Unfair and Risky? Profit Allocation in Closed-Loop Supply Chains by Cooperative Game Approaches

Abstract

Behavioral factors (i.e., risk aversion and fairness concern) are considered for profit allocation in a closed-loop supply chain. This paper studies a two-echelon closed-loop supply chain (CLSC) consisting of a risk-neutral manufacturer, a risk-averse fairness-neutral retailer, and a risk-neutral retailer having fairness concerns. Cooperative game analysis is used to characterize equilibriums under five scenarios: a centralized, a decentralized and three partially allied models. Analytical results confirm that even when factoring in retailers’ risk aversion and fairness concern, the centralized model still outperforms decentralized. This paper makes a numerical study on the effects of risk aversion and fairness concern on profit distribution under these five models. It reveals that the impact of the risk aversion parameter and fairness concern parameter is dynamic, not always positive or negative. These research results provide helpful insights for CLSC managers to find out available choices and feasible ways to achieve fair profit allocations.

1. Introduction

Under the increasingly stressful environment, more and more enterprises remanufacture and recycle preowned goods, forming a closed-loop supply chain. Circular economy has been recognized as more efficient and competitive than linear economy [1]. Circular closed-loop supply chains (CLSCs) management provides a new perspective to the sustainability [2,3]. There remain challenges in managing circular supply chains effectively and various strategies are proposed by researchers [4,5,6].

One of the primary objectives of CLSCs is to maximize channel profits. Decision makers in CLSCs are usually not absolutely rational [7,8]. Behavioral factors such as risk aversion and fairness concern have impacts on operational decisions and profit distribution [9]. Under the uncertainties of market demand or recycling profit, CLSC decision makers tend to avoid potential risks. CLSC members are often concerned with the proportion of their profits in the total CLSC profit. Price strategy is a primary methodology to coordinate CLSC management.

This research addresses two themes: (1) derivation of equilibrated production quantities, prices and profits for a CLSC with both risk aversion and fairness concern under the five models; (2) evaluation of risk aversion and fairness concern’s effect on profit allocations in cooperative game settings. Supply chains often have a single manufacturer and multiple retailers [10]. In this research, we incorporate retailers’ risk aversion and fairness concern into a two-echelon CLSC consisting of a manufacturer (M), a risk-averse retailer (L) without fairness concern, and a risk-neutral retailer (F) having fairness concern. By cooperative game approaches, we examine five scenarios: (1) a centralized model where one central decision maker plans for all agents (CC); (2) a decentralized model where each agent makes decisions independently (DC); (3) L and F compose an alliance (LF); (4) M and L compose an alliance (ML) and (5) M and F compose an alliance (MF).

2. Literature Review

Considerable attention has been attracted to CLSCs in academia and practice. Remarkable studies have been conducted regarding two-echelon CLSCs [11,12,13,14,15] and three-echelon CLSCs [16,17,18]. Retailers of a CLSC frequently cooperate with manufacturers to collect preowned goods [19]. For instance, the Procter & Gamble company collects and resells used products through Loop, an online shopping platform which also sells new products. By forming strategic partnerships, enterprises in CLSCs can improve the profitability of individuals and distribution channels.

In past research on CLSCs, agents were assumed to be of complete rationality. In actual fact, decision making agents often show different social preferences such as risk aversion. Research on CLSC coordination with risk aversion has been carried out. Ke et al. [20] examined pricing and remanufacturing issues in a CLSC which is consisted of a dominant manufacturer and a risk-averse retailer. Zeballos et al. [21] constructed and analyzed a risk-averse multi-stage model of a CLSC which include several functional entities. Ma et al. [22] examined a bike-sharing operation network to study a multi-product, multi-agent, single-stage CLSC system under risk-averse criterion. Das et al. [23] studied a two-period risk-averse model of a CLSC for reusable packaging materials.

Fairness concerns also widely exist in CLSCs. Along this research line, Ma et al. [24] investigated reverse-channel CLSCs where retailers serve as collectors of preowned products while retailers have distributional fairness concern. Zheng et al. [25] incorporated fairness concern into a three-echelon CLSC composed of one manufacturer, one distributor and one retailer, and examined the impact of the retailer’s fairness concern. Sarkar and Bhala [26] analyzed the coordination of a CLSC in the presence of fairness concerns with a constant wholesale price contract. Wang et al. [27] studied the impact of fairness concern in e-commerce CLSCs and showed that the fairness concern of the e-commerce platform reduces individual profit and systematic efficiency.

Limited research has been carried out on CLSCs regarding both risk aversion and fairness concern. He et al. [28] examined pricing strategies of a CLSC consisting of one manufacturer and one retailer considering the factors of risk aversion and fairness concern. Li et al. [29] investigated the price decisions in a CLSC with a risk-averse retailer with fairness concern and a risk-neutral manufacturer. Zhang and Zhang [30] studied a CLSC composed of two suppliers, a manufacturer, a risk-averse retailer and a fairness-concerned third-party under supply disruptions. These studies show that risk aversion and fairness concerns bring complexities to the coordination of a CLSC, and past research is limited to non-cooperative game setups. Differing from existing studies, this paper investigates the impact of risk aversion and fairness concern in a CLSC by cooperative game approaches. As summarized in

Table 1. An overview of characteristics of the literature and this paper.

