Optimization of bus stop placement for routes on uneven topography

https://doi.org/10.1016/j.trb.2015.01.006Get rights and content

Highlights

  • Development of a mathematical modelling approach to optimal bus stop placement.

  • Incorporation of the effects of slope on accessibility and operation of the service.

  • Development of an evolutionary algorithm to approximate an optimal solution.

  • Application of model and solution method to a bus route in Auckland CBD, NZ.

Abstract

The improvement and expansion of public transport is an increasingly important solution to the high congestion costs and worsening environmental impacts of the car dominated transport systems seen in many cities today. The intelligent design of stop locations is one way to improve the quality of PT and thereby improve its ridership. Stop placement is a relatively complex task as it involves a trade-off between two competing goals; accessibility and operation; however this trade-off can be made explicit using an appropriate mathematical model. Many such models have been developed in the literature, however none consider the effects of uneven topography. Topography is an important but often neglected factor in the design of public transportation systems, with the potential to have a significant impact on the accessibility, operation and planning of a transit service. In this work a mathematical modelling approach to bus stop placement is developed which includes considerations of uneven topography in three ways; (1) Its effect on walking speed; (2) Its impact on the attractiveness of an access path to a transit service; and (3) Its effect on acceleration rates at stops. Because of the complexity of the model developed, a heuristic evolutionary algorithm’ is employed to approximate an optimal solution to the model. Finally, the model and solution method are applied to a case study in the Auckland CBD area in New Zealand.

Introduction

In many modern cities today, transport systems designed with a disproportionate focus on accommodating private vehicles, have led to various social problems including increased congestion and adverse environmental impacts. In New Zealand for example, congestion on Auckland roads is estimated to cost between $250 million to $1.25 billion per year (Wallis, 2013) and the New Zealand vehicle fleet contributes 24.2% of the country’s total CO2 emissions (NZ MfE, 2012), of which light passenger vehicles account for 65.2% (NZ MoT, 2013). It seems that continual expansion of the road network has not historically been effective in mitigating such effects. As a result, the need for an efficient, reliable, and reasonably priced public transport system is becoming increasingly pressing.

The problem of improving public transport patronage is a multifaceted one; however, it seems relatively intuitive that improving the quality of a service will lead to an increase in its ridership. For a bus service, one important factor which affects its quality is the location of stops along its routes. The problem of determining bus stop placement is not trivial as it involves a tradeoff between accessibility and operation. Litman and Rickert (2005) summarise the tradeoff as follows. “Adding a number of stops onto the bus route can stimulate ridership because of reduced access time. However, the corresponding user in-vehicle time and supplier cost might increase due to the excess acceleration and deceleration delays incurred by buses serving the additional stops.”

Because of this complexity it is desirable to develop a mathematical modeling approach to solving the problem. A number of studies have been undertaken on the subject of bus stop placement, however none that were reviewed in the course of this study explicitly take into account the effects of topography.

Topographical effects are an important consideration in the design of bus stop placement for a number of reasons. Firstly, topography has a significant impact on acceleration and deceleration delay at stops, ranging from 5 to 11 s for grades greater than 3% (Furth and SanClemente, 2006). Secondly, it is faster to walk downhill and on flat surfaces, than uphill (Terrier et al., 2010), and therefore, stop placement can have an impact on route accessibility. Finally, the degree to which slope causes delay along an access path has been shown to reduce pedestrian demand (Rodríguez and Joo, 2004), suggesting that pedestrians perceive up-hill access paths as unattractive for reasons other than simply their tendency to increase travel times. For example, the higher level of energy consumption required to traverse them (oxygen consumption at a 15% grade is 2 times greater than at level terrain (Terrier et al., 2010)).

Further to these points it has been observed that transit agencies operating in hilly geographic locations tend to see slope as an important issue. For example the Auckland Regional Transport Authorities, Bus Stop Infrastructure Design Guidelines (ARTA, 2009) includes topography as one of ten primary factors to consider when locating new bus stops, noting that “In areas where the topography is hilly or very steep, closer spacing of bus stops may be required”, and that “Grade of road should not impede accessibility.” In addition to this, a recent survey published as part of the Transport Cooperative Research Program, Synthesis 109: System-Specific Spare Bus Ratios Update (Appendix D) shows that 10 out of 42 agencies surveyed across the US consider steep hills to be a significant factor affecting the operation of their fleet (Minkoff and Martin, 2013). Finally, Daniels (2012) identifies 3 ways in which topography could be accounted for in the public transportation network of Brisbane, Australia, to improve its operation, usage, and planning, including considering the impact of topography on walking accessibility.

The problem considered in this paper is the following. For a single bus route r, select from a predetermined set of potential stop locations, the optimal set of stops to serve demand accessing the route in both service directions while taking into account the effects of topography on access time and operational cost. In this study we assume headway to be a fixed input to the model and that the headway selected is adequate to serve the demand along the route with an acceptable level of bus occupancy. Furthermore, we assume that passenger arrivals at stops are uniformly distributed.

Section snippets

Related literature

The theme of this study is directly related to the design of optimal public transport routes which is known to be a highly complex task from a computational perspective (Guan et al., 2003, Bagloee and Ceder, 2011). Consequently, exact search/optimization methods are unable to address the design of actual-sized transit networks, and thus heuristic methods are employed. For instance, Bagloee and Ceder (2011) propose a heuristic methodology considering categorization of stops, multiple classes of

Bus stop placement design

Previous studies in bus stop design can be broadly categorized by the following design decisions; (1) Studies that apply to a single route and those that apply to a network of routes; (2) Studies which seek to determine an optimal (and continuous) stop spacing or density, and those which seek to select an optimal set of stop locations from a discrete set of possible locations; (3) Studies which consider demand as a continuously varying function along a route and those which take demand to

Definitions

The following section outlines the variables used in the model and their notation, which is also summarized in Table 1. The underlying street network is represented by a digraph G = {N, A} where A is a set of arcs representing streets, roads, footpaths etc. and N is a set of nodes representing intersections, potential stop locations, demand points, origin points, and terminal points. G can be used to represent the street network at varying levels of complexity, as appropriate for the a particular

Solution method

The model outlined in the previous section is a mixed integer, non-linear–linear, bilevel programming problem (BLPP). The term ‘Non-linear–linear’ refers to the linearity of the upper and lower level problems, respectively. Note that if in-vehicle time is included in the lower level objective function, the problem becomes a mixed integer, non-linear-non-linear BLPP. There is little treatment of such problems in the literature, with most of the research effort in the field of BLPPs being

Small example

The following small example, set out in Fig. 8, is used to demonstrate calculation of the upper and lower level objective functions. In this example, and the following larger example, we assume access costs have been pre-calculated according to Eq. (1), accounting for the effects of slope on walk speed and hill-cost. Parameters of the model are summarised in Table 2. The headway of the service is fixed at h=20min.

Firstly, the four components of cycle time are calculated in order to evaluate

Case study

The model and solution method described in previous sections was applied to segments of existing bus routes in the central business district (CBD) of Auckland City, New Zealand. We consider the AM peak demand scenario, (occurring between 7:00 and 9:00 am), which represents a peak transit demand period.

Data was obtained from a variety of sources and stored in a PostgreSQL database with PostGIS extension. SQL and Python scripts were written to process the data into a workable form. The Python

Conclusions

The model and solution method described in this paper supply a framework for the design of stop placement along a single bus route which allows the effects of uneven topography to be explicitly considered. We also suggest a number of ways in which slope impacts upon the user and operational costs of a bus service considering walking speed, access path attractiveness, and acceleration delay; all of which lead to the optimal stop placement.

Benefits of the model and solution methodology, over

References (28)

  • B. Colson et al.

    An overview of bilevel optimization

    Annals of Operation Research

    (2007)
  • Furth, P.G., Mekuria, M.C., 2005. New bus stop spacing analysis: a tool for evaluating and optimizing bus stop location...
  • P. Furth et al.

    Parcel-level modeling to analyze transit stop location changes

    Journal of Public Transportation

    (2007)
  • P.G. Furth et al.

    Near side, far side, uphill, downhill: impact of bus stop location on bus delay

    Transportation Research Record: Journal of the Transportation Research Board

    (2006)
  • Cited by (41)

    • Similarity-based bus services assignment with capacity constraint for staggered bus stops

      2023, Transportation Research Part E: Logistics and Transportation Review
    • Joint design of shared-bike and transit services in corridors

      2021, Transportation Research Part C: Emerging Technologies
      Citation Excerpt :

      Environmental impacts (e.g., greenhouse gas emission) can also be included in our objective function to examine how they would affect the optimal design (Sun, 2017). In addition, one may also consider the possibility that cyclists may take shortcuts while transit routes may have detours, and that the cyclists’ biking cost per km would be higher on roads with slopes (Ceder et al., 2015). Our model can be modified to incorporate these realistic factors, for example, by using a location-dependent cycling speed to account for the different distances and times traveled by bike between any two points in a corridor.

    • Can multi-modal integration provide enhanced public transport service provision to address the needs of vulnerable populations?

      2020, Research in Transportation Economics
      Citation Excerpt :

      The need to consider this is highlighted in a study by Daniels and Mulley (2012), which notes the importance of taking into account the topography of an area when planning and developing public transport provision. For example, where there is steep or hilly terrain with narrow lanes or tight curves, large capacity vehicles may be unable to cope with the required turning radii or roadway slopes (Ceder et al., 2015), necessitating smaller vehicles with lower capacity. The general lower population densities of such areas (Telbisz et al., 2014) is also a consideration, as these may make the use of conventional full-size public transport vehicles less financially viable (Daniels & Mulley, 2012).

    View all citing articles on Scopus
    View full text