Optimizing limited-stop services with vehicle assignment

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Highlights

  • Combines the limited-stop service design with vehicle allocation in a large-scale network.

  • Considers both passenger travel time and comfort as the objective.

  • Evaluates several proposed methods and guides service providers in choice to fit their priority.

  • Conducts a case study on the Melbourne train system showing a significant improvement in service.

Abstract

This paper studies limited-stop operations for rail transit systems and presents an optimization model, which aims to minimize passengers’ travel times and on-board crowdedness. The model combines the design of service itinerary for each vehicle and the allocation of vehicles of various types among multiple lines. Three methods are proposed including a mixed-integer program and two column generation algorithms. Numerical results on generic scenarios show that the column generation algorithm can provide sub-optimal solutions and is computationally efficient. In a case study on Melbourne’s railway network, its solution significantly improves in the passenger travel time compared to the current practice.

Introduction

In many cities, public transport networks, particularly rail transit networks, keep expanding to cope with the rapid growth in demand. Public transport commuters are experiencing overcrowding and travel time increases. Common ways to deal with these problems include upgrading the infrastructures such as constructing more tracks and acquiring more vehicles. These approaches do not only require lots of investment and thereafter maintenance costs, but may not solve the aforementioned problems. In some cases, when the network expansion is completed after spending millions of dollars, overcrowding is still prevalent and travel times are not improved. For example, in early 2010, after 38 new trains were introduced to the Melbourne’s train system, overcrowding was increased by almost 20%, compared to the year 2009 (PTV, 2015). Infrastructure expansion should not be considered in isolation from resource scheduling when solving public transport problems. Well-designed schedules can make use of network resources more effectively and efficiently and often offer better solutions before and/or after the expansion projects. In this paper we aim to design well-coordinated limited-stop schedules and vehicle assignment strategies for rail transit networks to improve quality of service provided.

The concept of limited-stop (sometimes known as skip-stop1) operations was first mentioned by Black (1962), where selected stations were skipped by some vehicles. Past studies have shown that they can improve passenger waiting and in-vehicle times (Lee et al., 2014, Sun et al., 2013, Zhang et al., 2016) and also save operating costs. Fuel (or power) is saved as a result of not accelerating or decelerating at skipped stations. Furthermore, skipping stops allows vehicles to return to their depots in a shorter period of time. As a result, they can be reused sooner. Vehicles are limited resources in a rail transit network, and may possess different capacities. In order to improve the overall network performance as well as the overcrowding problem, in this study, we develop the limited-stop service for each vehicle as well as the strategy for distributing vehicles over different lines of the network. That is, we aim to determine

  • (a)

    the vehicle-to-trip assignment for multiple lines, that is, to decide which vehicles with various capacities to service different lines;

  • (b)

    the stopping pattern of each trip, that is, to decide which stations on each line that each train service stops at.

The objectives are to minimize the passenger waiting time, in-vehicle time, and passenger excess load (i.e., in-vehicle crowding). To the best of our knowledge, no previous studies in the literature have considered combining the stop service and the vehicle assignment problems together. This study fills this gap. Whilst crowding and discomfort in public transport was investigated in the context of timetable scheduling (De Palma et al., 2015), its implication for scheduling of limited-stop services and distribution of various sized vehicles across multiple lines is first studied.

We explore several methods to solve this problem: Mixed Integer Program (MIP) and Column Generation (CG). In the CG, the original problem is divided into two smaller problems. The subproblem determines the limited-stop schedules whilst the master problem determines the vehicle assignment. Two variants of the CG with different subproblems are considered: one with a MIP subproblem and the other with a heuristic subproblem. The proposed methods are evaluated on 24 simulated instances with various sizes and settings. The best method for large instances is applied to solve the limited-stop service and train distribution problem for the Melbourne’s railway system.

The remainder of this paper is organized as follows. Section 2 reviews the existing work related to this study. Section 3 discusses the constraints, assumptions and objective functions of the problem. Section 4 presents the three methods, the MIP formulation and the two CG algorithms. Section 5 conducts numerical studies on the simulated instances to evaluate the performance of the proposed methods. Section 6 conducts a case study on Melbourne’s railway network. Finally, Section 7 concludes this paper with some discussion.

Section snippets

Limited-stop scheduling

A special type of limited-stop operations is known as ‘zone-stop’ (Chapter 2.4 in (Vuchic, 2017)) where each vehicle services a zone consisting of either a string of adjacent stations or a single station. For example, if two vehicles service five stops, one zone-stop operation is to have one vehicle service stations 1, 2 and 3 and the other service stations 4 and 5. See (Jordan and Turnquist, 1979, Sun et al., 2013, Eisele, 1968, Ghoneim and Wirasinghe, 1986, Salzborn, 1969, Sun et al., 2008).

Problem description

In this section, we describe the limited-stop service and vehicle assignment problem in a rail transit network, which consists of multiple lines. After introducing the network setting and notation, we make a few assumptions, which help to reduce the complexity of the large-scale problem, and then state some major constraints for the optimization.

Mathematical models

In this section, we present the solution approaches to the combo scheduling problem. We explore the MIP algorithm and the CG algorithms with two different subproblems.

Input

To evaluate the methods proposed in Section 4, we consider 24 instances, for which the parameters are summarized in Table 3. Small instances 1–12 have less than 13 stations per line, whereas large instances 13–24 have up to 29 stations per line. The passenger arrival rates are randomly chosen with a maximum of 320 and 500 passengers per minute for instances 1–12 and 13–24, respectively. For instances 1–3 (correspondingly instances 13–15), we increase the number of vehicles. For instances 4–6

Case study

Now we conduct a case study on the Melbourne’s railway network. We start with the explanation of the input parameters and then discuss the numerical results produced by the CG-NOV method.

Conclusions

This study was inspired by the growing importance of public transport, its planning and operations. It investigated the large-scale limited-stop scheduling problem combined with the vehicle assignment in a rail transit network with multiple lines. From the passengers’ perspective, the objective function included the passenger waiting time, in-vehicle time, and load excess. The problem considered in this paper is interesting and challenging from four aspects: (i) integrating vehicle assignment

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