## 3 Closed-Loop Supply Chain Decision
### Decentralized Decision
#### 3.2.1 M-R Decision  
In the M-R decision model, the manufacturer as a leader first considers the retailer's optimal reaction function to determine the wholesale prices (\(\omega_{n},\omega_{r},\omega_{s}\)) and service levels (\(s_{n},s_{r},s_{s}\)). Subsequently, the retailer determines its sales price according to the manufacturer's decision, i.e., \(p_{n},p_{r},p_{s}\), (e.g., Apple Inc. and its affiliates). The decision model is:  
\[\begin{split}\max_{\omega_{n},\omega_{n},\omega_{n},\omega_{n}, \omega_{n},\omega_{n},\omega_{n},\omega_{n},\omega_{n}}\Pi_{M  
To solve Equations (12) and (13), we introduce Proposition 2:  
Under the M-R decision, the optimal selling prices of new products, remanufactured products and refurbished products under the CLSC \(p_{i}\), optimal wholesale price \(\omega_{i}\) and optimal service level \(s_{i}\) are:  
\[p_{n}^{d1*}=\frac{c\eta\theta(\gamma+\theta)+3\eta\theta\left(4\Delta_{1}\eta \theta^{2}\phi+\Delta_{2}\theta\right)+3\eta\left(g\theta^{2}+Q\left(\gamma \theta+4\eta\left(\theta^{4}-1\right)\theta+\theta^{4}+\theta^{2}\right) \right)}{64\eta^{3}\theta^{3}\phi^{2}} \tag{14}\]  
\[p_{r}^{d1*}=\frac{c\eta\theta(\gamma+\theta)-\Delta_{1}\eta\theta(\gamma+4 \theta)+12\Delta_{2}\eta^{2}\theta^{2}\phi+3\eta\theta\left(4g\eta\theta\phi +\theta Q\left(\gamma+\theta^{2}\right)\right)}{64\eta^{3}\theta^{3}\phi^{2}} \tag{15}\]  
\[p_{s}^{d1*}=\frac{c\eta\theta(\gamma+5\theta)-16\Delta_{2}\eta^{3}\theta^{3} \phi^{2}+12\Delta_{1}\eta^{2}\theta^{2}\phi+\Delta_{2}+3\eta\theta^{2}\left(1 6g\eta^{2}\theta\phi^{2}+\theta Q(\gamma+\theta)\right)}{64\eta^{3}\theta^{3} \phi^{2}} \tag{16}\]  
\[\omega_{n}^{d1*}=\frac{\theta(c+g)+4c\eta\phi+8\eta^{2}\theta^{2}\phi^{2}(c+2Q )+\theta\left(4\Delta_{1}\eta\theta\phi+\Delta_{2}\right)+Q\left(4\eta\left( \theta^{4}-1\right)+\theta^{3}+\theta\right)}{32\eta^{2}\theta^{2}\phi^{2}} \tag{17}\]  
\[\omega_{r}^{d1*}=\frac{c(\gamma+\theta)+\Delta_{1}(\gamma+2\theta)+4\Delta_{2 }\eta\theta\phi+4g\eta\phi+\theta Q\left(\gamma+\theta^{2}\right)}{32\eta^{2} \theta^{2}\phi^{2}} \tag{18}\]  
\[\omega_{s}^{d1*}=\frac{\Delta_{2}(2\eta\theta(\gamma+\theta))+2\eta\theta \left(c(\gamma+\theta)+4\Delta_{1}\eta\theta\phi+\theta^{2}\left(16g\eta^{2} \phi^{2}+\theta Q(8\eta\phi\phi(2\eta\phi+1))\right)\right)}{64\eta^{3}\theta ^{3}\phi^{2}} \tag{19}\]  
\[s_{n}^{d1*}=\frac{-c+4\eta\theta^{2}\left(4\Delta_{1}\eta\theta\phi+\Delta_{2 }\right)+4\eta\theta\left(g\phi+Q\left(4\eta\theta^{2}\phi^{2}+\theta^{3}-1 \right)\right)}{64\eta^{3}\theta^{3}\phi^{2}} \tag{20}\]  
\[s_{r}^{d1*}=\frac{4\eta\phi\left(c+\theta^{2}(4g\eta+\theta Q)\right)+16\eta^ {2}\theta^{2}\left(\Delta_{1}\left(\theta^{2}-1\right)+\Delta_{2}\phi\right) +\theta Q}{4\eta\theta\phi\left(16\eta^{2}\theta^{2}\phi-1\right)} \tag{21}\]  
\[s_{s}^{d1*}=\frac{\theta\left(16c\eta^{2}\phi^{2}+\theta Q\right)+16\eta^{2} \theta\phi\left(\Delta_{1}\theta-\Delta_{2}\right)+\Delta_{2}-16g\eta^{2} \theta\phi+g}{64\eta^{3}\theta^{3}\phi^{2}} \tag{22}\]  
According to the Stackelberg game model, manufacturers as the market leader first determine their wholesale prices \(\omega_{n},\omega_{r},\omega_{s},s_{n},s_{r},s_{s}\); for retailers as followers, the manufacturer's decision model is referred to in order to determine the sales price \(p_{n},p_{r},p_{s}\). From Equation (13), the Hessian matrix of the retailer's profit function for \(p_{n},p_{r}\) and \(p_{s}\) is negatively determined. In other words, the retailer's profit function is a strictly concave function for \(p_{n},p_{r}\) and \(p_{s}\). There is a unique optimal solution.