A Three-Echelon Dual-Channel Supply Chain Model with Learning Effect Under the Return Policy

Companies and business organizations are striving to develop marketing strategies to sustain their existence [1]. Supply chain management (SCM) is one such effective strategy that presents an integrated way to resolve business-related issues to ensure better customer service and optimize business profit [2-7]. In practice, defective products are caused by machinery fault, defective raw materials, and the lethargy of laborers for continuous work. A recent study by Lai, et al. [8] proposed a vendor-buyer supply chain (SC) model by considering an imperfect production system that regards the quality improvement investment as a function of the defect rate of the products. Sarkar and Moon [9] and Manna, et al. [4] offered a model in an out-of-control situation due to a higher production rate and a long production time. Furthermore, a recent study by Dey [10] proposed a model to compensate for the production of faulty items by setting a larger production lot size. Other studies on imperfect production include Gautam and Khanna [11] and Manna, et al. [12]. ARTICLE INFO ABSTRACT

In today's competitive and technologically developed era, many retailers have adopted an e-channel to increase sales, in addition to the existing traditional retail channel. Although many researchers studied this issue, there is hardly any research that comprehensively considers the learning-effect and return-policy. Therefore, this research aimed to develop an imperfect production dual-channel supply chain model consisting of a supplier, a manufacturer, and a retailer. The manufacturer also has a refurbishment unit adjacent to its production hub, where it reworks all the defectives. The main objective is to maximize the supply chain profit by considering factors, such as inspection error, return policy, and learning-effect of the employees. Finally, this model is analyzed with the Leader-follower relationship strategy and an integrated approach. The research found that the integrated approach is profitable for the entire supply chain, while commodity prices can be minimized. Sensitivity analysis is also presented in this study. reworks all the defective items, including those received from the retailer in the refurbishment unit adjacent to the main production block, and sells these products directly to the customers in the market at a reduced price. The retailer sells the commodities from the retail shop and through an online channel. Moreover, the profit functions of each of the supply chain players, as well as the whole supply chain system, have been derived. Finally, the learning effect induced from the efficiency of the employees results in a decrease in the production cost. This proposed model will be analyzed using a leaderfollower relationship strategy and integrated approach. The main principle of SC is to provide commodities to the customers at a lower price while at the same time optimizing total profit. This study will investigate the production and pricing policies that may aid in achieving these goals. The findings of this study may provide some insights to the practitioners and future researchers.
The structure of this paper is presented as follows: notations and assumptions, mathematical models, solution methods, and numerical examples are presented in section 2. Section 3 presents results and discussion. The last section of this paper is the conclusion.

Notations
This study uses notations to facilitate mathematical modeling. The following notations are used to develop the model. : the demand rate of the manufacturer to the supplier. : the total demand rate of the customer to the retailer, where = 1 + 2 , where 1 is the retail shop's demand and 2 is the e-channel's demand.

′
: the demand rate of the customers for the refurbished products to the manufacturer.
: the demand rate of the retailer to the manufacturer. : the ordering cost of the supplier. : the setup cost of the manufacturer for the main production unit.

′
: the setup cost of the manufacturer for the refurbishment unit.
: the ordering cost of the retailer. ℎ : the holding cost per unit item per unit time of the supplier. ℎ : the holding cost per unit perfect item per unit time of the manufacturer (in the main production unit). ℎ ′ : the holding cost per unit imperfect item per unit time of the manufacturer (in the refurbishment unit). ℎ : the holding cost per unit item per unit time of the retailer. : the purchasing cost per unit of the raw materials of the supplier.

′
: the salvage value per unit of the defective raw materials of the supplier. : inspection cost per unit raw material of the supplier. : inspection cost per unit item of the manufacturer. : inspection cost per unit item of the retailer. : selling price per unit item of the retailer in the e-channel.
: the percentage of non-conforming raw materials received by the supplier. : the percentage of defective products produced by the manufacturer in the main production unit.
: the percentage of defective products received by the retailer. : production cost per unit item of the manufacturer. : refurbishment cost per unit defective items of the manufacturer. : selling price per unit raw material of the supplier. : selling price per unit perfect item of the manufacturer, i.e. purchasing cost per unit item of the retailer. Π : total profit of the supplier. Π : total profit of the manufacturer. Π : total profit of the retailer. Π : total profit of the supply chain while using the Leader-follower relationship strategy. Π : total profit of the supply chain while using an integrated approach.
Decision variables : the lot size of the supplier.
: the production rate of the manufacturer : retail price per unit item of the retailer.

Assumptions
In the proposed model, the assumptions used include: • The manufacturer's demand rate for raw material to the supplier is higher than the production rate of the items to get rid of the shortage of raw materials in the production period, i.e., ≥ . • Due to the learning effect of the employees, the production cost of the manufacturer decreases gradually with time. Here the research assumed the production cost as = ( + − ), where , > 0 are constant, and is the learning parameter.
• The retailer's every channel's demand rate is reversely affected by its own selling price and directly relates to the other channel's price. Therefore, this research assumed 1 = − 1 + 1 ′ and 2 = − 2 + 2 ′ , , b 1 , b 1 ′ , c, d are positive constants, where 1 > 2 ′ and 2 > 1 ′ , i.e., the individual channel prices impact more on respective demand than the cross channel prices.

Mathematical model
Mathematical formulation for the models for supplier, manufacturer, and retailer is presented in this section. The illustration of the problem in this supply chain system is depicted in Fig. 1.
The supplier purchases raw materials in a lot size Q. The supplier inspects the entire raw materials immediately after receiving them and supplies the good quality raw materials to the manufacturer. The fraction of defective items observed by the supplier is . Therefore, the amount of perfect quality raw materials = (1 − ) , which the retailer sells to the manufacturer at a rate during (0, 1 ). The manufacturer starts production at a rate at time = 0 and continues up to time = 1 . Its inventory level reduces due to retailer's demand and reaches the zero level at time = 3 . The manufacturer collects all the defective products from the main production unit and retailer and then sends them to the refurbishment unit. The refurbishment unit reworks these defective products and sells these directly to the customers at a rate ′ during (0, 1 ). The retailer purchases the commodity from the manufacturer during (0, 3 ) at a rate Dr. Its inventory level depletes due to the customers' demand at a rate Dc and reaches the zero level at time = .

Supplier's model
Here, the supplier sells (1 − ) units of raw materials to the manufacturer at a rate during (0, 1 ) in equation (1). Therefore, The mathematical model of holding the cost of the raw materials at retailers is presented as follows: Therefore, the total profit of the supplier can be seen in equation (2). With 1 (0) = 0 and 1 ( 3 ) = 0. Solving differential equations in equation (3), we get 1 ( ) which is presented in equation (4).
From the continuity condition of Subsequently, set up cost of the manufacturer is . The production cost of the manufacturer is formulated as∫ 1 0 . Therefore, the detailed Production cost of the manufacturer is modeled as The inspection cost is formulated by (1− ) . Moreover, holding cost for perfect quality items is modeled by ℎ [∫ 1 ( ) Therefore, the manufacturer's total cost in the main production unit is presented in equation (6).
The mathematical model for the refurbishment unit is expressed with differential equations in equation (7), describing the rate of change of the inventory level.
From the continuity condition of 2 ( ) at = 3 we have,( . detail formula is described in equation (9).
Set up cost is formulated as ′ , and the cost of refurbishment is Putting the values of 1 , 3 and from the equations (1), (5), and (9) respectively and simplifying them, we have the holding cost of the manufacturer in the refurbishment unit as Therefore, the manufacturer's total cost in the refurbishment unit can be seen in equation (10).
The total revenue of the manufacturer obtained from selling the products is

retailer's model
In this section, the model at the retail level is proposed. The following differential equations in equation (11) describe the rate of change of the inventory level of the retailer .
The ordering cost of the retailer is , Purchasing cost is . The inspection cost is modeled by 3 = . The total profit of the retailer can be seen in equation (14).

