Reliable Computing 10: 469–487, 2004.
2004 Kluwer Academic Publishers. Printed in the Netherlands.
Discrete Optimization Problems
with Interval Data: Pareto Set of Solutions
or Set of Weak Solutions?
Zaporozhye National Technical University, Zhukovsky Str., 64, Zaporozhye, 69600, Ukraine,
(Received: 27 January 1997; accepted: 30 September 2003)
Abstract. For optimization problems with interval uncertainty, traditionally researchers have con-
sidered deﬁnitions based on element-wise optimality: a feasible solution is a strong solution if it
optimizes the objective function
c)forall possible values of the parameters c within the given
intervals c, and it is a weak solution if it optimizes the objective function
c. In our previous papers, we introduced the alternative approach based on the idea of Pareto
In this paper, we analyze the relation between these two concepts of optimality, and provide
arguments in favor of the Pareto approach.
1. What Is a Discrete Optimization Problem?
An optimization problem is usually deﬁned as a computational problem in which
we are given the set of alternatives X, and the objective is to ﬁnd the best of these
alternatives. More formally, we know the objective function
: X →
want to ﬁnd the alternative x
X for which the value of this objective function
attains the smallest possible value:
In optimization, alternatives x
X are usually called feasible solutions.
When the set X of feasible solutions is discrete, the corresponding optimization
problem is called a discrete optimization problem. In most practical applications,
the corresponding discrete set X is ﬁnite; because of this fact, in this paper, we will
assume that the set X is ﬁnite.
In real life, the consequences of selecting each alternative x depend not only on
this alternative, but also on other factors c
. Let us denote the total number
of such factors (“parameters”) by n, and the real-valued vector (c
containing the values of these parameters by c. In these notations, the objective
c) is a function