Decision-Making and Coordination of Remanufacturing Closed-Loop Supply Chain with PIR under the Different Government Subsidy Strategies

Abstract

Government subsidies and process innovation for remanufacturing (PIR) have become effective measures to improve the recycling and remanufacturing efficiency of waste products and promote sustainable economic development. Under different government subsidy strategies, the PIR decision and coordination of the closed-loop supply chain (CLSC) of authorized competition remanufacturing are studied. This paper constructs five centralized and decentralized decision models of CLSC and analyzes the influence of government subsidy strategy and PIR input entity differences on the competition and cooperation relationship between manufacturers and remanufacturers and the performance of CLSC. A fixed license fee contract is designed to realize CLSC coordination and to improve the performance and operational efficiency of the CLSC. It is found that patent licensing fees can always play a role in sharing government subsidies between the manufacturer and remanufacturer, and making changes in government subsidy strategies only affects formulation of patent licensing fees. Manufacturer PIR input mode is more conducive to improving the market demand for new products, while remanufacturer PIR input mode is more conducive to improving the PIR input level, remanufactured product market demand, CLSC members and overall performance. Moreover, government subsidies can produce better efficiency and effects under remanufacturer PIR input mode. A fixed license fee contract can coordinate a CLSC effectively. The increase in government subsidies and PIR input effect can effectively expand the negotiation space between the manufacturer and remanufacturer, which is conducive to achievement of coordination contracts. Finally, the main conclusions are verified by numerical analysis.

1. Introduction

With the continuous development of intelligent manufacturing and other new technologies, the production and renewal cycle of products is getting shorter and shorter [1,2]. While meeting the needs of people’s lives, many waste products are also produced, which causes more serious problems of environmental pollution and resource shortage [3]. According to the National Development and Reform Commission, about 100 million to 120 million electrical appliances are being eliminated every year, and the annual growth rate is as high as 20 percent. How to deal with the huge amount of waste products effectively has become an urgent problem for the government to build a low-carbon society. On the one hand, development of the remanufacturing industry can alleviate the negative impact of waste products on the environment. On the other hand, compared with the production of new products, remanufacturing can save 60% of energy and 70% of raw materials and reduce the emissions of air pollutants by more than 80% [4]. Therefore, all countries and enterprises in the world are trying their best to promote the development of the remanufacturing industry to realize environmental protection and a circular economy.

From the perspective of industry practice, most remanufacturing businesses are still handled by professional remanufacturers [4,5]. Manufacturers are reluctant to engage in remanufacturing due to considerations of R&D and sales of new products, fixed investment in recycling and remanufacturing, brand influence and other factors [6,7]. However, the remanufactured products produced by remanufacturers will pose a certain threat to the competition of the new products produced by the manufacturers. Therefore, many manufacturers will choose authorized remanufacturing mode to alleviate the conflict of interest between the two sides [8].

Meanwhile, in order to improve the recycling and remanufacturing efficiency of waste products, many countries and enterprises have paid attention to the role of process innovation for remanufacturing (PIR) in promoting development of the remanufacturing industry. The Chinese government encourages enterprises to implement PIR by implementing a series of policies, such as the Notice on Deepening the Pilot Work of Remanufacture and the Extended Producer Responsibility System, to improve the efficiency of recycling and recycling of discarded products. Many enterprises begin to implement PIR technology to reduce the cost and difficulty of remanufacturing in reality [9,10]. For example, Lenovo (a manufacturer of China) and Greenland (a remanufacturer of China) have actively invested in PIR to effectively reduce the difficulty and cost of recycling and remanufacturing waste products. Fuji, Xerox and Kodak improved the disassembly level of copiers through PIR input, optimized the remanufacturing process, effectively reduced the remanufacturing cost and reduced the resource consumption and waste generation by three times. Caterpillar, a remanufacturer, has effectively reduced the cost of remanufacturing and saved considerable energy and resource consumption through PIR [10,11]. Although cases of manufacturers and remanufacturers engaging in PIR input have been observed, it remains to be discussed which manufacturer or remanufacturer responsible for PIR input is more conducive to improving the operational efficiency of a closed-loop supply chain (CLSC). As an external guide, the government also effectively promotes development of the remanufacturing industry by adopting incentive policies, such as financial subsidies [12]. However, it remains to be discussed which side (manufacturer or remanufacturer) the government can subsidize to obtain the best subsidy efficiency and effect.

Based on this, this paper will focus on the following three issues from the perspective of authorized competitive remanufacturing: first, how different government subsidy strategies affect the optimal PIR input decision of the CLSC and the pricing decision of new/remanufactured products; second, whether different PIR input patterns affect the optimal subsidy strategy of the government and the competition and cooperation relationship between manufacturers and remanufacturers; third, what are the effects of relevant parameters on the coordination effectiveness of fixed license fee contracts in different situations?

The main contributions of this research are summarized as follows. First of all, to the best of our knowledge, this paper makes the first attempt to bridge government subsidy and PIR input strategy under authorization mode of remanufacturing CLSC. Second, we establish five centralized and decentralized decision models of manufacturers and remanufacturers considering different government subsidy strategies and PIR input modes. We explore the optimal strategy from the perspective of each participant of the CLSC and the environment. Last, we design a fixed license fee contract that can resolve channel conflicts and improve the performance and operational efficiency of the CLSC.

The paper is organized as follows. First, we describe the research background and objectives of this paper and review the relevant literature. Second, we provide related model assumptions and a model considering government subsidies and PIR input. We further investigate the optimal decisions and profit of the CLSC. The optimal PIR input mode and government subsidy strategy are provided. Third, we design a fixed license fee contract to realize CLSC coordination. Finally, more managerial implications can be obtained through numerical analysis. Furthermore, the conclusions and future research directions of the paper are presented.

2. Literature Review

Our paper is closely related to two streams of CLSC: (1) authorized remanufacturing and government subsidy in CLSC, and (2) PIR decision in CLSC. Next, we review the relevant literature.

2.1. Authorized Remanufacturing and Government Subsidy in CLSC

CLSC is a complete supply chain system formed by adding reverse feedback process (namely reverse supply chain) to the traditional “forward” supply chain [13]. Remanufacturing CLSC is a supply chain system under the coexistence of remanufacturing and manufacturing, and it is also the most complex CLSC structure [14]. In order to solve the contradiction and conflict between remanufacturers and manufacturers, many manufacturers carry out remanufacturing business through patent licensing mode [8]. Simultaneously, many researchers have conducted extensive research on authorized remanufacturing in CLSC. Sun et al. compared the influence of the competition mode with and without patent authorization on the game between manufacturers and remanufacturers and pointed out that original product manufacturers prefer patent authorization mode [15]. Zhang et al. studied three modes of manufacturer recycling manufacturer remanufacturing, seller recycling manufacturer remanufacturing and fully authorizing the third party to remanufacture and believed that the authorized remanufacturing mode can improve the remanufacturing operation efficiency and is superior to the other two modes in most cases [16]. Hong et al. compared the fixed patent license fee model and the unit product patent license fee model in a CLSC and pointed out that, when the fixed license fee is lower than a certain threshold, the unit license fee model is better [17]. Zhao et al. discussed the impact of different patent licensing models on the performance of the CLSC when considering the retailer’s service level and pointed out that fixed licensing fees can produce better economic benefits at this time [18]. Liu et al. considered the quality of remanufactured products and consumers’ quality preferences, investigated the issue of authorized remanufacturing strategy in a CLSC and pointed out that the authorized remanufacturing strategy under the two tariff contracts is always the best strategy for manufacturers [19]. Xia et al. analyzed the game theory between manufacturers and remanufacturers under the carbon trading mechanism based on the perspective of authorized remanufacturing and designed the corresponding coordination mechanism [6]. Cao et al. pointed out that patent licensing under the government’s extended producer responsibility system plays a significant role in promoting development of the remanufacturing industry and can effectively improve the product recycling and remanufacturing rate [20].

Many researchers pay attention to government subsidies in a CLSC. Mitra and Webster pointed out that government subsidies can effectively promote development of the remanufacturing industry and improve the profits of manufacturers and remanufacturing [21]. Ma et al. consider two ways of recycling subsidy and remanufacturing subsidy, discuss the intensity of government subsidies and recognition of the remanufactured products on pricing decisions in a CLSC [22]. Liu et al. studied the impact of government incentives and price competition between two recycling channels on the recycling market of electronic and electrical products and pointed out that, when the quality of waste products is within a certain range, government subsidies can produce better results [23]. Meng et al. studied the impact of the government’s consumption subsidy for remanufactured products on the pricing decisions of new products and remanufactured products in a CLSC and pointed out that government consumption subsidies are always conducive to promoting the demand for remanufactured products, but the members of the CLSC may not benefit from them [24]. Lou et al. compared and analyzed the impact of government production subsidies and sales subsidies on pricing decisions of new products and remanufacturing products in a CLSC under different game types and pointed out that the sales subsidies and production subsidies are equivalent in terms of influencing sales prices and sales volumes [25]. Zhang and Yu pointed out that the government recovery subsidy rate decreases with an increase in a manufacturer’s altruism intensity but increases with an increase in a retailer’s altruism intensity [26].

Although the above studies analyzed the remanufacturing authorization decisions and government subsidies of CLSCs from multiple perspectives, few studies compared and analyzed the operational differences of CLSCs under different government subsidy strategies based on the perspective of authorized competitive remanufacturing and did not involve PIR input mode selection of the CLSC.

2.2. PIR Decision in CLSC

PIR refers to the consideration at the beginning of product design to facilitate product recycling and remanufacturing, or research and development behaviors (such as technological innovation, green innovation, remanufacturing design, remanufacturing technology, etc.) that improve products in the process of remanufacturing waste products and improve them in the design of new products to reduce remanufacturing costs.

As an effective measure to improve the efficiency and benefit of remanufacturing, PIR has received wide attention by many researchers at home and abroad. Reimann et al. compared the influence of manufacturer or retailer remanufacturing mode on the manufacturer’s optimal PIR input level and production decision of CLSC and pointed out that the retailer remanufacturing mode may produce a higher PIR input level [9]. Chai et al. built a game model based on one supplier and one manufacturer and analyzed the optimal PIR strategy of green product remanufacturing in CLSC and the impact of different PIR strategies on the environment and pointed out that PIR can effectively improve the remanufacturing performance while improving the recovery rate of remanufacturing enterprises [7]. Terjesen believes that process innovations are central to improving a firm’s productivity and contributing to efficiency and gross domestic product growth. As a result of the complexity of identifying, developing and implementing process innovations, firms increasingly draw on external sources of knowledge [27]. Wu studied the CLSC game problem when the manufacturer decided on the remanufacturability level and the remanufacturer decided on the recovery strategy and pointed out that the product design strategy is beneficial to both the manufacturer and the remanufacturer under certain circumstances [5]. Chen and Zhang proposed a bargaining strategy to realize supply chain coordination under the condition that manufacturers can design product remanufacturing capacity [28]. Wang et al. analyzed the impact of government take-back regulations on the products interchangeability design level of suppliers and pointed out that the government’s mandatory remanufacturing goal could not always encourage suppliers to implement product design conducive to remanufacturing [29]. Li et al. studied the interaction between remanufacturing design and advertising investment when manufacturers and remanufacturers advertise new products and remanufactured products, respectively [30]. Chen et al. pointed out that the cost parameters for manufacturers to implement modular design and the cost of remanufactured products affect the choice of recovery mode in a CLSC [31]. Xia et al. established a game model between the original manufacturer and the remanufacturer and analyzed the impact of active remanufacturing design on the boundary recovery rate of waste products, unit retail prices, sales volume and sales profit of the two products and pointed out that the original manufacturer’s active remanufacturing design can reduce the retail price of new products per unit and increase the sales volume and sales profit of new products [32]. Hu et al. studied forward and reverse channel intrusion in a closed-loop supply chain under different modular design strategies of manufacturers and pointed out that modular design can make up for loss of retailers’ sales channel encroachment [33]. Chen et al. established a game model considering PIR and its cost-sharing mechanism based on one manufacturer and one retailer and studied the PIR input level, pricing decision and cost-sharing mechanism under different power structures in a CLSC. The study showed that, when consumers are more satisfied with remanufactured products, the cost-sharing mechanism will reduce the profits of members with less power in the CLSC [34].

Most of the above studies only focus on PIR input decision-making of manufacturers, ignoring the reality that remanufacturers also have the motivation to engage in PIR input. At the same time, there is no study on the impact of different government subsidy policies on PIR input decision-making and coordination mechanism design of CLSC.

Based on the above analysis and practical problems, from the perspective of authorized remanufacturing, this paper constructs five centralized or decentralized decision-making models of CLSC when the manufacturer or remanufacturer implements PIR and the government subsidizes the manufacturer or remanufacturer. First, the selection and implementation effect of government subsidy strategy under different situations of PIR implementation are analyzed. Second, the influence of different government subsidy strategies on optimal PIR input level, pricing decision and the impact of the member firms’ performance in the CLSC are discussed. Finally, a fixed license contract is designed to implement the coordination of a decentralized CLSC under different decision situations, and the main conclusions of this paper are verified by numerical simulation analysis.

The research of this paper provides a certain theoretical reference for PIR input decision-making of CLSC and government subsidy policy formulation under authorized remanufacturing mode.

3. Model and Assumption

3.1. Model and Notations Description

Consider a remanufacturing CLSC composed of a manufacturer and a remanufacturer, which belong to the Stackelberg game under complete information in a single period and the manufacturer is the leader of the channel. The manufacturer is responsible for the production and sale of new products, and the remanufacturer is responsible for the recovery of used products and the production and sale of remanufactured products. At the same time, in order to prevent the erosion of the remanufactured product on the new product, the manufacturer will adopt the authorized remanufacturing mode of charging the remanufacturer a certain amount of remanufacturing patent license fee [6,8]. To promote the development of the remanufacturing industry and protect the environment, the government usually provides certain subsidies to the manufacturers or remanufacturers involved in the production and sales of the remanufacturing CLSC [12].

Simultaneously, considering that, in reality, there are not only Xerox, Kodak and other manufacturing enterprises carrying out PIR input, but also Caterpillar, Greenmeet and other remanufacturers carrying out PIR input cases. Therefore, in the authorized remanufacturing mode, this paper studies the CLSC decision-making and coordination mechanism design under different government subsidy policies and different situations of PIR implementation.

Table 1. Notations table.

Notations	Description
$a_n,a_r$	The maximum market demand size for new and remanufactured products
$c_n$	Unit production cost of a new product
$c_r$	Unit production cost of the remanufactured product
$g$	PIR input level
$d$	PIR input effect, the influence coefficient of unit PIR input level on unit remanufacturing cost
$\beta$	The mutual substitution coefficient between new products and remanufactured products reflects the degree of mutual substitution between them
$p_n$,$p_r$	Retail prices for new and remanufactured products
$q_n$,$q_r$	Market demand for new products and remanufactured products
$b$	A fee charged by a manufacturer to a remanufacturer for licensing a unit of a remanufactured product patent
$s$	The government gives units a subsidy for remanufacturing products
$h(g)$	Input cost of PIR
${\pi }_X^i$	$X$’s profit in model $i$, $i=\{C,MM,MT,TM,TT\}$, represents centralized decision-making, government subsidies to manufacturers under manufacturer PIR input mode, government subsidies to remanufacturers under manufacturer PIR input mode, government subsidies to manufacturers under remanufacturer PIR input mode and government subsidies to remanufacturers under remanufacturer PIR input mode; $X=\{M,T,S\}$ represents manufacturer, remanufacturer and CLSC system as a whole.

details the list of notations that are used in the paper.

3.2. Model Assumptions

Referring to relevant literature, this paper gives the following assumptions and explanations.

 Assumption 1. 

There exist two types of products in the market, new products and remanufactured products. The manufacturer sells the new product at a unit price $p_n$ and the remanufacturer sells the remanufactured product at a unit price $p_r$. We assume that there is price competition between the new product and the remanufactured product. Therefore, we refer to the form of market demand function faced by two oligopolists with price competition in microeconomics [35], assume the demand function of new products is $q_n=a_n-p_n+\beta p_r$, the demand function for the remanufactured product is $q_r=a_r-p_r+\beta p_n$, where $\beta$ is the coefficient of mutual substitution between new products and remanufactured products, $0<\beta <1$; since consumers’ recognition of remanufactured products is lower than that of new products in reality, $a_n>a_r$.

 Assumption 2. 

Referring to literature [35,36], we assume that all decisions are considered in a steady-state period, the recycling quantity of waste products in the CLSC is equal to the market demand for remanufactured products, that is, the waste products recovered by the remanufacturer can be remanufactured products through remanufacturing and just meet the market demand for remanufactured products.

 Assumption 3. 

The literature on remanufacturing typically assumes that remanufacturing of one used product does not cost more than manufacturing of one new product, in addition, manufacturers have opportunities to reduce the variable remanufacturing cost. According to references [9,10] et al., we assume that the remanufacturing cost $c_r$ is affected by the PIR input level $g$, that is, $c_r=c_n-dg>0$, where $d>0$. The parameter $d$ measures the maximum amount of cost reduction that can be attained via PIR.

 Assumption 4. 

According to references [7,9] et al., fixed input costs will be generated when enterprises carry out PIR input, we assume that input cost of PIR is a function of the PIR input level $g$, that is; $h(g)=k{g}^2$, where $k>0$. The parameter $k$ represents the scale parameter of PIR input by the enterprise. Hence, the input cost of PIR $h(g)$ increases convexly in $g$.

 Assumption 5. 

Similar to the research assumption in literature [17], the negative definite condition of the Hessian matrix of each profit function and the economic feasibility of related expressions, the scale parameter satisfies $k>\max \left\{\frac{a_r+\beta c_n+s}{4c_n},\frac{d^2}{4}\right\}$.

4. Decision Model Based on Government Subsidy and PIR Input

4.1. Construction and Solution of Centralized Decision Model

Under the centralized decision-making, the manufacturer and the remanufacturer take the overall profit maximization of the CLSC as the decision-making goal and jointly determine the PIR input level and the retail price of new and remanufactured products. In this case, the total profit function of the CLSC system is (the upper corner “$C$” represents the case under centralized decision-making)

$${\pi }_S^C=(p_n-c_n)(a_n-p_n+\beta p_r)+(p_r-c_n+dg+s)(a_r-p_r+\beta p_n)-k{g}^2$$

(1)

In Formula (1), the first item on the right is the profit of new product sales. The second is the profit from sales of remanufactured products under government subsidies. The third item is the input cost of PIR.

 Theorem 1. 

Under centralized decision, the optimal PIR input level of the CLSC is $g^{C\ast }=\frac{d{\Delta }_2}{{\Delta }_1}$; The optimal retail price of the new product and the remanufactured product is $p_n^{C\ast }=\frac{a_n+\beta a_r+(1-{\beta }^2)c_n}{2(1-{\beta }^2)}$, $p_r^{C\ast }=\frac{\beta a_n{\Delta }_1+a_r{\Delta }_3+(1-{\beta }^2){\Delta }_4}{2{\Delta }_1(1-{\beta }^2)}$, respectively; the optimal market demand for new and remanufactured products is $q_n^{C\ast }=\frac{4k{\Delta }_5-d^2{\Delta }_6}{2{\Delta }_1}$, $q_r^{C\ast }=\frac{2k{\Delta }_2}{{\Delta }_1}$, respectively. The total profit of CLSC is ${\pi }_S^{C\ast }=\frac{{\Delta }_1{\Delta }_7{a}_n+(4k-{\beta }^2{d}^2)a_r{}^2+(1-{\beta }^2){\Delta }_8}{4{\Delta }_1(1-{\beta }^2)}$, where ${\Delta }_1=4k-d^2$, ${\Delta }_2=a_r-(1-\beta )c_n+s$, ${\Delta }_3=4k-(2-{\beta }^2)d^2$, ${\Delta }_4=(4k-\beta d^2)c_n-4ks$, ${\Delta }_5=a_n-(1-\beta )c_n-\beta s$, ${\Delta }_6=a_n+\beta a_r-(1-{\beta }^2)c_n$, ${\Delta }_7=a_n+2(\beta a_r-(1-{\beta }^2)c_n)$, ${\Delta }_8=8k(c_n-s)((1-\beta )c_n-a_r)+d^2{c}_n(2\beta a_r-(1-{\beta }^2)c_n)+4ks$.

 Proof of Theorem 1. 

The third-order Hessian matrix of the profit function ${\pi }_S^C\left(p_n,p_r,g\right)$ of the CLSC system with respect to $p_n,p_r,g$ is

$$H^C=\left(\begin{array}{c}\frac{\partial {\pi }_s^2}{\partial p_n{}^2}\frac{\partial {\pi }_s^2}{\partial p_n\partial p_r}\frac{\partial {\pi }_s^2}{\partial p_n\partial g}\\ \frac{\partial {\pi }_s^2}{\partial p_r\partial p_n}\frac{\partial {\pi }_s^2}{\partial p_r{}^2}\frac{\partial {\pi }_s^2}{\partial p_r\partial g}\\ \frac{\partial {\pi }_s^2}{\partial g\partial p_n}\frac{\partial {\pi }_s^2}{\partial g\partial p_r}\frac{\partial {\pi }_s^2}{\partial g^2}\end{array}\right)=\left(\begin{array}{ccc}-2& 2\beta & d\beta \\ 2\beta & -2& -d\\ d\beta & -d& -2k\end{array}\right)$$

(2)

According to the assumptions of the relevant parameters, we get $H_1^C=-2<0$, $H_2^C=4(1-\beta )>0$, $H_3^C=-2(1-{\beta }^2)(4k-d^2)<0$. In this case, the profit function ${\pi }_S^C(p_n,p_r,g)$ of the CLSC system with respect to $p_n,p_r,g$ is a strictly joint concave function. Then the first-order conditions can be established together to obtain the optimal retail price $p_n^{C\ast }$, $p_r^{C\ast }$, $g_{}^{C\ast }$ in the CLSC system. Furthermore, by substituting $p_n^{C\ast }$ and $p_r^{C\ast }$ into the market demand function, the optimal market demand $q_n^{C\ast }$ of the new product and the optimal market demand $q_r^{C\ast }$ of the remanufactured product can be obtained. □

Finally, by substituting the above optimal equilibrium variables into Formula (1), the total profit of CLSC system under centralized decision-making can be obtained.

4.2. Construction and Solution of Decentralized Decision Model for Manufacturer Implementation of PIR

When the manufacturer implements PIR, the decision sequence of the Stackelberg game between the manufacturer and the remanufacturer is: lead the manufacturer to determine the PIR input level, the retail price of new products and the unit remanufacturing patent license fee charged to the remanufacturer; next, the remanufacturer determines the retail price of the remanufactured product.

When the government subsidizes the manufacturer, the profit function of the manufacturer and the remanufacturer is as follows (The upper corner “$MM$” indicates the situation where the government subsidizes the manufacturer when the manufacturer implements PIR)

$${\pi }_M^{MM}=(p_n-c_n)(a_n-p_n+\beta p_r)+(b+s)(a_r-p_r+\beta p_n)-k{g}^2$$

(3)

$${\pi }_T^{MM}=(p_r-b-c_n+dg)(a_r-p_r+\beta p_n)$$

(4)

In Formula (3), the first item on the right is the profit of new product sales. The second item is the license fee for remanufactured products obtained by manufacturers under government subsidies. The third item is the input cost of PIR. In Formula (4), the right side is the sales income of remanufactured products.

When the government gives subsidies to the remanufacturer, the profit function between the manufacturer and the remanufacturer is as follows (the upper corner “$MT$” indicates the situation where the government subsidizes the remanufacturer when the manufacturer implements PIR).

$${\pi }_M^{MT}=(p_n-c_n)(a_n-p_n+\beta p_r)+b(a_r-p_r+\beta p_n)-k{g}^2$$

(5)

$${\pi }_T^{MT}=(p_r-b-c_n+dg+s)(a_r-p_r+\beta p_n)$$

(6)

In Formula (5), the first item on the right is the profit of new product sales. The second item is the authorization fee for remanufactured products obtained by the manufacturer. The third item is the input cost of PIR. In Formula (6), on the right is the sales income of remanufactured products under government subsidies.

 Theorem 2. 

When the manufacturer implements PIR and the government adopts two different subsidy strategies, the optimal PIR input level of the manufacturer is $g^{MM\ast }=g^{MT\ast }=\frac{d{\Delta }_2}{{\Delta }_9}$; the optimal retail price of the new product and the remanufactured product is $p_n^{MM\ast }=p_n^{MT\ast }=\frac{a_n+\beta a_r+(1-{\beta }^2)c_n}{2(1-{\beta }^2)}$, $p_r^{MM\ast }=p_r^{MT\ast }=\frac{\beta a_n{\Delta }_9+a_r{\Delta }_{10}+(1-{\beta }^2){\Delta }_{11}}{2{\Delta }_9(1-{\beta }^2)}$, respectively; The optimal market demand for new and remanufactured products is $q_n^{MM\ast }=q_n^{MT\ast }=\frac{4k{\Delta }_{12}-d^2{\Delta }_6}{2{\Delta }_9}$, $q_r^{MM\ast }=q_r^{MT\ast }=\frac{2k{\Delta }_2}{{\Delta }_9}$, respectively; The optimal unit remanufacturing patent license fee is $b^{MM\ast }=\frac{{\Delta }_9\beta a_n+(8k-{\beta }^2{d}^2)a_r-(1-{\beta }^2){\Delta }_{13}}{2{\Delta }_9(1-{\beta }^2)}$, $b^{MT\ast }=\frac{{\Delta }_9\beta a_n+(8k-{\beta }^2{d}^2)a_r-(1-{\beta }^2){\Delta }_{14}}{2{\Delta }_9(1-{\beta }^2)}$, respectively. The total profit of manufacturer, remanufacturer and CLSC system is

$${\pi }_M^{MM\ast }={\pi }_M^{MT\ast }=\frac{{\Delta }_7{\Delta }_9{a}_n+(4k(1+{\beta }^2)-{\beta }^2{d}^2)a_r{}^2-(1-{\beta }^2)(2{\Delta }_{11}{a}_r-{\Delta }_{15})}{4{\Delta }_9(1-{\beta }^2)}$$

