# Coordinating Contracts for Two-Stage Fashion Supply Chain with Risk-Averse Retailer and Price-Dependent Demand

Authors: Minli Xu; Qiao Wang; Linhan Ouyang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

When the demand is sensitive to retail price, revenue sharing
contract and two-part tariff contract have been shown to be able
to coordinate supply chains with risk neutral agents. We extend
the previous studies to consider a risk-averse retailer in a
two-echelon fashion supply chain. Based on the classic
mean-variance approach in finance, the issue of channel
coordination in a fashion supply chain with risk-averse retailer
and price-dependent 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 two-part 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 two-part 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.


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## Body

## 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 stock-outs [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 risk-neutral agents by fighting against the issue of double marginalization. These traditional contracts include returns policy [4, 5], revenue-sharing contract [6], quantity flexibility contract [7, 8], two-part 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].Revenue-sharing 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 revenue-sharing 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 two-part 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 two-part 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 revenue-sharing contract and two-part tariff contract are able to coordinate the newsvendor with price-dependent 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 risk-averse fashion retailer and price-dependent demand. Specifically, we investigate a single period, one-manufacturer one-retailer 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 self-interest 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 revenue-sharing contract and two-part tariff contract in supply chains with risk neural retailer can still coordinate the fashion supply chain with risk-averse 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 revenue-sharing contract and two-part 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 mean-variance (MV) [24], Neumann-Morgenstern utility function (VNUM), mean-downside-risk (MDR) [25], Value-at-risk (VaR) [26, 27], and Conditional Value-at-risk (CVaR) [28, 29]. Since MV is simple, implementable and is easily understood by managers and practitioners compared with other measures, we adopt mean-variance 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 price-dependent demand [30, 31] in terms of a random and price-dependent demand. It is also correlated to studies of supply chain coordination with agents having risk preferences in which we consider a risk-averse retailer [32–36]. But our study is the first to investigate the issue of channel coordination for the supply chain with risk-averse retailer and price-dependent demand. We firstly investigate the problem of coordinating a two-stage fashion supply chain under single contracts including revenue-sharing contract, two-part tariff contract and sales rebate and penalty contract. After proving that revenue-sharing contract and two-part 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 revenue-sharing contract, sales rebate and penalty with two-part tariff contract, and revenue sharing with two-part 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 Section2. 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 two-echelon fashion supply chain with a risk-neutral manufacturer and a risk-averse 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.Letp be the retail price, c the production cost incurred by the manufacturer, w(w≥c) the wholesale price, v(v<c) the salvage value of unsold inventory, and q the production/order quantity. Use t>0 as the sales target level and u>0 as the rebate (and penalty) for sales rebate and penalty contract. Use λ∈(0,1) as the fraction of revenue earned by the retailer in revenue-sharing contract and G>0 as the fixed transfer payment from the retailer to the manufacturer in two-part tariff contract.In the literature, there are two fashions in which the demandx depends on the selling price p: (1) the additive form x=D(p)+ξ; (2) the multiplicative form x=D(p)ξ, where D(p)≥0 is a function of p representing the expected demand and ξ is a nonnegative variable representing the random proportion of the demand.  ξ is independent of selling price p with a probability density function f(·) and a cumulative distribution function F(·). It is assumed that f(·)>0 has a continuous   derivative  f′(·). F(·) is continuous, strictly increasing, and differentiable. Let F-1(·) be the reverse function of F(·),  and F¯(·)=1-F(·).  D(p) is strictly decreasing in p,  and  D′(p)<0. 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 ofD(p) and ξ.Definition 1.
By definition,e=-pD′(p)/D(p) is the price elasticity of D(p).  D(p) has an increasing price elasticity (IPE) in p, if
(1)dedp≥0.Price elasticitye 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 ofD(p). Let D(p)=a-kp, where a>0 is the base demand and k>0 is the price elasticity of demand. Thus, we have p∈[c,a/k].Definition 2.
Definer(ξ)=f(ξ)/(1-F(ξ)) as the failure rate of the ξ distribution then ξ has an increasing failure rate (IFR), if for ξ≥0(2)r′(ξ)≥0.It is noted that, in the literature, various random distributions exhibit IFR property, involving uniform and normal distributions.To capture the decision making of risk-averse fashion retailer, we adopt the same risk aversion decision model as in [34]:
(P-1)minVr(q,p)s.t.Er(q,p)≥kr,
where Er(q,p) and Vr(q,p) denote the mean and the variance of the retailer’s profit, respectively, and kr>0 denotes the retailer’s expected profit threshold. kr can be considered to be the indicator of the retailer’s risk aversion degree, since larger values of kr indicate that the retailer does not want to earn a low expected profit, leading to a more risk-averse retailer. Define E-r=Er(qr*,pr*) as the retailer’s attainable maximum expected profit. Then from (P-1), kr≤E-r must establish, otherwise, there is no feasible solution for (P-1).


## 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 levelq and retail price p, which provide us  with guidelines to the optimal policy for the whole system. Define Q=q-D(p), η(Q)=2Q∫0QF(ξ)dξ-2∫0QξF(ξ)dξ-(∫0QF(ξ)dξ)2. The integrated fashion supply chain’s profit, expected profit, and the variance of profit are given as follows:
(3)∏sc(Q,p)=(p-c)D(p)+(p-c)Q-(p-v)(Q-ξ)+,(4)Esc(Q,p)=(p-c)D(p)+(p-c)Q-(p-v)∫0QF(ξ)dξ,(5)Vsc(Q,p)=(p-v)2η(Q).Let(Qsc*,psc*) and qsc*=D(psc*)+Qsc* be the optimal joint decision and optimal production level for the integrated supply chain.Proposition 3.
Under the additive price-dependent demand, the integrated supply chain’s optimal joint decision(Qsc*,psc*) and optimal production quantity qsc* exist and are unique, satisfying
(6)(psc*-c)-(psc*-v)F(Qsc*)=0,(7)a-k(2psc*-c)+Qsc*-∫0Qsc*F(ξ)dξ=0,(8)qsc*=a-kpsc*+F-1(psc*-cpsc*-v).Proof.
For any givenp∈[c,a/k], from (4), by taking the first and second differentials of Esc(Q,p) with respect to Q, we get
∂
E
sc
(
Q
,
p
)
/
∂
Q
=
(
p
-
c
)
-
(
p
-
v
)
F
(
Q
),  ∂2Esc(Q,p)/∂Q2=-(p-v)f(Q)<0. Thus, for any given p∈[c,a/k],Esc(Q,p) is a concave function of Q, and Qsc* is finite and unique, satisfying
(9)(p-c)-(p-v)F(Qsc*)=0.
From (9), we know that Qsc* is a function of p. According to the implicit function theorem, we have
(10)dQsc*dp=-∂2Esc(Qsc*,p)/∂Q∂p∂2Esc(Qsc*,p)/∂Q2=1-F(Qsc*)(p-v)f(Qsc*).
Therefore, from (10), Qsc* is strictly increasing in p.
By takingQsc* into Esc(Q,p), we get Esc(Qsc*,p). Taking the first and second  derivatives  of Esc(Qsc*,p) with respect to p, we derive:
(11)dEsc(Qsc*,p)dp=a-k(2p-c)+Qsc*-∫0Qsc*F(ξ)dξ,(12)d2Esc(Qsc*,p)dp2=-2k+(1-F(Qsc*))dQsc*dp.
Substituting (10) into (12), we get
(13)d2Esc(Qsc*,p)dp2=-2k+(1-F(Qsc*))2(p-v)f(Qsc*).
DefineH(Q)=f(Q)/(1-F(Q))2, and taking the derivative of H(Q) with respect to Q, we have
(14)dH(Q)dQ=(1-F(Q))f′(Q)+2(f(Q))2(1-F(Q))3.
From Definition2, we know that H(Qsc*) is strictly increasing in p. Therefore, d2Esc(Qsc*,p)/dp2 is strictly decreasing in p. Let p0 satisfy d2Esc(Qsc*,p)/dp2=0. If p0 does not exist, then we can know d2Esc(Qsc*,p)/dp2<0, since limp→∞d2Esc(Qsc*,p)/dp2=-2k<0, and Esc(Qsc*,p) is a concave function of p. If p0 exists, for p<p0, d2Esc(Qsc*,p)/dp2>0, and, for p>p0, ∂2Esc(Qsc*,p)/∂p2<0. That is, Esc(Qsc*,p) is convex in p for p<p0 and concave in p for p>p0. Because dEsc(Qsc*,p)/dp|p=c=(a-kc)+∫0Qsc*(c)F-(ξ)dξ>0, Esc(Qsc*,p,e) is unimodal in p∈[c,a/k].
Hence, there exists a unique retail pricepsc*∈[c,a/k] that maximizes Esc(Qsc*,p) and is given by (7).
Sinceqsc*=D(psc*)+Qsc*, it is natural to conclude that the fashion supply chain’s optimal production quantity qsc* is unique and satisfies (8).Remark 4.
Proposition3 reveals the optimal solutions of the integrated fashion supply chain with the additive price-dependent demand. Correspondingly, the entire supply chain’s maximum expected profit is E-sc=Esc(Qsc*,psc*).


## 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 risk-averse with an expected profit thresholdkr>0 and kr<E-sc (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 pricing-inventory decisions of a risk-averse retailer in the decentralized fashion supply chain under single contracts such as revenue sharing contact, sales rebate and penalty contract, and two-part tariff contract. For the purpose of simplification, define Q=q-D(p), T=t-D(p) and T*=t-D(p*). Let (Qr*,pr*) and (Qr,mv*,pr,mv*) be the optimal joint decisions for the risk-neutral retailer and risk-averse retailer, respectively.


### 4.1. SRP Contract

With a SRP contractθSRP(w,t,u), the manufacturer offers a sales target t>0 to the retailer prior to the selling season. At the end of the selling season, for each unit sold above t, the manufacturer will give the retailer a unit rebate u>0, otherwise, the retailer must pay the manufacturer a penalty u.In this setting, the fashion retailer’s profit, expected profit and the variance of profit are(15)∏r(Q,p)=(p-w)D(p)+(p-w)Q-(p-v)(Q-ξ)++u(min(ξ,Q)-T),(16)Er(Q,p)=(p-w)D(p)+(p-w)Q-(p-v)∫0QF(ξ)dξ+u(Q-T-∫0QF(ξ)dξ),(17)Vr(Q,p)=E(∏r(Q,p)-Er(Q,p))2=(p-v)2E(∫0QF(ξ)dξ-(Q-ξ)+)2+u2E((min(ξ,Q)-T)-(Q-T-∫0QF(ξ)dξ))2+2u(p-v)E(∫0QF(ξ)dξ-(Q-ξ)+)×((min(ξ,Q)-T)-(Q-T-∫0QF(ξ)dξ))=(p-v)2(∫0Q(Q-ξ)2f(ξ)dξ-(∫0QF(ξ)dξ)2)+u2({(Q-T-∫0QF(ξ)dξ)2}∫0Q(ξ-T)2f(ξ)dξ+∫Q∞(Q-T)2f(ξ)dξ-(Q-T-∫0QF(ξ)dξ)2)+2u(p-v)(∫0QF(ξ)dξ(Q-T-∫0QF(ξ)dξ)-∫0Q(Q-ξ)(ξ-T)f(ξ)dξ{∫0QF(ξ)dξ(Q-T-∫0QF(ξ)dξ)})=(p-v+u)2({(∫0QF(ξ)dξ)2}2Q∫0QF(ξ)dξ-2∫0QξF(ξ)dξ-(∫0QF(ξ)dξ)2)=(p-v+u)2η(Q).Proposition 5.
For a given SRP contractθSRP(w,t,u) offered by the manufacturer, the risk-neutral retailer’s optimal joint decision (Qr*,pr*) is given by
(18)(pr*-w+u)-(pr*-v+u)F(Qr*)=0,(19)a-k(2pr*-w+u)+Qr*-∫0Qr*F(ξ)dξ=0.Proof.
For any givenp∈(w-u,p-], from (16), by taking the first and second differentials of Er(Q,p) with respect to Q, we can derive that ∂Er(Q,p)/∂Q=(p-w+u)-(p-v+u)F(Q), and ∂2Er(Q,p)/∂Q2=-(p-v+u)f(Q)<0. Thus, Er(Q,p) is a concave function of Q.  Qr*can be given by
(20)(p-w+u)-(p-v+u)F(Qr*)=0.
From (20), we can get to know that Qr* is a function of p. By making use of the implicit function theorem, we have
(21)dQr*dp=-∂2Er(Qr*,p)/∂Q∂p∂2Er(Qr*,p)/∂Q2=1-F(Qr*)(p-v+u)f(Qr*).
Thus, we know thatQr* is strictly increasing in p.
SubstitutingQr* into Er(Q,p), we get
(22)Er(Qr*,p)=(p-w)D(p)+(p-w)Qr*-(p-v)∫0Qr*F(ξ)dξ+u(Qr*-T-∫0Qr*F(ξ)dξ).
From (22), Er(Qr*,p) can be regarded as a function of variable p alone. Taking the first and second derivatives of Er(Qr*,p) with respect to p, we get
(23)dEr(Qr*,p)dp=a-k(2p-w+u)+Qr*-∫0Qr*F(ξ)dξ,(24)d2Er(Qr*,p)dp2=-2k+(1-F(Qr*))dQr*dp.
By taking (21) into (24), we have
(25)d2Er(Qr*,p)dp2=-2k+(1-F(Qr*))2(p-v+u)f(Qr*).
Similar to Proposition3, we know that Er(Qr*,p) is unimodal in p. If w-u>c,  dEr(Qr*,p)/dp|p=w-u=(a-k(w-u))+∫0Qr*(w-u)F¯(ξ)dξ>0. Thus, there exists a unique pr* which satisfies (19).Remark 6.
By Comparing (19) with (7) and (18) with (6), we find that (Qsc*,psc*) is the risk-neutral fashion retailer’s optimal joint decision if and only if u=0 and w=c. However, a SRP contract with u=0 and w=c gives the manufacturer zero profit. So SRP contract cannot coordinate the supply chain with risk-neutral fashion retailer and price-dependent demand.


### 4.2. Revenue-Sharing Contract

A revenue-sharing contractθRS(w,λ) stipulates that the fashion retailer pays the upstream manufacturer a unit wholesale price w for each unit ordered plus a proportion of his revenue from selling the product. Let λ∈(0,1) be the fraction of supply chain revenue earned by the retailer, and thus (1-λ)  is the fraction shared by the manufacturer. Under the revenue-sharing contract θRS(w,λ), the retailer’s expected profit and the variance of profit are given as follows:
(26)Er(Q,p)=(λp-w)D(p)+(λp-w)Q-λ(p-v)∫0QF(ξ)dξ,(27)Vr(Q,p)=λ2(p-v)2η(Q).Proposition 7.
For a given revenue-sharing contractθRS(w,λ) offered by the manufacturer, the risk-neutral fashion retailer’s optimal joint decision (Qr*,pr*) is given by
(28)(λpr*-w)-λ(pr*-v)F(Qr*)=0,(29)λa-k(2λpr*-w)+λQr*-λ∫0Qr*F(ξ)dξ=0.Proof.
Similar to Proposition5.Remark 8.
Comparing (28) with (6) and (29) with (7), we find that (Qsc*,psc*) can be the risk-neutral retailer’s optimal ordering quantity and retail price if and only if w=λc<c, 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 risk-neutral retailer.


