Discrete Optimization
Hybrid large neighborhood search for the bus rapid transit route design problem

https://doi.org/10.1016/j.ejor.2014.04.005Get rights and content

Highlights

  • We present a new model formulation for the bus rapid transit route design problem.

  • We propose a hybrid metaheuristic based for the problem at hand.

  • The proposed algorithm is able to obtain high quality in short run times.

Abstract

Due to an increasing demand for public transportation and intra-urban mobility, an efficient organization of public transportation has gained significant importance in the last decades. In this paper we present a model formulation for the bus rapid transit route design problem, given a fixed number of routes to be offered. The problem can be tackled using a decomposition strategy, where route design and the determination of frequencies and passenger flows will be dealt with separately. We propose a hybrid metaheuristic based on a combination of Large Neighborhood Search (LNS) and Linear Programming (LP). The algorithm as such is iterative. Decision upon the design of routes will be handled using LNS. The resulting passenger flows and frequencies will be determined by solving a LP. The solution obtained may then be used to guide the exploration of new route designs in the following iterations within LNS. Several problem specific operators are suggested and have been tested. The proposed algorithm compares extremely favorable and is able to obtain high quality solutions within short computational times.

Introduction

Due to an increasing demand for public transportation and intra-urban mobility, an efficient organization of public transportation has gained significant importance in the last decades. Cities have continued to grow and hence require more transport capacity and improved access to those. In this paper we will focus on a specific type of public transportation system: the design of routes and their frequencies for bus rapid transit (BRT) systems.

BRT systems enjoy great popularity. Currently more than 168 cities world-wide employ BRT systems, covering a network of 4424 km and providing service to approximately 31 million passengers on a daily basis. BRT systems are especially popular in Latin America, where BRT systems are currently in use in 56 cities. One of the largest among those is operated in Bogotá (Colombia), covering a network of 106 km and offering 1.98 million passenger trips per day (see WWW, 2013).

BRT systems deliver fast and cost-effective public transportation through busses. The route design problem for BRT systems involves the design of routes given the current infrastructure, as well as the determination of their frequencies they will be operated. The network under consideration consists of a single corridor and stations, which are located within the corridor in a predefined way. Busses may be operated on designated lanes, allowing them right-of-way with respect to regular traffic. For the problem under consideration the network of the BRT system is given. A typical network consists of corridors (composed of several individual lanes), as well as the sequence and the location of stations along them. The travel time of busses is assumed to be given. Additional time will be taken into account for ac- and deceleration of busses after or before stopping at a station. Similarly we assume waiting times at stations to be fixed and given.

The demand for the public transport system can be represented in terms of an origin–destination matrix, which provides us with an estimate of the number of passengers requiring transportation between any pair of stations within the time horizon under consideration. This matrix is assumed to be known beforehand and is assumed not depend on the set of routes offered. Depending on the variation of demand throughout the day, the demand during peak hours should be taken into account preferably. The actual demand will depend on the offered set of routes and their frequencies, as passengers may react upon the offer. For the purpose of this paper we assume the demand to be constant and independent of the set of routes offered. The reaction can be seen as a dynamic process (see Guihaire & Hao, 2008), which is beyond the scope of this paper.

We consider a homogeneous and limited fleet of busses. Passengers may enter, leave or transfer among routes, at any station a route stops. As passengers tend to become confused if the number of offered routes is too large, the total number of routes offered will be limited from above. Besides, due to managerial efficiency it is desirable to operate the system with a limited number of routes (see Walteros, Medaglia, & Riaño, (forthcoming)).

We refer to Levinson et al., 2002, Walteros et al., forthcoming for a more detailed overview on BRT systems.

The problem at hand can be defined as follows: given a connected network we try to design a set of routes and determine their frequencies, such that the total travel time by passengers can be minimized, while taking into account capacity restrictions. Frequencies determine how often a route will be served within the planning horizon, which might be in regular intervals (cyclic timetable) or in an aperiodic way. Routes do not necessarily need to stop at all stations among a specific corridor, but may skip stations along their way.

