Solving multi-criteria decision making with incomplete linguistic preference relations

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Solving multi-criteria decision making with incomplete linguistic

preference relations

Shu-Chen Hsu

a

_{, Tien-Chin Wang}

b,⇑

a

Department of Information Engineering, I-Shou University, Kaohsiung, Taiwan, ROC

b

Department of International Business, National Kaohsiung University of Applied Sciences, 415 Chien Kung Road, Kaohsiung 807, Taiwan, ROC

a r t i c l e

i n f o

Keywords:

Incomplete linguistic preference relations InLinPreRa

Fuzzy preference relations Analytic hierarchy process Multi-criteria decision making

a b s t r a c t

When there are n criteria or alternatives in a decision matrix, a pairwise comparison methodology of ana-lytic hierarchy process (AHP) with the time of n(n 1)/2 is frequently used to select, evaluate or rank the neighboring alternatives. But while the number of criteria or comparison level increase, the efﬁciency and consistency of a decision matrix decrease. To solve such problems, this study therefore uses horizontal, vertical and oblique pairwise comparisons algorithm to construct multi-criteria decision making with incomplete linguistic preference relations model (InLinPreRa). The use of pairwise comparisons will not produce the inconsistency, even allows every decision maker to choose an explicit criterion or alter-native for index unrestrictedly. When there are n criteria, only n 1 pairwise comparisons need to be car-ried out, then one can rest on incomplete linguistic preference relations to obtain the priority value of alternative for the decision maker’s reference. The decision making assessment model that constructed by this study can be extensively applied to every ﬁeld of decision science and serves as the reference basis for the future research.

1. Introduction

Analytic hierarchy process (AHP) has emerged as a practical tool for solving complex unstructured economics, social and

manage-ment problems for many years (Saaty, 1978, 1980). Based on the

use of traditional method as the selection basis of the evaluation alternative, pairwise comparison is necessary for all of the strate-gies, this is not only subjective, but also incomplete decision-making consideration might happen, which would not be tallied with the demand of actual problems and lack of efﬁciency. In order to understand the inconsistent problems in analytic hierarchy pro-cess that are caused by multi-decision-makers, multi-criteria, and

multi-alternatives,Herrera-Viedma, Herrera, Chiclana, and Luque

(2004)addressed fuzzy preference relations, if there are n evalua-tion items, only n 1 pairwise comparisons are needed as to ob-tain the relative weight for each item, while AHP needs to compare n(n 1)/2 times, it needs less time that can greatly im-prove the decision making efﬁciency. Several studies showed that when decision makers used linguistic preference relations to carry

out pairwise comparison (Xu, 2004a), it is possible that they are

not able to compare because of time pressure, lack of complete information, lack of such kind of professional knowledge, or even the provided information is insufﬁcient. In order to solve this

prob-lem,Xu (2006b)brought up pairwise comparison, each expert can select anyone of the explicit items as the standard according to his/ her preference or recognition, and then the pairwise comparison would be carried out between the adjoining items in order obtain the complete preference matrix. During the decision-making pro-cess, the good and bad alternatives can only be chosen through comparison, since the things that need to be considered always have many complicated and uncertain factors, and there exist too many factors, in incomplete linguistic preference relations that is

addressed byXu (2006b), even through the decision-makers can

have more choices and ﬂexibility during the comparison process, the multi-criteria, multi-alternatives, and multi-decision-makers of decision making problems have not been included. For the sake of conforming to the general and actual decision making more, this study would base on this framework to propose multi-criteria decision making with incomplete linguistic preference relations, one hopes to establish a reasonable and objective evaluation model that can be ﬁt in the actual problems, so that the application scope can be more extensive, and it can serve as the reference and basis for the relevant evaluation decision making model.

2. Literature review

2.1. Multi-criteria decision making

Saaty (1978, 1980)proposed analytic hierarchy process (AHP), a multi-criteria decision making (MCDM) approach that has been

⇑Corresponding author. Tel.: +886 7 3814526x6126.

E-mail addresses:[email protected](S.-C. Hsu),[email protected]

(T.-C. Wang).

Expert Systems with Applications 38 (2011) 10882–10888

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Expert Systems with Applications

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used in almost the applications related with decision-making. A hierarchy framework of analytic hierarchy process is shown in

Fig. 1. Multiple Criteria Decision-Making is the optimal choice, with different type depended on decision makers’ preference, sorted of Multiple Objective Decision Making (MODM) and

Multi-ple Attribute Decision Making (MADM). Yoon and Hwang, 1985

provided that Multiple Criteria Decision-Making is a possible eval-uation scale for many characters or quantities of decision-makers’ evaluation. MCDM has been extensively applied to various areas

such as society, economics, military, management, etc. (Basin &

Imad, 2003; Ding & Liang, 2005; Geldermann, Spengler, & Rentz, 2000; Wang & Chang, 2007; Wu & Lee, 2007).

Under several alternatives and several evaluation criteria, MCDM quantiﬁes each evaluation criterion and applies scientiﬁc methods and skills to carry on multi-criteria decision-making anal-ysis, so as to conduct a quality order and evaluation for each alter-native, then the best alternative that can conform to the decision maker’s ideal (Yoon & Hwang, 1985).

2.2. Fuzzy preference relations

Herrera-Viedma et al. (2004) proposed fuzzy preference relations to understand the inconsistent problems in analytic hierarchy process that are caused by multi-decision-makers, multi-criteria, and multi-alternatives. Preference relation means that the decision maker counters a set of criteria or alternatives according to the linguistic variables so as to carry out the pairwise comparison, then a mapping value can be derived; this value rep-resents the degree of preference of the ﬁrst alternative towards the

second alternative (Chang & Chen, 1994; Chiclana, Herrera, &

Herrera-Viedma, 1998; Chiclana, Herrera, & Herrera-Viedma, 2001; Chiclana, Herrera, & Herrera-Viedma, 2002; Cordon, Herrera, & Zwir, 2002; Herrera & Viedma, 2000; Herrera & Herrera-Viedma, 2000; Herrera, Herrera-Herrera-Viedma, & Chiclana, 2001; Herrera, Viedma, & Martinez, 2000; Herrera, Herrera-Viedma, & Verdegay, 1995; Herrera, Herrera-Herrera-Viedma, & Verdegay, 1995; Herrera, Herrera-Viedma, & Verdegay, 1996; Herrera, Herrera-Viedma, & Verdegay, 1996a; Herrera, Herrera-Viedma, & Verdegay, 1996b; Herrera, Herrera-Viedma, & Verdegay, 1997; Herrera, Martinez, & Sanchez, 2005). Wang and Chen (2007)

utilized fuzzy preference relations to criticize the fuzzy analytical

approach to partnership selection. Wang and Chang (2007)

forecasted the probability of successful knowledge management by consistent fuzzy preference relations. The basic deﬁnitions and propositions are described as below.

Multiplicative preference relation

A multiplicative preference relation A on a set of alternatives X is indicated by a matrix A X X, A = (aij), aijis the ratio of the

preference degree of alternative xiover xj, A is assumed

multi-plicative reciprocal, that is

aij aji¼ 1 ð1Þ

Additive fuzzy preference relation

Suppose a fuzzy preference relation P on a set of alternatives X is denoted by P = (pij), pij=

l

p(xi, xj). pijindicates the ratio of the

preference intensity of alternative xi to that of xj. If pij¼12

implies there is no difference between xiand xj, pij= 1 indicates

xiis absolutely preferred to xj, similarly pij= 0 indicates xj is

absolutely preferred to xi, pij>12indicates that xiis preferred

to xj(xi> xj). P is assumed additive reciprocal, that is

pijþ pji¼ 1 ð2Þ

Proposition. Suppose there is a set of alternatives X = {x1,. . .,xn},

which is associated with a multiplicative preference relation A = (aij)

with aij2 19;9

.Then the corresponding reciprocal additive fuzzy pref-erence relation P = (pij) with pij2 [0, 1] to A = (aij) is deﬁned as follows.

pij¼ gðaijÞ ¼

1

2 ð1 þ log9aijÞ ð3Þ

Additive transitivity consistency of fuzzy preference relation A reciprocal additive fuzzy preference relation P = (pij) is

consis-tent if

pijþ pjkþ pki¼

3

2 ð4Þ

Construct a consistent fuzzy preference relation

A consistent fuzzy preference relation P0 on X = {x1, x2, . . . , xn,

n P 2} from n 1 preference values {p12, p23, . . ., pn1n} can be

constructed as follow. Compute the set of preference values B as B ¼ fpij;i < j ^ pijRfp12;p23; . . . ;pn1ngg pji¼ j i þ 1 2 piiþ1 piþ1iþ2... pj1j ð5Þ a ¼ j minfB [ fp12;p23; . . . ;pn1nggj ð6Þ P ¼ fp12;p23; . . . ;pn1ng [ B[ f1 p12;1 p23; . . . ;1 pn1ng [ :B ð7Þ

The consistent fuzzy preference relation P0_{is obtained as P}0_{= f(P)}

f : ½a; 1 þ a ! ½0; 1; f ðxÞ ¼ x þ a

1 þ 2a ð8Þ

2.3. The decision making matrix of incomplete linguistic preference relations

Linguistic preference relations are usually used by decision

makers to express their linguistic preference information (Chiclana

et al., 1998; Chiclana et al., 2001; Chiclana et al., 2002; Herrera & Herrera-Viedma, 2000; Herrera & Herrera-Viedma, 2000; Herrera et al., 2001; Herrera et al., 2000; Herrera et al., 1995; Herrera et al., 1995; Herrera et al., 1996; Herrera et al., 1996a; Herrera et al., 1996b; Herrera et al., 1997; Pelaez, Dona, & Gomez-Ruiz, 2007; Wang & Chang, 2007; Wang & Chen, 2007; Xu, 2004b; Xu, 2005; Xu, 2006a; Xu, 2007a; Xu, 2007b; Wu & Lee, 2007; Xu, 2004a; Xu, 2006b) based on pairwise comparisons (Xu, 2005).Xu (2006b)proposed the incomplete linguistic preference relations, during the process of pairwise comparison, each expert can select anyone of the explicit items as the standard according to his/her preference or recognition, and then the pairwise comparison would be carried out between the adjoining items in order obtain the original preference matrix; complete linguistic preference rela-tion counters the fact that all of the attribute decision-making ex-perts carry out the pairwise comparison through preference matrix; when the decision maker uses pairwise comparison to compare the original preference values, and the remaining un-known variables add with the adjoining numbers that equals to 0 through the corresponding opposite numbers so as to obtain the

Criteria Criteria Criteria Criteria Goal

Alternative Alternative Alternative Alternative Alternative

Fig. 1. Typical structure analytic hierarchy process.

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5. Conclusion

The problems of social economics and environmental change are more complicated and uncertain in the real world. We could make decision by unique criterion, but need to consider relative factors as well. Therefore, multi-criteria decision making could sat-isfy the need. When utilizing AHP method to compare and the ma-trix is n n, then it is necessary to compare n(n 1)/2 times in order to obtain the complete matrix, the multi-criteria incomplete linguistic preference relations that this study established only need to compare n 1 times, which can derive fast, it does not produce the inconsistent problems so that the decision making process can be more efficient, this model can also be applied to evaluate the criterion weight and to carry out the method of optimal alternative evaluation. The use of pairwise comparison for decision making matrix can establish a decision making tool that is not going to pro-duce inconsistence, this tool can make the decision making prob-lems purer, simpler, and more efficient, it also possesses complex flexibility, compatible subjective recognition, harmonizing and objective factors, it provides diversification and extensiveness. The decision making model is full of flexibility, so it is not neces-sary to consider the professional backgrounds of the decision mak-ers; when apply this model onto the multi-criteria decision making, more experts of different fields can be expanded, since it is not necessary to consider the professional backgrounds of the experts, each expert can select the explicit index unrestrictedly for pairwise comparisons, which are the so-called horizontal, ver-tical, and oblique comparisons; for the remaining unknown vari-ables, they can be derived by adjoining addition and their corresponding opposite relation algorithms, then the complete ma-trix can be produced quickly. Using this algorithm rule of decision making matrix to establish multi-criteria incomplete linguistic preference relations is conformed to the decision making process in the real life; when the decision makers have to carry out pair-wise comparison, it can also avoid many problems, such as time pressure, lack of complete information, lack of this kind of profes-sional knowledge, or even the provided information is uncertain during the decision making process; hence, the decision making experts can expand to different fields. The approach fuses all the individual preferences into the collective ones and aggregative to get the ranking of alternatives. Furthermore we shall continue working in the study of incomplete linguistic preference with many decision-making problems.

Acknowledgements

The authors thank the National Science Council of the Republic of China Taiwan for ﬁnancially supporting this research under Project NSC 96-2416-H-214-005.

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