Algorithms for Bicriteria Combinatorial Optimization Problems

Abstract

In recent years, many types of interactive optimization methods have been developed in order to support multicriteria decision makings (see the book by Sawaragi, Nakayama and Tanino, 1985 and Wierzbicki and Lewandowski, 1987). Given a feasible decision set X ⊆ R ⁿ, and p objective functions, f _l, f ₂,...,f _p (all are assumed to be minimization for convenience), the following problem has been used in various situations of interactive multicriteria decision makings:

$$ \mathop {minimize}\limits_{x \in X} \;\mathop {\max }\limits_{1 \leqslant i \leqslant p} \left\{ {\alpha _i f_i (x) + \beta _i } \right\}, $$

((1))

where α _i and β _i are positive and real constants respectively, which are computed based on the information supplied by the decision maker and/or the decision support system. For example, α _i and β _i are determined from aspiration and reservation levels in the reference point method which is one of the well known methods used in interactive multicriteria decision support systems (see Wierzbicki and Lewandowski, 1987, for the survey of reference point methods).