The ELECTRE multicriteria analysis approach based on Atanassov’s intuitionistic fuzzy sets

Expert Systems with Applications 38 (2011) 12318–12327

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

The ELECTRE multicriteria analysis approach based on Atanassov’s intuitionistic fuzzy sets Ming-Che Wu a,1, Ting-Yu Chen b,⇑ a b

Graduate Institute of Business Administration, College of Management, Chang Gung University, 259, Wen-Hwa 1st Road, Kwei-Shan, Taoyuan 333, Taiwan Department of Industrial and Business Management, College of Management, Chang Gung University, 259, Wen-Hwa 1st Road, Kwei-Shan, Taoyuan 333, Taiwan

a r t i c l e

i n f o

Keywords: Intuitionistic fuzzy set ELECTRE method Multi-criteria decision making

a b s t r a c t In recent decades, intuitionistic fuzzy sets have been applied to many different fields; however, few current studies have used the ELECTRE method to solve multi-criteria decision-making problems with intuitionistic fuzzy information. The purpose of this paper is to develop a new method, the intuitionistic fuzzy ELECTRE method, for solving multi-criteria decision-making problems. Atanassov’s intuitionistic fuzzy set (A-IFS) characteristics are simultaneously concerned with the degree of membership, degree of non-membership, and intuitionistic index, and people can use A-IFS to describe uncertain situations in decision-making problems. We use the proposed method to rank all alternatives and determine the best alternative. The proposed method can also use imperfect or insufficient knowledge of data to deal with decision-making problems. Finally, two practical examples are given that illustrate the procedure of the proposed method.  2011 Elsevier Ltd. All rights reserved.

1. Introduction The main purpose of this paper is to extend the ELECTRE (Elimination et Choice Translating Reality) method to develop a new method for solving multi-criteria decision-making (MCDM) problems in Atanassov’s intuitionistic fuzzy (A-IF) environments. 1.1. Atanassov’s intuitionistic fuzzy set Atanassov’s intuitionistic fuzzy set (A-IFS) was first introduced in 1986 and is characterized by a membership function and a non-membership function. A-IFS generalizes the fuzzy set, which was introduced by Zadeh (1965). A-IFS has been found to be highly useful when dealing with vagueness. Vague data are more adequate than crisp data for modeling real-life situations under many conditions. Human judgments that include preferences are often vague and cannot be estimated with exact numerical values. In recent decades, A-IFS has been applied to many different fields, including logic programming (Atanassov & Georgiev, 1993), medical diagnosis (De, Biswas, & Roy, 2001), decision making (Atanassov, Pasi, & Yager, 2005; Boran, Genc, Kurt, & Akay, 2009; Cheng, Lin, Hsu, & Shu, 2009; Li, 2005; Li, Wang, Liu, & Shan, 2009; Szmidt & Kacprzyk, 2004; Wang, 2009; Xu, 2007c, 2010; Xu & Yager, 2006), evaluation function (Chen & Tan, 1994; Hong & Choi, ⇑ Corresponding author. Tel.: +886 3 2118800x5678; fax: +886 3 2118500. E-mail addresses: [email protected] (M.-C. Wu), [email protected] (T.-Y. Chen). 1 Tel.: +886 3 2118800x5860; fax: +886 3 2118500. 0957-4174/$ - see front matter  2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.04.010

2000; Lin, Yuan, & Xia, 2007; Liu & Wang, 2007), intuitionistic fuzzy (IF) preference relation (Xu, 2007a; Xu & Yager, 2009), and the TOPSIS method (Boran et al., 2009; Xu, 2007b). Atanassov and Georgiev (1993) presented a logic programming system using A-IFS to model various forms of uncertainty. De et al. (2001) has applied the notion of A-IFS theory in medical diagnosis. Atanassov et al. (2005) presented IF interpretations of processes of multi-person and multi-measurement tools. Hong and Choi (2000) provided new functions to measure the degree of accuracy in grades of membership of each alternative with respect to a set of criteria represented by vague values. Xu and Yager (2006) proposed some geometric aggregation operators, and presented an application of the IF hybrid geometric (IFHG) operator to multiattribute decision making based on A-IFS. Xu (2007b) developed some similarity measures of A-IFSs. Szmidt and Kacprzyk (2004) applied a new measure of similarity to analyze the extent of agreement in a group of experts. Lin et al. (2007) proposed a new method for handling multi-criteria fuzzy decision-making problems based on A-IFS. Li et al. (2009) developed a new methodology to solve multi-attribute group decision-making problems, and both ratings of alternatives on attributes and weights of attributes expressed using A-IFS. Boran et al. (2009) proposed the technique for order preference by similarity to ideal solution (TOPSIS) method combined with A-IFS. Xu (2007c) developed new IF aggregation operators for aggregating IF information. Li (2005) proposed decision-making methods in which multi-attribute decision making using A-IFS was investigated, wherein attributes were explicitly considered, and several corresponding linear programming models were constructed to generate optimal weights of attributes. Chen

