Elmar Swarat

Micro–macro transformation of railway networks

Abstract This paper presents a bottom-up approach to automatic railway network simplification. Starting from a detailed microscopic level as it is used in railway simulation, the network is transformed by an algorithm to an aggregated level, i.e., to a macroscopic network, that is sufficient for long-term planning and optimization. Running and headway times are rounded to a user defined precision by a special cumulative method. After this “macrotization” trains from a given set of requests are added to the existing timetable by solving an optimal train path allocation problem. The objective of this problem is to maximize a sum of utilities of the allocated trains; the utility can be a constant, some monetary value, etc. The optimized schedule is re-transformed back to the microscopic level in such a way that it can be simulated without any conflicts between the train paths. We apply this algorithm to macrotize a microscopic network model of the highly frequented Simplon corridor in the Alps between Switzerland and Italy. To the best knowledge of the authors and confirmed by several railway practitioners this was the first time that track allocations that have been produced in a fully automatic way on a macroscopic scale fulfill the requirements of the originating microscopic model and withstand an evaluation in the microscopic simulation tool OpenTrack . Our micro–macro transformation method allows for a much faster planning and provides solutions of a quality that are at least comparable to the most sophisticated manual schedules. In this way meaningful scenario analyses can be carried out that pave the way towards a new level of decision support in railway planning.

Optimizing the Simplon railway corridor

This paper presents a case study of a railway timetable optimization for the very dense Simplon corridor, a major railway connection in the Alps between Switzerland and Italy. The key to deal with the complexity of this scenario is the use of a novel aggregation-disaggregation method. Starting from a detailed microscopic representation as it is used in railway simulation, the data is transformed by an automatic procedure into a less detailed macroscopic representation, that is sufficient for the purpose of capacity planning and amenable to state-of-the-art integer programming optimization methods. This macroscopic railway network is saturated with trains. Finally, the optimized timetable is re-transformed to the microscopic level in such a way that it can be operated without any conflicts among the train paths. Using this micro-macro aggregation-disaggregation approach in combination with integer programming methods, it becomes for the first time possible to generate a profit maximal and conflict free timetable for the complete Simplon corridor over an entire day by a simultaneous optimization of all trains requests. In addition, this also allows us to undertake a sensitivity analysis of various problem parameters.

A Stackelberg game to optimize the distribution of controls in transportation networks

We propose a game theoretic model for the spatial distribution of inspectors on a transportation network. The problem is to spread out the controls so as to enforce the payment of a transit toll. We formulate a linear program to find the control distribution which maximizes the expected toll revenue, and a mixed integer program for the problem of minimizing the number of evaders. Furthermore, we show that the problem of finding an optimal mixed strategy for a coalition of N inspectors can be solved efficiently by a column generation procedure. Finally, we give experimental results from an application to the truck toll on German motorways.

Optimizing Toll Enforcement in Transportation Networks: a Game-Theoretic Approach

We present a game-theoretic approach to optimize the strategies of toll enforcement on a motorway network. In contrast to previous approaches, we consider a network with an arbitrary topology, and we handle the fact that users may choose their Origin-Destination path; in particular they may take a detour to avoid sections with a high control rate. We show that a Nash equilibrium can be computed with an LP (although the game is not zero-sum), and we give a MIP for the computation of a Stackelberg equilibrium. Experimental results based on an application to the enforcement of a truck toll on German motorways are presented.

An IP approach to toll enforcement optimization on German motorways

This paper proposes the first model for toll enforcement optimization on German motorways. The enforcement is done by mobile control teams and our goal is to produce a schedule achieving network-wide control, proportional to spatial and time-dependent traffic distributions. Our model consists of two parts. The first plans control tours using a vehicle routing approach with profits and some side constraints. The second plans feasible rosters for the control teams. Both problems can be modeled as Multi-Commodity Flow Problems. Adding additional coupling constraints produces a large-scale integrated integer programming formulation. We show that this model can be solved to optimality for real world instances associated with a control area in East Germany.

