Mathematical Modeling Research in Fashion and Textiles Supply Chains and Operational Control Systems
View this Special IssueResearch Article  Open Access
Minli Xu, Qiao Wang, Linhan Ouyang, "Coordinating Contracts for TwoStage Fashion Supply Chain with RiskAverse Retailer and PriceDependent Demand", Mathematical Problems in Engineering, vol. 2013, Article ID 259164, 12 pages, 2013. https://doi.org/10.1155/2013/259164
Coordinating Contracts for TwoStage Fashion Supply Chain with RiskAverse Retailer and PriceDependent Demand
Abstract
When the demand is sensitive to retail price, revenue sharing contract and twopart tariff contract have been shown to be able to coordinate supply chains with risk neutral agents. We extend the previous studies to consider a riskaverse retailer in a twoechelon fashion supply chain. Based on the classic meanvariance approach in finance, the issue of channel coordination in a fashion supply chain with riskaverse retailer and pricedependent demand is investigated. We propose both single contracts and joint contracts to achieve supply chain coordination. We find that the coordinating revenue sharing contract and twopart tariff contract in the supply chain with risk neutral agents are still useful to coordinate the supply chain taking into account the degree of risk aversion of fashion retailer, whereas a more complex sales rebate and penalty (SRP) contract fails to do so. When using combined contracts to coordinate the supply chain, we demonstrate that only revenue sharing with twopart tariff contract can coordinate the fashion supply chain. The optimal conditions for contract parameters to achieve channel coordination are determined. Numerical analysis is presented to supplement the results and more insights are gained.
1. Introduction
Fashion supply chain is characterized by short product life cycle, high volatile customer demand, and clients’ varying tastes [1]. Within such supply chains, it is difficult to predict the demand accurately. Because of the highly demand uncertainty, the fashion retailer must suffer risks from the trading off between overstocks and stockouts [2]. Besides, the complex features of fashion supply chain make supply chain coordination increasingly significant for supply chain agents in fashion industry.
Coordination among supply chain agents via setting incentive alignment contracts is a hot topic in supply chain management. Under the coordinating contracts, the incentives of supply chain agents are aligned with the objective of the whole supply chain so that the decentralized supply chain behaves as well as the vertically integrated supply chain. Without supply chain coordination, problems involving double marginalization will prevail [3], reducing the supply chain’s efficiency tremendously. Over the past two decades, many forms of contracts with reasonable contract parameters have been studied to achieve supply chain coordination with riskneutral agents by fighting against the issue of double marginalization. These traditional contracts include returns policy [4, 5], revenuesharing contract [6], quantity flexibility contract [7, 8], twopart tariff contract [9], and sales rebate contract [10–13]. For more detailed information of papers on these and some other supply chain contracts, please refer to [14].
Revenuesharing contract indicates that the newsvendor retailer pays the upstream manufacturer a unit wholesale price for each unit ordered plus a proportion of his revenue from selling the product. Both theoretical and empirical studies have been carried out on the effect of revenuesharing contract in the video cassette rental industry [15, 16]. Under the classic newsvendor models, such contracts have been shown to be capable of coordinating the newsvendor [6, 17, 18].
Under a twopart tariff contract, the retailer gives the manufacturer a fixed transfer payment apart from the unit wholesale price for each unit purchased. And it has also been shown that a twopart tariff contract coordinates the supply chain, when the optimal value of unit wholesale price equals the manufacturer’s unit production cost [9].
Sales rebate and penalty (SRP) is based on the retailer’s sales performance. With a SRP contract, the manufacturer will specify a certain sales target prior to the selling season. Different from sales rebate which executes rebate only, for each unit sold above the target level, the retailer will be granted a unit rebate, or else the retailer must pay the manufacturer a penalty. In supply chain management, both SRP contract and sales rebate contract have been demonstrated unable to coordinate the channel when the demand is sensitive to retail price or the retailer’s sales effort [10, 19–21].
Early studies considered retail price exogenous, leaving the retailer with the decision of order quantity alone in order to maximize expected profit. As retail price plays an important role in marketing channel, a new steam of research on supply chain coordination and contracting integrates pricing into the order quantity decision of the retailer under different demand models. Reviews of this work [12, 22] explicitly stated that revenuesharing contract and twopart tariff contract are able to coordinate the newsvendor with pricedependent demand, while many other traditional contracts aren’t. And there is an increasing interest in examining combined contracts consisting of two or more traditional contracts to achieve channel coordination [19–21, 23].
However, the common results derived from the previous studies may not be precise in operations management since, in the real world, different decision makers may have different degrees of risk aversion. in light of this, we extend the results of proceeding studies to explore the issue of supply chain coordination with riskaverse fashion retailer and pricedependent demand. Specifically, we investigate a single period, onemanufacturer oneretailer fashion supply chain with a variety of contracts. The manufacturer, acting as the leader in the Stackelberg game, offers the retailer a contract with a set of contract parameters. The fashion retailer, acting as the follower, sets selfinterest order quantity and retail price as a response. We propose both single contracts and combined contracts with the optimal values of contract parameters to achieve channel coordination within fashion supply chain.
