1. Introduction

2. Model Framework

3. Model Solutions and Contract Coordination Design

3.4. Coordination Contract Model

It can be seen from propositions 3 to 6 that it is necessary to design contract to coordinate the supply chain so that the supply chain profits can achieve 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, this section makes:

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

3.3.1. Shapley value method

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

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

(16)

Then $\left[I,u\right]$ is called 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 obvious 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:

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

Which

$$\begin{array}{c}{A}_3=(6{\theta }^2{\beta }^2\phi \mu -12{\theta }^3{\mu }^2\phi ){(c-x)}^2+2{\theta }^4{\mu }^2(c^4-4c^{3}x-4c{x}^3-4+6c^2{x}^2+x^4)\hfill \\ \hspace{1em}\hspace{1em}+20\theta \mu {\phi }^2(\theta \mu -{\beta }^2)+5{\beta }^4{\phi }^2\hfill \\ A_4={\phi }^2\mu (2\theta \mu -{\beta }^2){(\alpha -\theta c)}^2(3\phi (2\mu x-{\beta }^2)-2{\theta }^2\mu {(c-x)}^2)\hfill \end{array}$$

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 the decentralized decision-making. The total supply chain profit reaches the level under centralized decision-making.

Proof 7.

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

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

Among, the leasing price, maintenance effort level and product recovery rate are respectively equal to those under centralized decision-making, so:

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

(24)

The simultaneous equations can be obtained:

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

(25)

Which

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 get:

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

It can be seen from Proposition 7 that when formulas (24) and (25) are satisfied under the Shapley value method, the manufacturer profit and the leaser profit both increase by $\mathrm{G}$ compared with the decentralized decision-making. The total supply chain profit reaches the level under centralized decision-making. It indicates that the closed-loop supply chain reaches the Pareto optimal under the Shapley value method.

3.3.2. Two-part pricing contract

In the construction machinery closed-loop supply chain, it is manufacturer who determines the selling price and product recovery rate, operates the Industrial Internet platform and remanufactures the used product. So, 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 by 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 its profit. Which is recorded as $F$.

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

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

(26)

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

(27)

In order for supply chain profits under the two-part pricing contract to reach the level under centralized decision-making. The following conditions need to be met.

$$p_n^{\ast }=p_c^{\ast }\text{，}{\gamma }_n^{\ast }={\gamma }_c^{\ast }\text{，}{e}_n^{\ast }=e_c^{\ast }\text{，}{w}_n^{\ast }=0$$

(28)

Substituting Equation (28) into Equation (26) and Equation (27), the optimal values of the manufacturer profit and the leaser profit under the two-part pricing contract can be obtained:

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

Which

$$\begin{array}{c}{U}_1=\mu {\theta }^{2}c(c(c+2v-2x)+x(x-2v))+2\phi ({\beta }^{2}c-2\theta \mu c)\hfill \\ \hspace{1em}\hspace{1em}+\theta \alpha \mu ((c-x)(c-x-2v)))\hfill \\ U_2=\theta \phi ({\beta }^{2}c-2\alpha \mu )+2{\theta }^3\mu vc(c-x)+\phi \alpha {\beta }^2+2{\theta }^2\mu (\alpha (c-x)(c-x-v)-\phi c)\hfill \end{array}$$

Proposition 8.

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

, the supply chain achieves coordination.

$$F_1=\frac{\mu \phi {(\alpha -\theta c)}^2}{2g}-\frac{h_1\theta \phi \mu (\alpha -\theta c)}{2f^2}\text{，}{F}_2=-\frac{2{\phi }^2\mu (2\theta \mu -{\beta }^2){(\alpha -\theta c)}^2}{2g^2}-\frac{h_2\phi \mu (\alpha -\theta c)}{2f^2}$$

Proof 8.

If the two-part pricing contract is to be effective, it is necessary to satisfy that the manufacturer profit and the leaser profit under the two-part pricing contract are greater than those under the 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 $\left[F_1,F_2\right]$. We get:

$$F_1=\frac{\mu \phi {(\alpha -\theta c)}^2}{2g}-\frac{h_1\theta \phi \mu (\alpha -\theta c)}{2f^2}\text{，}{F}_2=-\frac{2{\phi }^2\mu (2\theta \mu -{\beta }^2){(\alpha -\theta c)}^2}{2g^2}-\frac{h_2\phi \mu (\alpha -\theta c)}{2f^2}$$

Find $F_2-F_1$. We can get:

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

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 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 Pareto optimal.

