AN INTEGRATED INTUITIONISTIC FUZZY MULTI CRITERIA DECISION MAKING METHOD FOR FACILITY LOCATION SELECTION

The facility location selection, which is one of the important activities in strategic planning for a wide range of private and public companies, is a multi-criteria decision making problem including both quantitative and qualitative criteria. Traditional methods for facility location selection can not be effectively handled because information can not be represented by precise information under many conditions. This paper proposes the integration of intuitionistic fuzzy preference relation aiming to obtain weights of criteria and intuitionistic fuzzy TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method aiming to rank alternatives for dealing with imprecise information on selecting the most desirable facility location. To illustrate the application of the proposed method, a practical application is given. Key WordsFacility location selection, Multi criteria decision making, Intuitionistic fuzzy set, Intuitionistic fuzzy preference relation, Intuitionistic fuzzy TOPSIS method


INTRODUCTION
Today's fierce competitive environment enforces companies to make right decisions on management activities.Perhaps, one of the most important decisions for companies is facility location selection, since it is a costly and difficult to reserve activity.Facility location selection has a great impact on output of operating and management activities in companies [1].A poor choice of location might result in unnecessary transportation costs, a lack of qualified labor, lost of competitive advantage, insufficient supplies of raw materials, or some similar conditions that would be detrimental to operations [2].On the other hand, a good choice of location might result in some advantages such as decrease in transportation cost, maximizing the usage of resources, higher logistic performance and efficiency in operations for companies.
Facility location selection is a typical multi-criteria decision making problem including conflicting criteria such as political environment, proximity to markets and customers, supplier networks, expansion potential, availability of transportation systems and utility, quality-of-life issues, culture issues, etc. [3,4].The majority of these attributes are evaluated with human perceptions and judgments which cannot be quantified precisely [5] and therefore, involves the imprecision and vagueness inherent in linguistic assessment and fuzzy multiple attributes decision-making (FMADM) [6].
There are large numbers of methods that have been developed for the facility location selection.Fuzzy set theory (FST) has been applied in the recent studies to deal with selecting facility location with respect to subjective factors.Liang and Wang [7] developed an algorithm based on FST and hierarchical structure in order to deal with selecting facility location.Kuo et al. [8] proposed a decision support system (DSS) based on FST and the analytic hierarchy process (AHP) to select a site for a new convenience store (CVS).Kuo et al. [9] developed a DSS to select location of new CVSs by combining fuzzy AHP and artificial neural network.Chen [10] developed a new approach based on FMADM with a stepwise ranking procedure to resolve the selection of distribution center location under fuzzy environment.Chu [11] proposed fuzzy TOPSIS model for facility location selection.Kahraman et al. [12] presented four fuzzy multi-attribute group decision-making approaches for evaluating facility locations.Cou et al. [6] applied fuzzy simple additive weighting method for facility location selection with objective and subjective factors.Kapoor et al. [13] used fuzzy C-Means clustering algorithm to select appropriate facility location.There are four well known methods are commonly used in the facility location selection; factor rating system, break-even analysis, center of gravity method and transportation method [12,14,15].This paper proposes an intuitionistic fuzzy multi-criteria decision making method with the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method for selecting facility location.The TOPSIS method considering positive and negative ideal solution is one of the most popular methods in multi-criteria decision making [16] and applied some decision problems [17][18][19].On the other hand, the impact of alternatives on criteria provided by decision makers is usually difficult to be precisely expressed by the crisp data in the facility location selection.Intuitionistic fuzzy sets introduced by Atanassov [20], is the extension of fuzzy sets introduced by Zadeh [21].An intuitionistic fuzzy set is characterized by three parameters: membership function, non-membership function and hesitation margin namely, which is a flexible way to deal with uncertainty, while a fuzzy set is only characterized by membership function.
The rest of this paper is organized as follows.In the next section, brief descriptions on intuitionistic fuzzy set are given.Section 3 gives a detailed description of the proposed method.A practical application is given to illustrate the application of the proposed method in Section 4. Finally, conclusions of the paper are presented.

INTUITIONISTIC FUZZY SETS
In this section, definitions of intuitionistic fuzzy set related to this study are given.Definition 1. [20] An intuitionistic fuzzy set A in a finite set X can be written as:

 
, ( ), ( ) where are membership function and nonmembership function respectively, such that: 0 ( ) ( ) 1 The third parameter of the IFS A is: which is known as the intuitionistic fuzzy index or hesitation degree of whether x belongs to A or not.It is obviously seen that for every x X  : great, then knowledge about x is more uncertain.Obviously, when ( ) 1 ( ) , for all elements of the universe, the traditional fuzzy set concept is recovered [22].Definition 2. Let A and B IFSs in defined as:

AN INTEGRATED INTUITIONISTIC FUZZY MULTI-CRITERIA DECISION-MAKING METHOD
In this section, the TOPSIS method is extended to intuitionistic fuzzy environment, which is a very suitable for solving decision-making problems. Let be a set of criteria.Intutionistic fuzzy TOPSIS method consists of the following steps which are given as follows: Step 1. Construct an intuitionistic fuzzy preference relation matrix:  be an intuitionistic preference matrix of criteria as follows: and satisfies the following condition [23,24]: where   , then we call B a multiplicative consistent intuitionistic fuzzy preference relation.
does not satisfy the condition , then we call B an inconsistent intuitionistic fuzzy preference relation.
Step 4. Determine the intuitionistic fuzzy positive ideal solution and the intuitionistic fuzzy negative ideal solution: Let 1 J be the set of benefit criteria, 2 J be the set of cost criteria, * A be the intuitionistic fuzzy positive ideal solution, and A  be the intuitionistic fuzzy negative ideal solution, then * A and A  can be determined respectively as: , , , , ..., ..., where 1 max min , 1 min max 1 min max , 1 max min Step 5. Calculate the weighted separation measures: The weighted Hamming distance is used to obtain separation measures [25,26].The weighted lower and upper separation measures   alternative from the intuitionistic fuzzy positive ideal solution and the intuitionistic fuzzy negative ideal solution are respectively calculated:   Step 6. Calculate the relative closeness coefficient of each alternative to the intuitionistic fuzzy positive and the negative ideal solutions: The relative closeness coefficient of an alternative i A with respect to the intuitionistic fuzzy positive-ideal solution *  A and the intuitionistic fuzzy negative-ideal solution A  is defined as follows: Step 7. Rank the alternatives according to the descending order of the relative closeness coefficients In order to rank alternatives, the possibility degree formula proposed by Xu and Da [27] is used.
that is a superior to b to degree of, donated by Similarly, the degree of possibility of b a  is defined as: that is b a superior to a to degree of, donated by Alternatives are ranked according to descending order of i p .

PRACTICAL APPLICATION
A manufacturing company is select to location for building new plant.There are four candidates place 1 2 3 , , A A A and 4 A are chosen for further evaluation.In order to evaluate candidate locations, expansion possibility (C 1 ), availability of acquirement material (C 2 ), community considerations (C 3 ), distance to market (C 4 ) and labour cost (C 5 ) are considered as evaluation factors.
Negative and positive separation measures based on the weighted lower and upper Hamming distance for each candidate have been calculated by utilizing Eq.( 19) -Eq.( 22) and given in Table 2.
0.066 0.073 , , 0.541, 0.670 0.049 0.073 0.043 0.066 0.047 0.053 , , 0.416,0.5300.060 0.053 0.053 0.047 0.047 0.054 , , 0.398, 0.514 0.064 0.054 0.058 0.047 Step 7. Rank the candidates according to the descending order of the relative closeness coefficients.Four candidate locations have been ranked according to the descending order of the relative closeness coefficients.The candidates have been ranked by using the possibility degree formula and the following matrix has been constructed as follows: 0.5 0 0 0.049 1 0.5 1 1 1 0 0.5 0.574 0.951 0 0.426 0.5  Summing all elements in each line of matrix P , then: A has been selected as the most desirable facility location among candidates.

CONCLUSION
The success of companies depends on their capability on making right strategic decisions.Facility location selection is one of these strategic decisions, which it is a costly and difficult to reverse activity for companies.Therefore, this paper has presented the integration of intuitionistic fuzzy preference relation and intuitionistic fuzzy TOPSIS method for selecting the most desirable facility location.The decision factors (attributes), expansion possibility, availability of acquirement material, community considerations, distance to market and labour cost have been taken into account and candidate locations have been evaluated by the proposed method with respect to the decision factors.The intuitionistic fuzzy preference relation has been applied to derive the weights of criteria and intuitionistic fuzzy TOPSIS method has been used to rank alternative.The integrated intuitionistic fuzzy multi criteria decision making method has enormous chances of success for multi-criteria decision making problems due to having great superiority on dealing with vagueness.Therefore, in the future, the proposed method can be used for dealing with uncertainty in a variety of multi-criteria decision making problems.Moreover, the proposed method may be extended to group decision environment and apply to important decision making problems.This is the issue for future researches.

ACKNOWLEDGMENT
The author is very grateful to the anonymous referees for their constructive comments and suggestions that led to an improved version of this paper.
element of   * B matrix, are the membership degree and the non-membership degree of the alternative i x over j x , respectively, and

Table 1 .
as follows: Construct the intuitionistic fuzzy decision matrix.The intuitionistic fuzzy decision matrix has been constructed in Table1as follows: The intuitionistic fuzzy decision matrix

Table 2 .
Separation measures of candidatesCandidates Calculate the relative closeness coefficient of each candidate to the intuitionistic fuzzy positive and negative ideal solutions.The relative closeness coefficients of each candidate to the intuitionistic fuzzy positive and negative ideal solutions have been calculated by using Eq.(23) as follows: