Solving Public Transit Scheduling Problems

Operational planning within public transit companies has been extensively tackled but still remains a challenging area for operations research models and techniques. This phase of the planning process comprises vehicle scheduling, crew scheduling and rostering problems. In this paper, a new integer mathematical formulation to describe the integrated vehicle-crew-rostering problem is presented. The method proposed to solve this multi-objective problem is a sequential algorithm considered within a preemptive goal programming framework that starts from the solution of an integrated vehicle and crew scheduling problem and ends with the solution of a driver rostering problem. Feasible solutions for the vehicle and crew scheduling problem are obtained by combining a column generation scheme with a branch-and-bound method. These solutions are the input of the rostering problem, which is tackled through a mixed binary linear programming approach. An application to real data of a Portuguese bus company is reported and shows the importance of integrating the three scheduling problems

Analyzing a Family of Formulations for Cyclic Crew Rostering

In this paper, we analyze a family of formulations for the Cyclic Crew Rostering Problem (CCRP), in which a cyclic roster has to be constructed for a group of employees. Each formulation in the family is based on a partition of the roster. Intuitively, finer partitions give rise to a formulation with fewer variables, but possibly more constraints. Coarser partitions lead to more variables, but might allow to incorporate many of the constraints implicitly. We derive analytical results regarding the relative strength of the different formulations, which can serve as a guideline for formulating a given problem instance. Furthermore, we propose a column generation approach, and use it to compare the strength of the formulations empirically. Both the theoretical and computational results demonstrate the importance of choosing a suitable formulation. In particular, for practical instances of Netherlands Railways, stronger lower bounds are obtained, and more than 90% of the roster constraints can be modeled implicitly

Is Equality always desirable?

In this paper, we analyze the trade-off between perceived fairness and perceived attractiveness in crew rostering. First, we introduce the Fairness-oriented Crew Rostering Problem. In this problem, attractive cyclic rosters have to be constructed, while respecting a pre-specified fairness level. Then, we propose a flexible mathematical formulation, able to exploit problem specific knowledge, and develop an exact Branch-Price-and-Cut solution method. The solution method combines Branch-and-Bound with column generation, where profitable columns are separated by solving resource constrained shortest path problems with surplus variables. We also derive a set of valid inequalities to tighten the formulation. Finally, we demonstrate the benefit of our approach on practical instances from Netherlands Railways, the largest passenger railway operator in the Netherlands. We are able to construct the explicit trade-off curve between fairness and attractiveness and show that a sequential approach can lead to suboptimal results. In particular, we show that focusing solely on fairness leads to rosters that are disproportionally less attractive. Furthermore, this decrease in attractiveness is heavily skewed towards the most exible employees. Thus, in order to generate truly fair rosters, the explicit trade-off between fairness and attractiveness should be considered

Crew Planning at Netherlands Railways: Improving Fairness, Attractiveness, and Efficiency

The development and improvement of decision support voor crew planning at Netherlands Railways (NS

Operations research in passenger railway transportation

In this paper, we give an overview of state-of-the-art OperationsResearch models and techniques used in passenger railwaytransportation. For each planning phase (strategic, tactical andoperational), we describe the planning problems arising there anddiscuss some models and algorithms to solve them. We do not onlyconsider classical, well-known topics such as timetabling, rollingstock scheduling and crew scheduling, but we also discuss somerecently developed topics as shunting and reliability oftimetables.Finally, we focus on several practical aspects for each of theseproblems at the largest Dutch railway operator, NS Reizigers.passenger railway transportation;operation research;planning problems

Bus driver rostering by hybrid methods based on column generation

Tese de doutoramento, Informática (Engenharia Informática), Universidade de Lisboa, Faculdade de Ciências, 2018Rostering problems arise in a diversity of areas where, according to the business and labor rules, distinct variants of the problem are obtained with different constraints and objectives considered. The diversity of existing rostering problems, allied with their complexity, justifies the activity of the research community addressing them. The current research on rostering problems is mainly devoted to achieving near-optimal solutions since, most of the times, the time needed to obtain optimal solutions is very high. In this thesis, a Bus Driver Rostering Problem is addressed, to which an integer programming model is adapted from the literature, and a new decomposition model with three distinct subproblems representations is proposed. The main objective of this research is to develop and evaluate a new approach to obtain solutions to the problem in study. The new approach follows the concept of search based on column generation, which consists in using the column generation method to solve problems represented by decomposition models and, after, applying metaheuristics to search for the best combination of subproblem solutions that, when combined, result in a feasible integer solution to the complete problem. Besides the new decomposition models proposed for the Bus Driver Rostering Problem, this thesis proposes the extension of the concept of search by column generation to allow using population-based metaheuristics and presents the implementation of the first metaheuristic using populations, based on the extension, which is an evolutionary algorithm. There are two additional contributions of this thesis. The first is an heuristic allowing to obtain solutions for the subproblems in an individual or aggregated way and the second is a repair operator which can be used by the metaheuristics to repair infeasible solutions and, eventually, generate missing subproblem solutions needed. The thesis includes the description and results from an extensive set of computational tests. Multiple configurations of the column generation with three decomposition models are tested to assess the best configuration to use in the generation of the search space for the metaheuristic. Additional tests compare distinct single-solution metaheuristics and our basic evolutionary algorithm in the search for integer solutions in the search space obtained by the column generation. A final set of tests compares the results of our final algorithm (with the best column generation configuration and the evolutionary algorithm using the repair operator) and the solutions obtained by solving the problem represented by the integer programming model with a commercial solver.Programa de Apoio à Formação Avançada de Docentes do Ensino Superior Politécnico (PROTEC), SFRH/PROTEC/67405/201

