Optimal Decisions in an Authorized Remanufacturing Closed-Loop Supply Chain under Dual-Fairness Concerns

Abstract

This paper studies optimal decisions in an authorized remanufacturing closed-loop supply chain (CLSC) consisting of a manufacturer, a retailer, and an authorized third-party remanufacturer with dual-fairness concerns (distributional fairness concerns and peer-induced fairness concerns). Four Stackelberg game models are developed: (i) the dual-fairness concerns are considered by the retailer (model F); (ii) the retailer does not consider both types of fairness concerns (model N); (iii) the retailer only considers the distributional fairness concerns (model D); (iv) the retailer only considers the peer-induced fairness concerns (model P). We use numerical analysis to examine the equilibrium outcomes under dual-fairness concerns. The results show that: (1) The increase in the coefficient of peer-induced fairness concerns will result in more profit for the manufacturer in most cases, while distributional fairness concerns always hurt the manufacturer; (2) In most parameter cases, the increase in the degree of distributional fairness concerns favors the retailer. The retailer considers only peer-induced fairness concerns when the degree of distributional fairness concerns is low and the degree of peer-induced fairness concerns is relatively high, whereas in other cases, two kinds of fairness concerns are ignored; (3) Model P is the most profitable and model D is most disadvantageous for the third party, however, for the manufacturer it is the opposite; (4) The impact of fairness concerns on the environment depends on the retailer’s attitude towards fairness concerns. Model P is the best for the environment, while model D has the highest environmental impact. This study introduces dual-fairness concerns into the authorized remanufacturing CLSC model and provides theoretical references for authorized remanufacturing and sustainability practices.

1. Introduction

With the growth of the industries around the world, the problems of environmental pollution and natural resource scarcity are becoming increasingly serious. It is urgent to promote the harmonization of environmental protection and economic development. In terms of enterprises’ production and operation, remanufacturing, which involves collecting end-of-life products, bringing them back to like-new condition, and reselling them again [1], is a good method to achieve sustainable development. Remanufacturing is widely considered economical because it consumes fewer raw materials and energy in the manufacturing process compared to producing a new product [2,3]. Many famous original equipment manufacturers (OEMs), including IBM, General Electric, Caterpillar, Xerox, and HP, have actively participated in remanufacturing [4].

In practice, due to considerations such as high remanufacturing costs, reduced goodwill, and limited funding, OEMs usually do not engage in self-remanufacturing but choose to authorize third-party remanufacturers (TPRs) [5,6,7]. In contrast, more and more waste product collectors are expanding their remanufacturing business [8], and third-party remanufacturing has thus taken over as the primary remanufacturing strategy [5,7]. As the largest car gearbox manufacturer in China, Xin Meifu has obtained remanufacturing authorization from Aisin and ZF Friedrichshafen AG [6]. Foxconn has signed an agreement with Apple to obtain an exclusive license to remanufacture and sell iPhones in China [9]. Agrawal et al. [10] validated the rationality of TPR remanufacturing from a market perspective through behavioral experiments. They found that OEM remanufacturing and selling remanufactured products reduce consumers’ perceived value of their new products by 8%, while the emergence of third-party remanufactured products can increase the perceived value of new products by 7%. Therefore, implementing a closed-loop supply chain (CLSC) with authorized remanufacturing is significant for enterprise operations.

It is noteworthy that fairness concern behavior widely exists in supply chain cooperation [11,12]. Supply chain enterprises not only care about their own profits but may also pay attention to the profits of other supply chain members. When a company feels unfairly treated, it will make decisions that are not conducive to cooperation [13]. For example, Chinese manufacturer BBK Electronics, which owns two well-known cell phone brands, Oppo and Vivo, had reduced the trade margins provided to large stores and standalone stores in India, which led to strong opposition from numerous mobile stores, ultimately resulting in the loss of about 1000 sales stores for each of these two brands [14]. Apple once accused its chip supplier Qualcomm of charging unreasonable patent fees, and the long-standing unresolved conflict has led to lawsuits between the two companies and facilitated large-scale cooperation between Apple and Intel, which is the main competitor of Qualcomm [15].

In the one-to-many supply chain setting, on the one hand, decision makers care about the benefit distribution between themselves and their trading partners and hate getting unfavorable shares, that is, distributional fairness concerns. On the other hand, decision makers also compare themselves to similar peers in terms of profits and hate being behind, that is, peer-induced fairness concerns. Many scholars have considered these two fairness concerns together in supply chain decision-making research [16,17,18,19]. In fact, in the authorized remanufacturing CLSC system, the third party authorized by the original manufacturer to produce and sell remanufactured goods will be concerned about the profit distribution with the manufacturer, exhibiting distributional fairness concerns. On the other hand, the third party views the new product retailer as a retail peer and thus may exhibit peer-induced fairness concerns. Some scholars have introduced authorized remanufacturer’s distributional fairness concerns into the CLSC [20,21], however, they have not taken peer-induced fairness concerns into account. Therefore, it is significant to study the third party remanufacturer’s dual-fairness concerns in authorized remanufacturing CLSC decision-making.

In addition, many studies conducted research on supply chain decision-making under different attitudes of a decision-maker towards another’s fairness concerns [20,22,23], but they did not consider combining fairness concerns and authorized remanufacturing. It is worth introducing the issue of attitude choice towards another’s dual-fairness concerns from the perspective of maximizing benefits in authorized remanufacturing CLSC.

Motivated by the above discussion, we investigate the optimal decisions in an authorized remanufacturing CLSC considering third-party remanufacturer’s dual-fairness concerns and the new product retailer’s different attributes towards the dual-fairness concerns. This paper develops Stackelberg game models of the authorized remanufacturing CLSC, which contain a manufacturer, a new product retailer, and an authorized third party with dual-fairness concerns. The manufacturer manufactures new products and sells them through the retailer and authorize the third party to produce and sell remanufactured products. The manufacturer is the leader of the game, the retailer is the first follower, and the third party is the second follower. In addition, we assume that the retailer can choose whether to consider the third party’s fairness concerns. This paper mainly studies the following questions:

(1)

How do the third party’s distributional fairness concerns and peer-induced fairness concerns influence optimal decisions of the authorized remanufacturing CLSC enterprises under the retailer’s different attitudes towards the third party’s fairness concerns?

(2)

How do the authorized remanufacturing CLSC members’ profits and environmental impact change with third party’s dual-fairness concerns? How should the retailer choose its attitude towards the dual-fairness concerns of the third party to maximize profit?

The main contributions of this paper are as follows. (1) We introduce distributional fairness concerns and peer-induced fairness concerns into the authorized remanufacturing CLSC game for the first time. (2) This paper examines the influence of the third party’s dual-fairness concerns on members’ optimal decisions, profits, and environmental impact under the new product retailer’s different attitudes towards the dual-fairness concerns, and we obtain the retailer’s choice of attitude towards the third party’s dual-fairness concerns under different fairness concerns parameters based on the purpose of profit. The results provide theoretical support for reasonable decision-making of authorized remanufacturing CLSCs members and for the government to formulate corresponding supply chain environmental protection policies in the dual-fairness concerns scenarios.

The rest of this paper is structured as follows: Section 2 briefly reviews related studies. Section 3 provides the question description and model assumptions. Section 4 provides solutions for four scenarios considering the retailer’s different attitudes towards third party’s fairness concerns. In Section 5, numerical analysis is conducted to illustrate the impact of the third party’s two types of fairness concerns on equilibrium results under different scenarios. Section 6 concludes this paper.

2. Literature Review

The related literature includes two streams: (1) authorization in Remanufacturing and (2) fairness concerns in the supply chain.

