A COLLABORATIVE EPQ INVENTORY MODEL FOR A THREE-ECHELON SUPPLY CHAIN WITH MULTIPLE PRODUCTS CONSIDERING THE EFFECT OF MARKETING EFFORT ON DEMAND

. This paper presents an inventory model for a three-echelon supply chain with multiple products and multiple members considering the demand as an increasing function of the marketing eﬀort. In the proposed inventory model, a collaborative approach is studied and an analytical method is applied to obtain the optimal production lot size and the optimal marketing eﬀort in order to achieve the maximum proﬁts. Some numerical examples are illustrated to justify the model. Moreover, a sensitivity analysis is well done in order to analysis the eﬀect of the changes of key parameters of inventory model on the the maximum beneﬁts of all members of the chain.


1.
Introduction. Typically, a supply chain includes manufacturers, distribution centers, retailers, and customers who interchange flows of material, products, information, and resources necessary to satisfy market requirements. Most industries currently have problems involved in the inventory management of the supply chain because fluctuations in demand can produce excessive or shortages of inventory which increases inventory cost. Therefore, it is necessary to develop an inventory model for supply chains that determines the lot sizes required to minimize costs and increase the benefits of all members of the chain. In this direction, Sana [44] has proposed an inventory model for a three-layer supply chain considering perfect and imperfect quality items. Pal, Sana and Chaudhuri [29] have developed an inventory model for a three-layer supply chain considering the remanufacturing of the products. Cárdenas-Barrón, Teng, Trevino-Garza, Wee and Lou [7], Ben-Daya, Asad and Seliaman [4] and Sana, Acevedo-Chedid and Salas-Navarro [43] have investigated an inventory model in a three-layer supply chain that optimizes the order sizes of deterministic demand, In this direction, Roy, Sana and Chaudhuri [39], Abdelsalam and Elassal [1], Giri and Bardhan [11], Dai, Aqlan and Gao [8] and Pervin, Roy and Weber [33] have included uncertain demand and Panda, Modak, Basu and Goyal [30] have considered contract-bargaining process. Salas-Navarro, Acevedo-Chedid, Mercado-Caruso and Sana [40] have shown that a collaborative approach in the supply chain helps the members of the chain to make joint strategies about production and distribution activities in order to satisfy the customers demand. A collaborative strategy between members of a supply chain is the marketing effort which consists of merchandising activities, publicity, promotions and others activities done by the retailer, distribution center or manufacturer with the objective of sharing the costs of retail merchandising activities. Karray [16] and Nagler [26]. Kim and Gilliland [17] have considered the marketing effort including financial performance, uncertainty, synergy communication and support among the members of the supply chain. Pu, Gong and Han [37] have observed that demand in the market stimulates retailers marketing efforts and retail price. As a result, Pervin, Roy and Weber [34,35,36] have studied the inventory models considering stock-dependent demand. Hou and Lin [14] have presented an EOQ (economic order quantity) model with stock-dependent of selling rates and deteriorating items. Soni and Shah [47] have showed the effect of progressive credit periods to the demand. Goyal and Chang [12] and Panda, Saha and Goyal [31] have examined the display stock level when the inventory is transferred to the retailer or a new storage is rented. Min, Zhou and Zhao [22] have studied the current stock with different credit periods in a supply chain. Modak, Panda and Sana [23,24] and Panda, Modak, Sana and Basu [32] have included the pricing policy and effect of the price and recycling aspects of the demand. Lee, Wang and Chen [19] have considered a vendor managed inventory model with stock-dependent demand. Ahmadi-Javid and Hoseinpour [2] have discussed a location-inventory model with price-sensitive demand.
According to Ma, Wang and Shang [21], the demand in a supply chain is affected by both manufacturers quality improvement effort and retailers sales effort.The inventory holding cost and production cost are not considered in this model but only the marketing effort affects the market demand of the supply chain. Dube [10], Naik, Raman and Winer [25], Sana [41,42] and Cárdenas-Barrón and Sana [6] have implemented different forms of promotional effort to increase market demand.Sana [41] has proposed the EOQ model for salesmens initiatives in a production-inventory system for multi-item products. Krishnan, Kapusciniski and Butz [18], Ma, Wang and Shang [20] and Zhao and Zhu [49] have considered some strategies of marketing effort in a supply chain like price cuts, free gifts, advertising, publicity, merchandising activities, discount on production cost, contracts, and others. Tsao and Sheen [48] have considered promotional activities in the service industry to optimize the retailers promotion and replenishment decisions under retailer competition and marketing effort. Song, Li, Wu, Liang and Dolgui [46] have studied decision making model based on innovation and advertising.
Goyal and Gunasekaran [13] have presented an EPQ (economic production quantity) inventory model for a supply chain considering marketing policies, market prices, and advertising. Sana [45] has analyzed stock and promotional effort sensitive demand in production control policy. De and Sana [9] and Alawneh and Zhang [3] have studied stochastic demand in a two-echelon supply chain. Roy, Sana and Chaudhuri [38] have presented demand function depending on price, promotional, and service efforts. Cárdenas-Barrón and Sana [6] and Johari, Hosseini-Motlagh, Nematollahi, Goh and Ignatius [15] have proposed delay period offered to retailer for paying the production cost for the finished products into a two-layer supply chain.Paksoy, zceylan and Weber [27] and Paksoy, Pehlivan and zceylan [28] have presented a mixed integer linear programming model considering four-echelon supply chain with multiple members in each level and multi-quality raw material.Cárdenas-Barrón and Sana [5] have investigated a production-inventory model for a two-echelon supply chain comprising of one manufacturer and one retailer for a single product. Their model optimizes production rate, production lot size, initiatives of the sales teams, and shortage level of a collaborative system between the manufacturer and retailer. Table 1 presents the contributions of some authors related to EPQ models connected with this research. Sana [45] Crdenas-Barrn and Sana [6] Tsao and Sheen [48] Roy et al. [38] Crdenas-Barrn and Sana [5] Our present paper This paper presents an EPQ model for a three-echelon supply chain with multiple members in each echelon (multiple manufacturers, multiple distribution centers, and multiple retailers) for multiple products and different cycle times. This consideration differentiates it from the paper of Cárdenas-Barrón and Sana [5] and adds complexity to the model as the mathematical formulation compatible to the reality of the companies is quiet different compared to the existing model. The proposed production-inventory model addresses the problem of collaboration in a three-echelon supply chain with multiple members and marketing effort dependent demand that has been unexplored in the mathematical modeling of supply chain and the inventory literature because of its complexity of formulation and optimization of decision variables. The rest of the article is structured as follows. Section 2 presents the assumptions and notation of the proposed inventory model. Section 3 formulates the inventory model for a three-echelon supply chain. Section 4 provides some numerical examples and their solution. Finally, Section 5 shows a sensitivity analysis of production-inventory. Section 6 gives some conclusions and future extensions of the production-inventory model.

Assumptions and notation.
2.1. Assumption. The assumptions used to formulate the collaborative EPQ inventory model are given below: 1. Multiple products are considered in the inventory model. 2. The members of the three-echelon supply chain are manufactures, distribution centers and retailers who exchange different products and take strategies jointly to maximize the profit of entire supply chain. 3. The demand rates for the retailers are an increasing function of the marketing effort for a group of products in the supply chain. 4. The production lot size at manufacturers for each product is considered as continuous decision variable and the production rate per unit time at manufacturers for each product is greater than the demand rate at retailers. 5. The marketing effort is a sharing factor of the marketing effort cost.
