Comparison of fuzzy numbers using a fuzzy distance measure

Fuzzy Sets and Systems 130 (2002) 331–341www.elsevier.com/locate/fss

Comparison of fuzzy numbers using a fuzzy distance measure

Liem Trana ; ∗, Lucien Ducksteinb

aCenter for Integrated Regional Assessment, Pennsylvania State University, 2217 Earth and Engineering Sciences Building,University Park, PA 16802, USA

bEcole Nationale du Genie Rural, des Eaux et des Forets, 75732 Paris 15, France

Received 30 May 2000; received in revised form 24 April 2001; accepted 12 September 2001

Abstract

A new approach for ranking fuzzy numbers based on a distance measure is introduced. A new class of distance measuresfor interval numbers that takes into account all the points in both intervals is developed -rst, and then it is used to formulatethe distance measure for fuzzy numbers. The approach is illustrated by numerical examples, showing that it overcomesseveral shortcomings such as the indiscriminative and counterintuitive behavior of several existing fuzzy ranking approaches.c© 2002 Elsevier Science B.V. All rights reserved.

Keywords: Fuzzy numbers; Fuzzy ranking; Distance measure

1. Introduction

The purpose of this paper is to introduce a newmethod for ranking fuzzy numbers (FNs). Fuzzy rank-ing is a topic that has been studied by many researchers(see e.g., [6–8,14,19,27] for an overview andcomparison of various methods). In a more recentreview, Wang and Kerre [22,23] proposed severalaxioms as reasonable properties to determine the ra-tionality of a fuzzy ordering or ranking method andsystematically compared a wide array of fuzzy rankingmethods. Almost each method, however, has pitfallsin some respect, such as inconsistency with human

∗ Corresponding author. Tel.: +1-814-865-1587; fax: +1-814-865-3191.E-mail address: [email protected] (L. Tran).

intuition, indiscrimination and diAculty of interpreta-tion [6,8,27]. The fuzzy ranking method suggested inthis paper overcomes many of the problems inherent toexisting methods. To formulate this ranking method,a new class of distance measures for interval numbers(INs), that takes into account all the points in bothintervals, is developed -rst. Then it is used to formthe distance measure for FNs. The method for rank-ing FNs suggested in this paper is based on compari-son of distance from FNs to predetermined crisp ide-als of the best and the worst, a concept which is verycommon to decision makers faced with multi-criteriadecision making problems. The paper is organized asfollows — The new class of distance measure forINs is introduced in Section 2. Then the new class ofdistance measure for FNs is developed in Section 3,

0165-0114/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S 0165 -0114(01)00195 -6

332 L. Tran, L. Duckstein / Fuzzy Sets and Systems 130 (2002) 331–341

leading to the fuzzy ranking in Section 4 and numer-ical examples in Section 5. Section 6 is devoted todiscussion and conclusions.

2. Distance measure for interval numbers

A new distance measure for two INs is presented inthis section. It will be used later to construct a distancemeasure for FNs.

Let F(R) be the set of INs in R, and the distancebetween two INs A(a1; a2) and B(b1; b2) be de-ned as

D2(A; B) =∫ 1=2

−1=2

∫ 1=2

−1=2

×{[(

a1 + a2

2

)+ x(a2 − a1)

]

−[(

b1 + b2

2

)+ y(b2 − b1)

]}2

dx dy

(1)

=[(

a1 + a2

2

)−(b1 + b2

2

)]2

+13

[(a2 − a1

2

)2

+(b2 − b1

2

)2]: (2)

It can be proved that D(A; B) =√

D2(A; B) is adistance on F(R). First, D(A; B)¿0. Symmetry istransparent. If D(A; B) = 0 then A=B. The triangleinequality follows from the fact that the functionto be integrated in (1) is the square of Euclideandistance.

Although only the lower and upper bound valuesof the two INs appear in the Eq. (2), which is derivedfrom Eq. (1) for operational purpose, the integral in(1) shows that this distance takes into account everypoint in both intervals when computing the distancebetween those two INs. It is diGerent from most ex-isting distance measures for interval numbers whichoften use only the lower and upper bound values(e.g., those used in BHardossy et al. [3], Diamond [9],Diamond and KIorner [10], and Diamond and Tanaka[11]). Bertoluzza et al. [5] proposed a distance mea-sure for intervals which also considers every point

of both intervals. Its general form, however, is toocomplicated and the authors later restricted the mea-sure to a particular case with a -nite number of con-sidered values for operational purpose.

As an example, consider the crisp number A(0; 0)and two INs B(−1; 3) and C(1; 3). Using the distancemeasure applied in Diamond [9] and BHardossy et al.[3], it is found that d2(A; B) =d2(A; C) = 10. With ourdistance measure, D2(A; B) = 7=3¡D2(A; C) = 13

3 .This result makes more sense as A is inside ofB and outside of C, leading to the expectationthat the distance from A to B should be smallerthan the distance from A to C. In case of crispnumbers, this new distance measure becomes theEuclidean distance. Hence it may be considered asa generalization of the usual Euclidean distance, afeature not seen in the common HausdorG distance orthe “dissemblance index” distance of Kaufman andGupta [17]. It should be mentioned that the distancemeasure proposed in this paper has similar character-istics but diGerent formulation than the one suggestedin Tran and Duckstein [21].

3. Distance measure for fuzzy numbers

To be able to deal with curvilinear member-ship functions, generalized left right fuzzy numbers(GLRFN) of Dubois and Prade [12] as described byBHardossy and Duckstein [2] are used in this section.A fuzzy set A= (a1; a2; a3; a4) is called a GLRFN ifits membership function satisfy the following:

�(x) =

L(

a2 − xa2 − a1

)for a1 6 x 6 a2;

1 for a2 6 x 6 a3;

R(

x − a3

a4 − a3

)for a3 6 x 6 a4;

0 else

(3)

where L and R are strictly decreasing functions de-nedon [0; 1] and satisfying the conditions:

L(x) = R(x) = 1 if x 6 0;

L(x) = R(x) = 0 if x ¿ 1:

L. Tran, L. Duckstein / Fuzzy Sets and Systems 130 (2002) 331–341 333

For a2 = a3, we have the classical de-nition of leftright fuzzy numbers (LRFN) of Dubois and Prade[12]. Trapezoidal fuzzy numbers (TrFN) are specialcases of GLRFN with L(x) =R(x) = 1 − x. Triangu-lar fuzzy numbers (TFN) are also special cases ofGLRFN with L(x) =R(x) = 1 − x and a2 = a3.

A GLRFN A is denoted as

A = (a1; a2; a3; a4)LA−RA (4)

and an �-level interval of fuzzy number A as

A(�) = (AL(�); AU (�))

= (a2 − (a2 − a1)L−1A (�);

a3 + (a4 − a3)R−1A (�)) (5)

Let F(R) be the set of GLRFNs in R. Using thedistance measure for interval numbers de-ned above, adistance between two GLRFNs A and B can be de-nedas

D2(A; B; f)

=

⟨∫ 1

0

{[(AL(�) + AU (�)

2

)

−(BL(�) + BU (�)

2

)]2

+13

[(AU (�) − AL(�)

2

)2

+(BU (�) − BL(�)

2

)2]}

×f(�) d�〉/∫ 1

0f(�) d�: (6)

Here f, which serves as a weighting function, is acontinuous positive function de-ned on [0; 1]. The dis-tance is a weighted sum (integral) of the distancesbetween two intervals at all � levels from 0 to 1.It is reasonable to choose f as an increasing func-tion, indicating greater weight assigned to the distancebetween two intervals at a higher � level.

It can be proved that D(A; B)=√

D2(A; B; f) isa distance on F(R). First, D(A; B)¿0. Symmetry istransparent. If D(A; B) = 0 then A=B by the continu-ity of f; L; and R. The triangle inequality follows fromthe fact that the function to be integrated in (6) is thesquare of the distance between two INs presented inthe previous section. Since D(A; B) = |a− b| if A andB are crisp numbers, this distance may be consideredas a generalization of the usual Euclidean distance.

The calculation of the distance can be reformulatedusing the following derivation:

D2(A; B; f)

=∫ 1

0

{(a2 + a3

2− b2 + b3

2

)2

+(a2 + a3

2− b2 + b3

2

)[(a4 − a3)R−1

A (�)

− (a2 − a1)L−1A (�) − (b4 − b3)R−1

B (�)

+ (b2 − b1)L−1B (�)]

+13

(a3 − a2

2

)2

+13

(a3 − a2

2

)

×[(a4 − a3)R−1A (�) + (a2 − a1)L−1

A (�)]

+13

(b3 − b2

2

)2

+13

(b3 − b2

2

)

×(b4 − b3)R−1B (�) + (b2 − b1)L−1

B (�)]

+13

[(a4 − a3)2(R−1A (�))2 + (a2 − a1)2

×(L−1A (�))2 + (b4 − b3)2(R−1

B (�))2

+ (b2 − b1)2(L−1B (�))2]

− 13

[(a2 − a1)(a4 − a3)L−1A (�)R−1

A (�)

+ (b2 − b1)(b4 − b3)L−1B (�)R−1

B (�)]

+12

[(a4 − a3)(b2 − b1)R−1A (�)L−1

B (�)


+ (a2 − a1)(b4 − b3)L−1A (�)R−1

B (�)

− (a4 − a3)(b4 − b3)R−1A (�)R−1

B (�)

− (a2 − a1)(b2 − b1)L−1A (�)L−1

B (�)]

}

× f(�) d�/∫ 1

0f(�) d�: (7)

By introducing:

∫ 1

0f(�) d� = S; (8)

∫ 1

0(L−1

A (�))nf(�) d� = Ln∗A ; (9)

∫ 1

0(R−1

A (�))nf(�) d� = Rn∗A ; (10)

∫ 1

0L−1A (�)L−1

B (�)f(�) d� = LL∗AB; (11)

∫ 1

0R−1A (�)R−1

B (�)f(�) d� = RR∗AB; (12)

∫ 1

0L−1A (�)R−1

B (�)f(�) d� = LR∗AB; (13)

∫ 1

0R−1A (�)L−1

B (�)f(�) d� = RL∗AB; (14)

(7) can be rewritten as

D2(A; B)

=(a2 + a3

2− b2 + b3

2

)2

+1S

(a2 + a3

2− b2 + b3

2

)[(a4 − a3)R∗

A

− (a2 − a1)L∗A − (b4 − b3)R∗

B

+ (b2 − b1)L∗B]

+1

3S

(a3 − a2

2

)2

+1

3S

(a3 − a2

2

)

×[(a4 − a3)R∗A + (a2 − a1)L∗

A]

+1

3S

(b3 − b2

2

)2

+1

3S

(b3 − b2

2

)

×[(b4 − b3)R∗B + (b2 − b1)L∗

B]

+1

3S[(a4 − a3)2R∗∗

A + (a2 − a1)2L∗∗A

+ (b4 − b3)2R∗∗B + (b2 − b1)2L∗∗

B ]

− 13S

[(a2 − a1)(a4 − a3)L∗AR

∗A

+ (b2 − b1)(b4 − b3)L∗BR

∗B]

+1

2S[(a4 − a3)(b2 − b1)R∗

AL∗B

+ (a2 − a1)(b4 − b3)L∗AR

∗B

− (a4 − a3)(b4 − b3)R∗AR

∗B

− (a2 − a1)(b2 − b1)L∗AL

∗B]: (15)

Table 1 gives the equations to compute distance forsome of the commonly used fuzzy numbers with twodiGerent weighting functions: f(�) = 1 representingequal weights for intervals at diGerent � levels, andf(�) = � indicating more weight given to intervalsat higher � level. As a numerical example, con-sider crisp number A(1), TFN B(2; 2; 4)T, and TrFNC(0; 0; 2; 4)Tr (Fig. 1). Let f(�) = �. Using the Haga-man distance (DH) measure applied in BHardossy et al.[3], it is found that D2

H(A; B) =D2H(A; C) = 2. With our

distance measure, D2(A; B) = 17=9¿D2(A; C) = 7=9.This result makes more sense as A is outside of B andinside of C, leading to the expectation that the dis-tance from A to B should be greater than the distancefrom A to C.

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335Table 1Distance functions for some commonly used FNs

Fuzzy numbers f(�) D2T (A; B; f)

Trapezoidal fuzzy �(a2 + a3

2− b2 + b3

2

)2

+13

(a2 + a3

2− b2 + b3

2

)[(a4 − a3) − (a2 − a1) − (b4 − b3) + (b2 − b1)]

numbers

A = (a1; a2; a3; a4)Tr23

(a3 − a2

2

)2

+19

(a3 − a2

2

)[(a4 − a3) + (a2 − a1)] +

23

(b3 − b2

2

)2

+19

(b3 − b2

2

)[(b4 − b3) + (b2 − b1)]

B = (b1; b2; b3; b4)Tr

+118

[(a4 − a3)2 + (a2 − a1)2 + (b4 − b3)2 + (b2 − b1)2] − 118

[(a2 − a1)(a4 − a3) + (b2 − b1)(b4 − b3)]

+112

[(a4 − a3)(b2 − b1) + (a2 − a1)(b4 − b3) − (a4 − a3)(b4 − b3) − (a2 − a1)(b2 − b1)]

1(a2 + a3

2− b2 + b3

2

)2

+12

(a2 + a3

2− b2 + b3

2

)[(a4 − a3) − (a2 − a1) − (b4 − b3) + (b2 − b1)]

13

(a3 − a2

2

)2

+16

(a3 − a2

2

)[(a4 − a3) + (a2 − a1)] +

13

(b3 − b2

2

)2

+16

(b3 − b2

2

)[(b4 − b3) + (b2 − b1)]

+19

[(a4 − a3)2 + (a2 − a1)2 + (b4 − b3)2 + (b2 − b1)2] − 19

[(a2 − a1)(a4 − a3) + (b2 − b1)(b4 − b3)]

+16

[(a4 − a3)(b2 − b1) + (a2 − a1)(b4 − b3) − (a4 − a3)(b4 − b3) − (a2 − a1)(b2 − b1)]

Triangular fuzzy � (a2 − b2)2 +13

(a2 − b2)[(a3 + a1) − (b3 + b1)] +118

[(a3 − a2)2 + (a2 − a1)2 + (b3 − b2)2 + (b2 − b1)2]numbers

A = (a1; a2; a3)T − 118

[(a2 − a1)(a3 − a2) + (b2 − b1)(b3 − b2)] − 112

(2a2 − a1 − a3)(2b2 − b1 − b3)

B = (b1; b2; b3)T

1 (a2 − b2)2 +12

(a2 − b2)[(a3 + a1) − (b3 + b1)] +19

[(a3 − a2)2 + (a2 − a1)2 + (b3 − b2)2 + (b2 − b1)2]

− 19

[(a2 − a1)(a3 − a2) + (b2 − b1)(b3 − b2)] +16

(2a2 − a1 − a3)(2b2 − b1 − b3)


x

µ(x)

1

0

C(0,0,2,4) Tr

B(2,2,4) T

A(1)

1 2 4

Fig. 1. Illustration of the distance measure for fuzzy numbers.

4. Ranking based on distance measure for fuzzynumbers

The method for ranking FNs suggested in this paperis based on comparison of distance from FNs to somepredetermined targets: the crisp maximum (Max) andthe crisp minimum (Min). The idea is that a FN isranked -rst if its distance to the crisp maximum (Dmax)is the smallest but its distance to the crisp minimum(Dmin) is the greatest. If only one of these conditionsis satis-ed, a FN might be outranked the others de-pending upon context of the problem (for example,the attitude of the decision-maker in a decision situa-tion). This point will be discussed further in the nextsection.

The Max and Min are chosen as follows:

Max(I) ¿ sup

(I⋃

i=1

s(Ai)

);

Min(I) 6 inf

(I⋃

i=1

s(Ai)

);

where s(Ai) is the support of FNs Ai; i = 1; : : : ; I .Then Dmax and Dmin of FN A can be computed asfollows:

D2(A;M) =(a2 + a3

2− M

)2

+1S

(a2 + a3

2− M

)

×[(a4 − a3)R∗A − (a2 − a1)L∗

A]

+1

3S

(a3 − a2

2

)2

+1

3S

(a3 − a2

2

)

×[(a4 − a3)R∗A + (a2 − a1)L∗

A]

+1

3S[(a4 − a3)2R∗∗

A + (a2 − a1)2L∗∗A ]

− 13S

[(a2 − a1)(a4 − a3)L∗AR

∗A] (16)

where M is either Max or Min. Hence, Dmax =√D2(A;Max) and Dmin =

√D2(A;Min). Table 2

gives the equations to compute Dmax and Dmin forsome of the commonly used fuzzy numbers withtwo diGerent weighting functions: f(�) = 1 andf(�) = �.

5. Numerical examples

Fig. 2 and Table 3 show several typical exam-ples to illustrate the current method and compare itwith some other ranking methods. Most of resultsfor other methods are adapted from the data in Bor-tolan and Degani [6], Lee and Li [19], and Chen andHwang [8].

For example (a), a fair ranking is given by allmethods, complying with human intuition. Suchagreeable ranking however is not seen in example (b),where two symmetrical FNs have the same mode butdiGerent supports. In this case, a FN may be prefer-able or equal to the other, depending in a decisionsituation for example, on the attitude of the decision-maker (e.g., risk-prone or risk-averse). Hence a suf--cient ranking method should provide meaningfulindices for all three alternatives (e.g., A1 =A2, A1¿A2,or A1¡A2). All methods except Dubois and Prade’s[9] and the current method give only one answer,either non-discrimination of two FNs or in favorof one over another. Note that the Fortemps andRoubens’s method of “area of compensation” [14],which is considered robust with compensation, lin-earity, and additivity properties [20], also gives anon-discriminative result for this example. Consid-ering that the FNs represent some measure of riskor gain. To explain the result from their method,Lee and Li [19] suggest that human intuition wouldfavor the FN with smaller spread. This explana-tion however might not be true for a risk-pronedecision-maker, who may favor the FN with largerspread. On the other hand, four indices in Duboisand Prade’s provide suAcient information for thedecision maker to choose one FN over the other, de-

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Table 2Ranking functions for some commonly used FNs

Fuzzy numbers f(�) D2(A;M∗; f)

Trapezoidal fuzzy �(a2 + a3

2− M

)2

+13

(a2 + a3

2− M

)[(a4 − a3) − (a2 − a1)]

numbers

A = (a1; a2; a3; a4)Tr23

(a3 − a2

2

)2

+19

(a3 − a2

2

)[(a4 − a3) + (a2 − a1)] +

118

[(a4 − a3)2 + (a2 − a1)2] − 118

[(a2 − a1)(a4 − a3)]

1(a2 + a3

2− M

)2

+12

(a2 + a3

2− M

)[(a4 − a3) − (a2 − a1)]

13

(a3 − a2

2

)2

+16

(a3 − a2

2

)[(a4 − a3) + (a2 − a1)] +

19

[(a4 − a3)2 + (a2 − a1)2] − 19

[(a2 − a1)(a4 − a3)]

Triangular fuzzy � (a2 − M)2 +13

(a2 − M)[(a3 + a1) − 2M ] +118

[(a3 − a2)2 + (a2 − a1)2] − 118

[(a2 − a1)(a3 − a2)]numbersA = (a1; a2; a3)T

1 (a2 − M)2 +12

(a2 − M)[(a3 + a1) − 2M ] +19

[(a3 − a2)2 + (a2 − a1)2] − 19

[(a2 − a1)(a3 − a2)]

∗M is either Max or Min.


(a)

0

0.5

1

0 0.2 0.4 0.6 0.8 1(b)

0

0.5

1

0 0.2 0.4 0.6 0.8 1

(c)

0

0.5

1

0 0.2 0.4 0.6 0.8 1(d)

0

0.5

1

0 0.2 0.4 0.6 0.8 1

(e)

0

0.5

1

0 0.2 0.4 0.6 0.8 1

A1

A2

A3

Fig. 2. Examples of FN comparison.

pending on what index is used dominantly. For thecurrent method, if Dmax is used, A2 is preferable sinceDmax(A2)¡Dmax(A1). Conversely, if Dmin is used, A1

is preferable as Dmin(A1)¿Dmin(A2). This conQict isunderstandable because A1 may be preferable or lesspreferable due to its larger right and left dispersion,respectively. On this account, a risk-prone decisionmaker may want to use Dmin in selecting an alterna-tive, while a risk-averse person may prefer Dmax. Ifboth Dmax and Dmin are used in conjunction, A1 andA2 may be ranked equally. Hence the current methodprovides an appropriate three-way explanation forranking in this case. This feature is not seen in othermethods except that of Dubois and Prade [13].

For example (c), three FNs A1, A2, and A3 arediGerent on the left side. Although intuition wouldyield A1¿A2¿A3, non-discriminative result is seen

using several methods. Example (d) is a mirror caseof example (c), where three FNs A1, A2, and A3 areonly diGerent on the right side. DiGerent from resultin previous example, all methods but the Bass andKwakernaak’s [4] give the same result here. Thisshows that several methods only consider a partial setof attributes of FNs being compared, especially thoseon the right-hand side.

Example (e) is very typical in comparing fuzzynumbers where intuition is not as obvious as in otherexamples. Some methods favor A1 over A2 while oth-ers give a reverse result. For the Dubois and Prade’smethod, as four indices are not identical, the decision-maker needs to select which index (or indices) touse in deriving ranking order. With the Lee and Li’smethod, there is conQict in results when two diGer-ent distributions — uniform and proportional — are

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339

Table 3Examples of FN comparison

Methods (a) (b) (c) (d) (e)

A1 A2 A3 A1 A2 A1 A2 A3 A1 A2 A3 A1 A2

Yager [16–18] F1 0.760 0.700 0.630 0.500 0.500∗ 0.700 0.630 0.570 0.620 0.560 0.500 0.610 0.530F2 0.900 0.760 0.660 0.610 0.540 0.750 0.750 0.750∗ 0.810 0.640 0.580 0.660 0.690F3 0.800 0.700 0.600 0.600 0.500 0.700 0.650 0.570 0.620 0.540 0.500 0.580 0.560

Bass & Kwakernaak [4] 1.000 0.740 0.600 1.000 1.000∗ 1.000 1.000 1.000∗ 1.000 1.000 1.000∗ 0.840 1.000Baldwin & Guild [1] 1 :p 0.420 0.330 0.300 0.270 0.270∗ 0.370 0.270 0.270∗ 0.450 0.370 0.270 0.420 0.330

g 0.550 0.400 0.340 0.300 0.240 0.420 0.350 0.350∗ 0.530 0.400 0.280 0.440 0.370r : a 0.280 0.230 0.220 0.200 0.230 0.270 0.190 0.190∗ 0.310 0.280 0.210 0.340 0.240

Kerre [13] 1.000 0.860 0.760 0.910 0.910∗ 1.000 0.910 0.750 1.000 0.850 0.750 0.960 0.890Jain [11,12] k = 1 0.900 0.760 0.660 0.730 0.670 0.820 0.820 0.820∗ 0.900 0.690 0.640 0.660 0.690

k = 2 0.840 0.650 0.540 0.600 0.480 0.710 0.710 0.710 0.820 0.560 0.450 0.530 0.510k = 1=2 0.950 0.860 0.780 0.830 0.800 0.890 0.890 0.890 0.940 0.800 0.770 0.780 0.810

Dubois & Prade [9] PD 1.000 0.740 0.600 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.840 1.000PSD 0.740 0.230 0.160 0.730 0.240 0.500 0.500 0.500 0.800 0.200 0.000 0.540 0.460ND 0.630 0.380 0.180 0.270 0.760 0.670 0.350 0.000 0.500 0.500 0.500 0.540 0.460NSD 0.260 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.160

Lee & Li [14] U · m 0.760 0.700 0.630 0.500 0.500 0.700 0.630 0.570 0.620 0.560 0.500 0.610 0.530U · G — — — 0.120 0.040 — — — — — — — —P · m 0.800 0.700 0.600 0.500 0.500 0.700 0.650 0.580 0.630 0.550 0.500 0.530 0.580P · G — — — 0.090 0.030 — — — — — — — —

Fortemps & Roubens [10] F0 0.800 0.700 0.600 0.500 0.500∗ 0.700 0.650 0.575 0.625 0.550 0.500 0.490 0.610Tran & Duckstein Dmax;f(x) = x 0.187 0.308 0.442 0.505 0.501 0.304 0.342 0.457 0.395 0.473 0.502 0.574 0.355

Dmin;f(x) = x 0.838 0.704 0.573 0.505 0.501 0.702 0.671 0.585 0.650 0.539 0.502 0.451 0.673Dmax;f(x) = 1 0.231 0.316 0.416 0.510 0.501 0.307 0.365 0.445 0.398 0.462 0.504 0.531 0.417Dmin;f(x) = 1 0.808 0.707 0.611 0.510 0.501 0.703 0.658 0.590 0.639 0.560 0.504 0.512 0.628

∗Cases of indiscrimination.


used. Lee and Li [19] suggest that when conQict oc-curs as in this example, the proportional distributionseems more reasonable. However, a diAculty is thatthe choice of either distribution is arbitrary [8], with-out a clear motivation for either choice. On the otherhand, the current approach prefers A2 to A1 with anagreeing result of Dmax and Dmin for both cases off(�) = � and f(�) = 1. As Dmax and Dmin reveal thedistance of FNs to the reference points, this result isunderstandable and plausible.

6. Discussion and conclusions

The criteria suggested by Zhu and Lee [27] are usedto evaluate the current ranking method. They includecomplexity, ease of interpretation, robustness, Qexi-bility, and transitivity.

• Complexity can be judged via the amount of com-putation to accommodate the ranking. Eq. (16) andthose in Table 2 show that the computation of Dmax

and Dmin in the current method is straightforwardand can be programmed easily. This is a very goodpractical aspect which is not seen in several othermethods. Furthermore, this is an absolute rankingand no pairwise comparison of FNs is necessary,making the computation process simple and trans-parent.

• Ease of interpretation is one of the most crucialcriteria for the decision-maker. With the currentmethod, the distance to the reference (crisp) pointsis a very common concept and can be easilyunderstood by decision maker(s). On the otherhand, some concepts in other methods, such asconcepts of possibility and necessity in Dubois andPrade’s [13], or probability measure in Lee and Li[19], are not easy to perceive by decision maker(s).

• Robustness refers to the ability of consistent rank-ing for a diversity of cases. Among the methodsin Table 3, only the Dubois and Prade’s [13] andthe current method show their ability of distinguishdiGerent FNs robustly and consistently. Note that amixed comparison of FNs and crisp numbers canbe done using the current method.

• Flexibility, according to Zhu and Lee [27], is theability of providing more than one index and=orallowing the participation of decision makers.

Regarding this criterion, the Dmax and Dmin in-dices of the current method are meaningfuland eAcient for ranking purpose. In general,these two indices are in agreement, providinga consistent ranking order. If conQict does oc-cur, they provide purposeful information for thedecision-maker to reach a decision, dependingupon his=her attitude towards “risk”. On the otherhand, the presence of the weighting function fallows the participation of the decision-maker ina Qexible way. For example, when the decision-maker is risk-neutral, f(�) = � seems to be rea-sonable. A risk-averse decision-maker might wantto put more weight on information at higher �level by using other functions, such as f(�) = �2

or a higher power of �. For a risk-prone decision-maker, a constant (f(�) = 1), or even a decreasingfunction f can be utilized.

• Transitivity refers to the ability of giving a consis-tent conclusion in the comparison of more than twoFNs. As the current method uses a distance functionto map FNs to real numbers which can be orderedlinearly, the transitive property of this method issatis-ed.

In general, the current ranking method possessesseveral good characteristics and advantages as com-pared to other existing ranking methods.

In conclusion, the distance measures for INs andFNs introduced in this paper are considered to be atleast as reasonable as other existing distance measures.They can be used in other applications, such as fuzzyregression or fuzzy goal programming. The currentranking method can be a valuable tool in a variety ofproblems (e.g., fuzzy scheduling, fuzzy control) due toits simplicity, ease of interpretation, and eGectivenessin ranking FNs.

References

[1] J.F. Baldwin, N.C.F. Guild, Comparison of fuzzy sets onthe same decision space, Fuzzy Sets and Systems 2 (1979)213–233.

[2] A. BHardossy, L. Duckstein, Fuzzy Rule-Based Modelingwith Applications to Geophysical, Biological and EngineeringSystems, CRC Press, Boca Raton, 1995.

[3] A. BHardossy, R. Hagaman, L. Duckstein, I. Bogardi, Fuzzyleast squares regression: theory and application, in: J.Kacprzyk, M. Fedrizi (Eds.), Fuzzy Regression Analysis,

L. Tran, L. Duckstein / Fuzzy Sets and Systems 130 (2002) 331–341 341

Omnitech Press, Warsaw and Physica-Verlag, Heidelberg,1992, pp. 181–193.

[4] S.M. Bass, H. Kwakernaak, Rating and ranking ofmultiple-aspect alternatives using fuzzy sets, Automatica 13(1977) 47–58.

[5] C. Bertoluzza, N. Corral, A. Salas, On a new class of distancesbetween fuzzy numbers, Mathware Soft Comput. 2 (1995)71–84.

[6] G. Bortolan, R. Degani, A review of some method for rankingfuzzy sets, Fuzzy Sets and Systems 15 (1985) 1–19.

[7] P.-T. Chang, E.S. Lee, Ranking fuzzy sets based on theconcept of existence, Comput. Math. Appl. 27 (1994) 1–21.

[8] S.-J. Chen, C.-L. Hwang, Fuzzy Multiple Attribute DecisionMaking, Springer, Berlin, 1992.

[9] P. Diamond, Fuzzy least squares, Inform. Sci. 46 (1988)141–157.

[10] P. Diamond, R. KIorner, Extended fuzzy linear models andleast squares estimates, Comput. Math. Appl. 33 (1997) 15–32.

[11] P. Diamond, H. Tanaka, Fuzzy regression analysis, in: R.S lowiHnski (Ed.), Fuzzy Sets in Decision Analysis, OperationsResearch and Statistics, Kluwer Academic Publishers, Boston,1998, pp. 349–387.

[12] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory andApplications, Academic Press, New York, 1980.

[13] D. Dubois, H. Prade, Ranking of fuzzy numbers in the settingof possibility theory, Inform. Sci. 30 (1983) 183–224.

[14] P. Fortemps, M. Roubens, Ranking and defuzzi-cationmethods based on area compensation, Fuzzy Sets and Systems82 (1996) 319–330.

[15] R. Jain, Decision-making in the presence of fuzzy variables,IEEE Trans. systems, Man Cybernet. 6 (1976) 698–703.

[16] R. Jain, A procedure for multi-aspect decision making usingfuzzy sets, Internat. J. System Sci. 8 (1977) 1318–1323.

[17] A. Kaufman, M. Gupta, Introduction to Fuzzy ArithmeticTheory and Application, Van Nostrand Reinhold Company,New York, 1985.

[18] E.E. Kerre, The use of fuzzy set theory in electrocardiologicaldiagnostic, in: M.M. Gupta, E. Sanchez (Eds.), ApproximateReasoning in Decision Analysis, North-Holland, Amsterdam,1982, pp. 277–282.

[19] E.S. Lee, R.-J. Li, Comparison of fuzzy numbers based onthe probability measure of fuzzy events, Comput. Math. Appl.15 (1988) 887–896.

[20] E.C. IOzelkan, L. Duckstein, Optimal fuzzy counterparts ofscheduling rules, European J. Oper. Res. 113 (1999) 593–609.

[21] L. Tran, L. Duckstein, Multiobjective fuzzy regression withcentral tendency and possibilistic properties, Fuzzy Sets andSystems 130 (2002) 21–31.

[22] X. Wang, E.E. Kerre, Reasonable properties for the orderingof fuzzy quantities(I), Fuzzy Sets and Systems 118 (2001)375–385.

[23] X. Wang, E.E. Kerre, Reasonable properties for the orderingof fuzzy quantities(II), Fuzzy Sets and Systems 118 (2001)387–405.

[24] R.R. Yager, On a general class of fuzzy connectives, FuzzySets and Systems 4 (1980) 235–242.

[25] R.R. Yager, On choosing between fuzzy subsets, Kybernetes9 (1980) 151–154.

[26] R.R. Yager, A procedure for ordering fuzzy subsets of theunit interval, Inform. Sci. 24 (1981) 143–161.

[27] Q. Zhu, E.S. Lee, Comparison and ranking of fuzzy numbers,in: J. Kacprzyk, M. Fedrizi (Eds.), Fuzzy RegressionAnalysis, Omnitech Press, Warsaw and Physica-Verlag,Heidelberg, 1992, pp. 21–44.

[28] H.J. Zimmermann, Fuzzy Sets, Decision Making, and ExpertSystems, Kluwer-NijhoG Publishing, Boston, 1987.

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Comparison of fuzzy numbers using a fuzzy distance measure