2.1. Authorization in Remanufacturing

In recent years, many scholars have studied authorization in remanufacturing using the supply chain decision-making model. Zou et al. [9] developed a game between an OEM and a TPR and compared two remanufacturing operation modes of the OEM, i.e., authorization remanufacturing and outsourcing remanufacturing. They found that outsourcing is more advantageous for the OEM, while third parties prefer the authorization method when consumers have lower perceived value for remanufactured products. Hong et al. [24] studied the fixed fee licensing and royalty fee licensing under the sales and recycling competition of the manufacturer and the remanufacturer. They found that fixed fee licensing is beneficial for consumers and environmental protection, while the optimal licensing model for the manufacturers is determined by the threshold of fixed fees. Huang and Wang [4] analyzed the distributor’s market demand information sharing issue under different remanufacturing modes, i.e., manufacturer-remanufacturing, authorized third party, or distributor remanufacturing. Liu et al. [25] studied OEM’s refurbishing strategies when facing competition from a TPR while considering customers are more willing to accept authorized remanufactured products. They found that the OEM tends to choose authorization refurbishing when consumers have low preferences for refurbished products, and they suggested setting appropriate authorization fees to achieve a win-win situation for both parties. Huang and Wang [26] developed a CLSC composed of a manufacturer and an authorized third party and analyzed the impact of strategic consumer behavior and the remanufacturing cost of the third party on three remanufacturing scenarios (no-remanufacturing, full-remanufacturing, and partial-remanufacturing). Zhao et al. [27] analyzed manufacturer remanufacturing and different authorization methods for retailers in a Stackelberg game. They found that fixed fee authorization can achieve CLSC coordination when consumers’ acceptance of remanufactured products reaches a certain level. Chai et al. [28] analyzed the different strategies of the OEM in facing carbon cap-and-trade regulation and competition from remanufacturers: fixed-fee licensing, royalty licensing, and remanufacturing. They found that both kinds of licensing can coordinate two enterprises from an economic and environmental perspective under certain conditions. Zhou et al. [6] studied a CLSC game consisting of one OEM and two competing independent remanufactures (IRs) while considering the role of authorization in reducing remanufacturing costs and enhancing the acceptance of remanufactured products. They determined the conditions for the OEM to implement refurbishing authorization, for one IR to accept authorization, and for both IRs to accept authorization. Qiao and Su [29] studied the manufacturer’s licensing strategy (the royalty and fixed fee) and analyzed the channel selection of the remanufacturer (direct sales or resale) while considering customer online comments. Zhou et al. [30] constructed the model consisting of one OEM, one unauthorized remanufacturer, and one authorized remanufacturer and analyzed different situations of the decision-making power on authorization fees. Zheng and Jin [31] examined members’ decisions and profits, as well as social welfare, under two OEM relicensing schemes (authorized retailer remanufacturing and authorized third-party remanufacturing) in a CLSC game. Qin et al. [32] built three remanufacturing models: authorized remanufacturing, outsourced remanufacturing, and integrated remanufacturing. They compared the product output, carbon emissions, and corporate profits of different remanufacturing scenarios (no remanufacturing, partial remanufacturing, and complete remanufacturing). Li et al. [33] considered the case where the manufacturer’s authorization of TRP affects consumers’ willingness to pay (WTP) for both new and remanufactured products. The effects of unit new product cost, unit remanufacturing cost, and WTP on the feasibility of authorization were analyzed.

The above papers provide a solid foundation for our study, but none of them considered fairness concerns.

2.2. Fairness Concerns in Supply Chain

Many existing studies have considered fairness concerns in supply chain operations. Fehr and Schmidt [34] modeled the fairness concept as inequity aversion to character people’s resistance to unfair outcomes. Cui et al. [35] introduced fairness concerns for the first time into supply chain management based on the modeling of Fehr and Schmidt [34]. They find that in a dyadic channel consisting of a manufacturer and a retailer, a wholesale price contract can achieve channel coordination when both members are fairness concerned. Wu and Niederhoff [36] studied fairness concerns in the newsvendor model consisting of one retailer and one supplier, and they indicate that fairness concerns of the retailer can be beneficial to both parties when the demand uncertainty and its ideal allocation parameter meet certain conditions. Ma et al. [37] pointed out that the distributional fairness concerns of the retailer will reduce its marketing efforts and the manufacturer’s waste collection rate, which has a negative impact on overall efficiency. Li et al. [22] developed a two-echelon CLSC consisting of a manufacturer and a fairness concerned retailer. They indicated that both parties would gain less profits when the manufacturer does not consider the retailer’s fairness concerns than when it does. Deng et al. [23] established a three-stage government-led CLSC model considering remanufacturing subsidies and retailer’s fairness concerns. They pointed out that when the degree of fairness concerns and unit waste pollution is high, the manufacturer ignoring fairness concerns can bring more benefits for both companies through consuming high subsidies. Zhang et al. [38] considered sales efforts in green CLSC and analyzed the impact of fairness concerns on corporate interests and environmental benefits. Furthermore, some studies incorporated fairness concerns into the supply chain with authorized remanufacturing. Cao et al. [20] considered a CLSC consisting of a supplier, an assembler, and a patent-protected remanufacturer and analyzed the pricing decisions and profits of the members of the supply chain under different attitudes toward the manufacturer’s fairness concerns using a Stackelberg game. Xia et al. [21] considered remanufacturer’s fairness concerns in the authorized remanufacturing supply chain and compared different financing strategies of the OEM.

In addition, some scholars considered two kinds of fairness concerns in the field of supply chain management. Ho and Su [39] were the first to propose the concept of distributional fairness concerns and peer-induced fairness concerns. They developed ultimatum games between a leader and two followers and concluded that the second follower’s preference for peer-induced fairness is much stronger than its preference for distributional fairness. Ho et al. [40] introduced the model of He and Su [39] into a supply chain consisting of one supplier and two retailers and studied the interaction of these two types of fairness concerns in contract design. Nie and Du [19] developed a tripartite supply chain game model under distributional fairness and peer-induced fairness to analyze the coordination role of quantity contracts. Shu et al. [41] analyzed the impact of collectors’ multiple fairness concerns on a CLSC pricing decisions in both symmetric and asymmetric fairness information scenarios. They further considered the collector’s dual-fairness concerns ignored by a manufacturer. Pan et al. [17] investigated a supply chain consisting of a leading retailer and two following manufacturers and pointed out that the manufacturer’s peer-induced fairness concerns would reduce their own profits, but it would be beneficial for the retailer and overall profits. Zhong and Sun [18] considered a low-carbon supply chain consisting of a manufacturer with distributional fairness concerns and two competing retailers (one of which has peer-induced fairness concerns) and analyzed the optimal solutions under different competitive behaviors of retailers.

In sum, fairness concerns and authorized remanufacturing have been widely studied in supply chain decision-making, but few studies incorporate dual-fairness concerns into authorized remanufacturing CLSC. In addition, some studies have analyzed situations where a supply chain member’s fairness concerns are not considered by another supply chain member. However, they have not analyzed the cases where one of the two types of fairness concerns of a decision-maker is ignored. To fill the gap, this paper considers the authorized remanufacturing CLSC with a manufacturer, a retailer, and a fairness-concerned authorized third party. We develop Stackelberg game models to study the impact of third party’s dual-fairness concerns on decision-making of the authorized remanufacturing CLSC under the new product retailer’s different attitudes towards fairness concerns from the third party. This paper provides a new valuable reference for enterprises’ decision-making in authorized remanufacturing CLSC under dual-fairness concerns.

3. Problem Description and Formulation

This paper considers a CLSC containing a manufacturer, a retailer, and a third party. The manufacturer produces new products and sells them through the retailer. The third-party is authorized by the manufacturer to perform remanufacturing and remarketing operations. The manufacturer is the leader of the Stackelberg game, while the retailer and third party are followers. The sequence of events is as follows: first, the manufacturer decides the wholesale price of the new product charged to the retailer, and the unit authorized fee charged to the third party. Then, the retailer determines the quantity of the new product. Finally, the third-party sets the quantity of remanufactured products. The retailer makes decision before the third party because it has greater market influence. As an authorized remanufacturer and retailer of remanufactured products, the third party cares about profit distribution with the manufacturer, as well as the profit situation of the new product retailer. That is, the third party has distributional fairness concerns and peer-induced fairness concerns. The retailer can decide whether to consider the third party’s fairness concerns. Thus, we propose four Stackelberg game models (model F, N, D, and P). In model F, the third party’s dual-fairness concerns are considered by the retailer; in model N, the retailer ignores both kinds of fairness concerns; in model D, the retailer only considers the distributional fairness concerns; in model P, the retailer only considers the peer-induced fairness concerns.

We provide notations involved in

Table 1. Notations.

Symbols	Description
$w$	Wholesale price of new products (decision variable)
$f$	Unit authorization fee (decision variable)
$q_n$	Quantity of new products (decision variable)
$q_r$	Quantity of remanufactured products (decision variable)
$p_n$	Price of new product
$p_r$	Price of remanufactured product
$c$	Unit cost of new product
$c_r$	Unit cost of remanufactured product
$\varphi$	Market capacity
$\alpha$	Consumer WTP for the new product
$\delta$	Consumer acceptance of the remanufactured product
$k$	Scaling parameter of collection cost
$\theta$	Coefficient of distributional fairness concerns
$\lambda$	Coefficient of peer-induced fairness concerns
$e_n$$,e_r$	Environmental impact per unit of new and remanufactured products during production, respectively
$e_d$	Environmental impact of unit product uncollected by third party
$E$	The total environmental impact
${\pi }_i$	$\mathrm{Supply} \mathrm{chain} \mathrm{member} i’\mathrm{s} \mathrm{profit}, i\in \left\{m,r,t\right\}$, m is the manufacturer, r is the retailer, t is the third party
$U_t$	The utility of the third-party

:

The basic assumptions are provided as follows to facilitate the research:

1.

The market capacity is $\varphi$. Consumers’ willingness-to-pay (WTP) for a new product is $\alpha$, which follows the uniform distribution in the interval $\left[0,\varphi \right]$. Consumers’ WTP for a remanufactured product is $\alpha \delta$, where $\delta \in \left(0,1\right)$ is the consumers’ acceptance of the remanufactured product relative to the new product. Consumers’ utility from purchasing new and remanufactured products are $u_n=\alpha -p_n$ and $u_r=\alpha \delta -p_r$, respectively. If $u_n>u_r$ and $u_n>0$, the consumers will buy new products; if $u_r>u_n$ and $u_r>0$, the consumers will buy remanufactured products. The demand functions for the new and remanufactured products can be derived as follows [9,42]:

$$q_n=\varphi -{\displaystyle \frac{p_n-p_r}{1-\delta }}$$

(1)

$$q_r={\displaystyle \frac{\delta p_n-p_r}{\delta \left(1-\delta \right)}}$$

(2)

Further, the inverse demand functions can be derived as

$$p_n=\varphi -q_n-\delta q_r$$

(3)

$$p_r=\delta \left(\varphi -q_n-q_r\right)$$

(4)

2.

The unit remanufacturing cost is lower than the unit manufacturing cost of the new product [29,31], i.e., $c>c_r.$

3.

Following Xia et al. [21] and Zheng et al. [31], the third party’s collection cost is $k{\left(q_r\right)}^2/2$, $k$ is the cost scaling parameter. All the collected used products will be remanufactured.

4.

All supply chain members’ decisions are made in a single period [9,21].

5.

Referring to Xia et al. [21], we consider environmental impacts of product production and product disposal. The unit environmental impacts during the production of new and remanufactured products are $e_n$ and $e_r$ respectively. The environmental impact of unit unreturned product disposed is $e_d$. And $e_d>e_n>e_r$. The overall environmental impact is expressed as

$$E=e_n{q}_n+e_r{q}_r+e_d\left(q_n-q_r\right)$$

(5)

From the descriptions, the profits functions of the manufacturer, the retailer, and the third party are as follows:

$${\pi }_m=\left(w-c\right)q_n+f{q}_r$$

(6)

$${\pi }_r=\left(p_n-w\right)q_n=\left(\varphi -q_n-\delta q_r-w\right)q_n$$

(7)

$${\pi }_t=\left(p_r-f-c_r\right)q_r-{\displaystyle \frac{k}{2}}{q_r}^2=\left(\delta \varphi -c_r-\delta q_n-f\right)q_r-\left(\delta +{\displaystyle \frac{k}{2}}\right){q_r}^2$$

(8)

Referring to Liu et al. [16] and Pan et al. [17], the utility function of fairness concerned third party can be expressed as follows:

$$U_t={\pi }_t-\theta \left({\pi }_{m-t}-{\pi }_t\right)-\lambda \left({\pi }_r-{\pi }_t\right)=\left(1+\theta +\lambda \right){\pi }_t-\theta {\pi }_{m-t}-\lambda {\pi }_r$$

(9)

where ${\pi }_{m-t}$ is the manufacturer’s profits from authorized remanufacturing, that is $f{q}_r$. The $\theta >0$, $\lambda >0$ are the distributional fairness concerns coefficient and peer-induced fairness concerns coefficient, respectively.

In our four game models, the decision goal of the manufacturer and the retailer is to maximize their profits, while the third party decides the quantity of remanufactured products to maximize the utility.

4. Model Solution

In this section, provide solution procedures for models.

4.1. The Retailer Considers the Third Party’s Dual-Fairness Concerns (Model F)

In this model, the retailer considers the third party’s both types of fairness concerns when making decision. According to backward induction, we first derive the third party’s optimal response to the decisions of the manufacturer and the retailer. Substituting Equations (6)–(8) and ${\pi }_{m-t}=f{q}_r$ into Equation (9), the utility function of the third party is shown as:

$$\begin{array}{l}{U}_t=\left(1+\theta +\lambda \right)\left(\left(\delta \varphi -c_r-\delta q_n-f\right)q_r-\left(\delta +{\displaystyle \frac{k}{2}}\right){q_r}^2\right)\\ -\theta f{q}_r-\lambda \left(\varphi -q_n-\delta q_r-w\right)q_n\end{array}$$

(10)

Proposition 1. 

The fairness concerned third party’s best response function is:

$$q_r={\displaystyle \frac{\left(1+\theta +\lambda \right)\left(\delta \varphi -c_r\right)-\left(1+\theta \right)\delta q_n-\left(1+2\theta +\lambda \right)f}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}$$

(11)

Proof. 

From Equation (10), since ${\displaystyle \frac{{\partial }^2{U}_t}{\partial {q_r}^2}}=-\left(1+\theta +\lambda \right)\left(2\delta +k\right)<0$, $U_t$ is concave in $q_r$. By solving ${\displaystyle \frac{\partial U_t}{\partial q_r}}=0$, Equation (11) can be obtained. □

By knowing the fairness concerned third party’s best response function, the retailer determines the number of new products that can maximize its profits as the best response to the manufacturer’s decision. The retailer’s profit function can be shown as:

$${\pi }_r^F=\left(\varphi -{\displaystyle \frac{\left(\delta \varphi -c_r\right)\delta }{\left(2\delta +k\right)}}-\left(1-{\displaystyle \frac{\left(1+\theta \right){\delta }^2}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}\right)q_n+{\displaystyle \frac{\left(1+2\theta +\lambda \right)\delta f}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}-w\right)q_n$$

(12)

Proposition 2. 

In model F, the retailer’s best response function is as follows:

$$q_n^F={\displaystyle \frac{\left(1+\theta +\lambda \right)A_1+\left(1+2\theta +\lambda \right)\delta f-\left(1+\theta +\lambda \right)\left(2\delta +k\right)w}{2A_2}}$$

(13)

where $A_1=\left(2\delta +k\right)\varphi -\left(\delta \varphi -c_r\right)\delta$; A₂ = (1 + θ + λ)(2δ + k) − (1 + θ)δ²

Proof. 

