Fachbereich Mathematik
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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.
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Transit Dependent Evacuation Planning for Kathmandu Valley: A Case Study
(2014)
- Due to the increasing number of natural or man-made disasters, the application of operations research methods in evacuation planning has seen a rising interest in the research community. From the beginning, evacuation planning has been highly focused on car-based evacuation. Recently, also the evacuation of transit depended evacuees with the help of buses has been considered. In this case study, we apply two such models and solution algorithms to evacuate a core part of the metropolitan capital city Kathmandu of Nepal as a hypothetical endangered region, where a large part of population is transit dependent. We discuss the computational results for evacuation time under a broad range of possible scenarios, and derive planning suggestions for practitioners.
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The Multi Terminal q-FlowLoc Problem: A Heuristic
(2011)
- In this paper the multi terminal q-FlowLoc problem (q-MT-FlowLoc) is introduced. FlowLoc problems combine two well-known modeling tools: (dynamic) network flows and locational analysis. Since the q-MT-FlowLoc problem is NP-hard we give a mixed integer programming formulation and propose a heuristic which obtains a feasible solution by calculating a maximum flow in a special graph H. If this flow is also a minimum cost flow, various versions of the heuristic can be obtained by the use of different cost functions. The quality of this solutions is compared.
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Teoría de localización en las clases de matemáticas
(2002)
- La Teoría de localización abarca las posibilidades, para que con la ayuda de modelos matemáticos se busquen localizaciones teniendo en cuenta que los intereses económicos y administrativos sean óptimos. Así por ejemplo se encuentra la mejor localización para el almacén central de una empresa, cuando la suma de los gastos de transporte y de almacenaje sean mínimos y cuando se utilice el almacén óptimo. Si por otro lado, la administración busca la localización de una nueva estación de bomberos o de un hospital, hay que tener en cuenta un importante criterio para la localización óptima y es que la distancia mayor no sobrepase un valor dado.
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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.
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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.
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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.
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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.
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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.
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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.
