PRICING AND INVENTORY DECISION IN A TWO-LAYER SUPPLY CHAIN UNDER THE WEIBULL DISTRIBUTION PRODUCT DETERIORATION: AN APPLICATION OF NSGA-II

. Academicians and practitioners have focused a lot of attention on the separate issues of pricing and inventory control in a competitive setting. However, integrating these choices in a competitive environment has received scant attention for deteriorating inventory systems from academics despite being crucial to practitioners. From this perspective, our research focuses on designing a supply chain model with inventory coordination to reflect time systems with improved accuracy and optimal control systems. In this research, we develop a two-layer supply chain model consisting of one manufacturer and one retailer incorporating the inventory classification of the retailer. Price-sensitive market demand and two-parameter time-varying Weibull distribution deterioration have been assumed to develop the mathematical model. First, a collective decision on price and inventory control of a dete-riorating product has been evaluated in a duopoly environment. Secondly, to explore the decentralized scenario, we have proposed the NSGA-II algorithm to solve the bi-objective programming problem of the two-layer supply chain. The paper aims to explore product collaborative pricing policies and the inventory decision of the deteriorating item in two-layer supply chain coordination. Finally, numerical research is conducted to execute the centralized supply chain and NSGA-II application in a decentralized supply chain. The research findings can provide valuable insights for members of the two-layer supply chain to make optimal product pricing and inventory scheduling decisions.


Introduction
According to researchers and experts in the manufacturing industry, the classification of inventory is an important part of supply chain management.Organizations must emphasize certain competitive measurements and expand manufacturing capacities to achieve the chosen measures in order to improve their market position according to the competitive structure.The most elementary function of inventory in supply networks may be to assist the balance of demand and supply.To actually control the forward and reverse flow in the store network, organizations should manage upstream supplier trades and downstream client needs.As a result, a company is required to work to achieve an equilibrium between a sufficient supply of materials and goods with meeting client expectations, which are usually difficult to predict precisely or accurately.A well-known system for achieving this equilibrium is stock control.Inventories are the source of a significant portion of a company's costs, which have an immediate impact on the pricing strategy.The ability of the enterprises to survive must thus be improved, especially when deteriorating goods are taken into account.Cost, profit, flexibility, delivery, and quality are examples of competitive metrics.These metrics pertain to the manufacturing process, control, planning, facility, capacity, and personnel.The level of inventory, manufacturing run time, product sales price, and system processing time is estimated in this study.In an inventory model, Ghare [21] first considered deterioration.Deterioration is a natural phenomenon of food items and vegetables that occurs after harvesting.Besides, a different Economic Order Quantity model has been developed for deteriorating products in [32,36,39,52].Some items have a limited shelf life, after which they begin to degrade.The nodeterioration phase is the name given to this period.In terms of stock level, during the no-deterioration period, stock levels fall only owing to market demand.In a real-life setting, most products (for example, fruit products, vegetables, electronic items, blood banks, and so on) would have a period of original condition during which no deterioration would occur.After this time period, some of the products will start to degrade or deteriorate.Non-instantaneous deterioration is the term for this phenomenon.When products are ordered from a third-party supplier, it is also crucial to consider the replenishment time in order to make the best supply chain decisions.Inventory models with product pricing, partial back-order, and quantity discount have been studied in [51].Raafat et al. [39] developed an alternate approach to attaining the ideal features of Mak's failing manufacturing system.A deterministic inventory model for degrading items with time-dependent market demand and the partial backlogging shortage is discussed in [34].Industrial managers must devise an effective method of rising stock goods while determining the optimal order size, cycle length, and stock levels.Recently, many researchers have focused on including environmental sustainability in green production technology [4,13] in the presence of government guidelines and tax policy.Investment in preservation technology to reduce product deterioration [4] and study of multi-item inventory system [3,6] all are recent studies in sustainable multi-layer supply chain coordination.
Many inventory management systems assume a steady demand rate across the planning horizon; however, this is not the case in practice.Selling price, discount level, advertisement, degradation level, stock availability, and other factors can all influence demand.In today's competitive market, sales prices highly influence the direct market demand by the customer.Suppose the producer and retailer may adjust their retail price by calculating all the investment costs in the price-sensitive demand.In that case, customer demand highly influences the profitability of the SC in the presence of integrated mode.Ruidas et al. [43] developed a production inventory system under price-sensitive demand incorporating the carbon emission taxes associated with the production process, setup process, and inventory storing process.In this paper, we have assumed price-sensitive customer demand; when the sales price of the product increases, then market demand decreases.Additionally, we have included product deterioration, which follows the Weibull distribution deterioration rate.Integrated and nonintegrated scenarios have been demonstrated in the formulated model.The Non dominated sorting genetic algorithm (NSGA-II) technique has been used to optimize the solution.The goals of this study are to investigate the following research questions: a.In a price-competition marketing system, what are the item's ideal retail and wholesale prices?b.What are the optimal order quantity to the manufacturer so that the market demand and product supply are in equilibrium?c.When market demand is not totally deterministic, what is the best inventory scheduling strategy to avoid losses?d.How does economic parameters, price elasticity parameters affects the manufacturer's, retailer's, and entire supply chain system's optimal profit?e.What are the advantage of NSGA-II, to optimize a decentralized scenario in a bi-objective supply chain problem?
To respond to the above questions, this paper proposes an inventory-based two-echelon SC model considering a manufacturer-retailer.It is considered that the manufacturer's responsibility is to produce finished products and make them available to retailers to meet their demands.Delivering the final goods to the consumer is the retailer's responsibility.The products are stored at the retailer's warehouse, which deteriorates as time elapses.Product deterioration follows the two-parameter Weibull distribution of the form  −1 ,  signifies the scale parameter where  stands the shape parameter.The main goal is to find out the product pricing, ordering, and replenishment strategies throughout the supply chain to maximize the company's overall profit.We assume that retailers face a price-sensitive market demand while with the increasing values of sales price, the market demand will decrease significantly, and while the sales price of the product decreases, then demand will increase.The mathematical formulation of this demand is  − ,  is initial market potential,  is price-elasticity, and  is the sales price of a unit item.The considered model is analyzed under two decision-making structures, (i) Integrated scenario and (ii) Non-integrated scenario.In an integrated scenario, a centralized marketing decision has been analyzed using the joint profit function of the retailer and manufacturer.In the non-integrated scenario, a meta-heuristic algorithm NSGA-II has been developed to find the optimal decision of the supply chain member individually.
The remainder of the paper is laid out as follows: we have described a brief literature review in Section in 2. The parameters and notations for SC are represented in 3. Therefore, the supply chain model under two different scenarios has been discussed.In Sections 4 and 6, we have discussed the integrated and non-integrated scenarios with a discussion of NSGA-II algorithm.In Section 7, a numerical analysis is performed on four possible situations, followed by a sensitivity analysis of some essential parameters.The step-by-step explanation of the paper has been addressed by a diagram in Figure 1.Finally, the research conclusions with some future perspectives are addressed in Section 9.

Research background
The relevant study for this research is divided into three sub-sections: (i) Supply chain model with inventory coordination, (ii) Weibull distribution product deterioration, and (iii) NSGA-II approach in bi-objective problem.

Supply chain with inventory coordination
Inventory management is the process of ordering, holding, using, and releasing stock.This involves the preservation and processing of such items as well as the management of raw materials, components, and final products.As a part of the SC, storage inventory is responsible for a wide range of tasks, including supervising and monitoring purchases from suppliers and customers, maintaining stock storage, restricting the number of products available for sale, and fulfilling orders.The three basic components of inventory management are buying inventory, keeping an inventory, and making money from inventory.
The parameter involved in inventory management are not always immutable in real-life situations, but they can be considered interval-valued.Industrial managers can calculate customer demand, product sales price, optimal stock in the retailer's warehouse, and inventory costs in interval forms to maintain stock levels competently by optimizing the ability to meet customers' demands on time while also optimizing the average cost or profit.Dong and Xu [20] have evaluated the role of the vendor-managed inventory (VMI) system in supply channel coordination.Sana [45] investigated a three-layer production inventory system involving manufacturer, retailer, and vendor to predict the rate of production and the number of delivery.An application of the -stagemulti-consumer SC with inventory model has been explored in [7] with the help of an algebraic approach.Dillon et al. [19] suggested a two-level stochastic programming approach to assist hospitals in their blood supply chain inventory management decisions.Recently, finding the optimal lot size of the supply chain participant has been an exciting research topic in inventory management.Sana [44] have investigated a production-inventory system for perfect and imperfect inventory systems like chemical, food, textile, footwear, etc.The use of preservation technology investment has been suggested in the research [53] to show the rate of deterioration in a manufacturerretailer SC coordinated with price-reflexivity market demand.Darom et al. [12] proposed a recovery model for a two-stage serial supply chain with safety stock and carbon emissions that is sensitive to supply disruption.Product pricing and inventory classification are analyzed in a food supply chain under controllable deterioration, and product disruption [25].Customers' purchasing behavior for perishable grocery items is influenced by expiration dates.In addition, Increasing the quantity of inventory on display may also increase customer demand.Assembling all these points, Sebatjane and Adetunji [48] developed a three-echelon SC model to estimate optimal lot sizing and shipments decision.Strategic analysis of optimal product pricing and cycle duration of inventory for a non-instantaneous inventory item is discussed in [5].The production and consumption process in a sustainable supply chain has been studied in [15] under the circular economic process of restore, recycle, and reduce.Kamna et al. [27] has addressed imperfect production processes with energy usage and volume agility.Indicators of a circular economy in food waste management are provided by the research of [40], which highlights the potential for recycling food waste to produce organic fertilizer (compost) as well as its economic and environmental evaluation.The stock-dependent demand and trade credit policy for breakable items has been addressed in a two-warehouse inventory system in [16].Therefore, they have extended the concept in fuzzy environments in [18] and [17].Mahata and Debnath [29] have addressed the profit maximization criteria in a two-layer supply chain incorporating inventory and preservation technology investment of deteriorating items.