References	Subject Areas
	CLSCs	Risk Aversion	Fairness Concern	Cooperative Game Approach
Refs. [11,12,13,14,16,17,18]	Yes	No	No	No
Ref. [15]	Yes	No	No	Yes
Refs. [19,20,21,22,23]	Yes	Yes	No	No
Refs. [24,26,27]	Yes	No	Yes	No
Ref. [25]	Yes	No	Yes	Yes
Refs. [28,29,30]	Yes	Yes	Yes	No
This paper	Yes	Yes	Yes	Yes

, this paper differs from other existing CLSC literature in subject areas.

3. Problem Description and Assumptions

This section describes a two-echelon supply chain model consisting of a risk-neutral manufacturer (M) and two competing retailers (L and F) who play a Stackelberg game. It is assumed that Retailer L has a higher market share than Retailer F, so that customers have higher willingness to purchase products from L than F. Given Retailer F’s relatively weaker position and the wide recognition that disadvantaged agents often care about fairness issues [31,32], this research assumes that Retailer F has a fairness concern with the upper manufacturer. We assume Retailer L is risk-averse and attempts to avoid risks. The sales channels provided by the two retailers are available to all customers. Customers are risk neutral and make decisions to maximize their utility, and we consider one-period interactions among CLSC members.

To derive characteristic functions of the cooperative game, all potential alliances and the corresponding equilibriums are investigated. Five models are presented in Figure 1: the centralized model, CC, with a central decision maker, the decentralized model, DC, where agents make individual decisions independently, and three partial-cooperation models, LF, ML, and MF, where three partial alliances form and make centralized decisions within the respective coalition.

On the basis of the problem description, this paper employs the following notation throughout the research as shown in

Table 2. Variables and parameters.

Notation	Definition
$c_n, c_r$	Unit production cost for a new or remanufactured product
$w_n^L, w_r^L$	Unit wholesale price for a new or remanufactured product offered by M to L
$w_n^F, w_r^F$	Unit wholesale price for a new or remanufactured product offered by M to F
$p_n^L, p_r^L$	Unit retail price for a new or remanufactured product offered by L
$p_n^F, p_r^F$	Unit retail price for a new or remanufactured product offered by F
$q_n, q_r$	Production quantity for new or remanufactured products
$q_n^L, q_r^L$	Quantity of new or remanufactured products transferred from M to L
$q_n^F, q_r^F$	Quantity of new or remanufactured products transferred from M to F
$t_n^L, t_r^L$	Utility of a consumer receiving from L for a new or remanufactured product
$t_n^F, t_r^F$	Utility of a consumer receiving from F for a new or remanufactured product
A	Exogenous unit cost for recycling a preowned product
$\lambda$	F’s fairness concern parameter
$\eta$	$\mathrm{L}’\mathrm{s} \mathrm{risk} \mathrm{aversion} \mathrm{parameter}, \mathrm{where} \eta$ ∊ (0,1)
$\theta$	Consumers’ willingness-to-pay for a new product
$\delta$	$\mathrm{Consumers}’ \mathrm{value} \mathrm{deduction} \mathrm{for} \mathrm{a} \mathrm{remanufactured} \mathrm{product}, \delta \in \left[0,1\right)$
$\rho$	The correlation coefficient between new products and remanufactured products
${\sigma }^2$	The variance of demand uncertainty
$\frac{\alpha }{\beta }$	The maximum risk-aversion parameter to participate in the game
$U\left(L\right)$	The utility function of Retailer L
$\mathsf{\Pi }\left(L\right)$	The profit function of Retailer L
$Var\left(L\right)$	The profit mean-variance function of Retailer L
${\pi }_j^i$	Profit functions of alliance j of model i, i ∊ {CC, DC, LF, ML, MF} and j = G (Model CC); M, L, F (Model DC); M, LF (Model LF); ML, F (Model ML); MF, L (Model MF), where G is the grand coalition.
$u_y^x$	$\mathrm{Utility} \mathrm{function} \mathrm{of} \mathrm{coalition} y$$$, x ∊ {CC, DC, LF, ML, MF} and y = G (Model CC); M, L, F (Model DC); M, LF (Model LF); ML, F (Model ML); MF, L (Model MF), where G is the grand coalition.

.

