Applied Mathematical Modelling 34 (2010) 3864–3870

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Applied Mathematical Modelling

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

Multicriteria fuzzy decision-making method using entropyweights-based correlation coefficients of interval-valuedintuitionistic fuzzy sets

Jun Ye *

Department of Mechatronics Engineering, Shaoxing College of Arts and Sciences, 508 Huancheng West Road, 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

Article history:Received 3 December 2008Received in revised form 14 March 2010Accepted 30 March 2010Available online 11 April 2010

Keywords:Interval-valued intuitionistic fuzzy setEntropy weightCorrelation coefficientMulticriteria fuzzy decision-making

0307-904X/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.apm.2010.03.025

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

A multicriteria fuzzy decision-making method based on weighted correlation coefficientsusing entropy weights is proposed under interval-valued intuitionistic fuzzy environmentfor the some situations where the information about criteria weights for alternatives iscompletely unknown. To determine the entropy weights with respect to a decision matrixprovided as interval-valued intuitionistic fuzzy sets (IVIFSs), we propose two entropy mea-sures for IVIFSs and establish an entropy weight model, which can be used to determinethe criteria weights on alternatives, and then propose an evaluation formula of weightedcorrelation coefficient between an alternative and the ideal alternative. The alternativescan be ranked and the most desirable one(s) can be selected according to the values ofthe weighted correlation coefficients. Finally, two applied examples demonstrate the appli-cability and benefit of the proposed method: it is capable for handling the multicriteriafuzzy decision-making problems with completely unknown weights for criteria.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

The theory of fuzzy sets (FSs) proposed by Zadeh in 1965 [1] has achieved a great success in various fields. Out of severalhigher order FSs, intuitionistic fuzzy sets (IFSs) introduced by Atanassov [2] have been found to be highly useful to deal withvagueness. The concept of IFSs is a generalization of that of FSs. The concept of vague sets (VSs) introduced by Gau and Bueh-rer [3] is another generalization of fuzzy sets. But, Bustince and Burillo [4] point out that the notion of VSs is the same as thatof IFSs. Another, well-known generalization of an ordinary fuzzy set is so-called interval-valued fuzzy sets (IVFSs). Generally,the idea of IVFSs was attributed to Gorzalczany [5] and Turksen [6], but actually there is a strong connection between IFSsand IVFSs. Atanassov and Gargov [7] introduced the concept of interval-valued intuitionistic fuzzy sets (IVIFSs) as a furthergeneralization of that of IFSs, as well as of IVFSs. Atanassov [8] defined some operational laws of IVIFSs. Bustince and Burillo[9] proposed the correlation and correlation coefficient of IVIFSs, which is equivalent generalization of the notion of IFSs andIVFSs. Much work has been done on operators over IVIFSs [10–12]. Recently, the operators for IVIFSs were applied to mul-ticriteria decision-making problems [13–16]. Luo et al. [13] proposed a multicriteria fuzzy decision-making method based onweighted correlation coefficients under interval-valued intuitionistic fuzzy environment with known criteria weight infor-mation. Ye [14] proposed a multicriteria fuzzy decision-making method based on a novel accuracy function under interval-valued intuitionistic fuzzy environment with known criteria weight information. Wang et al. [15] proposed an approach to

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J. Ye / Applied Mathematical Modelling 34 (2010) 3864–3870 3865

multiattribute decision-making with incomplete attribute weight information where individual assessments are provided asIVIFSs. Then by employing a series of optimization models, the proposed approach derives a linear program for determiningattribute weights. Furthermore, Park et al. [16] proposed an improved correlation coefficient of interval-valued intuitionisticfuzzy sets and its application to multiple attribute group decision-making problems with partially known attribute weightinformation. They established an optimization model for the score matrix to determine the weights of attributes.

