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Cost-sharing mechanism design for freight consolidation

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Cost-sharing mechanism design for freight consolidation

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Content Cost-Sharing Mechanism Design for Freight Consolidation by Wentao Zhang A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirement for the Degree of DOCTOR of PHILOSOPHY (INDUSTRIAL AND SYSTEMS ENGINEERING) May 2018 Committee: Dr. Maged M. Dessouky, Chair Dr. John Gunnar Carlsson Dr. Ketan Savla Contents 1 Introduction 5 2 Literature Review 11 2.1 Freight Consolidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Cost-sharing Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Cost Allocation in Transportation Collaborations . . . . . . . . . . . . . . . . . . . . 17 3 Moulin Mechanism for Freight Consolidation 21 3.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Moulin Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2.2 Truthfulness and Budget-Balance . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 A Cost-sharing Mechanism Using Approximate Costs . . . . . . . . . . . . . . . . . . 29 3.3.1 An Approximately Budget-Balanced Approach . . . . . . . . . . . . . . . . . 29 3.3.2 The Cost-Sharing Mechanism Proportional to Effective Demand for Sharing (PEDS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3.3 Economic Efficiency of Cost-Sharing Mechanism PEDS . . . . . . . . . . . . 37 4 Moulin Mechanism for Single Truck 44 4.1 Truthfulness and Budget-Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Economic Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.3 Numerical experiments for Economic Efficiency . . . . . . . . . . . . . . . . . . . . . 54 5 Acyclic Mechanism for Freight Consolidation 59 5.1 Acyclic Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.2 Cost-Sharing Mechanism Based on Bin Packing (BBP) . . . . . . . . . . . . . . . . . 61 5.3 Properties of the Cost-Sharing Mechanism BBP . . . . . . . . . . . . . . . . . . . . . 68 5.4 Budget-Balance of Cost-sharing Mechanism BBP . . . . . . . . . . . . . . . . . . . . 72 5.4.1 Theoretical Results on Budget-Balance Ratio . . . . . . . . . . . . . . . . . . 73 5.4.2 Numerical Results on Budget-Balance Ratio . . . . . . . . . . . . . . . . . . . 82 5.5 Economic Efficiency of Cost-Sharing Mechanism BBP . . . . . . . . . . . . . . . . . 85 1 6 Conclusions and Future Directions 92 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 References 95 2 List of Figures 1 Structure of consolidation system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2 Cost structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 An example of the cost approximation function . . . . . . . . . . . . . . . . . . . . . 30 3 List of Tables 1 Experiment parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2 Summary of values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 Cost-sharing example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4 Cost shares for proportional to actual demand . . . . . . . . . . . . . . . . . . . . . . 33 5 Cost shares for proportional to effective demand for sharing . . . . . . . . . . . . . . 34 6 Summary of results from optimization model . . . . . . . . . . . . . . . . . . . . . . 41 7 Summary of results from cost-sharing mechanism PEDS . . . . . . . . . . . . . . . . 42 8 Comparison of social cost gaps for cost-sharing mechanism PEDS . . . . . . . . . . . 43 9 b G and b E for overestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 10 b G and b E for underestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 11 Frequencies of the unknown economic efficiency cases . . . . . . . . . . . . . . . . . . 57 12 Social cost gap for unknown economic efficiency cases . . . . . . . . . . . . . . . . . 58 13 Demand and shipping cost of 3 suppliers . . . . . . . . . . . . . . . . . . . . . . . . . 67 14 Cost shares in Moulin mechanism with . . . . . . . . . . . . . . . . . . . . . . . . 67 15 Cost shares in Cost-sharing mechanism BBP . . . . . . . . . . . . . . . . . . . . . . 68 16 Worst-case cost ratio summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 17 Budget-balance ratio for 3 suppliers demand profiles . . . . . . . . . . . . . . . . . . 84 18 Budget-balance ratio for 6 suppliers demand profiles . . . . . . . . . . . . . . . . . . 84 19 Budget-balance ratio for 10 suppliers demand profiles . . . . . . . . . . . . . . . . . 84 20 Summary of results from the optimization model for unsplittable demand . . . . . . 89 21 Summary of results from the cost-sharing mechanism BBP . . . . . . . . . . . . . . . 90 22 Comparison of social cost gaps for cost-sharing mechanism BBP . . . . . . . . . . . 91 4 1 Introduction Truck, air, rail and water are the four major means of transporting goods in the United States. Their extensive network serves as the backbone that supports the economic activities, including the production, distribution and consumption of commodities. As economies prosper, especially with the rapid growth of e-commerce, the increasing demand for commodities mobility in a more reliable, efficient and secure logistics network has also stimulated the development of transportation in U.S. According to the National Transportation Statistics [74], the transportation sector contributed 9.1% to the Gross Domestic Product (GDP) of the U.S. economy in 2015. As one important component of the transportation functions, the logistics sector, however, operates neither efficiently nor sustainably [68]. For instance, statistics show that trailers on the road are only approximately 60% filled on average and about 20% of the trailers are traveling completely empty [68]. Moreover, approximately 27% of the total U.S. CO 2 emission, the biggest source of greenhouse gases (GHG) emissions, in 2015 came from transportation primarily due to burning fossil fuel for cars, trucks, ships, trainsandplanes[4]. Asaresult, cost-effective, efficientandenvironmentallyfriendlylogistics practices are urgently called for. Freight consolidation, which is a process to assemble and transport small shipments together to avail of lower freight rates, is an effective strategy to increase capacity utilization, improve cost-effectiveness of operations, and reduce energy consumption and carbon footprint. To this end, more and more companies consolidate and ship their demand orders using shared transportation capacities collaboratively. Both academic research and industry practices have demonstrated the effectiveness of freight consolidation in carbon footprint reduction. Using a discrete-time based shipment consolidation strategy, Ülkü [90] showed that the increased usage of transportation capacities and the reduced number of dispatches due to freight consolidation directly help reduce the emissions of CO 2 . At the strategic level, Pan et al. [78] proposed a freight consolidation by merging supply chains of two companies. Applying their model on real data of two French retailers, they concluded that the suggested freight consolidation reduces 14% of the CO 2 emissions and this reduction will be 52% if rail transport is also considered. Moreover, a successful implementation of freight consolidation betweentwopharmaceuticalcompanies,UCBandBaxter,achievedanapproximately50%reduction of the CO 2 emissions [65] [91]. The economic incentive to consolidate freight is huge. Transportation costs play an important role in the success of various industries because they often account for a substantial portion of 5 the product costs. For instance, U.S. manufacturing companies usually spend approximately 30% of products cost on logistics [87] and large portions of revenues have long been paid to transport products in the agriculture industry [72]. However, the mismatch of the demand and supply in the global economy [82] and the ever-increasing demand for logistics services with high service levels havecontributedtotheriseintransportationcost. Therefore, itisapriorityforcompaniestoreduce transportation costs in order to remain competitive in the market. Freight consolidation serves as an effective means to reduce transportation costs by taking advantage of economies of scales [50]. Significantcostsavingshavealsobeenreportedthroughcollaboratedfreightconsolidationinvarious industries, e.g. [28] [92]. These cost savings, which will eventually lead to lowered product sale prices, better product qualities, improved customer services, etc, help companies gain competitive advantages. In addition, as shipping volumes aggregate due to consolidation, the bargaining power of companies or a group of companies increases as well. This enables the corresponding companies to negotiate with the transportation providers for lower shipping rates, which in turn results in further cost reductions. A real example that shows the great economic potential of freight consolidation is the current plight of the California Cut Flower industry, which largely relies on trucks to transport products. Currently, this industry has been facing increasing competition from the cut flower growers in South America, especially Colombia. California’s share of the U.S. cut flower market has decreased from 64% to 20% in the last two decades while South America’s share has reached approximately 70% in 2007 [5]. A shared cross-docking and distribution facility located in Miami, Florida has enabled South American growers to compete effectively with Californian growers. The central planners in Miami organize and consolidate the products of South American growers in the distribution facility before sending them to the rest of the U.S. using trucks. The resulting large volume shipments allow them to obtain the cheaper full-truckload (FTL) rates and the corresponding cost savings on transportation provide them with a huge competitive advantage. In contrast, most of the California cut flower growers, who are currently sending their products individually using more expensive less-than-truckload (LTL) rates, are often of small to medium size and have no power to negotiate favorable transportation rates on their own. Nguyen et al. [72] evaluated the currenttransportationpracticesintheCaliforniaCutFlowerindustryandexploredthepossibilityof building a consolidation center in Oxnard, California. They concluded that a shipping consolidation center could reduce transportation costs by 35%, saving $20 million per year if all the California 6 Cut Flower growers were to participate in the consolidation. Although the environmental and economical initiatives to do freight consolidation are widely acknowledged, various concerns for establishing successful collaborations have slowed down the implementation of freight consolidation with self-interested companies. Cruijssen et al. [29] con- ducted a survey with approximately 1500 representative logistics service providers in Belgium and concluded that concerns about fair cost/profit allocations, reliable partner selections, different ne- gotiation powers among partners and information sharing are the four major impediments for consolidating their shipments. Among these four concerns, lack of a fair cost/profit sharing scheme is the main hurdle that makes individual logistics service providers hesitate about collaboration. Freight consolidations often take place among companies who produce similar products or provide similar services in the same geographical region and therefore they are also competitors in the market. Establishing cooperation among them is not possible unless there exists a perceived “fair” way to share the benefit of collaboration to ensure that each company maintains its competitive advantages over time in the collaboration. Considering this fact, offering a “fair” cost allocation scheme is a fundamental step to encourage freight consolidation with cooperation. Themajorityofcostallocationschemesthathavebeendevelopedintransportationcollaboration come from cooperative game theory. It provides a framework to allocate costs among a set of players so that a certain “fair” criteria are satisfied. This approach focuses on what can be achieved by cooperating the entire set of players and whether it is possible to coordinate these players to achieve the goal by a set of cost shares. For example, the core – one of the most well-studied solution concepts in cooperative game theory – consists of allocations that the total cost shares can recover the cost incurred by the players and no individual or a group of players can benefit by deviating from the current cooperation. The emptiness or nonemptiness of the core is often studied as a proxy for the possibility of cooperation of all the players. The cooperation in this context is often established based on a cost share solution in the core. A binding agreement is signed by the entire set of players to pursue a stable and long term collaboration. However, failing to take into account the economy and market changes, which influence the profit and operational cost of each player over time [29], may make the agreed cost share solution less appealing and thus lead to a fragile cooperation. Moreover, this form of cooperation is too restrictive for a large number of small to medium businesses, who often experience variable or seasonal demands and therefore prefer not to make a long term commitment in a cooperation. 7 Cost-sharing mechanism, alternatively, determines who forms the cooperation and how the costs/profits are shared each time when collaboration happens based on the bids submitted from a set of potential collaborators. In the context of freight consolidation, companies that are interested in participating in the consolidation submit their shipping volumes and corresponding maximum costs they are willing to pay for the shipping service at the planning phase of each consolidation. Then the central planner of the consolidation applies the cost-sharing mechanism to decide who participates and how much cost each participant is shared. This process allows the potential collaborators to effectively evaluate the cooperation opportunities with their accurate operational costinformation. Meanwhile, thisflexibleorganizationofcooperationavoidslongtermcommitment and allows potential collaborators to bid for service at will. Another advantage of cost-sharing mechanism over cooperative game theory is that its cooperation outcomes can also be evaluated from the social perspective, e.g. social welfare. This is particular important because we want to advocate the cooperation that not only benefits the participants but also the society at large. However, there has been very limited prior research on mechanism design to solve cost allocation problems in transportation collaborations. To fill this gap, we design cost-sharing mechanisms to incentivize a group of self-interested suppliers to participate in freight consolidation in this dissertation. In the environment we consider, there is a set of suppliers that could cooperate using a nearby consolidation center, which groups and ships their demands to a common faraway destination by trucks. Our proposed cost-sharing mechanisms decide both the set of suppliers who participate in consolidation and their corresponding cost shares. We design the cost-sharing mechanisms to possess certain desirable properties. First, a well-defined cost-sharing mechanism shouldbetruthful, meaningthatsuppliersareincentivizedtorevealtheirtruewillingnesstopayand no individual supplier or a group of suppliers can benefit from submitting false bids (overreporting or underreporting their willingness to pay). Second, an ideal cost-sharing mechanism is budget- balanced so that the shipping cost incurred by consolidation can be fully recovered with cost shares or prices charged. When budget-balance is impossible to achieve, we design mechanisms to recover the incurred cost as much as possible. Finally, the outcome of the proposed cost-sharing mechanism should maximize social welfare as much as possible to ensure that the mechanism is as economically efficient as possible. We first study a cost-sharing problem in the freight consolidation setting, in which demands of suppliers are packed aggregately in trucks, i.e. a new truck is used only when the current truck is 8 fullypacked. BasedontheMoulinmechanismframework,whichisaschemethatallowsthedesignof truthful and (approximately) budget-balanced cost-sharing mechanisms, we find that it is generally not possible to obtain a simultaneously truthful and budget-balanced cost-sharing mechanism under the transportation cost structure we study. Consequently, we propose an approximately budget- balanced Moulin mechanism with an accompanying proportional cost-sharing method. Analyzing the budget-balance guarantee of the proposed mechanism, we show that there exists a trade-off between the budget-balance guarantee and the level of incentives that can be given to the suppliers. The economic efficiency of the proposed mechanism is studied numerically using social cost as the means of comparison. The minimum social cost for the freight consolidation problem is obtained via a mixed integer program (MIP). As a special case of our general problem, the cost-sharing problem for the single truck demand profiles, in which the total demand of all suppliers’ fits into one truck, is also studied. Our results demonstrate that the proposed Moulin mechanism is not only truthful and budget-balanced for the single truck problem, but also economically efficient for the majority of the demand profiles in our extensive experiments. In the second half of this dissertation, we study a more realistic cost-sharing problem by incorpo- rating a practical constraint that less-than-truckload demands cannot be split and must be shipped in one truck. In this environment, suppliers want their entire demand to be delivered at the same time and avoid unnecessary damage during extra handling to ensure the quality of the products. Also, such operations save the costs of separating and combining shipments for the consolidation center. This added constraint greatly complicates the cost-sharing problem because the cost shares now largely depend on the demands packing decisions that we obtain by solving bin packing prob- lems, which are known to be NP-hard. We design an acyclic cost-sharing mechanism, which is a generalization of the Moulin mechanism, to solve this cost-sharing problem based on bin packing solutions yielded by the subset sum algorithm. Then we study the budget-balance guarantee of the proposed mechanism both theoretically and numerically with different parameter settings. Finally, we compare the social costs of the mechanism’s outcomes to the minimum social costs of the freight consolidation problem to investigate the economic efficiency of the mechanism numerically. The main contribution of this dissertation is two fold. (1) We advance the research on cost allocations in transportation collaborations by proposing a novel approach – cost-sharing mecha- nism design. This approach serves as a more suitable cost allocation scheme that incentivizes the transportation collaborations in which variability or long term commitment is the primary concern. 9 Applying cost-sharing mechanisms, we can also provide additional insights for each cooperation by quantifying its impact on social welfare at large. (2) We demonstrate the applicability of cost- sharing mechanism design for solving cost-sharing problems in freight consolidation. In particular, we show how to incorporate the complex transportation cost structure in mechanism design using approximations, what are the trade-offs in mechanism design for the central planners in freight consolidation, and how to derive the cost-sharing method and the offer function based on packing solutions yielded by an approximation algorithm in acyclic mechanism design. This dissertation is organized as follows. In Chapter 2, we review the existing research on freight consolidation, cost-sharing mechanisms and cost allocation approaches in transportation collabo- rations. We first formally define our problem in Chapter 3. Then we propose an approximately budget-balanced Moulin mechanism and investigate its truthfulness, budget-balance guarantee and economic efficiency. In Chapter 4, we analytically and numerically study the proposed Moulin mechanism for the special case in which the total demand of all suppliers fits into one truckload. An acyclic mechanism is proposed in Chapter 5 to solve the cost-sharing problem when it is assumed that the entire less-than-truckload demand of each supplier is packed in one truck. We conclude our work and present future directions in Chapter 6. 10 2 Literature Review In this chapter, we first review the research work in two major fields that are relevant to the problem we study: freight consolidation and cost-sharing mechanism design. In particular, we review the freight consolidation problems and their corresponding solution techniques. In the review of cost-sharing mechanism design, we focus on introducing the cost-sharing settings and the two cost-sharing mechanism framework we use: Moulin mechanism and acyclic mechanism. Then, we review the literature on transportation collaborations to show the general approaches that have been applied to solve cost allocation problems in this context. 2.1 Freight Consolidation With the passage of the Motor Carrier Act of 1980, more transportation options became available to companies to improve logistics efficiency and customer service levels. Freight consolidation is one of the strategies that has been applied by many companies. Among the 53 surveyed firms on freight consolidation practice [50], all of them regarded freight consolidation as an important strategy to remain competitive in terms of cost and 77% of them indicated that freight consolidation was also helpful for providing better service. Freight consolidation strategies can be broadly characterized into three categories: spatial, product and temporal [67]. Spatial consolidation strategies focus on when and where to consolidate, which demand to consolidate and how to pick up and deliver the goods that are consolidated. Product consolidation is more concerned with increasing the shipment quantity to each customer by combining different kinds of goods into a single shipment. Temporal consolidation aims to leverage the benefit of lower transportation rates with large volumes as the shipment quantities aggregate in time against the inventory costs and service levels. Spatial consolidation is usually studied in distribution networks with origins/suppliers, destina- tions/customers and consolidation terminals. Daganzo [30] compared in-vehicle and out-of-vehicle consolidation strategies in a so-called one-to-many network with a single origin and many desti- nations. Under certain assumptions, Daganzo pointed out that consolidation in terminal (out-of- vehicle) was worse than in-vehicle consolidation. Burns et al. [19] analyzed the trade-offs between direct shipping and in-vehicle consolidation in a one-to-many network. They concluded that trucks should always be dispatched full for in-vehicle consolidation and that in-vehicle consolidation was cheaper when the items are valuable. Operational consolidation guidance was provided by Da- ganzo [31] for a many-to-one distribution network. Given the different shapes, sizes and densities 11 of shipping items, he provided guidance on which items should be combined in a shipment, which route should the shipment follow, and the optimal composition of each shipment using a linear program and a matching algorithm. Hall [45] proposed an optimal procedure to decide where to apply direct shipping and where to route shipments through a terminal in a many-to-many network. His procedure decomposed a multicommodity flow problem with concave costs into subproblems in a one-to-many network. Campbell [21] analyzed three routing schemes: nearest terminal (NT) routing, minimum distance (MD) routing and minimum transportation cost (MC) routing in a many-to-many logistics network with consolidation terminals. In order to explore the interactions between consolidation, location, and routing, the average transportation costs for each routing scheme were compared in both one dimensional and two dimensional service regions. He found that MC routing was characterized as a combination of NT routing and MD routing and the location of terminals as a whole should follow the density of the customer demand. Product consolidation is often studied with spatial consolidation [31, 45]. For example, Blu- menfeld et al. [13] analyzed the trade-offs between transportation, inventory and production costs in a network where direct shipping, shipping via a consolidation center, and the combination of the two are possible. The optimal shipping schedule in this paper was obtained by decomposing a few-to-many network to a few-to-one network in which the optimal shipping strategy can be chosen independently from other sub-networks. Carlisle et al. [23] presented a logistics planning system design for Marshalls, a chain of American and Canadian off-price department stores. The logistics challenges for Marshalls required the planning system to provide decisions on routings for all kinds of products using direct shipping, cross-docking, and intermediate consolidation terminals. Temporal consolidation’s efficiency highly depends on shipment-release policies. Shipment- release policies can be based on quantity, time, or both time and quantity [46]. With a quantity- based policy, shipments are released whenever the accumulated volume exceeds the predetermined dispatch quantity threshold. Shipments are scheduled on a predetermined date when a time-based policy is applied. Under a time-quantity-based policy, shipments are released whenever the con- dition – a predetermined dispatch quantity threshold or a predetermined shipping date – is met. Higginson et al. [46] were among the first to evaluate the three shipment-release policies using discrete event simulation. They compared the three policies based on two measures: mean per-unit cost and mean order delay. Experimental results showed that there was no best policy since the performance was largely dependent on order arrival rate and customer service levels. Quantity- 12 based policies usually performed well in terms of cost, while time-quantity-based policies had the best mean delay. The large performance variation of time-based policies indicated that a pure time- based policy should be applied with caution. More recently, Cai et al. [20] used a tree-structured markovian model to study the dispatch policies of the consolidated shipments. By incorporating the delay penalty into the cost structure, they introduced a class of dispatch policies that have not been studied before. Measuring the dispatch policies using long-run performance, they proposed heuristics to obtain the parameter settings for some dispatch policies that are optimal in specific situations. The optimal dispatch quantity was studied using economic shipment quantity (ESQ) models in [1, 19, 16] with deterministic approaches. Analogous to an economic order quantity (EOQ) model in inventory control, the ESQ model determines the optimal dispatch quantity by minimizing the per-order total cost of transportation, inventory, and other costs. Gupta et al. [44] proposed a stochastic approach of calculating the most cost-effective quantity using clearing models in a just- in-time environment. Renewal theory was applied to derive the optimal shipping quantity and the optimal timing for dispatch respectively for a quantity-based policy and a time-based policy in [24]. A time-quantity-based policy was studied in [15], where a system clearing model was applied to decide the maximum holding time and the predetermined release quantity. Higginson and Bookbinder [47] used a Markovian decision process to decide whether to dispatch or to continue to hold each time a new order arrives. They specifically analyzed the different decisions for private carriers and public carriers. Further in [48], Higginson compared the decisions made by a recurrent approach, inwhichmarginalanalysiswasconductedeachtimeanorderarrived, withadeterministic optimal shipping weight for both the private carrier and common carrier cases. More recent research has focused on deciding inventory control parameters and the shipment dispatch policies simultaneously in a vendor-managed inventory system. Çetinkaya et al. [25] in- troduced a renewal theoretical model to jointly determine the optimal stock replenishment strategy and the optimal time policy for shipment dispatch. Çetinkaya et al. [26] studied and compared different shipment-release policies in terms of costs and customer waiting time while optimizing the inventory replenishment plan. Lee et al. [58] proposed a network based solution methodology to obtain the optimal replenishment period and outbound shipping schedule in a two-echelon dynamic lot sizing model. Marklund [63] analyzed an integrated inventory model where the time-based shipment-release policy and the stock replenishment were evaluated simultaneously with different 13 retailers who reviewed their inventory continuously. Freight consolidation also happens at the last mile delivery. In order to deliver