The Effect of Nonlinear Charging Function and Line Change Constraints on Electric Bus Scheduling

The recharging plans are a key component of the electric bus schedule. Since the real-world charging function of electric vehicles follows a nonlinear relationship with the charging duration, it is challenging to accurately estimate the charging time. To provide a feasible bus schedule given the nonlinear charging function, this paper proposes a mixed integer programming model with a piecewise linear charging approximation and multi-depot and multi-vehicle type scheduling. The objective of the model is to minimise the total cost of the schedule, which includes the vehicle purchasing cost and operation cost. From a practical point of view, the number of line changes of each bus is also taken as one of the constraints in the optimisation. An improved heuristic algorithm is then proposed to find high-quality solutions of the problem with an efficient computation. Finally, a real-world dataset is used for the case study. The results of using different charging functions indicate a large deviation between the linear charging function and the piecewise linear approximation, which can effectively avoid the infeasible bus schedules. Moreover, the experiments show that the proposed line change constraints can be an effective control method for transit operators

A hybrid algorithm for the multi-depot vehicle scheduling problem arising in public transportation

In this article, a hybrid algorithm is proposed to solve the Vehicle Scheduling Problem with Multiple Depots. The proposed methodology uses a genetic algorithm, initialized with three specialized constructive procedures. The solution generated by this first approach is then refined by means of a Set Partitioning (SP) model, whose variables (columns) correspond to the current itineraries of the final population. The SP approach possibly improves the incumbent solution which is then provided as an initial point to a well-known MDVSP model. Both the SP and MDVSP models are solved with the help of a mixed integer programming (MIP) solver. The algorithm is tested in benchmark instances consisting of 2, 3 and 5 depots, and a service load ranging from 100 to 500. The results obtained showed that the proposed algorithm was capable of finding the optimal solution in most cases when considering a time limit of 500 seconds. The methodology is also applied to solve a real-life instance that arises in the transportation system in Colombia (2 depots and 719 services), resulting in a decrease of the required fleet size and a balanced allocation of services, thus reducing deadhead trips

Essays on urban bus transport optimization

Nesta tese, nós apresentamos uma compilação de três artigos de otimização aplicados no contexto de transporte urbano de ônibus. O principal objetivo foi estudar e implementar heurísticas com base em Pesquisa Operacional para otimizar problemas de (re)escalonamento de veículos off-line e on-line considerando várias garagens e frota heterogênea. No primeiro artigo, foi proposta uma abordagem heurística para o problema de escalonamento de veículos múltiplas garagens. Acreditamos que as principais contribuições são o método de geração de colunas para grandes instâncias e as técnicas de redução do espaço de estados para acelerar as soluções. No segundo artigo, adicionamos complexidade ao considerar a frota heterogênea, denotada como multiple depot vehicle type scheduling problem (MDVTSP). Embora a importância e a aplicabilidade do MDVTSP, formulações matemáticas e métodos de solução para isso ainda sejam relativamente inexplorados. A principal contribuição desse trabalho foi o método de geração de colunas para o problema com frota heterogênea, já que nenhuma outra proposta na literatura foi identificada no momento pelos autores. Na terceira parte desta tese, no entanto, nos concentramos no reescalonamento em tempo real para o caso de quebras definitivas de veículos. A principal contribuição é a abordagem eficiente do reescalonamento sob uma quebra. A abordagem com redução de espaço de estados, solução inicial e método de geração de colunas possibilitou uma ação realmente em tempo real. Em menos de cinco minutos, reescalonando todas as viagens restantes.In this dissetation we presented a three articles compilation in urban bus transportation optimization. The main objective was to study and implement heuristic solutions method based on Operations Research to optimizing offline and online vehicle (re)scheduling problems considering multiple depots and heterogeneous fleet. In the first paper, a fast heuristic approach to deal with the multiple depot vehicle scheduling problem was proposed. We think the main contributions are the column generation framework for large instances and the state-space reduction techniques for accelerating the solutions. In the second paper, we added complexity when considering the heterogeneous fleet, denoted as "the multiple-depot vehicle-type scheduling problem" (MDVTSP). Although the MDVTSP importance and applicability, mathematical formulations and solution methods for it are still relatively unexplored. We think the main contribution is the column generation framework for instances with heterogeneous fleet since no other proposal in the literature has been identified at moment by the authors. In the third part of this dissertation, however, we focused on the real-time schedule recovery for the case of serious vehicle failures. Such vehicle breakdowns require that the remaining passengers from the disabled vehicle, and those expected to become part of the trip, to be picked up. In addition, since the disabled vehicle may have future trips assigned to it, the given schedule may be deteriorated to the extent where the fleet plan may need to be adjusted in real-time depending on the current state of what is certainly a dynamic system. Usually, without the help of a rescheduling algorithm, the dispatcher either cancels the trips that are initially scheduled to be implemented by the disabled vehicle (when there are upcoming future trips planned that could soon serve the expected demand for the canceled trips), or simply dispatches an available vehicle from a depot. In both cases, there may be considerable delays introduced. This manual approach may result in a poor solution. The implementation of new technologies (e.g., automatic vehicle locators, the global positioning system, geographical information systems, and wireless communication) in public transit systems makes it possible to implement real-time vehicle rescheduling algorithms at low cost. The main contribution is the efficient approach to rescheduling under a disruption. The approach with integrated state-space reduction, initial solution, and column generation framework enable a really real-time action. In less than five minutes rescheduling all trips remaining