According to Equation (12), ${\displaystyle \frac{{\partial }^2{\pi }_r^F}{\partial {q_r}^2}}=-{\displaystyle \frac{2A_2}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}<0$, thus, ${\pi }_r^F$ is concave in $q_n$. By solving ${\displaystyle \frac{\partial {\pi }_r^F}{\partial q_n}}=0$, the retailer’s best response in model F can be obtained. □

Substituting Equation (13) into Equation (11), we can rewrite the third party’s response function as:

$$q_r^F=\left(\begin{array}{l}\left(1+\theta +\lambda \right)\left(\delta \varphi -c_r\right)-\left(1+2\theta +\lambda \right)f\\ -{\displaystyle \frac{\left(1+\theta \right)\delta \left(\begin{array}{l}\left(1+\theta +\lambda \right)A_1+\left(1+2\theta +\lambda \right)\delta f\\ -\left(1+\theta +\lambda \right)\left(2\delta +k\right)w\end{array}\right)}{2A_2}}\end{array}\right){\displaystyle \frac{1}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}$$

(14)

By knowing the responses of two followers, the manufacturer determines the optimal wholesale price and the optimal authorization fee. Substituting Equations (13) and (14) into Equation (6), the manufacturer’s profits function is shown as:

$$\begin{array}{l}{\pi }_m^F=\left(w-c\right){\displaystyle \frac{\left(\begin{array}{l}\left(1+\theta +\lambda \right)A_1+\left(1+2\theta +\lambda \right)\delta f\\ -\left(1+\theta +\lambda \right)\left(2\delta +k\right)w\end{array}\right)}{2A_2}}\\ +\left(\begin{array}{l}\left(1+\theta +\lambda \right)\left(\delta \varphi -c_r\right)-\left(1+2\theta +\lambda \right)f\\ -{\displaystyle \frac{\left(1+\theta \right)\delta \left(\begin{array}{l}\left(1+\theta +\lambda \right)A_1+\left(1+2\theta +\lambda \right)\delta f\\ -\left(1+\theta +\lambda \right)\left(2\delta +k\right)w\end{array}\right)}{2A_2}}\end{array}\right){\displaystyle \frac{f}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}\end{array}$$

(15)

Proposition 3. 

In model F, the manufacturer’s optimal authorization fee and wholesale price are:

$$f^{F\ast }=R_1\left(1+\theta +\lambda \right)$$

(16)

$$w^{F\ast }={\displaystyle \frac{A_1+\left(2+3\theta +\lambda \right)\delta R_1}{2\left(2\delta +k\right)}}+{\displaystyle \frac{c}{2}}$$

(17)

where $R_1={\displaystyle \frac{4A_2\left(\delta \varphi -c_r\right)-\left(\theta +\lambda \right)\left(2\delta +k\right)\delta c+\left(\theta +\lambda \right)\delta A_1}{4\left(1+2\theta +\lambda \right)\left(2A_2+\left(1+\theta \right){\delta }^2\right)-{\left(2+3\theta +\lambda \right)}^2{\delta }^2}}$.

Proof. 

According to Equation (15), we formulate the Hessian matrix of ${\pi }_m^F$: $H=\left|\begin{array}{cc}{\displaystyle \frac{{\partial }^2{\pi }_m^F}{\partial f^2}}& {\displaystyle \frac{{\partial }^2{\pi }_m^F}{\partial f\partial w}}\\ {\displaystyle \frac{{\partial }^2{\pi }_m^F}{\partial w\partial f}}& {\displaystyle \frac{{\partial }^2{\pi }_m^F}{\partial w^2}}\end{array}\right|=\left|\begin{array}{cc}-\left(2+{\displaystyle \frac{\left(1+\theta \right){\delta }^2}{A_2}}\right){\displaystyle \frac{\left(1+2\theta +\lambda \right)}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}& {\displaystyle \frac{\left(2+3\theta +\lambda \right)\delta }{2A_2}}\\ {\displaystyle \frac{\left(2+3\theta +\lambda \right)\delta }{2A_2}}& -{\displaystyle \frac{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}{A_2}}\end{array}\right|$

Since $-\left(2+{\displaystyle \frac{\left(1+\theta \right){\delta }^2}{A_2}}\right){\displaystyle \frac{\left(1+2\theta +\lambda \right)}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}<0$ and $4\left(\left(1+\theta \right){\delta }^2+2A_2\right)\left(1+2\theta +\lambda \right)-{\left(2+3\theta +\lambda \right)}^2{\delta }^2>0$, the matrix is negative definite. Thus,${\pi }_m^F$ is jointly concave in $w$ and $f$. By solving ${\displaystyle \frac{\partial {\pi }_m^F}{\partial f}}=0$ and ${\displaystyle \frac{\partial {\pi }_m^F}{\partial w}}=0$ simultaneously, the manufacturer’s optimal decisions, $w^{F\ast }$ and $f^{F\ast }$ can be obtained. □

From Equations (13), (14), (16), and (17), we obtain the optimal decisions of two followers:

$$q_n^{F\ast }=R_2\left(1+\theta +\lambda \right)$$

(18)

$$q_r^{F\ast }={\displaystyle \frac{R_3}{\left(2\delta +k\right)}}$$

(19)

where $R_2={\displaystyle \frac{A_1+\left(\theta +\lambda \right)\delta R_1-\left(2\delta +k\right)c}{4A_2}}$; $R_3=\left(\delta \varphi -c_r\right)-\left(1+2\theta +\lambda \right)R_1-\left(1+\theta \right)\delta R_2$.

Based on optimal decisions, products’ prices, supply members’ optimal profits, and environmental impact in model F can be derived as follows:

$$p_n^{F\ast }=\varphi -R_2\left(1+\theta +\lambda \right)-{\displaystyle \frac{\delta R_3}{\left(2\delta +k\right)}}$$

(20)

$$p_r^{F\ast }=\delta \left(\varphi -R_2\left(1+\theta +\lambda \right)-{\displaystyle \frac{R_3}{\left(2\delta +k\right)}}\right)$$

(21)

$${\pi }_m^{F\ast }=\left({\displaystyle \frac{A_1+\left(2+3\theta +\lambda \right)\delta R_1}{2\left(2\delta +k\right)}}-{\displaystyle \frac{c}{2}}\right)R_2\left(1+\theta +\lambda \right)+{\displaystyle \frac{R_1{R}_3\left(1+\theta +\lambda \right)}{\left(2\delta +k\right)}}$$

(22)

$${\pi }_r^{F\ast }=\left(\varphi -R_2\left(1+\theta +\lambda \right)-{\displaystyle \frac{\delta R_3}{\left(2\delta +k\right)}}-{\displaystyle \frac{A_1+\left(2+3\theta +\lambda \right)\delta R_1}{2\left(2\delta +k\right)}}-{\displaystyle \frac{c}{2}}\right)R_2\left(1+\theta +\lambda \right)$$

(23)

$${\pi }_t^{F\ast }=\left(\delta \varphi -c_r-\left(1+\theta +\lambda \right)\left(\delta R_2+R_1\right)\right){\displaystyle \frac{R_3}{\left(2\delta +k\right)}}-\left(\delta +{\displaystyle \frac{k}{2}}\right){\left({\displaystyle \frac{R_3}{\left(2\delta +k\right)}}\right)}^2$$

(24)

$$E^F=\left(e_n+e_d\right)R_2\left(1+\theta +\lambda \right)+{\displaystyle \frac{\left(e_r-e_d\right)R_3}{\left(2\delta +k\right)}}$$

(25)

4.2. The Retailer Does Not Consider the Third Party’s Both Types of Fairness Concerns (Model N)

In this model, the third party’s both types of fairness concerns are ignored by the retailer. The retailer makes the decision with the assumption that the third party determines the quantity of remanufactured products to maximize its own profits. Hence, letting $\theta$ and $\lambda$ in Equation (13) be equal to zero, we obtain the retailer’s best response function in the mode N as follows:

$$q_n^N={\displaystyle \frac{A_1+\delta f-\left(2\delta +k\right)w}{2A_3}}$$

(26)

where $A_3=\left(2\delta +k\right)-{\delta }^2$

Substituting Equation (26) into Equation (11), we can rewrite the third party’s response function as:

$$q_r^N=\left(\begin{array}{l}\left(1+\theta +\lambda \right)\left(\delta \varphi -c_r\right)-{\displaystyle \frac{\left(1+\theta \right)\delta \left(A_1+\delta f-\left(2\delta +k\right)w\right)}{2A_3}}\\ -\left(1+2\theta +\lambda \right)f\end{array}\right){\displaystyle \frac{1}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}$$

(27)

By knowing the best responses of two followers, given by Equations (26) and (27), the manufacturer’s profit function by substituting them into Equation (6) is expressed as:

$$\begin{array}{l}{\pi }_m^N=\left(w-c\right){\displaystyle \frac{A_1+\delta f-\left(2\delta +k\right)w}{2A_3}}\\ +\left(\begin{array}{l}\left(1+\theta +\lambda \right)\left(\delta -c_r\right)-\left(1+2\theta +\lambda \right)f\\ -{\displaystyle \frac{\left(1+\theta \right)\delta \left(A_1+\delta f-\left(2\delta +k\right)w\right)}{2A_3}}\end{array}\right){\displaystyle \frac{f}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}\end{array}$$

(28)

Proposition 4. 

In model N, the manufacturer’s optimal authorization fee and wholesale price are:

$$f^{N\ast }=R_4\left(1+\theta +\lambda \right)$$

(29)

$$w^{N\ast }={\displaystyle \frac{A_1+\left(2+2\theta +\lambda \right)\delta R_4}{2\left(2\delta +k\right)}}+{\displaystyle \frac{c}{2}}$$

(30)

where $R_4={\displaystyle \frac{\left(\lambda \delta A_1-\lambda \left(2\delta +k\right)\delta c\right)+4\left(1+\theta +\lambda \right)A_3\left(\delta \varphi -c_r\right)}{4\left(1+\theta +\lambda \right)\left(\left(1+\theta \right){\delta }^2+2A_3\left(1+2\theta +\lambda \right)\right)-{\left(2+2\theta +\lambda \right)}^2{\delta }^2}}$

Proof. 

According to Equation (28), we formulate the Hessian matrix of ${\pi }_m^N$: $H=\left|\begin{array}{cc}{\displaystyle \frac{{\partial }^2{\pi }_m^N}{\partial f^2}}& {\displaystyle \frac{{\partial }^2{\pi }_m^N}{\partial f\partial w}}\\ {\displaystyle \frac{{\partial }^2{\pi }_m^N}{\partial w\partial f}}& {\displaystyle \frac{{\partial }^2{\pi }_m^N}{\partial w^2}}\end{array}\right|=\left|\begin{array}{cc}-{\displaystyle \frac{\left(1+\theta \right){\delta }^2+2\left(1+2\theta +\lambda \right)A_3}{\left(2\delta +k\right)\left(1+\theta +\lambda \right)A_3}}& {\displaystyle \frac{\left(2+2\theta +\lambda \right)\delta }{2A_3\left(1+\theta +\lambda \right)}}\\ {\displaystyle \frac{\left(2+2\theta +\lambda \right)\delta }{2A_3\left(1+\theta +\lambda \right)}}& -{\displaystyle \frac{\left(2\delta +k\right)}{A_3}}\end{array}\right|$

Since $-{\displaystyle \frac{\left(1+\theta \right){\delta }^2+2\left(1+2\theta +\lambda \right)A_3}{\left(2\delta +k\right)\left(1+\theta +\lambda \right)A_3}}<0$ and ${\displaystyle \frac{\left(1+\theta \right){\delta }^2+2\left(1+2\theta +\lambda \right)A_3}{\left(1+\theta +\lambda \right){A_3}^2}}-{\displaystyle \frac{{\left(2+2\theta +\lambda \right)}^2{\delta }^2}{4{A_3}^2{\left(1+\theta +\lambda \right)}^2}}>0$, the matrix is negative definite. Thus, ${\pi }_m^N$ is jointly concave in $w$ and $f$. By solving ${\displaystyle \frac{\partial {\pi }_m^N}{\partial f}}=0$ and ${\displaystyle \frac{\partial {\pi }_m^N}{\partial w}}=0$ simultaneously, the manufacturer’s optimal decisions, $w^{N\ast }$ and $f^{N\ast }$ can be obtained. □

From Equations (26), (27), (29) and (30), we obtain the optimal decisions of two followers:

$$q_n^{N\ast }={\displaystyle \frac{R_5}{4A_3}}$$

(31)

$$q_r^{N\ast }={\displaystyle \frac{R_6}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}$$

(32)

where $R_5=A_1+\delta R_4\lambda -\left(2\delta +k\right)c$; $R_6=\left(1+\theta +\lambda \right)\left(\delta \varphi -c_r\right)-{\displaystyle \frac{\left(1+\theta \right)\delta R_5}{4A_3}}-\left(1+2\theta +\lambda \right)\left(1+\theta +\lambda \right)R_4$.

Based on these optimal decisions, products’ prices, supply members’ optimal profits, and environmental impact in model N can be derived:

$$p_n^{N\ast }=\varphi -{\displaystyle \frac{R_5}{4A_3}}-{\displaystyle \frac{\delta R_6}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}$$

(33)

$$p_r^{N\ast }=\delta \left(\varphi -{\displaystyle \frac{R_5}{4A_3}}-{\displaystyle \frac{R_6}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}\right)$$

(34)

$${\pi }_m^{N\ast }=\left({\displaystyle \frac{A_1+\left(2+2\theta +\lambda \right)\delta R_4}{2\left(2\delta +k\right)}}-{\displaystyle \frac{c}{2}}\right){\displaystyle \frac{R_5}{4A_3}}+{\displaystyle \frac{R_4{R}_6}{\left(2\delta +k\right)}}$$

(35)

$${\pi }_r^{N\ast }=\left(\varphi -{\displaystyle \frac{R_5}{4A_3}}-{\displaystyle \frac{\delta R_6}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}-{\displaystyle \frac{A_1+\left(2+2\theta +\lambda \right)\delta R_4}{2\left(2\delta +k\right)}}-{\displaystyle \frac{c}{2}}\right){\displaystyle \frac{R_5}{4A_3}}$$

(36)

$${\pi }_t^{N\ast }=\left(\delta \varphi -c_r-{\displaystyle \frac{\delta R_5}{4A_3}}-R_4\left(1+\theta +\lambda \right)\right){\displaystyle \frac{R_6}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}-\left(\delta +{\displaystyle \frac{k}{2}}\right){\left({\displaystyle \frac{R_6}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}\right)}^2$$

(37)

$$E^N={\displaystyle \frac{\left(e_n+e_d\right)R_5}{4A_3}}+{\displaystyle \frac{\left(e_r-e_d\right)R_6}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}$$

(38)

4.3. The Retailer Only Considers the Third Party’s Distributional Fairness Concerns (Model D)

In this model, the third party’s peer-induced fairness concerns are ignored by the manufacturer. The retailer makes its decision with the assumption that the third party only has distributional fairness concerns. Hence, letting $\lambda$ in Equation (13) be equal to zero, we obtain the retailer’s best response function in the mode D as follows:

$$q_n^D={\displaystyle \frac{A_1\left(1+\theta \right)+\left(1+2\theta \right)\delta f-\left(1+\theta \right)\left(2\delta +k\right)w}{2\left(1+\theta \right)A_3}}$$

(39)

Substituting Equation (39) into Equation (11), we can rewrite the third party’s response function in model D as:

$$q_r^D=\left(\begin{array}{l}\left(1+\theta +\lambda \right)\left(\delta \varphi -c_r\right)-\left(1+2\theta +\lambda \right)f\\ -\delta {\displaystyle \frac{A_1\left(1+\theta \right)+\left(1+2\theta \right)\delta f-\left(1+\theta \right)\left(2\delta +k\right)w}{2A_3}}\end{array}\right){\displaystyle \frac{1}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}$$

(40)

Substituting Equations (39) and (40) into Equation (6), we can rewrite the manufacturer’s profit function as:

$$\begin{array}{l}{\pi }_m^D=\left(w-c\right){\displaystyle \frac{A_1\left(1+\theta \right)+\left(1+2\theta \right)\delta f-\left(1+\theta \right)\left(2\delta +k\right)w}{2\left(1+\theta \right)A_3}}\\ +\left(\begin{array}{l}\left(\delta \varphi -c_r\right)\left(1+\theta +\lambda \right)-\left(1+2\theta +\lambda \right)f\\ -{\displaystyle \frac{\left(A_1\left(1+\theta \right)+\left(1+2\theta \right)\delta f-\left(1+\theta \right)\left(2\delta +k\right)w\right)\delta }{2A_3}}\end{array}\right){\displaystyle \frac{f}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}\end{array}$$

(41)

Proposition 5. 

