i2S minfs+;Tg X t=s y istL L x sL ; 8s = 1:::T (3.2b) t X s=maxf1;tg y istF +y istL +y istU =d it ; 8i2S;t = 1:::T (3.2c) y istF 0; 8i2S;s = 1:::T;t = 1:::T y istL 0; 8i2S;s = 1:::T;t = 1:::T y istU 0; 8i2S;s = 1:::T;t = 1:::T (3.2d) x tF 0; 8x tF 2Z;t = 1:::T x tL 0; 8x tL 2Z;t = 1:::T The objective (Equation 3.1) minimizes the total transportation cost. Equations 3.2a and 3.2b ensure that the volume on the trucks do not exceed the truck capacity for the FTL rate and the LTL rate, respectively. The volume sent by trucks and courier must satisfy the demand and be shipped before its deadline (Equation 3.2c). The volume is in real numbers because the unit of measurement is cubic feet and allows for partial volume, while the number of trucks must be integer (Equation 3.2d). 37 3.2 Cost Function We next examine the cost function in detail since it play a major role in developing ecient procedures for solving the problem. The transportation cost in this model includes three options. The full truckload (FTL) rate is a per truck rate. The less-than-truckload (LTL) shipping option charges for each LTL unit sent. The courier services generally use a per weight rate such as per pound or per kg. These rates are destination-dependent but they follow a similar trend. We assume economies of scale (see Figure 3.2) where the following property holds: c F F < c L L <c U (3.3) Figure 3.2: The Shipping Cost Function Very small shipments are least expensive to ship at the courier rate. As the volume increases, the LTL rate becomes more advantageous. However, very large shipments are best sent at the FTL rate. Figure 3.2 illustrates the general cost structure for all suppliers in the same area that are shipping to the same break-bulk destination. Shipments that are ready to leave the consolidation center can be transported using a combination of the three shipping methods. First, if the shipments leaving can ll a truck 38 to capacity, F , then it will be shipped at the FTL rate c F . The remaining volume can be shipped in dierent possible combinations: at the FTL rate, LTL rate, LTL and courier rate, or the courier rate. Equation 3.4 calculates the least-cost combination of LTL and courier service. LU (x) = x L + min c L ;c U x x L L (3.4) The cost function(x) determines the least-cost shipping method by using Equation 3.4. For nonnegative values of x, the cost function is the cost of shipping at the full truckload rates (when x> F ) plus the least expensive shipping method for the remaining volume. (x) = x F c F + min c F x x F F ; LU x x F F (3.5) For volume up to F , two breakpoints are dened. b F signies the volume above which the FTL rate is the least-cost shipping method. Volumes less than b F are best sent using a combination of LTL and courier. The breakpoint b L is the volume above which the least expensive cost is the LTL rate, while below it is the courier rate. b F = c F c L L + c F c F c L L c L L 1 c U (3.6) b L = c L L c U (3.7) The next sections propose two solution approaches: dynamic programming and a heuristic. 3.3 Dynamic Programming The dynamic programming model nds the total cost for a time horizon of lengthT . At each time t, t = 0 t ;::: k t ::: t is a vector of nonnegative values of length + 1 that represents the state variable. The variable t is segmented by the shipping deadline: k t represents the 39 demand in inventory that must leave by period t +k, for k = 0:::. The decision rule is: for a given state variable at time t, 0 t must leave because t is its deadline, and the decision maker must decide how much of the remaining inventory should be added to the outgoing shipment to minimize the total cost. g t ( t ) = min n ( 0 t +) +g t+1 f t t ; t ;d t o (3.8) Let g t ( t ) be the optimal value function (see above). It is dened as the total cost for periods t to T , when the current inventory levels are t . The rst part of this equation calculates the least-expensive shipping method for the outgoing shipment using the cost function (x). The outgoing shipment is the combined amount of the demand with today's deadline, 0 t , plus what the decision maker decides to add, . The second part is the total cost-to-go for periodst + 1 toT , the optimal solution of the subproblem with given demand f t ( t ; t ;d t ). We let d t be the aggregate demand that arrives from all the suppliers (d t = P i2S d it ). t = ( 1 t ::: k t ::: t ) represents the inventory added to the outgoing shipment at time t, and is partitioned by deadline. This vector contains the inventory and its corresponding deadline, allowing the model to transition to the correct subproblem. is the total amount being added to the outgoing shipment ( = P k=1 k t ). is chosen such that the total cost is minimized. The transition function f t t ; t ;d t updates the state variable from period t to t + 1 by using the information in t . k t k t 8k = 1 to is true because the amount that can be added to today's shipment is limited by the inventory level. can be all or a portion of inventory with feasible values in the range of 0 k t k t . The initial calculation of the deterministic dynamic programming model is g T ( T ). At time T , the state variable contains only the demand that must leave at time T , 0 T . The 40 other values, k T 8k = 1 to , are equal to zero because it is beyond the time horizon. The boundary condition simplies to: g T ( T ) =( 0 T ) (3.9) The principle of optimality with equations 3.8 and 3.9 suggest an optimal policy can be constructed. 3.3.1 Transition Function The transition function updates the state variable from period t to t + 1 by adjusting each k t so that it references the correct time period after a shipment of size + 0 t . Three pieces of information are required for the transition function: the current state variable t , the inventory added to today's shipment t and the aggregate incoming demand d t . When t + 1 + T , then the transition function adjusts by shifting demand values forward one period. The demand values are updated according to what is shipped and incoming demand is added to the end of the state variable. t+1 =f t t ; t ;d t = 8 > < > : k t+1 = k+1 t k+1 t 8k = 0::: 1 t+1 =d (3.10) When t + 1 + > T , then the transition function updates the values of the vector for t +k<T and the remaining values are set to 0 because there are no new demands. t+1 =f t t ; t ;d t = 8 > < > : k t+1 = k+1 t k+1 t 8k = 0:::T 1 t+1 = 0 8k =T 1:::t + 1 + (3.11) 41 3.3.2 Complexity For large daily demands, the dynamic programming model becomes computationally expen- sive. The state variable t represents the demands that must leave today and in the next periods. The set of states are dependent on how the daily demand is discretized. Let ! be the discretization factor and D m be the maximum daily aggregate demand across all suppliers. The number of possible states, Y , for = 1 is Y =D m =! + 1, including zero. If ! = 0:5 and D m = 10, then the number of states is Y = 21. A small discretization factor yields a more accurate total cost for a time horizon of length T . However, as ! decreases, the runtime for the discretized dynamic programming model will increase. For large-scale problems with multiple suppliers and a long time horizon, using a larger discretization factor may be a good option because of the lower runtime. For any positive integer value of , the number of states in one day is Y +1 because the state vector is of length + 1. We are considering the states that cover all the possible inventories and their deadlines. TY +1 possible states exist for the entire time horizon. As increases, the number of states increases exponentially, and is the major factor for the dynamic programming model's computational complexity. As a result, we are motivated to develop an ecient heuristic that performs well. We develop a pruning algorithm to decrease the number of states in the dynamic pro- gramming model. At each time period t, the dynamic programming model enumerates all the possible states. For each given state, is enumerated for all possible inventory values that can be added. However, not all values of need to be considered. The FTL rate is the least-cost shipment method when the demand leaving today is greater than b F . The cost of outgoing shipment does not change for values corresponding to b F P i2S 0 it + F . The inventory is being added to the outgoing shipment at no additional cost. This can be easily extended for volumes beyond F . We show the dierences in runtime for the dynamic programming model with and without pruning. Other pruning strategies may also improve 42 the speed of the dynamic programming model, but they are not the focus of this research. 3.4 -based Consolidation Heuristic We develop a look-ahead heuristic that aims to decrease transportation costs by making consolidation decisions for up to periods. The cost function in Figure 3.2 illustrates that the lowest cost per volume is the FTL rate. The next lowest is LTL then courier service. Therefore, it is least expensive to ship large volumes at the FTL rate. The strategy of this heuristic is to consolidate demand across time in order to ship economically at the cheaper FTL rate. At time t, the product that arrives today must leave in period t +. From time t to t +, the product can stay in inventory or be added to a shipment that leaves earlier than its deadline. This heuristic uses the breakpoints b F and b L to determine whether product should be consolidated to ship at the FTL, LTL or courier service rate. The heuristic makes decisions in the following manner for = 1: if there is an outgoing shipment, then rst ship completely full trucks at the FTL rate. If the remaining volume is greater than zero, then we need to decide if we should consolidate. If today's volume plus tomorrow's volume is greater thanb F , then we add demand to ll the excess capacity of the partial truck. If the combined volume is less than b F , then the heuristic rst calculates the number of full LTL units to ship at the LTL rate. If the remaining volume plus tomorrow's demand is greater than b L , then we ll up the LTL unit with inventory by earliest deadline. Otherwise, we ship the remaining volume at the courier rate and do not consolidate. This is easily extended for general . The procedure to ll an LTL unit and a full truckload are shown in algorithms 1 and 2. The basic idea is that when there is excess capacity that can be all or partially lled at no additional cost, the inventory is added by earliest deadline. 43 Algorithm 1 : t = LTLFill(( L 0 t ), t ) Fill excess capacity ( L 0 t ) k = 1 k t = 0 8k = 0::: for k = 1 to do k t =min( L 0 t P k1 m=1 m t ; k t ) k =k + 1 end for return t Algorithm 2 : t = FTLFill(( F 0 t ), t ) Fill excess capacity ( F 0 t ) k = 1 k t = 0 8k = 0::: while k and F 0 t P m=0 m t > 0 do k t =min( F 0 t P k1 m=0 m t ; k t ) k =k + 1 end while return t As stated in Section 3.1, the planning horizon isT and represents the maximum number of periods product can stay in inventory. We use the following variables and notation from the dynamic programming model as well: t = 0 t ;::: k t ::: t to track inventory at the consolidation center, t = ( 1 t ::: k t ::: t ) tracks the amount to be added to today's shipment, and d t = P i2S d it is the aggregate demand. The transition function (Equations 3.10 and 3.11) is also used here to update t to the next period. Algorithm 3 describes the complete -based consolidation heuristic. 44 Algorithm 3 : -based Consolidation Heuristic Initialize t = 0 t ;::: k t ::: t to demands d 1 ;d 2 ;d 3 ;:::d +1 for t = 1 to T do if 0 t > 0 then Shipb 0 t F c at the FTL rate Calculate the remainder: 0 t = 0 t b 0 t F c F if 0 t + 1 t b F then t = FTLFill(( K 0 t ), t ). Ship one truck at FTL rate. else Shipb 0 t L c at the LTL rate. Calculate remainder: 0 t = 0 t b 0 t L c L if 0 t + k t b L then t = LTLFill(( L 0 t ), t ). Ship one LTL unit at the LTL rate. else Do not add anything. Ship at courier rate. end if end if end if Update the inventory: t+1 =f t ( t ; t ;d t++1 ) end for 3.5 Experimental Results This section contains two sets of experimental results. First, we study the California cut ower industry and compare the consolidation model with their current transportation prac- tices with = 1. Then we study the performance of the dynamic programming model, an optimal solution from the CPLEX solver, the -based consolidation heuristic and a simple + 1 policy when is generalized. 