Service operation design in a transit network with congested common lines

https://doi.org/10.1016/j.trb.2020.12.003Get rights and content

Highlights

  • We study a continuous transit network problem with explicit consideration of congested common lines.

  • The problem is formulated into a tri-level programming model.

  • We propose two solution methods to solve the proposed model. One is to transform the model into a mixed-integer linear programming (MILP). The other is the surrogate optimization based method.

  • We perform extensive numerical studies to test the proposed model formulation and solution methods.

Abstract

This paper focuses on a transit service operation design problem that primarily determines the optimal frequency settings with explicit consideration of congested common lines in a bus service network. Other than passengers’ transit route choices, the transit service line choices among congested common lines are also specified in the model formulation. A tri-level programming approach is applied to formulate this problem, wherein the upper-level program optimizes the transit frequency to minimize the total operating costs and passengers’ transit costs; the middle-level program describes passengers’ transit routing choices, in which passengers will select a sequence of transfer nodes to minimize their transit costs; and the lower-level program formulates the equilibrium strategy in the common line problem on the route sections (i.e., between two successive transfer nodes), whose equilibrium solution may have multiple strategies depending on the congestion level of the common lines. The tri-level model is then reformulated into a mathematical program with equilibrium constraints. Two solution methods are proposed to solve the problem. One is to transform the model into a mixed-integer linear program so that the global optimal solution of the linearized problem can be guaranteed, and the other employs a surrogate optimization approach to ensure high solution efficiency for large size problems without compromising solution quality. Finally, we conduct extensive numerical examples to demonstrate the validity of our model formulation and the performance of the proposed solution algorithms.

Introduction

Bus transit services play a pivotal role in urban transportation systems, improving mobility for all, and reducing car dependence. It is ideal if bus services are operationally and financially sustainable, with expedient service quality and reduced operating costs. It is common in practice that one bus transit service operator runs multiple service lines across the bus network; therefore it is imperative for the operators to know how to determine the best operation strategy within the bus service network when passengers’ choice behaviors are incorporated, which is indeed a transit network design problem (TNDP). In this study, we focus on the optimal frequency design problem in a busy transit network wherein congested common lines are explicitly considered.

To date, TNDP has received intensive research attention in the literature (Ibarra-Rojas et al., 2015). TNDP can be broadly classified as discrete, continuous, and mixed problems, depending on the respective design of network topology, network parameters, or both (An and Lo, 2015). In the discrete transit network design, many works focused on the optimization of transit lines. Ceder and Wilson (1986) summarized the bus network design problem formulations and proposed a heuristic approach to design new bus routes considering the costs of both passengers and operators. Baaj and Mahmassani (1991) proposed an AI-based solution approach for transit network design, which consists of three major components: 1) route generation design algorithm (RGA); 2) transit route analysis; 3) route improvement algorithm (RIA). Inspired by RGA, Mauttone and Urquhart (2009) developed a route set construction algorithm for the transit network design problem, named Pair Insertion Algorithm (PIA). To solve a similar problem, Chakroborty and Wivedi (2002) developed a procedure for providing an “optimal” or “efficient” transit route network (or route set) with given link travel time and demand. They adapted the genetic algorithm (GA) to solve this combinatorial optimization problem. Renken et al. (2018) determined the optimal operating lines with the objective of minimizing the operating cost and penalty for passengers. An integer programming formulation of an OD-based model was presented to develop demand-driven line plans. Further, a selfish route choice behavior considering vehicle capacity was explicitly considered. Recently, some research papers worked on the limited bus stops services in transit corridors, wherein most solution methods are heuristic algorithms (Liu, Yan, Qu, Zhang, 2013, Chen, Liu, Zhu, Wang, 2015).

The continuous TNDP focuses on optimizing the network parameters with known network topology. These problems are typically modeled as bi-level programming models, which are able to capture the interactions between operator’s decision-making and passengers’ choice behaviors. For example, Constantin and Florian (1995) formed the optimal frequency design problem as a bi-level Min-Min non-convex optimization problem and solved it by a projected (sub)gradient algorithm. Gao et al. (2004) formed the transit network design model as a bi-level programming model, in which the upper model is to determine the frequency while the lower model is a transit equilibrium assignment model. Huang et al. (2013) proposed a bi-level programming model for optimizing bus frequencies with uncertain demand, wherein GA was applied to solve the formulated problem.

In terms of the mixed design problems, Fan, Machemehl, 2006, Fan, Machemehl, 2006 combined route set generation, transit assignment, frequencies setting and performance measurement in their models and formed the Optimal Transit Route Network Design Problem (OTRNDP) into a multi-objective nonlinear mixed-integer model, which was solved by heuristic algorithms. Borndörfer et al. (2007) formulated the line planning and frequency design problem into a linear programming (LP) problem to minimize total passenger traveling time and the cost of the operator. Recently, Szeto and Jiang (2014b) and Szeto and Wu (2011) solved the transit route and frequency design simultaneously by applying the bi-level modeling approach. The authors included the number of passenger transfers in the objective function, and meta-heuristic methods were adopted to solve the problem. Considering the travel time uncertainty, Yan et al. (2013) proposed a robust optimization model and a simulated annealing based algorithm to determine a network configuration and the associated frequencies. An and Lo (2016) proposed a two-phase stochastic programming model to solve the transit line topology and the corresponding frequency considering the stochastic demand pattern. Leiva et al. (2010) presented an optimization approach to determine line itinerary, frequency as well as vehicle size with the objective to minimize total costs of the passengers and operator. Moreover, Jara-Díaz et al. (2012) and Fielbaum et al. (2016) developed an optimization model to determine the optimal frequency and vehicle size. They compared several line structures to serve the spatial demand.

In transit network design problems, it is essential to determine the passenger flow assignment, i.e., passengers’ routing choices and service line choices. By doing so, one typical issue is to solve the common-line problem. However, even though the common-line problem is an essential feature in transit network assignment, not all models were able to consider it explicitly (Shimamoto et al., 2012). As stated by Nguyen and Pallottino (1988), a major obstacle in transit assignment problems comes from the common-line problem, wherein passengers can choose more than one line to board whichever arrives earliest. Chriqui and Robillard (1975) were the first to propose methodologies to describe and solve common line problems. They gave a heuristic solution to derive the optimal subset of routes within a probabilistic context. Later, Spiess and Florian (1989) presented their work on an assignment model dealing with optimal travel strategies. They proposed that at every transfer node, a transit commuter traveling between an OD pair forms a strategy comprised of a set of common lines that he/she could take to travel towards the destination node. The commuter then takes the first arriving bus line within the set of lines that minimize his/her expected transit time. This approach has then been widely used in the transit design problems (Constantin, Florian, 1995, Yu, Yang, Yao, 2010, Huang, Ren, Liu, 2013, Szeto, Jiang, 2014). Besides, it is important to model congestion effect and incorporate it into the transit service network design. Gendreau (1984) was the first to model congestion on a transit network using passenger arrival distributions and waiting time through a bulk queue model and hyper paths. De Cea and Fernández (1993) later proposed a new equilibrium model for a congested transit network using a congestion function and the common lines for every route section were computed through a hyperbolic programming problem. This approach has been adopted by Gao et al. (2004) and Uchida et al. (2007) to determine the optimal frequency design. To avoid the drawbacks of heuristic formulation by De Cea and Fernández (1993), Cominetti and Correa (2001) proposed a queue theoretic approach to model the congestion in common lines. They suggested that when a route section is served by several transit lines, at equilibrium there could exist multiple common-line strategies for passengers depending on the congestion levels. To consider the seat capacity in the common line problem, Kurauchi et al. (2003), Schmöcker et al. (2008) and Schmöcker et al. (2011) introduced a “fail-to-board” probability and proposed frequency-based assignment through a Markov-type network loading process.

In this paper, we focus on a continuous transit network design problem that primarily determines the optimal frequency settings on a congested bus service network, wherein the congested common lines are explicitly considered. Specifically, other than bus passengers’ transit route choices between origin and destination pairs, their service line choices, i.e., the selection of a set of attractive lines as a strategy in presence of congested common lines, are also fully incorporated in the transit network design. At equilibrium, multiple common-line strategies may be adopted by the passengers, depending on the level of congestion. Indeed, no previous work has considered the equilibrium strategy choices with congested common lines into the transit network design problems thus far. The problem is firstly cast into a tri-level programming model, which is then reformulated into a mathematical program with equilibrium constraints. To solve this problem, we propose two solution methods. One is to transform the originally nonlinear and nonconvex problem into a mixed integer linear programming problem, which can be solved by applying existing efficient solution algorithms while the global optimal solution of the linearized problem can be guaranteed. Although this method can secure high-quality solutions, it may not be suitable for solving large-size problems. Therefore, another solution method based on surrogate optimization is proposed to solve the problem more efficiently. In summary, this study aims to contribute to the literature in the following aspects:

  • 1.

