90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING
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Viele Firmen nutzen für ihre eigenen Softwareentwicklungen verschiedene Server mit unterschiedlichen Konfigurationen. Manche Server werden dazu eingestzt das Verhalten einer Software in einer bestimmten Umgebung zu testen und andere dienen zur Bereitstellung der Software für den Endnutzer. Hierbei ist es wichtig, dass die Konfiguration der Server regelmäßig überprüft wird. Eine solche Sicherstellung der Deployment- und Umgebungs-Integrität wird meistens durch eine Mitarbeiter der Firma oder durch einen externen Dienstleister erbracht. D.h. die Firma muss sich auf die Zuverlässigkeit eines Mitarbeiters oder einer externen Dienstleistung verlassen, bie zunehmender Komplexität ist sie sogar abhängig.
Das Ziel dieser Masterarbeit ist es, zu untersuchen, ob die Sicherstellung der Deployment- und Umgebungs-Integrität durch automatisierte kryptografische Beweise, anstelle externer Dienstleistungen oder anderer Mitarbeiter, gewährleistet werden kann. Als Anwendungsfall dient die Toll Collect GmbH. Im ersten Teil dieser Arbeit wird das Matheamtische Modell einer Blockchain erläutert. die Blockchain wurde erstmals in einem Dokument, welches unter dem Pseudonym Satoshi Nakamoto veröffentlicht wurde, beschrieben. Die erste große Anwendungen der Blockchain ist das dezentrale Zahlungssystem Bitcoin. Im zweiten Teil dieser Arbeit wird die Softwareimplementierung vorgestellt, welche im Rahmen dieser Arbeit entstanden ist. Mithilfe dieses Programms kann die Deployment- und Umgebungs-Integrität durch eine heirführ entwickelte Blockchainlösung dezentralisiert werden. Es wird außerdem der Übergang vom Mathematischen Modell zur Implementierung gezeigt.
In this paper, we present a new, optimization-based method to exhibit cyclic behavior in non-reversible stochastic processes. While our method is general, it is strongly motivated by discrete simulations of ordinary differential equations representing non-reversible biological processes, in particular molecular simulations. Here, the discrete time steps of the simulation are often very small compared to the time scale of interest, i.e., of the whole process. In this setting, the detection of a global cyclic behavior of the process becomes difficult because transitions between individual states may appear almost reversible on the small time scale of the simulation. We address this difficulty using a mixed-integer programming model that allows us to compute a cycle of clusters with maximum net flow, i.e., large forward and small backward probability. For a synthetic genetic regulatory network consisting of a ring-oscillator with three genes, we show that this approach can detect the most productive overall cycle, outperforming classical spectral analysis methods. Our method applies to general non-equilibrium steady state systems such as catalytic reactions, for which the objective value computes the effectiveness of the catalyst.
Der Einfluss von Langzahlarithmetik auf das Gewichtsraumpolyeder in mehrkriterieller Optimierung
(2018)
Die Arbeit befasst sich mit einem gewichtsraumbasierten Algorithmus, der ganzzahlige und lineare Optimierungsprobleme mit mehreren Zielfunktionen
löst und die Menge der unterstützt nicht dominierten Punkte ermittelt. Die dabei erzeugten Gewichtsraumpolyeder sind das entscheidende Mittel, um die gesuchte Lösungsmenge zu bestimmen. Aus softwaretechnischer Sicht sind numerische Ungenauigkeiten potentielle Fehlerquellen, die sich negativ auf das Endergebnis auswirken. Aus diesem Grund untersuchen wir anhand von Zuweisungs- und Rucksackinstanzen mit 3 Zielfunktionen, inwieweit der Gebrauch von Langzahlarithmetik die Gewichtsraumpolyeder und die damit verbundene Menge der unterstützt nicht dominierten Punkte beeinflusst.
