Fuzzy if-then rules for modeling interdependences in
FMOP problems ∗
Christer Carlsson
Institute for Advanced Management Systems Research, Åbo Akademi
University, DataCity A 3210, SF-20520 Åbo, Finland
Robert Fullér †
Department of Computer Science, Eötvös Loránd University,
Muzeum krt. 6-8, H-1088 Budapest, Hungary
Abstract
There has been a growing interest and activity in the area of multiple criteria decision
making (MCDM), especially in the last 25 years. modeling and optimization methods have
been developed in both crisp and fuzzy environments.
The overwhelming majority of approaches for finding best compromise solutions to MCDM
problems do not make use of the interdependences among the objectives. However, as has been
pointed out by [1, 2], in modeling real world problems (especially in management sciences and
in group decisions) we often encounter MCDM problems with interdependent objectives.
In multiple objective programs (MOP), application functions are established to measure the
degree of fulfillment of the decision maker’s requirements (achievement of goals, nearness to an
ideal point, satisfaction, etc.) about the objective functions (see e.g. [5, 11]) and extensively
used in the process of finding ”good compromise” solutions.
In [3] we demonstrated that the use of interdependences among objectives of a MOP in the
definition of the application functions provides for more correct solutions and faster convergence.
In [4], generalizing the principle of application function to fuzzy multiple objective programs
(FMOP) with interdependent objectives, we defined a large family of application functions for
FMOP in order to provide for a better understanding of the decision problem, and to find
effective and more correct solutions.
In this paper we define interdependencies among the objectives of FMOP by using fuzzy if-then
rules.
Keywords: FMOP, interdependencies, fuzzy if-then rules, group decisions
1
Introduction
Decision making with interdependent multiple criteria is a surprisingly difficult task. If we have
clearly conflicting objectives there normally is no optimal solution which would simultaneously
∗
in: Proceedings of EUFIT’94 Conference, September 20-23, 1994 Aachen, Germany, Verlag der Augustinus
Buchhandlung, Aachen, 1994 1504-1508.
†
Presently visiting professor at Department of Management and Computer Science, University of Trento, Italy.
Partially supported by the Hungarian National Scientific Research Fund OTKA under the contracts T 4281, I/3-2152
and T 7598
1
satisfy all the criteria. On the other hand, if we have pair wisely supportive objectives, such that
the attainment of one objective helps us to attain the another objective, then we should exploit this
property in order to find effective optimal solutions. Carlsson [2] used the fuzzy Pareto optimal set
of nondominated alternatives as a basis for an OWA-type operator [9] for finding a best compromise
solution to MCDM problems with interdependent criteria.
Felix [8] presented a novel theory for multiple attribute decision making based on fuzzy relations
between objectives, in which the interactive structure of objectives is inferred and represented
explicitely. In [3] we demonstrated that the use of interdependences among objectives of a MOP in
the definition of the application functions provides for more correct solutions and faster convergence.
In [4], generalizing the principle of application function to FMOP with interdependent objectives,
we defined a large family of application functions for FMOP in order to provide for a better
understanding of the decision problem, and to find effective and more correct solutions.
The aim of this paper is to provide a new method for defining interdependencies for FMOP via
fuzzy if-then rules.
2
Interdependences in MOP
In [3] we defined interdependences between the objectives of a MOP as follows:
Consider the following problem
max{f1 (x), . . . , fk (x)}
x∈X
(1)
where fi : Rn → R are objective functions, x ∈ Rn is the decision variable, and X is a subset of
Rn .
Definition 2.1 Let fi and fj be two objective functions of (1). We say that
(i) fi supports fj on X if fi (x′ ) ≥ fi (x) entails fj (x′ ) ≥ fj (x), for all x′ , x ∈ X;
(ii) fi is in conflict with fj on X if fi (x′ ) ≥ fi (x) entails fj (x′ ) ≤ fj (x), for all x′ , x ∈ X;
(iii) fi and fj are said to be independent on X, otherwise.
Figure 1. A typical example of conflict on R.
Figure 2 .Supportive functions on R.
