1. INTRODUCTION
Managing the orderingdelivery processes between organizations has been a key issue in supply chain management. Despite the increasing application of JIT1, lean and agile production systems and new information systems that increase the transparency in supply chains, there remain still many problems. Research conducted at European companies indicates that there has been little improvement in the procurement and delivery process over the last decade. Vendor managed inventory (VMI) is a recent alternative to the orderingdelivery process (Kaipia et al., 2002). Different terms have been used for VMI in different industries, but the base is the same for all. VMI is one of the supply chain strategies whereby the responsibility for managing customer (or producers) inventories is delegated to the seller (or the supplier) (Disney and Towill, 2003).
In the past years, many models have been developed for the VMI system with various assumptions. These assumptions are based on the type and conditions of the supply chain that creates different situations. The number of vendors, buyers, products and components to be planned, inventory storage locations (warehouses), and scheduling time periods are factors that influence the creation of different states in the VMI system. However, for most developed models, the objective function has been to minimize the total costs of the supply chain. Designing of VMI models with the goal of optimizing the total costs of supply chain requires access to large financial data, including storage costs, ordering costs, shortage costs, etc. Unfortunately, in manufacturing companies due to the confidentiality of financial data, access to financial data is impossible or very costly. Therefore, we suggest using inventory turnover index as the objective function to design the VMI model in the supply chain which can be a new approach that requires only a kind of financial data (commodity price).
The contribution of this paper is to create a new approach to designing VMI models based on the objective function that seeks the optimal inventory turnover in the warehouse of the producer. In this work, we design a new model based on the master production schedule (MPS) and the bill of materials (BOM) for different production products to achieve optimal levels of inventory. The constraints of the model are to prevent the occurrence of shortages of materials and to observe the minimum and maximum storage limits in the warehouse of the producer.
The rest of the paper is organized as follows: Section 2 gives related literature. In this section, there is first a classification of previous studies in the field of VMI, and then the history of designing VMI models will be examined and finally, a summary of the literature review is presented. Problem description will be described in Section 3. Section 4 explains the proposed evolutionary algorithm and discusses computational studies and simulation results. Finally, Section 5 includes conclusions and future researches.
2. LITERATURE REVIEW
2.1. A Classification of Previous Research in the Field of VMI
VMI systems have attracted the significant attention of many researchers in the world over the last two decades due to its ability to effectively coordinate supply chain partners through the sharing of information and exchanges. In simple terms, the VMI allows suppliers to manage inventory of producers and to determine their replenishment policies. In general, the VMI literature is divided into two main sections: (1) Modeling of VMI systems (2) Practical implementation of VMI. By examining the researches on VMI, three areas can be distinguished: (1) Public domain (2) Field of modeling (3) Field of case studies. In a more detailed classification, VMI researches can be categorized in four parts: (1) Modeling (Mathematical/Simulation) (2) Case studies (3) Benefits of VMI (4) Challenges and success factors of the VMI (Salem and Elomri, 2017) In Figure 1, an illustration of the VMI field classification scheme is shown:
2.2. History of Designing VMI Models
The latest research in the field of designing VMI models in recent years has been as follows:
Akbari and colleagues developed a VMIbased supply chain for perishable products aimed at minimizing the cost of the entire supply chain, including ordering costs, storage costs, discounts, and perishment of products, considering replenishment cycle and order size and required production time to provide inventory by retailers (Kaasgari et al., 2017).
Bani Asadi and his colleagues presented a VMI mathematical model for the supply chain with three levels of central warehouses, multiple distribution centers, and multiple retailers which are integrated into a hybrid integer programming and uncertain input parameters are included (BaniAsadi and Zanjani, 2017).
Anna presented the subject of supply chain inventory management implementation by the supplier with a simple probabilistic inventory model to the goal of examining the effect of inventory management before and after the implementation of the VMI. She proved the benefits of implementation of VMI for the supplier and buyer with the constraints of the existence of a supplier and a buyer and random demand with the normal distribution and the decision variables of the order quantity and the ordering time to the supply. She applied three numerical examples concludes that the proposed analytical model can reduce the expected costs of the supply chain, improve the level Services and increase inventory replenishment (Anna, 2016).
Bodhankar and Rangari (2016) on Supply Chain ManagementA new approach to the business process, followed to create a sustainable model for a supply chain system, consisting of a supplier and a producer using the VMI system and known parameters of inventory performance, for Comparison of the performance of the proposed VMI system with traditional tractioncompression systems, using a broadbased solution solving method for participatory prediction and time limits for fixed supply and demand distribution functions, the best ordering policy between EOQ, monthly order, JIT and VMI Were determined. The findings of this study showed that under VMI policy, supply chain inventory levels have decreased, better forecasting of demand, lower costs, and better services are provided (Bodhankar and Rangari, 2016).
