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Wed, 28 Oct 2015 14:33:01 +0100Wed, 28 Oct 2015 14:33:01 +0100Minimizing the Number of Apertures in Multileaf Collimator Sequencing with Field Splitting
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4206
In this paper we consider the problem of decomposing a given integer matrix A into
a positive integer linear combination of consecutive-ones matrices with a bound on the
number of columns per matrix. This problem is of relevance in the realization stage
of intensity modulated radiation therapy (IMRT) using linear accelerators and multileaf
collimators with limited width. Constrained and unconstrained versions of the problem
with the objectives of minimizing beam-on time and decomposition cardinality are considered.
We introduce a new approach which can be used to find the minimum beam-on
time for both constrained and unconstrained versions of the problem. The decomposition
cardinality problem is shown to be NP-hard and an approach is proposed to solve the
lexicographic decomposition problem of minimizing the decomposition cardinality subject
to optimal beam-on time.Horst W. Hamacher; Ines M. Raschendorfer; Davaatseren Baatar; Matthias Ehrgottpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4206Wed, 28 Oct 2015 14:33:01 +0100Minimizing the Number of Apertures in Multileaf Collimator Sequencing with Field Splitting
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4197
In this paper we consider the problem of decomposing a given integer matrix A into
a positive integer linear combination of consecutive-ones matrices with a bound on the
number of columns per matrix. This problem is of relevance in the realization stage
of intensity modulated radiation therapy (IMRT) using linear accelerators and multileaf
collimators with limited width. Constrained and unconstrained versions of the problem
with the objectives of minimizing beam-on time and decomposition cardinality are considered.
We introduce a new approach which can be used to find the minimum beam-on
time for both constrained and unconstrained versions of the problem. The decomposition
cardinality problem is shown to be NP-hard and an approach is proposed to solve the
lexicographic decomposition problem of minimizing the decomposition cardinality subject
to optimal beam-on time.Davaatseren Baatar; Matthias Ehrgott; Horst W. Hamacher; Ines M. Raschendorferarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4197Fri, 16 Oct 2015 11:00:21 +0200A Finite Dominating Set Algorithm for a Dynamic Location Problem in the Plane
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4017
A single facility problem in the plane is considered, where an optimal location has to be
identified for each of finitely many time-steps with respect to time-dependent weights and
demand points. It is shown that the median objective can be reduced to a special case of the
static multifacility median problem such that results from the latter can be used to tackle the
dynamic location problem. When using block norms as distance measure between facilities,
a Finite Dominating Set (FDS) is derived. For the special case with only two time-steps, the
resulting algorithm is analyzed with respect to its worst-case complexity. Due to the relation
between dynamic location problems for T time periods and T-facility problems, this algorithm
can also be applied to the static 2-facility location problem.Andrea Maier; Horst W. Hamacherpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4017Thu, 05 Mar 2015 14:56:26 +0100Bicriteria approach to the optimal location of surveillance cameras
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3997
We consider the problem of finding efficient locations of surveillance cameras, where we distinguish
between two different problems. In the first, the whole area must be monitored and the number of cameras
should be as small as possible. In the second, the goal is to maximize the monitored area for a fixed number of
cameras. In both of these problems, restrictions on the ability of the cameras, like limited depth of view or range
of vision are taken into account. We present solution approaches for these problems and report on results of
their implementations applied to an authentic problem. We also consider a bicriteria problem with two objectives:
maximizing the monitored area and minimizing the number of cameras, and solve it for our study case.Horst W. Hamacher; Gross Aleksandrapreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3997Fri, 20 Feb 2015 13:54:12 +0100Bicriteria approach to the optimal location of surveillance cameras
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3979
We consider the problem of finding efficient locations of surveillance cameras, where we distinguish
between two different problems. In the first, the whole area must be monitored and the number of cameras
should be as small as possible. In the second, the goal is to maximize the monitored area for a fixed number of
cameras. In both of these problems, restrictions on the ability of the cameras, like limited depth of view or range
of vision are taken into account. We present solution approaches for these problems and report on results of
their implementations applied to an authentic problem. We also consider a bicriteria problem with two objectives:
maximizing the monitored area and minimizing the number of cameras, and solve it for our study case.Aleksandra Gross; Horst W. Hamacherpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3979Thu, 29 Jan 2015 08:18:53 +0100Transit Dependent Evacuation Planning for Kathmandu Valley: A Case Study
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3944
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.Urmila Pyakurel; Marc Goerigk; Tanka Dhamala; Horst W. Hamacherreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3944Wed, 10 Dec 2014 14:11:41 +0100Sink Location to Find Optimal Shelters in Evacuation Planning
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3934
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.Philipp Heßler; Horst W. Hamacherpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3934Mon, 24 Nov 2014 11:46:38 +0100Hierarchical Edge Colorings and Rehabilitation Therapy Planning in Germany
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3832
In this paper we give an overview on the system of rehabilitation clinics in Germany in general and the literature on patient scheduling applied to rehabilitation facilities in particular.
