Logistics cost sharing in supply chains involving a third-party logistics provider

Transcription

1 CEJOR (2016) 24: DOI /s ORIGINAL PAPER Logistics cost sharing in supply chains involving a third-party logistics provider Lin Jiang Yong Wang Dongmei Liu Published online: 10 April 2014 Springer-Verlag Berlin Heidelberg 2014 Abstract In many cases end customers are sensitive to a product s logistics service level which is provided by a third-party logistics (3PL) provider therefore the continuous improvement of the logistics service is imperative and valuable. However the problem is that improving the logistics service benefits all of them but is costly to only the 3PL provider. The 3PL provider is not willing to do this. Sharing the logistics cost is one solution to this problem. This study investigates cost sharing in two kinds of supply chains i.e. one manufacturer-one 3PL provider-one retailer supply chain and two manufacturers-one 3PL provider-one retailer supply chain. Two types of cost sharing mechanisms i.e. decentralized cost sharing mechanisms and centralized cost sharing mechanisms are explored. Decentralized cost sharing mechanisms are proposed as contracts that chain members separately decide their cost sharing portions to optimize their own profits ignoring the collective impacts of their decisions on the channel as a whole. Centralized cost sharing mechanisms are in the situation that chain members negotiate their cost sharing portions so that their profits are the shares of the entire supply chain s profit implying that the supply chain is coordinated perfectly. This study aims to analyse how cost sharing mechanisms affect supply chain performance and under what conditions chain members are willing to engage in cost L. Jiang Y. Wang (B) School of Economics and Business Administration Chongqing University Chongqing China wangyongkt@163.com L. Jiang School of Mathmatics and Information Engineering Chongqing University of Education Chongqing China jlin666@163.com D. Liu School of Economics and Business Administration Chongqing University of Education Chongqing China liudm1718@126.com

2 208 L. Jiang et al. sharing mechanisms. Conditions necessary for cost sharing mechanisms to achieve win-win outcomes are identified. Keywords 3PL provider Logistics service level Cost sharing mechanisms 1 Introduction In recent years third-party logistics (3PL) has received considerable attention (Knemeyer et al. 2003; Maltz and Ellram 1997; Razzaque and Sheng 1998). A 3PL provider is an external provider who manages controls and delivers logistics activities on behalf of a shipper (Hertz and Alfredsson 2003). The use of a 3PL provider to take over some or all of a firm s logistics responsibilities is becoming more prevalent and more than 70 % of companies in Western Europe USA and Asia Pacific have logistics outsourcing experience (Hsiao et al. 2010). In many cases the demand of product is sensitive to the logistics service level provided by the 3PL provider. One example that supports our model is Kunming Hongri Flower Plant Co. Ltd. one of the biggest flower plant bases in the world which is located in Kunming the Spring City of China. The firm is a specialized export company of fresh cut flowers and exports carnations roses lilies etc. to other countries through a specialized 3PL provider. The market demand of the fresh-cut flowers critically depends on the products freshness upon arriving at the destination markets. How to maintain the freshness of the flower is a key concern in their operations and the firm hopes that its logistics service level is as high as possible and asks the 3PL provider to provide continuous improvement of logistics service to maintain the freshness of the flower. Another example that supports our model is customization. A customer who orders an individualized product from a retailer does not want a long order cycle but prefers to obtain the product more quickly and conveniently. To quickly respond to the demand of end customers the 3PL provider can decrease the logistics time. Shorter logistics time improves market demand and benefits supply chain members. In this study the logistics service level that affects product s demand refers to the 3PL provider s investment in the technology to maintain the product s quality or decrease the logistics time. A higher logistics service level means lower product s deterioration/damage or shorter logistics time. In other words the higher the logistics service level is the more customers will buy the product. A higher service level improves market demand and benefits the supply chain in form of improved demand potential. Unlike Cai et al. (2013) we do not use an interval of [0 1] to describe the service level and we only assume that the service level is positive. Our model regarding a 3PL provider s effort level is similar to that in Wu (2011). To improve the logistics service the 3PL provider can hire more employees buy new equipments and employ new technology. All of those activities are costly. As a result a conflict exists between chain members and the 3PL provider i.e. no matter what level the 3PL provider dedicates toward those activities chain members prefer that the 3PL provider exerts even more effort. The problem is that those activities benefit all of them but are costly to only the 3PL provider. Therefore the 3PL provider is not willing to do this. Sharing the logistics cost is one solution to the effort coordination problem. Chain

3 Logistics cost sharing in supply chains involving a third-party logistics provider 209 members can pay the 3PL provider a portion of his/her logistics cost in exchange for an increase in logistics service level. It should be noted that several conditions are needed for cost sharing to be an effective strategy: the chain members must be able to observe that the 3PL provider actually engaged in the costly activity (so the chain members know how much to compensate the 3PL provider) the 3PL provider s effort must be verifiable and the activity must directly benefit chain members. In many cases those conditions are met and chain members generally can observe and verify whether or not the 3PL provider bought new equipments and employed new technology. This study investigates cost sharing mechanisms in two kinds of supply chains. The first one is a supply chain (see Fig. 1) consisting of a manufacturer (denoted by M) a 3PL provider (denoted by L) and a retailer (denoted by R). The second one is a supply chain (see Fig. 2) consisting of two competing manufacturers (denoted by M 1 and M 2 ) a 3PL provider (denoted by L) and a retailer (denoted by R). We assume the manufacturer and the retailer are major customers of the 3PL provider otherwise they would have insufficient influence over the 3PL provider. In Kunming China many specialty 3PL providers offer logistics services to several flower companies and these flower companies are their major customers. For example Jinyuan Logistics Co. Ltd. ships flowers to Thailand for several flower companies located in Kunming. In the one manufacturer-one 3PL provider-one retailer supply chain two types of logistics cost sharing mechanisms namely decentralized and centralized cost sharing mechanisms are explored. Decentralized cost sharing mechanisms (see Sect. 3.2 for details) are proposed as contracts that chain members separately decide their cost sharing portions to optimize their own profits ignoring the collective impacts of their decisions on the channel as a whole. Centralized cost sharing mechanisms (see Sect. 3.3 for details) are in the situation that chain members negotiate their cost sharing portions so that their profits are the shares of the entire supply chain s profit implying that the supply chain is coordinated perfectly. Negotiations over cost sharing portions are similar to those given in Cachon and Lariviere (2005). In their study chain members negotiate the revenue sharing portions in a revenue sharing contract so that the supply chain is coordinated perfectly. We analyse how cost sharing mechanisms affect the supply chain and under what conditions members are willing to engage in cost sharing mechanisms. In the two manufacturers-one 3PL provider-one retailer supply chain each manufacturer produces only one product which is a substitute for the product of the Fig. 1 Cost sharing in a supply chain consisting of a manufacturer a 3PL provider and a retailer M % of logistics cost % of logistics cost L R Fig. 2 Cost sharing in a supply chain consisting of two manufacturers a 3PL provider and a retailer M 1 % of logistics cost L % of logistics cost R M 2 % of logistics cost % of logistics cost