Weibull distribution product deterioration
Rau et al. [41] derived an inventory model in multi-layer supply chain environments for deteriorating items to minimize the supplier, producer, and buyer individual costs.They have assumed a time-varying rate of deterioration that follows two-parameter Weibull distribution of  and .Thereafter, Lo et al. [28] extends this model by including an imperfect production system under established the integrated production-distribution model by incorporating production batch size.They demonstrate that the buyer's discrete delivery lot amount must be an integer multiple of the supplier's production batch size.Their findings reveal that cycle duration and optimal production lot size both decrease with higher values of deterioration.From the mathematical viewpoint, Sarkar [47] established a manufacturing inventory system with continuous stochastic deterioration and reached minimum cost with an integer number of deliveries with ideal lot sizes.The classification of ramp-type demand with time-varying product deterioration in the product inventory system has been explored by Pal et al. [37].The application of three parameters Weibull distribution product deterioration and ramp type demand has been included in a two-warehouse problem with permissible delay in payments in [8].Their results also show that inflation has a substantial impact on increasing the cost of items and the total cost of the distributor.Applying the Weibull distribution, a hybrid EOQ model for perishable products with time-varying deterioration and closing days inclusion has been proposed in [11].They look at how Weibull parameter changes affect perishability characteristics to demonstrate the Weibull distribution's versatility and suitability for addressing deterioration processes.Huang et al. [26] considers a two-echelon supply chain model and develops a quantity discount coordination mechanism that can quickly find the best long-term ordering policy to maximize the profits of the SC.The research shows that synchronizing quantity discounts might reduce the uncertainty regarding demand, lowering the number of retailer's order while increasing the overall profits of the supply chain.

NSGA II approach
There is a limited number of literature on multi-objective algorithms dealing with the supply chain problem.Cheng and Ye [10] developed a multi-objective order placement model to optimize the various cost and balance the manufacturing loads of suppliers.They proposed the well-known algorithm that is the Non-dominated sorting genetic algorithm (NSGA-II), to optimize the Pareto optimal set of solutions.Rezaei and Davoodi [42] also showed that the overall performance of NSGA-II is better than other algorithms in the joint pricing and order quantity supplier selection model.Bandyopadhyay and Bhattacharya [2] applied NSGA-II for solving a two-echelon tri-objective SC model.Chan et al. [9] used NSGA-II for solving the three-echelon supply chain distribution problem.Park et al. [38] provides an effective GA with an improved process for the VMIRPL that predicts replenishment schedules and lot size as well as vehicle routes in retail supply chains.The suggested GA addresses synthetically essential elements of supply chain operation in a VMI context, such as lost sales and limited storage space.An application of a genetic algorithm has been exhibited to optimize location routing inventory problems in [24].Babaveisi et al. [1] compared various methods, including NSGA-II, to solve a closedloop mechanism of three objective supply chains.Gholizadeh and Fazlollahtabar [22] used a modified GA and a robust optimization approach to solve a closed-loop stochastic green supply chain.They have done a case study of the melting industry and optimized the profit maximization criteria in the reverse flow SC network having a melting process.Sang [46] has explored the application of the GA and the BP neural network to figure out the SC risk finance.The concept of designing and optimizing a supply chain network plays a crucial role in e-commerce systems.Based on the e-commerce environment, Moghadam et al. [35] presents a mixed integer nonlinear programming problem in multi-objective optimization for designing a closed-loop supply chain framework.