2.2. Notation. The following notation are used to develop the proposed model. p mj : production rate per unit of time at m-th manufacturer for j-th product (units/time unit) c ij : material cost per unit for the i-th raw material for j-th product ($/unit) L mj : labor/energy cost at m-th manufacturer for j-th product ($/cycle) α mj : tool/die cost per unit at m-th manufacturer for j-th product ($/unit) AM mj : setup cost at m-th manufacturer for j-th product ($/setup) w M mj : selling price per unit at m-th manufacturer for j-th product ($/unit) h RM mij : inventory holding cost per unit per unit of time at m-th manufacturer for i-th raw material ($/unit/time unit) h P mj : inventory holding cost per unit per unit of time at m-th manufacturer for the j-th product ($/unit/time unit) β mj expected percent (0 ≤ β mj ≤ 1)of the marketing effort cost at m-th manufacturer for sales team for j-th product k mj : marketing effort cost of the sales teams at m-th manufacturer for the j-th product ($/unit) : an elasticity parameter DEM D dj : demand rate per unit of time at d-th distribution center for the jth product (units/time unit) AD dj : setup cost at d-th distribution center for the j-th product ($/setup) w D dj : selling price per unit at d-th distribution center for the j-th product ($/unit) h D dj : inventory holding cost per unit per unit of time at d-th distribution center for the j-th product ($/unit/time unit) β d : expected percent (0 ≤ β d ≤ 1)of the marketing effort cost at d-th distribution center for sales team k dj : marketing effort cost of the sales teams at d-th distribution center for the j-th product($/unit) DEM R rj : demand rate per unit of time at r-th retailer for the j-th product which is not dependent of the initiatives of sales teams(ρ) (units/time unit) A rj : setup cost at r-th retailer for the j-th product ($/setup) w R rj : selling price per unit at r-th retailer for the j-th product ($/unit) h R rj : inventory holding cost per unit per unit of time at r-th retailer for the j-th product ($/unit/time unit) β r : expected percent(0 ≤ β r ≤ 1) of the marketing effort cost at r-th retailer for sales teams k rj : marketing effort cost of the sales teams at r-th retailer for the j-th product($/unit) τ rj : a scale parameter that varies with effort of sales teams at r-th retailer for the j th product (units/time unit) D rj (ρ): demand rate per unit of time dependent on initiatives of the sales teams (ρ) at the r-th retailer for products j-(units/time unit) C mj : production cost per unit at m-th manufacturer for the j-th product ($/unit) λ rj : setup costs of manufacturers m and distribution centers d for the j-th product ($/setup) ϕ rj : setup costs of retailers r and inventory holding costs of distribution centers d and retailers r for the j-th product ($/unit/time unit) δ rj : marketing effort cost of the sales teams of manufacturers m, distribution centers d and retailers r for the j-th product ($/unit) φ rj : production costs of the manufacturers and revenues of manufacturers m and distribution centers d for the j-th product ($/unit) π mj : benefit of manufacturer m for the j th product ($/time unit) π dj : benefit of distribution center d for the j-th product ($/time unit) π rj : benefit of retailer r for the j-th product ($/time unit) µ: joint benefit of the supply chain for the j-th product ($/time unit) T C(q M mj , ρ): total costs of the supply chain dependent on production lot size (q M mj ) initiatives of the sales teams (ρ) ($) AP M : benefit of manufacturers ($) AP D: benefit of distribution centers ($) AP R: benefit of retailers ($) 2.2.3. Decision Variables. q M mj : production lot size at m-th manufacturer for the j-th product (units) ρ: initiatives of the sales teams 3. Formulation of production-inventory model. In this production-inventory model, the manufacturer fabricates a production lot size q M mj of finished products during the time t M mj . The manufacturer covers the demand of distribution center at a rate DEM D dj until the time k T mj . The demand of retailer is satisfied by the distribution center with rate DEM R rj upto time kT dj . So, the demand of the market is estimated by the retailer with rate D rj upto time T rj (See Figure 1). The equations of the collaboration system are as following. In this stage, m manufacturers fabricate each