Pricing and Coordination for the Leasing and Recycling of Construction Machinery in a Supply Chain Based on Industrial Internet Platform

Abstract

Leasing and recycling are important methods for developing the low-carbon sustainability of the construction machinery industry. However, there are some dilemmas caused by the extreme limitations of product service life. There has been little research on construction machinery leasing thus far and a lack of description of its unique characteristics in previous models. Focusing on the problem of pricing and coordination, a two-stage model of a leasing closed-loop supply chain based on the practical application of an Industrial Internet platform was proposed. Under centralized and decentralized decision-making scenarios, the prices of leasing and selling, maintenance effort, and recovery rate were calculated, respectively. By using the Shapley value method to optimize the selling price, and the two-part pricing contract to calculate the compensation fee, global profits can be reasonably distributed, while the supply chain can be improved. The effects of different contracts on the improvement of supply chain profit were investigated. Additionally, the correlation among parameters was explored via sensitivity analysis and numerical simulation. The findings show that the maintenance and recycling of construction machinery can be improved by using an Industrial Internet platform while achieving supply chain coordination via contracts. The enhancement of maintenance effort can decrease remanufacturing costs, which are positively correlated with the selling price and leasing price of products, further promoting the recovery rate of used products. The findings of this paper show that manufacturers can take advantage of the Industrial Internet platform to improve recycling efficiency as well as to decrease product prices and remanufacturing costs by promoting remanufacturing technologies. On the other hand, leasers need to enhance the maintenance of construction machinery. Strengthening cooperation via contracts can jointly promote leasing and recycling in the construction machinery industry and help achieve low-carbon sustainable development.

1. Introduction

As a high-energy-consuming industry, the manufacturing of construction machinery usually involves high costs, a complex working environment, and high maintenance requirements that are closely related to safety risks and high carbon emissions [1,2]. As a sustainable development approach, the construction machinery leasing industry has become popular with the rapid increase in international market demand, through which construction companies can obtain relatively low-cost equipment and professional maintenance [3]. Forty percent of China’s construction machinery is exported via leasing, which can effectively relieve the economic pressure on construction companies, improve the utilization efficiency of equipment, and reduce the production scale of new products [4].

However, there are some current dilemmas that hinder the development of the construction machinery leasing industry caused by extreme limitations of service life. In order to maintain equipment quality to avoid safety accidents on construction sites, construction companies often prefer to lease newer machinery from leasers, thus resulting in a high level of waste because many products with longer service lives are abandoned. Since the actual available life of the equipment is shortened, the lease return correspondingly decreases and inadequate maintenance often occurs. Both the leaser and manufacturer of construction machinery may choose cheap and low-performance products, causing safety risks on construction sites to increase and creating a vicious circle.

The application of Industrial Internet platforms can not only help to solve this dilemma but also promotes recycling and remanufacturing through life cycle data acquisition and sharing [5]. In recent years, various Industrial Internet platforms, such as GE Predix, ABB Ability, Siemens Mind Sphere, and PTC Thing Worx, have emerged, supporting industry chain collaboration [6]. For construction machinery, the “Han Yun” Industrial Internet platform developed by XCMG China has been put into practice and has three major functions, including equipment access, data collection and fault diagnosis online [7]. This Industrial Internet platform is generally developed and operated by the manufacturer to provide the performance data of construction machinery for the leaser and the consumer. When equipment cannot be used normally, the manufacturer recycles it from the leaser and resells it after remanufacturing, thus restoring the end-of-life products to an initial or even better performance via professional treatment [8,9]. On this basis, a closed-loop supply chain for the machinery leasing industry formed. With powerful information support, one of the most urgent problems is how to take advantage of Industrial Internet platforms to promote leasing and recycling while improving supply chain profit. This has recently become a practical concern but has received limited research attention thus far.

Many studies have been conducted on the design of contracts to improve the collaboration of member enterprises and the performance of supply chains, achieving supply chain coordination [10]. Common supply chain contracts include: a wholesale price contract [11], a buyback contract [12], a revenue-sharing contract [13], a quantity–flexibility contract [14], a two-part pricing contract [15], a quantity discount contract [16], etc. As a supply chain incentive contract, a two-part pricing contract can effectively improve the cooperation revenue of a water supply chain [15], a closed-loop supply chain [17], a fresh supply chain [18], and other systems, effectively solving the problem of a double marginalization effect [19]. Moreover, the Shapley value method can be used to reasonably allocate the profits of participating parties in different scenarios, which can strengthen enterprise cooperation [20,21,22]. The above studies show that a double marginalization effect can be effectively addressed by designing contracts to improve supply chain performance.

At present, there are relatively few studies on the leasing of construction machinery, which can be regarded as durable products because of their high price and long service lives. For the leasing of durable products, two lines of research have been explored. One is channel selection, and the other is leasing pricing and coordination strategies. For the former, Bulow [23] established a dual-channel model for durable product leasing and sales. By comparing profits under sales and leasing, it was found that the latter method can make more profits. Lgal et al. [24] proposed that manufacturers who dominate the market can benefit from leasing contracts. Zhang et al. [25] found that under carbon trading and carbon tax policies, leasing can help manufacturers to obtain more profits while reducing carbon emissions. Cheng et al. [26] found that leasing can provide consumers with a better experience compared to purchases made by investigating the leasing of automobile manufacturers. Li et al. [27] found that overall profits increased significantly under a hybrid strategy when leasing dominated the market. In terms of research on the pricing and coordination of the leasing supply chain, Van et al. [28] studied the pricing of a video leasing supply chain and employed revenue-sharing contracts to optimize the supply chain. Long et al. [29] studied a car battery leasing supply chain under uncertain demand and deemed it necessary to consider recycling policy preferences to reasonably determine the repurchase and wholesale prices of old batteries. Sadat et al. [30] constructed a dual-channel supply chain model and found that the optimal dynamic leasing pricing is affected by customer consumption patterns, production costs, and sales prices. Sarang et al. [31] found that leasing terms affect the profits made by a leaser by influencing leasing prices and the inventory. Supply chain profits can be optimized when the lease term is within a certain range. Studies on the leasing of durable goods show that the selling and leasing prices interact with each other as decision variables, which in turn affects profits and demand throughout the supply chain.

The recycling of construction machinery discussed in this paper is related to research on closed-loop supply chains. Thus far, there have been few studies that have focused on construction machinery in this context. Yi et al. [32] found that the reasonable allocation of recycling efforts in dual recycling channels can reduce reversed logistics costs and thus enhance the market acceptance of remanufactured products. Deng et al. [9] discussed the influence of equipment quality on recycling prices and remanufacturing costs under the uncertainty of construction machinery quality, applying a modal interval algorithm for dynamic pricing and recycling strategies. Xu et al. [33] established a quality evaluation method for used parts of construction machinery via ontology and hierarchical analysis, which effectively improved recycling efficiency. These studies show that recycling efficiency and supply chain profit can be improved through rational pricing and recycling strategies.

Considering the contribution of Industrial Internet platforms to the construction machinery business, there are fewer challenges to tackle. Most studies on these platforms focus on their use as recycling channels. Wang et al. [34] considered a closed-loop supply chain by integrating a web-based recycling platform and found that a reasonable platform cost could better facilitate product recycling. Zhu et al. [35] constructed a dual-channel closed-loop supply chain model and designed a revenue–cost sharing contract. Therefore, with the use of advantageous Industrial Internet data, processing and sharing can help improve the recycling efficiency of used products.

There have been studies on the leasing of industrial durable goods, such as cars and car batteries, as well as the recycling of construction machinery, providing useful insights for this study. However, few significant findings regarding construction machinery leasing have been obtained thus far. Compared to industrial durable goods, construction machinery is used on construction sites, where safety requirements are crucial. Leasers need to provide maintenance services and pay for maintenance costs, which are influenced by concerns about equipment health from consumers (construction companies). These have not yet been described in previous models. Therefore, to practically solve this dilemma of the construction machinery industry, it is necessary to establish a relevant model and conduct a useful theoretical exploration.

Considering the practical operation of leasing and recycling construction machinery via an Industrial Internet platform, a leasing-oriented closed-loop supply chain model is proposed in the paper, reflecting the safety requirements via parameters such as maintenance effort level, maintenance cost, and maintenance sensitivity coefficient, which represent the level of concern from consumers about product maintenance. On this basis, the sales price, lease price, recovery rate, and maintenance effort level are calculated, respectively, for decentralized and centralized decision scenarios, comparing the supply chain performance. Finally, the Shapley value method and two-part pricing contract are used to address the double marginalization effect and achieve supply chain coordination.

This study focuses on construction machinery leasing and its innovation mainly lies in the following points: (1) A leasing-oriented closed-loop supply chain model was established based on the new operation mode under the Industrial Internet platform. (2) Considering the characteristics of construction machinery and platform operations, decision variables and parameters such as maintenance effort level, maintenance sensitivity coefficient, platform cost, and remanufacturing cost were proposed to investigate the impact on supply chain performance.

Research on the pricing and coordination of the leasing supply chain of construction machinery can enrich theoretical research on supply chain coordination and help promote the practical development of leasing and recycling in the construction machinery industry. The main contributions are as follows: (1) From the perspective of supply chains, this study quantifies the maintenance of the construction machinery as the decision variable, and relevant parameters are introduced, for which the impact on supply chain profits is investigated. (2) Considering scenarios in which an Industrial Internet platform can be operated by the manufacturer, the pricing and coordination of the leasing supply chain of construction machinery were studied. The Shapley value method and two-part pricing contract were applied to achieve Pareto optimality, improving leasing demand and recycling efficiency. This can help to promote the safety of construction machinery and economic benefits of member enterprises.

With the aim of exploring new perspectives on this issue via supply chain coordination, this study began with a two-stage leasing closed-loop supply chain model with an Industrial Internet platform as a member, as described in Section 2. On this basis, Section 3 describes a Stackelberg game model for centralized and decentralized decision making, respectively. The Shapley value method and two-part pricing contract were used to achieve supply chain coordination. In Section 4, the performance of different contracts on the improvement of closed-loop supply chain returns was investigated and the correlation among different parameters, decision variables, and profit functions were explored through sensitivity analysis and numerical simulation. Section 5 provides the conclusion to this study.

2. Model Framework

2.1. Model Description

In the closed-loop supply chain of construction machinery leasing based on an Industrial Internet platform, the manufacturer is the leader of the Stackelberg game, the leaser is the follower, and the decision making of both the manufacturer and leaser is based on complete information. In the forward supply chain, the manufacturer uses new materials and recycled old equipment for production activities. The unit production costs of new products and remanufactured products are $c$ and $x$, respectively, and the manufacturer spends $\phi$ to establish and operate the Industrial Internet platform. After the production of construction machinery is completed, the manufacturer sells the new product and remanufactured product to the leaser at a uniform price $w$, and the leaser leases the product to consumers at a uniform price $p$. At the end of each leasing period, the leaser maintains the construction machinery, adjusts the maintenance effort level $\gamma$, and pays the maintenance cost $\mu$ according to the status information of construction machinery provided by the Industrial Internet platform and the maintenance sensitivity coefficient $\beta$. In the reverse supply chain, the manufacturer spends $\phi$ to establish and operate the Industrial Internet platform, and uses the construction machinery status data provided by the Industrial Internet platform to recycle old equipment from the leaser at $v$, while the recovery rate is $e$. Material flow is represented by solid lines for the production, sale, leasing, recycling, and remanufacturing process of products, while information flow is represented by dotted lines regarding the product status information that is shared between manufacturer and leaser. Both parties are risk-neutral and perfectly rational. The flow chart of a leasing-oriented closed-loop supply chain for construction machinery based on an Industrial Internet platform is shown in Figure 1.

2.2. Definition of Symbols

The main notations are shown in

Table 1. Notation.

Notation	Definition
$w$	Product selling price
$c$	Unit product cost
$p$	Product leasing price
$\alpha$	Maximum market demand
$\theta$	Leasing price sensitivity coefficient
$\beta$	Maintenance sensitivity coefficient
$\gamma$	Maintenance effort level
$\mu$	Maintenance cost coefficient
$x$	Unit remanufacturing cost
$v$	Unit recovery cost
$\phi$	Industrial Internet platform cost coefficient
$e$	Product recovery rate

.

2.3. Related Assumptions

Assumption 1.

For comparison purposes, the leasing price and selling price of the product are averaged, which can be interpreted as the cost per unit period time that the consumer spends on the product. In order to ensure that the leaser and the manufacturer make profits, $p>w$, $c>x$.

Assumption 2.

The manufacturer produces a remanufactured product of the same quality as the new product and applies a uniform pricing strategy to both products.

Assumption 3.

From the problem description, it is clear that the Industrial Internet platform cost is related to the product recovery rate. Considering the literature [36], the Industrial Internet platform cost is assumed to be a quadratic function of the recovery rate, so the Industrial Internet platform cost is $S_1=\frac{1}{2}\phi e^2(0<e<1)$.

Assumption 4.

The leaser needs to maintain the product in daily work. The maintenance cost is assumed to be a quadratic function of the maintenance effort level, so the maintenance cost is $S_2=\frac{1}{2}\mu {\gamma }^2$.

3. Model Solutions and Contract Coordination Design

3.1. Demand Functions and Profit Functions

According to the symbol definition and problem assumptions, the product demand function of a leasing-oriented closed-loop supply chain for construction machinery can be expressed as:

$$Q_n=\alpha -\theta p+\beta \gamma$$

(1)

where $\beta$ is the maintenance sensitivity coefficient, which represents the level of concern from consumers about product maintenance. A greater $\beta$ means that consumers are more concerned about the performance and maintenance of the product, thus enhancing the maintenance level $\gamma$.

$\beta \gamma$ is the impact of daily maintenance by the leaser on supply chain demand. The higher the level of maintenance, the greater the demand for the supply chain.

The total market demand includes the demand for new products and remanufactured products, so the product recovery quantity is:

$$Q_r=e{Q}_n$$

(2)

It can be seen that the total profits of the manufacturer, leaser, and closed-loop supply chains can be expressed as:

$${\pi }_m^i=(w-c)(Q_n-Q_r)+(w-x-v)Q_r-\frac{1}{2}\phi e^2$$

(3)

$${\pi }_r^i=(p-w)Q_n+v_r{Q}_r-\frac{1}{2}\mu {\gamma }^2$$

(4)

$${\pi }_t^i=(p-c)Q_n+(c-x)Q_r-\frac{1}{2}\phi e^2-\frac{1}{2}\mu {\gamma }^2$$

(5)

Equation (3) is the profit function of the manufacturer, in which the manufacturer makes profits by selling new products and remanufactured products, and pays for the Industrial Internet platform cost.

Equation (4) is the profit function of the leaser, in which the leaser makes profits by leasing out new products and remanufactured products, and pays for the maintenance cost at the same time.

Equation (5) is the sum of the manufacturer’s profit function and leaser‘s profit function.

