## 3 Closed-Loop Supply Chain Decision
### Decentralized Decision
#### 3.2.2 R-M Decision
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Game Party** & **Parameter Determined** **by the Formula** & **Formula** \\ \hline \multirow{6}{*}{Retailer (following status)} & New product & \(\int\!\!\!\frac{p_{n}-s_{n}-p_{r}+s_{r}}{\theta(1-\theta)}\) \\  & sales profit & \(\int\!\!\!\frac{g+p_{r}-p_{s}-s_{r}+s_{s}}{\theta^{2}}\) \\ \cline{2-3}  & Remanufactured product & \(\int\!\!\!\frac{g+p_{s}-s_{r}+s_{s}}{\theta(1-\theta)}\) \\  & Sales profit & \(\int\!\!\!\frac{g+p_{s}-s_{s}}{\theta^{2}}\) \\ \cline{2-3}  & Refurbished product & \(\int\!\!\!\frac{p_{n}^{Q}-s_{n}-p_{r}+s_{r}}{1-\theta}\) \\ \hline \multirow{6}{*}{Manufacturer (Leadership)} & New product & \(\int\!\!\!\frac{p_{n}^{Q}-s_{n}-p_{r}+s_{r}}{1-\theta}\) \\  & Sales profit & \(\int\!\!\!\frac{p_{n}-s_{n}-p_{r}+s_{r}}{1-\theta}\) \\ \cline{2-3}  & Remanufactured product & \(\int\!\!\!\frac{p_{n}-s_{n}-p_{r}+s_{r}}{1-\theta}\) \\ \cline{2-3}  & Remanufactured product & \(\int\!\!\!\frac{g+p_{r}-p_{s}-s_{r}+s_{s}}{\theta(1-\theta)}\) \\ \cline{2-3}  & Refurbished product & \(\int\!\!\!\frac{g+p_{r}-p_{s}-s_{r}+s_{s}}{\theta^{2}}\) \\ \cline{2-3}  & Total service cost & \(\frac{1}{2}\eta\left(s_{n}^{2}+s_{r}^{2}+s_{s}^{2}\right)\) \\ \cline{2-3}  & Recycling and & \\ \cline{2-3}  & remanufacturing & \(F\) \\ \cline{2-3}  & fixed costs & \(F\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Objective function model interpretation. \[s_{n}^{d2*} =\frac{-c+2\eta\theta^{2}\left(4\Delta_{1}\eta\theta\phi+\Delta_{2} \right)+2\eta\theta^{2}\left(g+Q\left(4\eta\theta\phi^{2}+\theta^{2}+1\right) \right)}{2\eta\theta(\gamma+\theta)} \tag{43}\] \[s_{r}^{d2*} =\frac{4c\eta\phi+8\eta^{2}\theta^{2}\left(\Delta_{1}\left(\theta ^{2}-1\right)+\Delta_{2}\phi\right)+\theta(2\eta\theta\phi(4g\eta+\theta Q)+Q )}{2\eta\theta^{2}(8\eta\theta\phi(2\eta\phi+1)+1)}\] (44) \[s_{s}^{d2*} =\frac{\theta\left(16c\eta^{2}\phi^{2}+\theta Q\right)+16\eta^{2 }\theta\phi\left(\Delta_{1}\theta-\Delta_{2}\right)+\Delta_{2}-16g\eta^{2} \theta\phi+g}{4\eta\theta(\gamma+\theta)} \tag{45}\]  
Proof. : According to the Stackelberg game model, retailers, as market leaders, first determine their retail prices \(p_{n},p_{r}\) and \(p_{s}\); manufacturers as followers, according to the retailer decision model, determine their sales prices \((\omega_{n},\omega_{r},\omega_{s})\) and service levels \((s_{n},s_{r},s_{s})\). Next, we substitute \(p_{n}=\omega_{n}+A\), \(p_{r}=\omega_{r}+B\) and \(p_{s}=\omega_{s}+M\) into \(\Pi_{M}^{d2}\) and obtain the first-order partial derivatives of \(\omega_{n},\omega_{r},\omega_{s}\). After including the human intervention, we set the first derivative be zero and solve the simultaneous equations. The group can obtain the unique optimal solution, by substituting the optimal wholesale prices \(\omega_{n},\omega_{r},\omega_{s}\) of new products, remanufactured products and refurbished products into \(\Pi_{R}^{d2}\). According to the reverse induction method, the optimal decision result in the proposition can be obtained. Conclusion 4: Under the decentralized R-M decision,  
1. \(p_{n},\omega_{n},s_{n}\) decrease with the increase in \(\theta\);
2. \(p_{r},\omega_{r},s_{r}\) increase with the increase in \(\theta\);
3. \(p_{s},\omega_{s},s_{s}\) increase with the increase in \(\theta\);  
Conclusion 5: Under the decentralized R-M decision,  
1. \(p_{n}^{c*}<p_{n}^{d2*};p_{r}^{c*}<p_{r}^{d2*};p_{s}^{c*}<p_{s}^{d2*}\);
2. \(s_{n}^{c*}>s_{n}^{d2*};s_{r}^{c*}>s_{r}^{d2*};s_{s}^{c*}>s_{s}^{d2*}\);
3. \(\Pi_{M}^{d1*}+\Pi_{R}^{d1*}<\Gamma_{5}^{c*}\). Inference 3:  
1. With the increase in consumer acceptance, remanufactured products and new products, the best selling price, best wholesale price and optimal service level should be increased, and new products should be reduced accordingly;
2.