EFFECTS OF GREEN IMPROVEMENT AND PRICING POLICIES IN A DOUBLE DUAL-CHANNEL COMPETITIVE SUPPLY CHAIN UNDER DECISION-MAKING POWER STRATEGIES

. With the intensive growth of internet use, the customers choose the online market as the right preference. Hence, manufacturers are attracted to launch an online channel that includes a retail channel. To maintain the versatile demand types of products, a retailer is to stock more than one product of the same category, and consequently, he has to purchase products from different manufacturers. This article formulates a dual-channel supply chain model with two manufacturers and a standard retailer, where the optimal online prices, retail prices, wholesale prices, and level of green improvements are decided under different types of decision making power strategies such as Centralized, joint manufacturers Stackelberg, separate Stackelberg, Nash games are investigated. The optimal results are derived and compared with the help of a numerical example. Moreover, a sensitivity analysis is performed to scrutinize the effect of some important parameters. It is found that the green level is higher in a double dual-channel model than in a single dual-channel model. Moreover, the own-channel price sensitivity parameters affect the profit functions of the members negatively. The manufacturers must control the cost-coefficients of greening to increase the green level of the manufacturing products.

that, Section 6 states the managerial real insights and implementations. Finally, in Section 7, the findings and future directions of the research studies are presented.

Literature Review
This section mainly focused on the vital research conducted in this direction and visualized the research gaps. This section starts with the outlines of the recent literature on the DC SCN, observes various GI research, and finally proposes the overview and research gaps.

Dual-channel supply chain
Chen [4] calculates the effect of pricing decisions and advertising methods on the DC SCN under different power-making strategies. It also investigates the optimal level of advertisement, investment, and selling prices. Huang et al. [18] consider a DC SCN with disrupted production type and analyze the optimal pricing decisions, production quantity, and the corresponding profits. They also draw some managerial insights on the strength of numerical examples. Tiaojun and Shi [37] examine the channel priority approaches under shortage due to arbitrary yields. The optimal DVs and profit functions (PFs) are discussed under different channel coordinations. Batarfi et al. [2] examine the effect of introducing a DC with a two-level SCN, in which the manufacturer sells standard products through the RC, and a particular type of customized products are sold through the OC. Chen et al. [6] consider a DC supply along with pricing and quality decisions and also analyze the effect of adding a new channel on the existing SCN. Wang et al. [38] investigate the price and servicing decisions of products with a retailer and two manufacturers, and numerical examples are formed to examine the optimal results. All these articles have investigated a different kind of DC SCN consisting of non-green products. Jafari et al. [19] formulate a DC SCN with a manufacturer and more than one retailer where a linear discount policy is proposed. Also, they investigate the equilibrium decisions with respect to different decision-making powers of the members of the SCN. Nowadays, the awareness about the environment increases very fast, and consequently, some researchers [20,22] take an interest in the Green products in the DC SCN. Heydari et al. [14] consider a SCN where the manufacturer conducts GI, and the optimal pricing decisions and the channel coordination is discussed. Pi et al. [29] consider a DC SCN and service strategies involving two retailers and a common manufacturer and construct a numerical example with a hypothetical data set and find some essential managerial insights. Ranjbar et al. [32] formulate a DC three-level closed-loop (CL) SCN with a manufacturer, a retailer, and a collector. They also assess the players' optimal decisions in various leadership game models. Rahmani and Yavari [30] consider a DC SCN in two decision-making structures. Then the pricing, GI level, and profits are calculated under demand disruption. Aslani and Heydari [1] discuss the issues of pricing, green level, and the coordination of the channels in a DC SCN. Moreover, transshipment contrast is proposed to analyze its applications and fulfillments. Ranjan and Jha [31] examines the pricing decisions and the coordination phenomena in a SCN where the retailer sells the non-green product, but the OC sells GI products, and finally evaluates the optimal values of the DVs. Gao et al. [10] consider a DC SCN where the government set a minimum GI level to be maintained. Furthermore, they discuss the impact of Greenness on the optimal DVs of the members of the chain. Chen et al. [7] examine the optimal decisions of the members in a DC SCN under different game-theoretic frameworks. Also, they focus on the retail service, manufacturer service, and quality effort and obtain some important insights. After that, a multi-channel SC is considered by Sarkar and Pal [33] where a single manufacturer deals with two retailers and also opens an OC. Also, the manufacturer has provided direct service to the customers. Then, the equilibrium decisions are obtained, and the best profitable strategy is detected. Cao et al. [3] study the production quantity and pricing decisions in a closed-loop DC SCN under two types of subsidy policies. Also, they investigate the best profitable policy. Esmailnezhad and Saidi-mehrabad [8] propose a mix-integer non-linear model to study the manufacturing systems in a three-stage SCN to act with the customers' demand functions. Sensitivity and a real case are analyzed to provide important insights. Also, a DC SCN is composed of a retailer and a supplier by Yan et al. [41] to discuss the optimal decisions of the SCN members under decentralized models with demand disruption. The results show that the performance of SCN can be improved by enriching the revenue-sharing contract. Taleizadeh et al. [36] investigate the effect of the pricing and quality of the products under the return policy. A solution algorithm is described and performed to solve the numerical example and analyze the parameters' sensitivity. On the other hand, [35] and [5] have also studied in DC SC models and investigated the optimal decisions of the members.

