Robustness-Based Approach for Slack Time Optimization in Tram Timetables

Urban Rail Transit https://doi.org/10.1007/s40864-024-00232-6 http://www.urt.cn/ ORIGINAL RESEARCH PAPERS Robustness‑Based Approach for Slack Time Optimization in Tram Timetables Weixia Zhou1 · Jing Teng1 · Enhui Chen1 Received: 26 March 2024 / Revised: 20 July 2024 / Accepted: 10 August 2024 © The Author(s) 2024 Abstract Considering the unique interplay of trams with road traffic, this study explored the issue of instability in tram operations—a prominent medium-capacity rail transit. Our goal was to design a timetable slack time optimization method for scheduling slack time to improve the stability of tram operations. To facilitate this, we derived the travel/dwelling time distribution from historical data, which assisted in estimating interference times and evaluating the requisite slack time. We then developed an integer programming model to calculate both the punctuality rate and expected delay under varying travel times, enabling the creation of alternative slack time schemes. Using a unique tram operation simulation logic, we assessed the operational efficiency and reliability of these alternate schemes based on specific operational indicators. The results suggest that our novel approach to timetable optimization significantly enhances the tram’s adaptability to disruptions, directly improving the passenger experience and tram competitiveness. This work offers a robust framework for timetable optimization for semi-independent right-of-way public transportation. Keywords Tram · Slack time · Integer programming model · Schedule optimization · Robust timetable optimization * Weixia Zhou * Jing Teng 1 Tongji University, Shanghai, China Communicated by Baoming Han 1 Introduction As an enduring public transportation medium, trams are ubiquitously deployed across the globe. Metropolitan areas such as Melbourne and Toronto already boast large-scale tram systems [1]. With the escalation of travel demands in recent years, traffic congestion in Chinese cities has become a pressing concern. Rail transit, and particularly trams, has emerged as a promising solution to solve the traffic congestion problem, primarily owing to its cost-effectiveness, shorter construction time frame, expedited project approval process, and eco-friendly attributes such as low energy consumption and emissions. As such, trams have become the preferred choice for rail transit construction in small to medium-sized cities [2]. However, tram operations in numerous Chinese cities are still in their early stages [3]. Many tram systems grapple with operational hurdles, including low speeds, considerable travel time variations, and inferior punctuality rates. Consequently, they are less appealing than other transportation modes such as private cars, resulting in low ridership and limited efficacy in mitigating traffic issues. The primary contributors to the trams’ reduced speed and stability are their semi-independent rights of way [4]. The semi-independent right-of-way tram line is a tram line that has its own dedicated lane but needs to follow the directions of traffic signals at intersections. This necessitates an uncertain wait time at these intersections, leading to fluctuating travel times. Additionally, passenger boarding and alighting times can vary during peak hours. In the absence of automatic train control (ATC) systems, travel speed hinges on drivers’ experience, leading to different travel times for the same route based on varying driving habits. Therefore, the tram system discussed in this paper is characterized as a Vol.:(0123456789) Urban Rail Transit semi-independent right-of-way system, operating on designated lanes but subject to intersection light control. Compared with large-scale delays, the timetable in this paper focuses on relatively minor delays that exceed normally needed travel/dwelling time caused by uncertainties in operations. The purpose of the model proposed in this paper is to improve the ability of trams to withstand the impact of uncertain delay—that is, robustness—while ensuring a certain operating efficiency. By incorporating appropriate slack time into the timetable, the method can buffer against interferences encountered during tram operations and improve punctuality rates. The tram line is divided into sections, delineated by intersections and time control stations, with the slack time for each section acting as the decision variable in a mixed-integer programming model (MIP). Utilizing historical tram operation data, we generated 200 sets of interference time and constructed a tram operation logic. This gave rise to a tram operation simulation system, which evaluates the slack time of timetables through indicators such as the number of stops and punctuality rate. These evaluation results guide improvements to the mixed-integer programming model, allowing for further optimization. This study’s primary contribution is the design of a schedule slack time optimization method to enhance tram robustness, coupled with a schedule robustness evaluation method. Compared with previous studies on tram and bus schedules, this paper comprehensively considers factors such as stations and signalized intersections, integrates optimization objectives such as punctuality rate and cost, and takes operational efficiency, fleet size, signal timing, and other factors as constraint conditions, which is conducive to improving the robustness and efficiency of tram operation. 2 Literature Review 2.1 Robustness of Timetable The focus of this paper is on the robust optimization of tram timetables, necessitating a clear definition of schedule robustness. Robustness is a constituent term within the concept of resilience [5]. As defined by Allenby and Fink [6], resilience is the ability of a system to maintain its function and structure amidst internal and external changes, and to appropriately degrade its capabilities when required. Therefore, a study of system robustness necessitates an exploration of system performance under unstable conditions. Various researchers provide slightly differing definitions of robustness, but the general consensus revolves around the system’s capacity to respond to unexpected challenges without major modifications [7], resistance to imprecision [8], and the ability to resist or absorb interference and remain intact when disturbed (Wang et al. [9]). Within schedule optimization, robustness is primarily invoked to resist instability factors in train operation. Andersson et al. [10] defined robustness as the timetable’s capacity to manage minor delays, proposing a novel robustness measure based on timetable critical points, which are particularly sensitive to delays (RCP). Fischetti et al. [11] established the optimization objective as maximizing infrastructure usage efficiency, ensuring, through constraints, that the timetable can absorb as much delay/interference from railway operations as possible. Artan and Şahin [12] evaluated the system’s se (...truncated)