45 3.5.1 Case Study: California Cut Flower Industry In this section, we focus on the California cut ower industry. For this application, there are no holding costs, and the maximum time any product can spend in inventory is one day ( = 1). The California Cut Flower Commission (CCFC) plans to build a consolidation center in Oxnard, CA in an attempt to decrease its transportation costs and become more competitive in the cut ower industry. The quality of the owers is extremely important, but adding a consolidation center to the supply chain increases the time the cut ower spends in transit. Consolidation decreases transportation costs in theory, but we do not want to decrease costs at the expense of low-quality product. Therefore, is set to one time period. Two models are considered: (1) the Consolidation model and (2) a model re ecting cur- rent transportation practices of CCFC. Our case study for the California cut ower industry sets to one day. We apply the-based Consolidation Heuristic to this practical setting and assume the growers will use this heuristic if they operated a consolidation center jointly. Current Practices In order to determine if cost benets exist for the Consolidation model, it needs to be compared to the cost of CCFC's current transportation practices. Currently, California ower growers ship everything independently and no cooperation or consolidation takes place. When an order is received, the grower fullls the order in the same period and ships all the products for the same destination using the cheapest option available (full truck, LTL or FedEx/UPS). Inventory is never held at the farm, so we assume that the grower cuts the owers and ships the order on the same day. Each grower incurs their own transportation costs. The total cost is determined through direct calculations of the outgoing shipment. For each destination, the following calculations are considered: the total volume leaving grower i on day t is d it . 46 Then the total transportation cost for each destination of the current practice is X i2S X t=1:::T (d it ) Numerical Results 2010 shipping data was provided by 16 members of the CCFC and it represents 53% of the production volume. It includes shipment date, delivery date, volume, destination and whether the customer is a wholesaler or mass marketer for an entire year. Wholesale ship- ments cannot be consolidated with mass market shipments. Both models consider this constraint. The data shows increases in demand for the weeks leading up to Valentine's Day and Mother's Day. Four growers provided their data from a similar study conducted by CCFC in 2008 and did not update it to 2010 data. To determine whether we can use the data of the four growers who were unable to update, we test the data of those who provided both 2008 and 2010 data to see if there are any dierences in the two data sets. A Two-sample t-test compares the response from the two groups of data. The two groups of data is the demand in 2008 and 2010 for the growers who provided both sets of data. Three assumptions must be true about the two sets of data in order for this test to be informative and valid. The two sets (1) must be samples from a distinct population, (2) are independent of each other, and (3) have normal distributions. The two data sets satisfy all three criteria. Figures 3.3 and 3.4 are probability plots from MINITAB for Farm A. The outliers at zero indicate the days with no shipments. The outliers beyond 10,000 are shipments for the busy time of the year (Mother's Day and Valentine's Day), which appears for both years. In general, they follow a normal distribution for the rest of the year. This is true for all the farms tested in the Two-Sample t-test. The two test statistics are the average sales of 2008 data and the average sales of 2010 47 Figure 3.3: Probability Plot for 2008 data from Farm A data for every farm. i;2008 =Average sales of 2008 data for grower i i;2010 =Average sales of 2010 data for grower i The null and alternate hypotheses are: H 0 : i;2010 i;2008 = 0 H a : i;2010 i;2008 6= 0 MINITAB generates the average dierence, 95% condence interval and p-value for each farm. The summary of the results are in Table 3.1. Overall, most of the farms show a violation of the null hypothesis. This indicates that the 2008 and 2010 data are statistically signicantly dierent and the 2008 data must be extrapolated in order to have a representative sample for 2010. The average dierence in production volume for the growers that submitted both data sets is 4.6%. The data for the four farms who submitted 2008 data and were unable to update to 2010 was increased by 4.6% in order to be used in the analysis. 48 Figure 3.4: Probability Plot for 2010 data from Farm A Table 3.1: Two-Sample t-test Summary Farm Avg Dierence (2010-2008) 95% CI, Lower Bound 95% CI, Upper Bound P-Value A 1321.2 1219.9 1422.5 0.000 B 52.9 17.6 88.2 0.003 C 540.4 493.2 587.6 0.000 D -665.1 -754.7 -575.6 0.000 E 64.1 51.8 76.4 0.000 F -202.9 -282.3 -123.6 0.000 G 24.9 -9.5 59.4 0.156 We use 365 days worth of demand data. The four extrapolated sets plus the sixteen submitted data sets gave us a total of twenty data sets for the analysis. This represents 63% of the production volume. LTL rates were provided by the growers who provided shipping data and FTL rates were provided by Supply Chain Coach, a consulting company in California. The transportation costs are assumed to be time-independent. Both models return the volume sent in full trucks, the volume sent as LTL units and the volume sent using courier services. These shipping methods are also partitioned into wholesale and mass market shipments because they cannot be shipped together on the same truck. Scenario Construction 49 We construct the scenarios by using 2010 aggregate sales gures provided by CCFC and the sixteen raw data sets for 2010. The four extrapolated data sets are not used because we want to extrapolate the missing growers based on 2010 information. 50 members on this list did not provide shipping data and we extrapolate based on a level of participation of the 50 missing data sets. The 50 growers are sorted in decreasing order using annual aggregate sales gures, s i , for missing grower i. The average sales volume per production dollar, , was calculated using the sixteen growers who provided information. This value was used to estimate the total annual demand, s i , for each of the 50 missing growers. The total demand is distributed across 49 U.S. destinations over 365 days of the year. The demand needs to be split into wholesale and mass market volumes. Using the sixteen raw data sets, a mass market and wholesale proportion were calculated for each day and des- tination combination. The variablesp jtM andp jtW represent the mass market and wholesale proportion of the total demand on day t for destination j (e.g., p 2TXM =:02 is the propor- tion of the total demand that is a mass market shipment sent to Texas on January 2nd). Multiplying the estimated total volume, s i , to the proportions distributes the data based on the raw data's 2010 activity. To summarize, the following steps estimate and distribute the volume for each missing farm. Step 1: Estimate the average sales volume per production dollar. Calculate . = 1 16 P 16 i=1 v i s i where v i is the total sales volume for each farm i that provided 2010 data, and s i is the corresponding 2010 production sales gure. Step 2: Estimate the total volume for every missing farm. Calculates l for every missing farm l. Step 3: Calculatedistributionproportions. Using all of the raw data, calculate mass market and wholesale proportions,p jkM andp jkW , for every day and destination combination. 50 p jtM = 1 16 X i=1 m ijt (3.12) p jtW = 1 16 X i=1 w ijt (3.13) = 1 P 16 i=1 s i (3.14) 1 = 365 X t=1 X j2D (p jtM +p jtM ) (3.15) where d is the set of destinations, is the total annual demand of the 16 farms, m ijt is the mass market demand sent by farm i to destination j on day t, and w ijt is the wholesale demand sent by farm i to destination j on day t. The sum of the proportions must equal 1 (Equation 3.15). Step 4: Distribute the estimated demand for each missing farm. Estimate the transportation volume for each missing farm l, day t = 1:::T and destination j2D. m ljt =p jtM s l w ljt =p jtW s l This procedure extrapolates the mass market and wholesale demand for each of the 50 farms that did not submit 2010 raw shipping data. The estimated demand is distributed according to the transportation activity characterized by the raw data. With 50 more data sets, we construct additional scenarios. Scenario 1: Do not extrapolate for missing farms Scenario 2: Extrapolate for the 10 largest missing farms Scenario 3: Extrapolate for the 20 largest missing farms 51 Scenario 4: Extrapolate for the 30 largest missing farms Scenario 5: Extrapolate for the 40 largest missing farms Scenario 6: Extrapolate for the 50 largest missing farms These scenarios re ect a conservative level of participation and are not likely to have the same possibility of materializing. However, we construct and test these scenarios to understand the relationship between the level of participation and potential cost reductions. Analysis Each scenario has a corresponding table of results that includes (i) the volume sent via full truckload, partial truckload (LTL) and FedEx/UPS, (ii) the annual cost, and (iii) the cost dierence between the consolidation case and the current practice (labeled as Base Case). Since mass market and wholesale shipments cannot be shipped together, they are separated in the results. Table 3.4 shows the detailed results for scenario three, an intermediate case of all the scenarios. In this scenario, the LTL volume decreases from 57.4% to 4.3% of the total volume from the base case to the consolidation case. The volume sent in full trucks increases from 36.9% to 95.7% of the total volume. The total annual transportation cost decreases by approximately $17 million or 37.0%. The dierence shows the benet from consolidating the shipments. Cost savings are due to more full trucks being used than LTL trucks. 52 Table 3.2: Detailed Results for Scenario 1 Volume sent by: Annual Cost ($) FedEx/UPS LTL Full Truck Total Volume Base Case Mass Market $ 1,675,919.10 7,396.44 415,331.81 621,139.49 1,043,867.75 Wholesale $ 27,822,232.14 252,718.72 4,508,629.51 5,125,425.30 9,886,773.54 Total $ 29,498,151.24 260,115.16 4,923,961.33 5,746,564.80 10,930,641.28 Consolidation Mass Market $ 1,238,400.68 1,863.86 61,948.46 980,055.42 1,043,867.75 Wholesale $ 21,537,451.40 1,425.36 784,127.74 9,101,220.44 9,886,773.54 Total $ 22,775,852.08 3,289.22 846,076.20 10,081,275.86 10,930,641.28 Dierence $ (6,722,299.17) (256,825.94) (4,077,885.13) 4,334,711.06 Table 3.3: Detailed Results for Scenario 2 Volume sent by: Annual Cost ($) FedEx/UPS LTL Full Truck Total Volume Base Case Mass Market $ 4,189,472.76 42,451.64 1,273,200.79 701,605.17 2,017,257.60 Wholesale $ 37,760,924.95 641,102.14 7,006,316.27 5,380,923.43 13,028,341.84 Total $ 41,950,397.71 683,553.78 8,279,517.06 6,082,528.60 15,045,599.44 Consolidation Mass Market $ 2,789,508.69 1,558.78 83,922.75 1,931,776.07 2,017,257.60 Wholesale $ 25,241,621.85 1,012.34 639,728.61 12,387,600.89 13,028,341.84 Total $ 28,031,130.55 2,571.12 723,651.36 14,319,376.96 15,045,599.44 Dierence $ (13,919,267.17) (680,982.66) (7,555,865.70) 8,236,848.36 53 Table 3.4: Detailed Results for Scenario 3 Volume sent by: Annual Cost ($) FedEx/UPS LTL Full Truck Total Volume Base Case Mass Market $ 5,124,687.27 59,841.93 1,601,729.53 701,605.17 2,363,176.63 Wholesale $ 41,797,404.55 894,737.29 7,869,273.18 5,380,923.43 14,144,933.89 Consolidation Mass Market $ 2,810,701.27 1,443.31 91,661.21 2,270,072.12 2,363,176.63 Wholesale $ 26,917,723.84 1,028.24 616,449.66 13,527,456.00 14,144,933.89 Dierence $ (17,193,666.71) (952,107.68) (8,762,891.84) 9,714,999.52 Table 3.5: Detailed Results for Scenario 4 Volume sent by: Annual Cost ($) FedEx/UPS LTL Full Truck Total Volume Base Case Mass Market $ 5,614,389.05 71,258.16 1,763,161.70 701,605.17 2,536,025.03 Wholesale $ 43,925,351.09 1,028,208.81 8,293,739.01 5,380,923.43 14,702,871.25 Total $ 49,539,740.14 1,099,466.97 10,056,900.71 6,082,528.60 17,238,896.28 Consolidation Mass Market $ 3,171,506.22 1,308.36 98,995.85 2,435,720.82 2,536,025.03 Wholesale $ 28,248,466.73 1,022.69 603,173.42 14,098,675.14 14,702,871.25 Total $ 31,419,972.95 2,331.05 702,169.27 16,534,395.96 17,238,896.28 Dierence $ (18,119,767.19) (1,097,135.92) (9,354,731.44) 10,451,867.36 54 Table 3.6: Detailed Results for Scenario 5 Volume sent by: Annual Cost ($) FedEx/UPS LTL Full Truck Total Volume Base Case Mass Market $ 5,927,026.41 80,995.47 1,856,033.98 701,605.17 2,638,634.63 Wholesale $ 45,321,244.38 1,094,864.74 8,558,296.62 5,380,923.43 15,034,084.78 Total $ 51,248,270.78 1,175,860.21 10,414,330.60 6,082,528.60 17,672,719.41 Consolidation Mass Market $ 3,364,901.31 1,361.29 96,635.87 2,540,637.46 2,638,634.63 Wholesale $ 28,938,817.40 1,033.35 602,803.97 14,430,247.46 15,034,084.78 Total $ 32,303,718.71 2,394.65 699,439.84 16,970,884.92 17,672,719.41 Dierence $ (18,944,552.07) (1,173,465.56) (9,714,890.76) 10,888,356.33 Table 3.7: Detailed Results for Scenario 6 Volume sent by: Annual Cost ($) FedEx/UPS LTL Full Truck Total Volume Base Case Mass Market $ 6,020,728.03 81,514.44 1,860,960.69 701,605.17 2,644,080.30 Wholesale $ 45,916,415.29 1,095,148.82 8,575,590.62 5,380,923.43 15,051,662.87 Total $ 51,937,143.32 1,176,663.26 10,436,551.31 6,082,528.60 17,695,743.17 Consolidation Mass Market $ 3,527,508.25 1,364.10 96,903.96 2,545,812.24 2,644,080.30 Wholesale $ 28,974,705.56 996.61 602,546.85 14,448,119.40 15,051,662.87 Total $ 32,502,213.81 4,005.28 699,450.81 16,956,741.59 17,695,743.17 Dierence $ (19,434,929.51) (1,172,657.98) (9,701,555.02) 10,874,212.99 The annual cost for the current practice is more expensive than the annual cost for consolidating the cut owers for all the scenarios. Figure 3.5 shows the annual trend for all six scenarios. The cost dierence between the current practice (base case) and consolidation for each scenario grows as more missing growers are assumed to participate. In other words, the current practice costs grow more rapidly than consolidation costs. 