# Competing Reverse Channels' Performance with Sustainable Recycle Innovation Input
## 4 Model Formulation and Solution
### Basic System without Innovation Input (Model B)  
Now, we start with the simple supply chain described above, including one collector and one manufacturer. Figure 1 shows this basic supply chain system structure. As the basic model, we first study the vertically integrated manufacturer that directly sells its product to consumers and one collector without innovation input \(e\).  
The demand function of the basic system without innovation input (model \(B\)) is given as:  
\[D^{B}(p)=\alpha-\beta p \tag{3}\]  
Many manufacturers play as leaders in the respective supply chains. They have strong market power to influence the decisions of other supply chain participants. As such, we consider that the integrated manufacturer has the power to determine the retail price and the transfer price. At first, the manufacturer determines the transfer price \(b\) and the selling price \(p\). Afterward, the collector determines the return rate of used product \(\tau\).  
The profit function of the collector of the basic model (model \(B\)), \(\Pi_{\mathbb{C}}^{B}\) (the superscript \(B\) representing basic model without innovation input \(e\) and the subscript \(C\) representing the collector), is as follows,  
\[\Pi_{\mathbb{C}}^{B}(\tau)=b\tau D(p)-A\tau D(p)-\frac{H}{2}\tau^{2} \tag{4}\]  
Solving the stage two optimization problem first, the collector's response function can be obtained as,  
\[\tau(p,b)=\frac{(b-A)(\alpha-\beta p)}{H} \tag{5}\]  
Further, the profit function of the manufacturer (denoted with a subscript \(M\) ) of model \(B\) is as follows,  
\[\Pi_{M}^{B}(p,b) = \left[p-c_{m}+\left(c_{m}-c_{r}-b\right)\tau\right]D^{B}(p) \tag{6}\] \[= \left[p-c_{m}+\left(\Delta-b\right)\tau\right]D^{B}(p)\]  
where \(\Delta=c_{m}-c_{r}\).  
The following proposition describes the findings in the basic model without innovation input \(e\).  
**Theorem 1**.: _To solve the manufacturer's problem, we substitute the obtained response function (5) into the manufacturer's profit function (6) and jointly optimize over the manufacturer's decisions (\(p\), \(b\)). It can be obtained that \(b^{B*}=\frac{\Delta+A}{2}\), and \(p^{B*}=\frac{\alpha+\beta c_{m}-2\beta\frac{\left(\Delta-A\right)^{2}}{\beta -2\beta\left(\Delta-A\right)^{2}}}{2\beta-2\beta\left(\beta\right)\frac{ \left(\Delta-A\right)^{2}}{\beta-2\beta\left(\beta\right)\frac{\left(\Delta-A \right)^{2}}{\beta-2\beta\left(\beta\right)}}}\). Based on the optimal decisions, the collector's collection  
Figure 1: The closed-loop supply chain (CLSC) structure of model \(B\).  
effort \(\tau\) can also be obtained as, \(\tau^{\mathbb{B}*}=\frac{(\Delta-A)(\alpha-\beta\epsilon_{m})}{4H-\beta(\Delta-A )^{2}}\). The optimal profits can be obtained as, \(\Pi_{M}^{\mathbb{B}}(p,b)=\frac{H(\alpha-\beta\epsilon_{m})^{2}}{\beta[4H- \beta(\Delta-A)^{2}]},\Pi_{\mathbb{C}}^{\mathbb{B}}(\tau)=\frac{H}{2}\left[ \frac{(\Delta-A)(\alpha-\beta\epsilon_{m})}{4H-\beta(\Delta-A)^{2}}\right]^{2}\), and \(\Pi_{\mathbb{C}}^{\mathbb{B}}(\tau)+\Pi_{M}^{\mathbb{B}}(p,b)=\frac{H(\alpha- \beta\epsilon_{m})^{2}[8H-\beta(\Delta-A)^{2}]}{2\beta[4H-\beta(\Delta-A)^{2 }]^{2}}\)._  
**Lemma 1**.: _If the condition of \(H>\frac{\beta(\Delta-A)^{2}+(\Delta-A)(\alpha-\beta\epsilon_{m})}{4}\) is satisfied, then \(\tau^{B*}=\frac{(\Delta-A)(\alpha-\beta\epsilon_{m})}{4H-\beta(\Delta-A)^{2}}\) is between 0 and 1._  
From the above analysis, it can be seen that the cost coefficient of recycling efforts \(H\) has a minimum value. Below the minimum value does not meet the actual situation, nor does it conform to the assumptions of this article. In other words, if the recycling cost factor \(H\) of recycling efforts is small, that is to say, the difficulty of recycling is small and the recycling rate may exceed \(1\). It is also reflected from the negative side that the recycling cost in real life is high and the recycling difficulty is great. On the other hand, as the transfer price rises, it will indirectly increase the difficulty of recovery.