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The biggest disadvantage of online shopping is that it is impossible to accurately assess the suitability of goods prior to purchase. Customers usually check “fit” at home, and thus one third of internet sales are returned. We study the pricing and ordering of a dual-channel supply chain which composed of risk-neutral manufacturers and risk-averse retailers serving customers who differ in how they purchase products in store or online. Customers may return misfit products either to stores for a full refund or online as per the retailer’s return policy. At the beginning of the sale season, channels order from manufacturers and set prices to be identical across channels. According to the criterion of conditional value at risk (CVaR), we express the problem as a Stackelberg game model and obtain the equilibrium solution under the conditions of decentralization and centralization. Further, we explore the impact of the retailer’s risk indicator and consumer returns rate on the optimal retail price, the ordering quantities, the profits of dual channels, and the overall profits of the supply chain. We find that dual-channel supply should reduce the risk aversion level of retailers and consumer return rate. Finally, the improved risk-sharing contract is proposed to coordinate the dual-channel supply chain in the presence of customer returns and risk-averse, and it is proved that the contract can achieve Pareto improvement of supply chain members.

The biggest flaw of online shopping is that it can’t accurately evaluate the fit of before products [

While retailers expand sales through the dual channels and full refund policies, it needs to face conflicts and uncertainties between channels, such as random market demand and manufacturers’ uncertain output, and these uncertainties strongly affect the decision making of supply chain members that lead to the risk-avoiding characteristics of decision-makers [

The contract mechanism is a common supply chain coordination method [

Research on the coordination of a dual channel supply chain with consumers’ return focuses mainly on single channel companies. However, we analyze the decision making of the dual channel setting. This study considers the coordination of a dual-channel supply chain in which the risk-averse retailer adopts a dual-channel sales model and a full return strategy when the market demand is uncertain, assuming that the supplier is risk-neutral. Whereas customers purchase dual channel may or may not return misfit items depending on the firm’s return policy. Keeping with the practices of most dual channel firms, the firm sets identical prices across its channels and allows free returns to its channel.

Our work is most closely related to Li et al. [

The rest of this paper is organized as follows. Section 2 is a related literature review. Section 3 presents a description of the problem. Section 4 introduces the decentralized optimal decision models when the retailer is risk-neutral and risk aversion and the concentrated optimal decision models when the retailer is risk aversion. Section 5 proposes an improved risk-sharing contract to coordinate a dual-channel supply chain. Section 6 provides numerical experiments, and Section 7 concludes. All proofs of propositions proposed in this paper are presented in Appendix A.

Many scholars have examined a refund policy in the study of consumer returns, and the return policy is exogenously identified as a full refund or Money Back Guarantee (MBG) [

With the in-depth study of customer return policy in operation management, scholars began to pay attention to know how customer returns affect the order, pricing, and supply chain coordination decision-making. For example, McWilliams [

Several other investigations have considered customer returns in dual-channel retailing systems. Widodo et al. [

In recent years, cooperation and coordination in the supply chain have received great attention aroused. Contracts with various coordination mechanisms have been widely disseminated in supply chain coordination, for example, price discount contract [

As seen from the related literature and to the best of authors’ knowledge, no study has investigated the coordination of a dual-channel supply chain in which the retailer has risk-averse behavior under uncertainties of market demand and customers return unsuitable products. This study contributes to the literature in several aspects. First, we design a new improved risk-sharing contract that coordinates the dual-channel supply chain and enables a win-win outcome for both the supplier and the retailer. Second, we tempted to analyze the effects of customer return rate and retailer’s risk-averse coefficient on the dual-channel supply chain. Third, this study contributes to the growing literature in the area by considering risk-averse behavior under uncertainties of market demand and customer returns.

This study describes the coordination of a dual-channel supply chain involving risk-neutral manufacturers and risk-averse retailers under consumer returns, in which the manufacturer is the leader of the supply chain who sells product to retailers at wholesale price w. We assume that the manufacturer’s unit production cost is c, which is generally simplified to 0. The retailer will sell products to consumers through the online channel and the traditional retailer channel at the same prices p_{o} and p_{r}, respectively, the surplus value of the products at the end of the sales season is zero. At the same time, the dual channels are independent of each other, and there is only one ordering opportunity before the sales season.