A short-turning policy for the management of demand disruptions in rapid transit systems

Rapid transit systems timetables are commonly designed to accommodate passenger demand in sections with the highest passenger load. However, disruptions frequently arise due to an increase in the demand, infrastructure incidences or as a consequence of fleet size reductions. All these circumstances give rise to unsupplied demand at certain stations, which generates passenger overloads in the available vehicles. The design of strategies that guarantee reasonable user waiting time with small increases of operation costs is now an important research topic. This paper proposes a tactical approach to determine optimal policies for dealing with such situations. Concretely, a short-turning strategy is analysed, where some vehicles perform short cycles in order to increase the frequency among certain stations of the lines and to equilibrate the train occupancy level. Turn-back points should be located and service offset should be determined with the objective of diminishing the passenger waiting time while preserving certain level of quality of service. Computational results and analysis for a real case study are provided.Junta de Andalucía P09-TEP-5022Natural Sciences and Engineering Research Council of Canada (NSERC) 39682-1

Optimisation de roulements de chauffeurs d’autobus

RÉSUMÉ: Le problème de roulements de chauffeurs d’autobus vise à déterminer les horaires de travail des chauffeurs d’autobus sur un horizon donné. Il s’agit d’un problème où des séquences de jours de repos et de journées de travail sont construites. Les journées de travail sont générées lors de la résolution du problème de construction de journées de travail. Ce problème a pour objectif de générer des journées de travail anonymes afin d’assurer, à un coût minimum, la couverture complète des horaires d’autobus. Plusieurs règles doivent être respectées lors de la résolution, entre autres, l’amplitude maximale ou minimale d’une journée de travail, le temps maximal ou minimal de travail, etc. Lorsque les journées de travail sont déterminées, elles sont affectées aux différents chauffeurs disponibles et les roulements des chauffeurs sont construits à cette étape. Les journées de travail sont affectées en respectant un ensemble de règles dérivées des conventions collectives, et chaque journée de travail est effectuée par un chauffeur durant un jour de la semaine. Dans notre contexte, les roulements de chauffeurs d’autobus sont cycliques et définis sur une semaine pour un certain horizon de planification. Ainsi les journées de travail peuvent varier d’un jour à l’autre mais se répètent d’une semaine à une autre. Le problème de roulements avec jours de repos fixés vise à affecter les journées de travail aux différents chauffeurs dans les jours de travail (c’est-à-dire, les jours qui ne sont pas des jours de repos). Nous proposons, d’abord, une nouvelle formulation forte en nombres entiers du problème de roulements avec repos fixés. Les règles d’affectation des journées de travail sont diverses et compliquées, surtout qu’elles impliquent des contraintes de repos de nuit entre deux journées de travail et des contraintes qui s’étendent sur plusieurs jours et parfois sur plusieurs semaines. La fonction objectif vise à équilibrer le plus possible la charge de travail entre tous les chauffeurs. Ceci a été traduit par la minimisation des déviations positives par rapport à la moyenne des charges de travail totale par semaine de toutes les journées de travail. Différentes modélisations des contraintes de repos de nuit ont été proposées, ainsi qu’une deuxième formulation de la fonction objectif, mais qui vise aussi à équilibrer la charge de travail entre les chauffeurs d’autobus. Nous avons montré que la nouvelle formulation permet de reserrer l’espace de recherche lors du branchement, ce qui permet d’avoir des solutions entières plus rapidement. Ensuite, une approche est proposée pour résoudre le problème de roulements intégré de construction de jours de repos et d’affectation de journées de travail. Le problème est modélisé comme un programme linéaire mixte en nombres entiers. Étant donné que le problème ne contient pas de règles de quarts de travail ni de règles souples (des préférences par exemple), le problème présente beaucoup de symétrie. Le modèle s’est avéré très difficile à résoudre à l’optimalité avec le solveur commercial CPLEX malgré l’ajustement très poussé des paramètres et l’utilisation des méthodes avancées de programmation en nombres entiers (fixation de variables, branchement priorisé, ...). Sur la base de ce modèle, nous avons introduit une matheuristique à deux étapes qui permet de trouver des solutions de très bonne qualité. En utilisant une telle solution comme donnée d’entrée dans un solveur commercial, le modèle intégré peut être résolu beaucoup plus rapidement. Nos expériences de calcul testées sur des instances réelles de grande taille ont montré l’efficacité de la matheuristique. Des solutions optimales ont été obtenues dans des temps de calcul relativement courts (3.5 heures pour le cas impliquant jusqu’à 333 chauffeurs). En outre, en fournissant ces solutions comme solutions initiales au solveur CPLEX, de grandes accélérations (jusqu’à 99%) ont été obtenues pour résoudre le problème intégré avec une optimalité prouvée. L’article intitulé "Integrated and sequential