Contract design in a cross-sales supply chain with demand information asymmetry

European Journal of Operational Research 275 (2019) 939–956

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Production, Manufacturing, Transportation and Logistics

Contract design in a cross-sales supply chain with demand information asymmetry Xiaojing Li a, Jing Chen b, Xingzheng Ai c,∗ a

School of Business, Sichuan Normal University, Chengdu, China Rowe School of Business, Dalhousie University, Halifax B3H 4R2, Canada c School of Economics and Management, University of Electronic Science and Technology of China, Chengdu, China b

a r t i c l e

i n f o

Article history: Received 14 October 2017 Accepted 18 December 2018 Available online 22 December 2018 Keywords: Supply chain management Supply chain contract Cross-sales Game theory Information asymmetry

a b s t r a c t We examine a supply chain with two manufacturers in which each manufacturer implements a crosssales strategy by selling a substitutable product through two common retailers. Both retailers face uncertain demands, and their demand forecasts are private. The manufacturers, as Stackelberg leaders, can decide whether or not to acquire this private information from the retailers through contract design: either a wholesale price contract without information sharing, or a two-part tariff contract with information sharing. We derive the Bayesian–Nash equilibriums for contract choice. We find that the two-part tariff contract can be a dominant choice under certain conditions. Specifically, with a lump sum payment set by the two manufacturers, both manufacturers and retailers can benefit from a two-part tariff contract if product package substitutability is more intensive and the demand uncertainty level is relatively high, or if the product package competition is less intense. They do not benefit from the two-part tariff contract, however, if product package substitutability is more intense and the demand uncertainty is relatively low. In addition, by comparing to the case in which the demand is deterministic, we find that a two-part tariff contract with shared information is more likely to benefit both manufacturers and both retailers than a wholesale price contract without information sharing. The implications of the contract choices are discussed. © 2018 Elsevier B.V. All rights reserved.

1. Introduction ‘Cross-sales’ is the production and sales of substitutable products by multiple manufacturers through common retailers. As market competition intensifies, cross-sales are becoming common practice in many supply chains. For example, fashion manufacturers, such as Calvin Klein and Ralph Lauren, sell their apparel through the same retailers, Macy’s and Lord & Taylor, two of the largest department stores chains. P&G and Unilever sell their personal care products through both Wal-Mart and Carrefour. Ferrero Rocher and Lindt sell their chocolate products in several of the same supermarket chains. Demand uncertainty causes information asymmetry among supply chain members; since the retailers sell the products directly to consumers, they have the advantage of having better demand data than the manufacturers. The benefits of information sharing among supply chain members have been extensively studied. Economic theory shows that private information can benefit ∗

Corresponding author. E-mail address: [email protected] (X. Ai).

https://doi.org/10.1016/j.ejor.2018.12.023 0377-2217/© 2018 Elsevier B.V. All rights reserved.

a weaker party (Kreps, 1990). Private information about demand can increase the bargaining power of retailers, especially for retailers who sell products with highly uncertain demand, a short selling season, and a long lead time, as they can negotiate contracts on whether or not to share such information with manufacturers. Obviously, the retailers’ decisions on whether or not to share demand information can significantly affect the supply chain performance. Asymmetric demand information in the supply chain may lead to poor service level, high inventories, and frequent stock-outs (Holweg, Disney, Holmstrom, & Smaros, 2005). Sharing demand information has significant potential benefits throughout the supply chain, such as improving allocation and utilization of logistics resources, production planning, and customer service, and reducing inventory costs and lead times (Hill, Doran, & Stratton, 2012; Kembro, Näslund, & Olhager, 2017; Prajogo & Olhager, 2012; Zhou & Benton, 2007). Several papers have discussed the design of incentive contracts under information asymmetry (for example, Ha & Tong, 2008, 2011) and the approach of sharing demand information (Ryu, Tsukishima, & Onari, 2009), but none of the previous studies discusses the implications of information sharing for contract design with cross-sales in the supply chain.

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The objective of the present paper is to study the impact of information sharing on cross-sales from a contract design perspective. We consider two well-studied and widely-used contracts between manufacturers and retailers: a wholesale price contract without demand information sharing, and a two-part tariff contract with demand information sharing. In a two-part tariff contract, a manufacturer requires a retailer to pay a lump sum (such as a franchise fee), for eligibility to sell the products, and then sets a retail price to maximize the channel’s profit after an order is placed. By the mechanism of the two-part tariff, channel members can take advantage of the supply chain’s vertical integration in terms of the flexible allocation of the channel’s maximal profit among the participants. Two-part tariff contracts have been widely discussed in the context of the vertical coordination of supply chain. They not only reduce double marginalization of the supply chain, but also realize the vertical integration of the supply chain (Moorthy, 1988). Some studies show that a two-part tariff contract can be more efficient than a wholesale price contract (Cachon & Kök, 2010; Corbett, Zhou, & Tang, 2004; Feng & Lu, 2013; Özer & Raz, 2011), and under certain conditions a two-part tariff contract can be more advantageous to the manufacturer than the quantity discount contract (Lee & Yang, 2013; Raju & Zhang, 2005). A wholesale price contract, on the other hand, is the simplest and most common contract in practice; the contract designer may merely prefer to offer a simple contract (Cachon, 2003). In practice, retailers are increasingly required by manufacturers to share demand information through a two-part tariff contract. For example, the LiNing Company, which is a famous sports apparel company in China, has convinced its retailers to install point of sale (POS) machines to provide its company with real-time access to sales data: when a consumer pays for a product, the sales data are sent instantly to LiNing’s IT system. In the sports apparel industry, the manufacturers’ traditional approach is to produce a new product half a year ahead of receiving the order from the retailer. Once an order is received from the retailer, the manufacturer starts to produce the product, and then delivers the product to the retailer. At this point, the products are sold to consumers, to whom the manufacturer pays little attention. In recent years, however, many sport brands, including LiNing, have experienced difficulties, including inventory issues, due to information asymmetry in the traditional wholesale price contract. By contrast, under a two-part tariff contract, manufacturers and retailers form alliances, so that sales guide production. The manufacturers can start with small production batches and test the market reaction through the retailer’s market sales data. If the product does not sell well, production can be reduced or stopped. Otherwise, the manufacturers can implement a “fast order,” in which the factory breaks the normal production cycle to achieve fast production and fast delivery to the market. In the present paper, a wholesale price contract without sharing of the retailer’s demand information serves as a benchmark to illustrate the benefits of sharing demand information through a two-part tariff contract. To examine the impact of cross-sales on information sharing through contract design in the supply chain, we will answer the following questions:

1) How should manufacturers choose their contracts in a crosssales supply chain with uncertain demand? Do the manufacturers benefit more from a two-part tariff contract, sharing retailers’ demand information, or a simple wholesale price contract without sharing of demand information? 2) How can a two-part tariff contract be designed to allocate the channel’s profit among the participants in a cross-sales supply chain in the presence of demand uncertainty?

3) Facing asymmetric demand information, what is the value of information sharing in contract choice for a cross-sales supply chain? We develop a game-theoretic model for a competing supply chain with two manufacturers and two common retailers, in which each manufacturer distributes a substitutable product through both retailers. Facing uncertain demand, retailers may obtain private, imperfect data about the market, while manufacturers cannot. The two manufacturers, however, as Stackelberg leaders in the supply chain, can each independently and simultaneously offer a contract, either a wholesale price contract without sharing of retailers’ demand forecasting information, or a two-part tariff contract with sharing of retailers’ demand forecasting information. The retailers, as followers, will choose to accept or reject the two-part tariff contract. We find that a two-part tariff contract is the dominant choice under certain conditions. Specifically, with a lump sum payment set by manufacturers, all supply chain members can benefit from a two-part tariff contract under certain conditions: either when the product package substitutability is more intense, and the demand uncertainty is relatively high, or when the product competition is less intense. They will, however, suffer if the product package substitutability is more intensive and the demand uncertainty is relatively low. In addition, we show that in the presence of demand uncertainty, information sharing makes a two-part tariff contract more attractive, as compared to the traditional wholesale price contract without information sharing. The major contributions of our paper are as follows. First, although studies in the literature have discussed contracting with asymmetric information (e.g., Corbett et al., 2004; Ha & Tong, 2008, 2011; Li, 2002), none of the studies has examined how manufacturers should choose a supply chain contract. In this paper, we consider a choice of two contracts: either a two-part tariff contract with information sharing, or a wholesale price contract without information sharing. We identify the conditions under which manufacturers should offer each type of contract, and whether or not retailers should accept the contract. Second, although some studies have focused on the supply chain structure of cross-sales (e.g., Cai, Dai, & Zhao, 2012; Feng & Lu, 2013; Trivedi, 1998), to the best of our knowledge, there has been no theoretical investigation of contracting with asymmetric information in a cross-sales supply chain. Without loss of generality, this paper studies the design of vertical contracting with asymmetric information in the presence of a cross-sales supply chain, in which two competing manufacturers supply products to two competing retailers. Third, some studies of contract choice in the cross-sales supply chain focus on a symmetric contract choice for two manufacturers (e.g., Feng & Lu, 2013). That is, both manufacturers adopt the same contracts. Unlike these studies, we allow the two manufactures to choose different contracts simultaneously and analyze the equilibrium contract choice decisions. The two manufacturers compete in a Bertrand-Nash game. Each manufacturer decides which contract should be offered, anticipating the competing manufacturer’s choice. Fourthly, we examine the value of information sharing by comparing a two-part tariff contract with information sharing to a wholesale price contract without information sharing. We identify the range of the lump sum fee to achieve a win-win for the manufacturers and retailers, and the impact on contract choice of the lump sum fee in the cross-sales supply chain. Finally, we provide some new managerial insights based on our results. The remainder of this paper is organized as follows. Section 2 reviews the related literature. Section 3 presents our model. In Section 4, we study retailers’ and manufacturers’ optimal contracting strategy in a cross-sales supply chain. In Section 5, we investigate contract choice equilibriums for cross-sales supply chains with certain and uncertain demands. Section 6 expands the

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case of different packages with the same substitutability to the case with different substitutability. In Section 7 we make concluding remarks and propose some directions for future research. All proofs are in Appendix. 2. Literature review Although there are extensive studies of contract design in the literature, the majority of them focus on a one-to-one simple supply chain structure without market competition (for example, Caldentey & Haugh, 2009; Corbett et al., 2004; Li & Zhang, 2015; Özer & Wei, 2006), or a many-to-one supply chain structure with competition among manufacturers (for example, Lee & Yang, 2013; Özer & Raz, 2011; Wu, Zhai, & Huang, 2008b), or on a one-to-many supply chain structure with competition among retailers (for example, Li & Zhang, 2008; Shin & Tunca, 2010). Cachon (2003) provides a comprehensive review on this research stream. Although these studies discuss the contract design in a supply chain from different perspectives, the general conclusion is that these contracts, including two-part tariffs, revenue-sharing contracts, and buyback contracts, allow enable better system optimization than wholesale-price contracts. Yao, Leung, and Lai (2008) investigate a revenue-sharing contract for coordinating a supply chain comprising one manufacturer and two competing retailers. Cachon and Kök (2010) analyze three types of contracts (a wholesale-price contract, a quantity-discount contract, and a two-part tariff), where multiple manufacturers sell through a single retailer. They show that complicated contracts may harm the interests of manufacturers while benefitting retailers. Govindan and Popiuc (2014) consider reverse supply chain coordination with a revenue sharing contract. They show that that performance measures and total supply chain profits improve through coordination with revenue sharing contracts in reverse supply chains. Özer and Raz (2011) study a supply chain with two suppliers competing over a contract to supply components to a manufacturer. They show that the information on both the small supplier’s production cost and the manufacturer’s processing cost plays a critical role in contracting decisions. Differing from the above studies, we investigate contract design in a supply chain configuration that not only allows for multiple competing upstream manufacturers, but also for competition between downstream retailers. Supply chain contracts also have been discussed in the context of coordinating the supply chain and sharing supply chain information (Chen, 2003; Zhang & Chen, 2013). Our paper is also related to the literature on information sharing in supply chains. As pointed out by Zhang and Chen (2013), both the double marginal effect and the bullwhip effect reduce the efficiency and effectiveness of supply chain. A supply chain contract can guarantee cooperation and coordination in decision making and demand information sharing among the members, and this is an effective method to mitigate bullwhip effect. Fiala (2005) concludes that information asymmetry is a source of the bullwhip effect, and information sharing on customer demand has an impact on the bullwhip effect. Some studies examine the value of information sharing in improving operational performance (for example, Lee, So, & Tang, 20 0 0; Sahin & Robinson, 2002), and in reducing the bullwhip effect (for example, Agrawal, Sengupta, & Shanker, 2009; Ouyang, 2007). Other studies discuss the incentive for firms in a supply chain to share information (Ha & Tong, 20 08; Li, 20 02). In our paper, we examine the value of information sharing by comparing a two-part tariff contract with information sharing to a wholesale price contract without information sharing. We also discuss the contract design issue. That is, under what conditions should manufacturers offer either a two-part tariff contract or a wholesale price contract, and whether or not retailers should accept the contract that the manufacturer offers.