In this CLSC, consumers’ willingness to pay for a new product is assumed to be $\theta$, which is a uniform distribution from 0 to 1. For a remanufactured product, consumers’ willingness to pay is a portion $\delta$ of $\theta$ with $\delta \in \left[0,1\right)$. The utility of a consumer receiving from Retailer L for a new product is $t_n^L\left(\theta \right)=\theta -p_n^L$, and the utility of a consumer receiving from Retailer L for a remanufactured product is $t_r^L\left(\theta \right)=\delta \theta -p_r^L$. Following the principle of utility maximization, if $t_n^L\ge \max \left\{t_r^L, 0\right\}$, consumers will purchase a new product, leading to a new product demand function $q_n^L\left(p_n^L,p_r^L\right)=1-\frac{p_n^L-p_r^L}{1-\delta }$. If $t_r^L\ge \max \left\{t_n^L, 0\right\}$, consumers will purchase a remanufactured product, resulting in a remanufactured product demand function $q_r^L\left(p_n^L,p_r^L\right)=\frac{p_n^L-p_r^L}{1-\delta }-\frac{p_r^L}{\delta }$ [25,33,34]. Likewise, the utility of a consumer receiving from Retailer F for a new product is $t_n^F\left(\theta \right)=\theta -p_n^F$, and for a remanufactured product is $t_r^F\left(\theta \right)=\delta \theta -p_r^F$. If $t_n^F\ge \max \left\{t_r^F, 0\right\}$, a new product demand function is $q_n^F\left(p_n^F,p_r^F\right)=1-\frac{p_n^F-p_r^F}{1-\delta }$. If $t_r^F\ge \max \left\{t_n^F, 0\right\}$, a remanufactured product demand function is $q_r^F\left(p_n^F,p_r^F\right)=\frac{p_n^F-p_r^F}{1-\delta }-\frac{p_r^F}{\delta }$.

Retailer F’s fairness concern is a reaction towards adverse inequality related to the upstream agent [31]. Given this assumption, Retailer F’s fairness concern becomes unrelated in the models of CC and MF due to the absence of financial transaction between M and F. However, in Models DC, LF, and ML, Retailer F shows fairness concerns with M.

With similar arguments in studies [35,36,37,38,39,40], it is assumed that a risk-averse retailer evaluates his or her profit on the basis of a mean–variance function. The utility function $U\left(L\right)$ of Retailer L considering its profit is presented as below:

$$\begin{array}{ll}U\left(L\right)=\mathsf{\Pi }\left(L\right)-& \eta Var\left(L\right)/2\\ & =\left(p_n^L-w_n^L\right)\ast q_n^L+\left(p_r^L-w_r^L\right)\ast q_r^L-\eta {\sigma }^2[{\left(p_n^L-w_n^L\right)}^2\\ & +{\left(p_r^L-w_r^L\right)}^2+2\rho \left(p_n^L-w_n^L\right)\left(p_r^L-w_r^L\right)]/2\end{array}$$

(1)

where $\eta (>0)$ is the risk aversion parameter for Retailer L and ${\sigma }^2$ is the variance of demand uncertainty. A risk pooling effect exists [41] when the correlation coefficient between new products and remanufactured products is negative $(-1<\rho <0)$. If the risk aversion parameter is too high, the risk-averse Retailer L does not participate in the market. We assume that $0<\eta <\frac{\alpha }{\beta }$ to ensure L participates in the game.

4. The Equilibrium Analysis

The equilibrium analysis is carried out in the following and results are derived for the five base models, CC, DC, LF, ML, and MF.

4.1. The Centralized Case (Model CC)

In the centralized case, a central decision maker acts on behalf of all CLSC agents for maximization of the systematic profit as shown in Figure 1a. No financial transaction occurs between M and F, thus F’s fairness concern does not exist. The centralized decision maker sells products to the end consumers directly. All CLSC members share risks, and L does not have risk aversion in this case. The profit function is formulated as:

$$\underset{p_n^L,p_r^L,p_n^F,p_r^F}{\max }{\pi }_G^{CC}=p_n^L{q}_n^L+p_n^F{q}_n^F-c_n\left(q_n^L+q_n^F\right)+p_r^L{q}_r^L+p_r^F{q}_r^F-c_r\left(q_r^L+q_r^F\right)-A\left(q_r^L+q_r^F\right)$$

(2)

Equation (2) presents the channel profit of this CLSC as two parts: the profit for new products and the profit for remanufactured products. The centralized decision maker determines the pricing for two types of products sold to two retailers, in order to achieve the highest channel profit.

Proposition 1.

