Optimization with linguistic variables ∗
Christer Carlsson
christer.carlsson@abo.fi
Robert Fullér
rfuller@abo.fi
Abstract
We consider fuzzy mathematical programming problems (FMP) in which the
functional relationship between the decision variables and the objective function is
not completely known. Our knowledge-base is supposed to 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 is either a linguistic
value of the objective function or a linear combination of the crisp values of the
decision variables. In this paper we suggest the use of an adequate fuzzy reasoning method to determine the crisp functional relationship between the objective
function and the decision variables, and to solve the resulting (usually nonlinear)
programming problem to find a fair optimal solution to the original fuzzy problem.
Furthermore, we illustrate how the optimal solution may change if we are able to
refine the rule base by introducing some non-monotonicity (dependency) rules.
1
Introduction
After Bellman and Zadeh [1], and then Zimmermann[14] introduced fuzzy sets into
methods for handling optimization problems, they cleared the way for a new family
of methods to deal with problems which had been inaccessible to and not solvable
with standard mathematical programming (MP) techniques. These problems include a
number of limitations and simplifications, which have become apparent, as the use of
MP models became widespread and common.
The parameters used in MP models need to be valid and accurate, and the estimates
used to define them should be built from reliable data sets, which are extensive enough
to allow repeated verification and validation. This is not often the case as data may be
spotty and incomplete, the data sets are limited and quite often ad hoc, and the validation procedures need to be simplified out of necessity. Nevertheless, MP models are
used with the simplifying assumption, that the parameter estimates are good enough
for the actual problem - or for the time being (which includes a promise of later adjustments and modifications). The end-result appears to be that optimal solutions found
with MP models do not fare too well in the long run, nor do they survive for very long.
We need methods to deal with the problems of poor, incomplete data head-on.
∗ The final version of this paper appeared in: C. Carlsson and R. Fullér, Optimization with linguistic
variables, in: J.L.Verdegay ed., Fuzzy Sets based Heuristics for Optimization , Studies in Fuzziness and Soft
Computing. Vol.126, Springer Verlag, 2003 113-121.
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The traditional use of one-objective MP models is in many cases an over-simplification as the intentions formulated in decision problems cannot be adequately dealt
with without using multiple objectives. It is true that MP models with multiple objectives become complex and difficult to handle and work with, which is why they are
avoided. There is even more to multiple objective models: in the standard case we
assume that the objectives are independent, which has given us a set of standard tools
for finding optimal solutions in cases with multiple objectives. In practical cases, we
normally have interdependence among the objectives: some of the objectives may be
conflicting, other may be supportive; in cases with greater numbers of objectives, we
may have complex combinations of interdependencies [2, 7, 3]. Model builders shy
away from these more complex issues as the methods for finding identifiable optimal
solutions have been rather few and, in some cases, have not yet been developed. Thus,
we need methods for handling interdependence issues in multiple objective MP models.
In practical problem solving work, and in handling decision problems in the business world, it has become evident for us that probably a majority of the problem solving
and decision processes is not of the type, which can be resolved with the precise MPtype models and methods. These practical situations have actors, who are vague about
objectives and constraints in the beginning of a process, and then they become more
focused and precise as the process is progressing towards some results. This appears to
be more of a search-learning process than a systematic process of analysis progressing
in well-defined steps towards a predefined goal. Then, for the early stages of the searchlearning process we need tools, models and methods, which allow for knowledge-rich
imprecision but which have an inner core of methodological structure, such that we
in later stages of the process can become analytically more precise as objectives and
constraints become more focused. Nevertheless, we want to keep the knowledge-rich
substance of the models and methods as we progress to more MP-like modelling. This
has long been considered a methodological contradiction, but it has become apparent
in recent years [5] that this standard objection is not necessarily true. A combination of
linguistic variables and fuzzy logic is emerging as a good approach to have knowledgerich imprecision, a systematic and firm methodological structure, and effective and fast
analytical MP-algorithms.
In this paper, we explore the conditions for optimisation with linguistic variables.
Fuzzy sets were introduced by Zadeh[12] as a means of representing and manipulating data that was not precise, but rather fuzzy. Let X be a nonempty set. A fuzzy set
A in X is characterized by its membership function
µA : X → [0, 1]
and µA (x) is interpreted as the degree of membership of element x in fuzzy set A for
each x ∈ X. Frequently we will write simply A(x) instead of µA (x). The family of all
fuzzy (sub)sets in X is denoted by F(X). A fuzzy set A in X is called a fuzzy point if
there exists a u ∈ X such that A(t) = 1 if t = u and A(t) = 0 otherwise. We will use
the notation A = ū.
