Efficient Strategic Pricing in a Dual-Channel Stackelberg Supply Chain: Incorporating Remanufacturing and Sales Commissions across Multiple Periods

Abstract

The rise of e-commerce has significantly impacted consumer shopping habits, resulting in profit loss for traditional supply chains. In response to intense competition, numerous companies have transitioned their business models to embrace dual-channel configurations, seeking to captivate customers and increase their market share. Nonetheless, research on decentralized dual-channel supply chain configurations is scarce and predominantly concentrates on single-period pricing. This paper addresses this gap by employing Stackelberg’s game theory to investigate the multi-periodic pricing and remanufacturing decisions within a decentralized dual-channel supply chain with reverse logistics, specialized in the manufacturing and sales of pharmaceutical products. Moreover, this work considers that the online channel pays a sales commission to the pharmacy in return for the provided after-sales services, aiming to incorporate the aspect of sharing revenues. A mathematical formulation is proposed in a multi-periodic environment allowing us to simultaneously maximize the total profits of the manufacturer, the pharmacy and the online channel, by optimizing the pricing and remanufacturing strategies. Numerical analyses examine the customer purchasing preference’s effect on the demand and pricing decisions of each channel, the impact of the collection cost on the optimal remanufacturing strategy, and assess the break-even point of the total profits generated in both channels according to the sales commission. This study’s novelty lies in employing Stackelberg’s game theory to develop a mathematical formulation for the multi-periodic pricing and remanufacturing problem within a decentralized dual-channel supply chain, incorporating a sales commission between both distributors.

1. Introduction

For the last decades, the efficient management of supply chains has become a critical task for surviving worldwide competition, considering various customer’s behaviors and preferences and diverse national regulations. In addition, due to the continuous advancement in technology and the Internet, the number and the size of companies based on an online platform’s business model continue to increase [1], expending their revenues by offering a variety of products and shifting customers’ shopping habits. For example, online shopping is estimated to represent 13.6% of the retail market worldwide in 2019, 34.1% and 21.8% in China and the UK, respectively [2]. Nowadays, multinational companies tend to expand their market reach by employing a variety of distribution channels offering customers the choice of buying products from several available sellers [3]. Thus, enhancing the performance of dual-channel and multi-channel supply chains represent a major interest for managers and researchers. Furthermore, in the worldwide competition, implementing an appropriate pricing strategy is considered a primordial activity for businesses in order to expand their market by attracting new clients [4]. For instance, the influence of making more suitable pricing decisions on the demands and the profitability of real-life industries was examined in the case of Amazon [5], used cars [6], Netflix [7], the telecommunications industry [8], pharmaceuticals [9,10], etc. Hence, investigating pricing decisions represents an indispensable task and a huge axis of interest not only for researchers, but also for small, medium and multinational firms.

Moreover, designing and managing sustainable systems focus on enhancing the effectiveness of the environmental and economic challenges of supply chains [11] and the economic stability related to a sustainable infrastructure and energy projects [12]. As a matter of fact, the shift in customers’ environmental concerns encouraged researchers and supply chain managers to adopt an efficient remanufacturing process, reducing the waste and costs related to manufacturing activities and enhancing competitiveness. For the last decades, environmental legislation has become more stringent, emphasizing the need for sustainable and circular supply chains. Therefore, numerous studies have explored the impact of remanufacturing activities on supply chains, addressing various configurations [13] and considering factors such as competition [14,15]. Also, the collection mode of used products [16] and the environmental awareness related to remanufactured products [17] have been investigated. Hence, studying remanufacturing decisions constitute a major interest for researchers and managers of supply chains, encouraging the shift towards more sustainable systems. Also, several studies [18,19,20] have investigated the impact of remanufacturing activities on the pricing strategy and established a correlation between both decisions. Therefore, simultaneously examining the pricing and remanufacturing problem has attracted the attention of several researchers since it arises from the needs of real-life businesses of this era. Due to the high competition, proposing suitable prices is considered essential for increasing one’s market share, since wisely managing pricing decisions represents the best strategy to attract new consumers [4]. As a consequence, the pricing problem is fairly well studied in the literature. For example, several studies have examined the remanufacturing and pricing problem for single-channel supply chains under linear pricing constraints [21,22,23,24,25,26]. A variety of assumptions are considered in these studies such as a short life cycle [21], after-sales services [22], different centralized and decentralized configurations [23,24,25,26], sales and marketing efforts [23,24], fuzzy demand [26] and customers’ acceptance towards remanufactured products [27].

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 Stackelberg’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. In Section 2, an explicit description of the dual-channel supply chain with reverse logistics, specialized in the manufacturing and distribution of pharmaceutical products, is presented. Section 3 is dedicated to the development of the mathematical model for studying the pricing manufacturing and remanufacturing decisions. Numerical examples are presented in Section 4 to assess the variation in the obtained optimal solutions in a multi-periodic environment, while sensitivity analyses aim to evaluate the robustness and the efficiency of the developed mathematical model by examining the evolution of the proposed optimal solutions according to changes in the values of several key parameters. Section 5 presents the conclusion of this work.

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.