    This study works on a service operation design problem in a transit network while the equilibrium passenger flow assignment with congested common lines is explicitly considered. In the literature, most of the existing works on such transit network design problems either ignored the congestion effects or could not capture the solution of the equilibrium common-line strategies for passengers. In urban areas with a busy bus transit network, it is essential to consider congested common line problem in the transit operation design. Passengers choose a set of bus lines, i.e., common-line strategy, in order to minimize the travel time between two successive transfers. If the congestion effects onto passengers’ waiting time at bus stops are considered, the common-line strategy is not fixed, but flow dependent. With the assumption that the waiting time at one station will increase with the number of waiting passengers, an equilibrium common-line strategy solution would be achieved, wherein multiple strategies could be adopted by the passengers, i.e., it is not true that all the passengers share the same strategy (Cominetti and Correa, 2001).

  • 2.

    A tri-level programming approach is applied to formulate the transit service operation design problem in this study. In the model formulation, the upper level problem is to determine the optimal operation strategy of frequency setting to minimize the total costs of passengers and operators; the second level and third level problems combine to capture the passenger’ routing choice behavior in the bus transit network: the second level describes passengers’ choices of transit routes. Passengers choose their transit routes by selecting a sequence of transfer nodes to minimize their transit costs (including waiting time cost and in-vehicle travel time cost) from their origins to destinations; Meanwhile, the third level entails the common line problem between the successive transfer nodes (i.e., the route section). Specifically, there are multiple bus service lines between the successive transfer nodes, and passengers need to determine a set of bus service lines to board (i.e., the common-line strategy).

  • 3.

    Two solution methods are proposed to solve the problem. One method is developed to acquire the optimal solution by transforming the original problem into a mixed-integer linear program with applications of various linearization approaches. Another method is designed to solve large-size problems efficiently by applying a surrogate optimization approach. In the literature, most of the previous studies employed heuristic solution methods to solve transit network design problems. These methods, on one hand, cannot guarantee the solution quality; on the other hand, they had compromised solution efficiency due to their computationally expensive solution process. In this study, two solution methods are developed to cater to different requirements in practical applications. If high-quality solutions are required in a small-scale problem, one can apply the linearization based approach to obtain the optimal solution of the problem; if high solution efficiency is required to solve a large-size problem, the surrogate optimization based method could be used.

The remainder of this paper is structured as follows. Section 2 revisits the common line problem. In Section 3, we present the underlying assumptions and notation list before a tri-level programming model is built up for this transit operation design problem. In Section 4, we propose two solution methods. Section 5 reports the results of the numerical experiments. Finally, we summarize the paper in Section 6.

Section snippets

Basic definition and assumption

In this paper, a few terms are used to describe the bus transit service network, such as transit line, transit route, and route section, as was defined in Gao et al. (2004). A “transit line” depicts a group of vehicles that run between two nodes on the transit network.

We assume all vehicles in the same line have identical size, capacity, and operating characteristics. They travel on the network through the same sequence of links and nodes. A “transit route” is any path that a transit user can

Network representation

A transit network generally consists of a series of stations for boarding, alighting, or transferring and a set of transit lines. Some route sections may be served by paralleling lines with common stops. Consider a primitive transit network that is denoted by G=(N,L) in Fig. 3a, where N represents the station nodes and L represents the transit lines. If the concept of route section is used for network presentation, the example transit network in Fig. 3a can be cast into the network in Fig. 3b.