### 4.3. Two-Part Tariff Contract

With a two-part tariff contractθTPT(w,G), the fashion retailer gives the manufacturer a fixed transfer payment G>0 apart from the unit wholesale price for each unit ordered. And the retailer’s expected profit and the variance of profit are
(30)Er(Q,p)=(p-w)D(p)+(p-w)Q-(p-v)∫0QF(ξ)dξ-G,(31)Vr(Q,p)=E(∏r(Q,p)-Er(Q,p))2=(p-v)2η(Q).Proposition 9.
For a given two-part tariff contractθTPT(w,G) offered by the manufacturer, the risk-neutral fashion retailer’s optimal joint decision (Qr*,pr*) is given by
(32)(pr*-w)-(pr*-v)F(Qr*)=0,(33)a-k(2pr*-w)+Qr*-∫0Qr*F(ξ)dξ=0.Proof.
Similar  to Proposition5.Remark 10.
By comparing (32) with (6) and (33) with (7), it is easy to get w=c, such that the independent retailer’s optimal decisions (Qr*,pr*) are equal to the integrated fashion supply chain’s optimal solution (Qsc*,psc*). Hence, a two-part tariff contract θTPT(w,G) could perfectly achieve channel coordination for a fashion supply chain with risk-neutral retailer and price-dependent demand.
Now, by considering the risk aversion decision model, as given in(P-1), we establish the following propositions to attain the optimal joint decision for the risk-averse retailer under single contracts.Proposition 11.
Under single contracts, for any givenp∈[w,a/k], Vr(Q,p) is strictly increasing in Q. For any given Q≥0,Vr(Q,p) is strictly increasing in p.Proof.
From (17), (27), and (31), taking differentials of  Vr(Q,p) with respect to Q and p, and since dη(Q)/dQ=2(1-F(Q))∫0QF(ξ)dξ>0, it can be easily verified that, for any given p∈[w,a/k], Vr(Q,p) is strictly increasing in Q and, for any given Q≥0,  Vr(Q,p) is strictly increasing in p.Proposition 12.
Given the retailer’s expected thresholdkr≤E-r, the risk-averse fashion retailer’s optimal joint decision (Qr,mv*,pr,mv*) satisfies
(34)Er(Qr,mv*,pr,mv*)=kr,Qr,mv*≤Qr*,pr,mv*≤pr*.Proof.
From the proceeding analysis, we know that under single contracts, such as SRP contract, revenue-sharing contract and two-part tariff contract,Er(Q,p) is a concave function of Q and is unimodal in p∈[w,a/k]. Besides, Vr(Q,p) is strictly increasing in Q and p. Therefore, according to (P-1), the optimal pricing-inventory decision for the risk-averse fashion retailer is obtained by solving Er(Qr,mv*,pr,mv*)=kr. Moreover, since kr≤E-r, in each region of (Q≤Qr*,p≤pr*), (Q>Qr*,p≤pr*), (Q≤Qr*,p>pr*), and (Q>Qr*,p>pr*), there exists a corresponding decision pair (Q,p) that could make Er(Q,p)=kr established. Since Vr(Q,p) is strictly increasing in Q and p, the optimal solution (Qr,mv*,pr,mv*) for (P-1) can only fall in the region (Q≤Qr*,p≤pr*), otherwise, (Qr,mv*,pr,mv*) cannot be the risk-averse fashion retailer’s optimal joint decision. So we have Qr,mv*≤Qr* and pr,mv*≤pr*.Remark 13.
From Proposition12, we can know that the maximum expected profit  of  the risk-averse fashion retailer generated under single contracts is always no greater than that of a risk-neutral 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 price-dependent demand, the risk-averse fashion retailer tends to order less and charge a lower price in comparison with a risk-neutral retailer, which is consistent with the known results derived from the studies on joint pricing and inventory decisions of a risk-averse 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,(Qr,mv*,pr,mv*)=(Qsc*,psc*). Now we present the following proposition to explore the necessary conditions for a contract to achieve channel coordination.Proposition 14.
For any givenkr<E-sc, a contract achieves supply chain coordination if and only if the contract satisfies (1) Er(Qsc*,psc*)=kr; (2) ∂Er(Q,psc*)/∂Q|Q=Qsc*≥0; (3) ∂Er(Qsc*,p)/∂p|p=psc*≥0.Proof.
If a contract achieves supply chain coordination, then(Qr,mv*,pr,mv*)=(Qsc*,psc*) stands. According to Proposition 12, we have Er(Qsc*,psc*)=kr. On the other hand, since a contract coordinates the supply chain, from (P-1), we know that Er(Qsc*,psc*)≥kr. We know that Vr(Q,p) is strictly increasing in Q and p, and Er(Q,p) is a continuous function of Q and p. If Er(Qsc*,psc*)>kr establishes, then there always exists an optimal joint decision Q<Qsc* and p<psc* such that Er(Q,p)≥kr and Vr(Q,p)<Vr(Qsc*,psc*), which contradicts the fact that (Qsc*,psc*) is the optimal joint pricing and inventory decisions for the risk-averse fashion retailer. Therefore, we have Er(Qsc*,psc*)=kr.
If(Qr,mv*,pr,mv*)=(Qsc*,psc*), then according to Proposition 12, Qsc*≤Qr* and psc*≤pr*. Since Er(Q,p) is a concave function of Q and strictly increasing in Q∈(0,Qr*], from Qsc*≤Qr*, we have ∂Er(Q,psc*)/∂Q|Q=Qsc*≥0. Otherwise, if ∂Er(Q,psc*)/∂Q|Q=Qsc*<0, and, because for any given p∈[w,a/k], Vr(Q,p) is strictly increasing in Q, then there exists Q<Qsc* such that Er(Qsc*,psc*)<Er(Q,psc*) and Vr(Q,psc*)<Vr(Qsc*,psc*). It contradicts the fact that (Qsc*,psc*) is the optimal joint decision of the risk-averse fashion retailer. Similarly, Er(Q,p) is unimodal in p∈[w,a/k]; then, from psc*≤pr*, we have ∂Er(Qsc*,p)/∂p|p=psc*≥0. Otherwise, if ∂Er(Qsc*,p)/∂p|p=psc*<0 and because Vr(Q,p) is strictly increasing in p for any given Q≥0, then there exists p<psc* such that Er(Qsc*,psc*)<Er(Qsc*,p) and Vr(Qsc*,p)<Vr(Qsc*,psc*). It contradicts the fact that (Qsc*,psc*) is the optimal joint decision of the risk-averse fashion retailer. As a result, we have ∂Er(Q,psc*)/∂Q|Q=Qsc*≥0 and  ∂Er(Qsc*,p)/∂p|p=psc*≥0.Remark 15.
From Proposition14, it can be derived that when the supply chain is coordinated, the risk-averse fashion retailer’s expected profit is equal to kr, and hence the manufacturer’s expected profit is equal to E-sc-kr.
Next, we investigate in more detail whether the single contracts above could achieve supply chain coordination.Proposition 16.
For any givenkr<E-sc, SRP contract cannot achieve supply chain coordination.Proof.
From Proposition14, we can get that the supply chain achieves coordination if and only if SRP contract satisfies Er(Qsc*,psc*)=kr,∂Er(Q,psc*)/∂Q|Q=Qsc*≥0, and ∂Er(Qsc*,p)/∂p|p=psc*≥0.
From∂Er(Q,psc*)/∂Q|Q=Qsc*≥0, we can get (psc*-w)-(psc*-v)F(Qsc*)+uF-(Qsc*)≥0. And from (6), we have uF¯(Qsc*)≥w-c. From ∂Er(Qsc*,p)/∂p|p=psc*≥0, we have a-k(2psc*-w+u)+Qsc*-∫0Qsc*F(ξ)dξ≥0. And from (7), it can be obtained that u≤w-c. Since uF-(Qsc*)<u, there does not exist some value of u such that uF-(Qsc*)≥w-c and u≤w-c establish simultaneously. In other words, SRP contract θSRP(w,t,u) cannot achieve channel coordination.Proposition 17.
For any givenkr<E-sc, revenue-sharing contract and two-part 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 revenue-sharing contract,w=λc, λ=kr/E-sc;(2)
for two-part tariff contract,w=c, G=E-sc-kr.Proof.
According to Proposition14, for the revenue-sharing contract, from Er(Qsc*,psc*)=kr, we get the expression w=λpsc*-(kr+λ(psc*-v)∫0Qsc*F(ξ)dξ)/qsc*. From ∂Er(Q,psc*)/∂Q|Q=Qsc*≥0, we have λpsc*-w-λ(psc*-v)F(Qsc*)≥0. Substituting (6) into λpsc*-w-λ(psc*-v)F(Qsc*)≥0, it can be calculated that w≤λc. From ∂Er(Qsc*,p)/∂p|p=psc*≥0, we get λa-k(2λpsc*-w)+λQsc*-λ∫0Qsc*F(ξ)dξ≥0, and, taking (7) into it, we know that w≥λc. Combining w≤λc and w≥λc, we have w=λc, and, by taking w=λc into the expression of w, we have λ=kr/E-sc. Hence, revenue-sharing contract can still coordinate the supply chain, when the fashion retailer is risk averse.
Similarly, for two-part tariff contract, fromEr(Qsc*,psc*)=kr, we have w=psc*-(kr+(psc*-v)∫0Qsc*F(ξ)dξ+G)/qsc*. From ∂Er(Q,psc*)/∂Q|Q=Qsc*≥0, we get psc*-w-(psc*-v)F(Qsc*)≥0, and, from (6), w≤c is derived. From ∂Er(Qsc*,p)/∂p|p=psc*≥0, we get a-k(2psc*-w)+Qsc*-∫0Qsc*F(ξ)dξ≥0, and, from (7), we have w≥c. Thus, we have w=c. Comparing w=c and w=psc*-[(kr+(psc*-v)∫0Qsc*F(ξ)dξ+G)/qsc*], we can get G=E-sc-kr. Therefore, two-part tariff also could achieve supply chain coordination with risk sensitive retailer.Remark 18.
From Propositions16 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 revenue-sharing contract and two-part tariff contract (with two parameters) can.
From Proposition17, the values of λ and G can be regarded as indicators of the fashion retailer’s risk aversion level. Specifically, with a larger λ, the expected profit threshold of the retailer kr is greater, and the retailer is more risk averse. Contrarily, a larger value of G 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.


