Large-Scale Transit Schedule Coordination Based on Journey Planner Requests

A two-stage stochastic program that reoptimizes multi-modal transit schedules city-wide is presented. The model works by perturbing or offsetting the schedule such that the expected value of waiting times at all transfer points in the system is minimized. Probabilistic information on transfers is gathered from a prototypical journey planner, a public-facing tool that transit riders query to find optimal paths through a multi-modal network. Aggregating journey plans in this manner provides information on optimal transfers as perceived by the service operator, which are then targeted for improvements. The model is implemented on the large-scale transit network of Washington, D.C., where sampled journey plans representing 9% of the daily transit demand is employed to generate a modified schedule that leads to a reduction in passenger wait times by 26.38%. The results serve to demonstrate how operators can take a user-centric view of their system as a fabric of services, gain insights from user interaction, and achieve no-cost improvements from coordinating services while accounting for uncertainty. TRB 2013 Annual Meeting Paper revised from original submittal. Nair, Coffey, Pinelli, Calabrese 2 INTRODUCTION A two-stage stochastic program that reoptimizes multi-modal transit schedules city-wide is presented. The model works by perturbing or offsetting the schedule such that the expected value of waiting times at all transfer points in the system is minimized. Information on transfers is gathered from a prototypical journey planner, a public-facing tool that transit riders query to find optimal paths through a multi-modal network. Transfer volumes of users across space and time are aggregated based on the result of queries that users make to a journey planner and are known only probabilistically. Aggregating journey plans in this manner provides information on optimal transfers as perceived by the service operator, which are then targeted for improvements. These transfers represent the ‘best-case’ service offerings on part of the operator, or a group of operators. While users may not necessary follow travel itineraries suggested by the journey planner, a representative sample of journey plans reveal systematic delays that occur at transfer points. The journey planner output, in essence, is the service that operators seek to provide. The use of a journey planner to assign users to paths in the network can be viewed as being analogous to an all-or-nothing assignment (a procedure where all users are assumed to take the shortest path to their destinations) but has key differences to that and other classical assignment procedures. A collection of journey planning requests constitute revealed or intended demand for travel. A collection of journey results constitute the operator’s best supply offerings. Since journey planners typically account for user preferences only in a very limited manner, heterogeneity in preferences (cost and travel time trade-offs for example) are ignored. Thus the principles of traffic assignment where supply and demand interact are not applied. Rather, the system is viewed entirely from the supply perspective. Given that operators supply journeys to a population of users, what are necessary schedule adjustments to provide optimal service? Existing transit schedules for large multi-modal transit networks typically evolve over the long term, and operators may lack a high-level snapshot of how interconnected services are employed by the traveling public. Further, for the case of multi-modal networks, where different entities manage modes, the coordination may be hindered by organizational barriers (1). The aim of this work is to serve as a feedback loop within this overall transit planning process (2). There is considerable literature on schedule generation, schedule coordination and optimization for public transit networks. In a review paper, Guihaire and Hao (3), highlight the need for multi-modal considerations in transit planning and service coordination. For the purposes of this paper we focus on previous work that address the problem of aligning different services to minimize transfer time. Several works that have studied coordination of services (4, 5, 6) aim to coordinate arrivals of services at a single stop. Salzborn (7) presented rules to generate schedules for simple cases and Bookbinder and Désilets (8) considered the problem of minimizing waiting times when travel time between stops is random and services have fixed headways. Given the computational intractability of associated problems, several authors have used meta-heuristic approaches to solve the general problem of transit schedule optimization (9, 10, 11, 12, 13, 14, 15, 16) using methods such as genetic algorithms, tabu search, intelligent agent based optimization, local search, or a combination of approaches (17). Specialized heuristics have been proposed as well, such as Lagrangian-based methods that seeks to optimize one line at a time (18) or generate a joint route and schedule plan (19). Large-scale network design and evaluations have been proposed for Rome (15), Boston (20), London (1), and Miami-Dade County (21). Liebchen (22) presents results from optimizing Berlin’s subways using a periodic event scheduling model where shorter TRB 2013 Annual Meeting Paper revised from original submittal. Nair, Coffey, Pinelli, Calabrese 3 wait times are achieved using fewer trains. Some previous works have studied the stochastic nature of the transit optimization and its associated processes. Yan et al. (19) present a joint route-timetable design model that considers stochasticity in demand. Mesa et al. (23) aim to find robust fleet assignments such that frequencies on lines that face high demand are increased in a tactical manner. The model doesn’t consider stochasticity explicitly, but aims to find plans that robust against perturbation in demand. Including variability in problem inputs complicates problem design and solution approaches, given the combinatorial aspect of the decision variables. Liebchen and Stiller (24) seek timetables that are resistant to delays. The proposed work shares the spirit of work by Guo and Wilson (1) where the cost of transfer inconvenience is evaluated in London using a path choice model and focuses on a set of 303 transfer movements. They highlight the importance of paying attention to transfer penalties that multi-modal transport systems impose on users along with barriers to reducing transfer penalties. There are significant institutional issues, since different portions of the network are managed by different entities, who each may view their role as being limited in correcting asynchronous schedules. The study also points to the gap in tools available to planners in quantifying and assessing the magnitude of transfer penalties, a gap that this paper seeks to address. In previous work by the authors (25), the multimodal connectivity of a transit system was analyzed and a hierarchy of transfers determined for different modal combinations. The paper proposed a deterministic program to optimize a single route, as opposed to a stochastic, system-wide method taken in this paper. The approach presented herein differs from the previously studied models in the following ways. Using journey planning queries and results, transfer costs city-wide are quantitatively characterized. This characterization leads to a model that determines local changes in schedule, defined by the offset, such that globally there is a decrease in expected waiting time. Key elements that have been the target of previous system optimization efforts such as frequencies, routes, and travel times, are retained from the existing schedule. The model is considered strategic, in that operational issues such as uncertainties in travel times and variability in demand are not considered. The existing schedule is assumed to have adequate slack built in the travel times to absorb these uncertainties. The model keeps these slacks and existing travel time, but only seeks and optimal temporal shift, should one be available. The aim therefore is to remedy particularly egregious transfer delays, as identified by the trip plans. Additionally, any transfers that are currently possible and reflected in the trip planning sample is retained in the modified schedule. The high fidelity of journey plan data is leveraged to tune the schedule in a way that has not previously been feasible. The paper makes the following contributions. First, it provides a framework to operators to leverage journey planning information to provide a comprehensive view of how services are being offered. Second, it presents a stochastic model to optimize the schedules and suggest trip offsets such that expected waiting time is minimized. Third, it presents a large-scale implementation for Washington, D.C. using the Open Trip Planner (OTP) to generate a modified schedule for a typical Wednesday. The next section describes the two-stage optimization model in greater detail, followed by which the system in Washington, D.C. is described. The journey planner set up and query samples are presented next followed by an evaluation of the improved schedule. TRB 2013 Annual Meeting Paper revised from original submittal. Nair, Coffey, Pinelli, Calabrese 4 PROBLEM FORMULATION To clarify presentation of ideas, we use the following terminology which mirrors the General Transit Feed Specification (GTFS). The term route describes one transit service that serves a series of stops (e.g., Route 86). A route is made up of a collection of trips, where each trip represents one run of the route (e.g., 7:49am service of Route 86). A journey refers to a travel itinerary from the perspective of the user, and involves a set of trips and is the output of a journey planner (e.g. take 7:49am service of Route 86 till Stop b, and walk 10 minutes). A schedule describes the arrival and departure times of each trip at each stop along the route. While the route provides a spatial context, the trip information provides a spatio-temporal sense of the transit network. Transfers of passengers occur on this spatio-temporal network. Given (a) an existing schedule of a multimodal transit network, (b) a set of routes that need to be coordinated, (c) time headways for each trip for the route, (d) a representative set of user journey plans and (e) a probabilistic estimate of transfer volumes, we seek to find an optimal offset of each transit trip, such that the expected waiting times are minimized. The problem is strategic in nature, and the output is a modified schedule that aims to coordinate trips in a manner that users transferring at various points across the network experience minimal delays. In determining the optimal offset for each trip, the aim is to seek local improvements in departure time at terminal stations. Existing transfer opportunities are preserved, as are the number of trips for each route, frequency, and travel times between stops. Travel times between stops in the existing schedule are assumed to account for slack, variability in traffic conditions, and incorporate time-of-day and day-of-week effects. For large transit networks, where schedules are periodically modified and evolve to match existing conditions on the ground, planners are cognizant of service performance. The travel time estimates contained in the existing schedule are therefore valuable. As an example see Figure 2 for Washington, D.C. that shows the space-time diagrams for selected routes demonstrating the sensitivity of schedules to day-of-week effects. The model therefore seeks to preserve the travel time estimates. The model is defined on a general service network defined by a set of nodes N which represent stops, indexed by i. On this network, there are a set of trips Q indexed by p. Each trip p visits a subset of nodes Np. The service p arrives at node i at tpi. Each time a trip p arrives at stop i, users are presented with a set of transfer opportunities. Users transfer from and to trip p. Denote a set Qpi as a set of trips that users seek to connect to (the minus sign signifies that they deboard trip p) and a set Q+pi as a set of trips that users seek to connect from (the plus sign signifying that users board trip p). Each transfer opportunity has an associated volume of passengers that is only known probabilistically denoted by C− pqi(ξ) and C + pqi(ξ) depending on if users deboard or board from service p to service q at node i. Here, ξ denotes the uncertainty in second-stage problem input. The service network and associated notation is shown in Figure 1. Define ∆pqi as the minimum transfer time required to make a successful transfer at node i from service p to q, and hp as the time headway of trip p. Additionally, define a and b as parameters that serve to bound perturbation as a fraction of time headway. For example, for values of a = −0.5 and b = 0.5, a trip is bound within one time headway of its existing departure. There are two sets of decision variables. From the operator perspective, denote xp, p ∈ Q as the time offset for each trip. This offset is determined by waiting times experienced by users, which are uncertain. Denote wpi(xp, ξ) as the waiting time in passenger-minutes associated with trip p at stop i. With these definitions, the network-wide transit coordination problem can be expressed as a two-stage stochastic program as follows. TRB 2013 Annual Meeting Paper revised from original submittal. Nair, Coffey, Pinelli, Calabrese 5 FIGURE 1 Service Network Representation and Associated Notation min x Eξ [Q(xp, ξ)] (1) s.t. ahp ≤ xp ≤ bhp ∀p ∈ Q (2) [(tqi + xq)− (tpi + xp)] ≥ ∆qpi ∀p ∈ Q, q ∈ Qpi, i ∈ Np (3) [(tpi + xp)− (tqi + xq)] ≥ ∆pqi ∀p ∈ Q, q ∈ Q+pi, i ∈ Np (4) xp ∈ R ∀p ∈ Q, (5) where Q(xp, ξ) is the second-stage program defined by