After first order derivation, we get the optimal retail prices, the resulting sales quantities, and the optimal profit in the centralized model as following (Proof See Appendix A):

$$\begin{array}{c}{p}_n^L{}^{\ast CC}=p_n^F{}^{\ast CC}=\frac{1+c_n}{2}, p_r^L{}^{\ast CC}=p_r^F{}^{\ast CC}=\frac{c_r+A+\delta }{2}, q_n^{\ast CC}=q_n^L{}^{\ast CC}+q_n^F{}^{\ast CC}=\frac{1-\delta -\left(c_n-c_r-A\right)}{1-\delta }, \hfill \\ q_r^{\ast CC}=q_r^L{}^{\ast CC}+q_r^F{}^{\ast CC}=\frac{\delta c_n-c_r-A}{\delta \left(1-\delta \right)}, and {\pi }_G^{\ast CC}=\frac{{\left(1-c_n\right)}^2}{2}+\frac{{\left(C_r+A-c_n\delta \right)}^2}{2\delta \left(1-\delta \right)} \hfill \end{array}$$

4.2. The Decentralized Case (Model DC)

In the decentralized case as shown in Figure 1b, a risk-neutral M and a risk-averse L are supposed to have no fairness concern, while F is fairness-caring with M. Besides its own profit, F also cares about its profit comparative to that of M. Similar to a large body of literature [42,43], the utility function of F is given as

$$u_F^{DC}={\pi }_F^{DC}-\lambda \left({\pi }_M^{DC}-{\pi }_F^{DC}\right)$$

(3)

where $\lambda \ge 0$ is F’s fairness concern parameter.

In this model, M maximizes its profit while L and F pursue their utility maximization. M determines wholesale prices for L and F, then L and F decide their retail prices for products to end consumers. Therefore, the Stackelberg game composed of M, L, and F is expressed as

$$\begin{array}{c}\max {\pi }_M^{DC}=\left(w_n^L-c_n\right)q_n^L+\left(w_n^F-c_n\right)q_n^F+\left(w_r^L-c_r-A\right)q_r^L+\left(w_r^F-c_r-A\right)q_r^F\hfill \\ s.t.\{\begin{array}{c}\max u_L^{DC}=\left(p_n^L-w_n^L\right)\ast q_n^L+\left(p_r^L-w_r^L\right)\ast q_r^L-\frac{\eta {\sigma }^2\left[{\left(p_n^L-w_n^L\right)}^2+{\left(p_r^L-w_r^L\right)}^2+2\rho \left(p_n^L-w_n^L\right)\left(p_r^L-w_r^L\right)\right]}{2}\\ \max u_F^{DC}={\pi }_F^{DC}-\lambda \left({\pi }_M^{DC}-{\pi }_F^{DC}\right)\end{array}\hfill \end{array}$$

(4)

where ${\pi }_F^{DC}=\left(p_n^F-w_n^F\right)q_n^F+\left(p_r^F-w_r^F\right)q_r^F$. The equilibrium results are obtained subsequently.

Proposition 2. 

In the model DC, equilibrium wholesale prices are found as$w_n^{\ast L}=\frac{c_n+1}{2}$, $w_r^{\ast L}=\frac{c_r+A+\delta }{2}$, $w_n^{\ast F}=\frac{\lambda +c_n+3\lambda c_n+1}{4\lambda +2}$, $w_r^{\ast F}=\frac{c_r+A+\delta +3\lambda c_r+3\lambda A+\lambda \delta }{4\lambda +2}$(Proof See Appendix B). The resulting equilibrium prices, sales quantities, and profits$p_n^{\ast L}, p_r^{\ast L}, p_n^{\ast F}, p_r^{\ast F}, q_n^{\ast L}, q_r^{\ast L}, q_n^{\ast F}, q_r^{\ast F}, {\pi }_M^{\ast DC}, u_L^{\ast DC}, u_F^{\ast DC}$can be obtained respectively. The corresponding results are not presented here because the expressions are too long.

4.3. L and F form an Alliance (Model LF)

In the LF model as presented in Figure 1c, L and F form an alliance as one decision maker to decide $p_n^L$, $p_r^L$, $p_n^F$, and $p_r^F$. The coalition LF has fairness concerns with M, and it is also risk averse. Therefore, this LF model is formulated as

$$\begin{array}{c}\max {\pi }_M^{LF}=\left(w_n^{LF}-c_n\right)q_n^{LF}+\left(w_r^{LF}-c_r-A\right)q_r^{LF}\hfill \\ s.t. , \max u_{LF}^{LF}=\left(p_n^{LF}-w_n^{LF}\right)\ast q_n^{LF}+\left(p_r^{LF}-w_r^{LF}\right)\ast q_r^{LF}-\eta {\sigma }^2\left[{\left(p_n^{LF}-w_n^{LF}\right)}^2+{\left(p_r^{LF}-w_r^{LF}\right)}^2+2\rho \left(p_n^{LF}-w_n^{LF}\right)\left(p_r^{LF}-w_r^{LF}\right)\right]/2-\lambda \left({\pi }_M^{LF}-{\pi }_{LF}^{LF}\right)\hfill \end{array}$$

where ${\pi }_{LF}^{LF}=\left(p_n^{LF}-w_n^{LF}\right)\ast q_n^{LF}+\left(p_r^{LF}-w_r^{LF}\right)\ast q_r^{LF}$. The equilibrium results are characterized in the following proposition.

Proposition 3. 

In the model LF, equilibrium prices are found as

$$\begin{array}{c}{p}_n^{\ast \mathit{LF}}\left(w_n^{\mathit{LF}},w_r^{\mathit{LF}}\right)=\frac{{\mathit{Kw}}_n^{\mathit{LF}}+\left({\eta \sigma }^2\rho +{\eta \sigma }^2\delta \right)w_r^{\mathit{LF}}-{\eta \sigma }^2{\lambda \delta }^2+{\eta \sigma }^2{\rho \lambda c}_r+{\eta \sigma }^2\rho \lambda A-{\eta \sigma }^2{\lambda c}_n\delta +{\eta \sigma }^2{\lambda c}_r\delta +{\eta \sigma }^2\lambda A\delta -{\eta \sigma }^2{\rho \lambda c}_n\delta +4\lambda -2{\lambda c}_n-2{\lambda }^2{c}_n+2{\lambda }^2-{\eta \sigma }^2{\delta }^2+{\eta \sigma }^2\delta +{\eta \sigma }^2\lambda \delta +2}{{\eta }^2{\sigma }^4{\rho }^2{\delta }^2-{\eta }^2{\sigma }^4{\rho }^2\delta -{\eta }^2{\sigma }^4{\delta }^2+{\eta }^2{\sigma }^4\delta +4{\eta \sigma }^2\rho \lambda \delta +4{\eta \sigma }^2\rho \delta +2{\eta \sigma }^2\lambda \delta +2{\eta \sigma }^2\lambda +2{\eta \sigma }^2\delta +2{\eta \sigma }^2+4{\lambda }^2+8\lambda +4},\hfill \\ p_r^{\mathrm{*LF}}\left(w_n^{\mathit{LF}},w_r^{\mathit{LF}}\right)=\frac{\left(K+{\eta \sigma }^2\delta -{\eta \sigma }^2\right)w_r^{\mathit{LF}}+\left({\eta \sigma }^2\delta +{\eta \sigma }^2\rho \delta \right)w_n^{\mathit{LF}}-\left(2\lambda +2{\lambda }^2+{\eta \sigma }^2\rho \lambda \delta +{\eta \sigma }^2\lambda \right)c_r+\left({\eta \sigma }^2\lambda \delta +{\eta \sigma }^2\rho \lambda \delta \right)c_n+H}{{\eta }^2{\sigma }^4{\rho }^2{\delta }^2-{\eta }^2{\sigma }^4{\rho }^2\delta -{\eta }^2{\sigma }^4{\delta }^2+{\eta }^2{\sigma }^4\delta +4{\eta \sigma }^2\rho \lambda \delta +4{\eta \sigma }^2\rho \delta +2{\eta \sigma }^2\lambda \delta +2{\eta \sigma }^2\lambda +2{\eta \sigma }^2\delta +2{\eta \sigma }^2+4{\lambda }^2+8\lambda +4}\hfill \end{array}$$

where

$$\begin{array}{c}K=2+6\lambda +4{\lambda }^2+2\eta {\sigma }^2+\eta {\sigma }^2\delta +{\eta }^2{\sigma }^4\delta +2\eta {\sigma }^2\lambda +3\eta {\sigma }^2\rho \delta +2\eta {\sigma }^2\lambda \delta +{\eta }^2{\sigma }^4{\rho }^2{\delta }^2+4\eta {\sigma }^2\rho \lambda \delta -{\eta }^2{\sigma }^4{\delta }^2-{\eta }^2{\sigma }^4{\rho }^2\delta ,\hfill \\ H=2\delta +4\lambda \delta +2{\lambda }^2\delta -2\lambda A-2{\lambda }^{2}A-\eta {\sigma }^2\lambda A-\eta {\sigma }^2\rho \delta +\eta {\sigma }^2\rho {\delta }^2+\eta {\sigma }^2\rho \lambda {\delta }^2-\eta {\sigma }^2\rho \lambda \delta -\eta {\sigma }^2\rho \lambda A\delta . \hfill \end{array}$$

(Proof See Appendix C). The resulting equilibrium wholesale prices, sales quantities, and profits, $w_n^{\ast LF},w_r^{\ast LF}, q_n^{\ast LF}, q_r^{\ast LF}, {\pi }_M^{\ast LF}, u_{LF}^{\ast LF}$, can be obtained, respectively. The specific results are not presented here because the expressions are too long.