In model D, the manufacturer’s optimal wholesale price and the authorization fee are:

$$w^{D\ast }={\displaystyle \frac{A_1+A_4{R}_7}{2\left(2\delta +k\right)}}+{\displaystyle \frac{c}{2}}$$

(42)

$$f^{D\ast }=R_7\left(1+\theta +\lambda \right)\left(1+\theta \right)$$

(43)

where $A_4=\left(1+\theta +\lambda \right)\left(1+2\theta \right)\delta +{\left(1+\theta \right)}^2\delta$; $R_7={\displaystyle \frac{\left(\begin{array}{l}\left(\left(1+\theta +\lambda \right)\left(1+2\theta \right)-{\left(1+\theta \right)}^2\right)\left(\delta A_1-\left(2\delta +k\right)\delta c\right)\\ +4\left(\delta \varphi -c_r\right)\left(1+\theta +\lambda \right)\left(1+\theta \right)A_3\end{array}\right)}{\left(\begin{array}{l}8{\left(1+\theta \right)}^2\left(1+\theta +\lambda \right)\left(1+2\theta +\lambda \right)A_3\\ -{\left(\left(1+\theta +\lambda \right)\left(1+2\theta \right)-{\left(1+\theta \right)}^2\right)}^2{\delta }^2\end{array}\right)}}$.

Proof. 

According to Equation (41), we formulate the Hessian matrix of ${\pi }_m^D$: $H=\left|\begin{array}{cc}{\displaystyle \frac{{\partial }^2{\pi }_m^D}{\partial f^2}}& {\displaystyle \frac{{\partial }^2{\pi }_m^D}{\partial f\partial w}}\\ {\displaystyle \frac{{\partial }^2{\pi }_m^D}{\partial w\partial f}}& {\displaystyle \frac{{\partial }^2{\pi }_m^D}{\partial w^2}}\end{array}\right|=\left|\begin{array}{cc}-\left(\begin{array}{l}{\displaystyle \frac{2\left(1+2\theta +\lambda \right)}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}\\ +{\displaystyle \frac{\left(1+2\theta \right){\delta }^2}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)A_3}}\end{array}\right)& \left({\displaystyle \frac{\left(1+2\theta \right)}{\left(1+\theta \right)}}+{\displaystyle \frac{\left(1+\theta \right)}{\left(1+\theta +\lambda \right)}}\right){\displaystyle \frac{\delta }{2A_3}}\\ \left({\displaystyle \frac{\left(1+2\theta \right)}{\left(1+\theta \right)}}+{\displaystyle \frac{\left(1+\theta \right)}{\left(1+\theta +\lambda \right)}}\right){\displaystyle \frac{\delta }{2A_3}}& -{\displaystyle \frac{\left(2\delta +k\right)}{A_3}}\end{array}\right|$

Since $-\left(2\left(1+2\theta +\lambda \right)+{\displaystyle \frac{\left(1+2\theta \right){\delta }^2}{A_3}}\right){\displaystyle \frac{1}{\left(1+\theta +\lambda \right)\left(2\delta +C\right)}}<0$ and $\left(2\left(1+2\theta +\lambda \right)+{\displaystyle \frac{\left(1+2\theta \right){\delta }^2}{A_3}}\right){\displaystyle \frac{1}{\left(1+\theta +\lambda \right)A_3}}-{\left(\left({\displaystyle \frac{\left(1+2\theta \right)}{\left(1+\theta \right)}}+{\displaystyle \frac{\left(1+\theta \right)}{\left(1+\theta +\lambda \right)}}\right){\displaystyle \frac{\delta }{2A_3}}\right)}^2>0$, the matrix is negative definite. Thus, ${\pi }_m^D$ is jointly concave in $w$ and $f$. By solving ${\displaystyle \frac{\partial {\pi }_m^D}{\partial f}}=0$ and ${\displaystyle \frac{\partial {\pi }_m^D}{\partial w}}=0$ simultaneously, the manufacturer’s optimal decisions, $w^{D\ast }$ and $f^{D\ast }$ can be obtained. □

From Equations (39), (40), (42), and (43), we obtain the optimal decisions of two followers:

$$q_n^{D\ast }={\displaystyle \frac{R_8}{2A_3}}$$

(44)

$$q_r^{D\ast }={\displaystyle \frac{R_9}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}$$

(45)

where $R_8={\displaystyle \frac{1}{2}}\left(A_1-A_4{R}_7-\left(2\delta +k\right)c\right)+\left(1+2\theta \right)\delta R_7\left(1+\theta +\lambda \right)$; $R_9=\left(1+\theta +\lambda \right)\left(\delta \varphi -c_r\right)-{\displaystyle \frac{\delta \left(1+\theta \right)R_8}{2A_3}}-\left(1+2\theta +\lambda \right)R_7\left(1+\theta +\lambda \right)\left(1+\theta \right)$.

Based on the optimal decisions, products’ prices, supply members’ optimal profits, and environmental impact in model D can be derived:

$$p_n^{D\ast }=\varphi -{\displaystyle \frac{R_8}{2A_3}}-{\displaystyle \frac{\delta R_9}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}$$

(46)

$$p_r^{D\ast }=\delta \left(\varphi -{\displaystyle \frac{R_8}{2A_3}}-{\displaystyle \frac{R_9}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}\right)$$

(47)

$${\pi }_m^{D\ast }=\left({\displaystyle \frac{A_1+A_4{R}_7}{2\left(2\delta +k\right)}}-{\displaystyle \frac{c}{2}}\right){\displaystyle \frac{R_8}{2A_3}}+{\displaystyle \frac{\left(1+\theta \right)R_7{R}_9}{\left(2\delta +k\right)}}$$

(48)

$${\pi }_r^{D\ast }=\left(\varphi -{\displaystyle \frac{R_8}{2A_3}}-{\displaystyle \frac{\delta R_9}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}-{\displaystyle \frac{A_1+A_4{R}_7}{2\left(2\delta +k\right)}}-{\displaystyle \frac{c}{2}}\right){\displaystyle \frac{R_8}{2A_3}}$$

(49)

$$\begin{array}{l}{\pi }_t^{D\ast }=\left(\delta \varphi -c_r-{\displaystyle \frac{\delta R_8}{2A_3}}-R_7\left(1+\theta +\lambda \right)\left(1+\theta \right)\right){\displaystyle \frac{R_9}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}\\ -\left(\delta +{\displaystyle \frac{k}{2}}\right){\left({\displaystyle \frac{R_9}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}\right)}^2\end{array}$$

(50)

$$E^D={\displaystyle \frac{\left(e_n+e_d\right)R_8}{2A_3}}+{\displaystyle \frac{\left(e_r-e_d\right)R_9}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}$$

(51)

4.4. The Retailer Only Considers the Third-Party’s Peer-Induced Fairness Concerns (Model P)

In this model, the third party’s distributional fairness concerns are ignored by the manufacturer. The retailer makes its decision with the assumption that the third party only has peer-induced fairness concerns. Hence, letting $\theta$ in Equation (13) be equal to zero, we obtain the retailer’s best response function in the mode P as follows:

$$q_n^P={\displaystyle \frac{\left(1+\lambda \right)\left(A_1+\delta f-\left(2\delta +k\right)w\right)}{2A_5}}$$

(52)

where $A_5=\left(1+\lambda \right)\left(2\delta +k\right)-{\delta }^2$.

Substituting Equation (52) into Equation (13), we can rewrite the third-party’s response function in model P as:

$$q_r^P=\left(\begin{array}{l}\left(1+\theta +\lambda \right)\left(\delta \varphi -c_r\right)-\left(1+2\theta +\lambda \right)f\\ -\left(1+\theta \right)\delta {\displaystyle \frac{\left(1+\lambda \right)\left(A_1+\delta f-\left(2\delta +k\right)w\right)}{2A_5}}\end{array}\right){\displaystyle \frac{1}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}$$

(53)

Substituting Equations (52) and (53) into Equation (6), we can rewrite the manufacturer’s profit function as:

$$\begin{array}{l}{\pi }_m^P=\left(w-c\right){\displaystyle \frac{\left(1+\lambda \right)\left(A_1+\delta f-\left(2\delta +k\right)w\right)}{2A_5}}\\ +\left(\begin{array}{l}\left(1+\theta +\lambda \right)\left(\delta \varphi -c_r\right)-\left(1+2\theta +\lambda \right)f\\ -{\displaystyle \frac{\left(1+\theta \right)\left(1+\lambda \right)\delta \left(A_1+\delta f-\left(2\delta +k\right)w\right)}{2A_5}}\end{array}\right){\displaystyle \frac{f}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}\end{array}$$

(54)

Proposition 6. 