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Some recent studies do focus on contract design in the context of many-to-many supply chains. Ha and Tong (2008) investigate how contract type affects the value of information sharing in two exclusive supply chains, each consisting of one manufacturer and one retailer. Ha and Tong (2011) show the incentive for vertical information sharing in the same exclusive supply chain structure with production diseconomies. Ai, Chen, and Ma (2012) study contract design under supply chain competition and demand uncertainty. They show that a revenue-sharing contract may achieve the performance of a centralized system and is better than a wholesale-price contract. Fang and Shou (2015) examine how the levels of supply uncertainty and competition intensity affect the equilibrium decisions of order quantity and contract offers with an exclusive supply chain. These studies only focus on exclusive supply chains, in which each manufacturer distributes product through one retailer, and do not consider cross-sales in the supply chain. Unlike those studies, we consider the selection of supply chain contracts for a cross-sales supply chain with asymmetric demand information. A significant part of the literature on supply chain structure with multiple manufacturers and multiple retailers focuses on multiple manufacturers with multiple exclusive retailers. McGuire and Staelin (1983) investigate the impact of product substitutability on Nash equilibrium channel configurations for two manufacturers and two exclusive retailers in the market. Coughlan (1985) extends McGuire and Staelin’s model to the electrical industry. Moorthy (1988) extends the work of McGuire and Staelin (1983) by investigating the effect of strategic interaction on a manufacturer’s channel-structure decision. Gupta and Loulou (1998) and Gupta (2008) analyze the process innovation and knowledge spillovers of competing supply chains involving two manufacturers and two exclusive retailers. Wu, Baron, and Berman (2008a) discuss three possible supply chain structures (vertical integration, manufacturer’s Stackelberg, and bargaining on the wholesale price), and identify the equilibrium channel structure for competing supply chains consisting of two manufacturers and two exclusive retailers. Lin and Parlakturk (2014) compare the three strategies (forward integration, backward integration, and no vertical integration) for a market of two manufacturers and two exclusive retailers. Karray (2015) studies the equilibrium strategies of two competing supply chains for both horizontal and vertical joint promotions (cooperative advertising). All of the above work is based on a competitive model of two manufacturers and two exclusive retailers forming two separate supply chains, and only considers the case of symmetric forecast information. In practice, multiple products are often sold through multiple retailers, and the market demand information observed by supply chain members is usually different between upstream and downstream members. Our paper examines the impact of asymmetric information on a cross-sales supply chain. We provide new insights into the conditions under which manufacturers and retailers should cooperate and coordinate, sharing retailers’ demand information, and how they should do it, when the supply chain involves both vertical (between manufacturers and retailers) and horizontal (among manufacturers and among retailers) competition. The work described here is also closely related to the literature on the cross-sales supply chain. Trivedi (1998), Wu and Mallik (2010), and Cai et al. (2012) compare the exclusive supply chain with the cross-sales supply chain in the presence of deterministic demand. Li, Zhou, Li, and Zhou (2013) and Feng and Lu (2013) study contract choice in the cross-sales supply chain, assuming symmetric forecast information. Dong, Zhang, and Nagurney (2004) and Nagurney, Cruz, Dong, and Zhang (2005) develop a cross-sales model in which multiple manufacturers and multiple retailers face uncertain demands. Adida and DeMiguel (2011) study multiple manufacturers competing in quantity to supply a set of

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products to multiple risk-averse retailers who compete in quantity to satisfy uncertain consumer demand. They study the optimal behavior of the various decision makers. They do not, however, consider supply chain contracts. Our paper analyzes the strategic behavior of the supply chain members, examining supply chain contract choice in the cross-sales structure when demand uncertainty and asymmetric forecast information are present among the supply chain members. Our paper is also related to the study of Cai et al. (2012), who also consider competition in cross-sales supply chains. Our paper differs from theirs in the following aspects. First, their study focuses on the comparison between the two structures of exclusive supply chain and cross-sales supply chain. Different from their work, our paper focuses on the choice of contracts between the wholesale price contract and two-part tariff contract under the cross-sales structure. Second, they discuss the supply chain’s decision under the information symmetry, while we consider the case that demand information is asymmetric for two retailers. Third, Cai et al. (2012) show that the decisions of supply chain members are only dependent on the product substitution coefficient, while our analysis indicates that the contract design under the cross-sales supply chain is not only related to the product substitution coefficient, but also affected by information accuracy and demand uncertainty.

as well as products. ai j is known by both manufacturers and retailers in the cross-sales supply chain and is normalized to a. As a result, a11 = a12 = a21 = a22 = a. This symmetric setting has been widely adopted in previous studies to preserve the tractability of the problem (see Cai et al., 2012; McGuire & Staelin, 1983; Trivedi, 1998; Wu & Mallik, 2010; and Li et al., 2013). Like Vives (1984) and Raju and Roy (20 0 0), we assume that retailer j can observe a demand signal f j = a + ε j , where ε j follows a bivariate normal distribution with mean 0 and variance s j , which is independent of a, i = 1, 2 and j = 1, 2. ε j reflects each retailer’s ability to forecast the risk of the market. This implies that the larger the dispersion of a, the larger the forecasting error of retailer j on the potential market of the product supplied by manufacturer i. fj is retailer j’s private information and is not known to its manufacturers without an agreement, although the same information can be observed by the competing retailer. Since ε1 and ε2 are independent, by defining the signal accuracy as s u t = u/(u + s ), we can derive: E (a¯ | f j ) = u+ s a + u+s f j = a + t ( f j − a ) 2 and E[( f j − a¯ ) ] = u + s. Referring to Yao et al. (2005), when manufacturer i observes the two retailers’ forecasting information, the expected demand of the product produced by manufacturer i and sold by retailer j is given by:

3. Models

E (a| f1 , f2 ) = a + Jt ( f1 − a ) + Jt ( f2 − a ), where J =

We consider a two-echelon supply chain consisting of two symmetric manufacturers (M1 and M2) and two retailers (R1 and R2). Each manufacturer produces a substitutable product and sells it through both retailers, who face uncertain demand. We use i and j to index the manufacturers and the retailers, respectively. Consumers in the market can purchase the product from one of the four potential manufacturer-retailer dyads: (ij), where i = 1, 2 and j = 1, 2. We follow the consumer’s utility function in Ingene and Parry (2007) and Cai et al. (2012) to model demands for cross-sales in the supply chain:

U=





a¯ i j qi j − q2i j /2 − b



i j=mn

ij

(qi j qmn ) −



pi j qi j .

(1)

ij

where pi j denotes the retail price of product package i j, and qi j and qmn denote demands of product package i j and mn, respectively. a¯ i j represents the consumer’s preference for product package ij and can be considered as the initial base demand when all retail prices are set to be equal to zero. b ∈ (0, 1 ) denotes the degree of product package substitutability; if b = 0, the implication is that the product packages are completely non-substitutable, while if b approaches 1, the product packages become completely substitutable. Maximizing (1), the price of package ij is

pi j = a¯ i j − qi j − b



qmn .

(2)

mn=i j

Then, the demand for the product package ij can be derived as follows:

qi j = Ai j − β pi j + γ



pmn ,

(3)

mn=i j

a¯

ij b b where Ai j = 1+3 , β = (1−b1+2 b )(1+3b) , and γ = (1−b)(1+3b) . β , γ , and Ai j represent the price coefficient, the cross-price coefficient, and the “attractiveness” of package i j, respectively. Following the assumption of Raju and Roy (20 0 0) and Yao, Yue, and Wang (2005), a¯ i j is normally distributed and a¯ i j = ai j + e ai j (the prior value of the initial base demand) is the expected value of a¯ i j and e follows a normal distribution with mean 0 and variance u. As in Ha and Tong (2011), we assume that e reflects the uncertainty of the market and is independent of individual retailers

1 . 1+t

(4)

In this paper, we examine the impact of a two-part tariff contract on a cross-sales supply chain system. For comparison purposes, we use the widely-adopted wholesale price contract as a benchmark case. Under a two-part tariff contract, the manufacturer’s profit is wq + F for demand q, where w denotes the unit wholesale price set by the manufacturer and F is a lump sum fee, which is determined by the manufacturer for each product package. Notice that, in this study, the wholesale price w is set by maximizing the whole channel’s profit rather than set by maximizing the player’s own profit. To investigate the pure impact of contract choice on the members, and following studies in the literature (such as Cai et al., 2012; Dukes, Gal-Or, & Srinivasan, 2006; Horn & Wolinsky, 1988; O’Brien & Shaffer, 1992, 2005), we assume that the same fee F applies for each product package, if more than one supply chain offers a two-part tariff contract. While the manufacturer will share the retailer’s demand forecasting information through a two-part tariff contract, it will not be able to share this information if it decides to offer a wholesale price contract. Each manufacturer offers the same type of contract to its two retailers, either a wholesale price contract or a two-part tariff contract. Therefore, there are 4 possible contract configurations, as shown in Fig. 1. We use n to represent the combination of contracts offered by the two manufacturers, n = {WW, WT, TW, TT}, where W represents a wholesale price contract, T represents a twopart tariff contract, and the first and the second letters represent contracts offered by MM1 and MM2, respectively. To examine the impact of demand uncertainty on the contract configuration n in the supply chain system, we compare the contract design in the supply chain system without demand uncertainty (using a superscript 0) to those with demand uncertainty. We can now denote Min and Rnj as the profits of manufacturer i and retailer j, respectively, if a contract configuration n is offered with uncertain demand. Thus, Min0 and Rnj 0 denote the profits of manufacturer i and retailer j if a contract configuration n is offered when demands are deterministic. Zin denotes the total profit of manufacturer i’s channel with contract configuration n (i.e., Zin = pni j qni j + pni,3− j qni,3− j ). Notice that the contract configurations WT and TW are symmetric. We therefore present only contract configuration WT. The notation used in this paper is summarized in Table 1.

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M1

M2

M1

M2

M1

M2

M1

M2

R1

R2

R1

R2

R1

R2

R1

R2

(a) WW

(b) WT

(c) TW

(d) TT

Fig. 1. A cross-sales supply chain with two manufacturers and two retailers: contact configuration WW, WT, TW, TT.

Table 1 Notation. Indexes i j ij n Parameters a¯ i j a

β 2(1 + b)2 (2 − b2 + 3b) b v e (1 + b)(1 + 2b)2 (2 − b2 + 3b)/2 fj t F

Decision variables pni j wni j pni j0 wni j0 Other notation Min , Rnj qni j , Zinj Zin Min0 , Rnj 0

Subscript, index of manufacturer i or product i, i = 1, 2 Subscript, index of retailer j, j = 1, 2 The channel ij, which consists of manufacturer i and retailer i Superscript, the contract choice configuration, n = {WW, WT, TW, TT}, where W and T represent the wholesale price contract and the two-part tariff contract, respectively Consumer’s preference for channel ij Prior value of initial base demand Price coefficient Cross-price coefficient Degree of channel substitutability Demand uncertainty level Uncertainty portion of market demand, normally distributed with mean 0 and variance u, where u = va2 Retailer j’s ability to forecast the risk of the market, following a bivariate normal distribution with a mean 0 and a variance x2 Retailer j’s signal of demand information Signal accuracy, where t = u/(u + s ) Lump sum fee of a two-part tariff contract Retail price of channel ij for configuration n with uncertain demand Wholesale price of channel ij for configuration n with uncertain demand Retail price of channel ij for configuration n with certain demand Wholesale price of channel ij for configuration n with certain demand Profits of manufacturer i and retailer j for configuration n with uncertain demand Demand and profit of channel ij for configuration n with uncertain demand Profit of manufacturer i’s channel for configuration n with uncertain demand, where Zin = Zinj + Zi,n3− j Profits of manufacturer i and retailer j for configuration n with certain demand

We resolve the problem with a two-stage game. The manufacturers are Stackelberg leaders in the supply chain, while the two manufacturers compete in a Bertrand-Nash game. The game sequence is as follows: In the first stage, both manufacturers independently but simultaneously decide which type of contract to offer to the two retailers, either a wholesale price contract without information sharing or a two-part tariff contract with information sharing. If a wholesale price contract is offered, the manufacturer decides the wholesale price (w) to maximize its profit; if a twopart tariff contract is offered, the manufacturer decides the wholesale price (w) to maximize the whole channel’s profit and sets a lump sum fee F. In the second stage, the retailers decide whether or not to accept the contract offered by the manufacturers, after the demand is revealed. If a two-part tariff contract is offered and the retailer decides not to accept this contract, the manufacturer implements a wholesale price contract. If the manufacturer offers a wholesale price contract, then the retailer has to accept it. Based on the contracts given, the retailer determines the retail price (p) to maximize its own profit.