solution methods for the cyclic bus driver rostering problem" traitant cet objectif a été publié dans la revue "Journal of the Operational Research Society" Finalement, nous avons intégré des règles relatives aux préférences des chauffeurs dans le modèle de roulements. Le nouveau modèle vise à affecter les journées de travail aux différents chauffeurs sur un horizon prédéfini, tout en respectant les règles strictes d’affectation, en équilibrant la charge de travail entre les chauffeurs et en minimisant le plus possible les violations des règles souples (les préférences). Deux nouvelles matheuristiques ont été proposées. La première limite l’espace de recherche en pré-assignant les journées de travail aux roulements avec des jours de repos fixés. La deuxième matheuristique utilise un problème de partitionnement d’ensemble pour décomposer les roulements de grande taille en sous-roulements de tailles petites à moyennes. Dans une série d’expériences de calcul menées sur des instances réelles, nous montrons que ces matheuristiques peuvent être utilisées pour produire des solutions de bonne qualité pour des grandes instances (333 chauffeurs et 1509 journées de travail) dans des temps de calcul relativement courts. L’article intitulé "Preference-based bus driver rostering problem with fixed days off" traitant cet objectif a été soumis à la revue "Public Transport"---------ABSTRACT:The bus driver rostering problem aims at building the work schedules of bus drivers over a given period of time. Solving such problem results in sequences of days off and duties. The duties are constructed via the duty scheduling problem, which creates anonymous duties in order to ensure, at a minimum cost, complete coverage of a set of bus trips. Several rules must be respected while solving this problem, i.e. maximum or minimum span of a duty, maximum or minimum working time, etc. The resulting duties must then be assigned to the different available drivers, creating their rosters. This process complies with a set of rules derived from collective agreements. Every duty is performed by one driver on one day of the week. Here in this context, bus driver rosters are cyclic, and defined over a week for a certain planning horizon. Thus, duties may vary from a day to another, but they are repeated weekly. The rostering problem with fixed days off aims at assigning duties to drivers in working days. First, a new mixed integer formulation of the problem is proposed. The assignment rules are diverse and complicated, especially since they involve night rest constraints between two duties and constraints that are extended over several days, and sometimes over several weeks. The objective function is to balance the workload among all the drivers. This has been achieved by minimizing positive deviations from the average total workload per week. Furthermore, different formulations of the night rest constraints are presented, as well as, a second formulation of the objective function that minimizes the sum of the absolute values of the deviations from the average workload per week. It is shown that the first proposed formulation makes it possible to tighten the search space during the branch-and-bound process and, consequently, helps finding integer solutions more rapidly. Next, an approach is proposed to solve the integrated days off scheduling and rostering problem. First the problem is modeled as a mixed integer linear program. In this problem, there are no shifts, and therefore, no shift related rules that reduce the solution space, nor shift related preferences that can reduce symmetry in the branch-and-bound process and ease the search for integer solutions. This model turns out to be very hard to solve to optimality without providing an initial solution. Based on this model, we introduce a new two-step matheuristic that can compute high-quality solutions. Using such a solution as an input to a commercial solver, the integrated model can be solved much more rapidly. Our computational results obtained on real-world instances involving up to 333 drivers and 1509 duties show that these initial solutions are optimal in most cases and, consequently, that the proposed matheuristic is very efficient by itself. Finally, we integrated the bus driver preference rules to the rostering problem. The new model aims at assigning duties to different drivers over a predefined cyclic horizon, while respecting a set of rules (hard constraints), balancing the workload among the drivers and satisfying as much as possible the driver preferences (soft constraints). We first model the problem as a mixed integer linear program that minimizes the number of preference violations while maintaining the workload balance of the solutions within a certain margin relative to the optimal one. Since this model is hard to solve for large instances, we propose two new matheuristics. The first one restricts the search space by preassigning duties to rosters based on an optimal solution to the duty assignment problem with fixed days off. The second algorithm makes use of a set partitioning problem to decompose rosters consisting of a large number of positions into sub-rosters of smaller sizes. In a series of computational experiments conducted on real-world instances, we show that these matheuristics can be used to produce high-quality solutions for large instances of the problem, within short computational times