Solution method
In this section, two strategies are used to analyze the model: (1) Leader-follower relationship strategy emphasizes that the supply chain members make their decisions sequentially to optimize their profit; i.e., first, a player (leader) fixes the decision variable, then the other members (followers) take their decision following the decision taken by the leader. (2) Integrated approach signifies that all the members make their decisions in an integrated way to optimize the total profit of the supply chain. In this model, the decision variables are the lot size of the supplier, the production rate of the manufacturer, and the selling price of the retailer. The purpose is to find the strategy that will be more profitable for the SC.

leader-follower relationship strategy
In this model, the supplier first optimizes the decision variable. Then, following the decision, the manufacturer makes the next decision. Finally, the retailer optimizes the decision variable following both the supplier and the manufacturer. Since the total profit of the supply chain is the summation of the total profits of the supplier, manufacturer, and retailer, the concavity of their profit functions should be proven to optimize the chain's total profit. Therefore, Π can be seen in equation (15).
The next step is differentiating equation (15) with respect to Q. it is modeled in equation (16).
It proves that the supplier's profit function is concave. Therefore, the optimum value of Q by equating the right-hand side of equation (16) to zero is obtained.
Differentiating equation (17) with respect to , the following is the formula that is described in equation (18). It proves that the manufacturer's profit function is concave. Therefore, the optimum value of the manufacturer's production rate by equating the right-hand side of equation (18) to zero is obtained. Theorem 3. The retailer's profit function is concave. Proof: The retailer's total profit per unit time is Π = 1 (Π ) (equation (19)) where, Differentiating equation (19) with respect to Sr the following is the equation obtained (equation 20): and, It proves that the retailer's profit function is concave. The optimum value of the retailer's decision variable is obtained by equating the right-hand side of equation (20) to zero.

Integrated approach
In this section, this research explained the implementation of the integrated approach. The supplier, manufacturer, and retailer take their decisions together in an integrated way to optimize the overall profit of the supply chain. As a result, the total profit of the supply chain per unit time is formulated in equations (21) and (22).
Now, the first derivative with respect to Q is presented in equation (23). and, Therefore, the profit function, in this case, is concave with respect to Q. The first derivative with respect to is presented in equation (24). and, Therefore, the profit function is concave with respect to the production rate of the manufacturer, provided that, The first derivative respect to is presented in equation (25). And, Therefore, the profit function is concave with respect to the retailer's selling price.

Numerical example
This research used the following numerical example to illustrate the applicability of the proposed model This study also performs sensitivity analysis on variables y, β, α, and b1.

Results and Discussion
The optimum results are shown in Table 1. It can be seen that the supply chain's total profit was more in the integrated approach than the leader-follower relationship method, or in other words, the supply chain performed better when its players worked in an integrated way. This table also shows that the retailer's selling price was significantly lower in the integrated approach, which is a vital aim of a supply chain. Also, the lot size of the supplier was larger in the second approach. However, the manufacturer's production rate was lower in this method. This research studied the effect of change in some parameters on the supply chain profit and the optimum decision variables depicted in Table 2 to Table 4. The effect of changing y can be seen in Table 2. With the increase in the variable y, the lot size of the supplier increased. However, the total supply chain profit decreased in both approaches.
The effect of change in is described in Table 3. It shows that the increase in value was directly proportional to the increase in Π . As the value of the learning parameter increased, the profit of the supply chain increased, which was quite practical. A change in the variable yielded different scenario (Table 4), where the profit decreased sharply with the increase in this parameter. The profit of the supply chain in both approaches decreased sharply with the increase in parameter 1 (Table 5).

Conclusion
This supplier-manufacturer-retailer supply chain model includes some practical factors like inspection error, the learning effect of the employees on the production cost, and the use of dual-channel by the retailer for selling the products. Besides, the manufacturer offers a return policy to the retailer to entice them to purchase more products. The manufacturer reworks the defective products, including those returned from the retailer, and sells these directly to the customers as refurbished products at a reduced price (as these products may still fail to function, harming the brand reputation). In these scenarios, the derivation of the profit functions of all SC players and the entire SC system ensued. The Leader-follower relationship method and integrated approach were employed to solve this model.
The numerical result shows that when the supply chain players optimize their decisions together in an integrated way, the supply chain gains more profit. Also, it was possible to sell the products to the customers at a reasonable price when using the integrated approach. Therefore, this model found the beneficial impact of the learning effect on the SC profit.
Researchers can extend this model in various ways. Most importantly, researchers can extend this model by adding one or more supply chain players. Furthermore,