Solution algorithms

In this section, we propose two solution methods to solve the formulated model in the previous section. One is to transform the original problem into a linear programming problem by applying various linearization approaches, so that a global optimization solution can be obtained for the linearized problem. By adopting a finer linearization scheme, one can acquire more refined global optimal solution. In this way, we can contend that this method is able to find the global optimum of the original

Numerical studies

In this section, we conduct numerical studies to illustrate the validity of our proposed model, as well as the solution quality and efficiency of the solution methods. We first use a toy network to validate the accuracy of the proposed methods. Then we use a small-scale network to show that our model can effectively capture the strategy equilibrium in passengers’ service line choices into the transit network design problem. We finally use the Sioux Falls network to test the efficiency of our

Conclusion

This paper formulates a transit service operation design problem in which the congested common lines in the transit network are considered. The common-line strategy equilibrium is explicitly incorporated into the transit network design. In the solution of the common-line equilibrium, there may exist multiple strategies for the passengers, depending on the congestion levels on the common lines. A tri-level programming model is applied to formulate the problem. To handle the nonlinear and

CRediT authorship contribution statement

Qingyun Tian: Methodology, Formal analysis, Writing - original draft. David Z.W. Wang: Conceptualization, Methodology, Validation, Writing - review & editing. Yun Hui Lin: Software, Investigation.

Acknowledgement

This work is supported by Singapore Ministry of Education Academic Research Fund MOE2017-T2-2-093.

References (64)

  • A.B. Hosapujari et al.

    Development of a hub and spoke model for bus transit route network design

    Procedia-Soc. Behav. Sci.

    (2013)
  • O.J. Ibarra-Rojas et al.

    Planning, operation, and control of bus transport systems: a literature review

    Transp. Res. Part B

    (2015)
  • S.R. Jara-Díaz et al.

    Is public transport based on transfers optimal? A theoretical investigation

    Transp. Res. Part B

    (2012)
  • W.H.-K. Lam et al.

    A stochastic user equilibrium assignment model for congested transit networks

    Transp. Res. Part B

    (1999)
  • C. Leiva et al.

    Design of limited-stop services for an urban bus corridor with capacity constraints

    Transp. Res. Part B

    (2010)
  • H. Liu et al.

    Global optimization method for network design problem with stochastic user equilibrium

    Transp. Res. Part B

    (2015)
  • Z. Liu et al.

    Bus stop-skipping scheme with random travel time

    Transp. Res. Part C

    (2013)
  • P. Luathep et al.

    Global optimization method for mixed transportation network design problem: a mixed-integer linear programming approach

    Transp. Res. Part B

    (2011)
  • A. Mauttone et al.

    A route set construction algorithm for the transit network design problem

    Comput. Oper. Res.

    (2009)
  • S. Nguyen et al.

    Equilibrium traffic assignment for large scale transit networks

    Eur. J. Oper. Res.

    (1988)
  • C. Osorio et al.

    Efficient calibration of microscopic car-following models for large-scale stochastic network simulators

    Transp. Res. Part B

    (2019)
  • J.-D. Schmöcker et al.

    A quasi-dynamic capacity constrained frequency-based transit assignment model

    Transp. Res. Part B

    (2008)
  • J.-D. Schmöcker et al.

    Frequency-based transit assignment considering seat capacities

    Transp. Res. Part B

    (2011)
  • H. Spiess et al.

    Optimal strategies: a new assignment model for transit networks

    Transp. Res. Part B

    (1989)
  • W. Szeto et al.

    Transit assignment: approach-based formulation, extragradient method, and paradox

    Transp. Res. Part B

    (2014)
  • W. Szeto et al.

    Transit route and frequency design: bi-level modeling and hybrid artificial bee colony algorithm approach

    Transp. Res. Part B

    (2014)
  • W.Y. Szeto et al.

    A simultaneous bus route design and frequency setting problem for Tin Shui Wai, Hong Kong

    Eur. J. Oper. Res.

    (2011)
  • E.I. Vlahogianni

    Optimization of traffic forecasting: intelligent surrogate modeling

    Transp. Res. Part C

    (2015)
  • D.Z. Wang et al.

    A novel discrete network design problem formulation and its global optimization solution algorithm

    Transp. Res. Part E

    (2015)
  • D.Z. Wang et al.

    Global optimum of the linearized network design problem with equilibrium flows

    Transp. Res. Part B

    (2010)
  • D.Z. Wang et al.

    Optimal bus service design with limited stop services in a travel corridor

    Transp. Res. Part E

    (2018)
  • J. Agrawal et al.

    Transit route network design using parallel genetic algorithm

    J. Comput. Civil Eng.

    (2004)
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