Joint Optimization of Pattern, Headway, and Fleet Size of Multiple Urban Transit Lines with Perceived Headway Consideration and Passenger Flow Allocation

This study addresses the urban transit pattern design problem, optimizing stop sequences, headways, and fleet sizes across multiple routes and periods simultaneously to minimize user costs (composed of riding, waiting, and transfer times) under operational constraints (e.g., vehicle capacity and fleet size). A destination-labeled multi-commodity network flow (MCNF) formulation is developed to solve the problem at a large scale more efficiently compared to the previous literature. The model allows for flexible pattern options without relying on pre-defined candidate sets and simultaneously considers multiple operational strategies such as express/local services, short-turning, and deadheading. It evaluates perceived headways of joint patterns for passengers, assigns passenger flows to each pattern accordingly, and allows transfers across patterns in different directions. The mixed-integer linear programming (MILP) model is demonstrated with a city-sized network of metro lines in Chicago, IL, USA, achieving near-optimal solutions in hours. The total weighted journey times are reduced by 0.61% and 5.76% under single-route and multi-period multi-route scenarios respectively. The model provides transit agencies with an efficient tool for comprehensive service design and resource allocation, improving service quality and resource utilization without additional operational costs.

💡 Research Summary

This paper tackles the urban rail service design problem by jointly optimizing three interdependent decision variables: stop‑sequence patterns, headways (service frequencies), and fleet sizes across multiple lines and time periods. Unlike most prior work that either relies on a pre‑generated set of candidate patterns or treats frequency and fleet sizing as a separate sub‑problem, the authors formulate a single mixed‑integer linear programming (MILP) model that captures all three dimensions simultaneously.

The core modeling innovation is the use of binary variables to directly encode whether a given stop follows another within a pattern, thereby allowing any feasible stop sequence—including express, local, short‑turn, and dead‑heading configurations—without restricting the solution space to a limited candidate pool. Headways are selected from a discrete set (e.g., 5‑minute, 7‑minute intervals), which linearizes the otherwise nonlinear relationship between waiting time and required fleet size.

A second major contribution is the introduction of “perceived headway.” When multiple patterns serve the same origin‑destination (O‑D) pair, passengers typically board the first vehicle that arrives, resulting in a waiting time that is shorter than the headway of any individual pattern. The authors adopt the frequency‑share rule (proportional to the inverse of headway) to compute a weighted average headway for each O‑D‑pattern combination, and embed this linear expression into the objective function. This captures passenger behavior that earlier models have ignored.

The problem is cast as a destination‑labeled multi‑commodity network flow (MCNF). The network contains four types of arcs: entry arcs (passengers entering the system), boarding arcs (assigning passengers to a headway‑pattern combination), travel arcs (vehicle movement between consecutive stops on a pattern), and exit arcs (passengers alighting). Flow conservation constraints enforce that the number of passengers entering, traveling, transferring, and exiting are balanced for each commodity (origin‑destination pair). Capacity constraints link headways to fleet size: the number of vehicles required for a pattern equals the service period length divided by the chosen headway, and must not exceed the available fleet for that line and period. Binary variables also decide which headway is active for each pattern in each period.

The MILP objective minimizes the total weighted journey time, where waiting time is weighted by the perceived headway, riding time by travel time, and transfer time by a transfer‑cost coefficient. Additional linear constraints ensure logical consistency of patterns (e.g., a stop can have at most one successor), enforce bi‑directional operation, and allow short‑turning and dead‑heading arcs.

Computational experiments use a realistic case study of Chicago’s metro system: eight lines, up to 40 stops per line, and three time periods (peak, off‑peak, night). The model is solved with Gurobi. For a single‑line, single‑period instance the solution is obtained in seconds, achieving a 0.61 % reduction in total weighted journey time compared with a benchmark that optimizes headways only. For the full multi‑line, multi‑period instance, near‑optimal solutions are found within 3–5 hours, delivering a 5.76 % reduction. The fleet size remains within the existing budget, demonstrating that better service quality can be achieved without additional resources.

The paper’s practical implications are clear: transit agencies can use the proposed tool to explore flexible service patterns (including new express or short‑turn services) while simultaneously determining optimal frequencies and fleet allocations. By accounting for perceived headways, the model better reflects passenger experience, leading to more accurate assessments of waiting costs.

Limitations include the assumption of fixed, inelastic demand and the discretization of headways to whole‑minute intervals. The model also omits operational cost components such as energy consumption or labor. Future research directions suggested are: incorporating elastic demand, stochastic travel times, real‑time re‑scheduling, mixed fleets of conventional and autonomous trains, and multi‑objective formulations that balance passenger welfare with operating costs.