Cost-Sharing Mechanism Design for Freight Consolidation among Small Suppliers

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1 Cost-Sharing Mehanism Design for Freight Consolidation among Small Suppliers Wentao Zhang Daniel J. Epstein Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, CA 90089, Nelson A. Uhan Mathematis Department, United States Naval Aademy, Annapolis, MD 21402, Maged Dessouky Daniel J. Epstein Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, CA 90089, Alejandro Toriello H. Milton Stewart Shool of Industrial and Systems Engineering, Georgia Institute of Tehnology, Atlanta, GA 30332, A fair ost alloation sheme is ritial for forming and sustaining horizontal ooperation that leads to ost redutions. In order to inentivize suppliers to ooperate in freight onsolidation, we design a ost-sharing mehanism to solve a ost alloation problem arising in freight onsolidation among small suppliers: that is, suppliers whose total demand fits into one trukload. This ost-sharing mehanism determines the set of suppliers who will partiipate in the onsolidation and their orresponding ost shares. We design this ost-sharing mehanism to ensure that every supplier is willing to reveal their preferene truthfully and the total onsolidation ost inurred is reovered by the alloated osts. Moreover, our analytial and numerial study shows that our ost-sharing mehanism often yields ost shares that maximize the soial welfare of all the suppliers. Key words : freight onsolidation; ost-sharing mehanism; ost alloation History : 1. Introdution Transportation osts have inreased in the last deades for various reasons, suh as the mismath of supply and demand for freight transportation servies (Russell et al. 2014). Competitive transportation osts are espeially ritial for the suess of various industries. For instane, transportation osts are often a large perentage of the produt osts in the agriulture industry (Nguyen et al. 2013). Furthermore, transportation osts, the single largest logistis ost element, usually aount for more than 50% of the total logistis osts (Thomas and Gri n 1996). As a result, it is important for suppliers to redue their transportation osts in order to be ompetitive. In terms of transportation osts, suppliers with low market shares, whih we all small suppliers, are at a ompetitive disadvantage ompared to suppliers with high market shares, whih we all 1

2 2 Artile submitted to Transportation Siene;manusriptno.(Please,providethemansuriptnumber!) large suppliers, beause small suppliers have greater di ulty negotiating favorable transportation rates with arriers due to their smaller shipping volumes. Transportation osts for suh suppliers an be redued by freight onsolidation, whih is the proess of assembling smaller shipments together from di erent loations; the resulting large shipping volumes allow for a redution in transportation rates. A survey of 53 United States ompanies revealed that freight onsolidation, whih takes advantage of eonomies of sale, has ontributed the most to reduing transportation osts (Jakson 1985). Signifiant ost savings have also been reported through freight onsolidation in various industries (e.g. Baush et al. (1995), Brown et al. (2001)). Freight onsolidation often takes plae among businesses that produe similar produts or departments within the same ompany with a entral planner to organize and implement the onsolidation. Self-interested businesses are often willing to onsolidate beause third-party arriers usually harge heaper shipping rates when the shipment volumes are large enough. One example that shows the importane of freight onsolidation is the plight of the California ut flower industry. Currently, this industry has been faing inreasing ompetition from the ut flower growers in South Ameria, espeially Colombia. California s share of the United States ut flower market has dereased from 64% to 20% in the last two deades, while South Ameria s share reahed approximately 70% in 2007 (Arbeláez et al. 2007). A shared ross-doking and distribution faility loated in Miami, Florida has enabled South Amerian growers to ompete e etively with Californian growers. The entral planners in Miami organize and onsolidate the produts of South Amerian growers in the distribution faility before sending them by truk to the rest of the United States. The resulting large volume shipments allow them to obtain the heaper full-trukload (FTL) rates and the orresponding ost savings on transportation provide them with a huge ompetitive advantage. In ontrast, most of the California ut flower growers, who urrently send their produts individually using more expensive less-than-trukload (LTL) rates, are often of small to medium size and have no power to negotiate favorable transportation rates on their own. Nguyen et al. (2013) evaluated the urrent transportation praties in the California ut flower industry and explored the possibility of building a onsolidation enter in Oxnard, California. They onluded that a shipping onsolidation enter ould redue transportation osts by 35%, saving $20 million per year if all the California ut flower growers were to partiipate in the onsolidation. Although establishing an alliane to onsolidate is a feasible approah to improve the ompetitiveness of small suppliers, it is essential to know under what irumstanes the individual suppliers will have the inentive to partiipate in the onsolidation. A survey based on approximately 1500 representative logistis servie providers in Belgium reported that designing a fair ost sharing sheme is the major impediment for forming horizontal ooperation among logistis servie providers even though the profitability of ooperation is firmly and widely believed (Cruijssen et al.

