FUZZ-IEEE 2009, Korea, August 20-24, 2009
The ELECTRE Multicriteria Analysis Approach Based on Intuitionistic Fuzzy Sets Ming-Che Wu and Ting-Yu Chen
Abstract—Over the last decades, intuitionistic fuzzy sets have been applied to many different fields, such as logic programming, medical diagnosis, decision making, etc. The purpose of this paper is to develop a new methodology for solving multi-attribute decision-making problems with intuitionistic fuzzy information by using the concept of ELECTRE method. ELECTRE uses the concept of an outranking relationship. We also use TOPSIS method to rank all of the alternatives and to determine the best alternative. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.
I. INTRODUCTION
T
HE intuitionistic fuzzy set (IFS) was first introduced by Atanassov in 1986 [1], which is characterized by a membership function and a non-membership function. The IFS generalizes the fuzzy set and introduced by Zadeh in 1965[2], and has been found to be highly useful to deal with vagueness. Over the last decades, IFS has been applied to many different fields, such as logic programming [3], medical diagnosis [4-5], decision making [6-10,10-1-10-4], etc. Atanassov & Georgiev [3] presented a logic programming system which uses a theory of IFS to model various forms of uncertainty. De et al. [4] has applied in medical diagnosis using the notion of IFS theory. Atanassov et al. [6] presented IF interpretations of the processes of multi-person and of multi-measurement tool with multi-criteria decision making. Hong and Choi [7] provided new functions to measure the degree of accuracy in the grades of membership of each alternative with respect to a set of criteria represented by vague values. Xu and Ronald [8] presented an application of the intuitionistic fuzzy hybrid geometric (IFHG) operator to multiple attribute decision making based on IFS. Z. Xu [9] developed some similarity measures of IFS. He defined the notions of positive ideal IFS and negative ideal IFS and applied the similarity measures to multiple attribute decision making under intuitionistic fuzzy environment. Szmidt and Kacprzyk [10] applied a new measure of similarity to analyze the extent of agreement in a group of experts. The proposed measure takes into account not only a pure distance between intuitionistic fuzzy preferences but also examines if the compared preferences are more similar or more dissimilar to each other. Lin et al. [11] proposed a new method for M. C. Wu is with the Graduate Institute of Business Management, College of Management, Chang Gung University, Kwei-Shan, Taoyuan 333, Taiwan (e-mail:
[email protected]). T. Y. Chen is with the Department of Business Administration, College of Management, Chang Gung University, Kwei-Shan, Taoyuan 333, Taiwan (corresponding author to provide phone: 886-3-2118800x5678; fax: 8863-2118500; e-mail:
[email protected]). 978-1-4244-3597-5/09/$25.00 ©2009 IEEE
handling multi-criteria fuzzy decision-making problems based on IFS. It allows the degrees of satisfiability and non-satisfiability of each alternative with respect to a set of criteria to be represented by intuitionistic fuzzy sets, respectively. Chen and Wang [12]presented interval-valued fuzzy permutation (IVFP) methods for solving multi-attribute decision making problems based on interval-valued fuzzy sets. Li et al. [13] developed a new methodology to solve the multi-attribute group decision-making problems with multiple attributes being considered explicitly and both ratings of alternatives on attributes and weight of attributes being expressed using IFS. Boran et al. [14] proposed the technique for order preference by similarity to ideal solution (TOPSIS) method combined with IFS to select appropriate supplier in a group decision making. In this study, also intuitionistic fuzzy weighted averaging (IFWA) operator is used to aggregate all individual decision makers’ opinions for rating importance of criteria and alternatives. Nevertheless, the literatures show that none of current studies use the concept of ELECTRE method to solve multi-attribute decision-making problems with intuitionistic fuzzy information. The ELECTRE method is one of the methods of multiple criteria decision making [15].The multiple criteria decision making (MCDM) models have two classifications: multiple objective decision making (MODM) and multiple attribute decision making (MADM). MODM have decision variable values which are determined in a continuous or integer domain with either an infinitive or a large number of choices, the best of which should satisfy the decision maker’s constraints and preference priorities. MADM on the other hand are generally discrete, have a limited number of alternatives. They require both intra and inter attributes comparisons and involve explicit tradeoff which are appropriate for the problem explained [16]. ELECTRE method was first introduced by Benayoun et al. [15].The origins of ELECTRE methods go back to 1965 at the European consultancy company SEMA, which is still active today. At that time, a research team from SEMA worked on a concrete, multiple criteria, real-world problem regarding decisions dealing with the development of new activities in firms [17]. The method uses the concept of an ‘outranking relations’. Its first idea concerning concordance, discordance and outranking concepts originated from real-world applications [18]. The method uses concordance and discordance indexes to analyze the outranking relations among the alternatives [19]. In this paper, we develop a new methodology for solving
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multi-attribute decision-making problems with intuitionistic fuzzy information by using the concept of ELECTRE method. We also use the TOPSIS index to rank all of the alternatives and to determine the best alternative.