The main objectives of our study cover the following: firstly, to explore whether the coordinating revenuesharing contract and twopart tariff contract in supply chains with risk neural retailer can still coordinate the fashion supply chain with riskaverse retailer who has to choose retail price in addition to stocking quantity; secondly, to compare the performance of a more complicated sales rebate and penalty contract in supply chain coordination with the performances of revenuesharing contract and twopart tariff contract; finally, when joint contracts are got by taking advantages of the three single contracts, to probe whether the resulting combined contracts are useful to coordinate the supply chain.
In recent years, an increasing number of researchers have noticed the importance and the impact of risk aversion in supply chain contracting and coordination and sought in succession for the criteria to depict supply chain agents’ risk aversion attitude or preference. In the literature, the measures for describing risk aversion involve meanvariance (MV) [24], NeumannMorgenstern utility function (VNUM), meandownsiderisk (MDR) [25], Valueatrisk (VaR) [26, 27], and Conditional Valueatrisk (CVaR) [28, 29]. Since MV is simple, implementable and is easily understood by managers and practitioners compared with other measures, we adopt meanvariance formulation to capture the fashion retailer’s risk aversion in this paper.
This paper is closely linked to the literature on supply chain coordinating and contracting with pricedependent demand [30, 31] in terms of a random and pricedependent demand. It is also correlated to studies of supply chain coordination with agents having risk preferences in which we consider a riskaverse retailer [32–36]. But our study is the first to investigate the issue of channel coordination for the supply chain with riskaverse retailer and pricedependent demand. We firstly investigate the problem of coordinating a twostage fashion supply chain under single contracts including revenuesharing contract, twopart tariff contract and sales rebate and penalty contract. After proving that revenuesharing contract and twopart tariff contract could still achieve channel coordination in this context while a more complex sales rebate and penalty cannot, we further explore the role of combined contracts (sales rebate and penalty with revenuesharing contract, sales rebate and penalty with twopart tariff contract, and revenue sharing with twopart tariff contract) in supply chain coordination. By identifying the coordination conditions and mechanisms of various contracts, our work contributes to supplement the current literature on supply chain coordination and contracting. We also provide meaningful guidance to managers in real operations management on how to choose the type of contract and determine the optimal contract parameters in order to coordinate fashion supply chain in more complicated newsvendor frameworks.
The paper is organized as follows. Model formulation and notation definition are presented in Section 2. The benchmark case of integrated fashion supply chain is studied in Section 3. Supply chain coordination under single contracts and combined contracts is investigated in Sections 4 and 5. Numerical study to supplement the analytical results and gain more insights is given in Section 6. Section 7 provides managerial insights and concluding comments.
2. Model Formulation and Notation Definition
Consider a twoechelon fashion supply chain with a riskneutral manufacturer and a riskaverse retailer. The retailer sells a fashion product whose demand is sensitive to retail price. The upstream manufacturer produces the product and sells it through a vertically separated retailer. The sequence of events in the supply chain is as follows. The manufacturer, as the leader of a Stackelberg game, offers the retailer a contract. After knowing the details of the contract, the fashion retailer commits his order quantity and retail price. Then the manufacturer organizes the production and delivers the finished products to the retailer prior to the selling season. Afterwards, the selling season starts, and the demand is realized. At the end of the selling season, based on the agreed contract, both the manufacturer and the retailer perform the respective contract terms and achieve transfer payments between each other.
Let be the retail price, the production cost incurred by the manufacturer, the wholesale price, the salvage value of unsold inventory, and the production/order quantity. Use as the sales target level and as the rebate (and penalty) for sales rebate and penalty contract. Use as the fraction of revenue earned by the retailer in revenuesharing contract and as the fixed transfer payment from the retailer to the manufacturer in twopart tariff contract.
In the literature, there are two fashions in which the demand depends on the selling price : (1) the additive form ; (2) the multiplicative form , where is a function of representing the expected demand and is a nonnegative variable representing the random proportion of the demand. is independent of selling price with a probability density function and a cumulative distribution function . It is assumed that has a continuous derivative . is continuous, strictly increasing, and differentiable. Let be the reverse function of , and . is strictly decreasing in , and . In this paper, we only consider the additive demand model. For the multiplicative one, we believe similar results would be derived.
In order to ensure the existence and uniqueness of model results, we give the following definitions of and .
Definition 1. By definition, is the price elasticity of . has an increasing price elasticity (IPE) in , if
Price elasticity measures the percentage change in demand with respect to one percentage change in selling price. The IPE property is intuitive. In the literature, many demand forms own IPE property, such as the simplest linear demand, isoelastic demand, and exponential demand.
For the ease of position, in this paper, we suppose a linear demand of . Let , where is the base demand and is the price elasticity of demand. Thus, we have .
Definition 2. Define as the failure rate of the distribution then has an increasing failure rate (IFR), if for
It is noted that, in the literature, various random distributions exhibit IFR property, involving uniform and normal distributions.