4. Equilibrium of the two-stage game 4.1. Retailer’s optimal strategy We start with the second stage game, examining the retailer’s optimal strategy, given a contract offered by the manufacturers. Since the retailers’ optimal strategy with deterministic demand is a special case when demand is uncertain, setting f j − a = 0, we first discuss the case with demand uncertainty. With demand uncertainty, retailers forecast the demand and have private and imperfect information about it. Under a wholesale price contract, given the wholesale price w, the retailer sets its retail price to maximize its profit. Under a two-part tariff contract, given the lump sum fee F, which has been agreed to by both the manufacturer and the retailer, and the wholesale price w, the retailer determines retail price p. Because F will not influence the retail prices set by the retailers (the retailer’s profit function is modified by subtracting F under a two-part tariff contract), we can simplify the analysis by only analyzing the retailer’s decision under a wholesale price

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contract. The retailer j’s expected profit from selling product i is Ri j = E[( pi j − wi j )qi j | f j ], where i = 1, 2 and j = 1, 2. Let R j be the total expected profit of the retailer j from selling the product supplied by the two manufacturers:

R j = E[(( pi j − wi j )qi j + ( p3−i, j − w3−i, j )q3−i, j ) | f j ], i = 1, 2; j = 1, 2.

(5)

Substituting (10) into (9), we can derive the equilibrium wholesale price and retail price. We summarize the results in Lemma 2. Lemma 2. For i = 1, 2 and j = 1, 2, if retailers face demand uncertainty, when both manufacturers offer two-part tariff contracts, there exists a unique symmetric price equilibrium, given by:

wTi jT =

From (5), for given wi j , it can be proved that there exists a unique retail price pi j , which is given by setting first-order conditions to zero:

b2 (1 − b)(a + Jtg)

(1 + b)2 (2 − b)

and



2(1 + b − 2b2 )E (a f j ) + b(1 + b)(wi,3− j + w3−i,3− j ) + b2 w3−i, j + (2 + 4b + b2 )wi j

pi j =

4 ( 1 + 2b )

4.2. Manufacturer’s optimal problem

pTi jT =

We now solve the first-stage game. For contract configuration WW, both manufacturers choose a wholesale price contract without sharing demand forecasting with retailers. M1 maximizes its expected profit from supplying the product to two retailers by setting the wholesale price wi j and wi,3− j , anticipating the retail prices given in (6): W MW (wi j , wi,3− j ) = wi j E (qi j | f j ) + wi,3− j E (qi,3− j | f3− j ), i

i = 1, 2; j = 1, 2.

(7)

Under a wholesale price contract, since the manufacturers cannot access the retailers’ forecasting information, they make decisions according to prior demand information a. With (6), manufacW as: turer i predicts the retail price pW ij W pW = ij

2a(1 − b2 ) , 4 + 3b − 3b2 a(1 − b)(6 + 7b − b2 ) t (1 − b)[bg + ( f1 − a )] = + , and 2 ( 1 + 3b ) 2(4 + 3b − 3b2 )

W pW i2

If both manufacturers choose a two-part tariff contract with information sharing (TT case), retailer j sets the retail price to maximize its own profit, but manufacturer i should set the wholesale price to maximize the expected profit of its channel (ZiT T ).

ZiT T = E[( pi j qi j + pi,3− j qi,3− j )| f1 , f2 ], i = 1, 2; j = 1, 2.

(9)

Because it shares retailer j’s forecasting information, and it anticipates the wholesale price of its competitor, with (6), manufacturer i predicts retail price pi j :

pTi jT =

Lemma 3. For i = 1, 2 and j = 1, 2, facing demand uncertainty, if one manufacturer offers a wholesale price contract and the other offers a two-part tariff contract to the two retailers, there exists a unique

.

T wW 1j =

2(1 − b)[a(4 + 12b + 9b2 + 3b3 ) + J b3tg(3 + b)] , y0

T wW 2j =

2b(1 − b)[a(2 + 7b + 8b2 − b3 ) + 2Jbtg(2 + 3b − b2 )] , y0 1−b 2 1−b 2

 

2a(24+84b+70b2 +3b3 −b4 )+t b3 (10+11b−b2 )g 2y0

+

t (bg+( f j −a )) 1+2b

2a(2+b)(8+28b+19b2 −3b3 )+t b2 (8+16b+5b2 −b3 )g 2y0

+



and

t (bg+( f j −a )) 1+2b



,

1 where y0 = 16 + 56b + 38b2 − 19b3 − 12b4 + b5 , J = 1+ t , and g = f j + f 3− j − 2a. Lemmas 1–3 show that with demand uncertainty, the equilibrium retail price in each contract configuration is affected by the degree of product package substitutability b, the accuracy of the forecasting information t, and the retailer’ signal f j . If the retailer’s forecast information is fully inaccurate t = 0. If the demand is deterministic ( f j = a), the retail price decreases with the degree of product package substitutability b. Unlike the retail price, the wholesale price in contract configuration WW is independent of t and ( f j = a), since the manufacturer under a wholesale price contract cannot access the retailer’s private information on demand forecast. For the WT case, although M1 does not have the retailer’s

2(1 + b − 2b2 )E (a| f1 , f2 ) + b(1 + b)(wi,3 − j + w3 − i,3 − j ) +b2 w3−i, j + (2 + 4b + b2 )wi j 4 ( 1 + 2b )

(8)

symmetric price equilibrium, given by:

T pW = 2j

a(1 − b)(6 + 7b − b2 ) t (1 − b)[bg + ( f2 − a )] = + , 2 ( 1 + 3b ) 2(4 + 3b − 3b2 ) where g = f j + f3− j − 2a.

1 and g = f j + f3− j − 2a. 1+t

For contract configuration WT, M1 offers a wholesale price contract to its retailers, while M2 offers a two-part tariff contract to its retailers. We derive the equilibrium wholesale prices and retail prices in Lemma 3.

T pW = 1j

W wW = ij W pW i1

where J =

4 ( 1 + 2b )

Lemma 1. For i = 1, 2 and j = 1, 2, with demand uncertainty, if both manufacturers offer a wholesale price contract to both retailers, there exists a unique symmetric price equilibrium:

(6)

(1 − b)[2a(2 + b) + tJb2 g] t (1 − b)[bg − ( f j − a )] + , 4(1 + b)(2 − b) 2 ( 1 + 2b )

2(1 + b − 2b2 )a + b(1 + b)(wi,3− j + w3−i,3− j ) + b2 w3−i, j + (2 + 4b + b2 )wi j

Substituting (8) into (7), we can obtain unique optimal prices, which can be summarized as in Lemma 1.

.

.

(10)

X. Li, J. Chen and X. Ai / European Journal of Operational Research 275 (2019) 939–956

945

Fig. 2. Equilibrium wholesale prices.

Fig. 3. Equilibrium retail prices.

forecasting information (under the wholesale price contract), its wholesale price is affected by t and ( f j = a). This is because the wholesale price of its competitor (M2) influences its equilibrium decision. Next, we compare the equilibrium prices in four contract configurations for cases in which the supply chain faces both deterministic demand and uncertain demand. The equilibrium prices when the supply chain system faces deterministic demand can be obtained by setting f j − a = 0 for prices presented in Lemmas 1–3. We summarize the impact of contract configuration on the equilibrium prices for deterministic demands in Proposition 1.

allows the manufacturer to allocate the maximal channel profit among the participants, from the vertical integration perspective. We now discuss the impact of demand uncertainty on equilibrium prices in the four configurations. When demand uncertainty is present, the wholesale price and retail price not only depend on the degree of product package substitutability (b), but also on the retailer’s signal f j and the accuracy of forecast information (t). We use a numerical example to illustrate the comparison. We set a = 1 and t = 0.8. The impact of f j − a on the equilibrium wholesale prices as b increases are illustrated in Figs. 4 and 5. Figs. 4(i) and 5(i) show that with demand uncertainty, if the retailer’s signal f j is not far from actual a (i.e., f j − a = 0.1), then T < wW T < wW W and pT T < pW T < pW T < pW W , which is wTi jT < wW 2j 1j ij ij 2j 1j ij the similar to the case when demand is deterministic. However, Figs. 4(ii) and (iii), and 5(ii) and (iii) show that as the signal f j increases (i.e., from f j − a = 0.1 to f j − a = 3 and f j − a = 5), equilibrium wholesale prices and retail prices under a two-part tariff T , pT T pW T ) can be close to or even higher than contract (wTi jT , wW 2j ij 2j

Proposition 1. For j = 1, 2, without demand uncertainty, then wTi jT 0 < T 0 < wW T 0 < wW W 0 and pT T 0 < pW T 0 < pW T 0 < pW W 0 . wW 2j 1j ij ij 2j 1j ij

Proposition 1 illustrates implications when demand is deterministic. First, if a manufacturer (either M1 or M2) moves from using a wholesale price contract to a two-part tariff contract, the other manufacturer’s equilibrium wholesale T 0 , wW T 0 < price and retail price will decrease (i.e., wT21T 0 < wW 21 11 W 0 , pT T 0 < pW T 0 and pW T 0 < pW W 0 ). Second, a two-part tariff wW 11 21 21 11 11 contract always offers a lower wholesale price and retail price, whether the competitor is using a wholesale price contract or a T 0 , wW T 0 < wW W 0 , pT T 0 < two-part tariff contract. (i.e.,wT11T 0 < wW 11 21 21 11 T 0 , and pW T 0 < pW W 0 ). In addition, the wholesale price and repW 11 21 21 tail price decrease with the number of channels using a two-part T 0 = wW T 0 < wW W 0 ). These results tariff contract, (i.e., wTi jT 0 < wW 2j 1j ij imply that a two-part tariff can lead to a low wholesale price and a retail price. This is because, under a two-part tariff contract, the manufacturer sets the wholesale price to maximize the profit of the whole channel to reduce the double marginalization effect, which leads to lower wholesale and retail prices. These results are illustrated in Figs. 2 and 3. T 0 ) under Fig. 2 shows that the wholesale prices (wTi jT 0 and wW 2j a two-part tariff contract increase first and then decrease with b. Specifically, when there is no competition in the market (i.e., b = 0), the manufacturer sets the wholesale price to zero to maximize the channel’s profit. It charges a lump sum fee from the retailers. With the increase of the degree of product package substitutability (b), the wholesale price increases. When b reaches a certain degree, the wholesale price starts to decrease again. The intuition is that a two-part tariff mechanism can achieve a vertical integration that

T , wW W , pW T , pW W ). those under a wholesale price contract (wW 1j ij 1j ij This is because under a two-part tariff contract, the manufacturers can obtain the retailer’s signal f j through information sharing. They decide wholesale prices based on the signal f j . A high signal f j results in a high wholesale price. Accordingly, the retail prices increase as well. Notice that the wholesale price can be increasing in the substitutability when supply chain contract configuration is T ) or TT (wT T ). This is because that the two-part either WT (wW 2j ij tariff contract is used by the manufacturers not only to obtain the demand information from the retailers, but also to coordinate the supply chain. A wholesale price under a two-part tariff contract is set by maximizing the profit of the whole supply chain, which is different from the wholesale price contract that aims to maximize the player’s own profit.