55 Figure 3.5: Annual Transportation Costs for Current Practices and Consolidation The LTL volume sent in the base case increases more rapidly than for the consolidation model. However, the opposite happens for the full truck volumes; see Figure 3.6. The gure shows that most growers do not have enough sales volume to send out a full truck, which results in more costly LTL and courier service shipments. The consolidation case would benet the California cut ower industry in terms of decreasing their transportation costs. Figure 3.6: Volume sent by Less-Than-Truckload and Full truckload The benets of a consolidation center stems from having one location for consolidating shipments. Third-party carriers benet from making smaller runs to a more accessible facility to pick-up consolidated shipments. The large trucks have diculty navigating through rural 56 roads to pick-up cut owers from multiple farms. In the current practice, the transportation carriers consolidate the shipments but the farms can consolidate the shipments prior to pick-up at a consolidation facility. 3.5.2 Comparison of Solution Approaches In this section, we compare four solution approaches. The rst is the discretized dynamic program proposed in Section 3.3. The second is the -based Consolidation Heuristic. The third solution comes from the IBM Optimization solver CPLEX. The nal solution approach, Every + 1 days policy, is a simple policy where you ship every + 1 days. CPLEX can solve a small instance of the proposed model, however, the gap between the LP relaxation and the best integer solution after 30 minutes of runtime increases as the problem increases in complexity. The number of decision variables increases exponentially as increases. For the following experiments, we limit CPLEX's runtime to 30 minutes. If an optimal solution is not reached, CPLEX returns the best integer solution it could nd thus far and the LP relaxation. The dynamic programming model's solution quality depends on the discretization factor. For the following experiments, we use a discretization factor! = 10 cubic feet because of the large time horizon and the large number of suppliers, which yields a large maximum daily aggregate demand. The capacity of a truck is 2,600 cubic feet, and an LTL unit's capacity is 1 cubic foot. We use the California cut ower industry data to run a series of experiments to compare our solution approaches. The experiments are run with actual data from the case study presented in the previous section. We have 20 suppliers, 49 destinations and a time horizon T = 365 days. No missing growers were added to this analysis. We apply the solution approaches to the raw data for = 1:::15. The purpose of these tests is to show the eectiveness of our heuristic against an optimal 57 or near optimal solution, and to determine if a ! value of 10 is sucient for the dynamic programming model to return a good quality solution. We also want to see how well each solution approach performs for dierent values of . We test the Every + 1 days to see if it matters to check the opportunity of consolidation, which our proposed heuristic does. 58 Table 3.8: General Theta Results Using CA Cut Flower Deterministic Demand CPLEX -based Consolidation Every + 1 days DP Model Lower Bound (LB) Cost % Di to LB Cost % Di to LB Cost % Di to LB Cost % Di to LB 1 16,845,809 17,023,371 1.04% 17,474,323 3.60% 17,785,636 5.28% 16,983,574 0.81% 2 15,039,478 15,418,825 2.46% 15,793,562 4.77% 16,285,634 7.65% 15,234,364 1.28% 3 13,809,217 14,245,479 3.06% 14,528,667 4.95% 15,142,083 8.80% 13,972,267 1.17% 4 12,984,391 13,317,469 2.50% 13,623,987 4.69% 14,316,084 9.30% 13,087,196 0.79% 5 12,494,589 12,856,967 2.82% 12,958,032 3.58% 13,710,693 8.87% 12,622,639 1.01% 6 12,294,616 12,659,456 2.88% 12,529,586 1.88% 13,304,380 7.59% 12,391,736 0.78% 7 11,828,004 12,165,276 2.77% 12,116,657 2.38% 12,948,348 8.65% 11,910,271 0.69% 8 11,615,943 11,932,428 2.65% 11,884,911 2.26% 12,719,641 8.68% 11,683,239 0.58% 9 11,469,019 11,754,284 2.43% 11,685,359 1.85% 12,461,278 7.96% 11,536,419 0.58% 10 11,336,454 11,609,191 2.35% 11,536,830 1.74% 12,287,014 7.74% 11,404,633 0.60% 11 11,226,525 11,468,558 2.11% 11,409,634 1.60% 12,211,528 8.07% 11,299,168 0.64% 12 11,177,661 11,398,260 1.94% 11,334,828 1.39% 12,058,617 7.31% 11,233,585 0.50% 13 11,101,050 11,304,189 1.80% 11,235,443 1.20% 11,926,392 6.92% 11,184,560 0.75% 14 11,016,495 11,203,240 1.67% 11,153,110 1.22% 11,878,956 7.26% 11,073,327 0.51% 15 10,960,025 11,037,348 0.70% 11,090,064 1.17% 11,757,513 6.78% 11,016,855 0.52% 59 Table 3.9: Average Runtime per Destination for General Theta (in seconds) CPLEX -based Consolidation Heuristic Every + 1 Dynamic Programming 1 982.3 1.9 1.0 1.5 2 979.3 2.0 1.5 2.1 3 1094.3 2.0 2.0 2.7 4 1101.6 2.0 2.0 3.4 5 957.2 2.0 2.0 3.9 6 1177.6 3.0 2.0 4.5 7 946.5 3.0 2.0 5.2 8 1046.1 3.0 2.0 12.8 9 1087.2 3.0 2.0 14.1 10 1086.3 3.0 3.0 15.0 11 1153.9 4.0 4.0 16.8 12 992.4 4.0 4.0 18.0 13 992.5 4.0 4.0 19.0 14 1071.3 4.0 4.0 20.6 15 1051.5 5.0 4.0 21.4 Table 3.8 contains the annual cost and a comparison to the lower bound (LB) solution from CPLEX. The discretized dynamic programming model performs extremely well for all values of with a discretization factor of 10. The % dierence to the lower bound is extremely low. Only 3 instances have a % dierence greater than 1%, while the rest are lower. The quality of the solution appears to be better with large values. The -based Consolidation heuristic possesses a similar pattern. The % dierence from the LB decreases as increases, in general. It provides a reasonable shipping schedule that costs no more than 5% of the LB for any value. The Every + 1 days policy performs the worst. This poor performance indicates that 60 there is value in checking for an opportunity to consolidate. Our proposed heuristic pushes the shipment date to as late as possible to increase opportunities for consolidating with other shipments. It also checks the current inventory to see if the FTL rate can be achieved. Otherwise, it will ship only the demand that must leave. The Every + 1 days policy ships at specic intervals regardless if the volume that just arrived could still stay in inventory. The average runtimes per destination for the results in table 3.8 are summarized in table 3.9. All the values in the table are in seconds. The two heuristics have the lowest runtimes. The longest runtimes are for CPLEX, and that includes a runtime stopping criteria of 30 minutes for each destination. Some destinations had very small runtimes, but others were stopped by the solver after 30 minutes. Overall, the runtimes for CPLEX do not appear to have a pattern with respect to changing. The average runtime for the discretized dynamic programming and two heuristic increases as increases. In summary, we show that the -based heuristic performs very well with low runtimes. The optimization solver is unable to solve the problems to optimality after 30 minutes in some instances. The dynamic programming model nds a near-optimal solution with a rather large discretization factor of 10. If ! was lower, the discretized dynamic programming problem could calculate the optimal solution, but at the consequence of a higher runtime. 61 Chapter 4 Stochastic Direct Shipment Consolidation Model In the deterministic direct shipment consolidation model, we assumed the demands were given for T time periods. However, in a practical setting, demands are not given far in advance. In this chapter, we consider a direct shipping model with a consolidation center. All suppliers have the opportunity to consolidate by shipping their produce to the consol- idation center, where products with the same break-bulk destination are grouped together to achieve better shipping rates. However, now we consider stochstic demand. We aim to solve the stochastic direct shipping consolidation model and determine whether the -based consolidation heuristic introduced in chapter 3.4 can provide a good solution. In this chapter, we introduce the direct shipment consolidation model with stochastic demands. First, we dene the problem and the model. Next, we formulate the dynamic programming model. Next, we show the necessary changes to the -based consolidation heuristic (Chapter 3.4) for the stochastic environment. We develop a rolling horizon algo- rithm, Every + 1 policy and a cost-to-go policy to compare with our proposed heuristic. The experimental results are shown next and nally, we propose a cost allocation method. 62 4.1 Model Formulation A set of suppliers independently ship their product to a consolidation center, where products with the same break-bulk destination are consolidated before shipment. We assume only direct shipping methods are used to and from the consolidation center. Since products headed to dierent destinations do not share the same transportation resource, we consider only the single destination problem. Inventory at the consolidation center will be low because of the perishability. The products in this model are perishable. The time limit for each product to stay at the consolidation center is dened as . We assume the consolidation center is located close to the suppliers, and inventory is held at the consolidation center only. The agricultural prod- uct is harvested immediately before shipment to the consolidation center. Transportation costs from the suppliers to the consolidation center are not considered because we assume the supplier are located closely to the consolidation center. Transportation costs to the consolidation center would be very small compared to the long-haul transportation to a break-bulk destination. We assume the unit cost of the product is small relative to long-haul transportation costs. Therefore, we do not consider inventory costs in this model. They will be considered in chapter 5. The parameters for this problem include: S: Set of suppliers T : Time Horizon : Maximum number of time periods inventory can stay at the consolidation cen- ter D it : Demand random variable with discrete distribution at time t for supplier i, 8i2S;t = 1:::T c F : Transportation cost for a full truck, ($ per truck) c L : Transportation cost for an LTL unit, ($ per cubic foot) c U : Transportation cost for a courier service, ($ per pound) : Conversion factor (pounds per cubic foot) F : Maximum capacity for a truck, in cubic feet L : The volume in an LTL unit, in cubic feet 63 The demand probability distribution can change for each period t and supplier i. This allows the model to capture changing behaviors in demand. For example, the California cut ower industry's two primary demand distributions are peak and nonpeak. Peak probability distributions occur when demand is generally high. This includes the