To capture the decision making of riskaverse fashion retailer, we adopt the same risk aversion decision model as in [34]: where and denote the mean and the variance of the retailer’s profit, respectively, and denotes the retailer’s expected profit threshold. can be considered to be the indicator of the retailer’s risk aversion degree, since larger values of indicate that the retailer does not want to earn a low expected profit, leading to a more riskaverse retailer. Define as the retailer’s attainable maximum expected profit. Then from , must establish, otherwise, there is no feasible solution for .
3. The Integrated Fashion Supply Chain
First, we offer a benchmark by analyzing the case when the fashion supply chain is vertically integrated so that the manufacturer owns its own retailer. Note that the type of contract does not affect the performance of the integrated fashion supply chain. The optimal solutions to this model are production level and retail price , which provide us with guidelines to the optimal policy for the whole system. Define , . The integrated fashion supply chain’s profit, expected profit, and the variance of profit are given as follows:
Let and be the optimal joint decision and optimal production level for the integrated supply chain.
Proposition 3. Under the additive pricedependent demand, the integrated supply chain’s optimal joint decision and optimal production quantity exist and are unique, satisfying
Proof. For any given , from (4), by taking the first and second differentials of with respect to , we get
, . Thus, for any given , is a concave function of , and is finite and unique, satisfying
From (9), we know that is a function of . According to the implicit function theorem, we have
Therefore, from (10), is strictly increasing in .
By taking into , we get . Taking the first and second derivatives of with respect to , we derive:
Substituting (10) into (12), we get
Define , and taking the derivative of with respect to , we have
From Definition 2, we know that is strictly increasing in . Therefore, is strictly decreasing in . Let satisfy . If does not exist, then we can know , since , and is a concave function of . If exists, for , , and, for , . That is, is convex in for and concave in for . Because , is unimodal in .
Hence, there exists a unique retail price that maximizes and is given by (7).
Since , it is natural to conclude that the fashion supply chain’s optimal production quantity is unique and satisfies (8).
Remark 4. Proposition 3 reveals the optimal solutions of the integrated fashion supply chain with the additive pricedependent demand. Correspondingly, the entire supply chain’s maximum expected profit is .
4. The Decentralized Fashion Supply Chain under Single Contracts
Now we consider the case when the fashion supply chain is decentralized. In the decentralized supply chain, the manufacturer and the fashion retailer are independent and enter a Stackelberg game as described before. Specifically, the retailer is assumed to be riskaverse with an expected profit threshold and (otherwise, there would be no incentive for the manufacturer to offer a contact). As an extension of prior works, in the following sections, we will consider the optimal joint pricinginventory decisions of a riskaverse retailer in the decentralized fashion supply chain under single contracts such as revenue sharing contact, sales rebate and penalty contract, and twopart tariff contract. For the purpose of simplification, define , and . Let and be the optimal joint decisions for the riskneutral retailer and riskaverse retailer, respectively.
4.1. SRP Contract
With a SRP contract , the manufacturer offers a sales target to the retailer prior to the selling season. At the end of the selling season, for each unit sold above , the manufacturer will give the retailer a unit rebate , otherwise, the retailer must pay the manufacturer a penalty .
In this setting, the fashion retailer’s profit, expected profit and the variance of profit are
Proposition 5. For a given SRP contract offered by the manufacturer, the riskneutral retailer’s optimal joint decision is given by
Proof. For any given , from (16), by taking the first and second differentials of with respect to , we can derive that , and . Thus, is a concave function of . can be given by
From (20), we can get to know that is a function of . By making use of the implicit function theorem, we have
Thus, we know that is strictly increasing in .
Substituting into , we get
From (22), can be regarded as a function of variable alone. Taking the first and second derivatives of with respect to , we get
By taking (21) into (24), we have
Similar to Proposition 3, we know that is unimodal in . If , . Thus, there exists a unique which satisfies (19).
Remark 6. By Comparing (19) with (7) and (18) with (6), we find that is the riskneutral fashion retailer’s optimal joint decision if and only if and . However, a SRP contract with and gives the manufacturer zero profit. So SRP contract cannot coordinate the supply chain with riskneutral fashion retailer and pricedependent demand.
4.2. RevenueSharing Contract
A revenuesharing contract stipulates that the fashion retailer pays the upstream manufacturer a unit wholesale price for each unit ordered plus a proportion of his revenue from selling the product. Let be the fraction of supply chain revenue earned by the retailer, and thus is the fraction shared by the manufacturer. Under the revenuesharing contract , the retailer’s expected profit and the variance of profit are given as follows:
Proposition 7. For a given revenuesharing contract offered by the manufacturer, the riskneutral fashion retailer’s optimal joint decision is given by
Proof. Similar to Proposition 5.
Remark 8. Comparing (28) with (6) and (29) with (7), we find that can be the riskneutral retailer’s optimal ordering quantity and retail price if and only if , which is equal to the optimal conditions for the contract parameters to coordinate the supply chain when retail price is given exogenously. Therefore, consistent with the finding in the literature [12], when the random demand is sensitive to pricing, revenue sharing contact with reasonable contract parameters is sufficient to coordinate the supply chain with riskneutral retailer.
4.3. TwoPart Tariff Contract
With a twopart tariff contract , the fashion retailer gives the manufacturer a fixed transfer payment apart from the unit wholesale price for each unit ordered. And the retailer’s expected profit and the variance of profit are
Proposition 9. For a given twopart tariff contract offered by the manufacturer, the riskneutral fashion retailer’s optimal joint decision is given by
Proof. Similar to Proposition 5.
Remark 10. By comparing (32) with (6) and (33) with (7), it is easy to get , such that the independent retailer’s optimal decisions are equal to the integrated fashion supply chain’s optimal solution . Hence, a twopart tariff contract could perfectly achieve channel coordination for a fashion supply chain with riskneutral retailer and pricedependent demand.
Now, by considering the risk aversion decision model, as given in , we establish the following propositions to attain the optimal joint decision for the riskaverse retailer under single contracts.
Proposition 11. Under single contracts, for any given , is strictly increasing in . For any given is strictly increasing in .
Proof. From (17), (27), and (31), taking differentials of with respect to and , and since , it can be easily verified that, for any given , is strictly increasing in and, for any given , is strictly increasing in .
Proposition 12. Given the retailer’s expected threshold , the riskaverse fashion retailer’s optimal joint decision satisfies
Proof. From the proceeding analysis, we know that under single contracts, such as SRP contract, revenuesharing contract and twopart tariff contract, is a concave function of and is unimodal in . Besides, is strictly increasing in and . Therefore, according to , the optimal pricinginventory decision for the riskaverse fashion retailer is obtained by solving . Moreover, since , in each region of , , , and , there exists a corresponding decision pair that could make established. Since is strictly increasing in and , the optimal solution for can only fall in the region , otherwise, cannot be the riskaverse fashion retailer’s optimal joint decision. So we have and .
Remark 13. From Proposition 12, we can know that the maximum expected profit of the riskaverse fashion retailer generated under single contracts is always no greater than that of a riskneutral retailer. This is the loss of profit brought out by the retailer’s risk aversion attitude or preference. In addition, it can be seen from Proposition 12 that under the additive pricedependent demand, the riskaverse fashion retailer tends to order less and charge a lower price in comparison with a riskneutral retailer, which is consistent with the known results derived from the studies on joint pricing and inventory decisions of a riskaverse newsvendor [29, 33].
A contract provided by the upstream manufacturer is said to coordinate the supply chain if it is able to align the incentives of the manufacturer and the retailer so that the independent retailer makes the same decisions as the integrated supply chain, namely, . Now we present the following proposition to explore the necessary conditions for a contract to achieve channel coordination.
Proposition 14. For any given , a contract achieves supply chain coordination if and only if the contract satisfies (1) ; (2) ; (3) .
Proof. If a contract achieves supply chain coordination, then stands. According to Proposition 12, we have . On the other hand, since a contract coordinates the supply chain, from , we know that . We know that is strictly increasing in and , and is a continuous function of and . If establishes, then there always exists an optimal joint decision and such that and , which contradicts the fact that is the optimal joint pricing and inventory decisions for the riskaverse fashion retailer. Therefore, we have .
If , then according to Proposition 12, and . Since is a concave function of and strictly increasing in , from , we have . Otherwise, if , and, because for any given , is strictly increasing in , then there exists such that and . It contradicts the fact that is the optimal joint decision of the riskaverse fashion retailer. Similarly, is unimodal in ; then, from , we have . Otherwise, if and because is strictly increasing in for any given , then there exists such that and . It contradicts the fact that is the optimal joint decision of the riskaverse fashion retailer. As a result, we have and .
Remark 15. From Proposition 14, it can be derived that when the supply chain is coordinated, the riskaverse fashion retailer’s expected profit is equal to , and hence the manufacturer’s expected profit is equal to .
Next, we investigate in more detail whether the single contracts above could achieve supply chain coordination.
Proposition 16. For any given , SRP contract cannot achieve supply chain coordination.
Proof. From Proposition 14, we can get that the supply chain achieves coordination if and only if SRP contract satisfies , and .
From , we can get . And from (6), we have . From , we have . And from (7), it can be obtained that . Since , there does not exist some value of such that and establish simultaneously. In other words, SRP contract cannot achieve channel coordination.
Proposition 17. For any given , revenuesharing contract and twopart tariff contract can achieve channel coordination. Specifically, the optimal conditions satisfied by the contract parameters of these two contracts to coordinate the supply chain are as follows:(1)for revenuesharing contract, , ;(2)for twopart tariff contract, , .
Proof. According to Proposition 14, for the revenuesharing contract, from , we get the expression . From , we have . Substituting (6) into , it can be calculated that . From , we get , and, taking (7) into it, we know that . Combining and , we have , and, by taking into the expression of , we have . Hence, revenuesharing contract can still coordinate the supply chain, when the fashion retailer is risk averse.