Das Wind-Interpolation-Problem (WIP) ist ein bisher selten diskutiertes Problem der Flugplanungsoptimierung, bei dem es darum geht, Wind-Komponenten auf einer Luftstraße zu approximieren. Anhand von Winddaten, die vektoriell an den Gitterpunkten eines den Globus umspannenden Gitters vorliegen, soll bestimmt werden, wie viel Wind entlang der Luftstraße und quer zu ihr weht. Thema dieser Arbeit ist ein Spezialfall des WIP, nämlich das statische WIP auf einer Planfläche (SWIPP). Dazu wird zuerst ein Algorithmus besprochen, der das SWIPP zwar löst, aber einem Ansatz zugrunde liegt, der bei genauerem Hinsehen nicht sinnvoll erscheint: hier wird Wind zwischen vier Punkten interpoliert, wozu es keine triviale Methode gibt. Ähnlich zu diesem Algorithmus, der heute als State-of-the-Art gilt, wird als Ergebnis dieser Arbeit ein neuer Algorithmus vorgestellt, der das SWIPP akkurater und schneller löst. Hier wird deutlich seltener
auf die Interpolation zwischen vier Punkten zurückgegriffen - stattdessen wird fast immer linear zwischen zwei Punkten interpoliert. Die Algorithmen zum Lösen des SWIPP werden auf ihre Genauigkeit,
asymptotische Laufzeit und Geschwindigkeit untersucht und verglichen. Als Testareal dienen zum einen echte Wetterdaten sowie das Luftstraßennetz, das die Erde umspannt, und zum anderen ein eigens generiertes Windfeld und fiktive Luftstraßen. Es wird gezeigt, dass der hier vorgestellte Algorithmus die State-of-the-Art-Variante in allen genannten Aspekten übertrifft.
A Polyhedral Study of Event-Based Models for the Resource-Constrained Project Scheduling Problem
(2018)
We consider event-based Mixed-Integer Programming (MIP) models for the Resource-Constrained Project Scheduling Problem (RCPSP) that represent an alternative to the common time-indexed model (DDT) of Pritsker et al. (1969) for the case where the underlying time horizon is large or job processing times are subject to huge variations. In contrast to the time-indexed model, the size of event-based models does not depend on the time horizon. For two event-based formulations OOE and SEE of Koné et al. (2011) we present new valid inequalities that dominate the original formulation. Additionally, we introduce a new event-based model: the Interval Event-Based Model (IEE). We deduce linear transformations between all three models that yield the strict domination order IEE > SEE > OOE for their linear programming (LP) relaxations, meaning that IEE has the strongest linear relaxation among the event-based models. We further show that the popular DDT formulation can be retrieved from IEE by certain polyhedral operations, thus giving a unifying view on a complete branch of MIP formulations for the RCPSP. In addition, we analyze the computational performance of all presented models on test instances of the PSPLIB (Kolisch and Sprecher 1997).
We consider the Cumulative Scheduling Problem (CuSP) in which a set of $n$ jobs must be scheduled according to release dates, due dates and cumulative resource constraints. In constraint programming, the CuSP is modeled as the cumulative constraint. Among the most common propagation algorithms for the CuSP there is energetic reasoning (Baptiste et al., 1999) with a complexity of O(n^3) and edge-finding (Vilim, 2009) with O(kn log n) where k <= n is the number of different resource demands. We consider the complete versions of the propagators that perform all deductions in one call of the algorithm. In this paper, we introduce the energetic edge-finding rule that is a generalization of both energetic reasoning and edge-finding. Our main result is a complete energetic edge-finding algorithm with a complexity of O(n^2 log n) which improves upon the complexity of energetic reasoning. Moreover, we show that a relaxation of energetic edge-finding with a complexity of O(n^2) subsumes edge-finding while performing stronger propagations from energetic reasoning. A further result shows that energetic edge-finding reaches its fixpoint in strongly polynomial time. Our main insight is that energetic schedules can be interpreted as a single machine scheduling problem from which we deduce a monotonicity property that is exploited in the algorithms. Hence, our algorithms improve upon the strength and the complexity of energetic reasoning and edge-finding whose complexity status seemed widely untouchable for the last decades.
SCIP-JACK is a customized, branch-and-cut based solver for Steiner tree and related problems. ug [SCIP-JACK, MPI] extends SCIP-JACK to a massively par- allel solver by using the Ubiquity Generator (UG) framework. ug [SCIP-JACK, MPI] was the only solver that could run on a distributed environment at the (latest) 11th DIMACS Challenge in 2014. Furthermore, it could solve three well-known open instances and updated 14 best known solutions to instances from the bench- mark libary STEINLIB. After the DIMACS Challenge, SCIP-JACK has been con- siderably improved. However, the improvements were not reflected on ug [SCIP- JACK, MPI]. This paper describes an updated version of ug [SCIP-JACK, MPI], especially branching on constrains and a customized racing ramp-up. Furthermore, the different stages of the solution process on a supercomputer are described in detail. We also show the latest results on open instances from the STEINLIB.
Given a factorable function f, we propose a procedure that constructs a concave underestimor of f that is tight at a given point. These underestimators can be used to generate intersection cuts. A peculiarity of these underestimators is that they do not rely on a bounded domain. We propose a strengthening procedure for the intersection cuts that exploits the bounds of the domain. Finally, we propose an extension of monoidal strengthening to take advantage of the integrality of the non-basic variables.
The Steiner tree problem in graphs is a classical problem that commonly arises in practical applications as one of many variants. Although the different Steiner tree problem variants are usually strongly related, solution approaches employed so far have been prevalently problem-specific. Against this backdrop, the solver SCIP-Jack was created as a general-purpose framework that can be used to solve the classical Steiner tree problem and 11 of its variants. This versatility is achieved by transforming various problem variants into a general form and solving them by using a state-of-the-art MIP-framework. Furthermore, SCIP-Jack includes various newly developed algorithmic components such as preprocessing routines and heuristics. The result is a high-performance solver that can be employed in massively parallel environments and is capable of solving previously unsolved instances. After the introduction of SCIP-Jack at the 2014 DIMACS Challenge on Steiner problems, the overall performance of the solver has considerably improved. This article provides an overview on the current state.
In this thesis we study order picking optimization problems for a two-blocks rectangle warehouse layout. We present combinatorial formulations and linear programming models based on the Steiner graph representation for order batching, picker routing, and joint order batching and picker routing problems. A special case of the latter is
considered. This case assumes that each order contains exactly one item and each item can be picked from different possible locations in a warehouse. The underlying optimization problem is called joint multi-location order batching and picker routing problem (JMLOBPRP).
Since having only one-item orders turns the JMLOBPRP into a special case of a capacitated vehicle routing problem, we suggest to implement algorithmic approaches for those to solve the JMLOBPRP. In particular, we define the JMLOBPRP as a generalization of the resource constrained assignment problem, for which a regional search method exists. The intention of the thesis is to investigate how a relaxation of the JMLOBPRP, a so-called group assignment problem (GrAP), can be solved following the ideas of regional search. We present a mathematical model of the GrAP and prove that it is NP-hard. Furthermore, we propose a novel heuristic algorithm for the GrAP. We call this method a network search algorithm, as it is based on a Lagrangian relaxation
of the GrAP, which is solved by the network simplex method. On each its iteration network search examines a solution region suggested by the network simplex algorithm and improves the incumbent solution.
Numerical experiments are conducted to assess a performance of the network search method. We create more realistic problem instances. The proposed algorithm is compared to the integer optimal solution of the GrAP and optimal fractional solution of its linear relaxation. Both computed using the commercial linear solver Gurobi. Our experiments
show that the developed network search algorithm leads to the hight-quality solution within a short computing time. The results obtained testing large problem instances which cannot be solved by Gurobi within a reasonable computing time, show that the network search method provides a solution approach which can be used in practice.