The parameters are defined as follows:

According to Huang and Swaminathan [

D r = θ D ¯ − a r p r + σ r p o (1)

D o = ( 1 − θ ) D ¯ − a o p o + σ o p r (2)

We assume a i > σ i , where signifies that the effect of the cross-channel price is lower that of the self-channel price [

Referring to Kevin [

D r = θ D ¯ − r 1 p (3)

D o = ( 1 − θ ) D ¯ − r 2 p (4)

where r 1 = a r − σ r , r 2 = a o − σ o , r 2 > r 1 . They are the self-channel price sensitivity on the two channels. Without losing generality, we assume p ≥ w to avoid a trivial case.

Moreover, the number of customer returns has a strong positive linear relationship with the quantity sold which is proved by the results of Anderson et al. [

Further, risk measurement methods include the mean-variance method, the VaR method, and the CVaR method. Wu et al. [

risk. To clearly demonstrate the retailer’s risk aversion affects the channel members’ strategies, we consider two cases: 1) the retailers are risk-neutral and 2) the retailers are risk-averse under the CVaR criterion.

In the following, we suppose that the three agents play a Stackelberg game in a dual-channel supply chain, i.e., first, the supplier acts as the leader, decides the wholesale price w. After observing the supplier’s decisions, the traditional retailer acts as decider for optimal order quantity and optimal pricing. Afterwards, the online retailer observes the traditional retailer’s optimal pricing and decides the optimal order quantity.

Let’s consider the π r d , π o d and π m d to be the profit function of the traditional retailer, the online retailer and the supplier under the decentralized model. So, they are as following:

π r d = p min { q r , D r } − w q r − p R i (5)

π o d = p min { q o , D o } − w q o − p R o (6)

π m d = w ( q r + q o ) (7)

According to min { q i , D i } = q i − ( q i − D i ) + , ( q i − D i ) + = max { q i − D i , 0 } , i = r or o, we can obtain:

π r d = ( ( 1 − β r ) p − w ) q r − ( 1 − β r ) p ( q r − D r ) + (8)

π o d = ( ( 1 − β o ) p − w ) q o − ( 1 − β o ) p ( q o − D o ) + (9)

Then, we can obtain the following proposition 1.

Proposition 1: When the condition D > 3 U is satisfied,there exists a unique equilibrium solution ( q r d * , p d * , q o d * , w d * ) under the Stackelberg model.

1)The risk-neutral traditional retailer’s optimal order quantity and optimal pricing are noted q r d * and p d * satisfies the following equation

q r d * = ( U + D ) θ − r 1 p d * − 2 U θ w d * ( 1 − β r ) p d * (10)

2 r 1 p d * 3 ( 1 − β r ) 2 − r 1 w d * p d * 2 ( 1 − β r ) − D θ p d * 2 ( 1 − β r ) 2 + θ U w d * 2 = 0 (11)

2)The risk-neutral online retailer’s optimal order quantity is

q o d * = ( U + D ) ( 1 − θ ) − r 2 p d * − 2 U ( 1 − θ ) w d * ( 1 − β o ) p d * (12)

3)The supplier’s optimal wholesale price satisfies the following equation:

U + D − r 1 p d * − r 2 p d * − 4 U θ w d * ( 1 − β r ) p d * − 4 U ( 1 − θ ) w d * ( 1 − β o ) p d * = 0 (13)

The proof is given in Appendix 1.

Next, we use the CVaR measures to explore the impact of retailer’s risk-averse on decision-making.

Reference to Li et al. [

C V a R η ( π r c ) = E [ π r c ≤ v η ( π r d ) ] = max v ∈ ℝ { v + 1 η E [ min ( π r c − v , 0 ) ] }

Similarly, η is the retailer’s risk indicator and v is the target profit lever. This formula can measure the average profit below the level of η -quantile level, ignores the profit contribution beyond the prescribed quantile.