Similarly, for twopart tariff contract, from , we have . From , we get , and, from (6), is derived. From , we get , and, from (7), we have . Thus, we have . Comparing and , we can get . Therefore, twopart tariff also could achieve supply chain coordination with risk sensitive retailer.
Remark 18. From Propositions 16 and 17, we find that when the end demand depends on retail price and the fashion retailer is risk sensitive, a more complex SRP contract (with three parameters) cannot achieve supply chain coordination, whereas simpler revenuesharing contract and twopart tariff contract (with two parameters) can.
From Proposition 17, the values of and can be regarded as indicators of the fashion retailer’s risk aversion level. Specifically, with a larger , the expected profit threshold of the retailer is greater, and the retailer is more risk averse. Contrarily, a larger value of means a smaller expected profit threshold for the retailer, indicating a less risk sensitive retailer. As a result, if the fraction of sales revenue or the value of fixed transfer payment which the fashion retailer is willing to offer to the manufacturer is small, then the retailer is relatively more risk averse.
5. The Decentralized Fashion Supply Chain under Combined Contracts
In the above section, we investigate the role of three single contracts in coordinating fashion supply chains and find that a more complicated SRP contract fails to coordinate the supply chain while two other simpler contracts perfectly achieve channel coordination. In this section, we further explore contracts that combine the advantages of the above contracts. Specifically, we try to explore whether the resulting contracts are effective to coordinate the supply chain when the coordinating contracts and the failed contract combine with each other. Define similarly , , and .
5.1. SRP with RevenueSharing Contract
Under this contract, the fashion retailer’s profit, expected profit, and the variance of profit are
Proposition 19. For a given SRP with revenuesharing contract offered by the manufacturer, the riskneutral fashion retailer’s optimal joint decision is given by
Proof. Similar to Proposition 5.
Remark 20. By comparing (39) with (7) and (38) with (6), we find that when , , establishes. However, it contradicts the assumption of in SRP with revenuesharing contract. Thus, when the fashion retailer is riskneutral, SRP with revenuesharing contract cannot achieve channel coordination.
5.2. SRP with TwoPart Tariff Contract
In this setting, the fashion retailer’s expected profit and the variance of profit are given by
Proposition 21. For a given SRP with twopart tariff contract offered by the manufacturer, the riskneutral fashion retailer’s optimal joint decision is given by
Remark 22. Comparing (41) with (18) and (42) with (19), we discover that, under SRP with twopart tariff contract, the riskneutral fashion retailer’s optimal decisions are equal to those under a single SRP contract. Therefore, consistent with the analysis in Section 4, SRP with twopart tariff contract cannot coordinate the supply chain with riskaverse retailer and pricedependent demand.
5.3. Revenue Sharing with TwoPart Tariff Contract
Under a revenue sharing with twopart tariff contract, the fashion retailer’s expected profit and the variance of profit are as follows:
Similar to SRP with twopart tariff contract, the optimal joint orderingpricing decision for the riskneutral fashion retailer under revenue sharing with twopart tariff contract is equal to that under a single revenuesharing contract. Hence, revenue sharing with twopart tariff contract is able to achieve channel coordination in the fashion supply chain with riskaverse retailer and pricedependent demand.
As a result, it only remains uncertain whether SRP with revenuesharing contract could achieve supply chain coordination with riskaverse retailer. Following the similar approach as presented in Section 4, we now investigate the role of SRP with revenuesharing contract in channel coordination.
From (37), by some simple deductions, we know that, under SRP with revenuesharing contract, is strictly increasing in and . Therefore, with any given expected threshold , the riskaverse retailer’s optimal joint decision satisfies (34).
Proposition 23. For any given , SRP with revenuesharing contract cannot achieve supply chain coordination.
Proof. From Proposition 14, we know that channel coordination is obtained if and only if SRP with revenuesharing contract satisfies , , and . From , we have . Combining with (6), we can derive that . Nonetheless, from , we get , and, from (7), by some simplifications, we have . Since , there does not exist some value of that could satisfy and simultaneously. Thus, SRP with revenuesharing contract cannot coordinate the fashion supply chain.
Remark 24. It is interesting to discover that, although a single revenuesharing contract itself could coordinate the quantity and pricing decisions in the fashion supply chain with risk sensitive retailer, the combined SRP with revenuesharing contract cannot optimize the whole supply chain’s profit. To some extent, this means that, when faced with more intricate supply chain circumstance, perhaps a simpler contract is more effective and efficient to achieve channel coordination in comparison with a more complicated one.
Furthermore, when a single revenuesharing contract and a single twopart tariff contract can coordinate the supply chain with riskaverse retailer and pricedependent demand, a composite contract of these two contracts would still be effective to coordinate the supply chain. Instead, a single SRP contract cannot achieve channel coordination; thus when it combines with revenuesharing contract or twopart tariff contract, the resulting combined contract is still unable to coordinate the fashion supply chain.
6. Numerical Analysis
In this section, we present numerical analysis to gain more insights on supply chain coordination with contracts. We focus on the coordinating revenuesharing contract, twopart tariff contract, and the combined revenue sharing with twopart tariff contract here. Numerical analysis can be decomposed into two parts: one is to investigate how to determine the optimal values of contract parameters, and the other is sensitivity analysis to explore the impacts of some important parameters on supply chain coordination and objectives of supply chain members.
