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Volume 10, Issue 1, 2017, Pages 986 - 1001

Decentralized Channel Decisions of Green Supply Chain in a Fuzzy Decision Making Environment

Authors
Shengju Sang*, sangshengju@163.com
Department of Economics and Management, Heze University, No.2269 University Road, Heze, Shandong, 274015, China†
*

Department of Economics and Management, Heze University, Heze, Shandong, China

†

China, Heze University, E-mail: sangshengju@163.com

Received 20 November 2016, Accepted 19 June 2017, Available Online 4 July 2017.
DOI
10.2991/ijcis.2017.10.1.66How to use a DOI?
Keywords
Green supply chain; Game theory; Fuzzy theory; Fuzzy variables
Abstract

This paper considers the greening policies in a decentralized channel between one manufacturer and one retailer in a fuzzy decision making environment. We consider the manufacturing cost and the parameters of demand function as the fuzzy variables. Based on the different market structures, we develop three different fuzzy decentralized decision models. For each case, the expected value, optimistic value and pessimistic value models are formulated, and their optimal solutions are also derived through the fuzzy set theory. Finally, three numerical examples are solved to examine the effectiveness of fuzzy models. The effects of the confidence level of the supply chain member’s profits and the fuzziness of parameters on optimal prices, level of green innovation, and fuzzy expected profits of actors are also analyzed.

Copyright
© 2017, the Authors. Published by Atlantis Press.
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

In a traditional supply chain, chain members usually focus on their total costs and profits, and ignore their operations impacted on environment. This phenomenon is altering rapidly as the problems of environment are affecting the population’s living conditions of the world more severely. With the development of the green economy and low-carbon economy, more and more scholars and market administrators have applied the green principles and techniques to develop and solve the green supply chain (GSC) management problems.

Recently, some studies have been done on analyzing the game theoretic models in GSC management. Sheu1 considered the nuclear power generation problem by using a multi-objective optimization programming approach in GSC management. Using the method of asymmetrical Nash bargaining game and backward induction, Sheu2 analyzed the negotiations problem with government intervention between the GSC members. Ghosh and Shah3 developed a game theoretic model for analyzing the influence of channel structures on greening levels, prices and profits in both cooperation and non-cooperation situations. Ghosh and Shan4 also discussed the coordination issues of GMC by using the cost sharing contract. By using evolutionary dynamics, Barari et al.5 framed integrated and holistic conceptual framework on maximizing the total supply chain’s profits that took the practical aspects into account in a GSC. Swami and Shah6 established a two-part tariff contract for coordinating GSC, where the models contained price and non-price variables. Similar issues was studied by Swami and Shah7, they studied the coordination mechanism of GSC that took the shelf-space allocation of products into consideration. Mirzapour Al-e-hashem8 proposed a stochastic programming model to investigate the production–distribution planning system after demand fluctuation in a GSC. Tomasin et al.9 used multiple case studies to investigate the elements that contribute to increase sales of green products in Brazil. Zhang and Liu10 proposed three types of coordination mechanisms to coordinate the three stage green supply chain.

The works mentioned above discussed the game theoretic models of green supply chain with deterministic market demands and production costs. However, in real world, there exist many in deterministic factors for some new green products. The production costs and market demands of these green products are often lack of historical data. Moreover, with the market uncertainty, the exact data are derived more and more unavailable due to the short of product life cycles. Thus, the fuzzy set theory, instead of probability theory is fit well for dealing with greening policy problem. In recent years, fuzzy sets theory is used by many researches to deal with the modes of supply chain. For instance, Xu and Zhai11, 12 presented fuzzy newsboy models in which the market demands were fuzzy. Zhou et al.13 investigated the operations between one-manufacturer and one-retailer, where the customer’s demand was fuzziness. Yu and Jin14 studied the decentralized channel, integrated channel and return contract models with fuzzy retail prices and demands. Wei and Zhao15 proposed fuzzy models with pricing competition of two retailers in a closed-loop supply chain. Ye and Li16 investigated the Stackelberg models in which the demand was fuzzy. In addition, Zhao et al.17 studied the pricing competitive strategies in which one manufacturer sold his substitutable products to two competing retailers with fuzzy costs and demands. Zhao et al.18 also studied pricing competition problems with two manufacturers under fuzzy demand environments. Wei and Zhao19 discussed the problems of pricing decisions in a reverse channel. Zhao and Wang20 studied service and price competition problem with two competitive retailers in a distribution system with fuzzy demand. Yu et al.21 developed fuzzy newsboy models to obtain the optimal prices between the supplier and the retailer. Recently, Sang22 investigated the coordination mechanism in a multiple supply chain with fuzzy demands and costs. Chang and Yeh23 analyzed the decentralized and the centralized supply chain system with fuzzy demand under a return policy, and showed that the fuzziness of demand affected the optimal results of the supply chain members. Khamseh et al.24 proposed four different pricing models of complementary products with two competing manufacturers in fuzzy environments. Sang25 proposed one expected value model and two chance-constrained programming models between two competing manufacturers and one common retailer under fuzzy uncertainty.

To the best of our knowledge, there is no work that deals with the optimal decisions of GSC in a fuzzy decision environment. Therefore, in this paper, we will examine how the retailer and the manufacturer should make their own pricing and green level decisions in a fuzzy decision environment. We also explore the impacts of the confidence level and the fuzzy degree of parameters on the equilibrium prices, green level and profits of the supply chain members.

The paper makes three contributions to the extant literature. Firstly, our proposed models extend the study of Ghosh and Shah3 by considering the greening policies under fuzzy uncertainty. The manufacturing costs and market demand are all fuzzy. Secondly, we apply the different ranking measures of fuzzy variables to reflect the attitudes of the participants. They can choose the expected values model to derive their optimal decisions if they are risk neutral. They can choose the optimistic values model to derive their optimal decisions if they are risk preferable. And they can choose the pessimistic values model to derive their optimal decisions if they are risk averse. Thirdly, compared to the method used in conventional environment, our work has some main findings: the profit margin, wholesale price, retail price and level of green innovation are higher, while the manufacturer’s profit is lower, and the retailer’s profit and supply chain system’s profit are higher under fuzzy uncertainly.

The rest of this paper is as follows. Firstly, some useful concepts and propositions about fuzzy set theory are presented in Section 2. Section 3 introduces the notations of the models. Three fuzzy green supply chain models are proposed in Section 4. In Section 5, three numerical examples are given to elucidate the solutions of each model. Section 6 summarizes the work.

2. Preliminaries

The possibility measure Pos was introduced by Nahmias26 and three axioms were given:

Axiom 1.

Pos(Θ) = 1, for the nonempty set Θ.

Axiom 2.

Pos(ϕ) = 0, for the empty set ϕ.

Axiom 3.

Pos(∪i=1mAi)=sup1≤i≤m Pos(Ai), for any events Ai in P(Θ), where P denotes the power set of Θ.

Definition 1

(Liu and Liu27). If Pos{ξ ≤ 0} = 0, then the fuzzy variable ξ is called positive.

Definition 2

(Liu and Liu27). Let ξ be a fuzzy variable. Then, for any α ∈ (0,1], ξαR=sup{r|Pos{ξ≥r}≥α} and ξαL=inf {r|Pos{ξ≤r}≥α} are said to be the α-optimistic value and the α-pessimistic value of ξ, respectively.

Example 1.

Let ξ be a triangular fuzzy variable with ξ = (a, b, c), then ξαR and ξαL are given as

ξαR=αb+(1−α)c and ξαL=αb+(1−α)a.