This thesis deals with a new algorithm for finding Shortest Paths on Airway Networks. It is about a Bidirectional A* Search, a Greedy algorithm exploring a network from two sides instead of one. We will use it to solve the so-called 'Horizontal Flight Trajectory Problem', where one searches for an aircraft trajectory between two airports of minimal costs on an Airway Network. The given network will be modeled as a directed graph and in order to reflect reality we concentrate on the dynamic version. Here a timedependent cost function for all arcs is integrated, that shall represent the winds blowing. This way we model the Horizontal Flight Trajectory Problem mathematically as a Time-Dependent Shortest Path Problem.
The basic algorithm idea derives from the algorithm presented in 'Bidirectional A* Search on Time-Dependent Road Networks' [1], where a similar setting is elaborated for road networks. The algorithm procedure bears on a modified generalization of Dijkstra's algorithm, made bidirectional and improved in several aspects. As for the backwards search the arrival times are not known in advance, the reversed graph it occurs on has to be weighted by a lower bound. Contrary to the static case the forwards search still has to go on, when they 'meet' in one node. In the static case, the shortest path would have been found at this point.
For road networks the TDSPP is well-studied, for airway networks cannot be found as much in literature. In order to test efficiency, we implement Dijkstra's algorithm, unidirectional A* Search and Bidirectional A* Search. We draw up how potential functions for the static case could look like and that with a suitable potential A* Search with works approx. 7 times faster than Dijkstra in the dynamic case. Our computations lead also to the result, that the unidirectional A* Search works even better on the network than our new bidirectional approach does. On average it labels fewer nodes and also yields 1,7 times faster to the solutions.
For assessing the efficiency of the different algorithms we compare the running times and to exclude processor characteristics we consider also the set labels relative to the labels on the resulting optimal path. In addition, we present examples of routes visually
and explain shortly why there appear local differences regarding performance of A* Search and Bidirectional A* Search.
In dieser Arbeit wird die Platzierung von Ladestationen für Elektrobusse untersucht. Dabei soll für eine Menge an gegebenen Linien eine Menge an Ladestationen gefunden werden, sodass jede Linie mit Nutzung der Ladestationen befahren werden kann und gleichzeitig die Kosten minimal sind. Zunächst wird der Fall betrachtet, dass die Batterie an jeder Station komplett vollgeladen werden könnte. Dieses Problem stellt sich als NP-schwer heraus. Für einige einfachere Fällewerden zudem Algorithmen entwickelt und untersucht. Anschließend wird der Fall einer unbegrenzt großen Batterie betrachtet, wobei an jeder Station derselbe Wert geladen werden kann. Auch dieses Problem ist NP-schwer. Erneut werden Algorithmen zur Lösung vereinfachter Problemstellungen gegeben und analysiert. Wird zudem angenommen, an jeder Station würde ein individueller Wert geladen, so ist das Problem schon für nur eine einzige Linie NP-schwer. Dennoch werden zwei exakte und ein approximierender Algorithmus entwickelt. Schließlich wird eine Batteriekapazität hinzugefügt und die zuvor entwickelten Algorithmen
werden entsprechend angepasst. Für die abschließende Problemdefinition werden verschiedene Batteriegrößen betrachtet und es werden zwei gemischt-ganzzahlige Programme aufgestellt. Anhand von existierenden Buslinien aus Berlin werden diese untersucht. Dabei stellt sich heraus, dass die Batteriekosten einen deutlich größeren Teil der Kosten ausmachen als die Ladestationen. Zudem sollten kleinere Batterien statt größerer und mehr Ladestationen genutzt werden.