6.1. Determine the Values of Contract Parameters
First, we give the values of parameters used in this section. Suppose the base demand and the price elasticity of demand . The random variable follows a uniform distribution with a lower bound and an upper bound . The unit production cost , and the unit salvage value . With these parameters, the optimal joint decision that maximizes the expected profit of the integrated fashion supply chain is and , and the supply chain’s optimal production level is given by . The respective expected profit and the variance of profit for the fashion supply chain are and. Since the expected profit threshold for the riskaverse retailer must be smaller than the maximum expected profit gained by the fashion supply chain, we assume in the following analysis.
We consider six values of and to explore the optimal values of contract parameters for the coordinating contracts above. It should be noted that for the combined revenue sharing with twopart tariff contract, must establish to ensure that . The results are summarized in Table 1.

From Table 1, we can see the effectiveness of revenue sharing, twopart tariff, and their combined contract in coordinating the fashion supply chain. Consistent with Proposition 17, in revenuesharing contract and in twopart tariff contract can be used to judge the downstream fashion retailer’s risk aversion level. For revenuesharing contract, with a larger , the retailer is more risk averse, thus leading to a higher fraction of sales revenue kept by the retailer himself. By anticipating the retailer’s response, the manufacturer would react by setting a higher wholesale price . For twopart tariff contract, a wholesale price equaling to the unit production gives the manufacturer zero profit, but the fixed transfer payment ensures a positive profit for the manufacturer. And a higher value of which the retailer is willing to pay indicates a less riskaverse retailer. In combined contract, for the retailer’s same risk aversion degrees, the proposition of sales revenue kept by the retailer himself must be larger than that in the single revenuesharing contract, owing to the fact that the fashion retailer must pay the manufacturer an additional fixed payment in the combined contract.
6.2. Sensitivity Analysis
Now, we study the effects of some important parameters on supply chain coordination and objectives of supply chain members. Firstly, we focus on revenuesharing contract with parameters such as base demand , price elasticity of demand , and demand uncertainty. The results are given in Table 2.

From Table 2, we find that, with the increase of base demand , the optimal production quantity and pricing for the integrated fashion supply chain also increase, leading to larger expected profit and the variance of profit . This is consistent with Proposition 3 that is a concave function of and is unimodal in , and is strictly increasing in and . Taking into account , we consider the appropriate values of and find that the optimal values of and do not exhibit some rule of changes since changes randomly. However, when we fix the value of in the region for all cases of , we could intuitively reach the conclusion that the values of , and tend to decrease.
However, it can be found from Table 2 that with larger values of price elasticity , the entire supply chain’s optimal production quantity and retail price become smaller, so do the supply chain’s expected profit and the variance of profit . On the contrary, by fixing values of subject to , we discover that the values of , and all become larger.
In addition, we also try to illustrate the effect of different degrees of demand uncertainty. We define demand uncertainty as , where represents the mean and denotes the standard variance of the random demand. We could derive from Table 2, that when the level of demand uncertainty increases, the optimal joint quantity and pricing decisions for the fashion supply chain incline to firstly increase and then decrease. Thus, the supply chain’s expected profit and the variance of profit also have the same rule of changes. With respect to the values of and , they change toward the opposite direction. The combined changes of and cause the changes of .
Similarly, following the same method, we could also get the results of sensitivity analysis for the other two coordinating contracts—twopart tariff contract and revenue sharing with twopart tariff contract. They are summarized in Tables 3 and 4, respectively.


From Table 3, it can be easily discovered that no matter how the values of parameter , and change, the optimal values of are always equal to 15. But the optimal values of are dependent upon the change in values of those parameters. Specifically, the optimal equals . When parameters and change, the integrated fashion supply chain’s optimal expected profit also change as shown in Table 2. If the values of are fixed, changes positively with the changes of . That is, the values of decrease in the case of and firstly increase and then decrease in the case of .
Similar analysis could be realized for the joint revenue sharing with twopart tariff contract. What is worth noting here is that, in Table 4, we could find that the values of are larger than the corresponding values in Table 2 for the single revenuesharing contract, whereas the values of are smaller than the corresponding values in the single twopart tariff contract. This is because, in the combined contract, the fashion retailer cannot keep all the sales revenue but has to pay an additional fixed transfer payment to the manufacturer. Accordingly, the values of the variance of profit in revenue sharing with twopart tariff contract are always larger than the respective values in revenuesharing contract alone, while they obtain their largest values in twopart tariff contract since they are equal to the according values of .
