Risk-Averse Pricing Decisions Related to Recyclables’ Quality in a Closed-Loop Supply Chain

In a closed-loop supply chain, uncertainty of recyclables’ quality is a major factor of supply chain members’ decision-making. Because of this uncertainty, manufacturers must pay varying manufacturing costs for remanufacturing recyclables. Our study assumed that manufacturers are risk-averse towards uncertainty in manufacturing costs and constructed a retailer recycling model and a third-party recycling model to investigate pricing decisions in a decentralized closed-loop supply chain under uncertainty about recyclables’ quality. Our findings can be summarized as follows: (1) the higher the degree of consumer preference for remanufactured products, the higher the wholesale and retail prices of remanufactured products and the higher the recycling price of used products; (2) the two recycling models showed a U-shaped relationship between supply chain revenue and the degree of consumer preference for remanufactured products, and this supply chain revenue is related to the consumer preference coefficient; (3) there is a U-shaped relationship between the retailer’s expected revenue and the degree of consumer preference for remanufactured products in the R mode and an M-shaped relationship between them in the 3P mode; (4) in both recycling modes, the manufacturer’s risk aversion is inversely proportional to supply chain revenue, and supply chain revenue in the R mode is higher than that in the 3P mode; and (5) the higher the uncertainty of recyclables’ quality, the lower the recycling price of used products and the lower the manufacturer’s enthusiasm for recycling or for used products.


Introduction
Along with rapid economic and technological development, changes in consumer demand and product updates are increasingly rapidly. To gain better experiences from products, consumers choose to shorten the product life cycle and buy new products, resulting in a sharp increase in the number of waste products in the market. According to the White Paper on WEEE Recycling Industry in China 2017, the theoretical scrappage of electrical and electronic products in China amounted to 500.04 million items in 2017, an increase of approximately 33% from 2016. According to the Annual Report of 2017 on Prevention and Control of Environmental Pollution by Solid Waste in China's Large and Medium-sized Cities, 79.35 million electrical and electronic waste items were recycled and disposed of by 103 pollution treatment enterprises countrywide in 2016, with a recycling ratio of only 21%. In addition, the number of recycled electrical and electronic waste products in 2017 was virtually the same as that of 2016. Recycling of used products saves material and energy costs and is conducive to sustainable development [1]. Due to the differences in history and habits of product use, the quality of used products is uncertain and varies widely; therefore, their remanufacturing costs also differ [2,3]. e wide variation in the quality of used products brings about uncertainty in remanufacturing costs paid by manufacturers. Hence, it is a matter of importance to study which attitude manufacturers will take toward uncertainty in remanufacturing costs to protect their own interests.
ere have been a number of studies on the pricing of recycled used products. For example, Savaskan et al. [4] and Chuang et al. [5] studied pricing and coordination in closed-loop supply chains under three conditions: recycling by manufacturers, recycling by retailers, and recycling by third parties. On this basis, Gong and Jiang [6] studied two closedloop supply chain models (including one model for manufacturer and retailer hybrid recycling and one model for manufacturer, retailer, and third-party hybrid recycling) and proposed respective optimal decisions. Zhu et al. [7] investigated the pricing decisions in supply chains when the costs of recycling by traditional channels differ from those of online recycling. However, most of these studies focused on recycling channels and assumed that the quality of used products is consistent while ignoring the impact of the varying quality of used products in supply chains on remanufacturing.
e varying quality of used products may affect remanufacturing costs in supply chains, as well as the pricing of new and remanufactured products. Cheng et al. [8] studied the role of government subsidies in the pricing decisions in closed-loop supply chains when the quality of used products is uncertain. ey found that the recycling price of used products decreases when the threshold of remanufacturing quality increases but increases with a rise in the subsidy coefficient. Examining government subsidies, Deng et al. [9] also considered carbon taxes, finding the following: (1) the optimal recycling quality coefficient is affected by carbon taxes; (2) government subsidies combined with carbon taxes can encourage the recycling of used products, reduce total carbon emissions, and encourage manufacturers to remanufacture more effectively. Guo and Tan [10] investigated the role of governmental regulation, which is manifested as the used recycling ratio specified by governments. ey found that governmental regulation can effectively improve the quality of recycled used products; however, the difference between the degree of governmental regulation and the recycling payment coefficient at different levels must stay within a certain range to ensure the effectiveness of trading markets. Gao and Li [11] and Huang et al. [12] studied the pricing of recycled used products from the perspective of recycling channels, and both studies classified used products into different grades. In Gao and Li's work, used products were recycled from consumers at the same prices, and then the recycled used products were graded. According to Huang et al., when used products were recycled in a competitive manner between manufacturers, retailers, and third parties, used products were graded according to their quality and were then priced. Most of the aforementioned studies assume that supply chain members are risk-neutral. However, Jammernegg and Kischka [13] found that decision makers' risk appetite affects the pricing of used products in supply chains. By combining the risk appetite of supply chain members with the selection of recycling channels, Chen [14] investigated whether manufacturers resort to the direct selling channel when retailers are risk-averse. eir findings showed that (1) the direct selling channel does not always damage the revenue of retailers and that (2) manufacturers resort to the direct selling channel only when the costs of direct selling meet certain conditions; otherwise, manufacturers will sustain losses. Li et al. [15] constructed a two-level supply chain model based on dual-source channels. eir study showed that the optimal retail prices of products under a direct selling channel and those under traditional channels are not related to whether retailers cooperate with manufacturers, but rather to the degree of risk aversion on both sides. Most of the studies cited above focus on the risk resulting from fluctuations in new product demand in the market rather than the risk of uncertainty about the quality of used products.
In summary, this study investigated the pricing differences between new products and remanufactured products under two recycling modes, retailer recycling and thirdparty recycling. Our study considered the uncertainty of used products' quality and the risk appetite of manufacturers and assumed that manufacturers are risk-averse to the uncertainty of used products' quality. Accordingly, we discuss the following three issues: (1) how are recycled products and new products priced under different recycling modes? (2) How does the risk appetite of manufacturers affect the pricing of recycled products and new products? (3) How does the quality of used products affect closed-loop supply chains? Retailer recycling and third-party recycling are common in the market. erefore, this study considered two recycling modes, retailer recycling (referred to as R mode hereinafter) and third-party recycling (referred to as 3P mode hereinafter). e remainder of the paper is organized as follows. Section 2 presents the problem description and basic hypotheses. Section 3 constructs the two models of retailer recycling (R mode) and third-party recycling (3P mode), respectively, and their optimal pricing decisions are analyzed. Section 4 analyzes the property of the two models. Numeric analysis is given in Section 5. Finally, Section 6 concludes the paper.

