## 4 Coordination Mechanism Design
### M-R Decision: Revenue Sharing Contract  
According to [PERSON] et al. (2019), the revenue-sharing contract is superior to other contracts and can significantly increase the overall benefit of the supply chain system. Its decision model is:\[\max_{ \begin{subarray}{c}\mu_{w},\mu_{r}\end{subarray} }\Pi_{M}^{\mathrm{r1}}= \frac{p_{n}-s_{n}-p_{r}+s_{r}}{\frac{1-\theta}{ \theta+p_{r}-p_{s}-s_{r}+s_{s}}}\left(\omega_{r}-\left(c-\Delta_{1}\right) \right)\,\,d\alpha+\frac{\frac{g+p_{r}-p_{s}-s_{r}+s_{s}}{\theta(1-\theta)}} {\theta^{2}}\left(\omega_{s}-\left(c-\Delta_{2}\right)\right)\,\,d\alpha\] \[+\int_{\underline{p}_{n}-s_{n}-p_{r}+s_{r}}^{\mathcal{Q}}\left( \omega_{n}-c\right)\,\,d\alpha-F+(1-\delta)(p_{r}\left(\int_{\underline{g+p_{ r}-p_{s}-s_{r}+s_{s}}}^{\frac{1-\theta}{1-\theta}}1\,d\alpha\right) \tag{46}\] \[+p_{s}\left(\int_{\underline{-g+p_{s}-s_{r}+s_{s}}}^{\frac{g+p_{r} -p_{s}-s_{r}+s_{s}}{\theta^{2}}}1\,d\alpha\right)+p_{n}\left(\int_{\underline {p}_{n}-s_{n}-p_{r}+s_{r}}^{\mathcal{Q}}1\,d\alpha\right))-\frac{1}{2}\eta \left(s_{n}^{2}+s_{r}^{2}+s_{s}^{2}\right)\] \[s.t.p_{n},p_{r}\in argmax_{w_{n},w_{r}}\Pi_{R}^{\mathrm{r1}}= \int_{\underline{g+p_{r}-p_{s}-s_{r}+s_{s}}}^{\frac{1-\theta}{ \theta}}\left(\delta p_{r}-\omega_{r}\right)\,\,d\alpha\] \[+\int_{\underline{-g+p_{s}-s_{s}}}^{\frac{g+p_{r}-p_{s}-s_{r}+s_{ s}}{\theta^{2}}}\left(\delta p_{s}-\omega_{s}\right)\,\,d\alpha+\int_{\underline {p}_{n}-s_{n}-p_{r}+s_{r}}^{\mathcal{Q}}\left(\delta p_{n}-\omega_{n}\right) \,\,d\alpha\] \[\Pi_{R}^{\mathrm{r}}\geq\Pi_{R}^{d},\Pi_{M}^{\mathrm{r}}\geq\Pi_{M }^{d} \tag{48}\]  
When \(\delta=1\), no cooperation exists between the manufacturers and retailers. At this time, they return to the M-R decision making model of the CLSC. Manufacturers obtain all the benefits, and the revenue-sharing contract is invalid. **Proposition 4**. : _The optimal sales price pi for the CLSC new products, remanufactured products and refurbished products under a revenue-sharing contract is:_  
\[p_{n}^{r1*}=\frac{-c+2\eta\theta^{2}\left(2\Delta_{1}\eta\theta\phi+\Delta_{2 }\right)+2\eta\theta\left(g\theta+2\eta\theta^{2}\phi^{2}(2\eta\lambda+Q)+ \left(\theta^{3}-1\right)Q\right)}{16\eta^{3}\theta^{3}\phi^{2}} \tag{49}\]  
\[p_{r}^{r1*}=\frac{1}{2}\left(\frac{2\eta\phi\left(c+\theta^{2}(2g\eta+\theta Q )\right)+4\eta^{2}\theta^{2}\left(\Delta_{1}\left(\theta^{2}-1\right)+\Delta_{2 }\phi\right)+\theta Q}{8\eta^{3}\theta^{2}}+c-\Delta_{1}+\theta Q\right) \tag{50}\]  
\[p_{s}^{r1*}=\frac{1}{2}\left(\frac{\theta(4c\eta\phi(\eta\phi+1)+\theta Q)+4 \eta^{2}\theta\phi\left(\Delta_{1}\theta-\Delta_{2}\right)+\Delta_{2}-4g\eta^{ 2}\theta\phi+g}{8\eta^{3}\theta^{3}\phi^{2}}+c-\Delta_{2}+g+\theta^{2}Q\right) \tag{51}\]  
**Proposition 5**. : _Under the coordination of the revenue-sharing contract model, the necessary and sufficient conditions for the M-R game model to constitute the optimal decision are \(\omega_{n}=c\delta,\omega_{r}^{r1*}=\delta(c-\Delta_{1})\) and \(\omega_{s}^{r1*}=\delta(c-\Delta_{2})\). At this time, \(s_{n}^{r1*}=s_{n}^{c*},s_{r}^{r1*}=s_{r}^{c*}\) and \(s_{s}^{r1*}=s_{s}^{c*}\)._  
Proof. : To achieve the M-R supply chain coordination, it is necessary to perform equal decisions under both the decentralized and centralized decision making. The following must be met:  
\[\left\{ \begin{array}{l}p_{n}^{r1*}=p_{n}^{c*}\\ p_{r}^{r1*}=p_{r}^{c*}\\ p_{s}^{r1*}=p_{s}^{c*}\end{array} \right. \tag{52}\]  
The first-order partial derivatives of \(p_{n},p_{r}\) and \(p_{s}\) are obtained from \(\Pi_{R}\) and are equal to the centralized decision making pricing strategy. \[\frac{\partial\Pi_{R}}{\partial p_{n}}=-\frac{-\delta s_{n}-\omega_{n}+\delta(-Q)}{2 \delta}=p_{n}^{c*} \tag{53}\]  
\[\frac{\partial\Pi_{R}}{\partial p_{r}}=-\frac{\delta\theta(-Q)-\delta s_{r}- \omega_{r}}{2\delta}=p_{r}^{c*} \tag{54}\]  
\[\frac{\partial\Pi_{R}}{\partial p_{s}}=-\frac{-\delta g+\delta\theta^{2}(-Q)- \delta s_{s}-\omega_{s}}{2\delta}=p_{s}^{c*} \tag{55}\]  
The propositions are solved according to Equations (53)-(55).