In this paper we consider the problem of decomposing a given integer matrix A into
a positive integer linear combination of consecutive-ones matrices with a bound on the
number of columns per matrix. This problem is of relevance in the realization stage
of intensity modulated radiation therapy (IMRT) using linear accelerators and multileaf
collimators with limited width. Constrained and unconstrained versions of the problem
with the objectives of minimizing beam-on time and decomposition cardinality are considered.
We introduce a new approach which can be used to find the minimum beam-on
time for both constrained and unconstrained versions of the problem. The decomposition
cardinality problem is shown to be NP-hard and an approach is proposed to solve the
lexicographic decomposition problem of minimizing the decomposition cardinality subject
to optimal beam-on time.
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.
In this paper we give the definition of a solution concept in multicriteria combinatorial optimization. We show how Pareto, max-ordering and lexicographically optimal solutions can be incorporated in this framework. Furthermore we state some properties of lexicographic max-ordering solutions, which combine features of these three kinds of optimal solutions. Two of these properties, which are desirable from a decision maker" s point of view, are satisfied if and only of the solution concept is that of lexicographic max-ordering.
The topic of this paper are discrete decision problems with multiple criteria. We first define discrete multiple criteria decision problems and introduce a classification scheme for multiple criteria optimization problems. To do so we use multiople criteria optimization classes. The main result is a characterization of the class of lexicographic max-ordering problems by two very useful properties, reduction and regularity. Subsequently we discuss the assumptions under which the application of this specific MCO class is justified. Finally we provide (simple) solution methods to find optimal decisions in the case of discrete multiple criteria optimization problems.
The computational complexity of combinatorial multiple objective programming problems is investigated. NP-completeness and #P-completeness results are presented. Using two definitions of approximability, general results are presented, which outline limits for approximation algorithms. The performance of the well known tree and Christofides' heuristics for the TSP is investigated in the multicriteria case with respect to the two definitions of approximability.
Life is about decisions. Decisions, no matter if taken by a group or an individual, involve several conflicting objectives. The observation that real world problems have to be solved optimally according to criteria, which prohibit an "ideal" solution - optimal for each decisionmaker under each of the criteria considered - , has led to the development of multicriteria optimization. From its first roots, which where laid by Pareto at the end of the 19th century the discilpine has prospered and grown, especially during the last three decades. Today, many decision support systems incorporate methods to deal with conflicting objectives. The foundation for such systems is a mathematical theory of optimaztion under multiple objectives. With this manuscript, which is based on lectures I taught in the winter semester 1998/99 at the University of Kaiserslautern, I intend to give an introduction to and overview of this fascinating field of mathematics. I tried to present theoretical questions such as existence of solutions as well as methodological issues and hope the reader finds the balance not too heavily on one side. The interested reader should be able to find classical results as well as up to date research. The text is accompanied by exercises, which hopefully help to deepen students' understanding of the topic.
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.
Fragestellungen der Standortplanung sollen den Mathematikunterricht der Schule bereichern, dort behandelt und gelöst werden. In dieser Arbeit werden planare Standortprobleme vorgestellt, die im Mathematikunterricht behandelt werden können. Die Probleme Produktion von Halbleiterplatinen, Planung eines Feuerwehrhauses und das Zentrallagerproblem, die ausnahmlos real und nicht konstruiert sind, werden ausführlich durchgearbeitet, so dass es schnell möglich ist, daraus Unterrichtseinheiten zu entwickeln.
La Teoría de localización abarca las posibilidades, para que con la ayuda de modelos matemáticos se busquen localizaciones teniendo en cuenta que los intereses económicos y administrativos sean óptimos. Así por ejemplo se encuentra la mejor localización para el almacén central de una empresa, cuando la suma de los gastos de transporte y de almacenaje sean mínimos y cuando se utilice el almacén óptimo. Si por otro lado, la administración busca la localización de una nueva estación de bomberos o de un hospital, hay que tener en cuenta un importante criterio para la localización óptima y es que la distancia mayor no sobrepase un valor dado.