expect value of fuzzy variable and fuzzy expected value model

Expected Value of Fuzzy Variable and Fuzzy Expected Value Models

Baoding Liu,Senior Member,IEEE,and Yian-Kui Liu

Abstract—This paper will present a novel concept of expected values of fuzzy variables,which is essentially a type of Choquet integral and coincides with that of random variables.In order to calculate the expected value of general fuzzy variable,a fuzzy sim-ulation technique is also designed.Finally,we construct a spectrum of fuzzy expected value models,and integrate fuzzy simulation, neural network,and genetic algorithms to produce a hybrid intel-ligent algorithm for solving general fuzzy expected value models. Index Terms—Expected value,fuzzy programming,fuzzy simu-lation,genetic algorithm,neural network.

I.I NTRODUCTION

S INCE Zadeh’s pioneering work[24],possibility theory was being perfected and became a strong tool to deal with in-complete and uncertain situation[2],[5],[19],[21].On the other hand,many researchers such as Zimmermann[25],Luhand-jula[11],[12],Yazenin[22],[23],Sakawa[17],Inuiguchi and Ramík[4],Tanaka et al.[20]applied the theory successfully to optimization problems.

As the development of more effective computer and the appearance of new algorithms such as genetic algorithm, simulated annealing and neural networks,many complex optimization problems can be solved by computers.Recently, Liu[8]laid a foundation for optimization theory in uncertain (stochastic,fuzzy,fuzzy random,etc.)environments and called such a theory uncertain programming,in which numerous models and hybrid intelligent algorithms are documented. Especially,there are two known classes of fuzzy programming, one is the fuzzy chance-constrained programming[6],[7],the other is fuzzy dependent-chance programming[9],[10].In this paper,we will present a novel definition of expected value of fuzzy variable and propose a new class of fuzzy programming called fuzzy expected value models.The interested readers may also consult the related work[3],where fuzzy intervals were viewed as consonant random sets,and the mean value of a fuzzy number is thus defined as an interval based on Dempster and Shafer(D–S)Theory.

The paper is organized as follows.Section II introduces the definition of expected value of fuzzy variable and presents some basic properties of expected value operator.Three types of fuzzy expected value models are formulated in Section III and a con-vexity theorem is also proved in this section.Finally,we will

Manuscript received April30,2001;revised November7,2001and December26,2001.This work was supported by National Natural Science Foundation of China Grant69804006,and the Sino-French Joint Laboratory for Research in Computer Science,Control and Applied Mathematics(LIAMA). The authors are with the Department of Mathematical Sciences,Tsinghua University,Beijing100084,China(e-mail:liu@https://www.360docs.net/doc/2918535337.html,).

Publisher Item Identifier10.1109/TFUZZ.2002.800692.design a hybrid intelligent algorithm to solve general expected value models and provide three numerical examples to illustrate the effectiveness of the algorithm.

II.E XPECTED V ALUE O PERATOR

Let

is said to be normal if there exits a real

number.In this paper,we always assume that fuzzy variables involved are normal.

Let

(1) while the necessity

of is defined

by

and show the possibility and necessity degrees to what

extent

by,

then

and are two particular fuzzy measures(see[16]) defined

on.In

addition,

and are a pair of dual fuzzy measures in the sense

that

is the complement

of

(3) for

any

implies

for

any

is a triangular fuzzy variable

,we can

calculate

Before we present the expected value of a fuzzy variable, we first recall the definition of Choquet integral.This kind of integral was first introduced in[1],and later was restudied in the field of fuzzy measure theory by some researchers such as Murofushi and Sugeno[13],[14],Murofushi,Sugeno and Machida[15],and Narukawa,Murofushi and Sugeno[16]. Let

is defined

as

is a finite fuzzy measure,then the Choquet integral of a

measurable function with respect

to

,in

the sense

that.

From the measure-theoretic interpretation of Choquet inte-

gral,it is usually regarded as the generalization of usual math-

ematical expectation.Therefore,motivated by the idea of Cho-

quet integral,we present the following definition.

Definition1:

Let

is defined

by

is defined

by

(6)

The expected value

of

(7)

When the right-hand side of(7)is of

form,the expected

value is not defined.

Remark1:If the fuzzy

variable

)

and

.This

means that the representation of expected value of fuzzy vari-

able is identical to that of random variable.

Remark2:Just like the case of random variable(for ex-

ample,Cauchy distributed variable),the expected value does not

exist for some fuzzy variable.One of the referees of this paper

provided the

example

.

Let

if

if

if

.

It follows from(7)that the expected value

of

and satisfy the fol-

lowing

constraints:

is a normalized fuzzy variable.

According to Definition1,for triangular,trapezoidal,and

normal fuzzy variables,we have the following results.

Example2:

If

is is

a trapezoidal fuzzy

variable,then the expected

value

of is a normal fuzzy

variable with the possibility distribution

function

then the expected value

of

and

and

for

any

is a fuzzy vector,

then

and

has the following properties:

i)

if,

then