## 1 Introduction
Nevertheless, the studies previously mentioned focus solely on single-channel systems, leaving a gap in the exploration of the changing dynamics of contemporary supply chains. The emergence of e-commerce and direct-to-customer sales in recent years has prompted managers to implement dual-channel or multi-channel frameworks, posing challenges to the conventional single-channel models. Therefore, the pricing decisions within dual-channel supply chains have widely been examined in the literature due to their significance to real-life businesses. Thus, the pricing problem in a dual-channel supply chain has been studied under different configurations [18; 20; 28; 29], remanufacturing effort [18; 28; 30], after-sales services [19; 20], carbon tax policy [30], extended warranty [31], competing supply chains [32], etc. Also, other papers have formulated the pricing problem for dual-channel supply chains basing on [PERSON]'s game theory [18; 28; 31; 33], enabling the entity with more market power to make the first decisions as a leader [28], while the other members are considered followers and act in accordance with the leader's decisions. Nevertheless, the aforementioned studies only examine mono-periodic pricing and remanufacturing decisions for dual-channel supply chains, and this is considered a gap between the literature and practice. In fact, the pricing, manufacturing and remanufacturing decisions represent strategic decisions for supply chain managers; hence, they are typically planned over a time horizon. Therefore, various works have investigated the pricing and remanufacturing problem in a multi-periodic environment for dual-channel supply chains considering different configurations [34; 35; 36], leasing products and customer preference [19; 37], proposed warranty [38; 39] and online promotion [40]. As already discussed, the works mentioned above shed light on the impact of efficiently managing pricing and remanufacturing decisions on the performance of the supply chain. Thus, this paper addresses a multi-periodic pricing and remanufacturing problem for a dual-channel supply chain with reverse logistics specialized in the manufacturing and the distribution of pharmaceutical products. The studied system includes a manufacturer and two distribution channels represented by a pharmacy and an online selling platform. In addition, nowadays, supply chains integrate commissions between channels to reduce investment costs. To incorporate the aspect of sharing revenues and gains by associating online sellers with brick-and-mortar retailers to create a win-win partnership, only the pharmacy proposes after-sales services for returned products sold in both channels in return for a unit commission on each online sale. Therefore, the main contribution of this work relies on formulating the pricing and remanufacturing decisions for the decentralized configuration of the considered dual-channel supply chain following the Stackelberg game theory. The decentralized configuration investigated in this paper considers both distributors as external channels to the manufacturer, so that the manufacturer, the online selling platform and the pharmacy aim to optimize their profits regardless of the other entities. Therefore, the developed Stackelberg game theory supposes that only the manufacturer is the leader and both distribution channels are recognized as followers. Also, as far as we know, the impact of the sales commission in return for the provided after-sales services on the revenues of both distribution channels is not examined within dual-supply chains for a planning horizon, which represents a novelty in relation to the literature. Additionally, this paper's contributions include examining how the optimal solutions evolve with respect to various crucial parameters and identifying the break-even point for a unit sales commission that enables both distribution channels to yield comparable profits. The remainder of this paper is organized as follows.