3.2. Centralized Decision-Making Scenario Analysis

Under centralized decision making, the manufacturer and leaser do not consider their benefits. They always work in close cooperation, share information, and make joint decisions to maximize supply chain profits. At this time, the Hessian matrix is constructed to solve $p$, $\gamma$, and $e$.

$$H_1=\left|\begin{array}{ccc}\frac{{\partial }^2{\pi }_t^c}{\partial p^2}& \frac{{\partial }^2{\pi }_t^c}{\partial p\partial \gamma }& \frac{{\partial }^2{\pi }_t^c}{\partial p\partial e}\\ \frac{{\partial }^2{\pi }_t^c}{\partial \gamma \partial p}& \frac{{\partial }^2{\pi }_t^c}{\partial {\gamma }^2}& \frac{{\partial }^2{\pi }_t^c}{\partial \gamma \partial e}\\ \frac{{\partial }^2{\pi }_t^c}{\partial e\partial p}& \frac{{\partial }^2{\pi }_t^c}{\partial e\partial \gamma }& \frac{{\partial }^2{\pi }_t^c}{\partial e^2}\end{array}\right|=\left|\begin{array}{ccc}-2\theta & \beta & -\theta (c-x)\\ \beta & -\mu & -\theta (c-x)\\ \beta (c-x)& -\theta (c-x)& -\phi \end{array}\right|$$

The Hessian matrix is negative for $-\phi (2\theta \mu -{\beta }^2)+{\theta }^2\mu {(c-x)}^2<0$ and $\left|H_1\right|>0$. Therefore, with respect to $p$, $\gamma$, and $e$, ${\pi }_t^c$ is strictly a concave function. By solving $\frac{\partial {\pi }_t^c}{\partial p}=0$, $\frac{\partial {\pi }_t^c}{\partial \gamma }=0$, $\frac{\partial {\pi }_t^c}{\partial e}=0$, we can obtain the following equation:

$$p_c^{\ast }=\frac{\phi (\theta \mu c+\alpha \mu -{\beta }^{2}c)-\theta \alpha \mu {(c-x)}^2}{\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2}$$

(6)

$${\gamma }_c^{\ast }=\frac{\beta \phi (\alpha -\theta c)}{\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2}$$

(7)

$$e_c^{\ast }=\frac{\theta \mu (c-x)(\alpha -\theta c)}{\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2}$$

(8)

Because $e_c^{\ast }\in (0,1]$, $\phi$ needs to be satisfied in the following:

$$\phi \ge \frac{\theta \mu (\alpha -\theta x)(c-x)}{2\theta \mu -{\beta }^2}$$

By substituting Equations (6)–(8) into Equations (1), (2) and (5), the optimal values of the product demand, product recovery quantity, and total supply chain profit under centralized decision making can be obtained as follows:

$$\begin{gathered} Q_n^{c\ast }=\frac{\theta \phi \mu (\alpha -\theta c)}{\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2},Q_r^{c\ast }=\frac{{\theta }^2{\mu }^2\phi (c-x){(\alpha -\theta c)}^2}{{(\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2)}^2} \\ {\pi }_t^{c\ast }=\frac{\phi \mu {(\alpha -\theta c)}^2}{2(\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2)} \end{gathered}$$

Proposition 1.

Under a centralized decision, the total supply chain profit, product demand, and product recovery quantity are negatively correlated with the maintenance cost coefficient, unit remanufacturing cost, and Industrial Internet platform cost coefficient. Additionally, they are positively correlated with the maintenance sensitivity coefficient and maximum market demand.

Proof 1.

Find $\frac{\partial {\pi }_t^{c\ast }}{\partial \mu }$, $\frac{\partial {\pi }_t^{c\ast }}{\partial x}$, $\frac{\partial {\pi }_t^{c\ast }}{\partial \phi }$, $\frac{\partial {\pi }_t^{c\ast }}{\partial \beta }$, $\frac{\partial {\pi }_t^{c\ast }}{\partial \alpha }$ $\frac{\partial Q_n^{c\ast }}{\partial \mu }$, $\frac{\partial Q_n^{c\ast }}{\partial x}$, $\frac{\partial Q_n^{c\ast }}{\partial \phi }$, $\frac{\partial Q_n^{c\ast }}{\partial \beta }$, $\frac{\partial Q_n^{c\ast }}{\partial \alpha }>0$, $\frac{\partial Q_r^{c\ast }}{\partial \mu }$, $\frac{\partial Q_r^{c\ast }}{\partial x}$, $\frac{\partial Q_r^{c\ast }}{\partial \phi }$, $\frac{\partial Q_r^{c\ast }}{\partial \beta }$. We can obtain:

$$\begin{gathered} \frac{\partial {\pi }_t^{c\ast }}{\partial \mu }<0,\frac{\partial {\pi }_t^{c\ast }}{\partial x}<0,\frac{\partial {\pi }_t^{c\ast }}{\partial \phi }<0,\frac{\partial {\pi }_t^{c\ast }}{\partial \beta }>0,\frac{\partial {\pi }_t^{c\ast }}{\partial \alpha }>0 \\ \frac{\partial Q_n^{c\ast }}{\partial \mu }<0,\frac{\partial Q_n^{c\ast }}{\partial x}<0,\frac{\partial Q_n^{c\ast }}{\partial \phi }<0,\frac{\partial Q_n^{c\ast }}{\partial \beta }>0,\frac{\partial Q_n^{c\ast }}{\partial \alpha }>0 \\ \frac{\partial Q_r^{c\ast }}{\partial \mu }<0,\frac{\partial Q_r^{c\ast }}{\partial x}<0,\frac{\partial Q_r^{c\ast }}{\partial \phi }<0,\frac{\partial Q_r^{c\ast }}{\partial \beta }>0, \frac{\partial Q_r^{c\ast }}{\partial \alpha }>0 \end{gathered}$$

□

Proposition 1 shows that, under a centralized decision-making scenario, the total supply chain profit, product demand, and product recovery quantity decrease with the increase in the maintenance cost coefficient, unit remanufacturing cost, and Industrial Internet platform cost coefficient. At the same time, they also increase with the increase in the maintenance sensitivity coefficient and maximum market demand. This illustrates that consumers pay attention to the safety of construction machinery. As the market demand for construction machinery expands, both the manufacturer and leaser need to make improvements to increase the total supply chain profit. The manufacturer needs to continuously improve product remanufacturing technology to reduce the unit remanufacturing cost and upgrade Industrial Internet platform operations and maintenance technology to reduce the cost of the platform. The leaser needs to pay more attention to the daily maintenance of mechanical equipment to improve the maintenance efforts level.

Proposition 2.

Under centralized decision making, the leasing price is positively correlated with the unit remanufacturing cost, and the maintenance effort level and product recovery rate are negatively correlated with the unit remanufacturing cost. The recovery rate is negatively correlated with the Industrial Internet platform cost coefficient.

Proof 2.

Find $\frac{\partial p_c^{\ast }}{\partial x}$, $\frac{\partial {\gamma }_c^{\ast }}{\partial x}$, $\frac{\partial e_c^{\ast }}{\partial x}$, $\frac{\partial e_c^{\ast }}{\partial \phi }$. We can obtain:

$$\frac{\partial p_c^{\ast }}{\partial x}>0,\frac{\partial {\gamma }_c^{\ast }}{\partial x}<0,\frac{\partial e_c^{\ast }}{\partial x}<0,\frac{\partial e_c^{\ast }}{\partial \phi }<0$$

□

Proposition 2 shows that, in a centralized decision-making scenario, the leasing price increases as the unit remanufacturing cost increases; the maintenance effort level and product recovery rate decrease as the unit remanufacturing cost increases. Additionally, the recovery rate decreases as the Industrial Internet platform cost coefficient increases. The analysis of Proposition 1 shows that the total supply chain profit, product demand, and product recovery quantity continue to decline as the unit remanufacturing cost increases. Centralized decision making pursues the best global profits, which are the sum of the profits of the manufacturer and the leaser. The leaser has to increase the leasing price to compensate for the reduced costs and demand, meaning that the leaser does not have excess funds for the daily maintenance of construction machinery. Due to the increase in the unit remanufacturing cost and Industrial Internet platform cost, the manufacturer is unable to undertake more remanufacturing work, and thus the product recovery rate is reduced.

