a new fuzzy electre approach with an alternative fuzzy

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resources bring the method in the foreground. In this work, a fuzzy ELECTRE procedure with an alternative ranking approach is proposed for selection problems, ...
A NEW FUZZY ELECTRE APPROACH WITH AN ALTERNATIVE FUZZY RANKING METHOD A. ÇAGRI TOLGA Industrial Engineering, Galatasaray University, Ortakoy Istanbul, 34357, Turkey [email protected] CENGIZ KAHRAMAN Industrial Engineering, Istanbul Technical University, Macka Istanbul, 34367, Turkey Multi Criteria Decision Making methods are frequently used in selection problems. There are many approaches in MCDM considering both qualitative and quantitative data. ELECTRE requires lesser data than the others so savings on manpower and time resources bring the method in the foreground. In this work, a fuzzy ELECTRE procedure with an alternative ranking approach is proposed for selection problems, which could be applied to many industrial problems.

1. Introduction ELECTRE, an acronym for Elimination Et Choix TRaduisant la REalité in French, is one of the multicriteria decision-making (MCDM) methods based on outranking of alternatives using concordance and discordance indexes. The method structures duality between preferred and nonpreferred alternatives by establishing outranking relationships. ELECTRE is an adaptable method that allows combining both qualitative and quantitative data. This method requires less detailed data, thus careful use of time and manpower resources is the important factor in the choice of the method. The sorts of ELECTRE methods distinguish from each other while the degree of complexity, or the nature of the main problem, or the richness of the information required differentiate. There are six main versions of ELECTRE: 1, 2, 3, 4, Tri, and 1S. Roy [1] introduced the first ELECTRE method. Obtaining a subset or kernel N of project options such that any alternative which is not in N is outranked by at least one alternative in N was the aim of that work. ELECTRE 2, developed by Roy and Bertier [2], is dissimilar with its premise. It ranks the options from best to worst, rather than simply establishing an initial Kernel. Roy

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[3] presented ELECTRE 3 for ranking alternatives using fuzzy logic. ELECTRE 1S method was enhanced by Roy and Skalka [4] to adopt the ELECTRE 1 into the fuzzy logic. The best option is selected from other options within the kernel. ELECTRE methods are applied to many areas since they are stated. For example Raoot and Rakshit [5] used the method for the multiple criteria evaluation and ranking of efficient layout alternatives. In daily life real circumstances are very often uncertain and vague in several ways. When there is a lack of information or no voluntary financial data out the company, a system might not be known completely. Zadeh [6] suggested a strict mathematical outline named the fuzzy set theory that overcomes these inadequacies. Up to present many scientists applied fuzzy ELECTRE to variant fields like Montezar et al. [7] did. They discussed the architecture of a fuzzy system including modules. The investigation on ranking fuzzy numbers began early 70’s. Many researchers have classified fuzzy ranking methods since 1980. Dubois and Prade’s [8] Possibility of Dominance method, Yager’s [9] weighted mean method, and Baas and Kwakernaak’s [10] conditional fuzzy set approach are the most used fuzzy number ranking methods. The rest of this paper is organized as follows: firstly alternative fuzzy numbers ranking method will be illustrated. At the third section proposed fuzzy ELECTRE method will be explicated. Steps of the new fuzzy ranking method integrated ELECTRE will be denoted at that section. 2. An Alternative Fuzzy Ranking Method Fuzzy numbers are a particular kind of fuzzy sets. A fuzzy number is a fuzzy set R of real numbers set with a continuous, compactly supported, and convex membership function. Let X be a universal set; a fuzzy subset N of X is defined by a function  N (.) : X  [0,1] , called membership function. Throughout this paper, X is assumed to be the set of real numbers R and F the space of fuzzy sets. The fuzzy set N  F is a fuzzy number iff: i.   [0,1] the set N   {x  R : N ( x)  } , which is called  -cut of N , is a convex set. ii.  N (.) is a continuous function. iii. supp( N ) = {x  R : N ( x)  0} is a bounded set in R. iv. height N = max xX N ( x)  h  0 . By conditions i and ii, each  -cut is a compact and convex subset of R hence it is a closed interval in R, N   [ NL ( ), N R ( )] . If h = 1 we say that the fuzzy number is normal; we denote the set of normal fuzzy numbers by NFN.

