## 2 Problem Setting  
In this section, a detailed presentation of the dual-channel chain with reverse logistics dedicated to the selling of pharmaceutical products is presented. In this paper, the studied system included a manufacturer and two distribution channels allowing the sale of pharmaceutical products represented by a pharmacy and an online platform. Also, the pharmacy offered after-sales services (AS) for products purchased in both distribution channels. Moreover, this work took into consideration the different flows existing along the supply chain such as manufacturing, remanufacturing, collection and after-sales activities. The parameters below were employed for presenting, describing and developing a mathematical model for the studied pricing and remanufacturing problem within the considered dual-channel supply chain. \begin{tabular}{l l} \(\alpha_{p}\) & Self-price sensitivity of the pharmacy's demand \\ \(\alpha_{o}\) & Self-price sensitivity of online demand \\ \(\beta\) & Sensitivity of the demand in one channel to price changes in the other channel \\ \(R_{p}\) & Proportion of the demand sold by the pharmacy returned to AS \\ \(R_{o}\) & Proportion of the demand sold online returned to AS \\ \(Uas\) & Unit processing cost of a returned product \\ \(C_{o}\) & Unit sales commission between both distribution channels \\ \end{tabular}   
Adding to that, the mathematical model was developed based on the decision variables illustrated below. \begin{tabular}{l l} \(P_{p}(t)\) & Product's unit selling price proposed by the pharmacy in period \(t\) \\ \(P_{o}(t)\) & Product's unit selling price proposed by the online platform in period \(t\) \\ \(W_{p}(t)\) & Product's unit wholesale price from the manufacturer to the pharmacy during period \(t\) \\ \(W_{o}(t)\) & Product's unit wholesale price from the manufacturer to the online platform during \\ period \(t\) \\ \(D_{p}(t)\) & Pharmacy's demand during period \(t\) \\ \(D_{o}(t)\) & Demand of the online platform during period \(t\) \\ \(\tau(t)\) & Remanufacturing rate during period \(t\) \\ \end{tabular}   
Also, the following notations were used to separate the profits of the different actors of the dual channel supply chain. \begin{tabular}{l l} \(\pi_{M}\) & Total profit of the manufacturer during the planning horizon \\ \(\pi_{p}\) & Total profit of the pharmacy during the planning horizon \\ \(\pi_{O}\) & Total profit of the online platform during the planning horizon \\ \end{tabular}   
This paper studied the pricing and remanufacturing decisions for a dual-channel supply chain with reverse logistics specialized in the manufacturing and the sale of pharmaceutical products. The considered supply chain was composed of a manufacturer (M), an online selling platform (O) and a pharmacy (P). The forward logistics examined in this paper were the flows of the manufacturing of new products, the remanufacturing of used products and the sales via both distribution channels, while the reverse logistics were represented by the collection flow of used products for remanufacturing activities and the processing of unsatisfied clients by the pharmacy's AS. In practice, a variety of pharmaceutical products could be remanufactured such as medicine tablets, syrup bottles, sprays, inhalers, etc. In addition, the manufacturer (M) ensured the manufacturing of the demands of the pharmacy \(D_{p}(t)\) and the online platform \(D_{o}(t)\), for each period \(t\). However, new products were manufactured from the direct transformation of raw materials and from the integration of remanufactured parts and subcomponents. These subcomponents were obtained from remanufacturing used products already collected after the end of their life cycle. Hence, during period \(t\), the ratio of remanufactured subcomponents to the totality of the components used in the manufacturing of each new product was \(\tau(t)\). As a consequence, the unit manufacturing cost borne by the manufacturer for each period \(t\) depended on the remanufacturing rate \(\tau(t)\) in the same period.