product with production rate p mj . The production runtime is t M mj , and the inventory cycle time of m-th manufacturer for j-th product is k T mj . Then, The inventory level of j-th product at m-th manufacturer is given by The inventory holding cost of the j-th product is: Substituting Eq. (1), (2) and Eq. (3) into Eq. (4), we have The revenue from selling products is The setup cost of the m th manufacturer is AM mj /k T mj = AM mj DEM D dj /q M mj . The marketing effort cost of the sales teams is k mj β mj ρ and the production cost per unit is C mj p mj = c ij + L mj /p mj + α mj p mj where c ij is the material cost per unit.The energy/labor cost (L mj ) per cycle which is shared with production rate.The tool/die cost is α mj per unit; this cost is proportional to production rate (α mj p mj ). Here, C mj (p mj ) is a convex function of p mj and C mj (p mj ) is minimal at p mj = L mj /α mj ( for instance vide [42]).The benefit of manufacturer m for the jth product is obtained as follows: Then, the benefit for products at manufacturers is 3.2. Profit of the distribution centers. The d-th distribution center receives jth product from the manufacturers at a rate DEM D dj up to time k T mj .The inventory at d-th distribution center piles up to time k T mj after satisfying the demand rate DEM R rj of the retailers of j-th product.The inventory level reaches zero at time kT dj .The variation of the inventory level with respect to time is modeled by the following equation: Substituting Eq. (3) into Eq. (10), we have Therefore, the inventory holding cost of finished product j is determined as follows: Substituting Eq. (3) and Eq. (11) into Eq. (12), we obtain The revenue from selling products is The purchasing cost is w M mj DEM R rj and the marketing effort cost of the sales teams is k dj β d ρ . The profit of distribution center d for the j th product is computed as Then, the total profit from all products at all distribution centers is as follows: 3.3. Profit of the retailers. The demand of the market is considered as deterministic dependent variable of marketing effort of the sales teams (ρ) and it is expressed as D rj (ρ) = DEM R rj + τ rj (1 − 1/(1 + ρ)), the demand DEM R rj is independent of marketing effort of the sales teams (ρ). D rj → (DEM R rj + τ rj ) when ρ → ∞ and D rj → DEM R rj when ρ → 0. It is important to mention that DEM R rj goes to zero when products are introduced into the market. In this situation, the demand of retailer attains to zero when ρ → 0. This situation occurs while the quality and prices are not familiar to the clients. The unit of ρ is quantified by the number of marketing efforts done by the sales team (See [42]).
The r-th retailer receives j-th product from distribution centers at a rate DEM R rj up to time kT dj . The inventory at r-th retailer piles up to time T rj after satisfying the demand rate D rj of the market of j-th product. The inventory level reaches zero at time T rj . Then, the cycle is repeating itself.The behavior of the inventory level with respect to time is represented by the following equation: Substituting Eq. (11) into Eq. (17), we have The inventory holding cost of the finished product is determined as follows: Substituting Eq. (11) and Eq. (18) into Eq. (19), we have The revenue from selling products is The purchasing cost is w D dj DEM R rj and the marketing effort cost of the sales team is k rj β r ρ . The profit of retailer r for the j-th product is expressed as follows: Then, the total profit for the products of the retailers is given by 3.4. Centralized system. In the collaborative approach, the members of the supply chain have the same role in making decisions. Here, the manufacturers, the distribution centers and the retailers are agree to share efforts for optimizing the profit function of the all actors of the supply chain. In this case, the profit function is obtained as follows: subject to the constraint The optimal solution for this production-inventory model is obtained by maximizing the profit function µ. However, in the mathematical expressions of π mj , π dj and π rj are given by equations (7), (14) and (21) respectively. The decision variables q M mj and ρ appear only in the terms related to cost parameters.Therefore, in order to find the optimal solution, it is necessary to minimize the following total cost function.