Chen et al. (2017) on the topic of semiconductor supply chain planning, along with decoupling points decisions and the VMI scenario by the goal of helping decision makers to determine the production and delivery amounts at different locations, determining the decoupling point of the pullpush supply chain, determining the limit values of inventory and inventory management scenario by the seller with several production constraints, multiple factories, multiple products ranked in the semiconductor industry, and using the Lingo 12.0 and Microsoft Excel 2010 solution, resulted in the proposed integrated model for determining the decoupling point, selection of appropriate scenario for VMI and production planning with the goal of optimizing the total benefits supply chain were provided (Chen et al., 2017).
Krichanchai and Maccarthy (2016) on the subject of acceptance of inventory management by the seller for the hospital supply chain with the goal of identifying the factors influencing VMI acceptance in a supply chain, including 3 hospitals, one distributor and one supplier/producer, using three techniques, used to collect data, including interviewing, visiting the site, and reviewing documents to enhance the credibility and robustness of the research, led to identifying two types of VMI actions (public and private) in the hospital/supplier area. The public sector, including hospitals, emphasizes improvements in service levels, while the private sector, which includes suppliers, emphasizes strong relationships with key customers (Krichanchai and Maccarthy, 2016).
Sadeghi et al. (2016) presented an inventory model on the topic of optimizing with fuzzy demand, backorders, and discounts, using the hybrid algorithm of imperialist competitive in the supply chain with a vendor and several retailers, and the space constraints of the warehouse and the number of orders, with the goal of finding the nearoptimal answer included order size, the retailer’s replenishment rate, the maximum amount of backorders of retailer’s items and the appropriate price of items to minimize the total inventory costs. The proposed model was a complete NP problem and used a colonial compilation algorithm to find a nearoptimal solution. Finally, the effects of some cost parameters on the solution to achieve management views are investigated (Sadeghi et al., 2016).
Escuin et al. (2017) with the subject of comparing inventory policies for timevarying random demand in the paper industry, with the goal of developing a mathematical model for calculating the optimal composition of inventory for dealing with minimumcost random demand in a twotier supply chain with capacity and service level constraints provided a model for comparing maketoorder and vendor managed inventory strategies. Using the simulation method, the findings of this research are to improve the level of customer service and supply chain inventory costs through proper production planning and replenishment scheduling in VMI (Escuin et al., 2017).
Mateen et al. (2015) with the VMI theme for singleseller multiretailer supply chains under random demand with the goal of minimizing the total expected costs of the VMI system and assuming that all retailers are replenished at the same time by the vendor in while of shortage in vendors, allocating inventories to retailers based on the same probability of shortage and constraint of inventory storage space for retailers and allocating fines in case of shortages, used a simulation method to test the validity of the proposed model and investigated effective factors in the system’s operation (Mateen et al., 2015).
Niknamfar (2015) with the subject of Production Planning  Multi objective distribution based on vendor managed inventory plan in a threetier supply chain including several foreign suppliers at the first tier, a secondtier producer and a thirdtier retailer with the goal of minimizing production costs, retailers’ costs, and distribution time in the supply chain using the multiobjective genetic algorithm (NSGAII) and (NRGA) and the TOPSIS ranking method showed that NSGAII has a better performance (Niknamfar, 2015).
Taleizadeh et al. (2015) with simultaneous optimization of price, material replenishment rates, final product replenishment cycle and production rate in vendor managed inventory system with perishable items in a twotier supply chain, including a vendor and several retailers in both the raw material and the final product have different rates of deterioration, with the goal of maximizing total profit of supply chain and constraint of the definite market demand for the final product and price sensitivity with the Stackelberg approach, led to an analysis of the effects of retailers’ costs, perish rate of raw material and final product replenishment and number of retailers on optimal decision variables and profit of the entire of supply chain (Taleizadeh et al., 2015).
Yu with the subject of how much VMI in the global environment is better than RMI, With the goal of comparing the vendor managed inventory strategy with the retailer’s inventory management strategy, improving supply chain performance through better decision making on appropriate order quantities with random demand and rate exchange constraints led to the findings that, firstly, the VMI was not necessarily at all the conditions are not better than RMI; secondly, the effect of demand changes on VMI performance is unknown and depends on the value of the parameters; and thirdly, when the exchange rate increases, the VMI is initially worse than the RMI, but then it performs better (Yu et al., 2015).
2.3. A Summary of the Literature Review