7. Management Insights and Concluding Remark
In this paper, the issue of supply chain coordination with riskaverse retailer and pricedependent demand is studied. We extend in this paper the previous works to consider a riskaverse retailer in a twostage fashion supply chain. Adopting the additive pricesensitive demand model, we construct the benchmark solution to the integrated fashion supply chain. Using the classic MV formulation in portfolio management in finance to characterize the risk sensitive fashion retailer’s decision models, we propose both single contracts and combined contracts to achieve channel coordination. We find that, under single contracts, the coordinating revenuesharing contract and twopart tariff contract in supply chains with riskneutral agents could still coordinate the supply chain with risk sensitive retailer. However, a more complicated sales rebate and penalty contract fails to do so. Then we try to combine traditional single contracts to explore whether the resulting joint contracts are useful to coordinate the supply chain. It is found that only the joint revenue sharing with twopart tariff contract is able to achieve supply chain coordination. By presenting numerical analysis to illustrate analytical results, we discuss the determination of optimal values of contract parameters in coordinating contracts as well as sensitivity analysis to explore the effects of base demand, price elasticity, and the degree of demand uncertainty on supply chain coordination and objectives of supply chain members.
We highlight the managerial insights of our results in the following. Meanvariance formulation for risk analysis in supply chains is intuitive to decisions makers, making it easy and applicable for managers and practitioners in fashion supply chains to use proposed models to determine the type of contract and the optimal values of contract parameters to achieve channel coordination. This paper captures the fundamental feature of retailing fashion channel that the fashion retailer could influence the end demand by setting appropriate retail price. Our findings indicate that, in such complicated retailing situation, it may be more effective for real managers to adopt relatively simple contracts, so as to achieve coordination. Moreover, the theoretical results offer references to decision makers on the conditions a contract must satisfy to coordinate fashion supply chains, which is significant for improving efficiency in such supply chains with effective retailing channels and fashion retailer’s risk aversion preference or attitude.
In this paper, we just focus on channel coordination in single period, single manufacture, and single retailer fashion supply chain. Future research on supply chain coordination over multiple periods or with multiple competing riskaverse retailers would be a meaningful direction and could produce more insights.
Acknowledgments
The authors sincerely thank the EditorinChief and the three anonymous reviewers for their valuable comments and suggestions on the revision of the paper. This paper was supported in part by (1) the Fund for Humanity and Social Science of the Ministry of Education, China, under Grant 09YJC630230; (2) the Natural Science Foundation of Hunan Province, China, under Grant 10JJ3023.
References
 T. M. Choi, Fashion Supply Chain Management: Industry and Business Analysis, IGI Global, 2011. View at: Publisher Site  Zentralblatt MATH
 T. M. Choi and C. H. Chiu, “Meandownsiderisk and meanvariance newsvendor models: implications for sustainable fashion retailing,” International Journal of Production Economics, vol. 135, no. 2, pp. 552–560, 2010. View at: Google Scholar
 J. J. Spengler, “Vertical integration and antitrust policy,” The Journal of Political Economy, vol. 58, no. 4, pp. 347–352, 1950. View at: Publisher Site  Google Scholar
 B. A. Pasternack, “Optimal pricing and return policies for perishable commodities,” Marketing Science, vol. 4, no. 2, pp. 166–176, 1985. View at: Publisher Site  Google Scholar
 T. M. Choi, D. Li, H. Yan, and C. H. Chiu, “Channel coordination in supply chains with agents having meanvariance objectives,” Omega, vol. 36, no. 4, pp. 565–576, 2008. View at: Publisher Site  Google Scholar
 G. P. Cachon and M. A. Lariviere, “Supply chain coordination with revenuesharing contracts: strengths and limitations,” Management Science, vol. 51, no. 1, pp. 30–44, 2005. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 A. A. Tsay, S. Nahmias, and N. Agrawal, “Modeling supply chain contracts: a review,” in International Series in Operations Research and Management Science, pp. 299–336, 1999. View at: Google Scholar  Zentralblatt MATH
 Z. Lian and A. Deshmukh, “Analysis of supply contracts with quantity flexibility,” European Journal of Operational Research, vol. 196, no. 2, pp. 526–533, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 C. J. Corbett, D. Zhou, and C. S. Tang, “Designing supply contracts: contract type and information asymmetry,” Management Science, vol. 50, no. 4, pp. 550–559, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 T. A. Taylor, “Supply chain coordination under channel rebates with sales effort effects,” Management Science, vol. 48, no. 8, pp. 992–1007, 2002. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 W. K. Wong, J. Qi, and S. Y. S. Leung, “Coordinating supply chains with sales rebate contracts and vendormanaged inventory,” International Journal of Production Economics, vol. 120, no. 1, pp. 151–161, 2009. View at: Publisher Site  Google Scholar