customers’ goods efficiently, a distribution center needs to consolidate customers’ goods into different vehicles based on various factors, e.g. geographical information, delivery time windows, etc. Koskosidis et al. [56] provided an iterative optimization-based heuristic to solve a capacitated clustering problem in which customer orders were efficiently consolidated into vehicle loads to save transportation costs. Nowak et al. [73] proposed a recursive heuristic using savings algorithm as the building block. They tested their heuristic results against a real-world data and showed that the cost savings are significant via their heuristic. Freight consolidation has also been studied from various other perspectives to provide valuable insights into efficient consolidation operations. For instance, Bookbinder and Barkhouse [14] in- troduced a conceptual model of the information and decision system and discussed the required phases, algorithmsandperformancemeasuresinaconsolidationcontext. Tyanetal. [89]considered a freight consolidation system using flights. The authors evaluated three consolidation policies for a logistics company who cooperated with suppliers and shared information to achieve better flight capacity utilization and customer service levels. Mixed integer programming models were imple- mented to compare the policies that varied in package modes and service requirements. The forms of effective collaboration in freight consolidation were studied by Zhou et al. [95]. They examined the effects of strategic alliance, in which companies operated independently while cooperating, and full cooperation, in which companies operated as a single company, using a time-quantity-based shipment-release policy. Simulation was applied to discover the conditions under which each form of cooperation worked better. The effectiveness of LTL consolidation (in-vehicle consolidation) is confirmed by the study of Mesa-Arango and Ukkusuri in the combinatorial auction settings [66]. In the combinatorial auction, trucking companies provide bids of an individual lane or a collection of lanes to the suppliers and in the hope that suppliers are willing to fill in their capacities for the bid- ding prices. They obtained the bids by solving a multi-commodity one-to-one pickup-and-delivery vehicle routing problem using branch-and-price algorithm. The resulting bids with LTL consolida- tion not only increase the likelihood of winning the suppliers but also improve the profit margins for the trucking companies. Marcucci et al. [62] conducted a stated-preference study to explore the influential factors that affect the decision of implementing an urban freight consolidation center (UFCC). Based on the surveyed data in Fano, Italy, UFCC service cost, delivery time, annual cost 14 of the access permit to the congested area, and parking distance from the shop were found to be the major factors influencing the potential demand for UFCC. 2.2 Cost-sharing Mechanisms A cost-sharing setting consists of a set of players who are interested in receiving service from a provider. A binary demand setting restricts the decision of the service provider to either serve the customer or not at all, whereas a general demand setting allows the provider to offer service at various levels. Each player has a private valuation of the service. The objective of the service provider is to decide who to serve, at what levels, and how to share the cost among the selected players. The algorithm that service providers apply to make these decisions is called a cost-sharing mechanism. In a cost-sharing mechanism, these decisions are made based on bids that players submit to the service provider. The bids of players express their maximum willingness to pay for service. The study of cost-sharing mechanisms mainly focuses on three desired properties: (i) truthfulness, the idea that it is optimal for individual players or groups of players to bid their true valuations, (ii) budget balance, the notion that the mechanism charges the players the cost they incur, and (iii) economic efficiency, the idea that the welfare for all the players is maximized. Unfortunately, almost 40 years ago, Green et al. [39] and Roberts [80] proved that it is not possible for a cost-sharing mechanism to possess these three desired properties simultaneously. This has led to a cost-sharing mechanism design paradigm that relaxes the constraint either on budget-balance oreconomicefficiency. Furthermore, theimpossibilityresultsalsomotivateapproximationmeasures on budget-balance and economic efficiency. Roughgarden et al. [81] introduced a measure called social cost to quantify inefficiency in cost-sharing mechanisms so that the mechanisms, which could previously yield zero or negative social welfare, now become meaningfully comparable. As a result, we are allowed to identify with increasing fidelity the relatively more efficient mechanisms. Without the constraint of economic efficiency, Moulin [69], and Moulin and Shenker [70] pro- posed a framework, now known as the Moulin mechanism, that allows the design of truthful and approximately budget-balanced cost-sharing mechanisms. A Moulin mechanism decides on the ser- vice set and the cost shares through an iterative process with the help from a cost-sharing method, which calculates the cost allocation for any given set of players to be served. The process starts with all players being considered. In each iteration, costs are calculated and offered to the considered players, and only the players who accept the cost shares remain to be considered in the next iter- 15 ation. The iterations continue until all remaining players accept the cost allocation offered by the cost-sharing method. Using a so-called cross-monotonic cost-sharing method, a Moulin mechanism offers a nondecreasing sequence of costs to the players to guarantee that no individual or coalition of players can be better off by submitting false bids. Meanwhile, approximate budget-balance is achieved by offering costs at each iteration that would in total approximately cover the cost incurred if the current iteration were to be the last. Due to its flexibility and reasonable performance on economic efficiency, approximately budget-balanced Moulin mechanisms have been designed for a wide range of cost-sharing applications arising in scheduling [17, 12], network design [52, 6, 42, 41], facility location [32, 55, 59, 77], and logistics [94]. Another scheme that leads to truthful and approximately budget-balanced cost-sharing mech- anisms is the acyclic mechanism [64]. It is a strict generalization of the Moulin mechanism with a designer-specified ordering of players. Different from the Moulin mechanism, an acyclic mecha- nism offers the costs to the players in each iteration according to the pre-defined order rather than simultaneously. This added ordering protocol allows the construction of truthful mechanisms to be no longer dependent on cross-monotonic cost-sharing methods, which are necessary to induce truthful Moulin mechanisms. This relaxation embraces new families of cost-sharing mechanisms with non-cross-monotonic cost-sharing methods. Mehta et al. [64] pointed out that a large num- ber of primal-dual algorithms naturally induce acyclic cost-sharing mechanisms with non-ascending prices. For instance, the primal-dual algorithm [53] and dual fitting algorithm [51] induce truthful acyclic cost-sharing mechanisms for the metric uncapacitated facility location cost-sharing prob- lem. Meanwhile, acyclic mechanisms have better budget-balance and economic efficiency than the Moulin mechanisms for important classes of cost-sharing problems, e.g. vertex cover, set cover, no- metric/metric uncapacitated facility location. For details of the comparison, please refer to Table 1 in [64]. Finally, the acyclic mechanism framework can be extended to solve cost-sharing problems with general demand settings, in which every player bids for each of the service level it may receive, while Moulin mechanisms are primarily applied in binary demand settings. Balireddi and Uhan [10] designed acyclic mechanisms for scheduling cost-sharing games with general demand settings. Based on the singleton acyclic mechanism for binary demand cost-sharing games proposed by Bren- ner and Schäfer [18], they extended the framework to study the general demand cost-sharing games in the single machine, identical parallel machine and concurrent open shop settings. Although acyclic mechanisms have the aforementioned advantages compared to the Moulin mechanism, it 16 achieves a weaker notion of truthfulness than the Moulin mechanism. When economic efficiency is the primary concern together with truthfulness in cost-sharing mechanism design, the Vickrey-Clarke-Groves (VCG) mechanism [27, 40, 93] is a powerful frame- work. As a special case of VCG mechanisms, the marginal cost mechanism is often used to achieve efficient cost allocations. The cost shares in the marginal cost mechanism are given such that the welfare each player obtains is its marginal contribution to the overall social welfare. However, this class of mechanisms usually has no budget-balance guarantee and sometimes raises zero revenue [70]. 2.3 Cost Allocation in Transportation Collaborations Transportation collaboration has become more common due to increasing transportation costs. A strategic alliance is one typical form of collaboration, in which companies pool their resources and plan centrally in order to achieve benefits that are not possible by operating alone. For instance, cooperationamongseacargocarriershasallowedthemtoimprovethecapacityutilizationofshipsby exchanging capacities on specific routes. Companies that procure trucking services have integrated their transportation network to reduce the cost of repositioning empty containers [3]. In order to form and sustain this kind of beneficial cooperation, allocating costs in a way that satisfies every collaborator is extremely important. There has been a lot of work that applies cooperative game theory to study the cost allocation problems arising in transportation collaborations. Different cost allocation methods have been proposed in [85, 84, 7, 36, 61]. In addition, linear programming has been used to find cost allocations in [83, 2, 3, 49, 75]. Meanwhile, cost-sharing mechanisms have been studied in [37] to provide comparable cost-sharing solutions that incentivize the participants to make particular socially desirable choices. A cooperative game consists of a group of players, who have the potential to work together toward a common goal. We call a subset of the players a coalition and the set of all players the grand coalition. A characteristic function assigns the total cost for any given coalition. A cost allocation method assigns the cost shares to each participant in the given coalition. This is also sometimes called a solution concept. Cooperative games are often studied in a static environment in which we assume that all the players cooperate to form the grand coalition. The “best” cost allocation method is ill-defined because the choice of a cost allocation method largely depends on the details of the situation, the players’ understanding of fairness, the complexity of calculating 17 the cost shares, etc. [88]. Therefore, a variety of cost allocation methods have been studied based on a set of desirable properties that decision makers are commonly interested in. For example, a cost allocation method is efficient or budget-balanced if the total cost allocated equals to the cost incurred by the grand coalition. A cost allocation method is individually rational if the cost allocated to any player in the grand coalition is less than the cost of this player operating alone. A cost allocation method is stable if no individual or group can deviate from the grand coalition and be better off. The core contains all the cost allocations that are both budget-balanced and stable. Different cost allocation methods have been widely applied in cost allocation problems arising in transportation collaborations. For instance, Krajewska et al. [57] applied the Shapley value [85] to allocate the profit achieved by a group of freight carriers, who cooperate by pooling transportation resources. The Shapley value is a cost allocation method that views the formation of the grand coalition as a sequential process with one player joining at a time. The cost share for each player is calculated by averaging the player’s marginal contribution over all possible coalition formation sequences. The profit allocation by the Shapley value in [57] was in the core, although in general the Shapley value is efficient but not necessarily stable. Besides the Shapley value, Liu et al. [61] and Frisk et al. [36] also considered a cost allocation method called the nucleolus [84]. The nucleolus is unique and a refinement of the core if the core is not empty. Another cost allocation method, which is based on separable and nonseparable costs, was first explored by the Tennessee Valley Authority in the 1930’s [86]. The separable cost is distributed by charging each player its marginal contribution to the overall cost when it joins the grand coalition. The nonseparable cost is shared based on various rules, e.g. the equal charge method and the alternative cost avoided method [88]. For example, Audy et al. [7] used a modified alternative cost avoided method to share the additional costs incurred by special requests so that companies who impose more expensive requirements were allocated with more costs. Many studied cost/profit allocation problems are application based. For example, Liu et al. [61] studied a profit allocation problem in a less-than-truckload carrier alliance. They first considered some commonly used cost allocation methods, e.g. the Shapley value, the nucleolus. However, they argued that the Weighted Relative Savings Model (WRSM), which they proposed, is more favorable because it considers individual contributions to the coalition and the balance of savings for each player. The profit shares were calculated by a linear program which minimized the difference of the weighted relative savings among the players. Similarly, the concern of fairness also led Frisk et 18 al. [36] to a new method – the Equal Profit Method (EPM) – for the cooperation of eight forest companies in southern Sweden. EPM applied a linear program to obtain a cost allocation in which the profit ratios of the participants were kept as close as possible. They concluded that EPM was more acceptable than the Shapley value because similar relative savings provided greater incentives for players to join the coalition in the negotiation phase. Following [36], Audy et al. [7] employed EPM with two modifications to solve the cost allocation problem arising in the Canadian furniture industry. The first modification was to require a minimum cost saving percentage for all partici- pants. The second modification introduced three non-transferable costs: inbound shipping costs, special requirement costs, and flat rate costs for using the service. Later in [8], Audy et al. studied the cost allocation issues in the forest transportation problem using cooperative graph-restricted games. They were particularly interested in analyzing the influences of different behaviors with different business models on the formation of coalitions and the corresponding benefit. Empir- ical results showed that companies that conduct planning for cooperation earned an additional 6:7% 19:8% profit. They also showed that different business models with different behaviors may lead to coalitions that cannot exert the full economic potential of the companies. Besides the rule-based cost allocation methods, linear programming duality has also been em- ployed to provide cost sharing solutions. Owen [75] formulated an economic production game using a linear program. The author showed that duality theory can be applied to calculate a stable cost allocation for the grand coalition. Sánchez-Soriano et al. [83] introduced a class of transportation games in which a set of producers work together to fulfill the demands of a set of destinations to maximize their profit. They showed that the non-negative optimal dual solutions of the underlying linear program characterized the efficient and individually rational profit allocations in the core. Similarly, Agarwal et al. [2] used dual solutions to construct cost allocations for a multicom- modity flow game in a network where each arc is owned by several agents. Agents form an alliance and share their arc capacities in the network to maximize the profit of fulfilling demands. They aimed to obtain a cost allocation by pricing arcs to induce the optimal arc utilization of the whole system. Agents who use arc capacities that belong to other agents should pay for the costs accord- ing to the prices of the arcs. The dual solutions of the optimal arc utilization problem were applied to construct the efficient and stable cost allocations and the arc prices were obtained by inverse optimization. They proved that the mechanism proposed in this paper provided a cost allocation in the core when each arc has a unique owner. A similar problem involving air cargo alliances was 19 studied using similar techniques by Houghtalen et al. [49], in which capacity exchange prices were utilized to encourage carriers to use their resources optimally from a system point of view. They further analyzed the effects of different control behaviors on obtaining core allocations. It turned out that cost allocations in the core could always be found when the proportion of exchangeable capacity in each airplane was pre-determined by the owner. Özener et al. [76] considered a cost allo- cation problem in which shippers integrate their transportation network to reduce the cost of empty container movements. Besides the well-studied properties, they also examined cross-monotonicity of various cost allocation methods to make sure that each coalition member’s cost allocation does not increase when new players are included. They also introduced two new properties, minimum liability and positive benefit, to better satisfy the requirements of the shippers’ alliance. Different cost allocation methods were proposed to satisfy the new properties while relaxing stability and budget-balance. Some cost allocation problems are studied in the presence of other factors, such as perishability. Nguyen et al. [71] first proposed a consolidation scheme, in which perishability was considered, to determine the shipping quantity and time for each batch of consolidated perishable products and then shared the shipping cost of the perishable products proportional to each suppliers’ demand volume. Taking a different approach, Li et al. [60] incorporated the perishability using a linear and negative exponential decay function in the transportation facility choice game, respectively. Based on their simulation results, sharing the total cost proportional to the marginal cost each supplier incurs has the best performance among the three proposed cost-sharing methods in terms of its presence in the core. Most of the cost allocation problems in transportation collaboration have been studied using cooperative game theory. One the other hand, mechanism design has been rarely used to solve these problems. Furuhata et al. [37] designed an online cost-sharing mechanism to provide quotes to passengers who shared a door-to-door transportation service provided by a demand-responsive transport (DRT) system. The authors proposed a novel cost sharing mechanism – Proportional OnlineCostSharing(POCS)–and discussedpropertiessuchasonlinefairness, immediateresponse, and ex-post incentive compatibility to specifically address the issues involved with sharing costs without knowing the future demand. Their analysis showed that POCS satisfied all the desired properties and provided incentives to the passengers to submit their requests as early as possible. 20 3 Moulin Mechanism for Freight Consolidation Freight consolidation in general requires the cooperation among a set of product suppliers. In order to form and sustain such a cooperation, an incentivizing cost allocation scheme is particularly important. Self-interested suppliers are more likely to participate in the consolidation if they have cost savings and the total shipping cost is shared in a generally acknowledged “fair” manner. In this chapter, we address the cost allocation problem in the freight consolidation system we study by designing a cost-sharing mechanism using the Moulin mechanism framework. Inthischapter, wefirstformallydefinethecost-sharingprobleminafreightconsolidationsystem with one consolidation center and one common destination. Next, we briefly review the design of the Moulin mechanism and demonstrate that it is generally not possible to obtain a simultaneously truthful and budget-balanced Moulin mechanism under the transportation cost structure we study. As a result, we approach the problem by proposing an approximately budget-balanced Moulin mechanism and discuss the trade-off between the budget-balance guarantee of the mechanism and the level of incentives that can be given to large suppliers. Finally, we study the economic efficiency of the proposed cost-sharing mechanism numerically using social cost as the means of comparison. 3.1 Problem Definition We study a freight consolidation system that consists of a group of suppliers who produce similar products, all located in a certain geographical region, and ship to a common destination. All the companies in the group are interested in cost reduction through freight consolidation. A central planner operates a center that provides a consolidation service in the same region. We assume suppliers in our environment are self-interested. They want to ship their demand with the lowest transportation rate. However, we consider an environment where the consolidation center is not profit-driven. Instead, it aims to encourage the participation of the consolidation while financing itself as much as possible, i.e. recovering as much of the incurred shipping cost as possible. Thismeansweassumetheconsolidationcenterissubsidizedbythegovernment, associated organizations, etc. In the case of the California flower industry, the center could for example be run by the non-profit California Cut Flower Commission (CCFC), or the state government. Althoughconsolidationhappensrepeatedlyovertimeinpractice, e.g. daily, weekly, thetruthfulness guaranteed by the cost-sharing mechanism makes the behavior of suppliers predictable in this dynamicenvironment. Thispropertyallowsustorelyonthemechanismtosolicitsuppliers’truthful 21 bids instead of learning their preferences over time. Therefore, under these assumptions, we can formulate our problem as a one-time game without loss of generality. Let N denote the set of suppliers who are interested in consolidating their shipments to a common destination. Each supplier i2 N has a positive shipping demand d i measured in ft 3 and a valuation v i for the service provided by the consolidation center. The valuations reflect the suppliers’ opinion about how much the service from the consolidation center is worth. The service provided by the consolidation center is binary: either a supplier is not served at all or its entire demand is served. Figure 1: Structure of consolidation system Figure 1 shows the structure of the consolidation system we study. Suppliers in N have two shipping options. They can ship the demand either directly to the destination or through a consol- idation center. Suppliers express their willingness to consolidate by submitting a bid for service at the beginning of the consolidation process. We denote supplier i’s bid by q i . Based on these bids, the consolidation center selects a set of suppliers S N to serve. Selected suppliers have their products consolidated first and then shipped to the common destination. We call the shipment from the suppliers to the consolidation center “inbound shipping”, and the corresponding cost incurred by each supplier the “inbound shipping cost”. We call the shipment from the consolidation center to the destination “outbound shipping”, and the corresponding cost incurred by the consolidation center the “outbound shipping cost”. We call the shipment from the suppliers to the destination “direct shipping”, and the corresponding cost for each supplier the “stand-alone cost”. The suppliers and the consolidation center use trucks to ship their products. There are two 22 important parameters in the trucking cost structure. One is the less-than-truckload (LTL) rate, or the cost for shipping each cubic foot when the shipping demand is less than some threshold value. The other is the full-truckload (FTL) rate, or the fixed cost for using an entire truck when the shipping demand is greater than the threshold value. Let b denote this threshold value, which we call the FTL equivalent volume. Shipping demand b or more in one truck costs the same as if the full truckload is used. The FTL rate is usually priced per mile while the LTL rate is usually priced based on other factors besides distance, such as density, freight class, weight per cubic foot, etc. However, with similar products in the shipment, we can assume that these factors influence the price in the same way and thus the LTL rate and FTL rate only depend on the mileage between the origin and the destination. Given the distance between the origin and the destination, we denote the corresponding LTL rate and FTL rate by c L and c F , respectively; note that b = c F c L . The transportation cost is a function of the shipping volume d and its value depends on the number of trucks used, the LTL rate and the FTL rate. The cost structure is illustrated in Figure 2. In mathematical terms, c(d) = 8 > > < > > : b d k F cc F + (dk F b d k F c)c L ifdk F b d k F c<b; (b d k F c + 1)c F ifdk F b d k F cb; wherek F denotesthecapacityofasingletruck. Startingfromzero, thetransportationcostincreases linearly with the LTL ratec L as the demand increases until the demand reaches the FTL equivalent volume b; then the cost remains the same for any demand volume beyond b but less than or equal to k F and the cost is c F = c L b. When the current truck has no more capacity for the demand, another truck is used following the same cost function. As a result, the total transportation cost is the sum of the total cost for shipping some number of full truckloads and the shipping cost of the last necessary truck. We assume that the suppliers and the consolidation center face the same cost structure but not necessarily the same rates or FTL equivalent volume. We define the shipping cost functions for the suppliers and the consolidation center based on the following assumptions: Consolidation center location assumption: The suppliers are all close to the consolidation center and approximately the same distance away. Consequently, we assume all the suppliers have the same positive LTL rate g L0 for inbound shipping. Destination location assumption 1: The suppliers are all far away from the destination and 23 Figure 2: Cost structure approximately the same distance away. Consequently, we assume all the suppliers have the same positive LTL rate g L1 for direct shipping. Destination location assumption 2: The distances between the suppliers and the destination are larger than the distances between the suppliers and the consolidation center. As a conse- quence, g L1 >g L0 and g F1 >g F0 , where g F0 is the suppliers’ positive FTL rate for inbound shipping and g F1 is its positive FTL rate for direct shipping. Location assumption: The suppliers, consolidation center and destination are located such that the inbound shipping distances, outbound shipping distance and the direct shipping distances satisfy the strict triangle inequality, i.e. g L1 <g L0 +c L1 . Threshold value assumption: Thesuppliers’inboundanddirectshippingoptionshavethesame FTL equivalent volume b G (ft 3 ). In general, the above assumptions represent a situation where the suppliers and the consolidation center are located in the same region and the destination is sufficiently far away such that outbound shipping costs dominate the inbound shipping costs if suppliers send demand via the consolidation center. Meanwhile, we consider the group of suppliers as a small community in which each supplier is able to obtain the same transportation rate through negotiation with the carriers. For example, if carriers charge suppliers based on shipping zones, suppliers in the same zone share the same transportation rate for the same destination even though there may be small differences in distance. Following the definition of the cost structure and the assumptions above, the inbound shipping 24 cost for supplier i is G 0 i = 8 > > < > > : b di k F cg F0 + (d i k F b di k F c)g L0 ifd i k F b di k F c > < > > : b di k F cg F1 + (d i k F b di k F c)g L1 ifd i k F b di k F c<b G ; (b di k F c + 1)g F1 ifd i k F b di k F cb G : Suppliers are responsible for their own inbound shipping costs if selected for service by the consolidation center. We consider the outbound shipping cost as the only cost incurred by the consolidation center while providing the service and therefore only the outbound shipping cost will be shared among the selected suppliers. Given the destination, let c L1 and c F1 denote the LTL rate and FTL rate for outbound shipping at the consolidation center. We denote FTL equivalent volume byb C = c F1 c L1 . In mathematical terms, the cost of shipping demandd at the consolidation center is (d) = 8 > > < > > : b d k F cc F1 + (dk F b d k F c)c L1 ifdk F b d k F c<b C ; (b d k F c + 1)c F1 ifdk F b d k F cb C : As a result, the total cost C(S) incurred when consolidating and shipping the demand of suppliers in S is C(S) = X i2S G 0 i + X i2S d i : We assume that the consolidation center only has partial information about the transportation costs of the suppliers. In particular, the consolidation center knows that it has the same cost structure for trucking as the suppliers, but it does not know the exact parameters of the cost functions that apply to the suppliers. The information that the consolidation center solicits from the suppliers is their bid for their corresponding shipping demand. Selected suppliers receive the consolidation service for their reported demand. Therefore, suppliers have to report their demand truthfully. In other words, we can safely assume that the shipping demand of the suppliers is known to the consolidation center. 25 3.2 Moulin Mechanism 3.2.1 Preliminaries The Moulin mechanism [69, 70] is used to design truthful and budget-balanced or approximately budget-balanced cost-sharing mechanisms. It simulates an iterative ascending auction to determine which subset of players to serve by using a cost-sharing method, a function that assigns a non- negative cost share for each player i2S in everySN. The cost shares for the selected subset of players indicate the prices charged for service. The Moulin mechanism operates as follows: 1. Collect a bid q i from each player i2N. 2. Initialize S :=N. 3. If q i (i;S) for every i2S, then stop. Return the set S. Each player i2S is charged the price p i =(i;S). 4. If q j <(j;S) for a player j2S, then set S :=Snfjg and return to Step 3. In Steps 3 and 4, cost shares are offered to players in S simultaneously. An arbitrary player j is removed from S if multiple players have cost shares that are greater than their bids. Because only the players whose bids are greater than or equal to their cost share stay in S, the players selected by the Moulin mechanism are never charged more than what they bid. The cost-sharing method plays a very important role in the Moulin mechanism design. It is almost always required to be cross-monotonic, which means that the cost share of each player cannot decrease as other players are removed, i.e. for all STN and i2S; (i;S)(i;T ). This implies that each player in S is offered a sequence of nondecreasing cost shares through the iterations. When the cost-sharing method is cross-monotonic and nonnegative, the Moulin mechanism is group strategyproof. Group strategyproofness is a strong notion of truthfulness: it means that not only can an individual player not be better off by falsely bidding, but also a subset of players can never strictly increase the utility of one of its members without decreasing the utility of some other member by coordinating false bids. We split the possible outcomes of this mechanism – the set of players served S – into three categories: Total participation: All the players in N are served. Zero participation: None of the players in N is served. 26 Partial participation: A non-empty proper subset of N is served. Observation 1: Any Moulin mechanism yields total participation if and only if (i;N) q i for all i2N. Observation 2: Any Moulin mechanism yields zero participation if and only if in every iteration k = 1; 2:::;n, there exists at least one player i such that(i;S k )>q i , whereS k denotes the remaining set of players at the beginning of iteration k. If the Moulin mechanism yields zero participation, then one player is removed from S k in Step 4 of iteration k of the mechanism. In other words, in each iteration of the mechanism there exists at least one player that has a cost share that is strictly greater than its bid. Now if in each iteration k, there exists a player i that satisfies (i;S k ) > q i , then a player will be removed from S k until there are no more players left. 3.2.2 Truthfulness and Budget-Balance SinceGreen, Kohlbergetal. [39]andRoberts[80]provedtheimpossibilityofobtainingtruthfulness, budget-balance, and economic efficiency simultaneously in a cost-sharing mechanism, one natural approach to designing a cost-sharing mechanism, which is our approach, is to relax the constraint on economic efficiency. In particular, we first examine if there exists a cost-sharing method that results in a cost-sharing mechanism that is always truthful and budget-balanced for any demand profile of a set of suppliers. We initiate our examination by assuming that cost shares may be approximately budget- balanced. In particular, a cost-sharing method is -budget-balanced if C(S) P i2S i C(S) ( 1) for any outcome set S, where i are the cost shares for suppliers in S given by the cost- sharing method. For any demand profile, we solve for a set of cost shares using a linear program that maximizes while enforcing-budget-balance and cross-monotonicity. The model is presented below. Parameters: N :=f1; 2;:::;ng Set of suppliers. C(S) : Total cost for coalition S when all suppliers in S use consolidation service, SN. Decision variables: (i;S) : Cost share for supplier i in coalition S, SN; i2S. : Budget-balance guarantee. 27 Model: max (1) s:t: X i2S (i;S)C(S); 8SN (2) X i2S (i;S)C(S); 8SN (3) (i;S)(i;S[fjg); 8SNnfjg; i2S; j2N; i6=j (4) (i;S) 0; 8SN;8i2S (5) Constraints(2)and(3)ensurethatcostsharesare-budget-balanced. Constraint(4)guarantees the cross-monotonicity of cost shares, i.e. the cost share for a given supplier does not increase when additional suppliers join the coalition. Nonnegativity constraint (5) ensures, along with constraint (4), that the resulting Moulin mechanism is truthful. The numbers of decision variables and constraints of the model (1)-(5) both grow exponentially as thenumberof suppliers increases. TherearejNj2 jNj1 +1 variables, 2 jNj+1 2-budget-balance constraints, andjNj(jNj 1)2 jNj2 cross-monotonicity constraints when there arejNj suppliers. Therefore, it is not trivial to find the cross-monotonic cost-sharing method with the largest for even modest values ofjNj. As a result, it may not be practical to apply this approach to design cost-sharing methods. We solved the above model on thousands of demand profiles with different numbers of suppliers ranging from 3 to 10. For each number of suppliers, we generate 100 demand profiles, with the total demand of each ranging from 0 to 20k F . The parameters we used are presented in Table 1. With b C , b G , c F1 and g L1 g L0 , all the other cost parameters can be calculated accordingly. Table 1: Experiment parameters k F (ft 3 ) b C (ft 3 ) b G (ft 3 ) c F1 ($) g L1 g L0 4000 2000 2000 6000 9 We summarize the results of for different number of suppliers in Table 2. Column “ = 1 ” shows the number of demand profiles out of 100 for which we can find a both cross-monotonic and budget-balanced cost-sharing method. We found that the maximum possible is less than 1 for the majority of the demand profiles. Especially for the demand profiles with larger number of suppliers, the number of demand profiles for which = 1 reduces quickly. This indicates that cross- 28 monotonic cost shares generally are not budget-balanced. Therefore, we can conclude that looking for cross-monotonic cost-sharing methods will not result in both a truthful and budget-balanced Moulin mechanism for our cost-sharing problem. Table 2: Summary of values = 1 min average 3 suppliers 83 0.9398 0.9970 4 suppliers 37 0.8444 0.9822 5 suppliers 11 0.8154 0.9663 6 suppliers 9 0.8030 0.9498 7 suppliers 9 0.8146 0.9385 8 suppliers 5 0.8022 0.9173 9 suppliers 7 0.7943 0.9128 10 suppliers 9 0.7885 0.9025 3.3 A Cost-sharing Mechanism Using Approximate Costs 3.3.1 An Approximately Budget-Balanced Approach Since our numerical experiments indicate the impossibility of finding a budget-balanced cross- monotonic cost sharing method, our goal is to design a cross-monotonic cost-sharing method with a good budget-balance guarantee . We assume that suppliers are responsible for their own inbound shipping cost and they only share the outbound shipping cost incurred at the consolidation center. One intuitive approach is to approximate the true outbound shipping cost function using a concave piecewise linear function, and use this approximation to determine the cost shares. The concavity willhelpinfindingcross-monotoniccostshares. Ifwehaveanoutboundshippingcostapproximation that is at most a factor away from the true shipping cost and we share the approximate outbound shipping cost among the suppliers, then we have an -budget-balanced cost-sharing method. Assume the capacity of the consolidation center is mk F . Consider the two-piece outbound shipping cost approximation function (d;): (d;) = 8 > > < > > : [ c F1 b C ( k F b C 1)]d if 0<db C ; (dk F ) +c F1 ifb C <dmk F : Note that is the slope of the second piece of (d;) and the second piece passes through (k F ;c F1 ). Figure 3 shows an example of (d;) when m = 4. 29 Figure 3: An example of the cost approximation function In order for (d;) to be a valid cost approximation function for our purposes, it needs to satisfy two conditions: (1) (d;) must be concave ind, and (2) (d;)(d) for alld2 [0;mk F ]. Both conditions are satisfied when 0 c F1 k F . With 0 c F1 k F , the slope of the second line segment is no greater than the slope of the first line segment, i.e. c F1 b C ( k F b C 1). Since the approximate cost attk F ,t = 1; 2;:::n is (t 1)k F +c F1 , which is smaller than the true cost tc F1 when 0 c F1 k F , the approximate cost is always less than or equal to the true cost. As a result, we require 0 c F1 k F for (d;) to be a valid cost approximation function. To decide which (d;) to select – in particular, to decide the value of – we can try to maximize the cost recovered by the outbound shipping cost approximation function in the worst case. Define the cost recovery ratio (d;) = (d;) (d) . If we look at the cost recovery ratio for d2 [tk F ; (t + 1)k F ) for some t = 0; 1;:::m 1, we can easily see that within this interval, (d;) decreases first as d increases and the true cost function increases at rate c L1 ; (d;) then increases as the true cost function becomes flat. As a result, the cost recovery ratio always reaches its smallest value in [tk F ; (t + 1)k F ) when d = tk F +b C , t = 0; 1;:::m 1. Therefore, to find the worst cost recovery ratio of a given cost approximation function, we only need to consider d = tk F +b C , t = 0;:::m 1. In other words, the worst case cost recovery ratio is w () = minf (d;)jd2 [0;mk F ]g = minf (d;)jd =tk F +b C ;t = 0;:::m 1g. We summarize the worst case cost recovery ratio results in Proposition 1. Proposition 1. Suppose the capacity of the consolidation center is mk F . Then: w () = 8 > > > > > > < > > > > > > : 1 m + (m2)k F +b C mc F1 if 0< c F1 2k F b C ; 1 k F b C c F1 if c F1 2k F b C < c F1 k F ; 1 2 + b C 2(2k F b C ) if = c F1 2k F b C : 30 Proof. Weidentifytheworstcastcostrecoveryratiobystudyinghowthecostrecoveryratiochanges ast changes. Since the cost function is different whent = 0, we first consider the cost recovery ratio atd =tk F +b C ,t = 1; 2:::;m1 as a function oft. Letf(t) = (tk F +b C ;) = [(t1)k F +b C ]+c F1 (t+1)c F1 . The derivative of f with respect to t is f 0 (t) = k F (t + 1)c F1 c F1 [((t 1)k F +b C ) +c F1 ] (t + 1) 2 c 2 F1 = (2k F b C )c F1 (t + 1) 2 c F1 : set f 0 (t) = 0, we have = c F1 2k F b C . Consequently, when 0< c F1 2k F b C ,f(t) 0 < 0. The cost recovery ratio decreases ast increases. When c F1 2k F b C < c F1 k F , f(t) 0 > 0. The cost recovery ratio increases as t increases. When = c F1 2k F b C , the cost recovery ratio is the same for any t = 1; 2;:::n 1. Now we consider the cost recovery ratio at d =b C and d =k F +b C . The cost recovery ratio at d =b C is (b C ;) = 1 k F b C c F1 . The cost recovery ratio atd =k F +b C is (k F +b C ) = b C +c F1 2c F1 . Since both ratios are linear in , we can set (b C ;) = (k F +b C ;) to get the threshold that determines the relationships of these two ratios: = c F1 2k F b C . Therefore, when 0 < c F1 2k F b C , (b C ;)> (k F +b C ;); when c F1 2k F b C < c F1 k F , (b C ;)< (k F +b C ;); and when = c F1 2k F b C , (b C ;) = (k F +b C ;). Combining the two sets of results, we can conclude that when 0 < c F1 2k F b C , w () = 1 m + (m2)k F +b C mc F1 ; when c F1 2k F b C < c F1 k F , w () = 1 k F b C c F1 ; and when = c F1 2k F b C , w () = 1 2 + b C 2(2k F b C ) . From Proposition 1 we can see that when 0 < c F1 2k F b C , only 1 m of the outbound shipping cost is guaranteed to be recovered. When c F1 2k F b C < c F1 k F , the worst case cost recovery ratio depends on b C k F . When b C is arbitrarily small, b C k F can be arbitrarily small as well, thus leaving w () only bounded below by 0. In addition, w () in both these cases is bounded above by k F 2k F b C = 1 2 + b C 2(2k F b C ) . It is only when = c F1 2k F b C that we can reach the maximum worst case cost recovery ratio, which guarantees that at least half of the outbound shipping cost is recovered. Since the suppliers in S share the approximate outbound shipping cost (D S ;), where D S = P i2S d i , the actual budget-balance ratio of the corresponding cost-sharing method for a selected set of suppliers S is (D S ;)+ P i2S G 0 i (D S )+ P i2S G 0 i , which is greater than (D S ;) (D S ) . As a result, the worst cost recovery ratios in Proposition 1 are lower bounds of the budget-balance guarantees of a cost-sharing 31 method that is budget-balanced for the outbound shipping cost approximation function. 3.3.2 The Cost-Sharing Mechanism Proportional to Effective Demand for Sharing (PEDS) Now we have an approximate outbound shipping cost to share with the suppliers who participate in consolidation. How should we charge the suppliers so they are willing to participate and bid their true willingness to pay for the service? Arguably, the most intuitive way of sharing the outbound shipping cost is to share proportionally to each supplier’s actual demand. This means setting the cost share for supplier i2 S to di P j2S dj ( P j2S d j ), where S is the selected set of suppliers. This cost-sharing method tends to allocate more cost to suppliers with larger demand and thus such suppliers, without whom the total outbound shipping demand may not be large enough to benefit from consolidation, may not have incentive to consolidate. To illustrate, consider an example where there are three suppliers who can consolidate their demand. Their transportation costs of shipping directly and shipping through the consolidation center are given in Table 3. We assume k F = 10000 ft 3 , b C = b G = 5000 ft 3 , c F1 = $1000, g L1 =c L1 = $0:2=ft 3 , g L0 = $0:43=ft 3 and each supplier bids their direct shipping costs. Supplier 3’s demand is larger than that of supplier 1 and supplier 2. If every supplier ships its demand individually, the transportation cost is $1400 in total. However, if we consolidate all of their demand, the total transportation cost is C(N) = $1301. If we share the outbound shipping cost proportional to actual demand in the Moulin mechanism, the corresponding cost shares for each supplier in each iteration is shown in Table 4. Table 3: Cost-sharing example Supplier 1 Supplier 2 Supplier 3 Demand (1000 ft 3 ) 1 1 8 Direct shipping cost ($) 200 200 1000 Inbound shipping cost ($) 43 43 215 Table4showsthatsharingtheoutboundshippingcostproportionaltoactualdemandinthefirst iteration of the Moulin mechanism leads to a total cost that is greater than the direct shipping cost forthesupplierwiththelargedemand. Asaresult, supplier3declinestousetheservice. Thisleaves only supplier 1 and supplier 2 under consideration. However, without supplier 3, supplier 1 and supplier 2 end up with cost shares that are higher than their direct shipping costs. Consequently, none of the suppliers can benefit from consolidation, which could have saved a total of $99 if 32 Table 4: Cost shares for proportional to actual demand Supplier 1 Supplier 2 Supplier 3 Iteration 1 Outbound cost share ($) 100 100 800 Inbound shipping cost ($) 43 43 215 Total cost ($) 143 143 1015 Decision Accept Accept Decline Iteration 2 Outbound cost share ($) 200 200 N/A Inbound shipping cost ($) 43 43 N/A Total cost ($) 243 243 N/A Decision Decline Decline N/A implemented properly. The example above shows the deficiency of sharing outbound shipping cost proportional to actual demand and reveals the importance of having the suppliers with larger demand participate. Therefore, in order to incentivize suppliers with large enough demand to take advantage of their FTL rate, we discount their demand while sharing outbound shipping cost proportionally. Since the consolidation center does not know the suppliers’ FTL equivalent volumes, it reasonably estimates it as b E b C because it has more power to negotiate for favorable transportation costs compared to individual suppliers. For each supplier, we discount the part of demand that exceeds b E by a factor of , 0 1. represents how much incentive is provided to the large suppliers. The smaller is, the larger the discount. Suppliers whose demand is less than b E share the outbound shipping cost according to their true demand. Effective demand for sharing: For each supplier i2N, its effective demand for sharing is d 0 i = 8 > > < > > : d i if 0<d i b E ; (d i b E ) +b E ifb E <d i mk F : We propose a cost-sharing method that shares the approximate outbound shipping cost propor- tional to effective demand for sharing (PEDS). Let D S = P i2S d i and D 0 S = P i2S d 0 i . The price offered to supplier i2 S is equal to its inbound shipping cost plus the share of the approximate outbound shipping cost for set S proportional to supplier i’s effective demand for sharing; that is, for SN, i2S, we define the cost share (i;S;;) as (i;S;;) =G 0 i + d 0 i D 0 S D S ; : 33 If we apply cost-sharing method PEDS in the example above with = 0, the cost shares we obtain in the corresponding Moulin mechanism for each supplier are summarized in Table 5. Table 5: Cost shares for proportional to effective demand for sharing Supplier 1 Supplier 2 Supplier 3 Outbound cost share ($) 143 143 714 Inbound shipping cost ($) 43 43 214 Total cost ($) 186 186 928 Decision Accept Accept Accept Cost-sharing method PEDS charges less to supplier 3 than sharing proportional to actual de- mand. The total cost for each supplier from cost-sharing method PEDS is less than its direct shipping cost. Thus, all suppliers are willing to participate in the consolidation, which successfully saves $100. Additionally, with the participation of supplier 3, each supplier of this coalition is able to reduce its transportation costs. This simple example reveals the power of our proposed cost-sharing method in incentivizing consolidation. We will next examine the truthfulness of the Moulin mechanism using cost-sharing method PEDS. As mentioned above, cross-monotonic cost-sharing methods lead to truthful Moulin mech- anisms. The proposition below gives conditions under which our cost-sharing method PEDS is cross-monotonic. Proposition 2. Suppose b E b C , 0 c F1 k F and the capacity of consolidation center is mk F . If (mk F b E ) (m1)k F b E +c F1 1, then cost-sharing method PEDS is cross-monotonic. Proof. Leti be an arbitrary supplier whose cost share we observe and compare in different subsets. Since the inbound shipping cost G 0 i is always included in supplier i’s cost share in the cost-sharing method PEDS, we only focus on supplier i’s cost share of the outbound shipping cost to prove cross-monotonicity. Let S be an arbitrary set such that i2 S and S Nnfjg, where i6= j. We obtain T by augmenting S with supplier j, i.e. T =S[fjg. Let D S = P i2S d i , D T = P i2T d i , D 0 S = P i2S d 0 i and D 0 T = P i2T d 0 i . Let (S;T ) denote the total cost share of the outbound shipping cost for suppliers in S when T is served. We first prove that the total cost share of suppliers in S does not increase when more suppliers are served, i.e. (S;T ) (D S ). Because function and effective demand for sharing change at b C and b E , respectively, there are six different D S , D T and d j combinations to consider. Case 1: D S <D T b C , d j b E , D 0 S =D S and D 0 T =D T . In this case, 34 (S;T ) = D 0 S D 0 T (D T ) = D S D T c F1 b C k F b C 1 D T = (D S ): Case 2: D S b C <D T , d j b E , D 0 S =D S and D 0 T =D T . In this case, (S;T ) (D S ) = D S D T [(D T k F ) +c F1 ] c F1 b C k F b C 1 D S =D S 1 D T 1 b C (c F1 k F ) 0: Case 3: D S b C <D T , d j >b E , D 0 S =D S and D 0 T =D S +d 0 j . In this case, (S;T ) (D S ) = D S D 0 T [(D T k F ) +c F1 ] c F1 b C k F b C 1 D S =D S D 0 T D T 1 + 1 D 0 T 1 b C (c F1 k F ) < 0: The last equality is valid because D T >D 0 T > (d j b E ) +b E >b E b C . Case 4: b C <D S <D T , d j b E , D 0 S =D S and D 0 T =D T . In this case, (S;T ) (D S ) = D S D T [(D T k F ) +c F1 ] (D S k F )c F1 = d j D T (c F1 k F )< 0: Case 5: b C <D S <D T , d j >b E , D 0 S =D S and D 0 T =D S +d 0 j . In this case, (S;T ) (D S ) = D S D 0 T [(D T k F ) +c F1 ] (D S k F )c F1 = (d j d 0 j ) D 0 T D S d 0 j D 0 T (c F1 k F ) = d 0 j D 0 T d j d 0 j 1 D S c F1 +k F (6) = d 0 j D 0 T d j d j + (1)b E 1 D S c F1 +k F d 0 j D 0 T 1 1 D S c F1 +k F < d 0 j D 0 T 1 1 (mk F b E )c F1 +k F d 0 j D 0 T (c F1 k F c F1 +k F ) = 0: The third to last inequality is valid because (1)b E 0. For the second to last inequality, d j >b E , D S +d j mk F , so D S <mk F b E . Because (mk F b E ) (m1)k F b E +c F1 , the last inequality 35 is valid. Case 6: b C <D S <D T , d j >b E , D 0 S <D S and D 0 T =D 0 S +d 0 j . In this case, (S;T ) (D S ) = D 0 S D 0 T [(D T k F ) +c F1 ] (D S k F )c F1 = D 0 S D T D S D 0 T D 0 T + D 0 S D 0 T 1 (c F1 k F ) = 1 D 0 T [(D 0 S d j D S d 0 j )d 0 j (c F1 k F )] = d 0 j D 0 T d j d 0 j D 0 S D S (c F1 k F ) < d 0 j D 0 T d j d 0 j 1 D S (c F1 k F ) (7) < 0: The second to last inequality is valid becauseD 0 S <D S . The last inequality is valid because (7) is the same as (6). The six cases above show that the total cost share of the outbound shipping cost does not increase when one more supplier is served, i.e. (S;T ) D S . Since the total cost share (S;T ) does not increase, the share of the outbound shipping cost for supplier i does not increase as well. Together with the inbound shipping cost, (i;T ) does not increase either compared to (i;S). Therefore, wheni2S,SNnfjg,T =S[fjg, andi6=j, we have(i;S)(i;T ). This implies that for arbitrary STN, (i;S)(i;T ). Corollary 1. Sharing the approximate outbound shipping cost proportional to demand is always cross-monotonic. Corollary 1 directly follows Proposition 2 because when = 1, cost-sharing method PEDS corresponds to sharing proportional to demand. Suppose we use Proposition 2 as a guideline to guarantee that the cost-sharing method PEDS is cross-monotonic. Aswecansee,isboundedbelowbyavaluethatisdeterminedbytheparameters m and . So, the largest discount we can provide to large suppliers while maintaining cross- monotonicity depends on m and . Let c F1 =k F + for some 0. Then (mk F b E ) (m1)k F b E +c F1 = mk F b E mk F b E + . When m!1, mk F b E mk F b E + ! 1 and therefore ! 1 to guarantee cost-sharing method PEDS to be cross-monotonic. As the capacity of the consolidation center grows, our cost- sharing method PEDS converges to sharing proportional to demand. According to Proposition 2, 36 when the center has infinite capacity, we should share the approximate outbound shipping cost proportional to demand to maintain cross-monotonicity of the cost-sharing method. We can see that as increases, must increase to guarantee cross-monotonicity. If we follow Proposition 2, when the total demand is greater than two truckloads, increasing the recovered cost with greater decreases the maximum discount we can offer to the large suppliers while maintaining cross-monotonicity. On the other hand, as decreases, can decrease. Hence, there exists a trade- off in this more-than-two-truckload scenario between how much cost we want to recover and how much incentive we want to offer to the large suppliers. In other words, this is a trade-off between the benefit of the consolidation center and the cost savings of the suppliers. Corollary 2. When = 0, cost-sharing mechanism PEDS is both truthful and budget-balanced when the total demand fits into one truckload with any 0 1. This result is very intuitive. When = 0, any 0 1 guarantees a cross-monotonic cost- sharing method. Actually, when = 0, the cost approximation function becomes the true cost function for the first truckload. This means that when the total demand of all suppliers fits into one truckload, we can have a both truthful and budget-balanced cost-sharing mechanism. This interesting result motivates us to further study this case specifically in Chapter 4. Since our cost-sharing method PEDS is cross-monotonic when (mk F b E ) (m1)k F b E +c F1 1, the Moulin mechanism that applies our cost-sharing method PEDS under these conditions, is group strategyproof. We call this cost-sharing mechanism PEDS. In our study of this mechanism, we assume that supplier i’s valuation of the consolidation service v i is its stand-alone cost and thus is its bid q i submitted under cost-sharing mechanism PEDS. This assumption is reasonable because cost-sharing mechanism PEDS is group strategyproof. 3.3.3 Economic Efficiency of Cost-Sharing Mechanism PEDS We have shown that our cost-sharing mechanism PEDS is truthful and approximately budget- balanced. The remaining desired property that is left to explore is economic efficiency. In order to examine cost-sharing mechanism PEDS from an economic efficiency perspective, we first introduce an optimization model that calculates a social-welfare-maximizing solution for any given demand profile. We then compare the outcomes of mechanism PEDS with economically efficient solutions under different parameter settings. Typically, the economic efficiency of a cost-sharing mechanism is measured by social welfare. 