3.3. Decentralized Decision-Making Scenario Analysis

Under decentralized decision making, the manufacturer and leaser aim to maximize their profits. The decision is divided into two stages. In the first stage, the manufacturer decides the selling price $w$ and product recovery rate $e$. In the second stage, the leaser determines the leasing price $p$ and maintenance effort level $\gamma$ according to the decision made by the manufacturer. Based on the reverse solution method, the Hessian matrix is constructed to solve $p$ and $\gamma$.

$$H_2=\left|\begin{array}{cc}\frac{{\partial }^2{\pi }_r^s}{\partial p^2}& \frac{{\partial }^2{\pi }_r^s}{\partial p\partial \gamma }\\ \frac{{\partial }^2{\pi }_r^s}{\partial \gamma \partial p}& \frac{{\partial }^2{\pi }_r^s}{\partial {\gamma }^2}\end{array}\right|=\left|\begin{array}{cc}-2\theta & \beta \\ \beta & -\mu \end{array}\right|$$

The Hessian matrix is negative for $2\theta \mu -{\beta }^2>0$ and $\left|H_2\right|>0$. Therefore, with respect to $p$ and $\gamma$, ${\pi }_r^s$ is strictly a concave function. By solving $\frac{\partial {\pi }_r^s}{\partial p}=0$, $\frac{\partial {\pi }_r^s}{\partial \gamma }=0$, we can obtain the results of the following equations:

$$p_s^{\ast }=\frac{\alpha \mu -{\beta }^{2}w+\theta \mu w+{\beta }^{2}ve-\theta \mu ve}{2\theta \mu -{\beta }^2}$$

(9)

$${\gamma }_s^{\ast }=\frac{\beta (\alpha -\theta w+\theta ve)}{2\theta \mu -{\beta }^2}$$

(10)

By substituting Formulas (9) and (10) into Formula (3):

$${\pi }_m^{s\ast }=\frac{A_1}{2(2\theta \mu -{\beta }^2)}$$

(11)

where:

$$\begin{array}{cc}{A}_1=& 2{\theta }^2\mu (w(c-w+2ve+xe)+v{e}^2(c-v-x))\\ & +2\theta \mu (\alpha (w-c-ve-x+ce)-\phi e^2)\\ & +{\beta }^2\phi e^2-2{\theta }^2\mu ve(c+we))\end{array}$$

To analyze the impact of $w$ on ${\pi }_m^{s\mathrm{*}}$ and the impact of $e$ on ${\pi }_m^{s\mathrm{*}}$, the Hessian matrix is constructed.

$$H_3=\left|\begin{array}{cc}\frac{{\partial }^2{\pi }_m^{s\ast }}{\partial w^2}& \frac{{\partial }^2{\pi }_m^{s\ast }}{\partial w\partial e}\\ \frac{{\partial }^2{\pi }_m^{s\ast }}{\partial e\partial w}& \frac{{\partial }^2{\pi }_m^{s\ast }}{\partial e^2}\end{array}\right|=\left|\begin{array}{cc}-\frac{2{\theta }^2\mu }{2\theta \mu -{\beta }^2}& -\frac{{\theta }^2\mu (2v-c+x)}{2\theta \mu -{\beta }^2}\\ -\frac{{\theta }^2\mu (2v-c+x)}{2\theta \mu -{\beta }^2}& \frac{2{\theta }^2\mu v(c-v-x)-\phi (2\theta \mu -{\beta }^2)}{2\theta \mu -{\beta }^2}\end{array}\right|$$

The Hessian matrix is negative for $-2\phi (2\theta \mu -{\beta }^2)+{\theta }^2\mu {(c-x)}^2>0$ and $\left|H_3\right|>0$. With respect to $w$ and $e$, ${\pi }_m^{s\ast }$ is strictly a concave function. By solving $\frac{\partial {\pi }_m^s}{\partial p}=0$, $\frac{\partial {\pi }_m^s}{\partial \gamma }=0$, we can obtain the results of the following equations:

$$w_s^{\ast }=\frac{A_2}{\theta (2\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2)}$$

(12)

$$e_s^{\ast }=\frac{\theta \mu (c-x)(\alpha -\theta c)}{2\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2}$$

(13)

where:

$$\begin{array}{c}{A}_2=-({\beta }^2\phi (\alpha +\theta c)+{\theta }^2\alpha \mu (c^2+x^2)+{\theta }^3\mu vc(c-x)\\ -{\theta }^2\alpha \mu v(c+x)-2\theta \mu \alpha \phi -2{\theta }^2\mu c(\phi +\alpha x))\end{array}$$

Because $e_s^{\ast }\in (0,1]$, then $\phi$ needs to satisfy:

$$\phi \ge \frac{\theta \mu (\alpha -\theta x)(c-x)}{2(2\theta \mu -{\beta }^2)}$$

By substituting Formulas (12) and (13) into Formulas (9) and (10), we can obtain:

$$p_s^{\ast \ast }=\frac{-{\beta }^2\phi (\theta c+\alpha )+{\theta }^2\mu (2\alpha cx-\alpha c^2+\phi c-\alpha x^2)+3\theta \phi \alpha \mu }{\theta (2\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2)}$$

(14)

$${\gamma }_s^{\ast \ast }=\frac{\beta \phi (\alpha -\theta c)}{2\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2}$$

(15)

By substituting Formulas (12)–(15) into Formulas (1)–(5), the optimal values of the product demand, product recovery quantity, manufacturer profit, leaser profit, and total supply chain profit under decentralized decision making can be obtained:

$$\begin{gathered} Q_r^{s\ast }=\frac{{\theta }^2{\mu }^2\phi (c-x){(\alpha -\theta c)}^2}{{(2\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2)}^2} \\ {\pi }_m^{s\ast \ast }=\frac{\phi \mu {(\alpha -\theta c)}^2}{2(2\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2)},{\pi }_r^{s\ast \ast }=\frac{\mu {\phi }^2(2\theta \mu -{\beta }^2){(\alpha -\theta c)}^2}{2{(2\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2)}^2} \\ {\pi }_t^{s\ast }=\frac{\phi \mu {(\alpha -\theta c)}^2(3\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2)}{2{(2\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2)}^2} \end{gathered}$$

Proposition 3.

Total supply chain profit under centralized decision making is always greater than the level under decentralized decision making.

$${\pi }_t^{c\ast }>{\pi }_t^{s\ast }$$

Proof 3.

Find ${\pi }_t^{c\ast }-{\pi }_t^{s\ast }$. We can obtain:

$${\pi }_t^{c\ast }-{\pi }_t^{s\ast }=\frac{\mu {\phi }^3(2\theta \mu -{\beta }^2){(\alpha -\theta c)}^2}{{(2\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2)}^2(\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2)}$$

□

Proposition 3 shows that the total supply chain profit under centralized decision making is greater than the profit under decentralized decision making. This indicates that members of the supply chain compete with each other under the decentralized decision making. The leaser and manufacturer in the supply chain make decisions for their own profit maximization, ignoring the overall profit of the supply chain, which means that the supply chain is unable to achieve coordination. There is a double marginalization effect in the leasing-oriented closed-loop supply chain operation for construction machinery based on the Industrial Internet platform, and further design contracts are needed to optimize it.

Proposition 4.

The values of the maintenance effort level, product recovery rate, product demand, and product recovery quantity under centralized decision making are all greater than those under decentralized decision making, but the leasing price under centralized decision making is less than that under decentralized decision making.

Proof 4.