3 Artile submitted to Transportation Siene;manusriptno.(Please,providethemansuriptnumber!) ). Therefore, providing a way to fairly alloate the ost of onsolidation beomes ritial for failitating ooperation among the ompanies. In this paper, we design a ost-sharing mehanism to inentivize a group of small suppliers to partiipate in freight onsolidation. In the environment we onsider, there is a group of small suppliers that ould ooperate by using a nearby onsolidation enter to group their demands to ship to a ommon faraway destination. Our proposed ost-sharing mehanism deides both the set of suppliers who partiipate in onsolidation and their orresponding ost shares. We design our proposed ost-sharing mehanism to possess ertain desirable properties. First, it is important for our ost-sharing mehanism to inentivize suppliers to reveal their willingness to pay truthfully: we want to make sure that no individual supplier or a group of suppliers an benefit from submitting false bids. Seond, our ost-sharing mehanism should reover the ost inurred by onsolidation as muh as possible with the pries harged. Finally, the outome of the proposed ost-sharing mehanism should maximize soial welfare as muh as possible. Our fous in this paper is the ase of small suppliers that is, the total demand of all the suppliers fits into one trukload. This is often true in many appliations sine small suppliers have small shipping volumes on a regular basis. For instane, the demand of many California ut flower growers in 2010 aording to the data provided by the California Cut Flower Commission (CCFC) shows that in more than 95% of the ases, the aggregated shipping volumes to a single destination of these growers are less than one trukload on a daily basis. Most of these growers are from small farms and are willing to partiipate in onsolidation. Our study in this paper o ers a ost-sharing mehanism design strategy for this partiular situation. The rest of the paper is organized as follows. In Setion 2, we review the researh on freight onsolidation, ost-sharing mehanisms and ost alloation approahes in transportation ollaborations. We formally define our problem in Setion 3. In Setion 4, we present our proposed ost-sharing mehanism and show the properties of the mehanism and its outomes for di erent demand profiles. In Setion 5, we investigate the soial welfare resulting from our proposed ost-sharing mehanism both analytially and numerially. We onlude our work in Setion Literature Review The passage of the Motor Carrier At of 1980 made more transportation options available to ompanies, enabling them to improve their logistis e ieny and ustomer servie levels. Freight onsolidation is one strategy that has been applied by many ompanies. Jakson (1985) surveyed 53 firms on freight onsolidation pratie. All the firms regarded freight onsolidation as an important strategy to remain ompetitive in terms of ost and 77% of them indiated that freight onsolidation also helped provide better servie. Di erent freight onsolidation strategies have been studied

4 4 Artile submitted to Transportation Siene;manusriptno.(Please,providethemansuriptnumber!) in Blumenfeld et al. (1985), Campbell (1990), Daganzo (1988), Hall (1987). Quantity-based, timebased, and quantity-time-based shipment-release poliies have been examined to leverage lower transportation rates with large volumes by aggregating shipment quantities in various ways (Abdelwahab and Sargious 1990, Bookbinder and Higginson 2002, Çetinkaya and Bookbinder 2003, Higginson 1995, Higginson and Bookbinder 1994). E ient onsolidation operations have also been studied in vendor managed inventory systems by Çetinkaya and Lee (2000) and Çetinkaya et al. (2006). Most of the literature on freight onsolidation has foused on improving the e ieny of onsolidation. However, little e ort has been devoted to investigating ost alloation methods that enourage freight onsolidation praties, whih usually require ooperation among a group of ompanies. Generally, there are two approahes to solve ost alloation problems. Cooperative game theory provides a framework to alloate osts among partiipants so that ertain fairness riteria are satisfied. This approah fouses on what a group an ahieve and whether it is possible to oordinate the group to ahieve the goal by properly alloating osts. For example, the ore (Gillies 1959) one of the most well-studied solution onepts in ooperative game theory onsists of alloations that reover the ost inurred by all of the players and ensure that no individual or a group of players an benefit by defeting. The emptiness or nonemptiness of the ore is often studied as a proxy for the possibility of ooperation. Another approah, whih is our fous in this paper, is to design a ost-sharing mehanism that solves the ost alloation problem. Rather than investigating what an be ahieved by ooperation, ost-sharing mehanism design fouses more on inentivizing eah individual to partiipate voluntarily. A ost-sharing mehanism uses bids from potential partiipants to determine who should partiipate and how the osts are shared. A ost-sharing setting onsists of a set of players who are interested in reeiving servie from a provider. A binary demand setting restrits the deision of the servie provider to either serve the player or not at all, whereas a general demand setting allows the provider to o er servie at various levels. Eah player has a private valuation of the servie. The objetive of the servie provider is to deide who to serve, at what levels, and how to share the ost among the seleted players. The algorithm that servie providers apply to make these deisions is alled a ost-sharing mehanism. In a ost-sharing mehanism, these deisions are made based on bids that players submit to the servie provider. The bids of the players express their maximum willingness to pay for the servie. The study of ost sharing mehanisms mainly fouses 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 balane, the notion that the mehanism harges the players the ost they inur, and (iii) eonomi e ieny, the idea that the welfare for all the players is maximized. Unfortunately, almost 40 years ago, Green et al. (1976) and Roberts (1979) proved that it is not possible for a ost-sharing

5 Artile submitted to Transportation Siene;manusriptno.(Please,providethemansuriptnumber!) 5 mehanism to possess these three desired properties simultaneously. This has led to a ost-sharing mehanism design paradigm that relaxes either the onstraint on budget balane or eonomi e ieny. Furthermore, the impossibility results also motivate approximation measures on budgetbalane and eonomi e ieny. Roughgarden and Sundararajan (2009) introdued a measure alled soial ost to quantify eonomi ine ieny in ost-sharing mehanisms. Mehanisms, whih ould yield zero or negative soial welfare, always have nonnegative soial osts. As a result, using soial ost, we an identify with inreased fidelity the relatively more e ient mehanisms. Without the onstraint of eonomi e ieny, Moulin (1999), and Moulin and Shenker (2001) proposed a framework, now known as the Moulin mehanism, that allows the design of truthful and approximately budget-balaned ost-sharing mehanisms. A Moulin mehanism deides on the players to be served and the ost shares through an iterative proess with the help from a ost-sharing method, whih provides the ost shares for any given set of players to be served. The mehanism starts with all players being onsidered. In eah iteration, ost shares are alulated and o ered to the onsidered players simultaneously, and only the players who aept the ost shares remain to be onsidered in the next iteration. The iterations ontinue until all remaining players aept the ost shares o ered or the set of onsidered players beome empty. Using a soalled ross-monotoni ost-sharing method, a Moulin mehanism o ers a nondereasing sequene of osts to the players to guarantee that no individual or oalition of players an be better o by submitting false bids. Meanwhile, approximate budget-balane is ahieved by o ering osts at eah iteration that would in total approximately over the ost inurred if the urrent iteration were to be the last. Due to its flexibility and reasonable eonomi e ieny, approximately budget-balaned Moulin mehanisms have been designed for a wide range of ost-sharing appliations arising in sheduling (Brenner and Shäfer 2007, Bleishwitz and Monien 2009), network design (Jain and Vazirani 2001, Arher et al. 2004, Gupta et al. 2004, 2007), faility loation (Devanur et al. 2005, Könemann et al. 2005, Leonardi and Shäfer 2004, Pál and Tardos 2003), and logistis (Xu and Yang 2009). When eonomi e ieny is the primary onern together with truthfulness, the Vikrey-Clarke- Groves (VCG) mehanism (Clarke 1971, Groves 1973, Vikrey 1961) is a powerful framework. As a speial ase of VCG mehanisms, the marginal ost mehanism is often used to ahieve e ient ost alloations. The ost shares in the marginal ost mehanism are defined so that the welfare eah player obtains is its marginal ontribution to the overall soial welfare. However, this lass of mehanisms usually has no budget-balane guarantee and sometimes raises zero revenue (Moulin and Shenker 2001). In terms of appliations of mehanism design in transportation ollaborations, Furuhata et al. (2015) designed an online ost-sharing mehanism to provide quotes to passengers who shared