Modeling and solving the multimodal car- and ride-sharing problem

We introduce the multimodal car- and ride-sharing problem (MMCRP), in which a pool of cars is used to cover a set of ride requests, while uncovered requests are assigned to other modes of transport (MOT). A car's route consists of one or more trips. Each trip must have a specific but non-predetermined driver, start in a depot and finish in a (possibly different) depot. Ride-sharing between users is allowed, even when two rides do not have the same origin and/or destination. A user has always the option of using other modes of transport according to an individual list of preferences. The problem can be formulated as a vehicle scheduling problem. In order to solve the problem, an auxiliary graph is constructed in which each trip starting and ending in a depot, and covering possible ride-shares, is modeled as an edge in a time-space graph. We propose a two-layer decomposition algorithm based on column generation, where the master problem ensures that each request can only be covered at most once, and the pricing problem generates new promising routes by solving a kind of shortest path problem in a time-space network. Computational experiments based on realistic instances are reported. The benchmark instances are based on demographic, spatial, and economic data of Vienna, Austria. We solve large instances with the column generation based approach to near optimality in reasonable time, and we further investigate various exact and heuristic pricing schemes

Column generation based heuristic framework for the multiple-depot vehicle type scheduling problem

The multiple-depot vehicle-type scheduling problem (MDVTSP) is an extension of the classic multiple-depot vehicle scheduling problem (MDVSP), where heterogeneous fleet is considered. Although several mathematical formulations and solution methods have been developed for the MDVSP, the MDVTSP is still relatively unexplored. Large instances of the MDVTSP (involving thousands of trips and several depots and vehicle types) are still difficult to solve in a reasonable time. We introduce a heuristic framework, combining time-space network, truncated column generation (TCG) and state space reduction, to solve large instances of the MDVTSP. Extensive testing was carried out using random generated instances, in which a peak demand distribution was defined based on real-world data from public transportation systems in Brazil. Furthermore, experiments were carried out with a real instance from a Brazilian city. The framework has been implemented in several algorithm variants, combining different developed preprocessing procedures, such as state space reduction and initial solutions for the TCG. Computational results show that all developed algorithms obtained very good performances both in quality and efficiency. The best solutions, considering simultaneously quality and efficiency, were obtained in the heuristics involving state space reduction

Problème d’horaire d’autobus avec dépôts multiples et modification contrôlée des heures de début des voyages