Research gap and contribution
There is numerous research on SC coordination and deteriorating item inventory models.Zhi et al. [54] investigated in-transit inventory financing in a capacity-constrained supply chain model.Including product life-cycle with time-varying product deterioration and discount payment facility, Mashud et al. [33] has developed an EOQ model to optimize the product pricing, inventory length, etc. Halim et al. [23] have introduced an EOQ model for non-linear price and non-linear stock level dependent demand in an overtime production system in case of the deteriorating item.To consume carbon emissions, Sepehri et al. [49] have developed a sustainable production system under cap-and-trade regulation.Estimating optimal product pricing and replenishment scheduling are sensitive decision variables to maximize a supply chain profit.By now, the concept of product deterioration, time-varying product deterioration, retailer's inventory classification, inventory in the supply chain, and application of NSGA-II is frequently discussed in the existing literature.However, no researchers have examined the inventory coordination of supply chain phenomena using the NSGA-II algorithm when these three phenomena are combined in a single platform.In this research, we have optimized the joint pricing strategy, inventory replenishment policy, and profit of the supply chain participants.Firstly, we have designed a two-layer supply chain model for deteriorating items incorporating the inventory classification of the retailer.Secondly, we have analyzed integrated scenarios under the profit maximization criteria in a centralized marketing decision.Third, we have used the NSGA-II algorithm to optimize the non-integrated scenario in a decentralized mode and evaluated the Pareto optimal solution of sales price, inventory cycle length, profit of the retailer, profit of the manufacturer, and overall profit of the supply chain.Finlay, we observed some major elasticity parameters and their application in industry.

Notations
The mathematical model is developed using the notations below.

Assumptions
The presumptions made to demonstrate the model are addressed as follows: 1.A linear price-sensitive demand in the form of () =  −  represents the uncertain market demand of the customer, which signifies that when the product's sales price increases, then the demand for the product decreases. and  are demand elasticity parameters.The demand is non-negative, satisfying the condition  <   where  >> , [5,48,53].2. The unit retail price () is always higher than the unit wholesale price () of the products.(Barman et al. [5]). , where  > 0 and  > 0 signifies the scale parameter and shape parameter,  > 0 is the time deterioration.The assumptions are widely observed in most of the time-varying inventory models like [11,37].7. The on-hand-inventory at time t is ().It is assumed that a shortage in the retailer's inventory has not been considered.The retailer's inventory will be fulfilled by the next round order when the inventory label reaches to zero.8. Product replacement or repairing of deteriorating products during the inventory cycle is not included in the two-layer supply chain.9.The inventory considered only a single type of item.

Problem statement and mathematical model
The two-layer supply chain model has been addressed in (Fig. 2) with inventory coordination of the retailer(Fig.3).Considering product deterioration, our investigation signifies the optimal inventory scheduling and product pricing strategy, which maximize the supply chain profitability.
Manufacturers and retailers make up the two-stage supply chain for deteriorating inventory items.The manufacturer collects the raw materials from the supplier, and after preparing the final product, he sells the product to the retailer at a wholesale price of .Retailer stores the product up to the inventory time  .In the inventory cycle, due to demand and deterioration, the inventory goes to zero, and it will be finished at time  .Therefore, if () denotes the retailer's inventory status at time  mentioned in Figure 3, then at  = 0, ( ) = 0.At the final stage, the retailer sells the product to the customer with a sales price (), then  > .Initially, if the retailer store  number of products, then (0) = .To maximize the profit, the manufacturer sets his wholesale price (), retailer fixes his sales price () and inventory cycle ( ).Now, the optimal decisions are classified in two possible ways: (i) Integrated scenario and (ii) Non-integrated scenario.In an integrated supply chain system, only one decision-maker is responsible for the overall chain's desired outcome.In such a framework, the decision maker intelligently optimizes the total profit of the supply chain by selecting service levels and retail prices that is beneficial for all members of the supply chain.On the other hand, in the nonintegrated SC system, all chain members, including the manufacturer and retailer, make decisions individually and simultaneously to optimize individual profits rather than the profits of the entire chain.
Congruent with the earlier hypothesis, the initial inventory of the retailer starts from .Due to demand and deterioration, the inventory level of the retailer goes to zero at time  .Graphical illustration of the the whole inventory system is provided in Figure 3.During the inventory cycle [0,  ], the status of the retailer's inventory can be formulated by the following differential equation: with the boundary condition (0) =  and ( ) = 0.The deterioration rate follow time dependent Weibull distribution where () =  −1 .Therefore, the modified form of the differential equation ( 1) can be described as follows: Taking power series expansion and ignoring second and higher powers in exponential series (assuming  to be very small), with the help of boundary condition ( ) = 0, the solution of (2) yields: Using the condition (0) = , the optimal order quantity is determined as: The components of the total inventory cost per unit time consists the following components: By assuming the preceding consideration and notation along with an inventory level delivered to a retailer, the proposed mathematical model is extended with a unit manufacturing cost , fixed ordering cost , holding cost ℎ, deterioration cost , and sales revenue  per unit time.Therefore, the total profit of the retailer per unit time can be obtained by: The comprehensive analysis of each component in the equation ( 5) are presented below In equation ( 6), the first term indicates the sales revenue of the retailer; the second term is the ordering cost per unit time, the third term indicates the cost for the purpose of purchasing from the retailer, and the deterioration cost per unit time, the last term signifies the holding cost per unit time of the retailer.
Individual inventory costs of the manufacturer have been fully ignored.Manufacturers make the products with a manufacturing cost of  per unit and sell the product at a wholesale price of  to the retailer with negligible lead time.It has been considered that the products are stored for a short period in the manufacturer's inventory.Therefore, we can neglect the costs associated with deterioration and inventory during the manufacturer's inventory and optimize the overall profit of the manufacturer.
In equation ( 7), the term ( − ) indicates the profit of the manufacturer per unit product and ( − )   represents the profit of the manufacturer per unit time in an inventory cycle.Therefore, substituting the value of  from (4), we get the simplest form of    .