## 4.1. SRP Contract

With a SRP contractθSRP(w,t,u), the manufacturer offers a sales target t>0 to the retailer prior to the selling season. At the end of the selling season, for each unit sold above t, the manufacturer will give the retailer a unit rebate u>0, otherwise, the retailer must pay the manufacturer a penalty u.In this setting, the fashion retailer’s profit, expected profit and the variance of profit are(15)∏r(Q,p)=(p-w)D(p)+(p-w)Q-(p-v)(Q-ξ)++u(min(ξ,Q)-T),(16)Er(Q,p)=(p-w)D(p)+(p-w)Q-(p-v)∫0QF(ξ)dξ+u(Q-T-∫0QF(ξ)dξ),(17)Vr(Q,p)=E(∏r(Q,p)-Er(Q,p))2=(p-v)2E(∫0QF(ξ)dξ-(Q-ξ)+)2+u2E((min(ξ,Q)-T)-(Q-T-∫0QF(ξ)dξ))2+2u(p-v)E(∫0QF(ξ)dξ-(Q-ξ)+)×((min(ξ,Q)-T)-(Q-T-∫0QF(ξ)dξ))=(p-v)2(∫0Q(Q-ξ)2f(ξ)dξ-(∫0QF(ξ)dξ)2)+u2({(Q-T-∫0QF(ξ)dξ)2}∫0Q(ξ-T)2f(ξ)dξ+∫Q∞(Q-T)2f(ξ)dξ-(Q-T-∫0QF(ξ)dξ)2)+2u(p-v)(∫0QF(ξ)dξ(Q-T-∫0QF(ξ)dξ)-∫0Q(Q-ξ)(ξ-T)f(ξ)dξ{∫0QF(ξ)dξ(Q-T-∫0QF(ξ)dξ)})=(p-v+u)2({(∫0QF(ξ)dξ)2}2Q∫0QF(ξ)dξ-2∫0QξF(ξ)dξ-(∫0QF(ξ)dξ)2)=(p-v+u)2η(Q).Proposition 5.
For a given SRP contractθSRP(w,t,u) offered by the manufacturer, the risk-neutral retailer’s optimal joint decision (Qr*,pr*) is given by
(18)(pr*-w+u)-(pr*-v+u)F(Qr*)=0,(19)a-k(2pr*-w+u)+Qr*-∫0Qr*F(ξ)dξ=0.Proof.
For any givenp∈(w-u,p-], from (16), by taking the first and second differentials of Er(Q,p) with respect to Q, we can derive that ∂Er(Q,p)/∂Q=(p-w+u)-(p-v+u)F(Q), and ∂2Er(Q,p)/∂Q2=-(p-v+u)f(Q)<0. Thus, Er(Q,p) is a concave function of Q.  Qr*can be given by
(20)(p-w+u)-(p-v+u)F(Qr*)=0.
From (20), we can get to know that Qr* is a function of p. By making use of the implicit function theorem, we have
(21)dQr*dp=-∂2Er(Qr*,p)/∂Q∂p∂2Er(Qr*,p)/∂Q2=1-F(Qr*)(p-v+u)f(Qr*).
Thus, we know thatQr* is strictly increasing in p.
SubstitutingQr* into Er(Q,p), we get
(22)Er(Qr*,p)=(p-w)D(p)+(p-w)Qr*-(p-v)∫0Qr*F(ξ)dξ+u(Qr*-T-∫0Qr*F(ξ)dξ).
From (22), Er(Qr*,p) can be regarded as a function of variable p alone. Taking the first and second derivatives of Er(Qr*,p) with respect to p, we get
(23)dEr(Qr*,p)dp=a-k(2p-w+u)+Qr*-∫0Qr*F(ξ)dξ,(24)d2Er(Qr*,p)dp2=-2k+(1-F(Qr*))dQr*dp.
By taking (21) into (24), we have
(25)d2Er(Qr*,p)dp2=-2k+(1-F(Qr*))2(p-v+u)f(Qr*).
Similar to Proposition3, we know that Er(Qr*,p) is unimodal in p. If w-u>c,  dEr(Qr*,p)/dp|p=w-u=(a-k(w-u))+∫0Qr*(w-u)F¯(ξ)dξ>0. Thus, there exists a unique pr* which satisfies (19).Remark 6.
By Comparing (19) with (7) and (18) with (6), we find that (Qsc*,psc*) is the risk-neutral fashion retailer’s optimal joint decision if and only if u=0 and w=c. However, a SRP contract with u=0 and w=c gives the manufacturer zero profit. So SRP contract cannot coordinate the supply chain with risk-neutral fashion retailer and price-dependent demand.


## 4.2. Revenue-Sharing Contract

A revenue-sharing contractθRS(w,λ) stipulates that the fashion retailer pays the upstream manufacturer a unit wholesale price w for each unit ordered plus a proportion of his revenue from selling the product. Let λ∈(0,1) be the fraction of supply chain revenue earned by the retailer, and thus (1-λ)  is the fraction shared by the manufacturer. Under the revenue-sharing contract θRS(w,λ), the retailer’s expected profit and the variance of profit are given as follows:
(26)Er(Q,p)=(λp-w)D(p)+(λp-w)Q-λ(p-v)∫0QF(ξ)dξ,(27)Vr(Q,p)=λ2(p-v)2η(Q).Proposition 7.
For a given revenue-sharing contractθRS(w,λ) offered by the manufacturer, the risk-neutral fashion retailer’s optimal joint decision (Qr*,pr*) is given by
(28)(λpr*-w)-λ(pr*-v)F(Qr*)=0,(29)λa-k(2λpr*-w)+λQr*-λ∫0Qr*F(ξ)dξ=0.Proof.
Similar to Proposition5.Remark 8.
Comparing (28) with (6) and (29) with (7), we find that (Qsc*,psc*) can be the risk-neutral retailer’s optimal ordering quantity and retail price if and only if w=λc<c, 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 risk-neutral retailer.


