Expert Systems with Applications 38 (2011) 11730–11734

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Expert Systems with Applications

journal homepage: www.elsevier .com/locate /eswa

Expected value method for intuitionistic trapezoidal fuzzy multicriteriadecision-making problems

Jun Ye ⇑Department of Mechatronics Engineering, Shaoxing College of Arts and Sciences, Shaoxing, Zhejiang Province 312000, PR China

a r t i c l e i n f o a b s t r a c t

Keywords:Fuzzy numberIntuitionistic fuzzy numberIntuitionistic trapezoidal fuzzy numberExpected valueMulticriteria decision-making

0957-4174/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.eswa.2011.03.059

⇑ Tel.: +86 575 88327323.E-mail address: yejnn@yahoo.com.cn

In this paper, we introduce the expected values for intuitionistic trapezoidal fuzzy numbers and pre-sented a handling method for intuitionistic trapezoidal fuzzy multicriteria decision-making problems,in which the preference values of an alternative on criteria and the weight values of criteria take the formof intuitionistic trapezoidal fuzzy numbers. Simple and exact formulas are proposed to determine theexpected weight values of the criteria for an alternative and weighted expected value for an alternative.Therefore, the expected value method is extended for ranking alternatives and the most desirable one(s)can be selected according to the weighted expected values. Finally, an illustrative example shows that theproposed method is simple and effective.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In many real-life cases, decision makers usually cannot choosebut provide their preferences in the form of fuzzy information asa result of vague knowledge about the preference of alternatives.Accordingly, it is greatly necessary to study the fuzzy multicriteriadecision-making problems under intuitionistic fuzzy environment.Chen and Tan (1994), Hong and Choi (2000), Liu and Wang (2007)and Ye (2007, 2009, 2010) presented some new techniques forhandling fuzzy multicriteria decision-making problems based onvague set theory or intuitionistic fuzzy sets, where the characteris-tics of the alternatives are represented by vague sets or intuition-istic fuzzy sets and the criteria weights are given by fuzzynumbers. However, intuitionistic fuzzy sets is the same asfuzzy sets, the domains of which are discrete sets, intuitionisticfuzzy sets are used to indicate the extent to which the criteriondoes or does not belong to some fuzzy concepts. The notion of afuzzy number and the operation on fuzzy numbers were intro-duced by Dubois and Prade (1978, 1987). Nehi and Maleki (2005)proposed Intuitionistic trapezoidal fuzzy numbers and some oper-ators for them, which are the extending of intuitionistic triangularfuzzy numbers. Intuitionistic triangular fuzzy numbers and intui-tionistic trapezoidal fuzzy numbers are the extending of intuition-istic fuzzy sets in another way, which extends discrete set tocontinuous set, and they are the extending of fuzzy numbers.

Furthermore, the expected value method is also applied to rank-ing. Heilpern (1992) proposed the expected value of a fuzzy num-ber. Then Grzegrorzewski (2003) proposed the expected value and

ll rights reserved.

ordering method for intuitionistic fuzzy numbers by using the ex-pected interval of intuitionistic fuzzy numbers. Also Wang andZhang (2009) defined some aggregation operators, including intui-tionistic trapezoidal fuzzy weighted arithmetic averaging operatorand weighted geometric averaging operator, and proposed anintuitionistic trapezoidal fuzzy multicriteria decision-makingmethod with known weights based on expected values, score func-tion, and accuracy function of intuitionistic trapezoidal fuzzynumbers.

In this paper, we introduce the expected value for an intuition-istic trapezoidal fuzzy number based on Nehi and Maleki (2005)and presented a handling method for an intuitionistic trapezoidalmulticriteria fuzzy decision-making problem, in which the prefer-ence values of an alternative on criteria and the weight values ofcriteria take the form of intuitionistic trapezoidal fuzzy numbers.The simple and exact formulas are proposed to determine the ex-pected weight values of the criteria for an alternative and weightedexpected value for an alternative. To do so, this paper is arrangedas follows: in the next section, we firstly introduce some basicnotations and preliminary definitions of intuitionistic fuzzy sets,fuzzy numbers, intuitionistic fuzzy numbers, and intuitionistictrapezoidal fuzzy numbers. In Section 3, we introduce the expectedvalue of intuitionistic trapezoidal fuzzy numbers based on the ex-pected value of intuitionistic fuzzy numbers. In Section 4, the pre-sentation of the fuzzy multicriteria decision-making problem isgiven in which the preference values of an alternative and weightvalues for criteria take the form of intuitionistic trapezoidal fuzzynumbers, and then we propose an expected value method to obtainthe expected weight values of criteria and weighted expected val-ues for alternatives, so that we can rank the alternatives and selectthe most desirable one(s) according to the weighted expected

J. Ye / Expert Systems with Applications 38 (2011) 11730–11734 11731

values. In Section 5, an illustrative example is given to illustrate theapplication of the proposed approach to the fuzzy multicriteriadecision-making problem. The conclusions are given in Section 6.

2. Preliminaries

This section introduces some definitions and basic conceptsrelated to intuitionistic fuzzy sets (Atanassov, 1986), fuzzynumbers (Dubois & Prade, 1978), intuitionistic fuzzy numbers(Grzegrorzewski, 2003) and intuitionistic trapezoidal fuzzynumbers (Nehi & Maleki, 2005).

Definition 1. Let X be a universe of discourse. Then an intuition-istic fuzzy set (Atanassov, 1986) A in X is given by

A ¼ fhx;lAðxÞ; mAðxÞijx 2 Xg; ð1Þ

where lA(x) : X ? [0,1] and mA(x) : X ? [0,1], with the condition0 6 lA(x) + mA(x) 6 1. The numbers lA(x) and mA(x) represent,respectively, the membership degree and nonmembership degreeof the element x to the set A. For each A in X, we can compute theintuitionistic index of the element x in the set A, which is definedas follows:

pAðxÞ ¼ 1� lAðxÞ � mAðxÞ; x 2 X; ð2Þ

where pA(x) is also called a hesitancy degree of x to A. It is obviousthat 0 6 pA(x) 6 1, x 2 X.

Definition 2. Let A be an fuzzy number in the set of real numbersR, its membership function (Dubois & Prade, 1978) is defined as

lAðxÞ ¼

0; x < a1;

fAðxÞ; a1 6 x 6 a2;

1; a2 6 x 6 a3;

gAðxÞ; a3 6 x 6 a4;

0; a4 < x;

8>>>>>><>>>>>>:

ð3Þ

where a1, a2, a3, a4 2 R, fA : [a1,a2] ? [0,1] is a nondecreasing contin-uous function, fA(a1) = 0, fA(a2) = 1, called the left side of the fuzzynumber A and gA : [a3,a4] ? [0,1] is a nonincreasing continuousfunction, gA(a3) = 1, gA(a4) = 0, called the right side of the fuzzynumber A.

Definition 3 (Grzegrorzewski, 2003). Let A be an intuitionisticfuzzy number in the set of real numbers R, its membership func-tion is defined as

lAðxÞ ¼

0; x < a1;

fAðxÞ; a1 6 x 6 a2;

1; a2 6 x 6 a3;

gAðxÞ; a3 6 x 6 a4;

0; a4 < x;

8>>>>>><>>>>>>:

ð4Þ

while its nonmembership function is defined as

mAðxÞ ¼

1; x < b1;

hAðxÞ; b1 6 x 6 b2;

0; b2 6 x 6 b3;

kAðxÞ; b3 6 x 6 b4;

1; b4 < x;

8>>>>>><>>>>>>:

; ð5Þ

where 0 6 lA(x) + mA(x) 6 1 and a1, a2, a3, a4, b1, b2, b3, b4 2 R suchthat b1 6 a1 6 b2 6 a2 6 a3 6 b3 6 a4 6 b4, and four functions fA, gA,hA, kA : R ? [0,1] are called the side of a fuzzy number. The func-tions fA and kA are nondecreasing continuous functions and thefunctions hA and gA are nonincreasing continuous functions.

It is worth noting that each intuitionistic fuzzy numberA = {hx,lA(x),mA(x)ijx 2 R} is a conjunction of two fuzzy numbers:A+ with a membership function lA+(x) = lA(x) and A� with amembership function lA

� (x) = 1 � mA(x). It is seen that suppA+ # supp A�.

