Valentina Cacchiani

A column generation approach to train timetabling on a corridor

We propose heuristic and exact algorithms for the (periodic and non-periodic) train timetabling problem on a corridor that are based on the solution of the LP relaxation of an ILP formulation in which each variable corresponds to a full timetable for a train. This is in contrast with previous approaches to the same problem, which were based on ILP formulations in which each variable is associated with a departure and/or arrival of a train at a specific station in a specific time instant, whose LP relaxation is too expensive to be solved exactly. Experimental results on real-world instances of the problem show that the proposed approach is capable of producing heuristic solutions of better quality than those obtained by these previous approaches, and of solving some small-size instances to proven optimality.

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.

A set-covering based heuristic algorithm for the periodic vehicle routing problem

We present a hybrid optimization algorithm for mixed-integer linear programming, embedding both heuristic and exact components. In order to validate it we use the periodic vehicle routing problem (PVRP) as a case study. This problem consists of determining a set of minimum cost routes for each day of a given planning horizon, with the constraints that each customer must be visited a required number of times (chosen among a set of valid day combinations), must receive every time the required quantity of product, and that the number of routes per day (each respecting the capacity of the vehicle) does not exceed the total number of available vehicles. This is a generalization of the well-known vehicle routing problem (VRP). Our algorithm is based on the linear programming (LP) relaxation of a set-covering-like integer linear programming formulation of the problem, with additional constraints. The LP-relaxation is solved by column generation, where columns are generated heuristically by an iterated local search algorithm. The whole solution method takes advantage of the LP-solution and applies techniques of fixing and releasing of the columns as a local search, making use of a tabu list to avoid cycling. We show the results of the proposed algorithm on benchmark instances from the literature and compare them to the state-of-the-art algorithms, showing the effectiveness of our approach in producing good quality solutions. In addition, we report the results on realistic instances of the PVRP introduced in Pacheco et al. (2011) [24] and on benchmark instances of the periodic traveling salesman problem (PTSP), showing the efficacy of the proposed algorithm on these as well. Finally, we report the new best known solutions found for all the tested problems.

A tutorial on non-periodic train timetabling and platforming problems

In this tutorial, we give an overview of two fundamental problems arising in the optimization of a railway system: the train timetabling problem (TTP), in its non-periodic version, and the train platforming problem (TPP). We consider for both problems the planning stage, i.e. we face them from a tactical point of view. These problems correspond to two main phases that are usually optimized in close sequence by the railway infrastructure manager. First, in the TTP phase, a schedule of the trains in a railway network is determined. A schedule consists of the arrival and departure times of each train at each (visited) station. Second, in the TPP phase, one needs to determine a stopping platform and a routing for each train inside each (visited) station, according to the schedule found in the TTP phase. Due to the complexity of the two problems, an integrated approach is generally hopeless for real-world instances. Hence, the two phases are considered separately and optimized in sequence. Although there exist several versions for both problems, depending on the infrastructure manager and train operators requirements, we do not aim at presenting all of them, but rather at introducing the reader to the topic using small examples. We present models and solution approaches for the two problems in a didactic way and always refer the reader to the corresponding papers for technical details.

Optimal Solutions to a Real-World Integrated Airline Scheduling Problem

We study an integrated airline scheduling problem for a regional carrier. It integrates three stages of the planning process (i.e., fleet assignment, aircraft routing, and crew pairing) that are typically solved in sequence. Aircraft maintenance is also taken into account. The objective function aims at minimizing a weighted sum of the number of aircraft routes, the number of crew pairings, and the waiting times of crews between consecutive flights. In addition, it aims at maximizing the robustness of the solution by also minimizing the number of times that crews need to change aircraft. We present two mixed integer linear programming models for the integrated problem. The first formulation, called the path-path model, can be considered as the “natural model” in which both the crew pairings and the aircraft routes are represented by path-based variables. The other formulation, called the arc-path model, is a novel model in which the aircraft routes are represented by arc-based variables and the crew pairi...

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.

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 strengthened with valid inequalities derived as

Timetable Optimization for High-Speed Trains at Chinese Railways

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