In model P, the manufacturer’s optimal wholesale price and the authorization fee are:

$$f^{P\ast }=R_{10}\left(1+\theta +\lambda \right)$$

(55)

$$w^{P\ast }={\displaystyle \frac{A_1+\left(2+2\theta +\lambda \right)\delta R_{10}}{2\left(2\delta +k\right)}}+{\displaystyle \frac{c}{2}}$$

(56)

where $R_{10}={\displaystyle \frac{\lambda \left(1+\lambda \right)\delta A_1-\lambda \left(1+\lambda \right)\left(2\delta +k\right)\delta c+4A_5\left(1+\theta +\lambda \right)\left(\delta \varphi -c_r\right)}{4\left(1+\theta +\lambda \right)\left(2\left(1+2\theta +\lambda \right)A_5+\left(1+\theta \right)\left(1+\lambda \right){\delta }^2\right)-{\left(2+2\theta +\lambda \right)}^2\left(1+\lambda \right){\delta }^2}}$.

Proof. 

According to Equation (54), we formulate the Hessian matrix of ${\pi }_m^P$: $H=\left|\begin{array}{cc}{\displaystyle \frac{{\partial }^2{\pi }_m^P}{\partial f^2}}& {\displaystyle \frac{{\partial }^2{\pi }_m^P}{\partial f\partial w}}\\ {\displaystyle \frac{{\partial }^2{\pi }_m^P}{\partial w\partial f}}& {\displaystyle \frac{{\partial }^2{\pi }_m^P}{\partial w^2}}\end{array}\right|=\left|\begin{array}{cc}-\left({\displaystyle \frac{2\left(1+2\theta +\lambda \right)}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}+{\displaystyle \frac{\left(1+\theta \right)\left(1+\lambda \right){\delta }^2}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)A_5}}\right)& {\displaystyle \frac{\left(2+2\theta +\lambda \right)\left(1+\lambda \right)\delta }{2\left(1+\theta +\lambda \right)A_5}}\\ {\displaystyle \frac{\left(2+2\theta +\lambda \right)\left(1+\lambda \right)\delta }{2\left(1+\theta +\lambda \right)A_5}}& -{\displaystyle \frac{\left(1+\lambda \right)\left(2\delta +k\right)}{A_5}}\end{array}\right|$

Since $-\left({\displaystyle \frac{2\left(1+2\theta +\lambda \right)}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)}}+{\displaystyle \frac{\left(1+\theta \right)\left(1+\lambda \right){\delta }^2}{\left(1+\theta +\lambda \right)\left(2\delta +k\right)A_5}}\right)<0$ and $\left(\begin{array}{l}2\left(1+2\theta +\lambda \right)\\ +{\displaystyle \frac{\left(1+\theta \right)\left(1+\lambda \right){\delta }^2}{A_5}}\end{array}\right){\displaystyle \frac{\left(1+\lambda \right)}{\left(1+\theta +\lambda \right)A_5}}-{\left({\displaystyle \frac{\left(2+2\theta +\lambda \right)\left(1+\lambda \right)\delta }{2\left(1+\theta +\lambda \right)A_5}}\right)}^2>0$, the matrix is negative definite. Thus, ${\pi }_m^P$ is jointly concave in $w$ and $f$. By solving ${\displaystyle \frac{\partial {\pi }_m^P}{\partial f}}=0$ and ${\displaystyle \frac{\partial {\pi }_m^P}{\partial w}}=0$ simultaneously, the manufacturer’s optimal decisions, $w^{D\ast }$ and $f^{D\ast }$ can be obtained. □

From Equations (52), (53), (55), and (56), we obtain the optimal decisions of two followers:

$$q_n^{P\ast }={\displaystyle \frac{\left(1+\lambda \right)R_{11}}{4A_5}}$$

(57)

$$q_r^{P\ast }={\displaystyle \frac{R_{12}}{\left(2\delta +k\right)}}$$

(58)

where $R_{11}=\left(A_1+\lambda \delta R_{10}-\left(2\delta +k\right)c\right)$; $R_{12}=\left(\delta \varphi -c_r\right)-{\displaystyle \frac{\left(1+\lambda \right)\left(1+\theta \right)\delta R_{11}}{4\left(1+\theta +\lambda \right)A_5}}-\left(1+2\theta +\lambda \right)R_{10}$.

Based on the optimal decisions, products’ prices, supply members’ optimal profits, and environmental impact in model P can be derived:

$$p_n^{P\ast }=\varphi -{\displaystyle \frac{\left(1+\lambda \right)R_{11}}{4A_5}}-{\displaystyle \frac{\delta R_{12}}{\left(2\delta +k\right)}}$$

(59)

$$p_r^{P\ast }=\delta \left(\varphi -{\displaystyle \frac{\left(1+\lambda \right)R_{11}}{4A_5}}-{\displaystyle \frac{R_{12}}{\left(2\delta +k\right)}}\right)$$

(60)

$${\pi }_m^{P\ast }=\left({\displaystyle \frac{A_1+\left(2+2\theta +\lambda \right)\delta R_{10}}{\left(2\delta +k\right)}}-c\right){\displaystyle \frac{\left(1+\lambda \right)R_{11}}{8A_5}}+{\displaystyle \frac{R_{10}{R}_{12}\left(1+\theta +\lambda \right)}{\left(2\delta +k\right)}}$$

(61)

$${\pi }_r^{P\ast }=\left(\varphi -{\displaystyle \frac{\left(1+\lambda \right)R_{11}}{4A_5}}-{\displaystyle \frac{\delta R_{12}}{\left(2\delta +k\right)}}-{\displaystyle \frac{A_1+\left(2+2\theta +\lambda \right)\delta R_{10}}{2\left(2\delta +k\right)}}-{\displaystyle \frac{c}{2}}\right){\displaystyle \frac{\left(1+\lambda \right)R_{11}}{4A_5}}$$

(62)

$${\pi }_t^{P\ast }=\left(\delta \varphi -c_r-{\displaystyle \frac{\left(1+\lambda \right)\delta R_{11}}{4A_5}}-R_{10}\left(1+\theta +\lambda \right)\right){\displaystyle \frac{R_{12}}{\left(2\delta +k\right)}}-\left(\delta +{\displaystyle \frac{k}{2}}\right){\left({\displaystyle \frac{R_{12}}{\left(2\delta +k\right)}}\right)}^2$$

(63)

$$E^P={\displaystyle \frac{\left(e_n+e_d\right)\left(1+\lambda \right)R_{11}}{4A_5}}+{\displaystyle \frac{\left(e_r-e_d\right)R_{12}}{\left(2\delta +k\right)}}$$

(64)

5. Numerical Analysis

Due to the algebraic expressions are too complex, we conduct a numerical study with Origin 2024 to analyze the impact of the third party’s dual-fairness concerns on the equilibrium outcomes of different models. We assume $\delta =0.75$, $c=2$, $c_r=1.35$, $k=0.025$, ϕ = 7.15,$e_n=0.5,e_r=0.3$ .

The impact of fairness concerns on optimal decisions of manufacturer are provided in Figure 1 and Figure 2. Figure 1 shows that the wholesale price of new products is decreasing as $\theta$ increases in model N and model P. In model F and model D, $\theta$ and wholesale price are positively correlated when $\lambda$ is high and negatively correlated when $\lambda$ is low. On the other hand, when θ is not high, the wholesale price decreases with increasing $\lambda$ in all modes. When $\theta$ is higher, the wholesale price increases with $\lambda$ in the models N and D, and the opposite is true in mode F. The wholesale price in mode P increases and then decreases with $\lambda$ when third party exhibits a very high degree of distributional fairness concerns. The wholesale price is highest in model D and lowest in model P. Figure 2 shows that in all models, the increase in $\lambda$ causes the manufacturer to charge third party higher authorization fee. Conversely, a boost in $\theta$ results in a lower $f$. In addition, the authorization fee is highest in model D and lowest in model P.