Green innovation
SCN articles on green innovation discuss strategies to increase competitiveness and maximize market share to produce environmentally friendly products. According to [43], the GI level of a product can be improved by a manufacturer. Papagiannakis et al. [28] state that the Green innovated products can increase the profit of the SCN and enhance the environmental concerns. Ghosh and Shah [11] investigate a SCN with a retailer and a manufacturer. Under different channel strategies, the pricing decisions, GI level, and profits are discussed. Zhou and Ye [46] demonstrated a carbon-neutral DC SCN with a producer and a retailer. The optimal strategies are being evaluated and compared between the single-channel and DC SCN. Wang and Song [40] consider a DC SCN with uncertain demand, and the retailer provides promotional effort. The manufacturer decides the optimal level of GI, and the optimal pricing decisions are calculated. Zhang et al. [44] develops a two-stage DC SCN and investigates the best pricing and greening strategies under two different decision scenarios. Gao et al. [10] studies two types of green products in a DC SCN and obtains the corresponding players' optimal Decisions. The significant findings reveal the impact of the eco-level policy set by the government on various products. Yan et al. [42] explores a CL SCN model with a socially responsible manufacturer, a retailer, and a third-party recycler under four different decision-making strategies. As a result of the findings, the recycling rate of the wastes is increased, as well as the manufacturer's performance to become more responsible corporate social life. Taleizadeh et al. [36] study the effects of carbon emission and remanufacturing on a DC SCN in both the direction of logistics. The optimal pricing and collection strategies are also investigated. Li et al. [24] investigate the optimal DV and PF of the members of a green SCN under the Stackelberg game with different information patterns. Here, a manufacturer produces green products and sells them through two competitive retailers. Li and Liu [21] consider a two-echelon green SCN with government interventions with a supplier and a retailer under fuzzy uncertainties. Then a numerical example is presented, and also the sensitivity analysis is conducted to obtain the important managerial insights.

Research gaps and contributions
Beyond these literature reviews, the comparison table (see Tab. 1) is constructed to identify the research gaps. As shown in Table 1, the main research gaps and contributions of the present study are 1. A few articles exist considering the vertical and horizontal competition in a DC SCN. However, there is no literature considering both the competitions in a DC green SCN. This study tries to fill the gap by assuming both channel competition in a green SCN. 2. There are multiple kinds of literature on the concept of discount policy on green products, but no such literature discussed the effect of discount policy in a DC SCN consisting of two substitute products. In order to fill this gap in research, the DC green SCN is formulated where two manufacturers manufacture green products, and the manufacturers set the discount policy. 3. No literature exists that discusses all of the game-theoretic approaches the members may take. In this model, all the possible strategies are formulated and compared the profits of the members, which helps them find the best profitable strategy.

Problem description
The present research investigates a DC green SCN, including two manufacturers and one identical retailer (See Fig. 1). The manufacturers produce their desirable green improved products (say Product 1 (P1) and Product 2 (P2)) and sell them through either individual OC or the RC.   Maximum possible demand of the channels ( = 1, 2, 3, 4) Own-channel price sensitivities on demand rates ( = 1, 2) (> 0) Demand sensitivity co-efficient of the GI (> 0) Fractional part of recyclable product of one unit used product Decision variables ( = 1, 2) GI levels of P1 and P2 (0 < < ) ( = 1, 2) Retail price for P1 and P2 ($ per unit). ( = 1, 2) Wholesale price for P1 and P2 ($ per unit) ( = 1, 2) Online prices for P1 and P2 ($ per unit) Dependent variables ( = 1, 2) customer demand of P1 and P2 in OC ( = 1, 2) Customer demand of P1 and P2 in RC Π 1, Π 2 and Π M1's, M2's and retailer's PFs ($) Consequently, the retailer sells both the products (i.e., P1 and P2) at different retail prices. Manufacturers have the power to control the level of GI of their respective products. In either case, the customer will select the RC or the OC and then choose whether to purchase P1 or P2.
Following that, different decision-making mechanisms were considered, such as Centralized policy(CP), Nash game(NG), and three decentralized models (Manufacturers Stackelberg (MS), Manufacturer 1 Stackelberg (M1S), Manufacturer 2 Stackelberg (M2S)). Under the decentralized models, players take decisions separately, and the sequence of decisions is also different. In CP, the members will make their decisions collaboratively, and consequently, the total profit (TP) of the SCN would be maximum under CP. Furthermore, we will compare the DV and the profits of the members in CP with the different decentralized models. Thus, the five other leadership models are CP, MS, M1S, and M2S and NG. Here, Table 2 describes the used notation of the parameters, DVs, PFs throughout the paper. The following assumptions have been made to validate the proposed model.

Assumptions
-Manufacturer 1 (M1) produces P1 with the GI level 1 and sells it through the RC at wholesale price 1 and sells directly to the customers through OC at a price 1 . Consequently, P2 is produced by Manufacturer 2 (M2). M2 imposed the GI level 2 while producing the product and sold it at wholesale price 2 to the retailer and fixed OC price 2 . The retail prices of P1 and P2 are 1 and 2 respectively. -All the demands are taken to be linear dependence pricing decisions, namely online prices ( 1 and 2 ) as well as retail prices ( 1 and 2 ) and level of GIs [24,38,43,46]. It is assumed that the demands are downward directing of the own channel pricing, upward directing of the cross channel pricing, and upward directing of the products' GI. For the demand for simplicity, the production costs of 1 and 2 are considered 0. Let 1 and 2 be the customer demands through the OC of P1 and P2, respectively. The customer demands of P1 and P2 through RC be 1 and 2 respectively, and the demand functions can be formed as follows: The self-price sensitivity parameters of each of the demand function are more effective than the cross channel price sensitivity parameter i,e., > , = 1, 2, 3, 4 [12,16,31]. Furthermore, the impact of self-price sensitivity parameters is greater than the sensitivity parameters of GIs of demand rates (i.e., , = 1, 2) i.e., > . It is also considered that both the manufacturer maintain a fixed ratio of OC and RC so that online prices remain lesser than the retail price. Mathematically, = , where < 1 is the fixed number determined by the corresponding manufacturer.
-The manufacturers are not subjected to the effects of marginal cost. Moreover, it has some fixed cost for imposing the GI on the manufacturers. The cost functions of the GI is considered as a convex function ( ) = 1 2 2 ( = 1, 2), where is the green cost coefficient [11,38].