where D rj (ρ) = DEM R rj + τ rj (1 − 1/(1 + ρ)).Then, the total cost can be expressed in a compact form as follows: where The proof of the convexity of the cost function of the chain T C(q M mj , ρ) with respect to decision variables are presented in Appendix A. Algorithm: The optimization process is described as follows: For a given value of ρ, the total cost given by equation (26) is minimized when q M mj = 2λrj ϕrj . For the given values of q M mj , the optimal value of ρ can be obtained through an iterative process using equation (26) using ρ = 0, 1, 2, 3, . Later, the values of AP M , AP D, AP R, and µ are calculated for each ρ in order to find the maximum value of the joint profit of the supply chain. 4. Numerical examples. This section solves two numerical examples in order to illustrate the proposed production-inventory model. Two supply chains are examined with two manufacturers, two distribution centers and two retailers for two raw materials and two types of products. Each manufacturer produces two products that are delivered to the distribution centers according to the demand rates of the distribution centers. Each distribution center receives the products and satisfies the demand rate of the retailers. The retailers sell the products in the market for satisfying the demand of the customer. The members of the supply chain develop a collaborative decision making strategy to obtain optimal lot-sizes of the products establishing marketing efforts of the sales teams in each echelon. So, the demand of the retailers and the demand of the distribution centers are influenced by the marketing effort to satisfy the demand of the market.
The proposed algorithm was programmed using Microsoft Visual C# 2017 in an Intel(R) Core(TM) i7-model PC at 2.4 GHz (6.0 GB RAM),and the solution was obtained in approximately 1 second. The results of Example 1 and Example 2 are presented as below.
4.1. Example 1. The values for the parameters of manufacturers, distribution centers, and retailers are given in Table 2, Table 3 and Table 4. Table 5 presents the solution for different values of ρ (marketing effort of the sales teams). So, the optimal solution for the system of collaborative approach is Q      The values for the parameters of manufacturers, distribution centers, and retailers are given in Table 6, Table 7,and Table 8. Table 9 presents an optimal solution for example 2 for different values of ρ (marketing effort of the sales teams). So, the optimal solution for the system of collaborative approach is Q * 11 = 298.66 units, Q   Two numerical examples have been considered to illustrate the proposed production-inventory model. The example 1 and the example 2 show that an increase in the marketing effort of the sales team to some extent generates greater benefits of the supply chain. For example 1, the optimum value of joint profit of the supply chain is 152897.37 and for example 2,the optimum value is 117881.76 which attain at fiftyone initiatives of marketing and sixty-three initiatives of marketing, respectively. It means the members of the supply chain should implement merchandising activities, publicity, promotions and others activities done by the retailer, distribution center or manufacturer with aim of sharing the costs of retail merchandising activities.
The production-inventory model generates benefits to all members in three stages of the supply chain, but the marketing effort is greater in the retailer because of its expected percent of the marketing effort cost. The term β r represents the merchandising activities which have to do compete in the market. However, the distribution centers and the manufacturers share the cost of marketing effort to sell the products to the customer in smaller proportion than retailers. The initiatives of the sales teams( ρ) promote the sales of the products in the market throughout the supply chain as it stimulates the production, distribution, and commercialization of the same. In turn, it requires the implementation of a strategy of collaboration, teamwork and sharing information among members of the supply chain to earn more revenues while the members compete with each other.    5. Sensitivity analysis. A sensitivity analysis is presented to evaluate the effects of the key parameters like inventory holding cost, setup cost, selling price, production rate, and marketing effort cost of the sales teams of manufacturer, distribution center and retailer for numerical example 1. The results of the sensitivity analysis are shown in Table 10. This section shows the effects of the cost changes on the decision variables and profits of the members of the supply chain by increasing or decreasing the costs parameters, by 60%, 40%, and 20%.