By examining the background of the researches, it is clear that the main researches done in the field of design of VMI models was aimed at optimizing the objective function of total costs of supply chain (Kaasgari et al., 2017; Anna, 2016; Arora et al., 2010; BaniAsadi and Zanjani, 2017; Ben Daya et al., 2013; Bookbinder et al., 2010; Braglia et al., 2014; Escuin et al., 2017; Gou et al., 2008; Gümüş et al., 2008; Kannan et al., 2013; Kastsian and Mönnigmann, 2011; Kiesmüller and Broekmeulen, 2010; Lee and Ren, 2011; Liao et al., 2011; Mahamani and Rao, 2010; Mateen and Chatterjee, 2015; Omar et al., 2010; Pasandideh et al., 2010, 2014; Nia et al., 2014; Sadeghi et al., 2013, 2014a; Sari, 2008; Southard and Swenseth, 2008; Van Nyen et al., 2009; Wang et al., 2008; Wong et al., 2009; Yao et al., 2007; YongQiang et al., 2010; Zhang et al., 2007).

In the field of designing an optimal model of VMI system with the goal of finding the optimal level of inventory turnover, no research has been observed. In a few studies that have addressed the issue of inventory turnover in the VMI domain, it has been considered as one of the indicators of measuring the performance of the VMI system and in fact As a result and one of the benefits of VMI implementation is considered (Achabal et al., 2000; Bai and Zhong, 2008; Aihua, 2007; Halkola, 2014; Mcbeath, 2003; Swan et al., 2002). So far, optimization of inventory turnover has not been used as the goal of designing the VMI model.

In literature related to the design of VMI models, storage space constraint is one of the most important and most used constraints that has been respected by most researchers in most projects (Braglia et al., 2014; Aryanezhad et al., 2012; Pasandideh et al., 2010, 2014; Nia et al., 2014; Sadeghi et al., 2011, 2013, 2014a, 2014b).

The constraint of the illegality of shortage of parts in the VMI system is also included in some studies (Bookbinder et al., 2010; Jasemi et al., 2014; Pasandideh et al., 2010; Zhang et al., 2007).

In previous studies, the following decision variables are considered in the VMI models:

 The number of orders sent from the supplier to the manufacturer (Bookbinder et al., 2010; Darwish and Odah, 2010; Aihua, 2007; Gronalt and Rauch, 2008; Gümüş et al., 2008; Kiesmuller and Broekmeulen, 2010; Li, 2010; Liao et al., 2011; Seifbarghy and Gilkalayeh, 2012; Aryanezhad et al., 2012; Nachiappan and Jawahar, 2007; Pasandideh et al., 2010; Pasandideh et al., 2011; Rasay et al., 2013; Razmi et al., 2010; Nia et al., 2014; Sadeghi et al., 2011, 2013; Sadeghi et al., 2014a, 2014b; SueAnn et al., 2012; Wang et al., 2008; Wong et al., 2009; Yao et al., 2007).

 The optimal level and frequency of inventory replenishment (BenDaya et al., 2013; Bookbinder et al., 2010; Choudhary and Shankar, 2015; Aihua, 2007; Gou et al., 2008; Guan and Zhao, 2010; Hu et al., 2013; Razmi et al., 2010; Sadeghi et al., 2013; Sadeghi et al., 2014a, 2014b; Sari, 2007, 2008; Sarpola, 2007; Van Nyen et al., 2009; Wang et al., 2008; Wong et al., 2009; Yu et al., 2009b, 2012).

 The optimal level of inventory in the warehouse (BenDaya et al., 2013; Bergvall and Björkman, 2007; Gronalt and Rauch, 2008; Kastsian and Mönnigmann, 2011; Lin et al., 2010a; Mahamani and Rao, 2010; Omar et al., 2010; Yu et al., 2009a).
2.4. A Substantive Comparative Discussion to Discern the Merits of the Current Research
Although much research has been done in the field of designing VMI models, most of the research has been based on the goal of minimizing the costs of the entire supply chain. This is useful because of the totality principle of supply chain management, but it requires a lot of data that may in practice affect the applicability of VMI models. In this research, by changing the approach to the objective function of the VMI models, it uses a financial indicator called the number of inventory turnover and seeks to maximize this index in the producer’s inventory. The advantage of this is the lack of largescale financial data, which will help in modeling the model.
The next topic that has been given special attention in this research is the use of constraints and decision variables that have been considered in most of the previous research. Meanwhile, the solution methods used for the proposed model include the most applicable solving methods in previous research.
3. PROBLEM DESCRIPTION
A supply chain consisting of several suppliers and one company producing the final products required by customers according to Figure 2 is assumed.
The problem is to optimize inventory turnover in the producer’s warehouse, under the following conditions:

The manufacturer produces several products.

Each product has the same or different components with other products.