During the past years hospitals saw themselves confronted with increasing economical pressure (WB06, p. V). Therefore, optimizing the general operational procedures has gained in importance. The revenue of a hospital depends on the kinds and quantity of treatments performed and on the effcient use and utilization of the corresponding resources. About 25 − 50% of the treatment costs of a patient needing surgery incurs in the operating rooms (WB06, p. 58). Hence skillful management of the operating rooms can have a large impact on the overall revenue of a hospital. Belien and Demeulemeester (BD07) describe the planning of operating room (OR) schedules as a multi-stage process. In the first stage OR time is allocated to the hospitals specialties and capacities and resources are adjusted. In the second stage a master surgery schedule (MSS) is developed, that is a timetable for D days that specifies the amount of OR time assigned to the specialties on every individual day. After D days this schedule will be repeated without any changes. Hence, developing an MSS is a long-term problem. Finally, specialties will schedule specific surgeries within their assigned OR time. In this work we will focus on the development of the MSS that maximizes the revenue of the hospital. Our main focus will be to ensure that the capacities of the downstream resources, i.e. the bed capacities in the ICU and ward, will not be exceeded. Additionally, we hope that our formulation of the problem will lead to a leveled bed demand without significant peaks. We will incorporate the uncertainty of patient demand and case mix in our model. There have been several approaches on this subject, for example in (Fü15) and (BD07) and this work is in part inuenced by these advances.
In commodity transport networks such as natural gas, hydrogen and water networks, flows arise from nonlinear potential differences between the nodes, which can be represented by so-called "potential-driven" network models. When operators of these networks face increasing demand or the need to handle more diverse transport situations, they regularly seek to expand the capacity of their network by building new pipelines parallel to existing ones ("looping").
The paper introduces a new mixed-integer non-linear programming (MINLP) model and a new non-linear programming (NLP) model and compares these with existing models for the looping problem and related problems in the literature, both theoretically and experimentally.
On this basis, we give recommendations about the circumstances under which a certain model should be used. In particular, it turns out that one of our novel models outperforms the existing models.
Moreover, the paper is the first to include the practically relevant option that a particular pipeline may be looped several times.
Calculation of clinch and elimination numbers for sports leagues with multiple tiebreaking criteria
(2018)
The clinch (elimination) number is a minimal number of future wins (losses) needed to clinch (to be eliminated from) a specified place in a sports league. Several optimization models and computational results are shown in this paper for calculating clinch and elimination numbers in the presence of predefined multiple tiebreaking criteria. The main subject of this paper is to provide a general algorithmic framework based on integer programming with utilizing possibly multilayered upper and lower bounds.
The aim of multimodal routing is to extract the best integrated journey of multiple
transportation networks. The integration of bike rental networks is challenging
particularly with respect to recognizing a valid path dependent on real-time
availability of bike boarding and alighting places. In this work a common model
for station-based bike rental networks extended with boarding possibilities for
free floating bikes is presented. Moreover a new model for alighting inside a free
floating area is introduced. In addition, a prototype of multimodal routing with
a bike rental network in Berlin is developed by extending the OpenTripPlanner
software. Due to recent public dispute about bike rental networks in Berlin, an
examination about speed-up potential of an integrated bike rental network in the
public transit of Berlin is provided.
Improving branching for disjunctive polyhedral models using approximate convex decompositions
(2018)
Disjunctive sets arise in a variety of optimization models and much esearch has been devoted to obtain strong relaxations for them. This paper focuses on the evaluation of the relaxation during the branch-and-bound search process. We argue that the branching possibilities (\ie binary variables) of the usual formulations are unsuitable to obtain strong bounds early in the search process as they do not capture the overall shape of the the entire disjunctive set. To analyze and exploit the shape of the disjunctive set we propose to compute a hierarchy of approximate convex decompositions and show how to extend the known formulations to obtain improved branching behavior.
Gas networks are an important application area for optimization. When considering long-range transmission, compressor stations play a crucial role in these applications. The purpose of this report is to collect and systematize the models used for compressor stations in the literature. The emphasis is on recent work on simple yet accurate polyhedral models that may replace more simplified traditional models without increasing model complexity. The report also describes an extension of the compressor station data available in GasLib (http://gaslib.zib.de/) with the parameters of these models.
We consider the modeling of operation modes for complex compressor stations (i.e., ones with several in- or outlets) in gas networks. In particular, we propose a refined model that allows to precompute tighter relaxations for each operation mode. These relaxations may be used to strengthen the compressor station submodels in gas network optimization problems. We provide a procedure to obtain the refined model from the input data for the original model. This procedure is based on a nontrivial reduction of the graph representing the gas flow through the compressor station in an operation mode.
