90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING
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For this thesis we study the Constrained Horizontal Flightplanning Problem (CHFPP) for which one has to find the path of minimum cost between airports s and t in a directed graph that respects a set of boolean constraints. To this end we give a survey of three different multilabel algorithms that all use a domination subroutine. We summarize an approach by Knudsen, Chiarandini and Larsen to define this domination and afterwards present our own method which builds on that approach. We suggest different implementation techniques to speed up the computation time, most notably a Reoptimization for an iterative method to solve the problem. Furthermore we implemented the different versions of the algorithm and present statistics on their computation as well as an overview of statistics on the set of real-world constraints that we were given. Finally we present two alternative approaches that tackle the problem, a heuristic with similarities to a Lagrangian relaxation and an approach that makes use of an algorithm which finds the k shortest path of a graph such as the ones of Epstein or Yen.
Scheduling ist ein wichtiger Forschungsgegenstand im Bereich der diskreten Optimierung. Es geht darum, einen Schedule, d.h. einen Ablaufplan, für gegebene Ereignisse zu finden. Dieser soll optimal hinsichtlich einer Zielfunktion wie zum Beispiel minimaler Dauer oder Kosten sein. Dabei gibt es in der Regel Nebenbedingungen wie Vorrangbeziehungen zwischen den Ereignissen oder zeitliche Einschränkungen, die zu erfüllen sind. Falls die Ereignisse periodisch wiederkehren, spricht man von periodischem Scheduling. Beispiele sind das Erstellen von Zugfahrplänen, die Schaltungvon Ampelsignalen oder die Planung von Produktionsabläufen.
Mathematisch können diese Probleme mit dem Periodic Event Scheduling Problem (PESP) modelliert werden, das als gemischt-ganzzahliges Programm formuliert werden kann. In dieser Bachelorarbeit wird ein Ansatz zur Lösung des PESP mittels Zerlegung und Dualisierung entwickelt. In den Kapiteln 2 und 3 werden zunächst die notwendigen graphentheoretischen Grundlagen und das PESP eingeführt. In Kapitel 4 wird das PESP durch Fixierung der ganzzahligen Variablen in lineare Programme zerlegt. Dieses Unterproblem wird dualisiert und wieder in das PESP eingesetzt. Dafür ist eine weitere Nebenbedingung nötig.
Im fünften Kapitel behandeln wir die Lösung des teildualisierten PESP. Eine Möglichkeit ist es, sich auf eine Teilmenge der Nebenbedingungen zu beschränken. Eine weitere Möglichkeit ist ein Algorithmus, derähnlich wie BendersZerlegung die Nebenbedingungen dynamisch erzeugt. Dieser Algorithmus wird in Kapitel 6 implementiert und an vier Beispielen getestet.
We propose a tropical interpretation of the solution space of the Periodic Event Scheduling Problem as a collection of polytropes, making use of the characterization of tropical cones as weighted digraph polyhedra. General and geometric properties of the polytropal collection are inspected and understood in connection with the combinatorial properties of the underlying periodic event scheduling instance. Novel algorithmic
ideas are presented and tested, making use of the aforementioned theoretical results to solve and optimize the problem.
One of the fundamental steps in the optimization of public transport is line planning. It involves determining lines and assigning frequencies of service such that costs are minimized while also maximizing passenger comfort and satisfying travel demands. We formulate the problem as a mixed integer linear program that considers all circuit-like lines in a graph and allows free passenger routing. Traveler and operator costs are included in a linear scalarization in the objective. We apply said programming problem to the Parametric City, which is a graph model introduced by Fielbaum, Jara-Díaz and Gschwender that exibly represents different cities. In his dissertation, Fielbaum solved the line planning problem for various parameter choices in the Parametric City. In a first step, we therefore review his results and make comparative computations. Unlike Fielbaum we arrive at the conclusion that the optimal line plan for this model indeed depends on the demand. Consequently, we analyze the line planning problem in-depth: We find equivalent, but easier to compute formulations and provide a lower bound by LP-relaxation, which we show to be equivalent to a multi-commodity flow problem. Further, we examine what impact symmetry has on the solutions. Supported both by computational results as well as by theoretical analysis, we reach the conclusion that symmetric line plans are optimal or near-optimal in the Parametric City. Restricting the model to symmetric line plans allows for a \kappa-factor approximation algorithm for the line planning problem in the Parametric City.
