Andreas Bärmann

Solving network design problems via iterative aggregation

In this work, we present an exact approach for solving network design problems that is based on an iterative graph aggregation procedure. The scheme allows existing preinstalled capacities. Starting with an initial aggregation, we solve a sequence of network design master problems over increasingly fine-grained representations of the original network. In each step, a subproblem is solved that either proves optimality of the solution or gives a directive where to refine the representation of the network in the subsequent iteration. The algorithm terminates with a globally optimal solution to the original problem. Our implementation uses a standard integer programming solver for solving the master problems as well as the subproblems. The computational results on random and realistic instances confirm the profitable use of the iterative aggregation technique. The computing time often reduces drastically when our method is compared to solving the original problem from scratch.

A Decomposition Method for Multiperiod Railway Network Expansion—With a Case Study for Germany

In this work, we report about the results of a joint research project between Friedrich–Alexander–Universitat Erlangen–Nurnberg and Deutsche Bahn AG on the optimal expansion of the German railway network until 2030. The need to increase the throughput of the network is given by company-internal demand forecasts that indicate an increase in rail freight traffic of about 50% over the next two decades. Our focus is to compute an optimal investment strategy into line capacities given an available annual budget, i.e., we are to choose the most profitable lines to upgrade with respect to the demand scenario under consideration and to provide a schedule according to which the chosen measures are implemented. This leads to a multiperiod network design problem—a class of problems that has received increasing interest over the past decade. We develop a mixed-integer programming formulation to model the situation and solve it via a novel decomposition approach that we call multiple-knapsack decomposition. The method...

A comparison of performance metrics for balancing the power consumption of trains in a railway network by slight timetable adaptation

We investigate the problem of designing energy-efficient timetables for railway traffic. More precisely, we slightly adapt a given timetable draft before it is published by moderately shifting the departure times of the trains at the stations. To this end, we propose a mixed-integer programming model for feasible adaptations of the timetable draft and investigate its behaviour under different objective functions which fall into two classes: reducing the energy cost and increasing the stability of the power supply system. These tests are performed on real-world problem instances from our industry partner Deutsche Bahn AG. They show a significant potential for improvements in the existing railway timetables.

Polyhedral Approximation of Ellipsoidal Uncertainty Sets via Extended Formulations — a computational case study —

Robust optimization is an important technique to immunize optimization problems against data uncertainty. In the case of a linear program and an ellipsoidal uncertainty set, the robust counterpart turns into a second-order cone program. In this work, we investigate the efficiency of linearizing the second-order cone constraints of the latter. This is done using the optimal linear outer-approximation approach by Ben-Tal and Nemirovski (Math Oper Res 26:193–205, 2001) from which we derive an optimal inner approximation of the second-order cone. We examine the performance of this approach on various benchmark sets including portfolio optimization instances as well as (robustified versions of) the MIPLIB and the SNDlib.

Aggregation Methods for Railway Network Design Based on Lifted Benders Cuts

Rail freight traffic in Germany has experienced significant growth rates over the last decade, and recent forecasts expect this tendency to continue over the next 20 years due to the increases in national and international trade. Internal predictions of Deutsche Bahn AG, the most important German railway enterprise, assume a mean increase of 2% per year for rail freight traffic until 2030. At this pace, the German railway network in its current state would reach its capacity limit way before this date. As investments into the network infrastructure bear a very high price tag, it is crucial to use the available budget in the most efficient manner. Furthermore, the large size of the networks under consideration warrants the development of efficient algorithms to handle the complex network design problems arising for real-world data. This led us to the development of network aggregation procedures which allow for much shorter solution times by compressing the network information. More exactly, our framework works by clustering the nodes of the underlying graph to components and solving the network design problem over this aggregated graph. This kind of aggregation may either be used as a quick heuristic, or it can be extended to an exact method, e.g. by iterative refinement of the clustering, The latter results in a cutting plane algorithm, which also introduces new variables with each refinement. This idea developed in Barmann et al. (Math Program Comput 7(2):189–217, 2015) is extended in this chapter such that it is able to incorporate the costs for routing flow through the network via lifted Benders optimality cuts. Altogether, our algorithm can be seen as a novel kind of Benders decomposition which allows to shift variables from the subproblem to the master problem in the process. Computations on several benchmark sets demonstrate the effectiveness of the approach.

Staircase compatibility and its applications in scheduling and piecewise linearization

Abstract We introduce the Clique Problem with Multiple-Choice constraints (CPMC) and characterize a case where it is possible to give an efficient description of the convex hull of its feasible solutions. This special case, which we name staircase compatibility, generalizes common properties in several applications and allows for a linear description of the integer feasible solutions to (CPMC) with a totally unimodular constraint matrix of polynomial size. We derive two such totally unimodular reformulations for the problem: one that is obtained by a strengthening of the compatibility constraints and one that is based on a representation as a dual network flow problem. Furthermore, we show a natural way to derive integral solutions from fractional solutions to the problem by determining integral extreme points generating this fractional solution. We also evaluate our reformulations from a computational point of view by applying them to two different real-world problem settings. The first one is a problem in railway timetabling, where we try to adapt a given timetable slightly such that energy costs from operating the trains are reduced. The second one is the piecewise linearization of non-linear network flow problems, illustrated at the example of gas networks. In both cases, we are able to reduce the solution times significantly by passing to the theoretically stronger formulations of the problem.

An Optimal Expansion Strategy for the German Railway Network Until 2030

This article summarizes the findings of my Ph.D. thesis finished in 2015, whose topic are algorithmic approaches for the solution of network design problems. I focus on the results of a joint project with Deutsche Bahn AG on developing an optimal expansion strategy for the German railway network until 2030 to meet future demands. I have modelled this task as a multi-period network design problem and have derived an efficient decomposition approach to solve it. In a case study on real-world data on the German railway network, I demonstrate both the efficiency of by method as well as the high quality of the solutions it computes.

Strategic Infrastructure Planning in Railway Networks

This chapter presents the most important concepts in railway infrastructure planning as they are referred to throughout the thesis. After the introduction of some basic terminology, we explain the necessary knowledge in railway infrastructure development. For most of the technical descriptions, we lean on the work of Horl (1998).

Integration of Routing Costs into the Aggregation Scheme

The preceding chapter saw the development of an aggregation-based algorithmic scheme to solve large-scale network design problems. Its main idea is the aggregation of single nodes of the network graph to components. Apart from reducing the size of the network design problems, it also features a reduction of (continuous) symmetry in the problem by neglecting the routing decisions within the components.

Modelling the Expansion Problem

The motivation behind the models developed in this chapter is the expansion of the current German rail freight network to meet future demands. As outlined in Chapter 1, rail freight traffic is predicted to attain tremendous increases over the next two decades. On the other hand, we saw in Chapter 2 that investments into the railway network bear a very high price tag and need to be planned well in advance.