INTRODUCTION
One of the critical factors for successful production or services organization
is marketing. Perhaps, we can say the most important area in marketing
is analyzing product’s market and establish agents for organization’s
products in the best locations. This identify organization’s products
to customers and increase customer’s satisfy with increase in quality
of product.
For helping organization’s management in order to recognize various
locations of organization’s agents, we can use multi criteria decision
methods. Operation research science is one of the largest sciences to
solve production, services agriculture, etc. problems.
In a problem relatively similar to that of this study, Hultz
et al. (1981) studied on multiactivitymultifacility problems and
proposed an interactive solution method to compute nondominated solutions to
compare and choose each others. Fortenberry and Mitra (1986),
an application of integer goal programming for facility location with multiple
competing objectives are addressed. Brandeau and Chiu (1989)
present a survey of representative problems that have addressed in location
problems (Ghosh and Rushton, 1987). Schilling
et al. (1993) applies an approximation scheme to generate a set of
nondominated solutions to a biobjective location problem.
There are several problems that are accepted as classical ones: the point objective
problem (Wendell and Hurter, 1973), the continuous multi
criteria minsum facility location problem (Hamacher and
Nickel, 1997). Hamacher et al. (1999) worked
on the network multicriteria median location problem. Ogryczak
(1999) applied symmetrically efficient location patterns in a multi criteria
discrete location problem.
Nowadays, multiobjective combinatorial optimization (MOCO), provides an appropriate
framework to tackle various types of discrete multicriteria problems (Klamroth
and Wiecek, 2000). Within this emergent research area several methods are
known to handle different problems such as dynamic programming enumeration (Villarreal
and Karwan, 1981). It is necessary to note that most of multi objective
combinatorial optimization (MOCO) problems are NPhard and intractable (Gandibleux
et al., 2000). In the case of incapacitated plant location problem,
the singleobjective version is already NPhard (Krarup and Pruzan, 1983).
In this study, using TOPSIS technique which is one of the most important
methods of multi attribute decision making set and entropy for locations
indexes weight calculation and also with using heuristic method that is
based on fuzzygoal programming method, we select best location for facility
location.
USING MULTI CRITERIA DECISION MAKING MODEL
In recent century, researchers focused on multi criteria decision making
for complex decisions. In these decisions, we may use one optimum criterion
in replace multi criteria. There are two kinds of these models:
Multi objective models are used for designing and multi attribute models are
used for selecting better alternative (Yoon and Hwang, 1981).
But more exact investigations shows that some techniques like TOPSIS and ELECRE
are the best for this case, because we have both desirable directions (min and
max) (Yoon and Hwang, 1981). However, due to better results
of implementation, we have used TOPSIS.
TOPSIS METHOD
TOPSIS assumes that each attribute in decision matrix takes either monotonically
increasing or monotonically decreasing utility in other phase, the best alternative
should have the shortest distance from ideal solution and farthest from the
negative ideal solution (Yoon and Hwang, 1981). The details
of each step in TOPSIS method are presented in the following section:
Step 1: Constructing the decision matrix and changing to normalized
matrix can be calculated as:
Step 2: Constructing the weighted normalized decision matrix (V).
The matrix can be calculated by multiplying each column of the matrix
R with associated weight W.
w = (w_{1},w_{2},...,w_{n}) 
(2) 
Step 3: Determining ideal and negative ideal solution. Let the
two artificial alternatives which be defined as:
where, j = 1,2,...,n is associated with benefit criteria and j’
= 1,2,...,n is associated with cost criteria. Then it is certain that
the two created alternatives, A^{+} and A^{–}, indicate
the most preferable alternative (ideal solution) and the least preferable
alternative (negative ideal solution), respectively.
Step 4: Calculation the separation measure, the separation between
each alternative and the ideal alternative can be measured by Euclidean
distance:
i = 1,2,...,m
where, di^{+} is the separation of each alternative from the
ideal one and d_{i}^{–} is the separation from the
negative ideal one.
Step 5: Calculating relative closeness to ideal solution:
It is clear, if cli^{+} is close to 1, Vj is close to Vj(max)
and cli^{+} = 1 if Vj = Vj (max ).
Step 6: Ranking the preference order based upon cli^{–}
(Yoon and Hwang, 1981). However, here do not use ranked
alternatives in our approach and the value of relative closeness is enough to
continue the procedure.
CALCULATING WEIGHTS FOR DECISION MATRIX INDEXES WITH ENTROPY TECHNIQUE
Entropy is criteria that have been expressed for unreliability with discrete
probability distribution (P_{i}), this unreliability is as follows
(E):
Which k is a positive constant number: 0≤E≤1.
In MADM model, a decision matrix has information that entropy method has been
used as criteria for evaluation (Yoon and Hwang, 1981).
The decision matrix is as follows (Table 1) (Yoon
and Hwang, 1981):
Firstly, we normalized decision matrix as follow:
Table 1: 
The decision matrix 