$${\pi }_T^{MM\ast }={\pi }_T^{MT\ast }=\frac{4k^2{\Delta }_2{}^2}{{\Delta }_9{}^2}$$

$${\pi }_S^{MM\ast }={\pi }_S^{MT\ast }=\frac{(16k{\Delta }_1+d^4){\Delta }_7{a}_n+{\Delta }_{16}{a}_r{}^2-(1-{\beta }^2)({\Delta }_{17}{a}_r+{\Delta }_{18}{c}_n{}^2+4ks{\Delta }_{19})}{4{\Delta }_9{}^2(1-{\beta }^2)}$$

where ${\Delta }_9=8k-d^2$, ${\Delta }_{10}=2(6k-d^2)-{\beta }^2{\Delta }_1$, ${\Delta }_{11}=\beta c_n{\Delta }_1+4k(c_n-s)$, ${\Delta }_{12}=2a_n+\beta a_r+({\beta }^2+\beta -2)c_n-\beta s$, ${\Delta }_{13}=(8k-\beta d^2)c_n+2s{\Delta }_1$, ${\Delta }_{14}=(8k-\beta d^2)c_n-8ks$, ${\Delta }_{15}=4k(3-{\beta }^2-2\beta )-(1-{\beta }^2)d^2)c_n{}^2-4ks(2(\beta -1)c_n-s)$, ${\Delta }_{16}=4k(4k({\beta }^2+3)-(3{\beta }^2+1)d)+{\beta }^2{d}^4$, ${\Delta }_{17}=8k(\beta c_n(4k-3d^2)+(c_n-s)(12k-d^2))+2\beta d^4{c}_n$, ${\Delta }_{18}=4k(4k({\beta }^2+6\beta -7)-d^2(3{\beta }^2+2\beta -5))-(1-{\beta }^2)d^4$, ${\Delta }_{19}=(12k-d^2)(2(1-\beta )c_n-s)$.

 Proof of Theorem 2. 

When $\frac{\partial {\pi }_T^2}{\partial p_r{}^2}=-2<0$, the profit function ${\pi }_T^{MM}(p_r)$ of the remanufacturer with respect to $p_r$ strictly concave, we get the optimal feedback function of the remanufacturer on the retail price of the remanufactured product is $p_r^{MM}=\frac{a_r+\beta p_n+c_n+b-dg}{2}$. Then, substitute $p_r^{MM}$ into Formula (3), that is $p_n^{MM\ast }$, $g_{}^{MM\ast }$, $b_{}^{MM\ast }$. Furthermore, by substituting $p_n^{MM\ast }$, $g_{}^{MM\ast }$, $b_{}^{MM\ast }$ into $p_r^{MM}$, the remanufacturer’s optimal remanufactured product retail price $p_r^{MM\ast }$ can be obtained. Then, by substituting $p_n^{MM\ast }$, $p_r^{MM\ast }$ into the market demand function of the new product and the remanufactured product, the optimal market demand $q_n^{MM\ast }$ of the new product and the optimal market demand $q_r^{MM\ast }$ of the remanufactured product can be obtained. Finally, the profit of the manufacturer, the remanufacturer and the CLSC as a whole can be obtained.

4.3. Construction and Solution of Decentralized Decision-Making Model for Remanufacturer Implementation of PIR

When the remanufacturer implements PIR, the decision sequence of the Stackelberg game between the manufacturer and the remanufacturer is: the leading manufacturer first determines the retail price of the new product and the unit remanufacturing patent license fee charged to the remanufacturer; next, the remanufacturer determines the PIR input level and the retail price of the remanufactured product.

When the government subsidizes the manufacturer, the profit function of the manufacturer and the remanufacturer is as follows (the upper corner “$TM$” indicates the situation when the government subsidizes the manufacturer when the remanufacturer implements PIR)

$${\pi }_M^{TM}=(p_n-c_n)(a_n-p_n+\beta p_r)+(b+s)(a_r-p_r+\beta p_n)$$

(7)

$${\pi }_T^{TM}=(p_r-b-c_n+dg)(a_r-p_r+\beta p_n)-k{g}^2$$

(8)

In Formula (7), the first item on the right is the profit of new product sales. The second item is the license fee for remanufactured products obtained by manufacturers under government subsidies. In Formula (8), the first item on the right is the sales income of remanufactured products. The second item is the input cost of PIR.

When the government gives subsidies to the remanufacturer, the profit function between the manufacturer and the remanufacturer is as follows (the upper corner “$TT$” indicates the situation when the government subsidizes the remanufacturer when the remanufacturer implements PIR)

$${\pi }_M^{TT}=(p_n-c_n)(a_n-p_n+\beta p_r)+b(a_r-p_r+\beta p_n)$$

(9)

$${\pi }_T^{TT}=(p_r-b-c_n+dg+s)(a_r-p_r+\beta p_n)-k{g}^2$$

(10)

In Formula (9), the first item on the right is the profit of new product sales. The second item is the authorization fee for remanufactured products obtained by the manufacturer. In Formula (10), the first item on the right is the sales income of remanufactured products under government subsidies. The second item is the input cost of PIR.

 Theorem 3. 

When the remanufacturer implements PIR and the government adopts two different subsidy strategies, the optimal PIR input level of the manufacturer is $g^{TM\ast }=g^{TT\ast }=\frac{d{\Delta }_2}{2{\Delta }_1}$; the optimal retail price of the new product and the remanufactured product is $p_n^{TM\ast }=p_n^{TT\ast }=\frac{a_n+\beta a_r+(1-{\beta }^2)c_n}{2(1-{\beta }^2)}$, $p_r^{TM\ast }=p_r^{TT\ast }=\frac{\beta a_n{\Delta }_1+a_r{\Delta }_{20}+(1-{\beta }^2){\Delta }_{21}}{2{\Delta }_1(1-{\beta }^2)}$, respectively; the optimal market demand for new and remanufactured products is $q_n^{TM\ast }=q_n^{TT\ast }=\frac{2k{\Delta }_{12}-d^2{\Delta }_6}{2{\Delta }_1}$, $q_r^{TM\ast }=q_r^{TT\ast }=\frac{k{\Delta }_2}{{\Delta }_1}$, respectively. The optimal unit remanufacturing patent license fee is $b^{TM\ast }=\frac{\beta a_n+a_r-(1-{\beta }^2)(c_n+s)}{2(1-{\beta }^2)}$, $b^{TT\ast }=\frac{\beta a_n+a_r-(1-{\beta }^2)(c_n-s)}{2(1-{\beta }^2)}$, respectively. The total profit of manufacturer, remanufacturer and CLSC system is

$${\pi }_M^{TM\ast }={\pi }_M^{TT\ast }=\frac{{\Delta }_1{\Delta }_7{a}_n+(2k(1+{\beta }^2)-{\beta }^2{d}^2)a_r{}^2-(1-{\beta }^2)(2{\Delta }_{21}{a}_r-{\Delta }_{22})}{4{\Delta }_9(1-{\beta }^2)}, {\pi }_T^{TM\ast }={\pi }_T^{TT\ast }=\frac{k{\Delta }_2{}^2}{4{\Delta }_1}$$

$${\pi }_S^{TM\ast }={\pi }_S^{TT\ast }=\frac{{\Delta }_1{\Delta }_7{a}_n+(({\beta }^2+3)k-{\beta }^2{d}^2)a_r{}^2-(1-{\beta }^2)({\Delta }_{23}{a}_r-{\Delta }_{24})}{4{\Delta }_1(1-{\beta }^2)}$$

where ${\Delta }_{20}=2(3k-d^2)-{\beta }^2(2k-d^2)$, ${\Delta }_{21}=2k\left(c_n-s\right)+\beta c_n\left(2k-d^2\right)$, ${\Delta }_{22}=2k(3-{\beta }^2-2\beta )-(1-{\beta }^2)d^2)c_n{}^2-2ks(2(1-\beta )c_n-s)$, ${\Delta }_{23}=6k(c_n-s)+2\beta c_n(k-d^2)$, ${\Delta }_{24}=k(7-{\beta }^2-6\beta )-(1-{\beta }^2)d^2)c_n{}^2-3ks(2(1-\beta )c_n-s)$.

The Proof of Theorem 3 is the same as that of Theorem 2.

5. Equilibrium Result Analysis

Our analysis is based on the solutions of these models through backward induction. We compare the equilibrium decisions and profits obtained under these decentralized models.

 Proposition 1. 