That is:

C V a R η ( π r c ) = max v ∈ ℝ { v + 1 η E [ min ( ( ( 1 − β r ) p − w ) q r − ( 1 − β r ) p ( q r − D r ) + − v , 0 ) ] } (14)

According to Li et al. [

C V a R η ( π r c ) = { ( 1 − β r p ) q r − w q r − ( 1 − β r ) p θ η ∫ − U q r + r 1 p θ − D F ( x ) d x if q r ≤ θ D − R 1 P + θ F − 1 ( η ) ( 1 − β r ) p θ F − 1 ( η ) + ( 1 − β r ) p θ D − r 1 p 2 ( 1 − β r ) − w q r − ( 1 − β r ) p θ η ∫ − U F − 1 ( η ) F ( x ) d x if q r > θ D − R 1 P + θ F − 1 ( η )

This simplification and analysis process is given in Appendix 2.

If q r > θ D − r 1 p + θ F − 1 ( η ) ,

C V a R η ( π r c ) = ( 1 − β r ) p θ F − 1 ( η ) + ( 1 − β r ) p θ D − r 1 p 2 ( 1 − β r ) − w q r − ( 1 − β r ) p θ η ∫ − U F − 1 ( η ) F ( x ) d x

Because ∂ C V a R η ( π r c ) ∂ q r = − w < 0 , so can be obtained q r * = θ D − r 1 p + θ F − 1 ( η ) .

Obviously, this is also the boundary of the first case, so we only study the first case.

That is, when q r > θ D − R 1 P + θ F − 1 ( η ) ,

C V a R η ( π r c ) = ( ( 1 − β r ) p − w ) q r − ( 1 − β r ) p θ η ∫ − U q r + r 1 p θ − D F ( x ) d x

Similarly, the profit function of online retailers is

C V a R η ( π o c ) = ( ( 1 − β o ) p − w ) q o − ( 1 − β o ) p ( 1 − θ ) η ∫ − U q r + r 1 p 1 − θ − D F ( x ) d x

According to the above analysis, the decision-making problem of each member of the supply chain becomes:

π m c = w ( q r + q o ) (15)

max q r , p C V a R η ( π r c ) = ( ( 1 − β r ) p − w ) q r − ( 1 − β r ) p θ η ∫ − U q r + r 1 p θ − D F ( x ) d x (16)

max q o C V a R η ( π o c ) = ( ( 1 − β o ) p − w ) q o − ( 1 − β o ) p ( 1 − θ ) η ∫ − U q o + r 2 p 1 − θ − D F ( x ) d x (17)

Proposition 2: Under the Condition Value-at-Risk criterion,if the demand uncertainty x obeys the uniform distribution of [ − U , U ] , there is a unique equilibrium solution ( q r c * , p c * , q o c * , w c * ) under the Stackelberg model.

1) The traditional retailer’s optimal order quantity and optimal pricing are noted q r c * and p c * satisfies the following equation:

q r c * = ( D − U ) θ − r 1 p c * + 2 U θ η ( ( 1 − β r ) p c * − w c * ) ( 1 − β r ) p c * (18)

2 r 1 p c * 3 ( 1 − β r ) 2 − r 1 w c * p c * 2 2 ( 1 − β r ) − ( D − ( 1 − η ) U ) θ p c * 2 ( 1 − β r ) 2 + θ U η w c * 2 = 0 (19)

2) The risk-neutral online retailer’s optimal order quantity is:

q o c * = ( D − U ) ( 1 − θ ) − r 2 p c * + 2 u η ( 1 − θ ) ( ( 1 − β o ) p c * − w c * ) ( 1 − β o ) p c * (20)

3)The supplier’s optimal wholesale price satisfies the following equation:

U + D − r 1 p c * − r 2 p c * + 2 U η − 4 U η θ w c * ( 1 − β r ) p c * − 4 U η ( 1 − θ ) w c * ( 1 − β o ) p d * = 0 (21)

The proof is given in Appendix 3.

At this time, the optimal profit of the physical channel and the online channel are as follows:

π r c * = ( 1 − β r ) p c * ( ( D − U ) θ − r 1 p c * + U η θ ) − w ( D − U ) θ + r 1 p c * − 2 U η θ w + U η θ w 2 ( 1 − β r ) p (22)

π o c * = ( 1 − β o ) ( 1 − θ ) ( D − U + U η ) p c * − ( 1 − β o ) r 2 p c * 2 − ( D − U ) ( 1 − θ ) w + r 2 p c * w − 2 U η ( 1 − θ ) w + U η ( 1 − θ ) w 2 ( 1 − β o ) p c * (23)

Accordingly, the total optimal profit of the supply chain is

Π s c c * = w c * ( q r c * + q o c * ) (24)

Proposition 3: Under the CVaR criterion, the following properties are satisfied.

1) p c * increase as the rate return of physical channel β r increase,

2) q r c * decrease as β r increase,

3) π r c * decrease as β r increase,

4) q o c * is negatively correlated with β o ,

5) π o c * is negatively correlated to β o .

The proof is given in Appendix 4.

Proposition 3 explains that the rate return of physical channel affects the optimal price and demand of physical channels. Furthermore, the rate return of physical channel β r tends to 0, the optimal profit of physical channels will increase. The effect of the return rate of the online channel on the online channel is similar to the above. In general, channels increase their retail price to make up for the losses caused by customer returns, resulting in a reduction in channel demand and a corresponding decrease in channel profits.

Proposition 4: Under the CVaR criterion, when the condition r 1 r 2 > θ 2 ( 1 − θ ) and 1 − β o r 1 s ( q o * ) > ( 1 − β o ) p c * − w are satisfied,the following properties are satisfied.

With the increase of the retailer’s risk-averse level η in [ 0 , 1 ] ,the optimal pricing p c * and the optimal order quantity q r c * , q o c * and the optimal profit of dual channel π r c * , π o c * and the total optimal profit of the supply chain increase monotonically.

Where, s ( q o * ) = ( D − U + U η ) ( 1 − θ ) − r 2 p c * − U η ( − θ ) w 2 ( 1 − β r ) 2 p c * 2 , this is also the expected sales volume of online channels.

The proof is given in Appendix 5.

Proposition 4 indicates that the optimal retail pricing, order quantity and profits depend on its risk-averse. With the channels risk-averse level η increases, the retailer has an optimistic attitude towards the market, the traditional physical retailer will raise the retail price and dual channel will increase the order quantity accordingly. At the same time, the optimal profits of each channel are also increasing. Suppliers also get benefit from retailers’ positive attitude, which increases their operating profit. Then, the retailer risk-averse η can increase the total profit of the supply chain. The degree of risk aversion of retailers affects the degree of damage to the interests of all members of the supply chain.

In the centralized supply chain, the supply chain decides the optimal retail price and dual channel’s order quantity at the same time. Under centralized decisions, the profit function of the supply chain is as follow:

max p , q r , q o Π s c = π m + C V a R η ( π r ) + C V a R η ( π o ) = ( 1 − β r ) p q r − ( 1 − β r ) p θ η ∫ − U q r + r 1 p θ − D F ( x ) d x + ( 1 − β o ) p q o − ( 1 − β o ) p ( 1 − θ ) η ∫ − U q o + r 2 p 1 − θ − D F ( x ) d x (25)

with the formula (22), we have the following result:

Proposition 5: There exists a unique equilibrium solution of optimal retail prices and dual channel’s optimal order quantity ( p * , q r * , q o * ) under centralized decision mode.

p * = ( D − U + U η ) ( ( 1 − β r ) θ + ( 1 − β o ) ( 1 − θ ) ) 2 r 1 ( 1 − β r ) + 2 r 2 ( 1 − β o )

q r * = ( D − U ) θ − r 1 p * + 2 U η θ

q o * = ( D − U ) ( 1 − θ ) − r 2 p * + 2 U η ( 1 − θ )

The proof is given in Appendix 6.