4. Numerical Analysis

4.1. The Impact of Unit Remanufacturing Cost x

In order to analyze the impact of unit remanufacturing cost on decision variables and recovery quantity, the value range of unit remanufacturing cost is set to $(1,3)$ while other parameters are unchanged. Figures 2–5 can be obtained

From Figures 2–5, as the unit remanufacturing cost increases, the leasing price and selling price increase under the scenarios of contract coordination and decentralized decision, while the recovery rate and recovery quantity decrease. Under the same unit remanufacturing cost, the recovery rate and recovery quantity of the coordination contract are always higher than that under decentralized decision-making, while the leasing price and selling price are always lower than that under decentralized decision-making. It shows that the decision variables under the coordination contract have reached the level of centralized decision-making. Using the Shapley value method and the two-part pricing contract, the profits of manufacturer and leaser are redistributed and begin to cooperate. Through enhance manufacturing technology, the manufacturer can reduce the cost of production, recovery and remanufacturing and sell the product to the leaser at a lower price with the current recovery rate. Due to the deduction of purchase cost, the leaser can also lower the leasing price and pay more attention to the daily maintenance of the product. With the help of contracts, consumers can obtain high-quality products and maintenance services without excessive restriction of product service life, ensuring safety and avoiding waste.

4.2. The Impact of Industrial Internet Platform Cost Coefficient φ

In order to analyze the influence of the Industrial Internet platform cost coefficient on the selling price and recovery rate, the value range of the Industrial Internet platform cost coefficient is set to (200,250) when other parameters remain unchanged. We obtained Figures 6 and 7.

From Figures 6 and 7, it can be seen that with the increase of the Industrial Internet platform cost coefficient, the selling price increase under both the decentralized decision-making and coordination contract, while the recovery rate 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, and the recovery rate is always greater than that of the decentralized decision-making. This shows that the coordination contract promotes the cooperation between the manufacturer and the leaser. With the contract, the manufacturer can make full use of the Industrial Internet platform to recycle used product so that the recovery rate can be increased, while selling product to leaser at a cheaper price.

4.3. The Impact of Maintenance Sensitivity Coefficient β

In order to analyze the influence of maintenance sensitivity coefficient on maintenance effort level and product demand, the value range of maintenance sensitivity coefficient is set to (2, 4) while other parameters are unchanged. We obtained Figures 8 and 9

As can be seen from Figures 8 and 9,, with the enhancement of the maintenance sensitivity coefficient, the maintenance effort level and product demand increase under both coordination contracts and decentralized decision-making. With the same maintenance sensitivity coefficient, the maintenance effort level and product demand of the coordination contract are greater than those of decentralized decision-making. In the reality of construction machinery leasing industry, consumers are very concerned about the maintenance status and safety of products. The analysis in Figures 2–5 shows that when manufacturers and leasers cooperate by coordination contract, leaser can provide better maintenance service for product, which ultimately brings an increase in the product demand.

4.4. The Impact of the Coordination Contract on Manufacturer Profit and Leaser Profit

It can be calculated from above that there exists a scope for compensation fee, $F\in (444,484)$. In order to comprehensively analyze the changes in the profits of manufacturers and leasers under different decision-making. The unit remanufacturing cost is set to $(1,3)$ while other parameters are unchanged. We obtained Figures 10–13 as follows. $F=470$ in Figures 10 and 11, $F=447$ in Figure 12, and $F=484$ in Figure 13.

As can be seen in Figures 10 and 11 when compensating fee is $F=470$, manufacturer and leaser profits decrease in all three decision scenarios as unit remanufacturing cost increase. Under the two coordination contracts, the manufacturer profit and leaser profit are always greater than those under decentralized decision-making. And the change of unit remanufacturing cost does not affect the coordination of the Shapley value method and two-part pricing contract on the supply chain. The manufacturer profit is greater than that under the two-part pricing contract with the Shapley value method, so the manufacturer prefers the Shapley value method. When using the two-part pricing contract, the leaser 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

In Figure 12, the manufacturer receives a compensation fee that is close to the minimum of the range of compensation fee. The Shapley value method is always effective in this case. But the two-part price contract coordination mechanism fails when the unit remanufacturing cost increases to a certain value. There are two remedies available to the manufacturer, one is to reduce unit remanufacturing cost by improving remanufacturing technology; the other is to require the leaser to pay more compensation fee as much as possible. In Figure 13, the compensation paid by the leaser is close to the maximum of the range of compensation fee. The Shapley value method and the two-part pricing contract are always valid.

5. Conclusions

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