Lemma 1

(Liu and Liu27 and Zhao et al.28). Let ξ and η be positive independent fuzzy variables. Then, for any α ∈ (0,1]

  1. (a)

    (ξ+η)αR=ξαR+ηαR and (ξ+η)αL=ξαL+ηαL;

  2. (b)

    (ξ⋅η)αR=ξαR⋅ηαR and (ξ⋅η)αL=ξαL⋅ηαL;

  3. (c)

    (ξ−η)αR=ξαR−ηαL and (ξ−η)αL=ξαL−ηαR.

Lemma 2.

(Liu and Liu27). Let ξ be a fuzzy variable with finite expected value, then

E[ξ]=12∫01(ξαL+ξαR)d α.

Lemma 3

(Liu and Liu27). Let ξ and η be independent fuzzy variables with finite expected values. Then for any numbers m and n

E[mξ+nη]=mE[ξ]+nE[η].

Definition 2.

Let ξ and η be positive independent fuzzy variables, ξ > η if and only if for any α ∈ (0,1], ξαL>ηαL and ξαR>ηαR.

Definition 3.

Let ξ and η be positive independent fuzzy variables, if ξ > η, then E[ξ] > E[η].

3. Problem Descriptions

Consider a GSC with a manufacturer and a retailer, where the retailer orders greening products from the manufacturer, and then he retails them to end customers. The manufacturer is assumed to produce only one product and the retailer sells only single product.

Similar to Ghosh and Shah3, the market demand function faced by the manufacturer and the retailer is considered as a linear form of the retail price p and the level of greening innovation θ, the market demand is

q˜=D˜−β˜ p+γ˜θ.
where the fuzzy parameter D˜ denotes the market potential, the fuzzy parameter β˜ denotes the retail price sensitivity of the customer, and the fuzzy parameter γ˜ denotes the greening innovation sensitivity of the manufacturer to the demand. The fuzzy parameters D˜, β˜ and γ˜ are positive and mutually independent.

Further, let w denote the unit wholesale price of greening product, c˜ the unit fuzzy producing cost of greening product for the manufacturer and m the unit margin profit of product for the retailer. As the retail price p is the sum of margin profit m and wholesale price w, we consider retail price as p = m + w. The fuzzy demand function of the greening product is presented as follows

q˜=D˜−β˜(m+w)+γ˜θ.

It is assumed that the marginal cost of the manufacturer is not affected by the greening innovation. In addition, for achieving greening innovation, it needs fixed investment. The cost of the fixed investment is assumed as a quadratic function of the parameter θ, and is expressed as Ĩθ2, where Ĩ denotes fuzzy fixed investment coefficient. The demand is positive in the real world. Thus we have Pos{D˜−β˜(m+w)+γ˜θ<0}=0, where Pos is a possibility measure. The order quantity of the greening product can be presented as q=E[q˜].

The fuzzy profit functions of the manufacturer and the retailer can be derived as

∏M=(w−c˜)(D˜−β˜(m+w)+γ˜θ)−I˜θ2.
∏R=m(D˜−β˜(m+w)+γ˜θ).

4. Model Analysis

In this section, we will discuss the operations of the supply chain participants in a GSC, and examine the manufacturer and the retailer how to set their optimal policies with different power structures under fuzzy uncertainty. We study the equilibrium decisions under three non-cooperative games in the decentralized channel: the manufacturer dominates the channel (Manufacturer-Stackelberg game), the retailer leads the channel (Retailer-Stackelberg game), and the supply chain participants have an equal bargaining power (Vertical-Nash game).

4.1. Manufacturer Stackelberg (MS) game

In the MS game, the retailer has less bargaining power than the manufacturer. That is to say, the manufacturer is the leader in the GSC. Firstly, the manufacturer sets his wholesale price w and level of greening innovation θ condition on the retailer’s optimal reaction to his decisions. Then, the retailer sets is own margin profit m. Hence, in this case, we can formulate the fuzzy optimal model as follows

{maxw,θ E[∏M(w,θ,m*)]=E[(w−c˜)(D˜−β˜(m+w)+γ˜θ)−I˜θ2]s. t.Pos {w−c˜<0}=0where m*derives from problem:        {maxm E[∏R(m)]=E[m(D˜−β˜(m+w)+γ˜θ)]s. t.Pos{D˜−β˜(m+w)+γ˜θ<0}=0
where E[∏M (w, θ, m*)] and E[∏R (m)] represent the fuzzy expected profit of the manufacturer and the retailer.

We first solve the reaction function of the retailer.

Theorem 1.

If Pos{D˜−β˜(m*+w)+γ˜θ<0}=0, then the retailer’s optimal reaction function is given as

m*(w,θ)=E[D˜]−E[β˜]w+E[γ˜]θ2E[β˜].

Proof.

The fuzzy expected profit E[∏R (m)] is

E[∏R(m)]=−E[β˜]m2+(E[D˜]−E[β˜]w+E[γ˜]θ)m.

  • The first order condition

    dd mE[∏R(m)]=−2E[β˜]m+E[D˜]−E[β˜]w+E[γ˜]θ.

  • The second order condition

    d2d m2E[∏R(m)]=−2E[β˜]<0.

Thus, the retailer’s fuzzy expected profit E[∏R (m)] is concave in m.

Let the first order condition be zero, the result (6) can be derived.

Theorem 1 is proved.

Here, we consider m* (w, θ) as the optimal response to the decisions of the manufacturer. Having the follower’s information about the decision, the leader sets his optimal decisions. So, the following Proposition can be derived.

Theorem 2.

If (E[γ˜])2<8E[β˜]E[I˜], Pos{w*−c˜<0}=0 and Pos{D˜−β˜(m*+w*)+γ˜θ*<0}=0, then the optimal equilibrium decisions are

m*=E[γ˜](2E[β˜c˜]E[γ˜]−E[β˜]∫01(γ˜αLc˜αR+γ˜αRc˜αL)d α)2E[β˜](8E[β˜]E[I˜]−(E[γ˜])2) +2E[I˜](E[D˜]−E[β˜c˜])8E[β˜]E[I˜]−(E[γ˜])2.
w*=E[γ˜](E[β˜c˜]E[γ˜]−E[β˜]∫01(γ˜αLc˜αR+γ˜αRc˜αL)d α)E[β˜](8E[β˜]E[I˜]−(E[γ˜])2) +4E[I˜](E[D˜]+E[β˜c˜])8E[β˜]E[I˜]−(E[γ˜])2.
θ*=E[γ˜](E[D˜]+3E[β˜c˜])−2E[β˜]∫01(γ˜αLc˜αR+γ˜αRc˜αL)d α8E[β˜]E[I˜]−(E[γ˜])2.

Proof.

The fuzzy expected profit E[∏M] is

E[∏M]=12∫01((w−c˜)(D˜−β˜(m+w)+γ˜θ)−I˜θ2)αLd α +12∫01((w−c˜)(D˜−β˜(m+w)+γ˜θ)−I˜θ2)αRd α=−E[β˜]w2−E[I˜]θ2+E[γ˜]wθ         +(E[D˜]+E[β˜c˜]−E[β˜]m)w         −12θ∫01(γ˜αLc˜αR+γ˜αRc˜αL)d α+E[β˜c˜]m         −12∫01(D˜αLc˜αR+D˜αRc˜αL)d α.

Substituting m in Eq.(6) into Eq.(11), the first order conditions can be obtained as

∂∂wE[∏M]=−E[β˜]w+12E[γ˜]θ+12(E[D˜]+E[β˜c˜]).∂∂θE[∏M]=−2E[I˜]θ+12E[γ˜]w+E[β˜c˜]E[γ˜]2E[β˜]   −12∫01(γ˜αLc˜αR+γ˜αRc˜αL)d α.

Therefore, the Hessian matrix of E[∏M] is

H=[−E[β˜]12E[γ˜]12E[γ˜]−2E[I˜]].

Since E[β˜]>0, E[I˜]>0 and (E[γ˜])2<8E[β˜]E[I˜], thus H is a definite negative matrix and E[∏M] is jointly concave in w and θ.

Let the first order conditions be zero, we can have Eqs.(9) and (10).

Substituting w* in Eq. (9) and θ* in Eq. (10) into Eq.(6), we can get Eq.(8).

Theorem 2 is proved.

The chance-constrained programming proposed by Liu and Iwamura29, 30, has a big role in formulating fuzzy models. Given a confidence level α, the decision makers try to optimize the critical value subject to chance constraint. Hence, in this case, we can formulate the maximax chance-constrained programming model as follows

{maxw,θ ∏¯Ms. t.Pos{(w−c˜)(D˜−β˜(m*+w)+γ˜θ)−I˜θ2≥∏¯M}≥αPos{w−c˜<0}=0where m*derives from problem:         {maxm ∏¯Rs. t.Pos{m(D˜−β˜(m+w)+γ˜θ)≥∏¯R}≥αPos{D˜−β˜(m+w)+γ˜θ<0}=0.
where α is a given confidence level for manufacturer’s and retailer’s profits, ∏¯R is the maximum value that ∏R (m) obtains with at least possibility α, and ∏¯M is the maximum value that ∏M (w, θ, m*) obtains with at least possibility α. Therefore, model (12) equals to the following model (13), in which supply chain actors attempt to maximize the α-optimistic value of profits (∏M(w,θ,m*))αR and (∏R(m))αR by choosing the optimal solutions, that is
{maxw,θ(∏M(w,θ,m*))αR=((w−c˜)(D˜−β˜(m*+w)+γ˜θ)−I˜θ2)αRs. t.Pos{w−c˜<0}=0 where m*derives from problem:        {maxm (∏R(m))αR=(m(D˜−β˜(m+w)+γ˜θ))αRs. t.Pos{D˜−β˜(m+w)+γ˜θ<0}=0.

Theorem 3.

Let (∏R(m))αR be the retailer’s α-optimistic value of profit. The wholesale price w and the level of greening innovation θ set by manufacturer are fixed. If Pos{D˜−β˜(m*+w)+γ˜θ<0}=0, then optimal reaction function is given as

m*(w,θ)=D˜αR−β˜αLw+γ˜αRθ2β˜αL.

Proof.

The retailer’s profit is

(∏R(m))αR=−β˜αLm2+(D˜αR−β˜αLw+γ˜αRθ)m.

  • The first order condition

    dd m(∏R(m))αR=−2β˜αLm+D˜αR−β˜αLw+γ˜αRθ.

  • The second order condition

    d2d m2(∏R(m))αR=−2β˜αL<0.

Thus, for any α ∈ (0,1], (∏R(m))αR is concave in m.

Let dd m(∏R(m))αR=0, we can get the retailer’s optimal margin profit as in Eq. (14).

Theorem 3 is proved.

Theorem 4.

Let (∏M(w,θ,m*))αR be the manufacturer’s α-optimistic value of profit. If (γ˜αR)2<8β˜αLI˜αL, Pos{D˜−β˜(m*+w*)+γ˜θ*<0}=0 and Pos{w*−c˜<0}=0, then the optimal equilibrium decisions are

m*=2I˜αL(D˜αR−β˜αLc˜αL)8β˜αLI˜αL−(γ˜αR)2.
w*=4I˜αL(D˜αR+β˜αLc˜αL)−(γ˜αR)2c˜αL8β˜αLI˜αL−(γ˜αR)2.
θ*=γ˜αR(D˜αR−β˜αLc˜αL)8β˜αLI˜αL−(γ˜αR)2.

Proof.

The manufacturer’s profit is

(∏M(w,θ,m*))αR=((w−c˜)(D˜−β˜(m*+w)+γ˜θ)−I˜θ2)αR=(w−c˜αL)(D˜αR−β˜αL(m*+w)+γ˜αRθ)−I˜αLθ2.

Substituting m* (w, θ) in Eq. (14) into Eq. (19), the first order conditions can be obtained as

∂∂w(∏M(w,θ,m*))αR=−β˜αLw+12γ˜αRθ+12(D˜αR+β˜αLc˜αL).∂∂θ(∏M(w,θ,m*))αR=−2I˜αLθ+12γ˜αRw−12γ˜αRc˜αL.

Therefore, the Hessian matrix of (∏M(w,θ,m*))αR is

H=[−β˜αL12γ˜αR12γ˜αR−2I˜αL].

Since β˜, Ĩ are positive fuzzy variables and (γ˜αR)2<8β˜αLI˜αL, thus, for any α ∈ (0,1], H is a definite negative matrix and (∏M)αR is jointly concave in w and θ.

Let the first order conditions be zero, we can have Eqs.(16) and (17).

Substituting w* in Eq. (16) and θ* in Eq. (17) into Eq.(14), we can get Eq.(18).

Theorem 4 is proved.

The optimal optimistic profits of the manufacturer and the retailer can be easily obtained as follows

(∏M)αR=I˜αL(D˜αR−β˜αLc˜αL)28β˜αLI˜αL−(γ˜αR)2.   (∏R)αR=4β˜αL(I˜αL)2(D˜αR−β˜αLc˜αL)2(8β˜αLI˜αL−(γ˜αR)2)2.

Similarly, in the MS game, we can formulate the minimax chance-constrained programming model of GSC as follows

{maxw,θ min∏¯M∏¯Ms. t.Pos{(w−c˜)(D˜−β˜(m*+w)+γ˜θ)−I˜θ2≤∏¯M}≥αPos{w−c˜<0}=0where m*derives from problem         {maxm min∏¯R∏¯Rs. t.Pos{m(D˜−β˜(m+w)+γ˜θ)≤∏¯R}≥αPos{D˜−β˜(m+w)+γ˜θ<0}=0.
where α is a given confidence level for manufacturer’s and retailer’s profits, ∏¯R is the maximum value that ∏R (m) obtains with at least possibility α, and ∏¯M is the maximum value that ∏M (w, θ, m*) obtains with at least possibility α. Therefore, model (20) equals to the following model (21), in which supply chain actors attempt to maximize the α-pessimistic value of profits (∏M(w,θ,m*))αL and (∏R(m))αL by choosing optimal solutions, that is
{maxw,θ(∏M(w,θ,m*))αL=((w−c˜)(D˜−β˜(m*+w)+γ˜θ)−I˜θ2)αLs. t.Pos {w−c˜<0}=0 where m*derives from problem:         {maxm(∏R(m))αL=(m(D˜−β˜(m+w)+γ˜θ))αLs. t. Pos{D˜−β˜(m+w)+γ˜θ<0}=0.

Theorem 5.

Let (∏R(m))αL and (∏M(w,θ,m*))αL be the α-pessimistic value of the profits for the retailer and the manufacturer. If (γ˜αL)2<8β˜αRI˜αR, Pos{w*−c˜<0}=0 and Pos{D˜−β˜(m*+w*)+γ˜θ*<0}=0, then the optimal equilibrium decisions in the MS case are

m*=2I˜αR(D˜αL−β˜αRc˜αR)8β˜αRI˜αR−(γ˜αL)2.
w*=4I˜αR(D˜αL+β˜αRc˜αR)−(γ˜αL)2c˜αR8β˜αRI˜αR−(γ˜αL)2.
θ*=γ˜αL(D˜αL−β˜αRc˜αR)8β˜αRI˜αR−(γ˜αL)2.

Proof.

Similar to Theorems 3 and 4.

The optimal pessimistic profits of the manufacturer and the retailer are given by

(∏M)αL=I˜αR(D˜αL−β˜αRc˜αR)28β˜αRI˜αR−(γ˜αL)2.(∏R)αL=4β˜αR(I˜αR)2(D˜αL−β˜αRc˜αR)2(8β˜αRI˜αR−(γ˜αL)2)2.

Remark 1.

If α=1, it is clear the parameters c˜, D˜, β˜, γ˜ and Ĩ reduce to crisp numbers, then the optimal solutions in Theorems 4 and 5 are

m*=2I(D−βc)8βI−γ2.w*=4I(D+βc)−γ2c8βI−γ2.θ*=γ(D−βc)8βI−γ2.

4.2. Retailer Stackelberg (RS) game

In the RS game, the manufacturer has more bargaining power than the retailer. That is to say, the retailer is the leader in the GSC. In this case, firstly, the retailer sets his margin profit m condition on the manufacturer’s optimal reaction to his decisions. Then, the manufacturer sets his wholesale price w and level of greening innovation θ. Hence, in this case, we can formulate the fuzzy optimal model as follows

{maxm E[∏R(m,w**,θ**)]=E[m(D˜−β˜(m+w)+γ˜θ)]s. t.where w** and θ** derive from problem:        {maxw,θ E[∏M(w,θ)]=E[(w−c˜)(D˜−β˜(m+w)+γ˜θ)−I˜θ2]s. t.Pos{w−c˜<0}=0Pos{D˜−β˜(m+w)+γ˜θ<0}=0.

We first solve the reaction function of the manufacturer.

Theorem 6.

Let E[∏M (w, θ)] be the manufacturer’s fuzzy expected profit. If (E[γ˜])2<4E[β˜]E[I˜], Pos{D˜−β˜(m+w**)+γ˜θ**<0}=0 and Pos{w**−c˜<0}=0, then the manufacturer’s optimal reaction functions are

w**(m)=4E[I˜](E[D˜]+E[β˜c˜])2(4E[β˜]E[I˜]−(E[γ˜])2)−E[γ˜]∫01(γ˜αLc˜αR+γ˜αRc˜αL)d α+4E[β˜]E[I˜]m2(4E[β˜]E[I˜]−(E[γ˜])2).
θ**(m)=E[γ˜](E[D˜]+E[β˜c˜])4E[β˜]E[I˜]−(E[γ˜])2−E[β˜]∫01(γ˜αLc˜αR+γ˜αRc˜αL)d α+E[β˜]E[γ˜]m4E[β˜]E[I˜]−(E[γ˜])2.

Proof.

From Eq. (11), the first order conditions can be obtained as

∂∂wE[∏M]=−2E[β˜]w+E[γ˜]θ+E[D˜]+E[β˜c˜]−E[β˜]m.∂∂θE[∏M]=−2E[I˜]θ+E[γ˜]w−12∫01(γ˜αLc˜αR+γ˜αRc˜αL)d α.

Therefore the Hessian matrix of E[∏M] is

H=[−2E[β˜]E[γ˜]E[γ˜]−2E[I˜]].

Since E[β˜]>0, E[I˜]>0 and (E[γ˜])2<4E[β˜]E[I˜], thus H is a definite negative matrix and E[∏M] is jointly concave in w and θ.

Let the first order conditions be zero, we can have Eqs.(26) and (27).

Theorem 6 is proved.

Here, w(m) and θ(m) can be regarded as the manufacture’s response functions to the margin profit m. The retailer sets his optimal decisions. So, the following Proposition can be derived.

Theorem 7.

Let E[∏R (m)] be the retailer’s fuzzy expected profit. If (E[γ˜])2<4E[β˜]E[I˜], Pos{w**−c˜<0}=0 and Pos{D˜−β˜(m**+w**)+γ˜θ**<0}=0, then the optimal equilibrium decisions are

m**=E[γ˜](2E[β˜c˜]E[γ˜]−E[β˜]∫01(γ˜αLc˜αR+γ˜αRc˜αL)d α)8(E[β˜])2E[I˜]+(E[D˜]−E[β˜c˜])2E[β˜].
w**=E[I˜](E[D˜]+3E[β˜c˜])4E[β˜]E[I˜]−(E[γ˜])2−E[γ˜](2E[β˜c˜]E[γ˜]+E[β˜]∫01(γ˜αLc˜αR+γ˜αRc˜αL)d α)4E[β˜](4E[β˜]E[I˜]−(E[γ˜])2).
θ**=E[γ˜](E[D˜]+3E[β˜c˜])2(4E[β˜]E[I˜]−(E[γ˜])2)−(8E[β˜]E[I˜]−(E[γ˜])2)∫01(γ˜αLc˜αR+γ˜αRc˜αL)d α8E[I˜](4E[β˜]E[I˜]−(E[γ˜])2)−E[β˜c˜](E[γ˜])34E[β˜]E[I˜](4E[β˜]E[I˜]−(E[γ˜])2).

Proof.

Substituting Eqs. (26) and (27) into Eq. (7), the first order condition is

dd mE[∏R(m)]=−4(E[β˜])2E[I˜]4E[β˜]E[γ˜]−(E[I˜])2m      +2E[β˜]E[I˜](E[D˜]−E[β˜c˜])4E[β˜]E[γ˜]−(E[I˜])2      +E[γ˜](2E[β˜c˜]E[γ˜]−E[β˜]∫01(γ˜αLc˜αR+γ˜αRc˜αL)d α)2(4E[β˜]E[γ˜]−(E[I˜])2).

The second order condition is

d2d m2E[∏R(m)]=−4(E[β˜])2E[I˜]4E[β˜]E[I˜]−(E[γ˜])2.

Since E[β˜]>0, E[γ˜]>0, E[I˜]>0 and (E[γ˜])2<4E[β˜]E[I˜], then the second order condition is negative definite. Thus, E[∏R (m)] is concave in m.

Let the first order conditions be zero, we can have Eq.(28).

Substituting m** in Eq. (28) into Eqs. (26) and (27), we can get Eqs. (29) and (30).

Theorem 7 is proved.

In the RS game, the maximax chance-constrained programming model of GSC can be formulated as

{maxm(∏R(m))αR=(m(D˜−β˜(m+w**)+γ˜θ**))αRs. t.where w** and θ** derive from problem:        {maxw,θ (∏M(w,θ))αR=((w−c˜)(D˜−β˜(m+w)+γ˜θ)−I˜θ2)αRs. t.Pos{w**−c˜<0}=0Pos{D˜−β˜(m+w**)+γ˜θ**<0}=0.

Theorem 8.

Let (∏M(w,θ))αR be the manufacturer’s α-optimistic value of profit. If (γ˜αR)2<4β˜αLI˜αL, Pos{w**−c˜<0}=0 and Pos{D˜−β˜(m+w**)+γ˜θ**<0}=0, then the optimal reaction functions for manufacturer are

w**(m)=2I˜αL(D˜αR+β˜αLc˜αL)−2β˜αLI˜αLm−(γ˜αR)2c˜αL4β˜αLI˜αL−(γ˜αR)2.
θ**(m)=γ˜αR(D˜αR−β˜αLc˜αL−β˜αLm)4β˜αLI˜αL−(γ˜αR)2.

Proof.

From Eq. (19), the first order conditions can be obtained as

∂∂w(∏M(w,θ,m*))αR=−2β˜αLw+γ˜αRθ−β˜αLm+D˜αR+β˜αLc˜αL.∂∂θ(∏M(w,θ,m*))αR=−2I˜αLθ+γ˜αRw−γ˜αRc˜αL.