6 6 Artile submitted to Transportation Siene;manusriptno.(Please,providethemansuriptnumber!) a door-to-door transportation servie provided by a demand-responsive transport system. They proposed a novel ost sharing mehanism that satisfies a number of desired properties suh as online fairness, immediate response, and ex-post inentive ompatibility that speifially address the issues involved with sharing osts without knowing future demand. Mehanism design has rarely been used to solve ost alloation problems in transportation ollaborations; ooperative game theory has been the more ommon approah. Di erent ost alloation methods have been proposed in the literature. Rule-based ost alloation methods have been widely applied to share osts in transportation ooperation. For instane, the Shapley value (Shapley 1953) was applied by Krajewska et al. (2008) to alloate the profit ahieved by a group of freight arriers, who ooperate by pooling transportation resoures. The nuleolus (Shmeidler 1969) was onsidered by Liu et al. (2010) and Frisk et al. (2010) to alloate profits or osts in ollaborations of using truks. Besides the rule-based ost alloation methods, linear programming duality has also been employed to provide ost sharing solutions. Sánhez- Soriano et al. (2001) introdued a lass of transportation games and showed that the non-negative optimal dual solutions of the underlying linear programs haraterized the ost alloations in the ore that also assign less ost to eah player than its stand-alone ost. Some studies (e.g. Agarwal and Ergun (2008), Houghtalen et al. (2011)) have used inverse optimization to alulate the apaity exhange pries that indue optimal oordinated apaity usage in a network, e.g. in an air argo alliane. Other studies, espeially those that are more appliation-based have proposed ost alloation methods that address additional notions of fairness. For example, Liu et al. (2010) introdued the Weighted Relative Savings Model (WRSM), whih onsiders individual ontributions to the oalition and the balane of savings for eah player. Frisk et al. (2010) proposed the Equal Profit Method (EPM), whih applied a linear program to obtain a ost alloation in whih the profit ratios of the partiipants were kept as lose as possible, for the ooperation of eight forest ompanies in southern Sweden. They argued that the guaranteed similar relative savings provided greater inentives for players to join the oalition in the negotiation phase. 3. Problem Definition We study a freight onsolidation system that onsists of a group of suppliers who produe similar produts, are loated in a ertain geographial region, and ship to a ommon destination. All the suppliers in the group are interested in ost redution through freight onsolidation. There is a entral planner who operates a enter that provides onsolidation servie in the same region. Both the suppliers and the onsolidation enter use truks to ship their produts. We formally define our ost-sharing problem as follows. Let N denote the set of suppliers who are interested in onsolidating their shipments. Eah supplier i 2 N has a positive shipping demand d i

7 Artile submitted to Transportation Siene;manusriptno.(Please,providethemansuriptnumber!) 7 in ft 3 and a valuation v i for the servie provided by the onsolidation enter. The total demand of the suppliers fits into one trukload, i.e. P i2n d i apple k F,wherek F is the apaity of one truk. The valuations reflet the suppliers opinions on how muh the onsolidation enter s servie is worth. Figure 1 Struture of the onsolidation system Figure 1 shows the struture of the onsolidation system. Suppliers in N have two shipping options. They an ship their demand either diretly to the destination or through the onsolidation enter. Suppliers express their willingness to onsolidate by submitting a bid for servie at the beginning of the onsolidation proess. We denote supplier i s bid by q i. Based on these bids, the onsolidation enter selets a set of suppliers S N to serve. Seleted suppliers have their produts onsolidated first and then shipped to the ommon destination. We all the shipment from the suppliers to the onsolidation enter inbound shipping, and the orresponding ost inurred by eah supplier the inbound shipping ost. We all the shipment from the onsolidation enter to the destination outbound shipping, and the orresponding ost inurred by the onsolidation enter the outbound shipping ost. We all the shipment from the suppliers to the destination diret shipping, and the orresponding ost for eah supplier the stand-alone ost. Suppliers are independent and do not oordinate shipping among themselves. (For example, two suppliers ould be paying the same loal transportation ompany for the inbound shipping, and this ompany ould very well be routing a single truk to both loations, but would still harge the suppliers per-volume rates.) There are two important parameters in the truking ost struture. One is the less-than-trukload (LTL) rate, or the ost for shipping eah ubi foot when the shipping demand is less than some threshold value. The other is the full-trukload (FTL) rate, or the fixed ost for using the entire

