Contract preference for the dominant supplier subject to inventory inaccuracy

Computers & Industrial Engineering 141 (2020) 106323

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Contract preference for the dominant supplier subject to inventory inaccuracy

T

Feng Taoa, , Hao Yua, Tijun Fana, Kin Keung Laib ⁎

a b

Department of Management Science and Engineering, East China University of Science and Technology, 200237 Shanghai, China College of Economics, Shenzhen University, Shenzhen, China

ARTICLE INFO

ABSTRACT

Keywords: Inventory inaccuracy Contract preference Supply chain management Retail competition

Inventory inaccuracy is usually inevitable, and affects decision making. This paper investigates the operational decisions and contract preference in a supply chain with one supplier and two competing retailers that are subject to inventory inaccuracy. The supplier, acting as the Stackelberg leader, first decides the contract format and pricing policy. The two retailers are followers, which make their decisions simultaneously. We establish a newsvendor model to derive the equilibrium solutions analytically under the buyback contract, consignment contract and hybrid contract scenarios. Then we conduct a numerical study to illustrate the impact of inventory inaccuracy on the supplier chain members’ operational policy and the dominant supplier’s contract preference. The results show that inventory inaccuracy plays an important role in determining the retailers’ equilibrium order quantities and retail prices, and the supplier’s wholesale (consignment) price. In particular, one retailer is better off (worse off) if the other who adopts the consignment (buyback) contract improves the inventory availability. Importantly, inventory inaccuracy affects the supplier’s contracting strategy. If the inventory availability changes, then the optimal contract selection may, accordingly, be changed. The conclusions reveal that although inventory inaccuracy is assumed to have occurred on the retailer’s side, both the retailer and the supplier are greatly influenced by the inventory inaccuracy. Therefore, the supplier should fully consider inventory availability and retail competition before determining the contract format. The retailer should improve his inventory availability to enhance competitiveness, to receive a more profitable contract format from the supplier.

1. Introduction Competition over a single supplier selling products to multiple competing retailers is a significant and interesting issue, which has been widely discussed in practice and academic fields during the last decade (Chen & Guo, 2013; Yao, Leung, & Lai, 2008). In a decentralized supply chain, supply chain members seek to maximize their own profits. This objective results in independent actions, and causes the loss of total supply chain benefits, which is well-known as double marginalization (Spengler, 1950). Among the contracts that can coordinate a supply chain, the buyback contract is one of the most frequently used (Cachon, 2003). It allows the retailer to return unsold products at a predetermined price (Tsay, 1999). Buyback contracts have been adopted extensively in various retail businesses, such as fashion apparel, computers, publishing and cosmetics (Zhao, Meng, Wang, & Cheng, 2014). To reduce inventory costs, and further improve the profits of the total supply chain, attention is increasingly being paid to inventory

⁎

management. Accordingly, transferring inventory from the retailer to the supplier, as a new method, has been extensively adopted in practice (Fang, Wang, & Hua, 2016; Li & Hua, 2008). This transfer mechanism refers to the consignment contract, where the supplier keeps full ownership of the inventory, and receives payment from the retailer only after the products are sold. Consignment contracts exist in the live plant industry (The Home Depot), e-businesses (Amazon.com), the automotive market (Autozone), healthcare venues (the University of California San Francisco Medical Center) and so on (Matta, Lowe, & Zhang, 2014). Various studies have been conducted, in general, on contract design in supply chains when the inventory is assumed to be accurate (Pan, Lai, Leung, & Xiao, 2010; Zhao et al., 2014). However, inventory inaccuracy is generally neglected. Inventory inaccuracy refers to the difference between the available quantity of the inventory and the record in the inventory management system, which often is caused by inefficient inventory management (Sahin & Dallery, 2009). In the

Corresponding author. E-mail addresses: [email protected] (F. Tao), [email protected] (H. Yu), [email protected] (T. Fan), [email protected] (K.K. Lai).

https://doi.org/10.1016/j.cie.2020.106323 Received 26 March 2019; Received in revised form 20 January 2020; Accepted 23 January 2020 Available online 27 January 2020 0360-8352/ © 2020 Elsevier Ltd. All rights reserved.

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market, many firms suffer from inventory inaccuracy, which has considerable effects on the ordering policy in the supply chain (Fan, Tao, Deng, & Li, 2015). For example, investigating the inventory of about 500 stores, Kang and Gershwin (2004) found that only half of the inventory on record was equivalent to the real quantity for sale. Due to inventory inaccuracy, 10% of profits are lost, on average (Heese, 2007). As is known, the crucial decision making in a supply chain consists of the ordering policy and the pricing policy. These two operational decisions, often associated with the contracting strategy, determine the performance of the one-to-two supply chain (Yao et al., 2008; Zhang, Matta, & Lowe, 2010). However, inventory inaccuracy, an important and often unheeded factor, also has a great impact on the ordering policy and profits, which may further influence, or even change, the equilibrium ordering policy and selection of the optimal contract. Therefore, the following questions guided this research: (1) How does the inventory inaccuracy influence the equilibrium order quantity, retail price and wholesale price in a one-to-two supply chain? (2) Which kind of contract would the dominant supplier prefer under retailer differentiation, and more importantly, if two retailers’ inventory inaccuracies change, does the supplier’s contract preference change accordingly? (3) Depending on how large the difference between two retailers’ inventory availability is, will the retailer with the lower inventory rate give up the competition? To answer these questions, we consider a supply chain with one supplier and two retailers who are subject to inventory inaccuracy. The demand is stochastic and price-dependent during a single selling season. The consignment contract and the buyback contract are considered for two reasons. These two types of contracts are widely adopted in industries. More importantly, the leftover inventory on the retailer’s side in the buyback contract is sold to the supplier, and that in consignment contract is returned to the supplier. We investigate how the difference between the two kinds of contracts influences the corresponding optimal decisions, and furthermore, affects the contract preference when the supply chain suffers from inventory inaccuracy. The rest of this paper is organized as follows: In Section 2, related literature is reviewed. In Section 3, the key model notations and model assumptions are described. In Section 4, the formulations are developed, and the equilibrium solutions in different scenarios are analyzed. In Section 5, a numerical study is conducted, and the results of the optimal contracting strategy are discussed. In Section 6, the work is summarized, and conclusions presented.