Figure 3 and Figure 4 show the variation of quantity of new and remanufactured products as $\theta$ and $\lambda$ changes. It is easy to see from Figure 3 that the production of new products increases with $\lambda$ in mode N and model D and decreases with $\lambda$ in mode F and model P. $\theta$ facilitates the promotion of new product sales when taken into account by the retailer; conversely, in model N and model P, the increase in $\theta$ reduces the sales of new products as long as third party exhibits peer-induced fairness concerns. Figure 4 shows that the production of remanufactured products increases with $\lambda$ in all modes. An increase in $\theta$ cuts the number of remanufactured products except when the third party does not have peer-induced fairness concerns and the retailer ignores the third party’s distributional fairness concerns. Model D has the highest number of new products and the lowest number of remanufactured products, while the opposite is true in the P model. Accordingly, the products’ prices determined by the market demand are provided in Figure 5 and Figure 6.

Figure 7, Figure 8 and Figure 9 show the impact of fairness concerns on supply chain members’ profits. As can be seen in Figure 7, an increase in $\theta$ always adversely affects the manufacturer’s profit. In modes N and model D, the manufacturer’s profit increases with increasing $\lambda$. In modes F and P, the manufacturer’s profit decreases and then increases with $\lambda$ at higher $\theta$ and is positively correlated with $\lambda$ at lower $\theta$. Model D is the most favorable to the manufacturer, mainly because the wholesale price, licensing fees, and sales of new products are significantly higher in this mode than in the other scenarios under the influence of fairness concerns. Figure 8 shows that the retailer’s profit decreases and then increases with $\theta$ when $\lambda$ is high in model D, while it is positively correlated with $\theta$ in all other cases. When $\theta$ is high, $\lambda$ negatively affects the retailer’s profit, while when $\theta$ is low, an increase in $\lambda$ raises the retailer’s profit under Models F and P and causes the retailer’s profits to increase and then decrease under the other two models. Overall, when $\theta$ is low and $\lambda$ is relatively high, model P is more favorable to retailer relative to the other models due to the high unit profit. In other cases, model N brings the highest benefit to retailer. Therefore, for profit maximization reasons, the retailer chooses its attitude towards fairness concerns of third party depending on the degrees of both types of fairness concerns. Figure 9 reflects that the third party’s profit increases and then decreases with $\theta$ in model F and model D when $\lambda$ is low, while in the rest of the cases $\theta$ and its profit are positively correlated. In model F and model P, when $\theta$ is high, the third party‘s profit increases first and then decreases with increasing $\lambda$; in all other cases, increasing $\lambda$ is detrimental to the third party. For dual-fairness concerned third party, model P is always the most economically beneficial.

Figure 10 shows the impact of fairness concerns on environmental impacts for two cases, $e_d=0.55e_d=3$. An increase in $\lambda$ favors the reduction of environmental impact in the F and P models but $\lambda$ increases the environmental burden in the other two models. In models F and D, environmental impacts grow with $\theta$, while in the other two models $\theta$’s growth can reduce the impact of the products as long as the third party cares about peer-induced fairness. The reason for this is that, under dual-fairness concerns, environmental issues are significantly affected by the production of new products. The model P has the best environmental performance due to producing the lowest quantity of new products and the highest number of remanufactured products. On the contrary, model D has the highest number of new products but the lowest remanufacturing rate, which leads to extreme waste pollution and, therefore, has the highest environmental impact. In addition, comparing Figure 10a,b, we can see that an increase in $e_d$ can increase the range of parameters for which model N is more favorable to the environment than model F. This is because an increase in $e_d$ not only makes the environmental impact of new product production larger but also makes the production of remanufactured products significantly more helpful in reducing environmental impact, and the quantity of remanufactured products in model N is more than that in model F in most fairness concerns parameter cases.

6. Conclusions

This research investigates the authorized remanufacturing CLSC consisting of a manufacturer, a new product retailer, and an authorized third party with dual-fairness concerns (distributional fairness concerns and peer-induced fairness concerns). Based on the retailer’s different attitudes towards fairness concerns of the third party, we established and solved four game models: (i) the dual-fairness concerns are considered by the retailer (model F); (ii) the retailer ignores both kinds of fairness concerns (model N); (iii) the retailer only considers the distributional fairness concerns (model D); (iv) the retailer only considers the peer-induced fairness concerns (model P). we use numerical analysis to study the impact of dual-fairness concerns on optimal solutions. The following conclusions were obtained: (1) An increase in distributional fairness concerns coefficient can reduce authorization fee, but the opposite is true for peer-induced fairness concerns. The manufacturer charges the highest wholesale price and authorization fee when the retailer only considers distributional fairness concerns, while the wholesale price and authorization fee are the lowest when the retailer only considers peer-induced fairness concerns. (2) In models N and D, the increase in peer-induced fairness concerns coefficient increases the production of new products, while conversely, it decreases the sales of new products in other models. Distributional fairness concerns are beneficial in promoting the quantity of new products in model F and model D; in the other two models, as long as the third party shows peer-induced fairness concerns, the increase in the distributional fairness concerns coefficient will reduce the quantity of new products. (3) The distributional fairness concerns of dual-fairness concerned third party negatively affect the quantity of remanufactured products. The quantity of remanufactured products is positively correlated with the peer-induced fairness concerns coefficient. (4) Distributional fairness concerns and the profit of the manufacturer are negatively correlated, while an increase in peer-induced fairness concerns coefficient, in most cases, will increase the profits of the manufacturer. Model D is most advantageous for manufacturers. (5) The increase in distributional fairness concerns coefficient benefits retailer in most parameter cases. The retailer’s profit increases with peer-induced fairness concerns coefficient in model F and model P when distributional fairness concerns at low values. When the distribution fairness concerns coefficient is low and the peer-induced fairness concerns coefficient is relatively high, the retailer can choose to only consider peer-induced fairness concerns to maximize profit. In other cases, both fairness concerns will be ignored. (6) For the third party with dual-fairness concerns, the scenario where the retailer only considers peer-induced fairness concerns is always the most profitable. The increase in peer-induced fairness concerns degree is unfavorable to third party’s profit in models N and D, while in the other two models, the third party’s profit first increases and then decreases with the peer-induced fairness concerns factor’s increase when distributed fairness concerns degree is high. Although distributional fairness concerns coefficient is positively correlated with third party’s own profits in most cases, an increase in distributional fairness concerns degree will increase and then decrease the profit when the degree of peer-induced fairness concerns is low in models F and D. (7) Peer-induced fairness concerns are favorable to reducing environmental impact in model F and model P but increase environmental burden in other models. Distribution fairness concerns have a negative impact on the environment in models F and D, while in the other two models, the distribution fairness concerns can reduce the environmental impact as long as the third party cares about peer-induced fairness. Model P is the best for the environment, while model D has the highest environmental impacts. In addition, an increase in environmental impact of unit unreturned product disposed can increase the range of parameters for which model N is more favorable to the environment than model F. Therefore, we suggest that the government should formulate appropriate subsidies and regulations according to the capacity of waste treatment plants, the level of consumer awareness of proper waste discarding, and firms’ fairness concerns so as to balance economic development and environmental protection.

From a theoretical perspective, this paper introduces the third party’s dual-fairness concerns (distributional and peer-induced fairness concerns) as well as the retailer’s different attitudes towards fairness concerns of the third party into authorized remanufacturing CLSC. It can further enrich the authorized remanufacturing CLSC management. From the practical side, this paper investigates the impact of dual-fairness concerns on authorized remanufacturing CLSC decisions, profits, and environment and identifies how the retailer chooses its attitude toward fairness concerns under different parameter conditions. Therefore, our study facilitates authorized remanufacturing CLSC enterprises to make reasonable operational decisions under third party remanufacturer’s dual-fairness concerns and also provides a reference value for governments to set environmental regulations and subsidy policies to promote sustainability.

This paper only considers the unit authorization fee pattern. It would be meaningful to compare the fixed authorization fee model and the unit authorization fee model under the fairness concerns of CLSC members. Moreover, we only consider third party remanufacturing; retailer remanufacturing under fairness concerns is equally worthy of being incorporated into future research.