Applied Mathematical Modelling 36 (2012) 4466–4472

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Applied Mathematical Modelling

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

Multicriteria decision-making method using the Dice similaritymeasure based on the reduct intuitionistic fuzzy sets of interval-valuedintuitionistic fuzzy sets

Jun Ye ⇑Faculty of Engineering, Shaoxing University, 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 1 April 2011Received in revised form 10 November 2011Accepted 27 November 2011Available online 3 December 2011

Keywords:Interval-valued intuitionistic fuzzy setReduct intuitionistic fuzzy setDice similarity measureMulticriteria decision making

0307-904X/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.apm.2011.11.075

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

This paper proposes the concept of the reduct intuitionistic fuzzy sets of interval-valuedintuitionistic fuzzy sets (IVIFSs) with respect to adjustable weight vectors and the Dicesimilarity measure based on the reduct intuitionistic fuzzy sets to explore the effects ofoptimism, neutralism, and pessimism in decision making. Then a decision-making methodwith the pessimistic, optimistic, and neutral schemes desired by the decision maker isestablished by combining adjustable weight vectors and the Dice similarity measure forIVIFSs. The proposed decision-making method is more flexible and adjustable in practicalproblems and can determine the ranking order of alternatives and the optimal one(s), sothat it can overcome the difficulty of the ranking order and decision making when thereexist the same measure values of some alternatives in some cases. This adjustable featurecan provide the decision maker with more selecting schemes and actionable results for thedecision-making analysis. Finally, two illustrative examples are employed to show the fea-sibility of the proposed method in practical applications.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

The similarity measure is one of the important tools for degree of similarity between objects. Functions expressing thedegree of similarity of items or sets are used in physical anthropology, numerical taxonomy, ecology, information retrieval,psychology, citation analysis, and automatic classification. In fact, the degree of similarity or dissimilarity between the ob-jects under study plays an important role. In vector space, especially the Jaccard, Dice, and cosine similarity measures [1–3]are often used in information retrieval, citation analysis, and automatic classification. However, these similarity measures donot deal with the similarity measures for vague information. Therefore, Ye [4] proposed the cosine similarity measure be-tween intuitionistic fuzzy sets (IFSs) and applied it to pattern recognition and medical diagnosis. Then, Ye [5] proposedthe Jaccard, Dice, and cosine similarity measures between trapezoidal intuitionistic fuzzy numbers (TIFNs) that are treatedas continuous and applied them to multicriteria group decision-making problems. Thus, the mentioned decision-makingmethods are one single measure rather than the multimeasures of optimism, neutralism, and pessimism. The concepts ofoptimism and pessimism have brought a great deal of research interest in decision making because the decision-maker’smethods of evaluating alternatives and making decisions are guided by subjective judgments. Yager [6] developed an atti-tudinal fuzzy measure that is generated with the aid of cardinality index for the sake of attitudinal characterization, includ-ing optimistic, neutral, and pessimistic characters. The risk attitude of a decision maker, as proposed by Yager, is defined in

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J. Ye / Applied Mathematical Modelling 36 (2012) 4466–4472 4467

one dimension, in which optimism and pessimism are two relative extremes, where a person who does not have an optimis-tic inclination is regarded as a neutralist or pessimist. Recently, Chen [7] proposed a useful method of relating optimism andpessimism to multiple criteria decision analysis in the context of IFSs based on the unipolar bivariate model to determine thepreference order of decision alternatives. There is little doubt that optimism, neutralism, and pessimism are important forpractice; nevertheless, the specific nature of these concepts has not yet been clearly delineated in multicriteria decision mak-ing reality. To realize the flexible schemes for decision making problems in interval-valued intuitionistic fuzzy setting, weshould use an adjustable framework to solve interval-valued intuitionistic fuzzy multicriteria decision-making problems.On the other hand, in vector similarity measures, since the cosine formula [3] is defined as the inner product of thesetwo vectors divided by the product of their lengths, it is undefined when one vector is zero. In this case, the Dice similaritymeasure cannot induce this undefined situation, which is its advantage. Therefore, the purpose of this study is to propose thereduct IFS concept of interval-valued intuitionistic fuzzy sets (IVIFSs) with respect to adjustable weight vectors and the Dicesimilarity measure based on the reduct IFSs of IVIFSs to explore the effects of optimism, neutralism, and pessimism in deci-sion making. By combining adjustable weight vectors and the Dice similarity measure for IVIFSs, we establish a decision-making method with the three types of optimism, neutralism, and pessimism desired by the decision maker.