4 210 L. Jiang et al. other manufacturer. The two manufacturers sell their products through the common 3PL provider and the common retailer. Our study is therefore also related to previous research on competing manufacturers. There are many substitutable products in different market place for example GM and Ford in auto industry; Dell and HP in computer market; Pepsi and Coca-Cola in soft drink market. In most of the consumer goods market retailers sell multiple brands at the same location. This kind of competing channel structure represents numerous markets including those consisting of specialty stores (e.g. consumer electronics sporting goods automobile parts) department stores supermarkets. Furthermore the competing manufacturers are willing to outsource their logistics service to the same 3PL provider. For example Minsheng Shipping Co. Ltd. a 3PL provider located in Chongqing China offers logistics services to two automobile manufacturers. Competing manufacturers may choose the same 3PL provider for several reasons: (i) the 3PL provider is an excellent and reliable firm making it the top choice of both manufactures; (ii) the competing manufacturers products are similar and the 3PL provide is able to integrate shipments of the competing products to decrease the logistics cost. In recent years the competing channel structure has been the focus of growing academic interest. This study extends previous works by adding a third tier a 3PL provider to the competing supply chain. In the supply chain modelled in this study both decentralized cost sharing mechanisms and centralized cost sharing mechanisms are explored and we find out that the competition between the two manufacturers has an important influence on cost sharing mechanisms. We must point out that our model does not involve pricing decisions of products such as in Chao et al. (2009). We assume the retailer purchases products from the manufacturer for a fixed wholesale price per unit and sells them to end customers for a fixed retail price per unit. A fixed price per unit quoted by the supply chain member is the standard assumption used in the newsvendor setting (Qin et al. 2011). However this assumption might be a main drawback of this paper. An ideal analytical model for such logistics service level-based demand should incorporate pricing decisions. However allowing for pricing will make the model intractable. To focus on the impact of logistics service level dimensions we restrict our study to certain markets where price is not the focus and can be considered exogenous. For industries with mature technology price is not the focus. The contributions of the paper are as follows: (i) This study examines the effect of logistics service level on supply chains and proposes a generalization of previous models and approaches. It also discusses competition based on logistics service level and contributes to research streams concerning competing supply chains; (ii) Two cost sharing mechanisms namely the decentralized and the centralized cost sharing mechanisms are explored to improve supply chain performance. The conditions for the cost sharing mechanisms to achieve win-win outcomes are indentified. The rest of this paper is organized as follows: In Sect. 2 a brief review of related research is provided. In Sect. 3 cost sharing mechanisms are characterized in a supply chain consisting of one manufacturer one 3PL provider and one retailer. In Sect. 4 cost sharing mechanisms are characterized in a supply chain consisting of two competing manufacturers one 3PL provider and one retailer. In Sect. 5 some numerical examples are given. In Sect. 6 conclusions are presented.

5 Logistics cost sharing in supply chains involving a third-party logistics provider Literature review Three research areas related to this study are as follows. (i) Supply chain coordination in a three-level supply chain; (ii) Cost sharing contract mechanisms; (iii) Manufacturer(s) sell through retailer(s) in competing environment. As an example of works that have studied supply chain coordination in a three-level supply chain the literature can be found in Munson and Rosenblatt (2001) Giannoccaro and Pontrandolfo (2004) Ding and Chen (2008) Jaber et al. (2010) Seifert et al. (2012) and He and Zhao (2012). Munson and Rosenblatt (2001) considered a three-level supply chain and explored the benefits of using quantity discounts on both end of the supply chain to decrease costs from the view of the manufacturer. Giannoccaro and Pontrandolfo (2004) proposed a model of RS mechanism aimed at coordinating a three-stage supply chain this model allowed the system efficiency to be achieved as well as it could improve the profits of all the supply chain actors by tuning the contract parameters. Ding and Chen (2008) studied the coordination issue of a three-level supply chain selling short life cycle products in a single period model. The results demonstrated that the three-level supply chain can be fully coordinated with appropriate contracts and the total profit of the channel can be allocated with any specified ratios among the firms. Jaber et al. (2010) studied three-level supply chain coordination with learning-based continuous improvement. The results demonstrated that learning-based improvements allow retailers to order in progressively smaller lot sizes and the entire supply chain benefits from implementing learning-based continuous quality improvements. Seifert et al. (2012) studied coordination in a three-echelon supply chain and examined the impact of supply chain coordination. They discovered that both the supplier and the retailer would prefer to act alone rather than to coordinate with the manufacturer. He and Zhao (2012) studied coordination in a three-echelon supply chain. They proposed an agreement that includes a buyback policy between the manufacturer and the retailer but stipulates a price-only contract between the supplier and the manufacturer. The results demonstrated that such an agreement can achieve perfect supply chain coordination. Cost sharing contract mechanism is an effective mechanism to coordinate supply chains. Some literature studies cost sharing contract in supply chains made up of two levels. Corbett and DeCroix (2001) discussed the use of a shared savings contract to induce supplier and buyer effort that reduces indirect material consumption. Chao et al. (2009) discussed two contractual agreements by which product recall costs can be shared between a manufacturer and a supplier to induce quality improvement effort. Leng and Parlar (2010) considered a multiple-supplier single manufacturer assembly supply chain and introduced cost sharing contracts to coordinate this assembly supply chain. Balireddi and Uhan (2012) investigated cost sharing mechanisms for scheduling cost-sharing games. However in reality supply chains often contain three or even more firm levels and coordination should not be limited to two levels. Choi (1991) analyzed a channel structure with two competing manufacturers and one common retailer that sells both manufacturers products. He studied three noncooperative games of different power structures between the two manufacturers and the retailer i.e. two Stackelberg and one Nash games. Choi (1996) studied multiple manufacturer-multiple retailer channel a major contribution of this paper was to incor-

6 212 L. Jiang et al. porate store differentiation in the model in addition to the usual product differentiation. Trivedi (1998) analyzed a channel structure with two competing manufacturers and two retailers each of whom distributes both products in a noncooperative game with two Nash and one Stackleberg equilibrium. He showed that the presence of competitive effects at both retail and manufacturer levels of distribution had a significant impact on profits and prices. Lee and Staelin (1997) provided an in-depth discussion on why previous studies depended upon the choice of the form of demand functions and their investigation involved mathematical analysis of an industry model composed of two manufacturers selling competing products both carried by two competing retailers. This general model could be used to analyze three more restrictive industry settings often found in the channels literature i.e. a bilateral monopoly two competing manufacturers selling through competing retailers and two competing manufacturers selling through one common retailer. Cachon and Kök (2010) studied how the firms manage their contract negotiations when multiple manufacturers sell through a single retailer. They allowed the manufacturers to compete for the retailer s business using one of three types of contracts: a wholesale-price contract a quantity-discount contract or a two-part tariff. Sinha and Sarmah (2010) analyzed the coordination and competition issues in a two-stage supply-chain distribution system where two vendors compete to sell differentiated products through a common retailer in the same market. Pan et al. (2010) considered a supply chain with two manufacturers and one retailer they discussed the results of price-only contracts and RS contracts under different channel power structures to check whether it was beneficial for manufacturers to use RS contracts under different scenarios. Subsequently they considered a supply chain with one manufacturer and two retailers and analyzed the likely outcomes under different scenarios. 3 One manufacturer-one 3PL provider-one retailer supply chain We assume that the manufacturer produces only one product. The logistics service level-dependent demand is given by q = Q 0 + αl where Q 0 denotes the primary market demand l denotes the logistics service level provided by the 3PL provider and can be observed through the 3PL provider s investment on new equipments and technology and α(α > 0) denotes the parameter that influences consumer sensitivity to the logistics service level. In this study the product demand q increases in the logistics service level l. To focus on the impact of logistics service level dimensions our model does not involve pricing decisions of products. m M denotes the manufacturer s unit profit which is equal to the manufacturer s wholesale price minus manufacturing costs. Similarly m L and m R denote the 3PL provider and the retailer s unit profits respectively. We let m T = m R + m L + m M.c(l) denotes the logistics cost function of the 3PL provider and assume c(l) = kl 2 this kind of convex cost function is used in much of the literature (Wu 2011). The decision variable r 1 (0 r 1 < 1)denotes the manufacturer s sharing portion of logistics cost. The decision variable r 2 (0 r 2 < 1) denotes the retailer s sharing portion of logistics cost. All data are assumed to be common knowledge.

7 Logistics cost sharing in supply chains involving a third-party logistics provider 213 Superscript * is added to the relative variables to represent their corresponding optimal values in a decentralized supply chain. Superscript is added to the relative variables to represent their corresponding optimal values in a centralized supply chain. 3.1 No cost sharing We first characterize the optimal decision with no cost sharing and use it as a benchmark to evaluate the performance of cost sharing mechanisms. The retailer s profit function R the 3PL provider s profit function L and the manufacturer s profit function M can be expressed as R = m R (Q 0 + αl). (1) L = m L (Q 0 + αl) kl 2. (2) M = m M (Q 0 + αl). (3) Therefore the profit function of the entire supply chain can be expressed as T = M + L + R = m T (Q 0 + αl) kl 2. (4) The 3PL provider determines the logistics service level l to maximize its profit as given in Eq. (2). Solving the first-order condition L / l =0forl gives the optimal logistics service level as l =α m L /(2k). Hence chain members optimal profits are obtained by substituting l into corresponding equalities. The optimal results are reported as follows: R = m R(2kQ 0 + α 2 m L ) 2k M = m M(2kQ 0 + α 2 m L ) T 2k = m T L = m L(4kQ 0 + α 2 m L ) 4k ( Q 0 + α2 m L 2k ) α2 m 2 L 4k. However from the viewpoint of the entire supply chain s profit we know that when l = α m T /(2k) the entire supply chain s profi is maximized and the optimal total profit is T = m T Q 0 +α 2 m 2 T /(4k). Because there is no cost sharing the 3PL provider does not have sufficient incentive to improve its logistics service level hence l < l and T < T the suboptimal decisions and the loss of the total profit emerge. 3.2 Decentralized cost sharing mechanisms To encourage the 3PL provider to improve the logistics service level both manufacturer and retailer execute cost sharing mechanisms. In the decentralized cost sharing mechanisms the manufacturer and the retailer decide their cost sharing portions sequentially. The sequence of events taking place within this decentralized system is as follows: (i) the manufacturer decides the cost sharing portion r 1 ; (ii) the retailer decides the cost sharing portion r 2 ; (iii) according to r 1 and r 2 the 3PL provider decides a logistics