We apply a class-teacher model developed to this environment and then generalize it to meet some of the specific constraints of inpatient rehabilitation clinics. To this end we introduce a restricted edge coloring on undirected bipartite graphs which is called group-wise balanced. The problem considered is called patient-therapist-timetable problem with group-wise balanced constraints (PTTPgb). In order to specify weekly schedules further such that they produce a reasonable allocation to morning/afternoon (second level decision) and to the single periods (third level decision) we introduce (hierarchical PTTPgb). For the corresponding model, the hierarchical edge coloring problem, we present some first feasibility results. Ines M. Raschendorfer; Horst W. Hamacherpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3832Tue, 22 Jul 2014 11:16:14 +0200On the Generality of the Greedy Algorithm for Solving Matroid Base Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3535
It is well known that the greedy algorithm solves matroid base problems for all linear cost functions and is, in fact, correct if and only if the underlying combinatorial structure of the problem is a matroid. Moreover, the algorithm can be applied to problems with sum, bottleneck, algebraic sum or \(k\)-sum objective functions. Lara Turner; Matthias Ehrgott; Horst W. Hamacherpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3535Wed, 19 Jun 2013 08:27:31 +0200The Multi Terminal q-FlowLoc Problem: A Heuristic
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2290
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.Stephanie Heller; Horst W. Hamacherreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2290Wed, 02 Mar 2011 16:46:24 +0100Universal Shortest Paths
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2230
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.Lara Turner; Horst W. Hamacherpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2230Mon, 16 Aug 2010 11:00:30 +0200Interdisziplinäres Seminar: „Fachdidaktik Chemie und Mathematik“ - Ausarbeitungen und Unterrichtsentwürfe
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2079
Im Sommersemester 2008 führte die AG Optimierung, FB Mathematik zusammen mit dem FB Chemie und dem FB Pädagogik ein interdisziplinäres Seminar zur „Fachdidaktik Chemie und Mathematik“ durch. Durch dieses integrative Lehrveranstaltungskonzept sollte die Nachhaltigkeit der Ausbildung gestärkt und die Verknüpfung von Allgemeiner Didaktik mit der Fachdidaktik sowie zwischen verschiedenen Fachbereichen gefördert werden. In dieser speziellen Veranstaltung erarbeiteten sich die Teilnehmer Inhalte in der Schnittmenge von Chemie und Mathematik, nämlich Kristallgeometrie, Analysis und Titration sowie Graphentheorie und Trennverfahren. Ihre Erkenntnisse wurden im Rahmen von Seminarvorträgen präsentiert und ausgearbeitet. Im folgenden Report befinden sich die Ausarbeitungen, welche Lernziele und Kompetenzen, Sach-, Methodische und Didaktische Analysen sowie Unterrichtsentwürfe umfassen.Florentine Bunke; Horst W. Hamacher; Anne M. Schwahnreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2079Wed, 08 Apr 2009 15:18:45 +0200A Note On Inverse Max Flow Problem Under Chebyshev Norm
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2056
In this paper, we study the inverse maximum flow problem under \(\ell_\infty\)-norm and show that this problem can be solved by finding a maximum capacity path on a modified graph. Moreover, we consider an extension of the problem where we minimize the number of perturbations among all the optimal solutions of Chebyshev norm. This bicriteria version of the inverse maximum flow problem can also be solved in strongly polynomial time by finding a minimum \(s - t\) cut on the modified graph with a new capacity function.Cigdem Güler; Horst W. Hamacherpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2056Thu, 15 Jan 2009 15:19:44 +0100A new sequential extraction heuristic for optimising the delivery of cancer radiation treatment using multileaf collimators
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1930
Finding a delivery plan for cancer radiation treatment using multileaf collimators operating in ''step-and-shoot mode'' can be formulated mathematically as a problem of decomposing an integer matrix into a weighted sum of binary matrices having the consecutive-ones property - and sometimes other properties related to the collimator technology. The efficiency of the delivery plan is measured by both the sum of weights in the decomposition, known as the total beam-on time, and the number of different binary matrices appearing in it, referred to as the cardinality, the latter being closely related to the set-up time of the treatment. In practice, the total beam-on time is usually restricted to its minimum possible value, (which is easy to find), and a decomposition that minimises cardinality (subject to this restriction) is sought.Davaasteren Baatar; Natashia Boland; Robert Johnston; Horst W. Hamacherpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1930Fri, 25 Jan 2008 22:43:20 +0100Minimum Cut Bases in Undirected Networks
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1857
Given an undirected, connected network G = (V,E) with weights on the edges, the cut basis problem is asking for a maximal number of linear independent cuts such that the sum of the cut weights is minimized. Surprisingly, this problem has not attained as much attention as its graph theoretic counterpart, the cycle basis problem. We consider two versions of the problem, the unconstrained and the fundamental cut basis problem. For the unconstrained case, where the cuts in the basis can be of an arbitrary kind, the problem can be written as a multiterminal network flow problem and is thus solvable in strongly polynomial time. The complexity of this algorithm improves the complexity of the best algorithms for the cycle basis problem, such that it is preferable for cycle basis problems in planar graphs. In contrast, the fundamental cut basis problem, where all cuts in the basis are obtained by deleting an edge, each, from a spanning tree T is shown to be NP-hard. We present heuristics, integer programming formulations and summarize first experiences with numerical tests.Florentine Bunke; Horst W. Hamacher; Francesco Maffioli; Anne Schwahnpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1857Thu, 26 Apr 2007 16:10:59 +0200Scheduling and Location (ScheLoc): Makespan Problem with Variable Release Dates