37 Social welfare W (S) is defined as the savings incurred by the set of suppliers S selected by the mechanism. In mathematical terms, W (S) = V (S)C(S), where V (S) is the total valuation of the suppliers inS andC(S) is the total cost to serve the suppliers in S. The economically efficient solution is the one that maximizes the social welfare. Unfortunately, Feigenbaum et al. [35] showed that truthful and approximately budget-balanced cost-sharing mechanisms often yield outcomes with zero or negative social welfare even though outcomes with strictly positive social welfare exist. This makes it difficult to compare the relative economic efficiency of cost-sharing mechanisms with the same budget-balance guarantee. To sidestep this issue, Roughgarden et al. [81] introduced social cost, another measure of economic efficiency. The social cost (S) is defined as the summation of the cost incurred by serving S and the total valuation of suppliers who are not in S. In mathematical terms, (S) = C(S)+V (NnS), whereV (NnS) is the total valuation of the suppliers not inS. In fact, social cost can be constructed by an affine transformation from social welfare: (S) =W (S) +V (N). This implies that minimizing social cost is equivalent to maximizing social welfare, although social cost is always nonnegative. For these reasons, we use social cost as the measure of economic efficiency and determine outcomes with the maximum social welfare by minimizing social cost. In our problem, the social cost is actually equal to the total shipping cost of all the suppliers in N. Slightly different from our problem definition above, we allow suppliers to ship part of their demand to the consolidation center when solving for the minimum social cost. We believe that this is a more meaningful cost to compare with the outcome of our mechanism because it truly reveals what can be achieved within this consolidation system. We consider the following optimization model to minimize the total shipping cost of all the suppliers. The decision variables and the model are presented below. Decision variables: x i F0 : Number of trucks sent from grower i to the consolidation center by the FTL rate8i2N. x i L0 : Binary variable. If supplier i’s inbound shipping uses the LTL rate x i L0 = 1, otherwise 0. x i F1 : Number of trucks sent from grower i to the destination by the FTL rate8i2N. x i L1 : Binary variable. If supplier i’s direct shipping uses the LTL rate x i L1 = 1, otherwise 0. x CF : Number of trucks sent from the consolidation center to the destination by the FTL rate. x CL : Binary variable. If outbound shipping uses the LTL rate x CL = 1, otherwise 0. y i F0 : Amount of supplier i’s demand sent by the FTL rate to the consolidation center8i2N. 38 y i L0 : Amount of supplier i’s demand sent by the LTL rate to the consolidation center8i2N. y i F1 : Amount of supplier i’s demand sent by the FTL rate to the destination8i2N. y i L1 : Amount of supplier i’s demand sent by the LTL rate to the destination8i2N. y CF : Amount of demand sent by the FTL rate from the consolidation center to the destination. y CL : Amount of demand sent by the LTL rate from the consolidation center to the destination. Model: min X i2N (g F0 x i F0 +g F1 x i F1 +g L0 y i L0 +g L1 y i L1 ) +c F1 x CF +c L1 y CL (8) s.t. y i F0 k F x i F0 ; 8i2N (9) y i L0 b G x i L0 ; 8i2N (10) y i F1 k F x i F1 ; 8i2N (11) y i L1 b G x i L1 ; 8i2N (12) y CF k F x CF ; (13) y CL b C x CL ; (14) y i F0 +y i L0 +y i F1 +y i L1 =d i ; 8i2N (15) X i2N (y i F0 +y i L0 ) =y CF +y CL (16) x i F0 ; x i F1 2f0g[Z + ;8i2N (17) x i L0 ; x i L1 2f0; 1g;8i2N (18) x CL 2f0; 1g (19) x CF 2f0g[Z + (20) all other decision variables are nonnegative (21) Constraints (9), (11), (13) ensure that the shipping volumes do not exceed the number of truckload when shipping with the FTL rates. Constraints (10), (12), (14) ensure that the shipping volumes do not exceed the FTL equivalent volumes when shipping with the LTL rates. Constraints (15) make sure that each supplier ships all of its demand. Constraint (16) enforces that what ships into the consolidation center ships out. The rest of the constraints restrict decision variables to be binary, integers or nonnegative reals. The optimal solution of this model provides a shipping plan 39 for each supplier that minimizes the social cost of the system. In order to study the economic efficiency of cost-sharing mechanism PEDS, we conducted a set of computational experiments to compare the social cost of the mechanism’s solutions to the optimal social cost obtained from the optimization model (8)-(21) for the same demand profile. In particular, we want to know how the social cost gap changes with different number of suppliers and different relative distances between the consolidation center and the destination. We define the social cost gap as mechanism social costoptimal social cost optimal social cost : In the demand profiles we use in these computational experiments, each supplier has less-than- truckload demand. In fact, the consolidation center should only accept less-than-truckload demand from each supplier. On one hand, from the consolidation center’s point of view, full-truckloads cannot contribute to the consolidation because there is no more room to consolidate other demand. Therefore, there is no reason for the consolidation center to accept full-truckloads. On the other hand, from the suppliers’ point of view, if they have one or several full truckloads of demand that can be shipped via the lowest transportation rate, they may not want to ship this demand to the consolidation center to avoid delays and extra operations. Instead, they may be only interested in shipping their less-than-truckload demand to the consolidation to see if they can pay less for shipping. Therefore, the most intriguing demand profiles to study are the ones in which each supplier has less-than-truckload demand. For each given number of suppliers n, we randomly generate 100 demand profiles. Each supplier’s demand is randomly generated from the uniform distribution on (0;k F ). In these experiments, we change the number of suppliers and the ratio g L1 g L0 to study their influences on the gap in social cost. We use the same values for k F , b C , b G and c F1 as in Table 1. We setb E =b C andg L1 =c L1 . We calculate the remaining parameters based on these parameters. For example, we choose = c F1 2k F b C in order to achieve the maximum budget-balance guarantee and therefore, the cost-sharing mechanism PEDS we study in this experiment is 1 2 -budget-balanced. Based on this value of , we give the maximum participation incentive to the suppliers and thus choose = (mk F b E ) (m1)k F b E +c F1 . All related shipping rates can be calculated by the relationship betweentheFTLrateandtheLTLrate. Wechange g L1 g L0 tochangetheratiobetweenthedistancefor direct shipping and the distance for inbound shipping. A larger g L1 g L0 represents a farther destination 40 Table 6: Summary of results from optimization model 3 suppliers g L1 g L0 1:5 2:4 3:2 4:8 9 15 Num. of total par. 0 10 15 20 34 41 Num. of zero par. 100 80 55 32 13 10 Avg. par. ratio 0 0.1667 0.3500 0.5200 0.6933 0.7367 Avg. served demand 0 0.1480 0.3257 0.4716 0.6312 0.6796 6 suppliers g L1 g L0 1:5 2:4 3:2 4:8 9 15 Num. of total par. 0 1 5 10 18 27 Num. of zero par. 100 26 3 0 0 0 Avg. par. ratio 0 0.4167 0.5933 0.6867 0.7467 0.7967 Avg. served demand 0 0.3083 0.4562 0.5743 0.6676 0.7389 10 suppliers g L1 g L0 1:5 2:4 3:2 4:8 9 15 Num. of total par. 0 0 0 1 7 10 Num. of zero par. 100 3 0 0 0 0 Avg. par. ratio 0 0.4580 0.6140 0.7210 0.7930 0.8210 Avg. served demand 0 0.3190 0.4796 0.6143 0.7429 0.7779 15 suppliers g L1 g L0 1:5 2:4 3:2 4:8 9 15 Num. of total par. 0 0 0 3 7 9 Num. of zero par. 100 0 0 0 0 0 Avg. par. ratio 0 0.4920 0.6433 0.7447 0.8160 0.8520 Avg. served demand 0 0.3377 0.4988 0.6252 0.7344 0.7957 compared to the location of the consolidation center. Our choices of g L1 g L0 are 1.5, 2.4, 3.2, 4.8, 9, and 15. In order to study the influence of number of suppliers on the gap in social cost, we choose the number of suppliers n to be 3, 6, 9, and 15. We first present some statistics in Table 6 and 7 to characterize the solutions from the social cost optimization model and our cost-sharing mechanism PEDS. In Table 6, we summarize the number of total and zero participation, average participation ratio and average served demand ratio of the solutions from the social cost optimization model over 100 demand profiles for each combination of parameters. Overall, the number of total participation, the average participation ratio and the average served demand ratio increase as g L1 g L0 increases. This implies more suppliers’ participation in consolidation is required to maximize social welfare as the destination becomes farther away. Comparing across different number of suppliers, we find that the average participation ratio and the average served demand ratio increase as the number of suppliers increases as well. This indicate the more suppliers bid, the more participation is required to maximize the social welfare. In Table 7, besides the statistics in Table 6, we also summarize the average budget-balance 41 for the outcomes from the cost-sharing mechanism PEDS. Although the cost-sharing mechanism PEDS is 1 2 -budget-balanced theoretically, the average budget-balance ratio is well above 1 2 ranging from 0.7167 to 0.8956. These results numerically show that our cost-sharing mechanism guarantees 40% more cost recovery than the theoretical bound in general. The number of total and zero participation, the average participation ratio and the average served demand ratio in Table 7 change in a similar manner as those in Table 6. The cost-sharing mechanism PEDS yields the outcomes with greater participation ratio as the destination becomes farther away or there are more suppliers interested in participating in consolidation. Comparing the results in Tables 6 and 7, we observe that the solutions from the cost-sharing mechanism PEDS always have greater participation ratios than the solutions of the social cost optimization model for the same combination of parameters due to the subsidy. It is also because of the subsidy that the cost-sharing mechanism PEDS yields total participation for all demand profiles for 10 parameter settings out of the 24. Table 7: Summary of results from cost-sharing mechanism PEDS 3 suppliers g L1 g L0 1:5 2:4 3:2 4:8 9 15 Avg. budget-balance NA 0.8956 0.8183 0.8095 0.7896 0.7813 Num. of total par. 0 18 46 84 99 100 Num. of zero par. 100 67 9 0 0 0 Avg. par. ratio 0 0.2800 0.7067 0.9433 0.9967 1 Avg. served demand 0 0.2550 0.5948 0.8978 0.9912 1 6 suppliers g L1 g L0 1:5 2:4 3:2 4:8 9 15 Avg. budget-balance NA 0.8695 0.8221 0.7801 0.7543 0.7405 Num. of total par. 0 17 55 100 100 100 Num. of zero par. 100 14 0 0 0 0 Avg. par. ratio 0 0.6383 0.8850 1 1 1 Avg. served demand 0 0.5056 0.8067 1 1 1 10 suppliers g L1 g L0 1:5 2:4 3:2 4:8 9 15 Avg. budget-balance NA 0.8357 0.7993 0.7699 0.7420 0.7269 Num. of total par. 0 8 61 100 100 100 Num. of zero par. 100 1 0 0 0 0 Avg. par. ratio 0 0.7740 0.9480 1 1 1 Avg. served demand 0 0.6168 0.8997 1 1 1 15 suppliers g L1 g L0 1:5 2:4 3:2 4:8 9 15 Avg. budget-balance NA 0.8240 0.7888 0.7618 0.7326 0.7167 Num. of total par. 0 6 65 100 100 100 Num. of zero par. 100 0 0 0 0 0 Avg. par. ratio 0 0.8127 0.9747 1 1 1 Avg. served demand 0 0.6725 0.9495 1 1 1 42 Comparing the social cost of the outcomes from our mechanism to the minimum social cost, we summarize the average social cost gaps in Table 8. From Table 8, we can see that overall, the social costgapsarelessthan10%. When g L1 g L0 = 1:5, themechanism’ssolutionsarealwaystheeconomically efficient solutions. In fact, the solutions are always zero participation because the destination is too close to benefit from consolidation. The social cost gaps are the largest when g L1 g L0 = 3:2. In other words, under our parameter settings, the mechanism’s solutions are the most different from the optimal social cost solutions when g L1 g L0 = 3:2. This social cost difference is due to the trade-off between the inbound shipping costs and the savings from shipping via the consolidation center. Consolidation is attractive only when the savings can offset the inbound shipping cost. When g L1 g L0 is larger than 3.2, as g L1 g L0 increases, the social cost gap decreases. Our experimental results also show that the mechanism almost always yields total participation when g L1 g L0 4:8. For the optimization model, as g L1 g L0 increases, the savings from consolidation dominates the inbound shipping cost and therefore, suppliers ship more demand through consolidation. This effect makes the solution of the mechanism and the optimization model more and more similar as g L1 g L0 gets larger and larger. In terms of the number of suppliers, we see that the social cost gaps generally become smaller as the number of suppliers increases. This is a good indication that, in order to maximize social welfare, we should encourage more suppliers to consider consolidation. Table 8: Comparison of social cost gaps for cost-sharing mechanism PEDS g L1 g L0 1:5 2:4 3:2 4:8 9 15 3 suppliers 0% 6.97% 9.45% 8.30% 4.37% 2.66% 6 suppliers 0% 6.21% 7.32% 6.70% 3.28% 1.91% 10 suppliers 0% 5.14% 6.93% 5.18% 2.47% 1.42% 15 suppliers 0% 5.06% 7.30% 4.86% 2.25% 1.29% 43 4 Moulin Mechanism for Single Truck In this chapter, we study the special case where the total demand of the suppliers fits into one truckload, i.e. P i2N d i k F . Consequently, the demand of each supplier also fits into one truck- load, i.e. d i k F . This case is worth studying not only because we can provide a both truthful and budget-balanced cost-sharing mechanism, but also because the suppliers with small demand in this case deserve more attention. They need the consolidation more than the suppliers who have large enough demand to ship with the FTL rate. Moreover, the results for this scenario provide managerial insights for some practical applications. For instance, according to the data provided by the California Cut Flower Commission (CCFC), the demand of many California cut flower growers in 2010 shows that in more than 95% of the cases, the aggregated shipping volumes to a single destination for these growers is less than one truckload on a daily basis. Most of these growers are from small farms and are willing to participate in consolidation. 4.1 Truthfulness and Budget-Balance As we pointed out in Chapter 3.2, when = 0, the cost approximation function becomes the true outbound shipping cost function for the first truckload. Thus, we can now design a both truthful and budget-balanced cost-sharing mechanism for the “single truck scenario”. In this scenario, the outbound shipping cost for a selected supplier set S becomes (D S ) = (D S ; 0) = 8 > > < > > : c L1 D S if 0D S b C ; c F1 if D S b C : According to Proposition 2, when = 0, any 0 1 guarantees that cost-sharing method PEDS is cross-monotonic and the associated cost-sharing mechanism PEDS is group strategyproof. In addition, the cost-sharing method and cost-sharing mechanism is budget-balanced. For this section, we choose = 0. The reasons for choosing this value are twofold. First, when cross- monotonicityisguaranteed,wewanttoprovideasmuchincentiveaspossibletothelargesuppliersto participate. Second, = 0 provides a more intuitive explanation of the cost-sharing method. When = 0, suppliers with demand greater thanb E haveb E as their effective demand for sharing. On the otherhand, supplierswithdemandsmallerthanb E havetheirtruedemandastheireffectivedemand for sharing. In other words, the effective demand for sharing for supplier i is d 0 i = minfd i ;b E g. 44 Intuitively, cost-sharing method PEDS shares the costs proportional to the consolidation center’s estimate of each supplier’s stand-alone cost. When the estimated FTL equivalent volume is equal to the true FTL equivalent volume of suppliers, i.e. b E =b G , cost-sharing method PEDS shares the cost proportional to the actual stand-alone cost of each supplier. The estimated stand-alone cost is a better reflection of the true shipping costs of the suppliers. In particular, the cost of shipping demand larger than b G is not proportional to the demand volume. To simplify the notation, let (i;S) =(i;S; 0; 0) denote the cost share of supplier i when the service set is S. 4.2 Economic Efficiency We study the economic efficiency of the cost-sharing mechanism PEDS for the single truck scenario by analytically comparing the social cost from the mechanism’s solutions to the minimum social cost of the system. The optimization model (8)-(21) can be easily adapted to get the minimum social cost for the single truck scenario. The only change is that x i F0 , x i F1 ,8i2 N and x CF are restricted to be binary instead of integral, because both the demand of each supplier and the total demand are less than or equal to one truckload. All of the other decision variables and constraints remain the same. Because of the single truck constraint, if a shipping volume d from a supplier to the consolidation center or the destination is smaller than b G , then LTL is the optimal shipping method; ifd is greater than or equal tob G , then FTL is the optimal shipping method. Increasing the shipment volume when the total shipping demand exceeds b G does not incur extra cost. Therefore, either the FTL rate or the LTL rate is used to ship a supplier’s entire demand to the consolidation center or the destination in the optimal social cost solution. In mathematical terms, for every supplieri, ~ x i F0 ~ x i L0 = 0 and ~ x i F1 ~ x i L1 = 0 where ~ x is in the optimal solution. The same logic applies to the consolidation center as well, i.e. ~ x CF ~ x CL = 0. We analyze the structure of the optimal solutions to this model to understand how the minimum social cost is achieved. The findings are summarized in Proposition 3, Corollary 3 and Proposition 4. Proposition 3. There exists an optimal solution to the model (8)-(21) in which each supplier ships all its demand either to the consolidation center or directly to the destination. Proof. We prove this proposition by contradiction. First of all, we show that 0<y CF +y CL <b C is not optimal. Suppose 0<y CF +y CL <b C is in an optimal solution. WLOG, we assume that y CF = 0. Therefore, the consolidation center incurs a cost of y CL c L1 . If b G b C , then y CL <b G . The total shipping cost is y CL (c L1 +g L0 ). However, 45 shipping the same demand directly to the destination instead costs only y CL g L1 <y CL (c L1 +g L0 ). If b G < b C , it is possible that b G < y CL < b C . In this case, the total shipping cost is y CL c L1 + (ng F0 +g L0 ), where n 0;n2 Z denotes the number of suppliers who can send their demand by FTL rate and denotes the demand volume sent by LTL rate. The cost of shipping the same demand directly to the destination is ng F1 +g L1 . Since ng F1 +g L1 y CL c L1 (ng F0 +g L0 ) = (nb G +)(g L1 g L0 )y CL c L1 (nb G +)(g L1 g L0 ) (nb G +)c L1 = (nb G +)(g L1 g L0 c L1 ) < 0; shipping directly is cheaper than consolidating. The first inequality is valid because the actual demand sent via FTL rate by each of the n suppliers should be greater than or equal to b G . The validity of the second inequality lies in the assumption that g L1 < c L1 +g L0 . Therefore, 0<y CF +y CL 0 to the consolidation center and d D i > 0 to the destination in the optimal solution, i.e. d C i +d D i =d i . Since it is optimal to use either LTL or FTL rate to ship each supplier’s demand to the consolidation center or directly to the destination respectively, there are only four possible shipping plans for supplier i. (1) y i L0 =d C i , y i F1 =d D i . If supplier i ships by this plan, then d C i b G directly to the destination reduces the shipping cost of supplier i by g L0 d C i . Now we examine whether the shipping cost of consolidation is increased by shipping d C i directly instead of consolidating it first. If (y CF +y CL )b C and (y CF +y CL d C i )b C , then the optimal shipping cost of the consolidation center remains c F1 . If (y CF +y CL )b C and (y CF +y CL d C i )<b C , the cost of shipping decreases from c F1 to (y CF +y CL d C i )c L1 c L1 b C =c F1 . Therefore, shipping d i directly to the destination instead yields a decrease in the total cost by at least g L0 d C i , and so this plan cannot be optimal. (2) and (3) y i F0 =d C i , y i L1 =d D i or y i F1 =d D i . 46 If supplieri ships by either of these two plans, then d C i b G . Sinced C i b G , if we also shipd D i to the consolidation center first, no extra cost of inbound shipping is incurred. Therefore, shipping d i >b G to the consolidation center reduces the shipping cost of supplier i byg L1 d D i org F1 . As for the shipping cost of the consolidation center, since (y CF +y CL )d C i b C , the optimal shipping cost remains c F1 if d D i is shipped to the consolidation center. To summarize, shipping d i to the consolidation center ensures a decrease in the total cost byg L1 d D i org F1 , and so these plans cannot be optimal. (4) y L0 =d C i and y L1 =d D i . If supplier i ships by this plan, then d C i 0, we must have y CF +y CL b C . Therefore, increasing the shipment volume of the consolidation center does not incur any extra cost. Now if we ship d D i to the consolidation center as well, the total cost is decreased by (g L1 g L0 )d D i ifd D i <b G or (g F1 g F0 ) ifd D i b G . As a consequence, shippingd i to the consolidation center ensures a decrease in the total cost by (g L1 g L0 )d D i or (g F1 g F0 ), and so this plan cannot be optimal. In the analysis above, we show that shipping d i either to the consolidation center or directly to the destination yields a smaller total cost than shipping d C i > 0 directly to the consolidation center and d D i > 0 to the destination. This contradicts the assumption that shipping d C i to the consolidation center and d D i to the destination is optimal. Therefore, there exists an optimal solution, in which each supplier ships all its demand either to the consolidation center or to the destination. Following the above proposition, we can prove a stronger result on the structure of the optimal solution. Corollary 3. Every optimal solution to the model (8)-(21) shares the same structure: ~ x i F0 + ~ x i L0 + ~ x i F1 + ~ x i L1 = 1 where ~ x is in the optimal solution. In other words, in every optimal solution to the model, each supplier’s entire demand is shipped either to the consolidation center or directly to the destination. Proof. In the proof of Proposition 3, we have shown that by shipping each supplier’s entire demand to the consolidation center or directly to the destination, we are able to reduce the total cost at least by g L0 d C i in shipping plan (1), g L1 d D i in shipping plan (2), g F1 in shipping plan (3), and (g L1 g L0 )d D i or (g F1 g F0 ) in shipping plan (4). With shipping rates and demand being non- zero and g L1 > g L0 ;g F1 > g F0 , the reduced cost is strictly positive. Consequently, the total 47 cost of shipping each supplier’s entire demand either to the consolidation center or directly to the destination is strictly less than shipping some of a supplier’s demand to the consolidation center and the rest directly to the destination. Therefore, in every optimal solution to the model, each supplier’s entire demand is shipped either to the consolidation center or directly to the destination. Combined with the result that it is optimal for suppliers to ship either by the FTL rate or the LTL rate, we can conclude that ~ x i F0 + ~ x i L0 + ~ x i F1 + ~ x i L1 = 1 where ~ x is in the optimal solution. So far, we have shown the best practice for each supplier inN in an optimal social cost solution. Although suppliers have two shipping options, shipping the entire demand of one supplier using one option leads to the minimum social cost. The optimal system-wide shipping plan is given next. Proposition 4. The optimal solution to the model (8)-(21) is either zero participation or total participation. A solution in which a subset of suppliers S N; S6=; ships their demand to the consolidation center first while the rest of the suppliers ship their demand directly to the destination is not optimal. Proof. We prove this proposition by contradiction. Based on Corollary 3, we assume that in the optimal solution, a subset of suppliers S N ship their demand to the consolidation center first and the rest of the suppliers ship their demand directly to the destination. supplier i2NnS is one of the suppliers who ship the demand directly to the destination. Ify CF +y CL = 0 in the optimal solution, the optimal solution of the model is zero participation. If y CF +y CL b C in the optimal solution, the optimal shipping method for the consolidation center is by FTL which costs c F1 . If supplier i ships its demand to the consolidation center first, then its shipping cost reduces from g L1 d i or g F1 to g L0 d i or g F0 , respectively. However, shipping d i directly to the destination does not incur any extra cost. Therefore, the total cost decreases by (g L1 g L0 )d i or (g F1 g F0 ) if supplier i ships its demand to the consolidation center first. Both results contradict the assumption that a solution in which a subset of suppliers S N; S6=; ships their demand to the consolidation center first while the rest of the suppliers ship their demand directly to the destination is optimal. As a consequence, the optimal solution is either zero participation or total participation. Since the economically efficient solution is either zero participation or total participation, we can easily verify if an outcome of cost-sharing mechanism PEDS is economically efficient or not. If the 48 outcome is partial participation, it is not an economically efficient solution. If the outcome is total or zero participation, we can compare its total shipping cost to the total shipping cost under zero or total participation to see if the outcome is economically efficient or not. When the total shipping cost of total participation and zero participation are the same, we assume total participation as the solution of the optimization model. Next, we characterize the participation of the outcomes of the cost-sharing mechanism PEDS and present the comparisons in the following propositions and corollaries. Based on the protocols of cost-sharing mechanism PEDS, a sufficient condition for the mecha- nism to yield zero participation is summarized in Lemma 1. Lemma 1. If (i;N)>q i for all i2N, cost-sharing mechanism PEDS yields zero participation. Proof. Assume (i;N)>q i for all i2N. Suppose supplier j2N is removed in the first iteration of the mechanism, resulting in S :=Nnfjg in the second iteration. Because cost-sharing method PEDS is cross-monotonic, (i;S)(i;N)>q i for all i2S. As a result, another supplier will be removed from S in the second iteration. By the same argument, there exists at least one supplier i such that (i;S) > q i in each iteration. By Observation 2, zero participation is the outcome of cost-sharing mechanism PEDS. Proposition 5. When D N <b C , cost-sharing mechanism PEDS is economically efficient. Proof. We prove the claim by first proving that the cost-sharing mechanism PEDS yields zero participation when D N <b C . Because D N < b C b E , the demand of each supplier d i is smaller than b E . As a result, the effective demand for sharing of each supplier d 0 i =d i . Consequently, suppliers share the outbound shippingcostbypayingtheLTLratenomatterhowmanysuppliersparticipateintheconsolidation, i.e. di D N c L1 D N =d i c L1 . Thus, in the first iteration of the mechanism, any supplier i with d i d i g L1 =q i : The inequality is valid because g L1 <g L0 +c L1 . Any supplier i with d i b G has the cost share (i;N) =b G g L0 +d i c L1 b G (g L0 +c L1 )>b G g L1 =g F1 =q i : 49 Consequently, every supplier i has a cost share (i;N)>q i . Therefore, according to Lemma 1, cost-sharing mechanism PEDS yields zero participation for the set of suppliers whose total demand is less than b C . Based on the proved claim above, it is obvious that total participation incurs more shipping cost because the total cost share of all suppliers equals total shipping cost. Since each supplier pays less when shipping directly, zero participation is less expensive than total participation. Therefore, the economically efficient solution must be zero participation as well. Next, we consider the case when D N b C . The consolidation center decides the value of b E before collecting bids. Without knowing the exact value of b G for the suppliers, the consolidation center’s estimate can be above, below, or equal to the true b G . Let D 0 N denote the total effective demand for sharing of all the suppliers in N, i.e. D 0 N = P i2N d 0 i = P i2N minfd i ;b E g. Note that b E b C andD N b C , soD 0 N b C . WhenD N b C , certain conditions are necessary in order for the cost-sharing mechanism PEDS to yield zero or total participation. (For the proof of Proposition 6 below, see the Appendix.) Proposition 6. When D N b C , the conditions for cost-sharing mechanism PEDS to yield zero or total participation are summarized below: 1. b E >b G , D 0 N b E b G c F1 g L1 g L0 () total participation 2. b E >b G , D 0 N < c F1 g L1 g L0 =) zero participation 3. b E <b G , D 0 N c F1 g L1 g L0 () total participation 4. g L1 g L0 c L1 b G <b E <b G , D 0 N < b E b G c F1 g L1 g L0 =) zero participation Proof. Case 1: Based on the relationships among d i , b G and b E , we categorize the suppliers into three groups: supplier i with d i <b G , supplier i with b G d i <b E and supplier i with d i b E . “=)” Suppose we have total participation. Based on Observation 1, for supplier i withd i <b G , we have d i g L1 d i g L0 + d i D 0 N c F1 () g L1 g L0 c F1 D 0 N () D 0 N c F1 g L1 g L0 : For supplier i with b G d i <b E , we have g F1 g F0 + d i D 0 N c F1 () b G (g L1 g L0 ) d i D 0 N c F1 () D 0 N d i b G c F1 g L1 g L0 : 50 In order to have all suppliers in this group participate in the consolidation we needD 0 N d b G c F1 g L1 g L0 where d = maxfd i jb G d i <b E g. For supplier i with d i b E , we have g F1 g F0 + b E D 0 N c F1 () b G (g L1 g L0 ) b E D 0 N c F1 () D 0 N b E b G c F1 g L1 g L0 : From the three conditions above, if we have total participation, then D 0 N b E b G c F1 g L1 g L0 . “( =” Suppose we have D 0 N b E b G c F1 g L1 g L0 . For supplier i with d i < b G , the cost share in the first iteration of the mechanism is (i;N) =d i g L0 + d i D 0 N c F1 d i g L0 + b G b E d i (g L1 g L0 ) = b G b E d i g L1 + 1 b G b E d i g L0 < b G b E d i g L1 + 1 b G b E d i g L1 =d i g L1 =q i : For supplier i with b G d i <b E , the cost share in the first iteration of the mechanism is (i;N) =g F0 + d i D 0 N c F1 g F0 + b G b E d i (g L1 g L0 ) = d i b E g F1 + 1 d i b E g F0 < d i b E g F1 + 1 d i b E g F1 =g F1 =q i : For supplier i with d i b E , the cost share in the first iteration of the mechanism is (i;N) =g F0 + b E D 0 N c F1 g F0 +b G (g L1 g L0 ) =g F1 =q i : Therefore, by Observation 1, cost-sharing mechanism PEDS yields total participation. Case 2: Similar to the proof of case 1, we categorize the suppliers into three groups: supplier i with d i <b G , supplier i with b G d i d i g L0 +d i (g L1 g L0 ) =d i g L1 =q i : For supplier i with b G d i g F0 +d i (g L1 g L0 ) g F0 +b G (g L1 g L0 ) =g F1 =q i : For supplier i with d i b E , the cost share in the first iteration of the mechanism is (i;N) =g F0 + b E D 0 N c F1 >g F0 +b E (g L1 g L0 ) g F0 +b G (g L1 g L0 ) =g F1 =q i : Therefore, by Lemma 1, cost-sharing mechanism PEDS yields zero participation. Case 3 and case 4 can be proved following the same steps, we omit their proofs here. The lower bound g L1 g L0 c L1 b G on b E in case 4 is necessary for b E b G c F1 g L1 g L0 to be a valid upper bound for D 0 N . Since D 0 N b C , b E > g L1 g L0 c L1 b G guarantees that b E b G c F1 g L1 g L0 b C . Whether the consolidation center underestimates or overestimates the suppliers’ FTL equivalent, there is a range ofD 0 N , for example [ c F1 g L1 g L0 ; b E b G c F1 g L1 g L0 ) with overestimation, whose corresponding outcome of cost-sharing mechanism PEDS remains unknown. This ambiguity no longer exists when the consolidation center correctly estimates the suppliers’ FTL equivalent volume. Corollary 4. When b E =b G , cost-sharing mechanism PEDS yields either zero or total participa- tion. Proof. In Proposition 6, when b E =b G , the conditions for total and zero participation depend on the same critical value c F1 g L1 g L0 . Thus, the conditions for total and zero participation complement each other. Therefore, given any demand profile, the result of cost-sharing mechanism PEDS is either zero participation or total participation. Having proved the conditions under which cost-sharing mechanism PEDS yields zero or total 52 participation, we show that these outcomes are also economically efficient in Proposition 7. Proposition 7. When D N b C , cost-sharing mechanism PEDS is economically efficient under each of the following conditions: 1. b E >b G and D 0 N b E b G c F1 g L1 g L0 2. b E >b G and D 0 N < c F1 g L1 g L0 3. b E <b G and D 0 N c F1 g L1 g L0 4. g L1 g L0 c L1 b G <b E <b G and D 0 N < b E b G c F1 g L1 g L0 5. b E =b G Proof. Under the conditions 1 and 3, cost-sharing mechanism PEDS yields total participation, which means that each participant pays no more than its stand-alone cost. Then, the social cost of total participation is no more than the social cost of zero participation. Given the assumption that the minimum social cost solution is total participation when the social costs of total participation and zero participation are the same, cost-sharing mechanism PEDS produces the same economic efficient solutions as the optimization model under conditions a and c. Under conditions 2 and 4, cost-sharing mechanism PEDS yields zero participation induced by Lemma 1, which means that each participant pays strictly more than its stand-alone cost if total participation is enforced. Then, the social cost of total participation is strictly more than that of zero participation. Therefore, cost-sharing mechanism PEDS produces the same economic efficient solutions as the optimization model under conditions b and d. Under condition 5, b C b E =b G . According to Proposition 6, cost-sharing mechanism PEDS yields zero participation when D 0 N < c F1 g L1 g L0 and total participation when D 0 N c F1 g L1 g L0 . In the optimization model, the cost of total participation is ng F0 +g L0 +c F1 and the cost of zero participation is ng F1 +g L1 , where n denotes the number of suppliers whose demands are greater than or equal to b G and denotes the total demand of the suppliers whose demands are smaller than b G . Suppose D 0 N < c F1 g L1 g L0 . The cost difference between zero participation and total participation 53 is ng F1 +g L1 (ng F0 +g L0 +c F1 ) =nb G g L1 +g L1 nb G g L0 g L0 c F1 = (nb G +)(g L1 g L0 )c F1 =D 0 N (g L1 g L0 )c F1 < 0; the last equality holds becauseb E =b G . Therefore, the optimization model yields zero participation when D 0 N < c F1 g L1 g L0 . Similarly, the optimization model yields total participation when D 0 N c F1 g L1 g L0 . As a result, cost-sharing mechanism PEDS yields the same solution as the optimization model for any demand profile as well. From Propositions 5, 6, and 7, we can conclude that the cost-sharing mechanism PEDS yields economically efficient solutions under the demand profiles for which we know the mechanism’s outcome is zero or total participation. When b E = b G , the mechanism’s solutions are always economically efficient. 