Integrated decision
Under an integrated business strategy, all the channel members of the supply chain are worked together, consulting with each other.In this scenario, retailers selling price () and replenishment scheduling ( ) are determined by implementing joint decisions of manufacturer and retailer.Here, the main objective of the chain is to maximize the total profit, which is gained by ( 6) and (7).The joint profit function can be written as Therefore, substituting the value of    and    from ( 6) and ( 7) in (8), the simplest form of supply chain profit   sc (︀ ,  )︀ is as follows, such that 0 <  < The necessary condition for maximizing   sc (︀ ,  )︀ are )︀  = 0.This leads to a system of two dependent non-linear equations with two unknowns.Solving this system of equations, we get the values of  and  .To check the optimality of the profit function   SC , we should check the behavior of the Hessian matrix followed by (5.1).
Proof.The supply chain profit   sc (︀ ,  )︀ is an objective function of two independent variables  and  .The necessary condition of optimality gives the first-order differentiation in equations ( 11) and (12).Solving the first-order differential equation, we find out the  and  numerically.To show concavity, we summarized the second-order differentiation and the condition of concavity.
Again, (︁ Hence,   sc is jointly concave with respect to the decision variable  and  when the ( 13) and ( 14) holds.

Non-integrated decision: NSGA-II approach
The power position and strategic plans of each member of an SC determine the connection between different members of the SC.In some SC, members have equal power.As a result, they can work together to increase profits (or lower overall costs) by implementing integrated inventory procedures.In certain SC, a member's decision is made independently of the other members, or one member has a disproportionate amount of authority over the other members.As a result, members can either make their own decisions or follow the leader.We employ non-integrated inventory policies in these situations.The non-integrated scenario has been optimized with the help of the non-dominated sorting genetic algorithm (NSGA-II) algorithm.The non-dominated sorting genetic algorithm (NSGA) is one of the most popular multi-objective optimization methods due to its simplicity, effectiveness, and minimum user interaction.
The mathematical formulation of the multi-objective problem is mentioned below, max , ,
In the bi-objective problem, the retailer profit function    (,  ) is the function of two decision variables  and  , on the other hand, the manufacturer's profit function    () is a function of decision variable .To solve a multi-objective optimization problem, the most powerful algorithm known as NSGA was first proposed by Srinivas and Deb [50].The concept of Pareto optimality has been used to generalize the algorithm.Later, Deb et al. [14] built up an incredible meta-heuristic procedure known as NSGA-II, which has fewer parameters, less computational unpredictability, and efficient constraint handling compared to NSGA.These features have made the NSGA II effective and famous in a wide scope of management science, engineering management, and other optimization fields.The essential steps of NSGA II are described as follows:

Parameters used in NSGA II
First, all the parameters connected with NSGA II are listed here.These parameters are -population size (N) =100; -crossover probability=0.8;-mutation probability= 0.01; -maximum number of generation (G) =100000.

Chromosome representation
A three dimension vector  = (, ,  ) is used to represent a chromosome in the population (a solution).In this vector , ,  all are real numbers, i.e , ,  ∈  + .

Population initialization
At the beginning of the algorithm, there is no useful information regarding the location of the Pareto-optimal solution.So, an initial population ( 0 ) the size of N is randomly generated considering the limits of the relevant decision variable.In this research, the value of all the decision variables is chosen randomly from the set of real numbers.

Fitness function
After obtaining the first as well as successive generations, the quality of the solution should be checked.The quality of the solution indicates the value of the objective function for each solution.In this study, retailers' and suppliers' profits are calculated as the objective function corresponding to each chromosome.Good fitness implies a higher functional value with respect to the best chromosome.There is a requirement for adjustment to apply the traditional selection process as used by [30] and [31], as the objective function in their research works is profit maximization.

Selection
Prior to selection, each member of the population is given a rank based on non-dominance; all non-dominated individuals are grouped into a single category (with a dummy fitness value, which is proportional to the population size).In order to maintain a diversified front, the crowding distance is also calculated, making sure that each member maintains a crowding distance apart from the others.This supports population diversity and aids the algorithm's exploration of the fitness environment.
In the selection process, a three-chromosome tournament selection operator is used to find the best solutions for the next generation in which three chromosomes are selected randomly and then compared in terms of the front rank and the crowding distance.