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1835
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.Donatas Elvikis; Horst W. Hamacher; Marcel T. Kalschpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1835Sun, 11 Feb 2007 14:52:09 +0100Decomposition of Matrices and Static Multileaf Collimators: A Survey
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1810
Multileaf Collimators (MLC) consist of (currently 20-100) pairs of movable metal leaves which are used to block radiation in Intensity Modulated Radiation Therapy (IMRT). The leaves modulate a uniform source of radiation to achieve given intensity profiles. The modulation process is modeled by the decomposition of a given non-negative integer matrix into a non-negative linear combination of matrices with the (strict) consecutive ones property.Matthias Ehrgott; Horst W. Hamacher; Marc Nußbaumpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1810Tue, 21 Nov 2006 13:41:32 +0100Acquisition Prioritization: A Multicriteria Approach Based on a Case Study
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1798
Selection of new projects is one of the major decision making activities in any company. Given a set of potential projects to invest, a subset which matches the company's strategy and internal resources best has to be selected. In this paper, we propose a multicriteria model for portfolio selection of projects, where we take into consideration that each of the potential projects has several - usually conflicting - values.Horst W. Hamacher; Stefan Ruzika; Akin Tanatmispreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1798Wed, 08 Nov 2006 23:24:46 +0100Semi-Simultaneous Flows and Binary Constrained (Integer) Linear Programs
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1753
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.Alexander Engau; Horst W. Hamacherpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1753Wed, 12 Jul 2006 18:03:36 +0200Hub Cover and Hub Center Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1748
Using covering problems (CoP) combined with binary search is a well-known and successful solution approach for solving continuous center problems. In this paper, we show that this is also true for center hub location problems in networks. We introduce and compare various formulations for hub covering problems (HCoP) and analyse the feasibility polyhedron of the most promising one. Computational results using benchmark instances are presented. These results show that the new solution approach performs better in most examples.Horst W. Hamacher; Tanja Meyerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1748Mon, 03 Jul 2006 19:31:22 +0200Stop Location Design in Public Transportation Networks: Covering and Accessibility Objectives
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1727
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.Dwi Retnani Poetranto; Horst. W. Hamacher; Simone Horn; Anita Schöbelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1727Mon, 24 Apr 2006 19:02:34 +0200Multiple Objective Minimum Cost Flow Problems: A Review
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1677
In this paper, theory and algorithms for solving the multiple objective minimum cost flow problem are reviewed. For both the continuous and integer case exact and approximation algorithms are presented. In addition, a section on compromise solutions summarizes corresponding results. The reference list consists of all papers known to the autheors which deal with the multiple objective minimum cost flow problem.Horst W. Hamacher; Christian Pedersen; Stefan Ruzikapreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1677Thu, 27 Oct 2005 16:08:17 +0200Finding Representative Systems for Discrete Bicriteria Optimization Problems by Box Algorithms
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1655
Given a discrete bicriteria optimization problem (DBOP), we propose two versions of an approximation procedure, the box algorithm, which results in a representation of the complete set of nondominated solutions by a finite representative system Rep satisfying the following quality features.Horst W. Hamacher; Christian Roed Pedersen; Stefan Ruzikapreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1655Wed, 17 Aug 2005 16:14:06 +0200Decomposition of Integer Matrices and Multileaf Collimator Sequencing
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1582
In this paper we consider the problem of decomposing an integer matrix into a weighted sum of binary matrices that have to strict consecutive ones property.Davaatseren Baatar; Horst W. Hamacher; Matthias Ehrgott; Gerhard J. Woegingerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1582Mon, 08 Nov 2004 11:12:15 +0100Algorithms for Time-Dependent Bicriteria Shortest Path Problems (revised version)
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1581
In this paper we generalize the classical shortest path problem in two ways. We consider two objective functions and time-dependent data. The resulting problem, called the time-dependent bicriteria shortest path problem (TdBiSP), has several interesting practical applications, but has not gained much attention in the literature.Horst W. Hamacher; Stefan Ruzika; Stevanus A. Tjandrapreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1581Thu, 28 Oct 2004 13:21:33 +0200