weeks before Valentine's Day and Mother's Day. Nonpeak demand distributions are all other days. The cost function in this problem is the same as the deterministic direct shipping con- solidation model. There are three shipping methods: full truckload, less-than-truckload and courier services. We assume economies of scale exists (see equation 3.3). The cost function is a piecewise linear function (see gure 3.2), and the cost function (x) returns the least-cost shipping method for some volume x (see equations 3.4 and 3.5). Given some volume, the function will ship volumes up to a multiple of F at the FTL rate. Volumes less than that are shipped as LTL units or a combination of LTL and courier, depending on what is cheapest. There are two breakpoints for volume up to a full truck: b F and b L (see equations 3.6 and 3.7, respectively). Volumes above b F are best sent at the FTL rate. Volumes below b F are best sent using LTL shipments or a combination of LTL and courier service. Volumes up to a full LTL unit are shipped at the LTl rate. The remaining volume can be shipped at the LTL rate, even if it doesn't ll it, or at the courier rate. If the volume is below b L , it should be sent at the courier rate. Otherwise, it should be shipped at the LTL rate. 4.2 Stochastic Dynamic Programming The stochastic dynamic programming model nds a shipping schedule that corresponds to the minimum expected total cost for a time horizon of length T . The state variable, t = 0 t ;::: k t ::: t , is a vector of nonnegative values representing the inventory and its corresponding deadline for shipment. k t is the inventory at time t that must leave by time 64 t +k. The aggregate inventory, P k=1 k t , is partitioned by its deadline. The optimal value function, g t ( t ), is the expected total cost for periods t to T when the current inventory levels are at t . The volume that must leave is 0 t . The optimal value function aims to minimize the cost of the outgoing shipment plus the expected cost of time periods t + 1 to T . The outgoing shipment includes delta 0 t plus an amount from the inventory with later deadlines. The expected cost-to-go for periodst+1 toT is a subproblem of the main problem. The recursion of the stochastic dynamic program is (equations 4.1 and 4.2): g t ( t ) = min n ( 0 t +) +E D h g t+1 f t t ; t ;d t io (4.1) g t ( t ) = min 8 < : ( 0 t +) +g t+1 f t t ; t ;d t 0 @ Y d 1 2 ^ D 1;t+1+ :::d j Sj2 ^ D jSj;t+1+ p i;t+1+ (d i ) 1 A 9 = ; (4.2) The stochastic dynamic programming algorithm begins with the boundary condition, g T ( T ). The state variable vector elements 1 T ::: T are equal to 0 except for 0 T , which is the volume that must leave at timeT . The boundary condition simplies to the cost of shipping 0 T . g T ( T ) =( 0 T ) (4.3) The subproblems of the stochastic dynamic programming model are not as simple. Equa- tion 4.2 illustrates how the expected cost-to-go for periodst+1 toT depends on the demand probability distributions. ^ D i;t is a nite set of all possible demand values for supplier i at timeT . p i;t (d i ) is the probability of supplieri facing demandd i at timet. LetD t = P i2S D it be a random variable representing the aggregate demand across all suppliers for time period 65 t, and let d = P i2S d i be the aggregate arriving demand for the realizations d i . To track the volume being added to the current shipment, we let t = 1 t ::: k t ::: t be a vector of nonnegative values corresponding to the volume being added from inventory that must leave by time t +k. The transition function, t+1 = f t t ; t ;d , updates the inventory for all deadlines. It also determines the state for the subproblem after demand d i 2 ^ D i;t arrives. The new inventory level for deadlinet+k is the current inventory minus the outgoing inven- tory, t , for allk = 1 to. The transition function shifts all the values in t by 1 period. The following subproblem corresponds to the updated t+1 . The new inventory level also contains the incoming demand d from all suppliers. The transition function for the stochastic direct shipping model is the same as the deterministic direct shipping model. For further details, see section 3.3.1. The total amount being added to today's shipment is = P k=1 k t . is chosen such that the overall expected cost is minimized. The amount to be added to today's shipment is limited by the inventory levels, k t k t . This means is bounded by zero and P k=1 k t . The principle of optimality with equations 4.1, 4.2 and 4.3, suggests an optimal policy can be constructed. The complexity of the state space for the stochastic dynamic program is the same as the deterministic dynamic programming model. Let D m be the maximum possible daily demand and ! be the discretization factor. For any value, the number of states for one day is Y = Dm ! +1 because the state vector is of length + 1. There are a total of TY across the entire time horizon. The number of states increases exponentially as increases, and is the major factor for the increasing computational complexity. The dierence in the two models is the demand. Now we are considering demands from a probability distribution. The computational demands are even higher because we are calculating the expected cost-to-go value, meaning, at time periodt and for current state t , we must consider all possible subproblems corresponding to all possible incoming demands 66 for timet + 1. For the deterministic direct shipping model, there is one subproblem at time t + 1 that corresponds to current inventory levels because the demand arriving at time t + 1 is given. The pruning method introduced in section 3.3.2 can be applied in the stochastic case to decrease the runtime as well. 4.3 Heuristics The computational limitations of the stochastic dynamic programming become more restric- tive as and the time horizon increase. The discretization factor ! controls the quality of the solution, but as it decreases to provide a more accurate solution, the number of states increases. In this chapter, we modify the -based consolidation heuristic for the stochastic case. We also introduce a rolling horizon algorithm and two alternate heuristic strategies 4.3.1 -based Consolidation Heuristic The-based consolidation heuristic decides how much of periods of demand to consolidate to save on transportation costs. The most expensive transportation method is the courier service. Next is the LTL rate and the least expensive method is the FTL rate. Consolidating across time can yield enough volume to ship at the more economical FTL rate. An arriving shipment at time t must leave by t +. There are days where the volume can either wait in inventory or be shipped with another outgoing shipment to take advantage of lower rate. At time t +, any remaining demand that has not been added to an earlier shipment must be sent. The heuristic uses breakpoints b F and b L (see equations 3.6 and 3.7) to decide the best shipping option and the amount necessary to consolidate. This heuristic is very similar to the version for the deterministic direct shipping model. We assume that we have values of inventory, corresponding to the days that any volume can stay in inventory. However, we 67 have no further information. At the end of time t, we receive information for the demand that will arrive at time t + 1. In other words, demands d i are generated from ^ D i;t+1+ . Algorithm 4 illustrates the steps for the -based consolidation heuristic for the stochastic case. Algorithm 4 : -based Consolidation Heuristic Initialize t = 0 t ;::: k t ::: t to demands d 1 ;d 2 ;d 3 ;:::d +1 for t = 1 to T do if 0 t > 0 then Shipb 0 t F c at the FTL rate Calculate the remainder: 0 t = 0 t b 0 t F c F if 0 t + 1 t b F then t = FTLFill(( K 0 t ), t ). Ship one truck at FTL rate. else Shipb 0 t L c at the LTL rate. Calculate remainder: 0 t = 0 t b 0 t L c L if 0 t + k t b L then t = LTLFill(( L 0 t ), t ). Ship one LTL unit at the LTL rate. else Do not add anything. Ship at courier rate. end if end if end if Generate demand for each supplier at time t + 1 +: d i 2 ^ D i;t+1+ 8i2S Calculate the aggregate demand: d = P i2S d i Update the inventory: t+1 =f t ( t ; t ;d t++1 ) end for The algorithms for LTLFill and FTLFill are the same as the deterministic case because they deal with volumes that are given. For the pseudocode of LTLFill and FTLll, see Algorithms 1 and 2 in chapter 3.4. 68 4.3.2 Other Heuristics We compare the -based consolidation heuristic with three alternate heuristics. The -based consolidation heuristic holds inventory for days, essentially knowing de- mands up to days. We compare it to arolling horizon algorithm that uses the deterministic dynamic programming algorithm (see 3.3). Let R be the number of time periods we have of given demand, and the planning horizon for deterministic dynamic programming algorithm. LetM be the number of periods to implement the solution. The rst M days includes deci- sions forM + days. Therefore,M +R must be true. After implementing the decisions for M days, the horizon is shifted forward M days and the problem is solved again, until it reaches the end. The every + 1 policy consolidates the demand for days before shipping. If = 1, then the heuristic ships every 2 days. One day's demand waits until the next day, where it is shipped with the second day's demand. The cost-to-go policy creates a cost-to-go table for the entire horizon, based on the ex- pected demand for each time period. This policy performs the deterministic dynamic pro- gramming on the expected demand corresponding to the probability distribution for that period. The table includes the expected cost-to-go for each demand discretization and each time period. The decisions are based on the current demand and the values in the table, with the goal to minimize the expected cost. 4.4 Experimental Results The data used to perform the numerical experiments are from the California cut ower industry for the year 2010. We chose ve suppliers shipping to the same destination with average daily demand under a full truckload. Without a consolidation center, the suppliers would ship their product using a combination of courier and LTL rates for the majority of 69 their shipments. Figure 4.1: Demand distributions for two suppliers of the California cut ower industry. Figure 4.1 shows the empirical distributions for two suppliers in our numerical experi- ments. The size of each bin is approximately 100 cubes of demand, where a cube is a cubic foot. We estimate two probability distributions for each supplier, one for peak periods and the other for nonpeak periods. The peak periods correspond to the week before Valentine's Day and Mother's Day. The demand distribution for each supplier-destination pair possess a dierent patterns, so we use empirical distributions. 4.4.1 Comparison of Heuristics 100 sets of demand, each with a time horizon of 365 days, are generated from the empirical distributions for the ve suppliers. A lower bound was determined by solving the determin- istic dynamic program on each generated set. The-based consolidation heuristic and every + 1 heuristic assume that periods of demand are given. 