The use of fuzzy sets provides a basis for a systematic way for the manipulation
of vague and imprecise concepts. In particular, we can employ fuzzy sets to represent
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linguistic variables. A linguistic variable [13] can be regarded either as a variable
whose value is a fuzzy number or as a variable whose values are defined in linguistic
terms. If x is a linguistic variable in the universe of discourse X and u ∈ X then we
simple write ”x = u” or ”x is ū” to indicate that u is a crisp value of x.
Fuzzy optimization problems can be stated in many different ways [8]. Authors
usually consider fuzzy optimization problems of the form
max/min f (x); subject to x ∈ X,
where f or/and X are defind in fuzzy terms. Then they are searching for a crisp x∗
which (in a certain) sense maximizes f on X. For example, fuzzy linear programming
(FLP) problems can be stated as
max/min f (x) = c̃x; subject to Ãx . b̃,
(1)
where the fuzzy terms are denoted by tilde.
Unlike in (1) the fuzzy value of the objective function f (x) may not be known
for any x ∈ Rn . More often than not we are only able to describe the partial causal
link between x and f (x) linguistically using some fuzzy if-then rules. In [6] we have
considered constrained fuzzy optimization problems of the form
max/min f (x); subject to {<1 (x), . . . , <m (x) | x ∈ X ⊂ Rn },
(2)
<i (x) : if x1 is Ai1 and . . . and xn is Ain then f (x) is Ci ,
(3)
with
where Aij and Ci are fuzzy numbers with strictly monotone membership functions;
and we have suggested the use of the Tsukamoto fuzzy reasoning method [11] to determine the crisp values of f . In [4] we assumed that the knowledge base is given in
the form
Pn
<i (x) : if x1 is Ai1 and . . . and xn is Ain then f (x) = j=1 aij xj + bi
(4)
where Aij is a fuzzy number, and aij and bi are real numbers. Crisp values of f have
been determined by the Takagi and Sugeno[10] fuzzy reasoning method. In both cases
the firing levels of the rules have been computed by the product t-norm [9], T (a, b) =
ab, (to have a smooth output function), and then a solution to the original fuzzy problem
(2) has been defined as a solution to the resulting deterministic (usually nonlinear)
optimization problem
max/min f (u), subject to u ∈ X.
where the crisp value of the objective function f at u ∈ Rn , denoted also by f (u), has
been determined by Zadeh’s compositional rule of inference.
2
Optimization with linguistic variables
To find a fair solution to the fuzzy optimization problem
max/min f (x); subject to {<1 (x), . . . , <m (x) | x ∈ X},
3
(5)
with fuzzy if-then rules of form (3) or (4) we first determine the crisp value of the
objective function f at u ∈ Rn , denoted also by f (u), by the compositional rule of
inference
f (u) := (x is ū) ◦ {<1 (x), · · · , <m (x)}.
That is, in case (3) we apply the Tsukamoto fuzzy reasoning method as
f (u) :=
−1
α1 C1−1 (α1 ) + · · · + αm Cm
(αm )
α1 + · · · + αm
where the firing levels are computed according to
αi =
n
Y
Aij (uj ).
(6)
j=1
Furthermore, in case (4) we apply the Takagi and Sugeno fuzzy reasoning method
as
f (u) :=
α1 z1 (u) + · · · + αm zm (u)
.
α1 + · · · + αm
where the firing levels of the rules are computed by (6) and the individual rule outputs,
denoted by zi , are derived from the relationships
zi (u) =
n
X
aij uj + bi .
j=1
In this manner our constrained optimization problem (5) turns into the following
crisp (usually nonlinear) mathematical programmimg problem
max/min f (u); subject to u ∈ X.
If X is a fuzzy set with membership function µX (e.g. given by soft constraints as
in [14]) then following Bellman and Zadeh[1] we define the fuzzy solution to problem
(5) as
D = µX ∩ µf ,
(7)
where µf is an appropriate transformation of the values of f to the unit interval, and an
optimal solution to (5) is defined to be as any maximizing element of D.