+ | π A ( x j ) − π B ( x j ) |) .
--Intuitionistic Euclidean distance: dis2 ( A, B) = (
II. INTUITIONISTIC FUZZY SETS
n
1 2n
∑[(μ i =1
A. Solution Process All solution process in this paper are divided into two parts. 1.Determine the decision matrix
x j ∈ X → μ A ( x j ) ∈ [0,1] ,
2.Determine concordance & discordance set
(2)
Define the degree of membership and the degree of non-membership of the element x j ∈ X to the set A ⊆ X ,
3. Calculate the concordance matrix
IF ELECTRE Method M.
respectively, and for every x j ∈ X , (3)
π A (x j ) = 1 − μ A (x j ) −ν A (x j )
(4)
4. Calculate the discordance matrix 5.Determine the concordance dominance matrix 5b. Determine the concordance dominance matrix 6.Determine the discordance dominance matrix 6b. Determine the discordance dominance matrix
as the intuitionistic index of the element x j in the set A . It is
7.Determine the aggregate dominance matrix
the degree of indeterminacy membership of the element x j to
7b. Determine the aggregate dominance matrix
the set A . It is obvious that for every x j ∈ X ,
8.Eliminate the less favorable alternatives
8b. Determine the best alternative
(5)
Fig. 1. Intuitionistic fuzzy ELECTRE method solution process.
B. Operation Reference [1] [20] and [21] shows that some operations for IFS. For every two IFSs A and B the following operations and relations are valid: (1) A ⊂ B iff ∀x ∈ X, (μ A ( x) ≤ μ B ( x) & v A ( x) ≥ v B ( x)) ; (2) A = B iff A ⊂ B & B ⊂ A ; (3) A = {( x, v A ( x), μ A ( x))} . C. Some related distance measures In many practical and theoretical problems, in order to find the difference between two objects, the knowledge of distance between two fuzzy sets is necessary. Some popular distance measure formula were introduced between two IFSs A and B that take into account the membership degree μ , the non-membership degree ν , and the hesitation degree (or intuitionistic fuzzy index) π in X = { x1 , x 2 ,..., x n } . Some of the intuitionistic fuzzy distance measures are as follows [22-23]:
They are IF ELECTRE Method (Step 1 to 8) and Ranking Method (Step 5b to 8b), and all steps shown in Fig. 1. B. IF ELECTRE Method The IF ELECTRE method is included in eight steps. The steps are as follows. Step 1. Determine the decision matrix: Let X ij = (μ ij ,ν ij , π ij ) , μij is the degree of membership of the ith alternative with
respect
A (x j ) −
to
the
jth
attribute, ν ij
is
the
degree
of
non-membership of the ith alternative with respect to the jth attribute, π ij is the intuitionistic index of the ith alternative with respect to the jth attribute. M is an intuitionistic fuzzy decision matrix, where (8) 0 ≤ μij +ν ij ≤ 1 , i = 1,2,…m, j = 1,2,…n π ij = 1 − μ ij − ν ij
n
∑ (| μ
Ranking Method
0 ≤ μ A ( x j ) +ν A ( x j ) ≤ 1.
We call
1 dis1 ( A, B) = 2n
(7)
III. INTUITIONISTIC FUZZY ELECTRE METHOD
where the functions
--Intuitionistic Hamming distance:
μ B ( x j )) 2 + (ν A ( x j ) − ν B ( x j )) 2 1
A. Concept Let X = {x1 , x2 ,..., xn } be a finite universal set. An IFS A in X is defined as an object of the following form: (1) A = {< x j , μ A ( x j ),ν A ( x j ) >│x j ∈ Χ}
0 ≤ π A (x j ) ≤ 1 .
A (x j ) −
+ (π A ( x j ) − π B ( x j )) 2 ]) 2 .
In this section, the concept and operations of IFSs and some of the related distance measures are described.
x j ∈ X → v A ( x j ) ∈ [0,1] .
(6)
μ B (x j ) | + | v A (x j ) − vB (x j ) |
j =1
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(9)
FUZZ-IEEE 2009, Korea, August 20-24, 2009 x1
step 2 and w j are weight of attributes that are also defined in
xn
A1 ⎡ X 11 ⎢ M= ⎢ Am ⎣⎢ X m1
X 1n ⎤ ⎥. ⎥ X mn ⎦⎥
In the decision matrix M, have m of alternatives (from A1 to Am ) and n of attributes(from x1 to xn ). The subjective importance of attributes, W, are given by the decision maker(s). For example, attribute x1 has attribute
step 1. Step 4. Calculate the discordance matrix: The discordance index d kl is defined as follows: max wD∗ × dis( X kj , X lj )
d kl =
dis( X kj , X lj ) =
where
Step 2. Determine the concordance and discordance sets: It use the concept of IFS relation to identify (determine) concordance and discordance set. For example, we can classify different types of the concordance sets as strong concordance set or moderate concordance set or weak concordance set. It can also be classify the discordance sets by the same concept. Let X ij = (μij ,ν ij , π ij ) ,
max dis( X kj , X lj )
((
1 μ kj − μlj 2
(11)
The weak concordance set C ''kl is defined as C kl′′ = { j│μ kj ≥ μ lj and ν kj ≥ ν lj } .
(12)
The strong discordance set Dkl is composed of all criteria for which Ak is not preferred to Al . The strong discordance set Dkl can formulate as Dkl = { j│μ kj < μ lj ,ν kj ≥ ν lj and π kj ≥ π lj } .
'
of dominating Al , if its corresponding concordance index ckl exceeds at least a certain threshold value c i.e. , ckl ≥ c , and
The decision maker(s) give the weight in different sets. For example, the strong concordance set Ckl have it own weight
'
wD ''
are respectively
.
Step 3. Calculate the concordance matrix: The relative value of the concordance sets are measured by means of the concordance index. The concordance index is equal to the sum of the weights associated with those criteria and relation which are contained in the concordance sets. Therefore, the concordance index ckl between Ak and Al is defined as: (16) ckl = wC × wj + w ' × w j +w '' × wj
∑
j∈Ckl
C
∑
j∈C ' kl
C
kl
k =1, k ≠ l l =1,l ≠ k
m × ( m − 1)
.
(19)
On the basis of the threshold value, a Boolean matrix F can be constructed, the elements of which are defined as f kl = 1 , if ckl ≥ c ; f kl = 0 , if ckl < c . Then each element of 1 on the matrix F represents a dominance of one alternative with respect to another one. Step 6. Determine the discordance dominance matrix: This matrix is constructed in a way analogous to the F matrix on the basis of a threshold value d to the discordance indices. The elements of g kl of the discordance dominance
_
d=
(15)
''
c=
m
The weak discordance set D '' kl is defined as Dkl′′ = { j│μ kj < μ lj andν kj < ν lj } .
'
m
∑ ∑c
(13) (14)
wC , wC , wC , w D , wD and
''
matrix G are calculated as
The moderate discordance set D ' kl is defined as Dkl′ = { j│μ kj < μ lj ,ν kj ≥ ν lj and π kj < π lj } .
D''kl
(18)
is equal to wD or w or w that depend on the D D different types of discordance sets and defined in step 2. Step 5. Determine the concordance dominance matrix: This matrix can be calculated with the aid of a threshold value for the concordance index. Ak will only have a chance
m
The moderate concordance set C ' kl is defined as Ckl′ = { j│μ kj ≥ μ lj ,ν kj < ν lj and π kj ≥ π lj } .
wC . Weight of C kl , C kl′ , C '' kl , Dkl , D ' kl and
)2 + (ν kj −ν lj )2 + (π kj − π lj )2 )
wD ∗
where μij , ν ij , π ij are defined at 1st step. The strong concordance set C kl of Ak and Al is composed of all criteria for which Ak is preferred to Al . In other words, The strong concordance set Ckl can formulate as (10) C kl = { j│μ kj ≥ μ lj ,ν kj < ν lj and π kj < π lj } .
(17)
j ∈J
weight w1, xn has attribute weight wn and the sum of weight of all attributes from x1 to xn are equal to 1.
j∈Dkl
∑
j∈Ckl′′
m
∑ ∑d
kl
k =1, k ≠ l l =1,l ≠ k
m × ( m − 1)
if d kl ≤ d ; g kl = 0 , if d kl > d . Also the unit elements in the G matrix represent the dominance relationships between any two alternatives. Step 7. Determine the aggregate dominance matrix: This step is to calculate the intersection of the concordance dominance matrix F and discordance dominance matrix G. The resulting matrix, called the aggregate dominance matrix E, is defined by means of its typical elements ekl as: (21) ekl = f kl ⋅ g kl . Step 8. Eliminate the less favorable alternatives: The aggregate dominance matrix E gives the partial-preference ordering of the alternatives. If ekl = 1, then Ak is preferred to Al for both the concordance and discordance criteria, but Ak still has the chance of being dominated by the other alternatives. Hence the condition that Ak is not dominated by ELECTRE procedure is,
where wC , wC ' , wC '' are weight in different sets and defined in 1385
g kl = 1 ,
(20)
ekl = 1, for at least one l , l = 1,2,…,m, k ≠ l ;
FUZZ-IEEE 2009, Korea, August 20-24, 2009
eik = 0, for at all i , i = 1,2,…,m, i ≠ k , i ≠ l . This condition appears difficult to apply, but the dominated alternatives can be easily identified in the E matrix. If any column of the E matrix has at least one element of 1, then this column is ‘ELECTREcally’ dominated by the corresponding row(s). Hence we simply eliminate any column(s) which have an element of 1. C. Ranking Method with TOPSIS index Because of the IF ELECTRE Method can not rank all of the alternatives we utilize TOPSIS index to rank them. Yoon and Hwang [15] developed the TOPSIS method based on the concept that the chosen alternative should have the shortest distance from the ideal solution and the farthest from the negative-ideal solution. The steps are as follows. Step 5b. Determine the concordance dominance matrix: It use the positive-ideal solution of TOPSIS, if c* is the biggest value in the concordance matrix, then calculate c’kl = c* - ckl (22) and determine the concordance dominance matrix C’. Step 6b. Determine the discordance dominance matrix D’: Let d* is the biggest value in the discordance matrix, then calculate d’kl = d* - dkl (23) and determine the discordance dominance matrix D’. Step 7b. Determine the aggregate dominance matrix P:
A1 A2 A M= 3 A4 A5 A6
x1 ⎡(0.35,0.33,0.32) ⎢ ⎢(0.24,0.34,0.42) ⎢ (0.11,0.16,0.73) ⎢ ⎢(0.09,0.38,0.53) ⎢(0.37,0.56,0.07) ⎢ ⎣⎢ (0.25,0.34,0.41)
x2 x3 x4 (0.22,0.44,0.34) (0.23,0.59,0.18) (0.10,0.87,0.03)⎤ ⎥ (0.26,0.42,0.32) (0.31,0.28,0.41) (0.44,0.39,0.17)⎥ . (0.19,0.47,0.34) (0.11,0.34,0.55) (0.19,0.66,0.15)⎥ ⎥ (0.43,0.29,0.28) (0.44,0.24,0.32) (0.41,0.26,0.33) ⎥ (0.34,0.39,0.27) (0.37,0.35,0.28) (0.34,0.25,0.41) ⎥ ⎥ (0.29,0.35,0.36) (0.30,0.28,0.42) (0.45,0.48,0.07)⎦⎥
Assume that the subjective importance of attributes, W, is given by the decision maker, W=[w1,w2,w3,w4]= [0.1, 0.2, 0.3, 0.4]. Applying step 2, determine the concordance and discordance sets. The decision maker also give the relative weight (W’). 2 1 2 1 W ' = [ wC , wC ' , wC '' , wD , wD ' , wD '' ] = [1, , ,1, , ] . 3 3 3 3
The strong concordance set ⎡− 1 ⎢2 − ⎢ ⎢− − C kl = ⎢ ⎢ 2 2,3 ⎢2 2 ⎢ ⎣⎢− −
1⎤ − ⎥⎥ − − − −⎥ ⎥. 2,3 − − 2,3⎥ 2 − − −⎥ ⎥ 3,4 − − − ⎦⎥ For example, C 21 = {2}, which is in the 2nd (horizontal) row and 1st (vertical) column of strong concordance set is “2”. st C13 ={–}, which is in the 1 row and 3th column of strong p12 p1m ⎤ ⎡ − concordance set is “empty” , and so forth. ⎢p ⎥. (24) − p p 21 23 2m ⎥ The moderate concordance set ⎢ P= ⎢ ⎥ − 2 − − −⎤ ⎡ − ⎢ ⎥ − ⎥⎦ ⎢⎣ p m1 p m 2 p m ( m −1) ⎢ 3,4 − 4 − − − ⎥ ⎢ ⎥ The element( pkl ) of P is define as follows: ⎢ ⎥ 4 1 − − − − C ' kl = ⎢ ⎥. d kl ' (25) pkl = ' ⎢ 3,4 − 4 − 2,3 −⎥ ' ckl + d kl ⎢ 3,4 − 4 − − −⎥ ⎢ ⎥ where c’kl are the element of concordance dominance matrix ⎢⎣2,3,4 2 2 − − −⎥⎦ ’ and d kl are the elements of discordance dominance matrix. The weak concordance set Step 8b. Determine the best alternative: − ⎤ ⎡− − 1,3 − − Using the result of step 7b, we can calculate the mix ⎢ evaluation value of alternatives, the formula as below 3 ⎥⎥ ⎢− − 1 4 4 m 1 ⎢− − − − − − ⎥ (26) pk = pkl , k = 1,2,..., m . C ''kl = ⎢ ⎥. m − 1 l =1,l ≠ k − ⎥ ⎢− − − − 4 ⎢ 1 1,3 1,3 1 − 1,2,3⎥ Thus, the best alternative A * can be generated so that ⎢ ⎥ A* = max{ p k } (27) − ⎥⎦ ⎢⎣− 1,4 1 4 4 The strong discordance set and the alternatives are ranked according to the increasing order of A j . ⎡− 2 − 2 2 − ⎤ ⎢ 1 − − 2,3 2 1 ⎥ ⎢ ⎥ IV. Numerical example ⎢ 2 2,3 − 2,3 2 3,4⎥ ⎥. Dkl = ⎢ In this section, we present a numerical example connected ⎢1 1 − − − 1 ⎥ with a decision making problem. Suppose that the proposed ⎢− − − − − − ⎥ ELECTRE method with IFS evaluates the intuitionistic fuzzy ⎢ ⎥ ⎢⎣ 1 − 3 2,3 − − ⎥⎦ decision matrix which refers to 6 of alternatives on 4 of The moderate discordance set attributes. The intuitionistic fuzzy decision matrix M in step 1 is given .
∑
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− 1 − 2,3 1 −
FUZZ-IEEE 2009, Korea, August 20-24, 2009
⎡− 3,4 4 3,4 3,4 2,3,4⎤ ⎢− − − − 2 ⎥⎥ − ⎢ ⎢− 4 − 4 4 2 ⎥ D ' kl = ⎢ ⎥. − − ⎥ ⎢− − 1 − ⎢− − − 2,3 − − ⎥ ⎥ ⎢ − − ⎦⎥ ⎣⎢− − − − The weak discordance set 1 −⎤ ⎡− − − − ⎢ − − − − 1,3 4 ⎥ ⎢ ⎥ ⎢ ⎥ 1 , 3 1 1 , 3 1 − − D '' kl = ⎢ ⎥. 1 4⎥ ⎢− 4 − − ⎢− 4 − 4 4⎥ − ⎢ ⎥ ⎣⎢ − − − − 1,2,3 −⎦⎥ Applying step 3, calculate the concordance matrix is. 0.1 0.2667 ⎡ − ⎢0.6667 − 0.8 ⎢ ⎢0.2667 0 − C=⎢ 0 . 6667 0 . 5 0 . 7667 ⎢ ⎢ 0.7 0.3333 0.6 ⎢ 0. 3 0.8667 ⎣⎢ 0.6
1
1 dis ( X 14 , X 24 ) = ( ((0.1 − 0.44 )2 + (0.87 − 0.39 )2 + (0.03 − 0.17 )2 )) 2 2
= 0.4276
and 1 wD × dis( X 12 , X 22 ) = 1 × ( × ((0.22 − 0.26)2 + (0.44 − 0.42)2 2 1
+ (0.34 − 0.32)2 )) 2 = 0.0346 , wD' × dis( X 13 , X 23 ) =
1
+ (0.18 − 0.41)2 )) 2 = 0.1858 , wD' × dis( X 14 , X 24 ) =
∑w
j ∈C 23
j
1
Applying step 5, determine the concordance dominance matrix is. The average concordance index is 6
0.1 0 0.1⎤ 0.2333 0.1333 0.1⎥⎥ 0.0667 0 0 ⎥. ⎥ − 0.4667 0.5⎥ 0.0333 − 0.2⎥ ⎥ 0.2333 0.1333 − ⎦⎥
_
c=
and C51 =
∑w
j∈C51
j
2 1 + 0 .1 × = 0 .8 3 3
= 0.3244 .
6×5
0 0 0 0 − 0 1 1 0
1 − 1 1 1
0 0 − 0 0
0 0 1 − 0
0⎤ 0 ⎥⎥ 0⎥ . ⎥ 1⎥ 0⎥ ⎥ − ⎥⎦
Applying step 6, determine the discordance dominance matrix is. The average discordance index is 6
_
d=
1 2 2 = 0.2 × 1 + 0.3 × + 0.4 × + 0.1 × = 0.7 . 3 3 3
6
∑∑ d
kl
k =1 l =1
6×5
= 0.4228 .
The discordance dominance matrix is ⎡− ⎢1 ⎢ ⎢1 G=⎢ ⎢0 ⎢1 ⎢ ⎢⎣ 1
Applying step 4, calculate the discordance matrix is. 0.6667 0.3410 0.6667 0.6667 0.6667⎤ − 0 1 0.3333 0.4251⎥⎥ 0.6593 − 0.8447 0.4160 0.8301⎥ . ⎥ 0.8739 0.4048 − 0.3333 0.5948⎥ 0.2271 0 0.1601 − 0.3333⎥ ⎥ 0.1048 0 0.5151 0.3313 − ⎥⎦
0 1 − 0 0 1 1
1 − 1 1 1
0 0 0⎤ 1 0 ⎥⎥ 1 0⎥ . ⎥ 1 0⎥ − 1⎥ ⎥ 1 − ⎥⎦
0 0 − 1 0
Applying step 7, determine the aggregate dominance matrix (E) is.
For example:
⎡− ⎢1 ⎢ ⎢0 E=⎢ ⎢0 ⎢1 ⎢ ⎢⎣ 1
max dis ( X 1 j , X 2 j )
d12
kl
k =1 l =1
⎡− ⎢1 ⎢ ⎢0 F =⎢ ⎢1 ⎢1 ⎢ ⎢⎣ 1
= w2 × wC + w3 × wC' + w4 × wC' + w1 × wC''
⎡ − ⎢0.2464 ⎢ ⎢0.3333 D=⎢ ⎢0.4523 ⎢ 0 ⎢ ⎢⎣0.2567
6
∑∑ c
The concordance dominance matrix is
= w2 × wC + w3 × wC + w4 × wC ' + w1 × wC '' = 0 .2 × 1 + 0 .3 × 1 + 0 .4 ×
2 1 × ( × ((0.1 − 0.44)2 + (0.87 − 0.39)2 3 2 + (0.03 − 0.17)2 )) 2 = 0.2850 .
For example, C23 =
2 1 × ( × ((0.23 − 0.31)2 + (0.59 − 0.28)2 3 2
0.2850 = = = 0.6667 , max dis( X 1 j , X 2 j ) 0.4276 j∈D12 j∈J
where 1
1 dis( X 11 , X 21 ) = ( ((0.35 − 0.24)2 + (0.33 − 0.34)2 + (0.32 − 0.42)2 )) 2 2 = 0.1054 ,
0
0 0 − 1 0 0 − 0 0 1 − 1 1 0 0 1 0
0 0 0 1 − 0
0⎤ 0 ⎥⎥ 0⎥ . ⎥ 0⎥ 0⎥ ⎥ −⎥⎦
Applying step 8, eliminate the less favorable alternatives is.
1
1 dis( X 12 , X 22 ) = ( ((0.2 − 0.26)2 + (0.44 − 0.42)2 + (0.34 − 0.32)2 )) 2 2 = 0.0346 , 1
1 dis ( X 13 , X 23 ) = ( ((0.23 − 0.31)2 + (0.59 − 0.28 )2 + (0.18 − 0.41)2 )) 2 2 = 0.2787 ,
The matrix E renders the following overranking relationships: A2 → A1,A2 → A3,A4 → A3,A4 → A5,A5 → A1,A5 → A2,A5→A3,A6→A1,A6→A3(illustrated with Fig. 2).
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FUZZ-IEEE 2009, Korea, August 20-24, 2009
the multiple criteria decision making methods. A1
REFERENCES
A4
[1]
A2
[2] [3]
A3
[4]
A6 A5
[5]
Fig. 2. The overranking relationships that matrix E rendered.
Using the Ranking process. Applying step 5b, calculate concordance outranking matrix is(c*= 0.8667). The concordance outranking matrix ⎡ − ⎢ ⎢ 0.2 ⎢ 0.6 ' C =⎢ ⎢ 0.2 ⎢0.1667 ⎢ ⎢⎣0.2667
[7]
0.7667 0.8667 0.7667⎤ − 0.0667 0.6334 0.7334 0.7667⎥⎥ − 0.8667 0.8 0.8667 0.8887⎥ . ⎥ − 0.3667 0.1 0.4 0.3667⎥ − 0.5334 0.2667 0.8334 0.6667⎥ ⎥ − ⎥⎦ 0.5667 0 0.6334 0.7334
0.7667
0.6
[8] [9]
Applying step 6b, calculate discordance outranking matrix is(d*= 1). The concordance outranking matrix ⎡ − ⎢0.7536 ⎢ ⎢0.6667 D' = ⎢ ⎢0.5477 ⎢ 1 ⎢ ⎣⎢0.7433
0.3333 0.6590 − 1 − 0.3407 0.1261 0.5952 0.7729 1 0.8952 1
0.3333 0 0.1553 − 0.8399 0.4849
0.3333 0.6667 0.5840 0.6667 − 0.6687
0.3333⎤ 0.5749⎥⎥ 0.1699⎥ ⎥ 0.4052⎥ 0.6667⎥ ⎥ − ⎦⎥
. [12] [13]
0.2778 0.303 ⎤ 0.4762 0.4285⎥⎥ − 0.2822 0.1626 0.4026 0.1639⎥ . ⎥ − 0.2559 0.8562 0.6250 0.5249⎥ − 0.5917 0.7895 0.5019 0.5 ⎥ ⎥ − ⎦⎥ 0.6124 1 0.4336 0.4769 0.303 −
0.5234 0.9375
[10]
[11]
Applying step 7b, determine the aggregate outranking matrix is. ⎡ − ⎢ 0.7903 ⎢ ⎢ 0.5263 P=⎢ ⎢0.7325 ⎢ 0.8571 ⎢ ⎣⎢0.7359
[6]
[14]
0.303 0
[15] [16]
Applying step 8b, determine the best alternative is.
[17]
p1 = 0.342 , p 2 = 0.5265 , p 3 = 0.3075 ,
[18]
p 4 = 0.5989 , p 5 = 0.648 , p 6 = 0.6518 .
The optimal ranking order of the alternatives is given by A6 A5 A4 A2 A1 A3 . The best alternative is A6 . V. CONCLUDING REMARKS In this study, we have provided a new methodology for solving multi-attribute decision-making problems with intuitionistic fuzzy information by using the concept of ELECTRE method. The new approach integrate the concept of ‘outranking relationship’ of ELECTRE method. We also used the Ranking method with TOPSIS index to rank all of the alternatives and to determine the best alternative. We also illustrated numerical example to demonstrate its practicality and effectiveness. In a future research, we shall utilize the concept of interval-valued intuitionistic fuzzy sets to develop
[19]
[20] [21] [22] [23]
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