1. INTRODUCTION
In the 1960s and 1970s, businesses were focused on improving their competitiveness by standardizing and improving their internal processes in order to deliver better quality products and services at lower costs. In the 1980s, with increasing variety in expected demand patterns, businesses became increasingly interested in increasing flexibility in product lines and developing new products to meet customer needs. In the 1990s, with the improvements in production processes and the use of reengineering models, managers in many industries realized that having improved internal processes and flexibility in the company’s capabilities is not enough to stay in the market, since suppliers of parts and materials must also produce materials of the highest quality at lowest costs and distributors must be closely coordinated with the market development policies of the manufacturer (Hugos, 2006).
In today’s markets, proper supply chain management can act as a major competitive advantage by reducing the price of products and services. Supply chain management includes the management of procurement, support, transportation, and marketing and organizational behavior, and networking, and strategic management, information systems management, and operations management (Chen and Paulraj, 2004).
The main goal of supply chain management is to guide decisionmaking at different stages of this process and coordinating these stages. Given the intense competition between manufacturers, if any of the links in this chain performs poorly, the whole system will not be successful as it will not perform at the expected level. In recent decades, businesses and organizations in industrialized and developed countries have paid special attention to supply chain management practices, whereby they have achieved significant global success, as evidenced by the high trade volume and remarkable revenue generation of their chains (Donald and Waters, 2003;Sangaiah et al., 2020).
In a study by Achabal et al. (2000) they designed an inventory and distribution system and showed that careful analysis of order patterns, cost distribution over the supply chain, and internal service levels provides useful information for supply chain management (Achabal et al., 2000).
Simatupang and Sridharan (2005) studied the conflict between objectives of realworld supply chain optimization problems. They stated that while past works have mostly focused on minimizing costs or maximizing revenue for a singleobjective optimization problem, in reality, the supply chain management may require optimizing several conflicting objectives simultaneously, such as reducing costs and increasing customer satisfaction (Simatupang and Sridharan, 2005;Daryakenari and Nasiri, 2021;Nattassha et al., 2019).
Sajadieh et al. (2010) developed an integrated vendor buyer model for a twostage supply chain. In this model, the vendor delivers the products to the buyer in equalsized batches. The delivered items are presented to the end customers in a display area. The quantity of items displayed in this area determines the demand. The objective is to maximize the total profit of the supply chain. The numerical analysis of this study showed that vendorbuyer coordination is more profitable when demand is more stockdependent (Sajadieh et al., 2010;Gaikwad, 2021).
In a study by Duna and Liao (2013) they investigated inventory management by the vendor, which is a popular partnership method for increase supply chain productivity. In this method, the vendor monitors the retailer inventory level and determines the time and amount of products to be delivered, which results in increased profitability by reducing inventory costs for the vendor and retailer (Duna and Liao, 2013).
Shankar et al. (2013) designed a singleproduct, multiobjective optimization model for a fourechelon supply chain consisting of suppliers, manufacturers, distribution centers, and customers. The main decision variables in this model were the number and location of factories, the flow of raw materials from suppliers, the volume of products transported. The goal was to minimize transport costs while maximizing demand satisfaction. To optimize these two objectives simultaneously, the four echelon mathematical model was formulated according to the related constraints, capacity, production, and transportation costs and solved with a particle swarm optimization algorithm (Shankar et al., 2013).
In a study by Zhang et al. (2018) they analyzed the research related to inventory management in Chinese supply chains. In this study, the effect of nonCO2 greenhouse gas emissions as an important factor in supply chain analysis was investigated. These researchers studied several different industries including food production, manufacturing of electrical products, etc. (Zhang et al., 2018).
Jabilles et al. (2019) simulated the impact of inventory on supply chain resilience. For this purpose, they developed a dynamic model and implemented it in 5 scenarios. The results of this work provide some guidelines for decisionmakers to manage inventory in such a way as to minimize the overall inefficiency of the supply chain (Jabilles et al., 2019).
Dai et al. (2020) developed a hybrid heuristic method for cyclic inventory management in Vendor Managed Inventory (VMI) supply chains. This algorithm determines the supply chain inventory in the first stage and optimizes the distribution of products in the second stage. The sensitivity analysis showed that the demand has a great impact on the performance of this algorithm (Dai et al., 2020)
In a study by Emtehani et al. (2021) they introduced a hybrid inventory and finance management model in supply chains. For this purpose, they proposed a method for coordinating the joint decisionmaking about the physical and financial flows of a capitalconstrained supply chain. They also considered the financial constraints of the investor that lead to the loss of sales. They formulated a twoobjective mathematical model for this problem and optimized it using the Epsilon constraint method. The results of this study showed the efficiency of the developed mathematical model (Emtehani et al., 2021).
2. METHODOLOGY
This study presents a threeobjective model of vendor managed inventory for a threeechelon supply chain consisting of a central warehouse, a vendor, and multiple retailers, in which demand is probabilistic and the vendor manages central warehouse inventory subject to constraints related to the number of orders and available budget. In this model, the first objective function minimizes the total inventory cost, the second objective function minimizes the use of storage space, and the third objective function minimizes the bullwhip effect in the supply chain. In short, the vendor assumes full responsibility for the ordering cost and therefore the retailers only have to pay the inventory cost. Furthermore, based on the economic order quantity (EOQ) policy, it is assumed that inventory replenishment for vendors and retailers is instantaneous and the reorder point is zero. It is also assumed that the system will lose excess demand and excess inventory displacement is not allowed. Also, retailers are assumed to sell all products shipped by vendors, meaning that $D={\displaystyle {\sum}_{i=1}^{r}{\displaystyle {\sum}_{v=1}^{k}{d}_{iv}}}$, where D is the annual demand of the vendors. The demand for each unit of product in each retailer per unit of time is a random value with a normal distribution with mean μ and standard deviation σ. The inventory cost of each unit of product in each retailer in each period is a random value with a uniform distribution bounded between a and b (a <x <b) with the density function $f\left(x\right)=\frac{1}{ba}$.