H	Length of the planning horizon
t	Period index
A(t)	Market size at period t if the product is free
Mc	Unit manufacturing cost of a product entirely produced from raw materials
δ	Unit cost saving of an entirely remanufactured product
B	Collection cost’s scaling parameter
ρ	Customer preference for purchasing the product via the online platform
αp	Self-price sensitivity of the pharmacy’s demand
αo	Self-price sensitivity of online demand
β	Sensitivity of the demand in one channel to price changes in the other channel
Rp	Proportion of the demand sold by the pharmacy returned to AS
Ro	Proportion of the demand sold online returned to AS
Uas	Unit processing cost of a returned product
Co	Unit sales commission between both distribution channels

Adding to that, the mathematical model was developed based on the decision variables illustrated below.

Pp(t)	Product’s unit selling price proposed by the pharmacy in period t
Po(t)	Product’s unit selling price proposed by the online platform in period t
Wp(t)	Product’s unit wholesale price from the manufacturer to the pharmacy during period t
Wo(t)	Product’s unit wholesale price from the manufacturer to the online platform during period t
Dp(t)	Pharmacy’s demand during period t
Do(t)	Demand of the online platform during period t
τ(t)	Remanufacturing rate during period t

Also, the following notations were used to separate the profits of the different actors of the dual channel supply chain.

πM	Total profit of the manufacturer during the planning horizon
πP	Total profit of the pharmacy during the planning horizon
πO	Total profit of the online platform during the planning horizon

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 τ(t). As a consequence, the unit manufacturing cost borne by the manufacturer for each period t depended on the remanufacturing rate τ(t) in the same period. Hence, the unit manufacturing cost was the unit cost Mc of producing one new product entirely from the direct transformation of raw materials minus the savings generated by the remanufacturing process. Therefore, the unit manufacturing cost for each period was Mc − δ·τ(t) [18]. Also, the manufacturer bore the collection cost of used products. In addition, the collection cost of used products for each period t was considered in this work to be (B·τ(t)²)/2, where B is a scaling parameter [18,20,33]. Nevertheless, this work considered new products entirely manufactured from transforming raw materials and new products containing remanufactured subcomponents identical.

In addition, a client chose to purchase a product from the pharmacy or directly from the online platform and had it delivered to their address. However, in case of dissatisfaction or failure, the client returned the products already purchased in both channels to the AS of the pharmacy (P). In this case, the pharmacy’s AS bore a unit processing cost Uas for each returned product. In exchange, the online selling platform paid a commission C_o to the pharmacy for every sold product in return for the provided services by the AS of the pharmacy. In this paper, the commission C_o between both channels for provided services was determined by the manufacturer. For every period, the ratios of unsatisfied clients from the pharmacy’s demand and the online demand were R_p and R_o, respectively. The following figure illustrates the dual-channel supply chain examined in this work.

As presented in Figure 1, this work treated the decentralized configuration of the supply chain where both distribution channels were considered external channels to the manufacturer. In fact, this paper formulated the pricing and remanufacturing decisions following the Stackelberg game theory where the manufacturer was the leader and both distribution channels were the followers. Therefore, for each period t, the manufacturer determined the wholesale price W_p(t) for the pharmacy, the online wholesale price W_o(t) and the remanufacturing rate τ(t) that optimized its own profit π_M. After receiving the wholesale prices, both distribution channels determined their proposed prices P_p(t) and P_o(t) for the selling of the product during the same period that maximized the profit of the pharmacy π_P and the profit of the online platform π_O.

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:

$$D_p(t)=(1-\rho )·A(t)-{\alpha }_p·P_p(t)+\beta ·P_o(t)$$

(1)

$$D_o(t)=\rho \cdot A(t)-{\alpha }_o·P_o(t)+\beta ·P_p(t)$$

(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 − ρ) multiplied by the market size at the same period A(t), while the online market share is equal to ρ·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 α_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 β. Thus, the pharmacy’s demand decreases by α_p customers if its proposed price P_p(t) at period t increases by 1 money unit (mu) and drops by β 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 α_p, α_o and β are positive and independent. Also, the self-price sensitivity of the pharmacy, α_p, is higher than the online self-price sensitivity α_o due to the AS provided by the pharmacy.

(3)

For each period, the production process could include a ratio τ(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:

$${\pi }_O\left(P_o(t)\right)={\displaystyle \sum _{t=1}^H\left[\left((1-C_o)·P_o(t)-W_o(t)\right)·D_o(t)\right]}$$

(3)

The total profit π_O of the online platform is the revenues generated during the planning horizon H by the activity of selling products. 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·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:

$$\begin{array}{c}Max {\pi }_O\left(P_o(t)\right)={\displaystyle \sum _{t=1}^H\left[\left((1-C_o)\cdot P_o(t)-W_o(t)\right)·D_o(t)\right]}\\ subject to\\ D_o(t)\ge 0\\ (1-C_o)·P_o(t)\ge W_o(t)\end{array}$$

(4)

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.

$${\pi }_P\left(P_p(t)\right)={\displaystyle \sum _{t=1}^H\left[\left(P_p(t)-W_p(t)\right)\cdot D_p(t)+C_o·P_o(t)·D_o(t)-\left(R_p·D_p(t)+R_o·D_o(t)\right)·Uas\right]}$$