Improving relaxations for potential-driven network flow problems via acyclic flow orientations
(2018)
The class of potential-driven network flow problems provides important models for a range of infrastructure networks. For real-world applications, they need to be combined with integer
models for switching certain network elements, giving rise to hard-to-solve MINLPs. We observe that on large-scale real-world meshed networks the usually employed relaxations are rather weak due to cycles in the network.
We propose acyclic flow orientations as a combinatorial relaxation of feasible solutions of potential-driven flow problems and show how they can be used to strengthen existing relaxations. First computational results indicate that the strengthend model is much tighter than the original relaxation, thus promising a computational advantage.
State-of-the-art solvers for mixed integer programs (MIP) govern a variety of algorithmic components. Ideally, the solver adaptively learns to concentrate its computational budget on those components that perform well on a particular problem, especially if they are time consuming.
We focus on three such algorithms, namely the classes of large neighborhood search and diving heuristics as well as Simplex pricing strategies.
For each class we propose a selection strategy that is updated based on the observed runtime behavior, aiming to ultimately select only the best algorithms for a given instance.
We review several common strategies for such a selection scenario under uncertainty, also known as Multi Armed Bandit Problem.
In order to apply those bandit strategies, we carefully design reward functions to rank and compare each individual heuristic or pricing algorithm within its respective class.
Finally, we discuss the computational benefits of using the proposed adaptive selection within the \scip Optimization Suite on publicly available MIP instances.
Large Neighborhood Search (LNS) heuristics are among the most powerful but also most expensive heuristics for mixed integer programs (MIP). Ideally, a solver learns adaptively which LNS heuristics work best for the MIP problem at hand in order to concentrate its limited computational budget.
To this end, this work introduces Adaptive Large Neighborhood Search (ALNS) for MIP, a primal heuristic that acts a framework for eight popular LNS heuristics such as Local Branching and Relaxation Induced Neighborhood Search (RINS). We distinguish the available LNS heuristics by their individual search domains, which we call neighborhoods. The decision which neighborhood should be executed is guided by selection strategies for the multi armed bandit problem, a related optimization problem during which suitable actions have to be chosen to maximize a reward function. In this paper, we propose an LNS-specific reward function to learn to distinguish between the available neighborhoods based on successful calls and failures. A second, algorithmic enhancement is a generic variable fixing priorization, which ALNS employs to adjust the subproblem complexity as needed. This is particularly useful for some neighborhoods which do not fix variables by themselves. The proposed primal heuristic has been implemented
within the MIP solver SCIP. An extensive computational study is conducted to compare different LNS strategies within our ALNS framework on a large set of publicly available MIP instances from the MIPLIB and Coral benchmark sets. The results of this simulation are used to calibrate the parameters of the bandit selection strategies. A second computational experiment shows the computational benefits of the proposed ALNS framework within the MIP solver SCIP.
The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming framework SCIP. This paper discusses enhancements and extensions contained in version 6.0 of the SCIP Optimization Suite. Besides performance improvements of the MIP and MINLP core achieved by new primal heuristics and a new selection criterion for cutting planes, one focus of this release are decomposition algorithms. Both SCIP and the automatic decomposition solver GCG now include advanced functionality for performing Benders’ decomposition in a generic framework. GCG’s detection loop for structured matrices and the coordination of pricing routines for Dantzig-Wolfe decomposition has been significantly revised for greater flexibility. Two SCIP extensions have been added
to solve the recursive circle packing problem by a problem-specific column generation scheme and to demonstrate the use of the new Benders’ framework for stochastic capacitated facility location. Last, not least, the report presents updates and additions to the other components and extensions of the SCIP Optimization Suite: the LP solver SoPlex, the modeling language Zimpl, the parallelization framework UG, the Steiner tree solver SCIP-Jack, and the mixed-integer semidefinite programming solver SCIP-SDP.