Supplying of required parts to the manufacturer is based on the VMI system. In this system, suppliers are responsible to manage producer inventory and, on the basis of the inventory level and the estimated limits, replenish producer warehouse.

The inventory levels are reviewed daily and the inventory planning period is a oneyear period of 12 months and equivalent to 257 days.

The inventory levels are between the minimum and maximum warehouse limitations that were estimated.

Lack of parts in the production line is not allowed.
3.1. Notations and Their Definitions

J: Total number of products (j = 1, 2, 3 ... J)

K: Total number of parts (k = 1, 2, 3 ... K)

M: Total number of months (m = 1, 2, 3 ... 12)

T: Total number of working days (t = 1, 2, 3... 257)

D_{j,m} : Quantity of product demand j in month m

BOM_{k,j}: Consumption coefficient of goods in product j

C_{k,m} : Required quantity of good in month m
$({\text{C}}_{\text{k},\text{m}}={\text{BOM}}_{\text{k},\text{j}}\times {\text{D}}_{\text{j},\text{m}},\hspace{0.17em}\forall \text{j})$

DAYS_{m} : Total number of working days in month m

T: Total working days of the year

C_{k,t} : Required quantity of good k on day t for producing, $\left({\text{C}}_{\text{k},\text{t}}=\frac{{\text{C}}_{\text{k},\text{m}}}{{\text{DAYS}}_{\text{m}}},\hspace{0.17em}\hspace{0.17em}\forall \text{m}\right)$

P_{k}: The purchase price of good k

I_{(k,t)}: Inventory of good k at day t

X_{(k,t)}: Quantity of good k that is delivered by the supplier on day t

I_{(k,min)}: The minimum stock of good k that must always be available in the warehouse

I_{(k,max)}: The maximum stock of good k can be in the warehouse

I_{(k,0)}: Inventory quantity of good k in the first period (certain)

I_{(k,VMI)}: Optimum inventory for replenishing good k in the VMI model
3.2. Problem Formulation
Based on the notation and definitions described above, the proposed model is formulated as follows:
Subject to
Eq. (1) is associated with the proposed objective function to be maximized. The objective function of the model is to maximize the number of times inventory turnover in the manufacturer’s warehouse. The number of times inventories is one of the financial indicators that shows the effective use of inventories. How to calculate this index is as follows:
$\begin{array}{l}\text{Inventoryturnoverindexinagiventimeperiod}\\ \text{=}\frac{\text{Valueofconsumedgoodsinagiventimeperiod}}{\text{Valueofinventoryaverageinthesametimeperiod}}\end{array}$
Using this index, the goal of the model is defined with the minimum financial information required and based solely on the price of the goods, and the basis for optimizing the model to reach the optimum level of replenishing inventories in the VMI system.
Eq. (2) until Eq. (9) are problem constraints: Eq. (2) shows the determining of inventory values at the start of the planning period. Eq. (3) calculates the quantity that must be supplied by the supplier in the inventory checking intervals. This value is determined by the inventory level of items and its difference to the optimal quantity of inventory in the VMI system. Eq. (4) calculates the inventory for each period based on the inventory of the previous period, the quantity supplied by the suppliers, and the quantity consumed from inventory in the current period. Eq. (5) shows the minimum inventory limit. Inventory at different time intervals should not be less than the minimum stock of inventory. Equation (6) shows the maximum inventory limit. Inventory at different time intervals should not exceed the maximum inventory limit. Eq. (7) shows that the optimal quantity of inventory in the VMI system is determined by the minimum and maximum inventory levels. Eq. (8) is a condition for nonnegativity of inventory and control to not occur the lack of goods. Eq. (9) shows that the number of goods supplied by the suppliers should not be negative.
4. THE PROPOSED METHODOLOGY
Genetic algorithms are the family of computational models developed by Holland for the first time inspired by the gradual evolution of nature (Goldberg and Holland, 1988; Holland, 1975). These algorithms encode a potential solution to a specific problem on a simple data structure, such as a chromosome, and combinatory operators use these data structures to maintain vital information. GAs are often considered as functional enhancers and cover a wide range of issues (Shi et al., 2003) and are applied to successfully solve various issues such as optimal design, fuzzy logic control, neural networks, expert systems, Scheduling and other issues (Schwefel, 1993; Stender et al., 1993).
Particle swarm optimization algorithm is a populationbased randomization optimization technique introduced by Eberhart and Kennedy for the first time (Eberhart and Kennedy, 1995; Kennedy and Eberhart, 1995). The PSO includes a population based on the random optimizer that acts by imitating the behavior of the bird group and the fish species (Yang et al., 2007). In this algorithm, each solution is considered as a particle in the swarm of particles (problem solutions) (Robinson et al., 2002). Each particle has a random velocity position and vector, which moves in the space of solutions to the best particles (Yang et al., 2007).
The property of the PSO optimization algorithm is that it converges quickly, but near the optimum point, the search process slows down considerably. On the other hand, we know that the genetic algorithm is also very sensitive to the initial conditions. In fact, the random nature of genetic operators makes the algorithm susceptible to the initial population. This dependence on the initial condition is such that if the initial population is not well chosen, the algorithm may not be convergent (Shi and Eberhart, 1999).
In this paper, a global search strategy based on the combination of GA and PSO algorithms is used. The combined algorithm works more efficient and effective. Our goal is to simultaneously benefit from both the capability of a global and local search to achieve the best possible answer with better performance. In the proposed algorithm, GA first uses the overall response space to perform a global search. Then, the PSO is used for local searches near the best GA solution, to improve the ultimate solution. This process is retranslated into GA’s global search and then returned to the local search of the PSO. This cycle continues until the condition for stopping the operation of the combined algorithm is met and reaching the best possible solution. The general flowchart of the proposed algorithm is shown in Figure 3.
4.1. Problem Representation
In this type of vendor managed inventory problems, our goal is to maximize the inventory turnover of the manufacturer. In order to achieve this objective, the optimal level of stock inventory for all items should be determined. This optimal quantity is the amount that suppliers must increase inventory levels up to this level in every period of manufacturer inventory checking. Therefore, each of the inventory items of the producer has an optimal level for replenishment by the suppliers. We seek to determine these optimal levels.
In order to solve the VMI problem using the proposed hybrid algorithm, a mechanism for formulating the problem in term of solutions should be used. The number of optimization variables in each solution in this problem is equal to K, where K is the total number of goods. The values that need to be optimized are the limits of inventory replenishment. The purpose of the proposed model is to find the best value for I_{k,VMI} in such a way that inventory turnover is maximized. The constraint for I_{k,VMI} is:
We perform the following mathematical operations on the above inequality:
If $\widehat{x}$ is a random variable between zero and one ($\left(\widehat{x}\in \left[0,1\right]\right)$ ), according to above fraction, it can be written:
Consequently, I_{k,VMI} can be extracted from the above statement:
Thus, for random values $\widehat{x}\in \left[0,1\right]$ , we can obtain different values for I_{k,VMI}.
A solvable solution is a solution that meets the model constraints for created random values. Therefore, if the constraints of the model are not established, the following will occur:

Shortage: In the state of shortage, the amount of inventory is negative (I_{k,t} < 0).
To avoid the shortage, a violation penalty is considered as follows:
$$\text{V}{1}_{\left(\text{k},\text{t}\right)}=\mathrm{max}\left(0,\hspace{0.17em}{\text{I}}_{\text{k},\text{t}}\right)$$Therefore, only when VI_{(k,t)} for all k and t is equal to zero, there will be no shortage.

Violation of Minimum inventory Constraint: In cases where the inventory of any item in any period of time decreases the minimum inventory considered for that item, we will have:
To prevent this situation, a violation penalty is considered as follows:
$$\text{V}{2}_{(\text{k},\text{t})}=\mathrm{max}\left(0,{\text{I}}_{\mathrm{min},\text{k}}{\text{I}}_{\text{k},\text{t}}\right)$$Therefore, only when V2_{(k,t)} for all k and t is equal to zero, the minimum inventory constraint is established.

Violation of Maximum inventory Constraint: In cases where the inventory of any item in any period of time exceeds the maximum inventory considered for that item, we will have:
To prevent this situation, a violation penalty is considered as follows:
Therefore, only when V3_{(k,t)} for all k and t is equal to zero, the maximum inventory constraint is established.
To make the solution feasible, all of its constraints must be established. In view of the penalties that are taken to violate the constraints, the feasibility of the solution is obtained when the total of the penalties set is zero, in other words, V = 0, so that:
In this way, we introduce the penalties in the objective function, so that if the constraints are not established, the value of the objective function is reduced to the value of the deviation of constraints, and eventually removed from the selection process. If α is considered as a large number such as 100 and the average penalties of deviations of constraints is shown by of constraints is shown by V_{Mean}, then the competency of the solutions can be calculated from Eq. 10.
Therefore, a feasible solution in the GA, PSO algorithms, and their combination, for a set of data, including the K item in the manufacturer’s warehouse, is in Table 1.
4.2. Generation of the Initial Population
The optimal solution search in the proposed algorithm starts from a random initial population and follows to optimize initial population in a repetitive process. The initial population is a primary solution that it may be not feasible first, but it follows to gain feasible solution in the next iterations. An example of initial population is shown in Table 2:
4.3. Fitness Evaluation
In each iteration, the fitness of each chromosome (solution) is evaluated according to the proposed objective function with respect to penalties due to deviation from the constraints according to Eq.10. Then all the chromosomes sort from the best to the worst, and part of the best chromosomes are selected based on the strategy of choosing the fittest ones to update the population in the next stages.
4.4. Global Search via GA
The optimization process in GA consists of three operators, which are transmitted from one population to a better population. The first is selection operator that imitates the natural principle of the survival of the best generation. The second is crossover operator, which is the same as mating in biological populations. This operator will transfer the characteristics of a better generation from the current population to future generations and will improve the value of the objective function. The last is mutation operator, which causes diversity in the characteristics of the population. This operator allows a global search in the solution space and prevents the algorithm from being trapped in local optimal solution (Hassan et al., 2005).
4.4.1. Selection Operator
