Frequency and Vehicle Capacity Determination using a Dynamic Transit Assignment

We integrate for the first time, to our knowledge, a dynamic transit assignment model into the tactical planning phase. The settings of service frequencies and vehicle capacities determine line capacity and have significant consequences for level-ofservice and operational costs. The objective of this study is to determine frequency and vehicle capacity at the network level while accounting for the impact of service variations on users and operator costs. To this end, we propose a simulation-based optimization approach. The proposed model allows accounting for variations in service headways and crowding as well as their consequences for passenger flows distribution, all of which have not been accounted for in the tactical planning so far. Practical benefits of the model are demonstrated by an application to a bus network in the Amsterdam metropolitan area. This study contributes to the development of a new generation of methods that integrate reliability into the tactical planning phase to improve service quality. Line capacity – that is, the number of passengers that can be transported within a certain time interval – is largely determined by the product of line frequency and the capacity of the vehicles assigned for operating it. The determination of frequencies and vehicle capacities is thus a crucial service design decision when planning public transport services. These decisions are considered both at the strategic and tactical levels. At the strategic level, frequency setting interacts with passengers’ route choices and the designated line capacity has consequences for the choice of public transport technology (e.g., metro, light rail, train, or bus). At the tactical level, both service frequencies and vehicle capacity (e.g., number of train cars, ordinary or articulated bus) can be altered on a seasonal basis and vary by time of day and day of the week. Service unreliability can severely affect line capacity by reducing the effective frequency. However, deviations from planning are only handled at the operational level by deploying real-time management strategies. In this study, we propose to integrate the impact of service reliability on both service provider and service users into the service dimensioning decisions. Service providers can amend service frequency or vehicle capacity in response to service utilization levels, for example if passenger loads exceed the desired on-board occupancy. While both increased frequency and deploying larger vehicles inflict additional costs, the former requires the reallocation of drivers and rolling stock, whereas the latter requires changes in rolling stock composition. From the passengers’ perspective, higher frequency is preferred over larger vehicles. While they both solve the on-board crowding problem, higher frequency also yields shorter waiting times, leading to a lower generalized travel cost. The consideration of consequences of service uncertainty for resource allocation requirements has so far been confined to vehicle and crew schedule, that is the operational planning phase, in the public transport planning literature. Desaulniers and Hickman (1) and Ibarra-Rojas et al. (2) provide exhaustive reviews of the considerable scientific efforts that have been devoted to solving the large range of public transport planning optimization problems. Frequency and vehicle size were typically either solved separately or jointly for a single line, neglecting their interplay when distributing a limited amount of resources under uncertainty across the service network. Service variability is inherent to (urban) public Delft University of Technology, Faculty of Civil Engineering and Geosciences, Transport & Planning, Delft, the Netherlands Corresponding Author: Address correspondence to Oded Cats: o.cats@tudelft.nl transport services and stems from the stochasticity of traffic conditions, operations and passenger demand and their interactions. Recently, Gkiotsalitis and Cats (3) integrated travel time, headway, and demand variability into an exact optimization of service headways where the expected passenger flows are assumed exogenous. In practice, authorities or operators typically use predefined service standards such as maximum vehicle occupancy rates along with local experience and expert judgment as the basis for setting frequencies and vehicle capacities. Early studies formulated rule-based decision criteria for determining the service frequency on a line given passenger arrival rates, potential fleet size constraints and a desired maximum vehicle load factor (e.g., 4, 5). Starting from the 1990s, the problem was often solved in combination with assignment models that can forecast passengers’ behavior in response to a potential supply setting (6, 7). Public transport supply optimization is then solved using bi-level optimization models: a supply optimization model at the upper level and an assignment model at the lower level which computes passenger flows under equilibrium conditions which result from a certain supply given by the upper level model. More recent developments involve the consideration of additional decision variables and the use of metaheuristics (8–10). Several studies develop approaches for determining both frequencies and vehicle capacities simultaneously (11–13). These methods enable a more anticipatory planning and dimensioning of supply than if service was merely adjusted to prevailing demand distribution conditions. All the methods developed hitherto have used static assignment models for distributing passenger demand on the service network, assuming perfectly reliable supply conditions. Travelers are thus assumed to make decisions based on average supply conditions. Performance indicators are, therefore, computed based on the given supply and passenger flows without taking into account the dynamic interaction between demand and supply. However, especially in the context of dense metropolitan systems, the dynamic and stochastic interaction between demand and supply may lead to significant reliability and crowding issues that are not accounted for in static assignment models. The objective of this study is to determine frequency and vehicle capacity at the network level while accounting for the impact of service variations on users and operator costs. To this end, we propose a simulation-based optimization approach consisting of a metaheuristic technique which iteratively evaluates the consequences of selected solutions using an agent-based dynamic transit assignment model. The latter explicitly models passenger flow distributions which are dependent on the respective supply specifications. To the best of the authors’ knowledge, this is the first study to use a dynamic transit assignment in solving a tactical planning problem, allowing the capture of the impacts of stochastic variations in system supply and demand on the desired service dimensioning. The practical applicability and implications of the proposed model are demonstrated using data from a case study in Amsterdam, the Netherlands. The paper is structured as follows: the next section provides a review and synthesis of the literature on headway and vehicle size determination. We then present a modeling framework along with a description of its formulation and implementation. The model is examined and verified using a test network and is thereafter applied for a real-world bus network, the set-up and results of which are detailed in the subsequent section. We conclude with the key findings and implications for public transport planning and point out directions for further research in the final section.