- Preprint (40) (entfernen)
- Universal Shortest Paths (2010)
- We introduce the universal shortest path problem (Univ-SPP) which generalizes both - classical and new - shortest path problems. Starting with the definition of the even more general universal combinatorial optimization problem (Univ-COP), we show that a variety of objective functions for general combinatorial problems can be modeled if all feasible solutions have the same cardinality. Since this assumption is, in general, not satisfied when considering shortest paths, we give two alternative definitions for Univ-SPP, one based on a sequence of cardinality contrained subproblems, the other using an auxiliary construction to establish uniform length for all paths between source and sink. Both alternatives are shown to be (strongly) NP-hard and they can be formulated as quadratic integer or mixed integer linear programs. On graphs with specific assumptions on edge costs and path lengths, the second version of Univ-SPP can be solved as classical sum shortest path problem.
- Stücklisten und lineare Algebra (2001)
- Mit der vorliegenden Veröffentlichung soll der Versuch unternommen werden, mathematischen Schulstoff aus konkreten Problemen herzuentwickeln. Im Mittelpunkt der vorliegenden Arbeit stehen betriebswirtschaftliche Planungs- und Entscheidungsprobleme, wie sie von fast allen Wirtschaftsunternehmen zu lösen sind. Dabei wird im besonderen auf folgende Optimierungsprobleme eingegangen: Berechnung des Rohstoffbedarfs bei gegebenen Bestellungen, Aufarbeitung von vorhandenen Lagerbeständen und das Stücklistenproblem.
- 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.
- Standortplanung im Mathematikunterricht (2001)
- 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.
- Some Complexity Results for k-Cardinality Minimum Cut Problems (2000)
- Many polynomially solvable combinatorial optimization problems (COP) become NP when we require solutions to satisfy an additional cardinality constraint. This family of problems has been considered only recently. We study a newproblem of this family: the k-cardinality minimum cut problem. Given an undirected edge-weighted graph the k-cardinality minimum cut problem is to find a partition of the vertex set V in two sets V 1 , V 2 such that the number of the edges between V 1 and V 2 is exactly k and the sum of the weights of these edges is minimal. A variant of this problem is the k-cardinality minimum s-t cut problem where s and t are fixed vertices and we have the additional request that s belongs to V 1 and t belongs to V 2 . We also consider other variants where the number of edges of the cut is constrained to be either less or greater than k. For all these problems we show complexity results in the most significant graph classes.
- Solving nonconvex planar location problems by finite dominating sets (2000)
- It is well-known that some of the classical location problems with polyhedral gauges can be solved in polynomial time by finding a finite dominating set, i.e. a finite set of candidates guaranteed to contain at least one optimal location. In this paper it is first established that this result holds for a much larger class of problems than currently considered in the literature. The model for which this result can be proven includes, for instance, location problems with attraction and repulsion, and location-allocation problems. Next, it is shown that the approximation of general gauges by polyhedral ones in the objective function of our general model can be analyzed with regard to the subsequent error in the optimal objective value. For the approximation problem two different approaches are described, the sandwich procedure and the greedy algorithm. Both of these approaches lead - for fixed epsilon - to polynomial approximation algorithms with accuracy epsilon for solving the general model considered in this paper.
- Sink Location to Find Optimal Shelters in Evacuation Planning (2014)
- The sink location problem is a combination of network flow and location problems: From a given set of nodes in a flow network a minimum cost subset \(W\) has to be selected such that given supplies can be transported to the nodes in \(W\). In contrast to its counterpart, the source location problem which has already been studied in the literature, sinks have, in general, a limited capacity. Sink location has a decisive application in evacuation planning, where the supplies correspond to the number of evacuees and the sinks to emergency shelters. We classify sink location problems according to capacities on shelter nodes, simultaneous or non-simultaneous flows, and single or multiple assignments of evacuee groups to shelters. Resulting combinations are interpreted in the evacuation context and analyzed with respect to their worst-case complexity status. There are several approaches to tackle these problems: Generic solution methods for uncapacitated problems are based on source location and modifications of the network. In the capacitated case, for which source location cannot be applied, we suggest alternative approaches which work in the original network. It turns out that latter class algorithms are superior to the former ones. This is established in numerical tests including random data as well as real world data from the city of Kaiserslautern, Germany.
- Semi-Simultaneous Flows and Binary Constrained (Integer) Linear Programs (2006)
- Linear and integer programs are considered whose coefficient matrices can be partitioned into K consecutive ones matrices. Mimicking the special case of K=1 which is well-known to be equivalent to a network flow problem we show that these programs can be transformed to a generalized network flow problem which we call semi-simultaneous (se-sim) network flow problem. Feasibility conditions for se-sim flows are established and methods for finding initial feasible se-sim flows are derived. Optimal se-sim flows are characterized by a generalization of the negative cycle theorem for the minimum cost flow problem. The issue of improving a given flow is addressed both from a theoretical and practical point of view. The paper concludes with a summary and some suggestions for possible future work in this area.
- Scheduling and Location (ScheLoc): Makespan Problem with Variable Release Dates (2007)
- While in classical scheduling theory the locations of machines are assumed to be fixed we will show how to tackle location and scheduling problems simultaneously. Obviously, this integrated approach enhances the modeling power of scheduling for various real-life problems. In this paper, we present in an exemplary way theory and a solution algorithm for a specific type of a scheduling and a rather general, planar location problem, respectively. More general results and a report on numerical tests will be presented in a subsequent paper.
- Polyhedral Properties of the Uncapacitated Multiple Allocation Hub Location Problem (2000)
- We examine the feasibility polyhedron of the uncapacitated hub location problem (UHL) with multiple allocation, which has applications in the fields of air passenger and cargo transportation, telecommunication and postal delivery services. In particular we determine the dimension and derive some classes of facets of this polyhedron. We develop some general rules about lifting facets from the uncapacitated facility location (UFL) for UHL and projecting facets from UHL to UFL. By applying these rules we get a new class of facets for UHL which dominates the inequalities in the original formulation. Thus we get a new formulation of UHL whose constraints are all facet defining. We show its superior computational performance by benchmarking it on a well known data set.