# Competing Reverse Channels' Performance with Sustainable Recycle Innovation Input
## 3 Problem Description and Assumption
### Assumption  
In this reverse channel model, collected used products are transferred to the manufacturer by the collectors in return of a fixed per-unit buy-back payment, \(b_{i}\). Subscript \(i\) takes the values of 1 and 2, denoting the collector 1 and collector 2, respectively. Given the retail price \(p\) and the buy-back payment \(b_{1}\) and \(b_{2}\), each collector \(i\) (\(i=1,2\)) solves for the optimal innovation input \(e\) and the collection effort \(\tau_{i}\), taking into consideration the competition from the other collectors. The manufacturer chooses the retail price \(p\) and the buy-back price \(b\), taking into consideration the effect of these decisions on the strategic decisions of the collectors. Consistent with the extant literature ([7, 8]), we assume that in that model, the manufacturer has sufficient channel power over the collectors to act as a Stackelberg leader, and the collectors compete in a Bertrand pricing game. We characterize the supply chain performance for each model in terms of the pricing decisions of the manufacturer and collectors, the collection effort, total supply chain profits, and the allocation of total profits between the manufacturer and the collectors. In the rest of the paper, the following notation is used: \(c_{m}\) denotes the unit cost of manufacturing a new product, and \(c_{r}\) the unit cost of remanufacturing a returned product into a new one. \(\Pi_{j}\) denotes the profit function for channel member \(j\). Subscript \(j\) takes the values of \(M\), \(C\), \(C_{1}\), and \(C_{2}\), denoting the manufacturer, the collector, the collector 1, and collector 2, respectively. While the manufacturer takes back the used product from collector \(i\) at a price \(b_{i}(i=1,2)\). \(D(p,e)\) denotes the demand at market as a function of the selling price \(p\) and collector's innovation input \(e\). \(R\) denotes cost coefficient of recycling innovation input. Due to the vertical externality brought by the recycling innovation input, we assume that \(e\) plays a positive role in the demand of consumers. We denote the returned quantity as \(\tau D(p,e)\), and \(0<\tau<1\), where \(\tau\) represents the reverse channel performance and it is a function of the product collection effort. \(\tau\) also denotes the fraction of the current generation products remanufactured from the returned units [13]. We assume the market demand function as a deterministic linear function of the retail price \(p\) and the innovation input \(e\). \[D(p,e)=\alpha-\beta p+Ke \tag{1}\]  
where \(\alpha>0\), \(\beta>0\), the constant \(\alpha\) indicates the market potential while the coefficient \(\beta\) represents the sensitivity of demand to price changes. \(K>0\), and \(K\) represents the demand expansion effectiveness coefficient of the innovation input by the collector, also the coefficient \(K\) measures the marginal consumer demand with respect to the innovation input \(e\). When \(\beta\geq K\), it means that consumers are more sensitive to the sales price of the product than the innovative input coefficient provided by the collector. When \(\beta<K\), it is contrary to the above statement. \(p\) represents the retail price of the unit determined by the manufacturer, and \(e\) represents the recycling innovation investment of the collector \(\alpha>\beta p\), that is, this is a rational market, and reasonable pricing controls the basis of the general market. This form of demand function has been adapted separately by the research community ([34, 37, 38]). The average unit cost of manufacturing can hence be written as \(c=c_{m}(1-\tau_{i})+c_{r}\tau_{i}\) (c.f., [3,13]). We assume that \(\alpha-\beta c_{m}>0\). The reverse channel performance is characterized by \(\tau\), the return rate of used products from the customers. \(\tau\) denotes the fraction of the current generation of products supplied from returned and thus remanufactured units, i.e., \(0<\tau<1\).