•Retailers sell all products shipped by the vendor.

•The annual demand of vendors and retailers is the same.

•The demand for each unit of product in each retailer per unit of time is a random value with a normal distribution with mean μ and standard deviation σ

•The inventory cost of each unit of product in each retailer in each period is a random value with a uniform distribution bounded between a and b: a <x <b.

• The density function of the uniform distribution for inventory cost is $f\left(x\right)=\frac{1}{ba}$.

•The annual demand of vendors is $D={\displaystyle {\sum}_{i=1}^{r}{\displaystyle {\sum}_{v=1}^{k}{d}_{iv}}}$

•The quantity and time of shipments and the replenishment rate of retailers and vendors are determined by the total cost of the system.

•The vendor manages the inventory of the central warehouse.

•The vendor is limited in terms of the number of orders and available budget.

•The vendor assumes responsibility for the ordering costs and the cost of shipping to retailers.

•Retailers only have to pay the inventory costs.

•The model uses the economic order quantity (EOQ) policy.

•Shortages are not allowed.

•Delivery time is zero.

•Costs are fixed (i.e. discounts and inflation are not taken into account).

•Orders are known and inventory replenishment for vendors and retailers is instantaneous.

•The reorder point is zero.

i : Index of retailers; i = 1,2,…, r

j : Index of products; j = 1,2,…, N

v : Index of vendor; v = 1, 2,…, k

w : Index of central warehouse; w = 1,2, ..., z

r : The number of retailers

N : The number of products

k : The number of vendors

z : The number of central warehouses

n : The number of retailer replenishments by vendors

m : The number of retailer replenishments by the central warehouse

T : Cycle of vendor

T_{i} : Cycle of retailer i

T_{r} : Joint cycle of retailers

LT_{i} : Delivery time to retailer i

H : Inventory cost of the central warehouse

h_{i} : Inventory cost of retailer i

x : Inventory cost per unit of time

f(x) : Density function of the retailer’s inventory cost per unit of product with the uniform distribution $f\left(x\right)=\frac{1}{ba},a<x<b$

a : The upper bound of the inventory cost in the uniform distribution

b : The lower bound of the inventory cost in the uniform distribution

h_{v} : The inventory cost of vendor v

h_{w} : The inventory cost of central warehouse w

A_{i} : Ordering cost for retailer i

A_{v} : Ordering cost for vendor v

A_{w} : Ordering cost for central warehouse w

f_{j} : Storage space needed for product j

F : Available storage space

C_{j} : The purchase price of product j (per unit)

B : Budget available to the vendor

N_{v} : The maximum number of orders for the vendor

q : The total quantity of products shipped to retailers

D : Annual demand of vendors; $D={\displaystyle {\sum}_{i=1}^{r}{\displaystyle {\sum}_{v=1}^{k}{d}_{iv}}}$ div: Retail demand from the v vendor

μ : Mean of the normal distribution

σ : Standard deviation of the normal distribution

q_{iv} : The quantity of products shipped to retailer i from vendor v

q_{ij} : The size of the order of retailer i for product j

Q_{ij} : The total size of orders of all retailers for product j; ${Q}_{ij}={\displaystyle {\sum}_{i=1}^{r}{\displaystyle {\sum}_{j=1}^{N}{q}_{ij}}}$

Q^{i}_{j} : The total size of all orders of retailer i; ${Q}_{j}^{i}={\displaystyle {\sum}_{j=1}^{N}{q}_{ij}}$

Q_{v} : The quantity of products shipped to vendor v from the warehouse

Q_{w} : The size of orders of central warehouse w

THC_{r} : Total inventory cost of retailers

THC_{v} : Total inventory cost of vendors

THC_{w} : Total inventory cost of the central warehouse

TOC_{r} : Total ordering cost of retailers

TOC_{v} : Total ordering cost of vendors

TOCw : Total ordering cost of the central warehouse

TC_{VMI} : Total cost of VMI system

BV : The bullwhip effect of the VMI system
As formulated in Equations (1) and (2), retailers replenish their inventory at regular intervals.
The size of the order of the vendor in each cycle can be obtained from Equation (3).
Similarly, Equation (4) shows the total size of orders of the warehouse in each cycle. In this equation, m is the number of times the vendor’s inventory is replenished from the central warehouse.
The first objective function, formulated in Equation (5), minimizes the sum of inventory costs.
Taking the derivative of the first objective function gives Equation (6) for the total inventory cost of all retailers.
The ordering costs of the vendor and retailers are formulated in Equation (7).
It should be noted that TOC_{v} is multiplied by $\frac{D}{{Q}_{v}}$ to obtain the total annual cost.
From equation (3), we have Q_{v}=nq. Also, for the total inventory cost of the vendor in one period, we have:
After multiplying by $\frac{D}{nq}$, the average annual inventory cost will be obtained as Equation (10):
From Equation (11) we have $\frac{q}{d}=\frac{{q}_{1}}{{d}_{1}}=\frac{{q}_{iv}}{{d}_{iv}}$
The ordering cost of the central warehouse is given by Equation (12).
Like the inventory cost of the vendor, the inventory cost of the warehouse is obtained from Equation (13).
Finally, the mathematical model of the problem is formulated as follows:
Using the derivatives of Equations (312) to (314), the final model can be formulated as Equation (15):
The second objective function, shown in Equation (16), minimizes the use of storage space.
The third objective function, formulated in Equation (17), minimizes the total bullwhip effect in the VMItype supply chain.
The vendor can handle a limited number of orders from retailers. Assuming that the maximum number of orders is N_{v}, we have the following inequality as a constraint.
Equation (19) ensures that the size of the order of retailer r in period T_{i} is always equal to or smaller than the total amount of product required by that retailer.
Equation (20) shows that the total amount of products required in period T_{i} is always equal to or smaller than the total amount to be delivered to retailers.
The vendor also faces budget constraints. The total budget available is limited to B, which results in the following inequality.
The constraint ensuring that the space occupied in the warehouse does not exceed the available warehouse space is formulated in Equation (22).
Equation (23) shows that the demand of retailer i from vendor v is always equal to or smaller than the quantity of products shipped to retailer i from vendor v.
Equation (24) shows the acceptable range of the variable used in the model.
To test the accuracy of the proposed model, we solved a smallscale problem instance using the Epsilon constraint method in the operational research software GAMS. In this numerical example, the supply chain consists of four retailers, three vendors, and two central warehouses with two products with an average price of 840 and 777 respectively.
The input parameters of the problem were generated using the normal uniform distribution such that:

A) The demand rate per unit of time follows a uniform distribution bounded between a and b; a <x <b

B) The density function of the demand in the uniform distribution is $f\left(x\right)=\frac{1}{ba}$

C) The potential annual demand of the vendors follows a normal uniform distribution; $D={\displaystyle {\sum}_{i=1}^{r}{\displaystyle {\sum}_{v=1}^{k}{d}_{iv}}}$
Detailed information of this numerical example is given in Table 1.
The data were imported into GAMS as inputs and the outputs were obtained using the Epsilon constraint method. In the following tables, the results are presented systematically. First, to determine the Pareto points, the problems were solved separately based on each objective function, reaching the results presented in Table 2.
After obtaining the values of Table 2, in line with the third step of the Epsilon constraint method, eight epsilon values were considered for the second objective function.
Table 3 shows the epsilon values of the various breakpoints considered for the second objective function.
Table 4 shows the computational results related to the Pareto points for the eight considered feasible breakpoints. In the end, the results were plotted in diagrams of Figure 1 for easier analysis.
In the diagrams of Figure 1, it can be seen that as the value of the first objective function increases, the values of the second and third objective functions also change. In other words, the objectives are not optimized equally, because when one objective changes, it has a direct impact on other objectives.
The best Pareto solution was identified using the Simple Additive Weighting (SAW) method, which involves giving a weight to each objective. Here, since all objectives were of the minimization type, the solution with the lowest total weight would be the best solution. Also, all of the objective function values were normalized as shown in Table 5 to make sure that they are on the same scale.
The weights considered for the first, second, and third objectives were 0.5, 0.4, and 0.1, respectively. As shown in Table 6, the first Pareto solution was found to be the best solution.
In general, the measure for determining whether a problem is large, medium, or small is the time it takes to reach the optimal solution (or explore the solution space) by an exact method such as the branch and bound algorithm used by GAMS software. If the exact method can find the optimal solution of the model in less than 11.5 hours, then the problem is said to be a smallscale problem. If the exact method cannot produce the optimal solution within this time, then the problem is called a largescale problem. Table 7 shows the dimensions of other generated problem instances and the best Pareto solution for each instance.
In Table 8, it can be seen that as the problem dimension increases, the total number of variables also increases from 31 to 6383. A similar increasing trend can be observed in all objective functions. The objective function value has increased from 86122.16 to 139766.64 for the first objective, from 2373.2 to 3560.86 for the second objective, and from 303.19 to 479.73 for the third objective.
4. CONCLUSION
The design of the supply chain network has a great impact on the efforts to achieve efficient and effective supply chain management and is one of the most important issues in longterm strategic decisions in this field. In general, supply chain network design counts among the most important areas of optimization. This study developed a model for integrated procurement, production, and distribution planning in a threeechelon supply chain with probabilistic demand and the objectives of minimizing the supply chain cost, minimizing the use of warehouse space, and minimizing the bullwhip effect. The main assumptions of the model included the supply chain being multipleproduct, multiperiod, and threeechelon, and the price being a function of demand.
Given the breadth of supply chain network design problems and their models, they can be solved through a wide variety of solution methods. One may also expand these models by defining new assumptions to make them more realistic and practical. Accordingly, the recommendations of this study for future research are presented in two categories:
A) Recommendations for modifying the problem structure:

•Considering the location problem for factories and distribution centers.

•Considering a variety of sales policies such as discounts.

•Expanding the model to incorporate production planning and inventory control in factories and warehouses.

•Adding a vehicle routing phase to each stage of the integrated supply chain management

•Considering uncertainty in costs and using fuzzy or probabilistic parameters for this purpose.

•Considering processing times or preparation times.