IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 4, AUGUST 2002
445
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 simulation 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 intelligent algorithm for solving general fuzzy expected value models.
Index Terms—Expected value, fuzzy programming, fuzzy simulation, genetic algorithm, neural network.
I. INTRODUCTION
design a hybrid intelligent algorithm to solve general expected
value models and provide three numerical examples to illustrate
the effectiveness of the algorithm.
II. EXPECTED VALUE OPERATOR
Let be a fuzzy variable with possibility distribution function
. A fuzzy variable is said to be normal if there
. In this paper, we
exits a real number such that
always assume that fuzzy variables involved are normal.
Let be a real number. It is well known that the possibility of
is defined by
S
INCE Zadeh’s pioneering work [24], possibility theory was
being perfected and became a strong tool to deal with incomplete and uncertain situation [2], [5], [19], [21]. On the other
hand, many researchers such as Zimmermann [25], Luhandjula [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 convexity theorem is also proved in this section. Finally, we will
(1)
while the necessity of
is defined by
(2)
and
show the possibility and necessity
degrees to what extent is not smaller than .
,
If we denote the support of by
and
are two particular fuzzy measures (see [16])
then
, where
is the power set of . In addition,
defined on
and
are a pair of dual fuzzy measures in the sense that
with
is the complement of .
Furthermore, based on possibility measure and necessity
measure, we give the third index, called credibility measure, as
follows:
(3)
. It is easy to check that satisfies the following
for any
conditions:
, and
;
i)
implies
for any
.
ii)
is also a fuzzy measure defined on
. BeThus,
is self dual, i.e.,
for any
sides,
.
Example 1: Suppose that is a triangular fuzzy variable
, for any real number , we can calculate
as follows:
Manuscript received April 30, 2001; revised November 7, 2001 and
December 26, 2001. This work was supported by National Natural Science
Foundation of China Grant 69804006, 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, Beijing 100084, China (e-mail: [email protected]).
Publisher Item Identifier 10.1109/TFUZZ.2002.800692.
1063-6706/02$17.00 © 2002 IEEE
.
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 4, AUGUST 2002
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].
be a fuzzy measure space. The Choquet inLet
tegral of a nonnegative measurable function with respect to a
fuzzy measure is defined as
If is a finite fuzzy measure, then the Choquet integral of a
measurable function with respect to is defined as
Remark 2: Just like the case of random variable (for example, Cauchy distributed variable), the expected value does not
exist for some fuzzy variable. One of the referees of this paper
provided the example
if
if
whose expected value does not exist because it is of the form
.
Let be a normalized discrete fuzzy variable whose possibility distribution function is defined by
if
(4)
if
,
, and
is the dual of in
where
.
the sense that
From the measure-theoretic interpretation of Choquet integral, it is usually regarded as the generalization of usual mathematical expectation. Therefore, motivated by the idea of Choquet integral, we present the following definition.
Definition 1: Let be a normalized fuzzy variable. The upper
, of is defined by
expected value,
if
We assume without loss of generality that
It follows from (7) that the expected value of
.
is:
(8)
where the weights are given by
(5)
while the lower expected value,
, of
is defined by
for
(6)
The expected value of
is defined as
and satisfy the fol-
lowing constraints:
and
(7)
, the expected
When the right-hand side of (7) is of form
value is not defined.
Remark 1: If the fuzzy variable is replaced with a random
is replaced with
variable (whose density function is ) and
(whose dual is itself), then we have
the probability measure
which is exactly the expected value of random variable . This
means that the representation of expected value of fuzzy variable is identical to that of random variable.
since is a normalized fuzzy variable.
According to Definition 1, for triangular, trapezoidal, and
normal fuzzy variables, we have the following results.
,
Example 2: If is a triangular fuzzy variable
. If is
then the expected value of is
, then the expected
a trapezoidal fuzzy variable
. If is a normal fuzzy
value of is
variable with the possibility distribution function
then the expected value of is exactly .
and
be real-valued functions defined on
,
Let
we say that and are comonotonic if
for any
(see [16]). In addition, if
is a fuzzy vector, then
and
are also fuzzy variables. The following proposition gives some
properties about their expected values.
