Multiobjective optimization with linguistic
variables ∗
Christer Carlsson
IAMSR, Åbo Akademi University,
DataCity A 3210, SF-20520 Åbo, Finland
Robert Fullér †
Turku Centre for Computer Science,
DataCity A 4150, SF-20520 Åbo, Finland
Abstract
Generalizing our earlier results on optimization with linguistic variables
[3, 6, 7] we introduce a novel statement of fuzzy multiobjective mathematical programming problems and provide a method for findig a fair solution
to these problems. Suppose we are given a multiobjective mathematical programming problem in which the functional relationship between the decision variables and the objective functions is not completely known. Our
knowledge-base consists of a block of fuzzy if-then rules, where the antecedent part of the rules contains some linguistic values of the decision variables, and the consequence part consists of a linguistic value of the objective
functions. We suggest the use of Tsukamoto’s fuzzy reasoning method to determine the crisp functional relationship between the decision variables and
objective functions. We model the anding of the objective functions by a
t-norm and solve the resulting (usually nonlinear) programming problem to
find a fair optimal solution to the original fuzzy multiobjective problem.
Keywords: Tsukamoto’s fuzzy reasoning, multiobjective fuzzy optimization,
∗
in: Proceedings of the Sixth European Congress on Intelligent Techniques and Soft Computing
(EUFIT’98), Aachen, September 7-10, 1998, Verlag Mainz, Aachen, [ISBN 3-89653-500-5], Vol. II,
1998 1038-1042.
†
On leave from Department of Operations Research, Eötvös Loránd University, Muzeum körut
6-8, H-1088 Budapest, Hungary
1
1
Introduction
Fuzzy multiobjective optimization problems can be stated and solved in many different ways (for good surveys see [5, 9, 11, 12, 15]). Usually the authors consider
optimization problems of the form
max/min {G1 (x), . . . , GK (x)}; subject to x ∈ X,
where Gk , k = 1, . . . , K, or/and X are defind by fuzzy terms. Then they are
searching for a crisp x∗ which (in certain) sense maximizes the Gk ’s under the
(fuzzy) constraints X. For example, multiobjective fuzzy linear programming
(FMLP) problems can be stated as
max/min {c̃1 x, . . . , c̃K x}; subject to Ãx ≤ b̃,
(1)
where x ∈ IRn is the vector of crisp decision variables, Ã = (ãij ), b̃ = (b̃i ) and
c̃j = (c̃ij ) are fuzzy quantities, the inequality relation ≤ is given by a certain fuzzy
relation and the (implicite) X is a fuzzy set describing the concept ”x satisfies
Ãx ≤ b̃”.
In many important cases (e.g. in strategy formation processes) the values of the
objective functions are not known for all x ∈ IRn , and we are able to describe the
causal link between x and the Gk ’s linguistically using fuzzy if-then rules. In this
paper we consider a new statement of multiobjective fuzzy optimization problems
(FMOP), namely
max/minX {G1 , . . . , GK }; subject to {1 , . . . , m },
(2)
where x1 , . . . , xn are linguistic variables, and
i : if x1 is Ai1 and . . . and xn is Ain then G1 is Ci1 and . . . and GK is CiK ,
constitutes the only knowledge available about the values of G, and Aij and Cik
are fuzzy numbers.
In our earlier work [3] we interpreted FMLP problems (1) with fuzzy coefficients
and fuzzy inequality relations as multiple fuzzy reasoning schemes, where the antecedents of the scheme correspond to the constraints of the MFLP problem and
the facts of the scheme are the objectives of the MFLP problem. Generalizing the
fuzzy reasoning approach introduced in [6, 7] we determine the crisp value of the
Gj ’s at y ∈ X by Tsukamoto’s fuzzy reasoning method [13], and obtain an optimal
solution to (2) by solving the resulting (usually nonlinear) optimization problem
max/min t-norm (G1 (y), . . . , GK (y)); subject to y ∈ X.
We illustrate the proposed method by a simple example.
2
(3)
2
Multiobjective optimization under fuzzy if-then rules
Consider the FMOP problem (2) with continuous Aij representing the linguistic
values of xi , and with strictly monotone and continuous Cik , i = 1, . . . , m representing the linguistic values of Gk , k = 1, . . . , K. To find a fair solution to the
fuzzy optimization problem (2) we first determine the crisp value of the k-th objective function Gk at y ∈ IRn from the fuzzy rule-base using Tsukamoto’s fuzzy
reasoning method as
Gk (y) :=
−1
−1
(α1 ) + · · · + αm Cmk
(αm )
α1 C1k
α1 + · · · + αm
where
αi = t-norm(Ai1 (y1 ), . . . , Ain (yn ))
denotes the firing level of the i-th rule, i . To determine the firing level of the
rules, we suggest the use of the product t-norm (to have a smooth output function).
