A sustainable game strategic supply chain model with multi-factor dependent demand and mark-up under revenue sharing contract

In the current socio-economic situation, the daily demand for essential goods in the business sector is always changing owing to various unavoidable reasons. Choosing the right method for a profitable business has become quite tricky. The proposed study introduces different business strategies based on trade credit, revenue sharing contract, variable demand and production rate. As trade credit is one of the best policies to attract customers, there are two types of models based on it. In the first model, demand depends on average selling price, green degree, and products quality. An additional trade-credit factor is in the second model. However, considering coordination, non-coordination, and revenue sharing contracts, each model has three sub-cases. The main aim is to find the best strategy for the profit maximization of the supply chain members. Green investment, maintenance, and multi-factor dependent demand make the model more sustainable. The global optimization is established theoretically and different propositions are developed. Through numerical experiments, the global optimality is also verified. Some special cases, with a comparative graph, are provided for the validation of these results and to find the best strategy for profit maximization. Finally, some concluding remarks along with future extensions are discussed.


Introduction
For the development of different business strategies in the modern competitive environment, supply chain management (SCM) plays an essential role (Sarkar and Seo [1]). The green investment makes the SCM one more dimension for sustainability. Sustainable supply chain management (SSCM) stands on the three pillars of sustainability named economy, social, and environmental benefit (Taleizadeh et al. [2]). For the environmental benefit, here green degree-dependent demand and unit production cost for reducing carbon emissions are considered. From the economic and social point of view, different game strategy, trade credit, maintenance, and mark-up policy create jobs, reduces cost, and increase the company's reputation. This study develops an inherent relationship with the production inventory supply chain system. However, the production system becomes complex due to a variable production system with fluctuating demand to reach products for customers promptly (Sarkar et al. [3], Moon et al. [4]).
Based on various past studies, some new considerations have been reflected here from the current situation. Here, the combination of different situations has been shown in two models. Here Stackelberg's game policy has been used between manufacturers and retailers to control the market demand and build a sustainable supply chain management(SSCM). The managers of industries always try to maximize profit by satisfying their customers. Researchers are always busy for developing diverse inventory and supply chain models through their research of different demand patterns. Mondal et al. [5] and Mahapatra et al. [6] extended this direction with flexibility of production and promotional effort to reduce the cost as minimum as possible. Garai and Sarkar [7] concentrated on the customize products, ordered by customers, through the reverse logistics but without having the idea of trade-credit or game strategies. In the other direction, Sarkar et al. [8] considered the quality-based demand function in their research paper. However, a new type of demand is required in the current context such that considerable demand is based on selling price, green degree, as well as credit period. Keeping in mind these types of research, the model considers a demand function consisting of the average selling price, product quality, greening cost, and credit period. Besides manufacturer units, production cost(UPC) is a function of variable production rate, tool/die cost, labor cost, development cost, quality of the products, and greening cost. Furthermore, there are required for optimization in logistics problems for supply chain networks under variable demand (Kumar et al. [9]).

Research gaps
The following research gap can be drawn based in the existing literature.
• Studies on VPR and different game strategies under SCM with quality and green degree dependent production cost for sustainable development, maintenance through markup strategy, and trade credit policy had not been studied yet. • The primary focus in previous research has been on the effects of variable demand on profit (Sarkar et al. [3]). However, selling price, quality-assured, greening level, and trade credit-dependent demand provides additional scope for maximizing profit. Therefore, such type of demand pattern and the effect on the SSCM for selecting the game strategy is a novel contribution to the literature. • Many SCM models are developed under fixed production rate and constant demand through game strategy under mark-up policy. However, some of them considered different strategies for retailer profit (Choi et al. [10]). But, how a can retailer improves the profit structure, not only for the total supply chain but also for himself/herself, by taking his/her own risk makes a significant contribution to the literature.
• Several research papers on trade credit, and revenue sharing contracts through different proposed models already have been studied (Sarkar et al. [11]). But, which one of the two models gives better responses in the context of the greening level of a product and different pricing decisions of manufacturer and retailer have rarely been investigated in the literature.

