Compound Interdependences in MOP ∗
Christer Carlsson
e-mail: ccarlsso@ra.abo.fi
Robert Fullér
e-mail: rfuller@ra.abo.fi
Abstract
We consider multiple objective programming (MOP) problems with compound interdependences, i.e. the case when the states of some chosen objective are attained through supportive or inhibitory feed-backs from several
other objectives. MOP problems with independent objectives (i.e. when the
cause-effect relations between the decision variables and the objectives are
completely known) will be treated as special cases of the MOP in which we
have interdependent objectives.
Keywords: Multiple objective programming, interdependences, feed-backs
1
MOP Problems with Interdependent Objectives
In their classical text Theory of Games and Economic Behavior John von Neumann
and Oskar Morgenstern (1947) described the problem with interdependence; in
their outline of a social exchange economy They discussed the case of two or more
persons exchanging goods with each others (page 11):
. . . then the results for each one will depend in general not merely upon his
own actions but on those of others as well. Thus each participant attempts to
maximize a function . . . of which he does not control all variables. This is
certainly no maximum problem, but a peculiar and disconcerting mixture of
several conflicting maximum problems. Every participant is guided by another principle and neither determines all variables which affects his interest.
The interdependence is part of the economic theory and all market economies, but
in most modelling approaches in multiple criteria decision making there seems to
∗
in: Proceedings of EUFIT’96 Conference, September 2-5, 1996, Aachen, Germany, Verlag
Mainz, Aachen, 1996 1317-1322.
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be an implicit assumption that objectives should be independent. This appears to
be the case, if not earlier than at least at the moment when we have to select some
optimal compromise among the set of nondominated decision alternatives. Milan
Zeleny [8] - and many others - recognizes one part of the interdependence (page
1),
Multiple and conflicting objectives, for example, ”minimize cost” and ”maximize the quality of service” are the real stuff of the decision maker’s or
manager’s daily concerns. Such problems are more complicated than the
convenient assumptions of economics indicate. Improving achievement with
respect to one objective can be accomplished only at the expense of another.
but not the other part: objectives could support each others. We will in the following explore the consequences of allowing objectives to be interdependent.
2
Linear interdependences in MOP
Objective functions of a multiple objective programming problem are usually considered to be independent from each other, i.e. they depend only on the decision
variable x. A typical statement of an MOP with independent objective functions is
max{f1 (x), . . . , fk (x)}
x∈X
(1)
where fi is the i-th objective function, x is the decision variable, and X is a subset,
usually defined by functional inequalities. Throughout this paper we will assume
that the objective functions are normalized, i.e. fi (x) ∈ [0, 1] for each x ∈ X.
However, as has been shown in some earlier work by by Carlsson and Fullér
[1, 2, 3, 4], and Felix [6], there are management issues and negotiation problems,
in which one often encounters the necessity to formulate MOP models with interdependent objective functions, in such a way that the objective functions are determined not only by the decision variables but also by one or more other objective
functions.
First we state the multiobjective decision problem with independent objectives and
then adjust our model to reality by introducing interdependences among the objectives.
Interdependences among the objectives exist whenever the computed value of an
objective function is not equal to its observed value. In this paper we claim that
the real values of an objective function can be identified by the help of feed-backs
from the values of other objective functions.
2
Suppose now that the objectives of (1) are interdependent, and the value of an
objective function is determined by a linear combination of the values of other
objectives functions. That is
fi (x) = fi (x) +
k
αij fj (x), 1 ≤ i ≤ k
(2)
j=1, j=i
where αij is a real number denoting the grade of interdependency between fi and
fj .
If αij > 0 then we say that fi is supported by fj ; if αij < 0 then we say that fi is
hindered by fj ; if αij = 0 then we say that fi is independent from fj .
α ij f j ( x)
1
1
Fig. 1
f j ( x)
α ij f j ( x)
f j ( x)
Linear feed-back with αij > 0 and αij < 0
In such cases, i.e. when the feed-backs from the objectives are directly proportional
to their independent values, then we say that the objectives are linearly interdependent.
Taking into consideration the linear interdependences among the objective functions (2), (1) turns into the following independent problem
max{f1 (x), . . . , fk (x)}
x∈X
(3)
It is clear that the solution-sets of (1) and (3) are usually not identical.
3
Simple nonlinear interdependences in MOP
Suppose now that the objectives of (1) are interdependent, and the value of an
objective function is determined by an additive combination of the feed-backs of
other objectives functions
fi (x) = fi (x) +
k
αij [fj (x)], 1 ≤ i ≤ k
j=1, j=i
3
(4)
where αij : [0, 1] → [0, 1] is a - usually nonlinear - function defining the value of
feed-back from fj to fi . We emphasize here that the value of the feed-back from
fj to fi depends only on the value of fj (it is why we call it simple nonlinear
interdependence).
If αij (z) > 0, ∀z we say that fi is supported by fj ; if αij (z) < 0, ∀t then we say
that fi is hindered by fj ; if αij (z) = 0, ∀z then we say that fi is independent from
fj . If αij (z1 ) > 0 and αij (z2 ) < 0 for some z1 and z2 , then fi is supported by fj
if the value of fj is equal to z1 and fi is hindered by fj if the value of fj is equal
to z2 .
α ij( f j ( x) )
1
1
f j ( x)
f j ( x)
α ij( f j ( x) )
Fig. 2
Nonlinear unilateral support and hindering.
α ij( f j ( x) )
β
1
f j ( x)
Fig. 3 fj supports fi if fj (x) ≤ β and fj hinders fi if fj (x) ≥ β.
To explain the issue more exactly, consider a three-objective problem with nonlinear interdependences
(5)
max{f1 (x), f2 (x), f3 (x)}
x∈X
Taking into consideration that the objectives are interdependent, the interdependent
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values of the objectives can be expressed by
f1 (x) = f1 (x)+α12 [f2 (x)]+α13 [f3 (x)] f2 (x) = f2 (x)+α21 [f1 (x)]+α23 [f3 (x)]
f3 (x) = f3 (x) + α31 [f1 (x)] + α32 [f2 (x)]
α2 1
f2
α1 2
f1
Fig. 4
α2 3
α3 2
α1 3
α3 1
f3
A three-objective interdependent problem with linear feed-backs.
For example, depending on the values of αij we can have the following simple
interdependences among the objectives of (5)
• if α12 (z) = 0, ∀z then we say that f1 is independent from f2 ;
• if α12 (z) > 0, ∀z then we say that f2 unilaterally supports f1 ;
• if if α12 (z) < 0, ∀z then we say that f2 hinders f1 ;
• if α12 (z) > 0 and α21 (z), ∀z > 0 then we say that f1 and f2 mutually
support each others;
• if α12 (z) < 0 and α21 (z) < 0 for each z then we say that f1 and f2 are
conflicting;
• if α12 (z)+α21 (z) = 0 for each z then we say that f1 are f2 are in a trade-off
relation;
It is clear, for example, that if f2 unilaterally supports f1 then the bigger the value
of f2 (supporting objective function) the bigger its support to f1 (supported objective function).
However, we can have more complex relationships between two objective functions, e.g.
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• if for some β ∈ [0, 1]
α12 (z) =
positive if 0 ≤ z ≤ β
negative if β ≤ z ≤ 1
then f2 unilaterally supports f1 if f2 (x) ≤ β and f2 hinders f1 if f2 (x) ≥ β.
• if for some β, γ ∈ [0, 1]


 positive
if 0 ≤ z ≤ β
0
if β ≤ z ≤ γ
α12 (z) =

 negative if γ ≤ z ≤ 1
then f2 unilaterally supports f1 if f2 (x) ≤ β, f2 does not affect f1 if β ≤
f2 (x) ≤ γ and then f2 hinders f1 if f2 (x) ≥ γ.
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Compound interdependences in MOP
Suppose now that the objectives of (1) are interdependent, and the value of an
objective function is determined by an additive combination of the feed-backs of
other objectives functions
fi (x) =
k
αij [f1 (x), . . . , fk (x)], 1 ≤ i ≤ k
(6)
j=1
where αij : [0, 1]k → [0, 1] is a - usually nonlinear - function defining the value
of feed-back from fj to fi . We note that αij depends not only on the value of
fj , but on the values of other objectives as well (it is why we call it compound
interdependence).
Let us again consider the three-objective problem with nonlinear interdependences
max{f1 (x), f2 (x), f3 (x)}
x∈X
With the assumptions of (6) the interdependent values of the objectives can be
expressed by
f1 (x) = α11 [f1 (x), f2 (x), f3 (x)]+α12 [f1 (x), f2 (x), f3 (x)]+α13 [f1 (x), f2 (x), f3 (x)]
f2 (x) = α22 [f1 (x), f2 (x), f3 (x)]+α21 [f1 (x), f2 (x), f3 (x)]+α23 [f1 (x), f2 (x), f3 (x)]
f3 (x) = α33 [f1 (x), f2 (x), f3 (x)]+α31 [f1 (x), f2 (x), f3 (x)]+α32 [f1 (x), f2 (x), f3 (x)]
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α2 2
α2 1
f2
α2 3
α1 2
α3 2
α1 1
f1
α1 3
α3 1
f3
α3 3
Fig. 5
A three-objective interdependent problem with compound interdependences.
Here we can have more complicated interrelations between f1 and f2 , because the
feedback from f2 to f1 can depend not only on the value of f2 , but also on the
values of f1 (self feed-back) and f3 . Unfortunately, in real life cases we usually
have compound interdependences (cf [4]).
References
[1] C.Carlsson and R.Fullér, Fuzzy if-then rules for modeling interdependencies in FMOP problems, in: Proceedings of EUFIT’94 Conference,
September 20-23, 1994 Aachen, Germany, Verlag der Augustinus Buchhandlung, Aachen, 1994 1504-1508.
[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) 251260.
[4] C.Carlsson and R.Fullér, On linear interdependences in MOP, in: Proceedings of CIFT’95, June 8-10, 1995, Trento, Italy, University of
Trento, 1995 48-52.
[5] C.Carlsson and R.Fullér, Fuzzy multiple criteria decision making: Recent developments, Fuzzy Sets and Systems, 78(1996) 139-153.
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[6] R.Felix, Relationships between goals in multiple attribute decision making, Fuzzy sets and Systems, 67(1994) 47-52.
[7] J. von Neumann and O. Morgenstern, Theory of Games and Economic
Behavior, Princeton University Press, Princeton 1947.
[8] M.Zeleny, Multiple Criteria Decision Making, McGraw-Hill, NewYork, 1982.
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