## 4.3. Two-Part Tariff Contract

With a two-part tariff contractθTPT(w,G), the fashion retailer gives the manufacturer a fixed transfer payment G>0 apart from the unit wholesale price for each unit ordered. And the retailer’s expected profit and the variance of profit are
(30)Er(Q,p)=(p-w)D(p)+(p-w)Q-(p-v)∫0QF(ξ)dξ-G,(31)Vr(Q,p)=E(∏r(Q,p)-Er(Q,p))2=(p-v)2η(Q).Proposition 9.
For a given two-part tariff contractθTPT(w,G) offered by the manufacturer, the risk-neutral fashion retailer’s optimal joint decision (Qr*,pr*) is given by
(32)(pr*-w)-(pr*-v)F(Qr*)=0,(33)a-k(2pr*-w)+Qr*-∫0Qr*F(ξ)dξ=0.Proof.
Similar  to Proposition5.Remark 10.
By comparing (32) with (6) and (33) with (7), it is easy to get w=c, such that the independent retailer’s optimal decisions (Qr*,pr*) are equal to the integrated fashion supply chain’s optimal solution (Qsc*,psc*). Hence, a two-part tariff contract θTPT(w,G) could perfectly achieve channel coordination for a fashion supply chain with risk-neutral retailer and price-dependent demand.
Now, by considering the risk aversion decision model, as given in(P-1), we establish the following propositions to attain the optimal joint decision for the risk-averse retailer under single contracts.Proposition 11.
Under single contracts, for any givenp∈[w,a/k], Vr(Q,p) is strictly increasing in Q. For any given Q≥0,Vr(Q,p) is strictly increasing in p.Proof.
From (17), (27), and (31), taking differentials of  Vr(Q,p) with respect to Q and p, and since dη(Q)/dQ=2(1-F(Q))∫0QF(ξ)dξ>0, it can be easily verified that, for any given p∈[w,a/k], Vr(Q,p) is strictly increasing in Q and, for any given Q≥0,  Vr(Q,p) is strictly increasing in p.Proposition 12.
Given the retailer’s expected thresholdkr≤E-r, the risk-averse fashion retailer’s optimal joint decision (Qr,mv*,pr,mv*) satisfies
(34)Er(Qr,mv*,pr,mv*)=kr,Qr,mv*≤Qr*,pr,mv*≤pr*.Proof.
From the proceeding analysis, we know that under single contracts, such as SRP contract, revenue-sharing contract and two-part tariff contract,Er(Q,p) is a concave function of Q and is unimodal in p∈[w,a/k]. Besides, Vr(Q,p) is strictly increasing in Q and p. Therefore, according to (P-1), the optimal pricing-inventory decision for the risk-averse fashion retailer is obtained by solving Er(Qr,mv*,pr,mv*)=kr. Moreover, since kr≤E-r, in each region of (Q≤Qr*,p≤pr*), (Q>Qr*,p≤pr*), (Q≤Qr*,p>pr*), and (Q>Qr*,p>pr*), there exists a corresponding decision pair (Q,p) that could make Er(Q,p)=kr established. Since Vr(Q,p) is strictly increasing in Q and p, the optimal solution (Qr,mv*,pr,mv*) for (P-1) can only fall in the region (Q≤Qr*,p≤pr*), otherwise, (Qr,mv*,pr,mv*) cannot be the risk-averse fashion retailer’s optimal joint decision. So we have Qr,mv*≤Qr* and pr,mv*≤pr*.Remark 13.
From Proposition12, we can know that the maximum expected profit  of  the risk-averse fashion retailer generated under single contracts is always no greater than that of a risk-neutral 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 price-dependent demand, the risk-averse fashion retailer tends to order less and charge a lower price in comparison with a risk-neutral retailer, which is consistent with the known results derived from the studies on joint pricing and inventory decisions of a risk-averse 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,(Qr,mv*,pr,mv*)=(Qsc*,psc*). Now we present the following proposition to explore the necessary conditions for a contract to achieve channel coordination.Proposition 14.
For any givenkr<E-sc, a contract achieves supply chain coordination if and only if the contract satisfies (1) Er(Qsc*,psc*)=kr; (2) ∂Er(Q,psc*)/∂Q|Q=Qsc*≥0; (3) ∂Er(Qsc*,p)/∂p|p=psc*≥0.Proof.
If a contract achieves supply chain coordination, then(Qr,mv*,pr,mv*)=(Qsc*,psc*) stands. According to Proposition 12, we have Er(Qsc*,psc*)=kr. On the other hand, since a contract coordinates the supply chain, from (P-1), we know that Er(Qsc*,psc*)≥kr. We know that Vr(Q,p) is strictly increasing in Q and p, and Er(Q,p) is a continuous function of Q and p. If Er(Qsc*,psc*)>kr establishes, then there always exists an optimal joint decision Q<Qsc* and p<psc* such that Er(Q,p)≥kr and Vr(Q,p)<Vr(Qsc*,psc*), which contradicts the fact that (Qsc*,psc*) is the optimal joint pricing and inventory decisions for the risk-averse fashion retailer. Therefore, we have Er(Qsc*,psc*)=kr.
If(Qr,mv*,pr,mv*)=(Qsc*,psc*), then according to Proposition 12, Qsc*≤Qr* and psc*≤pr*. Since Er(Q,p) is a concave function of Q and strictly increasing in Q∈(0,Qr*], from Qsc*≤Qr*, we have ∂Er(Q,psc*)/∂Q|Q=Qsc*≥0. Otherwise, if ∂Er(Q,psc*)/∂Q|Q=Qsc*<0, and, because for any given p∈[w,a/k], Vr(Q,p) is strictly increasing in Q, then there exists Q<Qsc* such that Er(Qsc*,psc*)<Er(Q,psc*) and Vr(Q,psc*)<Vr(Qsc*,psc*). It contradicts the fact that (Qsc*,psc*) is the optimal joint decision of the risk-averse fashion retailer. Similarly, Er(Q,p) is unimodal in p∈[w,a/k]; then, from psc*≤pr*, we have ∂Er(Qsc*,p)/∂p|p=psc*≥0. Otherwise, if ∂Er(Qsc*,p)/∂p|p=psc*<0 and because Vr(Q,p) is strictly increasing in p for any given Q≥0, then there exists p<psc* such that Er(Qsc*,psc*)<Er(Qsc*,p) and Vr(Qsc*,p)<Vr(Qsc*,psc*). It contradicts the fact that (Qsc*,psc*) is the optimal joint decision of the risk-averse fashion retailer. As a result, we have ∂Er(Q,psc*)/∂Q|Q=Qsc*≥0 and  ∂Er(Qsc*,p)/∂p|p=psc*≥0.Remark 15.
From Proposition14, it can be derived that when the supply chain is coordinated, the risk-averse fashion retailer’s expected profit is equal to kr, and hence the manufacturer’s expected profit is equal to E-sc-kr.
Next, we investigate in more detail whether the single contracts above could achieve supply chain coordination.Proposition 16.
For any givenkr<E-sc, SRP contract cannot achieve supply chain coordination.Proof.
From Proposition14, we can get that the supply chain achieves coordination if and only if SRP contract satisfies Er(Qsc*,psc*)=kr,∂Er(Q,psc*)/∂Q|Q=Qsc*≥0, and ∂Er(Qsc*,p)/∂p|p=psc*≥0.
From∂Er(Q,psc*)/∂Q|Q=Qsc*≥0, we can get (psc*-w)-(psc*-v)F(Qsc*)+uF-(Qsc*)≥0. And from (6), we have uF¯(Qsc*)≥w-c. From ∂Er(Qsc*,p)/∂p|p=psc*≥0, we have a-k(2psc*-w+u)+Qsc*-∫0Qsc*F(ξ)dξ≥0. And from (7), it can be obtained that u≤w-c. Since uF-(Qsc*)<u, there does not exist some value of u such that uF-(Qsc*)≥w-c and u≤w-c establish simultaneously. In other words, SRP contract θSRP(w,t,u) cannot achieve channel coordination.Proposition 17.
For any givenkr<E-sc, revenue-sharing contract and two-part 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 revenue-sharing contract,w=λc, λ=kr/E-sc;(2)
for two-part tariff contract,w=c, G=E-sc-kr.Proof.
According to Proposition14, for the revenue-sharing contract, from Er(Qsc*,psc*)=kr, we get the expression w=λpsc*-(kr+λ(psc*-v)∫0Qsc*F(ξ)dξ)/qsc*. From ∂Er(Q,psc*)/∂Q|Q=Qsc*≥0, we have λpsc*-w-λ(psc*-v)F(Qsc*)≥0. Substituting (6) into λpsc*-w-λ(psc*-v)F(Qsc*)≥0, it can be calculated that w≤λc. From ∂Er(Qsc*,p)/∂p|p=psc*≥0, we get λa-k(2λpsc*-w)+λQsc*-λ∫0Qsc*F(ξ)dξ≥0, and, taking (7) into it, we know that w≥λc. Combining w≤λc and w≥λc, we have w=λc, and, by taking w=λc into the expression of w, we have λ=kr/E-sc. Hence, revenue-sharing contract can still coordinate the supply chain, when the fashion retailer is risk averse.
Similarly, for two-part tariff contract, fromEr(Qsc*,psc*)=kr, we have w=psc*-(kr+(psc*-v)∫0Qsc*F(ξ)dξ+G)/qsc*. From ∂Er(Q,psc*)/∂Q|Q=Qsc*≥0, we get psc*-w-(psc*-v)F(Qsc*)≥0, and, from (6), w≤c is derived. From ∂Er(Qsc*,p)/∂p|p=psc*≥0, we get a-k(2psc*-w)+Qsc*-∫0Qsc*F(ξ)dξ≥0, and, from (7), we have w≥c. Thus, we have w=c. Comparing w=c and w=psc*-[(kr+(psc*-v)∫0Qsc*F(ξ)dξ+G)/qsc*], we can get G=E-sc-kr. Therefore, two-part tariff also could achieve supply chain coordination with risk sensitive retailer.Remark 18.
From Propositions16 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 revenue-sharing contract and two-part tariff contract (with two parameters) can.
From Proposition17, the values of λ and G can be regarded as indicators of the fashion retailer’s risk aversion level. Specifically, with a larger λ, the expected profit threshold of the retailer kr is greater, and the retailer is more risk averse. Contrarily, a larger value of G 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 similarlyQ=q-D(p), T=t-D(p), and T*=t-D(p*).


### 5.1. SRP with Revenue-Sharing Contract

Under this contract, the fashion retailer’s profit, expected profit, and the variance of profit are(35)∏r(Q,p)=(λp-w)D(p)+(λp-w)Q-λ(p-v)(Q-ξ)++u(min(ξ,Q)-T),(36)Er(Q,p)=(λp-w)D(p)+(λp-w)Q-λ(p-v)∫0QF(ξ)dξ+u(Q-T-∫0QF(ξ)dξ),(37)Vr(Q,p)=[λ(p-v)+u]2η(Q).Proposition 19.
For a given SRP with revenue-sharing contract offered by the manufacturer, the risk-neutral fashion retailer’s optimal joint decision(Qr*,pr*) is given by
(38)(λpr*-w+u)-[λ(pr*-v)+u]F(Qr*)=0,(39)λa-k(2λpr*-w+u)+λQr*-λ∫0Qr*F(ξ)dξ=0.Proof.
Similar to  Proposition5.Remark 20.
By comparing (39) with (7) and (38) with (6), we find that when u=0, w=λc, (Qr*,pr*)=(Qsc*,psc*) establishes. However, it contradicts the assumption of u>0 in SRP with revenue-sharing contract. Thus, when the fashion retailer is risk-neutral, SRP with revenue-sharing contract cannot achieve channel coordination.