4.4. M and L form an Alliance (Model ML)

In the ML model, as displayed in Figure 1d, M and L form an alliance and there is no financial transaction between M and L. Hence, M and L make decisions as one entity and share risks together, eliminating L’s risk aversion in this model. The alliance of M and L, as the Stackelberg leader, interacts with F in a non-cooperative setup. Consequently, the ML model is expressed as

$$\begin{array}{c}\max u_{ML}^{ML}=\left(p_n^{ML}-c_n\right)q_n^{ML}+\left(p_r^{ML}-c_r-A\right)q_r^{ML}+\left(w_n^F-c_n\right)q_n^F+\left(w_r^F-c_r-A\right)q_r^F\hfill \\ s.t. \max u_F^{ML}={\pi }_F^{ML}-\lambda \left(u_{ML}^{ML}-{\pi }_F^{ML}\right)\hfill \end{array}$$

where ${\pi }_F^{ML}=\left(p_n^F-w_n^F\right)\ast q_n^F+\left(p_r^F-w_r^F\right)\ast q_r^F$.

The alliance of M and L removes the competition between M and L as well as L’s risk aversion. The ML coalition makes joint pricing decisions $\left(p_n^{ML}, p_r^{ML}\right)$ to the final customers. The equilibrium results are attained and described in Proposition 4.

Proposition 4. 

The pricing equilibriums of Model ML are obtained as$p_n^{\ast ML}=\frac{c_n+1}{2}$, $p_r^{\ast ML}=\frac{c_r+A+\delta }{2}$, $w_n^{\ast F}=\frac{\lambda +c_n+3\lambda c_n+1}{4\lambda +2}$and$w_r^{\ast F}=\frac{c_r+A+\delta +3\lambda c_r+3\lambda A+\lambda \delta }{4\lambda +2}$(Proof See Appendix D). The resulting equilibrium prices, sales quantities, and profits,$p_n^{\ast F}$, $p_r^{\ast F}$, $q_n^{\ast F}$, $q_r^{\ast F}$, $q_n^{\ast ML}$, $q_r^{\ast ML}$, $u_F^{\ast ML}$, $u_{ML}^{\ast ML}$,can be obtained, respectively. The corresponding results are not presented here because the expressions are too long.

4.5. M and F form an Alliance (Model MF)

In the MF model, M and F form an alliance as depicted in Figure 1e. There is no financial transaction as well as no profit transfer from M to F inside this alliance, making F’s fairness concern ignorable in this model. L engages with the Stackelberg leader MF in a non-cooperative manner. As a result, Model MF is then formulated accordingly as

$$\begin{array}{c}\max u_{MF}^{MF}=\left(w_n^L-c_n\right)q_n^L+\left(p_n^{MF}-c_n\right)q_n^{MF}+\left(w_r^L-c_r-A\right)q_r^L+\left(p_r^{MF}-c_r-A\right)q_r^{MF}\hfill \\ s.t. \max u_L^{MF}=\left(p_n^L-w_n^L\right)\ast q_n^L+\left(p_r^L-w_r^L\right)\ast q_r^L-\eta {\sigma }^2\left[{\left(p_n^L-w_n^L\right)}^2+{\left(p_r^L-w_r^L\right)}^2+2\rho \left(p_n^L-w_n^L\right)\left(p_r^L-w_r^L\right)\right]/2\hfill \end{array}$$

In this case, the coalition of M and F removes the competition between M and F so that F’s fairness concern is irrelevant. A joint pricing decision $\left(p_n^{MF}, p_r^{MF}\right)$ is made by MF to the end consumers. The equilibrium results are represented in Proposition 5.

 Proposition 5. 

The pricing equilibriums of Model MF are obtained as$p_n^{\ast MF}=\frac{c_n+1}{2}$, $p_r^{\ast MF}=\frac{c_r+A+\delta }{2}$, $w_n^{\ast L}=\frac{c_n+1}{2}$ and $w_r^{\ast L}=\frac{c_r+A+\delta }{2}$ (Proof See Appendix E). The equilibrated prices, sales quantities, and utilities, $p_n^{\ast L}$, $p_r^{\ast L}$, $q_n^{\ast L}$, $q_r^{\ast L}$, $q_n^{\ast MF}$, $q_r^{\ast MF}$, $u_L^{\ast MF}$, $u_{MF}^{\ast MF}$, can be obtained, respectively. The corresponding results are not presented here because the expressions are too long.

4.6. Analytical Comparison of Resulting Equilibriums

The resulting equilibriums in Propositions 1–5 are assessed comparatively and the conclusions are drawn as the following.

 Proposition 6. 

The wholesale pricing of the five models satisfy

(1)

${\left(w_n^{\ast L}\right)}^{DC}={\left(w_n^{\ast L}\right)}^{MF}$, ${\left(w_r^{\ast L}\right)}^{DC}={\left(w_r^{\ast L}\right)}^{MF}$, ${\left(w_n^{\ast F}\right)}^{DC}={\left(w_n^{\ast F}\right)}^{ML}$, ${\left(w_r^{\ast F}\right)}^{DC}={\left(w_r^{\ast F}\right)}^{ML}$, and $\frac{\partial {\left(w_n^{\ast L}\right)}^i}{\partial \eta }=\frac{\partial {\left(w_r^{\ast L}\right)}^i}{\partial \eta }=0$ where $i\in \left\{DC,MF\right\}$.