Find $p_c^{\ast }-p_s^{\ast \ast }$, ${\gamma }_c^{\ast }-{\gamma }_s^{\ast \ast }$, $e_c^{\ast }-e_s^{\ast }$, $Q_n^{c\ast }-Q_n^{s\ast }$, $Q_r^{c\ast }-Q_r^{s\ast }$, and we can obtain:

$$p_c^{\ast }-p_s^{\ast \ast }<0,{\gamma }_c^{\ast }-{\gamma }_s^{\ast \ast }>0,e_c^{\ast }-e_s^{\ast }>0,Q_n^{c\ast }-Q_n^{s\ast }>0,Q_r^{c\ast }-Q_r^{s\ast }>0$$

□

Proposition 4 shows that the values of the product recovery rate and product recovery quantity are greater under centralized decision making. From the perspective of product recovery quantity, a cooperative strategy is preferred. Since the maintenance effort level is higher under centralized decision making, but the leasing price is lower, the level of product demand inevitably increases. Under decentralized decision making, all parties in the supply chain pursue their own interests. The leaser tends to reduce maintenance effort level and increase leasing price, which in turn leads to a decrease in product demand. At the same time, the manufacturer has a negative attitude towards the recycling of old equipment, resulting in its decrease. Proposition 3 shows that the total supply chain profit is always greater under the centralized decision making. Therefore, there is room for improvement for the profits of both the manufacturer and leaser.

Proposition 5.

Under the decentralized decision-making scenario, the selling price and leasing price are positively correlated with the unit remanufacturing cost. The maintenance effort level and product recovery rate are negatively correlated with the unit remanufacturing cost. Additionally, the product recovery rate is negatively correlated with the Industrial Internet platform cost coefficient.

Proof 5.

Find $\frac{\partial w_s^{\ast }}{\partial x}$, $\frac{\partial p_s^{\ast }}{\partial x}$, $\frac{\partial {\gamma }_s^{\ast }}{\partial x}$, $\frac{\partial e_s^{\ast }}{\partial x}$, $\frac{\partial e_s^{\ast }}{\partial \phi }$. We can obtain:

$$\frac{\partial w_s^{\ast }}{\partial x}>0,\frac{\partial p_s^{\ast }}{\partial x}>0,\frac{\partial {\gamma }_s^{\ast }}{\partial x}<0,\frac{\partial e_s^{\ast }}{\partial x}<0,\frac{\partial e_s^{\ast }}{\partial \phi }<0$$

□

Proposition 5 shows that, in a decentralized decision-making scenario, the selling and leasing price increase as the unit remanufacturing cost increases, while the maintenance effort level and product recovery rate decrease. Additionally, the product recovery rate decreases as the Industrial Internet platform cost coefficient increases. Under decentralized decision making, the manufacturer has to increase the selling price of construction machinery due to the unit remanufacturing cost. Therefore, the leaser must also pay more to purchase construction machinery, which is a disadvantage regarding daily maintenance. Due to the costs of unit remanufacturing and the Industrial Internet platform, the recovery rate and quantity of the used products decrease, which has a negative effect on the sustainable development of the construction machinery leasing industry.

Proposition 6.

The profit of the leaser is negatively correlated with the maintenance cost coefficient and positively correlated with the maintenance sensitivity coefficient and maximum market demand. Manufacturer profit is negatively correlated with the Industrial Internet platform cost coefficient and positively correlated with the maximum market demand. Product demand and product recovery quantity are negatively correlated with the maintenance cost coefficient, unit remanufacturing cost, and Industrial Internet platform cost coefficient, and positively correlated with the maintenance sensitivity coefficient.

Proof 6.

Find $\frac{\partial {\pi }_r^{s\ast \ast }}{\partial \mu }$, $\frac{\partial {\pi }_r^{s\ast \ast }}{\partial \beta }$, $\frac{\partial {\pi }_r^{s\ast \ast }}{\partial \alpha }$, $\frac{\partial {\pi }_m^{s\ast \ast }}{\partial \phi }$, $\frac{\partial {\pi }_m^{s\ast \ast }}{\partial x}$, $\frac{\partial {\pi }_m^{s\ast \ast }}{\partial \alpha }$, $\frac{\partial Q_n^{s\ast }}{\partial \mu }$, $\frac{\partial Q_n^{s\ast }}{\partial x}$, $\frac{\partial Q_n^{s\ast }}{\partial \phi }$, $\frac{\partial Q_n^{s\ast }}{\partial \beta }$, $\frac{\partial Q_r^{s\ast }}{\partial \mu }$, $\frac{\partial Q_r^{s\ast }}{\partial x}$, $\frac{\partial Q_r^{s\ast }}{\partial \phi }$, $\frac{\partial Q_r^{s\ast }}{\partial \beta }$. We can obtain:

$$\begin{gathered} \frac{\partial {\pi }_r^{s\ast \ast }}{\partial \mu }<0,\frac{\partial {\pi }_r^{s\ast \ast }}{\partial \beta }>0,\frac{\partial {\pi }_r^{s\ast \ast }}{\partial \alpha }>0,\frac{\partial {\pi }_m^{s\ast \ast }}{\partial \phi }<0,\frac{\partial {\pi }_m^{s\ast \ast }}{\partial x}<0,\frac{\partial {\pi }_m^{s\ast \ast }}{\partial \alpha }>0 \\ \frac{\partial Q_n^{s\ast }}{\partial \mu }<0,\frac{\partial Q_n^{s\ast }}{\partial x}<0,\frac{\partial Q_n^{s\ast }}{\partial \phi }<0,\frac{\partial Q_n^{s\ast }}{\partial \beta }>0,\frac{\partial Q_r^{s\ast }}{\partial \mu }<0,\frac{\partial Q_r^{s\ast }}{\partial x}<0,\frac{\partial Q_r^{s\ast }}{\partial \phi }<0,\frac{\partial Q_r^{s\ast }}{\partial \beta }>0 \end{gathered}$$

□

Proposition 6 shows that the leaser profit and product demand decrease as the maintenance cost coefficient increases, while these factors increase as the maintenance sensitivity coefficient and the maximum market demand increase. The manufacturer profit and product recovery quantity decrease as the Industrial Internet platform cost coefficient and unit remanufacturing cost increase. The increase in the maintenance sensitivity coefficient indicates that consumers are more concerned about the product safety. The leaser has to enhance the maintenance cost, which leads to a decrease in profit and product demand. The profit of the manufacturer is also reduced due to the increase in the cost of the Industrial Internet platform and the unit remanufacturing, hindering the recycling of old products. Therefore, the leaser needs to improve maintenance technology to reduce maintenance cost, while the manufacturer needs to improve remanufacturing technology, industrial Internet platform operations, and construction technology to decrease related cost, thereby increasing profit.

From Propositions 3–6, it is clear that the supply chain requires further improvement on the basis of decentralized decision making. The objective of coordination in this chapter is to improve the performance of the closed-loop supply chain for construction machinery leasing with platform access; therefore, the Shapley value method and the two-part pricing contract are used.

3.4. Coordination Contract Model

It can be seen from Propositions 3–6 that it is necessary to design contracts to coordinate supply chains so that their profits have a rational distribution. This section studies the coordination effect of the Shapley value method and two-part pricing contract on the supply chain.

In order to make the subsequent analysis more intuitive, we use the following equation:

$$f=\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2, g=2\phi (2\theta \mu -{\beta }^2)-{\theta }^2\mu {(c-x)}^2$$

3.4.1. Shapley Value Method

In order to coordinate the leasing-oriented closed-loop supply chain of construction machinery, the Shapley value method is used to distribute the overall profit of the supply chain. Let $I=\left\{1,2,\dots ,n\right\}$, and any subset $z$ of $I$ correspond to a function $u(v)$, $i=1,2,\dots ,n$, if it satisfies the following:

$$\left\{\begin{array}{l}u(\varphi )=0\\ u(v_i\cup v_j)\ge u(v_i)+u(v_j)\\ v_i\cap v_j,v_i\in I,v_j\in I\end{array}\right.$$

(16)

Then, $\left[I,u\right]$ is a multi-player cooperative game, and the characteristic function $u(v)$ represents the maximum benefit obtained from the interaction in the alliance $v$. It means that there is no benefit without cooperation, which means that the benefits of cooperation are greater than the simple sum of their benefits, and member activities are non-confrontational.

