A DISSONANCE REDUCTION METHOD FOR INTUITIONISTIC FUZZY...

Pan-Pacific Management Review 2011, Vol.14, No.1: 1-27

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A DISSONANCE REDUCTION METHOD FOR INTUITIONISTIC FUZZY MULTI-CRITERIA

DECISION-MAKING PROBLEMS

TING-YU CHEN Department of Industrial and Business Management College of Management,

Chang Gung University

YI-JEN LI Graduate Institute of Business Administration College of Management,


HSIAO-PIN WANG Graduate Institute of Industrial and Business Management College of Management,


ABSTRACT Considering that cognitive dissonance is regarded as a momentous psychological

factor in the decision making, this study proposes new techniques not only to tackle

multi-criteria decision making problems but also to reduce the cognitive dissonance

yielded by the decision maker under intuitionistic fuzzy environment. By applying the

intuitionistic fuzzy weighted averaging (IFWA) operator and the score function, the

proposed method derives the optimal weights of criteria and generates the priority of

alternatives from the mathematical programming. In order to diminish the dissonance,

the attractiveness of alternatives is magnified as large as possible in the programming

problem based on the Euclidean distance. A numerical example is given to demonstrate

the detailed calculating procedure; besides, an empirical case concerning a digital

camera selection problem is employed to ascertain the feasibility of the developed

method. According to the empirical results, the optimal alternative calculated by the

mathematical programming satisfies decision makers and indeed reduces the

dissonance. This study comes up with a successful manner to effectively reduce the

dissonance when decision makers face a multi-criteria decision making problem.

2 Pan-Pacific Management Review January

Keywords: cognitive dissonance, multi-criteria decision making, intuitionistic fuzzy set,

intuitionistic fuzzy weighted averaging (IFWA), score function

INTRODUCTION

Problems for multi-criteria decision-making (MCDM) are common cases, including

individual matter in real life. The decision-making problem has been a subject of an extensive

research effort that has resulted in a multitude of models and approaches with roots in

different kind of areas ranging from social science, mathematical science to cognitive science

(Szmidt & Kacprzyk, 2008). Although there has existed substantial MCDM methods,

including AHP (analytic hierarchy process), SMART (simple multi-attribute rating

technique), VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje), ANP

(analytic network process), and TOPSIS (technique for order preference by similarity to ideal

solution), most of them mainly consider the problem itself, focusing on the relative

importance between criteria and the priority of alternatives. Little research has been devoted

to exploration of the decision maker’s psychological factors such as the cognitive dissonance

in the decision-making methods. Without a better skeleton of psychological factors, MCDM

methods may remain incomplete in the explanations.

The theory of cognitive dissonance was pioneered by Festinger (1957) and offered

insights into persuasion (attitude change) resistance. According to this perspective, when

individuals are receiving new information which disagrees with existing cognitions, they

experience the dissonance. As cognitive dissonance is psychology uncomfortable, individuals

are motivated to reduce or eliminate it and restore cognitive consonance.

Once a decision is made, subsequent preferences are revealed along with an increase on

the attractiveness of the chosen alternative and a decrease on the attractiveness of the rejected

alternative. It is because the positive aspects of the rejected alternatives and the negative

aspects of the chosen alternative are dissonant with the post decision, and are changed to

reduce the unpleasant states associated with dissonance. A difficult decision signifies that the

alternatives are close to each other in attractiveness, and an easy decision signifies that the

alternatives are remote from each other in attractiveness (Liberman & Forster, 2006). The

2011 Ting-Yu Chen, Yi-Jen Li, & Hsiao-Pin Wang 3

more approaching the alternatives to be chosen are, the higher the level of dissonance occurs

(Mittelstaedt, 1969). When evaluating the non-inferior/non-dominated alternatives over the

given conflicting criteria, the decision makers will search the information about the

alternatives according to intrinsic and extrinsic cues, and establish a preference order. If the

decision is characterized by difficulty and importance, or the decision makers have doubts

and anxieties about the final decision, the level of post-decision dissonance will result easily

(Hawkins, Best, & Coney, 2001). In addition, dissonance will increase with the relative

attractiveness of the rejected alternatives (Greenwald, 1969). As there is a deficiency in

MCDM problems which involve cognitive dissonance, the presence, magnitude, and affects

of the dissonance need studying. The main purpose of this study is to propose a MCDM

method to reduce the dissonance, especially under intuitionistic fuzzy environment.

Owing to lacks of knowledge and information processing capabilities, decision makers

would be incapable of evaluating with crisp values. In order to properly tackle the vague

human judgments, a great number of studies have developed several MCDM approaches base

on various imprecise sets. Unlike a fuzzy set proposed by Zadeh (1965) and merely giving a

membership degree to which an element belongs to a set, an intuitionistic fuzzy set (IFS)

provides both a membership degree and a non-membership degree. The IFS, developed by

Atanassov (1986) is a useful tool to deal with the vagueness. In addition to the membership

and non-membership degrees, the IFS is characterized by a supplementary hesitancy degree

to capture the human indeterminacy in the decision-making process. The IFSs display a richer

apparatus to grasp imprecision than the conventional fuzzy sets. Due to the applicable nature

of IFSs, a great deal of research has extended IFSs to MCDM issues (Boran, Gene, Kurt, &

Akay, 2009; Miao & Wang, 2008; Li, Wang, Liu, & Shan, 2009).

Chen and Tan (1994) presented the evaluation function and score function for handling

multicriteria fuzzy decision-making problems. Based on Chen and Tan’s work, Liu and Wang

(2007) employed intuitionistic fuzzy point operators to reduce the degree of uncertainty

before implementing the evaluation function; besides, a series of new score functions for

tackling the MCDM problems were also proposed. Lin, Yuan, and Xia (2007) utilized a

linear programming model which permits the decision maker to alter the evaluating weights

during the decision-making process. Miao and Wang (2008) adopted a mathematical


programming model combining the extended TOPSIS skill to conduct multi-attribute and

multi-person decision-making problems. Boran et al. (2009) used the intuitionistic fuzzy

weighted averaging operator to aggregate individual opinions of decision makers.