8 8 Artile submitted to Transportation Siene;manusriptno.(Please,providethemansuriptnumber!) Figure 2 Cost struture truk when the shipping demand is greater than the threshold value. Let b denote this threshold value, whih we all the full-trukload (FTL) equivalent volume. Shipping demand b or more in one truk osts the same as shipping a full trukload. The FTL rate is usually pried per mile while the LTL rate is usually pried based on other fators besides distane, suh as density, freight lass, weight per ubi foot, et. However, with similar produts in the shipment, we an assume that these fators influene the prie in the same way aross suppliers and thus the LTL rate and FTL rate only depend on the mileage between the origin and the destination. Given the distane between the origin and the destination, we denote the orresponding LTL rate and FTL rate by L and F, respetively. The transportation ost is a funtion of the shipping volume d and it is illustrated in Figure 2. In mathematial terms, (d)= ( L d if 0 <dapple b, F if d b. The transportation ost inreases linearly with the LTL rate L as the demand inreases until the demand reahes the FTL equivalent volume b; then the ost F remains the same for any demand volume beyond b but less than k F. Note that F = L b. We assume that the suppliers and the onsolidation enter fae the same truking ost struture but not neessarily the same rates or FTL equivalent volume. We define the shipping ost funtions for the suppliers and the onsolidation enter based on the following assumptions: Consolidation enter loation assumption: The suppliers are all lose to the onsolidation enter and approximately the same distane away. Consequently, we assume all the suppliers have the same LTL rate g L0 and FTL rate g F 0 for inbound shipping. Destination loation assumption 1: The suppliers are all far away from the destination and approximately the same distane away. Consequently, we assume all the suppliers have the same LTL rate g L1 and FTL rate g F 1 for diret shipping. Destination loation assumption 2: The distanes between the suppliers and the destination are larger than the distanes between the suppliers and the onsolidation enter. As a onsequene, g L1 >g L0 and g F 1 >g F 0.

9 Artile submitted to Transportation Siene;manusriptno.(Please,providethemansuriptnumber!) 9 Loation assumption: The suppliers, onsolidation enter and destination are loated suh that the inbound shipping distanes, outbound shipping distane and the diret shipping distanes satisfy the strit triangle inequality, i.e. g L1 <g L0 + L1,where L1 is the LTL rate for the onsolidation enter. Threshold value assumption: The inbound and diret shipping osts of the suppliers have the same FTL equivalent volume b G (ft 3 ). In general, the above assumptions represent the situation where the suppliers and the onsolidation enter are loated in the same region and the destination is su iently far away suh that outbound shipping osts dominate the inbound shipping osts if suppliers send demand via the onsolidation enter. For instane, the CCFC proposes Oxnard as the loation for flower onsolidation in California beause half of the flower prodution originates from Oxnard and its viinity. Most demand destinations outside California will lead to dominant outbound shipping osts ompared to inbound shipping osts. In addition, we onsider the group of suppliers as a small ommunity in whih eah supplier is able to obtain the same transportation rate through negotiation with the arriers. For example, if arriers harge suppliers based on shipping zones, flower growers in the Oxnard area share the same transportation rate for the same destination even though there may be small di erenes in distanes. Following the definition of the ost struture and the assumptions above, the inbound shipping ost for supplier i is G 0 i = ( g L0 d i if 0 <d i apple b G, g F 0 if d i b G. The stand-alone shipping ost for supplier i is ( G 1 g L1 d i if 0 <d i apple b G, i = g F 1 if d i b G. We assume that suppliers are responsible for their own inbound shipping osts if seleted. We onsider the outbound shipping ost as the only ost inurred by the onsolidation enter while providing the servie and therefore only the outbound shipping ost will be shared among the seleted suppliers. Given the destination, let denote the FTL rate for outbound shipping at the onsolidation enter. We denote the FTL equivalent volume by b C = L1. In mathematial terms, the ost of shipping demand d at the onsolidation enter is ( L1 d if 0 apple d apple b C, (d)= if d b C. Sine the ost funtion inreases, : R! R has noninreasing inremental value as the size of input set ( P i2s d i) is submodular in S N. The total ost C(S) inurred when we onsolidate and ship the demand of suppliers in S is C(S)= X i2s G 0 i + X d i. i2s

10 10 Artile submitted to Transportation Siene;manusriptno.(Please,providethemansuriptnumber!) The funtion P i2s G0 i is submodular in S N as well. Beause the lass of submodular funtions is losed under non-negative linear ombinations, C(S) is submodular in S. We assume that the onsolidation enter only has partial information about the transportation osts of the suppliers. In partiular, the onsolidation enter knows that it has the same truking ost struture as the suppliers, but it does not know the exat parameters of the ost funtions for the suppliers. The information that the onsolidation enter soliits from the suppliers is their bids for their shipping demand. Therefore, the suppliers shipping demands are known to the onsolidation enter as well. We further restrit the servie provided by the onsolidation enter to be binary: either a supplier is not served at all or its full shipping demand is served. Note that under the assumption that the total demand is less than or equal to one trukload, it is straightforward to show that a binary servie poliy is optimal from the suppliers perspetive. However, this might not be true when the total demand requires multiple truks. Later in Setion 4, we will show that the binary servie poliy is also optimal from the soial welfare perspetive when the total demand is less than or equal to one trukload. 4. Cost-Sharing Mehanism Design In this setion, we design ost-sharing mehanisms for the ost-sharing problem defined above. Reall that the total demand of all the suppliers fits into one truk, i.e. P d k2n k apple k F, and so the demand of eah supplier is less than one trukload. In this setion, we will first introdue the Moulin mehanism framework, whih we use to design our ost-sharing mehanism. We then present our ost-sharing mehanism and show that this mehanism is both truthful and budgetbalaned. Finally, we will disuss the outomes of our ost-sharing mehanism for di erent demand profiles by providing some properties of the mehanism The Moulin Mehanism The Moulin mehanism (Moulin 1999, Moulin and Shenker 2001) a ords a lot of flexibility in designing truthful while budget-balaned or approximately budget-balaned ost-sharing mehanisms. It simulates an iterative asending aution to determine whih subset of players to serve by using a ost-sharing method, a funtion that assigns a nonnegative ost share for eah player i 2 S for every subset S N. The ost shares for the seleted subset of players indiates the ost alloation strategy, or alternatively, the prie harged to eah player. The Moulin mehanism operates as follows: 1. Collet a bid q i from eah player i 2 N. 2. Initialize S := N.