Air freight is usually shipped in standardized unit load devices (ULDs).
The planning process for the consolidation of transit cargo from inbound flights or locally emerging shipments into ULDs for outbound flights is called build-up scheduling.
More specifically, outbound ULDs must be assigned a time and a workstation subject to both workstation capacity constraints and the availability of shipments which in turn depends on break-down decisions for incoming ULDs.
ULDs scheduled for the same outbound flight should be built up in temporal and spatial proximity.
This serves both to minimize overhead in transportation times and to allow workers to move freight between ULDs.
We propose to address this requirement by processing ULDs for the same outbound flight in batches.
For the above build-up scheduling problem, we introduce a multi-commodity network design model.
Outbound flights are modeled as commodities; transit cargo is represented by cargo flow volume and unpack and batch decisions are represented as design variables.
The model is solved with standard MIP solvers on a set of benchmark data.
For instances with a limited number of resource conflicts, near-optimal solutions are found in under two hours for a whole week of operations.
The Periodic Event Scheduling Problem (PESP) is the central mathematical model behind the optimization of periodic timetables in public transport. We apply Benders decomposition to the incidence-based MIP formulation of PESP. The resulting formulation exhibits particularly nice features: The subproblem is a minimum cost network flow problem, and feasibility cuts are equivalent to the well-known cycle inequalities by Odijk. We integrate the Benders approach into a branch-and-cut framework, and assess the performance of this method on instances derived from the benchmarking library PESPlib.
Balanced separators are node sets that split the graph into size bounded components. They find applications in different theoretical and practical problems. In this paper we discuss how to find a minimum set of balanced separators in node weighted graphs. Our contribution is a new and exact algorithm that solves Minimum Balanced Separators by a sequence of Hitting Set problems. The only other exact method appears to be a mixed-integer program (MIP) for the edge weighted case. We adapt this model to node weighted graphs and compare it to our approach on a set of instances, resembling transit networks. It shows that our algorithm is far superior on almost all test instances.
Periodic timetable optimization problems in public transport can be modeled as mixed-integer linear programs by means of the Periodic Event Scheduling Problem (PESP). In order to keep the branch-and-bound tree small, minimum integral cycle bases have been proven successful. We examine forward cycle bases, where no cycle is allowed to contain a backward arc. After reviewing the theory of these bases, we describe the construction of an integral forward cycle basis on a line-based event-activity network. Adding turnarounds to the instance \texttt{R1L1} of the benchmark library PESPlib, we computationally evaluate three types of forward cycle bases in the Pareto sense, and come up with significant improvements concerning dual bounds.
We investigate preprocessing for single-source shortest path queries in digraphs, where arc costs are only known to lie in an interval. More precisely, we want to decide for each arc whether it is part of some shortest path tree for some realization of costs. We show that this problem is solvable in polynomial time by giving a combinatorial algorithm, using optimal structures that we call forks. Our algorithm turns out to be very efficient in practice, and is sometimes even superior in quality to a heuristic developed for the one-to-one shortest path problem in the context of passenger routing in public transport.
The covering of a graph with (possibly disjoint) connected subgraphs is a fundamental problem in graph theory. In this paper, we study a version to cover a graph's vertices by connected subgraphs subject to lower and upper weight bounds, and propose a column generation approach to dynamically generate feasible and promising subgraphs. Our focus is on the solution of the pricing problem which turns out to be a variant of the NP-hard Maximum Weight Connected Subgraph Problem. We compare different formulations to handle connectivity, and find that a single-commodity flow formulation performs best. This is notable since the respective literature seems to have dismissed this formulation. We improve it to a new coarse-to-fine flow formulation that is theoretically and computationally superior, especially for large instances with many vertices of degree 2 like highway networks, where it provides a speed-up factor of 10 over the non-flow-based formulations. We also propose a preprocessing method that exploits a median property of weight constrained subgraphs, a primal heuristic, and a local search heuristic. In an extensive computational study we evaluate the presented connectivity formulations on different classes of instances, and demonstrate the effectiveness of the proposed enhancements. Their speed-ups essentially multiply to an overall factor of 20. Overall, our approach allows the reliabe solution of instances with several hundreds of nodes in a few minutes. These findings are further corroborated in a comparison to existing districting models on a set of test instances from the literature.