From the above analysis, the optimal profit function of the whole supply chain to be rewritten as follows:

Π s c * = ( 1 − β r ) p * s ( q r * ) + ( 1 − β o ) p * s ( q o * ) (26)

where, s ( q r * ) = ( D − U + U η ) θ − r 1 p * , s ( q o * ) = ( D − U + U η ) ( 1 − θ ) − r 2 p * .

Proposition 6: In the centralized situation, the following properties are satisfied.

When the coordination r 1 r 2 > max { θ 4 ( 1 − θ ) − 3 ( 1 − β o ) 4 ( 1 − β r ) , 1 − β o 2 ( 1 − β r ) ( θ 1 − θ − 1 ) }

is stratified,the optimal retail price p * and order quantity q r * , q o * of dual channel and the optimal profit of the supply chain Π s c * increase monotonically as the retailer’s risk-averse level η increase in [ 0 , 1 ] .

The proof is given in Appendix 7.

Proposition 6 shows that the optimal strategy of the dual-channel supply chain is closely related to the risk preference behavior of the retailer’s dual channel. When the retailer’s risk-averse indicator η tends to 1, it means that the retailer has a positive and optimistic attitude towards the market, thereby increasing the number of orders and the retail price, and the total profit of the supply chain is also improved. Therefore, manufacturers should mitigate the retailers’ risk-averse behavior to improve their own and total supply chain operating performance and increase operating profits.

Proposition 7: In the centralized situation,the following properties are satisfied.

1)the optimal retail price p * and the optimal profit of the supply chain Π s c * decrease monotonically as the physical channel’s customer returns rate β r increase in ( 0 , 1 ) ,but the optimal order quantity q r c * increases monotonically as it increases.

2)the optimal retail price p * increase monotonically as the online channel’s customer returns rate β o increases in ( 0 , 1 ) ,but the optimal order quantity q r c * and the optimal profit of the supply chain Π s c * decrease monotonically as it increases.

The proof is given in Appendix 8.

Proposition 7 indicates that the optimal retail price p * will be significantly affected by the consumer returns rate of dual channels. When the physical channel’s return rate increases, it will force the retailer to reduce the retail prices to attract more consumers and to make up for the losses caused by returns. But when the online channel’s return rate increases, the physical channel as the decision-maker of the retail price will increase the optimal retail price. Therefore, with the customer returns rate of physical channels increases, its optimal order quantity gradually increases. However, with the customer return rate of online channels increases, its optimal order quantity gradually decreases. From the perspective of operation, the consumer returns behavior must damage the retailer’s profit. Proposition 7 also verifies the correctness of this conclusion.

Corollary 1: The decentralized optimal pricing is not equal to the optimal pricing under the centralized decision-making,that is p c * ≠ p * ,the profit of the supply chain in the decentralized situation will deviate from the optimal profit in the centralized situation.

The proof is given in Appendix 9.

Corollary 1 expatiates that the overall optimal profit of the supply chain under decentralized decision-making is still lower than that under centralized decision-making. As we know, decentralized decision-making reduces the overall efficiency of the supply chain. Next, we design a new contract to improve the overall performance of the supply chain and increase the profits of its members.

Referring to Li et al. (2016) we propose an improved risk-sharing contract to coordinate the supply chain and to achieve the improved operation performance of the supply chain under centralized decision-making and achieve Pareto improvement of each member. This contract is composed of wholesale price contract, revenue-sharing contract and risk-return contract, and use ( w , λ , b ) expresses it, where w is the wholesale price of the supplier, λ is the retailer’s revenue-sharing ratio and b is the ratio of supplier’s repurchase to retailer’s expected surplus inventory. Thus the profit functions of the supplier and two channels of retailer are, respectively:

π m s ( λ 1 , λ 2 , b 1 , b 2 ) = w ( q r + q o ) + ( 1 − λ 1 ) ( 1 − β r ) p [ q r − θ η ∫ − U q r + r 1 p θ − D F ( x ) d x ] + ( 1 − λ 2 ) ( 1 − β o ) p [ q o − 1 − θ η ∫ − U q o + r 2 p 1 − θ − D F ( x ) d x ] − b 1 θ η ∫ − U q r + r 1 p θ − D F ( x ) d x − b 2 ( 1 − θ ) η ∫ − U q o + r 2 p 1 − θ − D F ( x ) d x (27)

C V a R η ( π r s ( λ 1 , b 1 ) ) = λ 1 ( 1 − β r ) p ( q r − θ η ∫ − U q r + r 1 p θ − D F ( x ) d x ) + b 1 θ η ∫ − U q r + r 1 p θ − D F ( x ) d x − w q r (28)

C V a R η ( π o s ( λ 2 , b 2 ) ) = λ 2 ( 1 − β o ) p ( q o − 1 − θ η ∫ − U q o + r 2 p 1 − θ − D F ( x ) d x ) + b 2 ( 1 − θ ) η ∫ − U q o + r 2 p 1 − θ − D F ( x ) d x − w q o (29)

Proposition 8: When the condition D > r 1 b 1 λ 1 ( 1 + β r ) θ + ( 2 + η 2 ) U η is satisfied,

there exists a unique equilibrium solution of the retail price and dual channel’s optimal order quantity and optimal wholesale price ( p s * , q r s * , q o s * , w s * ) when the supply chain adopts this contract.

1)The optimal retail price p s * satisfies the formula

( D − U ) θ − 2 r 1 p s * − θ η U A 2 B 2 + A B 2 U η θ + r 1 w s * λ 1 ( 1 − β r ) = 0

2)The traditional retailer’s optimal order quantity satisfies the following equation

q r s * = ( λ 1 ( 1 − β r ) p s * − w s * ) 2 U η θ λ 1 ( 1 − β r ) p s * − b 1 + ( D − U ) θ − r 1 p s *

3)The online retailer’s optimal order quantity satisfies the following equation

q o s * = ( λ 2 ( 1 − β o ) p s * − w s * ) 2 U η ( 1 − θ ) λ 2 ( 1 − β o ) p s * − b 2 + ( D − U ) ( 1 − θ ) − r 2 p s *

4)The supplier’s optimal wholesale price satisfies the following equation:

D − U − r 1 p s * − r 2 p s * + 2 U η θ λ 1 ( 1 − β r ) p s * λ 1 ( 1 − β r ) p s * − b 1 − 4 U η θ w s * ( 1 − β r ) p s * b 1 − 4 U η ( 1 − θ ) w s * ( 1 − β o ) p s * − b 2 + 2 U η ( 1 − θ ) λ 2 ( 1 − β o ) p s * λ 2 ( 1 − β o ) p s * − b 2 = 0

where, A = λ 1 ( 1 − β r ) p s * − w s * , B = λ 1 ( 1 − β r ) p s * − b 1 .

The proof is given in Appendix 10.

Proposition 9: Under an improved risk-sharing contract,if w = b 1 = b 2 = b and θ < θ ¯ ,then p * = p s * , q r * = q r s * , q o * = q o s * .At this point,the total profit of the supply chain is equal to that in the centralized situation.

Where,

b = ( 1 − β r ) ( D − U + U η ) r 1 ( 2 r 1 ( 1 − β r ) + 2 r 2 ( 1 − β o ) ) ( r 1 ( 1 − β o ) ( 1 − θ ) ( 1 + λ 1 ( 1 − β r ) ) − ( r 1 ( 1 − β r ) + 2 r 2 ( 1 − β o ) − r 1 λ 1 ( 1 − β r ) 2 ) θ ) ,

θ ¯ = r 1 ( 1 − β o ) ( 1 + λ 1 ( 1 − β r ) ) r 1 ( 1 − β r + 1 − β o ) + r 1 λ 1 ( 1 − β r ) ( 1 − β o ) + 2 r 2 ( 1 − β o ) − r 1 λ 1 ( 1 − β r ) 2 .