4.3 Numerical experiments for Economic Efficiency In Chapter 4.2, we showed in Proposition 7 that under certain conditions, cost-sharing mechanics PEDS yields economically efficient solutions. However, the economic efficiency of cost-sharing mechanism PEDS remains unknown for some cases; in particular: 1. b E >b G and c F1 g L1 g L0 D 0 N < b E b G c F1 g L1 g L0 2. g L1 g L0 c L1 b G <b E <b G and b E b G c F1 g L1 g L0 D 0 N < c F1 g L1 g L0 3. b E < g L1 g L0 c L1 b G <b G and D 0 N < c F1 g L1 g L0 We call these cases the unknown economic efficiency cases. We use numerical experiments to inves- tigate the economic efficiency of cost-sharing mechanism PEDS for the above cases. In particular, we want to know empirically how frequently such demand profiles occur, how often and how much the outcomes of cost-sharing mechanism PEDS deviate from the minimum social cost solutions, how overestimation and underestimation influence the economic efficiency of the mechanism, and how the variations in distances affect the mechanism’s economic efficiency. 54 We setk F andb C to the same values as in Table 1. We setg L1 = 1:28$=ft 3 andc L1 = 1:25$=ft 3 . All the other cost parameters can be calculated accordingly. In the experiments, we change the values of b G , b E and g L0 to study the influences of overestimation, underestimation and variations in distances. The values ofb G are chosen such that some of them are less thanb C and some of them are greater than b C . b E is chosen to be 10%, 20%, 50%, and 100% away from b G while being in its valid range, i.e. b C b E k F . The selected b G and the corresponding b E are displayed in Tables 9 and 10 for overestimation and underestimation, respectively. We change g L0 to change the ratio between the distance for direct shipping and the distance for inbound shipping, i.e. g L1 g L0 . A smaller g L0 induces a farther destination compared to the location of the consolidation center. Intuitively, we reduce the inbound shipping distance to make the destination relatively farther. Similarly, a larger g L0 induces a closer destination. The values we use for g L0 are $0:05/ft 3 , $0:1/ft 3 , and $0:3/ft 3 . Table 9: b G and b E for overestimation b G (ft 3 ) b E (ft 3 ) off percentage 1500 3000 "100% 2250 "50% 1800 3600 "100% 2700 "50% 2160 "20% 2000 4000 "100% 3000 "50% 2400 "20% 2200 "10% 2500 3750 "50% 3000 "20% 2750 "10% 3000 3600 "20% 3300 "10% “"” means overestimate 55 Table 10: b G and b E for underestimation b G (ft 3 ) b E (ft 3 ) off percentage 2500 2250 #10% 2000 #20% 3000 2700 #10% 2400 #20% 3500 3150 #10% 2800 #20% 4000 3600 #10% 3200 #20% 2000 #50% “#” means overestimate In order to obtain instances of the unknown economic efficiency cases with 5 suppliers, for each given b G , b E , and g L0 , we first generate a demand profile with 5 suppliers and then examine it to verify if it satisfies the conditions to be an unknown economic efficiency case. For each given b G , b E , and g L0 , we generate demand profiles until we obtain 10,000 unknown economic efficiency cases. The demand profiles with 5 suppliers are generated in the following way: the total demand D N is first generated from a uniform distribution on (b C ; k F ]. Then 5 more random numbers x i ; i = 1; 2; 3; 4; 5 are generated by a uniform distribution on (0; 1) to determine the demand for each supplier. The demand for supplier i is xi P i2N xi D N . For each givenb G ,b E , andg L0 , we regard the ratio between 10,000 and the number of required demand profiles to obtain 10,000 desired cases as the frequency of the unknown economic efficiency cases. The ratio is always in (0; 1) and a larger ratio implies a higher frequency. The frequencies of the unknown economic efficiency cases in our experiments are summarized in Table 11. From Table 11, we see that when b G is overestimated by b E , the more deviation b E has, the more frequent the unknown economic efficiency cases will happen. The frequencies are quite similar when b G is overestimated by b E with the same percentage. In addition, when g L0 = 0:05 and 0:1, the frequencies are nearly identical for all underestimating b E . Moreover, when g L0 = 0:05 and 0:1, the frequencies of unknown economic efficiency cases are much lower for underestimation than overestimation. Therefore, accurate estimation and underestimation help to decrease the occurrences of unknown economic efficiency cases. The fact that the frequencies for g L0 = 0:3 are always higher than that forg L0 = 0:05 and 0:1 whenb G >b C indicates that the unknown economic efficiency cases are more likely to happen with closer destinations when b G >b C . In terms of the economic efficiency of cost-sharing mechanism PEDS for the generated instances, only217instanceswerefoundtohavedifferentcost-sharingmechanismoutcomesfromtheminimum 56 Table 11: Frequencies of the unknown economic efficiency cases Overestimation b G b E off percentage g L0 = 0:05 g L0 = 0:1 g L0 = 0:3 1500 3000 "100% 0.984 0.937 0.725 2250 "50% 0.502 0.528 0.635 1800 3600 "100% 0.983 0.940 0.726 2700 "50% 0.508 0.524 0.637 2160 "20% 0.206 0.213 0.255 2000 4000 "100% 0.986 0.940 0.728 3000 "50% 0.510 0.530 0.641 2400 "20% 0.204 0.210 0.252 2200 "10% 0.101 0.104 0.128 2500 3750 "50% 0.511 0.533 0.642 3000 "20% 0.204 0.213 0.252 2750 "10% 0.102 0.103 0.128 3000 3600 "20% 0.205 0.214 0.257 3300 "10% 0.102 0.107 0.130 Underestimation b G b E off percent g L0 = 0:05 g L0 = 0:1 g L0 = 0:3 2500 2250 #10% 0.016 0.060 0.127 2000 #20% 0.016 0.059 0.255 3000 2700 #10% 0.016 0.060 0.129 2400 #20% 0.016 0.060 0.260 3500 3150 #10% 0.016 0.059 0.127 2800 #20% 0.016 0.060 0.253 4000 3600 #10% 0.016 0.060 0.127 3200 #20% 0.016 0.060 0.254 2000 #50% 0.016 0.060 0.281 social cost solutions among the 690,000 sampled unknown economic efficiency cases. The overall average gap for these 217 instances in social cost is 5.45%. The details are presented in Table 12. In Table 12, the “diff: cases” columns show the number of instances whose mechanism outcome is not economically efficient and the “avg: gap” columns show the average social cost gaps for these instances. The results in Table 12 suggest that the outcomes of cost-sharing mechanism PEDS for the unknown economic efficiency cases are rarely not economically efficient. For instance, although the frequency of unknown economic efficiency cases is 0:984 when b G = 1500, b E = 3000, and g L0 = 0:05, only 6 out of the 10,000 instances were found to have sub-optimal social cost solutions when applying cost-sharing mechanism PEDS. A social cost gap occurs most frequently whenb G = 1500 and 1800. However, it is uncommon for suppliers to have a FTL equivalent volume that is smaller than the FTL equivalent volume at the consolidation center. With larger aggregated volumes, the consolidation center should be able to negotiate a lower FTL equivalent volume, which leads to cheaper FTL rates with fixed LTL rates. Another observation is that the social cost gaps 57 Table 12: Social cost gap for unknown economic efficiency cases Overestimation g L0 = 0:05 g L0 = 0:1 g L0 = 0:3 b G b E off percentage diff. cases avg. gap diff. cases avg. gap diff. cases avg. gap 1500 3000 "100% 6 12.45% 8 8.07% 67 5.72% 2250 "50% 9 5.86% 17 6.68% 57 6.02% 1800 3600 "100% 0 0 0 0 11 3.46% 2700 "50% 0 0 0 0 10 4.68% 2160 "20% 1 0.07% 4 1.85% 15 2.04% 2000 4000 "100% 0 0 0 0 3 1.97% 3000 "50% 0 0 0 0 2 0.83% 2400 "20% 0 0 0 0 3 1.11% 2200 "10% 0 0 0 0 0 0 2500 3750 "50% 0 0 0 0 0 0 3000 "20% 0 0 0 0 0 0 2750 "10% 0 0 0 0 0 0 3000 3600 "20% 0 0 0 0 0 0 3300 "10% 0 0 0 0 0 0 Underestimation g L0 = 0:05 g L0 = 0:1 g L0 = 0:3 b G b E off percentage diff. cases avg. gap diff. cases avg. gap diff. cases avg. gap 2500 2250 #10% 0 0 0 0 1 6.56% 2000 #20% 0 0 0 0 2 0.82% 3000 2700 #10% 0 0 0 0 0 0 2400 #20% 0 0 0 0 0 0 3500 3150 #10% 0 0 0 0 0 0 2800 #20% 0 0 0 0 0 0 4000 3600 #10% 0 0 0 0 0 0 3200 #20% 0 0 0 0 0 0 2000 #50% 0 0 0 0 1 8.47% occur more frequently when g L0 = 0:3. This indicates that cost-sharing mechanism PEDS is more likely to yield minimum social cost solutions for the unknown economic efficiency cases with supplier destinations that are farther away. In summary, our experimental results show that cost-sharing mechanism PEDS seldom yields sub-optimal social cost solutions for the unknown economic efficiency cases. Combined with the results in Propositions 7, we can conclude that the outcomes of cost-sharing mechanism PEDS very often closely resemble the solution of a social cost minimizing central planner. 58 5 Acyclic Mechanism for Freight Consolidation In this chapter, we study a cost-sharing problem for freight consolidation under a slightly different problem setting from Chapters 3 and 4. Instead of allowing to split the demands of suppliers to fill a truck as much as possible at the consolidation center, we now restrict that the entire less-than- truckload demand of each supplier must be shipped in one truck at the consolidation center. There are multiple reasons for us to make this assumption. On one hand, suppliers want their demand to be delivered in one time and they prefer less handling of the products to avoid unnecessary damage. For instance, a lot of agricultural products are prone to damage during handling processes, e.g. flowers, eggs. On the other hand, the non-profit consolidation center wants to save extra handling costs of separating and combining demands of different suppliers. This small change in the problem setting greatly complicates the decisions that need to be made by the consolidation center compared to Chapters 3 and 4, in which demands are packed aggregately. In order to obtain the selected set of suppliers to serve in the current problem, the consolidation center needs to determine how to pack the demands first so that the cost shares can be correspondingly calculated and compared. Suppliers with multiple truckloads demand may not submit bids for their full truckloads demand because they can ship it at FTL rate and by shipping it directly they gain more shelf life for their products compared to shipping via the consolidation center. As a result, suppliers may only submit bids for their remaining less-than-truckload demand to see if they can save shipping cost by consolidation. From the consolidation center’s perspective, full truckloads do not contribute to consolidationatallbutrequireextrahandlingtoshipthemfromtheconsolidationcenter. Moreover, it is trivial to decide whether full truckloads should be shipped via the consolidation center because such operations are beneficial only when the outbound shipping cost savings can cover the inbound shipping cost of full truckloads. Based on the above discussions, the kind of demand profile that is worth studying is the one in which each supplier has less-than-truckload and consolidating or not makes a significant difference for them. In the following sections, we design an acyclic mechanism to solve the cost-sharing problem for a set of suppliers N, d i < k F ,8i2 N. We first briefly review the key concepts of the acyclic mechanism. We then propose the cost-sharing mechanism Based on Bin Packing (BBP) using the packing solution yielded by the subset sum algorithm for solving a bin packing problem. In particular, we show how the cost-sharing method and the corresponding offer function that induce the acyclic mechanism are derived from the packing solution and argue why we choose the subset 59 sum algorithm to obtain the critical packing solution. Next, we investigate the behavior of the cost- sharing mechanism BBP by analytically showing how demand profiles and the parameters influence the participation results of the mechanism. The budget-balance guarantee of the cost-sharing mechanism BBP is studied both theoretically and empirically. We prove different budget-balance guarantees for different demand profiles and parameter settings. Finally, we analyze the economic efficiencyofthecost-sharingmechanismBBPbycomparingitssolutions’socialcosttotheminimum social cost numerically. 5.1 Acyclic Mechanism Acyclic mechanisms were first introduced by Mehta et al. [64] as an alternative framework to the Moulin mechanism, that enables the design of truthful and approximately budget-balanced cost-sharing mechanisms. An acyclic mechanism is induced by a cost-sharing method and a corresponding valid offer function . An offer function (i;S) is a mapping from any given subset SN and player i2S to a nonnegative offer time. The offer time reveals the sequence that cost shares will be offered to the players in each iteration. Players with lower offer times are offered cost shares earlier than the players with higher offer times. Players with equal offer times are offered cost shares simultaneously. Although the cost-sharing method in an acyclic mechanism is defined in the same way as in Section 3.2, it is not required to be cross-monotonic to induce a truthful mechanism. This flexibility comes from the orders in which the players are offered the cost shares. In other words, the order of offers can be designed to suppress the non-cross-monotonicity of the cost-sharing method. A player is offered a sequence of nondecreasing cost shares as the iterations progress in a truthful acyclic mechanism. As a result, designing a valid offer function for a specific cost-sharing method is critical in acyclic mechanism design. For a subset S N and a player i2S, letL(i;S),E(i;S), andG(i;S) denote the players ofS whose offer time are strictly less than, equal to, and strictly greater than (i;S). A valid offer function for a cost-sharing method is defined in [64] as follows: Definition 1. An offer function is valid for the cost-sharing method, if for every subsetSN and a player i2S, (a) (i;SnW ) =(i;S) for every subset WG(i;S) (b) (i;SnW )(i;S) for every subset WG(i;S)[ (E(i;S)nfig) From the above definition, we can see that the cost shares for a player i cannot decrease if 60 the players in G(i;S) and E(i;S) are removed from the service set. These two conditions ensure that cost shares for player i are cross-monotonic when players in G(i;S) and E(i;S) are removed. However, the definition does not restrict how the cost shares change when players in L(i;S) are removed. When we have a cost-sharing method and a corresponding valid offer function defined, an acyclic mechanism can be defined as follows. Definition2. [64]Anacyclic mechanism isamechanismM(;)inducedbyacost-sharingmethod and an offer function that is valid for . M(;) operates as follows: 1. Collect a bid q i from each player i2N. 2. Initialize S :=N. 3. If q i (i;S) for every i2S, then stop. Return the set S. Each player i2S is charged the price p i =(i;S). 4. If there exists some players j2 J, J S such that q j < (j;S), choose j 2 J such that (j ;S)(j;S)8j2J, set S :=Snfj g and return to Step 3. Comparing the acyclic mechanism with the Moulin mechanism, we can see that only Step 4 is different. Instead of removing an arbitrary player whose cost share is greater than its bid, the acyclic mechanism removes the one with the earliest offer time and if there is a tie, then breaks the tie arbitrarily. That is to say, we can operate Step 4 by offering cost shares to a sequence of players in S ordered by the offer time, and as soon as we encounter a player j whose bid is greater than its cost share, we remove player j and return to Step 3. So any player k whose offer time are greater than that of player j is not offered a cost share in the iteration. This roughly explains why Definition 1 does not restrict how the cost share changes for player k when playerj is removed and the two conditions in Definition 1 are sufficient to induce the truthfulness of the acyclic mechanism. Observation 3: Any acyclic mechanism yields total participation if and only if (i;N) q i , 8i2N. 5.2 Cost-Sharing Mechanism Based on Bin Packing (BBP) In the problem we studied in Chapters 3 and 4, the total demand at the consolidation center is always shipped in the cheapest way, i.e. demands from different suppliers are split and consolidated 61 to fill the trucks to the fullest. Therefore, the outbound shipping cost entirely depends on the total demand of the select suppliers. In our current problem, we assume the demands of the suppliers cannot be split, and so the consolidation center has to decide which suppliers’ demands to pack together in a truck. This packing decision is crucial because it affects the outbound shipping cost and thus influences each supplier’s cost share and the selection of the suppliers to be served. We make this decision by solving a bin packing problem using the subset sum algorithm. Based on this packing decision, we derive our cost-sharing method and the offer function. Our packing problem can be interpreted as a bin packing problem. In a typical bin packing problem, given a list of items L, each with a nonnegative size, and the capacity of each bin H, we want to find a way to pack all the items using the minimum number of bins. Similarly, the consolidation center wants to pack suppliers’ demands into the minimum number of trucks of capacityk F . Solving the packing problem to minimize the number of trucks used serves the purpose of the consolidation center since the outbound shipping cost is closely related to the number of trucks used. On one hand, as a non-profit organization, the consolidation center needs to minimize its costs. On the other hand, a smaller outbound shipping cost leads to smaller cost shares for suppliers and thus encourages more suppliers to consolidate. The bin packing problem is known to be NP-hard. We solve it using a heuristic approach called the subset sum algorithm (ss). The subset sum algorithm is an intuitive way to solve the bin packing problem since it iteratively fills one bin to its fullest using the unpacked items. Mathematically, in each iteration we solve the optimization problem below: max X i2L x i h i s:t: X i2L x i h i H; x i 2f0; 1g; i2L: wherex i = 1 means to pack itemi in the current bin, 0 otherwise,h i denotes the size of itemi and L is the list of currently unpacked items. This is a special case of the 0-1 Knapsack problem, in which the value of each item equals to its size. This problem is the subset sum problem. Although the subset sum algorithm seems computationally expensive, it is shown in [79] that the subset sum problem can be solved to optimality efficiently even for lists with a very large number of items. Empirically, Gupta and Ho [43] showed that the subset sum algorithm outperforms the two well- 62 known bin packing algorithms: first-fit decreasing (FFD) and best-fit decreasing (BFD) algorithms [54]. The first-fit (FF) algorithm packs an item in the earliest opened bin that it fits. If the item does not fit in any opened bins, it is packed in a newly opened bin. FFD operates FF with the items sequenced in a non-increasing order of their sizes. Similarly, BFD also orders the items and then packs each item in the bin whose available capacity is the closest to the size of the item. If no such bin exists, the item is packed in a newly opened bin. Given a set of suppliers N each with a nonnegative demand and the capacity of a truck k F , we apply the subset sum algorithm to solve our packing problem as follows: Algorithm 1 Subset sum algorithm 1: U N 2: k 1 3: while U6=; do 4: T k arg max PU f P i2P d i : P i2P d j k F g 5: U UnT k 6: k k + 1 7: end while 8: return T 1 ;T 2 ;:::;T k1 Let ss(N) denote the number of trucks required for the suppliers in N using the subset sum algorithm. The output of the subset sum algorithm is a sequence of ordered sets T 1 ;T 2 ;:::;T ss(N) ; each set contains the indices of suppliers whose demands are assigned to the same truck. One natural property of the returned sets follows directly from the subset set algorithm. Lemma 2. The returned sets T 1 ;T 2 :::T ss(N) for any set of suppliers N are ordered such that D(T 1 ) D(T 2 )::: D(T ss(N) ), where D(T k ) denotes the total demand of suppliers in T k , k2 f1;:::;ss(N)g. The packing solution T k , k2f1;:::ss(N)g, can also be used as a shipping solution for the consolidation center. Each set T K of suppliers’ demands are shipped using one truck. Intuitively, suppliers should be responsible for the shipping cost of the truck in which their demands are packed. As a result, we share the shipping cost of each truck among its corresponding suppliers proportional to their demand. We formally define our cost-sharing method , which assigns a nonnegative cost share to each supplier i2S for every SN, as follows: (i;S) = 8 > > < > > : G 0 i +c L1 d i ; if D(T k )<b C ; i2T k G 0 i + di D(T k ) c F1 ; if D(T k )b C; i2T k : 63 Note that suppliers are responsible for their inbound shipping cost as well. With the results T k , k2f1;:::ss(S)g, from the subset sum algorithm for any set of suppliers S N, we formally define our offer function (i;S), which determines the sequence of cost-share revelation for the set of suppliers S: (i;S) =k s.t.i2T k ;8i2S: Because k indicates the iteration in which the supplier’s demand is packed, this offer function implies that suppliers whose demands are assigned in the earlier iterations in the subset sum al- gorithm are offered cost shares earlier than those assigned in the later iterations. The suppliers, whose demands are assigned to the same truck, are offered the cost shares at the same time. Based on our subset sum algorithm and the offer function , we show how the removal of some suppliers influences our packing solution in Lemma 3 and Lemma 4. Lemma 3. For every subset SN and supplier i2S, if any subset WG(i;S) is removed from S, every supplier j2L(i;S)[E(i;S) will be packed in the same truck before the removal. Proof. The above claim indicates that the removal of suppliers who are packed in iterations later than supplier i does not affect the packing solutions of the suppliers who are packed before any of the removed suppliers. This is true for packing solutions obtained by the subset sum algorithm. Because the demand of any supplier in W G(i;S) does not contribute to maximizing the total demandintheearlieriterations, theirexistencedoesnotaffectthepackingsolutionsforthesuppliers whose demands are packed in the earlier iterations. As a result, every supplierj2L(i;S)[E(i;S), including supplieri, ends up being packed in the same truck when we remove WG(i;S) fromS, i.e. L(i;SnW ) =L(i;S) and E(i;SnW ) =E(i;S). Lemma 4. For every subsetSN and supplieri2S, if any subsetW (E(i;S)nfig) is removed fromS, supplieri will be packed inT 0 such thatD(T 0 )D(T ), whereT denotes the truck in which supplier i is packed before the removal of W. Proof. We prove by contradiction. If D(T 0 )>D(T ) after the removal of W, then according to the subset sum algorithm, supplier i should end up being in T 0 instead of T before the removal of W. This contradicts the fact that supplier i is packed in T before the removal. Now that we have defined the cost-sharing method and the offer function , we next show that the offer function is valid for the cost-sharing method . 