Whether the manufacturer or the remanufacturer implements PIR, there is $p_n^{H\ast }=p_n^{L\ast }$, $p_r^{H\ast }=p_r^{L\ast }$, $q_n^{H\ast }=q_n^{L\ast }$, $q_r^{H\ast }=q_r^{L\ast }$, $g^{H\ast }=g^{L\ast }$, ${\pi }_M^{H\ast }={\pi }_M^{L\ast }$, ${\pi }_T^{H\ast }={\pi }_T^{L\ast }$, ${\pi }_S^{H\ast }={\pi }_S^{L\ast }$, $b_{}^{MT\ast }-b_{}^{MM\ast }=s$, $b^{TT\ast }-b^{TM\ast }=s$ when the government adopts different subsidy strategies, where $H=\{MM,TM\}$, $L=\{MT,TT\}$.

Proposition 1 shows that either the manufacturer or the remanufacturer implements PIR, different government subsidy policy (subsidize manufacturers or remanufacturers) only for two kinds of PIR model has influenced the unit remanufacturing patent licensing fees, and the level of PIR inputs, new products and the retail price of remanufacturing product and market demand, as well as members of the enterprise profits. Simultaneously, either the manufacturer or the remanufacturer implements PIR, the unit remanufacturing patent licensing fee is higher when the government subsidizes the remanufacturer than when the government subsidizes the manufacturer and the difference between the two is just the amount of government subsidy per unit $s$. This is because, when the government subsidizes the manufacturer or the remanufacturer, the manufacturer, as the dominant player, can share the government subsidy with the remanufacturer by adjusting the unit remanufacturing patent license fee so that different subsidy strategies of the government will not have different effects on the optimal pricing decision and performance of the CLSC.

By comparing the equilibrium results under government different subsidy strategies, we can find that patent licensing fees in authorized remanufacturing can be used as a lever to coordinate the competition and cooperation relationship between manufacturer and remanufacturer. Whether manufacturer or remanufacturer implemented PIR, both parties can share government subsidies by coordinating patent licensing fees for remanufacturing. Therefore, the difference in government subsidy strategy will not affect the optimal pricing decision and revenue of the manufacturer and the remanufacturer, but the government can still adjust the remanufacturing authorization cost between the two sides by adjusting the subsidy strategy.

Therefore, the government can affect the manufacturer’s optimal patent license fee decision by adjusting the subsidy strategy so as to promote the manufacturer and remanufacturer to better cooperate, improve the recycling and remanufacturing efficiency of waste products and achieve sustainable economic and environmental development.

 Proposition 2. 

Either the manufacturer or the remanufacturer implements PIR, (1) When the government subsidizes manufacturers, $\frac{\partial b^{TM\ast }}{\partial s}<\frac{\partial b^{MM\ast }}{\partial s}<0$; (2) When the government subsidizes remanufacturers, $\frac{\partial b^{TT\ast }}{\partial s}>\frac{\partial b^{MT\ast }}{\partial s}>0$.

Proposition 2 shows that either the manufacturer or the remanufacturer implements PIR; with the increase in government subsidies, when the government subsidizes the manufacturer, the manufacturer will gradually reduce the unit remanufacturing patent license fee so as to benefit the remanufacturer; however, when the government subsidizes the remanufacturer, the manufacturer will share the government subsidy with the remanufacturer in the form of gradually increasing the remanufacturing patent license fee. Furthermore, with the change in government subsidies, although the overall trend of patent licensing fees in the two different PIR implementation scenarios is similar, the difference in the different PIR implementation scenarios still affects the speed of change in this trend.

 Proposition 3. 

Whether the government subsidizes manufacturers or remanufacturers, PIR is implemented in different situations, (1) $\frac{\partial g^{T\ast }}{\partial s}>\frac{\partial g^{M\ast }}{\partial s}>0$, $\frac{\partial p_n^{T\ast }}{\partial s}=\frac{\partial p_n^{M\ast }}{\partial s}=0$, $\frac{\partial p_r^{T\ast }}{\partial s}<\frac{\partial p_r^{M\ast }}{\partial s}<0$, $\frac{\partial q_n^{T\ast }}{\partial s}<\frac{\partial q_n^{M\ast }}{\partial s}<0$, $\frac{\partial q_r^{T\ast }}{\partial s}>\frac{\partial q_r^{M\ast }}{\partial s}>0$; (2) $\frac{\partial {\pi }_M^{T\ast }}{\partial s}>\frac{\partial {\pi }_M^{M\ast }}{\partial s}>0$, $\frac{\partial {\pi }_T^{T\ast }}{\partial s}>\frac{\partial {\pi }_T^{M\ast }}{\partial s}>0$, $\frac{\partial {\pi }_S^{T\ast }}{\partial s}>\frac{\partial {\pi }_S^{M\ast }}{\partial s}>0$. Including, $M=\{MM,MT\}$, $T=\{TM,TT\}$.

Proposition 3 shows that either the manufacturer or the remanufacturer implements PIR; the increase in government subsidies will not affect the retail price of new products but will lead to the decrease in the market demand of new products. Simultaneously, it is beneficial to reduce the price of remanufactured products, increase the PIR input level and the market demand of remanufactured products and eventually play a role in increasing the total profits of manufacturers, remanufacturers and the system.

In fact, whatever government subsidy policy and PIR input mode, government subsidies were beneficial to increase the income of the remanufacturing products, and manufacturers and manufacturers can share the remanufacturing yield through patent licensing fees, so both sides have a profit motive increase PIR investment level, to reduce the manufacturing cost again, achieve the purpose of further improving the efficiency of remanufacturing and benefit. Simultaneously, the reduction in remanufacturing cost will also make the remanufacturer take the initiative to reduce the retail price of the remanufactured product to stimulate demand and obtain more profits, while the reduction in the retail price of the remanufactured product will invisibly weaken the market competitiveness of the new product and lead to the decline of the market demand for the new product. However, the manufacturer can adjust the patent license fee to transfer the profit from the production and sales of remanufactured products, and the profit is higher than the profit loss caused by the decrease in market demand for new products. Therefore, the members and the whole CLSC can benefit from the government subsidy.

Furthermore, compared with the manufacturer’s implementation of PIR, the remanufacturer’s PIR input level, retail price and market demand of new and remanufactured products, CLSC members and overall profit change with the change in government subsidies are more obvious. This is due to the remanufacturer implementing PIR; the remanufacturer bears the input cost pressure of PIR, and the profit from the production and sales of remanufactured products is the only source of profit for the remanufacturer. Therefore, with the increase in government subsidies, remanufacturers will be more motivated to increase the PIR input level to effectively improve the efficiency of remanufacturing, to obtain more remanufacturing benefits and government subsidies.

Through the equilibrium analysis of changes in government subsidies, we can find that, regardless of the implementation of PIR, government subsidies are always conducive to increasing the PIR input level, improving the efficiency and benefit of remanufacturing, promoting the sales of remanufacturing products and increasing the members and overall profits of CLSC. Especially when the remanufacturer implements PIR, the benefits brought by the increase in government subsidies will be more obvious, and the efficiency and effect of government subsidies will be more significant.

For the government, while implementing subsidies to promote the development of the remanufacturing industry, it can encourage remanufacturing enterprises to actively implement PIR investment so as to effectively improve the efficiency of government subsidies, give better play to the role of government subsidies and promote sustainable development.

 Proposition 4. 

Whether the government subsidizes manufacturers or remanufacturers, PIR is implemented in different situations, (1) $\frac{\partial g^{T\ast }}{\partial d}>\frac{\partial g^{M\ast }}{\partial d}>0$, $\frac{\partial p_n^{T\ast }}{\partial d}=\frac{\partial p_n^{M\ast }}{\partial d}=0$, $\frac{\partial p_r^{T\ast }}{\partial d}<\frac{\partial p_r^{M\ast }}{\partial d}<0$, $\frac{\partial q_n^{T\ast }}{\partial d}<\frac{\partial q_n^{M\ast }}{\partial d}<0$, $\frac{\partial q_r^{T\ast }}{\partial d}>\frac{\partial q_r^{M\ast }}{\partial d}>0$; (2) $\frac{\partial b^{M\ast }}{\partial d}>0$, $\frac{\partial b^{T\ast }}{\partial d}=0$; (3) $\frac{\partial {\pi }_M^{T\ast }}{\partial d}>\frac{\partial {\pi }_M^{M\ast }}{\partial d}>0$, $\frac{\partial {\pi }_T^{T\ast }}{\partial d}>\frac{\partial {\pi }_T^{M\ast }}{\partial d}>0$, $\frac{\partial {\pi }_S^{T\ast }}{\partial d}>\frac{\partial {\pi }_S^{M\ast }}{\partial d}>0$. Including, $M=\{MM,MT\}$, $T=\{TM,TT\}$.