A Case Study on Optimizing Toll Enforcements on Motorways

In this paper we present the problem of computing optimal tours of toll inspectors on German motorways. This problem is a special type of vehicle routing problem and builds up an integrated model, consisting of a tour planning and a duty rostering part. The tours should guarantee a network-wide control whose intensity is proportional to given spatial and time dependent traffic distributions. We model this using a space-time network and formulate the associated optimization problem by an integer program (IP). Since sequential approaches fail, we integrated the assignment of crews to the tours in our model. In this process all duties of a crew member must fit in a feasible roster. It is modeled as a Multi-Commodity Flow Problem in a directed acyclic graph, where specific paths correspond to feasible rosters for one month. We present computational results in a case-study on a German subnetwork which documents the practicability of our approach.

Railway Track Allocation - Simulation, Aggregation, and Optimization

Today the railway timetabling process and the track allocation is one of the most challenging problems to solve by a railway company. Especially due to the deregulation of the transport market in the recent years several suppliers of railway traffic have entered the market in Europe. This leads to more potential conflicts between trains caused by an increasing demand of train paths. Planning and operating railway transportation systems is extremely hard due to the combinatorial complexity of the underlying discrete optimization problems, the technical intricacies, and the immense size of the problem instances. In order to make best use of the infrastructure and to ensure economic operation, efficient planning of the railway operation is indispensable. Mathematical optimization models and algorithms can help to automatize and tackle these challenges. Our contribution in this paper is to present a renewed planning process due to the liberalization in Europe and an associated concept for track allocation, which consists of three important parts, simulation, aggregation, and optimization. Furthermore, we present results of our general framework for real world data.

Optimal Duty Rostering for Toll Enforcement Inspectors

We present the problem of planning mobile tours of inspectors on German motorways to enforce the payment of the toll for heavy good trucks. This is a special type of vehicle routing problem with the objective to conduct as good inspections as possible on the complete network. In addition, we developed a personalized crew rostering model, to schedule the crews of the tours. The planning of daily tours and the rostering are combined in a novel integrated approach and formulated as a complex and large scale Integer Program. The main focus of this paper extends our previous publications on how different requirements for the rostering can be modeled in detail. The second focus is on a bi-criteria analysis of the planning problem to find the balance between the control quality and the roster acceptance. Finally, computational results on real-world instances show the practicability of our method and how different input parameters influence the problem complexity.

Network spot-checking games: Theory and application to toll enforcing in transportation networks

We introduce the class of spot-checking games SC games. These games model problems where the goal is to distribute fare inspectors over a toll network. In an SC game, the pure strategies of network users correspond to paths in a graph, and the pure strategies of the inspectors are subset of arcs to be controlled. Although SC games are not zero-sum, we show that a Nash equilibrium can be computed by linear programming. The computation of a strong Stackelberg equilibrium SSE is more relevant for this problem and we give a mixed integer programming MIP formulation for this problem. We show that the computation of such an equilibrium is NP-hard. More generally, we prove that it is NP-hard to compute a SSE in a polymatrix game, even if the game is pairwise zero-sum. Then, we give some bounds on the price of spite, which measures how the payoff of the inspector degrades when committing to a Nash equilibrium. Finally, we report computational experiments on instances constructed from real data, for an application to the enforcement of a truck toll in Germany. These numerical results show the efficiency of the proposed methods, as well as the quality of the bounds derived in this article.

Designing Inspector Rosters with Optimal Strategies

We consider the problem of enforcing a toll on a transportation network with limited inspection resources. We formulate a game theoretic model to optimize the allocation of the inspectors, taking the reaction of the network users into account. The model includes several important aspects for practical operation of the control strategy, such as duty types for the inspectors. In contrast to a formulation in Borndorfer et al. (Networks, 65, 312–328, [1]) using flows to describe the users’ strategies we choose a path formulation and identify dominated user strategies to significantly reduce the problem size. Computational results suggest that our approach is better suited for practical instances.