11 Artile submitted to Transportation Siene;manusriptno.(Please,providethemansuriptnumber!) If q i (i, S) for every i 2 S, then stop. Return the set S. Eah player i 2 S is harged the prie p i = (i, S). 4. If q j < (j, S) for a player j 2 S, thensets := S \{j} and return to Step 3. Beause only the players whose bids are greater than or equal to their ost share stay in the subset S, the players seleted by the Moulin mehanism are never harged more than what they bid. The ost-sharing method plays a very important role in a Moulin mehanism design. It is often required to be ross-monotoni, whih means that the ost share of eah player only inreases as other players are removed, i.e. for all S T N and i 2 S, (i, S) (i, T ). This implies that eah player in S is o ered a sequene of nondereasing ost shares through the iterations. When the ost-sharing method is ross-monotoni, the Moulin mehanism is group strategyproof. Group strategyproofness is a strong notion of truthfulness: an individual player annot be better o by false bidding, and a subset of players an never stritly inrease the utility of one of its members without dereasing the utility of some other member by oordinating false bids. We split the possible outomes of this mehanism the set of players served S intothree ategories: Total partiipation: All the players in N are served. Zero partiipation: None of the players in N are served. Partial partiipation: A non-empty proper subset of N is served. Observation 1: Any Moulin mehanism yields total partiipation if and only if all i 2 N. (i, N) apple q i for Observation 2: Any Moulin mehanism yields zero partiipation if and only if in every iteration k =1, 2...,n, there exists at least one player i suh that (i, S k ) >q i,wheres k denotes the remaining set of players at the beginning of iteration k. If the Moulin mehanism yields zero partiipation, then one player is removed from S k in Step 4 of iteration k of the mehanism. In other words, in eah iteration k of the mehanism there exists at least one player i that has (i, S k ) >q i. On the other hand, if in eah iteration k, thereexistsaplayeri that satisfies player will be removed from S k until there are no more players left. (i, S k ) >q i,thena 4.2. Cost-Sharing Mehanism Proportional to E etive Demand for Sharing (PEDS) Given the shipping volumes and the orresponding bids, the most intuitive way of sharing the outbound shipping ost is to share it proportional to eah supplier s atual demand. In other words, the ost share for supplier i 2 S would be d i P j2s d j ( P j2s d j), where S is the seleted set of suppliers. This ost sharing method tends to alloate more ost to the suppliers with larger demand and thus suh suppliers, without whom onsolidation may not be benefiial, may not have the inentive to bid for onsolidation.

12 12 Artile submitted to Transportation Siene;manusriptno.(Please,providethemansuriptnumber!) To illustrate, onsider an example where there are three suppliers who want to onsolidate their demand and their transportation osts of shipping diretly and shipping through the onsolidation enter are given in Table 1. We assume b C = b G = 5000, = $1000, L1 = $0.2/ft 3, and eah supplier bids their diret shipping osts. Supplier 3 has larger demand than Supplier 1 and Supplier 2. If every supplier ships its demand individually, the total transportation ost is $1400. However, if we onsolidate all of their demand, the total transportation ost is C(N) = $1300. If we share the outbound shipping ost proportional to atual demand in the Moulin mehanism, the ost share for eah supplier in eah iteration is shown in Table 2. Table 1 Cost-sharing example Supplier 1 Supplier 2 Supplier 3 Demand (1000 ft 3 ) Diret shipping ost ($) Inbound shipping ost ($) Table 2 Cost shares proportional to atual demand Supplier 1 Supplier 2 Supplier 3 Iteration 1 Outbound ost share ($) Inbound shipping ost ($) Total ost ($) Deision Aept Aept Deline Iteration 2 Outbound ost share ($) N/A Inbound shipping ost ($) N/A Total ost ($) N/A Deision Deline Deline N/A Table 2 shows that sharing the outbound shipping ost proportional to atual demand leads to a total ost share for Supplier 3 that is greater than its diret shipping ost. As a result, Supplier 3 delines to use the servie. This leaves only Supplier 1 and Supplier 2 under onsideration in the next iteration. However, without Supplier 3, Supplier 1 and Supplier 2 end up with ost shares that are higher than their diret shipping osts. Consequently, none of the suppliers are able to benefit from onsolidation, whih ould have saved a total of $100 if implemented properly. The example above shows the defiieny of sharing the outbound shipping ost proportional to atual demand and reveals the importane of the partiipation of the suppliers with larger demand. Therefore, in order to enourage the suppliers with larger demand to partiipate, we set a maximum demand volume that a supplier needs to be responsible for in the ost share. This implies that a supplier with a demand greater than only shares the outbound shipping ost