64 Proposition 8. The offer function is valid for the cost-sharing method . Proof. We first prove part (a) of Definition 1. Part (a) indicates that supplieri’s cost share remains the same if we remove some suppliers who will be offered the cost shares later than supplier i. In other words, if we remove the suppliers who are packed in the later iterations, supplieri’s cost share does not change. According to Lemma 3, all suppliers j2 L(i;S)[E(i;S), including supplier i, are packed in the same truck even when WG(i;S) is removed from S. As a result, supplier i ’s cost share remains the same when we remove WG(i;S) from S. For part (b), when W G(i;S)[ (E(i;S)nfig), we assume there exist two supplier sets P and Q such that P G(i;S), Q (E(i;S)nfig), and P[Q = W. By the same argument as in the proof of Lemma 3, we can conclude that the removal of P results in L(i;SnP ) = L(i;S). Since the packing solution up to iteration L(i;SnP ) does not affect the packing of supplier i’s demand, we can restrict our attention to the setting in which Q is removed from SnP. According to Lemma 4, supplier i is packed in T 0 such that D(T 0 ) D(T ), where T denotes the truck in which supplier i is packed before the removal of Q. Since shares the outbound shipping cost proportional to demand, when D(T 0 ) D(T ), the cost share of supplier i should not decrease. Therefore, (i;SnW )(i;S). From the proofs of Lemma 3, Lemma 4 and Proposition 8, we can see that the structure of the packing solutions yielded by the subset sum algorithm helps satisfy the two conditions in Definition 1. If we pack the demands using the more commonly used algorithms for bin packing problems, such as FF, FFD, and BFD, the two conditions in Definition 1 can hardly be met. The packing solutions of FF, FFD, and BFD are all sequence dependent. If the order of the to-be-packed items changes, the solutions change accordingly. Although FFD and BFD pack items in order of non- increasing size, it is not guaranteed that some items will be packed so that the corresponding cost shares remain the same if one or more items are removed from the list. This uncertainty comes from the fact that FF, FFD, and BFD pack each item without knowing the whole list of items while the subset sum algorithm maximizes the packing size over all unpacked items. Therefore, we choose the subset sum algorithm to determine the packing solutions for the demands. With a valid offer function for the cost-sharing method , we define the cost-sharing mecha- nism Based on Bin Packing (BBP) as the acyclic mechanism M(;) induced by and . Mehta et al. [64] showed that every acyclic mechanism is weakly group strategyproof (WGSP). Therefore, our cost-sharing mechanism BBP is WSGP. Compared with the Moulin mechanism, the truthful- 65 ness guarantee of the acyclic mechanism is slightly weaker since the Moulin mechanism is group strategyproof (GSP). The difference is that when a mechanism is GSP, if a coordinated false bid can strictly increase the utility of one of its members, there must exist another member whose utility strictly decreases, while when a mechanism is WGSP, there only must exist another member whose utility remains the same, i.e. an indifferent member. In addition to the truthfulness result of our cost-sharing mechanism BBP, we can also show that the packing solution for the selected suppliers from the cost-sharing mechanism BBP is a strong Nash equilibrium from a non-cooperative game theory perspective. Consider a bin packing game in which each demand is controlled by a self-interested supplier. Each supplier decides how its demand is packed given a set of suppliersS. The cost of the outbound shipping for each truck is still shared proportional to demand among the suppliers whose demands are packed in the same truck. Such a game was first introduced and studied by Bilò [11] and is referred to as the selfish bin packing problem. In this game, each supplier has complete information about other suppliers and a set of strategies to determine in which truck to pack its demand for each possible packing of all the other suppliers’ demands. Although suppliers may be indifferent to packing their demands in multiple trucks if the resulting total demand is less than b C , overall they want to pack their demands in the most filled truck that still fits their demands to minimize the shipping cost shares. Bilò [11] proved that there always exists a pure Nash equilibrium to the bin packing game defined above. A strategy profile is a Nash equilibrium if no supplier can strictly reduce its shared shipping cost by moving its demand to another truck while the packing of other demands remains the same. A stronger notion of Nash equilibrium is strong Nash equilibrium [9], in which any subset of suppliers can not strictly reducethesharedshippingcostsofeverymemberbymovingtheirdemandswhiletheotherdemands are packed in the same way. Epstein and Kleiman [34] proved that the packing solutions yielded by the subset sum algorithm for bin packing games are always strong Nash equilibria. Relating this result to our cost-sharing mechanism BBP, whose cost-share-determining packing solution is yielded by subset sum algorithm, we can conclude that the packing solutions from our cost-sharing mechanism BBP are strong Nash equilibria in the setting where suppliers are allowed to pick the truck to pack their demands. As a result, even if we present the resulting packing solutions from the cost-sharing mechanism BBP to the suppliers and allow them to change the packing, no subset of suppliers can move their demands to benefit every member of the coalition. In other words, every supplier should be satisfied with the packing solutions provided by the cost-sharing mechanism 66 BBP. This outcome endorses the use of the subset sum algorithm to produce the packing solutions. It also helps to persuade the suppliers to participate in the consolidation because the demands are packed to incur the most cost savings for each supplier. Although the acyclic mechanism is a strict generalization of the Moulin mechanism, our cost- sharing mechanism BBP is not a Moulin mechanism in general because its cost-sharing method is not cross-monotonic. We illustrate this fact using an example with 3 suppliers. We assume k F = 4000 ft 3 , b C = b G = 3000 ft 3 , g L1 = c L1 = 1$/ft 3 and g L0 = 0:15$/ft 3 . The demand and shipping cost of each supplier are presented in Table 13. Table 13: Demand and shipping cost of 3 suppliers Supplier 1 Supplier 2 Supplier 3 Demand (1000 ft 3 ) 3.1 0.5 2.7 Direct shipping cost ($) 3000 500 2700 Inbound shipping cost ($) 450 75 405 If we apply the cost-sharing method in the Moulin mechanism framework, the demands of supplier 1 and supplier 2 will be packed in a truck and supplier 3’s demand will be packed in another truck in iteration 1. The corresponding cost shares are shown in Table 14. The cost shares of suppliers 1 and 3 are greater than their direct shipping cost. If suppliers bid truthfully, either supplier 1 or supplier 3 should be removed from the service set. Assume supplier 1 is removed. In iteration 2, the demands of supplier 2 and supplier 3 will be packed in a truck. We observe non-cross-monotonic cost shares for supplier 3 since its cost decreases as supplier 1 is removed. Table 14: Cost shares in Moulin mechanism with Supplier 1 Supplier 2 Supplier 3 Iteration 1 Cost share ($) 3033.33 491.67 3105 Iteration 2 Cost share ($) N/A 543.75 2936.25 Theaboveexampleindicatesthatanofferfunctionisindeedrequiredforourcost-sharingmethod to ensure that cross-monotonicity of the cost shares is perceived by the suppliers. If we solve for cost shares using cost-sharing mechanism BBP for the above example, the cost shares for each supplier are shown in Table 15. From Table 15 we see that supplier 3 is not offered a cost share in iteration 1 due to the offer sequence given by the offer function . Therefore, all cost shares offered to the suppliers are cross-monotonic. This simple example shows that our cost-sharing mechanism BBP is not a Moulin mechanism 67 Table 15: Cost shares in Cost-sharing mechanism BBP Supplier 1 Supplier 2 Supplier 3 Iteration 1 Cost share ($) 3033.33 491.67 N/A Iteration 2 Cost share ($) N/A 543.75 2936.25 in general and our offer function is necessary to guarantee the cross-monotonicity of the cost shares perceived by the suppliers under our cost-sharing method . Our cost-sharing mechanism BBP becomes a Moulin mechanism only when the total demand of all bidding suppliers fit into one truckload so that the cost shares are offered simultaneously to all suppliers. 5.3 Properties of the Cost-Sharing Mechanism BBP In this section, we will show the conditions under which the cost-sharing mechanism BBP yields zero or total participation, and the parameters that influence the result of the mechanism. This informationhelpsusbetterunderstandthebehaviorofthecost-sharingmechanismBBPfordifferent kinds of demand profiles and different location relationships. Now we assume supplier i’s valuation of the consolidation center service v i is its stand-alone cost, and each supplier bids truthfully with the same value q i =v i . This assumption is a natural outcome of a WGSP mechanism. We first study one kind of demand profile, in which all suppliers would have to ship their demand on their own directly with LTL rates, i.e. d i < b G ,8i2 N. Let T k , k2f1;:::;ss(N)g be the output of the subset sum algorithm for a set of suppliers N. For the sake of convenience, we define the cost of using truck T k , k2f1;:::ss(N)g, as: Z(T k ) = 8 > > < > > : D(T k )c L1 ; if D(T k )<b C ; c F1 ; if D(T k )b C : Proposition 9. Suppose d i <b G ;8i2N. Let T k , k2f1;:::ss(N)g, be the output of the subset sum algorithm for supplier set N. If D(T k ) c L1 g L1 g L0 b C ,8k2f1;:::;mg, m < ss(N), then the cost-sharing mechanism BBP yields partial participation with only suppliers in T k , k2f1;:::;mg. Proof. Since d i < b G ; 8i2 N, suppliers can only ship their demand on their own directly by the LTLrate. Theirshippingratesperunitofdemandtotheconsolidationcenterandtothedestination areg L0 andg L1 , respectively. Because suppliers bid truthfully in our mechanism, they are removed only when their cost shares from consolidating are more than their stand-alone shipping cost, i.e. 68 d i g L1 (i;S) for supplier i in T k . This indicates that Z(T k ) D(T k ) g L1 g L0 , the cost per unit of demand of using the truck is greater than g L1 g L0 . Otherwise, supplier i should remain in T k . Consider the suppliers inT 1 who are offered the cost shares first. BecauseD(T 1 ) c L1 g L1 g L0 b C > b C , the cost per unit of demand using this truck is at most c F1 D(T1) g L1 g L0 . Based on the above argument, all the suppliers in T 1 should remain in the service set. According to the protocols of our cost-sharing mechanism BBP, we continue to offer the cost shares to the suppliers in T 2 . Since D(T k ) c L1 g L1 g L0 b C ,8k2f1;:::;mg, the same evaluation for suppliers in T 1 is repeated until T m . All suppliers inT k ,k2f1;:::;mg, remain in the service set. When we start to offer cost shares to suppliers inT m+1 , we have two cases to consider. WhenD(T m+1 )g L1 , we havec L1 >g L1 g L0 . WhenD(T m+1 )b C , the cost per unit of demand of using the truck is c F1 D(T1) . Sinceb C D(T m+1 )< c L1 g L1 g L0 b C , cost per unit of demand is greater thang L1 g L0 . As a result, one of the suppliers inT m+1 will be removed in either case. Because of the removal of one supplier in T m+1 and Lemma 3, we can conclude that the total demand volume in any truck k, k>m in later iterations is no more than D(T m+1 ). Therefore, in each of the following iterations, suppliers in the first m trucks remain the same and one supplier is removed from the (m + 1)th truck until all suppliers in T k ,k2fm + 1;:::;ss(N)g, are removed. Consequently, our cost-sharing mechanism BBP yields partial participation with only suppliers in T k , k2f1;:::;mg. Corollary 5. If d i <b G ;8i2N the conditions for the cost-sharing mechanism BBP to yield zero or total participation are summarized below: 1. When D(T 1 )< c L1 g L1 g L0 b C , then the mechanism yields zero participation. 2. When D(T ss(N) ) c L1 g L1 g L0 b C , then the mechanism yields total participation. Proof. Because of Lemma 2, D(T 1 ) is the largest demand volume we can fill into a truck for the current demand profile. When D(T 1 ) < c L1 g L1 g L0 b C , based on the argument in Proposition 9, all the suppliers will be removed by the mechanism. However, D(T ss(N) ) c L1 g L1 g L0 b C is equivalent to setting m =ss(N) in Proposition 9. Therefore, when D(T ss(N) ) c L1 g L1 g L0 b C , the cost-sharing mechanism BBP yields an outcome with all the suppliers participating. From Proposition 5, we can see that when d i b G ,8i2N, c L1 g L1 g L0 b C is the threshold demand volume in one truck for the corresponding suppliers to accept the cost shares from the mechanism. Therefore, it is worth discussing how this threshold changes as c L1 , g L1 , g L0 and b C change. First 69 of all,b C b G and c L1 g L1 g L0 > 1. If c L1 g L1 g L0 b C b G and c L1 g L1 g L0 b C d i b G ,8i2N, even if each supplier’s demand is packed alone in a truck, shipping the demand via the consolidation center is still cheaper than direct shipping. When c L1 g L1 g L0 b C >b G , the threshold largely depends on c L1 g L1 g L0 . Fix the location of the consolidation center and the destination, i.e. fix g L1 g L0 . The lower c L1 is, the more possible it is that suppliers can accept the cost shares from consolidation. Consequently, the outcome of the cost-sharing mechanism BBP may have more suppliers participating. Now we fix c L1 , g L1 , and change g L0 to simulate the relative locations of the consolidation center and the destination. The smaller g L0 is, the larger g L1 g L0 is and therefore, the farther the destination is compared with the location of the consolidation center. In addition, as g L0 decreases, c L1 g L1 g L0 decreasesaswell. Inthiscase, thecost-sharingmechanismBBPismorelikelytohavemoresuppliers participating under the same demand profile in the long haul transportation scenario than the short haul transportation scenario. Based on the above discussion, the consolidation center should try its best to negotiate lower b C and c L1 for the given destinations to encourage consolidation. The second kind of demand profile we study are the ones with d i > 1 2 k F , 8i 2 N. One characteristic of this kind of demand profile is that each supplier’s demand is packed in a truck alone for any subset SN no matter which algorithm we apply to solve the bin packing problem. Consequently, each supplier’s participation decision entirely depends on its demand volume and the removal of any supplier from the service set does not affect the cost shares for the remaining suppliers. We characterize the participation decision of suppliers as follows: Proposition 10. Let c L1 =(g L1 g L0 ), > 1. Suppose d i > 1 2 k F ;8i2N. The conditions for supplier i to participate in consolidation or not are summarized below: 1. When 1 2 k F <d i b G b C , supplier i does not participation. 3. When b C d i <b G and di bc , supplier i participates. 4. When d i b G b C and b G b C , supplier i participates. Proof. Because each supplier’s demand is packed in a separate truck and their existences do not affect the others’ cost shares, we can analyze each supplier separately. WLOG, let supplier i be an arbitrary supplier from the demand profile. When 1 2 k F <d i g L1 , we havec L1 d i +g L0 d i g L1 d i > 0. Since the cost share of consolidating is greater than the stand-alone cost, supplier i does 70 not participate. When d i b C 1 2 k F , we have two cases to consider. If b C d i c L1 b C (g L1 g L0 )b G : Since c L1 g L1 g L0 > b G b C , wehavec L1 b C +g L0 d i g L1 d i >c L1 b C (g L1 g L0 )b G > 0. Ifd i b G b C , the difference between the two costs for supplieri isc L1 b C +g L0 b G g L1 b G =c L1 b C (g L1 g L0 )b G > 0. As a result, supplier i does not participate in either of the above cases. When b C d i < b G and c L1 g L1 g L0 di b C , the difference between the two costs for supplier i is c L1 b C +g L0 d i g L1 d i = c L1 b C (g L1 g L0 )d i 0. When d i b G b C and c L1 g L1 g L0 b G b C , the difference between the two costs for supplieri isc L1 b C +g L0 b G g L1 b G =c L1 b C (g L1 g L0 )b G 0. Consequently, supplier i participates in the consolidation due to its lower cost. Based on the results in the above proposition, we can derive further conditions for the cost- sharing mechanism BBP to yield zero or total participation. Corollary 6. Let c L1 = (g L1 g L0 ), > 1, d max = maxfd 1 ;:::d N g. Suppose d i > 1 2 k F ; 8i2 N. The conditions for the cost-sharing mechanism BBP to yield zero or total participation are summarized below: 1. When 1 2 <d max b G b C , cost-sharing mechanism BBP yields zero participation. 3. When b G b C , cost-sharing mechanism BBP yields total participation. Proof. The first claim is a direct outcome of result 1 in Proposition 10. When all the suppliers inN have demand smaller than b C , all of them will be removed from the service set by the mechanism. Combining results 1 and 2 in Proposition 10, we obtain the second claim, in which we do not further constrain the demand profiles; instead, adding > b G b C makes the cost-sharing mechanism BBP yield zero participation. Finally, we combine result 3 and 4 in Proposition 10 to obtain the third claim. In result 3, because d i 1 2 k F ,8i2N. 71 Although for the demand profiles d i > 1 2 k F , 8i 2 N, we cannot actually consolidate their demands at the consolidation center, it is still beneficial for the suppliers to ship their demand via the consolidation center under certain conditions. In particular, the consolidation center may have sufficient negotiation power to have small enough c L1 or b C such that c L1 g L1 g L0 b G b C . Of course we do not expect the demand profiles submitted to the consolidation center are always of this kind, but this shows the advantage of having negotiation power against the shipping companies. 5.4 Budget-Balance of Cost-sharing Mechanism BBP In this section, we study the budget-balance guarantee of the cost-sharing mechanism BBP under different conditions and problem settings. Before we delve into the details, we first refine our -budget-balance definition in Chapter 3.2.1. Recall that in Chapter 3.2.1, we stated that a cost- sharing method is -budget-balanced if C(S) P i2S i C(S), 0< 1, for any outcome setS. As we mentioned at the beginning of Chapter 5.2, the problem we studied using the Moulin mechanism in Chapters 3 and 4 allows the suppliers’ demands to be split so that the consolidated demands are always shipped using the smallest possible outbound shipping cost. However, the cost-sharing problem we solve using the cost-sharing mechanism BBP does not allow the suppliers’ demand to be split and the shipping solution yielded by the subset sum algorithm is not necessarily the minimum cost solution. Thus, we define a cost-sharing method as -budget-balanced if C M (S) P i2S i C(S), 1, for any outcome set S, where C M (S) is the cost of a feasible solution output by the mechanism and C(S) is the optimal cost of serving set S. This definition bounds the total cost share of suppliers in S from above by a factor of the optimal cost. These two definitions of approximate budget-balance can be easily modified to represent each other. For instance, if we scale in the -budget-balance condition by 1 , we obtain C(S) P i2S i 1 C(S), 1 1. Now 1 is equivalent to in the -budget-balance definition. Recall that in the discussion of our cost-sharing mechanism PEDS, we want to find an as large as possible such that C(S) P i2S i C(S) for any outcome set S. Similarly, in the study of the budget- balancedness of our cost-sharing mechanism BBP, we want to find a as small as possible such that C M (S) P i2S i C(S) for any outcome set S. In our case, we regard the total shipping cost associated with our subset sum algorithm solution for S as C M (S). Because our cost-sharing mechanism BBP directly charges each supplier what the cost-sharing method yields as its cost share, if our cost-sharing method is -budget-balanced, our cost-sharing mechanism BBP is also 72 -budget-balanced. As a result, we study the value of for our mechanism in this section. 5.4.1 Theoretical Results on Budget-Balance Ratio The total cost of serving S consists of two costs: the total inbound shipping cost P i2S G 0 i , which is fixed for S, and the outbound shipping cost of suppliers in S. The difference between the total cost share of cost-sharing mechanism BBP and the optimal shipping cost, in our case the minimum shipping cost, comes from the outbound shipping cost. The outbound shipping cost closely relates how the suppliers’ demands are packed together. Intuitively, shipping using a smaller number of trucks leads to a smaller cost. So the optimal outbound shipping cost should be induced by the optimal bin packing solution. However, this is not necessarily true with our trucking cost structure. For example, let c L1 = $1,b C = 13,k F = 14, soc F1 = $13. Assume we have 13 suppliers with 5 units of demand and 6 suppliers with 3 units of demand. One optimal packing solution, which uses 7 trucks, is to pack 5, 5 and 3 units in the first 6 trucks and 5 units in the last truck. The total outbound shipping cost of this optimal packing solution is $83. The packing solution of our cost-sharing mechanism, which we call the subset sum packing solution, uses 8 trucks. It packs 5, 3, 3 and 3 units in the first two trucks, 5 and 5 units in the next 5 trucks and 5 units in the last truck. The total outbound shipping cost of this packing solution is $81. From the above example, we can see that although the optimal packing solution uses one fewer truck, it costs more to ship the total demand. This phenomenon is due to our trucking cost structure, in which shipping 13 units or more in one truck costs the same. The subset sum packing solution ships two more units of demand in the first two trucks without paying more. Because of this phenomenon, it is hard to determine the minimum shipping cost and use it to study the budget-balance ratio. However, the outbound shipping cost for a set of suppliers when their demands can be split and therefore can be consolidated to fill a truck as much as possible, is a lower bound for the minimum shipping cost when suppliers’ demands cannot be split. We call this lower bound the lowest outbound shipping cost. If we compare the total shipping cost incurred by the subset sum packing solution to the lowest outbound shipping cost, we can obtain an upper bound on the budget-balance ratio for our cost-sharing mechanism BBP. Since the cost difference comes from the outbound shipping costs, we focus on comparing the outbound shipping cost Z ss (S) induced by the subset sum packing solution with the outbound 73 shipping cost Z(S) in the minimum shipping cost. In other words, C M (S) = Z ss (S) + P i2S G 0 i and C(S) = Z(S) + P i2S G 0 i . For the convenience of analysis, we define instance cost ratio as ss (S) = Zss(S) Z(S) and the worst-case cost ratio as ss = sup S Zss(S) Z(S) . Once we obtain ss for our cost-sharing mechanism BBP, we obtain an upper bound of based on the following proposition. Proposition 11. The best possible budget-balance ratio of the cost-sharing mechanism BBP is bounded by the worst-case cost ratio ss from the above, i.e. ss . Proof. Accordingtothedefinitionof-budget-balance, C M (S) C(S) = Zss(S)+ P i2S G 0 i Z(S)+ P i2S G 0 i foranysuppli- erssetS. Since isthebestpossiblebudget-balaceguarantee,wemusthave = sup S Zss(S)+ P i2S G 0 i Z(S)+ P i2S G 0 i . In addition, Zss(S)+ P i2S G 0 i Z(S)+ P i2S G 0 i Zss(S) Z(S) for any suppliers set S and the equality is valid only when Z ss (S) =Z(S). Consequently, = sup S Z ss (S) + P i2S G 0 i Z(S) + P i2S G 0 i sup S Z ss (S) Z(S) = ss : Let Z (S) denote the lowest outbound shipping cost for the suppliers set S and its value only depends on the total demand volume for a given set of cost parameters: Z (S) = 8 > > < > > : b P i2S di k F cc F1 + ( P i2S d i k F b P i2S di k F c)c L1 if P i2S d i k F b P i2S di k F c<b C ; (b P i2S di k F c + 1)c F1 if P i2S d i k F b P i2S di k F cb C : Next we find an upper bound of ss usingZ (S) . By definition,Z (S)Z(S) for any suppliers set S. Lemma 5. Given a set of suppliersS such that (m1)k F < P i2S d i mk F , thenss(S) 2m1. Proof. When m = 1, the subset sum algorithm packs all the demands in one truck. So the claim holds. Next, when m 2, we prove this claim in two steps. First, we prove the subset sum algorithm uses no more than 2m 1 trucks. In any subset sum packing solution, the sum of the demands in any two trucks exceeds k F . Assume the subset sum algorithm uses t trucks. Then there are t(t1) 2 different pairs of trucks in the solution. Summing over all these pairs, we have more than t(t1) 2 k F units of demand packed in these t(t1) 2 pairs of trucks. Each truck participates in exactly t 1 pairs. As a result, the total demand packed in these t trucks is strictly greater than 74 t 2 k F . Since t 2 k F < P i2S d i mk F , we have t < 2m. Second, we prove that 2m 1 is a tight upper bound. WLOG, let P i2S d i = (m 1 2 )k F +, 1 2 k F . We construct a demand profile with 2m 1 suppliers, each with demand 1 2 k F + 2m1 . Since each supplier’s demand is strictly greater than 1 2 k F , the subset sum algorithm will pack each supplier’s demand in a separate truck and thus use 2m 1 trucks. Proposition 12. The worst-case cost ratio ss < 2. Proof. WLOG, assume we have a set of suppliersS such that (m 1)k F (m1)c F1 . If the packing solution yielded by the subset sum algorithm for the same demand profile uses at most 2m 2 trucks, then ss (S) = Zss(S) Z(S) (2m2)c F1 Z (S) < (2m2)c F1 (m1)c F1 = 2. If the packing solution uses 2m 1 trucks, we can conclude that P i2S d i > (m 1 2 )k F . This is because the first 2m 3 trucks must at least be half filledandthetotaldemandofthelasttwotrucksmustexceedk F . Thetotaldemandmustbestrictly greater than (2m3) 1 2 k F +k F = (m 1 2 )k F . As a result, the last truck in the consolidated packing solution must have more than 1 2 k F . If b C 1 2 k F , then Z (S) = mc F1 . If 1 2 k F (m 1)c F1 + 1 2 k F c L1 (m 1 2 )c F1 . Consequently, when the subset sum packing solution uses 2m 1 trucks, ss (S) = Zss(S) Z(S) (2m1)c F1 Z (S) < (2m1)c F1 (m 1 2 )c F1 = 2. To summarize, ss = sup S f ss (S)g< 2. Consequently, we can easily derive the following corollary based on Proposition 11. Corollary 7. The cost-sharing mechanism BBP is 2-budget-balanced. Although we have obtained an upper bound on for our cost-sharing mechanism, this bound is not necessarily tight. We use the lowest outbound shipping cost Z (S) to obtain this upper bound and sometimes this cost can be much lower than the optimal outbound shipping cost for our problem. We could possibly obtain better bounds on the value of by using the optimal outbound shipping cost that is induced by a packing solution, in which the suppliers’ demands are not split. We cannot easily find the minimum shipping cost in general, but we can restrict our attention to special cases for which we can determine the minimum shipping cost. For example, we can look at certain input demand profiles that produce structured subset sum packing solutions. Proposition 13. For a given suppliers set S, if the subset sum packing solution uses no more than two trucks, then ss (S) = 1. 75 Proof. If the subset sum packing solution uses one truck to pack all demands, then obviously ss (S) = 1. If the subset sum packing solution uses two trucks to pack all the demands, then it must be true that any other packing solution will use at least two trucks to pack all the demands. The actual shipping cost for an arbitrary packing solution is ( P i2S d i D a )c L1 , where D a is the total demand volume that exceeds b C in all trucks. Since all the demands can be packed into two trucks and the subset sum algorithm maximizes the total demand packed in the first truck, then the subset sum packing solution has the largestD a . Consequently, the subset sum packing solution yields the minimum shipping cost. Lemma 6. [22] If no three suppliers’ demands fits in a truck from suppliers set S then ss(S) = opt(S). Proposition 14. Given a set of suppliers S, if no three supplier’s demands fit in one truck, then the packing solution yielded by the subset sum algorithm induces the minimal shipping cost. Proof. We prove the above claim by proving that the packing solution yielded by the subset sum algorithm does not cost more than any other packing solution for the same set of suppliers. WLOG, let T ss 1 ;:::;T ss M denote the packing solution from the subset sum algorithm and T 1 ;:::;T K denote any packing solution that is different from T ss 1 ;:::;T ss m . For the sake of analysis, we order the packing solution T 1 ;:::;T K so that D(T 1 ) D(T 2 )::: D(T K ). Because of Lemma 6, M K. Recall that each T k , k2f1; 2;:::;Kg, in the solution set contains the suppliers’ indices whose demands are packed in the kth truck. We define T ss m = T k , m2f1; 2;:::;Mg, k2f1; 2;:::;Kg, only if the two trucks contain the same set of suppliers’ demands. In order, we compare T ss m with T m form2f1; 2;:::Mg. Letm M be the smallest index such thatT ss m =T m ,m2f1; 2:::m g. Therefore, up to them th truck,T ss 1 ;:::;T ss M andT 1 ;:::;T K have the same exact packing solution, and thus incur the same shipping cost. Since T ss m = T m , m2f1; 2:::m g, the packing solutions T ss m +1 ;:::;T ss M andT m +1 ;:::;T K containthedemandsofthesamesetofsuppliers, whosedemands are not packed in the first m trucks. Consequently, D(T ss m +1 ) D(T m +1 ). We consider the following four cases based on the relationships between D(T ss m +1 ), D(T m +1 ) and b C . Case 1: when D(T m +1 ) D(T ss m +1 ) b C , two packing solutions incur the same shipping cost because the demands in the remaining trucks are all shipped at the LTL rate. Case 2: whenD(T ss m +1 )>b C D(T m +1 ), the shipping cost incurred byT ss m +1 ;:::;T ss M is strictly lower because it ships at leastD(T ss m +1 ) at the flat FTL rate whileT m +1 ;:::;T K ships all demands at the LTL rate. 76 Case 3: when D(T ss m +1 ) = D(T m +1 ) > b C , we prove that we can have T ss m +1 = T m +1 while retaining the same shipping cost of both packing solutions. WhenT ss m +1 =figandT m +1 =fjg, buti6=j, theremustexistT m + , 2f2;:::Km 1g so that T m + =fig. Otherwise, if supplier i’s demand is packed with another supplier’s demand, this contradicts the fact that D(T ss m +1 ) D(T m +1 ). If we switch T m +1 and T m + , we have T ss m +1 =T m +1 . WhenT ss m +1 =fig andT m +1 =fj;ug, buti6=j6=u, there must existT m + , 2f2;:::K m 1g so thatT m + =fig for the same reason in the above argument. If we switchT m + with T m +1 , we have T ss m +1 = T m +1 . Similarly, when T ss m +1 =fj;ug and T m +1 =fig, we can have T ss m +1 =T m +1 as well. When T ss m +1 =fi;jg and T m +1 =fu;vg butfi;jg6=fu;vg, consider the following. If T ss m +1 andT m +1 do not share suppliers, i.e.fi;jg\fu;vg =;, we can always have a truck that contains the demands of supplieru andv in the subset sum algorithm solution without changing the shipping cost. If there existsT ss m + , 2f2;:::Mm 1g so thatT ss m + =fu;vg, then the above claim holds. If the demands of supplier u and v are not packed in the same truck, then either d u or d v should be packed in a truck whose total demand equals d u +d v , otherwise the demands of supplier u and v should be packed together by the subset sum algorithm to yield a truck with greater total demand. Assume d l is packed with d u in T ss m + , 2f2;:::Mm 1g such that d l +d u = d u +d v . Therefore, d v = d l . If we switch d v and d l , we obtain a T ss m + that contains the demands of supplieru andv and retain the shipping cost of the packing solution. By switching T ss m +1 and T ss m + , we have T ss m +1 =T m +1 . If T ss m +1 and T m +1 share one common supplier – WLOG, we assume i =u – then d j =d v . d j must be packed in one of the trucks in T m +2 ;:::;T K . If we switch d v and d j in T m +1 ;:::;T K , we have T ss m +1 =T m +1 . Note that all the swaps in Case 3 only change the ordering of trucks with equal demand volume or the packings of equal demands. The resulting packing solutions are essentially equivalent to T ss m +1 ;:::;T ss M and T m +1 ;:::;T K . Therefore, their shipping costs do not change. Case 4: when D(T ss m +1 )>D(T m +1 )>b C , we prove that we can always change T m +1 ;:::;T K to have T ss m +1 =T m +1 without increasing the shipping cost of T m +1 ;:::;T K . First, we prove when D(T ss m +1 )>D(T m +1 )>b C , there must be two suppliers’ demands in T ss m +1 . If T ss m +1 contains only one supplier’s demand d i , d i must be packed in one of T m +1 ;:::;T K . This contradicts 77 D(T ss m +1 ) > D(T m +1 ). Therefore, T ss m +1 must contain two suppliers’ demands d i and d j . Since D(T ss m +1 )>D(T m +1 ),d i andd j are not packed in the same truck inT m +1 ;:::;T K . Let’s assume d i is packed with d p and d j is packed with d q somewhere in the packing solution T m +1 ;:::;T K . Because d i and d j are packed in T ss m +1 , d i +d j > d i +d p and d i +d j > d j +d q . Thus, d j > d p and d i >d q . Now suppose we modify T m +1 ;:::;T K to pack d i and d j together in one truck and pack d p and d q together in another truck. Because d i +d j is the largest demand volume that can be packed in one truck, T m +1 =fi;jg. If d i +d p b C , d j +d q b C , and d p +d q b C , then packing d i and d j together in T m +1 ;:::;T K does not change the shipping cost of T m +1 ;:::;T K . If d i +d p b C and d j +d q d p . Therefore, the cost for shipping d i , d j , d p , d q decreases from c F1 + (d j +d q )c L1 to c F1 + (d p +d q )c L1 after packing d i and d j together. Similarly, if d i +d p < b C and d j +d q b C , the cost of shipping T m +1 ;:::;T K decreases as well after packing d i and d j together. If d i +d p <b C , d j +d q <b C and d p +d q <b C , the cost of shipping d i , d j , d p , d q decreases from (d i +d p +d j +d q )c L1 to c F1 + (d p +d q )c L1 after packingd i andd j together. Thus, having T ss m +1 =T m +1 by switching demands in T m +1 ;:::;T K does not increase the shipping cost of the packing solution. Finally, if d i or d j is packed in a truck alone in T m +1 ;:::;T K , the above conclusion also holds because it can be seen as a special case of the above situation where d p = 0 or d q = 0. Comparing T ss m + and T m + , 2 f2;:::M m 1g, if at any time, D(T ss m + ) and D(T m + ) satisfy the conditions in cases 1 and 2, we can conclude that the shipping cost of T ss 1 ;:::;T ss M is no more thanT 1 ;:::;T K . IfD(T ss m + ) andD(T m + ) satisfy the conditions in cases 3 and 4, we setm =m + 1 and repeat the above procedures again. If D(T ss m + ) andD(T m + ) always fall into cases 3 and 4, we will end up with T ss m = T m , m2f1;:::;Mg. Because we do not increase the shipping cost every time we perform the above procedures to change T 1 ;:::;T K toward T ss 1 ;:::;T ss M , we can conclude that T ss 1 ;:::;T ss M costs no more than T 1 ;:::;T K . With the result in Proposition 14, we can easily draw the conclusion in Corollary 8. Corollary 8. If no three suppliers’ demands fits in one truck in suppliers set S , then ss (S) = 1. As a summary of the above results, we present Proposition 15. Proposition 15. Cost-sharing mechanism BBP is budget-balanced for the demand profiles in which (1) no three suppliers’ demands fit in one truck or (2) the corresponding subset sum packing solutions use no more than two trucks. 