In order to select chromosomes for mutation or crossover, we remove the percentage of the primary population and select one or two chromosomes randomly to perform crossover or mutation operation (Masoud et al., 2013). The selection operator is necessary to maintain the number of population, after each crossover and the creation of new offspring (Abdullah et al., 2012). Selection of the best chromosome takes place based on fitness function evaluation to participate in the next generation and to ignore the poor choices in the next steps of the algorithm (Thanushkodi and Deeba, 2011). This operator selects two major chromosomes to produce new strings (for example, offspring). In GA algorithm, parent strings are selected randomly (AbdElWahed and ElShorbagy, 2011). The roulette rotary wheel is usually used for random selection (Liu et al., 2010b). This method also called a random selection method with substitution, acts as the selection options on a continuum, so that the size of each section of this continuum is proportional to the value of the function of the corresponding option. A random number is generated, depending on which number in the continuum section it is selected, the option in that section is selected. This process repeats until to get the number of needed options (Premalatha and Natarajan, 2010).
4.4.2. Crossover Operator
To generate an offspring, two parents are selected randomly among all parents and a crossover operator is performed on them. In this paper, the uniform crossover operator is applied as shown in Figure 4. In this type of symmetric and uniform crossover, each gene of offspring is complemented by the gene of one of the parents with equal probability (Shokouhifar and Jalali, 2015).
Consider the crossover operator as C (x_{i}, x_{j}): (Yang et al., 2007)
$\begin{array}{l}{\text{Parent}}_{\text{a}}{\text{=x}}_{\text{i}}\\ {\text{Parent}}_{\text{b}}{\text{=x}}_{\text{j}}\\ {\text{r}}_{\text{c}}\text{=random}\left(\text{0,1}\right)\\ {\text{Child}}_{\text{a}}{\text{=r}}_{\text{c}}\hspace{0.17em}{\text{\xd7Parent}}_{\text{a}}\text{+}\left({\text{1r}}_{\text{c}}\right){\text{\xd7Parent}}_{\text{b}}\\ {\text{Child}}_{\text{b}}\text{=}\left({\text{1r}}_{\text{c}}\right){\text{\xd7Parent}}_{\text{a}}{\text{+r}}_{\text{c}}{\text{\xd7Parent}}_{\text{b}}\\ \text{C}\left({\text{x}}_{\text{i}}{\text{,x}}_{\text{j}}\right)\text{=}\left({\text{Child}}_{\text{a}}{\text{,Child}}_{\text{b}}\right)\end{array}$
4.4.3. Mutation Operator
The purpose of the mutation is to create diversity in the search path and prevent convergence towards local optimal solutions. A mutation is a general operator that changes one or more of the genes in a chromosome. This can lead to an increase in the value of the objective function using the new chromosome. Genetic algorithm with these new genetic values may achieve better solutions than previously possible. The mutation operator is an important part of a genetic search that helps prevent population to trap in local optimum solutions and increases the likelihood of finding globally optimal solutions (Premalatha and Natarajan, 2010). Generally, two types of mutation are used: exchange mutation and swap mutation. In the exchange mutation, the values of some of the genes are displaced and mutated chromosomes are generated. In the swap mutation, some genes are changed and mutated chromosomes are formed (Nikabadi and Naderi, 2016). Figure 5 shows how to operate two types of mutations.
In this paper, a swap mutation is applied that generates mutated chromosomes by changing the values of some genes.
4.5. Local Search via PSO
The PSO generally starts with a random primary solution. In the proposed hybrid algorithm, the best global GA solution is used as a primary solution to the PSO. In PSO, each particle is considered as a potential solution in the solution space. Each particle has a position as shown below:
Each particle saves its best previous position, which is as follows:
Due to the movement of particles in the solution space to the optimum solution, each particle has a velocity as follows:
The best value of each particle (so far) is specified (pbest). Also, the best value of total particles (so far) is known (gbest). Each particle changes its position using the following information:
In each PSO iteration, two steps are taken to update the velocity and position of the particle. After evaluating and selecting the best solutions, this cycle continues until the condition for stopping the operation is met.
4.5.1. Updating the Particle Velocity
The particle velocity is updated according to the eq. 11 with regards the current particle velocity and position, as well as the best position of the particle (so far) and the best position of the total particles (so far):
where,