According to q r * = q r s * and q o * = q o s * we can get w = b 1 = b 2 = b . Then bring p * into the equation

( D − U ) θ − ( 1 + λ 1 ( 1 − β r ) r 1 p * ) − + 2 U θ η A B − U η θ A B + r 1 w 1 − β r = 0 can get b.

Proposition 9 shows that the improved risk-sharing contract achieves the coordination of the two-channel supply chain. When the condition θ < θ ¯ is satisfied, for any λ 1 ∈ ( 0 , 1 ) , the b is always nonnegative. This phenomenon illustrates that no matter how much profit the retailer shares with the supplier, the manufacturer must bear the retailer’s loss caused by the risk aversion and it is worth noting that manufacturers bear the same proportion of double channel losses. This seems contrary to common sense, but the careful study is consistent with reality because we assume that the risk aversion of dual channels is the same, the resulting losses are directly related to this. So there is a phenomenon that suppliers share the same proportion of revenue to the dual channels.

This condition θ < θ ¯ indicates that only when the market share of traditional physical channels is low, the traditional physical channels would like to accept this contract. Otherwise, when θ ¯ < θ < 1 , there is w = b 1 = b 2 < 0 , this contract is unable to achieve the coordination of the dual-channel supply chain because the supplier will not accept this contract. But in the early stage of supplier development, if he wants retailers to help him to develop the market, this contract may be accepted by him and achieve the dual-channel supply chain coordination.

In this section, we do use numerical simulation to verify the validity and reliability of the above model. We first examine the effect of the improved risk-sharing contract by comparing the equilibrium solutions of centralized supply chains and the decentralized supply chains with the improved risk-sharing contract, we can get the following Corollary 2.

Corollary 2: Under the improved risk-sharing contract,there exists a group of { λ 1 * , λ 2 * , b * } to make members of the supply chain gain a win-win outcome. In other words,there are regions that satisfy the following conditions

C V a R η ( π r s * ( λ 1 * , b 1 * ) ) > C V a R η ( π r c * ) , C V a R η ( π o s * ( λ 2 * , b 2 * ) ) > C V a R η ( π o c * ) and π m s ( λ 1 * , λ 2 * , b 1 * , b 2 * ) > π m c * .

where C V a R η ( π r s * ( λ 1 * , b 1 * ) ) , C V a R η ( π o s * ( λ 2 * , b 2 * ) ) , π m s ( λ 1 * , λ 2 * , b 1 * , b 2 * ) shows the optimal profit of dual channels retailer and the supplier under the improved risk-sharing contract. Similarly, C V a R η ( π r c * ) , C V a R η ( π o c * ) and π m c * represents the optimal profit of dual channels retailer and the supplier under the decentralized situation (

Next, we will analyze the impact of the physical channel customer’s returns rate β r on the optimal profit of the supply chain Π s c c * . Let U = 100 , D = 600 , r 1 = 0.8 , r 2 = 1 , β o = 0.35 , w = 15 . It considers three cases as follows: 1) θ = 0.6 , η = 0.8 2) θ = 0.7 , η = 0.8 3) θ = 0.6 , η = 0.9 , the specific results are shown in

From

damages the interests of the supply chain. Therefore, retailers should actively look for the reasons for consumers’ return and draft the corresponding solutions to reduce the return rate and to increase the operating efficiency of themselves and the supply chain and increase the operating profit.

Similarly, we do discuss the impact of the online channel customer returns rate β o on the optimal profit of the supply chain below. Π s c c * . Let U = 100 , D = 600 , r 1 = 0.8 , r 2 = 1 , β r = 0.2 . It considers three cases as follows: 1) θ = 0.6 , θ = 0.6 2) θ = 0.55 , η = 0.8 3) θ = 0.6 , η = 0.9 . The results are shown in

From

Then, we do analyze the impact of the retailer’s risk-averse level η on the equilibriums solutions p c * , q r c * , π r c * , q o c * , π o c * , Π s c c * . Let D = 100 , r 2 = 1 , β r = 0.1 , β o = 0.2 , w = 13 . It considers three cases as follows: 1) U = 25 , θ = 0.5 , r 1 = 0.8 2) U = 25 , θ = 0.5 , r 1 = 1 3) U = 15 , θ = 0.53 , r 1 = 0.8 . The specific results are shown in Figures 5-7.