Crossover
The majority of the population in generations are produced by crossover operator.In this investigation one point crossover is utilized.In the wake of picking the parents by selection procedure, a random value has been generated by crossover operation which signify what part of these two solutions should be exchanged in order to have new solutions.This procedure is rehashed until reaching at the percentage of the population for latest generation.

Mutation
Mutation operation is used to maintain genetic diversity of one generation of population to next.As a result, a small percentage of new generation is generated.Mutation randomly takes one of the solutions from the chromosome obtained after crossover and allocate a new value to the solution within the bounds.After that the corresponding fitness value is calculated.These steps are replicated until a new population is obtained induced by mutation operator.

Non-dominated sorting (and Elitism)
After selection, crossover, and mutation, suppose a new offspring generation {  } of size  is generated.Therefore a new population   =   ∪   of size 2 has been initiated by combining parent's chromosome   .According to their rank and crowding distance, the next parent population  +1 of size  has been created.The sorting has been performed according to the better rank of the chromosome.For example, if a chromosome   has a better rank than   , then we select   .The front rank with the greatest crowding distance shall be picked if the front ranks are tied.Then a measure called the crowding distance (CD), which is defined in equation ( 15) had been performed to assess the populations' solution fronts in terms of the corresponding relative density of the individual solutions.Finally,  +1 will be sorted based on their ranks and crowding distances.After performing a fixed number of iterations, the algorithm will be terminated.

Performance measure
Pareto-based multi-objective optimization algorithms always give an approximate non-dominated front.In this manner, the performance measurements of this technique are not the same as single objective strategies.Therefore a new methodology is required to look at the exhibition of the optimization.To quantitatively compare the consequences of multi-objective algorithms the subsequent overall performance measures are used:

Number of non-dominated Pareto optimal solution(NPS)
This performance keeps track of all non-dominated solutions that were obtained using a given set of rules.

Crowding distance(CD):
The crowding distance  used in the proposed NSGA-II is essentially based on the cardinality of the solution sets and their distance to the solution boundaries.Suppose in a particular front there is  number of chromosomes.Then the measurement of the crowing distance of  ℎ chromosome is followed by where   ( max ) and   ( min ) are the maximum and minimum functional value of the  ℎ objective function function.In this study, there are two objective function i.e.  = 2.To measure the convergence of the Pareto fronts, we derive the Mean Ideal Distance ( ) of the corresponding Pareto fonts.Suppose  1 ,  2 , . . .  are the Pareto fronts comes after fixed iteration then the mean formulation is described as 6.9.4.Standard deviation(): From a statistical viewpoint, Standard deviation is the measurement of the spread between numbers in a data set.To analyze the closeness of the Pareto optimal solutions, we study the data set's Standard deviation ().The measurement of standard deviation is as follows:

. Multiple run performance evaluation
NSGA-II being a heuristic algorithms, as such, won't yield the exact solutions in multiple runs as because any evolutionary algorithm starts with a random generation of the initial population.The variations in results occur due to the use of random numbers initialization as well as mutation and cross-over stages.So we run the algorithms 15 times and list the result in Table 4.The outcomes of each run are analyzed by mean and variance coordination.
The step-wise procedure of the mentioned algorithm has been addressed in the flowchart (4).

Numerical illustration
The numerical and sensitivity analysis presented in this section are used to validate the generated model.We put the results of a large numerical experiment into action and examined the findings.The goal of this study is to come up with the best decisions a decision maker or manager could make.The following numerical data has been used to validate the model. = 1000 units,  = 1.2,  = $40,  = 5,  = 0.04,  = $100 per order,  = $3, ℎ = $8.Firstly, we have optimized the supply chain decision under an integrated scenario using an analytical approach.Therefore, to maximize the retailer's and manufacturer's profit individually, we conduct the decentralized technique (non-integrated scenario with the help of the NSGA-II technique).

