# Competing Reverse Channels' Performance with Sustainable Recycle Innovation Input
## 4 Model Formulation and Solution
### Basic System with Innovation Input (Model BE)  
Next, in Figure 2, we consider a similar basic supply chain structure consisting of a manufacturer and a collector, in which the collector invests in innovation input in the process of collection. As mentioned in the above assumptions, the demand function of the market can be expressed as, \(D^{BE}(p,e)=\alpha-\beta p+Ke\).  
The collector's profit function of the model \(BE\) can be written as,  
\[\Pi_{\mathbb{C}}^{BE}\left(\tau,e\right)=b\tau D^{BE}(p,e)-A\tau D^{BE}(p,e)- \frac{H}{2}\tau^{2}-\frac{R}{2}e^{2} \tag{7}\]  
From the concavity of the objective functions, the best response functions are determined by the simultaneous solution of first-order conditions for \(\tau^{BE},e^{BE}\).  
The optimal product return rate is \(\tau^{BE}\left(b,p\right)=\frac{R(b-A)(\alpha-\beta p)}{RH-K^{2}(b-A)^{2}}\). The optimal innovation input is \(e^{BE}\left(b,p\right)=\frac{K(b-A)^{2}(\alpha-\beta p)}{RH-K^{2}(b-A)^{2}}\).  
The manufacturer's profit function can be written as,  
\[\pi_{M}^{BE}(p,b)=(p-c_{m}+(c_{m}-c_{r}-b\tau)D^{BE}(p,e) \tag{8}\]  
Figure 2: The CLSC structure of model \(BE\).  
**Theorem 2**.: _We substitute the obtained the closed-form solutions of \(\tau^{BE}\) and \(e^{BE}\) into the manufacturer's profit function (8) and jointly optimize over the manufacturer's decisions (\(p\), \(b\))._  
\[b^{BE*}= \frac{\beta R(\Delta+A)-2K^{2}A}{2\beta R-2K^{2}} \tag{9}\] \[p^{BE*}= \frac{\left(\alpha+\beta c_{m}\right)M-2\alpha\beta RN\right.}{2 \beta M-2\beta^{2}RN} \tag{10}\]  
_where \(M=RH-\frac{K^{2}\beta^{2}R^{2}\left(\Delta-A\right)^{2}}{4\left(\beta R-K^{2 }\right)^{2}}\), \(N=\frac{\beta R\left(\beta R-2K^{2}\right)\left(\Delta-A\right)^{2}}{4\left( \beta R-K^{2}\right)^{2}}\). Substituting \(b^{BE*}\), \(p^{BE*}\) into the collector's profit function, then the optimal decisions are given as follows,_  
\[\tau^{BE*}= \frac{\beta R}{\beta R-K^{2}}*\frac{R\left(\Delta-A\right)\left( \alpha-\beta c_{m}\right)}{4M-4\beta RN} \tag{11}\] \[e^{BE*}= \frac{K}{2}\frac{\left[\beta R\right]^{2}}{\left[\beta R-K^{2} \right]^{2}}\frac{\left(\Delta-A\right)^{2}\left(\alpha-\beta c_{m}\right)}{4M -4\beta RN} \tag{12}\]  
_Thus, the optimal profit functions can be obtained as,_  
\[\Pi_{M}^{BE*}(p,b)= \frac{RH\left(\alpha-\beta c_{m}\right)^{2}}{\beta\left(4M-4\beta RN \right)}\] \[\Pi_{C}^{BE*}(p,b)= \left[\frac{H}{2}-\frac{R}{8}\left(\frac{\beta K(\Delta-A)}{ \left(\beta R-K^{2}\right)}\right)^{2}\right]\left[\frac{R*\beta R(\Delta-A) }{\beta R-K^{2}}*\frac{\left(\alpha-\beta c_{m}\right)}{4M-4\beta RN}\right] ^{2}\] \[\Pi_{T}^{BE*}(p,b)= \frac{RH\left(\alpha-\beta c_{m}\right)^{2}}{\beta\left(4M-4 \beta RN\right)}+\left[\frac{H}{2}-\frac{R}{8}\left(\frac{\beta K(\Delta-A) }{\left(\beta R-K^{2}\right)}\right)^{2}\right]\left[\frac{R*\beta R(\Delta- A)}{\beta R-K^{2}}*\frac{\left(\alpha-\beta c_{m}\right)}{4M-4\beta RN} \right]^{2}\]  
**Lemma 2**.: _If the condition of \(H>\frac{\beta\left(\Delta-A\right)^{2}}{4\left(\beta R-K^{2}\right)}\left[ \frac{R\left(\alpha-\beta c_{m}\right)}{\left(\Delta-A\right)}+\beta R\right]\) is satisfied, then \(\tau^{BE*}=\frac{\beta R}{\beta R-K^{2}}*\frac{R\left(\Delta-A\right)\left( \alpha-\beta c_{m}\right)}{4M-4\beta RN}\) is between 0 and 1._  
From the above analysis, after the integration of innovative investment, it can be seen that the threshold of the cost coefficient of recycling efforts \(H\) has increased compared with that in the basic model \(B\). In other words, when recyclers invest on the innovation, they also increase the entry barriers to the innovational recycling.  
**Lemma 3**.: _When \(\Delta>b\), we can get \(R>\frac{2K^{2}}{\beta}\). In this case, it means that the threshold of the recycling innovation cost coefficient is raised, that is, the barriers to recycling innovation are raised. Under the current conditions, with the increase of \(b\), it also indirectly improves the difficulty of recycling efforts. Such situations are more in line with the actual situation._