Proposition 4 shows that, regardless of the government subsidy strategy and the implementation of PIR, the increase in PIR input effect does not affect the retail price of new products but will lead to a subsequent decline in the market demand for new products. Simultaneously, it can also play a role in reducing the price of remanufactured products, increasing the PIR input level, remanufactured product market demand and the total profit of manufacturers, remanufacturers and systems. Compared with the PIR implemented by the manufacturer, the PIR input level, the retail price and market demand of new and remanufactured products, the members of the CLSC and the overall profit of the remanufacturer are more obvious to change with the PIR input effect. Additionally, when manufacturers implement PIR, the per-unit patent licensing fee increases with the PIR input effect; however, when the remanufacturer implements PIR, the increase in PIR input effect has no effect on the unit patent licensing fee.

In fact, the increase in PIR input effect will effectively increase the income of unit PIR input level to remanufacturing. Therefore, both manufacturers and remanufacturers have the profit motive to improve the PIR input level to effectively improve the efficiency and benefit of remanufacturing. At the same time, in the manufacturer’s PIR input mode, since the manufacturer bears the PIR input cost, it will share the income brought by the increase in PIR input level with the remanufacturer by adjusting the unit patent license fee to cover its own cost. Combined with Proposition 2, it can be found that the unit patent license fee can not only share the government subsidy effect, but also help the manufacturer transfer the remanufacturing income of the remanufacturer. This also further validates the conclusion that manufacturers and remanufacturers can share the production and sales revenue of remanufactured products and government subsidies by adjusting the unit patent licensing fee.

Through the equilibrium analysis of PIR input effect changes, we can find that, if either the manufacturer or the remanufacturer implements PIR, both sides have the profit motive to improve the PIR input effect so as to reduce the remanufacturing cost more effectively and promote the development of the remanufacturing industry. Therefore, when implementing PIR, enterprises can improve the PIR input effect by introducing cutting-edge technology or optimizing the PIR implementation process, with the aim of achieving better cost reduction and efficiency increase. Simultaneously, when the manufacturer implements PIR, the manufacturer can also transfer the remanufacturing income of the remanufacturer by adjusting the patent license fee.

Combine Propositions 3 and 4, for the government to effectively promote the remanufacturing industry, should be at the same time of subsidies but through the help enterprises to introduce advanced technology and improve the level of remanufacturing technology to further improve the PIR input effect of enterprises to effectively mobilize the innovation enthusiasm of enterprises in PIR input; thus, the efficiency and benefit of remanufacturing can be improved, and the coupling of policy and market can be formed so that government subsidies can play a more effective role.

 Proposition 5. 

Either the manufacturer or the remanufacturer is subsidized by the government, there are different situations in the implementation of PIR, $g^{C\ast }>g^{T\ast }>g^{M\ast }$, $p_n^{C\ast }=p_n^{T\ast }=p_n^{M\ast }$, $p_r^{M\ast }>p_r^{T\ast }>p_r^{C\ast }$, $q_n^{M\ast }>q_n^{T\ast }>q_n^{C\ast }$, $q_r^{C\ast }>q_r^{T\ast }>q_r^{M\ast }$. Including, $M=\{MM,MT\}$, $T=\{TM,TT\}$.

Proposition 5 shows that centralized decision-making is optimal in terms of increasing PIR input level, reducing the price of remanufactured products and increasing the market demand for remanufactured products, followed by remanufacturer implementation of PIR and finally by manufacturer implementation of PIR. Simultaneously, different situations of PIR implementation do not affect the retail price of new products, and the market demand for new products is the highest when the manufacturer implements PIR. In fact, although the manufacturer can adjust the patent licensing fees to transfer through remanufacturing yield, the manufacturer’s earnings are still mainly from the sale of new products, and new products compete with remanufactured products, and the profit of the remanufacturer is completely dependent on the production and sale of the remanufactured product, so compared with the manufacturers to implement PIR, When remanufacturers implement PIR, they will have more incentive to increase PIR input level.

Propositions 3 and 5 reveal that, compared with the manufacturer’s implementation of PIR, the remanufacturer’s implementation of PIR has higher PIR input level and market demand for remanufactured products, and the effect of government subsidy is also better. Therefore, when the government implements subsidies, it can actively combine the forces of industries and enterprises to promote the remanufacturing enterprises to actively invest in PIR, therefore effectively amplify the efficiency and effect of government subsidies and then rapidly promote the development of the remanufacturing industry.

 Proposition 6. 

Whether the government subsidizes the manufacturer or the remanufacturer, (1) ${\pi }_M^{T\ast }>{\pi }_M^{M\ast }$, (2) ${\pi }_T^{T\ast }>{\pi }_T^{M\ast }$, (3) ${\pi }_S^{C\ast }>{\pi }_S^{T\ast }>{\pi }_S^{M\ast }$ exists in different situations when PIR is implemented. Including, $M=\{MM,MT\}$, $T=\{TM,TT\}$.

Proposition 6 shows that compared with the manufacturer’s implementation of PIR, the manufacturer, the remanufacturer and the CLSC as a whole all obtain more profits when the remanufacturer implements PIR, but the total system profit is still lower than that when the decision is centralized. This is because, when the remanufacturer implements PIR, the PIR input level is higher and the remanufacturing benefit is more significant. Although the market demand for new products has decreased, the benefit brought by the increase in the market demand for remanufactured products is still higher than the loss of the revenue from the decline of the market for new products. Therefore, both CLSC members and the whole can benefit from the increased market demand for remanufactured products.

Proposition 6 reveals that under authorized remanufacturing mode, the manufacturer always prefers the remanufacturer to implement PIR input, but the remanufacturer does not prefer the manufacturer to implement PIR input. Both the members and the whole CLSC benefit more from the remanufacturer’s implementation of PIR model, which also well explains that many worlds leading remanufacturers such as Caterpillar begin to actively engage in PIR research and development investment in reality.

By comparing the model equilibrium results when the manufacturer implement PIR input with those when the remanufacturer implement PIR input, it can be found that the manufacturer’ implementation of PIR input is more conducive to improving the market demand for new products. The implementation of PIR input by remanufacturers is not only more conducive to improving the PIR input level and the market demand of remanufactured products, but also more conducive to improving the closed-loop supply chain members and the overall performance of the system and improving the efficiency of government subsidies.

Simultaneously, it can be found that the PIR input level, the market demand for remanufactured products and the overall profit of the CLSC are all higher under the centralized decision-making compared with the decentralized decision-making. This also shows that centralized decision-making is more conducive to promoting the development of remanufacturing industry and realizing the coordinated development of economy and environment, while decentralized decision-making has some efficiency loss. Therefore, we will design the corresponding contract in the next section to realize the coordination of the CLSC.

6. Design of Coordination Contract Based on Government Subsidy and PIR Input

Because of the difference in government subsidy policy on the CLSC optimal PIR, pricing decisions as well as members of the enterprise performance are no different. Therefore, in this section, we take the centralized decision-making model in Section 4 as a reference and, when the government subsidizes manufacturers, as an example, when the manufacturer or the remanufacturer implements PIR, a CLSC fixed license fee coordination contract considering both government subsidy and PIR input is designed and implemented.

6.1. Coordination Model for Manufacturer Implementation of PIR

The basic idea of the fixed license fee coordination contract: as the dominant player, the manufacturer can first improve the PIR input level to reduce the remanufacturing cost of the remanufacturer; moreover, a lower patent license fee $b^{MMR}$ is provided to the remanufacturer to motivate the remanufacturer to sell the remanufactured product at the remanufactured product price under centralized decision. Second, the remanufacturer transfers a fixed license fee $T^{MMR}$ to the manufacturer, which compensates the manufacturer for the lost revenue to achieve CLSC coordination. Therefore, under the fixed license fee coordination contract $(b^{MMR},T^{MMR})$, the revenue function of the manufacturer and the remanufacturer (the upper corner “$MMR$” represents the coordination model under the manufacturer’s PIR input mode)

$${\pi }_M^{MMR}=(p_n-c_n)(a_n-p_n+\beta p_r)+(b+s)(a_r-p_r+\beta p_n)-k{g}^2+T$$

(11)

$${\pi }_T^{MMR}=(p_r-b-c_n+dg)(a_r-p_r+\beta p_n)-T$$

(12)

 Theorem 4. 

