## 3 Mathematical Model  
This section is dedicated to developing a mathematical formulation allowing the optimization of the pricing and remanufacturing decisions for the manufacturer, the pharmacy and the online selling platform. The proposed mathematical model aimed to maximize the profit of each entity in the decentralized dual-channel supply chain specialized in the manufacturing and the sale of pharmaceutical products described above. Thus, the demands of products in both distribution channels are linear functions of the market share and the proposed prices [18,28] and are illustrated as follows:  
\[D_{p}(t)=(1-\rho)\cdot A(t)-\alpha_{p}\cdot P_{p}(t)+\beta\cdot P_{o}(t) \tag{1}\]  
\[D_{o}(t)=\rho\cdot A(t)-\alpha_{o}\cdot P_{o}(t)+\beta\cdot P_{p}(t) \tag{2}\]  
As presented in Equations (1) and (2), for each period, the pharmacy demand \(D_{p}(t)\) and the online demand \(D_{o}(t)\) depend on the market share and the proposed selling prices in both channels. For example, the market share of the pharmacy at period \(t\) is equal to the customer preference for purchasing the product directly from the pharmacy \((1-\rho)\)  
Figure 1: The considered dual-channel supply chain. multiplied by the market size at the same period \(A(t)\), while the online market share is equal to \(\rho\cdot A(t)\). In addition, the pricing strategy of both distribution channels impacts the demand of each channel. Hence, at period \(t\), the pharmacy demand \(D_{p}(t)\) fluctuates following the changes in the pharmacy's pricing decisions multiplied by its self-price sensitivity \(\alpha_{p}\), as well as following the prices proposed by the online platform according to the sensitivity of the demand to the other channel's proposed prices \(\beta\). Thus, the pharmacy's demand decreases by \(\alpha_{p}\) customers if its proposed price \(P_{p}(t)\) at period \(t\) increases by \(1\) money unit (\(mu\)) and drops by \(\beta\) customers whenever the online platform decreases its proposed price \(P_{o}(t)\) by \(1\)\(mu\). In practical terms, attracting customers in a competitive environment heavily relies on the effectiveness of the pricing strategy. Conversely, the demand of the online platform \(D_{o}(t)\) was obtained by implementing the same logic. In this work, the following assumptions were considered for modeling the studied problem:  
1. Only the pharmacy (P) provides AS for returned products sold in both channels in return for a commission on each sale from the online platform (O). The manufacturer (M) determines the unit sales commission \(C_{o}\) between both channels
2. All the price sensitivity parameters \(\alpha_{p}\), \(\alpha_{o}\) and \(\beta\) are positive and independent. Also, the self-price sensitivity of the pharmacy, \(\alpha_{p}\), is higher than the online self-price sensitivity \(\alpha_{o}\) due to the AS provided by the pharmacy. 3. For each period, the production process could include a ratio \(\tau(t)\) of remanufactured subcomponents in the manufacturing of new products
4. All the new products either entirely manufactured from transforming raw materials, containing remanufactured subcomponents and fully obtained from remanufacturing used products are considered identical
5. Even considering a decentralized configuration, the manufacturer (M), the pharmacy (P) and the online platform (O) operate under the assumption of possessing perfect information regarding the decisions of the other members of the supply chain. Their objective is to maximize their own profits. According to assumption 5, knowing the pricing decisions of the pharmacy and the manufacturer, the managers of the online platform propose their online selling prices that maximize the platform's profit presented in Equation (3) below:  
\[\pi_{O}(P_{o}(t))=\sum\limits_{t=1}^{H}[((1-C_{o})\cdot P_{o}(t)-W_{o}(t))\cdot D _{o}(t)] \tag{3}\]  
The total profit \(\pi_{O}\) of the online platform is the revenues generated during the planning horizon \(H\) by the activity of selling products.