Notation
The notations are used to develop the model are list in the following table.

Model formulation
With the help of above assumptions, we obtain the PFs of the players of the model as follows: where the notations are described previously and 1 , 2 , 1 and 2 are taken from (3.1), (3.2), (3.3) and (3.4) respectively. The subscripts 1, 2 and represent the M1, the M2 and the retailer respectively.
The TP of the SCN is the addition of the individual players' profits and is obtained as Now, We consider the following game theoretic models: Under each game-theoretic approach, the procedure to determine optimal decisions and consequently the profits of the players are discussed.

Centralized policy
All SCN players operate as a single-player under this strategy, i.e., there is a central decision-maker to make the decisions for the whole system. The results of this model are usually used as a benchmark for comparing with the decentralized models, and there is only one overall profit for the entire SCN. The TP of the SCN is obtained from the equation (4.4), and it depends on the retail prices ( 1 and 2 ) and the GI levels ( 1 and 2 ) of the P1 and P2. Simplifying the equation (4.4), we have Proposition 4.1. The PF of the CP (4.5) is maximum at the point: if the following conditions holds: The optimal profit of the SCN can be obtained by substituting the values of the DVs from the proposition 4.1 in the equation (4.5).

Manufacturers' Stackelberg
In the Stackelberg model, all the members of the SCN optimize their corresponding decisions one by one according to the decision-making power. Both the manufacturers act as a single-player in this game and lead the SCN. In the end, the retailer makes his decision following both the manufacturer. Member leadership mainly depends on the members' ability to make decisions and has a truthful effect on the SCN. In this study, the level of GI is controlled by the manufacturers to offer a new eco-friendly product to the customers, so the manufacturers have more ability to make decisions and are considered as a leader. According to the Stackelberg model principle, the optimal response to the follower (i.e., the retailer) is derived. Then, using these optimal responses in the leader's PFs (i.e., manufacturers), the optimal decisions of the leaders are determined. Therefore the formulation of the model according to the decision making power is as follows: {︂ 1 : Profit of M1 + M2 2 : Profit of retailer i.e., Proposition 4.2. The PF of the retailer Π is maximum at the point: if the following conditions holds: Then, substituting the values of 1 and 2 in Π , the PFs can be maximized.

Proof. See Appendix B
Using the optimal decisions of the members, the profits of each of the member can be maximized.

Manufacturer 1 Stackelberg
Here, we assume that M1 has the greatest ability to decide first, and that M2 is the follower. The retailer will follow both of the manufacturers. Therefore the M1 sets his/her corresponding DVs, and then other players will optimize their related profits accordingly by setting their DVs. We formulate the model as follows: The solution procedure of this model is discussed in Appendix D.

Manufacturer 2 Stackelberg
This section simply interchanges the manufacturer's decision-making powers of the previous Section 4.3 and then repeats the similar procedure as given above.

Nash game
To validate the NG in our proposed model, we have to construct an assumption as follows: the retail price is depended on the wholsale price and the relation is = for = 1, 2 and = 3, 4 provided 1 , 2 > 3 , 4 (using the assumption, < ). Putting the values = in the equations (4.1), (4.2) and (4.3), the following equations are obtained: When the players have same decision power and have set their respective decisions independently and simultaneously, then the game is called NG. We formulate the model as follows: The maximum profits of the players are maximum under NG when the values of the DVs are as follows: if the following conditions holds: Proof. See Appendix C The optimal decisions of the members are listed in the proposition 4.3 and the profits are maximum at that point.

Discussion of Results
We evaluate the sensitivity of the model's important parameters and explain the behavior of the proposed model with a numerical illustration.