(5)

The pharmacy’s total profit π_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·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:

$$\begin{array}{c}Max {\pi }_P\left(P_p(t)\right)={\displaystyle \sum _{t=1}^H\left[\begin{array}{c}\left(P_p(t)-W_p(t)\right)\cdot D_p(t)+P_o(t)·C_o·D_o(t)\\ -\left(R_p·D_p(t)+R_o·D_o(t)\right)·Uas\end{array}\right]}\\ subject to\\ D_p(t)\ge 0\\ P_p(t)\ge W_p(t)\end{array}$$

(6)

On the other hand, the manufacturer’s total profit is as follows:

$${\pi }_M\left(W_p(t),W_o(t),\tau (t)\right)={\displaystyle \sum _{t=1}^H\left[\begin{array}{c}{W}_p(t)\cdot D_p(t)+W_o(t)·D_o(t)-B·\tau {(t)}^2/2\\ -\left(D_p(t)+D_o(t)\right)·\left(Mc-\delta ·\tau (t)\right)\end{array}\right]}$$

(7)

The total profit π_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 Mc − δ·τ(t) and the costs related to the remanufacturing decisions. The remanufacturing constraint is as follows:

$$0\le \tau (t)\le 1$$

(8)

Inequality (8) illustrates the boundaries of the remanufacturing process. Based on assumption 3, τ(t) represents the ratio of remanufactured subcomponents to the totality of the subcomponents necessary or the manufacturing of a new product. 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 τ(t) is equal to 0. However, the remanufacturing rate τ(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 < τ(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.

$$\frac{\partial \left({\pi }_O(P_o(t))\right)}{\partial P_o(t)}={\displaystyle \sum _{t=1}^H\left[(1-C_o)\left(-2{\alpha }_o\cdot P_o(t)+\rho ·A(t)+\beta ·P_p(t)\right)+{\alpha }_o·W_o(t)\right]}$$

(9)

$$\frac{{\partial }^2\left({\pi }_O(P_o(t))\right)}{{\partial }^2{P}_o(t)}=-2H·{\alpha }_o(1-C_o)<0$$

(10)

As illustrated in Equation (10), the second-order derivative of the total profit of the online platform π_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 π_O(P_o(t)) (assumption 5) are as follows:

$${\displaystyle \sum _{t=1}^H{P_o}^{*}(t)}=\frac{1}{2{\alpha }_o}{\displaystyle \sum _{t=1}^H\left[\rho ·A(t)+\beta \cdot P_p(t)+\frac{{\alpha }_o}{1-C_o}{W}_o(t)\right]}$$

(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:

$$\frac{\partial \left({\pi }_P(P_p(t))\right)}{\partial P_p(t)}={\displaystyle \sum _{t=1}^H\left[\begin{array}{c}(1-\rho )·A(t)-2{\alpha }_p\cdot P_p(t)+{\alpha }_p\left(W_p(t)+R_p·Uas\right)\\ +\beta \left((1-C_o)P_o(t)-R_o·Uas\right)\end{array}\right]}$$

(12)

$$\frac{{\partial }^2\left({\pi }_P(P_p(t))\right)}{{\partial }^2{P}_p(t)}=-2H·{\alpha }_p<0$$

(13)

Similar to the online platform, the total profit of pharmacy π_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 π_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:

$${\displaystyle \sum _{t=1}^H{P_p}^{*}(t)}=\frac{1}{2{\alpha }_p}{\displaystyle \sum _{t=1}^H\left[\begin{array}{c}(1-\rho )\cdot A(t)+{\alpha }_p\left(W_p(t)+R_p·Uas\right)\\ +\beta \left((1-C_o)·P_o(t)-R_o·Uas\right)\end{array}\right]}$$

(14)

The pharmacy’s optimal responsive pricing strategy presented in Equation (14) is obtained by setting the first-order derivative of the pharmacy’s total profit during the planning horizon (Equation (14)) to zero and solving it. At this stage, knowing the optimal responsive pricing decisions of both distribution channels, the manufacturer determines its wholesale prices and its remanufacturing strategy using backward induction. Therefore, the manufacturer’s optimal decisions are the solutions to the following problem:

$$\begin{array}{c}Max {\pi }_M\left(W_p(t),W_o(t),\tau (t)\right)={\displaystyle \sum _{t=1}^H\left[\begin{array}{c}{W}_p(t)\cdot D_p(t)+W_o(t)·D_o(t)-B·\tau {(t)}^2/2\\ -\left(D_p(t)+D_o(t)\right)·\left(Mc-\delta ·\tau (t)\right)\end{array}\right]}\\ subject to\\ D_p(t)=(1-\rho )·A(t)-{\alpha }_p·P_p(t)+\beta ·P_o(t)\\ D_o(t)=\rho ·A(t)-{\alpha }_o·P_o(t)+\beta ·P_p(t)\\ {\displaystyle \sum _{t=1}^H{P}_p(t)}=\frac{1}{2{\alpha }_p}{\displaystyle \sum _{t=1}^H\left[\begin{array}{l}(1-\rho )·A(t)+{\alpha }_p\left(W_p(t)+R_p·Uas\right)\\ +\beta \left((1-C_o)P_o(t)-R_o·Uas\right)\end{array}\right]}\\ {\displaystyle \sum _{t=1}^H{P}_o(t)}=\frac{1}{2{\alpha }_o}{\displaystyle \sum _{t=1}^H\left[\rho ·A(t)+\beta ·P_p(t)+\frac{{\alpha }_o}{1-C_o}{W}_o(t)\right]}\\ D_p(t)\ge 0\\ D_o(t)\ge 0\\ P_p(t)\ge W_p(t)\\ \left(1-C_o\right)·P_o(t)\ge W_o(t)\\ 0\le \tau (t)\le 1\end{array}$$