Contribution
One of the aspects of this model is investment for maintenance. Maintenance is essential to overcome an out-ofcontrol situation and reduce imperfect production due to natural causes. For all these reasons, a fixed maintenance has been considered here with the greening cost. Thus, it is possible to avoid all kinds of unwanted things due to the type of investment. A special kind of credit period facility has been given in the case of payments. Two types of models based on that particular delay-in-payment are considered here. The first model is designed without a credit period offer, the second model with a credit period offer. Three types of cases have been discussed in each model. First, the manufacturer and the retailer work together to make a profit for the entire supply chain. Secondly, as the leader and follower, the manufacturer and retailer try to increase their own profits in different ways using their suitable strategies. Thirdly, there is a revenue-sharing agreement between the participating members. They try to increase their own profits with their strategies by acting as a manufacturer leader and retailer follower under the deal. The two models' different cases have been thoroughly discussed to find out the maximum profits jointly and separately. There are two special cases considering without quality function and without greening cost. All these special cases have been compared with original models, and it has been seen that the method described in the original model is the best method for the maximum profit. This is the first study dealing with SCM, game strategy, revenue sharing contract, green degree dependent demand, quality assured unit production cost, maintenance, and markup policy under sustainability concept with variable production rate. For the environmental issues, the green investment makes the model more eco-friendly. A mathematical model is proposed and more importantly, the global optimality of the resulting solution has been proved. This model maximizes the total profit as well as the separate profit of manufacturer and retailer by optimizing the markup set by the manufacturer and retailer, products quality, production rate, and greening investment. The classical optimization technique is utilized here for the optimal solution of the decision variable and for the optimization of the profit. QGOFD -"Quality and greening cost and other factors dependent". QGSTCD -"Quality, greening cost, selling price and trade credit period dependent". SSCM -"Sustainable supply chain management". GDD -"Green degre dependent". GCD -"Green level and credit period dependent". SCM -"Supply chain management". RODLD -"Retail price, online price, and delivery lead time dependent". QSD -"Quality and selling price dependent". SCPCAD -"Selling price, credit period and credit amount dependent". EPQ -"Economic production quantity". AD -"Advertisingdependent". CPD -"Credit period dependent". SPCED -"Selling price and carbon emission dependent demand". SPD -"Selling price dependent". SPQD -"Selling price and quality dependent". ADT -"Advertisement and time-dependent"

Structure of this study
The remainder of this paper is organized as follows. Section 2 describes the related literature review. Table 1 gives the research gap among previous authors. Section 3 presents the problem purpose, relevant symbols, and associated assumptions. Section 4 describes the model formulation, and Sect. 5 provides the methodology of the solution. The numerical experiments are described in Sect. 6, and the sensitivity analysis is given in Sect. 7. Section 8 provides managerial insights into the study, and Sect. 9 presents the conclusions.

Related literature review
Contributions of previous research concerning the model proposed in this study and the gaps in brief for the literature are discussed in this section. Furthermore, the novel contribution of this study is stated in the subsections. This model's importance is immense for the social development of the world by fasting from the horrible aggression of the current society. Through sustainable development, socio-economic conditions can be improved. For delivering the necessary goods before the customer's hand sustainable supply chain management (SSCM) is very much essential in our society. The contributions of previous researchers to SSCM research are described in the first subsection. Further, game strategy, revenue sharing contract, trade credit, and multifactor-dependent demand are important keywords related to this model. Existing research about the keywords is discussed in this section. However, for a better understanding of the research gap, a research gap determined in Table 1 is provided. A glimpse of them is repeated below.

Sustainable supply chain management
For coverage of the different needs of customers in the current market, the SCM plays an important role in the world. The SSCM is an extension of that approach. However, in today's competitive business world most commercial enterprises are tied up in a supply chain management system to get tensionless stable sources of market and also to avail the maximum profit from each other. For reducing waste and controlling the environmental effects of green production, there is a need of a sustainable approach (Moon et al. [4]). This concept automatically improves society. Another way of controlling environmental pollution is reducing deterioration and reducing carbon emission with the concept of circular economy. Sarkar et al. [12] introduced a new concept of circular economy to reduce waste and pollution. However, there are other approaches to a sustainable supply chain through a coordinated way of members. Thomas and Mishra [13] recently proposed an SSCM for minimizing the waste in the plastic industry. Another sustainable approach was developed by Sarkar et al. [14] through reduced carbon emissions and considering the substitutable product. They reduced the defective products for economic and environmental benefits. Sephari et al. [15] discussed a sustainable production model of deteriorating items through preservation and controllable carbon emissions. An SCM model with controllable lead time and transportation discount approach for the sustainable approach were introduced by Bhuniya et al. [16]. A sustainable biofuel manufacturing model for renewable energy through variable production and autonomation were developed by Sarkar et al. [17]. Dey and Giri [18] discussed a reverse SSCM model for the selection of the best business strategy. It is clear from these studies mentioned earlier that strong sustainability is based on special investment in the green environment. Every earlier research generally focussed on the sustainable development of the inventory management and SCM, but no earlier research can focus on the effect of green investment, products quality, mark up, and VPR on the SSCM. However, to find the best strategy among the SCM member through maintaining SSCM, game strategy can play an important role in avoiding the shortage as well as making a good reputation. Moreover, the a research gap in the literature on the effects of game policy on SSCM for business strategy selection. Thus, in the next section, the effects of game strategy with SSCM are described.

Game strategy
In recent times, SCM members have resorted to various approaches to maximize profits at minimal cost, with different game strategies. The concept of leader and follower among SCM members is replaced by a game strategy based on coordination and non-coordination. With the game strategy, it is easy for the members to run the business without any hassle. Many research papers have been published on different game strategies in the past. A new concept-based inventory model were developed by Jaggi et al. [19] in which credit-linked demand and permissible delay in payments were considered. They introduced little lead time and an instantaneous type of replenishment rate. An imperfect production-based model were proposed by Chen and Kang [20]. They considered coordination strategy between members of SCM through trade credit policy. A cost-sharing contract-based model were studied by Ghosh and Shah [21]. They introduced different game strategic planning with green-sensitive demand. Heydari et al. [22] discussed a stochastic model of remanufacturing in which coordination and revenue sharing contract(RSC) between SCM members help find the best strategy. Song and Gao [23] introduced an SCM with different game strategies and revenue-sharing contracts for green products. In the other direction, the mark-up rate-based SCM were developed by Panja and Mondal [24]. They applied the Stackelberg game policy, credit period, and revenue sharing agreement to maximize the SCM profit. Sarkar and Seo [1] considered an SCM of renewable energy by considering the coordination and non-coordination steps of the members. Three joint decision models were constructed by Cao et al. [25] for the maximum profit of the SCM members through the stackelberg game approach. Recently, Dey and Giri [18] discussed the effects of coordination strategy on the SSCM through quality uncertainty.
The previous research details stated in this section mainly focussed on the game strategy. The research considered different game policies among the SCM members only but the effects of revenue sharing contract (RSC) on the game strategy under SSCM through VPR is not done yet. However, the best way of the maximum profit earned in an SSCM with manufacturer FPR, green investment, and quality assured joint mark up, which is the novel concept, introduced in the proposed model. Moreover, the RSC is the main way of profit maximization in the business strategy. The effects of game strategy on the RSC in an SSCM system are described in the next section.

Revenue sharing contract
A successful supply chain is characterized by a consistent policy of distributing profits made through business to the supply chain members. Revenue sharing contracts also help to be transparently active among supply chain members. Much research has already been done depending on this topic. An SCM with revenue sharing agreement and coordination strategy were developed by Zhang et al. [26]. They introduced the demand fluctuations on the centralized SCM. Zhang et al. [27] extended the previous model by considering deteriorating items. They introduced demand as a function of selling price and introduced preservation technology to reduce deterioration in their model. An imperfect productionbased SCM model under maintenance policy were developed by Bhuniya et al. [28] with multi-factor-dependent demand. In the other direction, SCM with commitment and revenuesharing, and penalty contract were discussed by Shafiq and Savino [29]. They considered short life-span products with a make-to-order policy. The perishable products-based supply chain model was incorporated by Avinadav [30]. Chernonog [31] developed a consignment contract with revenue sharing among the SCM members under uncertainty. In an SCM, revenue sharing-commission coordination contracts for community groups were discussed by Lan and Yu [32]. Recently, Dey and Giri [18] published a research article about incorporating RSC in an SSCM. From the above research studies, it may be concluded that every research just included RSC in their SCM model. No research still now considers the effect of RSC on an SSCM under VPR and game strategy for quality assured mark-up policy. However, RSC transparently encourages all interested parties, which are connected with sustainability. The proposed study also considered green investment and quality assured unit production cost, which is very much helpful for environmental protection. The RSC is correlated with trade credit. In this sense, how the trade credit connects with RSC for the SSCM under VPR and game strategy, is discussed in detail in the next section.