Taking into account the human thoughts with uncertainty and the dissonance arousal

explicitly, this study proposes two mathematical programming models to solving MCDM

problems under the intuitionistic fuzzy decision situation. One is based on the intuitionistic

fuzzy weighted averaging operator and the score function to generate optimal weights of

criteria, and then to obtain preference order of alternatives. The other further adds the

maximal of alternative attractiveness to anticipate reducing the dissonance. Finally, an

empirical study was employed to support the feasibility and effectiveness of the proposed

methods.

DECISION ENVIRONMENT BASED ON IFSs

Atanassov (1986) generalized the concept of fuzzy set and defined the concept of IFSs.

Tizhoosh (2008) suggested that IFSs and interval-valued fuzzy sets (IVFSs), which were

presented by Gorzlczany (1987) and Turksen (1996), constitute a mathematical isomorphism,

but perform with different semantics. It is worthwhile to mention that IVFSs are mathematically equivalent to IFSs (Dubois, Gottwald, Hajek, Kacprzjk, & Prade, 2005).

More detailed information about the relationship between IFSs and other models of

imprecision was discussed by Deschrijver and Kerre (2007).

Definition 2.1. Atanassov (1986) let X be a finite universe of discourse. An IFS A in X

is an object having the following form:

XxxxxA AA )(),(, (1) where the function ]1,0[: XA and ]1,0[: XA define the degree of

membership and the degree of non-membership of the element Xx to the set XA ,

respectively, such that 1)()(0 xx AA . The value of

)()(1)( xxx AAA (2)

calls the intuitionistic index. It is the degree of uncertainty (or indeterminacy) (Atanassov,

1999) or the degree of hesitancy (Szmidt & Kacprzyk, 2000) associated with the membership


of element Xx in IFS A.

The presentation of a decision-making problem which contains multiple alternatives

and criteria under the intuitionistic fuzzy environment can be concisely expressed in a

decision matrix M. The values in the matrix are composed of the evaluations of the i-th

alternative, iA , with respect to the j-th criterion, jc . The evaluations of each alternative with

respect to each criterion are measured on a fuzzy concept of “excellence” based on similar

intuitionistic fuzzy definitions given by Li (2005) and Lin et al. (2007). Suppose that there

exists a non-inferior alternative set mAAAA ,...,, 21 . Each alternative is assessed on n

criteria, denoted by nCCCC ,...,, 21 . Assume that ij and ij are the degree of

membership and the degree of non-membership of the alternative AAi with respect to the

criterion CC j to the fuzzy concept “excellence”, respectively, where 10 ij ,

10 ij and 10 ijij . Denote that ijijijij xX ,, is one of IFSs in the finite

universe of discourse X. The intuitionistic index of the alternative iA in the set ijX is

defined by ijijij 1 . The intensity of the “excellence” of the alternative iA with

respect to the criterion xj is given by ij , but is affected by ij and ij . Especially, when

ij gets larger, a higher hesitation margin of the “excellence” occurs. The intuitionistic fuzzy

decision matrix M is defined as the following form:

. 1C 2C … nC

),(),(),(

),(),(),(),(),(),(

2211

2222222121

1112121111

2

1

mnmnmmmm

nn

nn

mA

AA

M

(3)

Further consider the relative importance of criteria, and let j and j be the degree

of membership and the degree of non-membership of the criterion CC j to the fuzzy

concept “importance”, respectively, where 10 j , 10 j and 10 jj . The


intuitionistic index jjj 1 . The intensity of the “importance” of the criterion

CC j is given by j , but is affected by j and j . Especially, when j gets larger, a

higher hesitation margin of the “importance” occurs. Since IFSs and IVFSs are

mathematically equivalent, the weights of criteria lie in the closed interval

],[],[ jjjjj ww ul , where jjw

l and jjjjw 1u . ljw and

ujw are the

lower bound and upper bound of the membership to the fuzzy concept “importance” in IVFSs,

respectively. The lower bound of membership in IVFSs is equal to the degree of membership

in IFSs. The length between lower bound and upper bound in IVFSs is equal to the degree of

uncertainty in IFSs. Obviously, 10 ul jj ww for each criterion Cc j . In addition,

11

n

jjwl and 1

1

n

jjwu are assumed in order to determine weights ]1,0[jw

) ,,2 ,1( nj satisfying ul jjj www and 11

n

jjw . An interval weight vector of all

criteria can be concisely expressed by W as follows:

],[,],,[],,[ 2211 unlnulul wwwwwwW ],[,],,[],,[ 222111 nnn . (4)

DECISION MAKING WITH DISSONANCE REDUCTION

Intuitionistic Fuzzy Aggregation Operator

Several approaches have been developed to aggregate sets under fuzzy environment.

Chen and Tan (1994) proposed new techniques in which the degrees that each alternative

satisfies and does not satisfy the decision-maker’s requirement can be assessed by the

evaluation function based on the maximum and minimum operators of the vague values.

Hong and Choi (2000) presented Max-min, Max-max, and Max-center methods to tackle

multicriteria fuzzy decision-making problems. Deschrijver and Kerre (2005) introduced

aggregation operators on the lattice L, which is the underlying lattice of both interval-valued

fuzzy sets and intuitionistic fuzzy sets, and considered some particular classes of binary

aggregation operators based on t-norms on the unit interval. Liu and Wang (2007) extended


the intuitionistic fuzzy point operators to the works proposed by Chen and Tan (1994) and

Hong and Choi (2000) in order to reduce the degree of uncertainty of the elements

corresponding to an IFS in advance. Xu (2007) developed a similarity measure based on the

set-theoretic approach, and defined the notions of positive and negative ideal set to further

calculate the scores of alternatives. In spite of the adequacy to compute the performance of

each alternative, these techniques are merely employed based on the maximum and minimum

operators. A simple boundary operation may lead to the loss of information and the

imprecision in the preference order of alternatives. To aggregate a collection of intuitionistic

fuzzy values without any loss of information, Xu (2007) developed the intuitionistic fuzzy

weighted averaging (IFWA) operator.