(2)

If $A+\delta <1$, ${\left(w_n^{\ast L}\right)}^i>{\left(w_r^{\ast L}\right)}^i$ where $i\in \left\{DC,MF\right\}$.

(3)

If $\left(3\lambda +1\right)\left(c_r+A-c_n\right)+\lambda \left(\delta -1\right)+\delta <1$, ${\left(w_n^{\ast F}\right)}^i>{\left(w_r^{\ast F}\right)}^i$ where $i\in \left\{DC,ML\right\}$.

Proposition 6(1) demonstrates that the wholesale price from M to L in Model DC equals that in Model MF. It also presents that the wholesale price from M to F in Model DC is equivalent to that in Model ML. This observation is explainable that M makes the same decision of wholesale price to the downstream single party (i.e., L or F). Thus we have ${\left(w_n^{\ast L}\right)}^{DC}={\left(w_n^{\ast L}\right)}^{MF}$, ${\left(w_r^{\ast L}\right)}^{DC}={\left(w_r^{\ast L}\right)}^{MF}$, ${\left(w_n^{\ast F}\right)}^{DC}={\left(w_n^{\ast F}\right)}^{ML}$, and ${\left(w_r^{\ast F}\right)}^{DC}={\left(w_r^{\ast F}\right)}^{ML}$. In addition, M considers L’s risk aversion only when M and L form an alliance. Therefore, M makes decision of wholesale prices independently of L’s risk aversion parameter $\eta$ in the models of DC and MF. Therefore, we have $\frac{\partial {\left(w_n^{\ast L}\right)}^i}{\partial \eta }=\frac{\partial {\left(w_r^{\ast L}\right)}^i}{\partial \eta }=0$, where $i\in \left\{DC,MF\right\}$.

Proposition 6(2) is a comparison of wholesale prices for new and remanufactured products in the DC and MF models. When $A+\delta <1$, L has higher willingness to pay for new products than remanufactured products.

Proposition 6(3) compares the wholesale prices of new and remanufactured products of the DC and ML models. When $\left(3\lambda +1\right)\left(c_r+A-c_n\right)+\lambda \left(\delta -1\right)+\delta <1$, F has lower willingness to pay for remanufactured products.

 Proposition 7. 

The retail pricing of new and remanufactured products in the five models satisfy

(1)

$p_n^{\ast ML}=p_n^{\ast MF}=p_n^L{}^{\ast CC}=p_n^F{}^{\ast CC}$, $p_r^{\ast ML}=p_r^{\ast MF}=p_r^L{}^{\ast CC}=p_r^F{}^{\ast CC}$, and $\frac{\partial p_n^{\ast i}}{\partial \eta }=\frac{\partial p_r^{\ast i}}{\partial \eta }=\frac{\partial p_n^{\ast i}}{\partial \lambda }=\frac{\partial p_r^{\ast i}}{\partial \lambda }=0$ where $i\in \left\{ML,MF,CC\right\}$.

(2)

$p_n^L{}^{\ast CC}<p_n^L{}^{\ast DC}$, $p_r^L{}^{\ast CC}<p_r^L{}^{\ast DC}$

(3)

$If A+\delta <1$, $p_n^{\ast i}>p_r^{\ast i}$ where $i\in \left\{ML,MF,CC\right\}$.

Proposition 7(1) shows that the retail prices of the ML coalition, MF coalition, and centralized model are the same. F’s fairness concerns and L’s risk aversion parameter do not affect the retail prices of the ML, MF, and MLF alliances. This is clear given that L or F partners with upstream agent M in the three models.

Proposition 7(2) concludes that the retail price of L in the CC model is always lower than that in the DC model. This is due to customers having higher willingness to purchase products from L than MLF.

Proposition 7(3) is a comparison of retail prices for new and remanufactured products in the three models. When $A+\delta <1$, consumers have higher willingness to buy new products than remanufactured products.

Proposition 8. 

The profits of the five models satisfy:

$${\pi }_G^{\ast CC}>{\pi }_M^{\ast DC}+u_L^{\ast DC}+u_F^{\ast DC}.$$

Proposition 8 compares the profits of the centralized case and decentralized case. With consideration of both fairness concern and risk aversion, the profit of the centralized model is higher than that of the decentralized model.