The rest of paper is organized as follows. Section 2 gives some necessary concepts of the Dice similarity measure [2] invector space. The weighted reduct IFS of IVIFS with respect to adjustable weight vectors is presented, specifically the pessi-mistic, optimistic, and neutral reduct IFSs are defined in Section 3. The Dice similarity measure and the weighted Dice sim-ilarity measure based on the reduct IFSs of IVIFSs are proposed in Section 4, which are derived from an extension of the Dicesimilarity measure [2] in vector space. In Section 5, a decision-making method with the adjustable schemes of optimism,neutralism, and pessimism desired by the decision maker is established by means of the Dice similarity measure basedon the reduct IFSs of IVIFSs. Two illustrative examples are presented to illustrate the developed approach in Section 6. Final-ly, some final remarks and future work are given in Section 7.

2. Dice similarity measure

As mentioned above, the Dice similarity measure cannot induce this undefined situation when one vector is zero, whichovercomes the disadvantage of the cosine similarity measure. Therefore, the concept of the Dice similarity measure is intro-duced in the section.

Let X = (x1 x2, . . . ,xn) and Y = (y1 y2, . . . ,yn) be the two vectors of length n where all the coordinates are positive. Then theDice similarity measure [2] is defined as follows:

D ¼ 2X � YkXk2

2 þ kYk22

¼ 2Pn

i¼1xiyiPni¼1x2

i þPn

i¼1y2i

; ð1Þ

where X � Y ¼Pn

i¼1xiyi is the inner product of the vectors X and Y and kXk2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn

i¼1x2q

and kYk2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn

i¼1y2q

are the Euclideannorms of X and Y (also called the L2 norms).

The Dice similarity measure takes value in the interval [0, l]. However, it is undefined if xi = yi = 0 (i = 1,2, . . . ,n). In thiscase, let the Dice measure value be zero when xi = yi = 0 (i = 1,2, . . . ,n).

3. Reduct IFSs of IVIFSs

3.1. Some concepts of IFSs and IVIFSs

As a generalization of an ordinary Zadeh fuzzy set, the notion of IFS was introduced for the first time by Atanassov [8]. AnIFS A in the universe of discourse X is given by

A ¼ fhx;lAðxÞ; mAðxÞijx 2 Xg;

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 non-membership 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,pA(x) is called Atanassov’s intuitionistic index of the element x in theset A [8]. It is a hesitancy degree of x to A. It is obvious that 0 6 pA(x) 6 1, x 2 X.

Then, the notion of IVIFS was introduced by Atanassov and Gargov [9] as a generalization of an IFS.Let D[0,1] be the set of all closed subintervals of the interval [0,1] and X(–Ø) be a given set. Following Atanassov and

Gargov [9], an IVIFS A in X is an expression given by

A ¼ fhx;lAðxÞ; vAðxÞijx 2 Xg;

where lA(x): X ? D[0,1], vA(x): 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) and vA(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,respectively, denoted by lAL(x), lAU(x), vAL(x), and vAU(x). We can denote by

4468 J. Ye / Applied Mathematical Modelling 36 (2012) 4466–4472

A ¼ fhx; ½lALðxÞ;lAUðxÞ�; ½mALðxÞ; mAUðxÞ�ijx 2 Xg;

where 0 6 lAU(x) + vAU(x) 6 1, lAL(x) P 0, vAL(x) P 0.For each element x we can compute the unknown degree (hesitancy degree) of an intuitionistic fuzzy interval of x 2 X in

the set A defined as follows:

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

3.2. Concept of the reduct IFSs of IVFSs

IVIFSs are the extensions of IFSs and interval valued fuzzy sets (IVFSs), then IFSs, IVFSs, and IVIFSs are all the generaliza-tions of the notion of fuzzy sets. Therefore, an algorithm is introduced is to transform IVIFSs into the reduct IFSs with respectto the opinion weighting vector. The concept of the reduct IFS of an IVIFS A is proposed as follows.

Let A be an IVIFS in a universe of discourse X and two weight vectors be P = (p1,p2) and Q = (q1,q2) for p1, p2, q1, q2 2 [0,1],p1 + p2 = 1, and q1 + q2 = 1. Then, the weighted reduct IFS of the IVIFS A with respect to the adjustable weight values of p1, p2,q1, and q2 is defined by

Aw ¼ fhx;p1lALðxÞ þ p2lAUðxÞ; q1mALðxÞ þ q2mAUðxÞijx 2 Xg: ð2Þ

By adjusting the value of p1, p2, q1, and q2, an IFIVS A can be converted into the reduct IFS desired by a decision maker.Specifically, if p1 = 1, p2 = 0, q1 = 0, and q2 = 1, we have the pessimistic reduct IFS defined by

A� ¼ fhx;lALðxÞ; mAUðxÞijx 2 Xg: ð3Þ

If p1 = 0, p2 = 1, q1 = 1, and q2 = 0, we have the optimistic reduct IFS defined by

Aþ ¼ fhx;lAUðxÞ; mALðxÞijx 2 Xg: ð4Þ

If p1 = p2 = q1 = q2 = 0.5, we have the neutral reduct IFS defined by

AN ¼ fhx; ðlALðxÞ þ lAUðxÞÞ=2; ðmALðxÞ þ mAUðxÞÞ=2ijx 2 Xg: ð5Þ

Thus the models, which can transform IVIFSs into the reduct IFSs, establish the optimistic, neutral, and pessimistic effects.Therefore, the optimism is defined as a bias in perceptions and expectancies in favor of the positive feature, while the pes-simism is defined as a negative bias, then the neutralism is defined as an unbiased feature. Accordingly, under some pref-erence situations, optimists expect positive subjective judgments and further reconstruct the IFS with more favorableoutcomes, whereas pessimists expect negative subjective judgments and reconstruct the IFS with more unfavorable out-comes. However, neutralists expect neutral subjective judgments and reconstruct the IFS with average outcomes.

4. Dice similarity measure based on the reduct IFSs of IVIFSs

In this section, the Dice similarity measure based on the reduct IFSs of IVIFSs is proposed as a generalization of the Dicesimilarity measure in vector space [2].

Let A and B be two IVIFSs in the universe of discourse X = {x1,x2, . . . ,xn} and two weight vectors be P = (p1,p2) andQ = (q1,q2), p1, p2, q1, q2 2 [0,1], p1 + p2 = 1, and q1 + q2 = 1. Then, the weighted reduct IFSs of the IVIFSs A and B with respectto the adjustable weight values of p1, p2, q1, and q2 are given as follows:

Aw ¼ fhxi;p1lALðxiÞ þ p2lAUðxiÞ; q1mALðxiÞ þ q2mAUðxiÞijxi 2 Xg;Bw ¼ fhxi;p1lBLðxiÞ þ p2lBUðxiÞ; q1mBLðxiÞ þ q2mBUðxiÞijxi 2 Xg:

Let lAi = p1lAL(xi) + p2lAU(xi), mAi = q1mAL(xi) + q2mAU(xi), lBi = p1lBL(xi) + p2lBU(xi), and mBi = q1mBL(xi) + q2mBU(xi), i = 1,2, . . . ,n.Thus, the weighted reduct IFSs of the IVIFSs A and B can be rewritten as

Aw ¼ fhxi;lAi; mAiijxi 2 Xg;Bw ¼ fhxi;lBi; mBiijxi 2 Xg:

By adjusting the value of p1, p2, q1 and q2, two IFIVSs A and B can be converted into the reduct IFSs Aw and Bw.Then, the IFSs Aw and Bw can be considered, respectively, as n vector representations with the two elements. Based on the

extension of the Dice similarity measure [2], the Dice similarity measure between A and B is proposed in the vector space asfollows:

DIVIFSðA;BÞ ¼1n

Xn

i¼1

2ðlAilBi þ mAimBiÞl2

Ai þ m2Ai þ l2

Bi þ m2Bi

: ð6Þ

The Dice similarity measure between two IVIFSs A and B satisfies the following properties:

(P1) 0 6 DIVIFS(A,B) 6 1;

J. Ye / Applied Mathematical Modelling 36 (2012) 4466–4472 4469

(P2) DIVIFS(A,B) = DIVIFS(B,A);(P3) DIVIFS(A,B) = 1 if and only if A = B, i.e., lA(xi) = lB(xi) and mA(xi) = mB(xi) for i = 1, 2, . . . ,n.

Proof

(P1) Let us consider the ith item of the summation in Eq. (6):

DiðAi;BiÞ ¼2ðlAilBi þ mAimBiÞ

l2Ai þ m2

Ai þ l2Bi þ m2

Bi

: ð7Þ

It is obvious that Di(Ai,Bi) P 0 and l2Ai þ m2

Ai þ l2Bi þ m2

Bi P 2ðlAilBi þ mAimBiÞ according to the inequality (a2 + b2 P 2ab). Thus,0 6 Di(Ai,Bi) 6 1.From Eq. (6), the summation of n terms is 0 6 DIVIFS(A,B) 6 1.

(P2) It is obvious that the property is true.(P3) When A = B, there are lAi = lBi and mAi = mBi for p1, p2, q1, and q2, i.e., lA(xi) = lB(xi) and mA(xi) = mB(xi) for i = 1,2, . . . ,n. So

there is DIVIFS(A,B) = 1. When Dvs(A,B) = 1, there are lAi = lBi and mAi = mBi for p1, p2, q1, and q2, i.e., lA(xi) = lB(xi) andmA(xi) = mB(xi) for i = 1,2, . . . ,n. So there is A = B.

Therefore, we have finished the proofs. h

Then, the Dice similarity measure between IVIFSs A and B degenerates to the Dice similarity measure between IFSs A andB without the adjustable weight values of p1, p2, q1 and q2 when lAL(xi) = lAU(xi), mAL(xi) = mAU(xi), lBL(xi) = lBU(xi), andmBL(xi) = mBU(xi) for i = 1,2, . . . ,n, which is given as follows:

DIFSðA;BÞ ¼1n

Xn

i¼1

2ðlAðxiÞlBðxiÞ þ mAðxiÞmBðxiÞÞl2

AðxiÞ þ m2AðxiÞ þ l2

BðxiÞ þ m2BðxiÞ

: ð8Þ

If we consider the weight of xi, the weighted Dice similarity measure between IVIFSs A and B is proposed as follows:

WIVIFSðA;BÞ ¼Xn

i¼1

wi2ðlAilBi þ mAimBiÞ

l2Ai þ m2

Ai þ l2Bi þ m2

Bi

; ð9Þ

where wi 2 [0,1], i = 1,2, . . . ,n, andPn

i¼1wi ¼ 1, and lAi = p1lAL(xi) + p2lAU(xi), mAi = q1mAL(xi) + q2mAU(xi), lBi = p1lBL(xi) +p2lBU(xi), and mBi = q1mBL(xi) + q2mBU(xi).

If we take wi = 1/n, i = 1,2, . . . ,n, then there is WIVIFS(A,B) = DIVIFS(A,B).Obviously, the weighted Dice similarity measure between IVIFSs A and B also satisfies the following properties:

(P1) 0 6WIVIFS(A,B) 6 1;(P2) WIVIFS(A,B) = WIVIFS(B,A);(P3) WIVIFS(A,B) = 1 if and only if A = B, i.e. lA(xi) = lB(xi) and mA(xi) = mB(xi) for i = 1,2, . . . ,n.

Similarly to the previous proof method, we can prove that the properties (P1)–(P3).By adjusting the values of p1, p2, p3 and p4, we can obtain a weighted Dice similarity measure between IVIFSs A and B

desired by a decision maker. Specifically let p1 = 1, p2 = 0, q1 = 0, and q2 = 1, p1 = 0, p2 = 1, q1 = 1, and q2 = 0, andp1 = p2 = q1 = q2 = 0.5, respectively, then we have three choosing schemes: pessimistic, optimistic, and neutral weighted Dicesimilarity measures between IVIFSs A and B desired by a decision maker.

However, when lAL(xi) = lAU(xi), mAL(xi) = mAU(xi), lBL(xi) = lBU(xi), and mBL(xi) = mBU(xi) for i = 1,2, . . . ,n, the weighted Dicesimilarity measure between IVIFSs A and B degenerates to the weighted Dice similarity measure between IFSs A and B with-out the adjustable weight value of p1, p2, q1, and q2, which is given as follows:

WIFSðA;BÞ ¼Xn

i¼1

wi2ðlAðxiÞlBðxiÞ þ mAðxiÞmBðxiÞÞ

l2AðxiÞ þ m2

AðxiÞ þ l2BðxiÞ þ m2

BðxiÞ: ð10Þ

5. Decision-making method with optimistic, neutral, and pessimistic schemes

The wide variety of possible relationships among the alternatives in decision-making problems motivates our interest inseeking flexible/adjustable methods that can be used to model these various possibilities. In this section, we present the Dicesimilarity measure based on the reduct IFSs of IVIFSs for decision-making problems with optimistic, neutral, and pessimisticschemes.

4470 J. Ye / Applied Mathematical Modelling 36 (2012) 4466–4472

Let A = {A1,A2, . . . ,Am} be a set of alternatives and C = {C1,C2, . . . ,Cn} be a set of criteria. Assume that the weight of the cri-terion Cj (j = 1,2, . . . ,n), entered by the decision maker, is wj, wj 2 [0,1] and

Pnj¼1xj ¼ 1. In this case, the characteristic of the

alternative Ai is presented by the following IVIFS:

Ai ¼ fhCj; ½lAiLðCjÞ;lAiU

ðCjÞ�; ½vAiLðCjÞ; vAiUðCjÞ�ijCj 2 Cg;

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 value that is the pairof intervals lAi

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

valued intuitionistic fuzzy value is usually elicited from the evaluated score to which the alternative Ai satisfies the criterionCj by means of a score law and data processing or from appropriate membership functions in practice. Therefore, we canelicit a decision matrix D = (dij)m�n.

For the ranking order of the alternatives in the decision-making problem, we define an ideal IVIFS for each criterion in theideal alternative A⁄ as d�j ¼ ð½1;1�; ½0; 0�Þ for ‘‘excellence’’. Then, by considering criteria weights and applying Eq. (6), we canobtain the weighted Dice similarity measure between the ideal alternative A⁄ and an alternative Ai (i = 1,2, . . . ,m):

WiðA�;AiÞ ¼Xn

j¼1

wj2ðp1aij þ p2bijÞ

1þ ðp1aij þ p2bijÞ2 þ ðq1cij þ q2dijÞ2; ð11Þ

where wj 2 ½0;1�;Pn

j¼1xj ¼ 1; p1; p2; q1; q2 2 ½0;1�; p1 þ p2 ¼ 1, and q1 + q2 = 1.The bigger the value of Wi(A⁄,Ai), the better the alternative Ai, as the alternative Ai is closer to the ideal alternative A⁄.

Therefore, the alternatives can be ranked according to the Dice similarity measure by adjusting the value of p1, p2, q1, andq2 or choosing a scheme for optimistic or pessimistic or neutral attitude desired by a decision maker so that the best alter-native can be selected.

In this decision-making method, there is a large variety of choosing weight vectors that can be used to find the optimalchoice, hence this handling method has great flexibility and adjustable capability. Specifically, if p1 = 1, p2 = 0, q1 = 0 andq2 = 1, p1 = 0, p2 = 1, q1 = 1 and q2 = 0, and p1 = p2 = q1 = q2 = 0.5, respectively, we have the pessimistic, optimistic, neutralschemes adopted by the decision maker. Usually the decision making with pessimistic, optimistic, neutral preferences isa useful method in practical applications [6]. Therefore, in this paper we discuss the Dice similarity measure based on thereduct IFSs of IVIFSs for the decision-making problem with optimistic, neutral, and pessimistic schemes.

As pointed out in [10], many decision making problems are essentially humanistic and subjective in nature, hence thereactually does not exist a unique or uniform criterion for decision making in an imprecise environment. Furthermore, the pro-posed decision-making method can overcome the difficulty of the ranking order and decision-making when there exist thesame evaluation values of some alternatives in the decision making process. This adjustable feature can provide the decisionmaker with more selecting schemes and can produce actionable results for the decision-making analysis.

6. Illustrative examples

For the comparative convenience, the two examples of multicriteria decision-making problems discussed in [11–13] areused as the demonstration of the applications of the proposed decision-making method in a realistic scenario. The influencesof optimism, neutralism, and pessimism on the ranking orders of alternatives are discussed in the decision-making process.

6.1. Example 1

Let us consider the decision-making problem discussed in [11,12]. There is an investment company, which wants to in-vest a sum of money in the best option. There is a panel with four possible alternatives to invest the money: (1) A1 is a carcompany; (2) A2 is a food company; (3) A3 is a computer company; (4) A4 is an arms company. The investment company musttake a decision according to the following three criteria: (1) C1 is the risk analysis; (2) C2 is the growth analysis; (3) C3 is theenvironmental impact analysis. The criteria weight is given as W = (0.35,0.25,0.40). The four possible alternatives Ai

(i = 1,2,3,4) are to be evaluated using the interval-valued intuitionistic fuzzy information by the decision maker underthe above three 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�Þð½0:6;0:7�; ½0:2;0:3�Þ ð½0:6;0:7�; ½0:2; 0:3�Þ ð½0:4;0:7�; ½0:1;0:2�Þð½0:3;0:6�; ½0:3;0:4�Þ ð½0:5;0:6�; ½0:3; 0:4�Þ ð½0:5;0:6�; ½0:1;0:3�Þð½0:7;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

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Then, we utilize the developed approach to obtain the ranking order of the alternatives and the most desirable one(s).By applying Eq. (11), according to three choosing schemes we can obtain the computing results of all the alternatives as

shown in Table 1. For the comparative convenience, the ranking orders of different methods are shown in Table 2. Therefore,from Table 2 we can see that the four ranking orders of them are the same and the alternative A2 is the best choice in theproposed method and Ye’s method [11]. But Ye’s method [12] is another ranking order and the alternative A4 is the bestchoice. The ranking orders may be different according to different methods because each algorithm focuses on different

Table 2Ranking orders of the alternatives for different methods.

Method Ranking order

Ye’s method [11] A2, A4, A3, A1

Ye’s method [12] A4, A2, A3, A1

Proposed method (Pessimist) A2, A4, A3, A1

Proposed method (Optimist) A2, A3, A4, A1

Proposed method (Neutralist) A2, A4, A3, A1

Table 1Computing results for three choosing schemes.

Choosing scheme W1(A⁄,A1) W2(A⁄,A2) W3(A⁄,A3) W4(A⁄,A4)

Pessimist 0.4220 0.7632 0.6438 0.7395Optimist 0.6546 0.9224 0.8469 0.8462Neutralist 0.5435 0.8573 0.7585 0.7980

J. Ye / Applied Mathematical Modelling 36 (2012) 4466–4472 4471

point of view. However, the proposed method has different orders and can provide the decision maker with more selectingschemes as compared with Ye’s methods [11,12].

6.2. Example 2

We consider the same problem discussed in [13]. Assume that a fund manager in a wealth management firm is assessingfour potential investment opportunities, i.e., the set of alternatives is A = {A1,A2,A3,A4}. The firm mandates that the fund man-ager has to evaluate each investment against four criteria: risk (C1), growth (C2), socio-political issues (C3), and environmen-tal impacts (C4), and then the criteria weight W= (0.2319,0.2562,0.2514,0.2605) are given by the decision-maker. Inaddition, the fund manager is only comfortable with providing his/her assessment of each alternative on each criterion asan IVIFS and the decision matrix [13] 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�Þð½0:4;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�Þð½0:3;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�Þð½0:2;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

26664

37775:

Each element of this matrix is an IVIFS, representing the fund manager’s assessment as to what degree an alternative isand is not an excellent investment as per a criterion. For instance, the top-left cell, ([0.42,0.48], [0.4,0.5]), reflects the fundmanager’s belief that alternative A1 is an excellent investment from a risk perspective (C1) with a margin of 42–48% and A1 isnot an excellent choice given its risk profile (C1) with a chance between 40% and 50%. Then, the proposed method is appliedto obtain the ranking order of the alternatives and the most desirable one(s).

By applying Eq. (11) and the three selecting schemes desired by the decision maker, we can obtain the computing resultsof all the alternatives as shown in Table 3. For comparison with different decision-making methods, we give the ranking or-ders of the proposed method and Ye’s methods [13] as shown in Table 4.

As we can see from Table 4, with the change of the choosing schemes, the ranking orders may be different, and then thedecision maker can properly select the desirable alternative according to his/her choosing scheme and actual need. Then, theproposed method can provide the decision maker with more selecting schemes as compared with Ye’s method [13].

The above examples demonstrate that implementing the proposed decision-making method only requires few computa-tion efforts; moreover, this method needs not complicated procedures to handle the influences of optimism, neutralism, andpessimism in decision making and produces actionable results in the decision-making analysis. The proposed decision-mak-ing method provides more selecting schemes for many practical applications because it is the multimeasures of optimism,neutralism, and pessimism rather than one single measure in interval-valued intuitionistic fuzzy decision-making problems.

7. Conclusion and future work

This paper presented the concept of the reduct IFSs of IVIFSs with respect to adjustable weighting vectors, specificallydefined the pessimistic, optimistic, and neutral reduct IFSs, and the Dice similarity measure and weighted Dice similaritymeasure based on the reduct IFSs of IVIFSs. A decision-making method with optimism, neutralism, and pessimism wasestablished by use of the Dice similarity measure based on the reduct IFSs. By choosing the pessimistic, optimistic, and neu-tral schemes desired by the decision maker, the proposed decision-making method is more flexible and adjustable in prac-tical problems and can determined the ranking order of alternatives and the best one(s), so that it can overcome the difficultyof the ranking order and decision making when there exist the same measure values of some alternatives in some cases.

Table 3Computing results for three choosing schemes.

Choosing scheme W1(A⁄,A1) W2(A⁄,A2) W3(A⁄,A3) W4(A⁄,A4)

Pessimist 0.8419 0.8641 0.7874 0.8431Optimist 0.7053 0.6664 0.5796 0.7099Neutralist 0.7814 0.7814 0.6909 0.7811

Table 4Ranking orders of the alternatives for different methods.

Method Ranking order

Ye’s method [13] A4, A1, A2, A3

Proposed method (Pessimist) A2, A4, A1, A3

Proposed method (Optimist) A4, A1, A2, A3

Proposed method (Neutralist) A1(A2), A4, A3

4472 J. Ye / Applied Mathematical Modelling 36 (2012) 4466–4472

Thus, the proposed decision-making method can provide the decision maker with more selecting schemes and can produceactionable results for the decision-making analysis because it is the multimeasures of optimism, neutralism, and pessimismrather than one single measure in interval-valued intuitionistic fuzzy decision-making problems. Two illustrative examplesdemonstrated the feasibility of the proposed method in practical applications.

However, this paper addresses the issue of the decision-making method using the Dice similarity measure based on thereduct IFSs of IVIFSs to treat the influences of optimism, neutralism, and pessimism on the multicriteria decision-makingproblem. Thus the present findings of this study also provide some evidence for the validity of assessing optimism, neutral-ism, and pessimism separately. For this purpose, future work should continue to investigate the influences of decision-mak-ing under various weight vectors and to find optimal weight vectors for some optimization models.

Acknowledgements

The author is very grateful to the anonymous referees for their insightful and constructive comments and suggestions.

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