70 Table 4.1: Numerical Results of Algorithms and Heuristics Stochastic Lower Bound -based consolidation heuristic Every + 1 Cost-to-go 1 202,903 195,303 1.01 1.01 1.24 2 - 141,594 1.04 1.09 1.62 3 - 119,408 1.04 1.19 1.75 4 - 110,475 1.04 1.26 1.80 5 - 107,309 1.04 1.25 1.79 The average of the 100 runs are displayed in table 4.1. The deterministic dynamic programming for the lower bound and the stochastic dynamic programming calculations are based on a demand discretization factor of 0.5. The pruning technique was used for all the tests. However, it was only computationally possible to nd the optimal expected cost for the stochastic dynamic program with = 1. Instead of real values, we calculate the ratios of the remaining three heuristics against the dynamic programming solutions as a comparison. For = 1, the stochastic dynamic programming algorithm is used for the comparison, and the lower bound is used for > 1. The-based consolidation heuristic performs well, nding a solution that is within 1% for = 1 and within 4% for > 1. As increases, the expected annual cost decreases because of the increased opportunity for consolidation. A larger means demand can be held at the consolidation center for a longer time period, allowing more product to consolidate. The -based consolidation heuristic considers days of demand when deciding if consolidation should take place today or not. If there is excess capacity, then inventory is added if there is no additional cost in doing so. This happens when an entire truck is paid for even though it is only partially lled with the current outgoing shipment. Shipping two trucks on one day versus two separate days yields no dierence in transportation costs. The same idea is true for LTL units. Since no inventory cost is being considered, the heuristic takes advantage 71 of and allows product to stay as long as is necessary to achieve enough volume for a full truckload. The every + 1 policy and cost-to-go policy perform worse as increases. The every + 1 policy does not check the inventory levels when it makes a decision. It will ship on the ( + 1)th day, starting its count on day 1. Once the oldest demand in inventory has been held for days, everything in the consolidation center leaves. The every + 1 policy is a time-based shipment release policy that consolidates even at additional cost. The cost-to-go policy performs the worst and deteriorates the fastest as increases. The table recommends an amount to consolidate with the outgoing shipment. The entire table was constructed based on expected values of the demand distributions. If the realized demand was greater than the expected value, the amount added to the outgoing shipment exceeded the excess capacity of a truck or LTL unit. Therefore, the total annual cost increases. If realized demand is less than the expected value, the inventory added to the outgoing shipment is less than the recommendation and the outgoing shipment is not shipped at the cheaper rate it is expected to. Increasing results in a larger dierence in the volume that the cost-to-go table recommends to consolidate and the volume of the actual demand that is consolidated. Therefore, transportation costs increase. While the solution from the -based consolidation heuristic is sub-optimal, the runtime is much smaller. On average, the heuristic is done within a minute. However, the stochastic dynamic programming model took 13.1 hours of CPU time on an Intel Core i3 CPU 530 processor @ 2.93 Ghz. The lower bound calculations for 100 sets of demand nished within 12 minutes. 72 Table 4.2: Average Runtime for the Dynamic Programming Model (in seconds) Without Pruning With Pruning 1 37.9 13.0 2 74.7 17.7 3 153.0 28.3 4 158.3 43.5 5 202.7 57.5 The pruning strategy for the dynamic program signicantly reduces the runtime. For = 5, the runtime for pruning is approximately one third of the original runtime. Additional pruning techniques may further reduce the runtime. Table 4.3: Rolling Horizon and -based Heuristic Comparison Ratio by Run Length -based consolidation heuristic 5 Days 10 Days 20 Days 30 Days 40 Days 50 Days 1 206,160 1.03 1.01 0.95 0.95 0.95 0.95 2 146,483 1.11 1.07 0.98 0.97 0.97 0.97 3 123,218 1.23 1.16 1.03 1.00 0.98 0.97 4 114,676 1.21 1.16 1.08 1.04 1.01 0.99 5 111,348 1.25 1.17 1.14 1.11 1.06 1.03 We choose an implementation period,M, of ve days to re ect a typical work week. The numerical tests are for run lengths (or planning horizons) ofR = 5; 10; 20; 30; 40 and 50 days of known demand. We compare the annual cost from the -based consolidation heuristic with the rolling horizon experiments. The results are summarized as ratio values in table 4.3. The ratios are the rolling horizon solution divided by the heuristic solution shown in column two. 73 The rolling horizon algorithm performs poorly compared to the heuristic. It requires 20 days of known demand before it performs better than the heuristic for = 1; 2. This result is most likely due to end-period eects because the rolling horizon forces the schedule to consider zero inventory at the end of the run horizon R. The decisions in the last days includes more outgoing demands because there were no incoming demands. By dening the ending inventory to be zero, the consolidation opportunities for the last days are restricted. While the rolling horizon algorithm improves as the run length increases, it is unlikely to know demand that far in advance. 4.4.2 Sensitivity to Consolidation In this section, we compare the benets of consolidating versus not consolidating. When there is no consolidation center, each supplier operates independently. For the following analysis, we use the -based consolidation heuristic to generate the solutions for the consolidation strategy. Figure 4.2: Ratio of consolidating versus operating independently Figure 4.2 illustrates the ratio of the consolidation strategy against the non-consolidation 74 strategy. The demands are scaled by 0.2, 0.5, 1, 1.5, 2, 5, 10, 30, 50 and 100. The pattern in the graph shows that at very low demands, the benets of consolidating are small be- cause even under a consolidation strategy, not enough volume is being shipped at the FTL rate. Most of the demands are shipped at the LTL rate for the consolidation and the non-consolidation strategy. As the expected demand increases, the aggregate demands are shipped at the more advantageous FTL rate (ratio values are closer to zero). However, as demand increases to very large values, the benets disappear because FTL rate shipments are made in both the consolidation case and the non-consolidation case. Figure 4.3: Average Number of Suppliers per Full Truckload Shipped Figure 4.3 graphs the number of suppliers that are consolidating their product in a full truckload for varying values of demand. This re-emphasizes how consolidating benets suppliers with low values of demand. As demand increases, less suppliers are consolidating with each other. With small demands, more suppliers' volume are needed to achieve the cheaper FTL rate. 75 4.4.3 Cost Allocation Consolidating across many suppliers with low demands yields lower system-wide costs, how- ever, individual suppliers will not cooperate unless a benet exists for every participating supplier. In this section, we brie y consider a cost allocation policy and determine whether it supports the continued cooperation of the suppliers at the consolidation center. We propose a proportional cost allocation policy. Each supplier is charged a proportion of the total cost equal to the proportion of the demand in the current shipment that belongs to the supplier. The cost is divided every time a shipment is sent. At time t, the cost of the total outgoing shipment is ( 0 t +), where is decided using the-based consolidation heuristic. If i is the proportion of the outgoing shipment's volume belonging to supplier i, then supplier i is charged i ( 0 t +). We performed this allocation on 100 samples of annual nonpeak demand. Full cooper- ation, known as the grand coalition, includes all 5 suppliers. A subset of the 5 suppliers is called a coalition. Table 4.4: Grand Coalition Average Daily Costs per Supplier Coalition size Supplier 1 Supplier 2 Supplier 3 Supplier 4 Supplier 5 5 147.37 100.38 150.43 73.70 153.57 The results in table 4.4 show the average daily cost for every supplier under the propor- tional cost allocation policy and in a grand coalition. We compare these results with the average cost of a coalition. In order to do this, we consider all possibilities to create coalition sizes of 4, 3, 2 and 1 of the original 5 suppliers. We applied the proportional cost allocation policy for each coalition and calculated the average daily cost across all coalitions of the same size. Table 4.5 contains the coalition average daily cost divided by the grand coalition average daily cost in table 4.4. 76 Table 4.5: Ratio Comparison Against the Grand Coalition Coalition size Supplier 1 Supplier 2 Supplier 3 Supplier 4 Supplier 5 4 1.17 1.21 1.15 1.21 1.17 3 1.49 1.62 1.43 1.69 1.47 2 2.04 2.33 1.92 2.52 1.99 1 2.77 3.03 2.57 3.00 2.72 All suppliers benet the most in the grand coalition because all the values in table 4.5 are greater than 1. The ratio value increases as the coalition size decreases. These results imply that under the proposed cost allocation policy, every supplier benets from a full cooperation of all the suppliers versus operating alone or in a coalition. However, the benet to each supplier is not the same. One supplier might benet more than the other suppliers if his demand is low and easily consolidated with the other supplier's partial LTL units or partial FTL. Overall, the -based consolidation heuristic performs very well and quickly versus the stochastic dynamic programming algorithm. It also performs better than the rolling horizon, every + 1 policy and the cost-to-go policy. We show in this section that consolidation is not benecial for very low values of demand per supplier and very high values of demand per supplier with respect to the capacity of a full truckload. And the proportional cost allocation policy encourages suppliers to participate in the grand coalition. 77 Chapter 5 Inventory Consolidation Model In this chapter, we relax the assumption that the unit cost is small compared to the shipping cost and now we must consider the inventory cost. Consolidating products across time allows us to ship larger shipments at more advantageous rates, however, holding product incurs an inventory cost. In this problem, we must consider the trade o between shipping a product versus holding it for the opportunity to consolidate it later. In some cases, holding a product might be a good decision because a FTL rate shipment can be sent instead of two LTL rate shipments. However, for very large inventory costs, the cost savings of consolidated shipments might not be enough to oset the inventory costs to hold any product. Inventory cost is also a soft constraint for perishability, while the maximum inventory time is a hard constraint. In this model, both constraints exist. We will introduce the model in the next section. Then we will formulate the dynamic programming model. We propose a myopic heuristic that examines the trade o between shipping and holding product. Finally, we summarize the numerical experiments for the solution approaches. 78 5.1 Model Formulation The inventory consolidation model examines direct shipping transportation strategies, sim- ilar to the deterministic and stochastic direct shipping models. A set of suppliers ship perishable product to a consolidation center, where products headed to the same break-bulk destination leave in combined shipments. Suppliers ship their perishable goods to the consol- idation center as soon as they are prepared, and so inventory is held only at the consolidation center. Goods from the consolidation center to each break-bulk destination are shipped using FTL, LTL or courier rates. Routing strategies are not considered for the transportation from the supplier to the consolidation center and from the consolidation center to the break-bulk destinations. Products with dierent destinations do not share the same transportation re- sources, therefore this model is formulated as a single break-bulk destination problem. We solve each destination separately. The model imposes a hard time constraint by limiting the number of time periods any item can stay in inventory. The inventory costs are a soft constraint for the perishable goods. Due to both of these constraints, inventory levels will be kept low, and capacity is not an issue. We assume the perishable products have a departure deadline after arrival at the consol- idation center. We dene as the maximum number of time periods the product can stay at the consolidation center. The consolidation center is closely located to the suppliers. There- fore, the transportation costs from the supplier to the consolidation center are signicantly smaller than the long-haul costs, and we do not consider them in the model. The following is a list of all the parameters for this problem: 79 S: Set of suppliers T : Time Horizon d it : Demand at period t from supplier i,8i2S;t = 1:::T c F : Transportation cost for a full truck, ($ per truck) c L : Transportation cost for an LTL unit, ($ per cubic foot) c U : Transportation cost for a courier service, ($ per pound) : Conversion factor (pounds per cubic foot) : Maximum inventory time at the consolidation center F : Maximum capacity for a truck, in cubic feet L : The volume in an LTL unit, in cubic feet h i : Unit holding cost for a product from supplier i The decision variables include the number of trucks sent at the FTL ratex tF , the number of LTL unitsx tL , and the volumes sent using the FTL, LTL and courier service rates (y istF , y istL , and y istU ). The