Mixed integer programming is a versatile and valuable optimization tool. However, solving specific problem instances can be computationally demanding even for cutting-edge solvers. Such long running times are often significantly reduced by an appropriate change of the solver's parameters. In this paper we investigate "algorithm selection", the task of choosing among a set of algorithms the ones that are likely to perform best for a particular instance.
In our case, we treat different parameter settings of the MIP solver SCIP as different algorithms to choose from. Two peculiarities of the MIP solving process have our special attention. We address the well-known problem of performance variability by using multiple random seeds. Besides solving time, primal dual integrals are recorded as a second performance measure in order to distinguish solvers that timed out.
We collected feature and performance data for a large set of publicly available MIP instances. The algorithm selection problem is addressed by several popular, feature-based methods, which have been partly extended for our purpose. Finally, an analysis of the feature space and performance results of the selected algorithms are presented.
The design of rolling stock rotations is an important task in large-scale railway planning. This so-called rolling stock rotation problem (RSRP) is usually tackled using an integer programming approach. Markus Reuther did so in his dissertation [15] for the ICE railway network of DB ("Deutsche Bahn"). Due to the size of the network and the complexity of further technical requirements, the resulting integer problems tend to become very large and computationally involved. In this thesis, we tackle the linear programming relaxation of the RSRP integer program. We will do so by applying a modified version of an algorithm recently proposed by Dan Bienstock and Mark Zuckerberg [2] for the precedence constrained production scheduling
problem that arises in open pit mine scheduling. This problem contains a large number of "easy" constraints and a relatively small number of "hard" constraints. We will see that a similar problem structure can also be found in the RSRP. The Bienstock-Zuckerberg algorithm relies on applying Lagrangian relaxation to the hard constraints as well as on partitioning the variable set. We propose three different partition schemes which try to exploit the specific problem structure of the RSRP. Furthermore, we will discuss the influence of primal degeneracy on the algorithm's performance, as well as possible merits of perturbating the right-hand side of the constraint matrix. We provide computational results to assess the performance of those approaches.
We establish a general computational framework for Chvátal’s conjecture based on exact rational integer programming. As a result we prove Chvátal’s conjecture holds for all downsets whose union of sets contains seven elements or less. The computational proof relies on an exact branch-and-bound certificate that allows for elementary verification and is independent of the integer programming solver used.
A Simple Way to Compute the Number of Vehicles That Are Required to Operate a Periodic Timetable
(2018)
We consider the following planning problem in public transportation: Given a
periodic timetable, how many vehicles are required to operate it?
In [9], for this sequential approach, it is proposed to first expand the periodic
timetable over time, and then answer the above question by solving a flow-based
aperiodic optimization problem.
In this contribution we propose to keep the compact periodic representation of
the timetable and simply solve a particular perfect matching problem. For practical
networks, it is very much likely that the matching problem decomposes into several
connected components. Our key observation is that there is no need to change any
turnaround decision for the vehicles of a line during the day, as long as the timetable
stays exactly the same.
Cycle inequalities play an important role in the polyhedral study of the periodic
timetabling problem. We give the first pseudo-polynomial time separation algo-
rithm for cycle inequalities, and we give a rigorous proof for the pseudo-polynomial
time separability of the change-cycle inequalities. Moreover, we provide several
NP-completeness results, indicating that pseudo-polynomial time is best possible.
The efficiency of these cutting planes is demonstrated on real-world instances of the
periodic timetabling problem.
All feasible flows in potential-driven networks
induce an orientation on the undirected graph underlying the network.
Clearly, these orientations must satisfy two conditions: they are acyclic and there are no "dead ends" in the network, i.e. each source requires outgoing flows, each sink requires incoming flows, and each transhipment vertex requires both an incoming and an outgoing flow. In this paper we will call orientations that satisfy these conditions acyclic source-transhipment-sink orientations (ASTS-orientation) and study their structure. In particular, we characterize graphs that allow for such an orientation, describe a way to enumerate all possible ASTS-orientations of a given graph, present an algorithm to simplify and decompose a graph before such an enumeration and shed light on the role of zero flows in the context of ASTS-orientations.