Consider the CLSC of government subsidy and manufacturer PIR input. When fixed license fee coordination contract parameters under condition $4k-d^2>0$ satisfy $b^{MMR\ast }=\frac{\beta {\Delta }_6-2s(1-{\beta }^2)}{2(1-{\beta }^2)}$, $T^{MMR\ast }\in \left[T_{\min }^{MMR},T_{\max }^{MMR}\right]$, the fixed license fee coordination contract $(b^{MMR},T^{MMR})$ can realize the perfect coordination of the CLSC, and both manufacturers and remanufacturers can accept the coordination contract. That is, satisfy $g^{MMR\ast }=g^{C\ast }$, $p_n^{MMR\ast }=p_n^{C\ast }$, $p_r^{MMR\ast }=p_r^{C\ast }$, $q_n^{MMR\ast }=q_n^{C\ast }$, $q_r^{MMR\ast }=q_r^{C\ast }$, ${\pi }_S^{MMR\ast }={\pi }_S^{C\ast }$. Furthermore, the profits of the manufacturer and the remanufacturer under this coordination contract are:

$${\pi }_M^{MMR\ast }=\frac{{\Delta }_7{\Delta }_{25}{a}_n+{\Delta }_{26}{a}_r{}^2-(1-{\beta }^2)({\Delta }_{27}{a}_r-{\Delta }_{28}{c}_n{}^2-2ks{d}^2{\Delta }_{29})}{4{\Delta }_1{}^2(1-{\beta }^2)}+T$$

(13)

$${\pi }_T^{MMR\ast }=\frac{4k^2{\Delta }_2{}^2}{{\Delta }_1}-T$$

(14)

where $T_{\min }^{MMR}=\frac{16k^3{\Delta }_2{}^2}{{\Delta }_1{}^2{\Delta }_9}$, $T_{\max }^{MMR}=\frac{32k^3(6k-d^2){\Delta }_2{}^2}{{\Delta }_1{}^2{\Delta }_9{}^2}$, ${\Delta }_{25}=8k(2k-d^2)+d^4$, ${\Delta }_{26}=4k{\beta }^2{\Delta }_1-d^2(4k-{\beta }^2{d}^2)$, ${\Delta }_{27}=8k(4k\beta c_n-d^2((1+\beta )c_n-s))+2\beta d^4{c}_n$, ${\Delta }_{28}=(1-{\beta }^2)(16k^2+d^4)+4k{d}^2({\beta }^2+2\beta -3)$, ${\Delta }_{29}=c_n(1-\beta )-2s$.

Theorem 4 shows that the fixed license fee coordination contract can perfectly realize the coordination of decentralized CLSC when manufacturers implement PIR. This contract enables PIR input levels, retail prices for the new and remanufactured products and market demand to reach the optimal level under centralized decision-making, and the total profit of CLSC system reaches the maximum profit under centralized decision. Second, there is a range of fixed license fees transferred from the remanufacturer to the manufacturer in the contract, and the specific execution fee can be a specific result of negotiation between the manufacturer and the remanufacturer within the range. When $T^{MMR\ast }$ is set to the upper limit value, the remanufacturer only obtains the same profit as the decentralized decision, while the manufacturer uses channel leadership to obtain all the profits brought by coordination; however, at this time, the PIR input level and the market demand for remanufactured products are improved, which is beneficial to consumers, the environment and the overall stable operation of the CLSC.

6.2. Coordination Model for Remanufacturer Implementation of PIR

The basic idea of the fixed license fee coordination contract: as the dominant player, the manufacturer can first offer a lower patent licensing fee $b^{TMR}$ to the remanufacturer, to motivate the remanufacturer to sell the remanufactured product at the remanufactured product price under the centralized decision, and improve the PIR input level to the optimal level under the centralized decision. Second, the remanufacturer transfers a fixed license fee $T^{TMR}$ to the manufacturer, which compensates the manufacturer for the lost revenue to achieve CLSC coordination. Therefore, under the fixed license fee coordination contract $(b^{TMR},T^{TMR})$, the revenue functions of the manufacturer and the remanufacturer are, respectively (the upper corner “$TMR$“ represents the coordination model under the input mode of the remanufacturer PIR):

$${\pi }_M^{TMR}=(p_n-c_n)(a_n-p_n+\beta p_r)+(b+s)(a_r-p_r+\beta p_n)+T$$

(15)

$${\pi }_T^{TMR}=(p_r-b-c_n+dg)(a_r-p_r+\beta p_n)-k{g}^2-T$$

(16)

 Theorem 5. 

Consider government subsidies and remanufacturers implementing PIR in a CLSC. When the parameters of the fixed license fee coordination contract meet $b^{TMR\ast }=\frac{\beta {\Delta }_6-2s(1-{\beta }^2)}{2(1-{\beta }^2)}$, $T^{TMR\ast }\in \left[T_{\min }^{TMR},T_{\max }^{TMR}\right]$, the fixed license fee coordination contract $(b^{TMR},T^{TMR})$ can achieve the perfect coordination of the CLSC, and both the manufacturer and the remanufacturer can accept the coordination contract. That is, satisfy $g^{TMR\ast }=g^{C\ast }$, $p_n^{TMR\ast }=p_n^{C\ast }$, $p_r^{TMR\ast }=p_r^{C\ast }$, $q_n^{TMR\ast }=q_n^{C\ast }$, $q_r^{TMR\ast }=q_r^{C\ast }$, ${\pi }_S^{TMR\ast }={\pi }_S^{C\ast }$. Furthermore, the profits of the manufacturer and the remanufacturer under this coordination contract are

$${\pi }_M^{TMR\ast }=\frac{{\Delta }_6{}^2}{4(1-{\beta }^2)}+T$$

(17)

$${\pi }_T^{TMR\ast }=\frac{k{\Delta }_{30}{}^2}{{\Delta }_1}-T$$

(18)

where $T_{\min }^{TMR}=\frac{k{\Delta }_2{}^2}{2{\Delta }_1}$, $T_{\max }^{TMR}=\frac{3k{\Delta }_2{}^2}{4{\Delta }_1}$, ${\Delta }_{30}=a_r-(1-{\beta }^2)c_n+s$.

Theorem 5 shows that the fixed license fee coordination contract can perfectly realize the coordination of decentralized CLSC when the remanufacturer implements PIR. This contract makes the PIR input level, retail price and market demand of the new and remanufactured products reach the best level under centralized decision, and the total profit of CLSC system reaches the maximum profit under centralized decision. Second, compared with the coordination contract in which the manufacturer implements PIR, the manufacturer resets the patent license fee in the coordination contract under the two cases (that is, $b^{MMR\ast }=b^{TMR\ast }$). The difference is that the fixed license fee in the coordination contract in this model is changed. This is because the remanufacturer decides the retail price of the remanufactured product and the PIR input level at the same time; thus, more decision-making power can be obtained in the game with the manufacturer, which will affect the manufacturer’s decision and the negotiation of fixed license fee with the manufacturer.

 Proposition 7. 

The effect of government subsidy on the feasible range of the fixed license fee coordination contract is (1) $\frac{\partial T_{\min }^{MMR}}{\partial s}>0$, $\frac{\partial T_{\max }^{MMR}}{\partial s}>0$, $\frac{\partial \Delta T_{}^{MMR}}{\partial s}>0$; (2) $\frac{\partial T_{\min }^{TMR}}{\partial s}>0$, $\frac{\partial T_{\max }^{TMR}}{\partial s}>0$, $\frac{\partial \Delta T_{}^{TMR}}{\partial s}>0$, where $\Delta T_{}^{MMR}=T_{\max }^{MMR}-T_{\min }^{MMR}$, $\Delta T_{}^{TMR}=T_{\max }^{TMR}-T_{\min }^{TMR}$.

Proposition 7 shows that, under the two PIR scenarios, the upper and lower limits of the fixed license fee in the fixed license fee coordination contract increase with the increase in government subsidies, and the feasible range of the fixed license fee coordination contract also increases. In fact, government subsidies can improve the benefits of remanufacturing to a certain extent, thereby relieving the cost pressure of manufacturers and remanufacturers. With the increase in government subsidies, the negotiation scope of the fixed license fee between the manufacturer and the remanufacturer will also become larger, thus effectively expanding the feasible area of the contract. Therefore, government subsidies can effectively improve the possibility of cooperation between member enterprises of the CLSC and then better improve the operational efficiency and performance of the CLSC.

 Proposition 8. 

The influence of PIR input effect on the feasible range of the fixed license fee coordination contract is (1) $\frac{\partial T_{\min }^{MMR}}{\partial d}>0$, $\frac{\partial T_{\max }^{MMR}}{\partial d}>0$, $\frac{\partial \Delta T_{}^{MMR}}{\partial d}>0$; (2) $\frac{\partial T_{\min }^{TMR}}{\partial d}>0$, $\frac{\partial T_{\max }^{TMR}}{\partial d}>0$, $\frac{\partial \Delta T_{}^{TMR}}{\partial d}>0$, where $\Delta T_{}^{MMR}=T_{\max }^{MMR}-T_{\min }^{MMR}$, $\Delta T_{}^{TMR}=T_{\max }^{TMR}-T_{\min }^{TMR}$.

Proposition 8 shows that, under the two PIR implementation scenarios, the upper and lower limits of the fixed license fee in the fixed license fee coordination contract increase with the increase in the PIR input effect, and the range of possible harmonized contracts for fixed license fees is growing. In fact, the increase in PIR input effect magnifies the benefits brought by PIR input to remanufacturing to a certain extent, which can help remanufacturers reduce remanufacturing costs more effectively, improve remanufacturing benefits and efficiency and relieve remanufacturers’ remanufacturing cost pressure. With the increase in PIR input effect, the remanufacturer’s acceptable range of fixed license fee will also become larger, thus effectively expanding the feasible area of this contract.