5.1. Sensitivity analysis of the manufacturers cost changes. From analysis of Table 10, the following realistic scenarios are observed: * A higher inventory holding cost of the manufacturer h P 22 reduces the optimum values of AP M, AP D,and AP R affecting the joint benefit of the supply chain µ. This situation decreases to fifty points of the initiative of the sales team, that means the members should make less promotional activities to decrease holding costs for storage of the products so that the total profit of the chain is maximum.Moreover, the optimum value q M 22 is sensitive to variations in inventory holding cost. So, the reduction in inventory holding cost of the manufacturer is the situation more convenient to the supply chain.The sensitivity analysis shows that the variation of the inventory holding cost generates significant changes of the benefit of the supply chain members. * A lower setup cost (AM 22 ) increases the optimum value of AP M affecting positively the joint benefit of the members of the supply chain. Also, the initiative of the sales teams is reduced to fifty points that means the members should make less marketing activities. Moreover, the optimum value q M 22 is sensitive to variations in setup cost. * The optimal solution (ρ, µ, AP R, q M 11 , q M 12 , q M 21 , q M 22 ) is unchanged with variations in w M 22 , whereas the increment of the selling price increases the benefit of the manufacturers but reduces the benefit of the distribution centers, because these should purchase the product at a higher selling price. * The reduction of the marketing effort cost of the manufacturer increases the optimum value of ρ and the increment of this parameter reduces the initiatives of the sales teams and the benefit of manufacturers. So, for the members of the supply chain it is more convenient to reduce the marketing effort cost for the manufacturers. * When the production rate increases, the initiatives of sales teams are reduced to fifty points and the benefits of members of the supply chain are contracted because a higher production rate without the increment of the demand rate increases the inventory holding cost for products affecting the benefits of the members. * The joint benefit of the supply chain and the benefits of manufacturers increase by decreasing the marketing effort cost of the sales teams. The decrease in production rate increases the benefits of distribution centers, retailers and joint benefit of the supply chain but it reduces the benefit of the manufacturers. * The joint benefit of supply chain increases by decreasing the inventory holding cost, setup cost, selling price, marketing effort cost of the sales teams, and production rate. But, the benefit of the manufacturers decreases by decreasing the inventory holding cost, selling price of the product, and production rate. So, to increase the benefit of manufacturers, it could be reduced the setup cost and marketing effort cost, or increase the selling price. 5.2. Sensitivity analysis of the distribution centers cost changes. According to the results shown in Table 10, the affect of the changes of costs of the distribution centers on the benefits of the members of the supply chain and decision variables is concluded as follows: * When the inventory holding cost of a distribution center increases by 20%, 40% and 60%, the optimum value of ρ, the joint benefit of the supply chain and the benefits of distribution centers and retailers increase, whereas it reduces the benefit of the manufacturers because they have to bear higher cost for storing products. The increase of inventory holding cost of distribution center affects the optimum value of q M 12 because the manufacturer 1 should increase the production lot size of the product 2. * A lower setup cost for the distribution center (AD 12 ) decreases the optimum value of ρ to fifty points and increases the benefits of the manufacturers and distribution centers. So, this situation affects positively the joint benefit of the supply chain.
* The optimal solution (ρ, µ, AP D, q M 11 , Q M 12 , q M 21 , q M 22 ) is unchanged with variations in w D 12 , but the benefits of distribution centers and retailers increase by increasing the selling price. * Higher marketing effort cost at the distribution center 1 for the product 2 (k 12 ) decreases the optimum value of (ρ) to fifty points and reduces the benefit of distribution centers and retailers because this case increases the cost of making promotional initiatives.

Sensitivity analysis of the retailers cost changes.
Analyzing the results of Table 10, the impact of the retailers cost changes on the benefit of the members of the supply chain is given by: * When the inventory holding cost of the retailer h R 22 decreases, the optimum value of ρ, µ, AP D and AP R decrease, but it increases the benefit of the manufacturers because they could store a higher number of products. Also, the optimum value of q M 22 increases by increasing the inventory holding cost of the retailer. * A higher setup cost for the retailer AR 22 decreases the optimum value of ρ, the benefits of the distribution centers and retailers. So, this situation affects the joint benefit of the supply chain as it is reduced. * A higher selling price w R 22 increases especially the benefit of the retailers because it increases its revenue, affecting positively the joint benefit of the supply chain. Also, with a higher selling price, the optimum value of ρ increases from 52 until 55 points But, the decrease of the selling price decreases the optimum value of ρ from ρ = 51 until ρ = 45 ,i.e., it is not necessary to make more promotional initiatives. * A lower marketing effort cost of the retailer k 22 increases the optimum value of initiatives of sales teams ρ and the benefit of the retailers. It stimulates the promotional activities of sales teams because more initiatives could be done at lower cost that results in more demand of the products in the market. Moreover, the increment of the marketing effort cost of the retailer decreases the joint benefit of the supply chain. Consequently, it is not recommended.