And for each Ej from set P_{ij}, we have:
Now, unreliability for each criterion is as follow:
Finally, weights of criterias will be:
If decision maker has a mind judgment (λ_{j}) as a relative
importance for each criterion (j), we can obtain the weights with Entropy
method are as follows:
MULTI OBJECTIVE DECISION MODEL IN RELATED TO LOCATION PROBLEMS AND
SOLUTION METHODOLOGY
In general, covering problems and proposed techniques for solving them may
be important to model the service facility location problems (Karasakal
and Karasakal, 2004). Set covering model is as follows (Francis
et al., 1992; Mirchandani and Francis, 1990):
where, X_{i} is a binary variable that is equal to one if the
feasible alternative, i is suitable for locating distribution centers,
otherwise it is equal to 0.
In this expression A = [a_{ij}] is called covering matrix; is
equal to 1 if a potential supportive center located in location i, is
able to cover the supported center located in location j.
However, the objective functions of the problem are as follows:
• 
Minimizing the number of distribution centers or warehouses 
• 
Maximizing the utility of the selected locations 
• 
Minimizing facility location cost 
In order to constructing objective function in relation to maximizing
the utility of the selected locations, we use cli^{–} which
are TOPSIS algorithm results:
where, C_{i} has been already defined as TOPSIS algorithm result.
S_{i} is facility locating cost for location i.
Constraint Eq. 19 ensures that all of the supported
centers are covered. Some optimum heuristics and metaheuristics like
GA have been proposed for solving covering problems. However, LINGO 8.00
as powerful software for solving problems based on the selected method
and the size of problem is used. It is in the nature of the MODM problems
to have conflicting objectives. Therefore, in this phase we face a MODM
problem. Note that the first and second objective functions must be maximized
and third objective function must be minimized. For solving this model,
firstly we solve model only with first objective function and obtain optimum
amount of Z_{1} (say Z_{1}*).
Then we have fixed the value of Z_{1} in a constraint to find
the value of Z_{2} and Z_{3}, then we repeat this procedure
for second and third objective functions. The results are presented in
Table 2.
Table 2: 
The procedure of proposed 

After obtain the optimum of the each objective function, in order to
use fuzzy approach, firstly we have to define membership degree for each
objectives as follow:
Also, for each of objective functions, we defined α_{i}
as follow:
α_{i} 
= 
The percent of utility that each function arrived to
optimum amount 
Now, we can formulate fuzzygoal model as follows:
With solving this model, optimum amounts of each objective (α_{i})
and distribution centers locations (x_{i}) can be obtained. With
changing w_{i}, we can propose various amounts of X_{i}
and α_{i} to decision maker to select the best X_{i}
as the best locations.
NUMERICAL EXAMPLE
A producer which is producing gassy cooler in suburb of Tehran wants
to establish agents in some city centers in the country, in order to products
marketing and increasing in sale’s amount.
This company selects only 15 cities after survey that be done to establish
agents and wants to establish minimum 8 agents in the best cities. For
selection the best cities, company selects indexes areas follows:
A 
: 
People population in city 
B 
: 
Level of people life in city 
C 
: 
Accumulation level and existing similar agent 
D 
: 
Distance between city and company 
E 
: 
Road availability for transportation 
F 
: 
Transportation cost 
G 
: 
City’s weather 
Also, the company allocates weight to each index that shows importance
of indexes. The allocated indexes are as follows:
The cities that company candidate for establish agents are:
(1) Tehran, (2) Mashhad, (3) Esfehan, (4) Shiraz, (5) Yazd, (6) Rasht,
(7) Qazvin, (8) Ardabil, (9) Kermanshah, (10) Lorestan, (11) Khoozestan,
(12) Hormozgan, (13) Golestan, (14) Booshehr and (15) Kohkilooye va booyerahmad
Also, this company is defined costs of establish agents in different
cities as Table 3.
The Table 4 is a decision matrix. It is necessary to
say, that we can obtain information of this matrix, with getting from
similar companies after recognize cities.
For solving this example, firstly weights with entropy technique in respect
to defined algorithm should be obtained:
Also, the mind judgments of decision maker are as follows:
Finally criteria’s weights are as follows:
Then, with using w’j and TOPSIS method, we obtain cli^{–}
that show quality of each city as Table 5.
Table 3: 
Costs of establish agents in different cities 

Table 5: 
The quantities of cli^{–} for each city 

Table 7: 
The quantities of variables 

In this problem, three objectives are as follows:
• 
Maximizing the number of agents 
• 
Maximizing the utility of the selected locations 
• 
Minimizing facility location cost 
CONSTRAINTS FOR THIS PROBLEM
All of the supported cities must be covered with agents. According to
company’s survey, there are 26 cities that must be covered. Also,
minimum 8 agents must be establishing in the best cities. At it was mentioned
earlier, firstly, the constrain table is defined as Table
6.
Now, with weights which are proposed from decision maker, we can formulate
fuzzygoal programming for solving this problem:
After solving this model via LINGO software, the quantities of variables are
obtained as Table 7.
Therefore, the cities: Tehran, Mashhad, Esfehan, Shiraz, Yazd, Rasht,
Qazvin, Khoozestan, Hormozgan, Booshehr and Kohkilooye va booyerahmad
were selected for establish agents.
CONCLUSION
In this study, an algorithm for modeling and solving location problems
using fuzzygoal programming is proposed. Also a case study about finding
the locations of the agents with the maximum number of agents, maximum
quality of the locations and minimum cost of the locations for increasing
in company’s sale was investigated and finally the best cities for
establishing agents are selected with fuzzygoal programming. However,
in such cases due to the size, complexity and large number of attributes
in the problem, it is solved sequentially and in each stage regarding
the situation we used different tools and models.