A fuzzy ranking method with range reduction techniques

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Decision Support

A fuzzy ranking method with range reduction techniques

Li-Ching Ma

a,*

, Han-Lin Li

b

a

Department of Information Management, National United University, No. 1, Lienda, Gongjing Village, Miaoli City, 36003, Taiwan, ROC

b

Institute of Information Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, ROC Received 25 May 2005; accepted 13 December 2006

Available online 11 January 2007

Abstract

For ranking alternatives based on pairwise comparisons, current analytic hierarchy process (AHP) methods are difficult to use to generate useful information to assist decision makers in specifying their preferences. This study proposes a novel method incorporating fuzzy preferences and range reduction techniques. Modified from the concept of data envelopment analysis (DEA), the proposed approach is not only capable of treating incomplete preference matrices but also provides reasonable ranges to help decision makers to rank decision alternatives confidently.

Keywords: Decision analysis; Ranking; Fuzzy; Range reduction; Preference

1. Introduction

This paper addresses the range computation for pairwise comparison preference rating. The motivation, purpose and advantages of the proposed approach are introduced ﬁrst. Then, the concept and insuﬃciencies of conventional fuzzy AHP models are described. Next, the range reduction and fuzzy ranking models are pro-posed and constructed. Finally, a numerical example is used to illustrate the solving process.

The analytic hierarchy process (AHP), developed bySaaty (1977), is a popular approach to rank tives. Through the ratio-scaled assessment of pairwise preferences between alternatives, the ranks of alterna-tives are found by computing the eigenvalues of the preference matrix. Conventional AHP, however, cannot treat incomplete preference matrices. In addition, AHP has been proven to be a mathematically ﬂawed system in deriving weights and synthesizing scores of attributes by several authors (Barzilai, 1997, 2001, 2005; Brugha,

2000, 2004; etc.).

Since fuzziness and vagueness commonly exist in many decision-making problems (Levary and Wan, 1998;

Ribeiro, 1996), numerous ranking methods (Graan, 1980; Laarhoven and Pedrycz, 1983; Boender et al., 1989;

doi:10.1016/j.ejor.2006.12.023

* _{Corresponding author. Tel.: +886 37381828; fax: +886 37371462.}

E-mail addresses:lcma@nuu.edu.tw(L.-C. Ma),hlli@cc.nctu.edu.tw(H.-L. Li).

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Chang, 1996; Ruoning and Xiaoyan, 1996; Leung and Cao, 2000; Yu, 2002) have been developed to solve fuzzy decision problems with pairwise comparison matrices. A major disadvantage of conventional fuzzy AHP methods is that no range information is provided to help a decision maker to specify preferences conveniently. Conventionally, Saaty (1980)ratio scale of [1/9, 9] is used as default upper and lower bounds, yet the ranges are usually too big for a decision maker to use as a useful reference. In addition, most of current AHP methods require a decision maker to specify a complete pairwise comparison matrix.

Data envelopment analysis (DEA) is another commonly used technique in ranking decision alternatives. The DEA technique is intended to evaluate the eﬃciency of each alternative using CCR models (Charnes

et al., 1978) or BCC models (Banker et al., 1984) based on the concept of maximizing the ratio of outputs

to inputs. However, there are some insufficiencies of conventional DEA models in ranking alternatives. First, current DEA models may generate too many efficient alternatives with the same rank. The lack of discrimi-nation among alternatives prohibits its applications in real cases (Angulo-Meza and Lins, 2002). In addition, most DEA methods do not incorporate the preferences specified by the decision maker.

This study proposes a novel ranking method with pair-wise preference comparisons. The proposed model first adopts a modified DEA model to generate reasonable upper and lower bounds of preference ratios. By referring to these ranges, a decision maker then specifies his/her fuzzy preferences partially. A goal-program-ming model with minimal approximation errors and maximal fulfillment of a decision maker’s preferences is proposed to solve the fuzzy decision problem.

The major advantages of the proposed approach are listed as follows:

(i) Reasonable upper and lower bounds are provided to help a decision maker to articulate related fuzzy preferences.

(ii) Incomplete preference matrix can be handled.

(iii) Various fuzzy preferences with convex, concave or mixed convex–concave features are treated to obtain a crisp optimal solution eﬃciently.

2. Conventional fuzzy AHP models

Consider a set of n alternatives A¼ fAiji ¼ 1; . . . ; ng for solving a decision problem. From the basis of AHP

(Saaty, 1980), the pairwise comparison of Aiover Aj, denoted as hi,j, is the preference speciﬁed by a decision

maker as the ratio of the weights of Aito Aj. Let hi;j¼wwijmeasure the relative dominance of Aiover Ajin terms of priority weighs w1>0; . . . ; wn>0. Following Saaty, hi,j are speciﬁed as 1–9 numerical rates. Denote

H¼ ðhi;jÞ, where hj;i¼_h1_i;j is assumed. A fuzzy AHP problem can be expressed as follows:

Min X n i¼1 Xn j>i wi wj ~hi;j Max X n i¼1 Xn j>i lð~hi;jÞ Subject to X n i¼1 wi¼ 1; wi; ~hi;jP0; ð2:1Þ

where ~hi;j is a fuzzy number representing how many times is Ai preferred over Aj, which is speciﬁed by the

decision maker. lð~hi;jÞ is the membership function of ~hi;j. The ﬁrst objective is to minimize the sum of

devia-tions resulted from approximation, and the second objective is to maximize the sum of membership funcdevia-tions of ~hi;j. Model(2.1)is in the form of goal-programming (Cooper, 2005). This model can be solved by weights

method (Taha, 2003) to optimize both objectives jointly.

A most commonly used membership function is a triangle type as shown in Fig. 1, where hi;j;1 and hi;j;3

are, respectively, the lower and upper bounds of hi,j, and hi;j;2 is the hi,j value which is the most likely to

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Many methods have been developed to solve fuzzy AHP problems. For examples,Graan (1980)generated a fuzzy priority vector by assigning fuzzy weights.Laarhoven and Pedrycz (1983) and Boender et al. (1989)

proposed logarithmic least squares methods to generate a priority vector under fuzzy environment. However, most conventional fuzzy AHP methods use repetitive extension principal processes or tedious arithmetic calculations to solve problems. Besides, the obtained fuzzy priority vector needs extra defuzziﬁcation tech-niques to generate a crisp solution.

Yu (2002)proposed a goal-programming (GP) AHP model for solving group decision-making fuzzy AHP

problems based on the work ofLi and Yu (1999). If there are e decision-makers in the group, the GP-AHP model is formulated as follows:

Min X n i¼1 Xn j>i XE e¼1 ðln wi ln wjÞ ln hei;j Max X n i¼1 Xn j>i XE e¼1 lðln he i;jÞ Subject to ln he_i;j¼ l ln he i;j þ se i;j;2 s e i;j;1 ln he_i;j;2 se i;j;2 s e i;j;1 de_i;jþ se i;j;1ln h e i;j;1 n o =se i;j;2; ln he i;j ln h e i;j;2þ d e i;jP0;

de_i;j; he_i;jP0 8i; j; wiP0 8i;

ð2:2Þ

where he

i;jindicates the eth decision-maker’s fuzzy preference of Aiover Aj. The deviation variable, dei;j, is used

to treat the absolute term. The triangular membership, l ln he i;j , is a function of ln he i;j , where ln he i;j;1 , ln hei;j;2 and ln hei;j;3

are lower, middle and upper values of ln ~he i;j

. The slopes of the two line segments in the triangular membership function, se

i;j;1 and sei;j;2, are given by sei;j;1¼ l ln he i;j;2 ð Þ ð Þl ln he i;j;1 ð Þ ð Þ ln he i;j;2 ð Þln he i;j;1 ð Þ and se i;j;2¼ l ln he i;j;3 ð Þ ð Þl ln he i;j;2 ð Þ ð Þ ln he i;j;3 ð Þln he i;j;2 ð Þ .

Yu applied a linearization technique to solve fuzzy AHP problems involving triangular, convex and mixed concave-convex fuzzy estimates under a group decision-making environment. Instead of tedious computa-tions, a GP-AHP approach can obtain a crisp solution eﬃciently.

A major disadvantage for current fuzzy AHP methods (Yu, 2002; Boender et al., 1989; Laarhoven and

Pedrycz, 1983; Graan, 1980) is that there is no bound information about the piecewise preferences. A core

issue of a fuzzy pairwise comparison model is how to specify the membership function of a preference. For instance, how to specify hi;j;1 and hi;j;3 in Fig. 1. All current methods assume that a decision maker can tell

these values. In fact, without additional information, it is quite diﬃcult for a decision maker to guess these values. If the range is too wide (as hi;j;1¼ 1=9 and hi;j;3¼ 9), it is meaningless to specify the preferences. If

the range is too narrow (as hi;j;1¼ 6 and hi;j;3¼ 7), then some good solutions may be eliminated from the

solu-tion space.

This study proposes a novel ranking method, which can provide reasonable range information to help a decision maker specifying their fuzzy preferences with an incomplete pairwise comparison matrix.

0 1

hi,j,1 hi,j,2 hi,j,3

si,j,2 si,j,1 hi,j ) (hi, j μ

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3. Proposed fuzzy ranking models

Given a set of n alternatives, A¼ ðA1; A2; . . . ; AnÞ, for solving a decision problem, where each alternative

contains m criteria Ai¼ Aiðci;1; ci;2; . . . ; ci;mÞ. Denote wk as the weight of criterion k. Since the experience has

shown multi-criteria syntheses are diﬃcult, all weights are assumed to be positive to avoid making them any more complicated. All criteria values are transformed to the same positive format by subtracting from upper bound, and normalized to a scale from 1 to 9 in advance.

Denote ci;kas the transformed kth criterion value of alternative Ai. Based on the concept ofBrugha (2000,

2004), relative measured weights and scores should be synthesized using a power function. Instead of an

arith-metic synthesis of score function by AHP, the score function of Aiis assumed to be in a non-linear Cobb–

Douglas (Cobb and Douglas, 1928) form with constant return to scale, expressed below SiðwÞ ¼ c w1 i;1c w2 i;2; . . . ; c wm i;m; ð3:1Þ where w1; . . . ; wmP0,P m k¼1wk¼ 1 and 1 6 Si69.

Deﬁne a relative dominance matrix R¼ ðri;jÞ as a n n matrix, where element ri;j¼Score_Scorei_jexpresses the ratio

of scores of Aiover Aj. rj;i¼_r1_i;jis assumed. Section3.1illustrates how to articulate the reduced ranges of ri,jto

help a decision maker to specify related preferences. Section3.2describes the proposed fuzzy ranking model. 3.1. Range reduction techniques

This study proposes a range reduction technique, a modiﬁed DEA ranking method with rank minimization, which is modiﬁed from the concept of multiplicative DEA models (Charnes et al., 1982, 1983, 1996). Denote Scorep

j as the score of Ajand wpk as the weight of criteria k while Apis chosen as the target alternative (i.e. the

score or rank of Apis optimized). Denote Rankpas the rank of Ap. 1 6 Rankp6n. Let Rankp¼ 1 if Apis the

best choice. Apis superior to Aj(denoted as Ap AjÞ if and only if Rankp<Rankj.

Remark 1. Rankj<Rankp if and only if Score p

j >Score p p. Since Scorep

pis the maximum score that Apcan have, Score

p

j >Score p

p implies that Score p

j >Score p p>eno matter how we specify wp_k. Ajtherefore is clearly superior to Ap. Denote Sup(p) as a superior set of Ap. Sup(p) is

a collection of Ajwhich are superior to Ap, expressed as

SupðpÞ ¼ fAjjScorepj >Score p

p for j¼ 1; 2; . . . ; ng: ð3:2Þ

Rankpcan then be computed as

Rankp¼ 1 þ SupðpÞk k; ð3:3Þ

wherekSup(p)k is the number of elements in Sup(p).

For a target alternative Ap, the proposed DEA ranking model with rank minimization is formulated

below.

Model 1 (a modiﬁed DEA model)

Min X

n

j¼1;j6¼p

tp;j ð3:4Þ

Subject to Scorep_pþ M tp;jPScorepj 8j ¼ 1; 2; . . . ; n; ð3:5Þ

tp;j2 f0; 1g; M is a large value; ð3:6Þ 1 6 Scorep_j6₉ 8j; ð3:7Þ Xm k¼1 wp_k ¼ 1; ð3:8Þ w1; . . . ; wmP0: ð3:9Þ

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The objective is to minimize the rank of Ap. If tp;j¼ 0 for all j then Scorepphas the maximal value. Expression

(3.5)means that if Scorep

pPScore p

j then tp;j¼ 0, and otherwise tp;j¼ 1. A superior set Sup(p) of Apcan be

obtained by checking all tp;j. If tp;j¼ 1, then Ajis in the superior set of Ap.

Model 1 can be converted directly into following linear 0–1 programs:

Min X n j¼1;j6¼p tp;j Subject to X m k¼1 wp_klnðcp;kÞ þ M tp;jP Xm k¼1 wp_klnðcj;kÞ 8j ¼ 1; . . . ; n; ð3:10Þ lnð1Þ 6X m k¼1 wp_klnðcj;kÞ 6 lnð9Þ 8j ¼ 1; . . . ; n; ð3:11Þ tp;j2 f0; 1g; M is a large value; Xm k¼1 wp_k¼ 1; w1; . . . ; wmP0:

Let ri;j and ri;jbe, respectively, the upper and lower bound of ri,jwith ri;j6ri;j6ri;j. Here ri;j is obtained by

maximizing ri,junder the constraint that no other alternative getting a score greater than 1. Similarly, ri;j is

found by minimizing ri,jsubjected to the same constraints, as described in Model 2.

Model 2 (range reduction model) MaxðMinÞ ri;j¼

Scorei

Scorej

ð3:12Þ

Subject to Scorep<Scoreq 8Aq2 SupðpÞ 8p ¼ 1; . . . ; n; ð3:13Þ

1 6 Scorei69 8i;

Xm k¼1

wk ¼ 1;

w1; . . . ; wmP0:

The restrictions ‘‘Scorep<Scoreq’’(3.13)are imbedded into the constraint set for all Aq2 SupðpÞ. By

incor-porating the superior sets obtained from Model 1, Model 2 can substantially tighten the ranges of ri,j. It is

important to note that both ri;j and ri;j are suggested bounds to assist the decision maker to articulate their

preferences. The decision make can still revise both bounds directly. Model 2 can also be converted into a lin-ear 0–1 program as Model 1.

3.2. Proposed fuzzy ranking model

By incorporating the reduced ranges of ri,j, a proposed fuzzy ranking model can then be formulated as

follows:

Model 3 (a fuzzy ranking model)

Min Obj1¼X ~ri;j Scorei Scorej ~ri;j ð3:14Þ Max Obj2¼X ~ri;j lð~ri;jÞ ð3:15Þ

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Subject to ri;j6 Scorei Scorej 6_r_i;j_; _ð3:16Þ 1 6 Scorei69 8i; Xm k¼1 wk¼ 1; w1; . . . ; wmP0;

where ~ri;jis a fuzzy number representing how many times alternative i is preferred over j, speciﬁed by the

deci-sion maker. The ﬁrst objective is to minimize the sum of deviations resulting from the approximation. The second objective tries to maximize the sum of membership functions, which indicate the fulﬁllment of the deci-sion maker’s preferences. Expresdeci-sion (3.16)sets the reduced ranges of ri,j.

A piecewise linear function with triangular membership function is illustrated here. Given a triangular fuzzy preference ~ri;j¼ ðri;j;1; ri;j;2; ri;j;3Þ, a piecewise linear function of lnð~ri;jÞ can be expressed below

lðlnðri;jÞÞ ¼ si;j;1 ðlnðri;jÞ lnðri;j;1ÞÞ þ

ðsi;j;2 si;j;1Þ

2 j lnðri;jÞ lnðri;j;2Þj þ lnðri;jÞ lnðri;j;2Þ

; ð3:17Þ

where si;j;1¼

lðlnðri;j;2ÞÞlðlnðri;j;1ÞÞ

lnðri;j;2Þlnðri;j;1Þ and si;j;2¼

lðlnðri;j;3ÞÞlðlnðri;j;2ÞÞ

lnðri;j;3Þlnðri;j;2Þ .joj is the absolute value of o.

After taking logarithms, Model 3 can then be transferred into a linear program as follows:

Min Obj1¼X

~ri;j

ðlnðScoreiÞ lnðScorejÞ lnðri;jÞ þ 2zi;jÞ

Max Obj2¼X

~ri;j

lðlnðri;jÞÞ

Subject to lðlnðri;jÞÞ ¼ si;j;1 ðlnðri;jÞ lnðri;j;1ÞÞ þ ðsi;j;2 si;j;1Þ ðlnðri;jÞ lnðri;j;2Þ þ di;jÞ 8~ri;j;

ð3:18Þ

lnðri;jÞ lnðri;j;2Þ þ di;jP0 8~ri;j; ð3:19Þ

di;jP0 8~ri;j; ð3:20Þ

lnðri;jÞ 6 lnðScoreiÞ lnðScorejÞ 6 lnðri;jÞ 8i; j > i; ð3:21Þ

X

~ri;j

ðlnðScoreiÞ lnðScorejÞ lnðri;jÞ þ zi;jÞ P 0 8~ri;j; ð3:22Þ

zi;jP0 8~ri;j; ð3:23Þ lnðScoreiÞ ¼ Xm k¼1 wklnðci;kÞ 8i; ð3:24Þ lnð1Þ 6 lnðScoreiÞ 6 lnð9Þ 8i; Xm k¼1 w¼_k1; w1; . . . ; wmP0:

Expressions(3.18)–(3.20) are based onYu (2002). Expression (3.21)is from(3.16). In order to linearize the absolute term in Obj1, constraints (3.22) and (3.23) are added into the model based on the work of Li

(1996). Expression (3.24)is from (3.1). Model 3 is a multi-objective linear optimization problem, which can

be solved by many techniques to get a global optimum. One of commonly used methods is formulated below:

Min Obj1 Obj2

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4. A numerical example

Considering the implications of a tendency of multicriteria decision-making,Brugha (2004)used screening, ordering and choosing phases to ﬁnd a preference. The solving process of the proposed approach is illustrated by these three phases as listed below:

(i) The screening phase: the DM speciﬁes upper and lower bounds of attributes to screen out of poor alternatives.

(ii) The ordering phase: the DM tries to put a preference order on the remaining alternatives.

(a) All criteria values are transformed to the same positive format by subtracting from upper bound, and then normalized to a scale from 1 to 9.

(b) Use the proposed DEA ranking model (Model 1) to get the superior set of each alternatives. (c) Apply range reduction model (Model 2) to provide reasonable upper and lower bounds of ri,j, where

ri,j represents a pairwise comparison of Aiover Aj.

(d) Decision makers specify fuzzy preferences based on the support of suggested ranges. Apply the fuzzy ranking model (Model 3) to get the weights of each criterion.

(e) Calculate the scores of each alternative and get a preference order.

(iii) The choosing phase: the DM makes a choice between two or three close alternatives.

The following example, modified from Harvard Business Review (Hammond et al., 1998), is applied to illustrate above concepts. The example describes a business problem for renting an office. A decision maker defines four major objectives to fulfill in selecting his/her office: (i) a short commute time from home to office, (ii) good access to his clients, (iii) sufficient space, and (iv) low costs. The commuting time is the average time in minutes needed to travel to work during rush hour. The percentage of his clients within an hour’s drive of the office is used to measure the access to clients. Office size is measured in square feet, and cost is measured by monthly rent. The DM hopes to keep monthly cost and commuting time as small as possible and remaining criteria larger. There are thirty available alternatives.

(i) The screening phase

Suppose the DM sets the upper bounds of monthly cost and commute time to be 2200 and 60, respec-tively, and the lower bounds of oﬃce size and customer access to be 500 and 50%, respectively. Twenty alternatives are screened out. The remaining 10 alternatives are listed inTable 1.

(ii) The ordering phase

(a) Monthly cost and commute time are transformed to the positive format by subtracting from upper bound: inexpensiveness instead of costs, convenience instead of commute time. Then, all criteria values are normalized to a scale from 1 to 9, as listed inTable 2.

Table 1

Original criteria values for renting an oﬃce

Alternative Minimization Maximization

Monthly cost ($) Commute time (minutes) Oﬃce size (square feet) Customer access (%)

A1 1850 45 800 50 A2 1700 25 700 80 A3 1500 20 500 70 A4 1900 25 950 85 A5 1750 30 700 75 A6 1950 40 950 65 A7 1800 60 850 60 A8 1600 45 1000 50 A9 2200 50 900 75 A10 2000 45 1050 85

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(b) Let M ¼ 1000 and e ¼ 0:1, solving the oﬃce-renting example by Model 1 yields the best rank and the corresponding score of each alternative in the last three columns of Table 3. Taking A1 for

instance, let p¼ 1, solving Model 1 yields Score1

1¼ 5:04. By the proposed model, there are four

alternatives better than A1, that is Supð1Þ ¼ fA2; A4; A5; A8g. The best rank of A1is 5.

In order to compare the proposed model with conventional DEA models, the optimal score and the corresponding rank of each alternative by a conventional DEA model (multiplicative CCR model) are listed in the second and third column ofTable 3. Taking A1for instance, the rank of A1is 7. The

proposed model can obtain a better rankðRank1¼ 5Þ for alternative A1than that of conventional

DEA modelðRank1¼ 7Þ.

(c) Next, in order to provide reasonable ranges of ri,j, a range reduction model is applied to the

exam-ple. Applying Model 2 to the example yields the upperðri;jÞ and lower ðri;jÞ bound of ri,j, as listed in

Table 4. For simplicity, only the upper-right parts of the matrix are shown. Each element is divided

into two parts, where upper and lower values indicate the upper and lower bounds, respectively. The ranges of ri,jare signiﬁcantly reduced by adding the superior set constraints. Taking r1,2for instance,

the original range of r1,2 is 1=9 6 r1;2 69 because 1 6 Scorei69; 8i. After taking Model 2, the

range of r1,2is reduced to 0:24 6 r1;260:98.

In order to help the decision makers specify preferences conveniently, the values of ri,jare

trans-ferred to a discrete numerical rating r0

i;jbased on the pairwise comparison scale (Saaty, 1980) listed

inTable 5. The reduced upper and lower bounds of ri,jinTable 4can then be transferred to a

cor-responding matrix in Saaty’s scale inTable 6. These reduced ranges provide reasonable upper and lower bounds to help the decision maker specify their preferences.

Table 2

Transformed criteria values with positive format

Alternative Maximization

Inexpensive Convenience Oﬃce size (square feet) Customer access (%)

A1 5.00 4.00 5.36 1.00 A2 6.71 8.00 3.91 7.86 A3 9.00 9.00 1.00 5.57 A4 4.43 8.00 7.55 9.00 A5 6.14 7.00 3.91 6.71 A6 3.86 5.00 7.55 4.43 A7 5.57 1.00 6.09 3.29 A8 7.86 4.00 8.27 1.00 A9 1.00 3.00 6.82 6.71 A10 3.29 4.00 9.00 9.00 Table 3

Results and comparisons of the oﬃce-renting example

Conventional DEA model Proposed DEA model

Maximal score Rank Best rank Score Superior set Sup(k)

A1 5.36 7 5.04 5 {A2, A4, A5, A8} A2 8.00 2 7.17 1 A3 9.00 1 7.23 1 A4 9.00 1 8.33 1 A5 7.00 4 5.69 2 {A2} A6 7.55 3 6.02 2 {A4} A7 6.09 6 5.58 2 {A8} A8 8.27 2 5.92 1 A9 6.82 5 6.74 3 {A4, A10} A10 9.00 1 7.40 1

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Table 4

The reduced ranges½ri;j; ri;j of ri,j

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A1 1 0.98 3.31 1.00 1.00 1.21 4.00 1.00 4.16 1.43 0.24 0.27 0.22 0.28 0.40 0.70 0.64 0.41 0.31 A2 1 3.38 1.47 1.16 1.75 8.00 4.18 6.13 2.04 0.77 0.56 1.02 0.62 0.77 0.68 0.63 0.48 A3 1 1.91 1.44 2.26 9.00 3.67 7.94 2.66 0.17 0.30 0.18 0.23 0.21 0.19 0.14 A4 1 1.83 1.82 8.00 4.49 4.16 2.00 0.74 1.09 0.85 0.64 1.14 0.86 A5 1 1.59 7.00 3.61 5.59 1.85 0.60 0.76 0.64 0.62 0.47 A6 1 5.00 2.47 3.52 1.25 0.72 0.55 1.00 0.75 A7 1 1.00 4.91 1.56 0.25 0.33 0.25 A8 1 6.38 2.10 0.41 0.31 A9 1 0.76 0.32 A10 1 Table 5

The mapping of ri,jand the pairwise comparison scale for AHP preferences (Saaty, 1980)

Numerical rating Verbal judgments of preferences Value of ri,j

9 Extremely preferred ri;jP8:5

8 Very strongly to extremely 7:5 6 ri;j<8:5

7 Very strongly preferred 6:5 6 ri;j<7:5

6 Strongly to very strongly 5:5 6 ri;j<6:5

5 Strongly preferred 4:5 6 ri;j<5:5

4 Moderately to strongly 3:5 6 ri;j<4:5

3 Moderately preferred 2:5 6 ri;j<3:5

2 Equally to moderately 1:5 6 ri;j<2:5

1 Equally preferred 1 6 ri;j<1:5

Table 6

The reduced ranges of ri,jin Saaty’s scale

Max A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A1 1 1 3 1 1 1 4 1 4 1 1/4 1/4 1/4 1/4 1/2 1 1/2 1/2 1/3 A2 1 3 1 1 2 8 4 6 2 1 1/2 1 1/2 1 1 1/2 1/2 A3 1 2 1 2 9 4 8 3 1/6 1/3 1/5 1/4 1/5 1/5 1/7 A4 1 2 2 8 4 4 2 1 1 1 1/2 1 1 A5 1 2 7 4 6 2 1/2 1 1/2 1/2 1/2 A6 1 5 2 4 1 1 1/2 1 1 A7 1 1 5 2 1/4 1/3 1/4 A8 1 6 2 1/2 1/3 A9 1 1 1/3 A10 1

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(d) Suppose the decision maker speciﬁes the membership functions as ~r1;4¼ 1₄;1₃;1₂

, ~r4;7 ¼ ð2; 3; 5Þ and

~

r2;7¼ ð3; 5; 6Þ, Model 3 can be formulated as follows:

Min Obj1 Obj2

Obj1¼ X 4 k1 wk lnðc1;kÞ X4 k1 wk lnðc4;kÞ lnðr1;4Þ þ 2 z1;4 ! þ X 4 k1 wk lnðc4;kÞ X4 k1 wk lnðc7;kÞ lnðr4;7Þ þ 2 z4;7 ! þ X 4 k1 wk lnðc2;kÞ X4 k1 wk lnðc7;kÞ lnðr2;7Þ þ 2 z2;7 ! Obj2¼ lðlnðr1;4ÞÞ þ lðlnðr4;7ÞÞ þ lðlnðr2;7ÞÞ Subject to lðlnðr1;4ÞÞ ¼ 3:48 lnðr1;4Þ ln 1 4 þ ð2:47 3:48Þ lnðr1:4Þ ln 1 3 þ d1;4 ; lðlnðr4;7ÞÞ ¼ 2:47 ðlnðr4;7Þ lnð2ÞÞ þ ð1:96 2:47Þ ðlnðr4;7Þ lnð3Þ þ d4;7Þ; lðlnðr2;7ÞÞ ¼ 1:96 ðlnðr2;7Þ lnð3ÞÞ þ ð5:48 1:96Þ ðlnðr2:7Þ lnð5Þ þ d2;7Þ; lnðr1;4Þ ln 1 3 þ d1;4P0; lnðr4;7Þ lnð3Þ þ d4;7 P0; lnðr2;7Þ lnð5Þ þ d2;7 P0 d1;4P0; d4;7P0; d2;7P0; X4 k1 wk lnðc1;kÞ X4 k1 wk lnðc4;kÞ lnðr1;4Þ þ z1;4 ! P0; X4 k1 wk lnðc4;kÞ X4 k1 wk lnðc7;kÞ lnðr4;7Þ þ z4;7 ! P0; X4 k1 wk lnðc2;kÞ X4 k1 wk lnðc7;kÞ lnðr2;7Þ þ z2;7 ! P0; z1;4P0; z4;7P0; z2;7 P0; lnðri;jÞ 6 X4 k¼1 wk lnðci;kÞ X4 k¼1 wk lnðcj;kÞ 6 lnðri;jÞ 8i ¼ 1; . . . ; n 1; j ¼ i þ 1; . . . ; n; lnð1Þ 6X 4 k¼1 wk lnðci;kÞ 6 lnð9Þ 8i; Xm k¼1 wk ¼ 1; w1; . . . ; wmP0:

Solving the above program by Lingo software yields a global optimal solution with obj1¼ 0:465, obj2¼ 0, w1¼ 0:24, w2¼ 0:36, w3¼ 0, w4¼ 0:4, lðlnðr1;4ÞÞ ¼ 1, lðlnðr4;7ÞÞ ¼ 1, lðlnðr2;7ÞÞ ¼ 1,

lnðr1;4Þ ¼ 1:099, lnðr4;7Þ ¼ 1:099, lnðr2;7Þ ¼ 1:609 and d1;4 ¼ d4;7¼ d2;7¼ 0.

Represented in Saaty’s ratio scale, r1;4 ¼13, r4;7¼ 3 and r2;7¼ 5. The approximation error, obj1, is

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(e) Substituting the values of w1, w2, w3, and w4into Expression(3.1)yields the score and rank of each

alternatives, as listed inTable 7. A2is the best choice, following by A3, A4, A5, A10, A6, A9and A8.

A1and A7are at the same score and ranked the worst.

(iii) The choosing phase

Since the scores of the top three alternatives A2, A3and A4are close to each other, the DM may make a

ﬁnal choice among these three alternatives.

This oﬃce-renting example demonstrates how proposed approach provides reasonable upper and lower bounds information of preferences based on the concept of DEA. By referring to these ranges, a decision maker can specify his/her fuzzy preferences partially, and obtain the optimized ranks of alternatives. 5. Concluding remarks

This study proposes a novel ranking method which incorporates fuzzy preferences specified by a decision maker. Based on a modified DEA model, reasonable upper and lower bounds are provided to assist a decision maker in articulating related preferences. A goal-programming model with minimal approximation errors and maximal fulfillment of a decision maker’s preferences is proposed to solve the fuzzy preference problem directly and efficiently.

A comparison with other ranking methods, such as AHP methods (Saaty, 1977, 1980, etc.) and Fuzzy AHP methods (Graan, 1980; Chang, 1996; Leung and Cao, 2000; Yu, 2002, etc.), indicates the following advantages of the proposed method:

(i) The proposed method provides reasonable upper and lower bounds information about specifying pref-erences, which are not provided by other methods.

(ii) The proposed method can treat incomplete pairwise comparison matrices; while most of the other meth-ods cannot deal with them.

(iii) The proposed fuzzy ranking method results in a crisp solution directly; however, most of fuzzy AHP methods require extra defuzziﬁcation techniques to obtain such a solution.

Two issues could be studied in the future research. First, to enhance the fuzzy rating of the proposed method, the fuzzy set gradual membership grid technique (Badiru and Cheung, 2002) can be incorporated in the proposed fuzzy ranking method. Second, in order to improve some restrictions resulting from linear programming methods whose solutions are found at corners of combinations of constraints, non-linear or fuzzy constraints can be applied in the range limits.

Acknowledgements

This research is supported by the National Science Council of the Republic of China under contract NSC 95-2416-H-239-009.

Table 7

The ﬁnal score and rank of each alternative

Score Rank A1 2.43 9 A2 7.62 1 A3 7.43 2 A4 7.28 3 A5 6.67 4 A6 4.48 6 A7 2.43 9 A8 2.70 8 A9 3.18 7 A10 5.27 5

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