78 We can also determine the minimum shipping cost when we have specific values ofb C . Recall the example that shows the optimal outbound shipping cost is not necessarily induced by the optimal bin packing solution for a given set of suppliers. If we change b C to 7 and c F1 to $7 in the above example, we can see now the shipping cost of the optimal bin packing solution is $47, which is smaller than that of the subset sum packing solution $54. This example seems to indicate that with a smaller b C the optimal bin packing solutions yield the minimum shipping cost. Next, we show a sufficient condition for the optimal bin packing solutions to yield the minimum shipping cost among all bin packing solutions under our trucking cost structure. Let B 1 ;:::B m denote a bin packing solution using m bins, D(B k ), k = 1;:::;m denote the total item size packed in B k andH denote the capacity of bins. For the sake of analysis, we define nontrivial bin packing solutions. Definition 3. A bin packing solution B 1 ;:::B m is nontrivial if there are at least m 1 bins half filled. The outcomes of bin packing algorithms are often nontrivial bin packing solutions. The optimal bin packing solutions must be nontrivial. If they are not, we can easily reduce the solution by one bin by simply combining the items in two bins that are both less than half filled, contradicting the fact that the packing solution is optimal. The subset sum packing solutions are also nontrivial. If there are two bins less than half filled, the subset sum algorithm should pack all the items in these two bins in one bin to maximize the total size instead of keeping them in separate bins. On the other hand, the packing solutions provided by the next-fit algorithm may not be nontrivial. The next-fit algorithm packs an item in the most recently opened bin if it fits, otherwise the item is packed in a newly opened bin and the previously opened bin is closed. Consider 3 ordered items l 1 < 1 2 H, l 2 > 1 2 H and l 3 < 1 2 H with l 1 +l 2 > H and l 2 +l 3 > H. According to the next-fit algorithm, each of the 3 items is packed in a separate bin. This is not a nontrivial bin packing solution since there are two bins less than half filled. However, by combining the bins that l 1 and l 3 are packed in, we are able to obtain a nontrivial bin packing solution. In fact, we can always obtain a nontrivial bin packing solutions by combining two less than half filled bins iteratively until there is at most one bin less than half filled. In addition, from the shipping cost perspective under our trucking cost structure, combining less than half filled bins never increases the shipping cost. Proposition 16. When b C 1 2 k F , the optimal outbound shipping cost Z(S) is induced by an optimal bin packing solution for suppliers set S. 79 Proof. Since the inbound shipping cost is fixed for a given set of suppliers, we only need to compare the differences in the outbound shipping cost among the packing solutions. Let opt(S) denote the number of trucks that the optimal bin packing solution uses for suppliers set S. Because optimal bin packing solutions are nontrivial bin packing solutions, based on Definition 3, there are at leastopt(S) 1 trucks in the optimal bin packing solutions half filled. Since b C 1 2 k F , we have (opt(S) 1)c F1 <Z(S)opt(S)c F1 . Other nontrivial bin packing solutions that are not optimal uses at least opt(S) + 1 bins and therefore, cost strictly more than opt(S)c F1 to ship all demands. As a result, the shipping cost induced by an optimal bin packing solution is the smallest among all packing solutions for suppliers set S. Note that the optimal bin packing solution for a suppliers set may not be unique. If there are multiple optimal bin packing solutions, the shipping cost we refer to as C(S) is always induced by the one whose least filled truck has the smallest total demand among all optimal bin packing solutions. As we see above, when b C 1 2 k F , the number of trucks used plays an important role in determining the total shipping costs. In order to study , we want to know how many more trucks the subset sum algorithm uses compared to the optimal bin packing solution for any set of suppliers. For the convenience of the analysis we define instance ratio R ss (S) for a given suppliers set S as the ratio between the number of trucks used by the subset sum algorithm ss(S) and the number of trucksusedbyanoptimalbinpackingsolutionopt(S), i.e. R ss (S) = ss(S) opt(S) . Theabsolute worst-case ratio is R ss = sup S fR ss (S)g. In order to find an upper bound for , we are interested in knowing the worst-case behavior of the subset sum algorithm. The absolute worst-case ratio is the greatest ratio between ss(S) and opt(S) for any suppliers set S. It helps us bound the number of trucks used by the subset sum algorithm from above using the optimal number of trucks. Then we can derive the worst-case cost ratio accordingly. To the best of our knowledge, the absolute worst-case ratio of the subset sum algorithm has been rarely studied. However, the subset sum algorithm can be seen as a refinement of the first-fit algorithm [54], whose absolute worst-case ratio has been thoroughly studied. If a set of items are given in the sequence of how they are packed by the subset sum algorithm, the first-fit algorithm provides the same packing solution as the one yielded by the subset sum algorithm. That is to say, for any set of items, there always exists an ordering of items such that the first-fit algorithm yields 80 the same packing solution as the subset sum algorithm. Therefore, the performance of the first-fit algorithm cannot be better than the subset sum algorithm, i.e. R ss R ff , whereR ff denotes the absolute worst-case ratio of the first-fit algorithm. The exact value of R ff has been proven to be 1.7 by Dósa and Sgall [33]. Therefore, R ss 1:7. With R ss 1:7, we can easily calculate the maximum possible number of trucks used by the subset sum algorithm when given the optimal number of trucks. For a given set of suppliers S, the possible number of trucks used by the subset sum algorithm ss(S) =fkjopt(S) k b1:7opt(S)c;k 2 Z + g. Next we analyze the worst-case cost ratio for suppliers set S for which opt(S) = 3; 4; 5; 6; 7; 8; 9. Proposition 17. For a given supplier setS, ifb C 1 2 k F and 3opt(S) 9, then ss (S)< 1:875. Proof. Assumeopt(S) = 3, thenss(S) =f3; 4; 5g. First we provess(S) 4. WLOG, letT opt 1 ;T opt 2 and T opt 3 denote the optimal packing solution ordered such that D(T opt 1 ) D(T opt 2 ) D(T opt 3 ) and let T ss 1 ;:::;T ss 5 denote the subset sum packing solution. Because of how the subset sum algorithm works,D(T opt 1 )D(T ss 1 ). Then it must be true thatD(T opt 2 )+D(T opt 3 ) P 5 i=2 D(T ss i ). Additionally, the total demand volume of any two trucks in a subset sum packing solution exceeds k F , e.g. D(T ss 2 ) +D(T ss 3 )>k F . We sum up all the inequalities that only include the last 4 trucks and obtain P 5 i=2 D(T ss i ) > 2k F . So it must be true that D(T opt 2 ) +D(T opt 3 ) > 2k F . This is a contradiction. Therefore, when opt(S) = 3, ss(S) 4. Next we prove ss (S) 3 2 . If opt(S) = ss(S) = 3 and b C 1 2 k F , then ss (S) = 2c F1 +Z(T ss 3 ) 2c F1 +Z(T opt 3 ) < 3c F1 2c F 1 = 3 2 . If opt(S) = 3 and ss(S) = 4, then using notation similar to above, we have D(T ss 2 ) +D(T ss 3 )>k F , D(T ss 3 ) +D(T ss 4 )>k F and D(T ss 2 )+D(T ss 4 )>k F . Addingthesethreeinequalities, weobtainD(T ss 2 )+D(T ss 3 )+D(T ss 4 )> 3 2 k F . Therefore, D(T opt 2 ) +D(T opt 3 ) > 3 2 k F . This indicates D(T opt 3 ) > 1 2 k F . As a result, if opt(S) = 3, ss(S) = 4 and b C 1 2 k F , then ss (S) = 3c F1 +Z(T ss 4 ) 3c F1 < 4c F1 3c F 1 = 4 3 . Finally, we can conclude that when opt(S) = 3, ss (S)< 3 2 . Applyingtheexactsametechniquewhen 4opt(S) 9, weareabletoobtainthecorresponding worst-case cost ratios that are summarized in Table 16: 81 Table 16: Worst-case cost ratio summary opt(S) ss(S) ss (S) 4 4; 5; 6 5 3 5 5; 6; 7; 8 7 4 6 6; 7; 8; 9; 10 9 5 7 7; 8; 9; 10; 11 11 6 8 8; 9; 10; 11; 12; 13 13 7 9 9; 10; 11; 12; 13; 14; 15 15 8 To summarize, when 3opt(S) 9 andb C 1 2 k F , ss (S) = maxf 3 2 ; 5 3 ; 7 4 ; 9 5 ; 11 6 ; 13 7 ; 15 8 g = 15 8 = 1:875. Proposition 18. For a given suppliers set S, if b C 1 2 k F and opt(S) 10, then ss (S) < b1:7opt(S)c opt(S)1 . Proof. Letss(S)bethepossiblemaximumb1:7opt(S)c. FollowingthesameargumentinProposition 17, ss (S) = (b1:7opt(S)c 1)c F1 +Z(T ss ss(S) ) (opt(S) 1)c F1 +Z(T opt opt(S) ) < b1:7opt(S)cc F1 (opt(S) 1)c F1 = b1:7opt(S)c opt(S) 1 Corollary 9. For any given suppliers set S, if b C 1 2 k F , then ss < 17 9 . Proof. Based on Proposition 18, ss (S)< b1:7opt(S)c opt(S)1 1:7opt(S) opt(S)1 . Note that, the value off(x) = x x1 decreases as x increases. Therefore, the maximum value of 1:7opt(S) opt(S)1 is obtained when opt(S) = 10: As a result, ss (S) < 17 9 1:89 when opt(S) 10. Combining this result in Proposition 17, we conclude ss < 17 9 . Proposition 19. When b C 1 2 k F , the cost-sharing mechanism BBP is 17 9 -budget-balanced. 5.4.2 Numerical Results on Budget-Balance Ratio Although we have proved that, generally, the cost-sharing mechanism BBP charges less than twice the minimum shipping cost, this upper bound on is not tight because the minimum shipping cost 82 used for comparison is induced by the packing solution, in which the suppliers’ demands are split and consolidated to fill a truck as much as possible. In order to reveal a more accurate picture of the budget-balance ratio of our cost-sharing mechanism, we study the ratio numerically. For each given demand profile, we obtain a budget-balance ratio by calculating the minimum shipping cost and the total shipping cost incurred by our cost-sharing mechanism. As we have mentioned before, the minimum shipping cost is not always induced by the optimal shipping solution, but this does not mean that we cannot obtain the minimum shipping cost numerically. The following proposition helps us find the minimum shipping cost for any demand profile using the first-fit algorithm. Proposition 20. For any given suppliers set S, the packing solution that induces the minimum shipping cost for set S can be obtained by applying the first-fit algorithm to a specific order of the demand profiles in S. Proof. Let T 1 ;:::;T k be the packing solution that induces the minimum shipping cost C(S). As- sume T m , m 2 f1;:::;kg are ordered such that D(T 1 ) ::: D(T k ). Because the inbound shipping cost is fixed for a given set of suppliers, we prove that the first-fit algorithm can be applied to obtain T 1 ;:::;T k for the outbound packing. First of all, the packing solution that induces the minimum shipping cost may not be unique. We prove that applying the first-fit algorithm on a specific order of demand profiles leads us to one such packing solution. Among all the packing solutions that induce the minimum shipping cost, there must be one T 1 ;:::;T k such that each demand in T m , m2f1;:::kg, cannot be moved to T m , 2f1;:::;m 1g without moving other packed demands. Otherwise, we can create a packing solution that induces less shipping cost. If we order the demand profiles of suppliers S in a sequence of how they are placed in T 1 ;:::;T k , the packing solution of the first-fit algorithm for this sequence of demand profiles is T 1 ;:::;T k by the definition of the first-fit algorithm. Because of Proposition 20, we can always obtain the minimum shipping cost for a supplier set by packing the demands using the first-fit algorithm on any possible ordering of the demand profiles. The number of possible ordering of demand profiles for suppliers set S isjSj! and the first- fit algorithm can be implemented inO(jSjlogjSj). Therefore, we can exactly compute the minimum shipping cost for moderatejSj. We compare the minimum shipping costs and the shipping costs incurred by the packing solu- tions yielded by the subset sum algorithm for the same set of demand profiles we used in Section 3.3.3 with 3, 6 and 10 suppliers. Each supplier has less than truckload demand. Because the value 83 ofb C also influences the minimum shipping cost, we run the experiments with threeb C values while settingb C =b G . Given the number of suppliers and the value of b C , we summarize the comparison results over 100 demand profiles by reporting the statistics of the budget-balance ratios, which we define as C M (S) C(S) for suppliers setS. The experimental results are presented in Tables 17, 18 and 19. Table 17: Budget-balance ratio for 3 suppliers demand profiles Max ratio Min ratio Avg. ratio # of same cost b C = 1 4 k F 1 1 1 100 b C = 1 2 k F 1 1 1 100 b C = 3 4 k F 1 1 1 100 Table 18: Budget-balance ratio for 6 suppliers demand profiles Max ratio Min ratio Avg. ratio # of same cost b C = 1 4 k F 1.0508 1 1.0005 99 b C = 1 2 k F 1.2060 1 1.0039 98 b C = 3 4 k F 1.0924 1 1.0020 94 Table 19: Budget-balance ratio for 10 suppliers demand profiles Max ratio Min ratio Avg. ratio # of same cost b C = 1 4 k F 1.1805 1 1.0118 93 b C = 1 2 k F 1.1771 1 1.0146 78 b C = 3 4 k F 1.0847 1 1.0101 62 In Tables 17, 18 and 19, we present the maximum, minimum and average budget-balance ratios among the 100 demand profiles results for each combination of numbers of suppliers and b C values. In addition, we also show the number of demand profiles, whose minimum shipping cost equals to the shipping cost of the packing solution yielded by the subset sum algorithm. From the results in Table 17, we empirically see that when there are three suppliers in the demand profiles, the subset sum packing solution always yields the minimum shipping cost. This aligns with our Proposition 13. If the demands of three suppliers fit in one or two trucks using the subset sum algorithm, then ss (S) = 1. If any two of their demands cannot be packed in one truck, then their demands will be packed in three trucks. This packing also yields the minimum shipping cost. Therefore, it is not surprising that the cost-sharing mechanism BBP is budget-balanced for 84 demand profiles with only three suppliers. In Tables 18 and 19, the maximum ratios are much smaller than the upper bounds we found for . Moreover, the minimum ratios are always 1. In fact, according to the tables, we see that the packing solutions yielded by the subset sum algorithm very often induce the minimum shipping costs, although such packing solutions may not be unique. Essentially, the minimum shipping cost should be induced by the packing solution that has the most units of demand exceeding b C in all trucks. In this regard, the subset sum algorithm greedily maximizes the units of demand that exceeds b C for one truck, iteratively. The packing solutions yielded by the subset sum algorithm induce the minimum shipping cost for the majority of demand profiles in our experiments. In addition, the average ratios are all bounded above by 1.015. This empirically demonstrates that our cost-sharing mechanism BBP has good budget-balance performance on average. It generally only charges slightly more than the minimum shipping cost. Finally, the # of same cost decreases as the value ofb C increases. This indicates that it is less likely for the subset sum packing solution to yield the minimum shipping cost when the value of b C increases. Additionally, comparing the # of same cost in Tables 18 and 19 for the sameb C values, we can conclude that it is also less likely to obtain the minimum shipping cost from the subset sum packing solution as the number of suppliers in demand profiles increases. These numerical results are a good complement to our theoretical results of budget-balance ratio, and contributes to a comprehensive analysis of the budget-balance ratio of our cost-sharing mechanism BBP. 5.5 Economic Efficiency of Cost-Sharing Mechanism BBP In this section, we study the economic efficiency of the cost-sharing mechanism BBP by comparing the social cost of the cost-sharing mechanism BBP to the minimum social cost of our problem. The social cost is still the total shipping cost of all suppliers. We minimize the social cost of our problem using an optimization model, in which each supplier ships its all demand either to the consolidation center or to the destination directly. Each supplier’s demand is delivered in its entirety, without splitting. The parameters, decision variables, and the model are presented below. Parameters: N: set of suppliers. T: set of trucks available at the consolidation center. G 0 i : Inbound shipping cost for the entire demand of supplier i8i2N. 85 G 1 i : Stand-alone shipping cost for the entire demand of supplier i8i2N. Decision variables: x j CF : Binary variable. If outbound shipping of truckj uses the FTL rate, thenx j CF = 1; otherwise, 08j2T. y j CF : Amount of demand of truck j shipped by the FTL rate from the consolidation center to the destination8j2T. y j CL : Amount of demand of truck j shipped by the LTL rate from the consolidation center to the destination8j2T. z ij : Binary variable. If the entire demand of supplier i is shipped using truck j, then z ij = 1; otherwise z ij = 0. Model: min X i2N X j2T z ij G 0 i + X i2N (1 X j2T z ij )G 1 i + X j2T (c F1 x j CF +c L1 y j CL ) (22) s.t. y j CF k F x j CF ; 8j2T (23) y j CL b C ; 8j2T (24) y j CF +y j CL k F ; 8j2T (25) X j2T z ij 1; 8i2N (26) X i2N z ij d i =y j CF +y j CL ; 8j2T (27) x j CF ; z ij 2f0; 1g; 8i2N;8j2T (28) y j CF 0; y j CL 0;8j2T (29) Constraints (23) and (24) ensure that trucks correctly incur the FTL rate or the LTL rate under our trucking cost structure, respectively. Constraint (25) ensures that the packed demand in each truck does not exceed k F . Constraint (26) allows each supplier’s demand to be packed in at most one truck at the consolidation center. Constraint (27) makes sure that the consolidation center ships all demands packed in each truck. Constraints (28) and (29) enforce the corresponding decision variables to be binary and nonnegative reals. For supplier i, if P j2T z ij = 0, then supplier 86 i ships all its demand directly. If P j2T z ij = 1, then supplier i’s demand is shipped using truck j at the consolidation center. Therefore, in the objective function (22), P i2N P j2T z ij G 0 i represents the total inbound shipping cost for the suppliers who ship their demands via the consolidation center, P i2N (1 P j2T z ij )G 1 i represents the total stand-alone cost for the suppliers who ship their demands directly to the destination, and P j2T (c F1 x j CF +c L1 y j CL ) is the total outbound shipping cost. Like our cost-sharing mechanism BBP, the optimization model (22) - (29) also decides which suppliers participate in consolidation and how their demands are packed in trucks. We compare the shipping decisions and social costs of our cost-sharing mechanism BBP to those of the optimization model (22) - (29) for the same set of demand profiles, first introduced in Chapter 3.3.3 for the study of economic efficiency for cost-sharing mechanism PEDS. The demand profiles have 3, 6, 10, and 15 suppliers, respectively. Each supplier has less than truckload demand. Using the same experimental scheme in Chapter 3.3.3, we fix the values of some parameters and study how different numbers of suppliers and different g L1 g L0 ratios influence the social cost of cost-sharing mechanism BBP compared to the minimum social cost. We use the same values for k F ,b C andb G as in Table 1 and set c L1 = g L1 = 1$/ft 3 . Our choices of g L1 g L0 are still 1.5, 2.4, 3.2, 4.8, 9 and 15. Other cost parameters are calculated accordingly for each selection of g L1 g L0 . We first present some summary statistics of the solutions from the optimization model and the cost-sharing mechanism BBP in Tables 20 and 21. In these tables, we summarize the number of solutions with total participation, zero participation and partial participation, the average partici- pation ratio, the average served demand ratio and the average number of consolidated trucks. We test 100 instances for each combination of number of suppliers and g L1 g L0 ratio. When g L1 g L0 = 1:5, solutions from the optimization model and the cost-sharing mechanism BBP are always zero par- ticipation. This demonstrates that consolidation is more likely to be beneficial for suppliers when the destination is far (long-haul transportation) compared with the location of the consolidation center. For the rest of the g L1 g L0 ratios, generally, the solutions of the optimization model result in more total and partial participation and fewer zero participation cases than those of the cost- sharing mechanism BBP. One way to explain this difference is that solutions to the optimization model only require that the total cost shares of participating suppliers to be less than their total stand-alone cost. It is possible that some participating suppliers benefit from the consolidation individually while others do not. However, in the solutions of the cost-sharing mechanism BBP, every participating supplier is required to benefit from or at least be indifferent to consolidation. 87 Due to this stricter requirement in the cost-sharing mechanism BBP, its solutions have lower av- erage participation ratio, lower average served demand ratio and thus, lower average number of consolidated trucks for the majority of cases. Nevertheless, this does not mean that shipping more demand via the consolidation center always leads to smaller social costs. For instance, the cost-sharing mechanism BBP solution for 6 suppliers at g L1 g L0 = 9 and 15 result in more total par- ticipation than the social-cost-minimizing solutions. In one of the demand profiles for which the cost-sharing mechanism BBP and the optimization model yield different solutions, suppliers have 2:27; 2748:25; 1188:12; 393:61; 385:39; and 1533:93 ft 3 of demands, respectively. When g L1 g L0 = 9, the cost-sharing mechanism BBP yields total participation which packsf2:27; 2748:25; 1188:12g and f393:61; 385:39; 1533:93g in two trucks and the optimization model yields partial participation that packsf2:27; 1188:12; 393:61; 385:39; 1533:93g in one truck and ships 2748:25 ft 3 demand directly. Both shipping solutions ship demands at the FTL rate but shipping 2748:25 ft 3 demand directly saves the $2000 inbound shipping cost since b C =b G and g L1 =c L1 . In addition, as g L1 g L0 increases, the average participation ratios increase in both solutions. This is because as the destination be- comes farther away, the increasing savings via consolidation can offset the inbound shipping cost and enable more suppliers to benefit from consolidation. We present the number of instances for which the optimization model and the cost-sharing mechanism BBP result in the same solution and the average social cost gap in Table 22. For each g L1 g L0 ratio, thefirstcolumnshowsthenumberofsamesolutionsandsecondcolumnshowstheaverage social cost gap. Based on the results in Tables 20 and 21, it is not surprising that the cost-sharing mechanism BBP is economically efficient when g L1 g L0 = 1:5. In terms of number of same solutions, roughly speaking, it is less likely for the cost-sharing mechanism BBP to yield the social cost minimizing solution as the number of suppliers and the g L1 g L0 ratio increase. In particular, the number of same solutions decreases more quickly as the number of suppliers increases because the feasible packing solutions grows exponentially. In terms of social cost gap, all gaps presented in Table 22 are smaller than 3.8%. One interesting phenomenon of the average social cost gap is that the values are not monotonic in the number of suppliers or the g L1 g L0 ratios. For the demand profiles with the same number of suppliers, we find that in some cases different solutions for two “consecutive” g L1 g L0 ratios (e.g. 1.5 and 2.4, 2.4 and 3.2) come from completely two different sets of demand profiles. That is, it is not always true for a demand profile that as g L1 g L0 increases, zero participation becomes partial 88 Table 20: Summary of results from the optimization model for unsplittable demand 3 suppliers g L1 g L0 1:5 2:4 3:2 4:8 9 15 Num. of total par. 0 6 10 12 14 14 Num. of zero par. 100 90 72 55 46 38 Num. of partial par. 0 4 18 33 40 48 Avg. par. rate 0 0.0867 0.2200 0.3400 0.4067 0.4600 Avg. served demand 0 0.0861 0.2082 0.3071 0.3612 0.4108 Avg. consolidated trucks 0 0.1 0.28 0.45 0.54 0.62 6 suppliers g L1 g L0 1:5 2:4 3:2 4:8 9 15 Num. of total par. 0 0 2 2 4 7 Num. of zero par. 100 42 23 12 8 6 Num. of partial par. 0 58 75 86 88 87 Avg. par. rate 0 0.3167 0.4283 0.5050 0.5467 0.5700 Avg. served demand 0 0.2283 0.3141 0.3745 0.4140 0.4350 Avg. consolidated trucks 0 0.6 0.89 1.11 1.24 1.31 10 suppliers g L1 g L0 1:5 2:4 3:2 4:8 9 15 Num. of total par. 0 0 0 0 1 1 Num. of zero par. 100 6 2 0 0 0 Num. of partial par. 0 94 98 100 99 99 Avg. par. rate 0 0.4110 0.5390 0.6020 0.6410 0.6450 Avg. served demand 0 0.2682 0.3865 0.4472 0.4925 0.4999 Avg. consolidated trucks 0 1.29 1.91 2.24 2.47 2.51 15 suppliers g L1 g L0 1:5 2:4 3:2 4:8 9 15 Num. of total par. 0 0 0 0 1 1 Num. of zero par. 100 2 0 0 0 0 Num. of partial par. 0 98 100 100 99 99 Avg. par. rate 0 0.4707 0.5800 0.6340 0.6600 0.6666 Avg. served demand 0 0.2983 0.4025 0.4626 0.4936 0.4996 Avg. consolidated trucks 0 2.17 2.99 3.46 3.70 3.75 89 Table 21: Summary of results from the cost-sharing mechanism BBP 3 suppliers g L1 g L0 1:5 2:4 3:2 4:8 9 15 Num. of total par. 0 4 10 11 14 14 Num. of zero par. 100 93 77 60 48 41 Num. of partial par. 0 3 13 29 38 45 Avg. par. rate 0 0.0600 0.1867 0.3033 0.3933 0.4400 Avg. served demand 0 0.0599 0.1779 0.2739 0.3540 0.3912 Avg. consolidated trucks 0 0.07 0.23 0.40 0.52 0.59 6 suppliers g L1 g L0 1:5 2:4 3:2 4:8 9 15 Num. of total par. 0 0 2 3 8 14 Num. of zero par. 100 52 30 15 9 6 Num. of partial par. 0 48 68 82 83 80 Avg. par. rate 0 0.2650 0.3850 0.4767 0.5467 0.5867 Avg. served demand 0 0.1925 0.2850 0.3595 0.4339 0.4793 Avg. consolidated trucks 0 0.49 0.79 1.07 1.30 1.44 10 suppliers g L1 g L0 1:5 2:4 3:2 4:8 9 15 Num. of total par. 0 0 0 1 2 2 Num. of zero par. 100 12 3 0 0 0 Num. of partial par. 0 88 97 99 98 98 Avg. par. rate 0 0.3610 0.4920 0.5740 0.6220 0.6430 Avg. served demand 0 0.2358 0.3473 0.4213 0.4811 0.5037 Avg. consolidated trucks 0 1.13 1.72 2.12 2.42 2.55 15 suppliers g L1 g L0 1:5 2:4 3:2 4:8 9 15 Num. of total par. 0 0 0 0 1 1 Num. of zero par. 100 3 0 0 0 0 Num. of partial par. 0 97 100 100 99 99 Avg. par. rate 0 0.4220 0.5533 0.6020 0.6353 0.6473 Avg. served demand 0 0.2599 0.3719 0.4270 0.4692 0.4859 Avg. consolidated trucks 0 1.88 2.79 3.22 3.54 3.66 90 Table 22: Comparison of social cost gaps for cost-sharing mechanism BBP g L1 g L0 1:5 2:4 3:2 4:8 9 15 3 suppliers 100 0% 97 0.72% 95 2.33% 95 1.97% 98 1.66% 97 0.33% 6 suppliers 100 0% 88 0.70% 86 1.51% 81 3.22% 80 3.79% 79 3.29% 10 suppliers 100 0% 72 0.54% 65 1.10% 64 2.19% 58 2.97% 55 3.17% 15 suppliers 100 0% 50 0.54% 47 1.27% 43 2.08% 45 3.44% 38 3.62% participation or partial participation becomes total participation. Moreover, as we have pointed out earlier, even if participation rates increase as g L1 g L0 increases for the same demand profile, the social cost changes may not be monotonic. A possible contributing factor of this phenomenon is that packing the demands in trucks is essentially a combinatorial problem, whose solution largely depends on the specific composition of the demand profiles rather than the number of suppliers. As a result, the changes in social cost gap may not be aligned with changes of the g L1 g L0 ratios and the number of suppliers. Although the social cost gap changes are not strictly monotonic, we can still expect that the social cost gap generally increases as the number of suppliers or the g L1 g L0 increases. 