Therefore, the Hessian matrix of (∏M(w,θ,m*))αR is

H=[−2β˜αLγ˜αRγ˜αR−2I˜αL].

Since β˜, Ĩ are positive fuzzy variables and (γ˜αR)2<4β˜αLI˜αL, thus, for any α ∈ (0,1], H is a definite negative matrix and (∏M(w,θ))αR is jointly concave in w and θ.

Let the first order conditions be zero, the results (32) and (33) can be derived.

Theorem 8 is proved.

Theorem 9.

Let (∏R(m))αR be the retailer’s α-optimistic value of profit. If (γ˜αR)2<4β˜αLI˜αL, Pos{w**−c˜<0}=0 and Pos{D˜−β˜(m**+w**)+γ˜θ**<0}=0, then the optimal equilibrium decisions are

m**=D˜αR−β˜αLc˜αL2β˜αL.
w**=I˜αL(D˜αR+3β˜αLc˜αL)−(γ˜αR)2c˜αL4β˜αLI˜αL−(γ˜αR)2.
θ**=γ˜αR(D˜αR−β˜αLc˜αL)2(4β˜αLI˜αL−(γ˜αR)2).

Proof.

Substituting w** (m) and θ** (m) in Eqs. (32) and (33) into Eq. (15), the first order condition is

dd m(∏R(m))αR=−4(β˜αL)2I˜αL4β˜αLI˜αL−γ˜αRm+2β˜αLI˜αL(D˜αR−β˜αLc˜αL)4β˜αLI˜αL−γ˜αR.

The second order condition is

d2 d m2(∏R(m))αR=−4(β˜αL)2I˜αL4β˜αLI˜αL−γ˜αR.

Since β˜, Ĩ and γ˜ are positive fuzzy variables, and (γ˜αR)2<4β˜αLI˜αL, then the second order condition is negative definite. Thus, for any α ∈ (0,1], (∏R(m))αR is concave in m.

Let dd m(∏R(m))αR=0, we can get the retailer’s optimal margin profit as in Eq. (34).

Substituting m** in Eq. (34) into Eqs. (32) and (33), we can get Eqs. (35) and (36).

Theorem 9 is proved.

The optimal optimistic profits of the manufacturer and the retailer can be easily obtained as follows

(∏M)αR=I˜αL(D˜αR−β˜αLc˜αL)24(4β˜αLI˜αL−(γ˜αR)2).(∏R)αR=I˜αL(D˜αR−β˜αLc˜αL)22(4β˜αLI˜αL−(γ˜αR)2).

In the RS game, the minimax chance-constrained programming model of GSC can be formulate as follows

{maxm (∏R(m))αL=(m(D˜−β˜(m+w**)+γ˜θ**))αLs. t.where w** and θ** derive from problem:     {maxw,θ(∏M(w,θ))αL=((w−c˜)(D˜−β˜(m+w)+γ˜θ)−I˜θ2)αLs. t.Pos{w**−c˜<0}=0Pos{D˜−β˜(m+w**)+γ˜θ**<0}=0.

Theorem 10.

Let (∏R(m))αL and (∏M(w,θ))αL be the α-pessimistic value of profit for the retailer and the manufacturer. If (γ˜αL)2<4β˜αRI˜αR, Pos {w**−c˜<0}=0 and Pos{D˜−β˜(m**+w**)+γ˜θ**<0}=0, then the optimal equilibrium decisions are

m**=D˜αL−β˜αRc˜αR2β˜αR.
w**=I˜αR(D˜αL+3β˜αRc˜αR)−(γ˜αL)2c˜αR4β˜αRI˜αR−(γ˜αL)2.
θ**=γ˜αL(D˜αL−β˜αRc˜αR)2(4β˜αRI˜αR−(γ˜αL)2).

Proof.

Similar to Theorems 8 and 9.

The optimal pessimistic profits for the supply chain members are given by

(∏M)αL=I˜αR(D˜αL−β˜αRc˜αR)24(4β˜αRI˜αR−(γ˜αL)2).(∏R)αL=I˜αR(D˜αL−β˜αRc˜αR)22(4β˜αRI˜αR−(γ˜αL)2).

Remark 2.

If α=1, then the parameters c˜, D˜, β˜, γ˜ and Ĩ reduce to crisp numbers, then the optimal solutions in Theorems 9 and 10 are

m**=D−βc2β.w**=I(D+3βc)−γ2c4βI−γ2.θ**=γ(D−βc)2(4βI−γ2).

4.3. Vertical Nash (VN) game

In the VN game, the supply chain participants have an equal bargaining power. Thus, in this condition, the manufacturer makes his wholesale price w and level of greening innovation θ, and the retailer makes his margin profit m simultaneously and independently. The assumption VN game means a Nash equilibrium among the manufacturer and the retailer, hence, in this case, we can formulate the fuzzy optimal model as follows

{maxw,θ E[∏M(w,θ)]=E[(w−c˜)(D˜−β˜(m+w)+γ˜θ)−I˜θ2]maxm E[∏R(m)]=E[m(D˜−β˜(m+w)+γ˜θ)] s. t.Pos{w−c˜<0}=0Pos{D˜−β˜(m+w)+γ˜θ<0}=0.

Theorem 11.

Let E[∏M (w, θ)] and E[∏R (m)] be the manufacturer’s and retailer’s fuzzy expected profits. If (E[γ˜])2<4E[β˜]E[I˜], Pos{D˜−β˜(m***+w***)+γ˜θ***<0}=0 and Pos{w***−c˜<0}=0, then the optimal equilibrium decisions are

m***=E[γ˜](2E[β˜c˜]E[γ˜]−E[β˜]∫01(γ˜αLc˜αR+γ˜αRc˜αL)d α)2E[β˜](6E[β˜]E[I˜]−(E[γ])2)+2E[I˜](E[D˜]−E[β˜c˜])6E[β˜]E[I˜]−(E[γ])2.
w***=4E[I˜](E[D˜]+2E[β˜c˜])−E[γ˜]∫01(γ˜αLc˜αR+γ˜αRc˜αL)d α2(6E[β˜]E[I˜]−(E[γ])2).
θ***=2E[γ˜](E[D˜]+2E[β˜c˜])−3E[β˜]∫01(γ˜αLc˜αR+γ˜αRc˜αL)d α2(6E[β˜]E[I˜]−(E[γ])2).

Proof.

From Eq. (7), the first order condition is

dd mE[∏R(m)]=−2E[β˜]m+E[D˜]−E[β˜]w+E[γ˜]θ.

The second order condition is

d2 d m2E[∏R(m)]=−2E[β˜]<0.

Thus, the retailer’s fuzzy expected profit E[∏R (m)] is strictly concave in m.

From Eq. (11), the first order conditions are

∂∂wE[∏M]=−2E[β˜]w+E[γ˜]θ+E[D˜]+E[β˜c˜]−E[β˜]m.∂∂θE[∏M]=−2E[I˜]θ+E[γ˜]w−12∫01(γ˜αLc˜αR+γ˜αRc˜αL)d α.

Therefore, the Hessian matrix of E[∏M] is

H=[−2E[β˜]E[γ˜]E[γ˜]−2E[I˜]].

Since E[β˜]>0, E[I˜]>0 and (E[γ˜])2<4E[β˜]E[I˜], thus H is a definite negative matrix, and E[∏M] is jointly concave in w and θ.

We can have Eqs. (42), (43) and (44) by setting the first order conditions above be zero.

Theorem 11 is proved.