inventory levels at every time period for a supplier is denoted as I it . The mixed integer programming model represents the consolidation model for the direct shipping model with inventory costs. The objective function minimizes the transportation and inventory costs (Eq. 5.1). Equations 5.2a and 5.2b are capacity constraints for the full truckload, sent using the FTL rate, and LTL units, sent using the LTL rate. Equation 5.2c satises the demand, and equations 5.2d and 5.2e are inventory balance constraints. x tF : Number of full trucks leaving the consolidation center at time t,8t = 1:::T x tL : Number of LTL units leaving the consolidation center at time t,8t = 1:::T y istF : Volume sent on a full truck with origin supplier i, sent on period s, that must leave the consolidation center by period t,8i2 S;s = 1:::T;t = s:::minfs + ;Tg y istL : Volume sent on an LTL truck with origin supplieri, sent on periods, that must leave the consolidation center by period t,8i2 S;s = 1:::T;t = s:::minfs + ;Tg y istU : Volume sent using a courier service with origin supplier i, sent on period s, that must leave the consolidation center by period t,8i2 S;s = 1:::T;t = s:::minfs +;Tg I it : Inventory at time t for product with origin supplier i,8i2S;t = 1:::T 80 Minimize: T X t=1 (c F x tF +c L x tL ) +c U X i2G T X s=1 minfs+;Tg X t=s y istU + X i2S T X t=1 h i I it (5.1) Subject to: X i2S minfs+;Tg X t=s y istF F x sF ; 8s = 1:::T (5.2a) X i2S minfs+;Tg X t=s y istL L x sL ; 8s = 1:::T (5.2b) t X s=maxf1;tg y istF +y istL +y istU =d it ; 8i2S;t = 1:::T (5.2c) I i1 =I i0 + +1 X k=1 d ik T X t=1 (y i1tF +y i1tL +y i1tU ); 8i2S (5.2d) I is =I i;s1 +d i;s+ T X t=s (y istF +y istL +y istU ); 8i2S;s = 2:::T (5.2e) y istF 0; 8i2S;s = 1:::T;t = 1:::T y istL 0; 8i2S;s = 1:::T;t = 1:::T y istU 0; 8i2S;s = 1:::T;t = 1:::T (5.2f) x tF 0; 8x tF 2Z;t = 1:::T x tL 0; 8x tL 2Z;t = 1:::T I it 0; 8i2S;;t = 1:::T (5.2g) 5.2 Dynamic Programming Model The dynamic programming model calculates the total transportation and inventory cost for a time horizon of length T . We assume deterministic demand for this model. The optimal value function,g t ( t ), takes a matrix of nonnegative values as its input (see equation 5.3). It 81 contains three parts: the transportation cost for the outgoing shipment, the inventory cost of holding the remaining inventory for one period, and the cost-to-go for time t + 1 to T . The third part is a subproblem of this problem. Let t be the inventory partitioned by supplier and deadline. k it is the inventory at time tfor product originating from supplier i with deadline t +k. k it corresponds to the ith row and kth column for values i2 S and k = 1:::. The ith row of the matrix corresponds to the inventory for supplier i and the deadlines. The decision rule is: for a given state variable at timet, P i2S 0 it must leave because time t is its deadline, and the decision maker must decide how much of the remaining inventory should be added to the outgoing shipment to minimize the total transportation and inventory cost. The product added to the outgoing shipment is represented by the matrix t . The value k it corresponds to the inventory originating from supplieri and with deadlinet+k that is added to the outgoing shipment at timet. The amount of product that can be added is bounded by what is available in inventory: 0 k it k it . The transition function, t = f t ( t ; t ;d t+1+ ) determines the state variables for the subproblem at time t + 1, where d t is a vector of all the incoming demands d it for all suppliers i2 S at time t. It updates the inventory levels to include the shipment decisions from t and shifts it one period to be considered at period t + 1. The transition function is easily extended from the transition function introduced in chapter 3.3.1. In the deterministic direct shipping case, the inventory levels are aggregated across all suppliers. For this problem, we keep track of the inventory for all suppliers. g t ( t ) =min ( X i2S i0 t + ! + X i2S X k=1 h i ( ik t ik t ) +g t+1 (f t ( t ; t ;d t )) ) (5.3) The initial calculations occur at time T . At time T , inventory contains only the volume 82 that must leave. There are no incoming demands. Therefore, the calculation is only the shipping cost of what remains in inventory. We dene the boundary condition as: g T ( T ) =( X i2S i0 T ) (5.4) The principle of optimality with equations 5.3 and 5.4 suggest an optimal policy can be constructed. 5.2.1 Complexity For large daily demands, the dynamic programming model with inventory costs becomes computationally expensive. Previously in the transportation only models, the demands from each supplier could be aggregated. However, this model requires that the state variable be partitioned into deadline and supplier because the inventory cost per supplier is not assumed to be equal. The state variable is a matrix of size + 1 timesjSj. The number of states depends on how the daily demand is discretized. Let ! be the discretization factor and D max i be the maximum daily demand for supplier i. The number of possible states, Y , for = 1 is Y = Q i2S (D max i =! + 1), including zero. For any positive integer value , the number of states in one day is Y +1 because the state vector is of length + 1. We are considering the states that cover all the possible inventories and their deadlines. TY +1 possible states exist for the entire time horizon. A small discretization factor yields a more accurate total cost for a time horizon of length T . However, as ! decreases, the runtime will increase. As or the number of suppliers increase, the number of states increases exponentially, and is the major factor for the dynamic programming model's computational complexity. As a result, we are motivated to develop an ecient heuristic that performs well. 83 5.3 Comparison Heuristics In the previous models (see chapters 3 and 4), there was no penalty for holding inventory across time. Therefore, we allowed product to wait in inventory for consolidation purposes. The motivation for consolidation was to decrease shipping costs, which increased in magni- tude as the shipping volume increased. However, the decision to allow product to stay at the consolidation center is more dicult with inventory costs. We must consider the trade-o between the cost for holding inventory and the cost savings from consolidation. In this section, we propose a myopic heuristic for the inventory consolidation model. At period t, we decide how much to add to the current outgoing shipment such that the inventory cost for one period plus tomorrow's potential outgoing shipment is minimized. c t ( t ) = X i2S 0 it + ! + X i2S X k=1 h i k it k it + X i2S X k=1 k it k it ! +d i;t+1+ ! (5.5) The heuristic considers the following: at time t, there are periods of demand in in- ventory. The decision maker decides how to partition the inventory into two shipments, given what must leave today, P i2S 0 it , and the demand that arrives tomorrow, d t+1+ . See equation 5.5. The steps for the heuristic are shown in algorithm 5. In this heuristic, we use the same variables dened for the dynamic programming model: t for the current inventory, t for the amount of volume to consolidate and d t for the incoming demands. We know that P i2S P k=1 k it . The heuristic begins with = 0 and increases in increments of size !. For each value of , we consider the inventory from the suppliers with the highest holding costs rst. In other words, if a product is going to be added to the outgoing shipment, we add by decreasing inventory cost. Using equation 5.5, we calculate the cost of shipping 84 Algorithm 5 : Comparison Heuristic 1 Initialize c min =M Initialize t , where k it = 0 8i2S;k = 0:::;t = 1:::T Initialize t , where k it = 0 8i2S;k = 0:::;t = 1:::T Initialize inventory k i1 =d i;1+k 8i2S;k = 0::: Given all demands d it for t = 1 to T do for = 0::: P i2S P k=1 k it do Add inventory, k it to today's shipment in order of decreasing h i Calculate c t ( t ) if c t ( t )<c min then Update the min cost value: c min =c t ( t ) Save the amount to add: k it = k it ,8i2S;k = 1::: end if end for Add = P i2S P k=1 k it to today's shipment Ship P i2S 0 it + using the least-cost method Update t+1 =f t ( t ; t ;d t ) end for today plus the inventory cost of the remaining product and the cost of the outgoing shipment tomorrow with all the remaining inventory plus the incoming demand. In addition, we introduce another simple comparison heuristic. The previous heuristic compares transportation and inventory costs. In Comparison Heuristic 2, we only look at the cost of the LTL rate and the holding cost. The heuristic is shown in algorithm 6. For every periodt, we check the total inventory to see if any full trucks can be shipped. If there is enough inventory to ll one or more entire trucks, then they are shipped immediately. The inventory loaded on those trucks are, rst, the inventory that must leave today, and then, the inventory in the order of decreasing inventory cost. After that, the heuristic checks the inventory and adds to the outgoing shipment any inventory where the holding cost is greater than the LTL rate (h i >c L ). 85 Algorithm 6 : Comparison Heuristic 2 Initialize k it = 0 8i2S;k = 0:::;t = 1:::T Initialize inventory k i1 =d i;1+k 8i2S;k = 0::: Given all demands d it for t = 1 to T do if P i2S P k=0 k it > F then The number of full truckloads = jP i2S P k=0 k it F k Ship all full truckloads at the FTL rate. Inventory on the full trucks are loaded in the following order: 1) Inventory that must leave today 2) By decreasing holding cost, h i Update t end if Decide what inventory to add to outgoing shipment: for i2S do if c L <h i then = + P k=1 k it end if end for Ship P i2S 0 it + using the least-cost method Update t+1 =f t ( t ; t ;d t ) end for 5.4 Experimental Results In this section, we examine the numerical experiments performed on the inventory consol- idation model. The proposed dynamic programming model includes more states than the transportation only model because it is necessary to keep track of the inventory for every supplier. In the transportation only model, the inventory was represented as aggregate val- ues, while in this model, they are partitioned by both deadline and supplier. This increased the complexity and running simulations was dicult for the problem size we are considering. For the following experiments, we compare our heuristic with the IBM optimization software, CPLEX. These tests were designed to provide insight on the performance of our heuristic as 86 well as insight on the types of policies that should be considered under certain scenarios. We ran experiments using demand data from the California cut ower industry. The suppliers from the California cut ower industry represent the type of suppliers studied in this model. The daily demands per supplier average less than a full truckload F . We generate one data set and perform several scenarios. We rst look at three cases where the inventory cost for all suppliers are the same. The experiments are for 365 days and 5 suppliers. We stop CPLEX after 1800 seconds and return the best integer solution and best bound if an optimal solution has not been found. In the following, Heuristic 1 refers to Comparison Heuristic 1 and Heuristic 2 refers to Comparison Heuristic 2. The % dierence in the table is from the lower bound. Table 5.1: Case 1: All Inventory Costs are $0.01 Best Integer Lower Bound Heuristic 1 % Dierence Heuristic 2 % Dierence 1 368,663 Optimal 383,857 4% 391,000 6% 2 295,978 284,120 326,646 15% 311,240 10% 3 285,246 274,717 321,960 17% 295,888 8% 4 280,740 274,677 321,899 17% 284,890 4% 5 279,780 274,685 321,870 17% 280,263 2% Table 5.2: Case 1: CPLEX Details Best Integer Lower Bound Transportation Inventory Runtime (secs) 1 368,663 Optimal 367,172 1,491 140 2 295,978 284,120 293,040 2,938 1,801 3 285,246 274,717 281,073 4,173 1,801 4 280,740 274,677 275,590 5,150 1,807 5 279,780 274,685 273,714 6,066 1,811 87 Table 5.3: Case 1: Comparison Heuristic 1 details Total Cost Transportation Inventory Runtime (secs) 1 383,857 381,931 1,926 0.092 2 326,646 323,749 2,897 0.129 3 321,960 318,922 3,037 0.15 4 321,899 318,852 3,047 0.146 5 321,870 318,852 3,019 0.196 Table 5.4: Case 2: All Inventory Costs are $0.50 Best Integer Lower Bound Heuristic 1 % Dierence Heuristic 2 % Dierence 1 430,757 Optimal 444,011 3% 468,312 9% 2 414,488 413,698 433,871 4% 489,989 18% 3 414,136 412,787 433,244 5% 521,751 26% 4 