Therefore, the government can create a good environment for enterprises to implement PIR and help enterprises to improve the PIR input effect by combining social forces, such as industries. From the perspective of enterprises, the input efficiency of PIR can be improved by learning from new technologies and introducing new equipment. By improving the PIR input effect, it can not only promote the cooperation between member enterprises of the CLSC more effectively, improve the operational efficiency and performance of the CLSC but also provide more environmental benefits.

7. Numerical Examples

To further explore some management implications behind the model, this section extends and verifies the above main conclusions through numerical simulation. The PIR input cost coefficient $k$ and the substitution coefficient $\beta$ of new products and remanufactured products are important influencing factors of PIR input and pricing decision of CLSC. Therefore, the influence of the change in $k,\beta$ on the optimal decision-making of CLSC will be analyzed.

Refer to the literature ([9,10] et al.); that is: $a_n=120$, $a_r=100$, $c_n=35$, $d=9$, $s=5$.

As can be seen from Figure 1, whether the manufacturer or the remanufacturer implements PIR, a reduction in the PIR input cost coefficient $k$ and an increase in the substitution coefficient $\beta$ between new products and remanufactured products are conducive to improvement in the PIR input level. This is because, in any PIR input mode, a decrease in $k$ means a decrease in the difficulty and cost of PIR input, while an increase in $\beta$ means enhancement in the substitution degree of remanufactured products for new products, that is, an enhancement in the market competitiveness of remanufactured products. Therefore, with the decrease in $k$ and the increase in $\beta$, both the manufacturer and the remanufacturer have the profit motive to increase the PIR input level to improve the efficiency and benefit of remanufacturing. It can be further found that, regardless of the size of $k,\beta$, compared with the manufacturer’s implementation of PIR, the remanufacturer’s implementation of PIR always has a higher level of PIR input, and the range of change with $k,\beta$ is always larger.

Figure 1 reveals that, no matter whether the manufacturer or the remanufacturer implements PIR, to effectively improve the PIR input level, for enterprises, on the one hand, they should increase innovation and R&D in PIR input so as to reduce the difficulty and the cost of PIR input and improve the efficiency of PIR input. On the other hand, efforts should be made to improve the quality of remanufactured products so that they can achieve the same effect as new products and strengthen the publicity and promotion of remanufactured products to improve the degree of remanufactured products replacing new products. Simultaneously, efforts can also be made to promote remanufacturing enterprises to actively carry out PIR input to improve the efficiency and benefit of PIR input more effectively.

It can be seen from Figure 2, Figure 3, Figure 4 and Figure 5 that, whether the manufacturer or the remanufacturer implements PIR, the reduction in PIR input cost coefficient $k$ is not conducive to the sales of new products, but it is beneficial to improve the market demand of remanufactured products and increase the profits of the manufacturer and the remanufacturer. With the increase in substitution coefficient $\beta$ between new products and remanufactured products, the market demand of new products and remanufactured products and the profits of manufacturers and remanufacturers are also increasing. Furthermore, compared with the manufacturer’s implementation of PIR, the market demand for remanufactured products and the profits of the manufacturer and the remanufacturer are higher when the remanufacturer implements PIR. Obviously, the remanufacturer’s implementation of PIR is more conducive to promoting the development of remanufactured products and improving the income of CLSC members.

Figure 2, Figure 3, Figure 4 and Figure 5 reveal that the government, through subsidies to promote development of recycling waste products remanufacturing industry and, at the same time, to better improve the efficiency and effect of government subsidy, on the one hand, can influence joint industry strength and promote remanufacturing enterprises to actively participate in PIR research and development and PIR inputs to create a good environment for the enterprise in order to reduce the difficulty of the enterprise for PIR regarding R&D and cost.

On the other hand, social media and other forces can also be combined to strengthen the publicity and promotion of remanufactured products, improve consumers’ recognition of remanufactured products and thus improve the market demand for remanufactured products more effectively.

In general, implementation of PIR input by manufacturers is more conducive to promoting sales of new products, which helps manufacturers to better focus on development and sales of new products and enhance their core competitiveness. However, the high sales volume of new products makes the sales volume of remanufactured products relatively low, which is not conducive to environmental protection and sustainable development, affects the image of environmental responsibility of enterprises and is not conducive to long-term sustainable development of enterprises and CLSC.

Implementation of PIR input by remanufacturers, on the one hand, can not only better promote recycling and remanufacturing of waste products and increase the sales of remanufactured products but also effectively improve the profits of manufacturers and remanufacturers; on the other hand, it is more conducive to alleviating the double marginal effect of CLSC, improving the membership and overall performance of CLSC and promoting the long-term stable development of CLSC. At this time, when the remanufacturer implements PIR inputs, the implementation of the government subsidy mechanism is more effective and also extremely beneficial to consumers, environmental protection, CLSC members and the whole system. Therefore, the government should give priority to remanufacturers when making policies to develop the remanufacturing sector.

However, implementation of PIR by the remanufacturer may put significant financial pressure on the remanufacturer. Therefore, as the dominant player, the manufacturer should be aware of the economic and environmental benefits of PIR input, take the initiative to reach certain cooperation agreements with the remanufacturer and encourage the remanufacturer to actively implement PIR input so as to better realize long-term sustainable development of the CLSC.

8. Conclusions

Remanufacturing CLSC is a hot research topic because of its sustainable profile. In order to explore its economic benefits, we analyze the influence of different government subsidy strategies and different situations of PIR implementation on the decision-making and coordination of a remanufacturing CLSC under authorized remanufacturing mode. From the perspective of authorized remanufacturing, this paper considers two kinds of government subsidy strategies and two kinds of PIR implementation modes, respectively, constructs a CLSC game model under centralized and decentralized decision-making and compares and analyzes the influence of different government subsidy strategies, different situations of PIR implementation and different decision-making modes on CLSC decision-making. A fixed license fee contract is designed to realize CLSC coordination. The conclusions are as follows:

• Whether the manufacturer or the remanufacturer implements PIR or the government subsidizes manufacturers or remanufacturers, it is always beneficial to improve the PIR input level, the market demand of remanufactured products and the profit of manufacturers and remanufacturers. Moreover, both sides can share the government subsidy by adjusting the patent license fee so that different government subsidy strategies only affect the patent license fee but have no difference in the influence of enterprise decision-making.
• The implementation of PIR by the manufacturer is more conducive to promoting the sales of new products, while the implementation of PIR by the remanufacturer is more conducive to improving the PIR level, the market demand for remanufactured products, the members of the CLSC and the overall performance.
• Compared with the manufacturer’s implementation of PIR, the government subsidy is more efficient in the remanufacturer’s implementation of PIR and can improve the PIR input level, the market demand for remanufactured products and the overall profit of the CLSC more effectively.
• The increase in the PIR input effect, the substitution effect of new products and remanufactured products and the decrease in the PIR input cost coefficient are beneficial to improve the PIR input level, the market demand of remanufactured products and the performance of member firms.
• Either the manufacturer or the remanufacturer implements PIR, the fixed license fee contract can achieve perfect coordination of the CLSC and the increase in the government subsidy and PIR input effect is conducive to expanding the feasible area of the coordination contract.

The results of this paper provide some management insights for government policy-making and PIR optimization of a CLSC. First of all, while the government promotes development of the remanufacturing industry through subsidies, it is necessary to cooperate with enterprises to publicize and promote remanufactured products, improve consumers’ awareness of remanufactured products, better promote the sales of remanufactured products and, through market means, encourage remanufacturers to actively implement PIR investment. Second, manufacturers and remanufacturers should be aware of the economic and environmental benefits from PIR input and actively improve the efficiency of PIR input so as to improve the recycling and remanufacturing efficiency of waste products. While shaping a good corporate social image and enhancing competitiveness, it can achieve a win–win situation of supply chain performance improvement and environmental protection. Finally, as the leader of the CLSC, the manufacturer should take the initiative to reach cooperation with the remanufacturer, promote the remanufacturer to actively implement PIR input and effectively improve the economic and environmental performance of the CLSC.

There are several potential directions worthy of further research. First of all, in this paper, we only consider a CLSC system consisting of a manufacturer and a remanufacturer. In reality, a CLSC system may contain more members. Therefore, future research may consider the CLSC decision-making problem when retailers or recyclers participate and members compete. Second, we focus on analysis of the impact of government subsidies on the PIR input of firms, but some countries also implement other environmental regulation policies. It would be interesting to consider the impact of different government environmental regulation policies and PIR inputs on CLSC decision-making. Finally, this research can be extended into the PIR decision-making of multi-cycle CLSCs under government regulation.