5. Numerical Experiment

In this section, we illustrate how L’s risk aversion parameter $\eta$ and F’s fairness concern parameter $\lambda$ affect profit distribution under the five models. To obtain the schemes of profit and utility in relation to parameters $\eta$ and $\lambda$ only, we set other variables and parameters at: the variance of demand uncertainty ${\sigma }^2=0.64$, the correlation coefficient between new products and remanufactured products $\rho =-0.5$, unit production cost for a new product $c_n=1$, unit production cost for a remanufactured product $c_r=0.5$, exogenous unit cost for recycling a preowned product $A=0.1$, consumers’ value deduction for a remanufactured product $\delta =0.5$. By plugging these values into the profit and utility functions in Section 4, we obtain individual profit and utility functions with variables $\eta$and $\lambda$. The effect of L’s risk aversion parameter $\eta$ and F’s fairness concern parameter $\lambda$ on profit utility is graphically illustrated in Figure 2 and Figure 3. In the DC model, the utility of F is negative, as shown in Figure 2c, due to $\frac{{\pi }_F^{DC}}{{\pi }_M^{DC}}<\frac{\lambda }{\lambda +1}$. Figure 2 demonstrates that in the DC model, L’s risk aversion parameter $\eta$ affects all parties in different ways. L’s utility decreases linearly as $\eta$ increases. M’s profit and F’s utility have non-linear relations with $\eta$and $\lambda$.

Figure 3 visually shows how $\eta$ and $\lambda$ affect individual profit and utility under different models. Figure 2a illustrates that M’s profit in the LF model is affected by both $\eta$ and $\lambda$. Figure 2b shows that L’s profit utility in the MF model is in a negatively linear relation to its risk aversion parameter $\eta$. Figure 2c shows that F’s profit utility in the ML model is in a positive linear relation to its fairness concern parameter $\lambda$. By comparing Figure 2a and Figure 3a, it can be seen that M’s profit is significantly higher in the LF model than in the DC model. By comparing Figure 2b and Figure 3b, it can be seen that L’s utility is slightly higher in the MF model than in the DC model. By comparing Figure 2c and Figure 3c, it can be seen that F’s utility is higher in the ML model than in the DC model.

These numerical studies confirm that the decentralized case is the worst for profit optimization among all models. M, L, and F achieve higher optimal profit even when the other two parties form a coalition than when no coalition exists. In addition, it is further verified that the impact of risk aversion parameter η and fairness concern parameter λ is dynamic, not always positive or negative. The larger λ does not always lead to more profit being transferred to F. This phenomenon is different from what has been observed in supply chains [18] which consider only fairness concern but not risk aversion. The larger $\eta$, the less profit is transferred to L. It is understandable that lower risk endurance brings lower profit return.

6. Managerial Insights

The model analysis provides a framework for sustainable operational management of a CLSC. The operational sustainability of a CLSC relies on positive and consistent interactions among members. Through the incorporation of two irrational behavior factors and cooperative game approaches, CLSC participants can have a more comprehensive assessment towards pricing strategies. The cooperation or competition of members would determine the profit allocation and sustainability of a CLSC. The equilibrium analyses and numerical studies show that the profits of all members under the four cases (CC, LF, ML, MF) are higher than those of the decentralized case. This implies that members should make more efforts to maintain cooperation instead of competition. This research also demonstrates that the effects of irrational behavior factors vary due to whether they are related to wholesale prices or retail prices. The retail prices of alliances in the models of CC, ML, and MF are not affected by the two irrational behavior factors. According to past research, if only one irrational factor is considered, the higher the fairness concern level, the more profit is transferred from M to F. The higher the risk aversion parameter, the less profit is transferred from M to L. By incorporating two factors together, the effects of risk aversion and fairness concern on M’s profit are complicated. As the degree of cooperation increases, the retail prices decrease. This indicates that the manufacturer should try to cooperate with retailers in order to maintain a stable market price without being affected by risk sensitivity and fairness concern level.

7. Conclusions and Future Research

On the basis of a two-echelon CLSC composed of M, L, and F, this research takes L’s risk aversion parameter η and F’s fairness concern parameter λ into account on five different occasions: a centralized (CC), a decentralized (DC), and three partially allied models (LF, ML, and MF). Analytical comparison of the resulting equilibriums reveals that the more decentralized the CLSC, the less profit it generates. In models LF and MF, L’s risk aversion parameter η acts as a functional tool to reallocate profit between L and M. In models DC, LF, and ML where F’s fairness concern is effective, the parameter λ plays a role in re-allocating the profits between F and M. Numerical studies are conducted to investigate how L’s risk aversion parameter η and F’s fairness concern parameter λ affect profit distribution under the five models. Numerical studies confirm that M, L, and F achieve higher optimal profit even when the other two parties form a coalition than when no coalition exists. The impact of risk aversion parameter η and fairness concern parameter λ is dynamic, not always positive or negative. The larger λ does not lead to more profit transferred to F. The larger η, the less profit is transferred to L. These research results provide helpful insights for CLSC managers to find out available choices and feasible ways to reach fair profit allocations. CLSC members should make more efforts to maintain cooperation instead of competition.

Great opportunities remain for future research. This research discusses a two-echelon CLSC with three agents. It would be meaningful to expand this study into a further complicated CLSC consisting of more agents and more echelons. In addition, this research assumes that one member has fairness concerns, and one other member has risk aversions. It would be very interesting to integrate more agents’ risk aversions and fairness concerns into the CLSC and investigate their effects on profit allocations.