413,295 411,914 432,330 5% 515,122 25% 5 413,172 411,887 432,238 5% 505,689 23% Table 5.5: Case 2: CPLEX Details Best Integer Lower Bound Transportation Cost Inventory Cost Runtime (secs) 1 430,757 Optimal 325,717 105,040 17 2 414,488 413,698 222,535 191,953 1,801 3 414,136 412,787 217,461 196,675 1,801 4 413,295 411,914 220,138 193,157 1,807 5 413,172 411,887 220,664 192,508 1,815 88 Table 5.6: Case 2: Comparison Heuristic 1 Details Total Cost Transportation Cost Inventory Cost Runtime (secs) 1 444,011 381,465 62,547 0.097 2 433,871 348,506 85,365 0.134 3 433,244 347,413 85,831 0.172 4 432,330 347,413 84,917 0.161 5 432,238 347,343 84,894 0.237 Table 5.7: Case 3: All Inventory Costs are $1.00 Best Integer Lower Bound Heuristic 1 % Dierence Heuristic 2 % Dierence 1 475,295 Optimal 478,349 0.64% 547,202 15% 2 474,771 Optimal 477,838 0.65% 672,386 42% 3 474,127 Optimal 477,176 0.64% 752,223 59% 4 473,047 Optimal 476,087 0.64% 750,052 59% 5 472,949 Optimal 475,722 0.59% 735,716 56% Table 5.8: Case 3: CPLEX Details Best Integer Lower Bound Transportation Cost Inventory Cost Runtime (secs) 1 475,295 Optimal 408,685 66,610 7 2 474,771 Optimal 408,122 66,649 13 3 474,127 Optimal 407,477 66,650 13 4 473,047 Optimal 406,396 66,651 23 5 472,949 Optimal 406,308 66,641 40 89 Table 5.9: Case 3: Comparison Heuristic 1 Details Total Cost Transportation Cost Inventory Cost Runtime (secs) 1 478,349 415,486 62,864 0.094 2 477,838 414,971 62,867 0.133 3 477,176 414,310 62,866 0.169 4 476,087 413,221 62,866 0.141 5 475,722 412,332 63,390 0.184 Three tables are displayed for case 1, case 2 and case 3, where all suppliers' inventory costs are $0.01, $0.50 and $1.00 respectively: side-by-side comparison of the heuristics and CPLEX, detailed results from CPLEX and detailed results from Heuristic 1. The rst case is when all the inventory costs are equal to $0.01. Table 5.1 shows that Heuristic 2 performs better than Heuristic 1. Both heuristics are compared to the best integer solution for = 1 and the lower bound for> 1. Overall, Heuristic 1's performance worsens as increases. However, Heuristic 1 performs better than Heuristic 2 for case 2 and 3. Case 1 re ects the case where inventory costs are very low, and therefore, the consequence of holding inventory are low compared to the opportunity of consolidating (see table 5.7). Heuristic 2 performs better because it forces more product to be held in inventory for the chance to consolidate. However, when inventory costs are high, holding inventory has a larger consequence. Heuristic 2 performs worse while Heuristic 1 yields a solution within 1% of the optimal solution by CPLEX. CPLEX solves case 3 faster than case 1 or 2. The solution for case 3 is trivial because for high inventory costs, it is better to ship as soon as possible. It is more dicult for CPLEX to solve case 1 due to the low inventory costs. Case 1 and 3 test the heuristics and CPLEX at the extremes. In case 3, the inventory level is stable even though changes. The inventory 90 being held would have been shipped as courier but are instead held to be consolidated with LTL and FTL shipments. The inventory cost is is low enough that courier shipments are undesirable, but high enough for LTL shipments to be shipped and not be held for long periods to consolidated into FTL shipments. Due to this, we look at case 2, where all the inventory costs are $0.50. Table 5.5 shows that CPLEX cannot nd an optimal solution after 1800 seconds for> 1. Comparison Heuristic 1 performs better with a much lower runtime. We studied the performance when all the inventory costs are the same. In the following case, we allow the inventory costs for all the suppliers to be dierent. Table 5.10 contains the inventory cost per item for each supplier. This case re ects situations where the suppliers are shipping product with dierent perishability measurements. Supplier 5 has the largest inventory cost, meaning its product has the highest perishability compared to the rest. Table 5.10: Case 4: Inventory Costs Supplier Inventory Cost 1 1 2 0.5 3 0.8 4 0.2 5 1.1 Table 5.11: Case 4: Dierent Inventory Costs Best Integer Lower Bound Heuristic 1 % Dierence Heuristic 2 % Dierence 1 457,623 Optimal 464,651 2% 514,827 13% 2 446,438 Optimal 462,288 4% 583,356 31% 3 442,667 Optimal 461,782 4% 593,435 34% 4 440,579 Optimal 460,851 5% 602,245 37% 5 440,078 Optimal 460,356 5% 581,859 32% 91 Table 5.12: Case 4: CPLEX Details Best Integer Lower Bound Transportation Cost Inventory Cost Runtime (secs) 1 457,623 Optimal 371,305 86,318 6 2 446,438 Optimal 385,569 60,869 45 3 442,667 Optimal 333,736 108,931 138 4 440,579 Optimal 412,169 28,410 468 5 440,078 Optimal 277,826 162,252 698 Table 5.13: Case 4: Comparison Heuristic 1 Details Total Cost Transportation Cost Inventory Cost Runtime (secs) 1 464,651 397,558 67,093 0.058 2 462,288 389,873 72,415 0.116 3 461,782 388,782 73,000 0.146 4 460,851 388,783 72,068 0.176 5 460,356 388,714 71,642 0.195 For Case 4, Comparison Heuristic 1 works very well and Comparison Heuristic 2 works poorly. CPLEX provides an optimal solution, but it's runtime increases as increases. The runtime for Heuristic 1 is approximately the same for all values and determines a delivery schedule that is within 5% of the optimal solution. In summary, we demonstrate the performance of Comparison Heuristic 1 with low run- times and high inventory costs. The optimization solver is unable to solve the problems to optimality after 30 minutes in some instances. Heuristic 1 performs very well while Heuris- tic 2 performs poorly in cases where the inventory cost is moderate compared to the LTL rate. Case 2 and 4 are a better measurement of performance for the inventory consolidation model. If inventory costs are low, we can refer to the direct shipping consolidation model, where the -based Consolidation Heuristic is proposed. The real issue is what to between 92 the two extreme cases. For case 2, 3, and 4, the Comparison Heuristic 1 performs very well. It considers the trade-o between inventory cost and shipping today and tomorrow. The heuristic aims to nd a good partition of the inventory for today's shipment and for the following time period. 93 Chapter 6 Conclusions In this research, we studied two consolidation models: direct shipping consolidation model and inventory consolidation model. In both cases, we consider an agricultural supply chain whose suppliers highly depend on trucking transportation services. The suppliers each have low demands and ship using LTL rates or courier services. They are unable to ship at the more advantageous FTL rate because they do not possess enough demand. As a result, the high transportation cost tends to be passed onto the customers, causing them to lose businesses to domestic and foreign competitors. We dene a parameter to represent a shipping deadline for all products that arrive at the consolidation center. For the direct shipping consolidation model, we consider a deterministic and stochastic case. In the direct shipping consolidation model, there are no inventory costs. Therefore, no penalties are incurred when product stays in inventory. CPLEX has a dicult time solving for the optimal solution. We develop a dynamic programming model to solve the problem optimally but its runtime and complexity depends on the discretization of the state vector demand values. However, in the deterministic case, a discretization factor of 10 still provides a good quality solution with a reasonable runtime. In a practical setting, a dynamic programming algorithm would not be used since it requires all the demands to be known. 94 We develop an ecient heuristic that nds a good solution with only days of information. We introduce a -based Consolidation heuristic that allows inventory to stay at the consolidation as long as possible to consolidate and ship at a better rate. It takes advantage of the opportunity for consolidation with inventory cost. In a comparison with CPLEX and the dynamic programming algorithm, it performs very well with a better runtime than both solution approaches. The California cut ower industry is an example of an industry with low demands per supplier and increasing competition from foreign imports. We analyze their transportation network and compare it to a consolidation model using the -based consolidation heuristic as a policy. Overall, the consolidation model could save up to 17 million dollars per year if participation increased. The savings from this model are entirely because the LTL shipments in their current network could be shipped under the FTL rate. The heuristic and dynamic programming algorithm are easily modied for the stochastic direct shipping consolidation model. The dynamic programming algorithm calculates the expected cost for a time horizon of length T and uses the demand probability distributions for each supplier. However, the complexity of the dynamic program increases and the runtime increases signicantly. The -based Consolidation Heuristic only needs days of demand information and is easily modied for the stochastic case. Numerical experiments show that the heuristic performs well compared to the dynamic programming algorithm and a few other heuristics. The advantage of this heuristic is low requirement for given demand information. We assumed unit cost per item was low for the direct shipping consolidation model, which justied the mathematical model's lack of inventory cost. In the last model of this research, we relax this assumption and consider perishable products there deteriorate over time, which cannot explicitly measure. In this model, a penalty exists if inventory is held at the consolidation center. Now, the benet of holding a product is not as apparent. A dynamic programming model is formulated but the computational complexity is very 95 high great. the computational limitations make it dicult to run numerical experiments that are insightful. We develop a comparison heuristic that considers the inventory cost, adding a product to today's outgoing shipment and the potential cost of shipping the remaining inventory tomorrow. It considers the trade-o and decides how much should be added to today's shipment. Dierent scenarios are studied to analyze the performance of the proposed comparison heuristic. The numerical experiments show that the Comparison Heuristic performs well when inventory costs are high. We tested it on an instance where inventory costs are low and it did not perform well. However, if inventory costs are low such that they are insignicant, then the -based Consolidation Heuristic could be applied. 6.1 Future Work All the consolidation models in this work assume direct shipping occurs between the supplier and the consolidation center and between the consolidation center and the destinations. Con- solidation strategies are shown to provide system-wide savings, however, it is a conservative approach. Routing strategies may further improve the system's costs by utilizing the excess capacity of the vehicles. Under the direct shipping model and with small to medium-sized growers, the majority of the incoming shipments are arriving at the LTL rate. A vehicle routing problem with pick-ups from the suppliers and a drop-o at the consolidation center could further improve system-wide savings. Our model places a deadline on products at the consolidation center and, in our last model, we consider inventory costs that are constant with respect to time. A more accurate measure of decay for the perishable products may motivate consolidation dierently. It also allows us to explore applications and industries of products that are normally not considered to be shipped together. 96 The travel time from the supplier and to the destination are not considered in these models. Interesting future work would be to consider the penalties for holding product at the consolidation center in order to achieve a more advantageous rate. Picking up multi- ple suppliers' product would not impose a large delay if the suppliers are located close to the consolidation center. A routing problem with suppliers located further away from the consolidation center would be interesting to study. 97 Bibliography [1] B. Abdul-Jalbar, A. Segerstedt, J. Sicilia, and A. Nilsson. A new heuristic to solve the one-warehouse n-retailer problem. Computers & Operations Research, 37(2):265{272, 2010. [2] N. Absi, B. Detienne, and S. Dauz ere-P er es. 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Abstract (if available)