(15)

The optimization problem in (15) allows the manufacturer to determine its optimal wholesale prices and remanufacturing effort for each period t, which maximize the total profit of the manufacturer. Also, the optimal responsive pricing strategies of both distribution channels are identified simultaneously with the pricing and remanufacturing decisions of the manufacturer. Therefore, knowing the optimal sale and wholesale prices P_p^*(t), P_o^*(t), W_p^*(t) and W_o^*(t), the optimal total profits of the online platform and the pharmacy are deduced according to Equations (3) and (5), respectively.

4. Numerical Results and Sensitivity Analysis

In this section, a detailed presentation of a numerical example and sensitivity analyses of the obtained solutions is provided for the considered problem of pricing and remanufacturing decisions. Therefore, this section is dedicated to examining the solutions obtained from the proposed model studying the decentralized configuration of the dual-channel supply chain with the considered reverse flows, focusing on the sales of pharmaceutical products. The variations during the planning horizon of the different obtained optimal decisions are discussed in the numerical example’s subsection, while the sensitivity analyses delve into assessing the robustness of the obtained solutions according to several parameters of interest. Thus, the proposed model was implemented and solved using Wolfram Mathematica 12.0, a software tool equipped with a range of global and local optimization techniques and capable of addressing integer, linear and nonlinear problems. The optimal solutions were obtained by solving the optimization problem (15) to obtain the total profit π_M of the manufacturer. Also, the online total profit π_O was calculated according to Equation (3) and the total profit π_P of the pharmacy was obtained from Equation (5). The data used in this optimization problem are presented in the following table, where P stands for product and mu denotes a money unit.

Table 1. Key parameters.

H = 12	A(1) = 6000 P	A(2) = 5000 P	A(3) = 7000 P	A(4) = 8000 P
A(5) = 6000 P	A(6) = 5000 P	A(7) = 10 000 P	A(8) = 9000 P	A(9) = 6000 P
A(10) = 4000 P	A(11) = 5000 P	A(12) = 6000 P	Mc = 100 mu/P	δ = 40 mu/P
αp = 12 P/mu	αo = 14 P/mu	β = 8 P/mu	Rp = 0.2	Ro = 0.2
	Uas = 20 mu/P	Co = 5%	B = 500,000 mu

illustrates the different values of the parameter used to conduct the numerical example and the sensitivity analyses below. The proposed data could describe a dual-channel supply chain specialized in the manufacturing, remanufacturing, wholesale and selling of pharmaceutical products. As listed, all the experiments in this paper were performed in a multi-periodic environment composed of 12 periods. The variation from one period to another in the size of the market when the product is free, denoted as A(t), intended to examine the evolution of the obtained pricing and remanufacturing strategies under different market sizes and the adaptability of the obtained optimal results according to different scenarios. In addition, the online selling process was considered more self-sensitive to changes in its proposed prices than the pharmacy’s selling process. This gap in the self-price sensitivities between both distribution channels was caused by the absence of AS in the online platform. These provided services rendered the pharmacy more attractive to the clients and less sensitive to fluctuations in its proposed selling prices P_p(t).

4.1. Numerical Example

This subsection is dedicated to presenting the obtained optimal solutions and the different optimal pricing and remanufacturing strategies proposed by the developed model in this paper. This numerical example aimed to present the different optimal pricing and remanufacturing decisions along the supply chain and to examine their evolution over the planning horizon H. For that, the results discussed in this subsection were obtained according to the data presented in Table 1 and for a customer preference for purchasing the products from the online platform ρ = 0.5. Figure 2 and Figure 3 below describe the obtained optimal unit wholesale and selling prices, the optimal pharmacy and online demands and the manufacturer’s optimal remanufacturing strategy during the planning horizon H.

