A neuro-fuzzy approach to FMOLP problems ∗
Robert Fullér †
Dept. of Computer Science, Eötvös Loránd University,
Muzeum krt. 6-8, H-1088 Budapest, Hungary
Silvio Giove
Dip. di Inform. e Studi Aziendali, University of Trento
via Inama 5-7, I-38100 Trento, Italy
Abstract
We propose the use of fuzzy neural networks for finding a good compromise solution to fuzzy multiple objective linear programs (FMOLP).
Key words: Fuzzy neural network, fuzzy multiple objective program
1
The statement of the problem
We consider fuzzy decision problems of the form
max (C1 x, . . . , Ck x)
x
(1)
where Ci = (Ci1 , . . . , Cin ) is a vector of fuzzy numbers and x ∈ Rn is the vector
of crisp decision variables.
Suppose that for each objective function of (1) we have two reference fuzzy numbers, denoted by mi and Mi , which represent undesired and desired levels for the
i-th objective, respectively.
We now can state (1) as follows: find an x∗ ∈ Rn such that Ci x∗ is as close as
possible to the desired point Mi , and it is as far as possible from the undisered
point mi for each i.
∗
in: Proceedings of CIFT’94, June 1-3, 1994, Trento, Italy, University of Trento, 1994 97-101.
Presently visiting professor at Dip. di Informatica e Studi Aziendali, University of Trento. Partially supported by the Hungarian National Scientific Research Fund OTKA under the contracts T
4281, T 7598, T 14144, 816/1991 and I/3-2152,
†
1
2
Fuzzy neural networks
Suppose that there is a fuzzy neural network with the set of input-output pairs
{(Ai , Bi ), i = 1, . . . , m}, where Ai = (Ai1 , . . . , Ain ) is a vector of fuzzy numbers and the output Bi is a fuzzy numbers. In computer applications we usually use
discrete versions of the continuous fuzzy sets. The discrete version of the above
system is
Inputs
Outputs
A1 (x1 ), . . . , A1 (xM )
B1 (y1 ), . . . , B1 (yN )
...
...
Am (x1 ), . . . , Am (xM )
Bm (y1 ), . . . , Bm (yN )
where (x1 , . . . , xM ) and (y1 , . . . , yN ) are well-chosen partitions of the input and
output spaces.
Another possibility is to input the α-level sets of fuzzy numbers [6]. We need to
find weights, such that for all input vectors Ai all the computed α-level sets are as
close as possible to the α-level sets of the target fuzzy numbers Bi . The number of
inputs and outputs depend on the number of α-level sets considered.
a11
a1M
A discretization of a fuzzy input Aij , where aij = Ai (xj ).
In the architecture [5] input vectors and target outputs were given by fuzzy numbers. Weights and biases, however, were given by real numbers as in the case of
standard backpropagation algorithm.
The following extension of backpropagation algorithm to train neural networks
from fuzzy training patterns was proposed by [5].
2
Οp
wi
1
k
i
wij
j
1
n
...........
Ap1
Apn
The architecture of fuzzy neural network proposed in [5].
The output of the i-th hidden unit is
Opi = f (
n
wij Apj )
j=1
For the output unit
k
Op = f (
wi Opi )
i=1
Let us denote the α-level sets of the computed output Op and the target output Bp
by
[Op ]α = [OpL (α), OpR (α)]
[Bp ]α = [BpL (α), BpR (α)]
The cost function to be minimized can be stated as follows
R
ep (α) := eL
p (α) + ep (α)
where
L
L
2
eL
p (α) = (Bp (α) − Op (α)) /2
R
R
2
eR
p (α) = (Bp (α) − Op (α)) /2
The cost function for the training pattern p is
ep =
α
3
αep (α)
From the cost function ep (α) the following learning rules can be derived
∆wi (t + 1) = Ωα(−∂ep (α)/∂wi + α∆wi (t)
∆wij (t + 1) = Ωα(−∂ep (α)/∂wij + α∆wij (t)
3
Fuzzy linear equations
Let us consider the system of fuzzy linear equations
A11 x1 + . . . + A1n xn = B1
...
...
(2)
Am1 x1 + . . . + Amn xn = Bm
where Aij and Bi are fuzzy numbers.
The problem is to find a crisp vector x ∈ Rn satisfying this system of equations as
far as possible. We use the following single layer fuzzy neural network for finding
an approximate solution to (2).
Οi
x2
x1
= ∑Aijx j ~ Bi
x n-1
xn
...
Ai1
Ain
Ai2
where the training set is {(Ai1 , . . . , Ain ; Bi ), i = 1, . . . , m} and the weights are
the coordinates of decision variable x.
It is clear that in the end of the learning we get an x satisfying bestly (in the sense
of closeness of α-level sets) the system of equations (2).
It is well-known that fuzzy expert systems and neural networks are universal approximators. However, as was pointed out by Buckley and Hayashi [1], fuzzy
neural networks can not approximate all continuous fuzzy functions, which means
that the extended backpropagation algorithm from [5] can not be always used in
the process of finding approximate solutions to systems of fuzzy equations.
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4
Application of fuzzy neural networks to FMOLP
Consider the following FMOLP
max {(C1 x, . . . , Ck x)|x ∈ Rn }
(3)
where Ci is a vector of fuzzy numbers i = 1, . . . , k.
Suppose that for each objective function of (3) we have two reference fuzzy numbers, denoted by mi and Mi , which represent undesired and desired levels for the
i-th objective, respectively.
Now we should find an x∗ ∈ Rn such that Ci x∗ is as close as possible to the
desired point Mi , and it is as far as possible from the undisered point mi for each
i.
Let di denote the maximal distance between the α-level sets of mi and Mi , and let
mi be the fuzzy number obtained by shifting mi by the value di in the direction of
Mi . Then we consider mi as the reference level for the biggest acceptable value
for the i-th objective function.
m1
M1
m '1 C1x*
C1 x∗ is too far from M1 .
It is clear that good compromise solutions should be searched between Mi and
mi , and we can introduce weigths measuring the importance of ”closeness” and
”farness”.
5
m1
C1x*
M1
C1 x∗ is close to M1 , but not far enough from m1 .
Let Ω ∈ [0, 1] be the grade of importance of ”closeness” to the disered level and
then (1 − Ω) denotes the importance of ”farness” from the undisered level.
Then we can use the following training set for our single layer fuzzy neural network
M1
Input
Output
C1
.....
Ck
ΩM1 + (1 − Ω)m1
.....
ΩMk + (1 − Ω)mk
ΩM1 + (1-Ω)m'1
C1x*
m'1
A good compromise solution
It is clear that in the end of training we get the optimal weights (decision variables),
for which the values of the fuzzy objectives are as close as possible to the desired
levels and and far from the undisered levels in the sense of the choosen importance
degrees.
6
References
[1] J.J.Buckley and Y.Hayashi, Can fuzzy neural nets approximate continuous fuzzy functions?, Fuzzy Sets and Systems, 6(1994) 43-51.
[2] T.Hashiyama, T.Furuhashi and Y.Uchikawa, A study on a multi-attribute
decision making process using a fuzzy neural network, In: Proceedings
of Fifth IFSA World Congress, 1993 810-813.
[3] T.Hashiyama, T.Furuhashi and Y.Uchikawa, A decison making model
using a fuzzy neural network, In: Proceedings of the 2nd International
Conference on Fuzzy Logic & Neural Networks, Iizuka, Japan, 1992
1057-1060.
[4] E.H.L.Aarts, J.Wessels and P.J.Zwietering, The applicability of neural
nets for decision support, In: Proceedings of First European Conference
on Fuzzy and Intelligent Technologies, Aachen, September 7-10, 1993,
Verlag der Augustinus Buchhandlung, Aachen, 1993 379-386.
[5] H.Ishibuchi, K.Kwon and H.Tanaka, Implementation of fuzzy if-then
rules by fuzzy neural networks with fuzzy weights, In: Proceedings
of First European Conference on Fuzzy and Intelligent Technologies,
Aachen, September 7-10, 1993, Verlag der Augustinus Buchhandlung,
Aachen, 1993 209-215.
[6] H.Ishibuchi, H.Tanaka and H.Okada, Interpolation of fuzzy if-then rules
by neural networks, Int. J. of Approximate Reasoning, 10(1994) 3-27.
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