Many researchers mainly focus on the multicriteria decision-making problems with known or incompletely knowncriterion weight information under interval-valued intuitionistic fuzzy environment. Therefore, there is little researchon the application of the correlation coefficient of IVIFSs to multicriteria decision-making problems with completelyunknown criterion weight information in the existing literature. Thus, a multicriteria fuzzy decision-making methodusing the entropy weights-based correlation coefficient of IVIFSs is proposed in this paper to identify the best alterna-tive in the application of multicriteria decision-making problems in which all the information provided by the deci-sion-makers or experts is represented as an interval-valued intuitionistic fuzzy decision matrix where each of theelements is characterized by an IVIFS, and the information about criteria weights is completely unknown. To do that,we organize the rest of this paper as follows. In Section 2, we briefly review IFSs, Ye’s method [17] for calculatingintuitionistic fuzzy entropy, IVIFSs, and correlation coefficients between IVIFSs. In Section 3, we present two measuresfor calculating the entropy of IVIFSs. In Section 4, we establish an entropy weight model to determine the criteriaweights from the interval-valued intuitionistic fuzzy decision matrix and then propose an evaluation formula ofweighted correlation coefficient between an alternative and the ideal alternative. The alternatives can be ranked andthe most desirable one(s) can be selected according to the values of the weighted correlation coefficients. Section 5provides two applied examples to illustrate the applicability of the proposed method. Finally, the conclusion is givenin Section 6.

2. Some preliminaries of IFSs and IVIFSs

2.1. IFSs and entropy measures for IFSs

In the following, we introduce some basic concepts related to IFSs and Ye’s method [17] for calculating entropy measuresfor IFSs.

Definition 1 [2]. An IFS 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 condition 0 6 lA(x) + mA(x) 6 1. The numbers lA(x) and mA(x) represent,respectively, the membership degree and nonmembership degree of the element x to the set A.

For each IFS A in X, if pA(x) = 1 � lA(x) � mA(x), x 2 X, then pA(x) is called the intuitionistic index of the element x in the setA. It is a hesitancy degree of x to A. It is obvious that 0 6 pA(x) 6 1, x 2 X.

The complementary set Ac of an IFS A is defined as

Ac ¼ fhx; mAðxÞ;lAðxÞijx 2 Xg: ð2Þ

Definition 2 [17]. Let A be an IFS in the universe of discourse X. Then, two entropy measures of the IFS A are defined asfollows:

E1ðAÞ ¼ sinp� ½1þ lAðxÞ � mAðxÞ�

4þ sin

p� ½1� lAðxÞ þ mAðxÞ�4

� 1� �

� 1ffiffiffi2p� 1

; ð3Þ

E2ðAÞ ¼ cosp� ½1þ lAðxÞ � mAðxÞ�

4þ cos

p� ½1� lAðxÞ þ mAðxÞ�4

� 1� �

� 1ffiffiffi2p� 1

: ð4Þ

Theorem 1 [17]. Let A and B be two IFSs in the universe of discourse X. Then, the above measures of the intuitionistic fuzzyentropy satisfy the following axiomatic requirements:

(P1) E1(A) = E2(A) = 0 (minimum), iff A is a crisp set;(P2) E1(A) = E2(A) = 1 (maximum), iff lA(x) = mA(x) for any x 2 X;(P3) E1(A) 6 E1(B) and E2(A) 6 E2(B) if A is less fuzzy than B, i.e., lA(x) 6 lB(x) and mA(x) P mB(x) for lB(x) 6 mB(x) or

lA(x) P lB(x) and mA(x) 6 mB(x) for lB(x) P mB(x) and any x 2 X;(P4) E1(A) = E1(Ac) and E2(A) = E2(Ac).

Thus, E1(A) and E2(A) are two measures of intuitionistic fuzzy entropy.

3866 J. Ye / Applied Mathematical Modelling 34 (2010) 3864–3870

2.2. IVIFSs and correlation coefficients between two IVIFSs

Definition 3 [7]. Let D [0,1] be the set of all closed subintervals of the interval [0,1] and X ( – Ø) be a given set. An IVIFS in Xis defined as

A ¼ fhx;lAðxÞ; vAðxÞijx 2 Xg; ð5Þ

where lA: X ? D[0,1], vA: X ? D[0,1] with the condition 0 6 sup(lA(x)) + sup(vA(x)) 6 1 for any x 2 X. The intervals lA(x) andvA(x) denote, respectively, the degree of belongingness and the degree of nonbelongingness of the element x to the set A.Thus for each x 2 X, lA(x) and vA(x) are closed intervals and their lower and upper end points are denoted by lAL(x), lAU(x),vAL(x), and vAU(x), respectively. We can denote by