Problem Description and Basic Hypotheses
With the recent changes in the market structure and the relative strength of supply chain members, manufacturers attach more importance to business innovation and R&D of new products. Moreover, the role of the recycler in a closedloop supply chain has gradually shifted from manufacturers to third-party recyclers and retailers. Compared with manufacturers, third-party recyclers possess a more stable recycling network, and retailers are closer to markets; therefore, commissioned third-party recycling and retailer recycling have gradually gained popularity. Considering the two recycling modes, R and 3P, this study presents a closedloop supply chain model comprising a single manufacturer, a single retailer, a single third-party recycler, and a single product (Figure 1). In a traditional logistics (forward logistics) process, the manufacturer sells the new product and the remanufactured product to the retailer at the wholesale prices of ω n and ω r , respectively; then, the retailer sells them to consumers at the retail prices of p n and p r . In a reverse logistics process, the retailer or third-party recycler recycles the used product from consumers at the price of b r , and the manufacturer purchases the used product from the retailer or third-party recycler at the transfer price of b m . e manufacturer turns the used product into a remanufactured product and sells the remanufactured product in a forward logistics process.
Under the condition of complete market information disclosure, the manufacturer is the market dominator, and optimal revenue models under respective recycling modes are built according to game theory. e symbolic variables and model hypotheses are described as follows.
e symbolic variables are introduced and described as follows: d n and d r , respectively, denote the market demand for the new product and remanufactured product, where d n > d r c n and c r , respectively, denote the unit manufacturing cost of the new product made from new materials and the unit manufacturing cost of the remanufactured product made from a used product, where c n > c r ω n and ω r , respectively, denote the unit wholesale prices of the new product and remanufactured product p n and p r , respectively, denote the unit retail prices of the new product and remanufactured product b m ( > 0) denotes the unit price at which the manufacturer transfers a used product from the retailer, namely, the transfer price b r ( > 0) denotes the unit price at which the retailer recycles a used product from consumers, namely, the recycling price q r denotes the quantity of used products recycled from consumers, andq r � h + cb r , where h > 0 indicates the quantity of used products that consumers return for recycling voluntarily and c denotes the price elasticity of consumers to recycled used products u n and u r , respectively, denote the net utility of the new product and the remanufactured product to consumers; consumers decide to purchase the new product or remanufactured product based on a comparison between u n and u r i j denotes the profit function of supply chain member j under recycling mode i in the closed-loop supply chain (i �R or 3P); the subscript j � R, 3P, T, respectively, denote the retailer, the third-party retailer, and the supply chain; the superscript * denotes the optimal decision

Model Hypotheses
e manufacturer is the market dominator and is risk-averse.

Hypothesis 2.
e quality of recycled used products is uncertain, and the quality of used products (x) follows normal distribution, in which the expectation is μ and the variance is σ 2 , namely, x ∈ N(μ, σ 2 ). Here, μ denotes the average quality of μ, and σ denotes the fluctuation in the quality of used products around the average quality. It is uncertain whether the σ value is positive or negative, and |μ| > |σ| > 0. e unit manufacturing cost of the remanufactured product (c r ) is related to the quality of used products. In other words, c r � c n − vx, where v(v > 0) denotes the cost-of-quality coefficient and vx denotes the cost saved by recycling and remanufacturing the used product with quality x. Hypothesis 3. At the transfer price of b m , the manufacturer transfers the used products recycled by the retailer; at the price of b r , the retailer recycles the used products from consumers. In other words, b r � (1 − r)b m , where r denotes the profit per unit of the used product recycled by the retailer.

Hypothesis 4.
To encourage the recycler to recycle a product with higher quality, the manufacturer "rewards" the retailer or third-party retailer with the amount of the cost saved from the remanufactured product in the proportion of θ(0 < θ < 1), namely, θvxd r ; kx 2 /2 denotes the effort cost that the retailer or third-party recycler pays to recycle a product with higher quality (k is the effort cost coefficient).

Hypothesis 5.
e recycled used products can all be remanufactured at varying costs, the market demand for remanufactured products is smaller than or equal to the quantity of recycled used products, and the manufacturer treats the value of non-remanufactured used products as the scrap value s. e scrap value is small enough so that it is not considered in any model used in this study.

Hypothesis 6.
Consumers' valuation of the new product (ϕ) obeys a uniform distribution in the range of [A, B], and the probability density function is as follows:  Consumers' utility valuation of a remanufactured product is a times as much as that of a new product; a denotes the coefficient of consumer preference for a remanufactured product (0 < a < 1).

Hypothesis 7.
e manufacturer is risk-averse about the quality of used products, and the expected utility of the closed-loop supply chain and its members is measured using the mean-variance method. e utility function is , in which λ denotes the degree of risk aversion (0 ≤ λ ≤ 1) [16,17].
Consumers decide whether to purchase new products or remanufactured products based on a comparison of utility between the products. is section focuses on the pricing decisions for new products and remanufactured products; therefore, we ignored the time and distance costs from the purchase of products. ese include the differences between product utility (i.e., product valuation) and retail prices. erefore, the utility of purchased new products is expressed as follows: and (p n − p r )/(1 − a) ≤ B, namely, when a ≤ 1− (p n − p r )/B, the quantity of purchased new products is expressed as follows: namely, when a ≤ 1 − (p n − p r )/B, the quantity of purchased remanufactured products is expressed as follows: Our study investigated the pricing decisions for new products and remanufactured products; we therefore assumed that there are both new products and remanufactured products in the market. From the perspective of consumer utility, equations (3) and (4) have been selected to represent the demand for new products and demand for remanufactured products, respectively.

Optimal Pricing Decisions under Retailer Recycling (R Mode).
Under the R mode, the manufacturer recycles used products through the retailer. e manufacturer's revenue includes the revenue from the wholesale of new products and remanufactured products, as well as the costs saved by the remanufacturing of used products. e manufacturer's costs include the manufacturing cost of new products and remanufactured products, the transfer cost of the used products recycled by the retailer, and the reward given to the retailer. e retailer's revenue includes the revenue from the sale of new products and remanufactured products to consumers, the manufacturer's transfer cost, and the reward for recycling of used products. e retailer's costs include the wholesale cost of new products and remanufactured products paid to the manufacturer, the recycling cost paid to consumers, and the cost of the effort to recycle used costs with higher quality. Accordingly, the revenue function models for the manufacturer and retailer under the R mode can be expressed as follows: In this model, the manufacturer is the leader of the supply chain, so the decision-making sequence is as follows: (1) the manufacturer decides the wholesale prices of new products and remanufactured products and the transfer prices of recycled used products; (2) the retailer decides the retail prices of new products and remanufactured products and the recycling prices of recycled used products. Using the backward induction method, the retailer's expected profit (E( R R )) is expressed as follows: We calculate the first-order partial derivative of the retailer's utility with respect to the retail price of new products (p R n ), retail price of remanufactured products (p R r ), and per-unit profit (r R ), construct their simultaneous equations, and calculate the solutions as follows: 4 Mathematical Problems in Engineering Next, we substitute equations (8) to (10) into the manufacturer's expected utility function and calculate the first-order partial derivative of the wholesale price of new products (ω R * n ), wholesale price of remanufactured products (ω R * r ), and transfer price (b R * m ). en, we determine the optimal solutions of simultaneous equations. e manufacturer is risk-averse, so the manufacturer's utility can be calculated according to the criterion Using the manufacturer's expected utility function (equation (5)), we calculate the manufacturer's expected utility as follows: We then substitute equations (8) to (10) into equation (11); thus, the manufacturer's expected utility is as follows: Before solving the above equation, we solve the Hesse matrix of U( R M ) and determine whether it has an optimal solution. e Hessian matrix is expressed as follows: As a < 1, |H R M | � − c/(1 − a)a < 0, the second-order principal minor is greater than 0, and the first-order principal minor is smaller than 0. erefore, U( R M ) is a strictly concave function, which has a unique optimal solution.
e demand for remanufactured products should be smaller than or equal to the quantity of recycled used products. erefore, the Lagrange multiplier ε 1 ≥ 0 is introduced to the constraint condition in U( R M ) to construct the following Lagrange function: According to the KT condition, two circumstances are considered (i.e., ε 1 � 0 and ε 1 ≠ 0).
; then, the solution is as follows: Mathematical Problems in Engineering e transfer price b R * m cannot be negative; therefore, it is omitted when ε 1 � 0.
When ε 1 ≠ 0, the solution is as follows: erefore, the optimal pricing decision under the R mode is as follows: We substitute equations (15) to (18) into equations (8) to (10); then, the retail prices of new products and remanufactured products and per-unit profit of recycled used products are as follows: We substitute equations (21) to (23) into equations (3) and (4); then, the demand for new products and demand for remanufactured products, respectively, are as follows: We substitute equations (18) and (23) r ; then, the recycling price of used products is as follows: erefore, the quantity of recycled used products is as follows:

Mathematical Problems in Engineering
We substitute equations (15) to (27) into equations (11) and (7); then, the manufacturer's expected utility and the retailer's expected profit, respectively, are as follows:

Optimal Pricing Decisions under ird-Party Recycling (3P Mode).
Under the 3P mode, the manufacturer commissions a third-party recycler to recycle used products. e financial gains realized by the manufacturer include revenue from the wholesale of new products and remanufactured products and manufacturing costs saved by the remanufacturing of used products. e manufacturer's costs include the manufacturing cost of new products and remanufactured products, the transfer cost of used products recycled by the third-party recycler, and the reward given to the thirdparty recycler. e retailer's revenue is the revenue from the sale of new products and remanufactured products to consumers. e retailer's cost is the wholesale cost of new products and remanufactured products paid to the manufacturer. e third-party recycler's revenue includes the transfer cost paid by the manufacturer and the reward given by the manufacturer for recycling used products. e thirdparty recycler's costs include the wholesale cost paid to consumers and effort cost paid to recycle used costs with higher quality. Accordingly, the revenue function models for the manufacturer, retailer, and third-party retailer are, respectively, expressed as follows: At this time, the manufacturer is the leader of the supply chain, so the decision-making sequence is as follows: (1) the manufacturer decides the wholesale prices of new products and remanufactured products and transfer prices of used products recycled from the third-party recycler; (2) after being informed of the manufacturer's pricing decisions, the retailer and third-party recycler, respectively, decide the retail prices of new products and remanufactured products and the recycling prices of used products based on the principle of self-interest maximization. Using the backward induction method, the retailer's expected profit (E( 3P R )) and the third-party recycler's expected profit (E( 3P 3P )) are calculated according to equations (31) and (32). E( 3P R ) and E( 3P 3P ) are expressed as follows: We calculate the first-order partial derivative of the retailer's utility with respect to the retail price of new products (p M n ) and retail price of remanufactured products (p M r ) and calculate the first-order partial derivative of the third-party recycler's utility with respect to the per-unit profit of recycled products (r 3P 3P ). We construct their simultaneous equations and calculate the solutions as follows: We substitute equations (35) to (37) into the manufacturer's expected utility function and calculate the firstorder partial derivatives of the wholesale price of new products (ω 3P * n ), the wholesale price of remanufactured products (ω 3P * r ), and the transfer price (b 3P * m ). We can then determine the optimal solutions of their simultaneous equations. e manufacturer is risk-averse, so the manufacturer's utility can be calculated according to the criterion . Using the manufacturer's expected utility function (equation (25)), we determine the manufacturer's expected utility as follows: We substitute the determined retail price of new products (p 3P * n ) and retail price of remanufactured products (p 3P * r )-namely, equations (35) and (36)-into equation (38); thus, the manufacturer's expected utility is as follows: (39) Before solving the above equation, we solve the Hessian matrix of U( 3P M ) and judge whether it has an optimal solution. e Hessian matrix is expressed as follows: As a < 1, |H 3P M | � − c/(1 − a)a < 0; the second-order principal minor is greater than 0, and the first-order principal minor is smaller than 0. erefore, U( 3P M ) is a strictly concave function, which has a unique optimal solution.
e demand for remanufactured products should be smaller than or equal to the quantity of recycled used products. According to this constraint condition, the Lagrange multiplier ε 2 is introduced to the constraint condition in U( M M ) to construct the following Lagrange function: According to the Kuhn-Tucker condition, we consider two circumstances (i.e., ε 2 � 0 and ε 2 ≠ 0).
; then, the solution is as follows: 8 Mathematical Problems in Engineering (42) In the above section, this study assumes that the recycling price (b 3P * r ) is greater than zero. erefore, it is omitted when ε 2 � 0.
When ε 2 ≠ 0, the solution is as follows: erefore, the optimal pricing decision under the 3P mode is as follows: We substitute equations (43)-(45) into equations (35)-(37); thus, the retail prices of new products and remanufactured products and per-unit profit of recycled used products are as follows: We then substitute equations (48) and (49) into equations (3) and (4); then, the demand for new products and demand for remanufactured products are, respectively, as follows: en, the quantity of recycled used products is as follows: We substitute equations (43)-(53) into equations (33), (34), and (38); then, the manufacturer's expected utility, the retailer's expected profit, and the third-party recycler's expected profit, respectively, are as follows: Mathematical Problems in Engineering 9 (56)