deterministic inventory inaccuracy, Xu, Wei, and Tian (2012) compared ordering decisions in different cases in which the retailer ignored inventory inaccuracy, estimated inventory inaccuracy or shared inventory error information with the supplier. In addition, studies have extended the single-period ordering quantity problem to multipleperiod inventory control issues (Mersereau, 2013). Assuming that misplaced inventory could be recovered in the next selling season, Tao, Fan, Lai, and Li (2017) established a dynamic multi-period planning model to analyze the impact of inventory inaccuracy on the optimal inventory control policy. Although ordering and pricing decisions are also examined in this paper, we emphasize the analysis of the impact of inventory inaccuracy on operational policy and contract preference when the supplier is the leader. Vertical competition and horizontal competition are considered in this research. In addition, with respect to retail competition, we investigate different levels of retailers’ inventory inaccuracy. In terms of the second stream of literature, contract design in the supply chain has attracted much attention (Chen et al., 2013; Wang, Leng, & Liang, 2018). Under the buyback contract, the retailer pays the wholesale price for each unit ordered, but is allowed to return unsold products at an agreeable price, which is exploited extensively in various retail sectors (Pan et al., 2010). Considering the demands are pricedependent, Zhao et al. (2014) analyzed the effects of demand uncertainty under the buyback contract, and found that the buyback contract is beneficial for all supply chain members only when the level of demand uncertainty is intermediate. The consignment contract is a unique contract, where the supplier owns the goods, and does not receive the revenue from the retailer until the goods are sold (Wang et al., 2004; Zhang et al., 2010). Because the unsold products are returned to the supplier at the end of the current selling period, the consignment contract has some similarities with the buyback contract. However, the ownership of the inventory and the timing of payment for the orders in the two types of contract are different (Li & Hua, 2008). Ru and Wang (2010) investigated who should control the inventory in the consignment contract. They found that vendor-managed consignment inventory (VMCI) outperforms retailer-managed consignment inventory (RMCI). However, if the supply chain suffers from inventory inaccuracy, Tao, Fan, and Lai (2018) proposed that RMCI is sometimes better than VMCI for supply chain members. Therefore, RMCI is considered in this research, and we extend their model to the case where vertical competition and horizontal competition are simultaneously examined. In addition, a body of academic literature has focused on contract preference, which has been shown to be influenced by many economic and market factors (Lariviere & Porteus, 2001; Wang et al., 2004). Matta et al. (2014) investigated the effects of controllable factors (i.e., cost share and price markup) and uncontrollable factors (i.e., price sensitivity and demand uncertainty) on the supplier’s and retailer’s contract selection. Considering symmetric and asymmetric supply chain systems, Fang et al. (2016) analyzed how price sensitivities and the cost sharing rate influence the supplier’s contract preference. Although the works contributed to the literature, the scholars did not consider the impact of inventory inaccuracy on pricing and ordering decisions, and on contract preference when different contract scenarios are adopted. With respect to the third stream of literature, retail competition is another critical aspect in the supply chain, and the literature on competition for a single supplier selling products to multiple competing retailers is very rich (Bernstein & Federgruen, 2005; Chen & Guo, 2013; Chen & Xiao, 2017; Yao et al., 2008; Zhao, Tang, & Wei, 2012). For example, Adida and Ratisoontorn (2011) found that the most beneficial contract for the supplier depends on the retail competition level, while retailers always benefit more from the consignment price contract than from the wholesale price contract regardless of the retail competition level. Assuming that the supplier’s production capacity was limited, Chen et al. (2013) compared the impacts of two different allocation mechanisms (proportional allocation and lexicographic allocation) on

2. Literature review This paper builds on previous research in three streams: inventory inaccuracy (Cannella, Framinan, Bruccoleri, Barbosa-Póvoa, & Relvas, 2015; Kök & Shang, 2014), contract design in the supply chain (Avinadav, Chernonog, & Perlman, 2015; Wang, Jiang, & Shen, 2004) and retail competition (Chen, Li, & Zhang, 2013; Zhao, Wu, Jia, & Shu, 2018). We discuss these three key streams. In the first stream, inventory inaccuracy has been studied from various perspectives over the last decade (Heese, 2007; Wang, Fang, Chen, & Li, 2016). When discussing the operational policy, it is traditional to assume that the inventory is accurate. However, inventory inaccuracy is usually inevitable, and exists extensively in practice. Research has confirmed the existence of inventory inaccuracy, and specifies the impact of inventory inaccuracy on profits through empirical analysis (Dehoratius & Raman, 2008; Hardgrave, Aloysius, & Goyal, 2013). Other literature established a model for analyzing the inventory inaccuracy problem (Fan et al., 2015; Rekik, Sahin, & Dallery, 2008). By developing the models, stochastic (Lei, Chen, Wei, & Lu, 2015) and deterministic (Camdereli & Swaminathan, 2010) types of inventory inaccuracy have been studied. In terms of stochastic inventory inaccuracy, Rekik (2011) associated the issue of inventory inaccuracy with the random yield problem, and derived the optimal ordering policy when the inventory records are inaccurate. With respect to 2

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retail competition. Considering the deterministic demand, Zhang, Li, and Fan (2018) focused on Radio Frequency Identification (RFID) technology adoption strategies in a supply chain with two competing retailers under the wholesale price contract. The existing literature has demonstrated that the performance of the one-to-two supply chain is influenced by the contract format and retail competition. However, we found that inventory inaccuracy also plays an important role in the performance of the supplier-dominant supply chain. The papers most related to the present work are those by Pan et al. (2010) and Tao, Fan, Wang, and Lai (2019). Pan et al. (2010) analyzed the dominant supplier’s optimal contracting strategy between the wholesale price contract and the revenue-sharing contract in the one-totwo supply chain. In Tao et al. (2019), the wholesale price contract and the consignment contract are considered in a supply chain subject to inventory inaccuracy. The important difference between the present paper and Pan et al. (2010) is that we elaborate on the impact of inventory inaccuracy on the contracting decision. However, there are two key differences between the proposed model and the model in Tao et al. (2019): (a) Only vertical competition was analyzed by Tao et al. (2019), but we consider vertical competition and horizontal competition in the proposed model. (b) Whatever the equilibrium achieved in Tao et al. (2019), only one contract format was adopted in the supply chain. However, in this paper, using difference contract formats with different retailers is also the equilibrium solution.