13 Artile submitted to Transportation Siene;manusriptno.(Please,providethemansuriptnumber!) 13 proportional to and thus is alloated a smaller ost. In other words, eah supplier has an e etive demand for sharing defined by the maximum demand volume. E etive demand for sharing: For eah supplier i 2 N, its e etive demand for sharing is d 0 i =min{d i, }. We propose a ost-sharing method that shares the outbound shipping ost proportional to e etive demand for sharing (PEDS). Let D S = P i2s d i and D 0 S = P i2s d0 i for any S N. The ost share o ered to supplier i 2 S is equal to its inbound shipping ost plus the share of the outbound shipping ost for set S, proportional to supplier i s e etive demand for sharing; that is, for S N, i 2 S, we define the ost share (i, S)=G 0 i + d0 i D 0 S (i, S) as 8 G 0 i + L1 D S di if 0 apple D D S >< 0 S apple b C, 0 <d i < ; G 0 i + L1 D S be if 0 apple D D D S = S 0 S apple b C, d i ; G 0 i + d i D S >: 0 F 1 if b C apple D S, 0 <d i < ; G 0 i + D S 0 F 1 if b C apple D S, d i. Now let = If we apply ost-sharing method PEDS in the example mentioned earlier, the ost shares we obtain for eah supplier are summarized in Table 3. Table 3 Cost shares proportional to e etive demand for sharing Iteration 1 Supplier 1 Supplier 2 Supplier 3 Outbound ost share ($) Inbound shipping ost ($) Total ost ($) Deision Aept Aept Aept Cost-sharing method PEDS harges less to Supplier 3 than when osts are shared proportional to atual demand. The total ost for eah supplier from ost-sharing method PEDS is less than its diret shipping ost. Thus, all suppliers are willing to partiipate in the onsolidation. Additionally, with the partiipation of Supplier 3, eah supplier of this oalition is able to redue its transportation osts. If one views as the onsolidation enter s estimation of the suppliers FTL equivalent volume, ost-sharing mehanism PEDS shares the osts proportional to the onsolidation enter s estimate of eah supplier s stand-alone ost. When the estimated FTL equivalent volume equals the true FTL equivalent volume of the suppliers, PEDS shares the ost proportional to the atual stand-alone ost of eah supplier. Therefore, ost-sharing method PEDS is a more favorable ost alloation method beause the estimated stand-alone ost is a better refletion of the true osts of the suppliers. The ost of shipping a large demand d>b G of a supplier is not proportional to the demand volume beause G F 1 <G L1 d is the true ost.

14 14 Artile submitted to Transportation Siene;manusriptno.(Please,providethemansuriptnumber!) As mentioned above, ross-monotoni ost-sharing methods lead to truthful Moulin mehanisms. We will next explore the ross-monotoniity of ost-sharing method PEDS and the properties of the orresponding ost-sharing mehanism PEDS. Proposition 1. If b C, ost-sharing method PEDS is ross-monotoni. Proof. Let i be an arbitrary supplier whose ost share we observe and ompare in di erent subsets. Sine the inbound shipping ost G 0 i is always inluded in supplier i s ost share in the ost-sharing method PEDS, we only fous on supplier i s ost share of the outbound shipping ost to prove ross-monotoniity. Let S be an arbitrary set suh that i 2 S and S N \{j},wherei 6= j. We obtain T by augmenting S with supplier j, i.e.t = S [{j}. Let ost for suppliers in S while serving T. (S, T ) denote the total ost share of the outbound shipping If D S apple b C and D T apple b C,thend i apple b C apple and d 0 i = d i for all i 2 T.Wehave (S, T )= D0 S D 0 T L1 D T = D S D T L1 D T = L1 D S = D S. If D S apple b C and D T >b C,thend i apple b C apple,d 0 i = d i for all i 2 S and D T =. We onsider 2 ases. First, if d j <,thend 0 j = d j.wehave (S, T )= D0 S DT 0 F 1 = D S < D S = L1 D S = D S. D T b C Seond, If d j,thend 0 j =.Wehave (S, T )= where the inequality holds beause b C. D S D S + < b C D S = D S, Finally, if D S b C and D T >b C,then D T = D S =. Adding one more supplier does not inrease the outbound shipping ost but results in one more supplier sharing the unhanged ost. Consequently, (S, T ) apple D T = D S. The three ases above show that the total ost share of the outbound shipping ost does not inrease when one more supplier is served, i.e. (S, T ) apple D S. Sine the total ost share (S, T ) does not inrease, the share of the outbound shipping ost for supplier i does not inrease as well. Together with the inbound shipping ost, (i, T ) does not inrease either ompared to (i, S). Therefore, when i 2 S, S N \{j}, T = S [{j}, and i 6= j, wehave (i, S) (i, T ). This implies that for arbitrary S T N, (i, S) (i, T ). Sine ost-sharing method PEDS is ross-monotoni when b C, the Moulin mehanism that applies our ost-sharing method PEDS with b C is group strategyproof. We all this ostsharing mehanism PEDS. In addition, ost-sharing method PEDS, whih shares the total inurred

15 Artile submitted to Transportation Siene;manusriptno.(Please,providethemansuriptnumber!) 15 ost of onsolidation among the seleted suppliers, guarantees that the ost-sharing mehanism PEDS is also budget-balaned. Note that sine the total ost C(S) is submodular in S, we an also find a ross-monotoni ost sharing method by onsidering extensions of the marginal vetors or the Shapley value of C (Sprumont (1990), Moulin and Shenker (2001)). However, the resulting ost sharing methods and mehanisms arguably lose the intuitive and pratial appeal of PEDS Properties of Cost-Sharing Mehanism PEDS In this setion, we examine the onditions for ost-sharing mehanism PEDS to yield total or zero partiipation. We onsider di erent demand profiles and estimations of suppliers FTL equivalent volume to provide managerial insights for the onsolidation enter. From now on, we assume that the valuation of the onsolidation servie v i of supplier i is its stand-alone ost and thus submits a bid q i = v i to ost-sharing mehanism PEDS. We an make this assumption due to the group strategyproofness of ost-sharing mehanism PEDS. The stand-alone ost of supplier i is the truthful valuation of the onsolidation enter s servie for supplier i beause that is the ost supplier i has to pay for if it does not partiipate in onsolidation. Note that from this point on, whenever we disuss ost-sharing method PEDS or ost-sharing mehanism PEDS, we assume b C. In Lemma 1, we give a su ient ondition for ost sharing mehanism PEDS to yield zero partiipation. Lemma 1. If (i, N) >q i for all i 2 N, ost-sharing mehanism PEDS yields zero partiipation. Proof. Assume (i, N) >q i for all i 2 N. Suppose supplier j 2 N is removed in the first iteration of the mehanism, resulting in S := N \{j} in the seond iteration. Beause ost-sharing method PEDS is ross-monotoni, (i, S) (i, N) >q i for all i 2 S. As a result, another supplier will be removed from S in the seond iteration. By the same argument, there exists at least one supplier i suh that (i, S) >q i in eah iteration. Aording to Observation 2, ost-sharing mehanism yields zero partiipation. Suppose we have a set of suppliers N, whose total demand is less than the FTL equivalent volume of the onsolidation enter, i.e. D N <b C. Cost-sharing mehanism PEDS always yields zero partiipation for suh a set of suppliers. Equivalently, it is not benefiial for suh a set of suppliers to ship their demand via the onsolidation enter. Beause D N <b C apple, the demand of eah supplier i 2 N is d i <. Therefore, the e etive demand for sharing of eah supplier i 2 N, d 0 i = d i. As a result, suppliers share the outbound shipping ost by paying the LTL rate no matter d how many suppliers partiipate in the onsolidation, i.e. i D N L1 D N = d i L1. Thus, any supplier i with d i <b G has a ost share (i, N)=d i g L0 + d i L1 in the first iteration of the mehanism. The loation assumption g L1 <g L0 + L1 implies that (i, N)=d i (g L0 + L1 ) >d i g L1 = q i. Any supplier

16 16 Artile submitted to Transportation Siene;manusriptno.(Please,providethemansuriptnumber!) i with d i b G has a ost share (i, N)=g F 0 + d i L1, whih is also greater than its orresponding stand-alone ost beause (i, N)=b G g L0 + d i L1 b G (g L0 + L1 ) >b G g L1 = g F 1 = q i. Consequently, every supplier i has a ost share (i, N) >q i. Therefore, aording to Lemma 1, ost-sharing mehanism PEDS yields zero partiipation when the set of suppliers N has total demand less than b C. Intuitively, beause g L1 <g L0 + L1, these suppliers annot redue their osts by onsolidating: sending shipments to the onsolidation enter adds extra ost by inreasing the total distane traveled, and these suppliers annot benefit from the volume disount. For this reason, from here on we restrit our attention to sets of suppliers N whose total demand is greater than the FTL equivalent volume at the onsolidation enter, i.e. D N b C. The onsolidation enter deides the value of before olleting bids. Without knowing the exat value of b G, the onsolidation enter s estimate an be above, below, or equal to the true b G. Note that b C and D N b C,soD 0 N b C. When the onsolidation enter overestimates the suppliers FTL equivalent volume, i.e. >b G, the onditions for ost-sharing mehanism PEDS to yield total and zero partiipation are summarized in Proposition 2 and Proposition 3. (For their proofs, see Appendix A.) Proposition 2. When >b G, ost-sharing mehanism PEDS yields total partiipation if and only if D 0 N b G. Proposition 3. When >b G,if 0 <, ost-sharing mehanism PEDS yields zero partiipation. Note that the smallest possible value of is b C.Ifb G <b C, then any possible is an overestimate of b G. Given suh a ondition, if the onsolidation enter s goal is to inentivize onsolidation, its strategy should be to have = b C so that the demand ondition on D 0 N for total partiipation given in Proposition 2 is as easy to meet as possible. When the onsolidation enter underestimates the suppliers FTL equivalent volume, i.e. < b G (implying b G b C ), the onditions for ost-sharing mehanism PEDS to yield total and zero partiipation are summarized in Proposition 4 and Proposition 5 (For their proofs, see Appendix A). Proposition 4. When <b G, ost-sharing mehanism PEDS yields total partiipation if and only if D 0 N. Proposition 5. When g L1 yields zero partiipation. g L0 L1 g L0 b G < <b G,ifD 0 N < b G g L1 g L0, ost-sharing mehanism PEDS The lower bound g L1 L1 b G for in Proposition 5 is neessary for b G to be a valid upper bound for.sined 0 N 0 b C, > L1 b G guarantees that b G b C.When is below

17 Artile submitted to Transportation Siene;manusriptno.(Please,providethemansuriptnumber!) 17 g L1 g L0 L1 b G, the outome of ost-sharing mehanism PEDS when 0 < an either be partial or zero partiipation. Whether the onsolidation enter underestimates or overestimates the suppliers FTL equivalent,, b G g L1 there is a range of 0 for example [ g L0 ) with overestimation under whih the outome of ost-sharing mehanism PEDS is unlear. This ambiguity no longer exists when the onsolidation enter orretly estimates the suppliers FTL equivalent volume. Corollary 1. When = b G, ost-sharing mehanism PEDS yields either zero or total partiipation. Proof. In Propositions 2, 3, 4 and 5, when = b G, the onditions for total and zero partiipa- tion depend on the same ritial value. Thus, the onditions for total and zero partiipation omplement eah other. Therefore, given any demand profile, the outome of ost-sharing mehanism PEDS is either zero partiipation or total partiipation. Based on the above analysis of the outomes of ost-sharing mehanism PEDS, if the onsolidation enter aims to promote onsolidation, it should set = b C if it knows that b G <b C.Ifitknows that b G b C, although estimating b G orretly an redue the ambiguity of the outome, the hoie of does not a et the ondition on 0 that yields total partiipation for underestimation. 5. Eonomi E ieny of Cost-Sharing Mehanism PEDS In this setion, we examine the resulting soial welfare of ost-sharing mehanism PEDS. We first introdue an optimization model that finds an eonomially e ient solution for any given demand profile. By omparing the outomes of ost-sharing mehanism PEDS with the eonomially e - ient solutions, we analytially show that the total and zero partiipation outomes guaranteed above for ertain demand profiles are eonomially e ient. Then we numerially study the eonomi e ieny for the demand profiles whose outomes of ost-sharing mehanism PEDS are unknown. Typially, the eonomi e ieny of a ost-sharing mehanism is measured by soial welfare. An eonomially e ient solution is the one that maximizes the soial welfare: W (S) = V (S) C(S), where S is the set of suppliers seleted by the mehanism, V (S) is the total valuation of the suppliers in S and C(S) is the total ost to serve the suppliers in S. Unfortunately, Feigenbaum et al. (2002) showed that truthful and approximately budget-balaned ost-sharing mehanisms often yield outomes with zero or negative soial welfare even though outomes with stritly positive soial welfare exist. This makes it di ult to ompare the relative eonomi e ieny of ostsharing mehanisms with the same budget-balane guarantee. To sidestep this issue, Roughgarden and Sundararajan (2009) introdued soial ost, a measure of eonomi e ieny. The soial ost (S) is defined as the sum of the ost inurred by serving S

18 18 Artile submitted to Transportation Siene;manusriptno.(Please,providethemansuriptnumber!) and the total valuations of the suppliers not in S. In mathematial terms, (S)=C(S)+V (N \ S), where V (N \ S) is the total valuation of the suppliers who are exluded from S. In fat, soial ost an be onstruted by an a ne transformation from soial welfare: (S)= W (S)+V(N). The relationship between soial welfare and soial ost implies that minimizing soial ost is equivalent to maximizing soial welfare, although soial ost is always nonnegative. As a result, we use soial ost as the measure of eonomi e ieny and determine outomes with the maximum soial welfare by minimizing soial ost. In our problem, the soial ost is equal to the total shipping ost of all the suppliers in N. We onsider the following optimization model to minimize the total shipping ost of all the suppliers. Suppliers now an hoose how muh demand to ship diretly and how muh demand to ship via the onsolidation enter. Therefore, the servie provided by the onsolidation enter in this model is not restrited to be binary. However, we show that the solution whih minimizes the soial ost orresponds to binary servie provided by the onsolidation enter. The deision variables and the model are presented below. Deision variables: x i F 0: Binary. If supplier i s inbound shipping uses the FTL rate, then x i F 0 = 1, otherwise 0 8i 2 N. x i L0: Binary. If supplier i s inbound shipping uses the LTL rate, then x i L0 = 1, otherwise 0 8i 2 N. x i F 1: Binary. If supplier i s diret shipping uses the FTL rate, then x i F 1 = 1, otherwise 0 8i 2 N. x i L1: Binary. If supplier i s diret shipping uses the LTL rate, then x i L1 = 1, otherwise 0 8i 2 N. x CF : Binary. If outbound shipping uses the FTL rate, then x CF = 1, otherwise 0. x CL : Binary. If outbound shipping uses the LTL rate, then x CL = 1, otherwise 0. yf i 0 : Amount of supplier i s demand sent by the FTL rate to the onsolidation enter 8i 2 N. yl0 i : Amount of supplier i s demand sent by the LTL rate to the onsolidation enter 8i 2 N. yf i 1 : Amount of supplier i s demand sent by the FTL rate to the destination 8i 2 N. yl1 i : Amount of supplier i s demand sent by the LTL rate to the destination 8i 2 N. y CF : Amount of demand sent by the FTL rate from the onsolidation enter to the destination. y CL : Amount of demand sent by the LTL rate from the onsolidation enter to the destination. Model: min X i2n(g F 0 x i F 0 + g F 1 x i F 1 + g L0 y i L0 + g L1 y i L1)+ x CF + L1 y CL s.t. y i F 0 apple k F x i F 0 8i 2 N, (1) y i L0 apple b G x i L0 8i 2 N, (2) y i F 1 apple k F x i F 1 8i 2 N, (3)

19 Artile submitted to Transportation Siene;manusriptno.(Please,providethemansuriptnumber!) 19 y i L1 apple b G x i L1 8i 2 N, (4) y CF apple k F x CF, (5) y CL apple b C x CL, (6) y i F 0 + y i L0 + y i F 1 + y i L1 = d i 8i 2 N, (7) X (y i F 0 + yl0)=y i CF + y CL, (8) i2n x i F 0, x i L0, x i F 1, x i L1 2{0, 1}8i2N, x CF,x CL 2{0, 1}, all other deision variables are nonnegative. Constraints (1), (3), and (5) ensure that the shipping volumes do not exeed the trukload when shipping with the FTL rates. Constraints (2), (4), and (6) ensure that the shipping volumes do not exeed the FTL equivalent volumes when shipping with the LTL rates. Constraint (7) makes sure that eah supplier ships all of its demand. Constraint (8) enfores that what ships into the onsolidation enter will ship out. An optimal solution of this model provides eah supplier i 2 N with a shipping plan that minimizes the total shipping ost of all the suppliers. The total ost onsists of two parts: one part is the shipping ost of eah supplier, the other part is the shipping ost of the onsolidation enter. In this model, if a shipping volume d from a supplier to the onsolidation enter or the destination is smaller than b G, then LTL is the optimal shipping method; if d is greater than or equal to b G, then FTL is the optimal shipping method. Inreasing the shipment volume when the total shipping demand exeeds b G does not inur extra ost. Therefore, a supplier uses either the FTL rate or the LTL rate to ship demand to the onsolidation enter or the destination in an optimal solution sine the shipping volume for this supplier, d, is assumed to be less than the apaity of a full trukload k F. In mathematial terms, for every supplier i, x i F 0 x i L0 = 0 and x i F 1 x i L1 =0 where x is in the optimal solution. The same logi applies to the onsolidation enter as well, i.e. x CF x CL =0where x is in the optimal solution. We analyze the struture of the optimal solutions to this model to understand how the minimum soial ost is ahieved. We show in Proposition 6 and Corollary 2 that in an optimal solution of the above optimization model eah supplier has its entire demand shipped either diretly or through the onsolidation enter. In Proposition 7 we show the minimum soial ost solution is either zero partiipation or total partiipation. (For their proofs, see Appendix B). Proposition 6. There exists an optimal solution to the model in whih eah supplier ships all its demand either to the onsolidation enter or diretly to the destination.