M.-C. Wu, T.-Y. Chen / Expert Systems with Applications 38 (2011) 12318–12327

and Tan (1994) proposed techniques for handling multi-criteria fuzzy decision-making problems based on vague set theory. Liu and Wang (2007) defined an evaluation function for a multi-criteria decision-making problem in an IF environment and defined the concept of IF point operators. The literature review demonstrates that most A-IFS are applied to decision-making problems with scoring or compromising models and few studies use the outranking model. The ELECTRE method is one of the most famous outranking models. 1.2. ELECTRE method The ELECTRE method was first introduced by Benayoun, Roy, and Sussman (1966), Roy (1968). Its first idea concerning concordance, discordance and outranking concepts originate from real-world applications. It also uses concordance and discordance indices to analyze outranking relations among alternatives. ELECTRE methods have been applied to problems in many areas (Figueira, Mousseau, & Roy, 2005), including energy (Beccali, Cellura, & Ardente, 1998; Cavallaro, 2010; Georgopoulou, Sarafidis, Mirasgedis, Zaimi, & Lalas, 2003; Karagiannidis & Perkoulidis, 2009), environment or water management (Ahrens & Kantelhardt, 2009; Hanandeh & El-Zein, 2010; Hokkanen & Salminen, 1997; Salminen, Hokkanen, & Lahdelma, 1998; Lahdelma, Salminen, & Hokkanen, 2002), finance (Doumpos & Zopounidis, 2001; Huck, 2009; Li & Sun, 2010, 2008; Martel, Khoury, & Bergeron, 1988), project selection (Blondeau, Sperandio, & Allard, 2002; Colson, 2000), decision analysis (Almeida, 2005; Arondel & Girardin, 2000; Hokkanen, Salminen, & Ettala, 1995; Karagiannidis & Moussiopoulos, 1997; Montazer, Saremi, & Ramezani, 2009; Mróz, 2008; Shanian & Savadogo, 2006; Srinivasa, Duckstein, & Arondel, 2000), and transportation (Roy, Présent, & Silhol, 1986). ELECTRE I is the first decision aid method using the concept of outranking relation; it should be applied only when all criteria have been coded in numerical scales with identical ranges. Almeida (2005) presented a decision model for a repair contract problem with four criteria, and the model combined the utility theory with the ELECTRE I method. ELECTRE II is a method that can rank alternatives from the best to the worst option. Hokkanen et al. (1995) applied the ELECTRE II method to the real choice process of a solid waste management system. ELECTRE III is a method that uses pseudo-criteria and fuzzy binary outranking relations. Norese (2006) used the ELECTRE III method to compare and rank all alternatives, with the aim of selecting the best waste-treatment plants. Karagiannidis and Moussiopoulos (1997) and Mróz (2008) also used the ELECTRE III method to support the decision-making process. ELECTRE IV is a method that is designed to rank alternatives without using relative criteria importance coefficients; it is the only ELECTRE method that does not use such coefficients. Shanian and Savadogo (2006) used the ELECTRE IV method for material selection of the bipolar plate in a polymer electrolyte fuel cell. The ELECTRE TRI method is designed to classify alternatives in various categories that are separated by reference alternatives. As the comparison is performed between alternatives and reference alternatives, this method permits the decision maker to deal with many alternatives. Srinivasa et al. (2000) used the ELECTRE TRI method as a screening procedure to reduce the large-size payoff matrix and to predefine categories for further usage of MCDM techniques for a sustainable water resources plan in Spain. Arondel and Girardin (2000) also used the ELECTRE TRI method to sort cropping systems and to assess the impact of agricultural practices on environmental components. As mentioned previously, the ELECTRE method is one of the most famous outranking models, and can be used to solve the MCDM problem. It is one of the best methods because of its simple logic, but few studies have ever used this method with A-IFS. The

12319

ELECTRE method is composed of a pair-wise comparison of alternatives based on evaluated information provided by the decision maker. This method is concerned with concordance, discordance, and outranking relationships. The decision maker uses concordance and discordance indices to analyze outranking relations among different alternatives and to choose the best alternative using the crisp data. At the same time, the decision-making data of ELECTRE methods include evaluation information, which are vague or fuzzy in nature; moreover, they are inadequate or insufficient to model real-life decision problems. The threshold values in the ELECTRE method, for example, are very sensitive for filtering alternatives, and different threshold values have different filtering results; that is one of the reasons why we propose the ELECTRE method with A-IFS. Furthermore, the decision maker can use A-IFS characteristics to evaluate different alternatives and use those data to classify different kinds of concordance and discordance sets to fit the real decision-making environment during the decision process. 1.3. Contribution In this paper, we develop a new method for solving MCDM problems using the ELECTRE method under A-IF environments. A-IFS characteristics are simultaneously concerned with the degree of membership, degree of non-membership, and intuitionistic index, and people can use A-IFS to describe uncertain situations in a decision-making problem. Decision makers utilize A-IFS data instead of single values in the evaluation process of the ELECTRE method, and they are given A-IFS data with different alternative criteria. In the proposed method, using the intuitionistic fuzzy ELECTRE (IF ELECTRE) method, we can classify different types of concordance and discordance sets using the concepts of score function, accuracy function, and intuitionistic index, and use concordance and discordance sets to construct concordance and discordance matrices, respectively. Moreover, decision makers can choose the best alternative using the concepts of positive and negative ideal points. The best alternative should simultaneously have the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. Decision makers are able to take into account purely evaluated information without complex conversion and can also use imperfect or insufficient knowledge of data to deal with decision-making problems using the IF ELECTRE method. The remainder of this paper is organized as follows. Section 2 introduces the concept of A-IFS, a Euclidean distance between two A-IFSs, and the decision environment based on A-IFSs data. Section 3 describes ELECTRE methods and identifies concordance and discordance sets using the score function, accuracy function, and hesitancy degree of the A-IF value, in addition to developing the IF ELECTRE method. Section 4 illustrates the procedures followed in the proposed method using two numerical examples with cardinal and ordinal data. A discussion is given in Section 5. 2. Decision environment based on A-IFSs data In this section, the definition and operations of A-IFSs, including the normalized Euclidean distance measure and construction of the IF decision matrix, are introduced. They will be used throughout the paper. 2.1. Preliminaries Let X = {x1, x2, . . . , xn} be a finite universal set. An A-IFS A in X is defined as an object of the following form:

A ¼ fhxj ; lA ðxj Þ; mA ðxj Þijxj 2 Xg;

ð1Þ

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where the functions xj 2 X ? lA(xj) 2 [0, 1] and xj 2 X ? vA(xj) 2 [0, 1] define the degrees of membership and non-membership, respectively, of the element xj 2 X to the set A # X, and for every xj 2 X, 0 6 lA(xj) + mA(xj) 6 1. We refer to

pA ðxj Þ ¼ 1  lA ðxj Þ  mA ðxj Þ

ð2Þ

as the intuitionistic index of the element xj in the set A. This is the degree of indeterminacy membership of the element xj to the set A. It is clear that for every xj 2 X, 0 6 pA(xj) 6 1. The operations of A-IFS (Atanassov, 1986, 1989, 1999; Szmidt & Kacprzyk, 2000) are defined as follows. For every A, B 2 A-IFS (X), (1) A  B iff "x 2 X, (lA(x) 6 lB(x) and (2) A = B iff A  B and B  A; (3) A ¼ fðx; v A ðxÞ; lA ðxÞÞg; (4)

vA(x) P vB(x));

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u X u1 n dðA; BÞ ¼ t ðl ðxj Þ  lB ðxj ÞÞ2 þ ðmA ðxj Þ  mB ðxj ÞÞ2 þ ðpA ðxj Þ  pB ðxj ÞÞ2 ; 2n j¼1 A

ð3Þ where d(A, B) is the normalized Euclidean distance between A and B; this is proposed to measure the difference between two of them.

2.2. Construction of the IF decision matrix An MCDM problem can be expressed in a decision matrix whose element indicates the evaluation or value of the ith alternative Ai with respect to the jth criterion xj. In the present paper, we extend the canonical matrix format to an IF decision matrix M; that is, decision makers are expected to assign the degrees of membership and non-membership that, according to their opinions, capture the degree of the alternative Ai that satisfies the criterion xj. They can provide evaluative information for alternatives of each criterion. Let X be the discussion universe containing the decision criteria in the MCDM problem setting. The set of all criteria is denoted as X = {x1, x2, . . . , xn}. An A-IFS Ai of the ith alternative on X is given   by Ai ¼ hxj ; X ij ijxj 2 X , where Xij = (lij, mij). Xij indicates the degrees of membership and non-membership of the i th alternative with respect to the jth criterion, lij and mijare the respective degrees of membership and non-membership of Xij, and where 0 6 lij + mij 6 1, i = 1, 2, . . . , m, j = 1, 2, . . . , n, pij = 1  lij  mij. The IF decision matrix, M, can be expressed as follows:

ð4Þ Because all criteria cannot be assumed to be of equal importance, a set of grades of importance, W, is given by the decision makers. An A-IFS W in X is defined as follows:

W ¼ fhxj ; wj ijxj 2 Xg;

ð5Þ Pn

where 0 6 wj 6 1 and j¼1 wj ¼ 1; wj is the degree of importance assigned to each criterion, i.e., the weights of the criteria. The decision maker must pay twice to collect the evaluated data, the degrees of membership and non-membership data, to construct the IF decision matrix as compared to IVFS data; however, it cannot be ensured that A-IFS data are usable; the sum of membership and non-membership degrees must be less than or equal to one. The A-IFS theory is mathematically equivalent to the interval valued fuzzy sets (IVFS) theory (Deschrijver & Kerre,

2003; Dubois, Gottwald, Hajek, Kacprzyk, & Prade, 2005; Montero, Gomez, & Bustince, 2007). The decision maker’s evaluation with IVFS data is easier than with A-IF data, due to the constraint of sum of membership and non-membership degrees. Let Int ([0, 1]) stand for the set of all closed subintervals of [0, 1]. IVFS Ai of the ith alternative on X is given by Ai = {hxj, Mijijxj 2 X}, where Mij: þ X ? Int([0, 1]), such that xj ! M ij ¼ ½M  ij ; M ij . Mij indicates the possible degree to which the alternative Ai satisfies the criterion xj. M  ij and M þ ij are the lower bound and the upper bound, respectively, of the interval Mij. The decision maker evaluates all alternatives in the closed þ  interval ½M  ij ; M ij , starting at the beginning. Let M ij ¼ lij and  þ Mþ ¼ 1  m ; therefore, ½M ; M  ¼ ð l ; 1  m Þ. An interval ij ij ij ij ij ij þ ½M ; M  can be mapped onto an A-IFS, ( l , 1  m ). We can use ij ij ij ij the concept, the mathematical equivalence of A-IFS and IVFS, to transform IVF data into A-IF data. Furthermore, large amounts of data are evaluated with IVFS data, and it is not easy for the decision maker to evaluate all alternatives on different criteria based on their knowledge and experience. Decision makers can provide ranking, incomplete, or missing information, and those data can be transformed into A-IF data. The method calculates the number of alternatives that are surely worse than and surely better than a particular alternative. It admits incomplete ordinal data, as not all alternatives can be ranked with respect to a criterion. Considering the situation with missing information or non-comparable outcomes, we define two functions, aij and bij for each Ai with respect to xj. Let aij denote the number of alternatives A1, A2, . . . , Ai1, Ai+1, Ai+2, . . . , Am that are surely worse than Ai, while bij denotes the number of alternatives A1, A2, . . . , Ai1, Ai+1, Ai+2, . . . , Am that are surely better than Ai. The degrees of membership and non-membership, respectively, are given as follows:

lij ¼ mij

aij

; m1 bij : ¼ m1

ð6Þ ð7Þ

3. ELECTRE methods based on A-IFS In this section, concordance and discordance sets and the IF ELECTRE method (including the algorithm) are introduced. We will use the IF ELECTRE method algorithm to demonstrate numerical examples. ELECTRE methods are modeled using binary outranking relations; the relationship is built by the decision maker and does not need to be transitive. The relationship allows partial ordering of non-dominant alternatives. For each pair of alternatives k and l (k, l = 1, 2, . . . , m and k – l), each criterion in the different alternatives can be divided into two distinct subsets. The concordance set Ekl of Ak and Al is composed of all criteria for which Ak is preferred to Al. In other words, Ekl = {jjxkj P xlj}, where J = {jjj = 1, 2, . . . , n}. The complementary subset, which is the discordance set, is F kl ¼ fjjxkj < xlj g. In the proposed IF ELECTRE method, we can classify different types of concordance and discordance sets using the concepts of score function, accuracy function, and intuitionistic index, and use concordance and discordance sets to construct concordance and discordance matrices, respectively. The decision makers can choose the best alternative using the concepts of positive and negative ideal points. 3.1. Concordance and discordance sets We can compare different alternatives to their IF values using the concepts of score function, accuracy function, and hesitancy degree of the IF value. The better alternative has the higher score degree or higher accuracy degree in cases where alternatives have

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the same score degree. A higher score degree refers to a larger membership degree or smaller non-membership degree, and a higher accuracy degree refers to a smaller hesitation degree. We classify different types of concordance sets as the concordance set, midrange concordance set, and weak concordance set with the concepts of score function and accuracy function. The different types of discordance sets can also be classified as the discordance set, midrange discordance set, and weak discordance set. Chen and Tan (1994) proposed the score function to evaluate the degree of suitability to which an alternative satisfies a decision maker’s requirement. Let Xij = (lij, mij) be an IF value, where lij 2 [0, 1], mij 2 [0, 1], lij + mij 6 1. The score of Xij can be evaluated by the score function S, S(Xij) = lij  mij, where S(Xij) 2 [1, 1]. A larger score S(Xij) correlates with a larger IF value Xij; however, we cannot evaluate alternatives using only the score function when they have the same score. Hong and Choi (2000) proposed the accuracy function to evaluate the degree of accuracy of vague values. The degree of accuracy of Xij can be evaluated by the accuracy function H, H(Xij) = lij + mij, where Xij = (lij, mij) is an IF value. A larger value of H(Xij) correlates with a larger degree of accuracy of the IF value membership grade. From (2) and the accuracy function, we know that a higher accuracy degree H(Xij) correlates with a lower hesitancy degree p(Xij). As mentioned earlier, Xij = (lij, mij). The concordance set Ckl of Ak and Al is composed of all criteria for which Ak is preferred toAl. We use the concepts of score function, accuracy function, and hesitancy degree of the IF value to classify concordance sets. The concordance set Ckl can be formulated as follows:

C kl ¼ fjjlkj P llj ; mkj < mlj and

pkj < plj g;

ð8Þ

where J = {jjj = 1, 2, . . . , n}, a larger score refers to a larger IF value, a higher accuracy degree refers to a lower hesitancy degree, and Eq. (8) is more concordant than (9) or (10). The midrange concordance set C 0kl is defined as

C 0kl ¼ fjjlkj P llj ; mkj < mlj and

pkj P plj g:

ð9Þ

The major difference between (8) and (9) is the hesitancy degree; the hesitancy degree at the kth alternative with respect to the jth criterion is higher than the lth alternative with respect to the jth criterion in the midrange concordance set. Thus, Eq. (8) is more concordant than (9). The weak concordance set C 00kl is defined as

C 00kl ¼ fjjlkj P llj and

mkj P mlj g:

ð10Þ

The degree of non-membership at the kth alternative with respect to the jth criterion is higher than the lth alternative with respect to the jth criterion in the weak concordance set; thus, Eq. (9) is more concordant than (10). The discordance set is composed of all criteria for which Ak is not preferred to Al. The discordance set Dkl using the aforementioned concepts can be formulated as follows:

Dkl ¼ fjjlkj < llj ; mkj P mlj and

pkj P plj g:

ð11Þ

The formula also uses the same concepts, that a larger score refers to a larger IF value, and a higher accuracy degree refers to a lower hesitancy degree. The midrange discordance set D0 kl is defined as follows:

D0kl ¼ fjjlkj < llj ; mkj P mlj and

pkj < plj g:

ð12Þ

Eq. (11) is more discordant than (12). The weak discordance set D00kl is defined as follows:

D00kl ¼ fjjlkj < llj and

mkj < mlj g:

ð13Þ

The degrees of both membership and non-membership at the k th alternative with respect to the jth criterion are lower than the lth

alternative with respect to the j th criterion in the weak discordance set; thus, Eq. (12) is more discordant than (13). We use the concept of concordance and discordance sets to calculate concordance and discordance matrices and use the proposed IF ELECTRE method to determine the aggregate dominance matrix. We then choose the best alternative. 3.2. IF ELECTRE method The IF ELECTRE method is an integrated A-IFS and ELECTRE method with evaluation information. The relative value of the concordance set of the IF ELECTRE method is measured through the concordance index. The concordance index is equal to the sum of the weights associated with those criteria and relations that are contained in the concordance sets. Therefore, the concordance index ckl between Ak and Al in this paper is defined as:

X

g kl ¼ wC 

wj þ wC 0 

X

wj þ wC00 

j2C 0kl

j2C kl

X

wj ;

ð14Þ

j2C 00kl

where wC ; wC 0 , and wC00 are the weights of the concordance, midrange concordance, and weak concordance sets, respectively, and wj is the weight of the criteria that are also defined in (5). The concordance index reflects the relative dominance of a certain alternative over a competing alternative, based on the relative weight attached to the successive decision criteria. The concordance matrix G is defined as follows:

2

—

6 g 6 21 6 G¼6 6 ... 6g 4 ðm1Þ1 g m1

g 12

...