Table 1 Definition of symbols.

i,

i = 1, 2 and j = 3

i,

a b cS c Ri Di pi Qi w yi zi

the the the the

potential market demand volume buyback price supplier’s unit manufacturing cost retailer i ’s unit holding cost, i = 1, 2

the the the the the the

sensitivity parameter with regard to the retailer’s own price sensitivity parameter with regard to the competitor’s price random scaling factor to capture the uncertainty of the demand Di inventory availability rate of product i supplier’s profit retailer i ’s profit

i

i i S

Ri

the demand for product i , which is stochastic and price-dependent the unit retail price of product i determined by retailer i the order quantity of product i the wholesale (consignment) price determined by the supplier the deterministic part of the demand function Di the stocking factor of product i and z i = Qi yi permanent inventory error (e.g. shrinkage) temporary inventory error (e.g. misplacement)

inaccuracy. Following the existing literature (Fan et al., 2015; Wang et al., 2016; Xu et al., 2012), inventory inaccuracy is considered to be composed of two parts: permanent inventory error i (e.g., shrinkage) and temporary inventory error i (e.g., misplacement). The main difference between the two inventory errors is whether the inventory can be recovered before the end of the current selling season. In this paper, we assume that the lost inventory, which cannot be recovered and sold, is valueless. This assumption is reasonable in the seasonal and perishable product industry, where damaged items and obsolete products have no salvage value, and cannot be sold. Accordingly, the inventory availability rate is calculated as i = 1 i . Note that the value of i can be estimated from empirical research or historical inventory data, and therefore, all the supply chain members have information about the inventory availability rate before making decisions. In addition, due to inventory inaccuracy, the available inventory ( i Qi ) on retailer i ’s side is lower than the order quantity (Qi ). The notations are presented in Table 1. We model the decision making during a single period in this supply chain as a Stackelberg-Nash game. Vertically, the supplier and the two retailers play a Stackelberg game, where the supplier is the Stackelberg leader, and the retailers are followers. Horizontally, the two retailers play a Nash game, where they determine their order quantities and retail prices simultaneously. The supply chain members are assumed to be risk-neutral. We relax this assumption in Section 5.3 to investigate the results when the two retailers are risk-averse. We find that the main conclusions of our paper do not change qualitatively. Regarding the contracting strategies, three different scenarios are investigated:

We consider a supply chain consisting of one supplier and two retailers who encounter inventory inaccuracy. The supplier manufactures two products, and sells them to customers through the two competing retailers. The two products are different but substitutable for each other. The two retailers and products are represented by i and j , where i = 1, 2 and j = 3 i . Accordingly, retailer i handles product i , and retailer j handles product j . Assume that it costs cS for the supplier to produce one unit of the product, and c Ri for retailer i to hold and handle one product. Consider that the demand for product i , denoted by Di , is stochastic and price-dependent. We adopt a multiplicative demand function to capture the randomness of the demand and the price sensitivity, which is widely used in the literature (Jiang & Wang, 2010; Petruzzi & Dada, 1999): pi + pj ·

Definition

i

3. Model assumptions

Di = yi (pi , pj )· i = ae

Symbol

(1)

where pi and pj are the retail prices of products i and j . yi (pi , pj ) is the deterministic part of the demand function, decreasing in pi and increasing in pj . a is the potential market demand volume. and are the price elastic coefficients, which represent the sensitivity of each product’s market demand to its own retail price and its competitor’s price. Note that the fraction reflects the competition intensity between the is close to 1, the two products are more subtwo retailers. When stitutable, which means the retail price of one product has a higher effect on the market demand of the other product, so the retail competition is intensifying. When approaches 0, the products are more diverse, and the retail competition is relatively mild. i is a random scaling factor, which is the stochastic part of the demand function Di , representing the demand uncertainty of product . To focus on inventory inaccuracy, we suppose that the random scaling factors of D1 and D2 , 1 and 2 , have the same distribution, with cumulative distribution function F ( ) and probability density function f ( ) that have support on [A, B] R + with B > A (Fang et al., 2016; Wang, Wang, & Wang, 2013). Note that in Section 5.3, we discuss the results when assuming i and j are correlated. We assume that inventory inaccuracy occurs on the retailer’s side, so only some of the products are available for sale. We define the inventory inaccuracy rate i (0< i 1) to model the impact of inventory

(1) Buyback contract scenario: the supplier provides buyback contracts for the two retailers. (2) Consignment contract scenario: the supplier offers the two retailers consignment contracts. (3) Hybrid contract scenario: the supplier provides a buyback contract to one retailer, and provides a consignment contract to the other. 4. Analysis In this section, we model the three types of contract scenarios, and find the equilibrium solutions under different contracting strategies. 4.1. Buyback contract scenario In the buyback contract, the retailers pay the supplier as soon as 3

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they receive the products, and then they own the inventory (Tsay, 1999). At the end of the selling season, the leftover inventory is returned to the supplier at the price of b per unit. To compare this contract scenario with the other contract scenarios intuitively, we assume that the buyback price b is not a decision variable (b is a parameter). The decisions under the buyback contract scenario are made in two sequential steps: (1) the dominant supplier determines the wholesale price for each unit ordered; (2) considering the inventory inaccuracy, each retailer decides the order quantity and the retail price simultaneously before the demand is realized. We derive the equilibrium solutions with backward induction. In the second stage of the decision sequence, retailer i determines the optimal order quantity and retail price to maximize the retailer’s own expected profit based on the wholesale price w given by the supplier. Therefore, the objective function is: B Ri

= E [pi ·min(Di , = E [(pi

i Qi )

(w + c Ri )·Qi + b·( i Qi

b)·min(Di ,

i Qi )

(w + c Ri

Di )+]

i b)·Qi]

,

responses of the two retailers, the supplier chooses the best wholesale price w to maximize her profit. Let z i = Qi yi (pi , pj ), and the supplier’s expected profit can be expressed as: B S

i = 1, 2 and j = 3

i.

= E [(pi

b)·min(yi i, yi i z i )

= yi [(pi

b)·( i z i

= yi [pi ·( i z i = yi [( i pi w

( i z i ))

i

= i

( i z i )) c Ri )·z i

i b)·z i ]

(w + c Ri )· z i + b · ( i z i )] (pi b)· ( i z i )]

(2)

C Ri

,

B i b) z i

(w + c Ri B i zi

( i z iB )

+ b,

(4)

i i b)

=

1

1 F ( i z iB )

B i zi

( i z iB )

.

yi ((w

= E [(pi

= yi [(pi

i b)· Qi ]

( i z i )) + (w

cS )· z i

cS

i b)· z i )

b · ( i z i ))

.