Numerical results under integrated scenario
Under the centralized scenario, the concavity graph of the profit function    (︀ ,  )︀ has been demonstrated in Figure 5.If the retailer sells the product at a sales price  = $574.76then the optimal inventory cycle will be  = 1.796 years.Therefore, the optimal profit of the manufacturer will be  = $81, 453, the optimal profit of the retailer will be  = $17, 034 and the supply chain system achieves its highest profit $98, 487.
The effects of demand elasticity(), deterioration cost(), holding cost (ℎ) and manufacturing cost () on product sales price (), inventory time ( ), manufacturer's profit (   ), retailer's profit (   ) and overall  supply chain profit (  sc ) are shown in Table 2. Based on Table 2, the following inferences can be made from a managerial view point: 1.When potential demand increases, the sales price  * and the profit of the supply chain members increase simultaneously, but inventory cycle duration  * reduces.As example, when  increases from 1000 to 1100 then  increases from $546.76 to $630.06, profit of the manufacturer increases from $81, 453 to $105, 135, profit of the retailer increases from $17034 to $59031 and the profit of the supply chain increases from $98487 to $164167, but the retailer's inventory duration reduce 1.796 years to 1.788 years.From the economic viewpoint, when the market demand increase, the profit of the supply chain increase, and the retailer's inventory will stock out immediately, so the inventory cycle duration reduce.The results also show that the supply chain profit increases steadily as demand increases.2. When there is an increase in the value of manufacturing cost (), then the sales price  * and inventory cycle length  * increase, whereas the optimal profit of the supply chain reduces significantly.The physical significance implies that in case of higher manufacturing cost retailer should optimize his order quantity by balancing sales price.3. When the holding cost increases, the length of the retailer's inventory cycle  * decreases.The supply chain cost will be reduced if the retailer can efficiently minimize the item's holding cost by enhancing storeroom equipment.From the managerial point of view, the store should order less.4. Higher deterioration cost () increases the inventory stock in period and sales price, and the overall profit of the supply chain, as well as the profit of the supply chain member, decreases as shown in 2. From the business viewpoint, it is clear that due to increasing deterioration cost, the retailer has to raise a minimum around their selling price (p) to overcome the high losses in optimal profit.5.In similar way, Table 3 presents the effect of parameter  and  on profit when  = {2, 3} and  = {0.2,0.3, 0.4, 0.5, 0.6}.The lower  value reduces the profit of the manufacturer but improves the profit of the retailer and the overall supply chain.Conversely, the lower  value improves the supply chain member's profit.The economic significance implies that a lower deterioration rate will benefit the supply chain members for high profitability.
A retailer must decide on a few key components to run a commercial enterprise successfully.It is crucial to fix the product's sales price according to the market's uncertain demand.To remain a positive progress in the business, organizations must recognize the significance of these components and use them as a competitive strategy.In an industry manager, the inventory cycle duration and different cycle lengths significantly impact the supply chain's optimal profit.

Numerical results under non-integrated scenario: NSGA-II technique
In solving the supply chain problem, the major findings of this model are to find out the optimal sales price, wholesale price, demand, inventory scheduling, etc.The research problem suggested in this work is a multiobjective model that employs a meta-heuristic algorithm, especially NSGA-II, to present the optimal decision of the supply chain members by selecting the most suitable outcomes based on the existing information.In addition, we generate twelve non-dominated solution sets (Solution 1 through Solution 12).However, it is evident that in a real-world circumstance, a decision-maker might examine more ideal options.
The above two objective functions mentioned in equations ( 6) and ( 7) are dealt with by the NSGA-II approach, and the first Pareto feasible solutions are listed in Table 4 (Fig. 5).Table 4 illustrates the results of the optimal sales price (), wholesale price (), inventory cycle length ( ), and the corresponding manufacturer's profit (   ) and retailer's profit (   ) by the first iteration.In Table 4, it is demonstrated that we have got a total of 16 (sixteen) Pareto optimal solution sets, which are plotted graphically in Figure 7a.According to the supply chain profit   sc , we have chosen the best and average optimal solution and listed them in Table 5.From the optimal supply chain profit comes from run 1 in  1 , we have calculated the mean (  sc ), variance  2 (  sc ), MID and classified the best optimal solutions.We separately run the algorithm 12 times and compare the results using the mean, standard deviation, and performance metrics recorded.In run one, there are sixteen Pareto optimal solutions mentioned in Table 4.For example, the first solution set among the 16 sets of solutions suggests that when  = $535.52, = $87.56 and  = 0.1001 years and    = $91, 535,    = $16, 782 and the corresponding supply chain profit is the sum of manufacturer's profit and retailer's profit,   sc =    +    =$108, 317.From the first observation (run 1), there are 16 Pareto optimal solutions which are plotted in (7a).Among the 16 Pareto optimal solutions, the highest supply chain profit comes from solution set 2. In that case,  = $530.65, = $88.10 and  = 0.1001 years and    = $94, 094,    = $14, 242 with supply chain profit   sc = $108, 336.The average solution set is calculated by calculating the mean difference of the supply chain profit   sc .As an example, the mean profit of the supply chain profit   sc mentioned in Table 4 are $108, 183.Therefore, the average optimal value is chosen from the set of solutions that provides the nearest profit value of the supply chain compared to the mean profit.In this example, the average optimal value fixed as  = $540.02, = $74.58and  = 0.1027 years and    = $65, 570,    = $42, 626 with supply chain profit   sc = $108, 183.The best and average optimal solutions set in run one are listed in 5. Similarly, we have conducted 12 observations, i.e., run 2 to run 12, and write down the best and average optimal solution in Table 6.We have plotted the position of the Pareto optimal solution in each iteration in (7a)-(7l).For example, in run 2, the best optimal solution set is  = $531.54, = $86.24 and  = 0.1000 years and    = $90, 184,    = $18, 153 with supply chain profit   sc = $108, 337 and the corresponding average optimal solution set is  = $522.86, = $86.30and  = 0.1000 years and    = $92, 903,    = $15, 344 with supply chain profit   sc = $108, 247.The results are listed systematically in Table 4-6.
Table 6 signifies the multiple set of Pareto Optimal Solutions based on the decentralized decision.It has been noticed that the supply chain profit is primarily similar in all successive iterations.However, for different sales prices () and wholesale prices (), the profit of the manufacturer and retailer will fluctuate.The retailer's profit will be higher with a high sales price of .The manufacturer's profit is higher with a higher wholesale price, but with the higher wholesale price, the manufacturer's profit will be affected by low profitability.
The suggested multi-objective strategy gives decision-makers the ability to choose from a variety of trade-off solutions.The NSGA-II outperforms the other algorithm in terms of generational distance when compared to SPEA2, IBEA, etc.

