## Report in Wirtschaftsmathematik (WIMA Report)

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#### Keywords

- Location Theory (6)
- Algebraic Optimization (2)
- Geometrical Algorithms (2)
- Multicriteria Optimization (2)
- Algebraic optimization (1)
- Applications (1)
- Approximation (1)
- Bisector (1)
- Convex Analysis (1)
- Convexity (1)

- 26
- A flexible approach to location problems (1999)
- In continous location problems we are given a set of existing facilities and we are looking for the location of one or several new facilities. In the classical approaches weights are assigned to existing facilities expressing the importance of the new facilities for the existing ones. In this paper, we consider a pointwise defined objective function where the weights are assigned to the existing facilities depending on the location of the new facility. This approach is shown to be a generalization of the median, center and centdian objective functions. In addition, this approach allows to formulate completely new location models. Efficient algorithms as well as structure results for this algebraic approach for location problems are presented. Extensions to the multifacility and restricted case are also considered.

- 27
- A geometric approach to global optimization (1999)
- In this paper we consider the problem of optimizing a piecewise-linear objective function over a non-convex domain. In particular we do not allow the solution to lie in the interior of a prespecified region R. We discuss the geometrical properties of this problems and present algorithms based on combinatorial arguments. In addition we show how we can construct quite complicated shaped sets R while maintaining the combinatorial properties.

- 14
- A unified approach to network location problems (1999)
- In this paper we introduce a new type of single facility location problems on networks which includes as special cases most of the classical criteria in the literature. Structural results as well as a finite dominationg set for the optimal locations are developed. Also the extension to the multi-facility case is discussed.

- 23
- An Interior Point Method for Multifacility Location Problems with Forbidden Regions (1999)
- In this paper we consider generalizations of multifacility location problems in which as an additional constraint the new facilities are not allowed to be located in a presprcified region. We propose several different solution schemes for this non-convex optimization problem. These include a linear programming type approach, penalty approaches and barrier approaches. Moreover, structural results as well as illustratrive examples showing the difficulties of this problem are presented

- 19
- 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.

- 17
- 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.

- 28
- 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.

- 29
- 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.

- 52
- Geometrical properties of generalized single facility location problems (2000)
- In this paper we deal with single facility location problems in a general normed space where the existing facilities are represented by sets. The criterion to be satis ed by the service facility is the minimization of an increasing function of the distances from the service to the closest point ofeach demand set. We obtain a geometrical characterization of the set of optimal solutions for this problem. Two remarkable cases - the classical Weber problem and the minmax problem with demand sets - are studied as particular instances of our problem. Finally, for the planar polyhedral case we give an algorithmic description of the solution set of the considered problems.

- 15
- 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.