In the VN game, we can also formulate the minimax chance-constrained programming model of GSC as

{maxw,θ(∏M(w,θ))αR=((w−c˜)(D˜−β˜(m+w)+γ˜θ)−I˜θ2)αRmaxm(∏R(m))αR=(m(D˜−β˜(m+w)+γ˜θ))αR s. t.Pos{w−c˜<0}=0Pos{D˜−β˜(m+w)+γ˜θ<0}=0.

Theorem 12.

Let (∏M(w,θ))αR and (∏R(m))αR be the manufacturer’s and the retailer’s α-optimistic value of profits. If (γ˜αR)2<4β˜αLI˜αL, Pos{w***−c˜<0}=0 and Pos{D˜−β˜(m***+w***)+γ˜θ***<0}=0, then the optimal equilibrium decisions are

m***=2I˜αL(D˜αR−β˜αLc˜αL)6β˜αLI˜αL−(γ˜αR)2.
w***=2I˜αL(D˜αR+2β˜αLc˜αL)−(γ˜αR)2c˜αL6β˜αLI˜αL−(γ˜αR)2.
θ***=γ˜αR(D˜αR−β˜αLc˜αL)6β˜αLI˜αL−(γ˜αR)2.

Proof.

From Eq. (15), the first order condition is

dd m(∏R(m))αR=−2β˜αLm+D˜αR−β˜αLw+γ˜αRθ.

The second order condition is

d2d m2(∏R(m))αR=−2β˜αL<0.

Thus, for any α ∈ (0,1], (∏R(m))αR is concave in m.

From Eq. (19), the first order conditions are

∂∂w(∏M(w,θ,m*))αR=−2β˜αLw+γ˜αRθ−β˜αLm+D˜αR+β˜αLc˜αL.∂∂θ(∏M(w,θ,m*))αR=−2I˜αLθ+γ˜αRw−γ˜αRc˜αL.

Therefore, the Hessian matrix of (∏M(w,θ,m*))αR is

H=[−2β˜αLγ˜αRγ˜αR−2I˜αL].

Since β˜, Ĩ are positive fuzzy variables and (γ˜αR)2<4β˜αLI˜αL, then the Hessian matrix H is negative definite. Thus, for any α ∈ (0,1], (∏M(w,θ))αR is jointly concave in w and θ.

We can have Eqs. (46) , (47) and (48) by setting the first order conditions above be zero.

Theorem 12 is proved.

The optimal optimistic profits of the manufacturer and the retailer can be easily obtained as follows

(∏M)αR=I˜αL(4β˜αLI˜αL−(γ˜αR)2)(D˜αR−β˜αLc˜αL)2(6β˜αLI˜αL−(γ˜αR)2)2.      (∏R)αR=4β˜αL(I˜αL)2(D˜αR−β˜αLc˜αL)2(6β˜αLI˜αL−(γ˜αR)2)2.

In the VN game, the minimax chance-constrained programming model of GSC can be formulated as follows

{maxw,θ (∏M(w,θ))αL=((w−c˜)(D˜−β˜(m+w)+γ˜θ)−I˜θ2)αL  maxm (∏R(m))αL=(m(D˜−β˜(m+w)+γ˜θ))αL s. t. Pos{w−c˜<0}=0Pos{D˜−β˜(m+w)+γ˜θ<0}=0.

Theorem 13.

Let (∏M(w,θ))αL and (∏R(m))αL be the manufacturer’s and retailer’s α-pessimistic value of profits. If (γ˜αL)2<4β˜αRI˜αR, Pos{w***−c˜<0}=0 and Pos{D˜−β˜(m***+w***)+γ˜θ***<0}=0, then the optimal equilibrium decisions are

m***=2I˜αR(D˜αL−β˜αRc˜αR)6β˜αRI˜αR−(γ˜αL)2.
w***=2I˜αR(D˜αL+2β˜αRc˜αR)−(γ˜αL)2c˜αR6β˜αRI˜αR−(γ˜αL)2.
θ***=γ˜αL(D˜αL−β˜αRc˜αR)6β˜αRI˜αR−(γ˜αL)2.

Proof.

Similar to Theorem 12.

The optimal pessimistic profits of the manufacturer and the retailer can be obtained as follows

(∏M)αL=I˜αR(4β˜αRI˜αR−(γ˜αL)2) (D˜αL−β˜αRc˜αR)2(6β˜αRI˜αR−(γ˜αL)2)2.(∏R)αL=4β˜αR(I˜αR)2(D˜αL−β˜αRc˜αR)2(6β˜αRI˜αR−(γ˜αL)2)2.

Remark 3.

If α=1, then the parameters c˜, D˜, β˜, γ˜ and Ĩ reduce to crisp numbers, then the optimal solutions in Theorems 12 and 13 are

   m***=2I(D−βc)6βI−γ2.w***=2I(D+2βc)−γ2c6βI−γ2.θ***=γ(D−βc)6βI−γ2.

5. Numerical Examples

Because the optimal equilibrium decisions obtained in above Section are in very complicated forms, we have to conduct numerical examples to illustrate the computational process of fuzzy models proposed in this paper. We will also perform the impacts of the fuzziness of parameters β˜, γ˜ and the confidence level α on these models.

Consider the case where the retailer orders new greening household appliances from the manufacturer, and then he retails them to the end customers. The data used in the numerical examples are estimated from the Chinese household appliances manufacturing industry. These data have been properly handled before being adopted. We think these data can represent the real condition duce to lack of historical data when supply chain participants make their optimal decisions.

The linguistic descriptions and its corresponding triangular fuzzy variables determined by the experiences of experts are shown in Table 1.

Parameter Linguistic description Fuzzy variable
D˜ Large (about 1000) (900, 1000, 1100)
Small (about 500) (400, 500, 600)
β˜ Very sensitive (about 50) (45, 50, 55)
Sensitive (about 40) (35, 40, 45)
γ˜ Very sensitive (about 40) (35, 40, 45)
Sensitive (about 30) (25, 30,35)
Ĩ Very sensitive (about 30) (25, 30, 35)
Sensitive (about 20) (15, 20,25)
c˜ High (about 13) (11, 13, 15)
Medium (about 10) (9, 10, 11)
Low (about 7) (6, 7, 8)
Table 1.

The triangular fuzzy variables

5.1. Discussion 1

We first assume that the parameter D˜ is large, (D˜ is about 1000), price elastic coefficient β˜, greening level elastic coefficient γ˜ and investment elastic coefficient Ĩ are all in sensitive level, (β˜ is about 40, γ˜ is about 30 and Ĩ is about 20), the producing cost c˜ is medium, (c˜ is about 10). The expected values, α-optimistic values and α-pessimistic values related to these triangular fuzzy numbers are listed in Table 2.

Triangular fuzzy variable Expected value α-optimistic value α-pessimistic value
D˜ 100 1100 − 100α 900 + 100α
β˜ 40 45 − 5α 35 + 5α
γ˜ 30 35 − 5α 25 + 5α
Ĩ = (15,20,25) 20 25 − 5α 15 + 5α
c˜ 10 11 − α 9 + α
Table 2.

Triangular fuzzy variables, and their expected, α-optimistic and α-pessimistic values

From Table 2, we have

E[β˜c˜]=12∫01(β˜αLc˜αL+β˜αRc˜αR)d α=12053,∫01(γ˜αLc˜αR+γ˜αRc˜αL)d α=17903,12∫01(D˜αLc˜αR+D˜αRc˜αL)d α=299003.

The results of the optimal expected solutions under three non-cooperative games are showed in Tables 3 and 4.

m w p θ
MS 4.367 18.777 23.144 3.348
RS 7.507 15.264 22.770 3.989
VN 6.159 16.201 22.360 4.692
Table 3.