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Let us show a fuzzy number N  (n1, n2 , n3 , n4 ) , n1  n2  n3  n4 in this form that will be utilized in the following definitions. In fuzzy literature, there are three types of fuzzy numbers displayed as follows: L-R fuzzy numbers, trapezoidal fuzzy numbers, and triangular fuzzy numbers. In this work, triangular normal fuzzy numbers will be used hereafter. The fuzzy number N is a so-called triangular fuzzy number N  (n1, n2  n3 , n4 ) , n1  n2  n3  n4 if its membership function N ( x) : R  [0,1] is equal to  x  n1    x   n1 , n2   n2  n1    n  x   N ( x)   4  x   n3 , n4   n4  n3   0 others  

(1)

In this work, a new area-based approach proposed by Kahraman and Tolga [11] for ranking fuzzy numbers will be utilized. This approach has many advantages that are counted in cited paper. An index that measures the possibility of one fuzzy number being greater than another will be determined. That preference index will be illustrated by I ()  [0,1] and it is determined by Eq. (2):

I ( ) 

S lfavor  S rfavor  S joint

(2)

S N  SU

Using the areas given in Kahraman and Tolga’s work [11], the preference index can be determined as in the following standard form, as given in Eq.(3): 0   (n4  u1 )2   (u2  u1  n3  n4 )  ( n  n   4 3 n2  n1 )  (u4  u3  u2  u1 )  n4  n3  u2  u1 I ( )    (n4  n3  n2  n1 )  (u4  u3  u2  u1 )  (n2  u3 ) 2  (n4  n3  u2  u1 )  (u4  u3  n2  n1 )   (n  n  n  n )  (u  u  u  u ) 4 3 2 1 4 3 2 1  1 

,

u1  n4

, u2  n3 , u1  n4 , u3  n2 , u2  n3

, u3  n2 , u4  n1 ,

u4  n1

(3)

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and the fuzzy preference relation ( PKT ) of the fuzzy numbers will be determined as following:

N U  PKT ( N , U )   N  U U N 

if I ( w)  (0.5,1] if I ( w)  0.5

(4)

if I ( w)  [0, 0.5)

Calculation of index I ( ) is the key factor in said method and that method can be applied to both triangular and trapezoidal fuzzy numbers. Using Eq. (3) in triangular fuzzy numbers is as easy as winking. 3. Fuzzy ELECTRE ELECTRE 1S is chosen for modification by using alternative ranking method because of its appropriateness to that procedure. The basic steps of proposed fuzzy ELECTRE method can be viewed as follows: Suppose a MCDM problem has m decision criteria (C1, C2, …, Cm) and u alternatives (A1, A2,…, Au). Alternatives are evaluated by m criteria separately. The decision matrix is formed by the fuzzy ratings assigned to the alternatives w. r. t. each criterion denoted by T  (tij )mu . The relative fuzzy weight vector about criteria is displayed by W  (w1 , w2 ,..., wm ) . Step 1: Constitute the decision matrix from the decision maker’s thoughts as below:

 t11 t T   21   t1u

t1m  t2 m    tum 

t12 t22 t2 u

(5)

Step 2: If the objective is a minimization like cost criteria, apply the normalization procedure as seen in Eq. (6), otherwise like benefit criteria apply the normalization in Eq. (7):

rij  1/ tij rij  tij

u  i 11/ tij

 i 1 tij u

2

2

(6) (7)

where i = 1, 2 ,…,u and j = 1, 2 ,…,m. At that point division of fuzzy numbers (shown in Section 2) has to be explained that one can easily calculate with Eq.(8)

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1/ N  1/ n4 ,1/ n2 ,1/ n1 

(8)

Step 3: Calculate the weighted fuzzy normalized decision matrix V  (vij )mu for i = 1,2 ,…, u and j = 1,2 ,…, m, where vij  w j  rij . Step 4: Designate the fuzzy concordance and discordance sets. The fuzzy concordance set, which is formed by all criteria for which alternative Ap is preferred to alternative Aq, can be written as

E ( p, q)   j v pj  vqj 

(9)

The discordance set, the complement of E ( p, q) which includes all criteria for which Ap, is worse than Aq, can be written as

D( p, q)   j v pj  vqj 

(10)

Here, utilization of the alternative fuzzy ranking method aforementioned in section 2 is proposed. All the fuzzy comparisons in Eq.s (9) and (10) are made with the conclusion of Eq. (3). Step 4: Calculate the concordance and discordance indexes. The degree of confidence in the pairwise judgments of (Ap→Aq) is represented by Epq and it is calculated by Eq. (11)

E pq   j* w j*

(11)

where j* are criteria contained in the concordance set E (p, q). The discordance index of D (p, q) can be defined as

Dpq  ( j0 v pj0  vqj0 w j* ) ( j v pj  vqj )

(12)

Because the absolute value clause in Eq. (12) all the compared values in Eq.s (9) and (10) should be defuzzified by equation below and these values should be used in Eq.s (11) and (12).

v( Ni )  n 4 x N ( x)dx n

1

n N ( x)dx  (n1  n2  n4 ) / 3 n4

(13)

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Step 5: Outrank the relationships. A higher concordance index Epq and a lower discordance index Dpq means the dominance relationship of alternative Ap becomes stronger over alternative Aq. When the Epq ≥ E and Dpq < D , that represents Ap outranks Aq. Here, E and D are the averages of Epq and Dpq, respectively.

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4. Conclusions Contemporary decision-making problems necessitate multi dimensional evaluation. Considering qualitative and quantitative data together is indispensable and in this study one of the MCDM methods named ELECTRE is proposed for selection problems in a fuzzy manner. The contribution of this work is to integrate an alternative easy way of ranking fuzzy numbers to ELECTRE process. Application of this work to R&D projects, health-care management systems could be the future studies. Acknowledgments This work is financially supported by Galatasaray University scientific research project funds. References B. Roy, Classement et choix en présence de points de vue multiples (la méthode ELECTRE), Revue Informatique et Recherche Opérationnelle 2, 8 (1968) 57-75. 2. B. Roy and P. Bertier, La méthode ELECTRE II: Une méthode de classement en présence de critères multiples, SEMA 142, (1971) 25. 3. B. Roy, ELECTRE III: Un algorithme de classements fonde sur une représentation floue des préférences en présence de critères multiples, Cahiers de CERO 20, 1 (1978) 3-24. 4. B. Roy and J. M. Skalka, ELECTRE 1S: Aspects méthodologiques et guide d’utilisation, Document de LAMSADE 30, (1985) 1-125. 5. A. D. Raoot and A. Rakshit, A Linguistic Pattern Approach for Multiple Criteria Facility Layout Problems, International Journal of Production Research 31, 1 (1993) 203-222. 6. L. Zadeh, Fuzzy Sets, Inform Control 8, (1965) 338-353. 7. G. A. Montezar, H. Q. Saremi and M. Ramezani, Design a New Mixed Expert Decision Aiding System Using Fuzzy ELECTRE III Method for Vendor Selection, Exp. Systems with Applications 36, (2009) 10837-10847. 8. D. Dubois and H. Prade, Ranking of fuzzy numbers in the setting of possibility theory, Information Sciences 30 (1983) 183-224. 9. R. R. Yager, On a general class of fuzzy connectives, Fuzzy Sets and Systems 4 (1980) 235-242. 10. S. M. Baas and H. Kwakernaak, Rating, and ranking of multiple aspect alternative using fuzzy sets, Automatica 13 (1977) 47-58. 11. C. Kahraman and A. Ç. Tolga, An Alternative Ranking Approach and Its Usage in Multi-Criteria Decision-Making, International Journal of Computational Intelligent Systems 2, 3 (2009) 219-235. 1.