8 214 L. Jiang et al. service level l. For convenience a subscript DCS is added on the left of a notation to represent the decentralized cost sharing mechanism scenario. The retailer s profit function DCS R the 3PL provider s profit function DCS L and the manufacturer s profit function DCS M can be expressed as DCS R = m R (Q 0 + α DCS l) r 2 k DCS l 2. (5) DCS L = m L (Q 0 + α DCS l) (1 r 1 r 2 )k DCS l 2. (6) DCS M = m M (Q 0 + α DCS l) r 1 k DCS l 2. (7) The model is solved through backwards induction. Given r 1 and r 2 the 3PL provider determines DCS l to maximize its profit as given in Eq. (6). Solving the first-order condition DCS L / DCS l =0for DCS l gives the best reaction function of the 3PL provider as DCSl = α m L 2(1 r 1 r 2 )k. (8) Given r 1 the retailer uses the foresight of the best reaction function of the 3PL provider to determine r 2 to maximize its profit as given in Eq. (5). Substituting Eq. (8) into Eq. (5) and applying the first-order condition DCS R / r 2 = 0 to the resulting profit function the best reaction function of the retailer can be derived as r 2 = (1 r 1 ) 2m R m L 2m R + m L. (9) Substituting Eq. (9) into Eq. (8) and simplifying Eq. (8) can be rewritten as DCSl = α(2m R + m L ). (10) 4(1 r 1 )k The manufacturer uses the foresight of the best reaction functions of the 3PL provider and retailer to determine r 1 to maximize its profit as given in Eq. (7). Substituting Eq. (10) into Eq. (7) we have [ ] DCS M = m M Q 0 + α2 (2m R + m L ) + α2 (2m R + m L ) 2 4(1 r 1 )k 16(1 r 1 )k α2 (2m R + m L ) 2 16(1 r 1 ) 2. k DCS M is concave in R = 1/(1 r 1 ). To simplify the computation process we first obtain the optimal R according to the first-order condition DCS M / R = 0 then solving R = 1/(1 r1 ) for r 1 we obtain the optimal cost sharing portion r 1. Hence the optimal cost sharing portion r2 the optimal logistics service level DCSl and the optimal profits of the chain members are obtained by substituting r1 into the corresponding equalities. The optimal cost sharing portions and the optimal logistics service level are reported as follow:

9 Logistics cost sharing in supply chains involving a third-party logistics provider 215 r 1 = 4m M 2m R m L 4m M + 2m R + m L r 2 = 2(2m R m L ) 4m M + 2m R + m L DCSl = α(4m M + 2m R + m L ). The optimal profits of the chain members are reported as follows: DCS R = m R Q 0 + α2 (2m R + m L )(4m M + 2m R + m L ) 32k [ ] DCS L = m L Q 0 + α2 (4m M + 2m R + m L ) DCS M = m M Q 0 + α2 (4m M + 2m R + m L ) 2 DCS T = m T Q 0 + α2 (4m M + 6m R + 7m L )(4m M + 2m R + m L ). Achieving a win-win outcome is a necessary requirement for implementing the cost sharing contract. Next we discuss the conditions for a win-win outcome of this decentralized cost sharing contract. Proposition 1 Let 1 ={(m R m L m M ) 0 4m M 2m R + m L 2m R m L }.Only if (m R m L m M ) are in 1 then r1 0 and r 2 0. Comparing the optimal profits with decentralized cost sharing and without it we find out that if (m R m L m M ) are in 1 then DCS M M DCS L L DCS R R. Proposition 1 shows that whether the decentralized cost sharing contracts perform well critically depends on chain members unit profits. Only if chain members unit profits are in 1 i.e. the manufacturer and retailer have sufficient profit margins to compensate the logistics cost of the 3PL provider then the cost sharing mechanisms enable the chain members profits to increase and the cost sharing becomes an effective strategy. On the contrary if chain members unit profits are out of 1 this type of cost sharing mechanism does not work. We also find out that if (m R m L m M ) is in 1 then r1 / m M = 8(2m R + m L )/(4m M + 2m R + m L ) 2 > 0 r2 / m R = 4(4m M 2m R + 3m L )/(4m M + 2m R + m L ) 2 > 0 i.e. r 1 is increasing in m M and r 2 is increasing in m R. When unit profits increase both manufacturer and retailer will increase their sharing portions of logistics cost a finding consistent with our intuition. 3.3 Centralized cost sharing mechanisms Though the decentralized cost sharing mechanisms mentioned above offer the opportunity to improve the profits of chain members but chain members decide the cost

10 216 L. Jiang et al. sharing portions sequentially and part profit will be lost. To enable the decentralized supply chain to achieve the same performance as the centralized one the centralized cost sharing mechanisms are developed. In the centralized cost sharing mechanisms chain members negotiate their cost sharing portions so that their profits are the shares of the entire supply chain s profit indicating that the supply chain is coordinated perfectly. For convenience a subscript CCS is added on the left of a notation to represent the centralized cost sharing mechanism scenario. Under the centralized cost sharing mechanisms the manufacturer s profit function CCS M the retailer s profit function CCS R and the 3PL provider s profit function CCS L are the same as in Sect Proposition 2 The centralized cost sharing mechanisms with r 1 = m M /m T and r 2 = m R /m T coordinate the supply chain perfectly i.e. if chain members sharing portions of logistics cost are equivalent to their revenue portions from the supply chain the entire supply chain will be coordinated perfectly. Given the cost sharing mechanisms with r 1 = m M /m T and r 2 = m R /m T chain members profit functions CCS M CCS R and CCS L satisfy CCS M = (m M /m T ) T CCS R = (m R /m T ) T CCS L = (m L /m T ) T. All these profits are the shares of the entire supply chain s profit. When we maximize the profits of the chain members the profit of the entire supply chain is also maximized and there is no gap between the decentralized supply chain and the centralized one. Thus this supply chain is perfectly coordinated by the proposed centralized cost sharing mechanisms. Under the centralized cost sharing mechanisms l is the 3PL provider s optimal logistics service level. The optimal profits of members are shown as follows: CCS R = (m R/m T ) T CCS M = (m M/m T ) T CCS L = (m L/m T ) T where T is the maximal profit of the entire supply chain (see Sect. 3.1). Achieving a win-win outcome is a necessary requirement for implementing the centralized cost sharing contract. To achieve a win-win outcome one should set CCS R R CCS M M CCS L L (11) where the right sides of the inequalities are the optimal profits without cost sharing. Expression (11) is equivalent to m M + m R m L. Let 2 ={(m R m L m M ) 0 m M + m R m L }. (12) Clearly the centralized cost sharing contract with (m R m L m M ) in 2 has the opportunity to achieve a win-win outcome. We proceed to show that 1 2 (4m M 2m R +m L and 2m R m L imply that m M +m R m L ). The conditions for the decentralized cost sharing contract to achieve a win-win outcome are more constrainted than