### 5.2. SRP with Two-Part Tariff Contract

In this setting, the fashion retailer’s expected profit and the variance of profit are given by(40)Er(Q,p)=(p-w)D(p)+(p-w)Q-(p-v)∫0QF(ξ)dξ+u(Q-T-∫0QF(ξ)dξ)-G,Vr(Q,p)=(p-v+u)2η(Q).Proposition 21.
For a given SRP with two-part tariff contract offered by the manufacturer, the risk-neutral fashion retailer’s optimal joint decision(Qr*,pr*) is given by
(41)(pr*-w+u)-(pr*-v+u)F(Qr*)=0,(42)a-k(2pr*-w+u)+Qr*-∫0Qr*F(ξ)dξ=0.Remark 22.
Comparing (41) with (18) and (42) with (19), we discover that, under SRP with two-part tariff contract, the risk-neutral fashion retailer’s optimal decisions are equal to those under a single SRP contract. Therefore, consistent with the analysis in Section 4, SRP with two-part tariff contract cannot coordinate the supply chain with risk-averse retailer and price-dependent demand.


### 5.3. Revenue Sharing with Two-Part Tariff Contract

Under a revenue sharing with two-part tariff contract, the fashion retailer’s expected profit and the variance of profit are as follows:(43)Er(Q,p)=(λp-w)D(p)+(λp-w)Q-λ(p-v)∫0QF(ξ)dξ-G,Vr(Q,p)=λ2(p-v)2η(Q).Similar to SRP with two-part tariff contract, the optimal joint ordering-pricing decision for the risk-neutral fashion retailer under revenue sharing with two-part tariff contract is equal to that under a single revenue-sharing contract. Hence, revenue sharing with two-part tariff contract is able to achieve channel coordination in the fashion supply chain with risk-averse retailer and price-dependent demand.As a result, it only remains uncertain whether SRP with revenue-sharing contract could achieve supply chain coordination with risk-averse retailer. Following the similar approach as presented in Section4, we now investigate the role of SRP with revenue-sharing contract in channel coordination.From (37), by some simple deductions, we know that, under SRP with revenue-sharing contract, Vr(Q,p) is strictly increasing in Q and p. Therefore, with any given expected threshold kr≤E-r, the risk-averse retailer’s optimal joint decision (Qr,mv*,pr,mv*) satisfies (34).Proposition 23.
For any givenkr<E-sc, SRP with revenue-sharing contract cannot achieve supply chain coordination.Proof.
From Proposition14, we know that channel coordination is obtained if and only if SRP with revenue-sharing contract satisfies Er(Qsc*,psc*)=kr, ∂Er(Q,psc*)/∂Q|Q=Qsc*≥0, and ∂Er(Qsc*,p)/∂p|p=psc*≥0. From ∂Er(Q,psc*)/∂Q|Q=Qsc*≥0, we have (λpsc*-w)-λ(psc*-v)F(Qsc*)+uF-(Qsc*)≥0. Combining with (6), we can derive that uF-(Qsc*)≥w-λc. Nonetheless, from ∂Er(Qsc*,p)/∂p|p=psc*≥0, we get λa-k(2λpsc*-w+u)+λ(Qsc*-∫0Qsc*F(ξ)dξ)≥0, and, from (7), by some simplifications, we have u≤w-λc. Since  uF-(Qsc*)<u, there does not exist some value of u that could satisfy uF-(Qsc*)≥w-λc and u≤w-λc simultaneously. Thus, SRP with revenue-sharing contract cannot coordinate the fashion supply chain.Remark 24.
It is interesting to discover that, although a single revenue-sharing contract itself could coordinate the quantity and pricing decisions in the fashion supply chain with risk sensitive retailer, the combined SRP with revenue-sharing 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 revenue-sharing contract and a single two-part tariff contract can coordinate the supply chain with risk-averse retailer and price-dependent 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 revenue-sharing contract or two-part tariff contract, the resulting combined contract is still unable to coordinate the fashion supply chain.


## 5.1. SRP with Revenue-Sharing Contract

Under this contract, the fashion retailer’s profit, expected profit, and the variance of profit are(35)∏r(Q,p)=(λp-w)D(p)+(λp-w)Q-λ(p-v)(Q-ξ)++u(min(ξ,Q)-T),(36)Er(Q,p)=(λp-w)D(p)+(λp-w)Q-λ(p-v)∫0QF(ξ)dξ+u(Q-T-∫0QF(ξ)dξ),(37)Vr(Q,p)=[λ(p-v)+u]2η(Q).Proposition 19.
For a given SRP with revenue-sharing contract offered by the manufacturer, the risk-neutral fashion retailer’s optimal joint decision(Qr*,pr*) is given by
(38)(λpr*-w+u)-[λ(pr*-v)+u]F(Qr*)=0,(39)λa-k(2λpr*-w+u)+λQr*-λ∫0Qr*F(ξ)dξ=0.Proof.
Similar to  Proposition5.Remark 20.
By comparing (39) with (7) and (38) with (6), we find that when u=0, w=λc, (Qr*,pr*)=(Qsc*,psc*) establishes. However, it contradicts the assumption of u>0 in SRP with revenue-sharing contract. Thus, when the fashion retailer is risk-neutral, SRP with revenue-sharing contract cannot achieve channel coordination.


## 5.2. SRP with Two-Part Tariff Contract

In this setting, the fashion retailer’s expected profit and the variance of profit are given by(40)Er(Q,p)=(p-w)D(p)+(p-w)Q-(p-v)∫0QF(ξ)dξ+u(Q-T-∫0QF(ξ)dξ)-G,Vr(Q,p)=(p-v+u)2η(Q).Proposition 21.
For a given SRP with two-part tariff contract offered by the manufacturer, the risk-neutral fashion retailer’s optimal joint decision(Qr*,pr*) is given by
(41)(pr*-w+u)-(pr*-v+u)F(Qr*)=0,(42)a-k(2pr*-w+u)+Qr*-∫0Qr*F(ξ)dξ=0.Remark 22.
Comparing (41) with (18) and (42) with (19), we discover that, under SRP with two-part tariff contract, the risk-neutral fashion retailer’s optimal decisions are equal to those under a single SRP contract. Therefore, consistent with the analysis in Section 4, SRP with two-part tariff contract cannot coordinate the supply chain with risk-averse retailer and price-dependent demand.