Numerical Example
Here, a numerical example illustrates and validates the proposed model. Due to the difficulty of collecting real-life industrial data, some of the data is gathered from the previous literature, and the remainder is assumed hypothetically to verify the proposed model. In this study, the hypothetical data are consistent with the published literature [45], and [38] to the largest extent possible; however, it is impossible to assume an identical data set with any previous literature as our model is uniquely formulated and have studied never before. Therefore, the hypothetical example has been constructed as follows: Let us consider a market with two manufacturers (M1 and M2) and a common retailer. The M1 and M2 produce P1 and P2 respectively and sells them through either their personal OC or common RC. Let the market potential of OCs of M1 and M2 are 425 units/unit time (i.e., 1 = 425) and 410 units/unit time (i.e., 2 = 410) respectively and the market potential of the RCs of M1 and M2 are 475 units/unit time (i.e., 3 = 475) and 490 units/ unit time (i.e., 4 = 490) respectively. The own channel price sensitivity of the OCs of M1 and M2 are 5.6 unit/ unit $ (i.e., 1 = 5.6) and 5.5 unit/ unit $ (i.e., 2 = 5.5) respectively and the RCs are 6 unit/ unit $ (i.e., 3 = 6) and 6.25 unit/ unit $ (i.e., 4 = 6.25) respectively. Let also consider the cross channel pricing sensitivity of the channels is 1 unit/ unit $. Next, let both the manufacturer provide a discount 10% on the retail price to sell the similar products through their corresponding OCs (i.e., 1 = 0.9 and 2 = 0.9). The demand sensitivity of coefficient of 1 and 2 be 0.75 and 0.73 (i.e., 1 = 0.75 and 2 = 0.73) respectively and the cost coefficients of the level of GI 1 and 2 are $ 1.85 and $ 1.82 (i.e., 1 = 1.85 and 2 = 1.82 ) respectively. Hence, the data set is as follows: 1 = 0.9; 2 = 0.9; 1 = 425; 2 = 410; 3 = 475; 4 = 490; 1 = 5.6; 2 = 5.5; 3 = 6; 4 = 6.25; = 1; 1 = 0.75; 2 = 0.73; 1 = 1.85; 2 = 1.82; Now we check the conditions of optimality and also find the optimal results under each strategies.
In Table 3, the eigenvalues of HMs of the PFs are listed, and the optimal results are stored in Table 4 under each strategy.
In Table 3, all the eigenvalues are negative, i.e., the optimality condition holds. Therefore, the values of DVs are optimal, and the PFs are maximum at the values. The optimal values of the DVs and each member's maximized PF have been listed in Table 4 for each strategic game structure.

Discussion regarding Numerical analysis:
In Table 4, it can be observed that the total SCN's profit is maximum under CP, and the TP can be arranged as Π > Π > Π 1 > Π 2 > Π . The level of GI is maximum under CP. Therefore the products become more eco-friendly if the members make decisions jointly; consequently, the demands under CP are higher than the others. According to both manufacturers, the best profitable strategy would be NG. Retailer able to make Notes. ( ) Here we consider that both the manufacturer gives a discount of 20% on the retail price the best profit under MS2 and the retailer's profit obeys the inequality Π 1 > Π 2 > Π > Π . The retail prices are maximum under MS and lowest under NG. The CP always gives maximum profit for the SCN. When the players settle a contract on the wholesale price, the manufacturers provide a discount on the selling price to the retailer and agree to jointly make the decision, i.e., they accept the CP strategy. The discussion on individual profits behavior is stated in Corollary 5.1. The total channel's profit of CP is larger than any other strategy, but the members' profits in CP do not need to be always greater than their individual profits in other strategies, which may be less or equal. To analyze the impacts of the parameters 1 and 2 , the parameters are taken fixed in the numerical example 5.1 and only the value of ( = 1, 2) will vary at a time. Figure 2 demonstrates the change profits of the Manufacturers, retailer, and the SCN with ( = 1, 2). In Figures 2a and 2b, one can observe that the TP of the SCN is larger in CP than any other strategies. In addition, the PF rises with each increment of .

Discussion on parameters' sensitivity
The individual profit of each of the manufacturer is maximum under CP and the curve of the PFs are in upward direction with increasing values of (See Fig. 2c, 2e, 2d, 2f). But it is shown in Figures 2g and 2h that the retailer's profit is higher under strategy-M2S than any other strategies. Hence it is more profitable to join in the M2S strategy from in retailer's point of view, but the manufacturers would like to join in CP to gain more profits. The demand coefficients of the level of GIs (i.e., ) have positive impacts on each channel's demands. Table 5 describes that if is high, then the members would like to fix their DVs to an upper value. Thus if the demand coefficients of GI are large, then the manufacturer will develop the level of GI, which improves the demand rates and allows the manufacturers and retailer to set higher prices of the products. Since 1 is directly connected with 1 , so with the increment of 1 , the optimal value of 1 increases faster than the increment of 2 . Also, the level of GI is always higher under CP than any other strategy. Thus, CP performs better as an eco-friendly product.     Figs. 3c, 3d). The effect of 4 on the M2's profit is same as M1's profit (See the Fig. 3f). The effect of 1 on the profit of M2 is described in Corollary 5.2.  Figs. 3g and 3h). The profits are very sensitive with these two own-channel price sensitivity parameters 1 and 4 . With the increment of 1 and 4 , the demand rates negatively impact. As a result, the players have to decrease their corresponding pricing decisions. Depending on the value of 1 , the M2 chooses the most profitable approach (See Cor. 5.2). Furthermore, the level of GI decreases with the increasing values of 1 and 4 . Therefore, to make more eco-friendly products, the values of the own-channel pricing sensitivity should reduce as much as possible.

5.2.
3. Impact of the Parameters 1 and 2 Figure 4 depicts the variation of the profits with respect to the costs coefficients of GI (i.e., 1 and 2 ) and the changing behavior of the DVs are listed in Tables 5 and 6. All the DVs decreases with the increment of 1 and 2 . In CP, the SCN's TP, M1's profit and M2's profit are maximum compare to the other strategies (See the Figs. 4a, 4b, 4c, 4d, 4e, 4f). In Figures 4g and 4h, it is clear that the retailer's profit are equal in M1S and M2S and higher than the other strategies. With the inclusion of 1 and 2 , the retailer's PFs decrease slightly, it is not very sensitive with 1 and 2 . Also all members' PFs decreases with the increment of these two parameters.
The increment of coefficients of GI (i.e., ) reduces the optimal level of GI, which has a direct negative impact on the demands of both the product (See the Tab. 5). Therefore to reduce the negative impact of the lesser level of GI, the players (the manufacturers and the retailer) have to reduce their corresponding pricing decisions. Also, the members' profits decrease with the increment of . Therefore, the manufacturers should decrease their corresponding cost coefficients of GI so that the level of GI and the profits increase.

Impact of the Parameters
In Tables 5 and 6, it can be shown that as the cross-channel pricing sensitivity parameter is increased, all of the DVs increase. Then, Figure 5 illustrates that the curves of the members' PFs arise upward very gradually for the increment of , i.e., the PFs are very sensitive with the parameter , and they are proportional to . The TP of the SCN is maximum under CP strategies. On the contrary, for M1 and M2, the profits under NG and CP are equal until the value of is less than some fixed value, but the profits are maximum under CP (See Figs. 5b and 5c).
According to Figure 5d, the retailer gains more profit under M1S and M2S than the rest strategies.
As positively impacts the demands, the members' profits are proportional to . Also, increasing helps manufacturers to impose more GI on their related products. So the members should try to increase the value of to gain more and make products more eco-friendly.

