A Fast Heuristic Algorithm for the Train Unit Assignment Problem

In this paper we study a railway optimization problem known as the Train Unit Assignment Problem. A train unit consists of a self-contained train with an engine and a set of wagons with passenger seats. Given a set of timetabled train trips, each with a required number of passenger seats, and a set of train units, each with a given number of available seats, the problem calls for the best assignment of the train units to the trips, possibly combining more than one train unit for a given trip, that fulfills the seat requests. We propose a heuristic algorithm based on the computation of a lower bound obtained by solving an Integer Linear Programming model that gives the optimal solution in a "peak period" of the day. The performance of the heuristic algorithm is computationally evaluated on real-world instances provided by a regional Italian Train Operator. The results are compared with those of existing methods from the literature, showing that the new method is able to obtain solutions of good quality in much shorter computing times

Recoverable Robustness for Railway Rolling Stock Planning

In this paper we explore the possibility of applying the notions of Recoverable Robustness and Price of Recoverability (introduced by [5]) to railway rolling stock planning, being interested in recoverability measures that can be computed in practice, thereby evaluating the robustness of rolling stock schedules. In order to lower bound the Price of Recoverability for any set of recovery algorithms, we consider an "optimal" recovery algorithm and propose a Benders decomposition approach to assess the Price of Recoverability for this "optimal" algorithm. We evaluate the approach on real-life rolling stock planning problems of NS, the main operator of passenger trains in the Netherlands. The preliminary results show that, thanks to Benders decomposition, our lower bound can be computed within relatively short time for our case study

Single-Commodity Robust Network Design with Finite and Hose Demand Sets

We study a single-commodity Robust Network Design problem (sRND) defined on an undirected graph. Our goal is to determine minimum cost capacities such that any traffic demand from a given uncertainty set can be satisfied by a feasible single-commodity flow. We consider two ways of representing the uncertainty set, either as a finite list of scenarios or as a polytope. We propose a branch-and- cut algorithm to derive optimal solutions to sRND, built on a capacity-based integer linear programming formulation. It is strenghtened with valid inequalities derived as {0,1/2 }-Chvátal-Gomory cuts. Since the formulation contains exponentially many constraints, we provide practical separation algorithms. Extensive computational experiments show that our approach is effective, in comparison to existing approaches from the literature as well as to solving a flow based formulation by a general purpose solver

Single-Commodity Robust Network Design with Finite and Hose Demand Sets

We study a single-commodity Robust Network Design problem (sRND) defined on an undirected graph. Our goal is to determine minimum cost capacities such that any traffic demand from a given uncertainty set can be satisfied by a feasible single-commodity flow. We consider two ways of representing the uncertainty set, either as a finite list of scenarios or as a polytope. We propose a branch-and- cut algorithm to derive optimal solutions to sRND, built on a capacity-based integer linear programming formulation. It is strenghtened with valid inequalities derived as {0,1/2 }-Chvátal-Gomory cuts. Since the formulation contains exponentially many constraints, we provide practical separation algorithms. Extensive computational experiments show that our approach is effective, in comparison to existing approaches from the literature as well as to solving a flow based formulation by a general purpose solver

Single-commodity robust network design problem: Complexity, instances and heuristic solutions.

We study a single-commodity Robust Network Design problem (RND) in which an undirected graph with edge costs is given together with a discrete set of balance matrices, representing different supply/demand scenarios. In each scenario, a subset of the nodes is exchanging flow. The goal is to determine the minimum cost installation of capacities on the edges such that the flow exchange is feasible for every scenario. Previously conducted computational investigations on the problem motivated the study of the complexity of some special cases and we present complexity results on them, including hypercubes. In turn, these results lead to the definition of new instances (random graphs with {-1,0,1} balances) that are computationally hard for the natural flow formulation. These instances are then solved by means of a new heuristic algorithm for RND, which consists of three phases. In the first phase the graph representing the network is reduced by heuristically deleting a subset of the arcs, and a feasible solution is built. The second phase consists of a neighborhood search on the reduced graph based on a Mixed-Integer (Linear) Programming (MIP) flow model. Finally, the third phase applies a proximity search approach to further improve the solution, taking into account the original graph. The heuristic is tested on the new instances, and the comparison with the solutions obtained by Cplex on a natural flow formulation shows the effectiveness of the proposed method

An iterative heuristic for passenger-centric train timetabling with integrated adaption times

In this paper we present a method to construct a periodic timetable from a tactical planning perspective. We aim at constructing a timetable that is feasible with respect to infrastructure constraints and minimizes average perceived passenger travel time. In addition to in-train and transfer times, our notion of perceived passenger time includes the adaption time (waiting time at the origin station). Adaption time minimization allows us to avoid strict frequency regularity constraints and, at the same time, to ensure regular connections between passengers’ origins and destinations. The combination of adaption time minimization and infrastructure constraints satisfaction makes the problem very challenging. The described periodic timetabling problem can be modelled as an extension of a Peri- odic Event Scheduling Problem (PESP) formulation, but requires huge computing times if it is directly solved by a general-purpose solver for instances of realistic size. In this paper, we propose a heuristic approach consisting of two phases that are executed iteratively. First, we solve a mixed-integer linear program to determine an ideal timetable that mini- mizes the average perceived passenger travel time but neglects infrastructure constraints. Then, a Lagrangian-based heuristic makes the timetable feasible with respect to infras- tructure constraints by modifying train departure and arrival times as little as possible. The obtained feasible timetable is then evaluated to compute the resulting average per- ceived passenger travel time, and a feedback is sent to the Lagrangian-based heuristic so as to possibly improve the obtained timetable from the passenger perspective, while still respecting infrastructure constraints. We illustrate the proposed iterative heuristic approach on real-life instances of Netherlands Railways and compare it to a benchmark approach, showing that it finds a feasible timetable very close to the ideal one

A railway timetable rescheduling approach for handling large scale disruptions

On a daily basis, relatively large disruptions require infrastructure managers and railway operators to reschedule their railway timetables together with their rolling stock and crew schedules. This research focuses on timetable rescheduling for passenger trains at a macroscopic level in a railway network. An integer programming model is formulated for solving the timetable rescheduling problem, which minimizes the number of cancelled and delayed trains while adhering to infrastructure and rolling stock capacity constraints. The possibility of rerouting trains in order to reduce the number of cancelled and delayed trains is also considered. In addition, all stages of the disruption management process (from the start of the disruption to the time the normal situation is restored) are taken into account. Computational tests of the described model on a heavily used part of the Dutch railway network show that we are able to find optimal solutions in short computation times. This makes the approach applicable for use in practice