5. Contract choice equilibrium With Lemmas 1–3, we can derive the profits of the manufacturers and the retailers when demand is uncertain for the four cases {WW, WT, TW, TT}. We summarize these results in Table 2. To examine the impact of information sharing on contract choice, we consider contract design with certain demand first, then analyze contract design when the supply chain faces uncertain demand. The profits of the players in the supply chain system with

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Fig. 4. The impact of demand uncertainty ( f j − a ) on the equilibrium wholesale prices.

Fig. 5. The impact of demand uncertainty ( f j − a ) on the equilibrium retail prices. Table 2 The profits of manufacturers and retailers with demand uncertainty. Profit

WW

WT

TT

M1

a2 (1−b)x0 (4+3b−3b2 )2 (1+3b)

(1−b)(a2 x2 +ut y2 ) (1+3b)y0 2

(1−b)(a2 x5 +ut y5 ) (2−b)2 (1+b)3 (1+3b)

+ 2F

M2

a2 (1−b)x0 (4+3b−3b2 )2 (1+3b)

(1−b)(a2 x3 +ut y3 ) +2F (1+3b)y0 2

(1−b)(a2 x5 +ut y5 ) (2−b)2 (1+b)3 (1+3b)

+ 2F

R1

(1−b)(a2 x1 +ut y1 ) (4+3b−3b2 )2 (1+3b)(1+2b)2

(1−b)(a2 x4 +ut y4 ) (1+3b)(1+2b)2 y0 2

−F

(1−b)(a2 x6 +ut y6 ) (2−b)2 (1+b)3 (1+2b)2 (1+3b)

− 2F

R2

(1−b)(a2 x1 +ut y1 ) (4+3b−3b2 )2 (1+3b)(1+2b)2

(1−b)(a2 x4 +ut y4 ) (1+3b)(1+2b)2 y0 2

−F

(1−b)(a2 x6 +ut y6 ) (2−b)2 (1+b)3 (1+2b)2 (1+3b)

− 2F

where: x0 ∼ x6 and y0 ∼ y6 are given in Table A1 in Appendix.

deterministic demand can be obtained by setting ut = 0 in their corresponding profits when demands are uncertain.

W =MW T and Let FIIM0 and FIIR0 be the boundary values of MW 2 2 W =RW T , respectively, then: RW 2 2

(1−b) 0 FIIM0 = 2a (1+3 [ − b) (4+3b−3b2 )2 2

a2 (1−b) x4 [y 2 1+3b 0

5.1. Contract choice equilibrium with deterministic demand For contract design, a contract be offered and accepted only when both the manufacturer and the retailer are better off. To easily compare the manufacturers and retailers’ profits under two contracts in a cross-sales supply chain with certain demand, we first identify several boundary values. Let FIM0 and FI R0 be the boundary values T W T T of M2 =M2 and RT2W =RT2 T , respectively, then: (1−b) FI M0 = a21+3 b 2



x2 y0 2

x

− (2−b)25(1+b)3







(1−b) 6 and FIR0 = a 1+3 − 4 . b (2−b)2 (1+b)3 y20 2

x

x

−

x

x1

2

(4+3b−3b2 )

x3 ] y0 2

]. Let FIM0 and FIII R0 be the boundary II

W =MT T and RW W =RT T , respectively, then: values of MW 1 1 1 1



FII R0 =

and



FIM0 II =

a2 ( 1 − b ) x0 x5 − 2(1 + 3b) (4 + 3b − 3b2 )2 (2 − b)2 (1 + b)3

FIII R0 =

x1 a2 ( 1 − b ) x6 − . 2(1 + 3b) (2 − b)2 (1 + b)3 (4 + 3b − 3b2 )2



and



Comparing the boundary values defined above, we summarize the results in Lemma 4.

X. Li, J. Chen and X. Ai / European Journal of Operational Research 275 (2019) 939–956

Lemma 4. (i) if 0 < b < 0.2037, then (ii) if 0.7470 < b < 1, then

FIM0 FIM0 II

< FIIM0 > FIIM0

< FIM0 II > FIM0

< FIR0 > FIIR0

< FIRII0 > FIRII0

< FIIR0 ; > FIR0 .

Lemma 4 implies that when the degree of product package substitutability is low (0 < b < 0.2037), the maximum fee that retailers can afford is greater than the minimum fee required by the manufacturers. The implication of the result is there is a possibility that a two-part tariff contract is a dominant choice, as long as the lump sum fee is within a certain range. When the degree of product package substitutability is relatively high (0.7470 < b < 1), however, if the lump sum fee is set in the range (FIIR0 , FI M0 ), neither the manufacturer nor the retailer can benefit from a two-part tariff contract. From Lemma 4, we can derive the contract choice equilibriums and summarize the results in Proposition 2. Proposition 2. for a cross-sales supply chain with certain demand, (i) a two-part tariff contract is a subgame-perfect Bayesian–Nash equilibrium for FIM0 < F < FIR0 and b ∈ (0, 0.2037 ); II (ii) a wholesale contract is a subgame-perfect Bayesian–Nash equilibrium for FIIR0 < F < FI M0 and b ∈ (0.7470, 1 ); (iii) no subgame-perfect Bayesian–Nash equilibrium exists for b ∈ (0.2037, 0.7470), as no contracts can enable both the manufacturers and the retailers to be better off. The degree of product package substitutability (b) is segmented into 3 distinct regions (0, 0.2037), (0.2037, 0.7470), and (0.7470, 1). Proposition 2 shows that when b ∈ (0, 0.2037), that is, when b is small, the two-tariff contract adopted by both manufacturers (TT) is not only a dominant equilibrium, but also the best option for all members, as long as FIM0 < F < FI R0 . The intuition II of the result is that the two manufacturers and the two retailers are more profitable in contract configuration (TT) than in the other configurations. In such a case, the manufacturers set very low wholesale prices, which allow the retailers to set lower retail prices (Proposition 1). Low retail prices can attract more demand. The benefit of greater demand can outweigh the disadvantage of low wholesale prices. Thus, the manufacturers are more profitable. The retailers also benefit from increasing demand to be more profitable. When b is large, on the other hand, that is in the range b ∈ (0.7470, 1), wholesale-price contracts will be preferred by both manufacturers (WW). WW is the Nash equilibrium and the optimal option for all members, as long as FIIR0 < F < FI M0 . The reason is that when b is sufficiently large, the intense price competition pushes the manufacturers to reduce the wholesale price, leading to low retail prices (Proposition 1). The manufacturers benefit from expansion of demand and retailers benefit from both higher demand and higher profit margin. Finally, when b ∈ (0.2037, 0.7470), the manufacturers and the retailers are unable to agree on a lump sum fee, as they cannot benefit simultaneously. Therefore, no equilibrium is achieved, and they must cooperate through a wholesale price contract. 5.2. Contract choice equilibrium with uncertain demand In this section, we examine contract choice in the cross-sales supply chain system with demand uncertainty and sharing of information on demand forecasts. Existing studies on supply chain contracts (for example, Feng & Lu, 2013; Li et al., 2013) have shown that in a cross-sales supply chain, the choice of contract depends on market competition. These studies, however, do not examine the members’ decision-making behavior in the face of demand uncertainty and the role of information sharing on demand forecasts in a cross-sales supply chain system. We first compare and analyze the profits of manufacturers and retailers for three cases.

947

Case I: When one manufacturer offers a two-part tariff contract. Without loss of generality, we assume M1 offers a two-part tariff contract. From Table 2, we see that if M2 also chooses a twopart tariff contract, its expected profit is M2T T =

(1−b)(a2 x5 +ut y5 ) + (2−b)2 (1+b)3 (1+3b)

2F ; if M2 chooses a wholesale price contract, its expected profit is M2T W =

(1−b)(a2 x2 +ut y2 ) . Let FIM and FIR be the threshold values for (1+3b)y0 2

M2T W = M2T T and RTj W = RTj T , respectively. Then we have

FIM = FIR =

 a2 x5 + ut y5 (1 − b) a2 x2 + ut y2 − and 2 ( 1 + 3b ) y0 2 (2 − b)2 (1 + b)3

 1−b

ax6 + ut y6

(1 + 3b)(1 + 2b)2 (2 − b)2 (1 + b)3

−

ax4 + ut y4 . y0 2

To simplify the presentation of the paper, let u = va2 , where v reflects the demand uncertainty level. Comparing FIM and FIR , we have the following result, as summarized in Lemma 5. Lemma 5. FIM < FIR if and only if (b ∈ (0, 0.4147)) or (b ∈ (0.4147, 1) and v > vI );FIM > FIR , otherwise, where vI is the threshold value for FIM = FIR , which is given in Table A2 in Appendix. Lemma 5 shows the relation of FIM and FIR . From the profits of the manufacturers and retailers in Table 2, we can conclude that when FIM < F < FIR , both two retailers and M2 prefer a two-part tariff contract; when FIM > F > FIR , however, both two retailers and M2 prefer a wholesale price contract. We summarize the result for the M2’s contract choice when M1 offers a two-part tariff contract, in Proposition 3. Proposition 3. when M1 offers a two-part tariff contract to both retailers, M2 will offer: 1) a two-part tariff contract if M2 can set its lump sum fee in the range of (FIM , FIR ), and both retailers are better off if (b ∈ (0, 0.4147 )) or (b ∈ (0.4147, 1) and v > vI ); 2) a wholesale price contract, if FIM > FIR . With demand uncertainty, the contract choice in a cross-sales supply chain system depends on the degree of product package substitutability (b), demand uncertainty level (v), and the accuracy of forecast information (t). Proposition 3 shows that when one manufacturer offers both retailers a two-part tariff contract with information sharing, if the degree of product substitutability is lower (b ∈ (0, 0.4147)), or the degree of product substitutability and the demand uncertainty level are both higher (b ∈ (0.4147, 1) and v > vI ), a two-part tariff contract with information sharing is a win-win strategy for the other manufacturer’s supply chain, as long as the lump sum fee can be set in the range of (FIM , FIR ). Otherwise, the other manufacturer and the two retailers are better off with a wholesale price contract without information sharing. The above results imply that when both manufacturers offer a two-part tariff contract with information sharing, each member in the cross-sales supply chain has incentive to switch to a wholesale price contract without information sharing, if and only if the degree of product substitutability is high (b ∈ (0.4147, 1)), the level of demand uncertainty is low (v ∈ (0, vI )), and FIM > FIR . Notice that when FIM > FIR , the maximum payment that can be made by the retailer (FIR ) is lower than the minimum payment required by a manufacturer (FIM ) to offer a two-part tariff contract, so the retailers will reject this contract. Case II: When one manufacturer offers a wholesale price contract Without loss of generality, we assume that M1 offers a wholesale price contract. With profits of the manufacturers and the retailers in Table 2, let FIIM and FIIR be

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W = MW T the threshold values for MW 2 2

spectively. We have 1−b

(1+3b)(1+2b)2

[

a2 x4 +ut y4 y0 2

FIIM −

=

W = RW T , reand RW 2 2

a2 x0 1−b [ 2(1+3b) (4+3b−3b2 )2

a2 x1 +ut y1

2

(4+3b−3b2 )

−

a2 x3 +ut y3 ] y0 2

and

FIIR

=

].