6. Conclusion. A collaborative approach, in a highly competitive environment, is the definition of joint decisions among members of a supply chain. The joint sharing of marketing efforts or sales teams initiatives is to aim to increase the demand of the products on the market. In a three-layer supply chain, the manufacturers, distribution centers, and retailers share the costs of marketing effort of the sales team for all products offered on the market and it allows the sharing of benefits. This paper develops a collaborative EPQ inventory model for a three-echelon supply chain.This paper is based on the two-echelon supply chain model for single product proposed by Crdenas-Barrn and Sana [5] including a three-echelon supply chain comprising of multiple manufacturers, distribution centers and retailers for multiple products. Each manufacturer produces several products and delivers to several distribution centers which receive the products and satisfy the demand rate of the retailers. Each retailer sells the products to the market and orders the products to the distribution center according to their market demand dependent on marketing effort of sales teams. Additionally, this study shows how the marketing effort affects the profit of supply chain members with a collaborative approach. Also, the affects of some parameters as inventory holding cost, setup cost, selling price, production rate, and marketing effort cost of the sales teams are examined.
According to the sensitivity analysis, the major increment of the joint benefit of the supply chain occurs when the selling price of the retailer increases by 60% without affecting the benefits of manufacturers and distribution centers. Followed by a 60% decrease in the manufacturers selling price, the benefit of the manufacturers reduces whereas the benefit of the distribution centers increases notably. Furthermore, the joint benefit of the supply chain is affected by the variation of inventory holding cost, setup cost, selling price, production rate, and marketing effort cost. Additionally, the optimum value of ρ is sensitive to changes the key parameters but some optimum values of q M mj vary with an increase or reduction of the inventory holding cost, setup cost, production rate, marketing effort cost or selling price of the retailers.
Generally, the proposed inventory model provides an indication of a supply chain problem that has not been investigated in the mathematical modeling in supply chain and inventory fields. The new contribution of this paper compared with existing literature is that this approach considers the problem of optimal ordering size and production lotsize of multiple products for a three-layer supply chain with multiple members in each layer focused on the supply and marketing activities of products. Furthermore, this paper considers a collaborative system when each member takes decisions jointly to obtain maximum profit for the system and the inventory model can help to the managers of a industry to determine optimal strategies, contracts, promotions, and sales team initiatives to achieve maximum profits of the members of the chain. Consequently, the EPQ inventory model with a collaborative approach allows companies to control inventory levels in order to meet market demand, generating cost reduction and increasing the overall benefits of the supply chain. Finally, the results of the model show that the greater efforts of the sales teams improve the performance of the supply chain because of greater sales of products in the market and consequently, the downstream levels must increase their production and inventory level of the available products to attend the market demand.
The proposed model considers deterministic parameters, a perfect production system, and infinite replenishment. The present research work can be extended further in different forms. One extension consists of considering uncertain in the parameters of the supply chain. In this way, another extension can address with random production rate with machine breakdown in the production system. Moreover, another extension of this paper may consider shortages with full or partial backlogging when demand is dependent on the sales teams initiative or lost sales. These are some interesting research directions that researchers can do. 7. Appendix-A: Proof of the convexity of T C(q M mj , ρ). T C(q M mj , ρ) is convex when X.H.X T > 0 where X = [q M mj ρ] and ∂T C(q M mj ,ρ) ∂ρ = τrj q M mj (1+ρ) 2 + δ rj ρ −1 ;