V_{ik} : Velocity of particle

X_{ik} : Current position of the particle

w : Inertia factor

c_{1} : Coefficient of the relative effect of cognitive component

c_{2} : Coefficient of the relative effect of social component

P_{ik} : The best position of the particle so far (pbest)

P_{gk} : The best position of all particles so far (gbest)

r_{1}, r_{2} : Random numbers
In Eq. 11, w is an inertial factor that controls the effect of the previous velocity on the new velocity, r_{1} and r_{2} are random numbers that maintain diversity in the population and have a uniform distribution between [0, 1]. c_{1} is a positive constant value that is cognitive component coefficient and c_{2} is the positive constant value called the social component coefficient. In Eq. 11, each particle moves to a new position, taking into account its best previous position and the best previous position of the whole particle.
4.5.2 Updating the Particle Position
The particle position is updated in accordance with Eq. 12:
All particles move towards better positions, so the best position (optimal solution) is finally obtained through the combined efforts of the whole population (Premalatha and Natarajan, 2009).
4.6. Computational Results
The modelsolving algorithms were coded in MATLAB 8.3 environment, and the experiments were executed on a PC Intel Core 2.6 GHz processor and 8 GB memory running on Windows 10.
4.7. Data Generation
To solve the proposed model, sample problems with different sizes are randomly generated. Created problems based on the number of products in the supply chain and the number of items per product have different sizes. Based on this, five types of problems are created according to Figure 6.
The required data for each type of problem are consist of production schedule of each month, coefficients of consumption of items per product, inventory at the beginning of the period for each item, the price of each unit, maximum and minimum inventory limits of each item in the warehouse and percentage of suppliers portion. We have used uniform distribution to generate the required data.
4.8. Parameter Setting
Setting parameters is an important issue that effects on the efficiency of proposed algorithm performance that means the values of the parameters to be adjusted can change the performance of the metaheuristic algorithms (Arjmand and Najafi, 2015). In order to better compare GA, PSO and GAPSO hybrid algorithm, different values for each parameter are evaluated and the best values for simulation are used. In this paper, the population size of chromosomes was set to 20. The number of iteration in GA and PSO was set to 5,000. In the GAPSO algorithm, the number of iteration was set to 500, and in each iteration, 5 GA iteration and 5 PSO iteration are performed. Parameter setting for GA, PSO, and GAPSO algorithms have been summarized in Table 3.
To set parameters of algorithms, five test problems are considered as program dataset. Problems are divided into several categories based on the number of products and the number of items, One problem with large size (J = 10, K = 500), three problems with medium size (J = 2, K = 500), (J = 10, K = 100) and (J = 5, K = 250), A small size problem (J = 2, K = 100). These parameters are shown in Table 4.
4.9. Evaluation Methods
Evaluating the quality of the solutions is an important aspect of the work, and various criteria have been defined to calculate the performance of the results from used algorithms, such as the Mean Ideal Distance (MID), Metric of Diversification (MD), Spacing Metric (SM) Number of found solutions (NOS) and CPU time, etc. (Nikabadi and Naderi, 2016). In evaluating solutions, we seek two main goals consist of convergence and appropriate diversity. From the above criteria, MID and CPU time show the convergence rate of algorithms, and the rest of the criteria represent the variety of the algorithms. Variety of algorithms is used to evaluate the spread of solutions, and MID measures the convergence rate of solutions to a specific point such as (0). The spacing metric evaluates standard deviation of distances between the solutions of each algorithm (Zitzler and Thiele, 1998). In this paper, in addition to general metrics consist of average worst solutions, the average mean of solutions, average best solutions and the average standard deviations of solutions, we use the following four metrics to evaluate the quality of the proposed algorithms:

1. Mean Ideal Distance (MID): The closeness between solutions and the ideal point (0) and it is clear that the lower value of this index is better.(13)

2. Spacing Metric (SM) that measures the uniformity of distributed solutions (the spread of solutions) and the less value of SM value is more valid and reflects the uniform distribution of solutions.(14)
$$SM=\frac{{{\displaystyle \sum}}_{i=1}^{n1}\left\overline{d}{d}_{i}\right}{\left(n1\right)\overline{d}}$$(14)where d is the difference between two solutions and d is the mean of all d_{i}.