Theorem 1: Let be a fuzzy vector. The expected value operator has the following properties:
, then
;
i) if
;
ii)
LIU AND LIU: EXPECTED VALUE OF FUZZY VARIABLE AND FUZZY EXPECTED VALUE MODELS
iii) if functions
and
are comonotonic, then for
and , we have
any nonnegative real numbers
.
Proof: Applying [16, Th. 2.4] to expected values of fuzzy
and
, assertions i) and iii) can be proved.
variables
Next, we prove assertion ii). By Definition 1, we have
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satisfy as many goals as possible in the order specified. In order
to balance the multiple conflicting objectives, we may employ
the following fuzzy expected value goal programming model:
subject to:
which complete the proof of assertion ii).
Remark 3: For detailed expositions about Chouqet integral
and its properties used but not provided in this paper, the readers
may consult [13]–[16], and [18].
B. Convexity
III. EXPECTED VALUE MODELS
A. General Formulations
There are several possibilities to construct fuzzy programming models, for example, in [6], [7], [9], and [10], fuzzy
chance-constrained programming and fuzzy dependent-chance
programming have been developed. In this section, we will
open a new door to formulate fuzzy decision problems, called
fuzzy expected value models.
In order to obtain the decision with maximum expected return, we may employ the following single-objective fuzzy expected value model:
subject to:
(9)
where and are decision vector and fuzzy vector, respectively,
is the objective function, and
are the constraint functions
.
for
In many applications, a decision maker may want to optimize
multiple objectives. A fuzzy expected value multiobjective programming model may be formulated as follows:
subject to:
A mathematical programming is called convex if it has both
convex objective function and convex feasible set. For the fuzzy
expected value model, if we add some conditions to objective
function and constraint functions, then we have the following
result on convexity.
be an -ary fuzzy vector. Suppose
Theorem 2: Let
and
that, for any fixed , the functions
are convex with respect to , and for any
and , the functions
and
[resp.
given
and
for
] are comonotonic
with respect to . Then the following fuzzy expected value
model:
(12)
subject to:
is a convex programming problem.
Proof: By supposition of the theorem, for any fixed , the
following inequality:
is valid for any
and
. By Theorem 1, we have
(10)
are objective functions for
and
are constraint functions for
.
In multiobjective decision-making problems, the decisionmaker may assign a target level for each goal and the key idea is
to minimize the deviations (positive, negative, or both) from the
target levels. In the real-world situation, the goals are achievable only at the expense of other goals and these goals are usually incompatible. Therefore, there is a need to establish a hierarchy of importance among these incompatible goals so as to
where
(11)
is the preemptive priority factor which expresses the
where
for all ;
relative importance of various goals and
(resp.
) is the weighting factor corresponding to positive
(resp. negative) deviation for goal with priority assigned;
and are the goal constraints and real constraints, respectively,
and
;
and
are the
for
positive deviations and negative deviations from the target of
, respectively.
goal for
which implies that the objective function is convex.
Now let and be any two feasible solutions. By the convexity of , we have
for any given
and
. It follows from Theorem 1 that:
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 4, AUGUST 2002
which implies that
is a feasible solution. Hence,
the feasible set is convex. The proof of the theorem is complete.
IV. HYBRID INTELLIGENT ALGORITHM
In order to solve general fuzzy expected value models, the
crux is to calculate uncertain functions such as
(13)
for any given decision vector .
can be expressed as linear form
If the fuzzy variable
and all s are noninteractive triangular
is
(resp. trapezoidal, or normal) fuzzy variables, then
also a triangular (resp. trapezoidal, or normal) fuzzy variable.
can be calculated
For this case, the expected value
easily.
In most cases, we cannot do so due to the complexity of the
function . In the following, we suggest a fuzzy simulation to
.
estimate the value of
we can estimate the expected value (13) by the formula (14)
provided that is sufficiently large.
According to (14), for any given decision vector , we can
by the following algoevaluate the expected value
rithm.
Step 1. Sample
points
uniformly
for
.
from the -cut
Step 2. Calculate the values
for
.
Step 3. Rearrange the subscript
of
and
such that
.
Step 4. Calculate the weights
according to formula (15) for
Step 5. Calculate the expected value
according to formula (14).
and
.
B. Uncertain Function Approximation
if
Thus far, for any given decision vector , we can use fuzzy
in
simulation to compute the uncertain function value
(13). However, it is clearly a time-consuming process.
It is known that an NN has the ability to approximate continuous function and the high speed of operation, thus we wish to
train a feedforward NN to approximate the uncertain function
in order to speed up the solution process.
In this paper, we will train the feedforward NN by the popular
backpropagation algorithm.
if
C. Hybrid Intelligent Algorithm
A. Fuzzy Simulation
In order to obtain the expected value of a fuzzy variable,
we design a fuzzy simulation for both discrete and continuous
cases.
Case I: Assume that is a function, and
is a discrete fuzzy vector whose joint possibility distribution
function is defined by
if
where
and
with
are the possibility distribution function of for
.
for
. We assume without
Let
(otherwise we may
loss of generality that
rearrange them to satisfy the condition), then the expected value
is given by
(14)
where
(15)
for
.
Case II: Assume that is a continuous fuzzy vector with a
possibility distribution function .
Let be a sufficiently small positive number, and the -cut
is a bounded subset
of . Also, we assume in this paper that
. Thus, we can sample
points uniformly from the set
of
and denote them by
.
and
for
Assume that
such that
(otherwise we rearrange these numbers to satisfy the condition), then
Generally speaking, fuzzy expected value models are neither
convex nor unimodal. Traditional algorithms are not applicable
to such problems, for example, we cannot obtain the derivative of objective function due to the fact that the expected value
is estimated by the fuzzy simulation. Thus, we have to apply
heuristic algorithms to solving general fuzzy expected value
models. In this paper, we integrate fuzzy simulations, NN and a
genetic algorithm (GA) to produce a powerful hybrid intelligent
algorithm. The procedure to solve general fuzzy expected value
models is summarized as follows.
Step 1) Generate training input–output data for uncertain functions by fuzzy simulations.
Step 2) Train an NN to approximate the uncertain functions by
the generated training data.
Step 3) Initialize pop_size chromosomes in which the trained
NN can be used to calculate the values of uncertain
functions.
Step 4) Update the chromosomes by crossover and mutation
operations and the trained NN may be employed to
check the feasibility of offsprings.
Step 5) Calculate the objective values for all chromosomes by
the trained NN.
Step 6) Compute the fitness of each chromosome by
rank-based evaluation function based on the objective values.
Step 7) Select the chromosomes by spinning the roulette
wheel.
LIU AND LIU: EXPECTED VALUE OF FUZZY VARIABLE AND FUZZY EXPECTED VALUE MODELS
Step 8) Repeat the fourth to seventh steps a given number of
cycles.
Step 9) Report the best chromosome as the optimal solution.
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A run of the hybrid intelligent algorithm (6000 sample points
in simulation, 3000 data in NN, 1000 generations in GA) shows
that the optimal solution is
V. NUMERICAL EXPERIMENTS
In this section, we will give some numerical examples to illustrate the procedure of solving fuzzy expected value models
by the hybrid intelligent algorithm.
Example 1: Consider first the following fuzzy expected
value model:
whose objective value is 3.4496.
Example 3: Consider the following fuzzy expected value
goal programming model:
lexmin
subject to:
subject to:
where , and are triangular fuzzy variables ( 4, 2, 0),
( 2, 0, 2), and (0, 2, 4), respectively.
In order to solve this model, as previously discussed, we first
generate input–output data for the uncertain function
by fuzzy simulation. Then we train an NN (three input neurons
representing decision variables, five hidden neurons, one output
neuron representing objective function) to approximate the uncertain function . Lastly, the trained NN is embedded into a
GA to produce a hybrid intelligent algorithm.
A run of the hybrid intelligent algorithm (6000 sample points
in simulation, 2000 data in NN, 400 generations in GA) shows
that the optimal solution (here the optimal solution is, in fact, a
satisfactory solution) is
where
and
are triangular fuzzy variables ( 3, 2, 1)
and (1, 2, 3), respectively, and , are normal fuzzy variables
whose possibility distributions are given as follows:
and
In this problem, we first generate input–output data for the
, where
uncertain function :
whose objective value is 3.0103.
Example 2: We now consider another fuzzy expected value
model
subject to:
by fuzzy simulation. Then for each , the values of objective
functions are calculated as follows:
and
where
is a triangular fuzzy variable (1, 2, 3),
is a fuzzy
,
variable with possibility distribution
and is a trapezoidal fuzzy variable (2, 3, 4, 8).
In order to solve the model, we first employ the fuzzy simulation technique to generate input–output data for the uncertain
, where
function :
Then we use the training data to train an NN (three input neurons, eight hidden neurons, two output neurons) to approximate
the uncertain function . Finally, the trained NN is embedded
into a GA to produce a hybrid intelligent algorithm.
Using the training data, we train an NN (four input neurons,
ten hidden neurons, three output neurons) to approximate the
uncertain function . After that, we embed the trained NN into
a GA to produce a hybrid intelligent algorithm.
A run of the hybrid intelligent algorithm (5000 sample points
in simulation, 2000 data in NN, 3000 generations in GA) shows
that the optimal solution is
which satisfies the first two goals, but the third objective value
is 0.8309.
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 4, AUGUST 2002
VI. CONCLUSION
In this paper, we contributed to the research area of fuzzy
optimization in the following four aspects: 1) we presented a
new concept of expected value operator of fuzzy variable; 2) we
designed a fuzzy simulation to estimate the expected value; 3)
we constructed a new class of fuzzy programming—fuzzy expected value models—in addition to fuzzy chance-constrained
programming and fuzzy dependent-chance programming; and
4) we integrated fuzzy simulation, neural networks, and genetic
algorithm to produce a hybrid intelligent algorithm for solving
the fuzzy expected value models.
ACKNOWLEDGMENT
The authors would like to thank the anonymous referees for
their valuable comments and suggestions.
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Baoding Liu (M’99–SM’00) graduated from the Department of Mathematics, Nankai University, Tianjin,
China, and received the M.S. and Ph.D. degrees from
the Institute of Systems Science, Chinese Academy
of Sciences, Beijing, China, in 1986, 1989, and 1993,
respectively.
He has been a Full Professor at the Department
of Mathematical Sciences, Tsinghua University,
Beijing, China, since 1998. His current research
interests include stochastic programming, fuzzy programming, uncertain systems, intelligent systems,
and applications in inventory, scheduling, reliability, project management,
and engineering design. He is the author of Uncertain Programming (New
York:Wiley, 1999), Decision Criteria and Optimal Inventory Processes
(Boston, MA: Kluwer, 1999), and Stochastic Programming and Fuzzy Programming (Beijing, China: Tsinghua Univ. Press, 1998). He has published over
60 papers in international conferences and premier journals. He is an Editorial
Board member of the international journal Information, and Associate Editor
of Fuzzy Optimization and Decision Making.
Dr. Liu is currently serving as Associate Editor of the IEEE TRANSACTIONS
ON FUZZY SYSTEMS.
Yian-Kui Liu received the B.S. and M.S. degrees
from the Department of Mathematics, Hebei University, Baoding, China, in 1989 and 1992, respectively.
Since 1992, he has been with the College of
Mathematics and Computer, Hebei University,
where he is an Associate Professor. He is currently
a Research Fellow at the Uncertain Systems
Laboratory, Department of Mathematical Sciences,
Tsinghua University, Beijing, China. His research
interests, previously within the areas of nonadditive
measure theory and multivalued analysis, have
extended to include theory of optimization under uncertainty and intelligent
systems. He has published more than 20 papers in national and international
conferences and premier journals.