## 2 Problem Description and Related Assumptions
### Symbol Introduction  
**Assumption 1**.: _The unit manufacturing cost of the new product is \(c\); \(\Delta_{1}\) is the remanufactured product unit cost savings; \(\Delta_{2}\) is the renewal product unit cost savings. Among them, \(c>(c-\Delta_{1})>(c-\Delta_{2}),\dfrac{(c-\Delta_{1})}{c}\in[40\%,60\%]\)._  
**Assumption 2**.: _The wholesale price of refurbished products, remanufactured products and new products is \(\omega_{i}\); the sales price is \(p_{i}\); the quantity of demand (i.e., production) is \(q_{i},i\in\{n,r,s\}\); \(n\) indicates a new product; \(r\) indicates remanufactured products; \(s\) indicates renovated products. The profit function is \(\Pi_{j},j\in\{S,R,M\}\); \(S\) is the overall system; \(R\) is the retailer; \(M\) is the manufacturer._  
**Assumption 3**.: _Let the product recovery and remanufacturing cost be \(F\)._  
**Assumption 4**.: _According to the study by [PERSON] et al. [23], the \"services\" that express the ability to satisfy random demands are quantified as the primary indicators for measuring customer satisfaction. Let \(s_{i}\) be the service level; subsequently, the service cost is expressed as \(\frac{1}{2}\eta s_{i},i\in\{n,r,s\}\) where \(\eta\) represents the service cost influencing factors._  
**Assumption 5**.: _The revenue-sharing ratio is \(\delta\), and the channel fee is set to \(S\)._  
**Assumption 6**.: _Assuming a constant market size of \(Q\), consumers will pay for a new product and obey the uniform distribution of \(U\in\{0,1\}\). Let the consumer's preference coefficient be \(\theta,\theta\in(0,1)\); the willingness of the consumer to pay for remanufactured products is \(\alpha_{n}\theta_{1}\), and the willingness to pay for refurbished products is \(\alpha_{r}\theta_{2}\). In addition to price and service levels, government policy subsidies also affect consumer demand. The government is designed to subsidize consumers to purchase refurbished products. The subsidy value is \(g\)._  
According to [PERSON] et al. [24], when new products, remanufactured products and refurbished products are simultaneously present in the market, consumers will use self-selection models to compare the magnitude of consumer utility and decide which products to buy (Figure 2). For example: Let \(\theta_{1}=\theta_{2}\) to obtain the consumer's utility function for new products as \(\mu_{n}=\alpha_{n}-p_{n}\), and the utility function for remanufactured products and refurbished products is \(\mu_{i}=\theta\alpha_{i}-p_{i}\), \((i\in\{r,s\})\). Hence, we can obtain the distribution map of the consumers' needs under different strategies.  
Figure 1: (**a**) Centralized decision model. (**b**) Decentralized M-R decision model. (**c**) Decentralized R-M decision model.  
According to the literature, when satisfied,  
\[\left\{\begin{array}{l}p_{n}-(1-\theta)<p_{r}<\theta p_{n}\\ p_{n}-(1-\theta^{2})<p_{s}<\theta^{2}p_{n}\end{array}\right. \tag{1}\]  
meaning new product, remanufactured product and refurbished product requirements exist in the market. The consumer demand function can be expressed as:  
\[q_{n}=\int\!\!\!\frac{(p_{n}-s_{n})-(p_{r}-s_{r})}{1-\theta}\;1\,d\alpha \tag{2}\]  
\[q_{r}=\int\!\!\!\frac{(p_{n}-s_{n})-(p_{r}-s_{r})}{1-\theta}\;1\,d\alpha \tag{3}\]  
\[q_{s}=\int\!\!\!\frac{(p_{r}-s_{r})-(-g+p_{s}-s_{s})}{\theta(1-\theta)}\;1\,d\alpha \tag{4}\]  
The superscript \"\(c\)\" represents a centralized decision model; \"\(d1\)\" represents a decentralized M-R decision model; \"\(d2\)\" represents a decentralized R-M decision model; \"\(r1\)\" represents a coordination mechanism M-R decision model; \"\(72\)\" represents a coordination mechanism R-M decision model; and \"\(*\)\" indicates the optimal decision. To simplify the model display, we define: \(\lambda=c+Q\), \(\phi=\theta-1\), \(\gamma=16\eta^{2}\theta^{2}\phi^{2}\).