Optimization of bus stop placement for routes on uneven topography
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. 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 .
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
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