Definition 3.1. Harsanyi (1955) let WA: RR n , if WA

n

jjjnw awaaa

121 ) ,, ,(WA (5)

then WA is called a weighted averaging operator, where Tnwwww ),,,( 21 is the weight

vector of ), ,...,2 ,1( nja j with ]1,0[ jw and ,11

n

jjw R is the set of all real

numbers.

According to the basic operations given by Atanassov (1986), De, Biswas, and Roy

(2000) defined the concentration, dilation, and normalization of IFSs and presented some new

operations on IFSs. Several operations pertaining to the IFWA operator are presented as

follows:

Definition 3.2. Let ) ,(~ ~~ aaa and ) ,(~

~~ bbb be two intuitionistic fuzzy values;

then

) ,(~~ ~~~~~~ babababa ; (6)

) ,(~~ ~~~~~~ babababa ; (7)

0 ), ,)1(1(~ ~~ aaa ; (8)

0 ),)1(1 ,(~ ~~ aaa . (9)


Xu (2007) proposed the properties of intuitionistic fuzzy values, including the

Commutativity, Distributive laws, and Associativity, and completed the proof.

Definition 3.3. Let ) ,(~ ~~ aaa and ) ,(~

~~ bbb be two intuitionistic fuzzy values,

and , 1 , 2 >0

abba ~~~~ ; (10)

baba ~~)~~( ; (11)

aaa ~)(~~ 2121 . (12)

The IFWA operator extends the WA to accommodate the situations where the input

arguments are intuitionistic fuzzy values. For MCDM problems, the IFWA operator

aggregates each weighted evaluating values of criteria, and scores the performance of the

alternative.

Definition 3.4. Let )(XIFSX ij , AAi , CC j . The ),( ijijijX is a

collection of intuitionistic fuzzy values given by j=1, 2,…, n, and the IFWA: n , if

inniiijw xwxwxwX 2211)(IFWA , (13)

then IFWA is called an intuitionistic fuzzy weighted averaging operator of dimension n,

where Tnwwww ),,,( 21 is the weight vector of ijx (j = 1, 2,…, n), with ]1,0[jw ,

11

n

jjw , and is the set of all intuitionistic fuzzy values. More specifically, based on

Definition 3.3, if Tnnnw )/1 ,...,/1 ,/1( , the IFWA operator degenerates into an

intuitionistic fuzzy averaging operator of dimension n, which is defined as follows:

)(1)(IFA 21 iniiijw xxxnX . (14)

According to Definition 3.2 and 3.4, the IFWA operator can be rewritten and defined as

follows:

Definition 3.5. Let )(XIFSX ij , AAi , CC j . The )1,( ijijijX is a

collection of intuitionistic fuzzy values given by j = 1, 2,…, n, and

inniiijw xwxwxwX 2211)(IFWA

n

j

wij

n

j

wij

jj

11

,)1(1 (15)


where Tnwwww ),,,( 21 is the weight vector of ijx (j = 1, 2,…, n), with ]1,0[jw ,

11

n

jjw . The aggregated value by the IFWA operator is also an intuitionistic fuzzy value.

A MCDM Method Based on IFWA and Score Function

By applying the IFWA operator, each alternative can gain a score which consists of the

membership degree and non-membership degree. Nevertheless, it is difficult to immediately

compare the priority of alternatives in terms of three elements of , , and . To

facilitate the comparison, Chen and Tan (1994) presented a score function S to evaluate the

degree of suitability that an alternative satisfies a decision maker’s requirement. Let

),( ijijijX be an intuitionistic fuzzy value; based on the score function S, the suitability

of an intuitionistic fuzzy value for the alternative can be measured by

ijijijXS )( , (16) where ]1,0[ij , ]1,0[ij , 1 ijij , ]1,1[)( ijXS . The definition of the score

function equals the membership degree minus the non-membership degree. Incorporating the

IFWA operator, the score function is redefined in Definition 3.6.

Definition 3.6. Let )(XIFSX ij , AAi , CC j . The ),( ijijijX is a

collection of intuitionistic fuzzy values given by j=1, 2,…, n. The score function of each

alternative based on the IFWA operator is defined as follows:

n

j

wij

n

j

wiji

jjAS11

)1(1)( (17)

where ]1,1[)( iAS , for each i = 1, 2,…, m.

Under the intuitionistic fuzzy environment, the weights of criteria are characterized by

interval values. The length of an interval accounts for the degree of uncertainty. That is, the

possible weight of criterion can be any value within an interval. The decision maker can

change the evaluating weights of criteria between the lower bound and upper bound of an

interval. In this condition of adjustable weights, the degree to which the alternative iA

satisfies the decision maker’s requirement can be measured by an optimal programming with

the weighted score function and the IFWA operator. Besides, since there are m alternatives in

the decision-making problem, m programming models have to be solved. In this way, each


alternative can individually obtain the optimal weight vector. However, the optimal solutions

would be different in general so that the corresponding optimal values of the degree of

suitability for m alternatives cannot be compared. As the decision maker cannot evidently

judge the preference relations among all non-inferior alternatives, assigning an equal weight

1/m is plausible to aggregate m mathematical programming problems. The aggregated

mathematical programming problem is presented as follows:

n

jj

ujj

lj

m

ii

w

njwww

m

ASZ

1

1

.1

), ,,2 ,1( s.t.

)(max

(18)

A MCDM Method with Dissonance Reduction

There have existed some effective manners to reduce the dissonance, including attitude

change, opinion change, seeking and recall of consonant information, avoidance of dissonant

information, perceptual distortion, and behavioral change (Harmon-Jones & Harmon-Jones,

2007; Soutar & Sweeney, 2003; Sweeney, Hausknecht, & Soutar, 2000). Facing multiple

non-inferior alternatives to be evaluated, a decision maker will yield more post-decision

dissonance as this decision reaches a certain level of difficulty (Festinger, 1964; Menasco &

Hawkins, 1978). The number and the attractiveness of alternatives determine the level of

difficulty. Specifically, the more approaching the alternatives perform, the more difficult the

decision is. Hence, the dissonance generates in accordance with the attractiveness of

alternatives. To reduce the relative attractiveness of alternatives, an approach is offered to

magnify the difference between alternatives. For the score function of each alternative, the

sum of the Euclidean distance between alternative iA and other alternatives kA is equal to

2)()( ki ASAS . Due to the purpose of reducing decisional dissonance, the total distance

between each two of m alternatives must be maximized. That is, )!2)!2(

!(2

mmC m different


combinations of distance between each two alternatives need accumulating. The larger

distance between alternatives results in less dissonance. Thus, based on Eq. (18), the

maximization of

m

i

m

kki ASAS

1 1

2)()(21 is added to the original objective.

