## 3 Closed-Loop Supply Chain Decision
### Decentralized Decision
#### 3.2.2 R-M Decision  
In the RM decision model, the retailer, as the market leader (e.g., Wal-Mart), first determines the sales price \(p_{n},p_{r}\) and \(p_{s}\) according to its own objective function and the manufacturer's optimal reaction function. Subsequently, the manufacturer, according to its own objective function and the retailer's decision, determines the wholesale prices (\(\omega_{n},\omega_{r},\omega_{s}\)) and service levels (\(s_{n},s_{r},s_{s}\)). The decision model is:  
\[ \begin{split}\max_{p_{n},p_{r},p_{s}}\Pi_{R}^{d}=\int_{\underline {p_{n}-s_{n}-p_{r}+s_{r}}}^{Q}\left(p_{n}-\omega_{n}\right)\,d\alpha+\int_{ \underline{g+p_{r}-p_{s}-s_{r}+s_{s}}}^{Q}\left(p_{r}-\omega_{r}\right)\,d \alpha\\ +\int_{\underline{g+p_{r}-p_{s}-s_{r}+s_{s}}}^{Q}\left(p_{s}- \omega_{s}\right)\,d\alpha\end{split}  \tag{35}\]  
\[ \begin{split} s.t.=\left\{\begin{array}{l}\max_{x_{p_{n}},x_{p_{ r}}}\Pi_{M}^{d}=\int_{\underline{g+p_{r}-p_{s}-s_{r}+s_{s}}}^{Q}\left(\omega_{r}- \left(c-\Delta_{1}\right)\right)\,d\alpha\\ \frac{g+p_{r}-p_{s}-s_{r}+s_{s}}{\theta(1-\theta)}\\ +\int_{\underline{g+p_{s}-s_{s}}}^{Q}\left(\omega_{s}-\left(c- \Delta_{2}\right)\right)\,d\alpha\\ +\int_{\underline{p_{n}-s_{n}-p_{r}+s_{r}}}^{Q}\left(\omega_{n}-c \right)\,d\alpha-F-\frac{1}{2}\eta s_{n}s_{n}-\frac{1}{2}\eta s_{r}s_{r}-\frac {1}{2}\eta s_{s}s_{s}\\ p_{n}=\omega_{n}+A\\ p_{r}=\omega_{r}+B\\ p_{s}=\omega_{s}+M\end{array}\right.\end{split}  \tag{36}\]  
Among them, the constraints \(p_{n}=\omega_{n}+A\),and \(p_{r}=\omega_{r}+B,p_{s}=\omega_{s}+M\) imply that the retailer sales price is more significant than or equal to the manufacturer wholesale price. The objective function model is explained in Table 3.  
Next, Equations (35) and (36) are solved and the proposition obtained. For the proposition R-M decision under the CLSC of new products, remanufactured products and refurbished products, we obtain the best sales price \(p_{i}\), the best wholesale price \(\omega_{i}\) and the optimal service level \(s_{i}\) as follows:  
\[p_{n}^{d2*}=\frac{4\eta^{2}\theta\phi^{2}(c+3Q)+c+2\Delta_{1}\eta\theta\phi+ \Delta_{2}+g+Q(\theta(2\eta(\theta(\theta+4)-2)+\theta)+3)}{16\eta^{2}\theta \phi^{2}} \tag{37}\]  
\[p_{r}^{d2*}=\frac{\eta\theta^{2}\left(4\eta^{2}\theta\phi^{2}(c+3\theta Q)+c-2 \Delta_{1}\left(2\eta^{2}\theta\phi^{2}+1\right)+2\Delta_{2}\eta\phi+2g\eta \phi+\theta(\theta+2)Q\right)}{2(2\eta\theta\phi+1)\left(2\eta\left(2\theta^{2 }(\eta\phi+1)+1\right)-1\right)} \tag{38}\]  
\[p_{s}^{d2*}=\frac{\eta\theta^{2}\left(4\eta^{2}\theta\phi^{2}+1\right)\left(c+3 \left(g+\theta^{2}Q\right)\right)-\eta\theta^{2}\left(2\eta\phi\left(2\Delta _{2}\eta\theta\phi-\Delta_{1}\right)+\Delta_{2}\right)+\Delta_{2}}{16\eta^{3} \theta^{2}} \tag{39}\]  
\[w_{n}^{d2*}=\frac{16\eta^{2}\theta\phi^{2}(3c+Q)+3c+4\Delta_{1}\eta\theta\phi+ \Delta_{2}+g+Q\left(4\eta\left(\theta^{3}+1\right)+\theta^{2}+2\right)}{64 \eta^{2}\theta\phi^{2}+4} \tag{40}\]  
\[w_{r}^{d2*}=\frac{\theta\left(c\left(48\eta^{2}\theta\phi^{2}+3\right)+4\eta \phi\left(\Delta_{2}-12\Delta_{1}\eta\theta\phi\right)+4\eta\phi(g+\theta(2 \theta+1)Q)+Q\left(\gamma+\theta^{2}\right)\right)}{4(\gamma+\theta)} \tag{41}\]  
\[w_{s}^{d2*}=\frac{\theta\left(16\eta^{2}\theta\phi^{2}(3c+g)+24c\eta\theta \phi+c+8\eta\phi\left(\Delta_{1}-6\Delta_{2}\eta\theta\phi\right)-3\Delta_{2} +\theta Q(\gamma+\theta)\right)}{4(\gamma+\theta)} \tag{42}\]