We assume passengers decide upon their route using a common objective function, such as the total travel time of all passengers spent in the system. This is a common objective function chosen in the literature in the sense of a system optimum (Borndörfer, Grötschel, & Pfetsch, 2008).

For a route to be feasible it needs to stop at least at two stations. Furthermore we assume that the provided routes are symmetric, i.e. will be operated in both directions. This should make the system easier to be used from a customers’ point of view. The underlying model and the proposed solution approach can be extended easily to cope with asymmetric route designs as well (see Walteros et al., forthcoming).

A graphical representation of a simple instance is shown below. Fig. 1 shows a simple network consisting of one corridor and four stations, labeled 1 to 4, as well as a selection of 5 feasible routes. Routes may operate on the given corridor and may only stop at stations in the designated order.

For a single corridor the number of routes grows exponentially and can be calculated as 2S-S-1, given the number of stations S. For systems with several corridors the number of different routes can be calculated in a similar way, but in addition the structure of the network, i.e. the number of segments and stations several corridors might share, also needs to be taken into account. It is important to mention that for any realistically sized instance the total number of routes is too large to be explored or enumerated, let alone to be offered. The large number of routes makes it practically impossible to solve instances of realistic size. Hence we propose an efficient hybrid metaheuristic to solve the problem at hand.

The contribution of this paper is threefold. In this paper we are going to introduce a new mixed integer problem formulation for the problem at hand. In order to solve the problem we decompose the problem into two interrelated subproblem, the design of routes and determining the frequencies together with the resulting passenger flows. We then propose an iterative hybrid metaheuristic. A metaheuristic component is focused on designing promising routes, which then can be (optimally) evaluated by determining their frequencies as well as the flow of passengers while ensuring feasibility. They key contribution in terms of the proposed algorithm is the notation of feedback, which will guide and bias the decisions of the metaheuristic in the route design phase. Feedback will be provided in terms of estimates upon the consequences on the (global) objective function, based on the solution obtained in the previous iteration.

The paper is structured as follows. We discuss literature on related problems and similar solution approaches in Section 2. We then introduce the mathematical model in Section 3. Our hybrid solution approach, as well as its core components, will then be sketched in Section 4. Results obtained on benchmark instances available in the literature will be presented in Section 5.

Section snippets

Related literature

Mathematical optimization has gained considerable attention for optimizing line planning problems in public transportation. We refer to Odoni et al., 1994, Bussieck et al., 1997 for an overview. Planning of public transportation services includes several steps that are usually performed in a sequential manner (see Ceder and Wilson, 1986, Liebchen and Möhring, 2007). The problems as such however are highly interrelated. Interactions made possible by handling several problems at the same time

Mathematical problem formulation

We extend the problem formulation introduced in Feillet et al., 2010, Gonzalez et al., 2012. Rather than relying on a given (sub) set of routes the model designs a given number of routes, by determining where they are supposed to stop. Simultaneously their frequencies are chosen and passenger flows are determined. To formulate the optimization problem mathematically we introduce the following notation: We consider a network consisting of a set of stations S={1,,S} along a single corridor. The

Hybrid solution framework

The main idea of our hybrid solution approach can be described as follows: We propose a hybrid metaheuristic based on Large Neighborhood Search (LNS) and linear programming (LP) for the problem at hand. The solution approach will decompose the problem into a route design and evaluation phase. Decisions upon the actual design of routes will be handled using LNS. Next the set of feasible routes Ω obtained in this stage will be evaluated. The resulting (optimal) passenger flows and frequencies are

Results

We evaluated the performance of the proposed algorithm using the data set proposed in Feillet et al. (2010). The set contains 17 instances considering a single corridor covering 3-19 stations. In the following, we refer to this data set as SC.