(7)

w )·min(Di ,

i Qi )

(8)

c Ri Qi],

w )·min(yi i, yi i z i ) w )·( i z i

c Ri yi z i]

( i z i ))

c Ri z i]

.

(9)

Note that the payment to the supplier in the consignment contract is not realized until the products are eventually sold. Thus, in contrast to the buyback contract scenario, price w in the consignment scenario is associated with the sold quantities, rather than the ordered quantities. Whereas, c Ri , which is the holding cost of retailer i , is calculated based on the order quantity, rather than the available inventory. We denote the best response for retailer i as (ziC , piC ) , which maximizes profit function RCi for a given consignment price w . We show that the optimal solutions can be obtained from the following lemma.

(5)

B i zi

yi (b ·( i z i

= E [(pi

C Ri

and

(w + c Ri

cS

where superscript C refers to consignment contract scenario. Retailer i ’s expected profit under the consignment contract consists of two terms: One refers to the expected revenues from the sold products; and the other is the holding costs of the inventory. Similarly, let z i = Qi yi (pi , pj ) . The function above can be rewritten as:

Lemma 1. Given the wholesale price w chosen by the supplier, under the buyback contract, retailer i decides the optimal stocking factor z iB and retail price piB , which are expressed as:

+

+ (w

In the consignment contract scenario, the supplier retains full ownership of the inventory until the items are sold. Therefore, the supplier has to bear all the risk of excessive inventory caused by the demand uncertainty. Similarly, decisions are made in two sequential steps in the consignment contract scenario. In the second stage of the decision sequence, retailer i determines the best order quantity and retail price to maximize the retailer’s own expected profit based on the consignment price w given by the supplier, which is formulated as:

where ( i z i ) = A ( i z i u ) f (u ) du . In the buyback contract scenario, wholesale price w is associated with the order quantity, as the retailers pay the supplier when they order products. Denoting the best response for retailer i under the buyback contract scenario as (ziB , piB ) , we obtain the retailer’s optimal solutions as the following lemma.

1

i Qi )

4.2. Consignment contract scenario

i zi

piB =

Di )+]

b·( i Qi

The supplier’s expected profit under the buyback contract scenario is composed of two parts: One part is the expected revenues from the ordered products, and the other part is the costs of buying back the unsold products. Note that Eqs. (5) and (6) show that z iB and piB are the implicit functions of w . It is infeasible to obtain the explicit function of the wholesale price. And it is also intractable to analytically discuss the monotonicity of the pricing decisions. Thus, we need to resort to a numerical study in Section 5.

i b)· yi z i]

(w + c Ri

E [b ·min(Di ,

=

(3)

( w + c Ri

c Si )·Qi

i

By this assumption, determining the optimal order quantity Qi is equivalent to determining z i . Therefore, the problem of choosing the best (Qi , pi ) is equal to choosing the best (z i , pi ) . Accordingly, Eq. (2) can be rewritten as: B Ri

E [(w i

=

where superscript B refers to the buyback contract scenario. The first term in the expected function refers to the expected revenue of the sold products, the second term represents the ordering and holding costs, and the third term is the expected revenue for the buyback inventory. Following Petruzzi and Dada (1999) and Wang et al. (2004), we define the stocking factor

z i = Qi yi (pi , pj ),

=

Lemma 2. Considering the consignment price w given by the supplier, under the consignment contract, retailer i decides the optimal stocking factor z iC and retail price piC , which are expressed as:

(6)

The proof of Lemma 1 is shown in Part A1 of Supplementary Material. Lemma 1 shows that the optimal stocking factor and retail price in the buyback contract are associated with the inventory availability rates. According to Eq. (5), retailer i ’s equilibrium retail price consists of three parts: The first part is associated with the retailer’s own price sensitivity parameter. The second part reflects the retailer’s purchasing cost and holding cost for each unit order. The third part is the compensation from the supplier for each unit unsold. Note that retailer i ’s equilibrium retail price and stocking factor are also influenced by his competitor’s inventory availability rate, because the best response price and inventory quantity are affected by the wholesale price w , which is related to both retailers’ ordering and pricing decisions. In the first stage of the decision sequence, considering the best

piC =

1

+

c Ri z iC C i zi

( i z iC )

+ w,

(10)

and i

c Ri

=

1

1 F ( i z iC )

C i zi C i zi

( i z iC )

.

(11)

The proof of Lemma 2 is shown in Part A2 of Supplementary Material. Analogously, the optimal stocking factor and retail price in the consignment contract are related to the inventory availability rates. According to Eq. (10), retailer i ’s equilibrium retail price is composed of 4

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three parts: One part is associated with the retailer’s own price sensitivity parameter, one part reflects the retailer’s holding cost for each unit order, and one part is the consignment price paid to the supplier for each unit sold. Interestingly, from Eq. (11), in contrast to the buyback contract scenario, the optimal stocking factor in the consignment contract is independent of the consignment price. Thus, the equilibrium stocking factor is not affected by the competitor’s inventory availability rate. Further, we obtain the following property of the stocking factor.

The proof of Corollary 2 is shown in Part A5 of Supplementary Material. Corollary 2 indicates that under the consignment contract, the equilibrium retail price and the consignment price are also decreasing in the inventory availability in the symmetric case. When the consignment contract is adopted, the supplier and the retailer share the risk of inventory inaccuracy, where the supplier takes the risk of losing wC per unit of the unavailable inventory, and the retailer takes the risk of losing pC w C per unit of the unavailable inventory. Because the supplier retains full ownership of the inventory, and does not receive the payment from the retailer until the items are sold, the increase in the inventory availability rate results in an increment in the sales, which encourages the supplier to reduce the consignment price, and further urges the retailer to reduce the retail price. Therefore, the decrease in the consignment price is an incentive for the retailer to improve the efficiency of inventory management. By substituting Eqs. (11), (14) and (15) into Eqs. (9) and (12), we obtain the optimal expected profits of retailer i and the supplier as follows:

Corollary 1. In the one-to-two supply chain, under the consignment contract, the optimal stocking factor is decreasing in the inventory availability rate. The proof of Corollary 1 is shown in Part A3 of Supplementary Material. It is intuitive that when the inventory availability rate increases, more inventories could be used for sold so that fewer products should be ordered, which drives the stocking factor to decrease. In the first stage of the decision sequence, considering the best responses of the two retailers, the supplier decides the best consignment price w , and the supplier’s expected profit under the consignment contract scenario can be expressed as: C S

= w·E [min(D1, 1 Q1) + min(D2 , = w·[y1 ·( 1 z1 ( 1 z1)) + y2 ·( 2 z2

cS ·(Q1 + Q2 ) . ( 2 z2))] cS ·(y1 z1 + y2 z2)

From Eq. (12), the first term in the supplier’s expected function represents the expected revenues of the sold products; and the second term refers to the manufacturing costs. With some algebra, we find the equilibrium solution from the following corollary.

wC =

( 1 z1C )) + (

(( ( )(a1 ( 1 z1C

(

C

c Ri ziC

+( ) cS z2C ) C ( 1 z1 )) + a2 ( 2 z2C

( 2 z2C )))

,

(13)

C

where ai = e i zi ( i zi ) j z j ( j z j ) . The proof of Lemma 3 is shown in Part A4 of Supplementary Material. Lemma 3 indicates that the supplier’s optimal consignment price depends on both retailers’ inventory availability rates. Apparently, it is too complicated to derive the monotonicity of the consignment price, as well as the selling price. Nonetheless, one can simplify the expression by assuming that the holding costs and the inventory availability rates are symmetric, namely, c Ri = c Rj and i = j . There-

1

+

C i zi

cS z iC ( i ziC )

(14)

Accordingly, retailer i ’s equilibrium retail price is

piC

=

1

+

c Ri z iC C i zi

( i z iC )

+

wC

=

piC + pC j

[ i z iC

( i ziC )],

1

+

1

+

(cS + c Ri ) z iC C i zi

( i z iC )

yiC [ i z iC

( i z iC )]

=

2

ae

piC + pC j

[ i z iC

(16)

( i z iC )].

In the hybrid contract scenario, the supplier provides one retailer with the buyback contract, and provides the other with the consignment contract. Without loss of generality, we assume that the supplier offers the buyback contract to retailer i , and the consignment contract to retailer j , where i = 1, 2 and j = 3 i . Similarly, we derive the equilibrium solutions with backward induction. In the second stage of the decision sequence, based on the wholesale (or consignment) price w given by the supplier, retailer i and retailer j decide the optimal order quantity and retail price, respectively, to achieve the maximum expected profits, which are formulated as:

fore, ai = aj = e i ziC ( i ziC ) . It follows from Lemma 3 that the equilibrium consignment price is

wC =

ae

4.3. Hybrid contract scenario

) c Ri ziC

(

1

The proof of Corollary 3 is shown in Part A6 of Supplementary Material. Corollary 3 indicates that under the consignment contract, in a supply chain with one leading supplier and two follower retailers, the supplier and the retailers always benefit from the improvement of the inventory availability. It is beneficial for the supplier because more inventory is available to sell; and it is profitable for the retailer because the consignment price decreases, which further increases the expected demand.

c Rj zC j C

=2

=

Corollary 3. In a symmetric case, with respect to the consignment contract, the supplier’s and the retailer’s optimal expected profits are increasing in their own inventory availability rates.

) cS z1C ) + a2

C 2 z2 ))

C

C S

( i z iC )]

(17)

Lemma 3. Under the consignment contract, considering inventory inaccuracy, the equilibrium consignment price given by the supplier is as follows: C 2 z2

=

2 Q2 )]

(12)

a1 (( 1 z1C

yiC [ i z iC

C Ri

HB Ri

= E [(pi

b)·min(Di ,

i Qi )

= E [(pi b)·min(yi i, yi i z i ) = yi [( i pi w c Ri )· z i (pi

. (15)

HC Rj

Based on this rearrangement, we have the following conclusions with respect to the prices. Corollary 2. In a symmetric case, under the consignment contract,

= E [(pj

w )·min(Dj ,

j Qj )

(w + c Ri

(18)

c Rj Qj]

= E [(pj

w )·min(yj j, yj j zj )

c Rj yj zj],

= yj [(pj

w )·( j zj

c Rj zj]

( j zj ))

i b)· Qi]

(w + c Ri i b)·yi z i ], b)· ( i z i )]

(19)

where superscript HB refers to the retailer who uses the buyback contract, and superscript HC refers to the retailer who uses the consignment contract in the hybrid contract scenario. Following the same procedures stated in Sections 4.1 and 4.2, we find the optimal stocking factors in the hybrid contract scenario satisfy:

(1) the equilibrium consignment price is decreasing in the inventory availability rate; (2) the equilibrium retail price is decreasing in the inventory availability rate. 5

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F. Tao, et al.

i

(w + c Ri j

c Rj

=

i b)

=

1

HB i zi

HC j zj

1 F ( j z jHC )

1

HB i zi

1 F ( i z iHB )

HC j zj

( j z jHC )

( i z iHB )