11 Logistics cost sharing in supply chains involving a third-party logistics provider 217 the conditions for the centralized cost sharing contract to achieve a win-win outcome i.e. the centralized cost sharing contract is easier to implement than the decentralized cost sharing contract. On the other hand CCS T DCS T in terms of the total profit the centralized cost sharing mechanisms are prior to the decentralized cost sharing mechanisms. Thus from the perspective of the entire supply chain implementing the centralized cost sharing contract is the better option. Next we analyse whether it is more profitable for the individual firm to implement the centralized cost sharing contract compared to the decentralized contract. Proposition 3 Comparing the optimal profits as summarized in the decentralized cost sharing mechanisms and the centralized cost sharing mechanisms we find out that (i) If 4m 2 R +4m Rm L 4m M m L m 2 L 0 then CCS R DCS R. In other words if the retailer s unit profit m R is sufficiently large compared to m M and m L so that the former formula is nonnegative the retailer has incentive to engage in the centralized cost sharing mechanisms compared to the decentralized cost sharing mechanisms. (ii) If 8m M m L (2m R + m L ) 2 0 then CCS M DCS M. In other words if the manufacturer s unit profit m M is sufficiently large compared to m R and m L so that the former formula is nonnegative the manufacturer has incentive to engage in the centralized cost sharing mechanisms compared to the decentralized cost sharing mechanisms. (iii) CCS L > DCS L. the 3PL provider has incentive to engage in the centralized cost sharing mechanisms compared to the decentralized cost sharing mechanisms. 4 Two manufacturers-one 3PL provider-one retailer supply chain We now consider a supply-chain-wide system consisting of two manufacturers one 3PL provider and one retailer. The demand for product i is assumed to increase in its own logistics service level and decrease in that of the competitor this is a reasonable assumption because end customers prefer to buy competitor s products when the competitor s logistics service level increases. Similar to Choi (1991) and Lee and Staelin (1997) the logistics service level-dependent demand function is assumed to be q i = Q 1 + α l i β l j i = 1 2 and j = 3 i (13) where Q 1 denotes the same primary market demand of two products [the symmetric demand reduces the complexity of the discussion but is not particularly restrictive (Lee and Staelin 1997)] α(α > 0) denotes the parameter that influences consumers sensitivity to logistics service level and β(β > 0) denotes the competitive factors which is a measure of the sensitivity of product i s sale to change in product j s logistics service level. We assume α > β which ensures the existence of the equilibrium solution this assumption is reasonable because the product s demand is more sensitive to its own logistics service level than that of the competitor. Here α and β capture the product differentiation and α β is related to the degree of product substitutability

12 218 L. Jiang et al. that is the smaller α β is the more substitutable the product becomes. However our model does not permit α = β (i.e. completely substitutable). In fact the two manufacturers may not have the ability to produce identical products because of patent protection technological and geographical constraints or customer preference. Similar to Sect. 3 m M denotes the identical manufacturers unit profits. m L and m R denote the 3PL provider and the retailer s unit profits respectively. We let m T = m R +m L +m M.c(l i ) = kl 2 i denotes the logistics cost function. The decision variables r 1 (0 r 1 < 1) and r 2 (0 r 2 < 1) denote manufacturer 1 and the retailer s cost sharing portion for product 1 respectively. The decision variables λ 1 (0 λ 1 < 1) and λ 2 (0 λ 2 < 1) denote manufacturer 2 and the retailer s cost sharing portion for product 2 respectively. All data are assumed to be common knowledge. We add subscript MM on the left of a notation to represent the two manufacturer scenario. 4.1 No cost sharing The optimal decision is characterized in the supply chain with no cost sharing and it is used as a benchmark to verify whether it is beneficial for the manufacturers to choose cost sharing mechanism. The retailer s profit function MM R the 3PL provider s profit function MM L the manufacturer 1 s profit function MM M1 and the manufacturer 2 s profit function MM M2 can be expressed as MM R = m R [2Q 1 + (α β)(l 1 + l 2 )]. (14) MM L = m L [2Q 1 + (α β)(l 1 + l 2 )] k(l1 2 + l2 2 ). (15) MM M1 = m M (Q 1 + α l 1 β l 2 ). (16) MM M2 = m M (Q 1 + α l 2 β l 1 ). (17) The profit function of the entire supply chain can be expressed as MM T = MM M1 + MM M2 + MM L + MM R = m T [2Q 1 + (α β)(l 1 + l 2 )] k(l1 2 + l2 2 ). (18) The 3PL provider determines the logistics service level l i to maximize its profit as given in Eq. (15). Solving the first-order conditions MM L / l i = 0 gives the optimal logistics service level as l 1 = l 2 = (α β)m L/2k. Hence the optimal profits of the chain members are obtained by substituting l i into corresponding equalities. The optimal results are reported as follows: MM R = m R[2kQ 1 + (α β) 2 m L ] k MM L = m L[4kQ 1 + (α β) 2 m L ] 2k MM M1 = MM M 2 = m M[2kQ 1 + (α β) 2 m L ] 2k

13 Logistics cost sharing in supply chains involving a third-party logistics provider 219 MM T = m T [ 2Q 1 + (α ] β)2 m L k (α β)2 m 2 L. 2k However from the viewpoint of the entire supply chain s profit we know that when l 1 = l 2 = (α β)m T /(2k) the profit of the entire supply chain is maximized and the optimal total profit is MM T = 2m T Q 1 +(α β) 2 m 2 T /(2k). Because there is no cost sharing the 3PL provider does not have sufficient incentives to improve the logistics service level hence l1 = l 2 < l 1 = l 2 and MM T < MM T the suboptimal decisions and the loss of the total profit emerge. 4.2 Only one of the manufacturers chooses a cost sharing mechanism We suppose only manufacturer 1 and the retailer choose a cost sharing mechanism for product 1 while manufacturer 2 and the retailer choose a price-only contract for product 2 (see Fig. 3). We investigate whether manufacturer 1 can benefit from this kind of partial cost sharing mechanisms and if so under what conditions. The sequence of events taking place within this decentralized system is as follows: (i) manufacturer 1 decides the cost sharing portion r 1 ; (ii) the retailer decides the cost sharing portion r 2 ; (iii) according to r 1 and r 2 the 3PL provider chooses the logistics service level l i. For convenience a subscript PCS is added on the left of a notation to represent this kind of partial cost sharing mechanism scenario. The retailer s profit function PCSMM R the 3PL provider s profit function PCSMM L the manufacturer 1 s profit function PCSMM M1 and the manufacturer 2 s profit function PCSMM M2 can be expressed as PCSMM R = m R [2Q 1 + (α β)( PCS l 1 + PCS l 2 )] r 2 k PCS l 2 1. (19) PCSMM L = m L [2Q 1 + (α β)( PCS l 1 + PCS l 2 )] (1 r 1 r 2 )k PCS l 2 1 k PCS l 2 2. (20) PCSMM M1 = m M (Q 1 + α PCS l 1 β PCS l 2 ) r 1 k PCS l 2 1. (21) PCSMM M2 = m M (Q 1 + α PCS l 2 β PCS l 1 ). (22) The model is solved through backwards induction. Given r 1 and r 2 the 3PL provider determines PCS l i to maximize its profit as given in Eq. (20). Solving the first-order condition PCSMM L / PCS l i = 0 gives the best reaction functions of the 3PL provider as PCSl 1 = (α β)m L 2(1 r 1 r 2 )k PCSl 2 = (α β)m L. (23) 2k Fig. 3 Only manufacturer 1 and the retailer choose a cost sharing mechanism M1 r 1 r 2 L R M 2

14 220 L. Jiang et al. Given r 1 the retailer uses the foresight of the best reaction functions of the 3PL provider to determine r 2 to maximize its profit as given in Eq. (19). Substituting Eq. (23) into Eq. (19) and applying the first-order condition PCSMM R / r 2 = 0 to the resulting profit function the best reaction function of the retailer is derived as r 2 = (1 r 1 ) 2m R m L 2m R + m L. (24) Substituting Eq. (24) into Eq. (23) Eq. (23) can be rewritten as PCSl 1 = (α β)(2m R + m L ) 4(1 r 1 )k PCSl 2 = (α β)m L. (25) 2k Manufacturer 1 uses the foresight of the best reaction functions of the 3PL provider and the retailer to determine r 1 to maximize its profit. Substituting Eq. (25) into Eq. (21) Eq. (21) can be rewritten as [ PCSMM M1 = m M Q 1 + α(α β)(2m R + m L ) β(α β)m ] L 4(1 r 1 )k 2k + (α β)2 (2m R + m L ) 2 16(1 r 1 )k (α β)2 (2m R + m L ) 2 16(1 r 1 ) 2. (26) k PCSMM M1 is concave in 1/(1 r 1 ). Similar to Sect. 3.2 the optimal cost sharing portion r1 is obtained by applying the first-order condition PCSMM M1 / [1/(1 r 1 )]= 0. Hence the optimal cost sharing portion r2 the optimal logistics service level PCSli and the optimal profits of chain members are obtained by substituting r1 into the corresponding equalities. The optimal cost sharing portions and the optimal logistics service levels are reported as follows: r 1 = 4αm M (α β)(2m R + m L ) 4αm M + (α β)(2m R + m L ) r 2 = 2(α β)(2m R m L ) 4αm M + (α β)(2m R + m L ) PCSl 1 = 4αm M + (α β)(2m R + m L ) PCSl2 = (α β)m L. 2k The optimal profits of the chain members are as follows: PCSMM R = 2m R Q 1 + 4α(α β)m M(2m R + m L ) + (α β) 2 (4m 2 R +m2 L + 20m Rm L ) 32k [ PCSMM L = m L 2Q 1 + 4α(α β)m M + (α β) 2 ] (2m R + 5m L ) [ ] PCSMM M1 = m M Q 1 + 2α2 m M + 2α(α β)m R + (α β)(α 4β)m L + (α β)2 (2m R + m L ) 2

15 Logistics cost sharing in supply chains involving a third-party logistics provider 221 PCSMM M2 = m M [ Q 1 + (α β)(4α β)m ] L 4αβm M 2β(α β)m R. Proposition 4 Let 3 = {(m R m L m M ) 0 4αm M (α β)(2m R + m L ) 2m R m L }.Onlyif(m R m L m M ) are in 3 then r1 0 and r 2 0. Comparing the optimal profits with the partial cost sharing mechanisms and without it we find out that if (m R m L m M ) are in 3 then PCSMM M1 MM M1 PCSMM L MM L PCSMM R MM R and PCSMM M2 MM M2. Proposition 4 shows that if (m R m L m M ) are in 3 both manufacturer 1 and the retailer can benefit from this kind of partial cost sharing mechanisms and they will prefer to engage in a cost sharing contract. However manufacturer 2 incurs loss because it does not execute cost sharing mechanisms to improve the logistics service level. On the contrary if (m R m L m M ) are out of 3 neither manufacturer 1 nor the retailer has sufficient profit margin to compensate the logistics cost of the 3PL provider they have no choice but to engage in a price-only contract. 4.3 Both manufacturers choose cost sharing mechanisms Decentralized cost sharing mechanisms In the decentralized cost sharing mechanisms both manufacturers choose cost sharing mechanisms (see Fig. 4) and the two manufacturers and the retailer decide the cost sharing portions sequentially. The sequence of events taking place within this decentralized system is as follows: (i) each manufacturer decides the cost sharing portion r 1 or λ 1 conditional on the competitor s cost sharing portion; (ii) the retailer decides the cost sharing portion r 2 and λ 2 ; (iii) according to r i and λ i the 3PL provider chooses the logistics service level l i. For convenience a subscript BCS is added on the left of a notation to represent the scenario when both manufacturers choose the decentralized cost sharing mechanisms. The retailer s profit function BCSMM R the 3PL provider s profit function BCSMM L the manufacturer 1 s profit function BCSMM M1 and the manufacturer 2 s profit function BCSMM M2 can be expressed as BCSMM R = m R [2Q 1 + (α β)( PCS l 1 + PCS l 2 )] r 2 k BCS l1 2 λ 2k BCS l2 2. (27) BCSMM L = m L [2Q 1 + (α β)( PCS l 1 + PCS l 2 )] (1 r 1 r 2 )k PCS l1 2 (1 λ 1 λ 2 )k PCS l2 2. (28) Fig. 4 Both manufacturers choose cost sharing mechanisms M 1 r 1 r 2 L R M 2 λ 1 λ 2