Some managerial insights
Our computational experiments have reflected to find out the optimal decision of a multi-layer supply chain problem.This research reaches the following goals i) assess to take the optimal product pricing and inventory decision of retailers in two, three, or multi-layer supply chain models.ii) this concludes finding out the optimal decision by creating a multi-objective optimization problem, iii) different meta-heuristic algorithms including Non-dominated sorting genetic algorithm can be used to solve multi-objective supply chain problems.In the integrated marketing scenario, organizations can keep an eye on their operational costs and work hard to stay   within a tight budget by monitoring the complete supply chain from a single location.From the managerial perspective, there is no doubt that the industry always prefers the integrated market scenario only.Based on the supply chain performance, marketing criteria, or warehousing policy, an industry chooses a decentralized scenario for the best business strategy.We have addressed the decentralized marketing scenario using the meta-heuristic algorithm NSGA-II to figure out the individual optimal decision supply chain members.Decentralization in business strategy has the potential to enhance the efficiency of the supply chain.It is simpler to motivate participants in a supply chain with decentralized decision-making and ongoing trade to collaborate on effective activities.Our analysis demonstrates that the widely accepted ef ficiency of decentralized decision-making can improve not only the overall profit of the supply chain but also the individual profit of each participant in the supply chain.

Conclusion
This research aims to develop a two-echelon SC model considering the retailer's inventory.The supply chain model has been considered with time-dependent deterioration, which follows two parameters Weibull distribution.The finding of this research is to evaluate the optimal product sale price, wholesale price, order quantity, and inventory scheduling policy in the retailer's warehouse so that the total profit of the supply chain member, as well as the supply chain, will be maximized.We consider a price-sensitive market demand to establish the model.Each stage's supply is transported off to the subsequent downstream stage.There is no need to wait until a complete packet is accessible because shipments are dispatched as soon as they are available.From this phenomenon, only the retailer's inventory has been analyzed under profit maximization criteria.Firstly, the optimal decision of the manufacturer and retailer are discussed under an integrated scenario under a centralized marketing decision.Secondly, we have discussed the non-integrated scenario using the meta-heuristic algorithm NSGA-II to figure out the individual decision of the supply chain member.In NSGA-II, a set of Pareto optimal solutions are listed and analyzed to choose the best optimal solutions.A numerical example followed by sensitivity analysis has been provided to demonstrate the significance of sales price and gross profit under the integrated and non-integrated scenarios.It has been illustrated by the numerical example that the NSGA-II approach gives better results compared to the centralized marketing decision.In the NSGA-II algorithm, the retailer's profit increased drastically in decentralized marketing decisions.The product sales price is also lower in the non-integrated marketing scenario.Our analysis shows that the supply chain member can make individual decisions on product pricing and inventory scheduling based on the outcomes of NSGA-II.
The limitation of this study is that the uncertain production system, delivery channel, and shortage periods have been ignored in this study.
In future, the present research can be re-designed to incorporate the shortage of retailers' inventory.For example, shortages can occur in airplane engines due to conditions of uncertainty.In addition, the model can be generalized, including multi-layer supply chain participants with green investment and government responsibility.The accuracy of the model can be judged under uncertain and probabilistic market environments.
Disclosure statement.No potential conflict of interest was reported by the author(s).

Figure 5 .
Figure 5.The concavity curve of profit function   sc (︀ ,  )︀ (or  sc ) with respect to  and  .
,  > 0 and  ∈ R + .For a fixed wholesale price , the total supply chain profit

Table 1 .
Optimal solution under centralized scenario.

Table 2 .
Parametric behavior under centralized scenario.

Table 3 .
Parametric behavior under centralized scenario.