Method for aggregating triangular fuzzy intuitionistic fuzzy information and its application to decision making/Numanomu neapibreztuju aibiu teorija ir jos taikymas priimant sprendimus.

Zhang, Xin ; Liu, Peide

1. Introduction

Decision-making is the process of finding the best option from all of the feasible alternatives. Sometimes, decision-making problems considering several criteria are called multi-criteria decision-making (MCDM) problems. The MCDM problems may be divided into two kinds of problem. One is the classical MCDM problems (Hwang and Yoon 1981; Kaklauskas et al. 2006; Zavadskas et al. 2008a, 2008b; Lin et al. 2008; Ginevicius et al. 2008), among which the ratings and the weights of criteria are measured in crisp numbers. Another is the fuzzy multiple criteria decision-making (FMCDM) problems (Bellman and Zadeh 1970; Liu and Wang 2007; Liu and Guan 2008; Liu 2009a, 2009b), among which the ratings and the weights of criteria evaluated on imprecision, subjective and vagueness are usually expressed by linguistic terms, fuzzy numbers or intuition fuzzy numbers. In 1965, the fuzzy set theory was proposed by Prof. Zadeh (Zadeh 1965), it was a very good tool to research the FMCDM problems. Gau and Buehrer (1993) proposed the concept of vague set. On the foundation of it, Chen (1994) and Hong and Choi (2000) researched the fuzzy multi-criteria decision-making problem based on the Vague sets. Bustine and Burillo (1996) points out that Vague set is the intuitionistic fuzzy set. Because the intuitionistic considered the membership degree, the non-membership degree and the hesitancy degree synchronously, it is more flexible and practical than the traditional fuzzy set at the aspect of dealing with the vagueness and uncertain. Atanassov et al. (1986) extended the intuitionistic fuzzy set and proposed the concept of the interval intuitionistic fuzzy set. Atanassov (1989) defined a lot of basic operation rules of the interval intuitionistic fuzzy set and they thought that the fuzzy set is the special condition of the intuitionistic fuzzy set. Bustince and Burillo (1995) researched the correlation degree and decomposition theorem of the interval intuitionistic fuzzy set. Hung, Wu (2002) calculated the correlation coefficient using centroid method. Deschrijver and Kerre (2003) researched the relation of interval intuitionistic fuzzy set, L-fuzzy set, intuitionistic fuzzy and interval fuzzy set. Xu (2007) proposed the methods for aggregating interval-valued intuitionistic fuzzy information. Most related literatures are all the researches of interval intuitionistic fuzzy set and concentrates on the basic theory. Lots of problems about the intuitionistic fuzzy are worth to research.

The triangular fuzzy number is easier to show the fuzzy problem than the interval number. So, using the triangular fuzzy number in the intuitionistic fuzzy set is easier to deal with the fuzzy and the uncertain information. The application of the triangular fuzzy number in the area of the intuitionistic fuzzy set is rare. This paper discusses the intuitionistic fuzzy set based on the triangular fuzzy number and proposed the triangular intuitionistic fuzzy number. And two integration operations of the triangular intuitionistic fuzzy number are given. The score function and accuracy function of the triangular intuitionistic fuzzy number are also proposed. Then this paper represents a simple priority method. At last, this theory is used to decision making area.

2. The intuitionistic fuzzy set

Suppose X is a nonempty set, A = {< x, [u.sub.A] (x), vA (x) > x [member of] X} is a intuitionistic fuzzy set (Atanassov 1986), where, [u.sub.A] (x) is the membership degree of x belongs to Xand [v.sub.A] (x) is the non-membership degree of x belongs to X, [u.sub.A] : X [right arrow][0,1], [v.sub.A] : X [right arrow]--[0,1] and 0 [less than or equal to] [u.sub.A] (x) + [v.sub.A] (x) [less than or equal to] 1, [for all] x [member of] X. In addition, 1 - [u.sub.A] (x) - [v.sub.A] (x) denotes the hesitancy degree of x belongs to X.

If A = {< x, [u.sub.A] (x), [v.sub.A] (x) > x [member of] X} and B = {< x, [u.sub.B] (x), [v.sub.B] (x) > x [member of] X} are intuitionistic fuzzy numbers, the operation rules of the intuitionistic fuzzy number are as follows (Atanassov 1989, 1994):

A + B = {< x,[u.sub.A] (x) + [u.sub.B] (x) - [u.sub.A] (x) [u.sub.B] (x), [v.sub.A] (x) [v.sub.B] (x) > x [member of] X}; (1)

A x B = {< x, [u.sub.A] (x) [u.sub.B] (x), [v.sub.A] (x) + [v.sub.B] (x) - [v.sub.A] (x) [v.sub.B] (x) > x [member of] X}; (2)

nA = {< x,1 -[1 - [[u.sub.A] (x)].sup.n],[[[v.sub.A](x)].sup.n] > x [member of] X}; (3)

[A.sup.n[ = {< x,[[[u.sub.A] (x)].sup.n], 1-[[1 - [v.sub.A](x)].sup.n] > x [member of] X,n > 0}. (4)

The intuitionistic fuzzy set is flexible and practical to deal with the fuzzy and uncertain information. But things are usually complex and uncertain, so it is hard to express the membership degree and the non-membership degree using the exact real number value. Using the triangular fuzzy number to show them is a feasible method.

3. The triangular intuitionistic fuzzy number

This paper extends the intuitionistic fuzzy set and uses the triangular fuzzy number to express the membership degree [[bar.u].sub.A] (x) and the non-membership degree [[bar.v].sub.A] (x). So an intuitionistic fuzzy number is got based on the triangular fuzzy number. And it is called triangular intuitionistic fuzzy number. The general form of the triangular intuitionistic fuzzy number is marked as ([[a.sup.L], [a.sup.M], [a.sup.U]],[[b.sup.L], [b.sup.M], [b.sup.U]]), [[a.sup.L], [a.sup.M], [a.sup.U]] and [[b.sup.L], [b.sup.M], [b.sup.U]]] are triangular fuzzy numbers.

Suppose [I.sub.1] = ([[a.sub.1.sup.L], [a.sub.1.sup.M], [a.sub.1.sup.U], [[b.sub.1.sup.L], [b.sub.1.sup.M], [b.sub.1.sup.U]) and [I.sub.2] = ([[a.sub.2.sup.L], [a.sub.2.sup.M], [a.sub.2.sup.U]],[[b.sub.2.sup.L], [b.sub.2.sup.M], [b.sub.2.sup.U]) are two triangular intuitionistic fuzzy numbers, according to formulas (1)-(4) and operation rules of triangular fuzzy numbers, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

It is easy to see that the calculation results in definition 1 are triangular intuitionistic fuzzy numbers. According to the Eq.(5)-(8), the following calculation rules can be gained:

[I.sub.1] + [I.sub.2] = [I.sub.2] + [I.sub.1]; (9)

[I.sub.1 x [I.sub.2] = [I.sub.2] x [I.sub.1]; (10)

[lambda]([I.sub.1] + [I.sub.2]) = [lambda] [I.sub.1] + [I.sub.2], X > [lambda] [greater than or equal to] 0; (11)

[lambda][I.sub.1] + [[lambda].sub.2][I.sub.2]) = [[lambda].sub.1] + [[lambda].sub.2] [I.sub.1] + [[lambda].sub.1], [[lambda].sub.2] [greater than or equal to] 0; (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

4. Aggregation operators of the triangular intuitionistic fuzzy number and its score function and accuracy function

4.1. Aggregation operators of the triangular intuitionistic fuzzy number

On the foundation of the above calculation rules, the weighted arithmetic average operator of the triangular intuitionistic fuzzy numbers is given.

Definition 1: suppose [I.sub.i](i = 1, 2, ..., n) is a set of triangular intuitionistic fuzzy numbers, and suppose f: [[OMEGA].sup.n] [right arrow] [OMEGA], if

[f.sub.[omega]]([I.sub.1], [I.sub.2], ..., [I.sub.n]) = [n.summation over (i=1)] [[omega].sub.i][I.sub.i. (14)

Where, [OMEGA] is the set of all the triangular intuitionistic fuzzy numbers, [omega] = [([[omega].sub.1], [[omega].sub.2], ..., [[omega].sub.n]).sup.T] is the weight vector of Ii(i = 1, 2, ..., n), [[omega].sub.i] [member of] [0,1], [n.summation over (i=1)] [[omega].sub.i] = 1. Then thefis called the weighted arithmetic average operator of the triangular intuitionistic fuzzy number. Specially, if [omega] = [(1 / n, 1 / n, ..., 1/n).sup.T], thef is called the arithmetic averaging operator of the triangular intui tionistic fuzzy number.

Definition 2: suppose Ii(i = 1, 2 , ..., n) is a set of triangular intuitionistic fuzzy numbers, and suppose g : [[omega].sup.n] [right arrow] [omega], if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

Where, [OMEGA] is the set of all the triangular intuitionistic fuzzy numbers, [omega] = [([[omega].sub.1], [[omega].sub.2], ..., [[omega].sub.n]).sup.T] is the weight vector of [I.sub.i](i = 1, 2, ..., n), [[omega].sub.i] [member of] [0,1], [n.summation over (i=1)] [[omega].sub.i] = 1. Then the g is called the weighted geometric average operator of the triangular intuitionistic fuzzy number. Specially, if [omega] = [(1 / n, 1 / n, ..., 1 / n).sup. T], the g is called the geometric averaging operator of the triangular intuitionistic fuzzy number.

Theorem 1: suppose [I.sub.i] = ([[a.sub.i.sup.L], [a.sub.i.sup.M], [a.sub.i.sup.U]], [[b.sub.i.sup.L], [b.sub.i.sup.M], [b.sub.i.sup.U]]) (i = 1,2, ..., n) is a set of triangular intuitionistic fuzzy numbers, then the result is triangular intuitionistic fuzzy number aggregated by Eq.(14), and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

Where, [omega] = [([[omega].sub.1], [[omega].sub.2], ..., [[omega].sub.n]).sup.T] is the weight vector of [I.sub.i](i = 1,2, ..., n), [[omega].sub.i] [member of] [0,1], [n.summation over (i=1)] [[omega].sub.i] = 1

Mathematical induction is used to prove the Eq.(16). The proving procedures are as follows:

(1) When n = 1, it is right obviously.

(2) When n = 2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(3) when n = k, Eq. (16) is tenable,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Namely, when n = k + 1, Eq.(16) is tenable. According to (1) (2) (3), Eq.(16) is tenable for all n.

Theorem 2: suppose [I.sub.i] = ([[a.sub.i.sup.L], [a.sub.i.sup.M], [a.sub.i.sup.U]], aM,aU],[biL,bM,bU]) [[b.sub.i.sup.L], [b.sub.i.sup.M], [b.sub.i.sup.U]], (i = 1, 2, ..., n) is a set of triangular intuitionistic fuzzy numbers, then the result is triangular intuitionistic fuzzy number aggregated by Eq.(15), and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

The proving procedures of the theorem 2 are similar to the theorem 1, so this paper omits them. The emphasis points of the arithmetic averaging operator of the triangular intuitionistic fuzzy number and the geometric averaging operator are different. The former emphasizes on the personal importance and the latter pays attention to the collective effect.

4.2. The score function and accuracy function of the triangular intuitionistic fuzzy number

Definition 3 (Chen and Tan 1994): suppose I = ([mu].sub.A], [V.sub.A]) is a intuitionistic fuzzy number, then S(I) = [[mu].sub.A] - [v.sub.A] is called the score function of I.

Definition 4 (Xu 2007): suppose I = ([[a.sup.L], [a.sup.U]],[[b.sup.L], [b.sup.U]]) is a interval-valued intuitionistic fuzzy number, then S(I) = ([a.sup.L] - [b.sup.L] + [a.sup.U] - [b.sup.U])/2 is called the score function of I.

Definition 5 (Hong and Choi 2000): suppose I = ([[mu].sub.A], [v.sub.A]) is a intuitionistic fuzzy number, then H(I) = [[mu].sub.A] + [v.sub.A] is called the accuracy function of I.

Definition 6 (Xu 2007): suppose I = ([[a.sup.L], [a.sup.U]],[[b.sup.L], [b.sup.U]]) is a interval-valued intuitionistic fuzzy number, then H(I) = ([a.sup.L] + [b.sup.L] + [a.sup.U] + [b.sup.U])/2 is called the accuracy function of I.

According to definition 3-6, we can be deduced the score function and accuracy function of the triangular intuitionistic fuzzy number.

Definition 7: suppose I = ([[a.sup.L], [a.sup.M], [a.sup.U]],[[b.sup.L], [b.sup.M], [b.sup.U]]) is a triangular intuitionistic fuzzy number, then S(I) is called the score function of I,

S(I) = ([a.sup.L] - [b.sup.L] + [a.sup.M] - [bs.up.M] + [a.sup.U] - [b.sup.U])/3, (18)

where S(I) [member of] [-1,1]. Obviously, if the bigger S(I), the bigger I, especially, if S(I) = 1, the value of I is the biggest one I = ([1,1,1], [0, 0, 0]). If S(I) = -1, the value of I is the smallest one I = ([0, 0, 0],[1, 1,1]).

But the score function can not compare two triangular intuitionistic fuzzy numbers in a special situation. For example, if [I.sub.1] = ([0.3, 0.4, 0.5], [0.3, 0.4, 0.5]), [I.sub.2] = ([0.2, 0.3, 0.4], [0.2, 0.3, 0.4]), then S([I.sub.1]) = S([I.sub.2]) = 0 and the score function can not compare [I.sub.1] and [I.sub.2]. In order to resolve this problem, an accuracy function is given in this paper.

Definition 8: suppose I = ([[a.sup.L], [a.sup.M], [a/.sup.U]],[[b.sup.L], [b.sup.M], [b.sup.U]]) is a triangular intuitionistic fuzzy number, then E(I) is called the accuracy function of I,

E(I) = ([a.sup.L] + [b.sup.L] + [a.sup.M] + [b.sup.M] + [a.sup.U] + [b.sup.U])/3 , (19)

where E(I)e [0,1].

In the example above, if [I.sub.1] = ([0.3, 0.4, 0.5], [0.3, 0.4, 0.5]), [I.sub.2] = ([0.2, 0.3, 0.4], [0.2, 0.3, 0.4]) then the accuracy functions E= 0.8, E(I2) = 0.6. The accuracy function can compare [I.sub.1] and [I.sub.2].

This paper thinks that the following conclusion is right: when the values of score function of the triangular intuitionistic fuzzy numbers are the same, the bigger the value of the accuracy function, the bigger the corresponding triangular intuitionistic fuzzy numbers. So [I.sub.1] is better than [I.sub.2].

On the foundation of the above analyses, a priority method is proposed for the triangular intuitionistic fuzzy numbers.

Definition 9: suppose [I.sub.1] and [I.sub.2] are any two triangular intuitionistic fuzzy numbers, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

5. Decision making method based on the triangular intuitionistic fuzzy number

In a problem of multi-attribute decision making, suppose S = {[s.sub.1], [s.sub.2], [s.sub.m]} is a set of projects. R = {[r.sub.1], [r.sub.2], ..., [r.sub.n]}is a set of attributes. [omega] = [([[omega].sub.1], con)T is the weighted [[omega].sub.2], ... [[omega].sub.n]).sup.T is the weighted vector of the attributes, where, [[omega].sub.i] [member of] [0,1], [n.summation over (i=1)] [[omega].sub.i] = 1. Suppose the characteristic information of project [s.sub.i] is denoted by the triangular intuitionistic fuzzy number [S.sub.i] = {< [r.sub.j], [u.sub.i] ([r.sub.j])>, [v.sub.i] ([r.sub.j])> [r.sub.j] [member of] A}, i = 1, 2,m, where, [u.sub.i]([r.sub.j]) denotes the satisfaction degree of project [S.sub.i] to attribute [r.sub.j] and [v.sub.i] ([r.sub.j]) denotes the non-satisfaction degree of project [S.sub.i] to attribute [r.sub.j]. Here [u.sub.i] ([r.sub.j]) and [v.sub.i] ([r.sub.j]) are triangular fuzzy numbers. [u.sub.i] ([r.sub.j]) = [[a.sub.ij.sup.L], [a.sub.ij.sup.M], [a.sub.ij.sup.U]], [v.sub.i] ([r.sub.j]) = [[b.sub.ij.sup.L], [b.sub.ij.sup.M] [b.sub.ij.sup.U]. Then the corresponding triangular intuitionistic fuzzy numbers is denoted as: [I.sub.j] = ([[a.sub.ij].sup.L], [a.sub.ij].sup.M] [a.sub.ij].sup.U]]), i = 1,2, ..., m, j = 1,2, ..., n. So the decision making matrix is obtained: D = ([I.sub.ij]).sub.m x n].

The steps of the decision making based on triangular fuzzy inuitionistic fuzzy numbers are as follows:

Step 1: according to the weighted arithmetic averaging operator or the weighted geometric average operator to integrate all the elements [I.sub.ij] (j = 1, 2, ..., n) of the i-th row in the decision making matrix D. Then the comprehensive triangular intuitionistic fuzzy value for [S.sub.i] shows as follows.

[I.sup.f.sub.i] = [f.sub.[omega]] ([I.sub.i1], [I.sub.i2], ..., [I.sub.in]) (for the weighted arithmetic averaging operator) or

[I.sup.g.sub.i] = [g.sub.[omega]]([I.sub.i1], [I.sub.i2], ..., [I.sub.in]) (for the weighted geometric averaging operator)

Step 2: calculate the value of the score function S([I.sub.i]) and the value of the accuracy function E([I.sub.i]) using the formulas of the score function and the accuracy function, where, i = 1, 2, ..., m.

Step 3: according to the definition 6, confirm the order of the project [s.sub.i](i = 1, 2, ..., m). Then the best project can be gained.

6. Example analysis

A company intends to select one person to take the department manager position from four candidates ([s.sub.1] - [s.sub.4]). Five indicators must be evaluated. They are shown as follows: ideological and moral quality(r1), professional ability([r.sub.2]), creative ability ([r.sub.3]), knowledge range ([r.sub.4]) and leadership ability ([r.sub.5]). The weights of the indicators are W = (0.10, 0.25, 0.25, 0.15, 0.25). The leaders and people evaluate each indicator of each candidate. Suppose the evaluation information ([I.sub.ij]) can be denoted by the triangular intuitionistic fuzzy number after transforming. The evaluation data is shown in Table 1.

(1) According to the Eq. (16), integrate all the element [I.sub.ij] (j = 1, 2, ..., 5) of the i-th row. So the comprehensive value [I.sup.f.sub.i] (i = 1, 2, 3, 4) of candidate [S.sub.i] (i = 1, 2, 3, 4) is calculated:

[I.sub.1.sup.f] = ([0.269, 0.388, 0.532], [0.192, 0.333, 0.455]);

[I.sub.2.sup.f] = ([0.413, 0.477, 0.542], [0.167, 0.239, 0.332]);

[I.sub.3.sup.f] = ([0.436, 0.508, 0.581], [0.103, 0.170, 0.227]);

[I.sub.4.sup.f] = ([0.338, 0.399, 0.471], [0.237, 0.288, 0.349]).

(2) According to (18), the value of the score function S ([I.sub.i.sup.f]) can be calculated: S([I.sub.1.sup.f]) = 0.07, S([I.sub.2.sup.f]) = 0.231, S([I.sub.3.sup.f]) = 0.341, S([I.sub.4.sup.f]) = 0.111.

(3) According to the value of the score function, the order of the candidates can be confirmed:

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So, the best candidate is [s.sub.3].

In addition, we can do decision based on the weighted geometric average operator. The computational results are as follows:

[I.sub.1.sup.g] = ([0.245, 0.350, 0.518], [0.220, 0.343, 0.474]);

[I.sub.2.sup.g] = ([0.399, 0.462, 0.533], [0.200, 0.262, 0.348]);

[I.sub.3.sup.g] = ([0.427, 0.498, 0.567], [0.123, 0.185, 0.245]);

[I.sub.4.sup.g] = ([0.314, 0.384, 0.455], [0.248, 0.298, 0.363]);

S([I.sub.1.sup.g]) = 0.025, S([I.sub.2.sup.g]) = 0.195, S([I.sub.3.sup.g]) = 0.313, S([I.sub.4.sup.g]) = 0.081.

According to the value of the score function, the order of the candidates can be confirmed:

[s.sub.3] > [s.sub.2] > [s.sub.4] > [s.sub.1].

So the best candidate is also [s.sub.3].

In this example, the condition that the values of the score functions are same does not appear, so the accuracy function is not used. If the condition that the values of the score functions are same appears, the accuracy function should be used to calculate the final result to confirm the order of the candidates.

7. Conclusion

On the foundation of the theory of the intuitionistic fuzzy set, this paper extends the traditional research. This paper uses the triangular fuzzy to denote the membership degree and the non-membership degree and proposes the triangular intuitionistic fuzzy number. Then the operation rules of triangular intuitionistic fuzzy numbers are defined. The weighted geometric averaging operator and the weighted arithmetic average operator are presented and are used to the decision making area. An effective solution is offered for multi-attitude decision making problem and an active try is made. The example proves that the integration methods proposed in this paper are feasible effective.

doi: 10.3846/tede.2010.18

Acknowledgment

This paper is supported by the Humanities and Social Sciences Research Project of Ministry of Education of China (No. 09YJA630088), and the Natural Science Foundation of Shandong Province (No. ZR2009HL022). The authors also would like to express appreciation to the managing editor, Dr Jonas Saparauskas and the anonymous reviewers for their very helpful comments on improving the paper.

Received 24 November 2009; accepted 27 April 2010

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Xin Zhang (1), Peide Liu (2)

(1, 2) Information Management School, Shandong Economic University, Ji'nan, 250014, China E-mail: 2Peide.liu@gmail.com (Corresponding author)

Xin ZHANG (China, 1967) obtained his doctor degree in information management in the Beijing Jiaotong University. At present, he is the dean of school of Information Management at Shandong Economic University and also is a professor of management science and engineering. His main research focuses on information management and technology, knowledge management, and knowledge-based systems.

Peide LIU (China, 1966) obtained his doctor degree in information management in the Beijing Jiaotong University. His main research fields are technology and information management, decision support and electronic-commerce. He was engaged in the technology development and the technical management in the Inspur company a few years ago. Now he is a full-time professor in Shandong Economic University and assistant director of the Enterprise's Electronic-commerce Engineering Research Center of Shandong.

Table 1. The transformed evaluation information of the indicators
of each candidate (the decision making matrix D)

 [r.sub.1] [r.sub.2]

[s.sub.1] ([0.4, 0.5, 0.6], ([0.3, 0.5, 0.6],
 [0.1, 0.2, 0.3]) [0.1, 0.3, 0.4])

[s.sub.2] ([0.25, 0.3, 0.4], ([0.45, 0.5, 0.55],
 [0.4, 0.45, 0.5]) [0.1, 0.2, 0.3])

[s.sub.3] ([0.5, 0.55, 0.6], ([0.4, 0.45, 0.5],
 [0.2, 0.25, 0.3]) [0.1, 0.2, 0.25])

[s.sub.4] ([0.3, 0.35, 0.4], ([0.2, 0.3, 0.4],
 [0.4, 0.45, 0.5]) [0.3, 0.35, 0.45])

 [r.sub.3] [r.sub.4]

[s.sub.1] ([0.2, 0.3, 0.5], ([0.4, 0.5, 0.6],
 [0.3, 0.4, 0.5]) [0.2, 0.3, 0.4])

[s.sub.2] ([0.4, 0.5, 0.55], ([0.55, 0.6, 0.65],
 [0.15, 0.2, 0.35]) [0.1, 0.15, 0.2])

[s.sub.3] ([0.45, 0.5, 0.55], ([0.3, 0.4, 0.5],
 [0.1, 0.15, 0.2]) [0.25, 0.3, 0.4])

[s.sub.4] ([0.4, 0.45, 0.5], ([0.5, 0.55, 0.65],
 [0.2, 0.25, 0.3]) [0.2, 0.25, 0.3])

 [r.sub.4]

[s.sub.1] ([0.15, 0.2, 0.4],
 [0.3, 0.4, 0.6])

[s.sub.2] ([0.35, 0.4, 0.5],
 [0.3, 0.35, 0.4])

[s.sub.3] ([0.5, 0.6, 0.7],
 [0.05, 0.1, 0.15])

[s.sub.4] ([0.3, 0.35, 0.4],
 [0.2, 0.25, 0.3])

Based on the weighted arithmetic averaging operator, the
procedures to confirm the best candidate are as follows: