## 3 The Models
### Optimal Decisions in Policy C  
The manufacturer produces MDIGPs and sells to the retailer at wholesale price \(w_{i}^{C}\). The retailer sells those to the customers at a price of \(p_{i}^{C}\). The demand function in Policy C is \(D_{i}^{C}=a-(p_{i}^{C}-\rho_{i}^{C})+\beta\rho_{i}^{C}\). The manufacturer collects used product directly from the consumers for re-manufacturing. The government organization decides the subsidy rate by maximizing SW. A consumer subsidy on electronic vehicles is common in countries like China, Canada, Germany, Japan, etc. [66]. The profit functions for the retailer and manufacturer, and the SW for the government in Scenarios MC and RC are obtained as follows:  
\[\pi_{ri}^{C}(p_{i}^{C})=(p_{i}^{C}-w_{i}^{C})D_{i}^{C}, \tag{1}\]  
\[\pi_{mi}^{C}(w_{i}^{C},\theta_{i}^{C},\tau_{i}^{C})=(w_{i}^{C}-\lambda_{1} \theta_{i}^{C})D_{i}^{C}+(c_{m}\delta-\alpha-c_{r})\tau_{i}^{C}D_{i}^{C}-\kappa \tau_{i}^{C}{}^{2}-\lambda_{2}\theta_{i}^{C}{}^{2}, \tag{2}\]  
\[\pi_{gi}^{C}(\rho_{i}^{C})=\pi_{ri}^{C}+\pi_{mi}^{C}+\frac{D_{i}^{C}{}^{2}}{2} -\rho_{i}^{C}D_{i}^{C}. \tag{3}\]  
The government's objective function includes the influence of profits for each member, the social aspect, i.e., consumer surplus (CS), and the total amount of the subsidy provided by the government organization [67; 68]. The CS is the area of the demand curve below a given price, which can be expressed as \(\frac{D_{i}^{C^{2}}}{2}\). Optimal decisions in Policy C under the MS and RS games are presented in Lemma 1,2, respectively. The detail derivations of optimal decisions are presented in Appendix A,B, respectively. Additional notations used throughout the study are presented at the end of Appendix A.  
Lemma 1: _Optimal decision in Scenario MC are obtained as follows: \(p_{m}^{C}=\frac{6(a-c_{m})\kappa\lambda_{2}}{\lambda_{1}}\); \(w_{m}^{C}=\frac{(aN_{2}-2c_{m})\lambda_{2}-YZ\kappa}{\lambda_{1}}\); \(p_{m}^{C}=\frac{(aN_{3}-4c_{m})\kappa\lambda_{2}-YZ\kappa}{\lambda_{1}}\); \(\theta_{m}^{C}=\frac{(a-c_{m})\kappa Z}{\lambda_{1}}\); \(\tau_{m}^{C}=\frac{(a-c_{m})X\lambda_{2}}{\lambda_{1}}\); \(\pi_{mm}^{C}=\frac{(a-c_{m})^{2}\kappa\lambda_{2}(M_{1}\kappa+N_{3}\lambda_{2})} {\Delta_{1}}\); \(\pi_{mm}^{C}=\frac{4(a-c_{m})^{2}\kappa\lambda_{2}^{2}}{\Delta_{1}}\); \(\pi_{gm}^{C}=\frac{(a-c_{m})^{2}\kappa\lambda_{2}}{\lambda_{1}}\); \(Q_{m}^{T}=\frac{2(a-c_{m})\kappa\lambda_{2}}{\lambda_{1}}\), where \(\Delta_{1}=M_{1}\kappa-X^{2}\lambda_{2}\)._  
Lemma 2: _Optimal decision in Scenario RC are obtained as follows: \(p_{r}^{C}=\frac{(a-c_{m})\kappa(M_{1}\kappa+N_{2}\lambda_{2})}{\Delta_{1}}\); \(w_{r}^{C}=\frac{aN_{1}\lambda_{2}+YZ\kappa}{\lambda_{1}}\); \(p_{r}^{C}=\frac{(a-c_{m})M_{1}\kappa-c_{m}N_{1}\lambda_{2}+YZ\kappa}{\lambda_ {1}}\); \(\theta_{r}^{C}=\frac{(a-c_{m})Z\kappa}{\Delta_{1}}\); \(\tau_{r}^{C}=\frac{(a-c_{m})X\lambda_{2}}{\Delta_{1}}\); \(\frac{r_{mr}^{C}}{\tau_{mr}^{C}}=\frac{(a-c_{m})^{2}\kappa\lambda_{2}(2\kappa \lambda_{2}+\Delta_{1})}{\Delta_{1}}\); \(\pi_{rr}^{C}=\frac{2(a-c_{m})^{2}\kappa\lambda_{2}(2\kappa\lambda_{2}+\Delta_ {1})}{\Delta_{1}}\); \(\pi_{gr}^{C}=\frac{(a-c_{m})^{2}\kappa\lambda_{2})}{\Delta_{1}}\); \(Q_{m}^{C}=\frac{2(a-c_{m})\kappa\lambda_{2}}{\Delta_{1}}\)._  
The concavity of profit functions for CLSC members and SW for the government in Scenarios MC and RC is ensured by condition \(\Delta_{1}>0\) and \(4\kappa>X^{2}\), respectively. It is found that feasible values of GLs (\(\theta_{m}^{C}\) and \(\theta_{r}^{C}\)) of the product exists if \(\beta>\lambda_{1}\). Therefore, if per unit investment efficiency for the manufacturer is too low, but consumer sensitivity with green product is less, then the manufacturer cannot produce MDIGPs. The unit cost of the product increased with \(\lambda_{1}\); in this circumstance, the manufacturer cannot compensate increasing cost. Therefore, results make sense. By comparing optimal decisions between the MS and RS games, the following theorem is proposed. Theorem 1: _In Policy C_  
1.