1.INTRODUCTION
Nowadays, combination of queuing theory and inventory control models has been widely investigated by many researchers. In this system, customers arrive to the system to take your service. In order to compliance the customer requirement, a product of inventory is required. After complete the customer service, they leave the system and onhand inventory is reduced. Inventory is provided by an external supplier. A system with mentioned characteristic is wellknown as queuinginventory system (Schwarz et al., 2006). In these models, satisfying each demand needs onhand inventory and involves a process or service that takes time. In these systems, an important issue is the reaction of inventory management to queuing of demands. It is also different from the traditional inventory management because the inventory is consumed at the serving rate rather than the customers’ arrival rate when there are customers queued up for service (Teimoury et al., 2010; Chang and Lu, 2010).
Berman and kim (1999) considered a queuinginventory system with Poisson arrival rate and exponential service times under zero lead time. Berman and Sapna (2000) considered queuinginventory systems with Poisson arrival and a conventional service rate and zero lead time. Berman and Sapna (2001) studied a capacitated queuing system under exponential arrival rate, service rate and lead time. Berman and Kim (2001) investigated a supply chain based on internet under Poisson arrivals, exponential service times and the Erlang lead times. The authors founded that the optimal ordering policy has a monotonic threshold structure. Berman and Kim (2004) studied a queuinginventory system with Poisson arrival rate and exponential service rate and lead time. They obtained a replacement policy with maximizing profit. Dong and Chen (2005) developed a queuinginventory system for analyzing performance of an integrated logistic. They considered ordering policy and lot size to use GI^{x} / G / 1 system under inventory control policy of (s, S). Isotupa (2006) analyzed an (s, Q) Markovian inventory system with lost sales and two demand classes. She developed an efficient algorithm for minimizing replenishing, lost sales and inventory costs. Schwarz et al. (2006) considered M/M/1 queuinginventory system with lost sales under various inventory management policies such as (r, Q) policy and (r, S) policy and derived stationary distributions of joint queue length and inventory processes in explicit product form. Schwarz and Daduna (2006) studied an M/M/1 queuinginventory system with considering backordering. Krishnamoorthy et al. (2006) considered an (s, S) inventory system that server continues processing the products even in the absence of customers. Krishnamoorthy et al. (2006) introduced a new policy called Npolicy in (s, S) inventory system. Hill (2007) considered continuousreview inventory models with no fixed order cost, lostsales, and a Poisson demand process to minimize the long run total cost and explore alternative approaches which might offer better solutions. Manuel et al. (2007) and Manuel et al. (2008) investigated a perishable queuinginventory system with Markovian arrival process. Haji et al. (2011) presented an integrated queuinginventory system in a Twolevel Supply Chain. Zhao and Lian (2011) considered a queuinginventory system with two classes of customers with Poisson arrival and exponential service rate. Teimoury et al. (2012) studied decision making on Order Penetration Point strategy with a queuing concept. Order Penetration Point is boundary between Maketoorder and Maketostock. Saffari et al. (2013) considered an M/M/1 queuing system under (r, Q) inventory policy and in presence of lost sales. They derived the stationary distributions of the joint queue length (number of customers in the system) and onhand inventory. Krishnamoorthy et al. (2015) considered a queuinginventory system, with the item given with probability γ to a customer at his service completion epoch. They discussed two inventory policy of (s, S) and (s, Q) and obtained the joint distribution of the number of customers and the number of items in the inventory as the product of their marginal under the assumption that customers do not join when inventory level is zero. Alaghebandha and Hajipour (2015) proposed a multiproduct continuous review inventory control problem within batch arrival queuing approach (M^{Qr} /M/1) to find the optimal quantities of maximum inventory. They proposed an efficient imperialist competitive algorithm (ICA) to solve the model. Baek and Moon (2014) investigated a queuing system integrated with a productioninventory system in which the stocks are delivered both by an outside supplier and an internal production. Güray Güler et al. (2014) modeled the inventory problem under M/M/1 queuing system by considering several classes of customers. Sajadi et al. (2015) proposed a mixed integer nonlinear programming model to address the locationinventory problem under queuing approach. They considered uncertain demand and leadtime, which follow Poisson and Exponential distributions, respectively. Baek and Moon (2016) extended a productioninventory model with an M/M/c service queue and lost sales.
Most queuinginventory models analyzed in certain situation while information in reallife cases are often uncertain and ambiguous. Therefore, in the mentioned literature, there was less attention to analyze the price in supply chain. Whereas often revenue management as maximizing the profit is one of the main aims in supply chain management. As price affects on demand quantity, and demand rate affects on queuing parameters, therefore analyzing the price variable can be an important task. In this paper, a queuinginventory model is presented in uncertainty environment using fuzzy set theory introduced by Zadeh (1965) to formulate the problem in presence of increasing accuracy and validity of the problem. Moreover, arrival rate behaviors as Poisson and depends on price are considered, simultaneously. The goal is to maximize the total profit to determine optimal price as well as the order quantities. Since the proposed model is Nondeterministic Polynomialtime hard; therefore, nondominated ranking genetic algorithm is presented and compared with the literature to solve and justified the proposed model.
Rest of the paper is organized as follow: Next section defines the problem, notations, decision variables, and proposed mathematical formulation of the problem. Section 3 described Paretobased approaches based on GA to solve the model. Section 4 analyzed the results and comparisons. Finally, conclusions and some future researches are given.
2.PROBLEM FORMULATION
2.1.Problem Description
In this paper a system is considered that contain an external supplier, warehouse, retailer and customers. Retailer demands act as Poisson probability distribution. Retailer orders customer demands to the warehouse in a stochastic time interval. Demand of retailers causes to be provided some orders constantly with predetermined time horizons into the warehouse. Warehouse provides his order from external supplier. It is also assumes that transportation time from warehouse to retailer and also lead time delivery of external supplier are constant. To make the model more realistic and increase accuracy in decision making, in this research, we have proposed a fuzzy mathematical model under uncertainty environment. Modeling and optimizing a model under uncertain environment with considering costs parameters is one of the contributions of this paper. It is valuable to mention that restrictions of warehouse space, number for the shortage, service level, and expected shortage are considered in mathematical formulation process of the problem.
2.2.The Assumptions

The retailer faces with Poisson demand;

The warehouse faces with stochastic demand;

Unsatisfied demand by the retailer is as lost sales;

Shortage is not allowed at the warehouse;

Shortage is allowed at the retailer;

There is no lotsplitting at the warehouse;

The transportation time for an order to arrive at the retailer from the warehouse is as exponential distribution;

The warehouse orders to an external supplier with infinite capacity;

The time between two consecutive orders is as exponential distribution;

The lead time for an order to arrive at the warehouse is constant;

Cost coefficients in objective function are as fuzzy number;
2.3.The Mathematical Model
Consider a supply chain consisting of external supplier, warehouse, retailer, and customers. In this system, retailer is dealt with customer’s demand as Poisson process. Arrival rate is λ_{j} = ξ_{j} α_{j}p_{j} which is dependent on price. Retailer orders Q_{r} to warehouse in interval T. Moreover orders of warehouse are provided by external supplier. Transportation time from warehouse to retailer is as Exponential distribution and it is assumed that lead time delivery from external supplier is fixed. Since arrival of products to retailer is stockpile, therefore system is faced with batch arrivals with random amount of Q_{r} . Time between batch arrivals is Exponential distribution with parameter ϕ_{j}(ϕ_{j} > 0). Whereas only one retailer is in considered system, therefore system has one server. Service rate of the server for product j is as an exponential random variable with parameter of μ_{j}(μ_{j} > 0). Arrival time and service time are independent. As an order is delivered a time is required to service it, if in this moment service time of previous order has not been completed or time of service has been longer than next order; therefore, we have congestions in the system. Since products arrive to the retailer as batch arrivals; thus, M^{Q} / M / 1 queuing system is considered. With using mentioned issue and considering continuous review inventory control as inventory control policy, average number of customers of queuing system in long time ( ℓ ) is used for calculating average inventory (I). In order for better understanding, the process of modelling is explained in two following subsections including the investigation of the model in crisp environment and fuzzy one, respectively.
2.3.1.Crisp Model
In this paper, warehouse applies continuous review inventory control policy and retailer uses periodic review inventory control policy. In this system, retailer orders to warehouse are in random interval time. It means that demand of warehouse is established by orders of retailer and is also a random variable. Random demand for warehouse causes safety stock is increased and also inventory control system is established. The crisp model is formulated as follow:(1)
2.3.2.The Proposed Fuzzy Model
In this paper, a fuzzy mathematical model is presented for modeling uncertainty costs parameters in the proposed system. Therefore, shortage cost, holding cost, and ordering cost are considered as fuzzy number of $\tilde{A},\text{\hspace{0.17em}}\tilde{h},\text{\hspace{0.17em}}\tilde{\pi}$. To do so, Lia and Hwang (1992) method is applied to formulate fuzzy mathematical model. Consider following single objective model with fuzzy coefficients.
$\begin{array}{l}Maximize\text{\hspace{0.17em}}\tilde{C}X\\ S.t.\\ x\in X=\left\{x\rightAx\le b\text{\hspace{0.17em}and\hspace{0.17em}}x\ge 0\}\end{array}$
where $\tilde{C}=\left({C}_{L},\text{\hspace{0.17em}}{C}_{M},\text{\hspace{0.17em}}{C}_{R}\right)$ are triangular fuzzy numbers.
For this category of problem a multiobjective approach is presented. In order to maximize fuzzy objective function, Figure 1 indicates how single fuzzy objective is converted to three crisp objectives.
In order to maximize fuzzy objective function following steps are required:

1) Maximize C_{M}

2) Minimize interval between C_{M} and C_{L}

3) Maximize interval between C_{M} and C_{R}
Therefore, multiobjective mathematical model is(3)
$\begin{array}{l}Minimize\text{\hspace{0.17em}}{\text{Z}}_{1}=\left({C}_{M}{C}_{L}\right)X\\ Maximize\text{\hspace{0.17em}}{\text{Z}}_{2}={C}_{M}X\\ Maximize\text{\hspace{0.17em}}{\text{Z}}_{3}=\left({C}_{r}{C}_{M}\right)X\\ S.t.\\ x\in X=\left\{xAx\le b\text{\hspace{0.17em}and\hspace{0.17em}}x\ge 0\right\}\end{array}$
Note that with considering $\tilde{A}=\left({A}^{L},\text{\hspace{0.17em}}{A}^{M},\text{\hspace{0.17em}}{A}^{R}\right),\text{\hspace{0.17em}\hspace{0.17em}}\tilde{h}=\left({h}^{L},\text{\hspace{0.17em}}{h}^{M},\text{\hspace{0.17em}}{h}^{R}\right),\text{\hspace{0.17em}}\tilde{\text{\pi}}=\left({\text{\pi}}^{L},\text{\hspace{0.17em}}{\text{\pi}}^{M},\text{\hspace{0.17em}}{\text{\pi}}^{R}\right)$ in proposed model, we have following multiobjective model:(2), (3), (4), (5), (6), (7)
Objective functions (24) defuzzified objectives of propose model. Constraints (57) are upper limits of expected shortage of each product, lower limit of service level, and upper limit of total shortage costs, respectively. Constraints (8) indicate stationary of queuing system. Constraints (9) ensure demand of each product is satisfied. Constraints (10) determine ranges of variables and constraints (11) indicates arrival rate of customers which is dependent on price.
3.PARETOBASED APPROACHES
Most exact methodologies such as Lagrangian relaxation (Fisher, 2004) and branch and bound (As’ad and Demirli, 2011) have been developed in the literature to solve less complicated inventory control models but they cannot be applied to find exact optimal solutions of the model. In this paper, a nondominated sorting genetic algorithm (NSGAII) and a nondominated ranked genetic algorithm (NRGA) are presented to find Pareto solutions.
3.1.The Proposed NSGAII
Nondominated sorting genetic algorithm (NSGA), introduced by Deb et al. (2001), is one of the bestdeveloped Paretobased metaheuristic algorithms. Using Pareto dominance solutions, it is a computationally efficient algorithm implementing the idea of a selection method based on classes of dominance of all the solutions. The NSGAII includes five main steps: initialization, fast nondominated sorting (FNDS), crossover, mutation, and the elitist crowdedcomparison operator. The main steps are illustrated in details as follow:
3.1.1.Initialization
In this research, a chromosome represents different number of products ordered in different periods as presented by Alaghebandha and Hajipour (2015). A vector is added to the chromosome structure to generate the prices. Therefore, upper bound on the price in each product is considered, in the initialization step of the proposed NSGAII algorithm, a uniform integer random number is generated for each chromosome.
3.1.2.Fast NonDominated Dorting
The computational complexity of NSGAII is O(MR^{2} ) where M and R are the number of objectives and the population size, respectively (Deb, 2001). In this step, the R populations that were generated in the previous step are compared and are sorted. To do this, all chromosomes in the first nondominated front are first found. By concept of domination, in which a solution x_{i} is said to dominate solution , x_{j} if ∀_{o}∈{1, 2} we have Z_{o}(xi) ≤ Z_{o} j ( x ) and ∃ o∈{1, 2} such that Z_{o}(xi) < Z_{o}(xj). In this case, we say x_{i} is the nondominated solution within the solution set {x_{i}, x_{j}}. Otherwise, it is not. Then, in order to find the chromosomes in the next nondominated front, the solutions of the previous fronts are disregarded temporarily. This procedure is repeated until all solutions are set into fronts.
3.1.3.Crowding Distance
Crowding distance operator evaluates solution fronts of populations in terms of the relative density of individual solutions. To do this, consider Z and f_{k}; k = 1, 2, …, M the number of nondominated solutions in a particular front (F) and the objective functions, respectively. Besides, let d_{i} and d_{j} be the value of crowding distance on the solution i and j. The crowding distance is obtained using the following steps:

(1) Set d_{i} = 0 for i = 1, 2, …, Z

(2) Sort all objective functions k ; =1, 2,…, f k M in ascending order

(3) The crowding distance for end solutions in each front (d_{1} and d_{Z} ) are d_{1} = d_{Z} = ∞

(4) The crowding distance for d_{j}; j = 2, 3, …, (Z1) are ${d}_{j}={d}_{j}+\left({f}_{{k}_{j+1}}{f}_{{k}_{j1}}\right)$
3.1.4.Selection Strategy
To choose individuals of the next generation, tournament selection operator “ ⊱ ” is applied (Deb, 2001). To do this, n individuals in the population is randomly selected. Then, nondominated rank of each individual is obtained. Following this, CD of the solutions with equal nondominated rank is computed. At end, the solutions with least rank are the selected ones. Moreover, if more than one individual share the least rank, the individual with highest crowding distance must be selected.
3.1.5.Crossover and Mutation Operators
In presented NSGAII, we implemented continues uniform crossover and swap strategy for crossover and mutation operators, respectively (Hajipour et al., 2014, MontazerGh and Mahmoodik, 2016). It should be mentioned that, after the crossover and mutation operations, the fitness function values are calculated for offspring.
3.1.7.Merging the Population
In this step, the parents and offspring population are merged to ensure the elitism. Since the combined population size is naturally greater than the original population size N, FNDS is performed. In fact, chromosomes with higher ranks are selected and added to the populations until the population size becomes N. The last front is also consisted of the population based on the crowding distance. The algorithm stops when a predetermined number of iterations are reached.
3.2.The Proposed NRGA
NRGA is a new multiobjective genetic algorithm to find feasible Pareto front solutions. NRGA is similar to NSGAII with the difference that in the selection operation the roulette wheel strategy is employed (Jadaan et al., 2008). In NRGA, a fitness value representing rank is assigned to each individual of the population. In this regard, two ranked based roulette wheel selection features including:
The selection probability of fronts, P_{f} , and the selection probability of solutions, P_{fs} , are obtained using Eq. (12) and (13).
Where NF and NS_{f} are the number of fronts and the number of solutions in front f, respectively. Eq. (12) ensures that a front with highest rank has the highest probability to be selected. Similarly, based on Eq. (13), solutions with more crowding distance are assigned higher selection probability. The roulette wheel selection is iterated until a desired number of solutions are selected. At the end, the algorithm stops when a predetermined number of iterations are reached. The process of NSGAII and NRGA are represented in Figure 2.
4.RESULT ANALYSIS
To evaluate the performances of the proposed Paretobased algorithms, five standard metrics of multiobjective algorithms are applied as follows (Zitzler and Thiele, 1998):

Diversity: measures the extension of the Pareto front in which bigger value is better.

Spacing: measures the standard deviation of the distances among solutions of the Pareto front in which smaller value is better.

Number of found solutions (NOS): counts the number of the Pareto solutions in Pareto optimal front in which bigger value is better.

Mean ideal distance (MID): measures the convergence rate of Pareto fronts to a certain point (0, 0).

The computational (CPU) time of running the algorithms to reach near optimum solutions
The experiments are implemented on 10 test problems. Furthermore, to eliminate uncertainties of the solutions obtained, each problem is used three times under different random environments. Then, the averages of these three runs are treated as the ultimate responses. The NSGAII as most applicable Paretobased MOEAs in the literature is applied to demonstrate capability of the proposed algorithms to solve the multiobjective optimization problems. It required to be mentioned that the input parameters of the algorithms are reported in Table 1.
In order to analyze the performance of the proposed Algorithms, Table 2 first generate some test problems and then Table 3 and Table 4 report the multiobjective metrics amounts on the 10 test problems. It is noticed that the MATLAB Software has been used to code the proposed metaheuristic algorithms, and the programs have been executed on a 2 GHz laptop with eight GB RAM.
The algorithms are statistically compared based on the properties of their obtained solutions via the analysis of variance (ANOVA) test. In order to clarify our statistical results, individual value plots are represented in Figure 3Figure 7.Figure 4Figure 5Figure 6
The results show that both algorithms work same and both are capable to be selected as solving methodologies for our problem. However, NRGA works better in largesize problems.
5.CONCLUSION
In this paper, a fuzzy queuinginventory system with continuous review inventory control policy and batch arrival queuing approach is presented. We assume that demand function is stochastic and price dependent. The goal was to maximize total profit of system. Since the problem was NPhard, NRGA and NSGAII are proposed and justified to solve the model. The results show that NRGA perform NSGAII in largesize problems. For future research, one can develop the model into multiechelon one.