Property Analysis
Property 1. Under R and 3P modes, both optimal wholesale price and optimal retail price of remanufactured products are directly proportional to the coefficient of consumer preference for remanufactured products.
Proof. Under the R mode, the first-order partial derivative of the optimal wholesale price of remanufactured products concerning the consumer preference coefficient is as follows: (57) e zooming method is used. zω R * r /za must be greater than or equal to the value determined when a � 0. us, the following result can be obtained: According to equation (3), the demand for new products is B − (p n − p r )/(1 − a) < B, and the demand for remanufactured products is smaller than the demand for new products: a) < B, proving that the quantity of used products voluntarily recycled by consumers (h) is far smaller than consumers' maximum valuation of new products (B); c n − vμ denotes the average manufacturing cost of remanufactured products. erefore, zω R * r /za ≥ B/2 + (c n + c(c n − vμ) − h)/2 > 0. Under the R mode, the optimal wholesale price of remanufactured products is directly proportional to the degree of consumer preference for remanufactured products. Likewise, it can be said that, under the 3P mode, the optimal wholesale price of remanufactured products is directly proportional to the consumer preference coefficient.
Property 1 shows that, under R and 3P modes, the optimal wholesale and retail prices of remanufactured products will increase if the degree of consumer preference for remanufactured products rises. A rise in the degree of consumer preference for remanufactured products implies an increase both in consumers' utility valuation and their degree of acceptance of remanufactured products; in this case, the manufacturer wants to earn more revenue by raising the optimal wholesale and retail prices of remanufactured products.
Under R and 3P modes, the first-order partial derivatives of the optimal wholesale price of remanufactured products concerning the manufacturer's risk aversion coefficient are as follows: Under R and 3P modes, the first-order partial derivatives of the total revenue of the supply chain concerning the manufacturer's risk aversion coefficient are as follows: (65) □ 4.1. QED. Property 4 shows that the degree of the manufacturer's risk aversion affects optimal decisions in the supply chain. When the degree of risk aversion increases, the manufacturer is less willing to assume risk related to the quality uncertainty of used products; therefore, the manufacturer will reduce the quantity of recycled used products by reducing their optimal recycling price. In this case, the cost saved in the remanufacturing process and the quantity of remanufactured products will be reduced, and the manufacturer will raise the optimal retail price of remanufactured products to maintain their revenue and to reduce consumers' demand for remanufactured products. As a result, the revenue accruing to the supply chain will decrease during the process of the remanufacturing and sale of remanufactured products, and the total revenue of the supply chain will decrease with the rise in the degree of the manufacturer's risk aversion.

Impact of Consumer Preference for Remanufactured
Products. To investigate the impact of consumer preference for remanufactured products on the revenue of the supply chain and on pricing decisions, this study assumes that the values of other variables remain constant. Specifically, As shown in Figure 2, with the rise in the degree of consumer preference for remanufactured products, the wholesale price of remanufactured products increases under both recycling modes, and the wholesale price of remanufactured products under the R mode is always higher than that under the 3P mode.
As shown in Figure 3, the recycling price of used products increases with the rise in the degree of consumer preference for remanufactured products under both recycling modes, and the recycling price of used products under the R mode is always higher than that under the 3P mode. However, the recycling price of used products is positive (i.e., the recycler is willing to recycle used products at their own cost) only when the degree of consumer preference for remanufactured products reaches a certain level; otherwise, the recycler's benefit will be reduced.
As shown in Figure 4, the manufacturer's expected utility shows a U-shaped variation under both recycling modes with the rise in the degree of consumer preference for remanufactured products. Specifically, a low degree of consumer preference for remanufactured products inhibits the manufacturer's expected utility, whereas a high degree of consumer preference for remanufactured products increases the manufacturer's expected utility.
As shown in Figure 5, under the R mode, there is a U-shaped relationship between the retailer's expected revenue and the degree of consumer preference for remanufactured products. e optimal decisions of the retailer responsible for recycling of used products are primarily affected by the remanufacturer, so the retailer's profit shows a U-shaped variation. However, under the 3P mode, there is an M-shaped relationship between the retailer's expected revenue and the degree of consumer preference for remanufactured products. In the two recycling models, the retailer is only responsible for the sale of products and does not participate in any reverse logistics process. Furthermore, the wholesale and retail prices of new products are only related to their manufacturing costs.
As shown in Figure 6, there is a U-shaped relationship between the total utility of the supply chain and the degree of consumer preference under two recycling modes for remanufactured products; moreover, the total utility of the supply chain is related to the consumer preference coefficient.    To investigate the impact of the manufacturer's risk aversion coefficient on the revenue of the supply chain and on pricing decisions, this study assumes that the values of other variables remain constant. Specifically,