Trade credit
In recent times it has become an attractive tool in the competitive market. This policy provides unique benefits to customers in terms of payment. One of the most common ways to reduce credit period through gift policies is that discount policies are widely used in the market. Besides, customers do not have to pay the bill and the purchase through trade credit, but if customers do not pay after such a regular period, he/she has to pay at an increased rate. Many studies have been published on trade credit such as Pramanik et al. [33] discussed a three-level trade credit policy based on SCM, where demand is a function of the selling price, credit period, and credit amount. The previous model were extended by Wu et al. [34] through the default risk of trade credit. They also considered a delay in payments policy a better comparison among different cases. Jani et al. [35] discussed a research paper of perishable products with trade-credit and shortages. Yan and He [36] studied trade credit with deferred payment. Xu and Fang [37] discussed an SCM with a Partial credit guarantee and capital constraint. Mashud and Singh et al. [38] considered a supply chain model through two-level trade credit under a carbon emission reduction policy. In an SCM, the effects of carbon emission and energy with a two-level trade credit policy were studied by Sarkar et al. [12]. A partial type of trade credit in a sustainable inventory model through carbon emission were developed by Taleizadeh et al. [2]. Recently, Rini et al. [39] introduced two-stage credit financing for strategic decisions in an imperfect quality and inspection scenario. From the above research studies, it is found that no research focussed on the effect of trade credit and RSC on the VPR and mark up. The proposed study fulfills this gap. However, the trade credit for RSC through game strategy and green investment creates a new research direction. Moreover, there needs further investigation of the effects of multi-factor-dependent demand (MFDD) on the trade credit, RSC, game strategy, and overall SSCM. This research gap, which is discussed in the next section, is crucial to the supply chain.

Multi-factor dependent demand
The demand for every necessary item in the market depends on a different manner, such as product quality, market price, green degree, advertisement. Nowadays, Product quality and the market price is the best technique for increasing demand. Moreover, market demand mostly follows the freshness of the product. Also, the various factors of demand are significant in terms of business methods and selection. In the past, among many research papers, different demand policies were considered by different authors. Reduction of pollution through sustainable approach under flexible production system were introduced by Yadav et al. [40]. On the other hand, the variable type of mark-up and selling price, quality base demand was considered by Maiti and Giri [41]. Selling price and time-dependent delivery demand-based SCM were studied by Modak and Kelle [42]. They also considered leading time-dependent and stochastic demand for comparison studies. Kianfar [43] discussed an advertisement and price elasticity-based demand for a supply chain to maximize profit. Sarkar et al. [8] proposed a dual channel retailing based on autonomation policy. In their model, demand is based on selling price and quality. An average selling price and service dependent demand under multi-retailer based SCM was discussed by Sarkar and Bhuniya [44]. They considered budget and space constraints with service level facility to the SCM members. Deng et al. [45] updated an SCM with uncertain demand. A three-echelon green supply chain management for biodegradable products were developed by Sarkar et al. [3]. Taleizadeh et al. [2] discussed selling price and carbon emission-dependent demand function in their sustainable inventory model. From the above discussion and Table 1, one can find that different researcher-developed supply chain models consider different practical strategies to minimize the total cost or maximize the total profit. There were several research models by considered separately VPR, game strategy, RSC, trade credit, and MFDD in an SSCM. Most of the models were optimized through constant production rates and constant demand. Sustainable SCM under trade credit, RSC, and MFDD is an interesting topic in the present socio-economic situation. However, the research on sustainability through green investment, the VPR for smart supply chain, variable demand for controlling the market demand without shortage, the best strategy selection through maximum profit for strong long-run business process, and RSC contains a big research gap in the literature. All these keywords together not considered in the previous research. Therefore, a sustainable SCM with variable production, trade credit, RSC, game strategy, and MFDD are considered in this proposed study to fulfill the research gap.

Problem definition, notation and assumption
The following subsections consider the research problem, the corresponding notation and assumptions.

Problem definition
The supply chain is an essential pillar of delivering produced goods from the production factory to the customers' doorsteps. Provision of business development with different game strategies based on the leader-follower relationship between the supply chain members. In that case, two models have been considered depending on different strategies. In the first part of the model, the manufacturer produces the goods with ϕ p unit production cost and the delivered to the retailer with ζ 1 (ζ 1 > 0) mark-up and wholesale price p m . The retailer then delivers the product to the customer with ζ 2 (ζ 2 > 0) mark on the manufacturer's full sale price and retail price p r . The manufacturer's unit production cost depends on the variable production rate, tool/die cost, development cost, labor cost, quality of the products, and greening cost. The first two cases have been considered in terms of group work (centralized case) and leader-follower concept (decentralized case). In case number three, the leading manufacturer and follower retailer business has been outlined through the revenue sharing agreement. Model II incorporates the concept of credit period to facilitate customer payments. In this system, the follower retailer keeps a delay-in-payments facility for customers as a result. Even if it is partially lost, the business gains as a whole. For the second model, which is entirely the same guideline as to the first model and demand, is based on the credit period. Furthermore, the profit function of the manufacturer, retailer, and the whole supply chain has been changed. Here demand considers as a function of credit period, quality of the products, greening cost, and average selling price.
Moreover, this study considers different examples to show the model's robustness. This model considers manufacturer mark-up, retailer mark-up, joint supply chain mark-up, quality of the products, greening cost, credit period, and production rate as decision variables and at their optimum values to find total expected profit. The optimality of the decision variables and the total expected profit of manufacturer, retailer, and whole supply chain in different cases are proved here analytically. The classical optimization technique helps here to obtain decision variables' global optimum values and total expected profit.

Notation
To formulate the mathematical model following symbols are considered.

Decision variables
Mark up fixed by the manufacturer on the basis of UPC ζ 2 Mark up fixed by the retailer on the basis of manufacturer's price ζ 1 ζ 2 Joint supply chain mark-up based on manufacturer UPC q Quality of the produced product P Production rate ($/cycle) ξ Credit period offer (time unit) η Greening cost ($/cycle) Centralized case in first model Decentralized case in first model I 3 Revenue sharing contract case in first model Centralized case in second model Decentralized case in second model Revenue sharing contract case in second model Minimum selling price from retailer side ($/unit) α Scaling parameter for the quality function β Scaling parameter for the credit period ψ Loss of interest per product from retailer's side ($) μ Shape parameter of quality function γ Credit period facility when double mark-up fixed by the retailer (time unit) θ 1 Variation constant of raw material cost ($) θ 2 Variation constant of development cost ($) θ 3 Variation constant of tool/die cost ($) φ 0 Scaling parameter of the quality function to the considerable UPC function λ 1 Scaling parameter of the green investment to the considerable UPC function λ 2 Scaling parameter of the green investment to the demand function λ 3 Scaling parameter of the greening cost ϑ Fixed maintenance cost ($/cycle) Scaling parameter of the demand function Assumptions 1. The market price of any recent product is largely dependent on quality. On the other hand, the unit production cost has been considered, keeping in mind the environmental protection and green demand. Besides the cost for labor payment, development and maintenance of tool/die are included with the UPC. Here per unit production cost of the manufacturer be such that In the case of daily demand, it has been observed that customers always try to buy the right quality products for their health. Again, it is better to think that the lower market price of the product. So in Model I, demand consider as a function of price, product quality, and greening cost be such that It is often in many cases that customers cannot buy for lack of money despite the need. With this problem in mind, arrangements have been made to facilitate delayin-payments to the Model II customers. Here demand additionally depends on trade credit period, and function formula is such D = ( p max − p r ) ( p r − p min ) + βξ + αq μ + λ 2 η. 4. In any production process, the manufacturer always sets a mark-up at his own unit production cost when pricing a product. Here wholesale price per unit at the manufacturer's side be such that p m = ζ 1 ϕ p , (ζ 1 > 1). 5. Similarly, when the retailer gives supply the same items to the market, he/she will charge the mark-up, which depends on the manufacturer's full sale price. Here the price formula is such that p r = ζ 2 p m , (ζ 2 > 1). 6. Delay-in-payments have become an essential part of the business. Here credit period relaxation for the payment of customer is a strategy for more popularity and maximum sales. Here credit period formula is considered to be such that ξ = γ (ζ 2 − 1).

Model formulation
Two types of models are described below. The first model would show in detail the impact on the business if there are no credit periods. The second model explains the effect of having a credit period. Both models use the Stackelberg game policy to capture the manufacturer's leader and retail's follower. Moreover, both the models are shown in three cases using RSC.
Here after producing the manufacturer of the product sends them to the market through a retailer on time. The UPC ϕ p of the manufacturer depends on the cost for labor, development, and maintenance of tool/die, quality of the products, and greening cost. Here ζ 1 . Where ζ 1 > 0 be the mark-up fixed by the manufacturer when supplies to the retailers and also depends on the UPC ϕ p . They are receiving products from the manufacturer, retailers handing them over to customers with mark-up ζ 2 . Where ζ 2 > 0 and depends on the manufacturer's full sale price. Figure 1 gives the description of the model framework. For the sustainable development of society, and to choose the best strategy between manufacturer and retailer, here are considered two models based on trade credit. In Model I and Model II, there are three subcases through three-game strategy such as coordination, non-coordination, and revenue sharing contract. In Model I, demand depend on average selling price, quality, and green degree but in Model II, demand additionally depends on trade credit. In the coordination game strategy, manufacturer and retailer both act as a single organization, and for the non-coordination game, strategy supply chain members act through a leader-follower relationship for maximizing their own profit.

Model I (without credit period)
This part of the model describes that the customers' demand depends on the quality of the products, greening cost, and average selling price. Where quality exponentially and greening cost linearly increases with the demand for the products. Hence demand function is such that D = ( p max − p r ) ( p r − p min ) + αq μ + λ 2 η. Delay-in-payments are not considered here. Hence the mathematical formula of total profit for the manufacturer, retailer, and the joint SCM be such that

Centralized case (I1)
Here manufacturer and retailer both participating members of the SCM in a coordinated way like a single organization. Here both SCM members are equally important in making a decision. Besides, market demand depends on the selling price, the quality of the products, and the greening cost. Besides, Fig. 1 Developed sustainable supply chain model the SCM profit is divided equally in proportion to their corresponding cost. Hence the decision variables are q, P, η and ζ 1 ζ 2 . Now from (3) we have For convenience, let us denote Here, to solve the mathematical model, the classical optimization method is considered analytically as well as numerically also. The decision variables P, q, ζ , and η are optimized using a discrete optimization technique. As there are multiple decision variables, the Hessian matrix is used to test the globality of the solution. At first, the total expected profit is partially differentiated w. r. to the decision making variables and equated to zero. Thus, the decision variables optimum values are P * , q * , ζ * , η * such as follows:

Proposition 1 The total expected profit function is concave at P
Proof see Appendix A1.

Decentralized case (I2)
Here both the members of the SCM do business with their own planned strategy. Here manufacturer and retailer work as a Stackelberg leader and Stackelberg follower, respectively. In this case, the manufacturer first selects the mark-up, quality of products, and greening cost with its strategy and maximum profit. Then the retailer selects the mark-up with its strategy and maximum profit. Then the individual mathematical profit formula of the SCM members be such that Here, to solve the mathematical model, the classical optimization method is considered analytically and numerically also. The decision variables ζ 2 for the retailer, P, q, ζ 1 , and η for the manufacturer are optimized using a discrete optimization technique. As there are multiple decision variables, the Hessian matrix is used to test the solution's globality. At first, the total expected profit is partially differentiated w. r. to the decision making variables and equated to zero. Thus, optimum expressions of the decision making variables ζ * 2 , P * , q * , ζ * 1 , and η * such as follows:

Revenue sharing case (I3)
In this case, the χ portion of the earned revenue taken the retailer, remaining (1-χ ) revenue for the manufacturer. According to this agreement, the manufacturer will supply products to the retailer according to the minimum mark-up. In that case, their profit expressions described below: Where D = ( p max − p r ) ( p r − p min ) +αq μ +λ 2 η and ϕ p = (θ 1 P + θ 2 P + θ 3 )+φ 0 e q +λ 1 η. Here, to solve the mathematical model, the classical optimization method is considered analytically and numerically also. The decision variables ζ 2 for the retailer, P, q, ζ 1 , and η for the manufacturer are optimized using a discrete optimization technique. As there are multiple decision variables, the Hessian matrix is used to test the solution's globality. At first, the total expected profit is partially differentiated w. r. to the decision making variables and equated to zero. Thus, optimum expressions of the decision making variables ζ * 2 , P * , q * , ζ * 1 , and η * such as follows:

Model II (with credit period)
The second model is an improved version of the previously described model. In this case, the retailer gives the credit period facility for added convenience to the customers. Hence the demand for the products depends on credit period, quality of the products, average selling price, greening cost, and expression be such that D = ( p max − p r ) ( p r − p min ) +βξ +αq μ +λ 2 η. Here demand increases exponentially based on quality and linearly increases based on greening cost and credit period. Hence, the revenue earned after ξ time from the customer. Hence total interest loss is ψξ ζ 1 ζ 2 ϕ p D, where the loss of interest is ψ per unit product. Hence the mathematical formula of total profit for the manufacturer, retailer, and the joint SCM be such that

Centralized case (II1)
Here manufacturers and retailers work together as a single organization to select business strategies for maximum SCM profit. Here the mark-up of both the members of the SCM is considered as the single mark-up. Therefore, the total profit expression of the SCM is such as: For Convenience, let us denote ζ 1 ζ 2 = ζ , D = ( p max − p r ) ( p r − p min ) + βξ +αq μ +λ 2 η and ϕ p = (θ 1 P + θ 2 P +θ 3 )+φ 0 e q +λ 1 η. Here, to solve the mathematical model, the classical optimization method is considered analytically and numerically also. The decision variables P, q, ζ , and η are optimized using a discrete optimization technique. As there are multiple decision variables, the Hessian matrix is used to test the solution's globality. At first, the total expected profit is partially differentiated w. r. to the decision making variables and equated to zero. Thus, optimum expressions of the decision-making variables P * , q * , ζ * , and η * such as follows:

Proposition 4
The total expected profit function is concave at P * , q * , ζ * , and η * if χ I I 1 Proof see Appendix B1.

Decentralized case (II2)
Here stackleberg game policy has been used. The manufacturer first selects products mark-up with its own strategy and then retailer decides the mark-up with its own strategy. In this case, the quality of the products, the greening cost, the credit period are considered by the manufacturer. Thus the profit function of the SCM members be such that Here, to solve the mathematical model, the classical optimization method is considered analytically as well as numerically also. The decision variables ζ 2 for the retailer, P, q, ζ 1 , and η for manufacturer are optimized using a discrete optimization technique separately. As there are multiple decision variables, the Hessian matrix is used to test the globality of the solution. At first, the total expected profit is partially differentiated w. r. to the decision making variables and equated to zero. Thus, optimum expressions of the decision making variables ζ * 2 , P * , q * , ζ * 1 , and η * such as follows:

Proposition 5 The total expected profit function is concave
< 0, and Proof see Appendix B2.

Revenue sharing case (II3)
Similarly In this case, χ portion of the earned revenue taken by the retailer, remaining (1-χ ) revenue to the manufacturer. According to this type of agreement, the manufacturer will supply products to Retailer according to the minimum marksup. In that case their profit expressions described below: where χ ∈ (0, 1) Here, to solve the mathematical model, the classical optimization method is considered analytically and numerically also. The decision variables ζ 2 for the retailer, P, q, ζ 1 , and η for the manufacturer are optimized using a discrete optimization technique. As there are multiple decision variables, the Hessian matrix is used to test the solution's globality. At first, the total expected profit is partially differentiated w. r. to the decision making variables and equated to zero. Thus, optimum expressions of the decision variables ζ * 2 , P * , q * , ζ * 1 , and η * such as follows: Proof see Appendix B3.

Numerical examples
Here, to solve the mathematical model, the classical optimization method is considered analytically and numerically. The decision variables ζ 1 , ζ 2 , ζ 1 ζ 2 , q, P,ξ , η are optimized using a discrete optimization technique in different cases.
Because there are multiple decision variables, the Hessian matrix is used to test the solution's globality. First, the profit function in different cases is partially differentiated w. r. to the decision making variables and is equal to zero. Then all the optimum values of the decision variables in different cases are shown in that section. Numerically and analytically, all the hessian matrix clearly shows that the corresponding profit is maximum. The numerical tools as Mathematica 11.3.0 was used for the numerical results and to prove global optimality. The following subsections provides some numerical examples to validate the mathematical model numerically.
In the Model I, Case 2, it is found that for the retailer In the Model II, Case 3, it is found that for the retailer Here a comparative study with the different cases based on profit is shown through Figure 2. The graph clearly shows the fluctuation of profit. Here, the offered credit period is an important factor for maximizing the profit separately because, for Model II, all cases profit higher than the corresponding profit in Model I. Besides the graph, one can say that the maximum profit occurs for Model II cases, Centralized case.

Special case (without quality case)
Here a particular case without quality function is considered based on the same input parameter and the same numerical value of Example 1. Hence all the output of the decision-  making variables and corresponding profit are shown in Table 3.

Special case (Without greening concept case)
Here a special case without greening investment is considered based on the same input parameter and the same numerical value of Example 1. Hence all the output of the decision variables and corresponding profit in a different manner are shown in Table 4. Here Fig. 3 gives the comparative study among the total supply chain profit of different cases in Example 1 and the exceptional cases. Here the trend of fluctuation of profit helps to find the strategy of the business line. Besides the figure, it is clear that this model will have additional cost for greening investment and profitable. Profits are maximum as well as customer demand depends on quality which is shown in that figure.
Here Fig. 4 gives the comparative study among the joint mark-up of different cases in Example 1 and the exceptional cases. Here the trend of fluctuation of joint mark-up helps to find the strategy of the business line. Besides the figure, it is clear that the mark-up is much higher in exceptional cases compare to the mentioned model. Although increasing the mark-up may increase profits, it increases the selling price of products and reduces market demand. This model shows that with less mark-up and more profit can be made depending on a different factor.
The optimal values of the decision making variables and the corresponding profit in different manner are as follows Table 5:

Sensitivity analysis
Significant observations for cost and scaling parameters are numerically calculated. The changes in these parameters are   Tables 6, 7, 8, 9, 10, and 11. These tables show how the cost and scaling parameters affect the different profit in different cases due to changes such as (-50%, -25%, + 25%, + 50% ). Here, from the following sensitivity table, such conclusions can be made: 1. Table 6 shows how the cost and scaling parameters affect the mark-up of the manufacturer in different cases. Here a small change in the scaling parameter θ 1 (scaling parameter of the raw material cost) has been observed a large change in manufacturer mark-up. 2. Table 7 shows how the cost and scaling parameters affect the retailer's mark-up in different cases. In this table, it is seen that increasing or decreasing the value of α has no significant changes on the mark-up of the retailer but increasing or decreasing the value of θ 3 changes the mark-up of the retailer very clearly. 3. Table 8 shows how the cost and scaling parameters affect the joint supply chain mark-up in different cases. For the "decentralized case" and "revenue sharing contract case" of the first and second model, this table shows that the paraments θ 1 and θ 2 are the same effects on the supply chain's joint mark-up. 4. Table 9 shows how the cost and scaling parameters affect the profit of the retailer differently. Although here retail is a follower, this table shows that retailer is more profitable in revenue sharing case of Model II. Another essential aspect to note from this table is that a slight change in θ 3 shows a more significant retail profit change. 5. Table 10 shows how the cost and scaling parameters affect the profit of the manufacturer in different cases. Looking at this table, it is clear that the decentralized case of Model II is more profitable for the manufacturer. In this case, the significance of α and θ 3 is fully affected by the manufacturer's profit. 6. Table 11 shows how the cost and scaling parameters affect the joint SCM profit of the retailer and manufacturer, both in different cases. Naturally, it is understood from this table that the profits of the joint supply chains are the maximum profitable in the decentralized case of Model II. It can also be seen from this table that one has to bear extra interest-based on credit period offer, but the whole supply chain faces the maximum profit.

Managerial insights
The following are recommendations for improving the industry: 1. The manager can avoid any uncertainties regarding customer issues using a variable production rate rather than a constant production rate. Further, the commanding manager can increase market demand by considering the selling price, greening cost, and quality-dependent demand. Even if the cost increases a little, the profit will be okay. 2. Managers should pay special attention to credit when buying and selling. In that case, as the popularity of the product increases, so does the customer's confidence in the company. In that case, the manager should be consciously controlled about the extra interest paid due to the offered credit period. Also, a credit period should be given to eligible customers who will repay later. Otherwise, the company may face losses. 3. A successful company can only rise to the top of popularity if it works actively from producer to customer. Even if the model is considered here based on manufacturer leader and retailer follower, their good relationship can make a deep impression on SCM development. The manager can face more profit through the excellent relationship among the participating members. 4. Revenue sharing is one of the unique features of this model. Naturally, the manufacturer leader will get more share, and the retailer follower will get less share, but it should be noted that no member of the supply chain is harmed because anyone is more likely to break the supply chain.

Conclusions and future research ideas
In the current social context, the members' relationship is a stepping stone to the development of SCM. The model gives an appropriate solution for ending the competition among themselves regarding who will be the most profitable. In the same way, here has shown different models that will benefit more.  A thorough discussion of this research revealed that greening cost and quality of the products increase customer demand and product popularity and increase profit margins, shown in the unique case portion. Moreover, it is concluded that the centralized case gain is higher in both the models as compared to the other cases. In the case of delay-in-Payments, the centralized case has shown a higher profit (33.20 %) than the decentralized case and higher profit (08.84 %)than the revenue sharing case.
The proposed research may be improved in many ways in the future. Here assumes that the manufacturer is the leader and the retailer is the follower. Nevertheless, the opposite situation is not considered here, so the paper can be extended with being considered a manufacturer follower and retailer leader. Besides, there are only one manufacturer and one retailer in our model, but in most of the situation, there may be multi-retailer, which is the new direction of updating this research. Another way of future research can also be extended by considering effects of variable setup cost, reliability in a reliable production system under environmental responsibilities (Moon et al. [4]). Another aspect is quality improvement, and setup cost reduction from the manufacturer side with carbon emission reduction concept (Sarkar et al. [14]). On the other hand, the demand in this model depends on different   factors. One of the new idea that can be considered here is the service facility, which will make the model more attractive and realistic (Sarkar and Bhuniya [44]). The lock-down situa-tion may arise due to different issues, so the demand becomes uncertain (Deng et al. [45]). Hence it may be considered to update this paper.

Conflict of interest The authors declare no conflict of interest.
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