## 5.3. Revenue Sharing with Two-Part Tariff Contract

Under a revenue sharing with two-part tariff contract, the fashion retailer’s expected profit and the variance of profit are as follows:(43)Er(Q,p)=(λp-w)D(p)+(λp-w)Q-λ(p-v)∫0QF(ξ)dξ-G,Vr(Q,p)=λ2(p-v)2η(Q).Similar to SRP with two-part tariff contract, the optimal joint ordering-pricing decision for the risk-neutral fashion retailer under revenue sharing with two-part tariff contract is equal to that under a single revenue-sharing contract. Hence, revenue sharing with two-part tariff contract is able to achieve channel coordination in the fashion supply chain with risk-averse retailer and price-dependent demand.As a result, it only remains uncertain whether SRP with revenue-sharing contract could achieve supply chain coordination with risk-averse retailer. Following the similar approach as presented in Section4, we now investigate the role of SRP with revenue-sharing contract in channel coordination.From (37), by some simple deductions, we know that, under SRP with revenue-sharing contract, Vr(Q,p) is strictly increasing in Q and p. Therefore, with any given expected threshold kr≤E-r, the risk-averse retailer’s optimal joint decision (Qr,mv*,pr,mv*) satisfies (34).Proposition 23.
For any givenkr<E-sc, SRP with revenue-sharing contract cannot achieve supply chain coordination.Proof.
From Proposition14, we know that channel coordination is obtained if and only if SRP with revenue-sharing contract satisfies Er(Qsc*,psc*)=kr, ∂Er(Q,psc*)/∂Q|Q=Qsc*≥0, and ∂Er(Qsc*,p)/∂p|p=psc*≥0. From ∂Er(Q,psc*)/∂Q|Q=Qsc*≥0, we have (λpsc*-w)-λ(psc*-v)F(Qsc*)+uF-(Qsc*)≥0. Combining with (6), we can derive that uF-(Qsc*)≥w-λc. Nonetheless, from ∂Er(Qsc*,p)/∂p|p=psc*≥0, we get λa-k(2λpsc*-w+u)+λ(Qsc*-∫0Qsc*F(ξ)dξ)≥0, and, from (7), by some simplifications, we have u≤w-λc. Since  uF-(Qsc*)<u, there does not exist some value of u that could satisfy uF-(Qsc*)≥w-λc and u≤w-λc simultaneously. Thus, SRP with revenue-sharing contract cannot coordinate the fashion supply chain.Remark 24.
It is interesting to discover that, although a single revenue-sharing contract itself could coordinate the quantity and pricing decisions in the fashion supply chain with risk sensitive retailer, the combined SRP with revenue-sharing 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 revenue-sharing contract and a single two-part tariff contract can coordinate the supply chain with risk-averse retailer and price-dependent 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 revenue-sharing contract or two-part 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 revenue-sharing contract, two-part tariff contract, and the combined revenue sharing with two-part 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 demanda=600 and the price elasticity of demand k=10. The random variable ξ follows a uniform distribution with a lower bound A=0 and an upper bound B=160. The unit production cost c=15, and the unit salvage value v=2. With these parameters, the optimal joint decision that maximizes the expected profit of the integrated fashion supply chain is Qsc*=106.74 and psc*=41.06, and the supply chain’s optimal production level is given by qsc*=296.18. The respective expected profit and the variance of profit for the fashion supply chain are E-sc=6326.70andVsc(Qsc*,psc*)=1931261.13. Since the expected profit threshold for the risk-averse retailer must be smaller than the maximum expected profit gained by the fashion supply chain, we assume kr<6326.70 in the following analysis.We consider six values ofkr=1000,2000,3000,4000,5000,and6000 to explore the optimal values of contract parameters for the coordinating contracts above. It should be noted that for the combined revenue sharing with two-part tariff contract, λ>kr/E-sc must establish to ensure that G>0. The results are summarized in Table 1.Table 1
Optimal values of contract parameters for different values ofkr.
k
r
Revenue-sharing contract
Two-part tariff contract
Revenue sharing with two-part tariff contract
w
λ
w
G
λ
w
G
1000
2.37
0.16
15
5326.70
0.2
3
265.34
2000
4.74
0.32
15
4326.70
0.4
6
530.68
3000
7.11
0.47
15
3326.70
0.5
7.5
163.35
4000
9.48
0.63
15
2326.70
0.7
10.5
428.69
5000
11.85
0.79
15
1326.70
0.8
12
61.36
6000
14.23
0.95
15
326.70
0.96
14.4
73.63From Table1, we can see the effectiveness of revenue sharing, two-part tariff, and their combined contract in coordinating the fashion supply chain. Consistent with Proposition 17, λ in revenue-sharing contract and G in two-part tariff contract can be used to judge the downstream fashion retailer’s risk aversion level. For revenue-sharing contract, with a larger kr, 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 w. For two-part tariff contract, a wholesale price w equaling to the unit production c gives the manufacturer zero profit, but the fixed transfer payment G ensures a positive profit for the manufacturer. And a higher value of G  which  the retailer is willing to pay indicates a less risk-averse 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 revenue-sharing 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 revenue-sharing contract with parameters such as base demanda, price elasticity of demand k, and demand uncertainty. The results are given in Table 2.Table 2
Sensitivity analysis for revenue-sharing contract.
Parameter
Q
sc
*
q
sc
*
p
sc
*
k
r
w
λ
E
-
sc
V
sc
V
r
a
400
87.47
180.69
30.68
2000
13.97
0.93
2147.18
676524.01
586954.45
500
98.67
239.54
35.91
3000
11.31
0.75
3977.72
1236932.08
703589.94
600
106.74
296.18
41.06
5000
11.85
0.79
6362.70
1931261.13
1206220.67
700
112.89
351.36
46.15
8000
13.06
0.87
9187.50
2751202.56
2085967.18
800
117.74
405.53
51.22
10000
11.95
0.80
12556.40
3691564.92
2341424.45
900
121.67
458.97
56.27
14000
12.78
0.85
16431.10
4748912.61
3447602.58
1000
124.93
511.85
61.31
20000
14.42
0.96
20810.09
5920873.15
5468870.88
k
10
106.74
296.18
41.06
200
0.47
0.03
6362.70
1931261.13
1929.95
15
84.57
240.96
29.57
200
1.04
0.07
2895.46
578223.79
2758.79
20
64.53
188.77
23.79
200
2.18
0.15
1375.39
185400.79
3920.33
25
46.29
138.99
20.29
200
4.89
0.33
613.05
54132.63
5761.4
30
29.57
91.15
17.95
200
13.33
0.89
225.07
11800.57
9317.77
CV
15%
118.25
290.64
42.76
4000
8.15
0.53
7537.51
269951.09
76023.73
35%
255.40
383.92
47.15
4000
6.30
0.42
9522.90
6935423.41
1223642.39
55%
296.77
424.47
47.23
4000
6.62
0.44
9059.38
18604230.72
3626893.02
75%
94.84
291.40
40.34
4000
10.12
0.67
5930.02
2154476.37
980278.33
95%
45.88
258.58
38.73
4000
11.21
0.75
5354.43
649949.19
362321.17From Table2, we find that, with the increase of base demand a, the optimal production quantity qsc* and pricing psc* for the integrated fashion supply chain also increase, leading to larger expected profit E-sc and the variance of profit Vsc. This is consistent with Proposition 3 that Esc is a concave function of q and is unimodal in p, and Vsc is strictly increasing in q and p. Taking into account kr<E-sc, we consider the appropriate values of  kr and find that the optimal values of w and Vr do not exhibit some rule of changes since λ changes randomly. However, when we fix the value of kr in the region kr<E-sc for all cases of a, we could intuitively reach the conclusion that the values of  λ,w, and Vr tend to decrease.However, it can be found from Table2 that with larger values of price elasticity k, the entire supply chain’s optimal production quantity and retail price become smaller, so do the supply chain’s expected profit E-sc and the variance of profit Vsc. On the contrary, by fixing values of kr subject to kr<E-sc, we discover that the values of w,λ, and Vr all become larger.In addition, we also try to illustrate the effect of different degrees of demand uncertainty. We define demand uncertainty asCV=θ/δ, where δ represents the mean and θ 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 E-sc and the variance of profit Vsc also have the same rule of changes. With respect to the values of w and λ, they change toward the opposite direction. The combined changes of λ and Vsc cause the changes of Vr.Similarly, following the same method, we could also get the results of sensitivity analysis for the other two coordinating contracts—two-part tariff contract and revenue sharing with two-part tariff contract. They are summarized in Tables3 and 4, respectively.Table 3
Sensitivity analysis for two-part tariff contract.
Parameter
k
r
w
G
V
r
a
400
2000
15
147.18
676524.01
500
3000
15
977.72
1236932.08
600
5000
15
1326.70
1931261.13
700
8000
15
1187.50
2751202.56
800
10000
15
2556.40
3691564.92
900
14000
15
2431.11
4748912.61
1000
20000
15
810.09
5920873.15
k
10
200
15
6126.70
1931261.13
15
200
15
2659.46
578223.79
20
200
15
1175.39
185400.79
25
200
15
413.05
54132.63
30
200
15
25.07
11800.57
CV
15%
4000
15
3537.51
269951.09
35%
4000
15
5522.90
6935423.41
55%
4000
15
5059.38
18604230.72
75%
4000
15
1930.02
2154476.37
95%
4000
15
1354.43
649949.19Table 4
Sensitivity analysis for revenue-sharing with two-part tariff contract.
Parameter
k
r
λ
w
G
V
r
a
400
2000
0.95
14.25
39.83
610562.93
500
3000
0.8
12
182.18
791636.53
600
5000
0.8
12
61.36
1236007.12
700
8000
0.9
13.5
268.75
2228474.07
800
10000
0.85
12.75
672.94
2667155.65
900
14000
0.9
13.5
787.99
3846619.22
1000
20000
0.98
14.7
393.89
5686406.57
k
10
200
0.1
1.5
432.67
19312.61
15
200
0.2
3
379.09
23128.95
20
200
0.4
6
350.15
29664.13
25
200
0.7
10.5
229.13
26524.99
30
200
0.9
13.5
2.56
9558.46
CV
15%
4000
0.7
10.5
1276.26
132276.03
35%
4000
0.6
9
1713.74
2496752.43
55%
4000
0.5
7.5
529.69
4651057.68
75%
4000
0.8
12
744.02
1378864.88
95%
4000
0.9
13.5
818.99
526458.84From Table3, it can be easily discovered that no matter how the values of parameter a,k, and CV change, the optimal values of w are always equal to 15. But the optimal values of G are dependent upon the change in values of those parameters. Specifically, the optimal G equals E-sc-kr. When parameters k and CV change, the integrated fashion supply chain’s optimal expected profit E-sc also change as shown in Table 2. If the values of kr are fixed, G changes positively with the changes of E-sc. That is, the values of G decrease in the case of kr and firstly increase and then decrease in the case of CV.Similar analysis could be realized for the joint revenue sharing with two-part tariff contract. What is worth noting here is that, in Table4, we could find that the values of λ are larger than the corresponding values in Table 2 for the single revenue-sharing contract, whereas the values of G are smaller than the corresponding values in the single two-part 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 Vr in revenue sharing with two-part tariff contract are always larger than the respective values in revenue-sharing contract alone, while they obtain their largest values in two-part tariff contract since they are equal to the according values of Vsc.


## 6.1. Determine the Values of Contract Parameters

First, we give the values of parameters used in this section. Suppose the base demanda=600 and the price elasticity of demand k=10. The random variable ξ follows a uniform distribution with a lower bound A=0 and an upper bound B=160. The unit production cost c=15, and the unit salvage value v=2. With these parameters, the optimal joint decision that maximizes the expected profit of the integrated fashion supply chain is Qsc*=106.74 and psc*=41.06, and the supply chain’s optimal production level is given by qsc*=296.18. The respective expected profit and the variance of profit for the fashion supply chain are E-sc=6326.70andVsc(Qsc*,psc*)=1931261.13. Since the expected profit threshold for the risk-averse retailer must be smaller than the maximum expected profit gained by the fashion supply chain, we assume kr<6326.70 in the following analysis.We consider six values ofkr=1000,2000,3000,4000,5000,and6000 to explore the optimal values of contract parameters for the coordinating contracts above. It should be noted that for the combined revenue sharing with two-part tariff contract, λ>kr/E-sc must establish to ensure that G>0. The results are summarized in Table 1.Table 1
Optimal values of contract parameters for different values ofkr.
k
r
Revenue-sharing contract
Two-part tariff contract
Revenue sharing with two-part tariff contract
w
λ
w
G
λ
w
G
1000
2.37
0.16
15
5326.70
0.2
3
265.34
2000
4.74
0.32
15
4326.70
0.4
6
530.68
3000
7.11
0.47
15
3326.70
0.5
7.5
163.35
4000
9.48
0.63
15
2326.70
0.7
10.5
428.69
5000
11.85
0.79
15
1326.70
0.8
12
61.36
6000
14.23
0.95
15
326.70
0.96
14.4
73.63From Table1, we can see the effectiveness of revenue sharing, two-part tariff, and their combined contract in coordinating the fashion supply chain. Consistent with Proposition 17, λ in revenue-sharing contract and G in two-part tariff contract can be used to judge the downstream fashion retailer’s risk aversion level. For revenue-sharing contract, with a larger kr, 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 w. For two-part tariff contract, a wholesale price w equaling to the unit production c gives the manufacturer zero profit, but the fixed transfer payment G ensures a positive profit for the manufacturer. And a higher value of G  which  the retailer is willing to pay indicates a less risk-averse 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 revenue-sharing 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 revenue-sharing contract with parameters such as base demanda, price elasticity of demand k, and demand uncertainty. The results are given in Table 2.Table 2
Sensitivity analysis for revenue-sharing contract.
Parameter
Q
sc
*
q
sc
*
p
sc
*
k
r
w
λ
E
-
sc
V
sc
V
r
a
400
87.47
180.69
30.68
2000
13.97
0.93
2147.18
676524.01
586954.45
500
98.67
239.54
35.91
3000
11.31
0.75
3977.72
1236932.08
703589.94
600
106.74
296.18
41.06
5000
11.85
0.79
6362.70
1931261.13
1206220.67
700
112.89
351.36
46.15
8000
13.06
0.87
9187.50
2751202.56
2085967.18
800
117.74
405.53
51.22
10000
11.95
0.80
12556.40
3691564.92
2341424.45
900
121.67
458.97
56.27
14000
12.78
0.85
16431.10
4748912.61
3447602.58
1000
124.93
511.85
61.31
20000
14.42
0.96
20810.09
5920873.15
5468870.88
k
10
106.74
296.18
41.06
200
0.47
0.03
6362.70
1931261.13
1929.95
15
84.57
240.96
29.57
200
1.04
0.07
2895.46
578223.79
2758.79
20
64.53
188.77
23.79
200
2.18
0.15
1375.39
185400.79
3920.33
25
46.29
138.99
20.29
200
4.89
0.33
613.05
54132.63
5761.4
30
29.57
91.15
17.95
200
13.33
0.89
225.07
11800.57
9317.77
CV
15%
118.25
290.64
42.76
4000
8.15
0.53
7537.51
269951.09
76023.73
35%
255.40
383.92
47.15
4000
6.30
0.42
9522.90
6935423.41
1223642.39
55%
296.77
424.47
47.23
4000
6.62
0.44
9059.38
18604230.72
3626893.02
75%
94.84
291.40
40.34
4000
10.12
0.67
5930.02
2154476.37
980278.33
95%
45.88
258.58
38.73
4000
11.21
0.75
5354.43
649949.19
362321.17From Table2, we find that, with the increase of base demand a, the optimal production quantity qsc* and pricing psc* for the integrated fashion supply chain also increase, leading to larger expected profit E-sc and the variance of profit Vsc. This is consistent with Proposition 3 that Esc is a concave function of q and is unimodal in p, and Vsc is strictly increasing in q and p. Taking into account kr<E-sc, we consider the appropriate values of  kr and find that the optimal values of w and Vr do not exhibit some rule of changes since λ changes randomly. However, when we fix the value of kr in the region kr<E-sc for all cases of a, we could intuitively reach the conclusion that the values of  λ,w, and Vr tend to decrease.However, it can be found from Table2 that with larger values of price elasticity k, the entire supply chain’s optimal production quantity and retail price become smaller, so do the supply chain’s expected profit E-sc and the variance of profit Vsc. On the contrary, by fixing values of kr subject to kr<E-sc, we discover that the values of w,λ, and Vr all become larger.In addition, we also try to illustrate the effect of different degrees of demand uncertainty. We define demand uncertainty asCV=θ/δ, where δ represents the mean and θ 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 E-sc and the variance of profit Vsc also have the same rule of changes. With respect to the values of w and λ, they change toward the opposite direction. The combined changes of λ and Vsc cause the changes of Vr.Similarly, following the same method, we could also get the results of sensitivity analysis for the other two coordinating contracts—two-part tariff contract and revenue sharing with two-part tariff contract. They are summarized in Tables3 and 4, respectively.Table 3
Sensitivity analysis for two-part tariff contract.
Parameter
k
r
w
G
V
r
a
400
2000
15
147.18
676524.01
500
3000
15
977.72
1236932.08
600
5000
15
1326.70
1931261.13
700
8000
15
1187.50
2751202.56
800
10000
15
2556.40
3691564.92
900
14000
15
2431.11
4748912.61
1000
20000
15
810.09
5920873.15
k
10
200
15
6126.70
1931261.13
15
200
15
2659.46
578223.79
20
200
15
1175.39
185400.79
25
200
15
413.05
54132.63
30
200
15
25.07
11800.57
CV
15%
4000
15
3537.51
269951.09
35%
4000
15
5522.90
6935423.41
55%
4000
15
5059.38
18604230.72
75%
4000
15
1930.02
2154476.37
95%
4000
15
1354.43
649949.19Table 4
Sensitivity analysis for revenue-sharing with two-part tariff contract.
Parameter
k
r
λ
w
G
V
r
a
400
2000
0.95
14.25
39.83
610562.93
500
3000
0.8
12
182.18
791636.53
600
5000
0.8
12
61.36
1236007.12
700
8000
0.9
13.5
268.75
2228474.07
800
10000
0.85
12.75
672.94
2667155.65
900
14000
0.9
13.5
787.99
3846619.22
1000
20000
0.98
14.7
393.89
5686406.57
k
10
200
0.1
1.5
432.67
19312.61
15
200
0.2
3
379.09
23128.95
20
200
0.4
6
350.15
29664.13
25
200
0.7
10.5
229.13
26524.99
30
200
0.9
13.5
2.56
9558.46
CV
15%
4000
0.7
10.5
1276.26
132276.03
35%
4000
0.6
9
1713.74
2496752.43
55%
4000
0.5
7.5
529.69
4651057.68
75%
4000
0.8
12
744.02
1378864.88
95%
4000
0.9
13.5
818.99
526458.84From Table3, it can be easily discovered that no matter how the values of parameter a,k, and CV change, the optimal values of w are always equal to 15. But the optimal values of G are dependent upon the change in values of those parameters. Specifically, the optimal G equals E-sc-kr. When parameters k and CV change, the integrated fashion supply chain’s optimal expected profit E-sc also change as shown in Table 2. If the values of kr are fixed, G changes positively with the changes of E-sc. That is, the values of G decrease in the case of kr and firstly increase and then decrease in the case of CV.Similar analysis could be realized for the joint revenue sharing with two-part tariff contract. What is worth noting here is that, in Table4, we could find that the values of λ are larger than the corresponding values in Table 2 for the single revenue-sharing contract, whereas the values of G are smaller than the corresponding values in the single two-part 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 Vr in revenue sharing with two-part tariff contract are always larger than the respective values in revenue-sharing contract alone, while they obtain their largest values in two-part tariff contract since they are equal to the according values of Vsc.


## 7. Management Insights and Concluding Remark

In this paper, the issue of supply chain coordination with risk-averse retailer and price-dependent demand is studied. We extend in this paper the previous works to consider a risk-averse retailer in a two-stage fashion supply chain. Adopting the additive price-sensitive 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 revenue-sharing contract and two-part tariff contract in supply chains with risk-neutral 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 two-part 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. Mean-variance 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 risk-averse retailers would be a meaningful direction and could produce more insights.



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*Source: 259164-2013-02-25.xml*