16 222 L. Jiang et al. BCSMM M1 = m M (Q 1 + α PCS l 1 β PCS l 2 ) r 1 k PCS l 2 1. (29) BCSMM M2 = m M (Q 1 + α PCS l 2 β PCS l 1 ) λ 1 k PCS l 2 2. (30) The model is solved through backwards induction. Given r i and λ i the 3PL provider determines BCS l i to maximize its profit as given in Eq. (28). Solving the first-order condition BCSMM L / BCS l i =0for BCS l 1 and BCS l 2 gives the best reaction functions of the 3PL provider as BCSl 1 = (α β)m L 2(1 r 1 r 2 )k BCSl 2 = (α β)m L 2(1 λ 1 λ 2 )k. (31) Given r 1 and λ 1 the retailer uses the foresight of best reaction functions of the 3PL provider to determine r 2 and λ 2 to maximize its profit as given in Eq. (27). Substituting Eq. (31) into Eq. (27) and applying the first-order conditions BCSMM R / r 2 = 0 BCSMM R / λ 2 = 0 to the resulting profit function the best reaction functions of the retailer are derived as r 2 = (1 r 1 ) 2m R m L 2m R + m L λ 2 = (1 λ 1 ) 2m R m L 2m R + m L. (32) Substituting Eq. (32) into Eq. (31) Eq. (31) can be rewritten as BCSl 1 = (α β)(2m R + m L ) 4(1 r 1 )k BCSl 2 = (α β)(2m R + m L ). (33) 4(1 λ 1 )k Manufacturer i uses the foresight of the best reaction functions of the 3PL provider and the retailer to determine r 1 (or λ 1 ) to maximize its profit conditional on the competitor s cost sharing portion. Substituting Eq. (33) into Eqs. (29) and (30) respectively the optimal cost sharing portions r1 and λ 1 are obtained by solving the first-order conditions BCSMM M1 / [1/(1 r 1 )]=0 PCSMM M2 / [1/(1 λ 1 )]=0tothe resulting profit function. Hence the optimal cost sharing portion r2 λ 2 the optimal logistics service level BCS li and the optimal profits of chain members are obtained by substituting r1 and λ 1 into the corresponding equalities. The optimal cost sharing portions and the optimal logistics service level are reported as follows: r 1 = λ 1 = 4m M 2m R m L 4m M + 2m R + m L r 2 = λ 2 = 2(2m R m L ) 4m M + 2m R + m L BCSl1 = BCSl2 = (α β)(4m M + 2m R + m L ). The optimal profits of the chain members are reported as follows: BCSMM R = 2m R Q 1 + (α β)2 (2m R + m L )(4m M + 2m R + m L ) [ BCSMM L = m L 2Q 1 + (α ] β)2 (4m M + 2m R + m L )

17 Logistics cost sharing in supply chains involving a third-party logistics provider 223 BCSMM M1 = BCSMM M2 = m M Q 1 + (α β)2 (4m M + 2m R + m L ) 2. Proposition 5 Only if 4m M 2m R + m L 2m R m L then ri 0 λi 0 and BCSMM M1 = BCSMM M2 MM M1 = MM M2 BCSMM L MM L BCSMM R MM R. The two manufacturers the 3PL provider and the retailer can all benefit from this kind of cost sharing mechanisms. Note that the degree of product substitutability between the two manufacturers is inversely related to the difference between α and β i.e. as β approaches α the two products become more substitutable. Most existing literature on competing supply chains (Choi 1991; Yao et al. 2008) focuses on the effects of parameters α and β on supply chain performance. Thus it is necessary to discuss how supply chain performance changes with different values for α and β. For comparative purposes the profit improvements of the manufacturers the 3PL provider and the retailer are defined as 1 = BCSMM M1 MM M1 = BCSMM M2 MM M2 2 = BCSMM L MM L and 3 = BCSMM R MM R respectively. Proposition 6 If 4m M 2m R + m L and 2m R m L the effects of α and β on i (i = 1 2 3) are as follows. 1 (α β) = 2(α β)[(4m M +2m R +m L ) 2 32m M m L ] 2(α β)(4m M 2m L ) 2 (i) increasing in α β. (ii) (iii) 3 is increasing in α β. 2 (α β) = 4m M +2m R 3m L > 0 2 is increasing in α β. 3 (α β) = 2(α β)[(4m M +2m R +m L )(2m R +m L ) 16m R m L ] 4(α β)(2m R m L ) 2 > 0 1 is > 0 Proposition 6 shows that when α β decreases i decreases i.e. everyone s profit improvement decreases as products become more substitutable and the cost sharing mechanisms work poorly when the degree of product substitutability between the two manufacturers is very high Centralized cost sharing mechanisms Although the abovementioned decentralized cost sharing mechanisms improve the profits of supply chain members but the two manufacturers and the retailer decide their cost sharing portions separately and ignore the collective impacts of their decisions on the channel as a whole hence part profit is lost. If all of them negotiate the cost sharing portions so that their profits are the shares of the entire supply chain s profit the supply chain can be coordinated. We add subscript CCS on the left of a notation to represent the scenario of the coordinated cost sharing mechanisms. Under the centralized cost sharing mechanisms the manufacturer 1 s profit function CCSMM M1 the manufacturer 2 s profit function CCSMM M2 the retailer s profit function CCSMM R and the 3PL provider s profit function CCSMM L are the same as in Sect Proposition 7 The centralized cost sharing mechanisms with r 1 = λ 1 = m M /m T r 2 = λ 2 = m R /m T coordinate the entire supply chain perfectly.

