TY - GEN A1 - Borndörfer, Ralf A1 - Danecker, Fabian A1 - Weiser, Martin T1 - Convergence Properties of Newton's Method for Globally Optimal Free Flight Trajectory Optimization N2 - The algorithmic efficiency of Newton-based methods for Free Flight Trajectory Optimization is heavily influenced by the size of the domain of convergence. We provide numerical evidence that the convergence radius is much larger in practice than what the theoretical worst case bounds suggest. The algorithm can be further improved by a convergence-enhancing domain decomposition. T3 - ZIB-Report - 23-19 KW - shortest path KW - flight planning KW - free flight KW - optimal control KW - global optimization KW - Newton's method Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-91309 SN - 1438-0064 ER - TY - CHAP A1 - Borndörfer, Ralf A1 - Schwartz, Stephan A1 - Surau, William T1 - Finding Minimum Balanced Separators - an Exact Approach T2 - Operations Research Proceedings 2021 N2 - 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. T3 - ZIB-Report - 21-25 Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-83138 SN - 1438-0064 SP - 154 EP - 159 ER - TY - GEN A1 - Reuther, Markus A1 - Borndörfer, Ralf A1 - Schlechte, Thomas A1 - Weider, Steffen T1 - Integrated Optimization of Rolling Stock Rotations for Intercity Railways N2 - This paper provides a highly integrated solution approach for rolling stock planning problems in the context of intercity passenger traffic. The main contributions are a generic hypergraph based mixed integer programming model and an integrated algorithm for the considered rolling stock rotation planning problem. The new developed approach is able to handle a very large set of industrial railway requirements, such as vehicle composition, maintenance constraints, infrastructure capacity, and regularity aspects. By the integration of this large bundle of technical railway aspects, we show that our approach has the power to produce implementable rolling stock rotations for our industrial cooperation partner DB Fernverkehr. This is the first time that the rolling stock rotations at DB Fernverkehr could be optimized by an automated system utilizing advanced mathematical programming techniques. T3 - ZIB-Report - 12-39 KW - Mixed Integer Programming KW - Railway Optimization KW - Rolling Stock Rostering Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-16424 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Fügenschuh, Armin A1 - Klug, Torsten A1 - Schang, Thilo A1 - Schlechte, Thomas A1 - Schülldorf, Hanno T1 - The Freight Train Routing Problem N2 - We consider the following freight train routing problem (FTRP). Given is a transportation network with fixed routes for passenger trains and a set of freight trains (requests), each defined by an origin and destination station pair. The objective is to calculate a feasible route for each freight train such that a sum of all expected delays and all running times is minimal. Previous research concentrated on microscopic train routings for junctions or inside major stations. Only recently approaches were developed to tackle larger corridors or even networks. We investigate the routing problem from a strategic perspective, calculating the routes in a macroscopic transportation network of Deutsche Bahn AG. Here macroscopic refers to an aggregation of complex real-world structures are into fewer network elements. Moreover, the departure and arrival times of freight trains are approximated. The problem has a strategic character since it asks only for a coarse routing through the network without the precise timings. We give a mixed-integer nonlinear programming~(MINLP) formulation for FTRP, which is a multi-commodity flow model on a time-expanded graph with additional routing constraints. The model's nonlinearities are due to an algebraic approximation of the delays of the trains on the arcs of the network by capacity restraint functions. The MINLP is reduced to a mixed-integer linear model~(MILP) by piecewise linear approximation. The latter is solved by a state of the art MILP solver for various real-world test instances. T3 - ZIB-Report - 13-36 KW - Mixed-Integer Nonlinear Programming KW - multi-commodity flows KW - freight train routing Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-18991 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Schwartz, Stephan A1 - Surau, William T1 - Vertex Covering with Capacitated Trees N2 - 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. T3 - ZIB-Report - 21-14 Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-82616 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Denißen, Jonas A1 - Heller, Simon A1 - Klug, Torsten A1 - Küpper, Michael A1 - Lindner, Niels A1 - Reuther, Markus A1 - Schlechte, Thomas A1 - Söhlke, Andreas A1 - Steadman, William T1 - Microscopic Timetable Optimization for a Moving Block System N2 - 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. T3 - ZIB-Report - 21-13 KW - Moving Block KW - Railway Track Allocation KW - Railway Timetabling KW - Train Routing Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-82547 SN - 1438-0064 ER - TY - GEN A1 - Euler, Ricardo A1 - Borndörfer, Ralf A1 - Strunk, Timo A1 - Takkula, Tuomo T1 - ULD Build-Up Scheduling with Dynamic Batching in an Air Freight Hub N2 - 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. T3 - ZIB-Report - 21-31 KW - Logistics KW - Airline Applications Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-83482 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Mehrgardt, Julika A1 - Reuther, Markus A1 - Schlechte, Thomas A1 - Waas, Kerstin T1 - Re-optimization of Rolling Stock Rotations N2 - The Rolling Stock Rotation Problem is to schedule rail vehicles in order to cover timetabled trips by a cost optimal set of vehicle rotations. The problem integrates several facets of railway optimization, i.e., vehicle composition, maintenance constraints, and regularity aspects. In industrial applications existing schedules often have to be re-optimized to integrate timetable changes or construction sites. We present an integrated modeling and algorithmic approach for this task as well as computational results for industrial problem instances of DB Fernverkehr AG. T3 - ZIB-Report - 13-60 Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-42569 SN - 1438-0064 ER - TY - GEN A1 - Schwartz, Stephan A1 - Borndörfer, Ralf A1 - Bartz, Gerald T1 - The Graph Segmentation Problem N2 - We investigate a graph theoretical problem arising in the automatic billing of a network toll. Given a network and a family of user paths, we study the graph segmentation problem (GSP) to cover parts of the user paths by a set of disjoint segments. The GSP is shown to be NP-hard but for special cases it can be solved in polynomial time. We also show that the marginal utility of a segment is bounded. Computational results for real-world instances show that in practice the problem is more amenable than the theoretic bounds suggest. T3 - ZIB-Report - 16-53 KW - toll billing KW - graph segmentation KW - combinatorial optimization Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-60750 SN - 1438-0064 ER - TY - GEN A1 - Blanco, Marco A1 - Borndörfer, Ralf A1 - Hoang, Nam-Dung A1 - Kaier, Anton A1 - Schlechte, Thomas A1 - Schlobach, Swen T1 - The Shortest Path Problem with Crossing Costs N2 - We introduce the shortest path problem with crossing costs (SPPCC), a shortest path problem in a directed graph, in which the objective function is the sum of arc weights and crossing costs. The former are independently paid for each arc used by the path, the latter need to be paid every time the path intersects certain sets of arcs, which we call regions. The SPPCC generalizes not only the classical shortest path problem but also variants such as the resource constrained shortest path problem and the minimum label path problem. We use the SPPCC to model the flight trajectory optimization problem with overflight costs. In this paper, we provide a comprehensive analysis of the problem. In particular, we identify efficient exact and approximation algorithms for the cases that are most relevant in practice. T3 - ZIB-Report - 16-70 Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-61240 SN - 1438-0064 ER -