Then we can derive vII (the threshold value for FIIM = FIIR , which is given in Table A2 in Appendix). Then we have: FIIM < FIIR if and only if (b ∈ (0, 0.5481 )) or (b ∈ (0.5481, 1 ) and v > vII ); FIIM > FIIR , otherwise. In a two-part tariff contract, the retailer prefers a smaller lump sum fee and the manufacturers prefer a larger lump sum fee. Therefore, only when the lump sum fee is set in the range (FIIM , FIIR ), can a two-part tariff contract be successfully offered by manufacturer. If FIIM > FIIR , neither the manufacturer nor the retailers can benefit from a two-part tariff contract. We can formally state the results as follows: Proposition 4. when M1 offers a wholesale price contract to both retailers, M2 will offer: 1) a two-part tariff contract if the M2can set a lump sum fee in the range of (FIIM , FIIR ). M2and both retailers will be better off if (b ∈ (0, 0.5481)) or if (b ∈ (0.5481, 1 ) and v > vII ); 2) a wholesale price contract if FIIM > FIIR . As in Proposition 3, when M1 offers a wholesale price contract, when b ∈ (0, 0.5481), or b ∈ (0.5481, 1 ) and v > vII , a two-part tariff is a win–win strategy for M2 and two retailers, if M2 can set a lump sum fee in the range of (FIIM , FIIR ); however, if FIIM > FIIR , M2 prefers a wholesale price contract, and both retailers are better off with this wholesale price contract. This means that when both manufacturers offer a wholesale price contract to retailers, each member in the cross-sales supply chain has incentive to switch to a two-part tariff contract with information sharing, if (F ∈ (FIIM , FIIR ) and b ∈ (0, 0.5481)) or if (F ∈ (FIIM , FIIR ), b ∈ (0.5481, 1 ), and v > vII ). Case III: When two manufacturers offer the same contract. W = Denoting FIM and FIRII as threshold values for MW II i T T W W T T Mi and Rj = Rj (i, j = 1, 2), respectively, we have

a2 x

(a2 x +ut y )

1−b 0 5 5 FIM II = 2(1+3b) [ (4+3b−3b2 )2 − (2−b)2 (1+b)3 ]

and

FIRII =

a2 x6 +ut y6 a2 x1 +ut y1 1−b [ − 2 ]. vIII can be derived by 2 (1+2b)2 (1+3b) (2−b)2 (1+b)3 (4+3b−3b2 ) M R setting FIII = FIII , as given in Table A2 in Appendix. Then, we have FIM < FIRII if and only if (b ∈ (0, 0.2684 )) or (b ∈ (0.2684, 1 ) II and v > vIII ); FIM > FIRII , otherwise. Similar to Cases I and II, when II

(b ∈ (0, 0.2684)) or when (b ∈ (0.2684, 1 ) and v > vIII ), a two-part tariff with information sharing offered by both manufacturers is the dominant contract, as compared to a wholesale price contract, for all supply chain members, if the lump sum fee can be set in R M R the range of (FIM II , FIII ). If FIII > FIII , a wholesale price contract is the dominant contract. With the three cases discussed above, we now discuss the cross-selling supply chain’s dominant contracts and the conditions for those contracts. We first compare the boundary values of the lump sum fee discussed in three cases and summarize the result in Corollary 1. Corollary 1.

(i) (FIM , FIR ) ∩ (FIIM , FIIR ) ∩ (FIM , FIRr ) = (FIM , FIR ), if (b∈ (0, 0.2037 )) II II II or if (b ∈ (0.2037, 1) and v > vIV ); (ii) (FIR , FIM ) ∩ (FIIR , FIIM ) ∩ (FIRII , FIM ) = (FIIR , FIM ), if b ∈ (0.7470, 1) II and v ∈ (0, vV ), where vIV and vV are given in Table A2 in Appendix) The relationship of boundary values for the lump sum fee is illustrated in Fig. 6 for (b ∈ (0, 0.2037) ) and for (b ∈ (0.2037, 1) and v > vIV ). Fig. 6 shows that for the range of (FIM , FIR ) (Region II

1 ), when both the degree of package substitutability (reflecting an intense competition) and the demand uncertainty level are high (i.e., b ∈ (0.2037, 1) and v > vIV ), or when the degree of package substitutability is low (or competition is less intense, i.e., b ∈ R M R M R M R (0, 0.2037 )), there exists (FIM II , FI ) ⊂ (FI , FI ), (FIII , FI ) ⊂ (FII , FII ), M R M R and (FIII , FI ) ⊂ (FIII , FIII ). This reveals that the retailers will always accept a two-part tariff contract, as long as the lump sum fee is R given in (FIM II , FI ). In addition, there exists a range of (FIIR , FIM ) (as shown in region 2 in Fig. 6(ii)), when the degree of package substitutability is high (b ∈ (0.7470, 1)) and the demand uncertainty level is low (v ∈ (0, vV )), in which (FIIR , FIM ) ⊂ (FIR , FIM ),(FIIR , FIM ) ⊂ (FIIR , FIIM ), and (FIIR , FIM ) ⊂ (FIRII , FIM ) are true. This result means that when the II lump sum fee is set in the range of (FIIR , FIM ), manufacturers and retailers have higher profits under a wholesale price contract than under a two-part tariff contract. With Corollary 1, we develop the following results for the manufacturers’ contract choice in the cross-sales supply chain. The results are summarized in Proposition 5 and illustrated in Fig. 7. Proposition 5. The manufacturers’ contract choice in a cross-sales supply chain is: 1) choosing a two-part tariff contract (with a lump sum fee FIM < F < FIR ) with information sharing if (b ∈ (0, 0.2037) ) II or (b ∈ (0.2037, 1) and v > vIV ); this contract is a subgameperfect Bayesian–Nash equilibrium for all members; 2) choosing a wholesale contract without information sharing if FIIR < FIM , given b ∈ (0.7470, 1) and v ∈ (0, vV ); such a contract is a subgame-perfect Bayesian–Nash equilibrium for all members. As shown in Proposition 5 and illustrated in Fig. 7, there exR ists a range (FIM II , FI ) in Regions L1 and L2. As long as the lump sum fee is set in this range of (FIM , FIR ), a two-part tariff contract II is the dominant strategy, as it is preferred by both manufacturers and retailers. It is a win-win situation, and both the manufacturers and retailers are more profitable. In Regions L3-L7, however, the manufacturers do not have a unique choice for supply chain contracting. They will choose a contract according to the lump sum fee and the competitor’s contract choice. They may prefer a wholesale price contract, because the retailer will refuse the offered lump sum fee. In Region L8 and if the lump sum fee exists FIIR < FIM , the wholesale price contract is the dominant strategy, as it is preferred by both manufacturers and retailers. In this case, with high b, the manufacturer has no incentive to offer a two-part tariff contract, because its profit will decrease due to intense price competition. Although the manufacturer can be compensated by the lump sum fee F, the retailer can be better off after paying the lump sum fee to the manufacturer. As a result, both the manufacturer and retailer will be better off with a wholesale price contract. From the above analysis, we see that as demand uncertainty increases and competition decreases, both manufacturers and retailers have an incentive to achieve agreement on a wholesale price contract. When the demand uncertainty is relatively high, and the competition is less intense, the adoption of a two-part tariff contract can result in a lower retail price, as manufacturers can capture more market demand. Especially as demand uncertainty decreases, manufacturers can extract more profit from retailers by offering a two-part tariff contract. That is, both manufacturers and retailers prefer a two-part tariff contract if they can set a lump sum fee in the range (FIM , FIR ). II These observations can be explained as follows. When the degree of package substitutability (b) is relatively low, the manufacturers who adopt a two-part tariff contract can induce the retailers to lower retail prices, as long as the retailers can be profitable after paying the lump sum fee to the manufacturers. The lower retail

X. Li, J. Chen and X. Ai / European Journal of Operational Research 275 (2019) 939–956

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Fig. 6. The relationship of threshold values (for a = 1, f j = 1.1, and t = 0.8).

uncertainty increases, a two-part tariff contract will be preferred by manufacturers and retailers, which is a win-win situation. This is because, under a two-part tariff contract, manufacturers can mitigate risk through access to retailers’ information about uncertain demand. The retailers can obtain more profit under a two-part tariff contract than under a wholesale price contract, due to mitigation of double marginalization. From Fig. 7, we see that the range of the lump sum fee under a two-part tariff contract between the R manufacturer and retailer (FIM II , FI ) decreases gradually until it disappears as the product package competition increases. In addition, an increase in the degree of product package substitution (b) leads to an increase in the manufacturer’s required lump sum fee and a decrease in the retailer’s acceptable lump-sum fee. This reduces the desirability of the two-part tariff contract. Comparing the manufacturers’ contract choice equilibrium for certain demand to that for uncertain demand, the impact of information sharing on contract choice is summarized in Corollary 2.

Fig. 7. Manufacturers’ contract choice in a cross-sales supply chain for a = 1 and t = 0.5.

prices will attract more customers to buy the products and expand the market demand. The value of a two-part tariff contract with information sharing is enhanced by high demand uncertainty. As the degree of package substitutability increases, however, the competition in channels becomes more intensive, and the advantage of a two-part tariff contract with information sharing will decrease until it disappears. Thus, to prevent the equilibrium price from dropping too much, the members in the supply chain prefer a wholesale price contract without information sharing, rather than a twopart tariff contract with information sharing. The value of a wholesale price contract, however, is weakened by high demand uncertainty. ∂ (FIR −FIM ) ∂ (FIR −FIM ) II II > 0 and > 0 imply that higher accuracy in re∂v ∂t

tailers’ forecasting information increases the likelihood that the retailers will accept a two-part tariff contract. In addition, as demand

Corollary 2. With uncertain demand, information sharing makes a two-part tariff contract more valuable, while reducing the value of the wholesale price contract.

From Corollary 1, we have that contract choice in a cross-sales supply chain with certain demand is only affected by the degree of package substitutability (b). The boundary values of the contract choice (b = 0.2037 and b = 0.7470) are the same in both cases with and without demand uncertainty (Propositions 2 and 5). When the supply chain faces demand uncertainty, however, the manufacturers are more likely to adopt a two-part tariff contract, as the range where the two-part tariff contracts are dominant equilibrium is broad (Region L2 in addition to Region L1, where Region L1 is the region in manufacturers adopt a two-part tariff contract if demand is deterministic). For b ∈ (0.7470, 1), the range of a wholesale price contract that is a dominant equilibrium reduces to Region L8. This reveals that when uncertain demand is present, a two-part tariff contract is more valuable than a wholesale price contract, for all members in the supply chain.

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(i) k1

(iii) k1

0.4, k2

0.2

(ii)

1

(iv) k1

1.2, k2

k1

0.8, k2

1.6, k2

0.6

1.2

Fig. 8. Contract choice for the case of different package with different substitutability with demand uncertainty.

6. Extension In this section, we consider packages with different levels of substitutability. We follow the consumer’s utility function in Cai et al. (2012) to model demands for cross-sales in the supply chain:

U =

 ij

−



a¯ i j qi j − q2i j /2 − qi j (b0 q3−i, j + b1 qi,3− j + b2 q3−i,3− j )



pi j qi j ,

(11)

ij

where parameter b0 ∈ [0, 1 ) represents the degree of product (or supplier) substitutability, b1 ∈ [0, 1 ) represents the degree of retailer substitutability, and b2 ∈ [0, 1 ) represents the degree of substitutability between two different products sold to different retailers. Maximizing (11), the price of package ij can be derived





pi j = a¯ − qi j + b0 q3−i, j + b1 qi,3− j + b2 q3−i,3− j ,

(12)

To simplify the expressions, we define b1 = k1 b0 and b2 = k2 b0 , where k1 > 0 and k2 > 0. Notice that when k1 = 1 and k2 = 1, the degree of product package substitutability is the same as b0 , which implies that the price of package ij is reduced to the main model we discussed in previous sections. Although the game sequence for packages with different substitutability is the same as in the main model, the solution is significantly more complicated with two additional parameters (k1 and k2 ). In this section, we focus on discussion of the main insights and implications, and include detailed derivations and analysis in the Appendix. Fig. 8 presents the impact of b0 on the boundary values of the lump sum fee for different combinations of k1 and k2 . We set four values of k1 , where k1 = 0.5 and 0.8 (i.e., b1 is lower than b0 ), and 1.2 and 1.6 (i.e., b1 is higher than b0 ). The degree of substitutability between the same products sold to two different retailers (k1 b0 ) is always higher than or equal to the degree of substitutability between two different products sold to different retailers (k2 b0 ), so

X. Li, J. Chen and X. Ai / European Journal of Operational Research 275 (2019) 939–956

951

k1 > k2 . We assume k2 = k1 − 0.2, and the other parameters are set as a = 1, v = 0.1, and t = 0.8. With different substitutability for different packages, we find that the major results derived in Section 5 still hold. Specifically, first, when the degree of product substitutability is small, a twopart tariff contract with information sharing is the dominant contract for all members in a cross-sales supply chain, as long as the lump sum fee is set in a certain range (Regions I, III, V, and VII in Fig. 8). In addition, the range of lump sum fee will decrease with the degree of product substitutability. Second, as the degree of product substitutability increases, manufacturers and retailers are unable to strike a balance on the fee in a two-part tariff contract and are forced to cooperate under a traditional wholesale contract without information sharing. Third, when the degree of product substitutability is high, all members can benefit from a wholesale contract without information sharing, as long as the lump sum fee is set in a certain range (Regions II, IV, VI, VII), where the minimal lump sum fee that the manufacturers require to be profitable is higher than the maximal lump sum fee that the retailer is willing to pay to be profitable. Some new observations can be obtained for different packages with different substitutability; some new forces can influence the contract design. Specifically, the range of the degree of product substitutability in which a two-part tariff contract with information sharing is a dominant equilibrium decreases with the degree of retailer substitutability (from 0.4596 to 0.1993 in Fig. 8). However, the range of the degree of product substitutability in which a wholesale contract without information sharing is a dominant equilibrium decreases with the degree of retailer substitutability (from 0.8805 to 0.7654 in Fig. 8). That is, as the degree of retailer substitutability increases, a two-part tariff contract with information sharing intensifies price competition, which results in a loss for the members in the supply chain. On the other hand, a wholesale contract without information sharing can avoid price competition, which could be beneficial to members. In addition, the speed of the decrease in prices under a two-part tariff contract is faster than the speed of the increase in wholesale prices, and this implies that a two-part tariff contract with information sharing is more sensitive to the degree of retailers’ substitutability than a wholesale contract without information sharing.

uncertainty is present, a two-part tariff contract is more valuable for all members due to information sharing, and a wholesale price contract is less valuable to all members because there is no information sharing. There are several interesting directions for future research on this issue. First, we only discuss the impact of asymmetric information under either a two-part tariff contract or a wholesale price contract. Other types of contract would be of interest, including revenue sharing contracts and two-part tariff discount contracts. Second, we consider a supply chain structure of two manufacturers and two retailers, so a natural extension would be to study a competing supply chain consisting of multiple manufacturers and multiple retailers. Third, it would be interesting to discuss a multiperiod game theoretical model instead of the single-period model presented here. Finally, we consider the symmetric supply chain setting in this paper. A nature extension is to examine the impact of asymmetric information on the decision of the symmetric supply chain setting.

