## Report in Wirtschaftsmathematik (WIMA Report)

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#### Keywords

- Minkowski space (1)
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- Standorttheorie (1)
- Verkehsplanung (1)
- center hyperplane (1)
- centrally symmetric polytope (1)
- common transversal (1)
- consecutive ones property (1)
- hyperplane transversal (1)
- location theory (1)

- 97
- Stop Location Design in Public Transportation Networks: Covering and Accessibility Objectives (2006)
- In StopLoc we consider the location of new stops along the edges of an existing public transportation network. Examples of StopLoc include the location of bus stops along some given bus routes or of railway stations along the tracks in a railway system. In order to measure the ''convenience'' of the location decision for potential customers in given demand facilities, two objectives are proposed. In the first one, we give an upper bound on reaching a closest station from any of the demand facilities and minimize the number of stations. In the second objective, we fix the number of new stations and minimize the sum of the distances between demand facilities and stations. The resulting two problems CovStopLoc and AccessStopLoc are solved by a reduction to a classical set covering and a restricted location problem, respectively. We implement the general ideas in two different environments - the plane, where demand facilities are represented by coordinates and in networks, where they are nodes of a graph.

- 92
- Integer programming approaches for solving the delay management problem (2004)
- In the delay management problem we decide how to react in case of delays in public transportation. More specific, the question is if connecting vhicles should wait for delayed feeder vehicles or if it is better to depart in time.

- 86
- Locating stops along bus or railway lines - a bicriterial problem (2003)
- In this paper we consider the location of stops along the edges of an already existing public transportation network, as introduced in [SHLW02]. This can be the introduction of bus stops along some given bus routes, or of railway stations along the tracks in a railway network. The goal is to achieve a maximal covering of given demand points with a minimal number of stops. This bicriterial problem is in general NP-hard. We present a nite dominating set yielding an IP-formulation as a bicriterial set covering problem. We use this formulation to observe that along one single straight line the bicriterial stop location problem can be solved in polynomial time and present an e cient solution approach for this case. It can be used as the basis of an algorithm tackling real-world instances.

- 91
- Set Covering With Almost Consecutive Ones Property (2003)
- In this paper we consider set covering problems with a coefficient matrix almost having the consecutive ones property, i.e., in many rows of the coefficient matrix, the ones appear consecutively. If this property holds for all rows it is well known that the set covering problem can be solved efficiently. For our case of almost consecutive ones we present a reformulation exploiting the consecutive ones structure to develop bounds and a branching scheme. Our approach has been tested on real-world data as well as on theoretical problem instances.

- 80
- Properties of 3-dimensional line location models (2002)
- We consider the problem of locating a line with respect to some existing facilities in 3-dimensional space, such that the sum of weighted distances between the line and the facilities is minimized. Measuring distance using the l_p norm is discussed, along with the special cases of Euclidean and rectangular norms. Heuristic solution procedures for finding a local minimum are outlined.

- 74
- Anchored hyperplane location problems (2001)
- The anchored hyperplane location problem is to locate a hyperplane passing through some given points P IR^n and minimizing either the sum of weighted distances (median problem), or the maximum weighted distance (center problem) to some other points Q IR^n . If the distances are measured by a norm, it will be shown that in the median case there exists an optimal hyperplane that passes through at least n - k affinely independent points of Q, if k is the maximum number of affinely independent points of P. In the center case, there exists an optimal hyperplane which isatmaximum distance to at least n - k + 1 affinely independent points of Q. Furthermore, if the norm is a smooth norm, all optimal hyperplanes satisfy these criteria. These new results generalize known results about unrestricted hyperplane location problems.

- 76
- Locating New Stops in a Railway Network (2001)
- Given a railway network together with information on the population and their use of the railway infrastructure, we are considering the e ffects of introducing new train stops in the existing railway network. One e ffect concerns the accessibility of the railway infrastructure to the population, measured in how far people live from their nearest train stop. The second effect we study is the change in travel time for the railway customers that is induced by new train stops. Based on these two models, we introduce two combinatorial optimization problems and give NP-hardness results for them. We suggest an algorithmic approach for the model based on travel time and give first experimental results.

- 63
- Linear Facility Location in Three Dimensions - Models and Solution Methods (2000)
- We consider the problem of locating a line or a line segment in three- dimensional space, such that the sum of distances from the linear facility to a given set of points is minimized. An example is planning the drilling of a mine shaft, with access to ore deposits through horizontal tunnels connecting the deposits and the shaft. Various models of the problem are developed and analyzed, and effcient solution methods are given.

- 18
- Median hyperplanes in normed spaces (1999)
- In this paper we deal with the location of hyperplanes in n-dimensional normed spaces. If d is a distance measure, our objective is to find a hyperplane H which minimizes f(H) = sum_{m=1}^{M} w_{m}d(x_m,H), where w_m ge 0 are non-negative weights, x_m in R^n, m=1, ... ,M demand points and d(x_m,H)=min_{z in H} d(x_m,z) is the distance from x_m to the hyperplane H. In robust statistics and operations research such an optimal hyperplane is called a median hyperplane. We show that for all distance measures d derived from norms, one of the hyperplanes minimizing f(H) is the affine hull of n of the demand points and, moreover, that each median hyperplane is (ina certain sense) a halving one with respect to the given point set.

- 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.