,

availability rate is greater than 0.6. We fix the inventory availability rate of retailer 2 at 0.9, and then increase the inventory availability rate of retailer 1 from 0.6 to 1.0 with 21 discrete values. We choose the price sensitivity parameters = 2 and = 1 to ensure that > when investigating the impact of inventory availability on the decisions. Figs. B1 to B6 in Part B of Supplementary Material show the impact of the inventory availability rates on the retailers’ equilibrium decisions and profits in different contract scenarios. Note that BB refers to the buyback contract scenario, CC refers to the consignment contract scenario, and BC (CB) refers to the hybrid contract scenario, in which the buyback contract is given to retailer 1 (2), and the consignment contract is given to retailer 2 (1). Fig. B1 shows that the retailer’s best response stocking factor is decreasing in its own inventory availability in the consignment contract (Corollary 1), but it is not monotonic with respect to the inventory availability in the buyback contract. We know that with the increase in , fewer products are needed to meet the demand, so the order quantities decrease. However, the increase of also decreases the retail price, which further decreases the wholesale price, and drives the retailer to order more products. Thus, the non-monotonicity between the stocking factor and inventory availability in the buyback contract is contingent on the aggregate effect. Fig. B2 shows the effect of inventory availability on the price. We find that the retailers’ response prices are all decreasing in their own inventory availability rates, as well as in the other retailer’s inventory availability except for the buyback contract. From Eq. (5), it is easy to see that the retail price in the buyback contract is associated with the wholesale price which depends on the inventory availability rates and operational decisions of both retailers. Thus, the monotonicity of the retail price in the buyback contract is not clear. Fig. B3 depicts the sensitivity of the inventory availability of the retailers’ profits. The retailer’s profit is always increasing in its own inventory availability. This observation illustrates that the retailer can benefit from her own improvement in the inventory availability. Interestingly, a competitor’s improvement of its own inventory availability is not universally beneficial. We can know that when retailer i uses the consignment contract, the improvement in retailer i ’s inventory availability benefits retailer j ; however, when retailer i uses the buyback contract, the improvement in retailer i ’s inventory availability makes retailer j worse off. This result directly indicates that inventory availability is a very important factor in the decision regarding which contract format to adopt. Fig. B4–B6 in Part B of Supplementary Material show the impact of inventory availability rate on the supplier’s pricing decisions and profits in different contract formats. We find that in the consignment contract scenario and the hybrid contract scenario, the equilibrium wholesale (or consignment) prices are all decreasing in one of the retailers’ inventory availability rates. However, in the buyback contract scenario, the monotonicity is not explicitly observed (see Fig. B4). Based on the figures, in Table 2 we summarize the effect of inventory inaccuracy on the decisions and profits. From Table 2, we find that the stocking factor in the consignment contract is larger than that in the buyback contract. Two effects account for this phenomenon: (1) In the risk effect, under the consignment contract, the supplier bears the whole inventory risk, and consequently, the retailer places the order regardless of the excessive inventory. (2) In the independent effect, the retailer’s optimal stocking factor in the consignment contract is independent of the competitor’s inventory availability (see Eqs. (11) and (21)), and thus, inventory inaccuracy on the competitor’s side has no impact on the retailer’s own stocking factor. In this supply chain structure, the supplier’s profit consists of two parts: the profit from retailer 1 and the profit from retailer 2. Fig. B6 depicts the impact of inventory availability rate on the supplier’s profits in the contract scenarios, where S1 refers to the profit received

(20)

,

(21)

and the optimal retail prices satisfy:

piHB = pjHC =

1

1

+

(w + c Ri HB i zi

(

HB i b) z i HB ) i zi

c Rj z jHC

+

HC j zj

( j z jHC )

+ b,

(22)

+ w.

(23)

Assume that the wholesale price given to retailer i is equal to the consignment price given to retailer j under the hybrid contract scenario. In the first stage of the decision sequence, considering the best responses of the two retailers, the supplier decides the best wholesale (or consignment) price w , and the supplier’s expected profit in the hybrid contract scenario can be expressed as: H S

= E [(w

c Si )· Qi

= yi (b·( i z i = yi ((w

cS )·z i

b·( i Qi

( i z i)) + (w

Di )+] + w·E [min(Dj , cS

i b)· z i )

b · ( i z i)) + yj (z j ·( j w

j Qj )]

+ wyj ·( j zj cS )

cS Qj ( j z j ))

cS yj zj .

w· ( j z j ))

(24) The supplier’s expected profit in the hybrid contract scenario is the sum of the retailer with the buyback contract and the retailer with the consignment contract. According to Eqs. (21)–(23), z iHB , piHB and pjHC are known only as the implicit functions of w , so it is infeasible to obtain the explicit function of price w . However, with the extensive numerical study discussed in Section 5, we can demonstrate the supplier’s decision, and get some insights. 5. Numerical study In this section, we conduct a numerical study to illustrate the impact of inventory inaccuracy on the results of equilibrium decisions in different scenarios. Then, we perform a comprehensive investigation to compare the performances of three contracting strategies under different degrees of retail competition and inventory availability rates. Finally, based on the previous analysis, we illustrate the supplier’s contract preference with respect to inventory inaccuracy. We assume that the stochastic part of the demand function i is uniformly distributed within the range [0, 2]. This assumption can ensure that i has a mean value of 1 so that the expectation of demand E (Di ) is equal to the deterministic part of demand yi . In addition, with this assumption, the two retailers can be taken as two symmetric retailers. Therefore, we can focus on the impact of inventory inaccuracy on the operational decisions. This assumption was also adopted by Adida and Ratisoontorn (2011). Based on the data in their research, the manufacturing cost on the supplier side is cS = 0.75. Similarly, to explicitly show the impact of inventory inaccuracy on the equilibrium decisions and profits, the holding costs of the two retailers are assumed to be identical, which is c R1 = c R2 = 0.125. Finally, the potential market size a is set as 100, and buyback price b is fixed at 0.6. All the data and computer program code (written by MATLAB, 2016b) are available upon request from the authors. 5.1. Sensitivity analysis We study the impact of inventory availability rate on the retailers’ equilibrium stocking factors, equilibrium retail prices, equilibrium profits, as well as the supplier’s equilibrium wholesale (or consignment) price and equilibrium profits. Note that it is unlikely that the retailer has a very low inventory availability rate in practice. Thus, in this numerical study, we mainly consider the case where the inventory 6

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Table 2 The impact of the inventory availability rate decisions and profits when fixing j