Using $x_i$ to represent the profit obtained by member $i$ of $I$ from $\left[I,u\right]$, it is clear that the establishment of the cooperation must satisfy:

$${\displaystyle \sum _{i=1}^n{x}_i}=u(I),x_i\ge u(i)$$

(17)

In the Shapley value method, the benefit value allocated by the member $i$ is the Shapley value, which can be expressed as:

$$x_i={\displaystyle \sum _{v\in V_i}k(\left|v\right|)\left[u(v)-u(v\backslash i)\right]}$$

(18)

$$k(\left|v\right|)=\frac{(n-\left|v\right|)!(\left|v\right|-1)!}{n!}$$

(19)

Among them, $\left|v\right|$ is the number of elements, $k\left(\left|v\right|\right)$ is the weighting factor, $u(v)$ represents the benefits of cooperation between both parties in the supply chain when $v$ companies participate, and $u(v\backslash i)$ represents the benefits of supply chain cooperation without the participation of $v$ companies. Therefore, $u(v)-u(v\backslash i)$ represents the contribution value made by company $v$ in supply chain cooperation.

$${\pi }_m^{v\ast }=\frac{{\pi }_m^{\ast \ast }}{2}+\frac{{\pi }_t^{\ast }-{\pi }_r^{\ast \ast }}{2}$$

(20)

$${\pi }_r^{v\ast }=\frac{{\pi }_r^{\ast \ast }}{2}+\frac{{\pi }_t^{\ast }-{\pi }_m^{\ast \ast }}{2}$$

(21)

According to the Shapley value method, the profit distribution values for manufacturers and leasers are:

$${\pi }_m^{v\ast }=\frac{A_3\phi \mu {(\alpha -\theta c)}^2}{4f{g}^2}, {\pi }_{rv}^{\ast }=\frac{A_4}{4f{g}^2}$$

where:

$$\begin{gathered} \begin{array}{l}{A}_3=(6{\theta }^2{\beta }^2\phi \mu -12{\theta }^3{\mu }^2\phi ){(c-x)}^2+2{\theta }^4{\mu }^2(c^4-4c^{3}x-4c{x}^3-4+6c^2{x}^2+x^4)\\ +20\theta \mu {\phi }^2(\theta \mu -{\beta }^2)+5{\beta }^4{\phi }^2\end{array} \\ {A}_4={\phi }^2\mu (2\theta \mu -{\beta }^2){(\alpha -\theta c)}^2(3\phi (2\mu x-{\beta }^2)-2{\theta }^2\mu {(c-x)}^2) \end{gathered}$$

Proposition 7.

Under the Shapley value method, when the selling price satisfies $w_v^{\ast }=K$, both the manufacturer profit and the leaser profit increase compared with decentralized decision making. The total supply chain profit reaches the optimum level under centralized decision making.

Proof 7.

${\pi }_m^{v\ast }$ and ${\pi }_r^{v\ast }$ are the profits coordinated by the Shapley value method. When $w_v^{\ast }$ satisfies the following equation, the closed-loop supply chain can be coordinated.

$$\left\{\begin{array}{l}{\pi }_m^{v\ast }=(w_v^{\ast }-c)(Q_n-Q_r)+(w_v^{\ast }-x)Q_r-\frac{1}{2}\phi e_v^{2\ast }\\ {\pi }_r^{v\ast }=(p_v^{\ast }-w_v^{\ast })Q_n-\frac{1}{2}\mu {\gamma }_v^{2\ast }\end{array}\right.$$

The factors of leasing price, maintenance effort level, and product recovery rate are equal to those factors under centralized decision making, respectively. Therefore:

$$p_v^{\ast }=p_c^{\ast }, {\gamma }_v^{\ast }={\gamma }_c^{\ast }, e_v^{\ast }=e_c^{\ast }$$

(22)

The following simultaneous equations can be obtained:

$$w_v^{\ast }=\frac{A_5}{A_6}=K$$

(23)

where:

$$\begin{gathered} \begin{array}{}\begin{array}{cc}{A}_5& =4{\theta }^6{\mu }^3(\alpha (c^6+x^6)+4v(\theta c-\alpha )(c^5-x^5)+5\alpha c^2{x}^2(3x^2-4cx+3c^2)\hfill \\ & +4\phi c^2{x}^2(2c-3x))+40{\theta }^7{\mu }^3{c}^3{x}^{2}v(c+x)+13{\theta }^3\mu {\phi }^{2}cx(4{\theta }^2{\mu }^{2}x+{\beta }^4)(c-2x)\hfill \\ & -6{\beta }^4{\phi }^3\theta \mu (5\alpha -11\theta c)+8{\theta }^6{\mu }^3\phi c^{4}x(4c^3-x^3)+4{\theta }^4{\mu }^{2}cx(13{\beta }^2{\phi }^2-5{\theta }^3\mu c{x}^{2}v)(2c-x)\hfill \\ & -64{\theta }^4{\beta }^2{\mu }^2\phi \alpha cx(c^2+x^2-\phi v)+8{\theta }^5{\beta }^2{\mu }^2\phi c^3{x}^2(3c-2x)+16\theta {\beta }^2\mu {\phi }^2\alpha \hfill \\ & +4{\theta }^3{\mu }^2\alpha \phi (32{\theta }^2\mu c{x}^3-19{\beta }^2\phi )(c^2+x^2)+8{\theta }^2\mu v(12{\theta }^3{\mu }^2{\phi }^2\alpha c{x}^2+6{\theta }^2{\beta }^2\mu \phi \alpha c^{2}x\hfill \\ & -8{\theta }^2{\mu }^2{\phi }^2\alpha -5{\theta }^4{\mu }^2\alpha c^2{x}^2-2\theta {\beta }^4{\phi }^2\alpha +12{\theta }^3{\mu }^2{\phi }^{2}c)(c-x)+32{\theta }^6{\mu }^3\phi c{x}^{3}v(x^2+3c^2)\hfill \\ & +76{\theta }^4{\mu }^3{\phi }^2\alpha c(c-2x+x^2)+4{\theta }^4{\mu }^2\alpha (-6{\theta }^2\mu cx-8\theta \mu \phi +4{\beta }^2\phi +\theta {\beta }^2\phi )(c^4+x^4)\hfill \\ & -16{\theta }^5{\beta }^2{\mu }^2\phi \alpha cxv(3c^2+x^2)+16{\theta }^5{\beta }^2{\mu }^2\phi c^4(v-x)+4{\theta }^5{\mu }^3{c}^3(13{\phi }^2-5{\theta }^2{c}^{2}xv)\hfill \\ & +19{\beta }^4{\theta }^2{\phi }^2\alpha \mu (c^2+x^2)+4{\theta }^4{\beta }^2{\mu }^2{\phi }^2{c}^2(24\alpha x^2-13c-16v)-32{\theta }^6{\mu }^3\phi c^2(\alpha c^2+3x^{2}v)\hfill \\ & +8{\theta }^3{\beta }^2\mu {\phi }^2\alpha cx(19\mu c-2{\beta }^{2}v)+{\beta }^4{\theta }^3{\phi }^2\mu (13c^3+16v)+4{\theta }^3{\phi }^3{\mu }^2(33{\beta }^{2}c-10\mu \alpha )\hfill \\ & +2{\theta }^2{\beta }^2{\phi }^2\mu \alpha (30\phi \mu -19{\beta }^{2}cx)+4{\theta }^4{\mu }^2\alpha v(5{\theta }^2\mu c^{4}x+8\theta \mu \phi -4{\beta }^2\phi )(c^3-x^3)\hfill \\ & +{\beta }^6\phi (5\alpha +11{\phi }^2\theta c)-8{\theta }^4{\mu }^3\phi c({\theta }^2{c}^4+11{\phi }^2)+48{\theta }^5{\mu }^2\phi \alpha x^2({\beta }^2\phi v-4\mu c^2)\hfill \end{array}\hfill & \\ \begin{array}{cc}{A}_6& =-4({\theta }^5{\mu }^{2}cx(20\phi {\beta }^2-15{\theta }^2\mu cx-40\theta \mu \phi )-8{\beta }^4{\theta }^3{\phi }^2\mu -27{\theta }^5{\mu }^3{\phi }^2)(c^2+x^2)\hfill \\ & -4{\theta }^5{\mu }^2(6{\theta }^2\mu cx-5\phi )(c^4+x^4)+4{\theta }^7{\mu }^3(c^6+x^6-20c^3{x}^3)-40{\theta }^5{\mu }^3\phi (\theta c^4+x^4)\hfill \\ & -256{\theta }^4{\mu }^3{\phi }^2(\theta {\mu }^3-{\beta }^2)-120{\theta }^5{\mu }^2{c}^2{x}^2(\theta \phi -{\beta }^2)-64{\theta }^3{\beta }^2{\phi }^2({\beta }^{2}cx-3\phi {\mu }^2)\hfill \\ & -128{\theta }^4{\phi }^2{\mu }^2({\beta }^2{x}^2+{\beta }^2+\phi \mu )-16{\beta }^4\theta {\phi }^3(6\theta \mu -{\beta }^2)\hfill \end{array}\hfill \end{array} \end{gathered}$$

When $w_v^{\ast }=K$, finding ${\pi }_m^{v\ast }-{\pi }_m^{s\ast \ast }$, ${\pi }_r^{v\ast }-{\pi }_r^{s\ast \ast }$, ${\pi }_r^{v\ast }+{\pi }_m^{v\ast }$. We can obtain:

$$\begin{gathered} {\pi }_m^{v\ast }-{\pi }_m^{s\ast \ast }=\frac{{\phi }^3\mu {(2\theta \mu -{\beta }^2)}^2{(\alpha -\theta c)}^2}{4f{g}^2}=G,{\pi }_r^{v\ast }-{\pi }_r^{s\ast \ast }=\frac{{\phi }^3\mu {(2\theta \mu -{\beta }^2)}^2{(\alpha -\theta c)}^2}{4f{g}^2}=G \\ {\pi }_r^{v\ast }+{\pi }_m^{v\ast }={\pi }_t^{c\ast } \end{gathered}$$

□

It can be seen from Proposition 7 that when Formulas (22) and (23) are satisfied under the Shapley value method, the profits of both the manufacturer and leaser increase by $\mathrm{G}$ compared with the decentralized decision making. The total supply chain profit reaches the optimum level under centralized decision making. This indicates that the closed-loop supply chain reaches the Pareto optimal level under the Shapley value method.

3.4.2. Two-Part Pricing Contract

In the closed-loop supply chain of construction machinery, it is the manufacturer who determines the selling price and product recovery rate, operates the Industrial Internet platform, and remanufactures used products. Therefore, the two-part pricing contract can be designed as follows. The manufacturer offers a contract $\overline{F}=\left(w_n,F\right)$ to the leaser and sells the product to the leaser via a finance lease. Since the leaser does not have to pay the product cost at the beginning, the manufacturer has no return at the beginning. Subsequently, the leaser is required to compensate the manufacturer at the end of a leasing cycle for a portion of their profit, which is recorded as $F$.

At this time, the profit functions of the manufacturer and the leaser are:

$${\pi }_m^n=(w_n-c)(Q_n-Q_r)+(w_n-x-v)Q_r-\frac{1}{2}\phi e^2+F$$

(24)

$${\pi }_r^n=(p-w_n)Q_n+v{Q}_r-\frac{1}{2}\mu {\gamma }^2-F$$

(25)

In order for supply chain profits under the two-part pricing contract to reach the optimum level under centralized decision making, the following conditions must be satisfied.

$$p_n^{\ast }=p_c^{\ast }, {\gamma }_n^{\ast }={\gamma }_c^{\ast }, e_n^{\ast }=e_c^{\ast }, w_n^{\ast }=0$$

(26)

By substituting Equation (26) into Equations (24) and (25), the optimal values of the manufacturer profit and the leaser profit under the two-part pricing contract can be obtained:

$${\pi }_m^{n\ast }=\frac{U_1\theta \phi \mu (\alpha -\theta c)}{2f^2}+F, {\pi }_r^{n\ast }=-\frac{U_2\phi \mu (\alpha -\theta c)}{2f^2}-F$$

where:

$$\begin{gathered} \begin{array}{cc}{U}_1& =\mu {\theta }^{2}c(c(c+2v-2x)+x(x-2v))+2\phi ({\beta }^{2}c-2\theta \mu c)\hfill \\ & +\theta \alpha \mu ((c-x)(c-x-2v)))\hfill \end{array} \\ \begin{array}{c}{U}_2=\theta \phi ({\beta }^{2}c-2\alpha \mu )+2{\theta }^3\mu vc(c-x)+\phi \alpha {\beta }^2+2{\theta }^2\mu (\alpha (c-x)(c-x-v)-\phi c)\\ \end{array} \end{gathered}$$

Proposition 8.

Under the two-part pricing contract, when the compensation fee is $F\in \left(F_1,F_2\right)$, the supply chain is coordinated.

$$F_1=\frac{\mu \phi {(\alpha -\theta c)}^2}{2g}-\frac{h_1\theta \phi \mu (\alpha -\theta c)}{2f^2},F_2=-\frac{2{\phi }^2\mu (2\theta \mu -{\beta }^2){(\alpha -\theta c)}^2}{2g^2}-\frac{h_2\phi \mu (\alpha -\theta c)}{2f^2}$$

Proof 8.

If the two-part pricing contract is to be effective, it is necessary that the manufacturer and leaser profits under the two-part pricing contract be greater than those under decentralized decision making, namely ${\pi }_m^{n\ast }>{\pi }_m^{s\ast \ast }$, ${\pi }_r^{n\ast }>{\pi }_r^{s\ast \ast }$. Then, the interval of $F$ is set to $\left[F_1,F_2\right]$ and we obtain:

$$F_1=\frac{\mu \phi {(\alpha -\theta c)}^2}{2g}-\frac{h_1\theta \phi \mu (\alpha -\theta c)}{2f^2},F_2=-\frac{2{\phi }^2\mu (2\theta \mu -{\beta }^2){(\alpha -\theta c)}^2}{2g^2}-\frac{h_2\phi \mu (\alpha -\theta c)}{2f^2}$$

Find $F_2-F_1$. We can obtain:

$$F_2-F_1=\frac{{\theta }^2{\mu }^2\phi {(c-x)}^2{(\alpha -\theta c)}^2}{2fg}>0$$

□

From the calculation, $F_2>F_1$ can be obtained. Therefore, the leaser needs to compensate the manufacturer, and the scope of the compensation fee is $\left[F_1,F_2\right]$. When $F=F_1$, the manufacturer does not profit at this time and the two-part pricing contract does not have an effect. When $F=F_2$, the leaser is not profitable at this time and the two-pricing contract does not coordinate the supply chain. Therefore, a reasonable range of compensation fee should be $\left(F_1,F_2\right)$ when the supply chain reaches the Pareto optimal.

4. Numerical Analysis

As the optimal expressions for the decision variables and profit functions are complex, a numerical simulation is used to demonstrate the simulation to verify the validity of the model. The relevant parameters are set as: $\phi =200$, $\mu =5$, $\theta =3$, $c=5$, $\beta =2$, $\alpha =85$, $x=1$, $v=1$.

4.1. The Impact of Unit Remanufacturing Cost x

In order to analyze the impact of the unit remanufacturing cost on decision variables and recovery quantity, from Assumption 3, we can obtain $0<e\le 1$. Therefore, the value range of the unit remanufacturing cost is set to $(1,5)$, while other parameters remain unchanged. Figure 2, Figure 3, Figure 4 and Figure 5 illustrate these processes.

