## Fachbereich Mathematik

### Filtern

#### Erscheinungsjahr

- 1999 (14) (entfernen)

#### Schlagworte

- Location Theory (5)
- Algebraic Optimization (2)
- Geometrical Algorithms (2)
- Multicriteria Optimization (2)
- Algebraic optimization (1)
- Applications (1)
- Approximation (1)
- Bisector (1)
- Convexity (1)
- Forbidden Regions (1)

- Weber s Problem with attraction and repulsion under Polyhedral Gauges (1999)
- Given a finite set of points in the plane and a forbidden region R, we want to find a point X not an element of int(R), such that the weighted sum to all given points is minimized. This location problem is a variant of the well-known Weber Problem, where we measure the distance by polyhedral gauges and allow each of the weights to be positive or negative. The unit ball of a polyhedral gauge may be any convex polyhedron containing the origin. This large class of distance functions allows very general (practical) settings - such as asymmetry - to be modeled. Each given point is allowed to have its own gauge and the forbidden region R enables us to include negative information in the model. Additionally the use of negative and positive weights allows to include the level of attraction or dislikeness of a new facility. Polynomial algorithms and structural properties for this global optimization problem (d.c. objective function and a non-convex feasible set) based on combinatorial and geometrical methods are presented.

- On the number of Criteria Needed to Decide Pareto Optimality (1999)
- 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

- On Bisectors for Different Distance Functions (1999)
- Let rC and rD be two convexdistance funtions in the plane with convex unit balls C and D. Given two points, p and q, we investigate the bisector, B(p,q), of p and q, where distance from p is measured by rC and distance from q by rD. We provide the following results. B(p,q) may consist of many connected components whose precise number can be derived from the intersection of the unit balls, C nd D. The bisector can contain bounded or unbounded 2-dimensional areas. Even more surprising, pieces of the bisector may appear inside the region of all points closer to p than to q. If C and D are convex polygons over m and m vertices, respectively, the bisector B(p,q) can consist of at most min(m,n) connected components which contain at most 2(m+n) vertices altogether. The former bound is tight, the latter is tight up to an additive constant. We also present an optimal O(m+n) time algorithm for computing the bisector.

- Multiple objective programming with piecewise linear functions (1999)
- An approach to generating all efficient solutions of multiple objective programs with piecewise linear objective functions and linear constraints is presented. The approach is based on the decomposition of the feasible set into subsets, referred to as cells, so that the original problem reduces to a series of lenear multiple objective programs over the cells. The concepts of cell-efficiency and complex-efficiency are introduced and their relationship with efficiency is examined. A generic algorithm for finding efficent solutions is proposed. Applications in location theory as well as in worst case analysis are highlighted.

- Multicriteria Ordered Weber Problems (1999)
- In this paper we deal with the determination of the whole set of Pareto-solutions of location problems with respect to Q general criteria.These criteria include median, center or cent-dian objective functions as particular instances.The paper characterizes the set of Pareto-solutions of a these multicriteria problems. An efficient algorithm for the planar case is developed and its complexity is established. Extensions to higher dimensions as well as to the non-convexcase are also considered.The proposed approach is more general than the previously published approaches to multi-criteria location problems and includes almost all of them as particular instances.

- Multicriteria network location problems with sumb objectives (1999)
- 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.

- Geometric Methods to Solve Max-Ordering Location Problems (1999)
- 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.

- General Continuous Multicriteria Location Problems (1999)
- In this paper we deal with the determination of the whole set of Pareto-solutions of location problems with respect to Q general criteria. These criteria include as particular instances median, center or cent-dian objective functions. The paper characterizes the set of Pareto-solutions of all these multicriteria problems. An efficient algorithm for the planar case is developed and its complexity is established. the proposed approach is more general than the previously published approaches to multicriteria location problems and includes almost all of them as particular instances.

- Error bounds for the approximative solution of restricted planar location problems (1999)
- Facility location problems in the plane play an important role in mathematical programming. When looking for new locations in modeling real-word problems, we are often confronted with forbidden regions, that are not feasible for the placement of new locations. Furthermore these forbidden regions may habe complicated shapes. It may be more useful or even necessary to use approcimations of such forbidden regions when trying to solve location problems. In this paper we develop error bounds for the approximative solution of restricted planar location problems using the so called sandwich algorithm. The number of approximation steps required to achieve a specified error bound is analyzed. As examples of these approximation schemes, we discuss round norms and polyhedral norms. Also computational tests are included.

- Classification of Location Problems (1999)
- 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.