RÉSUMÉ : Dans le domaine des transports en commun, et en particulier des réseaux d’autobus, organiser efficacement et à moindres coûts le réseau est crucial. A partir d’une grille horaire donnant les heures de départ des trajets de chaque ligne, la compagnie d’autobus doit organiser sa flotte de véhicules et son personnel afin d’assurer le service de transport. Dans ce problème de planification opérationnelle, nous nous intéressons à la gestion des véhicules. Il s’agit de construire le parcours de chacun des véhicules afin de couvrir tous les trajets désirés, et cela au coût le plus faible possible. Dès lors que la compagnie entrepose ses véhicules dans des dépôts différents, ce problème est très difficile à résoudre (classe des problèmes NP-difficiles). On l’appelle problème d’horaire d’autobus avec dépôts multiples ou MDVSP pour l’anglais Multiple Depot Vehicle Scheduling Problem. Dans ce mémoire, on ajoute au MDVSP la possibilité de modifier les heures de départ initialement prévues. Cela laisse espérer une réduction du nombre d’autobus nécessaires et des trajets à vide moins coûteux pour la compagnie d’autobus. Pour modéliser ce problème, nous introduisons un graphe hybride entre les deux réseaux classiques de modélisation du MDVSP, à savoir le réseau de connexions et le réseau espacetemps. Le premier modélise explicitement par un arc chacune des connexions possibles entre deux trajets, le second représente chaque mouvement envisageable d’un autobus par un arc de déplacement dans le temps et éventuellement l’espace. Le réseau hybride introduit permet de conserver l’adaptabilité du premier type de réseau et le faible nombre d’arcs du second. En terme de temps de calcul, ce nouveau réseau donne de très bonnes performances pour résoudre le MDVSP classique et le MDVSP avec modification des horaires de départ. Pour résoudre les modèles ainsi construits, on décompose le modèle en deux entités, l’une construit des itinéraires d’autobus admissibles, l’autre choisit une combinaison de ces itinéraires de coût minimal. On peut ainsi résoudre la relaxation linéaire de notre problème par génération de colonnes et donc le problème en nombres entiers par un algorithme de Branch and Price. On décrit et implémente dans ce travail plusieurs méthodes pour accélérer cet algorithme, ralenti entre autres par la dégénérescence des problèmes. Les résultats obtenus montrent qu’on peut ainsi trouver des solutions quasi-optimales en des temps de calcul fortement réduits. Sur des exemples de 300 ou 500 voyages où l’on permet de modifier les horaires de plus ou moins 2 minutes, on voit qu’il est possible de réduire le nombre de véhicules nécessaires de 1 à 3 unités. On étudie également une instance réelle de plus de 1000 voyages que l’on résout en un peu plus d’une heure. Les réductions de coût pour la compagnie d’autobus peuvent donc être importantes. Cependant, modifier les heures de départ ne doit pas détériorer la qualité des horaires pour les utilisateurs du réseau. On voudrait, en particulier, garder des espacements égaux entre les trajets consécutifs d’une même ligne. Un équilibre doit donc être trouvé entre réduction des coûts pour la compagnie et qualité des horaires pour les passagers. Pour cela, on propose d’adopter une approche en deux phases : la première résout le MDVSP avec modification d’horaires, la seconde optimise les horaires de départ compatibles avec les itinéraires trouvés dans la première phase. Pour coordonner ces deux étapes, on tire profit du réseau hybride introduit précédemment. Les résultats montrent qu’on peut ainsi trouver de bons compromis entre le coût des itinéraires d’autobus et la qualité des horaires pour les passagers.----------ABSTRACT : In the field of public transit, and especially of bus systems, it is crucial to organize the network efficiently and at a low cost. Given a timetable for each line, a bus company should determine the schedule of all its vehicles and drivers to provide the required service. In this operational planning problem, we will focus on vehicle scheduling. The aim is to build vehicle itineraries in order to cover all desired trips, and this at the lowest possible cost. If the company stores its buses in several depots, this problem is hard to solve (NP-hard problem). It is called Multiple Depot Vehicle Scheduling Problem, abbreviated by MDVSP hereafter. In this work, we add to the classical MDVSP the possibility of modifying the initial timetable, in other words, we allow trip shifting. This way, one can expect to reduce the required number of vehicles and to limit the cost of deadhead trips, i.e. trips without any passengers. To model this problem, we introduce a hybrid graph mixing the two classical graphs used to model MDVSP, namely the connection network and the time-space network. The first one represents explicitly every possible connection between two compatible trips, the second one models every possible bus movement through an arc representing a move in time and possibly in space. Our hybrid network keeps the modeling flexibility of the first type of network and the small number of arcs of the second one. In terms of computation time, this new network performs very well when solving both the classical MDVSP and the MDVSP with trip shifting. In order to solve this problem, we split the model into two entities, one builds feasible bus itineraries, the other chooses a combination of these itineraries at the lowest cost. The linear relaxation of our problem could then be solved using a column generation method and the integer problem thus using a Branch and Price algorithm. In this work, we describe and implement several ways to speed up this algorithm, which is otherwise slowed down by degeneracy (among other things). For our test instances, we manage to find solutions very close to optimality with a substantial reduction of computation time. On instances with 300 or 500 trips, where schedule shifts of up to two minutes were allowed, the bus company can save up to 3 vehicles. We studied also a real-life problem involving more than 1000 trips that we solved in a bit more than one hour. Costs reduction can therefore be very significant. However, shifting trips should not be detrimental to the timetable quality for passengers. In particular, we wish to keep constant headways, i.e. the time lapses between two consecutive departures on the same line. Hence, a tradeoff should be found between cost reduction for the bus company and timetable quality for passengers. To this end, we propose to adopt a two-step algorithm: the first step solves the MDVSP with trip shifting, the second one optimizes the departure schedules while taking into consideration the itineraries found in the first step. To coordinate these two phases, we take advantage of the previously introduced hybrid network. Computational results with this algorithm show that one can quickly find interesting tradeoffs between itineraries costs and timetable quality