Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy AHP

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Applying fuzzy linguistic preference relations to the improvement of

consistency of fuzzy AHP

Tien-Chin Wang

a,*

, Yueh-Hsiang Chen

b,c

a

Department of Information Management, I-Shou University, 1, Section 1, Hsueh-Cheng Road, Ta-Hsu Hsiang, Kaohsiung 840, Taiwan b

Department of Information Engineering, I-Shou University, Kaohsiung 840, Taiwan c_{Department of Information Management, Kao-Yuan University, Kaohsiung 821, Taiwan}

a r t i c l e

i n f o

Article history: Received 26 April 2007

Received in revised form 31 March 2008 Accepted 30 May 2008

Keywords: Fuzzy AHP Consistency

Fuzzy linguistic preference relations Fuzzy LinPreRa

Decision making

a b s t r a c t

The lack of consistency in decision making can lead to inconsistent conclusions. In fuzzy analytic hierarchy process (fuzzy AHP) method, it is difﬁcult to ensure a consistent pair-wise comparison. Furthermore, establishing a pairpair-wise comparison matrix requiresnðn1Þ

2 judgments for a level with n criteria (alternatives). The number of comparisons increases as the number of criteria increases. Therefore, the decision makers judgments will most likely be inconsistent. To alleviate inconsistencies, this study applies fuzzy linguistic pref-erence relations (Fuzzy LinPreRa) to construct a pairwise comparison matrix with additive reciprocal property and consistency. In this study, the fuzzy AHP method is reviewed, and then the Fuzzy LinPreRa method is proposed. Finally, the presented method is applied to the example addressed by Kahraman et al. [C. Kahraman, D. Ruan, I. Dog˘an, Fuzzy group decision making for facility location selection, Information Sciences 157 (2003) 135– 153]. This study reveals that the proposed method yields consistent decision rankings from only n 1 pairwise comparisons, which is the same result as in Kahraman et al. research. The presented fuzzy linguistic preference relations method is an easy and practical way to provide a mechanism for improving consistency in fuzzy AHP method.

1. Introduction

The Analytic Hierarchy Process (AHP) is one of the extensively used multi-criteria decision making methods. Though this method is easier to understand and it can model expert opinions, however, the conventional AHP still cannot process impre-cise or vague knowledge. Zadeh[51]introduced fuzzy sets theory, to rationalize uncertainty associated with impression or vagueness, and in a manner analogous to human thought. Thus, fuzzy AHP, a fuzzy extension of AHP[35], was developed to solve imprecise hierarchical problems. Of the many fuzzy AHP methods developed by various authors, most propose system-atic approaches to the alternative selection and justiﬁcation problem using the concepts of fuzzy set theory and hierarchical structure analysis[2,4–7,11,14,16,29,30,33,34].

The study of consistency is crucial for avoiding misleading solutions. As part of the AHP procedure, a consistency check is required to identify inconsistency matrix. For comparison matrix which fails the consistency test, the decision maker must redo the ratios. Unlike the AHP method, the ratios are point estimates and the comparison ratios in fuzzy AHP method are given by fuzzy numbers. The likelihood of having inconsistent ratios within the given fuzzy numbers is therefore far greater.

*Corresponding author. Tel.: +886 7 6577711x6568; fax: +886 7 6577056.

E-mail addresses:tcwang@isu.edu.tw,dr.deor@gmail.com,123@king.idv.tw(T.-C. Wang),chris@cc.kyu.edu.tw(Y.-H. Chen).

Contents lists available atScienceDirect

Information Sciences

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To expect the decision maker to provide the comparison ratios such that the fuzzy numbers include only consistency would be laborious and highly unrealistic. The critical void is not only the need to have a consistency test to accept consistent matrices, but also a mechanism to ﬁlter out inconsistent information within a consistent matrix. Thus, Leung and Cao

[37]proposed setting deviation tolerances when performing fuzzy consistency analysis and determining whether a fuzzy positive reciprocal matrix is consistent based on R I < 0.1[39]. However, establishing a pairwise comparison matrix requires

nðn1Þ

2 judgments for a level with n criteria (alternatives), the number of comparisons increases as the number of criteria

(alternatives) increases. Consequently, the decision makers judgments will most likely be inconsistent.