From Figure 2 and Figure 3, as the unit remanufacturing cost increases, the leasing price and selling price increase under the scenarios of contract coordination and decentralized decision making, while the recovery rate decreases. Additionally, when the unit remanufacturing cost reaches a certain value, the recovery rate drops to zero. Under the same unit remanufacturing cost, the recovery rate of the coordination contract is always higher than that under decentralized decision making, while the leasing price and selling price are always lower than those under decentralized decision making. As shown in Figure 4 and Figure 5, as the unit remanufacturing cost increases, the profits of the manufacturer, leaser, and total supply chain all decrease, while the profit of the manufacturer first decreases and then increases under the two-part pricing contract. The increase in the unit remanufacturing cost will lead to the manufacturer asking for a higher compensation fee from the leaser to make up for the loss. Under the same unit remanufacturing cost, the profits of the manufacturer, leaser, and total supply chain under the Shapley value method and the two-part pricing contract are always higher than those under decentralized decision making. This shows that, through the use of the Shapley value method and two-part pricing contract, the cooperation of supply chain members is strengthened, and the profits of the manufacturer and leaser are redistributed and improved.

The change in the unit remanufacturing cost does not affect the coordination of the Shapley value method and two-part pricing contract on the supply chain. The manufacturer’s profit is greater than that under the two-part pricing contract with the Shapley value method, and thus the manufacturer usually prefers the Shapley value method. When using the two-part pricing contract, the leaser’s profit is greater than that under the Shapley value method, so the leaser prefers the two-part pricing contract. Although the manufacturer and the leaser have different biases in the choice of contract, the total supply chain profit under both coordinated contracts reaches the level of centralized decision making. By using enhanced manufacturing technology, the manufacturer can reduce the costs of production, recovery, and remanufacturing and sell the product to the leaser at a lower price with the current recovery rate. Since the purchase cost is reduced, the leaser can also lower the leasing price, and both can gain higher profits.

4.2. The Impact of Industrial Internet Platform Cost Coefficient $\phi$

In order to analyze the influence of the Industrial Internet platform cost coefficient on the selling price and recovery rate, from Assumption 3, we can obtain $0<e\le 1$. Therefore, the value range of the Industrial Internet platform cost coefficient is set to $(200,250)$, while other parameters remain unchanged. Figure 6, Figure 7, Figure 8 and Figure 9 illustrate these processes.

Figure 6, Figure 7, Figure 8 and Figure 9 show that as the Industrial Internet platform cost coefficient increases, the selling price also increases under both decentralized decision making and the coordination contract, while the recovery rate, as well as the profits of the manufacturer, leaser, and total supply chain, all decrease. Under the same Industrial Internet platform cost coefficient, the selling price of the coordination contract is always lower than that of the decentralized decision making, while the recovery rate, and the profits of the manufacturer, leaser, and total supply chain are always greater than those under decentralized decision making. This shows that the coordination contract promotes cooperation between the manufacturer and the leaser. With the coordination contract, the manufacturer can make full use of the Industrial Internet platform to recycle used products to increase recovery rates, selling products to leasers at a cheaper price, while the profits of all parties in the supply chain increase.

4.3. The Impact of Maintenance Sensitivity Coefficient $\beta$

In order to analyze the influence of the maintenance sensitivity coefficient on maintenance effort level and product demand, we need to ensure that $Q_n>0$, $Q_r>0$. Therefore, the value range of the maintenance sensitivity coefficient is set to $(0,5)$, while other parameters are unchanged. Figure 10, Figure 11, Figure 12 and Figure 13 illustrate these processes.

Figure 10, Figure 11, Figure 12 and Figure 13 show that with the enhancement of the maintenance sensitivity coefficient, the maintenance effort level, product demand, leasing price, and total supply chain profit all increase under both coordination contracts and decentralized decision making. With the same maintenance sensitivity coefficient, the maintenance effort level, product demand, and total supply chain profit of the coordination contract are greater than those of decentralized decision making. However, the leasing prices under the coordinated contract are initially lower than those for decentralized decision making. When the maintenance sensitivity coefficient increases to a certain value, the leasing price under the coordinated contract is higher than the prices for decentralized decision making. The analysis, combined with Figure 2, Figure 3, Figure 4 and Figure 5, shows that when the manufacturer and leaser cooperate via a coordination contract, the leaser can provide a better maintenance service for products, which ultimately causes an increase in product demand and reduces the unit remanufacturing cost for the manufacturer, thus increasing supply chain profits. However, due to great concerns of consumers regarding product safety, the product maintenance cost increases. Therefore, the leaser has to increase the leasing price to ensure an increase in profits for the total supply chain.

4.4. The Impact of Compensation Fee on Each Supply Chain Player Profit

The impact of the compensation fee on the profits of the manufacturer and leaser is analyzed while other parameters remain unchanged. According to Proposition 8, by setting the relevant parameters in $F_1$ and $F_2$, the calculation results are $F_1=444$ and $F_2=484$, respectively. The running results are shown in Figure 14.

As shown in Figure 14, when the compensation fee increases, the manufacturer’s profits increase and the leaser’s profits decrease. The model simulation shows that when the compensation fee is $F=463.87$, the profits of the manufacturer are equal to those of the leaser: $273.4$. According to Proposition 8, when the compensation fee $F\in (444,484)$, the profits of both the manufacturer and leaser are greater than those achieved under decentralized decision making. The leaser obtains machinery via financial leasing without any payment and compensates the manufacturer with a portion of the profit. With the two-part pricing contract, long-term stable cooperation can be achieved and product leasing can be promoted.

5. Conclusions

To study the impact of Industrial Internet platform access on the closed-loop supply chain for the leasing industry of construction machinery, this paper analyzed three factors. First, an Industrial Internet platform was introduced and the economics of the leasing and recycling were demonstrated by comparing centralized and decentralized decision models. Secondly, coordination contracts with the practical operation of construction machinery finance leasing were designed, and the effects of different contracts on closed-loop supply chains were investigated. Thirdly, sensitivity analysis and numerical simulation were conducted to explore the impact of different parameters and decision variables on supply chain performance. The relevant insights into this management are described below:

(1) In the case of centralized decision making, decentralized decision making, and coordination contracts, the increase in the unit remanufacturing cost and the Industrial Internet platform cost coefficient can enhance the selling and leasing prices, resulting in a reduction in the used product recovery rate and profits in the supply chain. The increase in the consumer maintenance sensitivity coefficient will not only promote the improvement of the maintenance effort level of the leaser but also increase the product demand and profit of the whole supply chain. Therefore, the manufacturer needs to make full use of the Industrial Internet platform to recycle used products, continuously improve remanufacturing technology, and reduce the unit remanufacturing cost. The leaser needs to enhance the maintenance of mechanical equipment to ensure operational safety.

(2) There is a double marginalization effect in the closed-loop supply chain under decentralized decision making. The Shapley value method and two-part pricing contract can solve this problem and make the total profit of the supply chain close to that under centralized decision making. The selling price determined by the Shapley value method can lead to higher profits for the manufacturer and leaser than those made under decentralized decisions. The manufacturer’s profits are greater than those of the two-part pricing contract and the leaser shows an opposite trend. The supply chain can be coordinated with the two-part pricing contract when the compensation fee is within a certain range. The design of the coordination contract can strengthen the cooperation between the manufacturer and the leaser, improving the supply chain’s performance and promoting sustainable low-carbon development.

However, only a single channel for leasing and recycling was explored in this paper. Furthermore, a third-party recycling agency can be introduced at the recycling stage and more subjects can participate in game pricing, which is more relevant to real-life situations. This study analyzed the effect of different contracts on the improvement of maintenance effort level, recycling rate, and supply chain performance; however, the change in maintenance effort levels and recycling rates over time are yet to be explored. Therefore, in future research, time variables can be employed to study optimal decisions under dynamic situations.