Schedule-based transit assignment model with vehicle capacity and seat availability

Abstract

In this paper, we propose a new schedule-based equilibrium transit assignment model that differentiates the discomfort level experienced by sitting and standing passengers. The notion of seat allocation has not been considered explicitly and analytically in previous schedule-based frameworks. The model assumes that passengers use strategies when traveling from their origin to their destination. When loading a vehicle, standing on-board passengers continuing to the next station have priority to get available seats and waiting passengers are loaded on a First-Come-First-Serve (FCFS) principle. The stimulus of a standing passenger to sit increases with his/her remaining journey length and time already spent on-board. When a vehicle is full, passengers unable to board must wait for the next vehicle to arrive. The equilibrium conditions can be stated as a variational inequality involving a vector-valued function of expected strategy costs. To find a solution, we adopt the method of successive averages (MSA) that generates strategies during each iteration by solving a dynamic program. Numerical results are also reported to show the effects of our model on the travel strategies and departure time choices of passengers.

Introduction

System planners and transit operators are interested in the passengers’ choice of transit services for determining the performance and revenue generated from the transit system. Particularly in the peak hours, passengers are often unevenly distributed over different transit vehicles and time periods due to various reasons (e.g. double headings for the case of buses) causing an inefficient utilization of the service capacity. Therefore, a transit assignment model, which takes into account the temporal distribution of demand and congestion over the transit network, is useful in estimating how passengers utilize a given transit system. This model will also serve as a tool for system planners and transit operators to plan and schedule the transit services for optimizing their objectives (e.g. minimizing the total delay or maximizing the revenue).

In the literature of transit assignment studies, models could be classified into two different categories: static (frequency-based) and dynamic (schedule-based) transit assignment models. Similar to the traditional static user equilibrium assignment on road networks, static transit assignment models consider a constant transit passenger demand such that the strategy, or hyperpath, costs of individual passengers are minimized (Spiess and Florian, 1989, De Cea and Fernandez, 1993, Lam et al., 1999). In static transit assignment models, as there is no time dimension, all model characteristics are averaged out over the modeling period (e.g. the morning peak hour). As the average values are adopted, static transit assignment models could not reveal the bottleneck induced congestion problem (Schmoecker et al., 2008) and are not able to properly evaluate the transit network under dramatically changing network conditions (e.g. passenger arrival rate and loading of transit services) during the period of analysis. Despite its weaknesses, static transit assignment models are commonly adopted for the strategic and long-term planning/evaluation. In order to precisely model the dynamic characteristics in transit networks, dynamic transit assignment models have gained their importance in the past two decades. For the majority of these models, timetables of the transit services are assumed to be sufficiently reliable, which in contrast to the use of average frequencies in static models. In the dynamic case, transit passengers are not only choosing their strategies/hyperpaths, but also their departure and arrival time for minimizing their generalized cost. In order to incorporate this time dependent choice, a time-dependent transit network should be adopted for the dynamic (schedule-based) transit assignment studies. Poon et al. (2004) suggested to classify the time-dependent transit network into: (a) diachronic graph representation (Nuzzolo et al., 2001); (b) dual graph representation (Moller-Pedersen, 1999); (c) forward star network formulation (Tong and Wong, 1998), and; (d) space-time formulation (Nguyen et al., 2001, Hamdouch and Lawphongpanich, 2008).

Over the past few decades, many studies have proposed models for dynamic transit assignment. Sumi et al. (1990) proposed a stochastic approach to model departure times and route choices of passengers on a mass transit system. Alfa and Chen (1995) developed a transit assignment model for forecasting the temporal demand distribution along a corridor under a random assumption of passenger boarding. Their model, however, neglected the congestion effect at the transit station, which affects the passengers’ waiting time, and in-vehicle discomfort, which affects the passengers’ preference on that transit line. Tong and Wong, 1998, Poon et al., 2004 proposed a dynamic user equilibrium model that considers the effect of congestion at transit stations on the time-dependent demand distribution and accounts for the First-Come-First-Serve (FCFS) principle when loading passengers. In Poon et al. (2004), the authors model the user equilibrium transit assignment problem as an optimization problem with an unaccountably infinite number of decision variables and later discretize the problem in order to find an approximate solution. Tian et al. (2007a) improved the model of Alfa and Chen (1995) by introducing the in-vehicle congestion through a bulk-queue model and analyzed theoretical properties of the equilibrium flows. However, their model and analysis are limited to transit networks with a single destination.

Optimal strategy approach is one of the commonly adopted formulations for transit assignment problems. The core idea for optimal strategy is that a traveler will select at each node of the network, a set of attractive lines that allows him/her to reach his/her destination at a minimum expected cost. Spiess and Florian (1989) were the first to propose a transit assignment model based on the optimal strategy approach. In their study, the optimal strategy approach is used to solve a linear, many-to-one and uncongested transit network. In Spiess and Florian (1989), as departure time is not considered in the strategy, their model is only applicable to static transit assignment. Wu et al. (1994) further extended the model in Spiess and Florian (1989) to accommodate asymmetric cost that models the waiting and in-vehicle cost as a function of the transit flow. Although the proposed algorithm could effectively solve the transit assignment problem, authors of that paper suggested that unless decomposition approach is used, considerable amount of computer storage is needed for large scale problems. Marcotte et al. (2004) formulated the strategic flow problem of a capacitated and static transit network into a variational inequality (VI) format. A combined Frank-Wolfe and projection algorithm is proposed by the authors and is proven to be an efficient and robust algorithm for solving the strategic flow problem. Different from the previous static models, Hamdouch and Lawphongpanich (2008) proposed a dynamic schedule-based transit assignment where the choice of strategy is an integral part of user behavior. In that study, passengers specify their travel strategy by providing, at each transit station and each point in time, an ordered list of transit lines they prefer to use to continue their journey. For a given passenger, the user-preference sets at each time-expanded (TE) node collectively yield a set of potential paths that depart from the passenger’s origin at the same time and generally arrive at the destination at different times. Also, when loading a transit vehicle at a station, on-board passengers continuing to the next station remains on the vehicle and waiting passengers are loaded in a first-come-first-serve (FCFS) basis.