91 6 Conclusions and Future Directions 6.1 Conclusions In this dissertation, we study the cost-sharing problem in a freight consolidation system with one consolidation center. Self-interested suppliers in the same region have the option to use this nearby consolidation center to ship their demands to a common destination by trucks for lower transporta- tion rates. We design a Moulin mechanism and an acyclic mechanism to solve this shipping cost allocation problem for freight consolidation, respectively. At the planning phase of each consolida- tion, the central planner of the consolidation collects bids from all the suppliers. Then applying a cost-sharing mechanism, the central planner decides the set of suppliers who participate in the consolidation and their corresponding cost shares based on the bids. We first study a cost-sharing problem in which the demands of the suppliers are packed aggre- gately to fill a truck as much as possible at the consolidation center. The resulting nonconvex and nonconcave outbound shipping cost makes it difficult to develop a truthful and budget-balanced Moulin mechanism. Indeed, our numerical experiments show that it is generally not possible for us to have a both cross-monotonic and budget-balanced cost-sharing method, which would lead to a truthful and budget-balanced Moulin mechanism. As a result, we approach the problem by approximating the outbound shipping cost function with piecewise linear concave functions. In order to encourage the participation of the large suppliers – suppliers who have large enough demand to ship with the FTL rate – we share the approximate outbound shipping cost proportional to each supplier’s effective demand for sharing, which discounts the part of demand that exceeds the estimated suppliers’ FTL equivalent volume. The proposed cost-sharing mechanism PEDS applies this cost-sharing method proportional to effective demand for sharing (PEDS). We provide the conditionsunderwhichthecost-sharingmechanismPEDSisgroupstrategyproofandapproximately budget-balanced with different guarantees. We found that in order to retain truthfulness of the mechanism, there exists a trade-off between the consolidation center’s benefit and the suppliers’ cost savings in the choice of and . In particular, as the consolidation center recovers more shipping cost by the cost shares, the maximum discount it can provide to the large suppliers decreases; when the consolidation center wants to offer more incentives to the large suppliers, the total cost shares recover less shipping cost. Thus the values of and should be determined based on the specific goal of the consolidation center. We study the economic efficiency of our cost-sharing mechanism 92 PEDS computationally using social cost as the measure. Our experimental results indicate that social cost gaps decrease as more suppliers bid for the service and the destination is farther away. We also study the above cost-sharing problem in the single truck case, namely the total demand of all bidding suppliers fits into one truck. We show that the proposed cost-sharing mechanism PEDS is both truthful and budget-balanced for this single truck problem. In addition, we prove that when the demand profiles satisfy certain conditions, the solutions of the mechanism PEDS are also economically efficient. For the demand profiles that do not satisfy the conditions, we con- duct extensive experiments to study their solutions’ economic efficiency. The experimental results demonstrate that the outcomes of mechanism PEDS are economically efficient for the majority of these demand profiles. A more accurate estimation of the suppliers’ FTL equivalent volume helps reduce the social cost gap when the mechanism outcome is not economically efficient. We then study a cost-sharing problem with a practical constraint that each supplier’s entire less- than-truckload demand cannot be split and must be shipped in one truck. In this problem setting, a critical problem we need to solve first is how to pack a set of suppliers’ less-than-truckload demands since the packing solution directly influences the outbound shipping cost share of each supplier. We formulate the above problem as a bin packing problem and obtain the packing solution using a subset sum algorithm. Based on the obtained packing solution, we derive the cost-sharing method and the offer function that induce our acyclic mechanism – cost-sharing mechanism Based on Bin Packing (BBP). Our cost-sharing mechanism BBP is weakly group strategyproof and it is generally not a Moulin mechanism. Besides, we find that the packing solutions yielded by the subset sum algorithm are strong Nash equilibria from a non-cooperative game theory perspective. This outcome endorses our use of the subset sum algorithm because every selected supplier should be satisfied with the packing of its demand. To study the behavior of the cost-sharing mechanism BBP, we investigate and discuss the conditions under which the mechanism BBP yields zero or total participation for two kinds of demand profiles: 1) all suppliers ship directly using the LTL rate, 2) all suppliers have more than half truckload demands. The budget-balance ratio of the cost-sharing mechanism BBP is first studied theoretically. We prove that our cost-sharing mechanism BBP is generally 2-budget-balanced. However, the mech- anism BBP is budget-balanced when the demand profiles satisfy either of the two following con- ditions: 1) the corresponding subset sum algorithm uses no more than two trucks, or 2) no three suppliers’ demands can fit in one truck. Additionally, when b C 1 2 k F , the cost-sharing mechanism 93 BBP is 17 9 -budget-balanced. To accomplish a comprehensive analysis of its budget-balance ratio, we then investigate the ratio empirically. On average, the budget-balance ratio is only slightly above 1. This demonstrates that our cost-sharing mechanism BBP generally has good budget-balance performance. In fact, our cost-sharing mechanism BBP is practically “budget-balanced” because it recovers the total outbound shipping cost by the cost shares, though it is not theoretically budget- balanced. The economic efficiency of the mechanism BBP is studied numerically. Although the changes in social cost gap are not aligned with changes of the g L1 g L0 ratios and the number of sup- pliers, the social cost gap can be expected to increase as the g L1 g L0 ratio and the number of suppliers increase. 6.2 Future Directions In this dissertation, the variations among suppliers are only due to demand volumes. But in a real competitive market, variations can be caused by other factors. For instance, different distances to the consolidation center / destinations may make suppliers with the same demand volume pay different inbound / stand-alone costs. In addition, large-size companies are more likely to get better transportation rates when compared to the small to medium-size companies because they have more negotiation power. To what extent these variations affect the cost-sharing mechanism design remains unknown. A comparative study of cost-sharing mechanism design with different levels of variations in supplier costs can help us understand how variations influence the outcome of the cost-sharing mechanism and how we can make our cost-sharing mechanism more robust without significantly compromising the performance with respect to budget-balancedness and economic efficiency. Asadifferentapproachfromourdissertation, ablack-boxreductionofapproximationalgorithms can also be applied to obtain cost-sharing mechanisms based on the work from Georgiou and Swamy [38]. They showed that approximation algorithms for the social-cost-minimization problem can be transformed into a truthful, approximation and cost-recovering mechanisms via a generic technique – black-box reduction. Instead of focusing on truthfulness and budget-balance first, like what we do in our dissertation, they primarily pay attention to solving the social-cost-minimization problem well to optimize economic efficiency and then convert the corresponding algorithm to induce the truthful and cost-recovering mechanism. This approach may be preferred when the common service is subsidized by the government who generally wants to maximize the social welfare. 94 References [1] Walid M Abdelwahab and Michel Sargious. Freight rate structure and optimal shipment size in freight transportation. Logistics and Transportation Review, 26(3):271–292, 1990. [2] Richa Agarwal and Özlem Ergun. Mechanism design for a multicommodity flow game in service network alliances. Operations Research Letters, 36(5):520–524, 2008. [3] Richa Agarwal, Özlem Ergun, Houghtalen Lori, and Okan Orsan Ozener. Collaboration in cargo transportation. Springer, 2009. [4] U.S. Environmental Protection Agency. Sources of greenhouse gas emissions. 2017. https://www.epa.gov/ghgemissions/sources-greenhouse-gas-emissions Accessed 11-19-2017. [5] María Angélica Arbeláez, Marcela Meléndez, and Nicolás León. The emergence of new successful export activities in Colombia. Technical report, Fedesarrollo, 2007. [6] Aaron Archer, Joan Feigenbaum, Arvind Krishnamurthy, Rahul Sami, and Scott Shenker. Approxi- mation and collusion in multicast cost sharing. Games and Economic Behavior, 47(1):36–71, 2004. [7] J-F Audy, S D’Amours, and L-M Rousseau. Cost allocation in the establishment of a collabora- tive transportation agreement – an application in the furniture industry. Journal of the Operational Research Society, 62(6):960–970, 2011. [8] Jean François Audy, Sophie D’Amours, and Mikael Rönnqvist. An empirical study on coalition for- mation and cost/savings allocation. International Journal of Production Economics, 136(1):13–27, 2012. [9] Robert J Aumarm. Acceptable points in general cooperative n-person games. Contributions to the Theory of Games (AM-40), 4:287, 2016. [10] SindhuraBalireddiandNelsonA.Uhan. Cost-sharingmechanismsforschedulingundergeneraldemand settings. European Journal of Operational Research, 217(2):270–277, 2012. [11] Vittorio Bilò. On the packing of selfish items. 20th International Parallel and Distributed Processing Symposium, IPDPS, 2006. [12] Yvonne Bleischwitz and Burkhard Monien. Fair cost-sharing methods for scheduling jobs on parallel machines. Journal of Discrete Algorithms, 7(3):280–290, 2009. [13] D.E. Blumenfeld, L.D. Burns, J.D. Diltz, and C.F. Daganzo. Analysing trade-off between transporta- tion, inventory and production cost on freight networks. Transportation Research, 19(5):361–380, 1985. [14] J. Bookbinder and C. Barkhouse. An information system for simultaneous consolidation of inbound and outbound shipments. Transportation Journal, 32(4):5–21, 1993. 95 [15] James H. Bookbinder and James K. Higginson. Probabilistic modeling of freight consolidation by private carriage. Transportation Research Part E: Logistics and Transportation Review, 38(5):305–318, 2002. [16] Joseph James Brennan. Models and analysis of temporal consolidation. Dissertation Abstracts Inter- national Part B: Science and Engineering, 42(1):1981, 1981. [17] Janina Brenner and Guido Sch. Cost Sharing Methods for Makespan and Completion Time Scheduling. In Proceedings of Symposium on Theoretical Aspects of Computer Science, pages 670–681. Springer, 2007. [18] Janina Brenner and Guido Schäfer. Singleton acyclic mechanisms and their applications to scheduling problems. In SAGT, pages 315–326. Springer, 2008. [19] Lawrence D. Burns, Randolph W. Hall, Dennis E. Blumenfeld, and Carlos F. Daganzo. Distribution strategies that minimize transportation and inventory costs. Operations Research, 33(3):469–490, 1985. [20] Qishu Cai, Qi-Ming He, and James H Bookbinder. A tree-structured markovian model of the shipment consolidation process. Stochastic Models, 30(4):521–553, 2014. [21] James F. Campbell. Freight consolidation and routing with transportation economies of scale. Trans- portation Research Part B: Methodological, 24(5):345–361, 1990. [22] Alberto Caprara and Ulrich Pferschy. Worst-case analysis of the subset sum algorithm for bin packing. Operations Research Letters, 32(2):159–166, 2004. [23] David P. Carlisle, Kenneth S. Nickerson, Stephen B. Probst, Denise Rudolph, Yosef Sheffi, and War- ren B. Powell. Turnkey, microcomputer-based logistics planning system. Interfaces, 17(4):16–26, 1987. [24] SilaÇetinkayaandJamesH.Bookbinder. Stochasticmodelsforthedispatchofconsolidatedshipments. Transportation Research Part B: Methodological, 37(8):747–768, 2003. [25] Sila Çetinkaya and Chung-Yee Lee. Stock replenishment and shipment scheduling for vendor-managed inventory systems. Management Science, 46(2):217–232, 2000. [26] Sila Çetinkaya, Fatih Mutlu, and Chung Yee Lee. A comparison of outbound dispatch policies for inte- gratedinventoryandtransportationdecisions. EuropeanJournalofOperationalResearch, 171(3):1094– 1112, 2006. [27] Edward H. Clarke. Multipart pricing of public goods. Public Choice, 11:17–33, 1971. [28] Frans Cruijssen, Peter Borm, Hein Fleuren, and Herbert Hamers. Supplier-initiated outsourcing: A methodology to exploit synergy in transportation. European Journal of Operational Research, 207(2):763–774, 2010. 96 [29] Frans Cruijssen, Martine Cools, and Wout Dullaert. Horizontal cooperation in logistics: Opportunities and impediments. Transportation Research Part E: Logistics and Transportation Review, 43(2):129– 142, 2007. [30] Carlos F Daganzo. A comparison of in-vehicle and out-of-vehicle freight consolidation strategies. Transportation Research Part B: Methodological, 22B(3):173–180, 1988. [31] Carlos F. Daganzo. Shipment composition enhancement at a consolidation center. Transportation Research Part B, 22(2):103–124, 1988. [32] Nikhil R. Devanur, Milena Mihail, and Vijay V. Vazirani. Strategyproof cost-sharing mechanisms for set cover and facility location games. Decision Support Systems, 39(1):11–22, 2005. [33] György Dósa and Jiří Sgall. First Fit bin packing: A tight analysis. Symposium on Theoretical Aspects of Computer Science, i:1–15, 2013. [34] Leah Epstein and Elena Kleiman. Selfish bin packing. Algorithmica, 60(2):368–394, 2011. [35] Joan Feigenbaum, Arvind Krishnamurthy, Rahul Sami, and Scott Shenker. Hardness results for mul- ticast cost sharing. Theoretical Computer Science, 304(1-3):215–236, 2003. [36] M. Frisk, M. Göthe-Lundgren, K. Jörnsten, and M. Rönnqvist. Cost allocation in collaborative forest transportation. European Journal of Operational Research, 205(2):448–458, 2010. [37] Masabumi Furuhata, Kenny Daniel, Sven Koenig, Fernando Ordóñez, Maged Dessouky, Marc Eti- enne Brunet, Liron Cohen, and Xiaoqing Wang. Online cost-Sharing mechanism design for demand- Responsive transport systems. IEEE Transactions on Intelligent Transportation Systems, 16(2):692– 707, 2015. [38] Konstantinos Georgiou and Chaitanya Swamy. Black-box reductions for cost-sharing mechanism de- sign. Games and Economic Behavior, 2013. [39] Jerry Green, Elon Kohlberg, and Jean Jacques Laffont. Partial equilibrium approach to the free-rider problem. Journal of Public Economics, 6(4):375–394, 1976. [40] Theodore Groves. Incentives in teams. Econometrica: Journal of the Econometric Society, pages 617–631, 1973. [41] Anupam Gupta, Jochen Könemann, Stefano Leonardi, R Ravi, and Guido Schäfer. An efficient cost- sharing mechanism for the prize-collecting Steiner forest problem. Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms, pages 1153–1162, 2007. [42] Anupam Gupta, Aravind Srinivasan, and Eva Tardos. Cost Sharing Mechanisms for Network Design. InApproximation, Randomization, andCombinatorialOptimization.AlgorithmsandTechniques, pages 139–150. 2004. 97 [43] JatinderN.D.GuptaandJohnnyC.Ho. Anewheuristicalgorithmfortheone-dimensionalbin-packing problem. Production Planning and Control, 10(6):598–603, 1999. [44] Yash P Gupta and Prabir K Bagchi. Inbound freight consolidation under just-in-time procurement: application of clearing models. Journal of Business Logistics, 8(2):74–94, 1987. [45] Randolph W Hall. Consolidation strategy: inventory, vehicles and terminals. Journal of Business Logistics, 8(2):57–73, 1987. [46] James K Higginson and James H Bookbinder. Policy recommendations for a shipment consolidation program. Journal of Business Logistics, 15(1):86–112, 1994. [47] JamesK.HigginsonandJamesH.Bookbinder. Markoviandecisionprocessesinshipmentconsolidation. Transportation Science, 29(3):242–255, 1995. [48] James K Higginson and James K Higginson. Recurrent decision approaches to shipment-release tim- ing in freight consolidation. International Journal of Physical Distribution & Logistics Management, 25(5):3–23, 1995. [49] Lori Houghtalen, Özlem Ergun, and Joel Sokol. Designing mechanisms for the management of carrier alliances. Transportation Science, 45(4):465–482, 2011. [50] George C. Jackson. A survey of freight consolidation practices. Journal of Business Logistics, 6(1):13– 31, 1985. [51] Kamal Jain, Mohammad Mahdian, Evangelos Markakis, Amin Saberi, and Vijay V Vazirani. Greedy facility location algorithms analyzed using dual fitting with factor-revealing lp. Journal of the ACM (JACM), 50(6):795–824, 2003. [52] Kamal Jain and Vijay Vazirani. Applications of approximation algorithms to cooperative games. In Proceedings of the thirty-third annual ACM symposium on Theory of computing, pages 364–372. ACM, 2001. [53] Kamal Jain and Vijay V Vazirani. Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and lagrangian relaxation. Journal of the ACM (JACM), 48(2):274–296, 2001. [54] D.S. Johnson, A. Demers, J.D. Ullman, M.R. Garey, and R.L. Graham. Worst-case Performance Bounds for Simple One-dimensional Packing Algorithms. SIAM J. Comput., 3(4):299–325, 1974. [55] JochenKönemann, StefanoLeonardi, andGuidoSchäfer. Agroup-strategyproofmechanismforSteiner forests. In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 612– 619. 2005. 98 [56] Yiannis A. Koskosidis and Warren B. Powell. Clustering algorithms for consolidation of customer orders into vehicle shipments. Transportation Research Part B: Methodological, 26(5):365–379, 1992. [57] M. A. Krajewska, H. Kopfer, G. Laporte, S. Ropke, and G. Zaccour. Horizontal cooperation among freight carriers: request allocation and profit sharing. Journal of the Operational Research Society, 59(11):1483–1491, 2007. [58] Chung-Yee Lee, Sila Eetinkaya, Wikrom Jaruphongsa, Sila Çetinkaya, and Wikrom Jaruphongsa. A dynamic model for inventory lot sizing and outbound shipment scheduling at a third-party warehouse. Operations Research, 51(5):735–747, 2003. [59] Stefano Leonardi and Guido Schäfer. Cross-monotonic cost sharing methods for connected facility location games. Theoretical Computer Science, 326(1-3):431–442, 2004. [60] Jun Li, Xiaoqiang Cai, and Yinlian Zeng. Cost allocation for less-than-truckload collaboration among perishable product retailers. OR spectrum, 38(1):81–117, 2016. [61] Peng Liu. Allocating collaborative profit in less-than-truckload carrier alliance. Journal of Service Science and Management, 03(01):143–149, 2010. [62] Edoardo Marcucci and Romeo Danielis. The potential demand for a urban freight consolidation centre. Transportation, 35(2):269–284, 2008. [63] Johan Marklund. Inventory control in divergent supply chains with time-based dispatching and ship- ment consolidation. Naval Research Logistics, 58(1):59–71, 2011. [64] Aranyak Mehta, Tim Roughgarden, and Mukund Sundararajan. Beyond Moulin mechanisms. Games and Economic Behavior, 67(1):125–155, 2009. [65] Ludovic Mènèdéme. Successfully prepare you supply chain for horizontal collaboration. Presentation. 2nd Horizonal Collaboration in the Supply Chain Summit, Brussels, 2011. [66] RodrigoMesa-ArangoandSatishVUkkusuri. Benefitsofin-vehicleconsolidationinlessthantruckload freight transportation operations. Procedia-Social and Behavioral Sciences, 80:576–590, 2013. [67] H. Min and M Cooper. A comparative review of analytical studies on freight consolidation and back- hauling. Logistics and Transportation Review, 26(2):149–169, 1990. [68] Benoit Montreuil. Toward a physical internet: meeting the global logistics sustainability grand chal- lenge. Logistics Research, 3(2-3):71–87, 2011. [69] Hervé Moulin. Incremental cost sharing: Characterization by coalition strategy-proofness. Social Choice and Welfare, 16(2):279–320, 1999. [70] Hervé Moulin and Scott Shenker. Strategyproof sharing of submodular costs: budget balance versus efficiency. Economic Theory, 533:511–533, 2001. 99 [71] Christine Nguyen, Maged Dessouky, and Alejandro Toriello. Consolidation strategies for the delivery of perishable products. Transportation Research Part E: Logistics and Transportation Review, 69:108– 121, 2014. [72] Christine Nguyen, Alejandro Toriello, Maged Dessouky, and James E. Moore. Evaluation of trans- portation practices in the California cut flower industry. Interfaces, 43(2):182–193, 2013. [73] Curt Nowak, Felix Hahne, and Klaus Ambrosi. A new approach to freight consolidation for a real-world pickup-and-delivery problem. In Operations Research Proceedings 2014, pages 429–435. Springer, 2016. [74] Bureau of Transportation Statistics. National transportation statistics 2017. Technical report, U.S. Department of Transportation, 2017. [75] Guillermo Owen. On the core of linear production games. Mathematical Programming, 9(1):358–370, 1975. [76] Okan Örsan Özener and Özlem Ergun. Allocating costs in a collaborative transportation procurement network. Transportation Science, 42(2):146–165, 2008. [77] Martin Pál and Éva Tardos. Group Strategyproof Mechanisms via Primal-Dual Algorithms. In Foun- dations of Computer Science, 2003. Proceedings. 44th Annual IEEE Symposium on, pages 584–593. 2003. [78] Shenle Pan, Eric Ballot, and Frédéric Fontane. The reduction of greenhouse gas emissions from freight transportbymergingsupplychains. InInternationalConferenceonIndustrialEngineeringandSystems Management. IESM, volume 5, pages 13–15, 2009. [79] David Pisinger and Paolo Toth. Knapsack problems. In Ding-Zhu Du and Panos M. Pardalos, editors, Handbook of Combinatorial Optimization, pages 299–428. Kluwer Academic, 1998. [80] Kevin Roberts. The characterization of implementable choice rules. In Aggregation and revelation of preferences, volume 12, pages 321–349. 1979. [81] Tim Roughgarden and Mukund Sundararajan. Quantifying inefficiency in cost-sharing mechanisms. Journal of the ACM, 56(4):1–33, 2009. [82] D. Russell, J.J. Coyle, K. Ruamsook, and E.A. Thomchick. The real impact of high transportation costs. CSCMP’s Supply Chain Quarterly, Quarter 1, 2014. http://www.supplychainquarterly.com/topics/Logistics/20140311-the-real-impact-of-high- transportation-costs/ Accessed 01-26-2016. [83] J. Sánchez-Soriano, M.A. Lopez, and I. García-Jurado. On the core of transportation games. Mathe- matical Social Sciences, 41(2):215–225, 2001. 100 [84] David Schmeidler. The nucleolus of a characteristic function game. SIAM Journal on applied mathe- matics, 17(6):1163–1170, 1969. [85] Lloyd S Shapley. A value for n-person games. Contributions to the Theory of Games, 2(28):307–317, 1953. [86] Philip D Straffin and James P Heaney. Game theory and the tennessee valley authority. International Journal of Game Theory, 10(1):35–43, 1981. [87] Douglas J Thomas and Paul M Griffin. Coordinated supply chain management. European Journal of Operational Research, 94(1):1–15, 1996. [88] Stef H Tijs and Theo SH Driessen. Game theory and cost allocation problems. Management Science, 32(8):1015–1028, 1986. [89] Jonah C. Tyan, Fu Kwun Wang, and Timon C. Du. An evaluation of freight consolidation policies in global third party logistics. Omega, 31(1):55–62, 2003. [90] M Ali Ülkü. Dare to care: Shipment consolidation reduces not only costs, but also environmental damage. International Journal of Production Economics, 139(2):438–446, 2012. [91] A. Van Breedam, B. Vannieuwenhuyse, and S. Verstrepen. The world’s first showcase of true horizontal collaboration between two world-class pharmaceutical companies: Brand new case study. Presentation. 2nd Horizonal Collaboration in the Supply Chain Summit, Brussels, 2011. [92] Christine Vanovermeire, Kenneth Sörensen, Alex Van Breedam, Bart Vannieuwenhuyse, and Sven Verstrepen. Horizontal logistics collaboration: decreasing costs through flexibility and an adequate cost allocation strategy. International Journal of Logistics Research and Applications, 17(4):339–355, 2014. [93] William Vickrey. Counterspeculation, auctions, and competitive sealed tenders. The Journal of Fi- nance, 16:8–37, 1961. [94] Dachuan Xu and Ruichun Yang. A cost-sharing method for an economic lot-sizing game. Operations Research Letters, 37(2):107–110, 2009. [95] Guanghui Zhou, Yer Van Hui, and Liang Liang. Strategic alliance in freight consolidation. Transporta- tion Research Part E: Logistics and Transportation Review, 47(1):18–29, 2011. 101

Abstract (if available)

Abstract Freight consolidation is a logistics practice that improves cost-effectiveness of operations, reduces energy consumption and carbon footprint. A “fair” cost allocation scheme is an indispensable element to help establish and sustain the cooperation of a group of suppliers in freight consolidation. We study a cost-sharing problem in a freight consolidation system with one consolidation center and a common destination. Designing and applying cost-sharing mechanisms, we decide which suppliers’ demands ship via the consolidation center and their corresponding cost shares. First, we design a truthful and approximately budget-balanced Moulin mechanism when the suppliers’ demands can be assembled to fill a truck as much as possible. There exists a trade-off between the budget-balance guarantee and the level of incentives that can be given to suppliers in this proposed mechanism. We use social cost as the measure of economic efficiency. Comparing the social cost of our mechanism with the minimum social cost obtained from a mixed integer optimization model, we conclude that encouraging more suppliers to bid helps to increase the overall social welfare. As a special case, we also study the proposed Moulin mechanism when the total demand of all suppliers fits in one truck. The proposed mechanism is not only truthful and budget-balanced, but also economically efficient for the majority of cases. Second, we design a truthful acyclic mechanism for a more restricted cost-sharing problem, in which each supplier’s entire less-than-truckload demand must be shipped in one truck. This added constraint greatly complicates the cost-sharing problem because the cost shares now largely depend on how the demands are packed. The budget-balance guarantee of this mechanism is studied both theoretically and numerically under different parameter settings. The economic efficiency of this mechanism is then studied numerically with different number of suppliers and distance of the destination.

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Creator Zhang, Wentao (author)

Core Title Cost-sharing mechanism design for freight consolidation

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Industrial and Systems Engineering

Publication Date 01/19/2018

Defense Date 12/04/2017

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

Tag acyclic mechanism,cost allocation,freight consolidation,Moulin mechanism,OAI-PMH Harvest

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Dessouky, Maged (committee chair), Carlsson, John Gunnar (committee member), Savla , Ketan (committee member)

Creator Email wentao@usc.edu,zwtcts@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-464873

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