Definition 2: The expected value of a fuzzy variable is defined as:
Definition 3: The optimistic function of α is defined as:
PROBLEM DEFINITION Consider a periodic inventory control model for one provider in which the times required to order each of several available products are stochastic in nature. Let the timeperiods between two productreplenishments be identical and independent random variables; the purchasing price of the products to be triangular fuzzy variables, the demands are crisp and in case of shortage, a fraction are considered backorder and a fraction as lostsale. The costs associated with the inventory control system are holding (a percentage of the purchasing cost), backorder, lostsales and purchasing costs. Furthermore, the incremental discount policy is used, the service level of each product, warehouse space and budget are considered constraints of the problem and the decision variables are integer digits. We need to identify the inventory levels in each cycle such that the expected profit is maximized. PROBLEM MODELING For the problem at hand, since the timeperiods between two replenishments are independent random variables, in order to maximize the expected profit of the planning horizon we need to consider only one period. Furthermore, since we assumed that the costs associated with the inventory control system are holding and shortage (backorder and lostsale), we need to calculate the expected inventory level and the expected required storage space in each period. Before doing this, let us define the parameters and the variables of the model.
The parameters and the variables of the model: For i = 1,2,…,n, let
us define the parameters and the variables of the model as:
R_{i}: 
The inventory level of the i^{th} product 
T_{i}: 
A random variable denoting the timeperiod between two replenishments
(cycle length) of the i^{th} product 
F_{Ti}(t_{i}): 
The Probability density function of T_{i} 
q_{ij}: 
The j^{th} discount point for the i^{th} product 
W_{i}: 
The crisp purchasing cost per unit of the i^{th} product
without discount 
W_{ij}: 
The crisp purchasing cost per unit of the i^{th} product at the
j^{th} discount point 

The fuzzy purchasing cost per unit of the i^{th} product at the
j^{th} discount point 
: 
The weighted expected purchasing cost of the i^{th} product 
FI_{i}: 
A fraction of the purchasing cost of the i^{th} product
used to calculate its holding cost 
h_{i}: 
The holding cost per unit inventory of the i^{th} product in each
period 
: 
The crisp holding cost per unit inventory of the i^{th} product
in each period 
: 
The fuzzy holding cost per unit inventory of the i^{th} product
in each period 
Q_{ij}: 
The order quantity of the i^{th} product at the j^{th}
discount price 
p_{i}: 
The backorder cost per unit demand of the i^{th} product 
: 
The shortage cost for each unit of lost sale of the i^{th} product 
P_{i}: 
The sale price per unit of the i^{th} product 
D_{i}: 
The constant demand rate of the i^{th} product 
SL_{i}: 
The lower limit of the service level for the i^{th} product 
t_{Di}: 
The time at which the inventory level of the i^{th} product reaches
zero 
b_{i}: 
The percentage of unsatisfied demands of the i^{th} product that
is backordered 
I_{i}: 
The expected amount of the i^{th} product inventory
per cycle 
L_{i}: 
The expected amount of the i^{th} product lostsale in each cycle. 
B_{i}: 
The expected amount of the i^{th} product backorder in each cycle 
Q_{i}: 
The expected amount of the i^{th} product order in each cycle 
f_{i}: 
The required warehouse space per unit of the i^{th}
product 
F: 
Total available warehouse space 
TB: 
Total available budget 
C_{hi}: 
The expected holding cost per cycle of the i^{th} product. 
C_{bi}: 
The expected shortage cost in backorder state of the i^{th}
product 
: 
The expected shortage cost in lostsale state of the i^{th} product 
C_{pi}: 
The expected purchase cost of the i^{th} product 
r_{i}: 
The expected revenue obtained from sales 

The expected profit obtained in each cycle 
For sake of simplicity, we first consider a singleproduct problem in which the purchasing prices and holding costs are crisp and there is no discount. Then, we are devoted for a singleproduct problem with incremental and total discount policies, respectively. We discuss the cases in which the demands are fuzzy random variables. Finally, we extend the singleproduct to the multiproduct modeling. However, let us introduce the pictorial representation of the singleproduct problem. Inventory diagram: According to Ertogal and Rahim^{[20] }and considering the fact that the timeperiods between replenishments are stochastic variables, two cases may occur. In the first case the timeperiod between replenishments is less than the amount of time required for the inventory level to reach zero (Fig. 1) and in the second case, it is greater (Fig. 2). Single product modelback order and lost sales cases: In this section, we first model the costs, the profit and the constraint of a singleproduct inventory problem with crisp demand where there is no discount on purchasing products. The replenishments are stochastic and backorder and lostsales are allowed.
Calculating the costs and the profit: In order to calculate the expected
profit in each cycle, we need to evaluate all of the terms in Eq.
6^{[20]}:
 Fig. 1: 
Presenting the inventory cycle when 
 Fig. 2: 
Presenting the inventory cycle when 
 Fig. 3: 
Presenting shortages in two cases of back order and lost sales 
Based on Fig. 3, L_{i}, B_{i}, I_{i}
and Q_{i} are evaluated by the following equations:
Presenting the constraints: As the total available warehouse space is
F, the space required for each unit of the i^{th} product is f_{i}
and the inventory level of the i^{th} product is R_{i}, the
space constraint will be:
Since the total available budget is TB, the cost for each unit of product is
W and the order quantity is Q, the budget constraint is:
Knowing that the shortages only occur when the cycle time is more than t_{Di}
and that the lower limit for the service level is SL_{i}, then:
In short, the complete mathematical model of the single product inventory problem
with crisp demand and no discount is:
Single product modelback ordered and lost sales cases with discount: In this
section, we assume that an incremental discount policy is applicable to purchase
the product. In incremental discount policy, the purchasing cost for each unit
of the i^{th} product depends on its order quantity and is assumed to
be:
The purchasing cost associated with this policy is calculated as follows:
where, for j = 1,2,…,T, q_{ij} and W_{ij} are the discount points and the purchasing costs for each unit of the i^{th} product that corresponds to the j^{th} discount break point, respectively.
In order to include the discount policy in the inventory model, using Eq.
16, the purchasing cost will be modelled as:
By this modeling, the inventory model of the single product problem with incremental
discount policy becomes:
Singleproduct model with discount, fuzzy purchasing and holding costs: The
singleproduct inventory model with crisp purchasing and holding cost and incremental
discount of (18) can be easily extended to single product models with fuzzy
purchasing and holding cost as follows:
In the next section, we extend the models in (19) to multiproduct models.
Multiproduct models: The singleproduct inventory models of (19) can
be easily extended to a multiple product. In these models, we consider two probability
density functions for T_{i} as follow:
T_{i} follows a uniform distribution: In this case the probability
density function of T_{i} is .
Accordingly, (19) will change to (20) as:
T_{I} follows an exponential distribution: If T_{i} follows
an exponential distribution with parameter ,
then the probability density function of T_{i} will be .
In this case, the model is shown in (21) as:
In the next section, we will introduce a hybrid intelligent algorithm to solve the model. A HYBRID INTELLIGENT ALGORITHM Since the models in (20) and (21) are fuzzy mixed integernonlinear in nature, reaching an analytical solution (if any) to the problem is difficult^{[21]}. In order to solve the model under different criteria, we develop a hybrid intelligent algorithm of fuzzy simulation and genetic algorithm.
Fuzzy simulation: In order to estimate the uncertain purchasing price
and holding cost of the fuzzy model, since the holding cost is a function of
its corresponding purchasing cost, an estimate of the former cost will provide
an estimate of the latter cost. As a result, in the simulation technique used
for the estimation, denoting
by ,
μ as the membership function of
and μ_{ij} are the membership functions of ,
we randomly generate
from the αlevel sets of fuzzy variables ,
i = 1,2,…,n, j = 1,2,…,T and k = 1,2,…,K as
and ,
where α is a sufficiently small positive number.
Based on the definition in Eq. 11, the expected value of
the fuzzy variable is:
Then, provided O is sufficiently large, for any number
can be estimated by:
and for any number can
be estimated by:
However, the procedure of estimating
in (23) and (24) is shown in algorithm (1).
• 
Randomly generate
from αlevel sets of fuzzy variables
and set Set
a = ,
b = 
• 
Randomly generate r from Uniform [a,b] 
• 
If ,
otherwise, 
• 
Repeat 4 and 5 for O times 
• 
Calculate 
Algorithm (1): Estimating .
Genetic algorithm: The main information unit of any living organism is the gene, which is a part of a chromosome that determines specific characteristics such as eyecolor, complexion, haircolor, etc. The fundamental principal of Genetic Algorithms (GA) first was introduced by Holland^{[22]}. Since then many researchers have applied and expanded this concept in different fields of study. Genetic algorithm was inspired by the concept of survival of the fittest. In genetic algorithms, the optimal solution is the winner of the genetic game and any potential solution is assumed to be a creature that is determined by different parameters. These parameters are considered as genes of chromosomes that could be assumed to be binary strings. In this algorithm, the better chromosome is the one that is nearer to the optimal solution. In applied applications of genetic algorithms, populations of chromosomes are created randomly. The number of these populations is different in each problem. Some hints about choosing the proper number of population exist in different reports by Man et al.^{[23]}. Genetic algorithms imitate the evolutionary process of species that reproduce. They therefore do not operate on a single current solution, but on a set of current solutions called population. New candidates for the solution are generated with a mechanism called crossover that combines part of the genetic patrimony of each parent and then applies a random mutation. If the new individual, called child or offspring, inherits good characteristics from his parents the probability of its survival increases. This process will continue until a stopping criterion is satisfied. Then, the best offspring is chosen as a near optimum solution.
In this research, the chromosomes are strings of the inventory levels of the
products (R_{i}). Each population or generation of chromosomes has the
same size which is wellknown as the population size and is denoted by N. If
N is relatively small, then a small search space will be investigated and the
GA algorithm will be very slow. In this research, 10, 100 and 500 are chosen
as different population sizes. In a crossover operation, it is necessary to
mate pairs of chromosomes to create offspring. There are three types of crossover
operations: singlepoint, multipoint and uniform^{[21]}. In this research,
we employ the singlepoint crossover that is applied to parent chromosomes with
the possibility of P_{c} = 0.8, 0.85 and 0.9. Mutation is the second
operation in a GA method for exploring new solutions and it operates on each
of the chromosomes resulted from the crossover operation. In mutation, we replace
a gene with a randomly selected number within the boundaries of the parameter^{[21]}.
We create a random number RN between (0,1) for each gene. If RN is less than
a predetermined mutation probability P_{m}, then the mutation occur
in the gene. Otherwise, the mutation operation is not performed in that gene.
More precisely, assume that for a specific gene such as a_{j} in a chromosome
R_{j} the generated random number is less than P_{m} and hence
the gene is selected for mutation. Then, we change the value of a_{j}
to the new value according
to Eq. 25 and 26, randomly and with the
same probability:
where, l_{j} and u_{j }are the lower and upper limits of the
specified gene, r is a uniform random variable between 0 and 1, i is the number
of current generation and max gen is the maximum number of generations. Note
that the value of a_{j} is transferred to its right or left randomly
by Eq. 25 and 26 respectively and r is
this percentage. Furthermore,
is an index with a value close to one in the first generation and close to zero
in the last generation that makes large mutations in the early generations and
almost no mutation in the last generations. In this research, 0.076, 0.098 and
0.1 are employed as different values of the P_{m }parameter. Furthermore,
Algorithm (1) of section 5.1 is used to evaluate the objective function of this
research.
The last step in a GA method is to check if the algorithm has found a solution that is good enough to meet the user’s expectations. Stopping criteria is a set of conditions such that when satisfied a good solution is obtained. Different criteria used in literature are as follows: (1) Stopping of the algorithm after a specific number of generations, (2) no improvement in the objective function and (3) Reaching a specific value of the objective function. In this research, we stop when a predetermined number of consecutive generations is reached. The number of sequential generations depends on the specified problem and the expectations of the user. In short, the steps involved in the hybrid method of fuzzy simulation and GA algorithm used in this research are: • 
Setting the parameters P_{c}, P_{m} and N 
• 
Initializing the population randomly 
• 
Evaluating the objective function for all chromosomes based
on Algorithm (1) 
• 
Selecting individual for mating pool 
• 
Applying the crossover operation for each pair of chromosomes
with probability P_{c} 
• 
Applying mutation operation for each chromosome with probability
P_{m} 
• 
Replacing the current population by the resulting mating pool 
• 
Evaluating the objective function 
• 
If stopping criteria is met, then stop. Otherwise, go to step
5 
In order to demonstrate the proposed Hybrid intelligent algorithm and evaluate its performance, in the next section we bring a numerical example used in Ertogal and Rahim^{[20]}. In this example, two cases of the uniform and the exponential distributions for the timeperiod between two replenishments are investigated. NUMERICAL EXAMPLES
Consider a multiproduct inventory control problem with eight products and
general data given in Table 1. Table 2 shows
the parameters of both the exponential and uniform distributions used for the
timeperiod between two replenishments. The total available warehouse space
and total budget are F = 22000 and TB = 550000, respectively. Table
3 shows the best combination and different values of the GA parameters used
to obtain the solution. In this research, all the possible combinations of the
GA parameters (P_{c}, P_{m} and N) are employed and using the
max(max) criterion the best combination of the parameters has been selected.
Table 4 shows the best result for the uniform and exponential
distributions. Furthermore, the convergence paths of the best result of the
objective function values in different generations of the uniform and the exponential
distributions are shown in Fig. 4 and 5,
respectively.
Table 1: 
General data 

Table 2: 
Parameters of exponential and uniform distributions 

Table 3: 
The parameters and the best combination of the GA method 

Table 4: 
The best result forR_{i} 

 Fig. 4: 
The convergence path of the best result in uniform example 
 Fig. 5: 
The convergence path of the best result in exponential example 
CONCLUSION AND RECOMMENDATIONS FOR FUTURE RESEARCH In this research, a stochastic replenishment multiproduct inventory model with discount and fuzzy purchasing price and holding cost was investigated. Two mathematical modeling for two cases of uniform and exponential distribution of the time between two replenishments in case of incremental discount have been developed and shown to be fuzzy mixed integernonlinear programming problems. Then, a hybrid intelligent algorithm (fuzzy simulation+GA) has been proposed to solve the fuzzy integer nonlinear problems. Some recommendations for future works are (1) considering demands as fuzzy or random variables, (2) employing a total discount policy and (3) applying some other metaheuristic algorithms. " target="_blank">View Fulltext
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