i

Decisions / profits

BB

CC

HB

HC

Remark

zi

ziCC = ziHC > ziBB > ziHB

N

↘

↗

↘

zj

N

→

↗

→

pi

↘

↘

↘

↘

N

pj

N

w Ri

Rj

S

↗ ↘ N

↘ ↘ ↗ ↗ ↗

↘ ↘ ↗ ↗ ↗

↘ ↘ ↗ ↘ N

consignment contracts to both retailers. If both inventory availability rates are high, but one is a little bit larger than the other, then the supplier’s best contracting strategy is to provide buyback contracts to both retailers. If both inventory availability rates are not high, the supplier should provide the buyback contract to the retailer whose inventory availability is low, and the consignment contract to the retailer whose inventory availability is high. We believe that two effects account for this contracting strategy: the risk-taking effect and the leftover inventory effect. When both retailers’ inventory available rates are high and close to each other, the inventory lost due to inventory inaccuracy is small. Thus, it is profitable for the supplier to take the entire inventory risk by adopting consignment contracts with both retailers (the CC scenario). However, when both inventory available rates are high, but one (i.e., retailer 2) is a little larger than the other (i.e., retailer 1), the supplier should change its contract strategy. In this case, more inventory would be lost from retailer 1′s side due to inventory inaccuracy, and thus, taking the whole inventory risk is not profitable for the supplier. Therefore, it seems that the BC scenario is better than the CC scenario. Nonetheless, because both retailers’ inventory available rates are high (larger than 0.9 in the case), changing retailer 2′s contract format from consignment to buyback would result in two effects: decreasing the benefit from retailer 2 and increasing the benefit from retailer 1 (see the blue rectangle line and the cyan circle line in Fig. B6(d) and (a)). The increment effect outweighs the decrement effect. Consequently, using the buyback contract with both retailers (the BB scenario) is the optimal strategy. Furthermore, if one retailer’s (i.e., retailer 1’s) inventory availability keeps decreasing, apparently, keeping the buyback contract with the retailer to avoid inventory risk is beneficial for the supplier. This increment effect is inferior to the decrement effect. Therefore, using the consignment contract with the other retailer (i.e., retailer 2) is more profitable, and the BC scenario is the best contract strategy. According to literature, we know that competition is one of the key factors in determining the contract format (Adida & Ratisoontorn, 2011; Fang et al., 2016; Pan et al., 2010). To further investigate the impact of inventory availability on the contracting strategy, we reconduct the numerical study using different values of . For ease of exposition, we choose = 1.4 and = 0.6 as two representative cases to indicate different competition levels between the two retailers: intensively versus mildly. From Fig. 2, we can see that if the competition between two retailers is intensive, it is never optimal to provide buyback contracts to the retailers; but if the competition is mild, it is never optimal to provide consignment contracts to the retailers. Despite the importance of competition in determining the optimal contracting strategy, inventory inaccuracy is also a key and non-negligible factor in deciding the contract format. Inventory inaccuracy should be focused on carefully and seriously, as it directly influences pricing and ordering decisions, and indirectly determines the contracting strategy.

on the supply chain members’

HC z CC > z jBB > z jHB j = zj

piHB > piBB , piHC > piCC for small

i;

piHB > piCC , piHC > piBB for large

i

pjHB > pjBB , pjHC > pCC for small j

i;

pjHB > pjBB , pCC > pjHC for large j

i

w HC > w BB , w HB > wCC for small w HC > w BB , wCC > w HB for large HC Ri HC Ri HC Ri HC Ri

> > > >

CC Ri BB Ri BB Ri CC Ri

> > > >

BB Ri CC Ri CC Ri BB Ri

> > > >

HB Ri HB Ri HB Ri HB Ri

i; i

for small

i;

for large

i

for small

i;

for large

i

Depends on inventory availability and competition between retailers

N, →, ↗ and ↘ respectively refer to Not monotonic, Constant, Increasing and Decreasing.

from retailer 1, and S2 refers to the profit received from retailer 2. We can see that in the hybrid contract scenario, when the inventory availability of the retailer with the buyback contract is much smaller than that of the retailer with the consignment contract (the cyan circle line in Fig. B6(d)), the retailer with the buyback contract contributes little to the supplier’s profit. In addition, she also earns little by herself (retailer 1, the red asterisk line in Fig. B3(a)). Consequently, in this case, the retailer may have no incentive to remain in business in the market. 5.2. Contract selection Because the supplier is the leader in this supply chain, the optimal contracting strategy is selected by the supplier. We conduct a comprehensive numerical study to analyze the supplier’s best contracting strategies in different situations. To facilitate the numerical study, we choose = 2 and = 1, and then, we change inventory availability rates 1 and 2 from 0.7 to 1.0, to demonstrate the impact of inventory inaccuracy on contract selection. From Fig. 1, we find that inventory inaccuracy plays an important role in determining the optimal contracting strategy. Specifically, if both inventory availability rates are high and very close to each other, the optimal contracting strategy for the supplier is to provide

5.3. Extension In this section, we relax some assumptions to discuss the robustness of the main results. Due the limited space, we only present the main conclusions in this subsection; interested reader please refers to Part C of Supplementary Material for more details. (1) Correlated scaling factors

Fig. 1. The supplier’s optimal contracting strategy when

= 2,

In previous section, we assume that the random scaling factors have the same distribution. Now, we assume that i and j are correlated, which captures the correlation between the demands of the two products. Without loss of generality, we assume that i and j satisfy the linear correlation j = k i + h , and the cumulative distribution function of i is F ( ) , and its probability density function is f ( ) that have support on [A, B] R + with B > A . Under the buyback contract scenario, the two retailers’ expected

= 1. 7

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F. Tao, et al.

(a) β = 2 , γ = 1.4

(b) β = 2 , γ = 0.6

Fig. 2. The supplier’s optimal contracting strategy when changing the value of

profit under this assumption can be formulated as: B Ri

= yi [(pi

B Rj

= yj (pj

(

where

b)·( i z i

b)·

j zj

h

k

( i z i ))

j zj

)=

j zj

k

A

(w + c Ri

h

(w + c Rj

k

j zj h k

( (

B j zj h k

j zj

h

k j zj

h

k

j b)·z j

) u) f (u) du ,

u f (u) du ,

Choosing the same values of the parameters as those in Section 5.2, following the analogous procedure, we re-conduct the numerical study to illustrate the supplier’s best contracting strategies when i and j are correlated (please see Part C1 of Supplementary Material for more details). It is straightforward that the correlation influences the supplier’s contract selection and we also confirm this impact in the numerical study. Nonetheless, importantly, given the correlation of two demands, the results clearly demonstrate that inventory inaccuracy still plays a significant role in determining the optimal contracting strategy, which further verifies the main conclusions of our paper.

(25)

i b)· z i],

k>0 k<0

.

(26)

.

Given the wholesale price w chosen by the supplier, retailer i decides the optimal stocking factor z iB and retail price piB , which are expressed as:

piB =

1

+

B i zi

i

(w + c Ri

B i b) z i

(w + c Ri

( i z iB ) =

i b)

+ b, B i zi

1

B i zi

F(

B i zi

)

(2) Risk-averse retailers In this extension, we consider the case that the two retailers are riskaversion. Following the arguments in Ma, Liu, Li, and Yan (2012) and Zheng, Li, and Song (2017), we adopt CVaR model. Now, retailer i ’s utility under η-CVaR criterion can be defined as

(27)

1

B i zi

(

)

CVaR (

,

(28)

1

+

B j zj

j

(w + c Rj

where F

j zj

B jzj

k

k

)=

(29)

F j zj h k

A B j zj h k

CVaR B (

C j zj

1 1

h

+ b,

h

k

=

j b)

(

B j b) z j

(w + c Rj

C jzj

h

C j zj

k

f (u) du ,

k>0

f (u) du,

k<0

C jzj

k

k

h

= b· yi ·( i z i + (w

cS

( i z i )) + yj · i b)· yi z i

+ (w

j zj

cS

k

j b)·yj z j .

