# Competing Reverse Channels' Performance with Sustainable Recycle Innovation Input
## 4 Model Formulation and Solution
### Hybrid System (Model H)
The manufacturer's profit function is concave and has a unique maximum value._  
_It is straight forward to find that the optimal solutions for the manufacturer's profit function are_  
\[b_{1}^{H*}= \frac{\beta R(\Delta+A)-2K^{2}A}{2\beta R-2K^{2}} \tag{16}\] \[b_{2}^{H*}= \frac{\Delta+A}{2}\] (17) \[p^{H*}= \frac{(\alpha+\beta c_{m})\ M-2\alpha\beta RN-\frac{(\Delta-A)^{ 2}}{2}\alpha\beta R}{2\beta M-2\beta^{2}RN-\frac{(\Delta-A)^{2}}{2}\beta^{2}R} \tag{18}\]  
_where \(M=RH-\frac{K^{2}\beta^{2}R^{2}(\Delta-A)^{2}}{4\left(\beta R-K^{2}\right)^{2}}\), \(N=\frac{\beta R\left(\beta R-2K^{2}\right)(\Delta-A)^{2}}{4\left(\beta R-K^{ 2}\right)^{2}}\)._  
\(b_{2}\) is independent of \(p\), \(b_{1}\), while \(p\) and \(b_{1}\) affect each other. With the above analytical results, we can derive Theorem 4. **Theorem 4**. : \(H>0\)_, \(R>0\), \(RH-\frac{K^{2}\beta^{2}R^{2}(\Delta-A)^{2}}{4\left(\beta R-K^{2}\right)^{2}}>0\) are satisfied, the optimal decisions of collectors are given as follows._  
\[\tau_{1}^{H*}= \frac{\beta R}{\beta R-K^{2}}*\frac{R(\Delta-A)\ (\alpha-\beta c_{m})}{4M-4 \beta RN-\beta R(\Delta-A)^{2}} \tag{19}\] \[\tau_{2}^{H*}= \frac{R(\Delta-A)\ (\alpha-\beta c_{m})}{4M-4\beta RN-\beta R( \Delta-A)^{2}}\] (20) \[e^{H*}= \frac{K}{2}\frac{[\beta R]^{2}}{[\beta R-K^{2}]^{2}}\frac{( \Delta-A)^{2}\ (\alpha-\beta c_{m})}{4M-4\beta RN-\beta R(\Delta-A)^{2}} \tag{21}\]  
_where \(M=RH-\frac{K^{2}\beta^{2}R^{2}(\Delta-A)^{2}}{4\left(\beta R-K^{2}\right)^{2}}\), \(N=\frac{\beta R\left(\beta R-2K^{2}\right)(\Delta-A)^{2}}{4\left(\beta R-K^{ 2}\right)^{2}}\)._  
**Theorem 5**. : _The optimal profit functions can be obtained as,_  
\[\Pi_{C_{2}}^{H}\left(\tau_{2}\right)=b_{2}\tau_{2}D-A\tau_{2}D-\frac{H}{2} \tau_{2}^{2} \tag{14}\]0 \[\Pi_{C_{2}}^{H}\left(\tau_{2}\right)=b_{2}\tau_{2}D-A\tau_{2}D-\frac{H}{2} \tau_{2}^{2} \tag{14}\]1 (23) \[\Pi_{C_{2}}^{H}\left(\tau_{2}\right)=b_{2}\tau_{2}D-A\tau_{2}D-\frac{H}{2} \tau_{2}^{2} \tag{14}\]2  
_where \(M=RH-\frac{K^{2}\beta^{2}R^{2}(\Delta-A)^{2}}{4\left(\beta R-K^{2}\right)^{2}}\), \(N=\frac{\beta R\left(\beta R-2K^{2}\right)(\Delta-A)^{2}}{4\left(\beta R-K^{2} \right)^{2}}\)._  
**Lemma 4**. : _If the condition of \(H>\frac{\beta(\Delta-A)^{2}}{4\left(\beta R-K^{2}\right)}\left(\frac{(\alpha- \beta c_{m})\left(2\beta R-K^{2}\right)}{\beta(\Delta-A)}+2\beta R-K^{2}\right)\) is satisfied, then \(\tau_{1}^{H*}\), \(\tau_{2}^{H*}\), \(\tau_{1}^{H*}+\tau_{2}^{H*}\) are all between \(0\) and \(1\)._Compared with the previous two models, the cost coefficient of the collection effort \(H\) has the highest threshold in model \(H\), that is to say, the system has the highest difficulty. The hybrid system is also much more close to real life, which also reflects the high recycling cost and the difficulty of recycling in real life. In some cases, \(b_{1}^{H*}>\Delta\) will appear. In this case, although it will greatly stimulate the enthusiasm of the collector 1 for recycling, it is difficult for manufacturers to achieve for a long time. Unless the government or the manufacturer participates in the coordination of subsidies, or manufacturers may sacrifice part of their profits in the short term in order to incentivize recyclers to make innovative investments in the short term or change the collector's recycling business model. In this case, the profit of the collector 1 is much higher than the profit of the collector 2. The collector 1 is also more willing to invest in cost recovery to transform and upgrade, and invest in innovation.