Impact of the
As shown in Figure 7, the recycling price of used products is inversely proportional to the manufacturer's risk aversion coefficient. Furthermore, the recycling price of used products under the 3P mode is lower than that under the R mode.
As shown in Figure 8, the retail price of used products is directly proportional to the manufacturer's risk aversion coefficient, and the retail price of used products under the 3P mode is higher than that under the R mode. Figures 9 and 10 show that both the manufacturer's expected utility and the retailer's expected revenue are inversely proportional to the manufacturer's risk aversion coefficient, and the profit earned by the manufacturer is higher than the profit earned by the retailer. In addition, both the manufacturer's expected utility and the retailer's expected revenue under the R mode are higher than those under the 3P mode.

Impact of Quality Uncertainty of Used Products.
To investigate the impact of the manufacturer's reward ratio on supply chain revenue and pricing decisions, we assumed that the values of other variables remain constant. Specifically, B � 250, h � 5, c � 4, c n � 50, v � 40, θ � 0.2, μ � 0.8, a � 0.8, λ � 0.2, and k � 2. Considering that the quality of used products fluctuates around the average quality, the σ value can be either positive or negative. We further assumed that σ ∈ [− 0.5, 0.5].
As shown in Figure 11, the recycling price of used products is inversely proportional to the quality fluctuation margin of used products (|σ|); moreover, it is highest in the   centralized decision-making model but lowest under the 3P mode. As shown in Figure 12, both wholesale and retail prices of remanufactured products are directly proportional to the quality fluctuation margin of used products (|σ|); in the decentralized decision-making model, the wholesale price of remanufactured products is highest under the R mode, whereas the retail price of remanufactured products is highest under the 3P mode.
As shown in Figures 13 and 14, both the manufacturer's expected utility and the retailer's expected profit are inversely proportional to the quality fluctuation margin of used products (|σ|).

Conclusions
is study investigated risk-averse pricing decisions associated with product recycling in a closed-loop supply chain.
rough theoretical analysis combined with numerical analysis, we reach the following conclusions: (1) the degree of consumer preference for remanufactured products is directly proportional to the wholesale and retail prices of remanufactured products, as well as the recycling price of used products; under the same degree of consumer preference for remanufactured products, the optimal wholesale and retail prices of remanufactured products under the R mode are higher than those under the 3P mode. (2) Under these two recycling modes, there is a U-shaped relationship between the revenue of the supply chain and the degree of consumer preference for remanufactured products; when the coefficient of consumer preference for remanufactured products is high, the total revenue of the supply chain under the R mode is higher than that under the 3P mode; thus, R mode should be selected. When the coefficient of consumer preference for remanufactured products is low, 3P mode should be selected. (3) ere is a U-shaped relationship between the retailer's expected revenue and the degree of consumer preference for remanufactured products under the R mode, whereas there is an M-shaped relationship between them under the 3P mode. (4) Under both recycling modes, the degree of the manufacturer's risk aversion is inversely proportional to the revenue of the supply chain; however, the revenue of the supply chain under the R mode is higher than that under the 3P mode. (5) e higher the uncertainty of used products' quality, the lower the recycling price of used products, the lower the manufacturer's enthusiasm for recycling or for used products, and the lower the manufacturer's and retailer's revenue. However, the R mode is superior to the 3P mode.
Despite its significant contributions, this study has a few limitations. For example, this study only considers a closed- loop supply chain comprising a single manufacturer and a single retailer, not a closed-loop supply chain comprising multiple manufacturers and retailers. is also indicates a direction for subsequent studies.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.