In this manner the constrained optimization problem (2) turns into the crisp (usually nonlinear) mathematical programmimg problem (3). The same principle is
applied to constrained maximization problems.
Example 1 Consider the optmization problem
max{G1 , G2 }
(4)
subject to y ∈ X = {(y1 , y2 ) ∈ [0, 1] × [0, 1] | y1 + y2 = 3/4}
where
1 : if x1 is small and x2 is small then G1 is small and G2 is big,
2 : if x1 is small and x2 is big then G1 is big and G2 is small,
and the universe of discourse for the linguistic values of G1 and G2 is also the unit
interval [0, 1] (see Figure 1). We will compute the firing levels of the rules by the
product t-norm. Let the membership functions in the rule-base be defined by
small(t) = 1 − t,
big(t) = t.
Let y1 , y2 ∈ [0, 1] be an input to the fuzzy system. Then the firing leveles of the
rules are
α1 = (1 − y1 )(1 − y2 ), α2 = (1 − y1 )y2 ,
It is clear that if y1 = 1 then no rule applies because α1 = α2 = 0. So we can
exclude the value y1 = 1 from the set of feasible solutions. The individual rule
outputs are
z11 = 1 − (1 − y1 )(1 − y2 ), z21 = (1 − y1 )y2 ,
3
x 1 is small
G1 is small
x2 is small
G2 is big
α1
1
x 1 is small
y1
1
z11
1
x2 is big
y2
1
1
G1 is big
product
α2
z21
1
z12
G2 is small
1
z22
Figure 1: Illustration of Example 1.
z12 = (1 − y1 )(1 − y2 ),
z22 = 1 − (1 − y1 )y2 ,
and, therefore, the overall system outputs are
G1 (y) =
(1 − y1 )(1 − y2 )(1 − (1 − y1 )(1 − y2 )) + (1 − y1 )y2 (1 − y1 )y2
=
(1 − y1 )(1 − y2 ) + (1 − y1 )y2
y1 + y2 − 2y1 y2 ,
G2 (y) =
(1 − y1 )(1 − y2 )(1 − y1 )(1 − y2 ) + (1 − y1 )y2 (1 − (1 − y1 )y2 )
=
(1 − y1 )(1 − y2 ) + (1 − y1 )y2
1 − (y1 + y2 − 2y1 y2 ).
Modeling the anding of the objective functions by the minimum t-norm our original
fuzzy problem (4) turns into the following crisp nonlinear mathematical programming problem
max min{y1 + y2 − 2y1 y2 , 1 − (y1 + y2 − 2y1 y2 )}
(5)
subject to y1 + y2 = 3/4, 0 ≤ y1 < 1, 0 ≤ y2 ≤ 1.
which has the optimal solutions (1/2, 1/4) and (1/4, 1/2), and its optimal value
is (1/2, 1/2).
Remark 2.1 We can introduce trade-offs among the objectives function by using
an OWA-operator in (5). However, as Yager has pointed out in [14], constrained
OWA-aggregations are not easy to solve, because the usually lead to a mixed integer mathematical programming problem of very big dimension.
4
1
The rules represent our knowledge-base for the fuzzy optimization problem. The
fuzzy partitions for lingusitic variables will not ususally satisfy ε-completeness,
normality and convexity. In many cases we have only a few (and contradictory)
rules. Therefore, we can not make any preselection procedure to remove the rules
which do not play any role in the optimization problem. All rules should be considered when we derive the crisp values of the objective function. We have chosen
Tsukamoto’s fuzzy reasoning scheme, because the individual rule outputs are crisp
numbers, and therefore, the functional relationship between the input vector y and
the system outputs (G1 (y), . . . , GK (y)) can be relatively easily identified (the only
thing we have to do is to perform inversion operations).
3
Summary
We have introduced a novel statement of multiobjective fuzzy mathematical programming problems and priveded a method for findig a fair solution to these problems. We addressed multiobjective mathematical programming problems in which
the functional relationship between the decision variables and the objective functions are not completely known. Our knowledge-base is assumed to consist only of
a block of fuzzy if-then rules, where the antecedent part of the rules contains some
linguistic values of the decision variables, and the consequence part consists of a
linguistic value of the objective functions. We suggested the use of Tsukamoto’s
fuzzy reasoning method to determine the crisp functional relationship between the
the decision variables and objective functions, and solve the resulting (usually nonlinear) programming problem to find a fair optimal solution to the original fuzzy
problem.
We can refine the fuzzy rule-base by introducing new lingusitic variables modeling
the linguistic dependencies between the variables and the objectives [1, 2, 4, 8, 10].
This will be the subject of our future research.
References
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