18 224 L. Jiang et al. Given the cost sharing mechanisms with r 1 = λ 1 = m M /m T r 2 = λ 2 = m R /m T chain members profit functions CCSMM M1 CCSMM M2 CCSMM R and CCSMM L satisfy CCSMM M1 = CCSMM M2 = (m M /2m T ) MM T CCSMM R = (m R /m T ) MM T CCSMM L = (m L /m T ) MM T. All these profits are the shares of the entire supply chain s profit. When we maximize the profits of the chain members the profit of the entire supply chain is also maximized and there is no gap between the decentralized supply chain and the centralized one. Thus this supply chain is perfectly coordinated by the centralized cost sharing mechanisms. Under the centralized cost sharing mechanisms l 1 = l 2 is the 3PL provider s optimal logistics service level. The optimal profits of members are shown as follows: CCSMM M1 = CCSMM M2 = (m M/2m T ) MM T CCSMM R = (m R/m T ) MM T CCSMM L = (m L/m T ) MM T where MM T is the maximal profit of the entire supply chain (see Sect. 4.1). We discover that if m M + m R m T then CCSMM M1 = CCSMM M2 MM M1 = MM M2 CCSMM R MM R and CCSMM L > MM L i.e. a win-win outcome is obtained by the centralized cost sharing mechanisms. Similar to Sect. 3the conditions for the decentralized cost sharing contract to achieve a win-win outcome are more constrainted than the conditions for the centralized cost sharing contract to achieve a win-win outcome i.e. the centralized cost sharing contract can be more easily implemented than the decentralized cost sharing contract. Next we analyse whether it is more profitable for the individual firm to implement the centralized cost sharing contract compared to the decentralized one. Proposition 8 Comparing the optimal profits as summarized in the decentralized cost sharing mechanisms and the centralized cost sharing mechanisms we find out that (i) If 4m 2 R + 4m Rm L 4m M m L m 2 L 0 then CCSMM R BCSMM R.In other words if the retailer s unit profit m R is sufficiently large so that the former formula is nonnegative the retailer has incentive to engage in the centralized cost sharing mechanisms compared to the decentralized cost sharing mechanisms. (ii) CCSMM L > BCSMM L the 3PL provider has incentive to engage in the centralized cost sharing mechanisms compared to the decentralized cost sharing mechanisms. (iii) If 8m M m L (2m R + m L ) 2 0 then CCSMM M1 BCSMM M1. In other words if the manufacturer s unit profit m M is sufficiently large so that the former formula is nonnegative the manufacturer has incentive to engage in the centralized cost sharing mechanisms compared to the decentralized cost sharing mechanisms.

19 Logistics cost sharing in supply chains involving a third-party logistics provider 225 Table 1 Optimal solutions under the price-only contract and the decentralized cost sharing mechanisms when β changes from 3.5 to 6 β MM M1 MM L MM R BCSMM M1 BCSMM L BCSMM R Numerical examples This section provides a numerical example that compares the performance of a competing supply chain with cost sharing and without it. Similar numerical solutions can be observed in one manufacturer-one 3PL provider-one retailer supply chain but for brevity we omit it here. Our example is generated as follows. The demand function is given by q i = l i 3.5l j. The unit profits are given by m M = 11 m L = 4 m R = 7. The logistics cost function is given by c(l) = 2.5l 2. When there is no cost sharing the optimal profits of chain members are given as follows: MM M1 = MM M2 = MM L = MM R = Case 1 We examine the effects of two types of cost sharing mechanisms. The optimal profits of chain members under the decentralized cost sharing mechanisms are given as follows: BCSMM M1 = BCSMM M2 = BCSMM L = BCSMM R = Under the decentralized cost sharing mechanisms the optimal profits of the manufacturers the 3PL provider and the retailer are improved by and 25.4 % respectively. The optimal profits of chain members under the centralized cost sharing mechanisms are given as follows: CCSMM M1 = CCSMM M2 = CCSMM L = CCSMM R = Under the centralized cost sharing mechanisms the optimal profits of the manufacturers the 3PL provider and the retailer are improved by and 29.9 % respectively. Case 2 The effect of product substitutability on the decentralized cost sharing mechanisms is examined. The demand function is given by q i = l i βl j where

20 226 L. Jiang et al. Table 2 Contrasted outcome between the price-only contract and the decentralized cost sharing mechanisms when β changes from 3.5 to 6 β BCSMM M1 MM M1 MM M1 (%) BCSMM L MM L MM L (%) BCSMM R MM R MM R (%) Table 3 Optimal solutions under the price-only contract and the centralized cost sharing mechanisms when β changes from 3.5 to 6 β MM M1 MM L MM R CCSMM M1 CCSMM L CCSMM R β changes from 3.5 to 6 at intervals of 0.5. The optimal profits under the priceonly contract and the decentralized cost sharing mechanisms are reported in Table 1. The contrasted outcomes are reported in Table 2. Table 2 shows that for a fixed α = 6.5 when β α the performance of the decentralized cost sharing mechanisms decreases. Particularly when the difference between α and β is very small the improvements of chain members profits by the decentralized cost sharing mechanisms are very small i.e. when the degree of product substitutability between the two manufacturers is very high the decentralized cost sharing mechanisms work poorly. Case 3 The effect of product substitutability on the centralized cost sharing mechanisms is examined. The demand function is given by q i = l i βl j where β changes from 3.5 to 6 at intervals of 0.5. The optimal profits under the priceonly contract and the centralized cost sharing mechanisms are reported in Table 3. The contrasted outcomes are reported in Table 4. Table 4 shows that for a fixed α = 6.5 when β α the performance of the centralized cost sharing mechanisms decreases. Especially when the difference between α and β is very small the improvements of chain members profits by the centralized cost sharing mechanisms are very small i.e. when the degree of product substitutability between the two manufacturers is very high the centralized cost sharing mechanisms work poorly.