# Competing Reverse Channels' Performance with Sustainable Recycle Innovation Input
## 4 Model Formulation and Solution
### Hybrid System (Model H)  
In the hybrid system, there is an integrated manufacturer and two collectors. The structure of the game played between the manufacturer and the collectors in the reverse channels is as follows. In this reverse channel model, collectors engage only in the recovering of the product while the integrated manufacturer participates in both manufacturing and distribution. The integrated manufacturer decides on the retail price of the product \(p\), transfer price \(b_{1}\) (per unit returned) reimbursed to the collector 1 and transfer price \(b_{2}\) (per unit returned) reimbursed to the collector 2. We consider that one of the collectors with innovation input (collector 1) decides on the investment in product collection effort \(\tau_{1}\) as well as the innovation input \(e\) of the recycling process. While the other one without innovation input (collector 2) determines the product collection effort \(\tau_{2}\) by taking into account the retail price charged by the manufacturer and the competition from the other collector 1 with innovationinput. Both collectors reimburse each consumer a fixed fee \(A\), per unit returned. We assume that in the model, the manufacturer has sufficient channel power over the retailers to act as a Stackelberg leader, and the collectors compete in a Bertrand pricing game. Figure 3 shows this hybrid supply chain system structure. When obsoleted products are returned to the collectors directly, the manufacturer has strong market power to influence the decisions of other supply chain members, so he firstly determines the transfer price \(b_{i}\)\((i=1,2)\), the subscript 1, 2 represents collector 1 and collector 2, and the selling price \(p\). Secondly, the collectors determine the optimal return rates of used product \(\tau_{1}\), \(\tau_{2}\) and the optimal innovation input \(e\). For example, the recycling mode of electronic equipment products such as mobile phones is similar to the recycling mode described above. Mobile phones can be recycled from the recycling channels of professional recyclers, from smart recycling bins, and from petty dealers. The profit function of the collector 1 with innovation input \(e\) in the model \(H\) is \(\Pi_{C_{1}}^{H}\)(the superscript \(H\) representing the hybrid system and the subscript \(C_{1}\) representing the collector 1). Given \(p\) and \(b_{1}\), the collector 1 solves the following profit function to determine the investment in product collection effort and the innovation input. \[\Pi_{C_{1}}^{H}\left(\tau_{1},e\right)=b_{1}\tau_{1}D-A\tau_{1}D-\frac{H}{2} \tau_{1}^{2}-\frac{R}{2}e^{2} \tag{13}\]  
From the concavity of the objective functions, the best response functions are determined by the simultaneous solution of first-order conditions for \(\tau_{1}^{H},e^{H}\). The optimal product return rate is \(\tau_{1}^{H}\left(b_{1},p\right)=\frac{R\left(b_{1}-A\right)\left(a-\beta p \right)}{RH-K^{2}\left(b_{1}-A\right)^{2}}\). The optimal innovation input is \(e^{H}\left(b_{1},p\right)=\frac{K\left(b_{1}-A\right)^{2}\left(a-\beta p \right)}{RH-K^{2}\left(b_{1}-A\right)^{2}}\). Hence the collector 2 without innovation input optimizes the following problem (the subscript \(C_{2}\) representing the collector 2). \[\Pi_{C_{2}}^{H}\left(\tau_{2}\right)=b_{2}\tau_{2}D-A\tau_{2}D-\frac{H}{2} \tau_{2}^{2} \tag{14}\]  
From the concavity of the objective function, we can easily show that, \(\tau_{2}^{H}\left(b_{1},b_{2},p\right)=\frac{R\left(b_{2}-A\right)\left(a- \beta p\right)}{RH-K^{2}\left(b_{1}-A\right)^{2}}\). Figure 3: The CLSC structure of model \(H\). Furthermore, the manufacturer's profit function is given below,  
\[\Pi_{M}^{H}(p,b_{1},b_{2})=(p-c_{m})D(p,e)+(c_{m}-c_{r}-b_{1})\tau_{1}D(p,e)+(c_{ m}-c_{r}-b_{2})\tau_{2}D(p,e) \tag{15}\]  
**Theorem 3**. : _When the parameters satisfy \(H>\frac{\beta(\Delta-A)^{2}}{4}*\frac{2\beta R-K^{2}}{\beta R-K^{2}}\), \(\beta R-K^{2}>0\).