...

—

g 23

...

...

—

...

...

...

—

g m2

...

g mðm1Þ

g 1m

3

g 2m 7 7 7 ... 7 7: g ðm1Þm 7 5 —

ð15Þ

where the maximum value of gkl is denoted by g⁄, which is the positive ideal point, and a higher value of gkl indicates that Ak is preferred to Al. Evaluations of a certain Ak are worse than evaluations of a competing Al. In this paper, the discordance index is defined as follows:

hkl ¼

maxj2Dkl wD  dðX kj ; X lj Þ ; maxj2J dðX kj ; X lj Þ

ð16Þ

where d(Xkj, Xlj) is defined in (3), and wD is equal to wD, wD0 , or wD00 depending on the different types of discordance sets. These sets include the weight of discordance, midrange discordance, and weak discordance sets, respectively. The discordance matrix H is defined as follows:

2

—

6 h 6 21 6 H¼6 6 ... 6 4 hðm1Þ1 hm1

h12

...

...

—

h23

...

...

—

...

...

...

—

hm2

...

hmðm1Þ

h1m

3

h2m 7 7 7 ... 7 7: 7 hðm1Þm 5 —

ð17Þ

where the maximum value of hkl is denoted by h⁄, which is the negative ideal point, and a higher value of hkl indicates that Ak is less favorable than Al. The concordance dominance matrix calculation process is based on the concept that the chosen alternative should have the shortest distance from the positive ideal solution; thus, the concordance dominance matrix K is defined as follows:

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2

—

6 k 6 21 6 K¼6 6 ... 6 4 kðm1Þ1 km1

k1m

k12

...

...

—

k23

...

...

—

...

...

...

—

km2

...

kmðm1Þ

3

k2m 7 7 7 ... 7 7; 7 kðm1Þm 5 —

can be calculated to determine the ranking order of all alternatives. The aggregate dominance matrix R is defined as follows:

ð18Þ

where

kkl ¼ g   g kl ;

—

6 l 6 21 6 L¼6 6 ... 6 4 lðm1Þ1 lm1

— 6 r 6 21 6 R¼6 6 ... 6 4 r ðm1Þ1 r m1

ð19Þ

which refers to the separation of each alternative from the positive ideal solution. A higher value of kkl indicates that Ak is less favorable than Al. The discordance dominance matrix calculation process is based on the concept that the chosen alternative should have the farthest distance from the negative ideal solution; thus, the discordance dominance matrix L is defined as follows:

2

2

l12

...

...

—

l23

...

...

—

...

...

...

—

lm2

. . . lmðm1Þ

l1m

3

l2m 7 7 7 ... 7 7; 7 lðm1Þm 5 —

ð20Þ

where 

lkl ¼ h  hkl ;

ð21Þ

which refers to the separation of each alternative from the negative ideal solution. A higher value of lkl indicates that Ak is preferred to Al. In the aggregate dominance matrix determine process, the distance of each alternative to both positive and negative ideal points

r 12 —

... r 23

... ...

...

—

...

...

...

—

r m2

...

r mðm1Þ

rkl ¼

lkl ; kkl þ lkl

ð23Þ

kkl and lkl are defined in (19) and (21), respectively, and rkl refers to the relative closeness to the ideal solution, with a range from 0 to 1. A higher value of rkl indicates that the alternative Ak is simultaneously closer to the positive ideal point and farther from the negative ideal point than the alternative Al; thus, it is a better alternative. In selecting the best alternative process,

Tk ¼

m X 1 r kl ; m  1 l¼1;l–k

k ¼ 1; 2; . . . ; m;

A ¼ maxfT k g;

ð25Þ

⁄

where A is the best alternative. The entire IF ELECTRE method algorithm will be defined below.

Identify the potential alternatives Receive a set of grades of importance for decision criteria Construct the decision matrix cardinal data

ordinal data

Aggregation stage Identify the concordance and discordance sets

Choose the best alternative

ð24Þ

and T k is the final value of evaluation. All alternatives can be ranked according to T k . The best alternative A⁄, which is simultaneously the shortest distance to the positive ideal point and the farthest distance from the negative ideal point, can be generated and defined as follows:

Choose the relevant criteria

Selection stage

ð22Þ

where

Evaluation stage

Provide decision maker’s opinions

3 r 1m r 2m 7 7 7 ... 7 7; 7 r ðm1Þm 5 —

Calculate the concordance (discordance) matrix Construct the concordance (discordance) dominance matrix Determine the aggregate dominance matrix

Fig. 1. Intuitionistic fuzzy ELECTRE method solution procedure.

M.-C. Wu, T.-Y. Chen / Expert Systems with Applications 38 (2011) 12318–12327

3.3. Algorithm This section describes a new MCDM approach, the IF ELECTRE method, for decision making by integrating the A-IFS and the ELECTRE methods with evaluation information. The algorithm for the proposed approach will be developed in the following three major stages: (1) evaluation stage, (2) aggregation stage, and (3) selection stage. Fig. 1 illustrates a conceptual model of the proposed method. In the evaluation stage, the decision makers choose the relevant criteria, identify alternatives (given the weights of different criteria), and determine the decision matrix by the evaluated information with A-IFS. In the aggregation stage, we compare each alternative to the others to confirm the dominance relationship and use the proposed method to construct concordance and discordance dominance matrices. We then determine the aggregate dominance matrix. In the selection stage, we use the IF ELECTRE method to choose the best alternative and to rank all alternatives. An algorithm and decision process of the IF ELECTRE method can be summarized in the following eight steps. Step 1. Construct the decision matrix with evaluative information: the decision makers give data either as IF values or as compared information between different alternatives. This step is divided into the following three steps. Step 1-1. Choose the relevant criteria and non-inferior alternatives: these criteria can be chosen according to the problem, i.e., different MCDM problems should have their own criteria, and most of them can be divided into two groups: subjective and objective criteria. The decision makers identify potential alternatives. Step 1-2. Receive a set of grades of importance for decision criteria. Criteria weights are defined in P (5) and nj¼1 wj ¼ 1. Step 1-3. Construct the decision matrix: the decision makers construct the IF decision matrix M with cardinal information using (4). If the decision makers give comparison data, then we can use (6) and (7) to transform them. Step 2. Identify the concordance and discordance sets: we utilize the concepts of score function, accuracy function, and hesitancy degree of the IF value to distinguish the different kinds of concordance and discordance sets. We can find C kl ; C 0kl ; C 00kl ; Dkl ; D0kl , and D00kl for pair-wise comparisons of alternatives using (8)–(13). Step 3. Calculate the concordance matrix G: the concordance matrix index is the operational result of different kinds of concordance sets and their weights, defined in (14) and (15). Step 4. Calculate the discordance matrix H: the discordance matrix index is the operational result of different kinds of discordance sets and their weights, defined in (16) and (17). Step 5. Construct the concordance dominance matrix K: the concordance dominance matrix index is the difference between the maximum index of the concordance matrix and its own index, defined in (18) and (19). Step 6. Construct the discordance dominance matrix L: the discordance dominance matrix index is the difference between the maximum index of the discordance matrix and its own index, defined in (20) and (21). Step 7. Determine the aggregate dominance matrix R: the aggregate dominance matrix is constructed from the indices of concordance and discordance dominance matrices, defined in (22) and (23).

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Step 8. Choose the best alternative: calculate the final value of evaluation using (24) and (25). The ranking order of all alternatives is generated, and the alternative with the maximum value is the best alternative. 4. Numerical example In this section, we present two practical examples that are connected to a decision-making problem in human resource management. One problem uses cardinal data, whereas the other uses ordinal data. 4.1. Illustration of cardinal evaluations We know that human resource management is very important during the recruiting and hiring stages of employment. Suppose the committee of a company intends to choose a project manager from a group of candidates. Project management is the application of knowledge, skills, tools, and techniques to the implementation of project activities for the purpose of meeting project requirements. The requirements of a project manager are not only morale, but also proficiency in project management. Suppose that four criteria, x1 (self-confidence), x2 (personality), x3 (past experience), and x4 (proficiency in project management), are taken into consideration in the selection problem. Suppose that there exist six candidates, named A1, A2, A3, A4, A5, and A6 (note that Step 1-1 has been completed). The subjective importance of the criteria, W, is given by the decision makers; W = [w1, w2, w3, w4] = [0.1, 0.2, 0.3, 0.4] (note that Step 1-2 has been completed). The decision makers also give the relative weights as follows:

  2 1 2 1 W 0 ¼ ½wC ; wC 0 ; wC 00 ; wD ; wD0 ; wD00  ¼ 1; ; ; 1; ; : 3 3 3 3 The IVF decision matrix decision M is given and transformed into the IF matrix decision with cardinal information (Step 1-3).

Applying Step 2, identify the concordance and discordance sets using the result of Step 1-3. The concordance set, applying (8), is:

3 — 1 — 1 — 1 64 — 4 1 — — 7 7 6 64 — — — — — 7 7 6 C kl ¼ 6 7: 6 2 3 3 — 3 2; 3 7 7 6 4— 2 2 — — — 5 4 — 3 1 — — 2

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For example, C21 = {4}, which is in the 2nd (horizontal) row and the 1st (vertical) column of the concordance set, is ‘‘4.’’ C13 = {–}, which is in the 1st row and 3rd column of the concordance set, is ‘‘empty,’’ and so forth. The midrange concordance set, applying (9), is:

2

—

—

2

6 6 — — 6 6 6 6 3 — 6 0 C kl ¼ 6 6 3; 4 2 6 6 6 6 3; 4 — 4 3

4

— — —

3

— 2 —

0:6667

3

— 3 1; 3 — — — 7 6 6 2 — 1 4 4 2 7 7 6 7 6 7 6 — 3 — 4 4 — 7 6 00 7: C kl ¼ 6 7 6 6 — — 1 — — — 7 7 6 7 6 6 1; 2 1; 3 1; 3 1; 4 — 1; 2; 3 7 5 4 4

4

—

The discordance set, applying (11), is:

2

— 4 6 61 — 6 6 6 — 2; 4 6 Dkl ¼ 6 6 61 1 6 6 6— — 4 1 —

4 — — — — —

2

—

4

1 —

— — 1; 2; 3 —

3 — 0:2 0:2667 0:1 0 0:1 6 0:4667 — 0:5667 0:2333 0:1333 0:0667 7 7 6 7 6 6 0:6 0:1 — 0:1333 0:1333 0 7 7: G¼6 6 0:6667 0:4333 0:4667 — 0:4333 0:5 7 7 6 7 6 4 0:5667 0:3333 0:3333 0:1667 — 0:2 5

2

1

1; 3

3 2 —7 7 7 17 7: 47 7 7 45

2

2; 4 — — —

1; 3

2

1; 2 1; 3

Applying Step 3, the concordance matrix is calculated.

The weak concordance set, applying (10), is:

2

— 2 — — 63 — 3 — 6 6 61 1 — 1 D00kl ¼ 6 6— 4 4 — 6 6 4— 4 4 — —

7 — — —7 7 7 7 — — —7 7 7: — 2 —7 7 7 7 — — —7 5

2

2

3

7 3 2 1; 3 7 7 7 3 2 3 7 7 7: 7 — 4 1 7 7 7 3 — — 7 5 2; 3 — —

The midrange discordance set, applying (12), is:

0:4

0:7333 0:2333 0:1333

—

For example,

g 23 ¼ wC  w4 þ wC 0  w2 þ wC00  w1 ¼ 1  0:4 þ

2 1  0:2 þ  0:1 ¼ 0:5667: 3 3

Applying Step 4, the discordance matrix is calculated.

2

—

1

0:5123 0:6667 0:6667

1

3

6 0:3369 — 0:0539 0:6578 0:3333 0:8542 7 7 6 7 6 6 0:3333 0:3333 — 0:6070 0:3333 0:3748 7 7: H¼6 6 0:4171 0:3333 0:3333 — 0:5290 0:3333 7 7 6 7 6 4 0 0:2976 0:1425 0:4693 — 0:1888 5 0:2558 0:1976

0

0:4873 0:3333

—

For example:

h12 ¼

maxj2D12 wD  dðX 1j ; X 2j Þ 0:2536 ¼ 1; ¼ 0:2536 maxj2J dðX 1j ; X 2j Þ

where dðX 11 ; X 21 Þ ¼ dðX 14 ; X 24 Þ ¼

 

1 ðð0:33  0:24Þ2 þ ð0:33  0:34Þ2 þ ð0:34  0:42Þ2 Þ 2 1 ðð0:15  0:44Þ2 þ ð0:57  0:39Þ2 þ ð0:28  0:17Þ2 Þ 2

12 12

¼ 0:0854; ¼ 0:2536;

and

wD  dðX 14 ; X 24 Þ  12 1 ðð0:15  0:44Þ2 þ ð0:57  0:39Þ2 þ ð0:28  0:17Þ2 Þ ¼ 0:2536; ¼1 2 wD0  dðX 12 ; X 22 Þ  12 1 1 ¼  ðð0:22  0:26Þ2 þ ð0:34  0:40Þ2 þ ð0:44  0:33Þ2 Þ ¼ 0:0291: 3 2

2

— — — 3; 4 3; 4

6— 6 6 62 D0kl ¼ 6 6— 6 6 4—

— —

2

—

— —

2

—

— —

—

—

— —

2

—

— — —

—

—

3

3

4 7 7 7 2; 4 7 7: — 7 7 7 — 5 —

The weak discordance set, applying (13), is:

Applying Step 5, the concordance dominance matrix is constructed. The concordance dominance matrix is as follows:

3 — 0:5333 0:4666 0:6333 0:7333 0:6333 6 0:2666 — 0:1666 0:5 0:6 0:6666 7 7 6 7 6 6 0:1333 0:6333 — 0:6 0:6 0:7333 7 7: 6 K¼6 0:3 0:2666 — 0:3 0:2333 7 7 6 0:0666 7 6 4 0:1666 0:4 0:4 0:5666 — 0:5333 5 2

0:0666 0:3333

0

0:5

0:6

—

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A1

A2

A1

A3

A6

A5

A2

A3

A6

A4

A5

A4

(1) criterion x1

(2) criterion x2

A1

A1

A2

A3

A6

A5

A4

A2

A3

A6

A5

A4

(4) criterion x4

(3) criterion x3

Fig. 2. Graphical representation of different criteria for the ordinal data example.

Applying Step 6, the discordance dominance matrix is constructed. The discordance dominance matrix is as follows:

2

— 6 0:6631 6 6 0:6667 6 L¼6 6 0:5829 6 4 1 0:7442

0 — 0:6667 0:6667 0:7024 0:8024

0:4877 0:9461 — 0:6667 0:8575 1

0:3333 0:3422 0:393 — 0:5307 0:5127

0:3333 0:6667 0:6667 0:4710 — 0:6667

3 0 0:1458 7 7 0:6252 7 7 7: 0:6667 7 7 0:8112 5 —

Applying Step 7, the aggregate dominance matrix is determined.

2

— 0 0:5111 0:3448 6 0:7132 — 0:8503 0:4063 6 6 6 0:8334 0:5128 — 0:3958 R¼6 6 0:8975 0:6897 0:7143 — 6 6 4 0:8572 0:6372 0:6819 0:4836 0:9179 0:7065

1

3 0:3125 0 0:5263 0:1795 7 7 7 0:5263 0:4602 7 7: 0:6109 0:7408 7 7 7 — 0:6033 5

0:5063 0:5263

—

Applying Step 8, the best alternative is chosen.

T 1 ¼ 0:2337;

T 2 ¼ 0:5351;

T 3 ¼ 0:5457;

T 4 ¼ 0:7306;

T 5 ¼ 0:6526;

T 6 ¼ 0:7314:

The optimal ranking order of alternatives is given by A6  A4  A5  A3  A2  A1. The best alternative is A6. 4.2. Illustration of ordinal evaluations This is the case where the decision maker only gives information about the same human resources management problem with ordinal data. The following outranking relationship is shown. Take criterion x1, for example, since alternative A1 is better than A2 with respect to x1, denoted as A1 > A2. In the same way, A1 > A4, A1 > A6, A2 > A4, A6 > A2, and A6 > A4 for criterion x1. A1 > A3, A2 > A3,

A4 > A1, A4 > A2, A4 > A3, A4 > A5, A4 > A6, A5 > A2, A5 > A3, and A6 > A3 for criterion x2. A3 > A1, A4 > A1, A4 > A2, A4 > A3, A4 > A5, A4 > A6, A5 > A1, A6 > A1, A6 > A2, and A6 > A3 for criterion x3. A2 > A1, A2 > A3, A3 > A1, A4 > A1, A5 > A1, A6 > A1, A6 > A2, and A6 > A3 for criterion x4. The over-ranking relationships between different alternatives are illustrated by the graph in Fig. 2. Each node is represented by circled letters. Each node corresponds to the non-dominant alternative in Fig. 2. In the graphical representation, the arrows emanating from the nodes are called directed paths and correspond to the outranking relation; they are analogous to a preference relationship. Take, for example, criterion x1. Alternative A1 is better than A2 with respect to x1 and is denoted as A1 > A2. In the same way, A1 > A4 and A1 > A6, but no alternative is better than A1 with respect to x1. We applied (6) and (7) to get l11 = a11 /(m  1) = 3/(6  1) = 0.6 and m11 = 0/(6  1) = 0. Since alternative A6 is better than A2 with respect to x1, A6 > A4, and A1 > A6, we get l61 = a61/(m  1) = 2/(6  1) = 0.4 and m61 = 1/ (6  1) = 0.2. In the meantime, criterion x1 has tied criterion-wise rankings between A1 and A3, A1 and A5, A2 and A3, A2 and A5, A3 and A4, A3 and A5, A3 and A6, A4 and A5, and A5 and A6. This method can also be applied to a situation with missing information; for criterion x4, for example, the decision maker cannot obtain complete information about A3 and A4, A4 and A5, and so on. We can use the outranking relationship and the same equations to construct the new IF decision matrix M0 :

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We can also use the IF ELECTRE method with ranking data to select the best alternative. In this case, we demonstrate Step 8, and choose the best alternative.

T 1 ¼ 0:3038;

T 2 ¼ 0:3608;

T 3 ¼ 0:2790;

T 4 ¼ 0:8490;

T 5 ¼ 0:4687;

T 6 ¼ 0:6579:

The optimal ranking order of alternatives is given by A4  A6  A5  A2  A1  A3. The best alternative is A4. 5. Discussion In this study, we provide a new method, the IF ELECTRE method, for solving MCDM problems with A-IFS information. This method can be also applied to situations with missing information or imperfect data. Decision makers utilize A-IFS data instead of single values in the evaluation process of the ELECTRE method and use those data to classify different kinds of concordance and discordance sets to fit a real decision environment. This new approach integrates the concept of the outranking relationship of the ELECTRE method. In the IF ELECTRE method, we can classify different types of concordance and discordance sets using the concepts of score function, accuracy function, and intuitionistic index, and use concordance and discordance sets to construct concordance and discordance matrices. Furthermore, decision makers can choose the best alternative using the concepts of positive and negative ideal points. We used the IF ELECTRE method to rank all alternatives and determine the best alternative. We illustrated two examples to demonstrate the process. Meanwhile, for the distance measured in this paper, we used the definition of a normalized Euclidean distance. We can replace different kinds of distances, such as the Hamming distance and normalized Hamming distance, in the process of discordance index calculation and then test the result of the alternative selection. This paper is the first step in using the IF ELECTRE method to solve MCDM problems. In a future study, we will apply the IF ELECTRE method to predict consumer behavior using a questionnaire in an empirical study about market research with different kinds of products. Furthermore, we will simulate 1,000 data sets to test the IF ELECTRE method for its ranking regularity. We will also develop different kinds of ELECTRE methods with IVIF information, interval valued intuitionistic fuzzy sets, to solve MCDM problems. References Ahrens, H., & Kantelhardt, J. (2009). Accounting for farmers’ production responses to environmental restrictions within landscape planning. Land Use Policy, 26(4), 925–934. Almeida, A. T. (2005). Multicriteria modelling of repair contract based on utility and ELECTRE I method with dependability and service quality criteria. Annals of Operations Research, 138(1), 113–126. Arondel, C., & Girardin, P. (2000). Sorting cropping systems on the basis of their impact on groundwater quality. European Journal of Operational Research, 127(3), 467–482. Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy sets and Systems, 20(1), 87–96. Atanassov, K. T. (1989). More on intuitionistic fuzzy sets. Fuzzy sets and Systems, 33(1), 37–45. Atanassov, K. T. (1999). Intuitionistic fuzzy sets: Theory and applications. New York: Physica-Verlag. Atanassov, K., & Georgiev, C. (1993). Intuitionistic fuzzy prolog. Fuzzy Sets and Systems, 53(2), 121–128. Atanassov, K., Pasi, G., & Yager, R. (2005). Intuitionistic fuzzy interpretations of multi-criteria multi-person and multi-measurement tool decision making. International Journal of Systems Science, 36(14), 859–868. Beccali, M., Cellura, M., & Ardente, D. (1998). Decision making in energy planning: The ELECTRE multicriteria analysis approach compared to a fuzzy-sets methodology. Energy Conversion and Management, 39(16–18), 1869–1881. Benayoun, R., Roy, B., & Sussman, B. (1966). ELECTRE: Une méthode pour guider le choix en présence de points de vue multiples. Note de travail 49, SEMA-METRA international, direction scientifique.

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