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Mon, 03 Apr 2000 00:00:00 +0200Mon, 03 Apr 2000 00:00:00 +0200Multicriteria network location problems with sumb objectives
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/487
In this paper network location problems with several objectives are discussed, where every single objective is a classical median objective function. We will lock at the problem of finding Pareto optimal locations and lexicographically optimal locations. It is shown that for Pareto optimal locations in undirected networks no node dominance result can be shown. Structural results as well as efficient algorithms for these multi-criteria problems are developed. In the special case of a tree network a generalization of Goldman's dominance algorithm for finding Pareto locations is presented.Horst W. Hamacher; Stefan Nickel; Martine Labbepreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/487Mon, 03 Apr 2000 00:00:00 +0200Classification of Location Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/491
There are several good reasons to introduce classification schemes for optimization problems including, for instance, the ability for concise problem statement opposed to verbal, often ambiguous, descriptions or simple data encoding and information retrieval in bibliographical information systems or software libraries. In some branches like scheduling and queuing theory classification is therefore a widely accepted and appreciated tool. The aim of this paper is to propose a 5-position classification which can be used to cover all location problems. We will provide a list of currentliy available symbols and indicate its usefulness in a - necessarily non-comprehensive - list of classical location problems. The classification scheme is in use since 1992 and has since proved to be useful in research, software development, classroom, and for overview articles.Horst W. Hamacher; Stefan Nickel; Anja Schneiderpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/491Mon, 03 Apr 2000 00:00:00 +0200Planar Location Problems with Barriers under Polyhedral Gauges
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/494
The Weber problem for a given finite set of existing facilities {cal E}x = {Ex_1,Ex_2, ... ,Ex_M} subset R^2 with positive weights w_m (m = 1, ... ,M) is to find a new facility X* in R^2 such that sum_{m=1}^{M} w_{m}d(X,Ex_m) is minimized for some distance function d. In this paper we consider distances defined by polyhedral gauges. A variation of this problem is obtained if barriers are introduced which are convex polygonal subsets of the plane where neither location of new facilities nor traveling is allowed. Such barriers like lakes, military regions, national parks or mountains are frequently encountered in practice.From a mathematical point of view barrier problems are difficult, since the prensence of barriers destroys the convexity of the objective function. Nevertheless, this paper establishes a descretization result: One of the grid points in the grid defined by the existing facilities and the fuundamental directions of the gauge distances can be proved to be an optimal location. Thus the barrier problem can be solved with a polynomial algorithm.Horst W. Hamacher; Kathrin Klamrothpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/494Mon, 03 Apr 2000 00:00:00 +0200Geometric Methods to Solve Max-Ordering Location Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/502
Location problems with Q (in general conflicting) criteria are considered. After reviewing previous results of the authors dealing with lexicographic and Pareto location the main focus of the paper is on max-ordering locations. In these location problems the worst of the single objectives is minimized. After discussing some general results (including reductions to single criterion problems and the relation to lexicographic and Pareto locations) three solution techniques are introduced and exemplified using one location problem class, each: The direct approach, the decision space approach and the objective space approach. In the resulting solution algorithms emphasis is on the representation of the underlying geometric idea without fully exploring the computational complexity issue. A further specialization of max-ordering locations is obtained by introducing lexicographic max-ordering locations, which can be found efficiently. The paper is concluded by some ideas about future research topics related to max-ordering location problems.Matthias Ehrgott; Horst W. Hamacher; Stefan Nickelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/502Mon, 03 Apr 2000 00:00:00 +0200A Note on Center Problems with forbidden Polyhedra
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/522
The problem of finding an optimal location X* minimizing the maximum Euclidean distance to existing facilities is well solved by e.g. the Elzinga-Hearn algorithm. In practical situations X* will however often not be feasible. We therefore suggest in this note a polynomial algorithm which will find an optimal location X^F in a feasible subset F of the plane R^2Horst W. Hamacher; Anita Schöbelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/522Mon, 03 Apr 2000 00:00:00 +0200Inverse radiation therapy planning a multiple objective optimisation approach
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/847
For some decades radiation therapy has been proved successful in cancer treatment. It is the major task of clinical radiation treatment planning to realise on the one hand a high level dose of radiation in the cancer tissue in order to obtain maximum tumour control. On the other hand it is obvious that it is absolutely necessary to keep in the tissue outside the tumour, particularly in organs at risk, the unavoidable radiation as low as possible. No doubt, these two objectives of treatment planning high level dose in the tumour, low radiation outside the tumour have a basically contradictory nature. Therefore, it is no surprise that inverse mathematical models with dose distribution bounds tend to be infeasible in most cases. Thus, there is need for approximations compromising between overdosing the organs at risk and underdosing the target volume. Differing from the currently used time consuming iterative approach, which measures deviation from an ideal (non-achievable) treatment plan using recursively trial-and-error weights for the organs of interest, we go a new way trying to avoid a priori weight choices and consider the treatment planning problem as a multiple objective linear programming problem: with each organ of interest, target tissue as well as organs at risk, we associate an objective function measuring the maximal deviation from the prescribed doses. We build up a data base of relatively few efficient solutions representing and approximating the variety of Pareto solutions of the multiple objective linear programming problem. This data base can be easily scanned by physicians looking for an adequate treatment plan with the aid of an appropriate online tool.Horst W. Hamacher; K.-H. Küferpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/847Mon, 03 Apr 2000 00:00:00 +0200Fixed Cardinality Combinatorial Optimization Problems - A Survey
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/971
Matthias Ehrgott; Horst W. Hamacherpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/971Fri, 18 Feb 2000 00:00:00 +0100