Figure 2 presented above illustrates the changes and the evolution, during the planning horizon H, of the obtained optimal unit wholesale prices proposed by the manufacturer W_p^*(t) and W_o^*(t) to the pharmacy and to the online platform, respectively, and the optimal unit selling prices to the customers proposed by the pharmacy P_p^*(t) and by the online platform P_o^*(t). As presented above, for all periods, the unit wholesale prices W_p^*(t) and the unit selling prices P_p^*(t) of the pharmacy were slightly higher than those proposed to and by the online selling platform, W_o^*(t) and P_o^*(t). The difference in the pricing decisions related to the pharmacy and the online platform was caused by the gap between the self-price sensitivities α_p and α_o. According to assumption 2, since the online channel was more self-sensitive to the variation in its proposed prices (α_o > α_p), the obtained optimal online unit prices W_o^*(t) and P_o^*(t) were marginally lower than those proposed by the pharmacy W_p^*(t) and P_p^*(t). In addition, as presented in Figure 2 and Figure 3, the obtained optimal pricing strategies and demands fluctuated during the planning horizon H following the variation in the market size A(t). For example, the highest demands, unit wholesale and selling prices in both distribution channels were offered during period 7 (largest market size A(7) = 10,000 P), while the lowest demands and pricing decisions were obtained for period 10 (smallest market size A(10) = 4000 P). Also, by studying Figure 3, the pharmacy’s demand D_p^*(t) and the online demand D_o^*(t) were very similar for each period t, since both distribution channels shared the market (ρ = 0.5) equally. In fact, the pharmacy sold a total of 11,666 P during the planning horizon while the online platform’s total demand was equal to 11,843 P. This slight difference between the pharmacy and the online platform in terms of total demand generated during the planning horizon is explained by the low gap in the pricing strategies of both channels. On the other hand, as presented in Figure 3, the obtained optimal solution indicated that each new product included for each period t a percentage τ^*(t) of remanufactured parts to reduce the manufacturing costs. Hence, the mean during the planning horizon of the optimal remanufacturing rates τ^*(t) was 0.16. Nevertheless, the obtained optimal remanufacturing rate τ^*(t) fluctuated from one period to another following the optimal total demand (D_p^*(t)+ D_o^*(t)) of the dual-channel supply chain, which in turn depended on the market size A(t). Therefore, the maximal optimal remanufacturing rate τ^*(t) was achieved during the 7th period while the lowest remanufacturing effort was obtained during the 10th period. Thus, the market size was an essential parameter in determining not only the demand but also the impacts of the remanufacturing effort and pricing strategy. Adding to that, the optimal total profits generated during the planning horizon by the manufacturer, the pharmacy and the online platform were π_M^* = 6,743,220 mu, π_P^* = 1,410,746 mu and π_O^* = 876,647 mu, respectively. However, for the same market (1 − ρ = ρ = 0.5), the online platform generated 38% less total profit than the pharmacy. In fact, even for similar demands and pricing strategies, the online channel paid a total commission of 273,585 mu in return for the AS provided by the pharmacy, which explained the gap in the total profit between both distribution channels.

4.2. Sensitivity Analyses According to the Customer Preference ρ

In this subsection, a detailed analysis of the impact of the changes in customer preferences toward purchasing a product from one channel or the other on the proposed optimal solutions is presented. In reality, customer preferences towards a particular channel, a specific product or proposed services represent a crucial parameter in attracting more customers and eventually increase the demand and the profit. Therefore, the value of the customer preference for purchasing the product from the online platform ρ was increased progressively and the evolution of the obtained optimal solutions was examined. The following results were obtained by using the parameter values illustrated in Table 1. Figure 4 displays the mean of the optimal unit prices for the manufacturer, the pharmacy and the online platform, the mean of the obtained optimal demands in both distribution channels, while Figure 5 below presents the optimal total profits generated by the pharmacy and the online platform for the different values of customer preferences.

As illustrated in the figure above, the progressive increase in the value of the customer preference for online purchasing, ρ, enhanced the attractiveness of the online channel to customers, leading to an expansion in the online market share ρ·A(t). This resulted in an increase in the optimal online demand D_o^*(t) for each period t and consequently, an increase in the mean online demand during the planning horizon H. As a matter of fact, the total demand of the online platform grew from 7 568 P for ρ = 0.2 to 16,119 P for ρ = 0.8, which represents a rise of 113%. Furthermore, as the value of the customer online preference ρ increased, both the mean of the obtained optimal unit online selling prices P_o^*(t) and the mean optimal wholesale prices W_o^*(t) proposed by the manufacturer to the online platform increased. This rise in the pricing decisions related to the online channel was a direct outcome of the increase in the online market share and demand. In fact, by quadrupling the online market share, the online platform could afford an increase in its proposed prices without greatly impacting its demand. Thus, the mean of the optimal unit online sale prices P_o^*(t) witnessed a rise of 34%, and the mean of the optimal wholesale prices W_o^*(t) increased by 22% due to the increase in the customer preference for online purchasing. Nonetheless, the customer preference towards purchasing the products from the pharmacy (1 − ρ) dropped whenever ρ increased. Hence, for each period t, the pharmacy’s market share ((1 − ρ)·A(t)), the obtained optimal demand D_p^*(t) and the mean of the demands of the pharmacy during the planning horizon H decreased, as a consequence of the increase in the customer online preference ρ. Consequently, the optimal results showed that the total demand of the pharmacy during the planning horizon dropped by 57% from 16,282 P for ρ = 0.2 to 7 051 P for ρ = 0.8. In addition, as a consequence of the increase in value of the customer online preference ρ, the mean of the obtained pharmacy’s optimal unit selling prices P_p^*(t) and the mean of the optimal wholesale prices W_p^*(t) proposed by the manufacturer to the pharmacy decreased by 31% and 26%, respectively, due to a decrease in the pharmacy’s optimal demand D_p^*(t) for all the periods. Thus, as presented in Figure 5, the increase in the value of the customer preference for purchasing the product via the online platform significantly impacted the total profits generated by both distribution channels during the planning horizon H. Specifically, the shift in the customer preference ρ from 0.2 to 0.8 led to a remarkable increase in the optimal total profit generated by the online platform π_O^* of 341% and a corresponding increase in the total commission paid by the online platform in return for the provided AS by the pharmacy of 183%. These results were directly linked to the rise in the optimal demands D_o^*(t) and the optimal unit selling prices P_o^*(t) of the online platform for all the periods of study. However, the pharmacy’s optimal total profit π_P^* experienced a 63% decrease due to the drop in both optimal pharmacy’s demands D_p^*(t) and optimal unit selling prices P_p^*(t) proposed by the pharmacy. Notably, for ρ = 0.8, half of the total profit of the pharmacy was generated through the received commission from the online channel for the AS. Thus, the sensitivity analyses on the obtained solutions conducted in this subsection show that the changes in the value of the customer preference towards purchasing products online significantly affects the optimal pricing decisions and demands in both distribution channels, resulting in an important impact on the optimal total profits generated in each channel and the total sales commission between the online platform and the pharmacy. Hence, precisely studying the customer behavior and preferences is an essential activity for managers in order to adopt an adequate pricing strategy that optimizes their total profits.

4.3. Sensitivity Analyses following the Variations in the Scaling Parameter of the Collection Cost B

This subsection is dedicated to assessing and examining the impact of a variation in the value of the collection cost’s scaling parameter B on the proposed optimal solutions, specifically the optimal remanufacturing strategy. In fact, designing and wisely managing the remanufacturing process rely heavily on the investments and the costs borne by the supply chain. However, in this paper, the costs related to the remanufacturing activities were only represented by the costs of collecting used products by the manufacturer. Thus, the evolution of several optimal decisions was studied according to the increase in the value of the scaling parameter B of the cost of collecting used products for the remanufacturing activities from B = 200,000 mu to B = 900,000 mu. Also, the customer preference towards purchasing products online was fixed at ρ = 0.5, in order to examine the evolution of the optimal solution under a market equitably balanced between both channels in terms of demands. Hence,

Table 2. Optimal remanufacturing strategy τ^*(t) according to the scaling parameter B.

B (mu)	τ*(1)	τ*(2)	τ*(3)	τ*(4)	τ*(5)	τ*(6)	τ*(7)	τ*(8)	τ*(9)	τ*(10)	τ*(11)	τ*(12)	mean τ*(t)
200,000	0.36	0.29	0.44	0.52	0.37	0.29	0.67	0.59	0.37	0.22	0.29	0.37	0.4
300,000	0.24	0.19	0.29	0.34	0.24	0.19	0.44	0.39	0.24	0.14	0.19	0.24	0.26
400,000	0.18	0.14	0.22	0.25	0.18	0.14	0.33	0.29	0.18	0.11	0.14	0.18	0.2
500,000	0.14	0.11	0.17	0.2	0.14	0.12	0.26	0.23	0.14	0.09	0.12	0.14	0.15
600,000	0.12	0.1	0.14	0.17	0.12	0.1	0.22	0.19	0.12	0.07	0.1	0.12	0.13
700,000	0.1	0.08	0.12	0.14	0.1	0.08	0.19	0.17	0.1	0.06	0.08	0.1	0.11
800,000	0.09	0.07	0.11	0.13	0.09	0.07	0.16	0.14	0.09	0.05	0.07	0.09	0.1
900,000	0.08	0.06	0.1	0.11	0.08	0.06	0.14	0.13	0.08	0.05	0.06	0.08	0.09

below illustrates the variation in the obtained optimal remanufacturing rate for the different values of the scaling parameter B of the cost of collecting used products.

As presented in Table 2, the obtained optimal remanufacturing rates τ^*(t) dropped for each period t whenever the value of the collection cost’s scaling parameter increased gradually from B = 200,000 mu to B = 900,000 mu. Thus, by increasing the value B, the mean during the planning horizon H of the obtained optimal remanufacturing rates τ^*(t) gradually decreased from 0.4 for B = 200,000 mu to 0.09 for B = 900,000 mu. Hence, as a consequence of the rise in the value of the scaling parameter B, new products manufactured during the planning horizon contained less and less remanufactured components. As explained above, the manufacturer bore for each period t the costs related to the remanufacturing activities represented by the collection cost B·τ(t)²/2. Therefore, for the same values of remanufacturing decisions τ(t), the augmentation in the value of B resulted in a rise in the collection cost, leading to a drop in the total profits generated by the manufacturer. Accordingly, by increasing the scaling parameter B, the optimal remanufacturing rate τ^*(t) decreased during all the periods in order to reduce the costs related to collecting used products. Thus, the costs generated by the remanufacturing process dictated the remanufacturing strategy adopted by the manufacturer. Also, as mentioned in Section 4.1, the obtained optimal remanufacturing rate τ^*(t) fluctuated from one period to another following the demands and the market size A(t). Hence, for all the values of B considered in this study, the highest and the lowest remanufacturing efforts were achieved during the 7th and the 10th periods, respectively. Figure 6 below describes the evolutions of key financial indicators according to the different values of the scaling parameter B of the cost of collecting used products.

As already elaborated, as a consequence of the increase in the scaling parameter B, the optimal remanufacturing rate τ^*(t) obtained by the proposed model in this study significantly decreased for each period t. Therefore, the manufacturing process of new products required less remanufactured components causing a rise in the optimal unit manufacturing cost (Mc−δ·τ^*(t)). Essentially, this increase was caused by the decrease in the optimal unit saving related to the remanufacturing activities δ·τ^*(t), due to the drop in the remanufacturing rates. Hence, the total savings generated by the remanufacturing process cumulated by the manufacturer during the planning horizon H significantly decreased whenever the scaling parameter of the collection cost increased from B = 200,000 mu to B = 900,000 mu. As a matter of fact, the optimal total savings related to the remanufacturing process dropped by almost 79%, as presented in Figure 6. The increase in the unit optimal manufacturing cost due to the decrease in the unit savings led to a drop in the optimal total profit of the supply chain of 2.8%. On the other hand, despite the rise in the scaling parameter B, the collection cost B·τ^*(t)²/2 decreased for each t due to the heavy drop in the obtained optimal remanufacturing effort. Therefore, the total collection cost of used products during the planning horizon H experienced a drop of 79% in response to the rise of the scaling parameter B. However, for each scaling parameter B, the total savings related to the remanufacturing strategies were significantly higher than the total collection costs, proving the necessity and the profitability of the remanufacturing activities. In addition, the increase in the unit manufacturing cost of new products evoked a rise in the mean of the optimal unit wholesale prices W_p^*(t) and W_o^*(t) proposed by the manufacturer by 1.7% and 1.4%, respectively, leading to an increase in the mean of the optimal unit selling prices P_p^*(t) and P_o^*(t) proposed by the pharmacy and the online platform of 1% each. Moreover, the rise in the pricing strategies of both distribution channels caused a decrease in the optimal total demand of the pharmacy and the online channel during the planning horizon H of 238 P and 324 P, respectively. Since the online platform was more self-price-sensitive than the pharmacy, the lost demand in the online channel was larger than the lost demand in the pharmacy’s channel. Therefore, the optimal total profit generated by the manufacturer π_M^* dropped by 2.3% due to the increase in the manufacturing costs. Also, both optimal total profits π_P^* and π_O^* generated, respectively, by the pharmacy and the online platform experienced a decrease of 3.2% and 5.3%, respectively, a direct outcome of the increase in the unit wholesale prices and the lost demands. In summary, efficiently designing and operating the remanufacturing activities impact the manufacturing costs, the pricing strategies, the demands and eventually the total profit of the supply chain.

4.4. Sensitivity Analyses According to the Sales Commission Co

A detailed sensitivity analysis examining the impact of the unit sales commission Co paid by the online platform to the pharmacy in return of the provided AS on the optimal obtained solutions is illustrated in this subsection. In this work, the obtained optimal total profits of both distribution channels were studied in order to determine the sales commission Co^* that represented the break-even point for the profits of both channels. Therefore, Co^* denotes to the break-even point of the unit sales commission that allows the pharmacy and the online platform to generate equal total profits. Hence, the variation in the optimal total profits of both distribution channels was examined for different values of unit sales commission Co and customer preference for online purchasing ρ. Also, the following results were conducted on the proposed model according to the data presented in Table 1. Figure 7 below illustrates the evolution of the optimal total profit of the pharmacy π_P^* and the online optimal total profit π_O^* generated during the planning horizon H for different unit sales commissions Co and different customer preferences ρ.

As presented above, the variation in the values of the unit sales commission Co impacted the optimal total profit generated in each distribution channel. In particular, for each customer preference towards purchasing products via the online channel ρ, the increase in the value of the unit sales commission C_o led to an increase in the optimal total profit π_P^* of the pharmacy and a decrease in the optimal online total profit π_O^* generated during the planning horizon H. In fact, following the rise in the value of the unit sales commission C_o, the total commission paid by the online platform to the pharmacy in return for the provided AS for products sold online (i.e., C_o·P_o^*(t)·D_o^*(t)) increased for each period t. Hence, the total paid commission during the planning horizon H increased for each customer preference for online purchasing ρ. For instance, for ρ = 0.6, the total paid commission during the planning horizon H quadrupled from 128,468.5 mu for C_o = 2% (representing 14% of the pharmacy’s total profit and 11% of the online total profit) to 513,382 mu for C_o = 8% (which corresponds to 35% of the pharmacy’s total profit and 48% of the online total profit). Therefore, for each value of the customer preference towards purchasing products from the online platform ρ, the increase in the total commission during the planning horizon H due to the rise in the value of the unit sales commission C_o generated more revenues for the pharmacy and less profits for the online channel. These outcomes explain the variations in the optimal total profit π_P^* of the pharmacy and the optimal online total profit π_O^*. On the other hand, as illustrated in Figure 7, the value of the break-even point according to the unit sales commission C_o^* increased whenever the customer online preference ρ increased. Furthermore, the break-even point C_o^* indicated the value of the unit sales commission C_o allowing the pharmacy and the online platform to generate equal total profits during the planning horizon H. For ρ = 0.5, since both distribution channels shared the market and generated similar demands, the pharmacy’s optimal total profit π_P^* was slightly higher than the online optimal total profit π_O^* due to the gap between the self-price sensitivities α_p and α_o. In addition, no unit sales commission C_o could achieve the break-even point condition related to the total profits for ρ = 0.5. However, the break-even point rose from C_o^* = 4.1% for ρ = 0.6 to Co^* = 8% for ρ = 0.7 until reaching Co^* = 11.2% for ρ = 0.8. As already discussed in Section 4.2, the increase in the value of the customer online preference ρ caused a shift in the demands of the supply chain towards the online platform, resulting in an increase in the online optimal total profit and a decrease in the pharmacy’s optimal total profit. Thus, for a high customer preference towards purchasing products online ρ, the online channel paid a higher unit sales commission to the pharmacy in order to achieve equal total profits for both channels. For example, in the case of the break-even point C_o^*, the total paid commission represented 46% and 64% of the optimal total profits generated by the pharmacy during the planning horizon H for a customer online preference ρ equal to 0.7 and 0.8, respectively. Hence, in-depth analyses of the market shares and the unit sales commission is considered essential activity for managers in order to allow both distribution channels to coexist, share profits and revenues and to establish a dynamic of cooperation profitable for everyone.