A ¼ fhx; ½lALðxÞ;lAUðxÞ�; ½vALðxÞ; vAUðxÞ�ijx 2 Xg; ð6Þ

where 0 6 lAU(x) + vAU(x) 6 1, lAL(x) P 0, and vAL(x) P 0.For each element x we can compute the so-called intuitionistic fuzzy interval of x 2 X in A defined as follows:

pAðxÞ ¼ 1� lAðxÞ � vAðxÞ ¼ ½1� lAUðxÞ � vAUðxÞ;1� lALðxÞ � vALðxÞ�: ð7Þ

The complementary set Ac of an IVIFS A is defined as

Ac ¼ fhx; mAðxÞ;lAðxÞijx 2 Xg ¼ fhx; ½vALðxÞ; vAUðxÞ�; ½lALðxÞ;lAUðxÞ�ijx 2 Xg: ð8Þ

Definition 4 [9]. Let two IVIFSs A and B in the universe of discourse X = {x1,x2, . . . ,xn} be A = {hxi ,[lAL(xi),lAU(xi)],[vAL(xi),vAU(xi)]ijxi 2 X} and B = {hxi, [lBL(xi),lBU(xi)], [vBL(xi),vBU(xi)]ijxi 2 X}. Then the correlation coefficient of A and B isdefined by

kðA;BÞ ¼ CðA;BÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEðAÞ � EðBÞ

p ; ð9Þ

where the correlation of two IVIFSs A and B is given by

CðA;BÞ ¼ 12

Xn

i¼1

ðlALðxiÞlBLðxiÞ þ lAUðxiÞlBUðxiÞ þ vALðxiÞvBLðxiÞ þ vAUðxiÞvBUðxiÞÞ" #

; ð10Þ

and the informational intuitionistic energies of two IVIFSs A and B are given, respectively, by

EðAÞ ¼Xn

i¼1

l2ALðxiÞ þ l2

AUðxiÞ þ v2ALðxiÞ þ v2

AUðxiÞ2

; ð11Þ

EðBÞ ¼Xn

i¼1

l2BLðxiÞ þ l2

BUðxiÞ þ v2BLðxiÞ þ v2

BUðxiÞ2

: ð12Þ

The correlation coefficient of two IVIFSs A and B satisfies the following properties:

(i) 0 6 k(A,B) 6 1;(ii) k(A,B) = k(B,A);

(iii) k(A,B) = 1 if A = B.

3. Entropy measures of IVIFSs

To obtain entropy measures of IVIFS, we will transform each IVIFS into an IFS. In the following, we will denote the set ofall the IVIFSs in X by IVIFS(X).

Let p, q 2 [0,1] be two fixed numbers. For any A 2 IVIFS(X) we define the operator Hp,q [9] through

Hp;qðAÞ ¼ fhx;lALðxÞ þ pWlAðxÞ; mALðxÞ þ qWmAðxÞijx 2 Xg; ð13Þ

where WlA(x) = lAU(x) � lAL(x) and WmA (x) = mAU(x) � mAL(x).Based on the two entropy measures for IFSs [17] in the above section, for any A 2 IVIFS(X) we propose the following two

entropy measures of A:

I1ðAÞ ¼ sinp� ½1þlALðxÞþpWlAðxÞ� mALðxÞ�qWmAðxÞ�

4þ sin

p�½1�lALðxÞ�pWlAðxÞþmALðxÞþqWmAðxÞ�4

�1� �

� 1ffiffiffi2p�1

;

ð14Þ

I2ðAÞ ¼ cosp�½1þlALðxÞþpWlAðxÞ�mALðxÞ� qWmAðxÞ�

4þ cos

p�½1�lALðxÞ�pWlAðxÞþmALðxÞþ qWmAðxÞ�4

�1� �

� 1ffiffiffi2p�1

:

ð15Þ

J. Ye / Applied Mathematical Modelling 34 (2010) 3864–3870 3867

Theorem 2. For A, B 2 IVIFS(X) the above two measures of the interval-valued intuitionistic fuzzy entropy also satisfy thefollowing axiomatic requirements [17]:

(P1) I1(A) = I2(A) = 0 (minimum), iff A is a crisp set;(P2) I1(A) = I2(A) = 1 (maximum), iff lA(x) = mA(x), i.e., lAL(x) = mAL(x) and lAU(x) = mAU(x) for any x 2 X and p = q for p,q 2

[0,1];(P3) I1(A) 6 I1(B) and I2(A) 6 I2(B) if Hp,q(A) is less fuzzy than Hp,q(B), i.e., lAL(x) + pWlA(x) 6 lBL(x) + pWlB(x) and mAL(x) +

qWmA(x) P mBL(x) + qWmB(x) for lBL(x) + pWlB(x) 6 mBL(x) + qWmB(x) or lAL(x) + pWlA(x) P lBL(x) + pWlB(x) and mAL(x) +qWmA(x) 6 mBL(x) + qWmB(x) for lBL(x) + pWlB(x) P mBL(x) + qWmB(x) and any x 2 X;

(P4) I1(A) = I1(Ac) and I2(A) = I2(Ac).

Proof

(P1) When A is a crisp set, i.e., lAL(x) = lAU(x) = 0, mAL(x) = mAU(x) = 1 or lAL(x) = lAU(x) = 1, mAL(x) = mAU(x) = 0 for any x 2 X. Nomatter in which case, we have I1(A) = I2(A) = 0.

(P2) Let lA(x) = mA(x), i.e., lAL(x) = mAL(x) and lAU(x) = mAU(x) for x 2 X and p = q for p,q 2 [0,1]. From Eqs. (14) and (15) weobtain I1(A) = I2(A) = 1.

(P3) In order to show that Eqs. (14) and (15) fulfill the requirement of (P3), they suffice to prove that the followingfunctions:

F1ðx; yÞ ¼ sinp� ½1þ x� y�

4þ sin

p� ½1� xþ y�4

� 1� �

� 1ffiffiffi2p� 1

; ð16Þ

F2ðx; yÞ ¼ cosp� ½1þ x� y�

4þ cos

p� ½1� xþ y�4

� 1� �

� 1ffiffiffi2p� 1

; ð17Þ

where x,y 2 [0,1] and two functions F1 and F2 are increasing with respect to its first argument x and decreasing for y. Takingthe partial derivative of F1 and F2 with respect to x and y, respectively, yields

@F1ðx; yÞ@x

¼ p4

ffiffiffi2p� 1

� � cospð1þ x� yÞ

4� cos

pð1� xþ yÞ4

� �; ð18Þ

@F1ðx; yÞ@y

¼ p4

ffiffiffi2p� 1

� � cospð1� xþ yÞ

4� cos

pð1þ x� yÞ4

� �; ð19Þ

@F2ðx; yÞ@x

¼ p4

ffiffiffi2p� 1

� � sinpð1� xþ yÞ

4� sin

pð1þ x� yÞ4

� �; ð20Þ

@F2ðx; yÞ@y

¼ p4

ffiffiffi2p� 1

� � sinpð1þ x� yÞ

4� sin

pð1� xþ yÞ4

� �: ð21Þ

In order to find the critical point of F1 and F2, we set @F1ðx;yÞ@x ¼ 0; @F1ðx;yÞ

@y ¼ 0; @F2ðx;yÞ@x ¼ 0, and @F2ðx;yÞ

@y ¼ 0. By solving the criticalpoint xcp, we obtain

xcp ¼ y: ð22Þ

From Eqs. (18), (20), and (22), we have

@F1ðx; yÞ@x

P 0 for x 6 y and@F1ðx; yÞ

@x6 0 for x P y; ð23Þ

@F2ðx; yÞ@x

P 0 for x 6 y and@F2ðx; yÞ

@x6 0 for x P y: ð24Þ

For any x,y 2 [0,1], F1 and F2 are increasing with respect to x for x 6 y and decreasing when x P y. Similarly, we obtain

@F1ðx; yÞ@y

6 0 for x 6 y and@F1ðx; yÞ

@yP 0 for x P y; ð25Þ

@F2ðx; yÞ@y

6 0 for x 6 y and@F2ðx; yÞ

@yP 0 for x P y: ð26Þ

Let us now consider Eqs. (14) and (15) with A 6 B. Assume that the finite universe of discourse X is partitioned into two dis-joint sets X1 and X2 with X1 [ X2 = X. Let us further suppose that for x 2 X1, lA(x) + pWlA(x) 6 lB(x) + pWlB(x) 6 mB(x) +qWmB(x) 6 mA(x) + qWmA(x), while for x 2 X2, lA(x) + pWlA(x) P lB(x) + pWlB(x) P mB(x) + qWmB(x) P mA(x) + qWmA(x). Then,from the monotonicity of F1,F2, and Eqs. (14) and (15), we obtain that I1(A) 6 I1(B) and I2(A) 6 I2(B) when A 6 B.

3868 J. Ye / Applied Mathematical Modelling 34 (2010) 3864–3870

From the proposed interval-valued intuitionistic fuzzy entropy, the nearer lA(x) + pWlA(x) to mA(x) + qWmA(x) for x 2 X, thegreater the interval-valued intuitionistic fuzzy entropy of I1(A) and I2(A), and then when lA(x) + pWlA(x) is equal tomA(x) + qWmA(x) for x 2 X, the interval-valued intuitionistic fuzzy entropy reach the maximum entropy, i.e., I1(A) = I2(A) = 1.

(P4) It is clear that Ac = {hx,mA(x),lA( x)ijx 2 X} for x 2 X, i.e., lAc ðxÞ ¼ mAðxÞ ¼ ½mALðxÞ; mAUðxÞ� and mAc ðxÞ ¼ lAðxÞ ¼½lALðxÞ;lAUðxÞ�. By applying Eqs. (14) and (15), we have I1(A) = I1(Ac) and I2(A) = I2(Ac). h

4. Multicriteria fuzzy decision-making method

In this section, we present a handling method for multicriteria fuzzy decision-making problems with unknown informa-tion of criteria weights for alternatives.

Let A = {A1,A2, . . . ,Am} be a set of alternatives and let C = {C1,C2, . . . ,Cn} be a set of criteria. The evaluation value of criteriaon an alternative Ai is represented by the following IVIFS:

Ai ¼ fhCj; ½lAiLðCjÞ;lAiU

ðCjÞ�; ½vAiLðCjÞ; vAiUðCjÞ�ijCj 2 Cg; ð27Þ

where 0 6 lAiUðCjÞ þ vAiUðCjÞ 6 1; lAiL

ðCjÞP 0; vAiLðCjÞP 0; j ¼ 1;2; . . . ; n, and i = 1,2, . . . ,m. The IVIFS that is the pair ofintervals lAi

ðCjÞ ¼ ½aij; bij�; vAiðCjÞ ¼ ½cij; dij� for Cj 2 C is denoted by aij = ([aij,bij], [cij, dij]) for convenience. Here, the IVIFS is

usually elicited from the evaluated score to which the alternative Ai satisfies the criterion Cj by means of a score law anddata processing or from appropriate membership functions in practice. Therefore, we can elicit an interval-valued intuition-istic fuzzy decision matrix D = ( aij)m�n.

If the information about weight wj of the criterion Cj(j = 1,2, . . . ,n) is completely unknown, for determining the criterionweight from the decision matrix D = (aij)m�n we can establish an exact model of entropy weights [18]:

wj ¼1� Hj

n�Pn

j¼1Hj; ð28Þ

where wj 2 ½0;1�;Pn

j¼1wj ¼ 1;Hj is calculated by

Hj ¼1m

Xm

i¼1

sinp�½1þaijþpðbij�aijÞ� cij�qðdij� cijÞ�

4þ sin

p�½1�aij�pðbij�aijÞþ cijþqðdij� cijÞ�4

�1� �

� 1ffiffiffi2p�1

� �; ð29Þ

or Hj ¼1m

Xm

i¼1

cosp�½1þ aijþpðbij� aijÞ� cij�qðdij� cijÞ�

4þ cos

p�½1�aij�pðbij�aijÞþ cijþ qðdij� cijÞ�4

�1� �

� 1ffiffiffi2p�1

� �; ð30Þ

and 0 6 Hj 6 1(j = 1,2, . . . ,n).According to the entropy theory [18], if the entropy value for each criterion is smaller across alternatives, it should pro-

vide decision-makers with the useful information. Therefore, the criterion should be assigned a bigger weight; otherwise,such a criterion will be judged unimportant by most decision- makers. In other words, such a criterion should be evaluatedas a very small weight.

In multicriteria decision-making environments, the concept of ideal and anti-ideal points has been used to help identifythe best alternative in the decision set. Although the ideal alternative does not exist in real world, it does provide a usefultheoretical construct against which to evaluate alternatives. Here we define the ideal alternative A� ¼ fhCj;a�j ðCjÞijCj 2 C¼ fhCj; ½1;1�; ½0;0�ijCj 2 Cg for ‘‘excellence”. Then based on the correlation coefficient between IVIFSs proposed by Bustinceand Burillo [9], the correlation coefficient between an alternative Ai and the ideal alternative A* with entropy weights for cri-teria can be measured by the weighted correlation coefficient Wi(i = 1,2, . . . ,m) [13]:

WiðA�;AiÞ ¼CiðA�;AiÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE�ðA�ÞEiðAiÞ

p ¼Pn

j¼1wjðaijþbijÞ

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPnj¼1

wjða2ijþb2

ijþc2ijþd2

ijÞ2

r : ð31Þ

The larger the value of weighted correlation coefficient Wi, the better the alternative Ai, as the alternative Ai is closer to theideal alternative A*. Therefore, all the alternatives can be ranked according to the value of the weighted correlation coeffi-cients so that the best alternative can be selected.

It is easy to check that weighted correlation coefficient Wi(i = 1,2, . . . ,m) of A* and Ai has the following properties:

(i) 0 6Wi(A*,Ai) 6 1,(ii) Wi(A*,Ai) = Wi(Ai,A*),

(iii) Wi(A*,Ai) = 1 if A* = Ai.

The decision procedure for the proposed method can be summarized as follows:

Step 1: Calculate the entropy weights of criteria by using Eqs. (28) and (29) or (30) from the decision matrix D = (aij)m �n

given by experts.Step 2: Calculate the weighted correlation coefficient Wi(A*,Ai)(i = 1,2, . . . ,m) by using Eq. (31).Step 3: Rank the alternatives according to the obtained correlation coefficients, and then obtain the best choice.

J. Ye / Applied Mathematical Modelling 34 (2010) 3864–3870 3869

5. Applied examples

In this section, we adopt two decision-making problems that were discussed in [14,15] to demonstrate how to apply theproposed approach.

5.1. Example 1

Suppose there is an investment company, which wants to invest a sum of money in the best option [14]. There is a panelwith four possible alternatives to invest the money: (1) A1 is a car company; (2) A2 is a food company; (3) A3 is a computercompany; (4) A4 is an arms company. The investment company must take a decision according to the following three crite-ria: (1) C1 is the risk analysis; (2) C2 is the growth analysis; (3) C3 is the environmental impact analysis. The four possiblealternatives are to be evaluated using the interval-valued intuitionistic fuzzy information by the expert under the abovethree criteria, as listed in the following decision matrix D:

D ¼

ð½0:4;0:5�; ½0:3;0:4�Þ ð½0:4;0:6�; ½0:2;0:4�Þ ð½0:1;0:3�; ½0:5;0:6�Þ

ð½0:6;0:7�; ½0:2;0:3�Þ ð½0:6;0:7�; ½0:2;0:3�Þ ð½0:4;0:7�; ½0:1;0:2�Þ

ð½0:3;0:6�; ½0:3;0:4�Þ ð½0:5;0:6�; ½0:3;0:4�Þ ð½0:5;0:6�; ½0:1;0:3�Þ

ð½0:7;0:8�; ½0:1;0:2�Þ ð½0:6;0:7�; ½0:1;0:3�Þ ð½0:3;0:4�; ½0:1;0:2�Þ

2666664

3777775:

Each element of this matrix is an IVIFS, representing the expert’s assessment as to what degree an alternative is and is notan excellent investment as per an criterion. For instance, the top-left cell, ([0.4,0.5], [0.3,0.4]), reflects the expert’s belief thatalternative A1 is an excellent investment from a risk perspective (C1) with a margin of 40–50% and A1 is not an excellentchoice given its risk profile (C1) with a chance between 30% and 40%. The proposed method is applied to solve this problemaccording to the following computational procedure:

Step 1: Take p = q = 0.5 (medium value). By using Eqs. (28) and (29) or (30), we can obtain the following entropy weights ofthe three criteria:

w1 ¼ 0:3404; w2 ¼ 0:3434; and w3 ¼ 0:3163:

Step 2: By applying Eq. (31), we can compute Wi(A*, Ai)(i = 1,2,3,4) as

W1ðA�;A1Þ ¼ 0:6577; W2ðA�;A2Þ ¼ 0:9262; W3ðA�;A3Þ ¼ 0:8381; and W4ðA�;A4Þ ¼ 0:9186:

Step 3: From the weighted correlation coefficients between the alternatives and the ideal alternative, the ranking order is A2

�A4�A3�A1.

Therefore, we can see that the ranking order is in agreement with Ye’s result [14] and the alternative A2 is the best choice.

5.2. Example 2

Assume that a fund manager in a wealth management firm is assessing four potential investment opportunities, i.e., theset of alternatives is A = (A1,A2,A3,A4). The firm mandates that the fund manager has to evaluate each investment againstfour criteria: risk (C1), growth (C2), socio-political issues (C3), and environmental impacts (C4). In addition, the fund manageris only comfortable with providing his/her assessment of each alternative on each criterion as an IVIFS and the decision ma-trix [15] is as follows:

D ¼

ð½0:42;0:48�; ½0:4; 0:5�Þ ð½0:6; 0:7�; ½0:05; 0:25�Þ ð½0:4;0:5�; ½0:2;0:5�Þ ð½0:55;0:75�; ½0:15;0:25�Þ

ð½0:4;0:5�; ½0:4;0:5�Þ ð½0:5; 0:8�; ½0:1;0:2�Þ ð½0:3;0:6�; ½0:3;0:4�Þ ð½0:6;0:7�; ½0:1;0:3�Þ

ð½0:3;0:5�; ½0:4;0:5�Þ ð½0:1; 0:3�; ½0:2;0:4�Þ ð½0:7;0:8�; ½0:1;0:2�Þ ð½0:5;0:7�; ½0:1;0:2�Þ

ð½0:2;0:4�; ½0:4;0:5�Þ ð½0:6; 0:7�; ½0:2;0:3�Þ ð½0:5;0:6�; ½0:2;0:3�Þ ð½0:7;0:8�; ½0:1;0:2�Þ

2666664

3777775:

Each element of this matrix is an IVIFS, representing the fund manager’s assessment as to what degree an alternative is and isnot an excellent investment as per an criterion. For instance, the top-left cell, ([0.42,0.48], [0.4,0.5]), reflects the fund man-ager’s belief that alternative A1 is an excellent investment from a risk perspective (C1) with a margin of 42–48% and A1 is notan excellent choice given its risk profile (C1) with a chance between 40% and 50%. The proposed method is applied to solvethis problem according to the following computational procedure:

Step 1: Take p = q = 0.5 (medium value). By using Eqs. (28) and (29) or (30), we can obtain the following entropy weights ofthe four criteria:

w1 ¼ 0:2319; w2 ¼ 0:2562; w3 ¼ 0:2514; and w4 ¼ 0:2605:

3870 J. Ye / Applied Mathematical Modelling 34 (2010) 3864–3870

Step 2: By applying Eq. (31), we can compute Wi(A*, Ai)(i = 1,2,3,4) as

W1ðA�;A1Þ ¼ 0:8501; W2ðA�;A2Þ ¼ 0:8466; W3ðA�;A3Þ ¼ 0:7966; and W4ðA�;A4Þ ¼ 0:8560:

Step 3: From the weighted correlation coefficients between the alternatives and the ideal alternative, the ranking order is A4

�A1�A2�A3.

Therefore, we can see that the alternative A4 is the best choice, which is the same result as [15].

6. Conclusion

In this paper, we have investigated the multicriteria fuzzy decision making problems with unknown information on cri-teria weights to which the valuation values of criteria on an alternative are given in terms of IVIFSs. For dealing with theunknown information about criteria weights, two entropy measures of IVIFSs and a model based on the entropy weightswere established so as to determine the entropy weights with respect to a decision matrix provided as IVIFSs. Then, we pro-posed an evaluation formula of the weighted correlation coefficient between an alternative and the ideal alternative forranking the alternatives and selecting the most desirable one(s). Finally, two applied examples were given to demonstratethe applicability and benefit of this proposed mehtod: it is capable for handling the multicriteria fuzzy decision-makingproblems with completely unknown weights for criteria. The techniques proposed in this paper can efficiently help thedecision-maker.

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