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2011 Elsevier Ltd. All rights reserved.

1. Introduction

Decision making expert systems are oftodelingeria debe veakingory ofhave b

the best alternative a ranking process is required. Extensivelyadopted in MCDM, the analytic hierarchy process (AHP) has suc-cessfully been applied to the ranking process of decision makingproblems (Saaty, 1988). The main advantage of the AHP is itsinherent ability to handle intangibles, which are present in anydecision making process. Also, the AHP less cumbersome mathe-matical calculations and, it is more easily comprehended incomparison with other methods. Triantaphyllou and Lin (1996)

erences and to provide exact pairwise comparison judgments.AHP is thus ineffective when applied to ambiguous problems. Sincethe real world is highly ambiguous, some researchers apply fuzzyAHP as an extension of conventional AHP and employ fuzzy settheory to handle uncertainty and overcome this limitation.

Recently, Mikhailov (2003) proposed a new fuzzy AHP methodnamely fuzzy preference programming (FPP) approach for derivingpriorities from fuzzy comparison judgments. Cakir and Canbolat(2008) developed a web-based package based on the Mikhailovsprioritization method. However, the steps in the FPP method makeit a little complicated from a computational point of view. This

Corresponding author. Tel.: +81 75 383 7499; fax: +81 75 383 3253.

Expert Systems with Applications 39 (2012) 960966

Contents lists availab

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.eE-mail address: javanbarg@quake.kuciv.kyoto-u.ac.jp (M.B. Javanbarg).dling multicriteria decision making systems (Beynon, Cosker, &Marshall, 2001; Chen & Chen, 2005; Chen & Lee, 2010; Chen &Wang 2009; Fu, 2008; Hua, Gong, & Xu, 2008; Kahraman & Cebi,2009; Kulak, 2005; Kwon & Kim, 2004; Kwon, Kim, & Lee, 2007;Lin, Hsu, & Sheen, 2007; Mikhailov, 2003; Tacker & Silvia, 1991;Yager, 1991, 1992).

Multicriteria decision making deals with the problem of choos-ing the best alternative, that is, the one with the highest degree ofsatisfaction for all the relevant criteria or goals. In order to obtain

bined with well-known operation research techniques to handlemore difcult problems; (5) AHP is easier to understand and caneffectively handle both qualitative and quantitative data. However,in many practical situations, the human preference model is uncer-tain and decision makers might be reluctant or unable to assign ex-act numerical values to the comparison judgments. Although theuse of the discrete scale of 19 for performing pairwise compara-tive analysis has the advantage of simplicity, a decision makermay nd it extremely difcult to express the strength of his pref-faceted. In recent years, tools for mimproved signicantly, and multicritmodels are widely considered toconicts related to the decision mand Zadeh (1970) developed the thefuzzy environment, various methods0957-4174/$ - see front matter 2011 Elsevier Ltd. Adoi:10.1016/j.eswa.2011.07.095en complex and multi-decision making havecision making (MCDM)ry useful in resolvingprocess. Since Bellmandecision behavior in aeen developed for han-

and Duran and Aguilo (2007) summarized the following advanta-ges for AHP: (1) it is the only known MCDM model that can mea-sure the consistency in the decision makers judgments; (2) theAHP can also help decision makers to organize the critical aspectsof a problem in a hierarchical structure, making the decision pro-cess easy to handle; (3) pairwise comparisons in the AHP are oftenpreferred by the decision makers, allowing them to derive weightsof criteria and scores of alternatives from comparison matricesrather than quantify weights/scores directly; (4) AHP can be com-equations. Several illustrative examples using existing fuzzy AHP methods are given to demonstratethe effectiveness of the proposed method.Fuzzy AHP-based multicriteria decision mswarm optimization

Mohammad Bagher Javanbarg , Charles Scawthorn,Department of Urban Management, Kyoto University, Kyoto 615-8540, Japan

a r t i c l e i n f o

Keywords:Fuzzy AHPMCDMPSO

a b s t r a c t

This paper presents a fuzzybased on a fuzzy analytic hsion makers, a fuzzy AHPcomparison judgments arethe proposed method drivewhich eliminate the need oforms a fuzzy prioritizationticle swarm optimization

Expert Systems

journal homepage: wwwll rights reserved.king systems using particle

nji Kiyono, Babak Shahbodaghkhan

timization model to solve multicriteria decision making (MCDM) systemsarchy process (fuzzy AHP). To deal with the imprecise judgments of deci-ision making model is proposed as an evaluation tool, where the expertsnslated into fuzzy numbers. Unlike the conventional fuzzy AHP methods,xact weights from consistent and inconsistent fuzzy comparison matrices,ditional aggregation and ranking procedures. The proposed method trans-oblem into a constrained nonlinear optimization model. An improved par-O) is applied to solve the optimization model as a nonlinear system of

le at ScienceDirect

ith Applications

lsevier .com/locate /eswa

ideaderiving crisp priorities from fuzzy judgment matricesis anew way to deal with the prioritization problem from fuzzy reci-procal comparisons in the fuzzy AHP. We describe a simple prior-itization method which can derive exact priorities from fuzzycomparison judgments by applying an efcient nonlinear optimi-zation tool for solving the fuzzy optimization model.

There are three objectives of this paper. First, we propose amodied fuzzy optimization model to deal with shortfalls of theAHP method in handling the uncertainties and imprecision of mul-ticriteria decision making systems. Secondly, we construct a fuzzyprioritization method which can derive exact priorities from con-sistent and inconsistent fuzzy comparison matrices. Thirdly, we

judgments, which are further used to construct a positive reciprocal

tization method. Several methods for deriving the local weights ofcriteria and the local scores of alternatives from judgment matriceshave been developed (Mikhailov, 2000). At this stage the consis-tency of each pairwise comparison is checked. A nal aggregationof local priorities is performed to rank the alternatives.

In summary, AHP is consistent, structured and intuitive. How-ever, the AHP is criticized for its inability to accommodate uncer-tainty in the decision making process. Critics argue that it wouldbe cognitively demanding to ask a decision maker to express his/her preference as a discrete numerical value in the pairwise com-parison matrices. The main problem lies in the fact that, since allcomparisons are done by importance comparisons, it loses the pos-sible nuances of describing criteria as fuzzy sets. Only the relativeimportance of each criterion to all others is measured. In other

M.B. Javanbarg et al. / Expert Systems with Applications 39 (2012) 960966 961comparison matrix A faijg 2 Rnn, where aji = 1/aij, and aij > 0, forj = 1, 2, . . . , n, i = 1, 2, . . . , n. A priority vector w = (w1, w2, . . . , wn)T

may be obtained from the comparison matrix by applying a priori-apply an improved particle swarm optimization (PSO) method tosolve the fuzzy optimization model as a system of nonlinearequations.

2. Preliminaries

Multicriteria decision making provides techniques for compar-ing and ranking different outcomes, even though a variety of crite-ria with different units are used. This is a very important advantageover traditional decision makingmethods where all criteria need tobe converted to the same unit. Another signicant advantage ofmost MCDM techniques is that they have the capacity to analyzeboth quantitative and qualitative evaluation criteria together. Sev-eral methods exist for MCDM (Vincke, 1992). Among these meth-ods, the most popular ones are ELECTRE (Roy, 1991), PROMETHEE(Brans & Vincke, 1985), TOPSIS (Hwang & Yoon, 1981), and theAHP methods (Saaty, 1980).

Given the advantages of integral structure, simple theory, andease-of-operation, AHP is a popular tool forMCDMwherein compli-cated decision problems and non-structural situations are dividedinto hierarchical elements. The decision problem is constructed asa hierarchical structure in which the overall goal of decision is lo-cated at the highest level, and m alternatives, A1, A2, . . . , Am, are atthe lowest level (Fig. 1). The n criteria, C1, C2, . . . , Cn, (sub-criteria)form the internal layers of the hierarchy. The preferences of deci-sion makers are elicited in the form of ratios using pairwise com-parison matrices. The judgment matrices of criteria oralternatives are dened by rating the relative importance of ele-ments based on a standard scale (where 1 = equally important;3 = weak importance; 5 = strong importance; 7 = demonstratedimportance; 9 = absolute importance). Comparing any two ele-ments at the same level of hierarchy, a decision maker can providea numerical value aij for the ratio of the elements importance. Eachset of comparisons for a level with n elements requires n(n 1)/2Fig. 1. Hierarchical structure for AHP-based MCDM.word, decision makers often face uncertain and fuzzy cases whenconsidering the relative importance of one element to another.Hence, it is difcult to derive the crisp ratios from pairwise com-parison matrices stated above. Fuzzy set based (van Laarhoven &Pedrycz, 1983) approach has been suggested to overcome theinability of AHP to handle uncertainties.

2.1. Fuzzy judgments

Fuzzy sets have been applied as an important tool to representand treat the uncertainty in various situations (Zadeh, 1965). A ma-jor contribution of fuzzy set theory is its capability of representingvague or uncertain data in a natural form. This capability is the rea-son for its success in many applications. A fuzzy set is a class of ob-jects with a continuum of grades of membership. Linguistic termsare represented by membership functions, valued in the real unitinterval, which translate the vagueness and imprecision of humanthought related to the proposed problem. In the literature, triangu-lar and trapezoidal fuzzy numbers are usually used to capture thevagueness of the parameters related to the topic. The arithmeticoperations of these types of fuzzy numbers can be found in Zim-mermann (1994). In this study, the triangular fuzzy numbers(TFNs) were used to represent the fuzzy relative importance. ATFN is graphically shown in Fig. 2 and can be described as:

l~Nx xlml ; l 6 x 6 muxum ; m 6 x 6 u0; otherwise

8>: 1

in which the parameters, l, m, and u respectively denote the small-est possible value, the most promising value, and the largest possi-ble value that describe a fuzzy event. The TFN ~N is often representedas (l,m, u).

Assume that the fuzzy comparison judgments are decided withrespect to the linguistic (non-numerical) judgments of decisionFig. 2. Triangular fuzzy number.

makers. Their imprecise and uncertain assessments can be trans-lated into corresponding triangular fuzzy numbers. Introductionof fuzzy linguistic variables instead of exact, crisp values will helpa decision maker to use non-numerical terms for his/her highlysubjective judgments and it can incorporate the imprecision re-lated to the decision makers preference. Hence, it will eliminatethe drawback of the static structure of the fundamental scale incapturing uncertainty in the comparisons. In this paper, we usethe fu

(3)

mentsinto atransf

962 M.B. Javanbarg et al. / Expert Systems wiconsistency and extracts the priorities from the pairwisecomparison matrices. In existing fuzzy AHP methods, onlya few past studies have addressed the issue of checking forinconsistencies in pairwise comparison matrices. Accordingto Buckley (1985), a fuzzy comparison matrix ~A f~aijg isconsistent if ~aik ~akj ~aij, where i, j, k = 1, 2, . . . , n, and isfuzzy multiplication, and denotes fuzzy equal to. Oncethe pairwise comparison matrix, ~A, passes the consistencycheck, fuzzy priorities ~wi can be calculated with conven-tional fuzzy AHP methods. Then, the priority vector(w1, w2, . . . , wn)T can be obtained from the comparisonmatrix by applying a prioritization method.

(4) Aggregation of priorities and ranking the alternatives. The nalstep aggregates local priorities obtained at different levels ofthe decision hierarchy into composite global priorities forthe alternatives based on the weighted sum method. If thereare i alternatives and j criteria, then the nal global priorityof alternative i is given as:

Table 1Fuzzy judgment scores in fuzzy AHP.

Fuzzy judgments Fuzzy score

About equal (1/2, 1,2)About x times more important (x 1, x, x + 1)About x times less important (1/(x + 1), 1/x, 1/(x 1)Between y and z times more important (y, (y + z)/2, z)

Betw

x = 2, 3,~A f~aijg

~a21 ~a22 : : : ~a2n: : :

: : :

: : :

~an1 ~an2 : : : ~amn

BBBBBBBB@

CCCCCCCCA2

Consistency check and deriving priorities. This step checks fordepicted in Table 1.

2.2. Fuzzy AHP

Fuzzy AHP uses fuzzy set theory to express the uncertain com-parison judgments as a fuzzy numbers. The main steps of fuzzyAHP are as follows:

(1) Structuring decision hierarchy. Similar to conventional AHP,the rst step is to break down the complex decision makingproblem into a hierarchical structure.

(2) Developing pairwise fuzzy comparisonmatrices. Consider a pri-oritization problem at a level with n elements, where pair-wise comparison judgments are represented by fuzzytriangular numbers ~aij lij;mij;uij. As in the conventionalAHP, each set of comparisons for a level requires n(n 1)/2judgments, which are further used to construct a positivefuzzy reciprocal comparison matrix ~A f~aijg, such that:

~a12 ~a12 : : : ~a1n0 1zzy judgments employed in Wang, Chu, and Wu (2007) aswhichresponadditi

een y and z times less important (1/z, 2/(y + z), 1/y)

9 & y, z = 1, 2, . . . , 9 & y < z.. By using a-cuts, the initial fuzzy judgments are transformedseries of interval judgments. The FPP method is employed toorm the prioritization problem into a fuzzy linear programAi Xnj1

wjaij 3

wherewj is the weight of criterion j and aij is the evaluation of alter-native Ai against criterion j. The higher the value Ai, the more pre-ferred the alternative. However, if the priorities are fuzzy as inthe conventional fuzzy AHP, then an appropriate ranking procedureshould be applied to defuzzify the rank of alternatives.

The existing fuzzy AHP methods mainly differ on the employedfuzzy judgments in above-stated step 2 or the developed fuzzy pri-oritization method in step 3, or both. van Laarhoven and Pedrycz(1983) used a triangular membership function, and developed afuzzy version of the logarithmic least squares method. Buckley(1985) proposed fuzzy priorities of comparison ratios similar tothe proposed approach of Wagenknecht and Hartmann (1983),and Buckley employed trapezoidal membership functions, claim-ing that such numbers are more easily understood by experts.Boender, de Grann, and Lootsma (1989) proposed an approachfor local priority normalization. Chang (1996) introduced an extentanalysis method for the synthetic extent values of the pairwisecomparisons and applied a simple arithmetic mean algorithm tond fuzzy priorities from comparison matrices, whose elementsare represented by triangular fuzzy numbers. Wang, Luo, andHua (2008), using several numerical examples, showed that thepriority vectors determined by the extend analysis method donot represent the relative importance of decision criteria or alter-natives. Rather it is a method for showing to what degree the pri-ority of one decision criterion or alternative is bigger than those ofall others in a fuzzy comparison matrix. These methods have somecommon characteristics.

First, they derive priorities from fuzzy comparison matrices.However, the approach of constructing fuzzy reciprocal matrices,taken by analogy from the crisp prioritization methods leads tosome problems, as demonstrated in the next section. In addition,in some cases the decision-maker might be unwilling or unableto provide all fuzzy comparisons necessary to construct full com-parison matrices.

Secondly, except for Chang (1996), all these methods derive fuz-zy priorities and, after aggregating, the nal scores of the alterna-tives are also represented as fuzzy numbers or fuzzy sets. Due tothe large number of multiplication and additional operations, theresulting fuzzy scores have wide supports and overlap over a largerange. As shown by Boender et al. (1989) and Gogus and Boucher(1997), the normalization procedure used in some of these meth-ods may even result in irrational nal fuzzy scores, where the nor-malized upper value is less than the normalized mean value, whichis less than the normalized lower value.

Finally, the fuzzy prioritization methods mentioned above re-quire an additional fuzzy ranking procedure in order to comparethe nal fuzzy scores. The different ranking procedures, however,often give different ranking results (Bortolan & Degani, 1985).

To overcome the shortcomings of the fuzzy prioritization meth-ods above, Mikhailov (2003) proposed an FPP for deriving prioritiesfrom fuzzy comparison judgments that eliminates some of thedrawbacks of the existing fuzzy prioritization methods. This ap-proach does not require the construction of complete fuzzy com-parison matrices and it can derive priorities from an incompleteset of fuzzy judgments. The proposed approach is also invariantto the specic form of the fuzzy sets used to represent the judg-

th Applications 39 (2012) 960966can derive crisp priorities from the interval judgments, cor-ding to each a-cuts level, thus eliminating the need for anonal fuzzy ranking procedure. An aggregation process of

s withe optimal priorities derived at the different a-level is also neededfor obtaining overall crisp scores of the prioritization elements.However, the steps in the FPP method are complicated from a com-putational point of view. This ideaderiving crisp priorities fromfuzzy judgment matricesshows a new way to deal with the prior-itization problem from fuzzy reciprocal comparisons in the fuzzyAHP. In the subsequent sections, we describe a simple fuzzy opti-mization model which can derive exact priorities from fuzzy com-parison judgments.

3. Fuzzy optimization model

Suppose that a fuzzy judgment matrix ~A is constructed as inEq. (2). The elements of the judgment matrix are pairwise compar-ison ratios represented by fuzzy triangular numbers~aij lij;mij;uij, where i and j = 1, 2, . . . , n. Moreover, it is assumedthat lij 0; wj > 0; i j and the symbol ~6 means fuzzy lessthan or equal to. To measure the degree of satisfaction for differentcrisp ratios wi/wj with respect to the double side inequality of Eq.(4), a new membership function can be dened as:

lijwi=wj mijwi=wj

mijlij ; 0