5. Conclusions

This paper delved into the study of pricing and remanufacturing strategies within a multi-period environment for a decentralized dual-channel supply chain configuration incorporating reverse logistics, focusing on the pharmaceutical sector. The supply chain in question comprised a manufacturer and two distribution channels: a physical pharmacy and an online sales platform. The manufacturer was responsible for both manufacturing and remanufacturing processes, while the distribution channels handled sales to end-consumers. Notably, the pharmacy also offered after-sales services (AS) for products returned from both channels, receiving a unit commission for each online sale in compensation. This investigation covered the forward processes of manufacturing and sales, alongside reverse logistics activities including the collection and refurbishment of used products. A central goal of this paper was to develop the pricing and remanufacturing problem within the framework of Stackelberg’s game theory, identifying the manufacturer as the leader who set wholesale prices and remanufacturing rates, followed by each distribution channel setting their respective retail prices. The key contributions of this study included the creation of a mathematical model for the pricing and remanufacturing issue in a completely decentralized dual-channel supply chain context, taking into account AS and sales commissions between the channels, and exploring the changes and variability of the optimal solutions across multiple periods for a range of significant parameters.

A mathematical formulation was elaborated and solved to explore the variations in the obtained optimal pricing and remanufacturing decisions of the manufacturer along with the optimal responsive pricing strategies of both distribution channels according to the parameters of interest. Several numerical results were presented, and sensitivity analyses were conducted to examine the robustness of the proposed model. Numerical results indicated that the pricing and remanufacturing decisions severely depended on the market size and the sensitivity of the demands. In addition, customer preference towards purchasing products via the online channel or the pharmacy dictated the demands, the pricing strategies and consequently the total profits generated by each distribution channel. Furthermore, the optimal remanufacturing strategy adopted by the remanufacturer was extremely affected by the costs related to collecting used products, causing an evolution in the savings, the unit wholesale and selling prices and eventually the total profit of the manufacturer, the pharmacy and the online platform. Moreover, the paper considered a sales commission between the distributors, examined its effect on the transfer of revenues from the online channel to the pharmacy, and investigated the unit sales commission considered as a break-even point allowing both distribution channels to coexist and generate similar profits for different scenarios. This approach highlighted the importance of sharing gains and revenues between online sellers and physical stores, establishing a dynamic of cooperation profitable for everyone.

This work endeavored to provide support for decision makers managing either a manufacturer or a vendor in a dual-channel supply chain, allowing them to enhance their operational capabilities. In fact, the developed model offers insights for business managers on their optimal pricing strategies over a planning horizon in response to the pricing decisions of their competitors or the other entities of the supply chain. Moreover, the results of this study constitute a dashboard enabling managers to make more suitable and profitable decisions over a tactical horizon according to several parameters of interest. Furthermore, this work allows decision makers to assess their performance and to evaluate the impact of proposing more services in order to increase their benefits. As a matter of fact, the results of Section 4.4 enable the less attractive distributor to evaluate the possibility of investing and providing more services for all the market in return for sales commission allowing to share the revenues with its more attractive competitors. The sales commission between the channels assists in the aspect of sharing revenues and profits by associating sellers with brick-and-mortar retailers to create a win–win partnership, representing a beneficial relationship for all stakeholders.

Nonetheless, this study examined only the deterministic approach for the multi-periodic pricing problem in a dual-channel supply chain under linear pricing constraints and made no distinction between new and remanufactured products and subcomponents. Therefore, investigating distinct pricing decisions for products manufactured entirely from raw materials versus those incorporating remanufactured components represents a focus for future studies. Moreover, examining different pricing policies such as logit demand equations constitutes a major axis for future research. Additionally, exploring the nature of proposed services, such as customized products and proposing and extended warranty or leasing contracts, constitutes an avenue for improvement. Furthermore, developing a statistical model taking into account the uncertainties in market size and sensitivities represents a major advancement in our work.