Since there are two objective functions, we transform the two-objective problem into a

single optimization problem according to the decision maker’s preference structure. Let

be a parameter which reflects the decision maker’s preference pertaining to the importance of

dissonance reduction when compared to the objective of maximal suitability degree, where

]1,0[ . To reduce dissonance, the following mathematical programming problem with an

adjustable parameter is considered.

n

jj

ujj

lj

m

i

m

kki

m

ii

w

njwww

ASAS

m

AS

1

1 1

2

1

.1

), ,,2 ,1( .t.s

2

)()()()1(max

(19)

In Eq. (19), the parameter α can be regarded as a function of involvement due to the fact

that the decision-making process is affected by the level of involvement (Rothschild, 1979).

More specifically, Brehm and Cohen (1962) indicated that involvement is a prerequisite for

the extent of dissonance. The dissonance would get large when decision makers are under

high-involvement decision-making environment (Anderson, 1973). Thus, it is necessary to

magnify the attractiveness of alternatives for reducing dissonance as the degree of

involvement is high. The parameter α determines the relative importance between the degree

of suitability and dissonance reduction. A large α will increase the salience of reducing

dissonance but impair the reachable levels of suitability. A small α causes the opposite effect.

Moreover, it is worthwhile to mention that the constraints of criterion weights in (18) and (19)

are equivalent, but the solutions of criterion weights may be unequal because of different

objectives in which the former is a straightforward model and the later further considers the

viewpoint of involvement to enlarge the distance among alternative and to reduce the

dissonance.


Numerical Example

In this section, a multi-criteria decision-making problem is used to illustrate our

proposed methods. Suppose that there are five alternatives 54321 ,,,, AAAAAA , which is

assessed by four criteria 4321 ,,, ccccC . Assume that the degrees ij of membership

and the degrees ij of non-membership for the alternative AAi with respect to the

criterion CCi to the fuzzy concept “excellence” are represented by an intuitionistic fuzzy

decision matrix as follows:

1C 2C 3C 4C

)30.0 ,28.0()45.0 ,17.0()18.0 ,13.0()26.0 ,65.0()04.0 ,51.0()38.0 ,11.0()27.0 ,28.0()20.0 ,32.0()20.0 ,19.0()32.0 ,08.0()37.0 ,58.0()09.0 ,55.0()28.0 ,13.0()16.0 ,71.0()81.0 ,01.0()39.0 ,58.0()04.0 ,44.0()25.0 ,41.0()06.0 ,36.0()52.0 ,30.0(

5

4

3

2

1

AAAAA

D

The degrees j of membership and the degrees j of non-membership for the four

criteria CCi to the fuzzy concept “importance” are assumed below:

1C 2C 3C 4C

)88.0 ,03.0(),38.0 ,16.0(),36.0 ,49.0(),63.0 ,18.0()),(( 41 jj . The criteria weights also lie in the closed intervals and are expressed by W as follows:

1C 2C 3C 4C

]12.0 ,03.0[],62.0 ,16.0[],64.0 ,49.0[],37.0 ,18.0[W .

It should be noted that

4

1186.0

j

ljw and

4

1175.1

j

ujw .

By applying (18), we can obtain the following mathematical programming:

530.045.018.026.072.083.087.035.01(

)04.038.027.020.049.089.072.068.01()20.032.037.009.081.092.042.045.01()28.016.081.039.087.029.099.042.01()04.025.006.052.056.059.064.070.01(

max

43214321

43214321

43214321

43214321

43214321

wwwwwwww

wwwwwwww

wwwwwwww

wwwwwwww

wwwwwwww


.1,12.003.0,62.016.0,64.049.0,37.018.0

s.t.

4321

4

3

2

1

wwwwwwww

The optimal solution of the above programming is Tw )03.0 ,16.0 ,49.0 ,32.0( . The

optimal objective value of Z reaches 0.11. Taking the optimal weights to calculate the degrees

of the suitability for each alternative by (17), we can obtain

)( 1AS =0.20, )( 2AS =－0.09, )( 3AS =0.27, )( 4AS =0.03, )( 5AS =0.12.

According to the degrees of suitability, the priority of the five alternatives is given by

24513 AAAAA . Based on the decision matrix and our proposed method, the

alternative 3A is the best choice for the decision maker.

In order to expand the attractiveness of alternatives and reduce the dissonance, the

second proposed method in (19) is used to help the decision maker determine the best

alternative. Assume that the degree of involvement is 0.5. By applying (19), the optimal

solution of the programming is Tw )03.0 ,16.0 ,63.0 ,18.0( . The optimal objective value of

is 1.19. In the same way, the degrees of the suitability for each alternative can be

calculated by using (17). The results which take into account the distance between

alternatives are )( 1AS =0.25, )( 2AS =－0.22, )( 3AS =0.23, )( 4AS =0.01, )( 5AS =0.05. The

priority of the alternatives is ordered by 24531 AAAAA . Instead of 3A , 1A

becomes the best choice for the decision maker. The outcome explains that the latter method

which magnifies the distance between alternatives may help the decision maker obtain better

choice satisfying the intrinsic requirement; meanwhile, due to the explicit judgment on the

difficult option of approaching alternatives, it is likely to reduce the decisional dissonance. In

order to manifest that the proposed method is capable of diminishing the dissonance, an

empirical study was employed to support the effectiveness, and presented in the following.


AN EMPIRICAL STUDY

There are two method proposed in this study to tackle MCDM problems. One is derived

from an optimization model with weighted score functions and IFWA; the other further

deliberate the idea of involvement and the distance between alternatives. The former can be

regarded as an applicable approach to the MCDM analysis; the latter is developed to

anticipate diminishing the decision maker’s cognitive dissonance, and properly calculate the

best alternative corresponding to decision maker’s choice. For simplicity, the two methods

are labeled as non-dissonance method and dissonance method, respectively. In order to

ascertain that the method was competent to reduce the cognitive dissonance, an empirical

study was employed to observe the variation of cognitive dissonance and to manifest the

effectiveness and feasibility of the method.

Procedure and Measures

Measures for the empirical study were divided into five parts, including the

demographics, weights of criteria, decision matrix for a MCDM problem, involvement, and

cognition dissonance. The investigation adopted a two-step design to collect data. In the first

step, subjects were asked to respond all questions except the cognitive dissonance. When

obtaining the preliminary data, we calculated the scores of each alternative and determined

the best alternative according to the two proposed methods simultaneously. The subjects were

unqualified if the preliminary data they provided generated the equivalent top rank of

alternatives by applying different methods. The eligible subjects proceeded to answer the

questions pertaining to the cognitive dissonance in the second step. The extent of cognitive

dissonance was measured with the specific condition in which subjects needed to react the

attitude toward the case that they purchased or selected the alternative we gave by two

mathematical outcomes, respectively.

Since MCDM problems often involve imprecise, uncertain and subjective elements, it is

very appropriate to assess decisional process under fuzzy environment. Considering the

decision making is complex and vague, we utilized IFSs to capture the degrees of alternative

with respect to each criterion which were yielded by subjects. Due to the fact that


interval-valued fuzzy sets are mathematically equivalent to IFSs (Dubois et al., 2005),

subjects can give an interval score to represent the evaluating performance. The lower bound

of interval equals the membership degree in the IFS. The length of interval accounts for the

hesitancy degree. Obviously, the non-membership degree is calculated by one minus the sum

of the membership degree and hesitancy degree. The following is an instance to illustrate

how to construct a decision matrix in IFSs. Suppose that there are three alternatives

321 ,, AAAA which are evaluated by means of three criteria 321 ,, CCCC in the

fuzzy concept “excellence”. Assume that one decision maker conceives that the performance

on the 1A with respect to 1x is 89~99. It implies that he/she is sure that the extent in which

1A has excellent performance on 1x is 89. The extent in which he/she is not sure is

represented by the length of interval. The hesitancy degree amounts to 10. In order to achieve

the definition of IFSs, the interval score can be expressed by 1A = ( 1x , 0.89, 0.01, 0.10). The

rest evaluating performance of other alternatives with respect to criteria use the same way to

form the decision matrix.

Involvement is defined as the decision maker’s perceived relevance of the decision

based on inherent needs, values, and interests. The scale for involvement was developed

based on the RPII (Revised Personal Involvement Inventory) proposed by Zaichkowsky

(1994). Ten items were measured on 5-point Likert scales anchored by “strongly disagree

(0)” and “strongly agree (4)”.

The operational definition of cognitive dissonance is given by a psychologically

uncomfortable state derived from the discrepancy between the previous expectation and final

consequences after purchase. The scale for cognitive dissonance was developed based on

Sweeney et al. (2000). Twenty-two items were measured on 5-point Likert scales anchored

by “strongly disagree (0)” and “strongly agree (4)”.

Subjects and Stimulus

The research subjects employed in this study focuses on the homogeneity. Calder et al.

(1981) suggested that research subjects had better possess high homogeneity due to the fact

that high homogeneity can obtain more correct inference and reduce the covariance problem


yielded from heterogeneous subjects. The homogeneous subjects were very crucial to a

fundamental research on testing the reliability of proposed approaches. Pinto (2001) indicated

that compared with women, men had larger cognitive dissonance while making a decision

because men were more prone to dichotomous thinking and cognitive distortions than women.

The age can also lead to different level of cognitive dissonance; specifically, younger

decision makers are believed to be more sophisticated and yield higher expectations

(Thompson, Pitts, & Schwankovsky, 1993). That is, they have a higher desire for the

involvement and higher associated dissonance (Anderson, 1973). Taking the above factors

into consideration, we enrolled 220 male college students by the convenience sampling in our

investigation. On one hand, college students have many aspects in common to maintain the

homogeneity we request in this study. On the other hand, the target of younger men who may

generate lager dissonance helps us examine the variation of cognitive dissonance easily.

Of the 220 subjects, 10 responses were eliminated owing to incomplete data. This

resulted in a usable sample of 210 responses and a valid rate of 95.45%. Of the 210 subjects,

145 sets of data calculated equal top rank of alternatives by the two proposed methods. Thus,

there were only 65 subjects qualified for the second step to further respond the questionnaire

of cognitive dissonance.

A digital camera selection was designed as a MCDM problem. Digital cameras are one

of favorable and prevalent products that a majority of college students have an experience of

usage. As s stimulus, the digital camera itself has various criteria to be evaluated by decision

makers. Moreover, the digital camera is categorized as durable goods that people always

possess higher involvement when making a decision. Since high involvement is related to

high dissonance cognitive, the digital camera is an appropriate stimulus to inspect the

dissonance cognitive for a decision. Because Hunt (1970) indicated that the dissonance was

greater for high priced than for the low priced purchases, the price of stimulus products was

set up from NT$15,000 to NT$20,000. There were four evaluating criteria, including the

appearance, pixels, screen size, and optical zoom. The detailed information for the designed

MCDM problem is presented in Table 1.


TABLE 1 The evaluated alternatives and criteria Alternatives Criteria

A B C D E Appearance

Pixels 10 million 7.2 million 9 million 15 million 8 million Screen size 2.5” 3.5” 2.7” 2.5” 2.8” Optical zoom 12x 18x 7x 10x 20x

Empirical Results

Of the 210 subjects, there are 65 male students accomplishing the two-step investigation.

Table 2 shows that the percentage of major in social science (58.5%) is more than in natural

science (41.5%). The subjects enrolled in this study live in North Taiwan mostly, and the

percentage occupies 81.5%. As to the monthly income, most of subjects have

NT$12,501-17,500 to dominate in a month, followed by “NT$7,501-12,500”, “NT$17,501+”,

and “Less than NT$7,500”.

TABLE 2 Respondent demographics

Demographic Profile Category Frequency Percentage

Major Social science 38 58.5%

Natural science 27 41.5%

Permanent address North 51 81.5%

Others (Middle, South,

and East) 12 18.5%

Monthly income Less than NT$7,500 4 6.2%

NT$7,501-12,500 26 40.0%

NT$12,501-17,500 30 46.2%

NT$17,501+ 5 7.6%


The magnitude of post-decision dissonance has been viewed as an increasing function

in important (high involvement) decisions. The involvement is a parameter in the dissonance

method to adjust the priority of the best alternative. We expect that the adjusted method is

capable of assisting decision makers to select an ideal alternative and reduce the cognition

dissonance. A paired-sample T Test was implemented to examine the extent of cognitive

dissonance between the non-dissonance method and the dissonance method. Table 3 indicates

that there is indeed a salient difference in cognitive dissonance by the two methods. The

dissonance method succeeds in reducing the cognitive dissonance. When a decision maker

faces a MCDM problem, the dissonance method can calculate the best alternative which

meets the decision maker’s requirements, and effectively prevents the cognitive dissonance

from expanding. It is worthwhile to notice that although involvement and cognitive

dissonance were measured by the Likert scales anchored by 0 to 4, the scores were

aggregated and normalized within 0 and 1 in order to fit the axiom of IFSs.

TABLE 3 Paired-sample t test of cognitive dissonance Methods Mean SD t-value p-value

Non-dissonance method 0.630 0.128 10.3 0.00 Dissonance method 0.390 0.156

Table 4 demonstrates the detailed alternation of optimal alternatives by the two methods.

According to the mathematical calculation, the highest scores account for the best alternative

for the decision maker. The values of cognitive dissonance occur when the decision makers

select and purchase the specific alternative. The numerical results shows if choosing the

alternative which is given by the non-dissonance method, the decision maker yields relatively

high cognitive dissonance due to the inconsistency between the calculating alternative and the

real alternative in the decision maker’s mind. Nevertheless, the alternative calculated by the

dissonance method is close to the decision maker’s need. There is a prominent reduction in

the cognitive dissonance. Table 4 also reveals that the Alternative A is the optimal selection

by applying non-dissonance method. Of 65 decision matrices, the number of Alternative A

which is viewed as the best choice by mathematically calculating amounts to 38. But, only 2


Alternative A maintain the best choice by applying the dissonance method. By contrast,

Alternative E becomes the optimal choice for the decision makers. Getting a further look in

each cells of the table, we can observe the scores of 1st and 2nd place alternatives are so

approaching by the non-dissonance method that the 2nd place alternative improves the

position to the top by the dissonance method. It implies that our proposed dissonance method

is appropriate to deal with hardly discriminative alternatives and efficiently reduce the

dissonance when decision maker are tackling a MCDM problem.

TABLE 4 The optimal results by two methods Non-dissonance Method Dissonance Method

Alternatives Alternatives Subject No.

A B C D E Disa

A B C D E Disa

1 0.170* 0.139 0.128 0.135 0.143 0.57 0.231 0.705 0.705 0.705 0.706* 0.382 0.143* 0.134 0.137 0.140 0.134 0.69 0.126 0.186 0.187 0.188* 0.185 0.283 0.141* 0.097 0.103 0.076 0.102 0.36 0.222 1.144 1.142 1.141 1.145* 0.334 0.135* 0.102 0.106 0.095 0.109 0.70 0.181 0.694 0.695 0.693 0.696* 0.265 0.123* 0.097 0.064 0.109 0.120 0.76 0.412 0.790 0.774 0.796 0.801* 0.346 0.135 0.153 0.144 0.157* 0.114 0.85 0.152 0.905* 0.904 0.902 0.900 0.477 0.135* 0.108 0.118 0.129 0.124 0.58 0.072 0.598 0.599 0.599 0.600* 0.258 0.161* 0.151 0.134 0.160 0.147 0.78 0.218 0.476 0.470 0.479* 0.474 0.369 0.160* 0.154 0.140 0.144 0.154 0.66 0.201 0.332 0.330 0.332 0.337* 0.2310 0.124* 0.123 0.113 0.116 0.118 0.73 0.116 0.217* 0.213 0.214 0.215 0.2611 0.139* 0.129 0.132 0.118 0.090 0.65 0.365 0.531 0.532* 0.521 0.515 0.2712 0.085* 0.072 0.083 0.071 0.064 0.50 0.175 0.376 0.381* 0.376 0.375 0.3213 0.162* 0.140 0.135 0.138 0.131 0.74 0.229 0.462 0.464 0.466* 0.459 0.3314 0.172* 0.137 0.128 0.136 0.132 0.67 0.300 0.335* 0.327 0.331 0.322 0.2615 0.160 0.161 0.172* 0.171 0.171 0.64 0.185 0.215 0.221 0.222* 0.221 0.6816 0.121* 0.056 0.060 0.107 0.085 0.95 0.410 1.262 1.263 1.277* 1.270 0.3417 0.117 0.112 0.123* 0.121 0.118 0.72 0.111 0.222 0.225 0.226* 0.224 0.7218 0.167* 0.146 0.138 0.137 0.141 0.65 0.097 0.626 0.626 0.626 0.627* 0.1819 0.134* 0.100 0.105 0.097 0.111 0.68 0.278 0.498 0.501 0.497 0.503* 0.2420 0.083* 0.080 0.077 0.078 0.086 0.58 0.086 0.165 0.164 0.165 0.168* 0.2621 0.143* 0.115 0.115 0.115 0.124 0.63 0.197 0.459 0.459 0.459 0.462* 0.3122 0.125 0.125* 0.119 0.119 0.124 0.75 0.109 0.155 0.153 0.153 0.156* 0.8023 0.114 0.114 0.116* 0.115 0.114 0.60 0.093 0.097 0.097 0.098* 0.097 0.5724 0.121 0.121 0.129 0.130* 0.121 0.59 0.133 0.196 0.201* 0.200 0.196 0.2725 0.161* 0.146 0.138 0.137 0.141 0.83 0.233 0.327* 0.322 0.322 0.324 0.3326 0.165* 0.151 0.136 0.158 0.141 0.50 0.137 0.649 0.646 0.650* 0.648 0.3327 0.159* 0.152 0.138 0.142 0.153 0.70 0.199 0.374 0.368 0.370 0.375* 0.3128 0.137* 0.117 0.127 0.132 0.122 0.68 0.188 0.315 0.319 0.322* 0.317 0.2829 0.132* 0.100 0.104 0.097 0.112 0.50 0.142 0.698 0.699 0.698 0.700* 0.5030 0.147* 0.120 0.087 0.072 0.085 0.88 0.386 1.428* 1.420 1.416 1.419 0.6831 0.112 0.124* 0.103 0.101 0.119 0.61 0.035 0.600 0.600 0.600 0.601* 0.2332 0.118 0.117 0.112 0.122* 0.123 0.53 0.116 0.219 0.217 0.222 0.222* 0.3533 0.131* 0.117 0.129 0.118 0.117 0.51 0.140 0.266 0.271* 0.267 0.265 0.5134 0.140* 0.125 0.115 0.125 0.089 0.65 0.355 0.527* 0.521 0.526 0.506 0.3535 0.173* 0.140 0.148 0.162 0.138 0.60 0.326 0.475 0.480 0.488* 0.474 0.3636 0.125* 0.125 0.071 0.071 0.115 0.64 0.456 0.682* 0.649 0.649 0.677 0.19


Non-dissonance Method Dissonance Method Alternatives Alternatives Subject No. A B C D E Dis

a

A B C D E Disa

37 0.114* 0.096 0.087 0.093 0.101 0.70 0.200 0.397 0.394 0.396 0.400* 0.2538 0.133* 0.130 0.108 0.100 0.100 0.49 0.306 0.519* 0.507 0.503 0.503 0.0339 0.057 0.071 0.123* 0.123 0.121 0.74 0.489 0.871 0.900 0.899* 0.899 0.4540 0.170* 0.116 0.130 0.135 0.143 0.52 0.335 0.827 0.832 0.834 0.837* 0.3141 0.170* 0.159 0.159 0.160 0.162 0.25 0.129 0.217 0.216 0.217 0.218* 0.0142 0.071 0.061 0.106* 0.062 0.104 0.76 0.364 0.688 0.700 0.689 0.710* 0.2743 0.141 0.105 0.076 0.143 0.153* 0.49 0.191* 0.139 0.136 0.187 0.190 0.0344 0.158* 0.039 0.101 0.114 0.112 0.35 0.577 1.778 1.797 1.801* 1.800 0.0145 0.053* 0.075 0.090 0.090 0.101 0.56 0.302 0.652 0.659 0.659 0.664* 0.0046 0.023 0.017 0.079 0.082* 0.082 0.78 0.464 1.143 1.168 1.169 0.160* 0.0747 0.096 0.117* 0.099 0.094 0.116 0.57 0.209 0.421 0.413 0.411 0.422* 0.3348 0.070 0.098 0.138 0.142 0.143* 0.56 0.487 0.770 0.796 0.798* 0.797 0.0249 0.074 0.095 0.134 0.142* 0.140 0.73 0.480 1.036 1.054 1.057 1.056* 0.1550 0.067 0.113 0.124 0.136 0.138* 0.53 0.221 1.423 1.424 1.426* 1.425 0.1951 0.047 0.080 0.135 0.137* 0.137 0.56 0.351 1.971 1.980 1.980 1.981* 0.2352 0.170* 0.116 0.128 0.135 0.143 0.45 0.373 0.756 0.761 0.764 0.768* 0.1953 0.047 0.125 0.138 0.139 0.140* 0.64 0.530 1.076 1.082 1.083* 1.082 0.0254 0.049 0.117 0.144 0.157 0.158* 0.75 0.381 2.148 2.152 2.155* 2.154 0.0755 0.063 0.101 0.139 0.140* 0.130 0.64 0.441 1.328 1.340* 1.338 1.337 0.3356 0.027 0.068 0.123 0.111 0.126* 0.57 0.653 1.379 1.407* 1.399 1.405 -0.0557 0.068 0.074 0.092 0.083 0.093* 0.45 0.174 0.486 0.499* 0.495 0.495 0.0058 0.065 0.086 0.098 0.122 0.124* 0.61 0.361 1.073 1.077 1.085* 1.081 0.0559 0.007 0.120 0.160 0.127 0.161* 0.74 0.700 2.573 2.584* 2.575 2.584 0.2460 0.015 0.057 0.085* 0.079 0.084 0.63 0.420 0.943 0.955 0.953 0.956* 0.0061 0.016 0.020 0.068 0.075 0.076* 0.53 0.394 0.581 0.613 0.618* 0.617 0.0962 0.170* 0.116 0.130 0.135 0.143 0.72 0.393 0.581 0.589 0.592 0.596* 0.4463 0.170* 0.159 0.159 0.160 0.162 0.67 0.144 0.209 0.209 0.210 0.211* 0.5564 0.071 0.069 0.104* 0.069 0.099 0.64 0.618* 0.543 0.615 0.565 0.616 0.2765 0.148* 0.093 0.090 0.143 0.146 0.80 0.453 0.356 0.632 0.365 0.765* 0.52

Note: “a” denote the extent of cognitive dissonance. “*” denotes the optimal alternatives.

Although the dissonance method can reduce the cognitive dissonance, it is impossible to

make the extent of dissonance approach none. There are only five alternatives and four

criteria for the decision makers to evaluate. The experimental alternatives and criteria may

not satisfy all subjects’ anticipation. The optimal alternative appearing in this study is merely

a relative selection rather than an absolute one. Therefore, a little bit dissonance in the

decision is acceptable.

CONCLUSIONS

Discussions

The MCDM approaches have been developed for a long while without any suspension

because of the importance of decision making in various fields. Dissimilar to the well-known

skills such as the AHP, SMART, VIKOR, ANP, and TOPSIS which cope with multiple


criteria, this study formulated the MCDM problem as two types of mathematical

programming structures based on the intuitionistic fuzzy concept. Since the indeterminate

factor always emerges when the decision maker is evaluating criteria and making a decision,

it is very proper to quantify human thoughts by applying IFSs in which in addition to the pros

and cons, the uncertain opinions are also well-expressed. In virtue of the uncertainty, the

weights of each criterion can be constructed within an interval and solved by the

mathematical programming.

Assumed all non-inferior alternatives to be of equal importance, the first programming

model was proposed based the IFWA operator and the score function to add another

technique for the MCDM analyses under intuitionistic fuzzy environment. The second

proposed model further considered the extent of involvement and the distance between

alternatives to effectively reduce the cognitive dissonance during the decision. An empirical

study was employed to manifest the effort of dissonance reduction. The consequent results

indicate that our proposed method is competent to diminish the cognitive dissonance as the

decision maker select or purchase the specific alternative. It is because the specific alternative

calculated by the dissonance method is in accordance with what the decision maker demands.

Superior to the non-dissonance method, the method further considering the dissonance tries

to enlarge the distance between alternatives so that it can assist in distinguishing the

difference among alternatives and reducing dissonance after a post-purchase decision.

However, involvement is a moderator between the two proposed methods. More specifically,

the two methods are equal when involvement approaches none. Dealing with

low-involvement decisions such as the routine decision, users may obtain similar outcomes

based on the non-dissonance or dissonance method. Relatively, the dissonance method is

suitable for the extensive decision and the limited decision due to higher involvement

demand in the process of decision making.

Managerial Implications

After the transactions are accomplished, what the enterprises are concerned is to look

forward to an increase on consumer satisfaction and consumer loyalty. Controlling the

magnitude of post-decision dissonance is a plausible manner to decrease the consumers’


negative opinions and to increase the positive preference for products or services. Facing a

difficult, important, and inalterable decision, consumers would experience the post-decision

dissonance easily. In this study , the proposed method helps consumers make a relatively

proper decision and effectively diminish the dissonance. As long as the post-decision

dissonance can be reduced, the brand loyalty and the possibility of repurchase will enhance

(Mittelstaedt, 1969).

In order to confront the tough competition, enterprises keeps on developing new

products, especially in the high-tech industry. The product line becomes various through the

line stretching and extension. Although the alternatives are abundant, it is so hard for

consumers to choose an appropriate product without any efforts that the dissonance occurs

after the decision is made. The marketers can utilize the presented method to magnify the

attractiveness of alternatives and assist consumers in selecting the best alternative and in

reducing the dissonance. In practice, the telecom companies in Taiwan have designed a web

page in which the users need to evaluate several criteria. Based on the evaluating values, the

web page will execute mathematical calculation and advise the users of a suitable phone rate.

We suggest that companies, which sell high-tech merchandise such as digital cameras, mobile

phones, notebooks, etc., can make use of our proposed method to create an auxiliary web

page for consumers to choose the best alternative.

Limitations

In order to examine a significant decrease on the cognitive dissonance, this study

mainly focused on the male and younger subjects who easily possess larger dissonance.

Hence, we do not ascertain whether the proposed method has the same effect on female and

elder decision makers. The future work can investigate a cross of different gender or age.

Moreover, the future stimulus can be employed more extensively to achieve the completeness

of the dissonance method for various products. The selection of experimental stimulus may

be picked up dependence on the decision category such as the extend problem solving,

limited problem solving, and routinized problem solving.


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Biographical Sketch

Ting-Yu Chen is currently an Associate Professor of the Department of Industrial and

Business Management at Chang Gung University in Taiwan. She received her B.S. degree in

Transportation Engineering and Management, M.S. degree in Civil Engineering, and Ph.D.

degree in Traffic and Transportation from National Chiao Tung University in Taiwan. Her

current research interests include multiple criteria decision making, fuzzy set theory, and

consumer decision analysis. She has published over 190 papers in peer-reviewed journals and

conference proceedings. She has received several awards, including the Distinguished

Research Award from the Chinese Institute of Transportation, the Outstanding Faculty Award

of Academic Research from Chang Gung University, the Distinguished Research Award

from the Chinese Management Association, Research Award from Chung Yuan Management

Review, and the Distinguished Young Scholar Award from Academia Sinica.

Yi-Jen Li received her M.S. degree in Business Administration in 2009 from Chang

Gung University in Taiwan. She was a part-time research assistant in the Graduate Institute

of Business Administration at Chang Gung University from September 2008 to July 2009.

Her research interests include fuzzy multiple criteria decision making and marketing

research.

Hsiao-Pin Wang received the B.S. degree in Statistics and Information Science in 2005

from Fu Jen University in Taiwan. He received his M.S. degree in Business Administration in

2007 from Chang Gung University in Taiwan. He was a full-time research assistant in the

Graduate Institute of Business Administration at Chang Gung University from November

2008 to February 2010. His research interests include fuzzy systems and marketing research.


直覺模糊多準則決策問題之降低失調方法

陳亭羽

長庚大學工商管理學系

李宜珍

長庚大學企業管理研究所

王曉斌

長庚大學工商管理研究所

中文摘要

考量認知失調於決策中為一項重要的心理變數，本研究藉由直覺模糊集合發展一套新的

多準則決策方法，設法降低決策者的認知失調情況。本方法利用直覺模糊加權平均運算

子與計分函數建構數學規劃模型，以規劃求解計算方式取得準則的權重，並排列方案優

劣順序。為了降低決策者期望與實際方案的失調程度，額外使用歐幾里得距離測度擴大

決策方案間的距離，以幫助決策者找出最佳方案。本研究提供一個數值例以便解釋詳細

的計算過程，此外經由一個實證研究以驗證研究方法的可行性。根據實證結果顯示，以

本研究方法為基礎所計算出的最佳方案，確實能夠滿足決策者同時亦能降低決策者的失

調程度。本研究成功發展一套有效降低認知失調的技術，幫助決策者解決多準則決策的

問題。

關鍵詞：認知失調、多準則決策、直覺模糊集合、直覺模糊加權平均運算、計分函數

通訊地址：陳亭羽，長庚大學工商管理學系副教授，333 桃園縣龜山鄉文化一路 259 號。聯絡電話：886-3-2118800 分機 5678 E-mail address: [email protected]

A DISSONANCE REDUCTION METHOD FOR INTUITIONISTIC FUZZY...

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