The influence of risk aversion level on the optimal profit of online channels is also closely related to other parameters. From

It explains that when the price sensitivity of consumers to the physical channel increases, retailers will reduce the retail price to attract more consumers.

From

From

With the risk-averse indicator

In this study, we investigated a distribution channel that includes a risk-neutral manufacturer and a risk-averse retailer who implements the dual-channel sales model. We have concluded that they play a Stackelberg game and both face uncertain market demand. Through analysis, we solved the equilibrium solution and proposed a contract that can coordinate the supply chain. Propositions and numerical analysis indicated our conclusions and further explored the impact of some parameters on the equilibrium solutions. We showed that when the retailer’s channels become more risk-averse, the optimal retail price will decrease and dual channels will decrease the optimal order quantity. The risk behavior of the channels always damages the interests of all members and the supply chain. Further, the return rate of physical channels reduces the optimal retail prices, while the return rate of online channels contributes to increasing the optimal retail prices. Then, we consciously proposed a proper risk-sharing contract to coordinate the dual-channel supply chain. Moreover, we proved that it is impossible to coordinate the supply chain with only wholesale price contracts. Finally, an improved risk-sharing contract is proposed that enables the coordination of the supply chain and Pareto improvement of members.

The authors declare no conflicts of interest regarding the publication of this paper.

Zhang, A.F., Ren, J.B., Guan, Z.Z. and Farooq, U. (2021) Decision and Coordination in the Dual-Channel Supply Chain Considering the Risk-Averse and Customer Returns. Journal of Mathematical Finance, 11, 48-83. https://doi.org/10.4236/jmf.2021.111003

According to formula (8), we have

We can know that the risk-neutral traditional retailer’s optimal solutions satisfy the following equations:

Then can get:

Note the Hessian matrix

where,

Because

According to formula (9), we have

Then, we can get the risk-neutral online retailer’s optimal order quantity is as following the equation:

that is

From this we can get the profit function of suppliers as follows:

Then, we can get

Simplify the formula (14), we can obtain

1) If

According to

2) If

Let

Thus,

So, we gain

Comprehensive to is it can know that CVaR is simplified as fowling:

According to formula (16), we have

Note the Hessian matrix

because,

satisfied,

Then can get:

According to formula (17), we have

Then, we can get the risk-averse online retailer’s optimal order quantity is as following the equation:

that is

From this we can get the profit function of suppliers as follows:

Then, we can get

1) Let

Then

when the condition

2) According to the formula

Because

3) According to the formula (21), we can have

4) According to the formula (20) can get

5) According to the formula (22) can get

Therefore, Proposition 3 holds.

Appendix 5. Proof of Proposition 41) Let

Then

Because

2) According to formula (18), we can know that

Simplify the above formula to get

Because

3) By the formula (20) can be obtained

According to

when the condition

4) Since the formula (21), we can have

Because

5)

where

6) The above studies proved that

Therefore, Proposition 4 holds.

Appendix 6. Proof of Proposition 5According to formula (22), we have

We can know that the risk-averse traditional retailer’s optimal solutions satisfy the following equations:

Then can get

Note the Hessian matrix

On the equilibrium point

Meanwhile, we can find

1) Let

2) According to the formula

3) Taking the first-order partial derivative of

when the coordination

4) The partial derivative of

It is easy to get

Because

1) Let

It can be seen from the above,

Because

2) Let

the partial derivative of

It easily obtains

We assume that

From the above formulas can be obtained,

From formula (27), we can see that

We can know that the risk-averse traditional retailer’s optimal solutions satisfy the following equations:

Solving the above equations can be obtained:

where,

Note the Hessian matrix

where

According to

From formula (25), we can see that

From the above formula, it can be see that

From this we can get the profit function of suppliers as follows:

Then, we can get

Summarizing the above formulations, we can get Proposition 8.