Abstract Freight transportation supports the economic activity of the United States. Trucking is the most frequently used method of transportation in the agriculture industry. The majority of suppliers in the agriculture industry are small farmers with little demand. We investigate a supply chain system with small farmers. Two models are considered in this research for the shipping of perishable goods. Both models consider a transportation network with direct shipping modes to and from a single consolidation center. We represent perishability as a hard constraint on the total time products are allowed to stay at the consolidation center. In the first model, we assume inventory costs are negligible, and only consider transportation costs. We develop a consolidation heuristic that is efficient and easy‐to‐implement in practice. The consolidation strategy exploits economies of scale from three shipping methods: full truckload rates, less‐than‐truckload rates, and courier rates. A dynamic programming model is developed to calculate an optimal solution. Both solution approaches solve the deterministic and stochastic cases of the direct shipping consolidation model with multiple suppliers and seasonal demand. The second model adds inventory costs, a soft constraint for perishability, to the first model. A dynamic programming algorithm and a heuristic that calculates the trade‐off between shipping and holding inventory are developed. Numerical results show that the heuristic performs well for high inventory costs. However, if the inventory costs are significantly large, the best solution is to ship product everyday and hold no inventory.

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Asset Metadata

Creator Nguyen, Christine V. (author)

Core Title Supply chain consolidation and cooperation in the agriculture industry

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Industrial and Systems Engineering

Publication Date 08/05/2014

Defense Date 08/05/2014

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Agriculture,consolidation,OAI-PMH Harvest,perishable products,supply chain

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Dessouky, Maged M. (committee chair), Toriello, Alejandro (committee chair), Gordon, Peter (committee member), Moore, James Elliott, II (committee member)

Creator Email cvinguyen@gmail.com,nguyen7@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-452204

Unique identifier UC11287147

Identifier etd-NguyenChri-2763.pdf (filename),usctheses-c3-452204 (legacy record id)

Legacy Identifier etd-NguyenChri-2763.pdf

Dmrecord 452204

Document Type Dissertation

Format application/pdf (imt)

Rights Nguyen, Christine V.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA