On weighted possibilistic mean and variance offuzzy numbers ∗

Robert Fullerrfuller@abo.fi

Peter Majlenderpeter.majlender@abo.fi

Abstract

Dubois and Prade defined an interval-valued expectation of fuzzy num-bers, viewing them as consonant random sets. Carlsson and Fuller definedan interval-valued mean value of fuzzy numbers, viewing them as possibil-ity distributions. In this paper we shall introduce the notation of weightedinterval-valued possibilistic mean value of fuzzy numbers and investigate itsrelationship to the interval-valued probabilistic mean. We shall also intro-duce the notations of crisp weighted possibilistic mean value, variance andcovariance of fuzzy numbers, which are consistent with the extension princi-ple. Furthermore, we show that the weighted variance of linear combinationof fuzzy numbers can be computed in a similar manner as in probability the-ory.

1 Weighted possibilistic mean values

In 1987 Dubois and Prade [2] defined an interval-valued expectation of fuzzy num-bers, viewing them as consonant random sets. They also showed that this expecta-tion remains additive in the sense of addition of fuzzy numbers.

In this paper introducing a weighting function measuring the importance ofγ-level sets of fuzzy numbers we shall define the weighted lower possibilisticand upper possibilistic mean values, crisp possibilistic mean value and varianceof fuzzy numbers, which are consistent with the extension principle and with thewell-known defintions of expectation and variance in probability theory. We shallalso show that the weighted interval-valued possibilistic mean is always a subset(moreover a proper subset excluding some special cases) of the interval-valuedprobabilistic mean for any weighting function.

∗The final version of this paper appeared in: Fuzzy Sets and Systems, 139(2003) 297-312.

1

The theory developed in this paper is fully motivated by the principles intro-duced in [2, 4, 6] and by the possibilistic interpretation of the ordering introducedin [5].

A fuzzy number A is a fuzzy set of the real line R with a normal, fuzzy convexand continuous membership function of bounded support. The family of fuzzynumbers will be denoted by F . A γ-level set of a fuzzy number A is defined by[A]γ = {t ∈ R|A(t) ≥ γ} if γ > 0 and [A]γ = cl{t ∈ R|A(t) > 0} (the closureof the support of A) if γ = 0. It is well-known that if A is a fuzzy number then[A]γ is a compact subset of R for all γ ∈ [0, 1].

Definition 1.1 Let A ∈ F be fuzzy number with [A]γ = [a1(γ), a2(γ)], γ ∈ [0, 1].A function f : [0, 1] → R is said to be a weighting function if f is non-negative,monotone increasing and satisfies the following normalization condition∫ 1

0f(γ)dγ = 1. (1)

We define the f -weighted possibilistic mean (or expected) value of fuzzy number Aas

Mf (A) =∫ 1

0

a1(γ) + a2(γ)2

f(γ)dγ. (2)

It should be noted that if f(γ) = 2γ, γ ∈ [0, 1] then

Mf (A) =∫ 1

0

a1(γ) + a2(γ)2

2γdγ =∫ 1

0[a1(γ) + a2(γ)] γdγ = M(A).

That is the f -weighted possibilistic mean value defined by (2) can be considered asa generalization of possibilistic mean value introduced in [1]. From the definitionof a weighting function it can be seen that f(γ) might be zero for certain (unim-portant) γ-level sets of A. So by introducing different weighting functions we cangive different (case-dependent) importances to γ-levels sets of fuzzy numbers.

Remark 1.1 The f -weighted possibilistic mean value of fuzzy numberA coincideswith the value of A (with respect to reducing function f ) introduced by Delgado,Vila and Woxman in ([4], page 127). In fact, their paper has inspired us to intro-duce the notation of f -weighted possibilistic mean.

In the following we will consider two weighting functions f1 and f2 to be equalif the integral of their absolute difference |f1 − f2| is zero, that is∫ 1

0|f1(γ)− f2(γ)|dγ = 0.

2

Let us introduce a family of weighting function 1 : [0, 1]→ R defined as

1(γ) ={

1 if γ ∈ (0, 1]a if γ = 0

where a ∈ [0, 1] is an arbitrary real number. It is clear that 1 is a weighting functionand

M1(A) =∫ 1

0

a1(γ) + a2(γ)2

1(γ)dγ =∫ 1

0

a1(γ) + a2(γ)2

dγ. (3)

Remark 1.2 We note here that the 1-weighted possibilistic mean value definedby (3) coincides with the generative expectation of fuzzy numbers introduced byChanas and M. Nowakowski in ([3], page 47).

Definition 1.2 Let f be a weighting function and let A be a fuzzy number. Thenwe define the f -weighted interval-valued possibilistic mean of A as

Mf (A) = [M−f (A),M+f (A)],

where

M−f (A) =∫ 1

0a1(γ)f(γ)dγ =

∫ 10 a1(γ)f(γ)dγ∫ 1

0 f(γ)dγ

=

∫ 10 a1(γ)f(Pos[A ≤ a1(γ)])dγ∫ 1

0 f(Pos[A ≤ a1(γ)])dγ

=

∫ 10 min[A]γf(Pos[A ≤ a1(γ)])dγ∫ 1

0 f(Pos[A ≤ a1(γ)])dγ,

and

M+f (A) =

∫ 1

0a2(γ)f(γ)dγ =

∫ 10 a2(γ)f(γ)dγ∫ 1

0 f(γ)dγ

=

∫ 10 a2(γ)f(Pos[A ≥ a2(γ)])dγ∫ 1

0 f(Pos[A ≥ a2(γ)])dγ

=

∫ 10 min[A]γf(Pos[A ≥ a2(γ)])dγ∫ 1

0 f(Pos[A ≥ a2(γ)])dγ.

Here we used the relationships

Pos[A ≤ a1(γ)] = supu≤a1(γ)

A(u) = γ,

3

andPos[A ≥ a2(γ)] = sup

u≥a2(γ)A(u) = γ.

So M−f (A) is the f -weighted average of the minimum of the γ-cuts and it iswhy we call it the f -weighted lower possibilistic mean value of fuzzy number A.Similarly, M+

f (A) is the f -weighted average of the maximum of the γ-cuts and itis why we call it the f -weighted upper possibilistic mean value of fuzzy numberA.

The following two theorems can directly be proven using the definition of f -weighted interval-valued possibilistic mean.

Theorem 1.1 Let A,B ∈ F and let f be a weighting function, and let λ be a realnumber. Then

Mf (A+B) = Mf (A) +Mf (B), Mf (λA) = λMf (A).

Remark 1.3 The f -weighted possibilistic mean of A, defined by (2), is the arith-metic mean of its f -weighted lower and upper possibilistic mean values, i.e.

Mf (A) =Mf−(A) +M+

f (A)

2. (4)

Theorem 1.2 Let A and B be two fuzzy numbers, and let λ ∈ R. Then we have

Mf (A+B) = Mf (A) + Mf (B), Mf (λA) = λMf (A).

2 Relation to upper and lower probability mean values

In this Section we will show an important relationship between the interval-valuedexpectation E(A) = [E∗(A), E∗(A)] introduced by Dubois and Prade in [2] andthe f -weighted interval-valued possibilistic mean Mf (A) = [M−f (A),M+

f (A)]for any fuzzy number with strictly decreasing shape functions.

An LR-type fuzzy number A can be described with the following membershipfunction:

A(u) =

L

(q− − uα

)if q− − α ≤ u ≤ q−

1 if u ∈ [q−, q+]

R

(u− q+β

)if q+ ≤ u ≤ q+ + β

0 otherwise

4

where [q−, q+] is the peak of fuzzy number A; q− and q+ are the lower and up-per modal values; L,R : [0, 1] → [0, 1] with L(0) = R(0) = 1 and L(1) =R(1) = 0 are non-increasing, continuous functions. We will use the notation A =(q−, q+, α, β)LR. Hence, the closure of the support ofA is exactly [q−−α, q++β].If L and R are strictly decreasing functions then the γ-level sets of A can easily becomputed as

[A]γ = [q− − αL−1(γ), q+ + βR−1(γ)], γ ∈ [0, 1].

The lower and upper probability mean values of the fuzzy number A are computedby [2]

E∗(A) = q− − α∫ 1

0L(u)du, E∗(A) = q+ + β

∫ 1

0R(u)du. (5)

The f -weighted lower and upper possibilistic mean values are computed by

M−f (A) =∫ 1

0

(q− − αL−1(γ)

)f(γ)dγ

=∫ 1

0q−f(γ)dγ −

∫ 1

0αL−1(γ)f(γ)dγ

= q− − α∫ 1

0L−1(γ)f(γ)dγ,

M+f (A) =

∫ 1

0

(q+ + βR−1(γ)

)f(γ)dγ

=∫ 1

0q+f(γ)dγ +

∫ 1

0βR−1(γ)f(γ)dγ

= q+ + β

∫ 1

0R−1(γ)f(γ)dγ.

(6)

We can state the following theorem.

Theorem 2.1 Let f be a weighting function and let A be a fuzzy number of typeLR with strictly decreasing and continuous shape functions. Then, the f -weightedinterval-valued possibilistic mean value of A is a subset of the interval-valuedprobabilistic mean value, i.e.

Mf (A) ⊆ E(A).

Furthermore, Mf (A) is a proper subset of E(A) whenever f 6= 1.

5

Proof 1 see Fuzzy Sets and Systems, 139(2003) 297-312.

Example 2.1 Let f(γ) = (n + 1)γn and let A = (a, α, β) be a triangular fuzzynumber with center a, left-width α > 0 and right-width β > 0 then a γ-level of Ais computed by

[A]γ = [a− (1− γ)α, a+ (1− γ)β], ∀γ ∈ [0, 1].

Then the power-weighted lower and upper possibilistic mean values of A are com-puted by

M−f (A) =∫ 1

0[a− (1− γ)α](n+ 1)γndγ

= a(n+ 1)∫ 1

0γndγ − α(n+ 1)

∫ 1

0(1− γ)γndγ

= a−α

n+ 2,

and,

M+f (A) =

∫ 1

0[a+ (1− γ)β](n+ 1)γndγ

= a(n+ 1)∫ 1

0γndγ + β(n+ 1)

∫ 1

0(1− γ)γndγ

= a+β

n+ 2,

and therefore,

Mf (A) =

[a−

α

n+ 2, a+

β

n+ 2

].

That is,

Mf (A) =12

(a−

α

n+ 2+ a+

β

n+ 2

)= a+

β − α2(n+ 2)

.

So,

limn→∞

Mf (A) = limn→∞

(a+

β − α2(n+ 2)

)= a.

6

Example 2.2 LetA = (a, b, α, β) be a fuzzy number of trapezoidal form with peak[a, b], left-width α > 0 and right-width β > 0, and let f(γ) = (n + 1)γn, n ≥ 0.A γ-level of A is computed by

[A]γ = [a− (1− γ)α, b+ (1− γ)β], ∀γ ∈ [0, 1],

then the power-weighted lower and upper possibilistic mean values of A are com-puted by

M−f (A) =∫ 1

0[a− (1− γ)α](n+ 1)γndγ

= a(n+ 1)∫ 1

0γndγ − α(n+ 1)

∫ 1

0(1− γ)γndγ

= a−α

n+ 2,

and,

M+f (A) =

∫ 1

0[b+ (1− γ)β](n+ 1)γndγ

= b(n+ 1)∫ 1

0γndγ + β(n+ 1)

∫ 1

0(1− γ)γndγ

= b+β

n+ 2,

and therefore,

Mf (A) =

[a−

α

n+ 2, b+

β

n+ 2

]That is,

Mf (A) =12

(a−

α

n+ 2+ b+

β

n+ 2

)=a+ b

2+

β − α2(n+ 2)

.

So,

limn→∞

Mf (A) = limn→∞

(a+ b

2+

β − α2(n+ 2)

)=a+ b

2.

Example 2.3 Let f(γ) = (n+ 1)γn, n ≥ 0 and let A = (a, α, β) be a triangularfuzzy number with center a, left-width α > 0 and right-width β > 0 then

Mf (A) =[a−

α

n+ 2, a+

β

n+ 2

]⊂ E(A) =

[a−

α

2, a+

β

2

]7

and for n > 0 we have

Mf (A) = a+β − α

2(n+ 2)6= E(A) = a+

β − α4

.

Example 2.4 Let A = (a, b, α, β) be a fuzzy number of trapezoidal form and let

f(γ) = (n− 1)

(1

n√

1− γ− 1

),

where n ≥ 2. It is clear that f is a weighting function with f(0) = 0 and

limγ→1−0

f(γ) =∞.

Then the f -weighted lower and upper possibilistic mean values of A are computedby

M−f (A) =∫ 1

0[a− (1− γ)α](n− 1)

[1

n√

1− γ− 1

]dγ

= a−(n− 1)α2(2n− 1)

,

and

M+f (A) =

∫ 1

0[b+ (1− γ)β](n− 1)

[1

n√

1− γ− 1

]dγ

= b+(n− 1)β2(2n− 1)

,

and therefore

Mf (A) =

[a−

(n− 1)α2(2n− 1)

, b+(n− 1)β2(2n− 1)

].

That is,

Mf (A) =12

(a−

(n− 1)α2(2n− 1)

+ b+(n− 1)β2(2n− 1)

)=a+ b

2+

(n− 1)(β − α)4(2n− 1)

.

So,

limn→∞

Mf (A) = limn→∞

(a+ b

2+

(n− 1)(β − α)4(2n− 1)

)=a+ b

2+β − α

8.

8

Remark 2.1 When A is a symmetric fuzzy number then the equation Mf (A) =E(A) holds for any weighting function f . In the limit case, whenA = (a, b, 0, 0) isthe characteristic function of interval [a, b], the f -weighted possibilistic and prob-abilistic interval-valued means are equal, E(A) = Mf (A) = [a, b].

3 Weighted possibilistic variance

Definition 3.1 Let A and B be fuzzy numbers and let f be a weighting function.We define the f -weighted possibilistic variance of A by

Varf (A) =∫ 1

0

(a2(γ)− a1(γ)

2

)2

f(γ)dγ, (7)

and the f -weighted covariance of A and B is defined as

Covf (A,B) =∫ 1

0

a2(γ)− a1(γ)2

·b2(γ)− b1(γ)

2f(γ)dγ. (8)

It should be noted that if f(γ) = 2γ, γ ∈ [0, 1] then

Varf (A) =∫ 1

0

(a2(γ)− a1(γ)

2

)2

2γdγ

=12

∫ 1

0[a2(γ)− a1(γ)]2 γdγ = Var(A),

and

Covf (A,B) =∫ 1

0

a2(γ)− a1(γ)2

·b2(γ)− b1(γ)

2f(γ)dγ

=12

∫ 1

0(a2(γ)− a1(γ)) · (b2(γ)− b1(γ)) 2γdγ = Cov(A,B).

That is the f -weighted possibilistic variance and covariance defined by (7) and (8)can be considered as a generalization of possibilistic variance and covariance intro-duced in [1]. It can easily be verified that the weighted covariance is a symmetricalbilinear operator.

Example 3.1 Let A = (a, b, α, β) be a trapezoidal fuzzy number and let f(γ) =

9

(n+ 1)γn be a weighting function. Then,

Varf (A) = (n+ 1)∫ 1

0

[a2(γ)− a1(γ)

2

]2

γndγ

=n+ 1

4

∫ 1

0[(b− a) + (α+ β)(1− γ)]2 γndγ

=[b− a

2+

α+ β

2(n+ 2)

]2

+(n+ 1)(α+ β)2

4(n+ 2)2(n+ 3).

So,

limn→∞

Varf (A) = limn→∞

([b− a

2+

α+ β

2(n+ 2)

]2

+(n+ 1)(α+ β)2

4(n+ 2)2(n+ 3)

)=b− a

2.

The following theorem can be proven in a similar way as Theorem 4.1 from[1].

Theorem 3.1 Let f be a weighting function, let A, B and C be fuzzy numbers andlet x and y be real numbers. Then the following properties hold,

Covf (A,A) = Varf (A),Covf (A,B) = Covf (B,A),Varf (x) = 0,Covf (x, A) = 0,

Covf (xA+ yB,C) = |x|Covf (A,C) + |y|Covf (B,C),

Varf (xA+ yB) = x2Varf (A) + y2Varf (B) + 2|x||y|Covf (A,B),

where x and y denote the characteristic functions of the sets {x} and {y}, respec-tively.

Example 3.2 Let A = (a, b, α, β) and B = (a′, b′, α′, β′) be fuzzy numbers oftrapezoidal form. Let f(γ) = (n + 1)γn, n ≥ 0, be a weighting function then thepower-weighted covariance between A and B is computed by

Covf (A,B)

=∫ 1

0

(b− a) + (1− γ)(α+ β)2

·(b′ − a′) + (1− γ)(α′ + β′)

2(n+ 1)γndγ

=

[b− a

2+

α+ β

2(n+ 2)

][b′ − a′

2+

α′ + β′

2(n+ 2)

]+

(n+ 1)(α+ β)(α′ + β′)4(n+ 2)2(n+ 3)

.

10

So,

limn→∞

Covf (A,B) = limn→∞

([b− a

2+

α+ β

2(n+ 2)

][b′ − a′

2+

α′ + β′

2(n+ 2)

]

+(n+ 1)(α+ β)(α′ + β′)

4(n+ 2)2(n+ 3)

)=b− a

2·b′ − a′

2.

If a = b and a′ = b′, i.e. we have two triangular fuzzy numbers, then theircovariance becomes

Covf (A,B) =(α+ β)(α′ + β′)2(n+ 2)(n+ 3)

.

4 Summary

In this paper we have introduced the notations of the weighted lower possibilisticand upper possibilistic mean values, crisp possibilistic mean value and varianceof fuzzy numbers, which are consistent with the extension principle and with thewell-known defintions of expectation and variance in probability theory. We havealso showed that the weighted interval-valued possibilistic mean is always a subset(moreover a proper subset excluding some special cases) of the interval-valuedprobabilistic mean for any weighting function.

References

[1] C. Carlsson, R. Fuller, On possibilistic mean value and variance of fuzzynumbers, Fuzzy Sets and Systems, 122(2001) 315-326.

[2] D. Dubois, H. Prade, The mean value of a fuzzy number, Fuzzy Sets andSystems, 24(1987) 279-300.

[3] S. Chanas and M. Nowakowski, Single value simulationof fuzzy variable,Fuzzy Sets and Systems, 25(1988) 43-57.

[4] M. Delgado, M. A. Vila and W. Woxman, On a canonical reprsentation offuzzy numbers, Fuzzy Sets and Systems. 93(1998) 125-135.

[5] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets andSystems, 18(1986) 31-43.

[6] S. Heilpern, The expected value of a fuzzy number, Fuzzy Sets and Systems,47(1992) 81-86.

11

5 Citations

[A9] Robert Fuller and Peter Majlender, On weighted possibilistic mean andvariance of fuzzy numbers, FUZZY SETS AND SYSTEMS, 136(2003) 363-374.[MR1984582]

in journals

A9-c22 Martin O, Klir G J, Defuzzification as a special way of dealing with re-translation, International Journal of General Systems, 36: (6), pp. 683-701.2007

http://dx.doi.org/10.1080/03081070701456088

A9-c21 Zhang W -G, Wang Y -L, A comparative analysis of possibilistic vari-ances and covariances of fuzzy numbers Fundamenta Informaticae, 79: (1-2), pp. 257-263. 2007

A9-c20 Xinwang Liu and Hsinyi Lin, Parameterized approximation of fuzzy num-ber with minimum variance weighting functions, MATHEMATICAL ANDCOMPUTER MODELLING, 46 (2007) 1398-1409. 2007

http://dx.doi.org/10.1016/j.mcm.2007.01.011

Apart from the weighting function that put different emphasis onthe possible values of a fuzzy number, we can also put differentemphasis on the cut-level set of a fuzzy number. This methodwas proposed by Fuller and Majlender [A9]. The method wasextended to a more general form without the monotonic condi-tion in the weighting function definition by Liu recently [19], anda parameterized weighting function with maximum entropy wasproposed. (page 1406)

A9-c19 Thavaneswaran A, Thiagarajah K, Appadoo SS Fuzzy coefficient volatil-ity (FCV) models with applications MATHEMATICAL AND COMPUTERMODELLING, 45 (7-8): 777-786 APR 2007

http://dx.doi.org/10.1016/j.mcm.2006.07.019

A9-c18 Vercher E, Bermudez JD, Segura JV Fuzzy portfolio optimization underdownside risk measures FUZZY SETS AND SYSTEMS 158 (7): 769-782APR 1 2007

http://dx.doi.org/10.1016/j.fss.2006.10.026

12

Weighted mean values introduced in [A9] are a generalization ofthose possibilistic ones and allow us to incorporate the impor-tance of α-level sets. It shows that for LR-fuzzy numbers, any f-weighted interval-valued possibilistic mean value is a subset ofthe interval-valued mean of a fuzzy number in the sense of Duboisand Prade. (page 770)

A9-c17 Supian Sudradjat, The Weighted Possibilistic Mean Variance and Co-variance of Fuzzy Numbers, JOURNAL OF APPLIED QUANTITATIVEMETHODS, Volume 2, Issue 3, pp. 349-356, September 30, 2007

A9-c16 Thiagarajah, K., Appadoo, S.S., Thavaneswaran, A. Option valuationmodel with adaptive fuzzy numbers, COMPUTERS AND MATHEMATICSWITH APPLICATIONS, 53 (5), pp. 831-841. 2007

http://dx.doi.org/10.1016/j.camwa.2007.01.011

In the next section, in the line of Fuller and Majlender [A9] wediscuss weighted possibilistic moment of the adaptive fuzzy num-ber. (page 834)

Viewing the fuzzy numbers as random sets, Dubois and Prade [5]defined their interval-valued expectation. However, Carlsson andFuller [A14] defined a possibilistic interval-valued mean valueof fuzzy numbers by viewing them as possibility distributions.Furthermore, weighted possibilistic mean, variance and weightedinterval-valued possibilistic mean value of fuzzy numbers are allintroduced in Fuller and Majlender [A9].

...Furthermore, weighted possibilistic mean, variance and weightedinterval-valued possibilistic mean value of fuzzy numbers are allintroduced in Fuller and Majlender [A9]. (page 835)

A9-c15 Dubois D Possibility theory and statistical reasoning COMPUTATIONALSTATISTICS & DATA ANALYSIS, 51 (1): 47-69 NOV 1 2006

http://dx.doi.org/10.1016/j.csda.2006.04.015

Fuller and colleagues (Carlsson and Fuller, 2001; Fuller and Ma-jlender, 2003) consider introducing a weighting function on [0, 1]in order to account for unequal importance of cuts when comput-ing upper and lower expectations.

13

...The notion of variance has been extended to fuzzy random vari-ables (Koerner, 1997), but little work exists on the variance of afuzzy interval. Fuller and colleagues (Carlsson and Fuller, 2001;Fuller and Majlender, 2003) propose a definition as follows

V (M) =∫ 1

0

(supMγ − inf Mγ

2

)2

f(γ) dγ

(page 63)

A9-c14 Stefanini L, Sorini L, Guerra ML Parametric representation of fuzzy num-bers and application to fuzzy calculus FUZZY SETS AND SYSTEMS 157(18): 2423-2455 SEP 16 2006

http://dx.doi.org/10.1016/j.fss.2006.02.002

Other functions can be calculated numerically; examples are thecovariance and variance of fuzzy numbers u, v (see [3, A9]);(page 2437)

A9-c13 Matia F, Jimenez A, Al-Hadithi BM, et al. The fuzzy Kalman filter: Stateestimation using possibilistic techniques, FUZZY SETS AND SYSTEMS,157 (16): 2145-2170 AUG 16 2006

http://dx.doi.org/10.1016/j.fss.2006.05.003

A9-c12 Xinwang Liu, On the maximum entropy parameterized interval approxi-mation of fuzzy numbers, FUZZY SETS AND SYSTEMS, 157(2006) 869-878. 2006

http://dx.doi.org/10.1016/j.fss.2005.09.010

Fuller and Majlender [A9] further extended these results to thenotion of weighted possibilistic mean with the weighting functionmethod.

...Based on this interval aggregation idea, the definition of the weight-ing function in [A9] is extended without the monotonic increas-ing assumption. In fact, the increasing weighting function in [A9]can be seen as the weighted average of the level sets which placesmore emphasis on the level sets with higher membership grades.(page 870)

14

The property of the f-weighted interval-valued possibilistic meanis also extended directly:Theorem 1 (Fuller and Majlender [A9]). Let A,B ∈ F and let fbe a weighting function, and let µ be a real number, then,

Mf (A+B) = Mf (A) +Mf (B), Mf (µA) = µMf (A).

(page 871)

Regarding the weighting function as an aggregation operator forthe fuzzy number level sets, the paper extends the concept of aweighting function proposed by Fuller and Majlender to a generalform without the monotonic increasing assumption. (page 878)

A9-c11 Ayala G, Leon T, Zapater V, Different averages of a fuzzy set with anapplication to vessel segmentation, IEEE TRANSACTIONS ON FUZZYSYSTEMS, 13(3): 384-393 JUN 2005

http://dx.doi.org/10.1109/TFUZZ.2004.839667

A9-c10 Cheng CB, Fuzzy process control: construction of control charts withfuzzy numbers, FUZZY SETS AND SYSTEMS, 154 (2): 287-303 SEP 12005

http://dx.doi.org/10.1016/j.fss.2005.03.002

Fuller and Majlender [A9] further generalized the above defini-tion by introducing a weighting function, ω(γ), to measure theimportance of the γ-level set of F , and defined the ω-weightedpossibilistic variance of F as

...However, to allow the use of the given scores in this estimation,the present study re-defines the γ-weighted possibilistic varianceof a triangular fuzzy number based on the definitions of Carlssonand Fuller [A14] and Fuller and Majlender [A9],

...In Eq. (4), the mode of the fuzzy number, m, replaces the arith-metic mean of the γ-level set used in the definition of Carlssonand Fuller [A14]. When F is symmetric, the γ-weighted possi-bilistic variance defined in Eq. (4) equals to the variance definedby Fuller and Majlender [A9]. (pages 291-292)

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A9-c9 Bodjanova S, Median value and median interval of a fuzzy number, IN-FORMATION SCIENCES, 172 (1-2): 73-89 JUN 9 2005

http://dx.doi.org/10.1016/j.ins.2004.07.018

Weighted possibilistic mean, variance, and covariance of fuzzynumbers can be found in the work of Fuller and Majlender [A9].(page 74)

A9-c8 Hong DH, Kim KT, A note on weighted possibilistic mean, FUZZY SETSAND SYSTEMS, 148 (2): 333-335 DEC 1 2004

http://dx.doi.org/10.1016/j.fss.2004.04.011

in proceedings

A9-c7 Cheng, Jao-Hong Lee, Chen-Yu , Product Outsourcing under Uncertainty:an Application of Fuzzy Real Option Approach, IEEE International FuzzySystems Conference, 2007 (FUZZ-IEEE 2007), 23-26 July 2007, London,[ISBN: 1-4244-1210-2], pp. 1-6. 2007

http://dx.doi.org/10.1109/FUZZY.2007.4295675

A9-c6 Wang X, Xu WJ, Zhang WG, et al., Weighted possibilistic variance offuzzy number and its application in portfolio theory, in: Fuzzy Systemsand Knowledge Discovery, LECTURE NOTES IN ARTIFICIAL INTEL-LIGENCE, vol. 3613, pp. 148-155. 2005

http://dx.doi.org/10.1007/11539506 18

Abstract. Dubois and Prade defined an interval-valued expecta-tion of fuzzy numbers, viewing them as consonant random sets.Fuller and Majlender then proposed an weighted possibility meanvalue, variance and covariance of fuzzy numbers, viewing themas weighted possibility distributions. In this paper, we define anew weighted possibilistic variance and covariance of fuzzy num-bers based on Fuller and Majlenders’ notations.

...In 2003, Fuller and Majlender proposed an weighted possibilitymean value of fuzzy numbers, viewing them as weighted possi-bility distributions [A9]. (page 148)

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A9-c5 Zhang WG, Wang YL, Portfolio selection: Possibilistic mean-variancemodel and possibilistic efficient frontier, in: Megiddo, N.; Xu, Y.; Alonstioti,N.; Zhu, B. (Eds.) Algorithmic Applications in Management First Interna-tional Conference, AAIM 2005, Xian, China, June 22-25, 2005, ProceedingsSeries: Lecture Notes in Computer Science , Vol. 3521 Sublibrary: Infor-mation Systems and Applications, incl. Internet/Web, and HCI, Springer,[ISBN: 978-3-540-26224-4], 2005, pp. 203-213. 2005

http://dx.doi.org/10.1007/11496199 23

Fuller and Majlender [A9] defined the weighted possibilistic meanvalue of A as

Mf (A) =∫ 1

0f(γ)

a(γ) + b(γ)2

dγ =MLf (A) +MU

f (A)2

.

(page 204)

A9-c4 Zhang, J.-P., Li, S.-M. Portfolio selection with quadratic utility functionunder fuzzy enviornment 2005 International Conference on Machine Learn-ing and Cybernetics, ICMLC 2005, pp. 2529-2533. 2005

http://dx.doi.org/10.1109/ICMLC.2005.1527369

A9-c3 Blankenburg, B., Klusch, M. BSCA-F: Efficient fuzzy valued stable coali-tion forming among agents Proceedings - 2005 IEEE/WIC/ACM Interna-tional Conference on Intelligent Agent Technology, IAT’05, 2005, art. no.1565632, pp. 732-738. 2005

http://dx.doi.org/10.1109/IAT.2005.48

A9-c2 Garcia, F.A.A. Fuzzy real option valuation in a power station reengineeringproject, Soft Computing with Industrial Applications - Proceedings of theSixth Biannual World Automation Congress, pp. 281-287. 2004

http://ieeexplore.ieee.org/iel5/9827/30981/01439379.pdf?

A9-c1 Wang, X., Xu, W.-J., Zhang, W.-G. A class of weighted possibilistic mean-variance portfolio selection problems Proceedings of 2004 International Con-ference on Machine Learning and Cybernetics, 4, pp. 2036-2040. 2004

http://ieeexplore.ieee.org/iel5/9459/30104/01382130.pdf?arnumber=1382130

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