## 3 Closed-Loop Supply Chain Decision  
### Centralized Decision  
Under the centralized decision making model, manufacturers and retailers form an overall joint decision (for example: [PERSON] electric manufacturing-sales model), whose profit function is expressed as:  
\[\max_{p_{n},p_{r},p_{s},s_{n},s_{r},s_{s}}\Pi_{s}^{c}=\int\!\!\! \frac{p_{n}-s_{n}-p_{r}+s_{r}}{1-\theta}\;(p_{n}-c)\;d\alpha+\int\!\!\!\frac{ \frac{p_{n}-s_{n}-p_{r}+s_{r}}{1-\theta}}{\theta(1-\theta)}\;(p_{r}-(c-\Delta _{1}))\;d\alpha \tag{5}\] \[+\int\!\!\!\frac{\frac{g+p_{r}-p_{s}-s_{r}+s_{s}}{\theta(1-\theta )}}{\theta^{2}}\;(p_{s}-(c-\Delta_{2}))\;d\alpha-F-\frac{1}{2}\eta\left(s_{n} ^{2}+s_{r}^{2}+s_{s}^{2}\right)\]  
Figure 2: Consumer utility floating range. Solving Proposition 1 according to first-order linear conditions:  
**Proposition 1**. : _The optimal sales prices of new products, remanufactured products and refurbished products under CLSC decisions \(p_{i}^{c*}\) and the optimal service level \(s_{i}^{c*}\) are:_  
\[p_{n}^{c*}=\frac{2c\eta^{2}\phi\left(2\theta^{2}(\eta\phi+1)+\theta\right)+\eta \theta\left(2\Delta_{1}\eta\theta^{2}\phi+\Delta_{2}\theta\right)+\eta\left(g \theta^{2}+Q\left(2\eta\left(2\eta\theta^{2}\phi^{2}+\theta^{4}\right)+\theta ^{3}\right)\right)}{4\eta\theta^{2}(\eta\phi+1)(2\eta\theta\phi+1)} \tag{6}\]  
\[p_{r}^{c*}=\frac{c\theta^{2}+2c+\theta^{2}\left(2\Delta_{2}\eta\phi-4\Delta_{1 }\eta^{2}\theta\phi^{2}\right)+4\eta^{2}\theta^{3}\lambda\phi^{2}+2\eta\theta \phi\left(\theta(g+\lambda)+\theta^{2}\lambda+Q\right)+\theta^{3}Q}{4\eta \theta\phi\left(2\theta^{2}(\eta\phi+1)+1\right)} \tag{7}\]  
\[p_{s}^{c*}=\frac{4c\eta^{3}\theta^{3}\mu-\Delta_{2}\eta\theta\left(4\eta^{2} \theta^{2}\mu+\theta\right)+2\Delta_{1}\eta^{2}\theta^{2}\phi+\eta\theta^{2} \left(4g\eta^{2}\theta\mu+\theta Q\left(4\eta^{2}\theta^{2}\mu+\theta\right) \right)}{4\eta^{2}\theta^{2}\phi(2\eta\theta\phi+1)} \tag{8}\]  
\[s_{n}^{c*}=\frac{2\eta\theta\left(2\Delta_{1}\eta\theta^{2}\phi+\Delta_{2} \theta\right)+2g\eta^{2}+Q\left(4\eta^{2}\theta^{3}\mu+2\eta\left(\theta^{3}- 1\right)\theta\right)}{8\eta^{2}\theta^{3}\phi(\eta\phi+1)} \tag{9}\]  
\[s_{r}^{c*}=\frac{2c\eta\phi+4\Delta_{2}\eta^{2}\theta^{2}\phi+4\Delta_{1}\eta ^{2}\left(\theta^{2}-1\right)\theta^{2}+2\eta\theta^{2}(2g\eta\phi+\theta Q \phi)+\theta Q}{4\eta\theta^{2}\left(2\eta\theta^{2}+1\right)\left(\eta\phi+1 \right)} \tag{10}\]  
\[s_{s}^{c*}=\frac{4c\eta^{2}\theta\mu+4\Delta_{1}\eta^{2}\theta^{2}\phi-4 \Delta_{2}\eta^{2}\theta\phi+\Delta_{2}-4g\eta^{2}\theta\phi+g+\theta^{2}Q}{8 \eta^{3}\theta^{3}\phi^{2}} \tag{11}\]  
The objective function model is explained in Table 1.  
It is proven that the Hessian matrix of the objective function under a centralized decision can be determined as negative by Equation (5). That is, the target profit function is a strictly concave function about \(p_{n},p_{r}\) and \(p_{s}\), and an optimal solution exists. The first-order partial derivatives of \(p_{n}\), \(p_{r}\) and \(p_{s}\) are solved for Equation (5) and set to zero. The simultaneous equations can subsequently be solved optimally. Conclusion 1: Under the centralized decision:  
1. \(p_{n}^{c*},s_{n}^{c*}\) decreases as the consumption preference increases;
2. \(p_{r}^{c*},p_{s}^{c*},s_{r}^{c*},s_{s}^{c*}\) increase with the increase in consumer preference;
3. \(p_{s}^{c*},g_{s}^{c*}\) increase with the increase in government subsidy degree \(g\); at this time, the producer surplus and consumer surplus increase.