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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.

Multiobjective combinatorial optimization problems have received increasing attention in recent years. Nevertheless, many algorithms are still restricted to the bicriteria case. In this paper we propose a new algorithm for computing all Pareto optimal solutions. Our algorithm is based on the notion of level sets and level curves and contains as a subproblem the determination of K best solutions for a single objective combinatorial optimization problem. We apply the method to the Multiobjective Quadratic Assignment Problem (MOQAP). We present two algorithms for ranking QAP solutions and nally give computational results comparing the methods.

In this paper we prove a reduction result for the number of criteria in convex multiobjective optimization. This result states that to decide wheter a point x in the decision space is pareto optimal it suffices to consider at most n? criteria at a time, where n is the dimension of the decision space. The main theorem is based on a geometric characterization of pareto, strict pareto and weak pareto solutions

In this paper we consider the problem of finding in a given graph a minimal weight subtree of connected subgraph, which has a given number of edges. These NP-hard combinatorial optimization problems have various applications in the oil industry, in facility layout and graph partitioning. We will present different heuristic approaches based on spanning tree and shortest path methods and on an exact algorithm solving the problem in polynomial time if the underlying graph is a tree. Both the edge- and node weighted case are investigated and extensive numerical results on the behaviour of the heuristics compared to optimal solutions are presented. The best heuristic yielded results within an error margin of less than one percent from optimality for most cases. In a large percentage of tests even optimal solutions have been found.

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 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.

Convex Operators in Vector Optimization: Directional Derivatives and the Cone of Decrease Directions
(1999)

The paper is devoted to the investigation of directional derivatives and the cone of decrease directions for convex operators on Banach spaces. We prove a condition for the existence of directional derivatives which does not assume regularity of the ordering cone K. This result is then used to prove that for continuous convex operators the cone of decrease directions can be represented in terms of the directional derivatices . Decrease directions are those for which the directional derivative lies in the negative interior of the ordering cone K. Finally, we show that the continuity of the convex operator can be replaced by its K-boundedness.

The notion of the balance number introduced in [3,page 139] through a certain set contraction procedure for nonscalarized multiobjective global optimization is represented via a min-max operation on the data of the problem. This representation yields a different computational procedure for the calculation of the balance number and allows us to generalize the approach for problems with countably many performance criteria.

In this paper we deal with an NP-hard combinatorial optimization problem, the k-cardinality tree problem in node weighted graphs. This problem has several applications , which justify the need for efficient methods to obtain good solutions. We review existing literature on the problem. Then we prove that under the condition that the graph contains exactly one trough, the problem can be solved in ploynomial time. For the general NP-hard problem we implemented several local search methods to obtain heuristics solutions, which are qualitatively better than solutions found by constructive heuristics and which require significantly less time than needed to obtain optimal solutions. We used the well known concepts of genetic algorithms and tabu search with useful extensions. We show that all the methods find optimal solutions for the class of graphs containing exactly one trough. The general performance of our methods as compared to other heuristics is illustrated by numerical results.