Managerial insights
The present study discusses the theoretical results and the sensitivity of the parameters. Based on them, the following managerial insights can be drawn: The TP of the SCN is maximum under CP, and the products' GI level is higher in CP than any other strategy. However, it does not imply that CP is the most lucrative strategy for each of the members. For example, in the given numerical example, the Manufacturer's best strategy is NG, and the retailer makes its best in M2S. The GI level is affected by the demand coefficient of GI (i.e., 1 and 2 ) and the cost coefficient of GI (i.e., 1 and 2 ). If the value of ( = 1, 2) increases, the GI's optimal level decreases, and the demands also decrease. Consequently, the selling prices should also drop down to push the demand up, and as a result, the members' profits decrease. Furthermore, the best profitable strategy for individual members may differ on the values of the own-price sensitivity parameters. So, it is crucial to investigate the parameters' values to choose the best profitable strategy and obtain optimal decisions. The SCN's profit is lowest in MS-game. Therefore, the SCN's TP can be increased by introducing double DC with a retailer. An important observation is that the optimal GI level of the products in double DC is greater than the single DC. Therefore, it is necessary to introduce a double DC in a SCN to reduce the product's harmfulness compared to a single DC.

Conclusion
This paper formulates a DC SCN with the concept of green improved products, where two manufacturers produce green products and sells through either by an identical retailer or through their corresponding OCs. The demand rates are linear functions depending on the retail prices, online prices, and product's GI levels. The individual manufacturer determines the level of GIs and wholesale prices, but the retailer's DV is the retail price. The manufacturer provides a fixed discount on the retail price to sell products through an OC.
Five decision models (CP, MS, M1S, M2S, NG) are formulated considering the GI to answer the research queries mentioned in the introduction. The decision models are distingue according to the power of decisionmaking of the members. The Stackelberg theory and the classical optimization technique obtain the members' optimal decisions under each decision model.
The discussion of the results expresses that CP is the best profitable strategy for the SCN, so the CP helps to accomplish the economic targets. The products' GI level is maximum in the CP, which is beneficial to the climate. If a manufacturer offers a high GI level, the optimal pricing decisions increase, and the optimal profits increase faster than the other players.
The own-channel price sensitivity parameters negatively impact the profits of the corresponding members and TP of the SCN. Also, there exists a fixed value of 1 , which determines the best profitable strategies. Also, the increment of the GI's cost coefficients decreases the optimal level of GIs and the players' profits.
Furthermore, an important conclusion is that the single DC SCN's profit is lesser than any strategies in a double DC SCN. Moreover, the optimal GIs of both the products is higher in double DC than a single one. The relation between the parameters is determined under the strategies so that the members' can achieve their corresponding optimal profits. With the help of a hypothetical data set, the optimal results of the various models will be compared, and the sensitivity of the parameters will be analyzed, and key insights will be gained.
Although the proposed model has some limitations, such as the demand functions are deterministic and linear, but in reality, it would be uncertain. In this article, a retailer with multiple manufacturers is considered within a DC SCN, but it is possible that the manufacturers have multiple retailers, which makes channel coordination more difficult. Finally, the SCN's return policies or the promotional effort is not introduced in this model. Future research can be done in several directions where this topic can be explored. Firstly, one can modify the present model by assuming an uncertain demand function. Secondly, it would be advisable for further study to consider the traditional off-line competitive situations where multiple retailers compete for the customers. Lastly, it would be interesting to study the effects of the return policies of the SCN members and the promotional effort on the member's optimal decisions and PFs.

Appendix A.
The 1st order conditions of (4.5) is given by: Solving the above equation with respect to the DVs 1 , 2 , 1 and 2 , we have Hence the function Π is maximum at the point ( * 1 , * 2 , * 1 , * 2 ) if the HM, 1 is '-ve' definite at the same point. Since 1 is independent of the DVs, we have to show that the principle minors of 1 are alternatively negative and positive. Hence, if the following conditions hold, then Π is maximum at that point: 1. -2 < 0 (which is always true as 2 is assumed to be positive), 2. 1 2 > 0 (which is always true as 1 and 2 are assumed to be positive), 3. 2 2 2 + 2 5 < 0, 4.