Expected values of the equilibrium prices under three non-cooperative games

E[∏R] E[∏M] E[∏SC]
MS 762.976 1386.471 2149.447
RS 1567.936 859.074 2427.010
VN 1517.424 1165.755 2683.179
Table 4.

Expected values of the equilibrium profits under three non-cooperative games

From Tables 3 and 4, we can find that

  • (1)The optimal green innovation’s level θ is the highest in the VN case when the actors have equal bargaining power. The MS case provides the lowest level of green innovation this is because under this case the full costs of greening are afforded by manufacturer.

  • (2)The wholesale price w under the MS case is the highest, followed by VN and then RS cases, this is because that the manufacturer incurs green costs and has the dominant power in pricing of the green product. The profit margin of the retailer m is highest in the RS case this is because in this case manufacturer charges a low wholesale price w. Under MS case, The retail price p is the highest, followed by RS and then VN cases.

  • (3)The retailer makes the smallest profit under MS case, and the largest under RS case. On the other hand, the manufacturer’s fuzzy expected profits are in the reverse order. It shows that the more power the actor has the more fuzzy expected profits he can derive. That is, the retailer’s fuzzy expected profit is the largest when the retailer is the leader, and the manufacturer’s largest when the manufacturer is the leader in the channel.

  • (4)The integrated system obtains his largest expected profit in the VN case when no actor is a channel leader. However, the manufacturer or the retailer has an incentive to be a leader this is because as a leader he can obtain more fuzzy expected profit.

  • (5) Under VN case, the retail price p is the lowest, and the level of green innovation θ is highest. For the customers, this means VN case is a preferred policy.

5.2. Discussion 2

In this subsection, we discuss the impacts of the fuzziness of parameters β˜ and γ˜ on prices, level of green innovation and expected profits under MS, RS and VN cases. Decreasing the fuzzy degree of the parameters β˜, γ˜ and observing their impacts. We use the same values of other parameters as in Discussion 1. Tables 5, 6, 7 and 8 give the solutions as follows.

β˜ m w p θ
MS (35, 40, 45) 4.367 18.777 23.144 3.348
(36, 40, 44) 4.368 18.770 23.139 3.343
(37, 40, 43) 4.370 18.764 23.134 3.338
(38, 40, 42) 4.371 18.758 23.128 3.332
(39, 40, 41) 4.372 18.751 23.123 3.327
RS (35, 40, 45) 7.507 15.264 22.770 3.989
(36, 40, 44) 7.508 15.257 22.765 3.984
(37, 40, 43) 7.510 15.249 22.760 3.979
(38, 40, 42) 7.512 15.242 22.754 3.973
(39, 40, 41) 7.514 15.235 22.749 3.968
VN 35, 40, 45) 6.159 16.201 22.360 4.692
(36, 40, 44) 6.161 16.194 22.355 4.687
(37, 40, 43) 6.162 16.187 22.349 4.682
(38, 40, 42) 6.164 16.180 22.344 4.677
(39, 40, 41) 6.165 16.174 22.339 4.672
Table 5.

The change of the expected value of the prices for three fuzzy models with the fuzzy degree of parameter β˜

β˜ E[∏R] E[∏M] E[∏SC]
MS (35, 40, 45) 762.976 1386.471 2149.447
(36, 40, 44) 763.346 1378.757 2142.104
(37, 40, 43) 763.717 1371.045 2134.762
(38, 40, 42) 764.088 1363.335 2127.423
(39, 40, 41) 764.459 1355.626 2120.085
RS (35, 40, 45) 1567.936 859.074 2427.010
(36, 40, 44) 1568.698 851.104 2419.802
(37, 40, 43) 1569.459 843.136 2412.595
(38, 40, 42) 1570.221 835.169 2405.391
(39, 40, 41) 1570.984 827.204 2398.188
VN 35, 40, 45) 1517.424 1165.755 2683.179
(36, 40, 44) 1518.161 1157.934 2676.095
(37, 40, 43) 1518.898 1150.114 2669.013
(38, 40, 42) 1519.636 1142.297 2661.933
(39, 40, 41) 1520.373 1134.481 2654.854
Table 6.

The change of the expected value of the profits for three fuzzy models with the fuzzy degree of parameter β˜

γ˜ m w p θ
MS (25, 30, 35) 4.367 18.777 23.144 3.348
(26, 30, 34) 4.366 18.773 23.138 3.339
(27, 30, 33) 4.364 18.769 23.133 3.329
(28, 30, 32) 4.362 18.766 23.128 3.319
(29, 30, 31) 4.360 18.762 23.122 3.310
RS (25, 30, 35) 7.507 15.264 22.770 3.989
(26, 30, 34) 7.503 15.261 22.765 3.979
(27, 30, 33) 7.500 15.259 22.760 3.969
(28, 30, 32) 7.497 15.257 22.754 3.959
(29, 30, 31) 7.494 15.255 22.749 3.950
VN (25, 30, 35) 6.159 16.201 22.360 4.692
(26, 30, 34) 6.157 16.198 22.355 4.682
(27, 30, 33) 6.154 16.196 22.350 4.672
(28, 30, 32) 6.151 16.193 22.345 4.662
(29, 30, 31) 6.149 16.191 22.340 4.651
Table 7.

The change of the expected value of the prices for three fuzzy models with the fuzzy degree of parameter γ˜

γ˜ E[∏R] E[∏M] E[∏SC]
MS (25, 30, 35) 762.976 1386.471 2149.447
(26, 30, 34) 762.341 1385.356 2147.697
(27, 30, 33) 761.706 1384.245 2145.951
(28, 30, 32) 761.071 1383.137 2144.208
(29, 30, 31) 760.437 1382.032 2142.469
RS (25, 30, 35) 1567.936 859.074 2427.010
(26, 30, 34) 1566.631 858.399 2425.030
(27, 30, 33) 1565.326 857.726 2423.052
(28, 30, 32) 1564.022 857.057 2421.079
(29, 30, 31) 1562.718 856.391 2419.109
VN (25, 30, 35) 1517.424 1165.755 2412.595
(26, 30, 34) 1516.161 1164.824 2680.985
(27, 30, 33) 1514.898 1163.896 2678.794
(28, 30, 32) 1513.636 1162.972 2676.608
(29, 30, 31) 1512.374 1162.050 2674.425
Table 8.

The change of the expected value of the profit for three fuzzy models with the fuzzy degree of parameter γ˜

From Tables 5, 6, 7 and 8, we can find that

  • (6)When the fuzziness of the parameter β˜ decreases, the profit margin m and expected profit of retailer increase slightly in the three game cases. Therefore, the retailer should seek as low fuzziness of parameter β˜ as possible.

  • (7)The wholesale price w, retail price p, level of green innovation θ, expected profit for manufacturer and whole supply chain drop slightly in three cases, as the fuzziness of parameter β˜ falls.

  • (8)When the fuzziness of the parameter γ˜ decreases, the profit margin m, prices w and p, level of green innovation θ and profits for supply chain members will all increase.

5.3. Discussion 3

Thirdly, we present the results of the α-optimistic values and α-pessimistic values under three decentralized channel decisions shown in Tables 9, 10, 11 and 12, respectively. In Tables 9, 10, 11 and 12, the 2th, 8th and 14th rows show the optimal equilibrium decisions under three cases in crisp environment at α=1.

α m w p θ
MS 1.00 4.364 18.727 23.091 3.273
0.95 4.487 18.924 23.411 3.436
0.90 4.614 19.128 23.742 3.608
0.85 4.745 19.339 24.084 3.790
0.80 4.879 19.559 24.438 3.981
0.75 5.019 19.787 24.806 4.182
RS 1.00 7.500 15.217 22.717 3.913
0.95 7.667 15.360 23.026 4.142
0.90 7.835 15.512 23.347 4.389
0.85 8.005 15.675 23.680 4.652
0.80 8.177 15.850 24.027 4.936
0.75 8.351 16.038 24.389 5.240
VN 1.00 6.154 16.154 22.308 4.615
0.95 6.343 16.293 22.637 4.858
0.90 6.540 16.440 22.979 5.114
0.85 6.743 16.593 23.336 5.386
0.80 6.954 16.754 23.709 5.673
0.75 7.174 16.924 24.099 5.979
Table 9.

α-optimistic value of the equilibrium prices under three cases with respect to α

α (∏R)αR (∏M)αR (∏SC)αR
MS 1.00 761.653 1309.091 2070.744
0.95 800.296 1367.387 2167.683
0.90 840.898 1427.903 2268.801
0.85 883.602 1490.754 2374.357
0.80 928.565 1556.068 2484.633
0.75 975.963 1623.981 2599.943
RS 1.00 1565.217 782.609 2347.826
0.95 1648.542 824.271 2472.713
0.90 1736.704 868.352 2605.056
0.85 1830.141 915.071 2745.212
0.80 1929.353 964.677 2894.029
0.75 2034.907 1017.453 3052.360
VN 1.00 1514.793 1088.757 2603.550
0.95 1599.432 1133.361 2732.793
0.90 1689.237 1179.204 2868.440
0.85 1784.665 1226.302 3010.967
0.80 1886.233 1274.671 3160.903
0.75 1994.521 1324.319 3318.840
Table 10.

α-optimistic value of the equilibrium profits under three cases with respect to α

α m w p θ
MS 1.00 4.364 18.727 23.091 3.273
0.95 4.244 18.537 22.781 3.117
0.90 4.127 18.354 22.480 2.969
0.85 4.103 18.176 22.189 2.828
0.80 3.902 18.004 21.906 2.694
0.75 3.794 17.838 21.631 2.566
RS 1.00 7.500 15.217 22.717 3.913
0.95 7.335 15.084 22.420 3.698
0.90 7.172 14.960 22.132 3.496
0.85 7.011 14.842 21.853 3.307
0.80 6.851 14.732 21.584 3.130
0.75 6.693 14.629 21.322 2.962
VN 1.00 6.154 16.154 22.308 4.615
0.95 5.971 16.021 21.992 4.386
0.90 5.794 15.894 21.687 4.169
0.85 5.622 15.772 21.394 3.963
0.80 5.456 15.656 21.111 3.767
0.75 5.294 15.544 20.838 3.581
Table 11.

α-pessimistic value of the equilibrium prices under three cases with respect to α

α (∏R)αL (∏M)αL (∏SC)αL
MS 1.00 761.653 1309.091 2070.744
0.95 724.840 1252.907 1977.747
0.90 689.740 1198.737 1888.477
0.85 656.246 1146.490 1802.735
0.80 624.261 1096.081 1720.342
0.75 593.696 1047.434 1641.129
RS 1.00 1565.217 782.609 2347.826
0.95 1486.341 743.171 2229.512
0.90 1411.570 705.785 2117.355
0.85 1340.596 670.298 2010.895
0.80 1273.147 636.574 1909.721
0.75 1208.977 604.488 1813.465
VN 1.00 1514.793 1088.757 2603.550
0.95 1434.910 1045.374 2480.328
0.90 1359.416 1003.189 2362.605
0.85 1287.985 962.181 2250.165
0.80 1220.320 922.327 2142.646
0.75 1156.155 883.604 2039.759
Table 12.

α-pessimistic value of the equilibrium profits under three cases with respect to α

From Tables 9, 10, 11 and 12, we can find that

  • (9) In the three different fuzzy models, the α-optimistic values of profit margin, prices, level of green innovation and profits for supply chain participants increase, as the confidence level α decreases. That is to say, if the confidence level of the supply chain participant’s profits α=1, then the supply chain participants derive their smallest profits. It means that the less responsible of supply chain participants to risk, the lower profits will be when they are risk preferable.

  • (10) In the three different fuzzy models, the α-pessimistic values of profit margin, prices, greening level and profits for supply chain actors increase, as the confidence level α increases. That is to say, if the confidence level of the supply chain participant’s profits α=1, then the supply chain participants derive their largest profits. It means that the more responsible of supply chain participants to risk, the larger profits will be when they are risk averse.

    In practice, the supply chain participants can alter the value of the parameter α to obtain the different equilibrium decisions under the different level of the supply chain participant’s profits. The equilibrium decision reflects the different risk attitude of the supply chain participant to market demand uncertainty and different prediction of possibility level.

  • (11) It is interesting to compare the solutions of this paper with the work of Ghosh and Shah3. When α=1, the solutions in this study will be the conventional solutions according to the method of Ghosh and Shah3. In Tables9, 10, 11 and 12, the 2th, 8th and 14th rows show the optimal equilibrium decisions under three cases in crisp environment. Compared these optimal equilibrium decisions in crisp environment to those in fuzzy decision making environment showed in Table 3, we observe that our profit margin, wholesale price, retail price and level of green innovation are all higher than those in crisp results. The manufacturer’s profit is lower, while the retailer’s profit and supply chain system’s profit are higher when they face fuzzy uncertainty. It means that the retailer could benefit from the fuzzy environment, while the manufacturer could suffer from this environment.

6. Conclusions

This paper deals with the coordination strategy in a m In this paper, we considered three different fuzzy models in green supply chain, in which supply chain participants pursued three different power balance scenarios. For each model, we provided the optimal equilibrium decisions of the manufacturer and the retailer when they were risk neutral, risk preferable and risk averse. We also found that under certain circumstances, the formulated fuzzy supply chain models can degenerate into crisp models. Besides, we show that the different power structures, the confidence level of the supply chain participant’s profits and fuzzy degree of parameters affect the final optimal solutions.

The current fuzzy green supply chain models have some limitations. The fuzzy demand for supply chain is assumed as a linear function. Another limitation is that we only consider one manufacturer and one retailer in GSC. Further research can focus on analyzing some more complicated greening policies in which some other kinds of demand functions or multiple supply chain participants exist. Still, we will develop the channel coordination in GSC under a fuzzy decision environment.

Acknowledgements

This work was supported by the Shandong Provincial Natural Science Foundation, China (No. ZR2015GQ001), and the Project of Shandong Provincial Higher Educational Humanity and Social Science Research Program (No. J15WB04).

References

15.J Wei and J Zhao, Pricing decisions with retail competition in a fuzzy closed-loop supply chain, Expert Syst Appl, Vol. 38, No. 9, 2011, pp. 11209-11216. http://dx.doi.org/10.1016/j.eswa.2011.02.168
24.AA Khamseh, F Soleimani, and B Naderi, Pricing decisions for complementary products with firm’s different market powers in fuzzy environments, J Intell Fuzzy Syst, Vol. 27, No. 5, 2014, pp. 2327-2340.
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Journal
International Journal of Computational Intelligence Systems
Volume-Issue
10 - 1
Pages
986 - 1001
Publication Date
2017/07/04
ISSN (Online)
1875-6883
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risenwbib
TY  - JOUR
AU  - Shengju Sang
PY  - 2017
DA  - 2017/07/04
TI  - Decentralized Channel Decisions of Green Supply Chain in a Fuzzy Decision Making Environment
JO  - International Journal of Computational Intelligence Systems
SP  - 986
EP  - 1001
VL  - 10
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.2017.10.1.66
DO  - 10.2991/ijcis.2017.10.1.66
ID  - Sang2017
ER  -