3. The spread of nondominance solutions (SNS) which represents the diversity in obtained solutions. The higher value of SNS means the better quality solution.(15)

4. The last metric is CPU time of the solutions which indicates the duration of solving algorithms, and the less time is better.
4.10. Simulation Results
Comparison of the results is based on the proposed GAPSO algorithm with GA and PSO algorithms. Solving algorithms were performed according to the parameters in Table 3 for all possible states of Type 1 to 5 problems, as shown in Table 4. Then, based on the obtained results, the metrics mentioned in the previous section are calculated for each set of data and for each solving algorithm. Table 5 shows the results of the calculations for the set of data. These calculations can be applied to a wider range of data and more diverse types of problems in future studies.
According to the last row of Table 5, the average of each metric is calculated in five possible problems. Based on these values, the proposed algorithm has the best performance in the average criteria of the answers, the best answer, and the average ideal distance (MID). For the criteria for the worstcase response criterion (SM), the GA algorithm’s performance is better. The PSO algorithm with the lowest value in the standard deviation of the answers and processing time, and the highest value for the Dominant Solutions Solution (SNS), has a better performance than the other two algorithms. The average values with better performance in the last row of Table 3 are highlighted.
The results of Table 5 in Figure 7 are shown based on each metrics. According to this figure, the proposed GAPSO algorithm has the highest value for the best solutions and has the lowest mean ideal distance (MID) than other algorithms. The GA algorithm has the smallest SM value. However, the least average CPU time is related to the PSO algorithm.
Regarding metrics of mean solutions, the worst solution, the standard deviation of solution and SNS, none of the algorithms have the best results for all types of problems, but the GAPSO algorithm has the highest mean solution in type 1, 2, 3, and 5 problems. In the worst solution metric, the GA algorithm works better on types 2, 4, and 5 than in other algorithms. The PSO algorithm has the lowest standard deviation for the type 1, 2, 4, and 5 problems, and has the highest SNS value for Type 3, 4, and 5 problems.
dSince none of the algorithms show the best performance in all the criteria, the best algorithm cannot be identified. Therefore, we use the TOPSIS approach to prioritize the solutions presented by the GA, PSO algorithms, and the proposed GAPSO algorithm. The basic idea in TOPSIS is to find the best solution so that the closest situation has the best positive solution (${d}_{l}^{+}$) and the farthest state of the worst negative solution (${d}_{l}^{}$). For each solution, the ratio is calculated below and the solution with a closer relationship to 1 is a better solution.
In this paper, this method is used to analyze the results on different problems. The results are presented in Table 6. We apply the TOPSIS method in terms of all kinds of problems in the last row of this table. According to this table, the proposed algorithm has the best performance when problems with the number of products and the number of items are appropriate, especially in the medium to medium position. Only in a situation where the number of products is too high and the number of items is small, the GA algorithm works better. In sum, the proposed solution algorithm performs better than the other two algorithms.
5. CONCLUSION
Several articles have been written in the field of designing VMI models, most of which have been considered to minimize supply chain costs as a system goal. The contribution of this paper is to maximize the inventory turnover of the producer as the objective function in VMI model. Accordingly, a general model of VMI was designed in terms of stock space constraints and lack of shortage of parts in production lines. To solve the designed model, a genetic algorithmbased hybrid algorithm (GA) and a local search algorithm, such as particle swarm algorithm (PSO), were proposed to find the optimal limits of inventory stock replenishment. In a global search through GA, three operators of selection, crossover, and mutation were used. Selection of the best chromosomes for generating the next generation was based on a fitness function, and the crossover operator was a uniform type and the mutation operator was a swap type. The best global solution from GA was used as the initial solution for the PSO. In a local search through the PSO, each particle was considered as a potential solution in the solution space, having position and velocity that updated in each iteration of PSO, and the best solutions were selected. To investigate the performance of the proposed hybrid algorithm, five types of small, medium and large test problem were generated based on the number of products and the number of items in the supply chain. Four general criteria in addition to four specific criteria were used to analyze and compare the performance of GA, PSO, and GAPSO hybrid algorithms. According to the results, the proposed algorithm shows the best solutions and the minimum mean ideal distance (MID). The GA algorithm has the lowest value in the spacing metric (SM). However, the least average CPU time is related to the PSO algorithm. To determine the most efficient algorithm, the TOPSIS method was used and the results indicate that the proposed algorithm is the most efficient algorithm than other algorithms.
Future research could include VMI models in multipleobjective situations and be adding more constraints to the model from the supplier area. Also, the results of the hybrid algorithm can be compared with other evolutionary algorithms.