Preference relations are the most common representation of information used for solving decision making problems due to their effectiveness in modeling decision processes. Generally, these preference relations can be categorized into multipli-cative preference relations[15,19,44], fuzzy preference relations[1,8–10,13,15,24,38,41–47]and linguistic preference rela-tions[17,18,20–23,25,26,48–50,52]. Herrera-Viedma et al.[24]proposed a new concept of consistency based on the additive transitivity property of fuzzy preference relations (Fuzzy LinPreRa) to avoid misleading conclusions. This new characteriza-tion simpliﬁes the analysis of consistency among expert opinions. Based on this new characterizacharacteriza-tion, this study proposes a method for constructing consistent fuzzy preference relations from a set of n 1 preference data. However, all of the above attempts focus on the decision matrix with crisp values, which cannot reﬂect expert opinions when modeling imprecise judgments. Crisp data are inadequate to model real-life situations. Since human judgments including preferences are often vague and cannot estimate human preference with an exact numerical value.

To solve the above problems, Wang and Chen[42]proposed a method using fuzzy linguistic assessment variables to con-struct fuzzy linguistic preference relation (Fuzzy LinPreRa) matrices based on consistent fuzzy preference relations[24]. In

[42], this paper only illustrates how to construct a Fuzzy LinPreRa matrix, not explains the applications of the proposed method. Thus, this study applies Fuzzy LinPreRa method to enhance the consistency of the fuzzy AHP method. The proposed method yields decision matrices for making pairwise comparisons using additive reciprocal property and consistency. In addition, it required only n 1 comparison judgments to ensure consistency for a level with n criteria (alternatives).

This study is organized as follows: Section2reviews consistent fuzzy preference relations. Section3describes triangular fuzzy numbers. Section4reviews the fuzzy AHP method. Section5proposes the Fuzzy LinPreRa method. Section6presents an illustrative example. Finally, Section7draws the conclusions.

2. Consistent fuzzy preference relations

For a set of criteria and a set of alternatives, fuzzy preference relations provide decision makers with values representing varying degrees of preference for one alternative over another. The following brieﬂy describes some deﬁnitions and propo-sitions presented in[8,9,17–23,25].

Given A = {a1, a2, . . . , an}, with n P 2 is a ﬁnite set of alternatives to be pairwise assessed by experts (E = {e1, e2, . . . , em}, with

m P 2). Expert preferences for the set of alternatives A may be expressed as follows[10]:

(a) Multiplicative preference relations: a multiplicative preference relation R in terms of a set of alternatives A, repre-sented by a matrix R : A A ! R, rij= R(ai, aj), where rijis the preference ratio of alternative aito aj. Saaty[39,40]

sug-gests measuring rijusing a ratio scale, and the deﬁned 1–9 scale. Herein, rij= 1 represents the absence of a difference

between aiand aj; rij= 9 denotes that aiis maximally better than aj. In this case, the preference relation R is typically

assumed to be a multiplicative reciprocal, aij aji= 1 "i, j 2 {1, . . ., n}.

(b) Fuzzy preference relations: a fuzzy preference relation P on a set of alternatives A is a fuzzy set on the product set A A with membership function P : A A ? [0, 1]. The preference relation is represented by the n n matrix P = (pij), where pij= P(ai, aj) "i,j 2 {1, . . . , n}. Herein, pijis the preference ratio of alternative aito aj: pij= 1/2 means that

no difference exits between aiand aj, pij= 1 indicates that aiis absolutely better than aj, and pij> 1/2 indicates that ai

is better than aj. In this case, the preference matrix P is generally assumed to be an additive reciprocal of

pij+ pji= 1 "i, j 2 {1, . . . , n}.

Herrera-Viedma et al.[24]proposed consistent fuzzy preference relations to construct the decision matrices of pairwise comparisons based on additive transitivity. Wang and Chen[43]applied consistent fuzzy preference relations to partnership selection. Wang and Chang[41]applied consistent fuzzy preference relations to forecast the probability of successful know-ledge management. In[24], they developed some important propositions given below.

Proposition 2.1[9,17,24,27,28]. Consider a set of alternatives, X = {x1, . . . , xn}, associated with a reciprocal multiplicative

preference relation A = (aij) for aij2 [1/9, 9]. Then, the corresponding reciprocal fuzzy preference relation, P = (pij) with pij2 [0, 1]

associated with A is given as pij¼ gðaijÞ ¼12ð1 þ log9aijÞ.

log9aijis considered when aijis between 1/9 and 9. If aijis between 1/7 and 7, then log7aijis used.

Proposition 2.2[24]. For a reciprocal fuzzy preference relation P = (pij), the following statements are equivalent;

(a) pijþ pjkþ pki¼32 8i; j; k.

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Proposition 2.3 [24]. For a reciprocal fuzzy preference relation P = (pij), the following statements are equivalent;

(a) pijþ pjkþ pki¼32 8i < j < k.

(b) piðiþ1Þþ pðiþ1Þðiþ2Þþ þ pðiþk1ÞðiþkÞþ pðiþkÞi¼kþ12 8i < j.

Proposition 2.3is very important because it can be used to construct a consistent fuzzy preference relation from the set of n 1 values {p12, p23, . . . , pn1n}. Accordingly, experts are able to express consistent preferences in decision processes. A

deci-sion matrix with entries in the interval [k, 1 + k], k > 0 besides the interval [0, 1], can be constructed by transforming the obtained values using a transformation function that preserves reciprocity and additive consistency. The transforming func-tion is f : [k, 1 + k] ? [0, 1], f ðxÞ ¼ xþk

1þ2k.

The drawback of consistent fuzzy preference relations is that the values in consistent fuzzy preference relation matrix are crisp, which cannot reﬂect expert opinions when modeling imprecise judgments.

3. Triangular fuzzy numbers

The fuzzy set theory[51]is designed to deal with the extraction of the primary possible outcome from a multiplicity of information vaguely and imprecisely. Fuzzy set theory treats vague data as possibility distributions in terms of set member-ships. Once determined and deﬁned, the sets of memberships in possibility distributions can be effectively used in logical reasoning. Triangular fuzzy numbers are one of the major components. According to the deﬁnition of Laarhoven and Pedrycz

[35], a triangular fuzzy number (TFN) should possess the following basic features. Deﬁnition 3.1. A fuzzy number A on R to be a TFN if its membership function

l

_e

AðxÞ : R ! ½0; 1 is equal to

l

_e AðxÞ ¼ ðx lÞ=ðm lÞ; l 6 x 6 m; ðu xÞ=ðu mÞ; m 6 x 6 u; 0; otherwise: 2 4 ð1Þ

where l and u represent the lower and upper bounds of the fuzzy number eA, respectively, and m is the median value. The TFN is denoted as eA ¼ ðl; m; uÞ and the following is the operational laws of two TFNs eA1¼ ðl1;m1;u1Þ and eA2¼ ðl2;m2;u2Þ, as

shown[31,32]:

Fuzzy number addition :

eA1 eA2¼ ðl1;m1;u1Þ ðl2;m2;u2Þ ¼ ðl1þ l2;m1þ m2;u1þ u2Þ: ð2Þ

Fuzzy number subtraction :

eA1 eA2¼ ðl1;m1;u1Þ ðl2;m2;u2Þ ¼ ðl1 u2;m1 m2;u1 l2Þ: ð3Þ

Fuzzy number multiplication :

eA1 eA2¼ ðl1;m1;u1Þ ðl2;m2;u2Þ ﬃ ðl1 l2;m1 m2;u1 u2Þ for li>0; mi>0; ui>0: ð4Þ

Fuzzy number division Ø:

eA1ØeA2¼ ðl1;m1;u1ÞØðl2;m2;u2Þ ﬃ ðl1=u2;m1=m2;u1=l2Þ for li>0; mi>0; ui>0: ð5Þ

Fuzzy number logarithm:

lognðeAÞ ﬃ ðlognl; lognm; lognuÞ n is base: ð6Þ

Fuzzy number reciprocal:

ðeAÞ1¼ ðl; m; uÞ1ﬃ ð1=u; 1=m; 1=lÞ for l; m; u > 0: ð7Þ

Note that the results of Eqs.(4)–(7)are not TFNs, but these results can be approximated by TFNs[31,32].

4. Fuzzy AHP method

With the AHP not being able to overcome the deﬁciency of the fuzziness during decision making, Laarhoven and Pedrycz

[35]have evolved Saaty’s AHP into the fuzzy AHP, bringing the triangular fuzzy number of the fuzzy set theory directly into the pairwise comparison matrix of the AHP. The purpose is to solve vague problems, which occur during the analysis of criteria and judgment process. The procedure of the fuzzy AHP is described as follows:

Step 1: Hierarchical structure construction.

Placing the goal of the desired problem on the top layer of the hierarchical structure, and the evaluation criteria on the second layer, then the alternatives lay on the bottom layer.

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consistency. When the number of criteria or alternatives increases, the number of comparisons required can be reduced by the method introduced here. For instance, seven criteria require six comparisons, which the number of comparisons can be reduced by ðC72 6Þ ¼ 15 times. The method not only enhances the quality of decision making in imprecise or vague

envi-ronments, but also resolves the problem of consistency of the fuzzy AHP. This study provides a set of mechanisms for iden-tifying consistent fuzzy rankings.

In the future research, we may apply this method to different real-life problems. For example, the proposed method, may be particularly useful in formulating a national energy policy given the uncertainty and vagueness of information related to such decisions.

Acknowledgements

The authors would like to thank the editor, Professor Witold Pedrycz, and the anonymous referees for their constructive comments and suggestions that led to an improved version of this paper.

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