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An outcome-oriented approach to multicriteria decision analysis with intuitionistic fuzzy


Expert Systems with Applications 37 (2010) 7762–7774

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Expert Systems with Applications
journal homepage: www.elsevier.com/locate/eswa

An outcome-oriented approach to multicriteria decision analysis with intuitionistic fuzzy optimistic/pessimistic operators
Ting-Yu Chen *
Department of Industrial and Business Management, College of Management, Chang Gung University, 259, Wen-Hwa 1st Road, Kwei-Shan, Taoyuan 333, Taiwan

a r t i c l e

i n f o

a b s t r a c t
The concept of Atanassov’s intuitionistic fuzzy sets is a generalization of ordinary fuzzy sets. Under the intuitionistic fuzzy decision environment, this paper develops optimistic and pessimistic point operators to draw the in?uences of optimism and pessimism on multiple criteria decision analysis. We propose the optimistic and pessimistic score functions based on point operators to measure evaluations of the alternative with respect to each criterion. The suitability function can then be established to determine the degree to which the alternative satis?es the decision maker’s requirement. An optimization model with suitability functions is propounded on account of ill-known membership grades to generate optimal weights for criteria. Finally, an outcome-oriented approach is adopted to validate the proposed methods. Several multiattribute evaluation cases in consumer decision making reality are addressed to examine the feasibility and applicability of the current method. From the empirical results of effectiveness examination, the proposed intuitionistic fuzzy multicriteria decision making method with optimistic/pessimistic score functions performed signi?cantly better than the method without consideration of optimism/ pessimism. Therefore, we can conclude that the proposed methods by using intuitionistic fuzzy optimistic/pessimistic point operators have desirable potentials in practice. ? 2010 Elsevier Ltd. All rights reserved.

Keywords: Intuitionistic fuzzy set Point operator Optimism Pessimism Multiple criteria decision analysis Score function Optimization model

1. Introduction The notion of Atanassov’s intuitionistic fuzzy sets (A-IFSs) characterized by three functions expressing the degrees of membership, non-membership and hesitation, is an extension of the fuzzy set theory (Atanassov, 1986). In recent years, A-IFSs have been found useful in diverse ?elds of logic programming (Atanassov & Georgiev, 1993), topology (Lupia?ez, 2006; Mondal & Samanta, 2001), medical diagnosis (De, Biswas, & Roy, 2001), drug selection (Kharal, 2009), pattern recognition (Mitchell, 2005; Wang & Xin, 2005), microelectronic fault analysis (Shu, Cheng, & Chang, 2006), machine learning (Liang & Shi, 2003), decision making problems (Lin, Yuan, & Xia, 2007; Liu & Wang, 2007) and so on. It is worthwhile to mention that vague sets are A-IFSs showed by Bustince and Burillo (1996). In addition, the interval-valued fuzzy set theory is mathematically equivalent to A-IFS theory (Deschrijver & Kerre, 2003; Dubois, Gottwald, Hajek, Kacprzyk, & Prade, 2005). Although A-IFS and the interval-valued fuzzy set constitute an isomorphism (Tizhoosh, 2008), they are based on different semantics, such as weighing/modeling preferences versus imprecise membership (Dubois et al., 2005; Grzegorzewski & Mrówka, 2005). Furthermore, the semantic is crucial for real applications
* Tel.: +886 3 2118800x5678; fax: +886 3 2118500. E-mail address: tychen@mail.cgu.edu.tw 0957-4174/$ - see front matter ? 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.04.064

(Tizhoosh, 2008). In the present paper, we will focus on optimistic and pessimistic estimations based on A-IFSs to develop multiple criteria decision analysis (MCDA) methods and to illustrate the applications of consumer decision problems. There exist many useful methods for handling MCDA problems on a basis of A-IFSs (Li, 2005; Lin et al., 2007; Liu & Wang, 2007; Pankowska & Wygralak, 2006; Xu, 2007; Xu & Yager, 2008), while there is little research in the effect of dispositional optimism and pessimism. Optimism and pessimism, pioneered by Scheier and Carver (1985), are fundamental constructs that re?ect how people respond to their perceived environment and how to construe and affect subjective judgments. Although theories differ in speci?cs, common is the idea that optimists and pessimists diverge in the ways in which they explain and predict future events (Fischer & Chalmers, 2008). Optimists interpret their lives positively and anticipate desirable outcomes, whereas pessimists construe their lives negatively and expect unfavorable outcomes (Sanna & Chang, 2003). The previous studies have shown that, relative to individuals who are pessimistic about general outcomes of events, optimists report better adjustment to a variety of stressors including dietary habit and smoking behavior (Kelloniemi, Ek, & Laitinen, 2005), pregnancy (Park, Moore, Turner, & Adler, 1997), caregiving for soberly ill relatives (Given et al., 1993), serious health threats of undergoing bypass surgery (Scheier et al., 1989), idiosyncratic personal goal-directed activities (Jackson, Weiss, Lundquist,

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& Soderlind, 2002), coping strategy and stress reduction (Iwanaga, Yokoyama, & Seiwa, 2004), inevitability in the sting of failure (Sanna & Chang, 2003), economic theory such as utility adjustment (Brunnermeier & Parker, 2005) and portfolio and saving choices (Puri & Robinson, 2007), and ?nancial intermediation (Coval & Thakor, 2005). Although the power of dispositional optimism or pessimism has been examined in several areas, only few attempts have been made at the manner in which optimists and pessimists organize thinking and conceptualize MCDA problems. Hence there is a need to include the in?uences of optimism and pessimism on multicriteria decision making processes. Hey (1979) pointed out that the Hurwicz procedure can handle the decision problem with optimistic or pessimistic nature. By means of a pessimism-optimism index, the Hurwicz procedure is an amalgamation of the maximin and maximax methods, in that it takes into account both the worst and the best outcomes. However, it is inadequate when considering the whole multiple criteria problem because of the noncompensatory nature of the selection process (Hwang & Yoon, 1981). Yager (1988) introduced the ordered weighted averaging (OWA) operators. Yager (1992) suggested a pseudo probability distribution to re?ect the decision maker’s attitude by applying OWA operators, and he demonstrated that the Hurwicz approach is a special case of his method. Using the similar idea, Yager (1993) proposed that the orness measure with an OWA operator can be interpreted as a measure of the optimism of the decision making, while the andness is a measure of pessimism. Furthermore, Yager (2002) developed an attitudinal fuzzy measure which is generated with the aid of cardinality index for the sake of attitudinal characterization, including optimistic, neutral, and pessimistic characters. The risk attitude of the decision makers proposed by Yager is de?ned on one dimension in which the optimism and pessimism are two relative extremes. In other words, if a person who does not have optimistic inclination will be regarded as a pessimist. However, several studies indicated that optimism and pessimism belong to diverge dimensions (Chang, Maydeu-Olivares, & D’Zurilla, 1997, 2003; Scheier, Carver, & Bridges, 1994). That is, human beings can have both optimistic and pessimistic tendencies at the same time. Following the discussion above, a bi-dimensional measure will be employed to handle optimism and pessimism. This paper will present a useful method of relating optimism and pessimism to MCDA under Atanassov’s intuitionistic fuzzy decision environment. We develop several new score functions based on optimistic and pessimistic point operators for the sake of quanti?cation of optimism and pessimism. Furthermore, the suitability function to determine the degree to which an alternative satis?es the decision maker’s requirement is assessed according to weighted score functions. Then, we construct an optimization model to generate optimal weights for criteria to solve a multicriteria decision making problem. Finally, we conduct an outcome-oriented approach to test the accuracy of our proposed method. The outcome-oriented approach is based on the view of the decision outcome and its correct prediction (Zeleny, 1982). That is, if we can correctly predict the outcome of the decision process, then we obviously understand the decision process. We conduct speci?c ?eld studies concerning consumer decision problems. According to the comparative analysis of empirical results, we can demonstrate that the proposed method using intuitionistic fuzzy optimistic/pessimistic estimations is feasible and applicable in the real world.

alternative, Ai, with respect to the jth criterion, xj. The present paper extends the canonical matrix format to intuitionistic fuzzy decision matrix D. By similar de?nitions of Li (2005) and Lin et al. (2007), the evaluations of each alternative with respect to each criterion on a fuzzy concept ‘‘excellence” are given using AIFSs. Suppose that there exists a non-inferior alternative set A = {A1, A2, . . ., Am}. Each alternative is assessed on n criteria, denoted by X = {x1, x2, . . ., xn}. Assume that lij and mij are the degree of membership and the degree of non-membership of the alternative Ai 2 A with respect to the criterion xj 2 X to the fuzzy concept ‘‘excellence”, respectively, where 0 6 lij 6 1, 0 6 mij 6 1 and 0 6 lij + mij 6 1. Denote that X ij ? fhxj ; lij ; mij ig. The intuitionistic index of the alternative Ai in the set Xij is de?ned by pij = 1 ? lij ? mij. The larger pij the higher a hesitation margin of the decision maker as to the ‘‘excellence” of the alternative Ai with respect to the criterion xj whose intensity is given by lij. The intuitionistic fuzzy decision matrix D is de?ned as the following form:

?1?

In a similar way, let /j and uj be the degree of membership and the degree of non-membership of the criterion xj 2 X to the fuzzy concept ‘‘importance”, respectively, where 0 6 /j 6 1, 0 6 uj 6 1 and 0 6 /j + uj 6 1. The intuitionistic index sj = 1 ? /j ? uj. The larger sj the higher a hesitation margin of the decision maker as to the ‘‘importance” of the criterion xj 2 X whose intensity is given by /j. Since A-IFSs and interval-valued fuzzy sets are mathematically equivalent, the decision maker’s weight lies in the closed interval ?wlj ; wu ? ? ?/j ; /j ? sj ?, where wlj ? /j and wu ? /j ? sj ? 1 ? uj . j j Obviously, 0 6 wlj 6 wu 6 1 for each criterion xj 2 X. In addition, Pn Pn j u l j?1 wj 6 1 and j?1 wj P 1 are assumed in order to determine weights wj 2 [0, 1](j = 1, 2, . . ., n) satisfying wlj 6 wj 6 wu and j Pn j?1 wj ? 1. Atanassov (1986) established a way of transforming an A-IFS into a fuzzy set with a ?xed parameter. Burillo and Bustince (1996) extended Atanassov’s operator and de?ned a point operator. On the basis of the point operator, Bustince (2000) further studied the construction of intuitionistic fuzzy relations with determined properties using the construction theorem (Burillo & Bustince, 1996). Following the same perspective, we will de?ne several point operators for each point in Xij and aij, bij 2 [0, 1] in order to draw the in?uences of optimism and pessimism in decision making reality. Atanassov, Pasi, and Yager (2005) have discussed the possibility to use measurement tool estimations, accounting experts’ opinions about the separate tools. Based on the measurement tool estimations and the measurement tool scores, Atanassov et al. deformed the measurement tool estimations, including optimistic estimation, optimistic estimation with restrictions, pessimistic estimation, pessimistic estimation with restrictions, estimation with decreasing of uncertainty, and estimation with increasing of uncertainty. Furthermore, they applied these estimations on group decision making. In the following, we will extend the concept of measurement tool estimations to develop optimistic and pessimistic point operators in the cause of obtaining optimistic and pessimistic score functions. 3. Optimistic and pessimistic point operators

2. Intuitionistic fuzzy decision environment A MCDA problem can be concisely expressed in a decision matrix, whose element indicates the evaluation or value of the ith We relate optimism and pessimism to multicriteria decision making behavior and establish appropriate assessment tools for measuring them in decision analysis. Our assessment is rooted in

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the concepts of dispositional optimism and pessimism. That is, optimists construe their lives and future states of the world positively, whereas pessimists construe their lives and future states of the world negatively. In addition, optimists expect greater overall utility or favorable outcomes, but pessimists expect less overall utility or unfavorable outcomes. The above rationale coincides with several Atanassov’s operators (Atanassov, 1999). By extension we will develop point operators denoted by J ? ij ;bij ; J aij ;bij ; P aij ;bij ; H? ij ;bij ; Haij ;bij a a and Q aij ;bij for each point in Xij and aij, bij 2 [0, 1]. These estimations on A-IFSs are called as optimistic or pessimistic point operators. Optimism will be assessed using optimistic point operators, a revision of Atanassov’s measurement tool estimations. On the other hand, pessimism will be assessed using pessimistic point operators. 3.1. Optimistic estimations on A-IFSs De?nition 3.1. For each Ai 2 A and xj 2 X, taking aij, bij 2 [0, 1] we de?ne optimistic point operators J ? ij ;bij ; J aij ;bij ; Paij ;bij : IFS?X? ! a IFS?X? (the class of A-IFSs on the universe X) as follows, for Xij 2 IFS(X):

Proof. See Appendix A. h ?1 ?; The inclusion relation J ?;ijg;bij ?X ij ? & J aijg;bij ?X ij ? is obviously inferred a from Theorem 3.1. In addition, the limit of J ?;ijg;bij ?X ij ?, denoted by a limg!1 J ?;ijg;bij ?X ij ?, is a crisp set with membership degree of one. a The intuitionistic index of J aij ;bij ?X ij ? is:

pJaij ;bij ?Xij ? ?xj ? ? ?1 ? aij ?pij ? ?1 ? bij ?mij :
For short, we write

?8?

lJ ?xj ? ? lJa ;b

ij ij

?X ij ? ?xj ?; ?

mJ ?xj ? ? mJaij ;bij ?Xij ? ?xj ?, and

pJ ?xj ? ? pJaij ;bij ?Xij ? ?xj ?. By analogy to Jaij ;bij , the point operator Jaij ;bij
has the following properties. Theorem 3.2. Let Xij 2 IFS(X), Ai 2 A, xj 2 X, and aij, bij 2 [0, 1]. Let g be a positive integer and let J aij ;bij ?X ij ? ? J aij ;bij ?J aij ;bij ?X ij ??, where J aij ;bij ?X ij ? ? X ij .
0

g

g?1

?i?

lJg ?xj ? ? lij ? ?1 ? lij ? ? ?1 ? ?1 ? aij ?g ? ? aij ? mij
? ! g?1 X ?1 ? aij ?k ? bg?1?k ; ij
k?0

?9?

J ? ij ;bij ?X ij ? ? fhxj ; lij ? aij ? ?1 ? lij ? bij ? mij ?; bij ? mij ijxj 2 Xg; a J aij ;bij ?X ij ? ? fhxj ; lij ? aij ? pij ; bij ? mij ijxj 2 Xg;

?2? ?ii? ?3?

pJg ?xj ? ? ?1 ? lij ? bg ? mij ? ? ?1 ? lij ? ? ?1 ? ?1 ? aij ?g ? ij
! g?1 X k g?1?k : ?1 ? aij ? ? bij ? aij ? mij ?
k?0

?10?

Paij ;bij ?X ij ? ? fhxj ;max?lij ; aij ?;min?mij ; bij ?ijxj 2 Xg for aij ? bij 6 1: ?4?
Using J aij ;bij ; J aij ;bij and Paij ;bij on Xij separately will enhance the evaluation of alternative Ai with respect to criterion xj on the fuzzy concept ‘‘excellence” because of the increasing degree of membership. Thus, J ? ij ;bij ; J aij ;bij and Paij ;bij can re?ect the fact that optimistic a decision makers construe the decision situation positively and estimate favorable outcomes. From De?nition 3.1 we know that the membership degree of J aij ;bij ?X ij ? is the sum of the membership part of Xij and a part of the intuitionistic index. The repeated usage of J aij ;bij will lead to the highest membership degree of one (ultra-optimistic) if the time of redistributing the intuitionistic index is suf?ciently large. The operator J ? ij ;bij also demonstrates a similar a phenomenon. In contrast, the deformation of optimistic estimation by P aij ;bij has the limitation of max (lij, aij). Namely, the operator Paij ;bij represents an optimistic estimation with restrictions. The optimistic point operator J ? ij ;bij transforms an A-IFS Xij into a another A-IFS J ? ij ;bij ?X ij ? with the intuitionistic index: a
?

Proof. We can analogously prove (9) and (10) by using mathematical induction on g. h From the above theorem we know that J g?1ij ?X ij ? & J gij ;bij ?X ij ?, a aij ;b where J 0ij ;bij ?X ij ? ? X ij . Furthermore, limg!1 lJg ?xj ? ? 1 and a limg!1 pJg ?xj ? ? 0. The intuitionistic index of P aij ;bij ?X ij ? is:

pPaij ;bij ?Xij ? ?xj ? ? 1 ? max?lij ; aij ? ? min?mij ; bij ?:

?11?

Let lP ?xj ? ? lPa ;b ?Xij ? ?xj ?; mP ?xj ? ? mPaij ;bij ?Xij ? ?xj ?, and pP ?xj ? ? pPaij ;bij ?X ij ? ij ij ?xj ? for convenience. The operator Paij ;bij has the following properties. Theorem 3.3. Let Xij 2 IFS(X), Ai 2 A, xj 2 X,aij,bij 2 [0, 1] and aij + bij 6 1. Let g be a positive integer and let Pgij ;bij ?X ij ? ? a g?1 Paij ;bij ?P aij ;bij ?X ij ??, where P0ij ;bij ?X ij ? ? X ij . a

?i? ?ii?

lPg ?xj ? ? max?lij ; aij ?;
mPg ?xj ? ? min?mij ; bij ?;

?12? ?13?

pJ?aij ;bij ?Xij ? ?xj ? ? ?1 ? aij ? ? ?1 ? lij ? bij ? mij ?:
For convenience, noted by

?5?

?iii? P 0ij ;bij ?X ij ? & P1ij ;bij ?X ij ? ? P2ij ;bij ?X ij ? ? ?? ? ? Pg?1ij ?X ij ? ? Pgij ;bij ?X ij ?: a aij ;b a a a ?14?
Proof. (i) and (ii) are obvious. (iii) is evident since lij 6 max (lij,aij) and mij P min (mij,bij). 3.2. Pessimistic estimations on A-IFSs De?nition 3.2. For each Ai 2 A and xj 2 X, taking aij, bij 2 [0, 1] we de?ne pessimistic point operators H? ij ;bij ; Haij ;bij ; Q aij ;bij : IFS?X? ! a IFS?X? as follows, for Xij 2 IFS(X):

lJ? ;b a

ij ij

?X ij ? ?xj ?;

mJ? ij ;bij ?Xij ? ?xj ?, and pJ? ij ;bij ?Xij ? ?xj ? are dea a

lJ? ?xj ?; mJ? ?xj ?, and pJ? ?xj ?, respectively. The operator J? ij ;bij a
h

has the following properties. Theorem 3.1. Let Xij 2 IFS(X), Ai 2 A, xj 2 X, and aij, bij 2 [0, 1]. Let g
?1 be a positive integer and let J ?;ijg;bij ?X ij ? ? J ? ij ;bij ?J ?;ijg;bij ?X ij ??, where a a a

J ?;0;bij ?X ij ? aij

? X ij .

?i?

lJ?;g ?xj ? ? lij ? ?1 ? lij ? ? ?1 ? ?1 ? aij ?g ? ? aij ? bij ? mij
! g?1 X k g?1?k ; ? ?1 ? aij ? ? bij
k?0

?6?

H? ij ;bij ?X ij ? ? fhxj ; aij ? lij ; mij ? bij ? ?1 ? aij ? lij ? mij ?ijxj 2 Xg; a Haij ;bij ?X ij ? ? fhxj ; aij ? lij ; mij ? bij ? pij ijxj 2 Xg;

?15? ?16?

?ii?

pJ?;g ?xj ? ? ?1 ? lij ? bg ? mij ? ? ?1 ? lij ? ? ?1 ? ?1 ? aij ?g ? ij
? aij ? bij ? mij ? ! g?1 X g ?1 ? aij ?k ? bij?1?k :
k?0

?7?

Q aij ;bij ?X ij ? ? fhxj ; min?lij ; aij ?;max?mij ;bij ?ijxj 2 Xg for aij ? bij 6 1: ?17?

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Applying H? ij ;bij ; Haij ;bij and Q aij ;bij will lessen the intensity of the a fuzzy concept ‘‘excellence” in virtue of the decreasing degree of membership, whereby they can ful?ll that pessimistic decision makers construe the decision situation negatively and validate unfavorable outcomes. Since the separate membership degrees of H? ij ;bij ?X ij ? and Haij ;bij ?X ij ? equal a part of the membership degree of a Xij from De?nition 3.2, we will obtain the lowest membership degree of zero (ultra-pessimistic) by using the two operators suf?ciently large times. The point to be emphasized is that the operator Q aij ;bij represents pessimistic estimations with restrictions, and it has limited optimism on min (lij, aij). The pessimistic point operator H? ij ;bij , de?ned in (15), transa forms the A-IFS Xij into another A-IFS H? ij ;bij ?X ij ? with intuitionistic a index:

Theorem 3.6. Let Xij 2 IFS(X), Ai 2 A, xj 2 X,aij,bij 2 [0, 1] and aij + bij 6 1. Let g be a positive integer and let Q gij ;bij ?X ij ? ? Q aij ;bij a ?Q g?1ij ?X ij ??, where Q 0ij ;bij ?X ij ? ? X ij . aij ;b a

?i? ?ii?

lQ g ?xj ? ? min?lij ; aij ?;
mQ g ?xj ? ? max?mij ; bij ?;
g?1

?25? ?26?

?iii? Q 0ij ;bij ?X ij ? ' Q 1ij ;bij ?X ij ? ? Q 2ij ;bij ?X ij ? ? ? ? ? ? Q aij ;bij ?X ij ? a a a ? Q gij ;bij ?X ij ?: a
4. Methods for multicriteria decision analysis 4.1. Optimistic/pessimistic score functions The evaluation value of alternative Ai with respect to criterion xj can be determined by the score function S, which has been conceptualized and applied to multicriteria decision making problems by Chen and Tan (1994), Hong and Choi (2000), and Liu and Wang (2007). The de?nition of the score function equals the membership degree minus the non-membership degree, also called core or degree of support (Li, Olson, & Qin, 2007). By applying J ? ij ;bij ; J aij ;bij and a Paij ;bij , De?nition 4.1 presents optimistic score functions. On the other hand, De?nition 4.2 is suitable for pessimistic cases on the basis of H? ij ;bij ; Haij ;bij and Q aij ;bij . a De?nition 4.1. xj 2 X. Let g be g g SJ? ?X ij ?; SJ ?X ij ?; Let Xij 2 IFS(X) and aij,bij 2 [0, 1] for each Ai 2 A and a positive integer. The optimistic score functions g SP ?X ij ? based on the J ? ij ;bij ; J aij ;bij ; Paij ;bij point operaa

?27?

pH?aij ;bij ?Xij ? ?xj ? ? ?1 ? bij ? ? ?1 ? aij ? lij ? mij ?:
In the
ij ij

?18?

mH? ;b a

?X ij ? ?xj ?,

following, we write lH? ?xj ? ? lH? ?X ij ? ?xj ?; mH? ?xj ? ? aij ;bij and pH? ?xj ? ? pH? ;b ?Xij ? ?xj ? for short. a
ij ij

Theorem 3.4. Let Xij 2 IFS(X), Ai 2 A, xj 2 X, and aij,bij 2 [0, 1]. Let g be ?;g ?;g?1 a positive integer and let Haij ;bij ?X ij ? ? H? ij ;bij ?Haij ;bij ?X ij ??, where a H?;0;bij ?X ij ? ? X ij . aij

?i?

mH?;g ?xj ? ? mij ? ?1 ? mij ? ? ?1 ? ?1 ? bij ?g ? ? aij ? bij ? lij
! g?1 X k g?1?k ; ? ?1 ? bij ? ? aij
k?0

?19?

?ii?

pH?;g ?xj ? ? ?1 ? mij ? ag ? lij ? ? ?1 ? mij ? ? ?1 ? ?1 ? bij ?g ? ij
? aij ? bij ? lij ?
g?1 X
k?0

!

tors, respectively, are de?ned as follows:

g ?1 ? bij ?k ? aij?1?k :

?20?

?; Haijg;bij ?X ij ?

&

?1 H?;ijg;bij ?X ij ?, a

where H?;0;bij ?X ij ? ? X ij . Moreover, limg!1 aij

Sg? ?X ij ? ? lij ? ?1 ? lij ? ? ?1 ? ?1 ? aij ?g ? ? aij ? bij ? mij J ! g?1 X k g?1?k ? bg ? mij ; ? ?1 ? aij ? ? bij ij
k?0

?28?

mH?;g ?xj ? ? 1 and limg!1 pH?;g ?xj ? ? 0.
The intuitionistic index of Haij ;bij ?X ij ? is:

pHaij ;bij ?Xij ? ?xj ? ? ?1 ? aij ?lij ? ?1 ? bij ?pij :
For convenience, we write

?21?

Sg ?X ij ? ? lij ? ?1 ? lij ? ? ?1 ? ?1 ? aij ?g ? ? aij ? mij J ! g?1 X g?1?k g ? ?1 ? aij ?k ? bij ? bij ? mij ;
k?0

?29?

lH ?xj ? ? lHaij ;bij ?Xij ? ?xj ?; mH ?xj ? ? mHaij ;bij ?Xij ? ?xj ?,

and pH ?xj ? ? pHaij ;bij ?Xij ? ?xj ?. Furthermore, the operator Haij ;bij has the following properties. Theorem 3.5. Let Xij 2 IFS(X), Ai 2 A, xj 2 X, and aij, bij 2 [0, 1]. Let g be a positive integer and let Hgij ;bij ?X ij ? ? Haij ;bij ?Hg?1ij ?X ij ??, where aij ;b a H0ij ;bij ?X ij ? ? X ij . a

Sg ?X ij ? ? max?lij ; aij ? ? min?mij ; bij ? for g 2 f1; 2; 3; . . .g; P
where SJ? ?X ij ?; SJ ?X ij ?; SP ?X ij ? 2 ??1; 1? and
g g g

?30?
? 1.

S1 ?X ij ? J?

?

S1 ?X ij ? J

De?nition 4.2. Let Xij 2 IFS(X) and aij,bij 2 [0, 1] for each Ai 2 A and xj 2 X. Let g be a positive integer. The pessimistic score functions Sg ? ?X ij ?; Sg ?X ij ?; Sg ?X ij ? based on the H? ij ;bij ; Haij ;bij ; Q aij ;bij point a H Q H operators, respectively, are measured as follows:

?i?

mHg ?xj ? ? mij ? ?1 ? mij ? ? ?1 ? ?1 ? bij ?g ? ? bij ? lij
?
g?1 X g ?1 ? bij ?k ? aij?1?k ;
k?0

!

?22?

Sg ? ?X ij ? ? ag ? lij ? mij ? ?1 ? mij ? ? ?1 ? ?1 ? bij ?g ? ? aij ? bij ? lij ij H ! g?1 X k g?1?k ; ? ?1 ? bij ? ? aij
k?0

?31?

?ii?

pHg ?xj ? ? ?1 ? mij ? ag ? lij ? ? ?1 ? mij ? ? ?1 ? ?1 ? bij ?g ? ij
! g?1 X g?1?k k ? bij ? lij ? ?1 ? bij ? ? aij :
k?0
0

?23?

Sg ?X ij ? ? ag ? lij ? mij ? ?1 ? mij ? ? ?1 ? ?1 ? bij ?g ? ? bij ? lij H ij ! g?1 X g ? ?1 ? bij ?k ? aij?1?k ;
k?0

?32? ?33?

Hgij ;bij ?X ij ? a

&

mHg ?xj ? ? 1 and limg!1 pHg ?xj ? ? 0.

Hg?1ij ?X ij ?, aij ;b

where Haij ;bij ?X ij ? ? X ij . Moreover, limg!1

Sg ?X ij ? ? min?lij ; aij ? ? max?mij ; bij ? for g 2 f1; 2; 3; . . .g; Q
g g g

The intuitionistic index of Q aij ;bij ?X ij ? is:

pQ aij ;bij ?Xij ? ?xj ? ? 1 ? min?lij ; aij ? ? max?mij ; bij ?:
Let lQ ?xj ? ? lQ a ;b ?Xij ? ?xj ?, ij ij ?xj ? for short.

?24?

mQ ?xj ? ? mQ aij ;bij ?Xij ? ?xj ?, and pQ ?xj ? ? pQ aij ;bij

where SH? ?X ij ?; SH ?X ij ?; SQ ?X ij ? 2 ??1; 1? and ? ? ?1. Recall that w1, w2, . . ., wn are the weights of the criteria x1, x2, . . ., xn, respectively, where w1,w2,. . .,wn 2 [0, 1] and w1 + w2 + ? ? ? + wn = 1. Based on the simple additive weighting (SAW) method (Harsanyi, 1955), we can obtain a total score for

S1? ?X ij ? H

S1 ?X ij ? H

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T.-Y. Chen / Expert Systems with Applications 37 (2010) 7762–7774

each alternative simply via multiplying the optimistic or pessimistic score function for each criterion by the importance weight assigned to the criterion and then summing these products over all criteria. Let zJ? ; zJ ; zP ; zH? ; zH ; zQ be the total scores of alternative Ai i i i i i i by using the point operators J ? ij ;bij ; J aij ;bij ; P aij ;bij ; H? ij ;bij ; Haij ;bij and a a Q aij ;bij , respectively. They are de?ned as follows:

(iii) For the pessimistic condition:

( max zH? i ?

n X j?1

) SH? ?X ij ? ? wj ?j ? 1; 2; . . . ; n?; ?43?
g

zJ? ? w1 ? Sg? ?X i1 ? ? w2 ? Sg? ?X i2 ? ? ? ? ? ? wn ? Sg? ?X in ?; i J J J zJi ? w1 ? SJ ?X i1 ? ? w2 ? SJ ?X i2 ? ? ? ? ? ? wn ? SJ ?X in ?; zP ? w1 ? Sg ?X i1 ? ? w2 ? Sg ?X i2 ? ? ? ? ? ? wn ? Sg ?X in ?; i P P P zH? ? w1 ? Sg ? ?X i1 ? ? w2 ? Sg ? ?X i2 ? ? ? ? ? ? wn ? Sg ? ?X in ?; i H H H zH ? w1 ? Sg ?X i1 ? ? w2 ? Sg ?X i2 ? ? ? ? ? ? wn ? Sg ?X in ?; i H H H zQ i ? w1 ? SQ ?X i1 ? ? w2 ? SQ ?X i2 ? ? ? ? ? ? wn ? SQ ?X in ?:
g g g g g g

?34? ?35? ?36? ?37? ?38? ?39?

8 l > wj 6 wj 6 wu j < n s:t: P > wj ? 1 :
j?1

for each i = 1, 2, . . ., m; or

( max zH ? i

n X j?1

) Sg ?X ij ? ? wj H ?j ? 1; 2; . . . ; n?; ?44?

We denote these total scores as the suitability functions to determine the degrees to which an alternative satis?es the decision maker’s requirement. After comparing the total scores for all alternatives, the alternative with the highest score is the one prescribed to the decision maker. However, the information of multiple criteria corresponding to decision importance may be incomplete or lack of knowledge in real applications, and thus we will develop an optimization model for multicriteria decision making under the intuitionistic fuzzy environment with criteria being explicitly taken into account. 4.2. Optimization model with weighted score functions

8 l > wj 6 wj 6 wu j < n s:t: P > wj ? 1 :
j?1

for each i = 1, 2, . . ., m. (iv) For the pessimistic condition with restrictions:

( max zQ i ?

n X j?1

) SQ ?X ij ? ? wj ?j ? 1; 2; . . . ; n?; ?45?
g

8 l > wj 6 wj 6 wu j < n s:t: P > wj ? 1 :
j?1

for each i = 1, 2, . . ., m. Based on the SAW method, the suitability function zi to determine the degree to which the alternative Ai satis?es the decision maker’s requirement can be measured by an optimization model with weighted score functions. Several linear programming models will be constructed to determine optimal weights for criteria and further to acquire the optimal suitability degree. For each alternative Ai 2 A, we compute the optimal value of the suitability function zi by using the following programming models: (i) For the optimistic condition: Take Haij ;bij for example to explain the solving process of the programming model. Since there are m alternatives in the set A, we have to solve m linear programmings, in total, by the Simplex method. Although we can obtain the optimal weight vector for each alternative, the optimal solutions may be generally different and thus the corresponding optimal values of the suitability functions for all m alternatives cannot be compared. In view that the decision maker cannot easily or evidently judge the preference relations among all non-inferior alternatives, it is reasonable to assume all non-inferior alternatives to be of equal importance. Hence, by assigning an equal weight 1/m, we can aggregate m linear programmings into the following programming model:

( max zJ? ? i

n X j?1

) Sg? ?X ij ? ? wj J ?j ? 1; 2; . . . ; n?; ?40?

8 l > wj 6 wj 6 wu j < n s:t: P > wj ? 1 :
j?1

( max zH ?

for each i = 1, 2, . . ., m; or

( max zJi ?

n X j?1

) SJ ?X ij ? ? wj ?j ? 1; 2; . . . ; n?; ?41?
g

8 l > wj 6 wj 6 wu j < n s:t: P > wj ? 1: :
j?1

m n 1 XX g S ?X ij ? ? wj m i?1 j?1 H

!)

?j ? 1; 2; . . . ; n?;

?46?

8 l > wj 6 wj 6 wu j < n s:t: P > wj ? 1 :
j?1

    The optimal solution w ? ?w1 ; w2 ; . . . ; wn ?T can be obtained by solving (46) with the Simplex method. Correspondingly, the optimal value of suitability functions for the alternative Ai 2 A can be computed as follows:

for each i = 1, 2, . . ., m. (ii) For the optimistic condition with restrictions:

H ? zi

n X j?1

 Sg ?X ij ? ? wj H

?47?

( max zP ? i

n X j?1

) Sg ?X ij ? ? wj P ?j ? 1; 2; . . . ; n?; ?42?

8 l > wj 6 wj 6 wu j < n s:t: P > wj ? 1 :
j?1

for each i = 1, 2, . . ., m. The higher H the better the alternative Ai. zi Thus, the best alternative Ai* 2 A can be generated such that

zi Ai? ? fAi 2 AjmaxiH g:

?48?

for each i = 1, 2, . . ., m.

Moreover, the m alternatives can be ranked according to the decreasing order of H ’s for all Ai 2 A. zi

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5. Effectiveness examination on consumer decision reality In order to realize the effectiveness of the proposed MCDA method with intuitionistic fuzzy optimistic/pessimistic estimations in the real world, we have implemented an empirical study on consumer decision problems, consisting of selection of utilitarian products and hedonic products, to demonstrate the applicability of our proposed methods. 5.1. Pilot studies We conducted two pilot studies that one is con?rmation of the product category divided into utilitarian and hedonic products and the other is to choose an appropriate optimistic and pessimistic scale for the formal empirical study. In the ?rst pilot study, the digital camera and bank service were selected to represent utilitarian products. The chocolate and amusement park were chosen to represent hedonic products. The test was held to check whether the four products could representatively stand for the utilitarian and hedonic products. The 50 graduate students who participated in the pretest completed a four-item questionnaire which derived from Drolet, Williams, and Lau-Gesk (2007) and was measured on 5-point scales anchored by extremely disagree and extremely agree. Results of the mean test showed that the digital camera (t = 12.02, p < 0.001) and bank service (t = 8.94, p < 0.001) undoubtedly belonged to utilitarian products; the chocolate (t = ?9.37, p < 0.001) and amusement park (t = ?4.47, p < 0.001) belonged to hedonic products. A thought-list test was also employed to ask respondents to write down the criteria that they concerned when purchasing the four products separately. The results indicated that the most important three criteria were function, price, and ?gure for camera; brand awareness, service attitude, and deposit interest rate for bank service; ?avor, price, and packing for chocolate; amusement facility, ticket price, transportation for amusement park. Due to the dispositional optimism and pessimism that we want to observe in decision making problems, it is necessary to employ a proper scale for measuring the extent of optimism and pessimism to which decision makers have. According to the past literature, there were three popular scales to gain the optimistic and pessimistic degree. The Life Orientation Test (LOT) was pioneered by Scheier and Carver (1985) for measures of optimism and pessimism. Based on the LOT, Scheier et al. (1994) proposed a reevaluation of the LOT and called LOT-R. Chang et al. (1997) increased items and used factor analysis to determine 15-item Extended Life Orientation Test (ELOT). In our second pilot study, another 33 graduate students were enrolled to be participants and responded the three scales on a 5-point Likert scale. The reliability of scales was an index to decide which scale can be employed in the following main study. The pretest results demonstrated that optimistic and pessimistic constructs of LOT had higher and more stable reliability than the other two scales (Cronbach’s a = 0.692 in optimistic construct and Cronbach’s a = 0.635 in pessimistic construct). Although the reliability values were not high enough, we believed it is acceptable to meet the requisite of reliability because numerous researches showed that Cronbach’s a of exploring the dispositional optimism and pessimism was merely within 0.5–0.7 (Chang, Sanna, & Tang, 2003; Conway, Magai, Springer, & Jones, 2008; Hart & Hittner, 1995; Nicholls, Polman, Levy, & Backhouse, 2008). As a consequence, the LOT was viewed as the formal scale to measure optimistic and pessimistic degree in this study. 5.2. Questionnaire design and survey The aim of our empirical study is to examine the validity of the proposed intuitionistic fuzzy MCDA methods with optimism and

pessimism. Since A-IFSs and interval-valued fuzzy sets are mathematically equivalent, respondents will be asked to give an interval score within 0–100 when they evaluate the criterion importance and the outcomes of alternatives with respect to each criterion. The lower bound of the interval score represents the membership degree of A-IFS. The width of the interval stands for the intuitionistic index. Moreover, the non-membership degree equals one minus the upper bound of the interval score. The whole questionnaire can be divided into three parts. The ?rst part is the demographic characteristics of respondents which are capable of ?guring out the sample pro?le. In the second part, the importance of criteria and the evaluation of alternatives are investigated by using interval scores. Then, ask respondents to designate the preference rank of alternatives. In the third part, the eight-item LOT inventory is used to measure respondents’ degrees of optimism and pessimism on 5-point Likert scales anchored by ‘‘extremely disagree” and ‘‘extremely agree”. In order to generate a decision matrix, ?ve alternatives are determined in this study. The ?ve alternatives of four products are separately as follows. (i) digital camera: Canon, Sony, Olympus, Nikon, and Pentax; (ii) bank service: Bank of Taiwan, Chinatrust Commercial Bank, Cathay United Bank, First Bank, and HSBC; (iii) chocolate: Ferrero Rocher, Dove, M& M’s, Tappl, and Godiva; and (iv) amusement park: Leofoo Village Theme Park, Janfusun Fancyworld, Formosan Aboriginal Culture Village, Hualien Farglory Ocean Park, and Yamay Recreation World. The convenience sampling was adopted to enroll 330 respondents in this study. After the elimination of invalid samples, the number of ?nal samples amounted to 258. The valid rate reaches 78.2%. The proportion of female respondents occupies more than that of male (56.6% vs. 43.4%). The distribution of age is concentrated within 21–30 years old and approached 79.5%. In addition, because the samples exceeding 41 years old are de?cient, 31–40 years old, 41–50 years old, and over 51 years old respondents are viewed as category of ‘‘exceeding 31 years old”. As to the average monthly income, most of respondents have 5001–10,000 NT dollars to dominate in a month because the samples enrolled in this investigation are students in the majority. The detailed ?gures are indicated in Table 1. The reliability analysis can examine the internal consistency of scales. Guilford (1965) indicated that Cronbach’s a which exceeds 0.7 stands for high reliability. The optimistic and pessimistic constructs derived from the LOT surpass the standard of high reliability in our empirical study. The Cronbach’s a values are found to be 0.816 and 0.790 for optimism and pessimism, respectively. The results are better than other studies using the LOT to measure dispoTable 1 Pro?le of respondents. Demographic variable Total Gender Male Female Age 0–20 years old 21–30 years old Over 30 years old Average monthly income NT$0–NT$5000 NT$5001–NT$10,000 NT$10,001–NT$15,000 NT$15,001–NT$20,000 NT$20,001–NT$25,000 NT$25,001–NT$30,000 NT$30,001–NT$35,000 Over NT$35,000 Number of samples 258 112 146 36 205 17 70 109 24 17 13 6 8 11 Percentage (%) 100 43.4 56.6 14.0 79.5 6.5 27.1 42.3 9.3 6.6 5.0 2.3 3.1 4.3

7768 Table 2 Summary of ANOVA signi?cance levels. Variable Dependent variable: optimism Gender Age Average monthly income Dependent variable: pessimism Gender Age Average monthly income

T.-Y. Chen / Expert Systems with Applications 37 (2010) 7762–7774

F-value 0.182 2.790 0.399 3.062 0.042 0.382

p-value 0.670 0.063 0.902 0.081 0.958 0.912

The degree /j of membership and the degree uj of non-membership for the three criteria xj 2 X to the fuzzy concept ‘‘importance” are given as follows:

The criterion weights also lie in the closed intervals, respectively. Namely,

sitional optimism and pessimism (Hart & Hittner, 1995; David, Montgomeryand, & Bovbjerg, 2006; Nicholls et al., 2008). The optimistic construct has a standardized mean of 0.690 and the pessimistic construct gets a mean of 0.495. Males and females have no signi?cant difference on the mean of optimistic scores (F = 0.182, p > 0.05) and pessimistic scores (F = 3.062, p > 0.05). Both the age and average monthly income also have no salient difference on the mean of optimistic and pessimistic scores. Table 2 shows some statistics of ANOVA. According to the mean of optimistic and pessimistic scores, samples will be categorized into one of quadrants because we have to discriminate each respondent who belongs to an optimist or a pessimist. Table 3 demonstrates the classi?cation result. The category of high optimism and low pessimism (Type II) occupies the largest proportion which amounts to 35.3%. The second largest proportion amounting to 26.7% is the category of low optimism and high pessimism (Type III). We will conduct an examination approach to demonstrate the validity of our proposed method by employing the empirical data, especially test outcomes of Types II and III respondents. 5.3. Illustrative example We take the investigation data of a Type III respondent as an illustrative example to demonstrate how to apply the proposed method in consumer decision analysis and how to examine the effectiveness of the method with intuitionistic fuzzy optimistic/ pessimistic estimations. The respondent has been asked to consider a decision problem of bank service if he is going to do some ?nancial events. Five alternatives are denoted by A = {A1, A2, . . ., A5} for him to select, including Bank of Taiwan (A1), Chinatrust Commercial Bank (A2), Cathay United Bank (A3), First Bank (A4), and HSBC (A5). Three criteria of bank service are given to evaluate, including brand awareness (x1), service attitude (x2), and deposit interest (x3). Denote the set of criteria by X = {x1, x2, x3}. We convert all interval scores into A-IFS values to construct an intuitionistic fuzzy decision matrix, where lij and mij represent the membership and non-membership degree, respectively, for Ai 2 A with respect to xj 2 X to the fuzzy concept ‘‘excellence”:

P P where 3 wlj ? 0:70 6 1 and 3 wu ? 1:32 P 1. In addition, this j?1 j?1 j respondent was asked to give his ranking order regarding overall evaluations of ?ve banks according to three criteria, and his response is as follows: A2 1 A4 1 A5 1 A3 1 A1. In our examination approach, we consider two methods to observe the difference between the results. One employs the original score function which equals the membership degree minus the non-membership degree; the other takes the pessimistic score function based on the pessimistic point operator. Case I: employment of original score functions The decision matrix D has been transferred into D0 by means of original score functions as follows:

The suitability functions for separate alternative based on SAW procedures are

z1 ? 0:00w1 ? 0:51w2 ? 0:55w3 ; z2 ? 0:27w1 ? 0:30w2 ? 0:44w3 ; z3 ? 0:32w1 ? 0:14w2 ? 0:28w3 ; z4 ? ?0:06w1 ? 0:38w2 ? 0:37w3 ; z5 ? 0:50w1 ? 0:76w2 ? 0:22w3 :
According to the ?ve suitability functions, a linear programming model can be obtained as follows:

  1:03wi ? 2:09w2 ? 1:86w3 max z ? 5 8 > 0:44 6 w1 6 0:64; > > < 0:01 6 w 6 0:21; 2 s:t: > > 0:25 6 w3 6 0:47; > : w1 ? w2 ? w3 ? 1:
Using the Simplex method to solve the above linear program ming, its optimal solution can be acquired as w ? ?0:44; 0:21; 0:35?T . In addition, we can then acquire z1 ? 0:2996; z2 ? 0:3358; z3 ? 0:2686; z4 ? 0:1829, and z5 ? 0:2366. Hence, the optimal ranking order of ?ve alternatives is given by A2 1 A1 1 A3 1 A5 1 A4. The Spearman rank correlation coef?cient (q) can help us to capture the relationship between the ranking order given by the respondent and the estimated ranking based on original score functions. In Case I, q = ?0.3. Case II: employment of pessimistic score functions Consider the pessimistic point operator Haij ;bij and let aij and bij be ?xed values for simplicity. The parameter values are designated as follows: aij = 0.56, bij = 0.96 for each Ai 2 A and xj 2 X, and g = 1.

Table 3 Sample classi?cation by optimistic and pessimistic constructs. Type I II III IV Description Low optimism and low pessimism High optimism and low pessimism Low optimism and high pessimism High optimism and high pessimism Number 47 91 69 51 Percentage (%) 18.2 35.3 26.7 19.8

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Then, the decision matrix D00 can be derived through the pessimistic score function Sg ?X ij ? in (32). For example, the pessimistic score H value of X23 is given by S1 ?X 23 ? ? 0:561 ? 0:44 ? 0? H 1 ?1 ? 0? ? ?1 ? ?1 ? 0:96? ? ? 0:96 ? 0:44 ? ?0:2912.

We have the suitability functions zH of each alternative by using i (38) and further get the objective zH in (46). Then, we can obtain the following linear programming model:

  ?1:6952wi ? 1:2236w2 ? 1:4844w3 max zH ? 5 8 > 0:44 6 w1 6 0:64; > > < 0:01 6 w 6 0:21; 2 s:t: > > 0:25 6 w3 6 0:47; > : w1 ? w2 ? w3 ? 1:
The optimal solution of the above programming model is  w ? ?0:44; 0:21; 0:35?T . By applying (48), we can obtain z1 H ? ?0:2885; z2 H ? ?0:3754; z3 H ? ?0:4869; z4 H ? ?0:2670, and z5 H ? ?0:1045. Hence, the optimal ranking order of ?ve alternatives can be generated by A5 1 A4 1 A1 1 A2 1 A3. In Case II, the Spearman rank correlation coef?cient (q) between the investigated ranking order and the estimated ranking based on pessimistic score functions is 0.9. Comparing the two q-values, we conclude that the proposed method with pessimistic score functions has desirable determination results on ?nal rankings of alternatives for the selected Type III respondent. 5.4. Effectiveness examination and empirical results We use the aforementioned approach to examine the effectiveness of the proposed method with optimistic/pessimistic score functions. A simple and intuitively reasonable algorithm for comparing ranking orders yielded by investigation data and by optimistic/pessimistic estimations is developed. The algorithm of effectiveness examination given here is suitable for individual or group decision makers. In addition, we provide another algorithm to compare the investigated ranking order and the estimated ranking based on original score functions in Appendix B. The detailed implementation procedure of the examination algorithm is as follows: 5.4.1. Algorithm of effectiveness examination (EE)

mism types according to their LOT scores. The sample size of Types I, II, III, IV are j1, j2, j3, j4, respectively, where j1 + j2 + j3 + j4 = j. Each iteration in this algorithm will be labeled f, where f = 0, 1, 2, . . .. Initialize the total number of iterations, fmax, as desired. Step 1: Set the parameter values For Types II and III respondents, set the parameter values aij, bij 2 [0, 1] for each Ai 2 A and xj 2 X. For Types I and IV respondents, separately set the parameter values aij,bij 2 [0, 1] and aij + bij 6 1 for each Ai 2 A and xj 2 X. Set the g value as a positive integer. Set the iteration counter: f = 0. Step 2: While the stopping condition is false, do Steps 3–10 Step 3: Construct the decision matrix based on score functions For each Type II respondent nk (k = 1, 2, . . ., j2), apply (28) or (29) to determine the decision matrix D00 by using k optimistic score functions as follows:

?49?

?50?

For each Type III respondent nk (k = 1, 2, . . ., j3), apply (31) or (32) to determine the decision matrix D00 by using pessimistic score funck tions as follows:

?51?

?52?
Step 0: Input data through questionnaire survey For each Ai 2 A and xj 2 X, evaluate alternatives in terms of criteria on the fuzzy concept ‘‘excellence”, where the values of criterion functions are expressed by A-IFSs. Denote X k ? fhxk ; lk ; mk ig for each respondent nk ij j ij ij (k = 1, 2, . . ., j), where j is the number of valid samples. Evaluate each criterion xj 2 X on the fuzzy concept ‘‘importance” that is expressed by a closed interval. Denote the criterion weight ?wlj;k ; wu ? ? ?/k ; /k ? sk ? for j j j j;k each respondent nk (k = 1, 2, . . ., j), where 0 6 wlj;k 6 Pn Pn l u wu 6 1; j?1 wj;k 6 1 and j?1 wj;k P 1 for each criterion j;k xj 2 X. Investigate the respondent’s subjective judgment on the preference priority toward all alternatives in A. The respondents are classi?ed into four optimism/pessi-

For each Type I respondent nk (k = 1, 2, . . ., j1) and Type IV respondent nk (k = 1, 2, . . ., j4), apply (30) or (33) to determine the decision matrix D00 by using restricted score functions as follows: k

?53?

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T.-Y. Chen / Expert Systems with Applications 37 (2010) 7762–7774

For each Type I respondent nk (k = 1, 2, . . ., j1) and Type IV respondent nk (k = 1, 2, . . ., j4), apply (42) or (45) to construct the optimization model:

?54?

( max zP ? k

Step 4: Determine the suitability function by weighted score functions For each Ai 2 A and each Type II respondent nk (k = 1, 2, . . ., j2), determine the suitability function by applying (34) or (35) as follows:

8 l > wj;k 6 wk 6 wu j j;k < n s:t: P k > wj ? 1; :
j?1

m n 1 XX g k S ?X ? ? wk j m i?1 j?1 P ij

!)

?j ? 1; 2; . . . ; n?;

?65?

or

zJ? ? wk ? Sg? ?X k ? ? wk ? Sg? ?X k ? ? ? ? ? ? wk ? Sg? ?X k ?; 1 2 n i1 i2 in i;k J J J zJi;k ? wk ? Sg ?X k ? ? wk ? Sg ?X k ? ? ? ? ? ? wk ? Sg ?X k ?: 1 2 n i1 i2 in J J J

?55? ?56? max

( zQ k

For each Ai 2 A and each Type III respondent nk (k = 1, 2, . . ., j3), determine the suitability function by applying (37) or (38) as follows:

8 l > wj;k 6 wk 6 wu j j;k < n s:t: P k > wj ? 1: :
j?1

m n 1 XX g k ? S ?X ? ? wk j m i?1 j?1 Q ij

!)

?j ? 1; 2; . . . ; n?;

?66?

zH? ? wk ? Sg ? ?X k ? ? wk ? Sg ? ?X k ? ? ? ? ? ? wk ? Sg ? ?X k ?; i1 i2 in i;k 1 2 n H H H zH ? wk ? Sg ?X k ? ? wk ? Sg ?X k ? ? ? ? ? ? wk ? Sg ?X k ?: i1 i2 in i;k 1 2 n H H H

?57? ?58?

For each Type I respondent nk (k = 1, 2, . . ., j1) and Type IV respondent nk (k = 1, 2, . . ., j4), determine the suitability function for each Ai 2 A by applying (36) or (39) as follows:

Step 6: Derive the optimal value of suitability functions For each Type II respondent nk (k = 1, 2, . . ., j2), we solve  (61) or (62) to obtain the optimal solution wk ? 1 2 n ?wk ; wk ; . . . ; wk ?T , and the optimal value of suitability functions for Ai 2 A can be derived by:

zP i;k zQ i;k

?

wk 1

?

SP ?X k ? i1 SQ ?X k ? i1
g

g

?

wk 2

?

SP ?X k ? i2
g

g

? ??? ?

wk n

?

SP ?X k ?; in SQ ?X k ?: in
g

g

?59? ?60?

J? ? zi;k Ji;k ? z

n X j?1 n X j?1

j Sg ?X k ? ? wk ; or ij J? j Sg ?X k ? ? wk : ij J

?67? ?68?

? wk ? 1

? wk ? 2

SQ ?X k ? i2

? ? ? ? ? wk ? n

Step 5: Establish the optimization model with maximal suitability For each Type II respondent nk (k = 1, 2, . . ., j2), apply (40) or (41) to construct the optimization model:

( max zJ? ? k

8 l > wj;k 6 wk 6 wu j j;k < n s:t: P k > wj ? 1; :
j?1

m n 1 XX g k S ? ?X ? ? wk j m i?1 j?1 J ij

!)

For each Type III respondent nk (k = 1, 2, . . ., j3), the optimal value of suitability functions for Ai 2 A can be acquired through solving (63) or (64) as follows:

?j ? 1; 2; . . . ; n?;

?61?

H? ? zi;k H ? zi;k

n X j?1 n X j?1

j Sg ?X k ? ? wk ; or ij H? j Sg ?X k ? ? wk : ij H

?69? ?70?

or

( max zJk

8 l > wj;k 6 wk 6 wu j j;k < n s:t: P k > wj ? 1: :
j?1

m n 1 XX g k ? S ?X ? ? wk j m i?1 j?1 J ij

!)

For each Type I respondent nk (k = 1, 2, . . ., j1) and Type IV respondent nk (k = 1, 2, . . ., j4), the optimal value of suitability functions for Ai 2 A can be acquired through solving (65) or (66) as follows:

?j ? 1; 2; . . . ; n?;

?62? P ? zi;k Q ? zi;k

n X j?1 n X j?1

j Sg ?X k ? ? wk ; or ij P j Sg ?X k ? ? wk : ij Q

?71? ?72?

For each Type III respondent nk (k = 1, 2, . . ., j3), apply (43) or (44) to construct the optimization model:

( max
? zH k

8 l > wj;k 6 wk 6 wu j j;k < n s:t: P k > wj ? 1; :
j?1

m n 1 XX g k ? S ? ?X ? ? wk j m i?1 j?1 H ij

!)

?j ? 1; 2; . . . ; n?;

?63?

or

( max zH ? k

8 l > wj;k 6 wk 6 wu j j;k < n s:t: P k > wj ? 1: :
j?1

m n 1 XX g k S ?X ? ? wk j m i?1 j?1 H ij

!)

?j ? 1; 2; . . . ; n?;

?64?

Step 7: Ranking order of alternatives For each Type II respondent nk (k = 1, 2, . . ., j2), rank m alternatives according to the decreasing order of J? ’s or zi;k J ’s. zi;k For each Type III respondent nk (k = 1, 2, . . ., j3), rank m alternatives in descending order of H? ’s or H ’s. zi;k zi;k For each Type I respondent nk (k = 1, 2, . . ., j1) and Type IV respondent nk (k = 1, 2, . . ., j4), rank m alternatives in descending order of P ’s or Q ’s. zi;k zi;k Step 8: Compute correlation between ranking orders For each respondent nk(k = 1, 2, . . ., j), compute the Spearman rank correlation coef?cient (qk) between the ranking order given by the respondent and the estimated ranking yielded by optimistic/pessimistic score functions. The average Spearman rank correlation coef?cients P  can then be derived by q ? j qk =j. k?1

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Step 9: Update the iteration counter Let f(new) = f(old) + 1. Step 10: Test for the stopping condition  If q P e, where e is a designated threshold value, or if f > fmax, then stop; otherwise, reset the values of aij, bij, g and continue. The point we would like to emphasize is different setting of aij and bij between Types I and IV respondents in the present algorithm. Although the optimistic/pessimistic operators with restrictions have been used to treat the investigation data of Types I and IV respondents, the parameter settings are not the same. The setting values of aij and bij for Type IV respondents lie in a relatively wide extent, while the values of aij and bij for Type I respondents are close to lij and mij, respectively, on account of the nature of low optimism and low pessimism. In our empirical study, we set g = 1, fmax = 2, and e = 0.9 in all cases for simplicity. For Type II respondents, we adopt the point operator J aij ;bij , where the values of aij and bij are initialized as folP P lows: aij ? j2 lk =j2 and bij ? j2 mk =j2 for each Ai 2 A and k?1 ij k?1 ij xj 2 X. For Type III respondents, we use the point operator Haij ;bij , Pj3 k P where initial aij ? k?1 lij =j3 and bij ? j3 mk =j3 for each Ai 2 A k?1 ij and xj 2 X. The speci?cation of initial aij and bij for Types I and IV respondents is in a similar way, but the employment Paij ;bij or Q aij ;bij will depend on the LOT scores on optimistic and pessimistic constructs. If the respondent’s pessimistic score is larger than the optimistic score, the point operator Q aij ;bij will be used; otherwise, P aij ;bij will be employed. We conduct the algorithm of effectiveness examination on the aforementioned 258 samples to validate the proposed method with optimistic/pessimistic estimations. Furthermore, the algorithm in Appendix B will be also implemented for the sake of comparing results with and without consideration of optimism/ pessimism. Table 4 summarizes the major results of average Spearman rank correlation coef?cients by using original score functions and optimistic/pessimistic score functions. Moreover, the rates of change of average Spearman correlation coef?cients are listed in Table 5. On the average, the comparison results show medium correlations of the investigated rankings and estimated rankings by using the optimization model without consideration of optimism/pessimism. Besides, the standard deviations of Spearman correlation coef?cients are very large in four product categories, and thus the discrepancy of Spearman correlation coef?cients is strikingly obvious among respondents. As depicted in the ?rst part of Table 4, among the comparison results by Algorithm OS in four types, Type I has the largest Spearman correlation coef?cients; whereas Type IV has the lowest correlation coef?cients, except for the digital

Table 5 Improvement rates (%) of average Spearman correlation coef?cients. Type Utilitarian product Digital camera Total I II III IV 72.32 47.98 71.00 96.79 72.70 Bank service 88.27 53.32 78.51 108.03 133.28 Hedonic product Chocolate 146.82 117.06 174.14 109.36 208.94 Amusement park 118.15 74.91 114.22 141.03 158.64

camera category. Taking into account the fact that Algorithm OS was carried out under neutral condition, the results were reasonably reliable. These results appear to furnish evidence that the SAW approach is more appropriate for neutral respondents, and thus the results in Type I (low optimism and low pessimism) have relatively high correlations in comparison to other types. The ?ndings re?ected in the second part of Table 4 demonstrate that our proposed MCDA method with intuitionistic fuzzy optimistic/pessimistic operators has reasonably valid and reliable results on account of high correlations and low standard deviations. Of the 16 correlations of Algorithm EE results in Types I–IV, it can be seen that 11 are above 0.9 and 5 are above 0.8, especially the ob tained q is as large as 0.9722 for Type I respondents in the digital camera category. The resulting correlation coef?cients for total samples turn out to be very high as follows: 0.9526 in digital camera, 0.9229 in bank service, 0.8772 in chocolate, and 0.9016 in amusement park. These correlations present a very strong relationship between the investigated rankings provided by respondents and estimated rankings yielded by the proposed MCDA method. In addition, the results reveal that Algorithm EE has more accurate effects on utilitarian products than on hedonic products. The same observation can be covered in the results by Algorithm OS. The overall accuracy in the digital camera category is highest, second by bank service, third by amusement park, and the lowest by chocolate. Table 5 highlights the rates of improvement on average Spearman correlation coef?cients between the results by two algorithms. Three points are worth making about Table 5. First, the results of the optimization model by using optimistic/pessimistic score functions have bring about great improvement on validity.  The improvement rates of q values are as follows: 72.32% in digital camera, 88.27% in bank service, 146.82% in chocolate, and 118.15% in amusement park. Second, the method with optimistic/pessimistic operators generally produces a marked effect on accuracy enhancement in Types III and IV respondents. Third, the results on hedonic products show that their improvement effects are signi?cantly higher than those on utilitarian products. As a whole, the

Table 4  Results of average Spearman correlation coef?cients ?q?. Type Utilitarian product Digital camera Comparison results by algorithm OS Total 0.5528 I 0.6570 II 0.5555 III 0.4791 IV 0.5517 Comparison results by Algorithm EE Total 0.9526 I 0.9722 II 0.9499 III 0.9428 IV 0.9528
a

Hedonic product Bank service 0.4902 0.6311 0.5137 0.4359 0.3918 0.9229 0.9676 0.9170 0.9068 0.9140 (0.4750) (0.3935) (0.4658) (0.4950) (0.5030) (0.1774) (0.0898) (0.1879) (0.2101) (0.1686) Chocolate 0.3554(0.5189) 0.4204 (0.4601) 0.3244 (0.4981) 0.4157 (0.5331) 0.2695 (0.5815) 0.8772 0.9125 0.8893 0.8703 0.8326 (0.2215) (0.1544) (0.1714) (0.2741) (0.2677) Amusement park 0.4133 0.5501 0.4227 0.3673 0.3329 0.9016 0.9622 0.9055 0.8853 0.8610 (0.4910) (0.4204) (0.4877) (0.5191) (0.5046) (0.1944) (0.0863) (0.1977) (0.2133) (0.2228)

(0.4527)a (0.3478) (0.4656) (0.5008) (0.4391) (0.1269) (0.1112) (0.1286) (0.1529) (0.0969)

Standard deviations are in parentheses.

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above-mentioned results appear to support the superiority of the proposed MCDA method with optimistic/pessimistic estimations. We can be fairly certain that the proposed MCDA method is feasible and practical for consumer choice problems. The empirical results support the validity of using optimistic/pessimistic point operators to restructure the decision matrix and to determine the suitability function. These ?ndings indicate that if the decision maker tends toward optimism/pessimism, we can take advantage of the optimistic/ pessimistic point operators to adjust the values in the original decision matrix and determine a more accurate ranking order corresponding to what the decision make yield. 6. Conclusions Under the intuitionistic fuzzy environment, this research handles the decision making problem with optimism and pessimism in nature by setting appropriate values of aij’s and bij’s within optimistic/pessimistic point operators. The optimistic or pessimistic score functions can then be acquired by using the information provided by the corresponding point operators. Next, the degree of suitability to which an alternative satis?es the decision maker’s requirement will be determined by aggregating score functions, where the optimal weights for criteria are determine by linear programming models. Finally, the bigger the degree of suitability is, the more preferred the alternative is. Since membership functions and operations on fuzzy sets are generally context-dependent, the appropriateness of intuitionistic fuzzy optimistic/pessimistic point operators must be examined in the context of various particular applications. Therefore, we adopted an outcome-oriented approach to validate practical usefulness of these operators. Feasibility and effectiveness of the proposed MCDA methods based on optimistic/pessimistic point operators are illustrated by empirical studies on consumer decision problems, including utilitarian and hedonic products. In view of average Spearman correlation coef?cients, our proposed optimization model with optimistic/pessimistic score functions performs much better than the method with original score functions. In addition, the overall accuracy of utilitarian products is greater than that of hedonic products, while a contrast observation has been found on the improvement rates of correlation coef?cients. The empirical results lead us to believe that we can reasonably conclude that the usage of optimistic/pessimistic point operators can improve estimation validity in real-world decision problems. In addition to a signi?cant improvement in overall accuracy through the present model has been obtained, this article points to new possibilities for further research of real-world applications by using other operators, such as a universal operators Xa,b,c,d,e,f proposed by Atanassov (1999). Appendix A

lJ?;1 ?xj ? ? lJ?;0 ?xj ? ? aij ? ?1 ? lJ?;0 ?xj ? ? bij ? mJ?;0 ?xj ?? ? lij ? ?1 ? lij ? ? aij ? aij ? bij ? mij ; pJ?;1 ?xj ? ? 1 ? lJ?;1 ?xj ? ? mJ?;1 ?xj ? ? ?1 ? lij ? bij ? mij ? ? ?1 ? lij ? ? aij ? aij ? bij ? mij :
Taking g = 1 in (6) and (7), we obtain

lJ?;1 ?xj ? ? lij ? ?1 ? lij ? ? aij ? aij ? bij ? mij ; pJ?;1 ?xj ? ? ?1 ? lij ? bij ? mij ? ? ?1 ? lij ? ? aij ? aij ? bij ? mij :
Thus, (6) and (7) hold for g = 1. (b) For g = 2, by (A.1) we have

lJ?;2 ?xj ? ? lJ?;1 ?xj ? ? aij ? ?1 ? lJ?;1 ?xj ? ? bij ? mJ?;1 ?xj ?? ? lij ? ?1 ? lij ? ? aij ? aij ? bij ? mij ? aij ? ?1 ? lij ? ?1 ? lij ? ? aij ? aij ? bij ? mij ? b2 ? mij ? ij ? lij ? ?1 ? lij ? ? ?2aij ? a2 ? ij ? aij ? bij ? mij ? ??1 ? aij ? ? bij ?; pJ?;2 ?xj ? ? 1 ? lJ?;2 ?xj ? ? mJ?;2 ?xj ? ? ?1 ? lij ? b2 ? mij ? ? ?1 ? lij ? ? ?2aij ? a2 ? ij ij ? aij ? bij ? mij ? ??1 ? aij ? ? bij ?:
From (6) and (7), let g = 2, we get

lJ?;2 ?xj ? ? lij ? ?1 ? lij ? ? ?2aij ? a2 ? ? aij ? bij ? mij ? ??1 ? aij ? ? bij ?; ij pJ?;2 ?xj ? ? ?1 ? lij ? b2 ? mij ? ? ?1 ? lij ? ? ?2aij ? a2 ? ij ij ? aij ? bij ? mij ? ??1 ? aij ? ? bij ?:
Thus, (6) and (7) hold for g = 2. (c) If (6) and (7) hold for g = w, that is

lJ?;w ?xj ? ? lij ? ?1 ? lij ? ? ?1 ? ?1 ? aij ?w ? ? aij ? bij ? mij
? ! w?1 X ?1 ? aij ?k ? bw?1?k ; ij
k?0

pJ?;w ?xj ? ? ?1 ? lij ? bw ? mij ? ? ?1 ? lij ? ? ?1 ? ?1 ? aij ?w ? ij
? aij ? bij ? mij ?
w?1 X k?0

!

?1 ? aij ? ?

k

bw?1?k ij

;

then, when g = w + 1, by (A.1), we have

lJ?;w?1 ?xj ? ? lJ?;w ?xj ? ? aij ? ?1 ? lJ?;w ?xj ? ? bij ? mJ?;w ?xj ?? ? lij ? ?1 ? lij ? ? ?1 ? ?1 ? aij ?w ?

! w?1 X k w?1?k ? aij ? bij ? mij ? ?1 ? aij ? ? bij k?0 n ? aij ? 1 ? lij ? ?1 ? lij ? ? ?1 ? ?1 ? aij ?w ? ! ) w?1 X ?aij ? bij ? mij ? ?1 ? aij ?k ? bw?1?k ? bw?1 ? mij ij ij
k?0

Proof. Theorem 3.1 will be proven by using mathematical induction on g. Let

2

3 2 3 lJ?;g?1 ?xj ? ? aij ? ?1 ? lJ?;g?1 ?xj ? ? bij ? mJ?;g?1 ?xj ?? lJ?;g ?xj ? 6 m ?;g ?x ? 7 6 7 bij ? mJ?;g?1 ?xj ? 4 J 5; j 5 ? 4 1 ? lJ?;g ?xj ? ? mJ?;g ?xj ? pJ?;g ?xj ?

? lij ? ?1 ? lij ? ? ?1 ? ?1 ? aij ?w?1 ? ! w X k w?k ?1 ? aij ? ? bij ; ? aij ? bij ? mij ?
k?0

pJ?;w?1 ?xj ? ? 1 ? lJ?;w?1 ?xj ? ? mJ?;w?1 ?xj ?
? ?1 ? lij ? bw?1 ? mij ? ? ?1 ? lij ? ? ?1 ? ?1 ? aij ?w?1 ? ij ! w X k w?k ? aij ? bij ? mij ? ?1 ? aij ? ? bij :
k?0

lJ?;0 ?xj ? lij 5. mij where g = 0, 1, 2, . . . , with 4 mJ?;0 ?xj ? 5 ? 4 pJ?;0 ?xj ? 1 ? lij ? mij
(a) From (A.1), let g = 1, we have

2

3

2

3

?A:1?

Since (6) and (7) hold for g = w + 1, they hold for all g. h

T.-Y. Chen / Expert Systems with Applications 37 (2010) 7762–7774

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Appendix B In a similar manner of Algorithm EE, we provide the following algorithm to determine the ranking results of alternatives by using the proposed optimization model with original score functions. B.1. Algorithm with original score functions (OS) Step 0: Input data through questionnaire survey Collect ratings regarding the decision matrix and criterion weights in intuitionistic fuzzy data form, and investigate the respondent’s preference order of alternatives. See Step 0 of Algorithm EE for further details. Step 1: Construct the decision matrix based on score functions For each respondent nk (k = 1, 2, . . ., j), determine the decision matrix D0k by using original score functions as follows:

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?A:2?

Step 2: Determine the suitability function by weighted score functions For each respondent nk (k = 1, 2, . . ., j), determine the suitability function by using weighted score functions as follows:

z0i;k ? wk ? ?lk ? mk ? ? wk ? ?lk ? mk ? ? ? ? ? ? wk ? ?lk ? mk ?: 1 i1 i1 2 i2 i2 3 in in ?A:3?
Step 3: Establish the optimization model with maximal suitability For each respondent nk (k = 1, 2, . . ., j), we construct the following programming model to determine the optimal weights and optimal suitability function:

( max z0k

m n  1 X X k ? lij ? mk ? wk ij j m i?1 j?1

!)

8 l > wj;k 6 wk 6 wu j j;k > < s:t: P n > > wk ? 1: : j
j?1

?j ? 1; 2; . . . ; n?;

?A:4?

Step 4: Derive the optimal value of suitability functions For each respondent nk (k = 1, 2, . . ., j), we solve (A.4) to  1 2 n obtain the optimal solution wk ? ?wk ; wk ; . . . ; wk ?T and the optimal value of suitability functions for Ai 2 A can be derived by:

0i;k ? z

n X j ?lk ? mk ? ? wk : ij ij j?1

?A:5?

Step 5: Ranking order of alternatives For each respondent nk (k = 1, 2, . . ., j), rank m alternatives according to the decreasing order of 0i;k ’s for all Ai 2 A. z Step 6: Compute correlation between ranking orders For each respondent nk (k = 1, 2, . . ., j), calculate the Spearman rank correlation coef?cient (qk) between the ranking order given by the respondent and the estimated ranking yielded by original score functions. Then, compute average  Spearman rank correlation coef?cients q.

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