We present an optimization model which is capable of routing and ordering trains on a microscopic level under a moving block regime. Based on a general timetabling definition (GTTP) that allows the plug in of arbitrarily detailed methods to compute running and headway times, we describe a layered graph approach using velocity expansion, and develop a mixed integer linear programming formulation. Finally, we present promising results for a German corridor scenario with mixed traffic, indicating that applying branch-and-cut to our model is able to solve reasonably sized instances with up to hundred trains to optimality.
In practice, transient gas transport problems frequently have to be solved for large-scale gas networks. Gas network optimization problems typically belong to the class of Mixed-Integer Nonlinear Programming Problems (MINLP). However current state-of-the-art MINLP solvers are not yet mature enough to solve large-scale real-world instances. Therefore, an established approach in practice is to solve the problems with respect to a coarser representation of the network and thereby reducing the size of the underlying model. Two well-known aggregation methods that effectively reduce the network size are parallel and serial pipe merges. However, these methods have only been studied in stationary gas transport problems so far. This paper closes this gap and presents parallel and serial pipe merging methods in the context of transient gas transport. To this end, we introduce the concept of equivalent and heuristic subnetwork replacements. For the heuristic methods, we conduct a huge empirical evaluation based on real-world data taken from one of the largest gas networks in Europe. It turns out that both, parallel and serial pipe merging can be considered as appropriate aggregation methods for real-world transient gas flow problems.
Growing demand, distributed generation, such as renewable energy sources (RES), and the increasing role of storage systems to mitigate the volatility of RES on a medium voltage level, push existing distribution grids to their limits. Therefore, necessary network expansion needs to be evaluated to guarantee a safe and reliable electricity supply in the future taking these challenges into account. This problem is formulated as an optimal power flow (OPF) problem which combines network expansion, volatile generation and storage systems, minimizing network expansion and generation costs. As storage systems introduce a temporal coupling into the system, a multiperiod OPF problem is needed and analysed in this thesis. To reduce complexity, the network expansion problem is represented in a continuous nonlinear programming formulation by using fundamental properties of electrical engeneering. This formulation is validated succesfully against a common mixed integer programming approach on a 30 and 57 bus network with respect to solution and computing time. As the OPF problem is, in general, a nonconvex, nonlinear problem and, thus, hard to solve, convex relaxations of the power flow equations have gained increasing interest. Sufficient conditions are represented which guarantee exactness of a second-order cone (SOC) relaxation of an operational OPF in radial networks. In this thesis, these conditions are enhanced for the network expansion planning problem. Additionally, nonconvexities introduced by the choice of network expansion variables are relaxed by using McCormick envelopes. These relaxations are then applied on the multiperiod OPF and compared to the original problem on a 30 and a 57 bus network. In particular, the computational time is decreased by an order up to 10^2 by the SOC relaxation while it provides either an exact solution or a sufficient lower bound on the original problem. Finally, a sensitivity study is performed on weights of network expansion costs showing strong dependency of both the solution of performed expansion and solution time on the chosen weights.
We propose a new mixed integer programming based heuristic for computing new benchmark primal solutions for instances of the PESPlib. The PESPlib is a collection of instances for the Periodic Event Scheduling Problem (PESP), comprising periodic timetabling problems inspired by real-world railway timetabling settings, and attracting several international research teams during the last years. We describe two strategies to merge a set of good periodic timetables. These make use of the instance structure and minimum weight cycle bases, finally leading to restricted mixed integer programming formulations with tighter variable bounds. Implementing this timetable merging approach in a concurrent solver, we improve the objective values of the best known solutions for the smallest and largest PESPlib instances by 1.7 and 4.3 percent, respectively.
Secure energy transport is considered as highly relevant for the basic infrastructure of nowadays society and economy. To satisfy increasing demands and to handle more diverse transport situations, operators of energy networks regularly expand the capacity of their network by building new network elements, known as the expansion planning problem.
A key constraint function in expansion planning problems is a nonlinear and nonconvex potential loss function. In order to improve the algorithmic performance of state-of-the-art MINLP solvers, this paper presents an algebraic description for the convex envelope of this function. Through a thorough computational study, we show that this tighter relaxation tremendously improve the performance of the MINLP solver SCIP on a large test set of practically relevant instances for the expansion planning problem. In particular, the results show that our achievements lead to an improvement of the solver performance for a development version by up to 58%.
We report our progress on the project for solving larger scale quadratic assignment problems (QAPs). Our main approach to solve large scale NP-hard combinatorial optimization problems such as QAPs is a parallel branch-and-bound method efficiently implemented on a powerful computer system using the Ubiquity Generator(UG) framework that can utilize more than 100,000 cores. Lower bounding procedures incorporated in the branch-and-bound method play a crucial role in solving the problems. For a strong lower bounding procedure, we employ the Lagrangian doubly nonnegative (DNN) relaxation and the Newton-bracketing method developed by the authors’ group. In this report, we describe some basic tools used in the project including the lower bounding procedure and branching rules, and present some preliminary numerical results.
Our next target problem is QAPs with dimension at least 50, as we have succeeded to solve tai30a and sko42 from QAPLIB for the first time.
The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. We describe a substantial revision and extension of this framework that integrates symbolic presolving, features an exact repair step for solutions from primal heuristics, employs a faster rational LP solver based on LP iterative refinement, and is able to produce independently verifiable certificates of optimality. We study the significantly improved performance and give insights into the computational behavior of the new algorithmic components. On the MIPLIB 2017 benchmark set, we observe an average speedup of 6.6x over the original framework and 2.8 times as many instances solved within a time limit of two hours.
A decision support system relies on frequent re-solving of similar problem instances. While the general structure remains the same in corresponding applications, the input parameters are updated on a regular basis. We propose a generative neural network design for learning integer decision variables of mixed-integer linear programming (MILP) formulations of these problems. We utilise a deep neural network discriminator and a MILP solver as our oracle to train our generative neural network. In this article, we present the results of our design applied to the transient gas optimisation problem. With the trained network we produce a feasible solution in 2.5s, use it as a warm-start solution, and thereby decrease global optimal solution solve time by 60.5%.
While graph covering is a fundamental and well-studied problem, this field lacks a broad and unified literature review. The holistic overview of graph covering given in this article attempts to close this gap. The focus lies on a characterization and classification of the different problems discussed in the literature. In addition, notable results and common approaches are also included. Whenever appropriate, our review extends to the corresponding partioning problems.
We investigate polyhedral aspects of the Periodic Event Scheduling Problem (PESP), the mathematical basis for periodic timetabling problems in public transport. Flipping the orientation of arcs, we obtain a new class of valid inequalities, the flip inequalities, comprising both the known cycle and change-cycle inequalities. For a point of the LP relaxation, a violated flip inequality can be found in pseudo-polynomial time, and even in linear time for a spanning tree solution. Our main result is that the integer vertices of the polytope described by the flip inequalities are exactly the vertices of the PESP polytope, i.e., the convex hull of all feasible periodic slacks with corresponding modulo parameters. Moreover, we show that this flip polytope equals the PESP polytope in some special cases. On the computational side, we devise several heuristic approaches concerning the separation of cutting planes from flip inequalities. These produce better dual bounds for the smallest and largest instance of the benchmarking library PESPlib.
Public transportation networks are typically operated with a periodic timetable. The Periodic Event Scheduling Problem (PESP) is the standard mathematical modelling tool for periodic timetabling. Since PESP can be solved in linear time on trees, it is a natural question to ask whether there are polynomial-time algorithms for input networks of bounded treewidth. We show that deciding the feasibility of a PESP instance is NP-hard even when the treewidth is 2, the branchwidth is 2, or the carvingwidth is 3. Analogous results hold for the optimization of reduced PESP instances, where the feasibility problem is trivial. To complete the picture, we present two pseudo-polynomial-time dynamic programming algorithms solving PESP on input networks with bounded tree- or branchwidth. We further analyze the parameterized complexity of PESP with bounded cyclomatic number, diameter, or vertex cover number. For event-activity networks with a special -- but standard -- structure, we give explicit and sharp bounds on the branchwidth in terms of the maximum degree and the carvingwidth of an underlying line network. Finally, we investigate several parameters on the smallest instance of the benchmarking library PESPlib.
This paper investigates the estimation of the size of Branch-and-Bound (B&B) trees for solving mixed-integer programs. We first prove that the size of the B&B tree cannot be approximated within a factor of~2 for general binary programs, unless P equals NP. Second, we review measures of the progress of the B&B search, such as the gap, and propose a new measure, which we call leaf frequency.
We study two simple ways to transform these progress measures into B&B tree size estimates, either as a direct projection, or via double-exponential smoothing, a standard time-series forecasting technique. We then combine different progress measures and their trends into nontrivial estimates using Machine Learning techniques, which yields more precise estimates than any individual measure. The best method we have identified uses all individual measures as features of a random forest model.
In a large computational study, we train and validate all methods on the publicly available MIPLIB and Coral general purpose benchmark sets. On average, the best method estimates B&B tree sizes within a factor of 3 on the set of unseen test instances even during the early stage of the search, and improves in accuracy as the search progresses. It also achieves a factor 2 over the entire search on each out of six additional sets of homogeneous instances we have tested. All techniques are available in version 7 of the branch-and-cut framework SCIP.
The SCIP Optimization Suite provides a collection of software packages for
mathematical optimization centered around the constraint integer programming frame-
work SCIP. This paper discusses enhancements and extensions contained in version 7.0
of the SCIP Optimization Suite. The new version features the parallel presolving library
PaPILO as a new addition to the suite. PaPILO 1.0 simplifies mixed-integer linear op-
timization problems and can be used stand-alone or integrated into SCIP via a presolver
plugin. SCIP 7.0 provides additional support for decomposition algorithms. Besides im-
provements in the Benders’ decomposition solver of SCIP, user-defined decomposition
structures can be read, which are used by the automated Benders’ decomposition solver
and two primal heuristics. Additionally, SCIP 7.0 comes with a tree size estimation
that is used to predict the completion of the overall solving process and potentially
trigger restarts. Moreover, substantial performance improvements of the MIP core were
achieved by new developments in presolving, primal heuristics, branching rules, conflict
analysis, and symmetry handling. Last, not least, the report presents updates to other
components and extensions of the SCIP Optimization Suite, in particular, the LP solver
SoPlex and the mixed-integer semidefinite programming solver SCIP-SDP.
In a recent paper, Conte et al. [CGMR2017] presented an algorithm for enumerating all acyclic orientations of a graph G=(V,E) with a single source (and related orientations) with delay O(|V||E|). In this paper we revisit the problem by going back to an early paper by de Fraysseix et al. [FMR1995], who proposed an algorithm for enumerating all bipolar orientations of a graph based on a recursion formula. We first formalize de Fraysseix et al.'s algorithm for bipolar orientations and determine that its delay is also O(|V||E|). We then apply their recursion formula to the case of Conte et al.'s enumeration problem and show that this yields a more efficient enumeration algorithm with delay O(\sqrt(|V|)|E|). Finally, a way to further streamline the algorithm that leads to a particularly simple implementation is suggested.
Maximal Quadratic-Free Sets
(2019)
The intersection cut paradigm is a powerful framework that facilitates
the generation of valid linear inequalities, or cutting planes, for a potentially complex set S. The key ingredients in this construction are a
simplicial conic relaxation of S and an S-free set: a convex zone whose
interior does not intersect S. Ideally, such S-free set would be maximal
inclusion-wise, as it would generate a deeper cutting plane. However, maximality can be a challenging goal in general. In this work, we show how
to construct maximal S-free sets when S is defined as a general quadratic
inequality. Our maximal S-free sets are such that efficient separation of
a vertex in LP-based approaches to quadratically constrained problems is
guaranteed. To the best of our knowledge, this work is the first to provide
maximal quadratic-free sets.
Improved optimization models for potential-driven network flow problems via ASTS orientations
(2019)
The class of potential-driven network flow problems provides important models for a range of infrastructure networks that lead to hard-to-solve MINLPs in real-world applications. On large-scale meshed networks the relaxations usually employed are rather weak due to cycles in the network. To address this situation, we introduce the concept of ASTS orientations, a generalization of bipolar orientations, as a combinatorial relaxation of feasible solutions of potential-driven flow problems, study their structure, and show how they can be used to strengthen existing relaxations and thus provide improved optimization models. Our computational results indicate that ASTS orientations can be used to derive much stronger bounds on the flow variables than existing bound tightening methods and to yield significant performance improvements for an existing state-of-the-art MILP model for large-scale gas networks.
The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global epsilon-optimality with spatial branch and bound is a tight, computationally tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solver can often successfully handle a moderate presence of nonconvexities, which opens the door for the use of potentially tighter nonconvex relaxations. In this work, we exploit this fact and make use of a nonconvex relaxation obtained via aggregation of constraints: a surrogate relaxation. These relaxations were actively studied for linear integer programs in the 70s and 80s, but they have been scarcely considered since. We revisit these relaxations in an MINLP setting and show the computational benefits and challenges they can have. Additionally, we study a generalization of such relaxation that allows for multiple aggregations simultaneously and present the first algorithm that is capable of computing the best set of aggregations. We propose a multitude of computational enhancements for improving its practical performance and evaluate the algorithm’s ability to generate strong dual bounds through extensive computational experiments.
Urban transportation systems are subject to a high level of variation and fluctuation in demand over the day. When this variation and fluctuation are observed in both time and space, it is crucial to develop line plans that are responsive to demand. A multi-period line planning approach that considers a changing demand during the planning horizon is proposed. If such systems are also subject to limitations of resources, a dynamic transfer of resources from one line to another throughout the planning horizon should also be considered. A mathematical modelling framework is developed to solve the line planning problem with transfer of resources during a finite length planning horizon of multiple periods. We analyze whether or not multi-period solutions outperform single period solutions in terms of feasibility and relevant costs. The importance of demand variation on multi-period solutions is investigated. We evaluate the impact of resource transfer constraints on the effectiveness of solutions. We also study the effect of line type designs and question the choice of period lengths along with the problem parameters that are significant for and sensitive to the optimality of solutions.
In this dissertation, we study matchings and flows in hypergraphs using combinatorial methods. These two problems are among the best studied in the field of combinatorial optimization. As hypergraphs are a very general concept, not many results on graphs can be generalized to arbitrary hypergraphs. Therefore, we consider special classes of hypergraphs, which admit more structure, to transfer results from graph theory to hypergraph theory. In Chapter 2, we investigate the perfect matching problem on different classes of hypergraphs generalizing bipartite graphs. First, we give a polynomial time approximation algorithm for the maximum weight matching problem on so-called partitioned hypergraphs, whose approximation factor is best possible up to a constant. Afterwards, we look at the theorems of König and Hall and their relation. Our main result is a condition for the existence of perfect matchings in normal hypergraphs that generalizes Hall’s condition for bipartite graphs. In Chapter 3, we consider perfect f-matchings, f-factors, and (g,f)-matchings. We prove conditions for the existence of (g,f)-matchings in unimodular hypergraphs, perfect f-matchings in uniform Mengerian hypergraphs, and f-factors in uniform balanced hypergraphs. In addition, we give an overview about the complexity of the (g,f)-matching problem on different classes of hypergraphs generalizing bipartite graphs. In Chapter 4, we study the structure of hypergraphs that admit a perfect matching. We show that these hypergraphs can be decomposed along special cuts. For graphs it is known that the resulting decomposition is unique, which does not hold for hypergraphs in general. However, we prove the uniqueness of this decomposition (up to parallel hyperedges) for uniform hypergraphs. In Chapter 5, we investigate flows on directed hypergraphs, where we focus on graph-based directed hypergraphs, which means that every hyperarc is the union of a set of pairwise disjoint ordinary arcs. We define a residual network, which can be used to decide whether a given flow is optimal or not. Our main result in this chapter is an algorithm that computes a minimum cost flow on a graph-based directed hypergraph. This algorithm is a generalization of the network simplex algorithm.
In this paper, we consider the Cyclic Crew Rostering Problem with Fairness Requirements (CCRP-FR). In this problem, attractive cyclic rosters have to be constructed for groups of employees, considering multiple, a priori determined, fairness levels. The attractiveness follows from the structure of the rosters (e.g., sufficient rest times and variation in work), whereas fairness is based on the work allocation among the different roster groups. We propose a three-phase heuristic for the CCRP-FR, which combines the strength of column generation techniques with a large-scale neighborhood search algorithm. The design of the heuristic assures that good solutions for all fairness levels are obtained quickly, and can still be further improved if additional running time is available. We evaluate the performance of the algorithm using real-world data from Netherlands Railways, and show that the heuristic finds close to optimal solutions for many of the considered instances. In particular, we show that the heuristic is able to quickly find major improvements upon the current sequential practice: For most instances, the heuristic is able to increase the attractiveness by at least 20% in just a few minutes.
In this paper we introduce a technique to produce tighter cutting planes for mixed-integer non-linear programs. Usually, a cutting plane is generated to cut off a specific infeasible point. The underlying idea is to use the infeasible point to restrict the feasible region in order to obtain a tighter domain. To ensure validity, we require that every valid cut separating the infeasible point from the restricted feasible region is still valid for the original feasible region. We translate this requirement in terms of the separation problem and the reverse polar. In particular, if the reverse polar of the restricted feasible region is the same as the reverse polar of the feasible region, then any cut valid for the restricted feasible region that \emph{separates} the infeasible point, is valid for the feasible region. We show that the reverse polar of the \emph{visible points} of the feasible region from the infeasible point coincides with the reverse polar of the feasible region. In the special where the feasible region is described by a single non-convex constraint intersected with a convex set we provide a characterization of the visible points. Furthermore, when the non-convex constraint is quadratic the characterization is particularly simple. We also provide an extended formulation for a relaxation of the visible points when the non-convex constraint is a general polynomial. Finally, we give some conditions under which for a given set there is an inclusion-wise smallest set, in some predefined family of sets, whose reverse polars coincide.
The Periodic Event Scheduling Problem is a well-studied NP-hard problem with applications in public transportation to find good periodic timetables. Among the most powerful heuristics to solve the periodic timetabling problem is the modulo network simplex method. In this paper, we consider the more difficult version with integrated passenger routing and propose a refined integrated variant to solve this problem on real-world-based instances.
In the planning process of public transportation companies, designing the timetable is among the core planning steps. In particular in the case of periodic (or cyclic) services, the Periodic Event Scheduling Problem (PESP) is well-established to compute high-quality periodic timetables.
We are considering algorithms for computing good solutions for the very basic PESP with no additional extra features as add-ons. The first of these algorithms generalizes several primal heuristics that had been proposed in the past, such as single-node cuts and the modulo network simplex algorithm. We consider partitions of the graph, and identify so-called delay cuts as a structure that allows to generalize several previous heuristics. In particular, when no more improving delay cut can be found, we already know that the other heuristics could not improve either.
The second of these algorithms turns a strategy, that had been discussed in the past, upside-down: Instead of gluing together the network line-by-line in a bottom-up way, we develop a divide-and-conquer-like top-down approach to separate the initial problem into two easier subproblems such that the information loss along their cutset edges is as small as possible.
We are aware that there may be PESP instances that do not fit well the separator setting. Yet, on the RxLy-instances of PESPlib in our experimental computations, we come up with good primal solutions and dual bounds. In particular, on the largest instance (R4L4), this new separator approach, which applies a state-of-the-art solver as subroutine, is able to come up with better dual bounds than purely applying this state-of-the-art solver in the very same time.
About 20% of the German energy demand is supplied by natural gas. Ad-
ditionally, for about twice the amount Germany serves as a transit country. Thereby, the German network represents a central hub in the European natural gas transport network. The transport infrastructure is operated by so-called transmissions system operators or TSOs. The number one priority of the TSOs is to ensure security of supply. However, the TSOs have no knowledge of the intentions and planned actions of the shippers (traders).
Open Grid Europe (OGE), one of Germany’s largest TSO, operates a high-
pressure transport network of about 12.000 km length. Since flexibility and security of supply is of utmost importance to the German Energy Transition (“Energiewende”) especially with the introduction of peak-load gas power stations, being able to predict in- and out-flow of the network is of great importance.
In this paper we introduce a new hybrid forecast method applied to gas
flows at the boundary nodes of a transport network. The new method employs optimized feature minimization and selection. We use a combination of an FAR, LSTM DNN and mathematical programming to achieve robust high quality forecasts on real world data for different types of network nodes.
Keywords: Gas Forecast, Time series, Hybrid Method, FAR, LSTM, Mathematical Optimisation
One of the most fundamental ingredients in mixed-integer nonlinear programming solvers is the well- known McCormick relaxation for a product of two variables x and y over a box-constrained domain. The starting point of this paper is the fact that the convex hull of the graph of xy can be much tighter when computed over a strict, non-rectangular subset of the box. In order to exploit this in practice, we propose to compute valid linear inequalities for the projection of the feasible region onto the x-y-space by solving a sequence of linear programs akin to optimization-based bound tightening. These valid inequalities allow us to employ results from the literature to strengthen the classical McCormick relaxation. As a consequence, we obtain a stronger convexification procedure that exploits problem structure and can benefit from supplementary information obtained during the branch-and bound algorithm such as an objective cutoff. We complement this by a new bound tightening procedure that efficiently computes the best possible bounds for x, y, and xy over the available projections. Our computational evaluation using the academic solver SCIP exhibit that the proposed methods are applicable to a large portion of the public test library MINLPLib and help to improve performance significantly.
We propose a simple and general online method to measure the search progress within the Branch-and-Bound algorithm, from which we estimate the size of the remaining search tree. We then show how this information can help solvers algorithmically at runtime by designing a restart strategy for Mixed-Integer Programming (MIP) solvers that decides whether to restart the search based on the current estimate of the number of remaining nodes in the tree. We refer to this type of algorithm as clairvoyant.
Our clairvoyant restart strategy outperforms a state-of-the-art solver on a large set of publicly available MIP benchmark instances.
It is implemented in the MIP solver SCIP and will be available in future releases.
We introduce a concurrent solver for the periodic event scheduling problem (PESP). It combines mixed integer programming techniques, the modulo network simplex method, satisfiability approaches, and a new heuristic based on maximum cuts. Running these components in parallel speeds up the overall solution process. This enables us to significantly improve the current upper and lower bounds for all benchmark instances of the library PESPlib.
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.
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.
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.
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.
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 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.
In dieser Arbeit betrachten wir das Problem, für den Fahrplan eines (Nah-) Verkehrsnetzes schnellste Wege zu berechnen. Da die Verkehrsmittel zu unterschiedlichen Zeiten von den einzelnen Haltestellen/Bahnhöfen abfahren, kann das Problem nicht ohne Weiteres mit einem „statischen“ Graphen modelliert werden. Es gibt zwei unterschiedliche Ansätze für dieses zeitabhängige Problem: Erstens können die verschiedenen An-/Abfahrtereignisse an einem Halt durch „Kopien“ dargestellt werden, das ist das zeit-expandierte Modell. Zweitens können die Gewichte der Kanten zeitabhängig sein, das ist
das zeitabhängige Modell. Wir untersuchen in dieser Arbeit, wie der „klassische“ Dijkstra-Algorithmus und der A* Algorithmus mit einer geeigneten Heuristik im Vergleich abschneiden. Die gewählte Heuristik ist der Abstand zum Zielknoten, wenn die Abfahrtszeiten ignoriert werden. Nach unseren Untersuchungen zeigt sich, dass der A* Algorithmus dem Dijkstra-Algorithmus weit überlegen ist für genügend große Nahverkehrsnetze. Wir testen anhand der echten Verkehrsnetze von Berlin und Aachen. Unsere Berechnungen zeigen, dass die gewählte Heuristik besonders gut ist für Start- und Zielknoten, welche unabhängig von ihrer Distanz nur 1–2 verschiedene mögliche kürzeste Pfade für alle Zeitschritte haben. Dort ist der A* Algorithmus bis zu 20-mal schneller. Dies kommt aber nicht häufig in unseren Testinstanzen vor. Die einzelnen Laufzeitvergleich zeigen, dass
der A* Algorithmus durchschnittlich 7-mal so schnell ist wie der Dikstra-Algorithmus.
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.
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.
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.