 G. P. Cachon, “Supply chain coordination with contracts,” Handbooks in Operations Research and Management Science, vol. 11, no. 1, pp. 229–340, 2003. View at: Google Scholar
 C. H. Chiu, T. M. Choi, H. T. Yeung, and Y. X. Zhao, “Sales rebate contracts in fashion supply chains,” Mathematical Problems in Engineering, vol. 2012, Article ID 908408, 19 pages, 2012. View at: Publisher Site  Google Scholar
 J. D. Dana and K. E. Spier, “Revenue sharing and vertical control in the video rental industry,” The Journal of Industrial Economics, vol. 49, no. 3, pp. 223–245, 2001. View at: Google Scholar
 J. H. Mortimer, “Vertical contracts in the video rental industry,” Review of Economic Studies, vol. 75, no. 1, pp. 165–199, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 E. Cao, C. Wan, and M. Lai, “Coordination of a supply chain with one manufacturer and multiple competing retailers under simultaneous demand and cost disruptions,” International Journal of Production Economics, vol. 141, no. 1, pp. 425–433, 2013. View at: Publisher Site  Google Scholar
 D. Hu and Q. Li, “Supply chain coordination under revenuesharing contract with sales effort effect,” in Proceedings of the International Conference on Computer and Management (CAMAN '11), pp. 1–4, May 2011. View at: Publisher Site  Google Scholar
 K. Chen and T. Xiao, “Ordering policy and coordination of a supply chain with twoperiod demand uncertainty,” European Journal of Operational Research, vol. 215, no. 2, pp. 347–357, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 C. H. Chiu, T. M. Choi, and C. S. Tang, “Price, rebate, and returns supply contracts for coordinating supply chains with pricedependent demands,” Production and Operations Management, vol. 20, no. 1, pp. 81–91, 2011. View at: Publisher Site  Google Scholar
 Y. He, X. Zhao, L. Zhao, and J. He, “Coordinating a supply chain with effort and price dependent stochastic demand,” Applied Mathematical Modelling, vol. 33, no. 6, pp. 2777–2790, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 H. Krishnan, R. Kapuscinski, and D. A. Butz, “Coordinating contracts for decentralized supply chains with retailer promotional effort,” Management Science, vol. 50, no. 1, pp. 48–63, 2004. View at: Publisher Site  Google Scholar
 H. Emmons and S. M. Gilbert, “Note. The role of returns policies in pricing and inventory decisions for catalogue goods,” Management Science, vol. 44, no. 2, pp. 276–283, 1998. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 Y. Qin, H. Tang, and C. Guo, “Channel coordination and volume discounts with pricesensitive demand,” International Journal of Production Economics, vol. 105, no. 1, pp. 43–53, 2007. View at: Publisher Site  Google Scholar
 H. M. Markowitz, Portfolio Selection: Efficient Diversification of Investments, 1959. View at: MathSciNet
 P. C. Fishburn, “Meanrisk analysis with risk associated with belowtarget returns,” The American Economic Review, vol. 67, no. 1, pp. 116–126, 1977. View at: Google Scholar
 S. Choi and A. Ruszczyński, “A riskaverse newsvendor with law invariant coherent measures of risk,” Operations Research Letters, vol. 36, no. 1, pp. 77–82, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 C. H. Chiu and T. M. Choi, “Optimal pricing and stocking decisions for newsvendor problem with valueatrisk consideration,” IEEE Transactions on Systems, Man, and Cybernetics A, vol. 40, no. 5, pp. 1116–1119, 2010. View at: Publisher Site  Google Scholar
 J. Y. Gotoh and Y. Takano, “Newsvendor solutions via conditional valueatrisk minimization,” European Journal of Operational Research, vol. 179, no. 1, pp. 80–96, 2007. View at: Publisher Site  Google Scholar
 Y. H. Chen, M. H. Xu, and Z. G. Zhang, “A riskaverse newsvendor model under the CVaR criterion,” Operations Research, vol. 57, no. 4, pp. 1040–1044, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 K. S. Sundar, S. Narayanan, and D. Nagaraju, “Coordination in two level supply chain with price dependent demand,” Applied Mathematical Sciences, vol. 6, no. 74, pp. 3687–3703, 2012. View at: Google Scholar
 J.M. Chen, H.L. Cheng, and M.C. Chien, “On channel coordination through revenuesharing contracts with price and shelfspace dependent demand,” Applied Mathematical Modelling, vol. 35, no. 10, pp. 4886–4901, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 F. J. Arcelus, S. Kumar, and G. Srinivasan, “Evaluating manufacturer's buyback policies in a singleperiod twoechelon framework under pricedependent stochastic demand,” Omega, vol. 36, no. 5, pp. 808–824, 2008. View at: Publisher Site  Google Scholar
 V. Agrawal and S. Seshadri, “Impact of uncertainty and risk aversion on price and order quantity in the newsvendor problem,” Manufacturing and Service Operations Management, vol. 2, no. 4, pp. 410–423, 2000. View at: Publisher Site  Google Scholar
 C.H. Chiu, T.M. Choi, and X. Li, “Supply chain coordination with risk sensitive retailer under target sales rebate,” Automatica, vol. 47, no. 8, pp. 1617–1625, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 X. Gan, S. P. Sethi, and H. Yan, “Coordination of supply chains with riskaverse agents,” Supply Chain Coordination under Uncertainty, vol. 3, no. 1, pp. 135–149, 2004. View at: Google Scholar
 Y. Wei and T. M. Choi, “Meanvariance analysis of supply chains under wholesale pricing and profit sharing schemes,” European Journal of Operational Research, vol. 204, no. 2, pp. 255–262, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH
Copyright
Copyright © 2013 Minli Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.