In a first step we evaluate the performance of our proposed LNS depending on the choice of operators used within the destroy and repair phase. Hence we set up the following experiment: We test several possible combinations of our proposed operators, either

Discussion and conclusions

In this paper we presented a model formulation for the bus rapid transit (BRT) route design problem, given upper bounds on number of routes to be offered. We proposed a hybrid metaheuristic based on Large Neighborhood Search (LNS) for the problem at hand. We decompose the problem into a route design and evaluation phase. Decisions upon the actual design of routes have been handled by LNS. We proposed and tested several problem-specific destroy and repair operators to be used within the LNS

Acknowledgments

Special thanks go to Jaime E. González and Andrés L. Medaglia for fruitful discussions and the provision of the instances.

References (34)

  • A.R. Odoni et al.

    Models in urban and air transportation

  • S.N. Parragh et al.

    Hybrid column generation and large neighborhood search for the dial-a-ride problem

    Computers & Operations Research

    (2013)
  • G. Schrimpf et al.

    Record breaking optimization results using the ruin and recreate principle

    Journal of Computational Physics

    (2000)
  • L. Silman et al.

    Planning the route system for urban buses

    Computers & Operations Research

    (1974)
  • Borndörfer, R., & Karbstein, M. (2012). A direct connection approach to integrated line planning and passenger routing....
  • R. Borndörfer et al.

    A column-generation approach to line planning in public transport

    Transportation Science

    (2007)
  • R. Borndörfer et al.

    Models for line planning in public transport

  • Cited by (27)

    • Hybridizations of evolutionary algorithms with Large Neighborhood Search

      2022, Computer Science Review
      Citation Excerpt :

      In other words, even if an exact technique is used for exploring the respective large neighborhood, the returned solution is often not the best one possible. Numerous applications of this type of LNS can be found in the related literature, including for example [19–21]. However, there are many well-known alternative ways of defining large neighborhoods as seen, for example, in local branching [22], the corridor method [23], and POPMUSIC [24].

    • Two-stage robust railway line-planning approach with passenger demand uncertainty

      2021, Transportation Research Part E: Logistics and Transportation Review
      Citation Excerpt :

      A train with more stops can meet the passenger travel demand better; however, it may enhance the total travel time of long-distance passengers. Some studies attempt to optimize the stop schedule and other elements of the LPPDPP simultaneously because a separate optimization can result in undesirable suboptimal solutions (Lin and Ku, 2014; Park et al., 2013; Walteros et al., 2013; Schmid, 2014; Qi et al., 2018a). However, it remains a big challenge to obtain a high-quality solution for the comprehensive optimization model of the LPPDPP because of its complexity (Fu et al., 2015).

    • Comparing traveler preferences for BRT and LRT systems in developing countries: Evidence from Multan, Pakistan

      2020, Journal of Traffic and Transportation Engineering (English Edition)
      Citation Excerpt :

      Given growing demand for transport, high capacity modes, such as light rail transit (LRT) and bus rapid transit (BRT), are considered, while introduction of metro systems is often prohibitive due to the financing difficulties (Kassa, 2014). The literature discusses comparative advantages and disadvantages of LRT and BRT (Hinebaugh, 2004; Lavery and Kanaroglou, 2012; Stojanovski et al., 2013; Stutsman, 2002; Vuchic, 2003), as well as planning, design and implementation issues (Canca et al., 2017; Currie and Delbosc, 2014; Ruano-Daza et al., 2018; Schmid, 2014). BRT systems are often preferable in the developing world, because of their low costs and their independence from electric power supply, which is important for countries with power shortages (Hinebaugh, 2004).

    • Integrated Railway Rapid Transit Network Design and Line Planning problem with maximum profit

      2019, Transportation Research Part E: Logistics and Transportation Review
      Citation Excerpt :

      The all-or-nothing procedure supposes that for each OD pair, passengers follow a shortest path, and therefore, the operator is forced to respond to the demand by increasing the capacity of the services (frequencies or lengths of trains), thus incurring an increase in operating costs. In Schmid (2014) the author presents a model for the bus rapid transit route design problem, for a fixed number of lines. The problem is solved through a decomposition technique.

    View all citing articles on Scopus
    View full text