Mit dem Voranschreiten der Technologie erhalten die öffentlichen Verkehrsmittel eine größere Bedeutung. Die Beförderung mehrerer Personen eröffnet der Gesellschaft viele Möglichkeiten, unter Anderem den Vorteil der Zeitersparnis. Die Dauer des Verkehrswegs mit öffentlichen Verkehrsmitteln ist häufig geringer, als die mit individuellen Verkehrsmitteln. Jedes öffentliche Transportmittel ist mit einem Fahrplan versehen. Dieser bietet Passagieren, die öffentliche Verkehrsmittel öfter nutzen, eine Strukturierung und Planung ihrer Zeit. Dabei lassen sich Taktfahrpläne aufgrund ihres periodischen Verhaltens leicht einprägen. Dieses periodische Verhalten ist durch mathematische Modellierungen darstellbar. Das persönliche Nutzverhalten vieler Bürger im Personenverkehr ist auf die öffentlichen Verkehrsmittel beschränkt. Diese beinhalten im Gegensatz zum individuellen Verkehrsmittel eine Wartezeit. Dabei stellt sich die Frage, ob man anhand mathematischer Modelle diese Wartezeit minimieren kann. Eine bekannte mathematische Modellierung dieses Problems ist das Periodic Event Scheduling Problem (PESP). Die optimale Planung eines periodischen Taktfahrplanes steht im Vordergrund.
Während ich dieses Problem betrachtet habe, wurde ich auf das Rechnen mit linearen Gleichungssystemen modulo T aufmerksam. Bei periodischen Taktfahrplänen wird ein einheitliches zeitliches Muster, welches sich nach T Minuten wiederholt, betrachtet. Das dabei zu betrachtende Lösungsproblem eröffnet ein Teilgebiet der Mathematik, welches bislang nicht im Vordergrund stand: Das Lösen linearer Gleichungen modulo T, wobei T für die Zeit in Minuten steht und somit 60 ist. Da 60 keine Primzahl ist, kann – wie im Laufe der Arbeit präsentiert – das lineare Gleichungssystem nicht mehr über einen Körper gelöst werden. Lineare Gleichungssysteme werden nun über Nicht-Körpern betrachtet. Die Literatur weist sowohl im deutschsprachigem als auch im englischsprachigen Raum wenig Umfang bezüglich linearer Gleichungssysteme über Nicht-Körper auf. Der Bestand an Fachliteratur bezüglich den Themen lineare diophantische Gleichungssysteme, Hermite-
Normalform und Smith-Normalform ist zurzeit gering, dennoch erreichbar, beispielsweise in [1], welches in dieser Bachelorarbeit genutzt wurde. Insbesondere wurde ich bei der Suche nach geeigneter Literatur zu linearen Gleichungssystemen über Restklassenringe, die keinen Körper bilden, nicht fündig. Dabei recherchierte ich sowohl in den Universitätsbibliotheken als auch in webbasierenden Suchmaschinen.
Aufgrund dem geringen Bestand an Fachliteratur in diesem Kontext, war ich gezwungen, an vielen Stellen eigene logische Verknüpfungen zu konzipieren und zu beweisen. Dies brachte viele Schwierigkeiten mit sich, die mit bestmöglichem Verständnis bearbeitet wurden.
Abseits der Zugänglichkeit der Literatur, finde ich es sehr überraschend, dass sich viele Professoren der Mathematik mit diesem Themenbereich nicht beschäftigten. Insbesondere gingen von den Dozenten, die ich um Literaturempfehlung bat, kein Werk aus. Damit
wurde das Thema "Lineare Gleichungssysteme Modulo T" einerseits eine große Herausforderung, andererseits eine große Motivation, da ich mit dieser Bachelorarbeit vielen Interessenten der Mathematik als Sekundärliteratur dienen kann.