3
Examples
Example 1 Consider the optimization problem
min f (x); subject to {x1 + x2 = 1/2, 0 ≤ x1 , x2 ≤ 1},
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(8)
where
<1 (x) : if x1 is small and x2 is small then f (x) = x1 + x2 ,
<2 (x) : if x1 is small and x2 is big then
f (x) = −x1 + x2 .
Let u = (u1 , u2 ) be an input to the fuzzy system. Then the firing levels of the rules are
α1 = (1 − u1 )(1 − u2 ),
α2 = (1 − u1 )u2 ,
It is clear that if u1 = 1 then no rule applies because α1 = α2 = 0. So we can exclude
the value u1 = 1 from the set of feasible solutions. The individual rule outputs are
computed by
z1 = u1 + u2 ,
z2 = −u1 + u2 .
and, therefore, the overall system output, interpreted as the crisp value of f at u is
f (u) =
(1 − u1 )(1 − u2 )(u1 + u2 ) + (1 − u1 )u2 (−u1 + u2 )
=
(1 − u1 )(1 − u2 ) + (1 − u1 )u2
u1 + u2 − 2u1 u2 .
Thus our original fuzzy problem turns into the following crisp nonlinear mathematical
programming problem
min (u1 + u2 − 2u1 u2 )
subject to {u1 + u2 = 1/2, 0 ≤ u1 < 1, 0 ≤ u2 ≤ 1}.
which has the optimal solution u∗1 = u∗2 = 1/4 and its optimal value is f (u∗ ) = 3/8.
Even though the individual rule outputs are linear functions of u1 and u2 , the computed
input/output function f (u) = u1 + u2 − 2u1 u2 is a nonlinear one.
Example 2 Consider the problem
max f
(9)
X
where X is a fuzzy susbset of the unit interval with membership function
µX (u) = 1 − (1/2 − u)2 ,
for u ∈ [0, 1], and the fuzzy rules are
<1 (x) : if x is small then f (x) = 1 − x,
<2 (x) : if x is big then
f (x) = x.
Let u ∈ [0, 1] be an input to the fuzzy system {<1 (x), <2 (x)}. Then the firing levels of
the rules are α1 = 1 − u, α2 = u. The individual rule outputs are z1 = (1 − u)(1 − u),
z2 = u2 and, therefore, the overall system output is
f (u) = (1 − u)2 + u2 = 2u2 + 2u + 1.
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Then according to (7) our original fuzzy problem (9) turns into the following crisp
biobjective mathematical programming problem
max min{2u2 + 2u + 1, 1 − (1/2 − u)2 }; subject to u ∈ [0, 1],
which has the optimal value of 0.8333 and two optimal solutions
{0.09, 0.91}.
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 the Takagi and
Sugeno and the Tsukamoto fuzzy reasoning scheme, because the individual rule outputs are crisp functions, and therefore, the functional relationship between the input
vector u and the system output f (u) can be easily identified.
4
Extensions
Assume that besides {<1 , . . . , <m } we are able to justify some monotonicity properties
in the functional link between x and f (x), for example, ”if xi is very Aij then f (x) is
very Ci ”, where ”very Aij ” and ”very Ci ” are new values of linguistic variables xi and
Ci , respectively. Consider the following very simple optimization problem
max f (x); subject to {<1 (x), <2 (x) | x ∈ X = [0, 1]},
(10)
where
<1 (x) : if x is small then f (x) is small
<2 (x) : if x is big then
f (x) is big
Let small(x) = 1 − x and big(x) = x, and let u be an input to the rule base < =
{<1 , <2 } then the firing levels of the rules are computed by
α1 = 1 − u,
α2 = u.
Then we get
f (u) = (1 − u)u + u × u = u.
Thus our original fuzzy problem turns into the following trivial crisp problem
max u; subject to u ∈ [0, 1].
(11)
which has the optimal solution u∗ = 1.
Assume that besides {<1 , <2 } we are able to justify the following monotonicity
properties between x and f (x), ”if x is very small then f (x) is very small” and ” if x
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is very big then f (x) is very big”. Assume further that these monotonicity rules are
implemented by
<3 (x) : if x is very small then f (x) is very small
<4 (x) : if x is very big then
f (x) is very big
where
(
1 − 2u if 0 ≤ u ≤ 1/2,
(very small)(u) =
0
otherwise,
and
(
(very big)(u) =
2u − 1
if 1/2 ≤ u ≤ 1,
0
otherwise,
Then the fuzzy problem
max f (x); subject to {<1 (x), <2 (x), <3 (x), <4 (x) | x ∈ X = [0, 1]},
where
<1 (x) : if x is small then
f (x) is small
<2 (x) : if x is big then
f (x) is big
<3 (x) : if x is very small then f (x) is very small
<4 (x) : if x is very big then
f (x) is very big
turns into the same crisp optimization problem (11), and therefore, the optimal solution
remains the same, u∗ = 1. If, however, <2 does not entail <4 , but we have
if x is very big then f (x) is not very big
instead, then the optimal solution changes. Really, the problem
max f (x); subject to {<1 (x), <2 (x), <3 (x), <4 (x) | x ∈ X = [0, 1]},
where
<1 (x) : if x is small then
f (x) is small
<2 (x) : if x is big then
f (x) is big
<3 (x) : if x is very small then f (x) is very small
<4 (x) : if x is very big then
f (x) is not very big
has the following crisp objective function (see Figure 2)
(
f (u) =
u
if 0 ≤ u ≤ 1/2,
5u−2u2 −3/2
2u
otherwise,
and the solution to the resulting crisp optimization problem
max f (u); subject to u ∈ [0, 1].
is u∗ = 0.865 and its optimal value is f (u∗ ) = 0.766.
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Figure 1: Illustration of Example 2.
5
Summary
We have addressed FMP problems where the functional relationship between the decision variables and the objective function is known linguistically. We have suggested
the use of an appropriate fuzzy reasoning method to determine the crisp functional
relationship between the decision variables and the objective function and solve the resulting (usually nonlinear) programming problem to find a fair optimal solution to the
original fuzzy problem.
We have shown that the refinement of the intial fuzzy rule base (by introducing
some non-monotonicity rules) can result in a substantial change of the solution.
References
[1] R. E. Bellman and L.A.Zadeh, Decision-making in a fuzzy environment, Management Sciences, Ser. B 17(1970) 141-164.
[2] C. Carlsson and R. Fullér, Interdependence in fuzzy multiple objective programming, Fuzzy Sets and Systems, 65(1994) 19-29.
[3] C. Carlsson and R. Fullér, Multiple Criteria Decision Making: The Case for Interdependence, Computers & Operations Research, 22(1995) 251-260.
[4] C. Carlsson, R. Fullér and S. Giove, Optimization under fuzzy rule constraints,
The Belgian Journal of Operations Research, Statistics and Computer Science
38(1998) 17-24.
[5] C. Carlsson and R. Fullér, Multiobjective linguistic optimization, Fuzzy Sets and
Systems, 115(2000) 5-10.
[6] C. Carlsson and R. Fullér, Optimization under fuzzy if-then rules, Fuzzy Sets and
Systems, 119(2001) 111-120.
[7] R.Felix, Relationships between goals in multiple attribute decision making, Fuzzy
sets and Systems, 67(1994) 47-52.
[8] M.Inuiguchi, H.Ichihashi and H. Tanaka, Fuzzy Programming: A Survey of Recent Developments, in: Slowinski and Teghem eds., Stochastic versus Fuzzy
8
Approaches to Multiobjective Mathematical Programming under Uncertainty,
Kluwer Academic Publishers, Dordrecht 1990 45-68.
[9] B.Schweizer and A.Sklar, Associative functions and abstract semigroups, Publ.
Math. Debrecen, 10(1963) 69-81.
[10] T.Takagi and M.Sugeno, Fuzzy identification of systems and its applications to
modeling and control, IEEE Trans. Syst. Man Cybernet., 1985, 116-132.
[11] Y. Tsukamoto, An approach to fuzzy reasoning method, in: M.M. Gupta, R.K.
Ragade and R.R. Yager eds., Advances in Fuzzy Set Theory and Applications
(North-Holland, New-York, 1979).
[12] L.A.Zadeh, Fuzzy Sets, Information and Control, 8(1965) 338-353.
[13] L.A.Zadeh, The concept of linguistic variable and its applications to approximate
reasoning, Parts I,II,III, Information Sciences, 8(1975) 199-251; 8(1975) 301-357;
9(1975) 43-80.
[14] H.-J. Zimmermann, Description and optimization of fuzzy systems, Internat. J.
General Systems 2(1975) 209-215.
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