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The balance space approach (introduced by Galperin in 1990) provides a new view on multicriteria optimization. Looking at deviations from global optimality of the different objectives, balance points and balance numbers are defined when either different or equal deviations for each objective are allowed. Apportioned balance numbers allow the specification of proportions among the deviations. Through this concept the decision maker can be involved in the decision process. In this paper we prove that the apportioned balance number can be formulated by a min-max operator. Furthermore we prove some relations between apportioned balance numbers and the balance set, and see the representation of balance numbers in the balance set. The main results are necessary and sufficient conditions for the balance set to be exhaustive, which means that by multiplying a vector of weights (proportions of deviation) with its corresponding apportioned balance number a balance point is attained. The results are used to formulate an interactive procedure for multicriteria optimization. All results are illustrated by examples.

In this paper we investigate the problem offending the Nadir point for multicriteria optimization problems (MOP). The Nadir point is characterized by the component wise maximal values of efficient points for (MOP). It can be easily computed in the bicriteria case. However, in general this problem is very difficult. We review some existing methods and heuristics and propose some new ones. We propose a general method to compute Nadir values for the case of three objectives, based on theoretical results valid for any number of criteria. We also investigate the use of the Nadir point for compromise programming, when the goal is to be as far away as possible from the worst outcomes. We prove some results about (weak) Pareto optimality of the resulting solutions. The results are illustrated by examples.

In this paper we address the question of how many objective functions are needed to decide whether a given point is a Pareto optimal solution for a multicriteria optimization problem. We extend earlier results showing that the set of weakly Pareto optimal points is the union of Pareto optimal sets of subproblems and show their limitations. We prove that for strictly quasi-convex problems in two variables Pareto optimality can be decided by consideration of at most three objectives at a time. Our results are based on a geometric characterization of Pareto, strict Pareto and weak Pareto solutions and Helly's Theorem. We also show that a generalization to quasi-convex objectives is not possible, and state a weaker result for this case. Furthermore, we show that a generalization to strictly Pareto optimal solutions is impossible, even in the convex case.

This paper provides an annotated bibliography of multiple objective combinatorial optimization, MOCO. We present a general formulation of MOCO problems, describe the main characteristics of MOCO problems, and review the main properties and theoretical results for these problems. One section is devoted to a brief description of the available solution methodology, both exact and heuristic. The main part of the paper is devoted to an annotation of the existing literature in the field organized problem by problem. We conclude the paper by stating open questions and areas of future research. The list of references comprises more than 350 entries.

Many polynomially solvable combinatorial optimization problems (COP) become NP when we require solutions to satisfy an additional cardinality constraint. This family of problems has been considered only recently. We study a newproblem of this family: the k-cardinality minimum cut problem. Given an undirected edge-weighted graph the k-cardinality minimum cut problem is to find a partition of the vertex set V in two sets V 1 , V 2 such that the number of the edges between V 1 and V 2 is exactly k and the sum of the weights of these edges is minimal. A variant of this problem is the k-cardinality minimum s-t cut problem where s and t are fixed vertices and we have the additional request that s belongs to V 1 and t belongs to V 2 . We also consider other variants where the number of edges of the cut is constrained to be either less or greater than k. For all these problems we show complexity results in the most significant graph classes.