1

v

B Ri )

E [min(

= max vB +

1

vB

v, 0)] ,

Ri

E [min((pi

( w + c Ri

(30)

(32)

i b ) Qi

b)·min(Di ,

i Qi )

vB , 0)] .

(33)

Note that according to Zheng et al. (2017), risk-aversion is the behavior of decision makers to attempt to alleviate the negative impact of uncertainty. In our model, this uncertainty comes from the demand, which would unnecessarily change the property of the buyback contract. Therefore, in the risk-averse case, p > b is still an implicit assumption. With some algebra, we obtain

.

j zj

= max v +

,

The supplier’s expected profit under this assumption can be expressed as: B S

Ri )

1) is defined as a risk-averse indicator. = 1 represents where (0< that the retailer i is risk-neutral, and 0< < 1 means that retailer i is risk-averse. Retailer i tends to be more risk-averse when decreases towards 0. According to Eq. (32), retailer i ’s utility in the buyback contract is

and retailer j ’s optimal stocking factor z jB and retail price pjB are:

pjB =

.

h

k

(pi

(31)

CVaR B (

However, unfortunately, due to the implicit relationship between the wholesale price and the stocking factors, as well as the selling price, it is infeasible to obtain the explicit function of the optimal wholesale price and the supplier’s profit. Therefore, we resort to the numerical study.

B Ri

)=

b) yi

yi (pi

F 1( ) A

b)

(

(w + c Ri

f ( )d

i zi

1

i b) z i

(w + c Ri ( i z i)

)

i b) yi z i ,

F

1(

)

i zi

F

1(

) >

i zi

,

(34) Given the wholesale price w chosen by the supplier, retailer i 8

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F. Tao, et al.

decides the optimal stocking factor z iB and retail price p iB , which are expressed as:if F 1 ( ) i zi,

p iB =

z iB = if F

1

1(

F

(w + c Ri F 1( )

1

)

i b) z i

+ b,

xf (x ) dx

(35)

,

(36)

i

1(

p iB =

+

) >

1

+

i zi,

(w + c Ri

i b)

i b) z i

1

i zi

i

(w + c Ri

inventory availability rates, but not approximate. Otherwise, the hybrid contract is the best choice. (2) When the retail competition is intensifying (mild), the consignment (buyback) contract is the optimal contract policy if both retailers have high and approximate inventory availability rates; otherwise, the hybrid contract is the best selection. Although inventory inaccuracy is assumed to occur on the retailer’s side, the retailer and the supplier are greatly affected by inventory inaccuracy. Therefore, the supplier should fully consider the inventory availability rates and retail competition of the two retailers before determining the contract scenario. If inventory inaccuracy is ignored, the supplier may fail to choose the best contract format so that she would never achieve maximum profit. For the retailer, it is beneficial for him to improve the efficiency of the inventory management, which not only can directly increase his profit but also would make the supplier more willing to take the inventory risk, and offer a profitable contract format. This research can be extended in several ways. As the results are based on the assumption that the two retailers have symmetric holding costs and make decisions simultaneously, it would be of interest to investigate the case where the holding cost is asymmetric or decision making is sequential. It would also be valuable to explore the supplier’s contract preference when two retailers have private information on inventory inaccuracy. Finally, considering the competition between multiple competing suppliers and a single retailer is another interesting topic.

=

( i zi )

+ b,

(37)

1 1

1

F(

B i zi )

B izi

B i zi 1

( i z iB )

.

(38)

Comparing with Lemma 1, we find that Eq. (37) and (38) are quite similar to Eqs. (5) and (6), except that there is a risk-averse indicator in the equilibrium stocking factor and retail price when the retailer is risk-averse. Following the same procedure, we could derive the optimal ordering and pricing decisions under the consignment contract scenario and the hybrid contract scenario when the retailer is risk-averse. Using the same parameter set defined in Section 5.2, we re-conduct a numerical study to illustrate the impact of inventory inaccuracy on the supplier’s contracting strategies when retailers are risk-averse. Although we find that when the degree of risk-aversion increases ( decreases), the supplier becomes more inclined to provide the consignment contracts to both retailers, inventory inaccuracy is also an essential factor in determining the supplier’s optimal contracting strategies (please refer to Part C2 of Supplementary Material for the details).

Acknowledgements This work was supported in part by the Shanghai Pujiang Program under Grant 18PJC025; in part by the National Natural Science Foundation of China under Grant 71872064, Grant 71972071, Grant 71431004, Grant 71772063; in part by the Natural Science Foundation of Shanghai under Grant 18ZR1409400.

6. Conclusion

Appendix A. Supplementary material

In this paper, we develop a newsvendor model to investigate the optimal contracting strategy in a competitive supply chain, in which one supplier acts as the leader, and two retailers, subject to inventory inaccuracy, act as followers. We analyze the impact of inventory inaccuracy on the decisions and performance of supply chain members with the buyback contract and the consignment contract in three scenarios, and further conduct a numerical study to specify the optimal contract selection in different situations. The results indicate that inventory inaccuracy plays an important role in determining retailers’ equilibrium stocking factors and retail prices, and the supplier’s wholesale (consignment) price. In addition, inventory inaccuracy has a considerable effect on the dominant supplier’s contract preference. Specifically, the retailer’s optimal stocking factor is decreasing in its own inventory availability in the consignment contract scenario, but is not monotonic with respect to inventory availability in the buyback contract scenario. In addition, one retailer would have a higher optimal stocking factor when using the consignment contract than the buyback contract, no matter what kind of contract the competitor is given. Moreover, the retailer’s equilibrium expected profits in the three scenarios are all increasing in the retailer’s own inventory availability. However, it is not always beneficial for the competitor. When one retailer uses the consignment contract, the improvement in the retailer’s inventory availability increases the other retailer’s profit. When one retailer uses the buyback contract, the improvement of the inventory availability reduces the other retailer’s profit. With respect to contract preference, the optimal contracting strategies for the supplier are (1) when retail competition is moderate, the consignment contract is the optimal contract policy if both retailers have high and approximate inventory availability rates. The buyback contract becomes the optimal selection if both retailers have high

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