## 3 Mathematical Model
Therefore, the ratio of the used remanufactured subcomponents is a real number between 0 and 1. For example, if a new product is obtained entirely from transforming raw materials, the remanufacturing rate \(\tau(t)\) is equal to 0. However, the remanufacturing rate \(\tau(t)\) of the new products produced during period \(t\) from entirely assembling remanufactured subcomponents is equal to 1. On the other hand, at the end of period \(t\), new products containing transformed raw materials and remanufactured subcomponents from used products already collected are characterized by a remanufacturing rate \(0<\tau(t)<1\). In this decentralized dual-channel supply chain, following the Stackelberg game theory, the manufacturer, as a leader, determines its wholesales pricing strategy \(W_{p}(t)\) and \(W_{o}(t)\), and then the pharmacy and the online platform propose their selling prices \(P_{p}(t)\) and \(P_{o}(t)\), respectively. However, as presented above, knowing the pricing decisions of both distribution channels is obligatory for determining the pricing strategy of the manufacturer and vice versa. Therefore, backward induction is used to calculate the optimal unit selling prices of one channel for the given pricing decisions of the other channel and the manufacturer. Considering the online channel, the first- and second-order partial derivatives of the online total profit are presented below. \[\frac{\partial(\pi_{O}(P_{o}(t)))}{\partial P_{o}(t)}=\sum_{t=1}^{H}\bigl{[}(1-C_ {o})\bigl{(}-2\alpha_{o}\cdot P_{o}(t)+\rho\cdot A(t)+\beta\cdot P_{p}(t)\bigr{)} +\alpha_{o}\cdot W_{o}(t)\bigr{]} \tag{9}\]  
\[\frac{\partial^{2}(\pi_{O}(P_{o}(t)))}{\partial^{2}P_{o}(t)}=-2H\cdot\alpha_{o }(1-C_{o})<0 \tag{10}\]  
As illustrated in Equation (10), the second-order derivative of the total profit of the online platform \(\pi_{O}(P_{o}(t))\) is strictly negative. Consequently, the online total profit during the planning horizon is concave with respect to the proposed online prices \(P_{o}(t)\). Given the wholesale prices of the manufacturer \(W_{o}(t)\) and the selling prices of the pharmacy \(P_{p}(t)\), the optimal responsive online pricing decisions allowing us to maximize its own profit \(\pi_{O}(P_{o}(t))\) (assumption 5) are as follows:  
\[\sum_{t=1}^{H}{P_{o}}^{*}(t)=\frac{1}{2\alpha_{o}}\sum_{t=1}^{H}\biggl{[}\rho \cdot A(t)+\beta\cdot P_{p}(t)+\frac{\alpha_{o}}{1-C_{o}}W_{o}(t)\biggr{]} \tag{11}\]  
The pricing constraint (11) related to the optimal online proposed prices is obtained by setting the first-order derivative Equation (9) to zero and solving it. On the other hand, as for the online platform, the first- and second-order partial derivatives of the total profit generated by the pharmacy during the planning horizon \(H\) are presented in the following equations:  
\[\frac{\partial\bigl{(}\pi_{P}(P_{p}(t))\bigr{)}}{\partial P_{p}(t)}=\sum_{t=1 }^{H}\biggl{[}(1-\rho)\cdot A(t)-2\alpha_{p}\cdot P_{p}(t)+\alpha_{p}\bigl{(} W_{p}(t)+R_{p}\cdot\mathit{Uas}\bigr{)}\biggr{]} \tag{12}\]  
\[\frac{\partial^{2}(\pi_{P}(P_{p}(t)))}{\partial^{2}P_{p}(t)}=-2H\cdot\alpha_{p }<0 \tag{13}\]  
Similar to the online platform, the total profit of pharmacy \(\pi_{P}(P_{p}(t))\) is a concave function with respect to the proposed prices \(P_{p}(t)\) set by the pharmacy, since the second-order derivative of the total profit of the pharmacy illustrated in Equation (13) is strictly negative. Consequently, the pharmacy's optimal pricing strategy allowing it to maximize its own total profit \(\pi_{P}(P_{p}(t))\) in response to the manufacturer's wholesale prices \(W_{p}(t)\) and to the online selling prices \(P_{o}(t)\) is illustrated below:  
\[\sum_{t=1}^{H}{P_{p}}^{*}(t)=\frac{1}{2\kappa_{p}}\sum_{t=1}^{H}\biggl{[} \begin{array}{c}(1-\rho)\cdot A(t)+\alpha_{p}\bigl{(}W_{p}(t)+R_{p}\cdot \mathit{Uas}\bigr{)}\\ +\beta((1-C_{o})\cdot P_{o}(t)-R_{o}\cdot\mathit{Uas})\end{array}