A useful tool for dealing with fuzzy numbers is their a-cuts.Every a-cut of a fuzzy number is a closed interval and a family ofsuch intervals describes completely a fuzzy number under study.In the case of intuitionistic fuzzy numbers it is convenient todistinguish following a-cuts: (A+)a and (A�)a. It is easily seen that

ðAþÞa ¼ fx 2 RjlAðxÞP ag ¼ Aa; ð6Þ

ðA�Þa ¼ fx 2 Rj1� mAðxÞP ag ¼ fx 2 RjmAðxÞ 6 1� ag

¼ A1�a: ð7Þ

According to the definition it is seen at once that every a-cut(A+)a or (A�)a is a closed interval. Hence we haveðAþÞa ¼ AþL ðaÞ;A

þUðaÞ

� �and ðA�Þa ¼ A�L ðaÞ;A

�UðaÞ

� �, respectively,

where

AþL ðaÞ ¼ inf x 2 RjlAðxÞP a� �

; ð8Þ

AþUðaÞ ¼ supfx 2 RjlAðxÞP ag; ð9Þ

A�L ðaÞ ¼ inffx 2 RjmAðxÞ 6 1� ag; ð10Þ

A�UðaÞ ¼ supfx 2 RjmAðxÞ 6 1� ag: ð11Þ

In particular, if the increasing functions fA and kA and decreasingfunctions gA and hA are linear, then we have intuitionistic trapezoi-dal fuzzy numbers, which are preferred in practice.

Definition 4 (Nehi and Maleki, 2005). An intuitionistic trapezoidalfuzzy number A with parameters b1 6 a1 6 b2 6 a2 6 a3 6 b3 6

a4 6 b4 is denoted as A = h(a1,a2,a3,a4), (b1,b2,b3,b4)i in the set ofreal numbers R. In this case, its membership function and non-membership function can be given as

lAðxÞ ¼

0; x < a1;x�a1

a2�a1; a1 6 x 6 a2;

1; a2 6 x 6 a3;x�a4

a3�a4; a3 6 x 6 a4;

0; a4 < x;

8>>>>>><>>>>>>:

; ð12Þ

mAðxÞ ¼

1; x < b1;x�b2

b1�b2; b1 6 x 6 b2;

0; b2 6 x 6 b3;x�b3

b4�b3; b3 6 x 6 b4;

1; b4 < x:

8>>>>>><>>>>>>:

ð13Þ

If b2 = b3 (hence a2 = a3) in an intuitionistic trapezoidal fuzzynumber A, the intuitionistic triangular fuzzy numbers are consid-ered as special cases of the intuitionistic trapezoidal fuzzynumbers.

The following properties for intuitionistic trapezoidal fuzzynumbers have been given in Nehi and Maleki (2005).

Let A1 = h(a11,a12,a13,a14), (b11,b12,b13,b14)i and A2 = h(a21,a22,a23,a24), (b21,b22,b23,b24)i be two intuitionistic trapezoidal fuzzynumbers and r be a positive scalar number. Then,

A1 þ A2 ¼ hða11 þ a21; a12 þ a22; a13 þ a23; a14 þ a24Þ; ðb11

þ b21; b12 þ b22; b13 þ b23; b14 þ b24Þi; ð14Þ

rA1 ¼ hðra11; ra12; ra13; ra14Þ; ðrb11; rb12; rb13; rb14Þi: ð15Þ

11732 J. Ye / Expert Systems with Applications 38 (2011) 11730–11734

3. Expected values of trapezoidal intuitionistic fuzzy numbers

In this section we introduce a definition of the expected valuefor the intuitionistic fuzzy number proposed by Grzegrorzewski(2003) and then give the expected value of an intuitionistic trape-zoidal fuzzy number.

Grzegrorzewski proposed an ordering method for intuitionisticfuzzy numbers by using the expected interval of an intuitionisticfuzzy number. The expected interval of an intuitionistic fuzzynumber A = h(a1,a2,a3,a4), (b1,b2,b3,b4)i is a crisp interval EI(A) gi-ven by

EIðAÞ ¼ ½E�ðAÞ; E�ðAÞ�; ð16Þ

where

E�ðAÞ ¼b1 þ a2

2þ 1

2

Z b2

b1

hAðxÞdx� 12

Z a2

a1

fAðxÞdx; ð17Þ

E�ðAÞ ¼ a3 þ b4

2þ 1

2

Z a4

a3

gAðxÞdx� 12

Z b4

b3

kAðxÞdx: ð18Þ

The expected value of an intuitionistic fuzzy number is definedas follows.

Definition 5 (Grzegrorzewski, 2003). Let A = h(a1,a2,a3,a4), (b1,b2,b3,b4)i be an intuitionistic fuzzy number in the set of real numbersR. Then the expected value of A for the center of the expectedinterval of A is defined by

EVðAÞ ¼ E�ðAÞ þ E�ðAÞ2

: ð19Þ

Therefore, the expected value of an intuitionistic trapezoidalfuzzy number based on the expected value of intuitionistic fuzzynumbers can be obtained as the following theorem.

Theorem 1. Let A = h(a1,a2,a3,a4), (b1,b2,b3,b4)i be an intuitionistictrapezoidal fuzzy number in the set of real numbers R. Thus, whenfAðxÞ ¼ x�a1

a2�a1; gAðxÞ ¼ x�a4

a3�a4; hAðxÞ ¼ x�b2

b1�b2, and kAðxÞ ¼ x�b3

b4�b3, a1, a2,

a3, a4, b1, b2, b3, b4 2 R, its expected value is obtained by

EVðAÞ ¼ 18ða1 þ a2 þ a3 þ a4 þ b1 þ b2 þ b3 þ b4Þ: ð20Þ

Especially, there is (a1,a2,a3,a4) = (b1,b2,b3,b4). In this case, theexpected value is as follows:

EVðAÞ ¼ 14ða1 þ a2 þ a3 þ a4Þ: ð21Þ

Thus, Eq. (21) degenerates to the expected value of trapezoidalfuzzy numbers.

The following orders were proposed by Grzegrorzewski (2003).

Table 1Linguistic values of intuitionistic trapezoidal fuzzy numbers for linguistic terms.

Linguistic Linguistic values of intuitionistic trapezoidal fuzzy

Theorem 2. Let �L and �U denote the quasi-order with respect to thelower and upper horizon, respectively, based on the metric d1 (i.e. dp

for p = 1). Then for two intuitionistic fuzzy numbers A and B, we getthe following orders:

A�LB() EVðAÞP EVðBÞ; ð22Þ

A�UB() EVðAÞP EVðBÞ: ð23Þ

terms numbers

Absolutely low h(0.0,0.0,0.0,0.0), (0.0,0.0,0.0,0.0)iLow h(0.0,0.1,0.2,0.3), (0.0,0.1,0.2,0.3)iFairly low h(0.1,0.2, 0.3,0.4), (0.0,0.2,0.3,0.5)iMedium h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)iFairly high h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)iHigh h(0.7,0.8, 0.9,1.0), (0.7,0.8,0.9,1.0)iAbsolutely high h(1.0,1.0, 1.0,1.0), (1.0,1.0,1.0,1.0)i

4. Multicriteria decision-making method based on expectedvalues of intuitionistic trapezoidal fuzzy numbers

In this section, we present a handling method for intuitionistictrapezoidal fuzzy multicriteria decision-making problems withintuitionistic trapezoidal fuzzy weights.

Let A = {A1,A2, . . . ,Am} be a set of alternatives and letC = {C1,C2, . . . ,Cn} be a set of criteria. The preference value of analternative on a criterion Cj (j = 1,2, . . . ,n) is a intuitionistic trape-zoidal fuzzy number tij = h(aij1,aij2,aij3,aij4), (bij1,bij2,bij3,bij4)i, a1, a2,a3, a4, b1, b2, b3, b4 2 R, j = 1,2, . . . ,n, and i = 1,2, . . . ,m, which indi-cates the degree that the alternative Ai satisfies or does not satisfiesthe criterion Cj given by decision makers or experts according tolinguistic values of intuitionistic trapezoidal fuzzy numbers for lin-guistic terms. Therefore, we can elicit a decision matrix D = (tij)m�n,which is expressed by intuitionistic trapezoidal fuzzy numbers.

For the weight of a criterion Cj (j = 1,2, . . . ,n), it is a trapezoidalintuitionistic fuzzy weight wi = h(c1,c2,c3,c4), (d1,d2,d3,d4)i(j = 1,2, . . . ,n). The expected weight value wj (j = 1,2, . . . ,n) for anintuitionistic trapezoidal fuzzy weight is obtained by Eq. (20). Thenwe normalize the expected weight value wj (j = 1,2, . . . ,n) by usingthe following formula:

xj ¼EVðwjÞPnj¼1EVðwjÞ

: ð24Þ

Therefore, we give the weighted expected value for an alterna-tive Ai (i = 1,2, . . .,m):

WEVðAiÞ ¼Xn

j¼1

xjEVðtijÞ: ð25Þ

Thus by using Eq. (25), we calculate the weighted expected va-lue for an alternative Ai (i = 1,2, . . . ,m) to rank alternatives and thento select the best one(s) in all the alternatives.

In summary, the decision procedure for the proposed methodcan be summarized as follows:

Step 1. Give decision matrix and weights according to linguisticvalues of intuitionistic trapezoidal fuzzy numbers for lin-guistic terms given by experts.

Step 2. Calculate the expected weight value for a criterion Cj

(j = 1,2, . . . ,n) by using Eqs. (20) and (24).Step 3. Calculate the weighted expected value for an alternative Ai

(i = 1,2, . . . ,m) by using Eqs. (20) and (25).Step 4. Rank the alternatives and select the best one(s) in accor-

dance with the weighted expected value WEV(Ai)(i = 1,2, . . . ,m).

5. Illustrative example

In this section, an illustrative example for a multicriteria deci-sion-making problem of alternatives is used as a demonstrationof the application of the proposed fuzzy decision-making methodin a realistic scenario, as well as the effectiveness of the proposedmethod.

There is a panel with four possible alternatives to invest themoney (adapted from Herrera and Herrera-Viedma (2000)): (1)A1 is a car company; (2) A2 is a food company; (3) A3 is a computercompany; (4) A4 is a television company. The investment companymust take a decision according to the following three criteria: (1)C1 is the risk analysis; (2) C2 is the growth analysis; (3) C3 is the

Table 2Preference values of alternatives and criteria weights given from five experts by linguistic values of intuitionistic trapezoidal fuzzy numbers.

k C1 C2 C3

A1 1 h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i h(0.1,0.2, 0.3,0.4), (0.0,0.2,0.3,0.5)i2 h(0.1,0.2, 0.3,0.4), (0.0,0.2,0.3,0.5)i h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i h(0.0,0.1,0.2,0.3), (0.0,0.1,0.2,0.3)i3 h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.1,0.2, 0.3,0.4), (0.0,0.2,0.3,0.5)i4 h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i5 h(0.1,0.2, 0.3,0.4), (0.0,0.2,0.3,0.5)i h(0.1,0.2, 0.3,0.4), (0.0,0.2,0.3,0.5)i h(0.1,0.2, 0.3,0.4), (0.0,0.2,0.3,0.5)i

A2 1 h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i2 h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.7,0.8, 0.9,1.0), (0.7,0.8,0.9,1.0)i h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i3 h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i4 h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i5 h(0.7,0.8, 0.9,1.0), (0.7,0.8,0.9,1.0)i h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i

A3 1 h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i2 h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.1,0.2, 0.3,0.4), (0.0,0.2,0.3,0.5)i3 h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i4 h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i5 h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i h(0.7,0.8, 0.9,1.0), (0.7,0.8,0.9,1.0)i h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i

A4 1 h(0.7,0.8, 0.9,1.0), (0.7,0.8,0.9,1.0)i h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.1,0.2, 0.3,0.4), (0.0,0.2,0.3,0.5)i2 h(0.7,0.8, 0.9,1.0), (0.7,0.8,0.9,1.0)i h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.1,0.2, 0.3,0.4), (0.0,0.2,0.3,0.5)i3 h(0.7,0.8, 0.9,1.0), (0.7,0.8,0.9,1.0)i h(0.7,0.8, 0.9,1.0), (0.7,0.8,0.9,1.0)i h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i4 h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.7,0.8, 0.9,1.0), (0.7,0.8,0.9,1.0)i h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i5 h(0.7,0.8, 0.9,1.0), (0.7,0.8,0.9,1.0)i h(0.7,0.8, 0.9,1.0), (0.7,0.8,0.9,1.0)i h(0.1,0.2, 0.3,0.4), (0.0,0.2,0.3,0.5)i

Weights 1 h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i h(0.1,0.2, 0.3,0.4), (0.0,0.2,0.3,0.5)i h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i2 h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i3 h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.1,0.2, 0.3,0.4), (0.0,0.2,0.3,0.5)i h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i4 h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i5 h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i h(0.3,0.4, 0.5,0.6), (0.2,0.4,0.5,0.7)i h(0.5,0.6, 0.7,0.8), (0.4,0.6,0.7,0.9)i

J. Ye / Expert Systems with Applications 38 (2011) 11730–11734 11733

environmental impact analysis. The four possible alternatives areto be evaluated under the above three criteria by correspondingto linguistic values of intuitionistic trapezoidal fuzzy numbers forlinguistic terms, as shown in Table 1.

Steps using the proposed method are as follows:

Step 1. Suppose we invite k experts (here k = 5) to make thejudgement. They are expected to give linguistic values ofintuitionistic trapezoidal fuzzy numbers for linguisticterms. Then, the preference values of an alternative Ai

(i = 1,2,3,4) on a criterion Cj (j = 1,2,3) and the criteriaweights are given from five experts, as listed in Table 2.An intuitionistic trapezoidal fuzzy number tij = h(aij1,aij2,aij3,aij4), (bij1,bij2,bij3,bij4)i (i = 1,2,3,4; j = 1,2,3) in the deci-sion matrix and weights can be calculated by

D¼

hhhh

26664

tij ¼P5

k¼1aij1ðkÞ5

;

P5k¼1aij2ðkÞ

5;

P5k¼1aij3ðkÞ

5;

P5k¼1aij4ðkÞ

5

!;

*

�P5

k¼1bij1ðkÞ5

;

P5k¼1bij2ðkÞ

5;

P5k¼1bij3ðkÞ

5;

P5k¼1bij4ðkÞ

5

!+:

Thus the decision matrix D = (tij)4�3 and the weight value are givenas follows:

ð0:26;0:36;0:46;0:56Þ;ð0:16;0:36;0:46;0:66Þi hð0:34;0:44;0:54;0:64Þ;ð0:24;0:44;0:54;0:74Þi hð0:12;0:22;0:32;0:42Þ;ð0:04;0:22;0:32;0:50Þið0:50;0:60;0:70;0:80Þ;ð0:42;0:60;0:70;0:88Þi hð0:50;0:60;0:70;0:80Þ;ð0:42;0:60;0:70;0:88Þi hð0:34;0:44;0:54;0:64Þ;ð0:24;0:44;0:54;0:74Þið0:38;0:48;0:58;0:68Þ;ð0:28;0:48;0:58;0:78Þi hð0:54;0:64;0:74;0:84Þ;ð0:46;0:64;0:74;0:92Þi hð0:26;0:36;0:46;0:56Þ;ð0:16;0:36;0:46;0:66Þið0:66;0:76;0:86;0:96Þ;ð0:64;0:76;0:86;0:98Þi hð0:62;0:72;0:82;0:92Þ;ð0:58;0:72;0:82;0:96Þi hð0:18;0:28;0:38;0:48Þ;ð0:08;0:28;0:38;0:58Þi

37775;

w ¼ fhð0:38;0:48;0:58;0:68Þ; ð0:28; 0:48; 0:58;0:78Þi;hð0:22;0:32;0:42;0:52Þ; ð0:12; 0:32; 0:42;0:62Þi;hð0:46;0:56;0:66;0:76Þ; ð0:36; 0:56; 0:66;0:86Þig:

Step 2. Calculate the expected weight value for a criterion Cj

(j = 1,2,3) by using Eqs. (20) and (24), thus we obtainthe following expected weight value xj (j = 1,2,3): x1 =0.351, x2 = 0.245, and x3 = 0.404.

Step 3. Calculate the weighted expected value for an alternativeAi (i = 1,2,3,4) by using Eqs. (20) and (25) to obtain thefollowing weighted expected value WEV(Ai) (i = 1,2,3,4):WEV(A1) = 0.3730, WEV(A2) = 0.5854, WEV(A3) = 0.5207,and WEV(A4) = 0.6063.

Step 4. Rank the alternatives as follows:

A2 � A4 � A3 � A1:

Thus the most desirable alternative is A2 in accordance with theabove result.

6. Conclusion

In this paper, we have proposed an expected value methodfor intuitionistic trapezoidal fuzzy multicriteria decision-makingproblems, in which the preference values for an alternative oncriteria and the weight values of criteria take the form of intui-tionistic trapezoidal fuzzy numbers. By using expected values ofintuitionistic trapezoidal fuzzy numbers, we have establishedsimple and exact formulas, which are derived to determine theexpected weight values for criteria and the weighted expected

values for alternatives. So the expected value method has beenextended for ranking alternatives. Finally, an illustrative exampleshows that the proposed method is simple and effective.

11734 J. Ye / Expert Systems with Applications 38 (2011) 11730–11734

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