7. Conclusions We consider a cross-sales supply chain model consisting of two competing manufacturers and two competing retailers with information asymmetry. Each manufacturer produces a substitutable product and distributes it through two common retailers. We present the supply chain members’ strategy decisions for different contract configurations and develop the impact of product package competition and asymmetric information on the manufacturer’s equilibrium contract choice to provide a new perspective on supply chain contracting. Our analysis suggests that with demand uncertainty, the contract choice of the two manufacturers depends on product package competition and demand uncertainty. When either (i) product package substitutability is more intensive and demand uncertainty is relatively high, or (ii) product package competition is less intensive, both manufacturers and both retailers are better off under two-part tariff contracts, as long as the lump sum fee is set in a certain range, which we have identified. When product package substitutability is more intensive and demand uncertainty is relatively low, a two-part tariff contract is not valuable to members in a cross-sales supply chain. Then a wholesale price contract can be a dominant equilibrium outcome. Otherwise, the manufacturer and retailer will implement a wholesale price contract, due to failure of the lump sum fee negotiation. In addition, when demand

Acknowledgments The authors gratefully acknowledge financial support from the National Natural Science Foundation of China (Grants Nos. 71372140, 71432003, 71671081, 71331004, and 71531003) and the Natural Sciences and Engineering Research Council of Canada. Appendix Proof of demand function for Eq. (3): Taking the first derivative of Eq. (1) with respect to qi j :

∂ U /∂ qi j = a¯ i j − qi j − b(q3−i, j + qi,3− j + q3−i,3− j ) − pi j = 0 Then, we can derive:

q11 = a¯ 11 − bq12 − bq21 − bq22 − p11 ,

(A1)

q12 = a¯ 12 − bq11 − bq21 − bq22 − p12 ,

(A2)

q21 = a¯ 21 − bq12 − bq11 − bq22 − p21 , and

(A3)

q22 = a¯ 22 − bq12 − bq21 − bq11 − p22 .

(A4)

For simplicity, we assume that the initial base demand of each package is the same as a¯ . We solve Eqs. (A1)–(A4):



qi j = a¯ − a¯ b − pi j − 2b pi j + b pi,3− j + b p3−i, j + b p3−i,3− j







−3b2 + 2b + 1 .

(A5)

Simplifying (A5), we can derive Eq. (3) as follows: qi j =

A i j − β pi j + γ pmn , where Ai j = a¯ /(1 + 3b), β = (1 + 2b)/[(1 − mn=i j

b)(1 + 3b)], γ = b/[(1 − b)(1 + 3b)]. Proof of Lemma 1: For the case WW with demand uncertainty, take the first derivative of Eq. (4) w.r.t. pi j and p3−i, j :

∂ RWj W = E (a¯ | f j )/(3b + 1 ) − β (2 pi j − wi j ) + γ ( p3−i, j − w3−i, j ) W ∂ pW ij + γ ( pi,3− j + p3−i, j + p3−i,3− j ) = 0, and (A6)  ∂ RWj W = E (a¯  f j ) /(3b + 1 ) − β (2 p3−i, j − w3−i, j ) W W ∂ p3−i, j + γ ( pi j − wi j ) + γ ( p3−i,3− j + pi j + pi,3− j ) = 0.

(A7)

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X. Li, J. Chen and X. Ai / European Journal of Operational Research 275 (2019) 939–956 Table A1 Expressions used in Table 1. x0

2(1 + b)2 (2 − b2 + 3b)

x1

(1 + b)(1 + 2b)2 (2 − b2 + 3b)/2

x2

2(2 − b2 + 3b)(4 + 12b + 9b2 + 3b3 )2

x3

2(8 + 28b + 19b2 − 3b3 )(2 + 7b + 8b2 − b3 )(2 − b2 + 3b)

x4

(40 + 224b + 388b2 + 192b3 − 11b4 + 3b5 )(1 + 2b)2 (2 − b2 + 3b)2 /2

x5

b2 (2 − b2 + 3b)

x6

(1 + 2b)2 (2 − b2 + 3b)2 /2

y0

16 + 56b + 38b2 − 19b3 − 12b4 + b5

y1

(4 − 3b2 + 3b)(1 + b)(1 + 2b + 2b2 )/2

y2

2J (b − 1 )(3 + b)y0 + 4J 2 b3 (3 + b)(32 + 112b + 76b2 − 32b3 − 13b4 + 2b5 − b6 )

y3

2(2 − b2 + 3b)Jb[(b − 1 )y0 + 2Jb(32 + 112b + 68b2 − 62b3 − 25b4 + 20b5 − b6 )]

2

J b4 (32 + 128b + 42b2 − 210b3 − 27b4 + 117b5 − 19b6 + b7 )(1 + 2b)2 −J b2 (1 − b2 )(4 + 9b − b2 )(1 + 2b)2 y0 + (1 + b)(1 + 2b + 2b2 )y0 2 /2

y4

The Hessian matrix is:

⎡

H=⎣

∂ 2 RWW j W2 ∂ pW ij

∂

∂

∂2R j ∂ pWW ∂ pWW 3−i, j ij

2 WW Rj pWW pWW ij 3−i, j 2 WW Rj W2 pW 3−i, j

∂

∂ ∂



y5

−tJ b2 (1 − b)(2 − b)(1 + b) + 2t J 2 b2 (4 + 2b − 3b2 + b3 )

y6

[−J b2 (1 + 2b)(1 − b) + (1 + b)3 (2 − b)]2 /2 + [−J b2 (1 + 2b)(1 − b) + b(1 + b)2 (2 − b)]2 /2

⎤

⎦ = −2β 2γ



2γ . and |H | = 4(b + 1 )/[(3b + 1 )(b − 1 )2 ] > 0. −2β

Therefore, we can show that the Hessian matrix H is a diagonally dominant matrix, which guarantees joint concavity of the profit W ( pW W , pW W ). With (A1) and (A2), we have: function RW j ij 3−i, j

p11 = (E (a¯ | f j )/(1 + 3b) + β w11 + γ ( p21 − w21 ) + γ ( p12 + p21 + p22 ))/(2β ),

(A8)

p12 = (E (a¯ | f j )/(1 + 3b) + β w12 + γ ( p22 − w22 ) + γ ( p11 + p22 + p21 ))/(2β ),

(A9)

p21 = (E (a¯ | f j )/(1 + 3b) + β w21 + γ ( p11 − w11 ) + γ ( p22 + p11 + p12 ))/(2β ), and

(A10)

p22 = (E (a¯ | f j )/(1 + 3b) + β w22 + γ ( p12 − w12 ) + γ ( p21 + p12 + p11 ))/(2β ),

(A11)

Solving (A8)–(A11), we obtain the retail price pi j , which is a function of the wholesale prices:

pi j ( p, w ) =

(1 − b)E (a¯ | f j ) + (1 + 2b)wi j + bpi,3− j + 2bp3−i, j + bp3−i,3− j − bw3−i, j 4b + 2

.

(A12)

Solving Eq. (A12), we obtain the retail price pi j , which is a function of the wholesale prices,

pi j ( w ) =

(2 + 2b − 4b2 )E (a¯ | f j ) + b(1 + b)(wi,3− j + w3−i,3− j ) + b2 w3−i, j + (b2 + 4b + 2 )wi j . 4 ( 2b + 1 )

(A13)

Without retailer’s forecasting information, manufacturer i predicts the retail price pi j as

pi j ( w ) =

(2 + 2b − 4b2 )a + b(1 + b)(wi,3− j + w3−i,3− j ) + b2 w3−i, j + (2 + 4b + b2 )wi j . 4 ( 1 + 2b )

(A14)

Manufacturer i uses the prediction of the retail price [Eq. (A14)] in profit function [Eq. (7)], and takes the first derivatives w.r.t. wi j and wi,3− j : W ∂ MW 2aβ /(1 + 3b) + (2β 2 + γ 2 − 4βγ )wi j + γ (β − γ )wi,3− j + γ 2 w3−i, j + γ (β − γ )w3−i,3− j i = = 0 and W W 4β ( β − 2γ ) ∂ wi j W ∂ MW 2aβ /(1 + 3b) + (2β 2 + γ 2 − 4βγ )wi,3− j + γ (β − γ )wi j + γ 2 w3−i,3− j + γ (β − γ )w3−i, j i = =0 W W 4β ( β − 2γ ) ∂ wi,3− j

The Hessian matrix is: ⎡ ∂ 2 MWW i ⎢ ∂ wWi j W 2 H=⎣ ∂ 2 Mi ∂ wWW ∂ wWW i,3− j ij

∂ 2 MWW i ∂ wWW ∂ wWW ij i,3− j ∂ 2 MWW i W2 ∂ wW i,3− j

⎤

2β 2 + γ 2 − 4βγ ⎥ = ⎦ γ (β − γ )

 γ (β − γ ) and |H | = (4 + 15b + 15b2 + 6b3 + 15b4 + 3b5 − 9b6 ) > 0. 2β 2 + γ 2 − 4βγ

X. Li, J. Chen and X. Ai / European Journal of Operational Research 275 (2019) 939–956

953

Therefore, we  can show that the Hessian matrix H is a diagonally dominant matrix, which guarantees the joint concavity of the profit W wW W , wW W . Thus, we get function MW i ij i,3− j

2(1 + 2b)(1 − b2 )a + b(2 + 5b + b2 )w3−i, j + b(1 + b) (w3−i,3− j + 2wi,3− j ) 2

W wW = ij

2(2 + 8b + 7b2 − b3 )

.

(A15)

W in Eq. (A15), we can derive the equilibrium wholesale price as wW W = Solving wW ij ij equilibrium retail prices as shown in Lemma 1. The proof of Lemma 1 is completed.

2a (1−b2 ) . 4−3b2 +3b

W into Eq. (A14), we get Substituting wW ij

Proof of Lemma 2: In the TT case, the manufacturer has R1 and R2’s forecasting information simultaneously. From Eq. (A14), we can obtain the manufacturer’s predicting retail price as

 (2 + 2b − 4b2 )E (a¯  f j , f3− j ) + b(1 + b)(wi,3− j + w3−i,3− j ) + b2 w3−i, j + (2 + 4b + b2 )wi j pi j ( w ) = . 4 ( 1 + 2b )

(A16)

Manufacturer i uses the prediction of the retail price [Eq. (A16)] in profit function [Eq. (9)], and takes the first derivatives w.r.t. wi j and wi,3− j :

 ∂ ZiT T = ∂ wTi jT

= 0 and

8(1 + 2b) (1 + 2b − 3b2 ) 2

 ∂ ZiT T = ∂ wTi,3T− j





2b2 (1 + 3b − 4b3 )E (a¯  f j , f3− j ) − (4 + 24b + 47b2 + 31b3 − 2b5 )wi,3− j −2b2 (1 − 2b − b3 )wi j + b(4 + 18b + 11b2 + 4b3 + 2b4 )w3−i,3− j + 2b3 (1 + 2b + b2 )w3−i, j

= 0.

8(1 + 2b) (1 + 2b − 3b2 ) 2

The Hessian matrix is: ⎡

⎢ H=⎣





2b2 (1 + 3b − 4b3 )E (a¯  f j , f3− j ) − (4 + 24b + 47b2 + 31b3 − 2b5 )wi j +2b2 (b3 − 2b − 1 )wi,3− j + b(4+18b+11b2 +4b3 + 2b4 )w3−i, j + 2b3 (1 + 2b + b2 )w3−i,3− j

∂ 2 ZiT T ∂ wTi jT 2

∂ 2 ZiT T ∂ wTi jT ∂ wTi,3T − j

∂ 2 ZiT T ∂ wTi,3T − j ∂ wTi jT

∂ 2 ZiT T ∂ wTi,3T−2 j

⎤

 (2b5 − 31b3 − 47b2 − 24b − 4 ) ⎥ ⎦= (2b5 − 4b3 − 2b2 )

(2b5 − 4b3 − 2b2 )



(2b5 − 31b3 − 47b2 − 24b − 4 )

and

|H | = (2b5 − 31b3 − 47b2 − 24b − 4 )2 − (2b5 − 4b3 − 2b2 )2 > 0. Therefore, we can show that the Hessian matrix H is a diagonally dominant matrix, which guarantees the joint concavity of the profit function ZiT T (wTi jT , wTi,3T − j ). Thus, we get

b[2b(1 − 4b3 + 3b)E (ai j | f1 , f2 ) + (2b4 + 4b3 + 11b2 + 9b + 2 )w3−i, j + 2b2 (1 + b) w3−i,3− j − 2b2 (−b2 + 2 )wi,3− j ] . −2b5 + 31b3 + 47b2 + 24b + 4 2

wTi jT =

(A17)

W in Eq. (A17), we can derive the equilibrium wholesale price in Lemma 2. Substituting the equilibrium wholesale price Solving wW ij into Eq. (A16), we get equilibrium retail prices as shown in Lemma 2. The proof of Lemma 2 is completed.

Proof of Lemma 3: T andwW T are the same as those in case WW with uncertain demand; wW T andwW T are the In case WT with demand uncertainty, wW 11 12 21 22 same as those in case TT with uncertain demand. Then, we can derive:

T wW 11 =

2(2b + 1 )(1 − b2 )a + b(b2 + 5b + 2 )w21 + b(1 + b) (w22 + 2w12 ) , 2(−b3 + 7b2 + 8b + 2 )

T wW 12 =

2(2b + 1 )(1 − b2 )a + b(b2 + 5b + 2 )w22 + b(1 + b) (w21 + 2w11 ) , 2(−b3 + 7b2 + 8b + 2 )

T wW 21 =

b(2b(1 − 4b3 + 3b)E (a¯ | f1 , f2 ) + (2b4 + 4b3 + 11b2 + 9b + 2 )w11 + 2b2 (1 + b) w12 − 2b2 (−b2 + 2 )w22 ) , −2b5 + 31b3 + 47b2 + 24b + 4

T wW 22 =

b(2b(1 − 4b3 + 3b)E (a¯ | f1 , f2 ) + (2b4 + 4b3 + 11b2 + 9b + 2 )w12 + 2b2 (1 + b) w11 − 2b2 (−b2 + 2 )w21 ) ., −2b5 + 31b3 + 47b2 + 24b + 4

2

2

2

2

Solving the above equations, we obtain the equilibrium wholesale price in Lemma 3. Substituting the equilibrium wholesale price into Eq. (13), we get the equilibrium retail price in Lemma 3. The proof of Lemma 3 is completed.

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X. Li, J. Chen and X. Ai / European Journal of Operational Research 275 (2019) 939–956 Table A2 Thresholds of v.

vI vII vI I I

2

2

vV

2

2

2

2

[x0 (1+2b) +2x1 ]y20 −[x3 (1+2b) +2x4 ] (4+3b−3b2 ) 2

2

2t y20 y1 −t [y3 (1+2b) +2y4 ] (4+3b−3b2 ) 2

2

3

2

2

[x1 +x0 (1+2b) ] (2−b) (1+b) −[x6 +x5 (1+2b) ] (4+3b−3b2 ) 2 2 3 2 t y1 (2−b) (1+b) −t[y6 +y5 (1+2b) ] (4+3b−3b2 ) 2

vIV

3

[x2 (1+2b) +2x4 ] (2−b) (1+b) −[x5 (1+2b) +2x6 ]y0 2 2 2 3 2 t [y2 (1+2b) +2y4 ] (2−b) (1+b) −t [y5 (1+2b) +2y6 )y0 2 ]

2

2

3

2

2

[2x4 (4+3b−3b2 ) +x0 (1+2b) y20 ] (2−b) (1+b) −[2x6 −x5 (1+2b) ]y20 (4+3b−3b2 ) 2

2

3

2

2

t [2y4 (4+3b−3b2 ) ] (2−b) (1+b) −t[2y6 y20 −y5 (1+2b) y20 ] (4+3b−3b2 ) 2

2

2

3

2

2

2

3

2

2

2

[2x1 y20 +x2 (2b+1 ) (4+3b−3b2 ) ] (2−b) (1+b) −[2x4 (4+3b−3b2 ) −x5 (2b+1 ) y20 ] (4+3b−3b2 ) 2

2

2

t[2y1 y20 +y2 (2b+1 ) (4+3b−3b2 ) ] (2−b) (1+b) −t[2y4 (4+3b−3b2 ) −y5 (2b+1 ) y20 ] (4+3b−3b2 )

Proof of Proposition 1: W0 wW ij T0 wW 1j T0 wW 1j T0 wW 2j T0 wW 2j

wTi jT 0

=

(1 + b)(b5 − 12b4 − 19b3 + 38b2 + 56b + 16 ) b(b5 − 2b4 − 19b3 + 16b2 + 34b + 12 ) =1+ > 1, 2 3 2 (4 − 3b + 3b)(3b + 9b + 12b + 4 ) (4 − 3b2 + 3b)(3b3 + 9b2 + 12b + 4 )

=

3b3 + 9b2 + 12b + 4 > 1, and −b4 + 8b3 + 7b2 + 2b

=

b6 − 12b5 − 19b4 + 38b3 + 56b2 + 16b + (b6 − 4b5 − b4 + 2b3 + 18b2 + 24b + 8 ) > 1. b6 − 12b5 − 19b4 + 38b3 + 56b2 + 16b

T 0 < wW T 0 < wW W 0 . Similarly, we can derive pT T 0 < pW T 0 < pW T 0 < pW W 0 . The proof of Then, we can derive that wTi jT 0 < wW 2j 1j ij ij 2j 1j ij Proposition 1 is completed.

Proof of Proposition 2: Contract choice equilibrium with certain demand can be obtained by setting f j − a = 0 in contract choice equilibrium when demands are uncertain. Then, please refer to the proof of Proposition 3. Proof of Proposition 3: From expression M2T W , M2T T , RT2W , RT2 T , we get that 1) if FIM < FIR , then M2T W < M2T T , RT2W < RT2 T ; 2) if FIM > FIR , then M2T W > M2T T , RT2W > RT2 T . Comparing FIM andFIR , we can derive:

FIM − FIR =

=

(1 − b)[a2 (x2 (1 + 2b)2 + 2x4 ) + ut (y2 (1 + 2b)2 + 2y4 )] (1 − b)[a2 (x5 (1 + 2b)2 + 2x6 ) + ut (y5 (1 + 2b)2 + 2y6 )] − 2 2 3 2 2 ( 1 + 3b ) ( 1 + 2b ) y0 2 2 ( 2 − b ) ( 1 + b ) ( 1 + 3b ) ( 1 + 2b )   [a2 (x2 (1 + 2b)2 + 2x4 ) + ut (y2 (1 + 2b)2 + 2y4 )](2 − b)2 (1 + b)3 (1 − b ) −[a2 (x5 (1 + 2b)2 + 2x6 ) + ut (y5 (1 + 2b)2 + 2y6 )]y0 2 2 ( 1 + 3b )y0 2 ( 2 − b ) ( 1 + b ) ( 1 + 2b ) 2

 (1 − b ) = =

where vI =

vI is the

3



a2 [ ( x2 ( 1 + 2 b )2 + 2 x4 ) ( 2 − b )2 ( 1 + b )3 − ( x5 ( 1 + 2 b )2 + 2 x6 )y0 2 ] +ut[(y2 (1 + 2b)2 + 2y4 )(2 − b)2 (1 + b)3 + (y5 (1 + 2b)2 + 2y6 )y0 2 ] 2 ( 1 + 3b )y0 2 ( 2 − b ) ( 1 + b ) ( 1 + 2b ) 2



3

( 1 − b )a2 ( 1 − v/v1 )  2 2 3 2 2 ( 1 + 3b ) ( 2 − b ) ( 1 + b ) ( 1 + 2b ) y0 2 [ ( x2 ( 1 + 2b ) + 2x4 ) ( 2 − b ) ( 1 + b ) − ( x5 ( 1 + 2b ) + 2x6 )y0 2 ] 2

2

3

2

[x2 (2b+1 )2 +2x4 ] (2−b)2 (1+b)3 −[x5 (2b+1 )2 +2x6 ]y0 2

t [y2 (2b+1 )2 +2y4 ] (2−b)2 (1+b)3 −t [y5 (2b+1 )2 +2y6 )y0 2 ] threshold value of FIM =FIR , and when b > 0.4147,

there exists vI < 0. Then we can derive:

1) if b ∈ (0, 0.4147 ), or b ∈ (0.4147, 1 ) and v > vI , then FIM FIR . Therefore, with demand uncertainty, when M1 adopts a two-part tariff contract with information sharing: 1) for any mation 2) for any as long

given b ∈ (0, 0.4147 ), or b ∈ (0.4147, 1 ) and v > vI , M2 and the two retailers all prefer a two-part tariff contract with inforsharing, as long as the lump sum fee set in the range (FIM , FIR ), where vI is given in Table A2 in Appendix. given b ∈ (0.4147, 1 ) and v ∈ (0, vI ), M2 and the two retailers prefer a wholesale price contract without information sharing, as FIR < FIM . The proof of Proposition 3 is completed.

Proof of Proposition 4: The proof of Proposition 4 is similar to the proof of Proposition 3.

X. Li, J. Chen and X. Ai / European Journal of Operational Research 275 (2019) 939–956

955

Proof of Corollary 1: (i) FIM − FIM = II

2

(1−b)[x0 y0 2 −(4−3b2 +3b) (x2 +vy2 )] (1−b)[2(b+1 )3 (b−2 )2 (x3 +vy3 )−(x5 +vy5 )y0 2 ] > 0 and FIM − FIIM = > 0. 2 3 2 II 2 2y0 2 (3b+1 ) (4−3b2 +3b)  2y0 (3b+1)(b+1) (b−2) 2

(1−b)[(4−3b2 +3b) (x4 +vy4 )−(x1 +vy1 )y0 2 ] > 0, FIIR − FIR = 2 2(3b+1 ) (2b+1 )2 (4−3b2 +3b) y0 2 (1−b) 0, where 5 = > 0. 2 2y0 2 (3b+1 ) (4−3b2 +3b) (b−2 )2 (b+1 )3 (2b+1 )2

(ii) FIRII − FIR =

(iii) (iv) (1) (2) (3)

5

2(4 − 3b2 + 3b)2 (b − 2 )2 (b + 1 )3 (x4 + vy4 ) −(x1 + vy1 )(b − 2 )2 (b + 1 )3 y0 2 − (x6 + vy6 )(4 − 3b2 + 3b)2 y0 2

 >

R Comparing vI , vII and vIII , we can obtain that when v > vIII , there exists FIM < FIR , FIIM < FIIR , and FIM II < FIII . R , and vIV is the threshold value of FIM = F II I when v > vIV , there exists FIM < FIR ; II R when v < vIV , there exists FIM II > FI . if b > 0.2026, then vIV > 0.

