## 3 Mathematical Model
In fact, for each period \(t\), the demand of the online platform is determined according to the proposed online prices \(P_{o}(t)\) and by the pharmacy \(P_{p}(t)\). In addition, for each period \(t\), the online platform acquires the product from the manufacturer for a unit wholesale price \(W_{o}(t)\) and sells it to the customer for a unit selling price \(P_{o}(t)\). Also, in return for AS, the online platform pays the pharmacy a commission for each sale equals to \(C_{o}\cdot P_{o}(t)\). Thus, the online selling prices are determined by ensuring that the sale activity of products is still profitable after paying the wholesale price to the manufacturer and the commission to the pharmacy. Therefore, the optimal total profit of the online platform is obtained by solving the following optimization problem:  
\[\begin{split} Max\ \pi_{O}(P_{o}(t))=\sum\limits_{t=1}^{H}[((1-C_{o}) \cdot P_{o}(t)-W_{o}(t))\cdot D_{o}(t)]\\ \text{  
Also, according to assumption 5, having perfect information regarding the pricing strategies of the online platform and the manufacturer, the pharmacy establishes its pricing decisions in order to maximize its own profit.  
\[\pi_{P}\big{(}P_{p}(t)\big{)}=\sum\limits_{t=1}^{H}\big{[}\big{(}P_{p}(t)-W_{p}( t)\big{)}\cdot D_{p}(t)+C_{o}\cdot P_{o}(t)\cdot D_{o}(t)-\big{(}R_{p}\cdot D_{p}(t)+R_{o} \cdot D_{o}(t)\big{)}\cdot\textit{Ulas}\big{]} \tag{5}\]  
The pharmacy's total profit \(\pi_{P}\) is equal to the combined revenues of directly selling products to customers and the obtained commission on each sold product in the online channel minus the costs generated by the pharmacy's AS related to the treatment of returned products already sold in both channels. Adding to that, the pharmacy demand \(D_{p}(t)\) is also determined according to the selling prices \(P_{p}(t)\) and \(P_{o}(t)\) proposed, respectively, by the pharmacy and the online platform. Similar to the online platform, the pharmacy acquires the product from the manufacturer for a unit wholesale price \(W_{p}(t)\), sells it to the customer for a unit selling price \(P_{p}(t)\) and obtains a commission \(C_{o}\cdot P_{o}(t)\) for each product sold online. Hence, the pharmacy determines its pricing strategy while guaranteeing the profitability of the sale activity after paying the wholesale price \(W_{p}(t)\) to the manufacturer. Therefore, the pharmacy's optimal total profit is the solution to the following optimization problem:  
\[ \begin{array}{c}Max\ \pi_{P}\big{(}P_{p}(t)\big{)}=\sum\limits_{t=1}^{H} \begin{bmatrix}\big{(}P_{p}(t)-W_{p}(t)\big{)}\cdot D_{p}(t)+P_{o}(t)\cdot C_{o }\cdot D_{o}(t)\\ -\big{(}R_{p}\cdot D_{p}(t)+R_{o}\cdot D_{o}(t)\big{)}\cdot\textit{Ulas}\\ \textit{subject to}\\ D_{p}(t)\geq 0\\ P_{p}(t)\geq W_{p}(t)\end{bmatrix}\end{array}  \tag{6}\]  
On the other hand, the manufacturer's total profit is as follows:  
\[\pi_{M}\big{(}W_{p}(t),W_{o}(t),\tau(t)\big{)}=\sum\limits_{t=1}^{H} \begin{bmatrix} W_{p}(t)\cdot D_{p}(t)+W_{o}(t)\cdot D_{o}(t)-B\cdot\tau(t)^{2}/2\\ -\big{(}D_{p}(t)+D_{o}(t)\big{)}\cdot(\textit{Mc}-\delta\cdot\tau(t))\end{bmatrix}  \tag{7}\]  
The total profit \(\pi_{M}\) of the manufacturer is the sum during the planning horizon \(H\) of the revenues minus the cost of the different activities. The manufacturer generates revenues from selling the product to the pharmacy and the online platform with unit wholesale prices \(W_{p}(t)\) and \(W_{o}(t)\), respectively. However, the costs borne by the manufacturer are related to the manufacturing process of the demand of both channels with a unit manufacturing cost of \(\textit{Mc}-\delta\cdot\tau(t)\) and the costs related to the remanufacturing decisions. The remanufacturing constraint is as follows:  
\[0\leq\tau(t)\leq 1 \tag{8}\]  
Inequality (8) illustrates the boundaries of the remanufacturing process. Based on assumption 3, \(\tau(t)\) represents the ratio of remanufactured subcomponents to the totality of the subcomponents necessary or the manufacturing of a new product.