Among the transit assignment models developed in the literature, capacity constrained transit assignment have been widely considered (De Cea and Fernandez, 1993, Lam et al., 1999, Kurauchi et al., 2003, Hamdouch et al., 2004, Cepeda et al., 2006, Hamdouch and Lawphongpanich, 2008) for replicating the actual capacity constraint of transit vehicles. De Cea and Fernandez (1993) had indirectly incorporate the capacity constraint of transit vehicles in the waiting time of passengers at transit stops that leaded to the definition of effective frequency of transit services. In their study, the passenger waiting time will increase as the volume to capacity ratio of the transit services increases. However, as this setup will only increase the waiting time as the transit vehicle is more congested, it still allows the capacity to be exceeded. Different from the study of De Cea and Fernandez (1993), the capacity constraint are strictly enforced in the transit assignment models formulated in Kurauchi et al., 2003, Hamdouch et al., 2004. In Kurauchi et al. (2003), the absorbing Markov chain is adopted to solve the capacity constrained transit assignment problem. The strict capacity constraint is incorporated through the consideration of failure-to-board probabilities, which is dependent on the residual capacity of the transit vehicles. Similar to Kurauchi et al., 2003, Hamdouch et al., 2004 had considered the boarding probability to incorporate the capacity constraint. Hamdouch and Lawphongpanich (2008) extended the model to the dynamic setting where the fail-to-board passengers are assigned to the waiting arc to wait for their next preferred transit services with residual capacities.

Despite the rich literature in capacity constrained transit assignment, few studies had drawn attention to the issue of differentiating between sitting and standing capacities. As mentioned in the second edition of the Transit Capacity and Quality of Service Manual (Kittelson & Associates et al., 2003), the seat capacity is one of the critical factors for assessing the transit quality of service. The treatment of seat allocation can allow transit planners to consider the allocation of space in vehicle to seat and standing passengers. This is an important issue in countries with high utilization of transit service (e.g. Hong Kong and Japan). Some passengers, particularly those with a long-distance journey, may prefer to wait for a transit service with more available seats or decide to arrive at stations earlier to increase their chances of getting a seat. Tian et al. (2007b) extended their previous model to differentiate the in-vehicle congestion effects of sitting and standing passengers in many-to-one transit networks. Schmoecker et al. (2009) proposed a static frequency-based assignment model that considers travelers probability of finding a seat in their perception of route choice and used Markov chain process to find an equilibrium solution. In their model, standing on-board passengers have priority to get available seats and waiting passengers are loaded on a random manner when they mingle on stations. Leurent (2010) incorporated the seat capacity into the hyperpath-based transit assignment model and considered different discomfort cost for the seating and standing passengers. In their model, priority rules are adopted to govern the probability of getting a seat and are used to assign the passengers to sit, or stand, along each of the transit segment. Leurent and Liu (2009) had further applied the model in Leurent (2010) to Paris for demonstrating its applicability in large-scale network. Apart from the aforementioned static models, Sumalee et al. (2009) formulated a dynamic transit assignment model with explicit consideration of stochastic seat allocation among the standing and boarding passengers, and proposed a solution algorithm based on Monte–Carlo simulation. In their seat allocation process, the probability of getting a seat depends on the time that passengers spent and is going to spend on the transit line.

The objective of the paper is to extend the schedule-based transit assignment model in Hamdouch and Lawphongpanich (2008) to differentiate the discomfort level experienced by the sitting and standing passengers. As compared to the model in Sumalee et al. (2009), we propose an analytical model that captures the stochastic nature of the standing and boarding passengers to get a seat. Each class of passengers, grouped by their remaining journey lengths and times already spent on-board, is assigned success-to-sit, success-to-stand and failure-to-board probabilities. These probabilities are computed by performing a dynamic network loading. When loading a vehicle, standing on-board passengers continuing to the next station have priority to get available seats and waiting passengers are loaded on a First-Come-First-Serve (FCFS) principle. The stimulus of a standing passenger to sit increases with his/her remaining journey length and time already spent on-board. When a vehicle is full, passengers unable to board must wait for the next vehicle to arrive.

For the remainder, Section 2 presents the notation, network representation and assumptions of the proposed model. Travel strategies and the dynamic loading process are described in Section 3. Section 4 shows the calculation of the expected cost associated with an optimal strategy. Section 5 formulates the transit assignment problem as a variational inequality and discusses the existence of an equilibrium solution. To illustrate the effects of our model on the strategy choices and departure times of passengers, Section 6 presents numerical results from two transit networks. Finally, Section 7 concludes the paper.