Therefore, if b ∈ (0, 0.2026 ), or b ∈ (0.2026, 1 ) and v > vIV , then FIM < FIR . Based on the above results, we can derive the first point of II Corollary 1. Similarly, we can derive the other two points of Corollary 1 easily. The proof of Corollary 1 is completed. Proof of Proposition 5: From Corollary 1, we can get that: W < MT T , MW W < MW T , and 1) If either i) b ∈ (0, 0.2684 ) or ii) b ∈ (0.2684, 1 ) and v > vIII (Regions L1, L2, and L3 in Fig. 7), then MW 2 2 i i W T T T W W T T W W W T W T T T W W T T W W W T W T T T M1 < M1 ; R1 < R1 , R1 < R1 , and R1 < R1 ; R2 < R2 , R2 < R2 , and R2 < R2 . W > MT T , 2) If either i) b ∈ (0.2684, 1 ) and v ∈ (0, vIII ) or ii) b ∈ (0.4147, 1 ) and min(1, vIII ) > v > max(vI , vII ) (Region L4 in Fig. 7), then MW i i W W W T W T T T W W T T W W W T W T T T W W T T W W W T W T T T M2 < M2 , and M1 < M1 ; R1 < R1 , R1 < R1 , and R1 < R1 ; R2 < R2 , R2 < R2 , and R2 < R2 . W > MT T , MW W > MW T , and MW T < MT T ; RW W < RT T , RW W < RW T , and 3) If b ∈ (0.5481, 1 ) and vII > v > vI (Region L5 in Fig. 7), then MW 2 2 1 1 1 1 1 1 i i W T T T W W T T W W W T W T T T R1 < R1 ; R2 < R2 , R2 < R2 , and R2 < R2 . W > MT T , 4) If either i) b ∈ (0.5481, 1 ) and v ∈ (0, vI ) or ii) b ∈ (0.5481, 1 ) and vI > v > vII (as shown in Region L6 in Fig. 7), then MW i i W W W T T T W T W W T T W W W T W T T T W W T T W W W T W T T T M2 < M2 , and M1 < M1 ; R1 < R1 , R1 < R1 , and R1 < R1 ; R2 < R2 , R2 < R2 , and R2 < R2 . W > MT T , MW W > MW T , and MW T > 5) If b ∈ (0.5481, 1 ) and min(vI , vII ) > v > 0 (as shown in the Regions L7 and L8 in Fig. 7), then MW 2 2 1 i i T T W W T T W W W T W T T T W W T T W W W T W T T T M1 ; R1 < R1 , R1 < R1 , and R1 < R1 ; R2 < R2 , R2 < R2 , and R2 < R2 .

In addition, 1) if and only if either b ∈ (0, 0.2037 ), or b ∈ (0.2037, 1 ) and v > vIV (as shown in Regions L1 and L2 in Fig. 7), then (FIM , FIR ) ∩ (FIIM , FIIR ) ∩ (FIM , FIRII ) = (FIM , FIR ); II II 2) if and only if b ∈ (0.7470, 1 ) and v ∈ (0, vV ) (as shown in Region L8 in Fig. 7), then (FIR , FIM ) ∩ (FIIR , FIIM ) ∩ (FIRII , FIM ) = (FIIR , FIM ). II Therefore, we can derive that (i) for any given b ∈ (0, 0.2037 ), or b ∈ (0.2037, 1 ) and v > vIV , there exist FIR and FIM . A two-part tariff contract with information II sharing is a subgame-perfect Bayesian–Nash equilibrium for all members, as long as FIm < F < FIr . II R M (ii) for any given b ∈ (0.7470, 1 ) and v ∈ (0, vV ), there exists FII and FI such that a wholesale contract without information sharing is a subgame-perfect Bayesian–Nash equilibrium for all members, as long as FIIr < FIm . The proof of Proposition 5 is completed. References Adida, E., & DeMiguel, V. (2011). Supply chain competition with multiple manufacturers and retailers. Operations Researc, 59(59), 156–172. Agrawal, S., Sengupta, R. N., & Shanker, K. (2009). Impact of information sharing and lead time on bullwhip effect and on-hand inventory. European Journal of Operational Research, 192(2), 576–593. Ai, X. Z., Chen, J., & Ma, J. H. (2012). Contracting with demand uncertainty under supply chain competition. Annals of Operations Research, 201(1), 17–38. Cachon, G. P. (2003). Supply chain coordination with contracts. In S. Graves, & T. Kok (Eds.), Handbooks in operations research and management science (11, pp. 227–339). North-Holland: Boston. Cachon, G. P., & Kök, A. G. (2010). Competing manufacturers in a retail supply chain: On contractual form and coordination. Management Science, 56(3), 571–589. Cai, G. S., Dai, Y., & Zhao, S. X. (2012). Exclusive channels and revenue sharing in a complementary goods market. Management Science, 31(1), 172–187. Caldentey, R., & Haugh, M. B. (2009). Supply contracts with financial hedging. Operations Research, 57(1), 47–65. Chen, F. (2003). Information sharing and supply chain coordination. In Ton G. de Kok, & Stephen C. Graves (Eds.), Handbooks in operations research and management science: Supply chain management. North-Holland: Boston. Coughlan, A. T. (1985). Competition and cooperation in marketing channel choice: Theory and application. Marketing Science, 4(1), 110–129. Corbett, C. J., Zhou, D., & Tang, C. S. (2004). Designing supply contracts: Contract type and information asymmetry. Management Science, 50(4), 550–559. Dong, J., Zhang, D., & Nagurney, A. (2004). A supply chain network equilibrium model with random demands. European Journal of Operational Research, 156(1), 194–212. Dukes, A., Gal-Or, E., & Srinivasan, K. (2006). Channel bargaining with retailer asymmetry. Journal of Marketing Research, 43(1), 84–97. Fang, Y., & Shou, B. (2015). Managing supply uncertainty under supply chain Cournot competition. European Journal of Operational Research, 243(1), 156–176.

Feng, Q., & Lu, L. X. (2013). Supply chain contracting under competition: Bilateral bargaining vs. Stackelberg. Production and Operations Management, 22(3), 661–675. Fiala, P. (2005). Information sharing in supply chains. Omega, 33, 419–423. Govindan, K., & Popiuc, M. N. (2014). Reverse supply chain coordination by revenue sharing contract: A case for the personal computers industry. European Journal of Operational Research, 233, 326–336. Gupta, S., & Loulou, R. (1998). Process innovation, product differentiation, and channel structure: Strategic incentives in a duopoly. Marketing Science, 17(3), 301–316. Gupta, S. (2008). Research note—Channel structure with knowledge spillovers. Marketing Science, 27(2), 247–267. Ha, A. Y., & Tong, S. (2008). Contracting and information sharing under supply chain competition. Management Science, 54(4), 701–715. Ha, A. Y., & Tong, S. (2011). Sharing demand information in competing supply chains with production diseconomies. Management Science, 57(3), 566–581. Hill, A., Doran, D., & Stratton, R. (2012). How should you stabilise your supply chains? International Journal of Production Economics, 135(2), 870–881. Holweg, M., Disney, S., Holmstrom, J., & Smaros, J. (2005). Supply chain collaboration: Making sense of the strategy continuum. European Management Journal, 23(2), 170–181. Horn, H., & Wolinsky, A. (1988). Bilateral monopolies and incentives for merger. RAND Journal of Economics, 19(3), 408–419. Ingene, C. A., & Parry, M. E. (2007). Bilateral monopoly, identical competitors/distributors, and game theoretic analyses of distribution channels. Journal of the Academy of Marketing Science, 35(4), 586–602. Karray, S. (2015). Cooperative promotions in the distribution channel. Omega, 51, 49–58.

956

X. Li, J. Chen and X. Ai / European Journal of Operational Research 275 (2019) 939–956

Kembro, J., Näslund, D., & Olhager, J. (2017). Information sharing across multiple supply chain tiers: A Delphi study on antecedents. International Journal of Production Economics, 193(11), 77–86. Kreps, D. (1990). A course in microeconomics theory. New Jersey: Princeton University Press. Lee, H., So, K., & Tang, C. (20 0 0). The value of information sharing in a two-level supply chain. Management Sciences, 46, 626–643. Lee, G. Y., & Yang, R. (2013). Supply chain contracting with competing suppliers under asymmetric information. IIE Transactions, 45(1), 25–52. Li, L. (2002). Information sharing in a supply chain with horizontal competition. Management Science, 48(9), 1196–1212. Li, L., & Zhang, H. (2008). Confidentiality and information sharing in supply chain coordination. Management Science, 54(8), 1467–1481. Li, T., & Zhang, H. T. (2015). Information sharing in a supply chain with a make-to– stock manufacturer. Omega, 50, 115–125. Li, B. X., Zhou, Y. W., Li, J. Z., & Zhou, S. P. (2013). Contract choice game of supply chain competition at both manufacturer and retailer levels. International Journal of Production Economics, 143(1), 188–197. Lin, Y. T., & Parlakturk, A. K. (2014). Vertical integration under competition: Forward, backward, or integration. Production and Operations Management, 23(1), 19–35. McGuire, T., & Staelin, R. (1983). An industry equilibrium analysis of downstream vertical integration. Marketing Science, 2(2), 161–191. Moorthy, K. S. (1988). Strategic decentralization in channels. Marketing Science, 7(3), 335–355. Nagurney, A., Cruz, J., Dong, J., & Zhang, D. (2005). Supply chain networks, electronic commerce, and supply side and demand side risk. European Journal of Operational Research, 164(1), 120–142. O’Brien, D. P., & Shaffer, G. (1992). Vertical control with bilateral contracts. RAND Journal of Economics, 23(3), 299–308. O’Brien, D. P., & Shaffer, G. (2005). Bargaining, bundling, and clout: The portfolio effects of horizontal mergers. Rand Journal of Economics, 36(3), 573–595. Ouyang, Y. (2007). The effect of information sharing on supply chain stability and the bullwhip effect. European Journal of Operational Research, 182(3), 1107–1121. Özer, Ö., & Wei, W. (2006). Strategic commitments for an optimal capacity decision under asymmetric information. Management Science, 52(8), 1238–1257. Özer, Ö., & Raz, G. (2011). Supply chain sourcing under asymmetric information. Production and Operations Management, 20(1), 92–115.

Prajogo, D., & Olhager, J. (2012). Supply chain integration and performance: The effects of long-term relationships, information technology and sharing, and logistics integration. Journal of Production Economics, 135(1), 514–522. Raju, J., & Roy, A. (20 0 0). Market information and firm performance. Marketing Science, 46(8), 1075–1084. Raju, J., & Zhang, Z. J. (2005). Channel coordination in the presence of a dominant retailer. Marketing Science, 24(2), 254–262. Ryu, S. J., Tsukishima, T., & Onari, H. (2009). A study on evaluation of demand information-sharing methods in supply chain. International Journal of Production Economics, 120(1), 162–175. Sahin, F., & Robinson, E. P. (2002). Flow coordination and information sharing in supply chains: Review, implications, and directions for future research. Decision Sciences, 33(4), 505–536. Shin, H., & Tunca, T. I. (2010). Do firms invest in forecasting efficiently? The effect of competition on demand forecast investments and supply chain coordination. Operations Research, 58(6), 1592–1610. Trivedi, M. (1998). Distribution channels: An extension of exclusive retailership. Management Science, 44(7), 896–909. Vives, X. (1984). Duopoly information equilibrium: Cournot and Bertrand. Journal of Economic Theory, 34(1), 71–94. Wu, C. Q., & Mallik, S. (2010). Cross-sales in supply chains: An equilibrium analysis. International Journal of Production Economics, 126(2), 158–167. Wu, D. S., Baron, O., & Berman, O. (2008a). Bargaining in competing supply chains with uncertainty. European Journal of Operational Research, 6(3), 1–9. Wu, J. H., Zhai, X., & Huang, Z. M. (2008b). Incentives for information sharing in duopoly with capacity constraints. Omega, 36(6), 963–975. Yao, D., Yue, X., & Wang, X. (2005). The impact of information sharing on a return policy with the addition of a direct channel. International Journal of Production Economics, 97(2), 196–209. Yao, Z., Leung, S. C. H., & Lai, K. K. (2008). Manufacturer’s revenue-sharing contract and retail competition. European Journal of Operational Research, 186(2), 637–651. Zhang, J., & Chen, J. (2013). Coordination of information sharing in a supply chain. International Journal of Production Economics, 143, 178–187. Zhou, H., & Benton, W. C. (2007). Supply chain practice and information sharing. Journal Operation of Management, 25(6), 1348–1365.