If the objective functions are differentiable on X then we have
(i) fi supports fj on X iff ∂e fi (x)∂e fj (x) ≥ 0 for all e ∈ Rn and x ∈ X,
(ii) fi is in conflict with fj on X iff ∂e fi (x)∂e fj (x) ≤ 0 for all e ∈ Rn and x ∈ X,
where ∂e fi (x) denotes the derivative of fi with respect to the direction e ∈ Rn at x ∈ Rn . If for a
given direction e ∈ Rn ,
∂e fi (x)∂e fj (x) ≥ 0 [∂e fi (x)∂e fj (x) ≤ 0]
holds for all x ∈ X then we say that fi supports fj [fi is in conflict with fj ] with respect to the
direction e on X.
3
Interdependencies in FMOP
Consider a fuzzy multiple objective program
max{f1 (x), . . . , fk (x)}
x∈X
(2)
where the values fi (x) are given by fuzzy if-then rules of the form
R1 :
if x1 is A11 and . . . and xn is A1n then f1 (x) is B11 and . . . and fk (x) is B1k
R2 :
if x1 is A21 and . . . and xn is A2n then f1 (x) is B21 and . . . and fk (x) is B2k
············
············
Rm : if x1 is Am1 and . . . and xn is Amn then f1 (x) is Bm1 and . . . and fk (x) is Bmk
where Aij and Bij are fuzzy sets representing our knowledge about the values of the decision
variable and the objective functions.
Suppose that after normalization we have the following fuzzy partition with 7 primary fuzzy sets
(linguistic terms): NB negative big; NM negative medium; NS negative small; ZE zero; PS positive
small; PM positive medium; and PB positive big (see Figure 3.). We shall use the natural inequality
relation between linguistic terms
N B ≤ N M ≤ N S ≤ ZE ≤ P S ≤ P M ≤ P B.
(3)
It should be noted that by using 7 linguistic terms for the possible values of x1 , . . . , xn and the
objective functions we can have 7n+k rules, which is a very large number, e.g. even in the simple
case, n = 2 and k = 2, there can be 2401 rules.
Definition 3.1 Let fi be an objective function of (2) and let the inequality relation between
the linguistic terms be defined in the sense of (3). We say that (i) fi is increasing if x ≤ x′ entails
that f (x) ≤ f (x′ ); (ii) fi is decreasing if x ≤ x′ entails that f (x) ≥ f (x′ ).
Table 1: Example for an increasing function.
x
f (x)
NB
NM
NM
NM
NS
ZE
ZE
ZE
PS
ZE
PM
ZE
PB
PM
In the following we define interdependencies among the objective functions of (2).
Definition 3.2 Let fi and fj be two objective functions of (2). We say that
(i) fi supports fj if fi (x′ ) ≥ fi (x) entails fj (x′ ) ≥ fj (x), for all linguistic terms;
(ii) fi is in conflict with fj if fi (x′ ) ≥ fi (x) entails fj (x′ ) ≤ fj (x), for all linguistic terms;
(iii) fi and fj are said to be independent, otherwise.
It is easy to see that two objective functions support each other if they simultaneously increasing/decreasing and they are in conflict if their behaviour is contradictory.
NB
NM
NS
-1
ZE
PS
PM
PB
1
Figure 3. A fuzzy partition with seven linguistic terms.
Let us consider a five-objective problem, where the rules are given by the following table:
Table 2: Look-up table for interdependencies.
x
NB
NM
NS
ZE
PS
PM
PB
f1 (x)
f2 (x)
f3 (x)
f4 (x)
f5 (x)
PB
PM
PS
ZE
NS
NM
NB
NB
NM
NS
ZE
PS
PM
PB
PB
NS
PS
ZE
ZE
NM
PM
PS
PS
ZE
ZE
NS
NS
NM
ZE
ZE
ZE
NS
NS
NS
NM
It is easy to see that f1 , f4 and f5 support each other, f1 and f2 are in conflict and f3 is in no
relation with the others.
The explicit use of interdependencies in the process of finding a ”good compromise” solution to (2)
will be the subject of further research.
A promising area of application of the principle of interdependencies is the consensus reaching
modul in group decisions, where the experts’ knowledge is represented by fuzzy if-then rules ([6, 7]).
It is quite natural to suppose that the experts do not modify their ratings for alternatives on each
of the criteria independently. On the contrary, if the expert modifies his rating on a criterium then
he should modify his ratings on each of related (conflicting, supporting) criteria.
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