The impact of using a naïve approach in the limited-stop bus service design problem

Abstract

The proven benefits of limited-stop services have captured the attention of researchers, especially during the last decade. However, to solve the limited-stop service design problem many existing works directly impose a capacity constraint to a total social cost objective function. This “naïve approach” implicitly assumes that passengers behave altruistically, basing their decisions on what is best for the whole system. Although this issue has been identified in earlier works, the magnitude of the error induced by this simplification has not been studied yet. The objective of this work is to measure this error and to understand how it misrepresents passenger flows and bus occupation rates. To measure this error gap, we optimize a set of test scenarios by applying a naïve approach, and then take the resulting design and obtain a benchmark passenger assignment using a simple behavioral model. We propose two main indicators to compare both passenger assignment: the total passenger deviation, and the total capacity deficit. This comparison reveals that the assignment of the naïve approach may indeed be unrealistic, and raises concerns that a network design based on the naïve approach might have severe problems when implemented. Thus, the work highlights the importance of taking the results of the naïve approach with caution and verify them with a passenger assignment model before their implementation.

Introduction

The first city that captured worldwide attention with its bus rapid transit (BRT) system was Curitiba, Brazil, which implemented a network of such services in the period of 1974–1991. Since then, 170 cities have implemented BRT worldwide (www.brtdata.org). Of these systems, 80% have been implemented since the year 2000, and 40% in the last 10 years. BRT systems are generally recognized as a high-capacity and cost-effective alternative to rail-based systems. They can provide the quality of service of rail-based systems with the cost and flexibility of buses. To fully exploit their potential, BRT often relies on the operation of limited-stop services, i.e., services that skip some selected stops along their route to provide a faster connection between specific stops. These services, when designed properly, can reduce users’ travel times and increase the capacity of a corridor simultaneously. This characteristic allows the Transmilenio system of Bogotá (Colombia) to cater to around 45,000 commuters per hour in one direction, with an average speed of 25 km/h (Hidalgo & Muñoz, 2014) which is similar to the performance of many high-capacity rail-based transit systems. The benefits of limited-stop bus services stem from the fact that a faster ride reduces cycle times, allowing buses to carry more passengers and thus increase the productivity of the system (El-Geneidy and Surprenant-Legault, 2010, Larrain and Muñoz, 2016, Tétreault and El-Geneidy, 2010).

The tangible benefits of limited-stop services have attracted the attention of researchers, especially during the last decade. The question of how to design this type of services has been tackled under different assumptions and using diverse methodologies. One particularly challenging aspect of the limited-stop service design problem (LSDP) comes from the fact that the level of service of any given solution is a direct consequence of how passengers behave, i.e., of how they choose their routes given the available services (we will later discuss exactly what we understand as a route in this context). One way to deal with this issue is to define a separate passenger assignment problem to predict how users choose their routes, and use it to evaluate the performance of the network for any given solution. Then, a solution can be found with a multi-level approach, iterating between a design and a passenger assignment phase. This approach is sensible, but it usually converges to suboptimal solutions. To deal with this issue it seems reasonable to attempt to solve the design problem and a passenger assignment problem simultaneously, using a social cost function as the objective function. Intuitively, this approach makes sense, because minimizing social costs should lead to passengers individually minimizing their own travel times as well. Unfortunately, this argument is invalid when limited bus capacity forces some passengers to settle for a route they would not choose as their first option.

To find the optimal service design using a simultaneous approach, many existing works either leave vehicle capacity out of the analysis, or directly impose a capacity constraint to a total cost objective function. The first approach makes sense in systems where buses rarely reach their capacity, which is not the case in most BRT systems in practice. The second approach, imposing a capacity constraint, equals to assuming that passengers behave altruistically, basing their decisions on what is best for the whole system. This “naïve” approach may provide service designs (i.e., solutions to the frequency optimization problem) that would work under a distorted picture of passenger assignment, since in the resulting solution some passengers might be assigned by the model to services they would not find attractive given their options. We call these services, in the context of this work, “undesired” services. Thus, the resulting assignment may not reflect reality, and will have a tendency to underestimate total user costs.

Although some previous works (Larrain and Muñoz, 2016, Leiva et al., 2010, Soto et al., 2017) have repeatedly pointed out this issue, the magnitude of the error induced by this simplification has not been studied yet. Many works in the related literature implement this naïve approach as part of their methodologies without discussing its implications. For instance, Sun et al. (2008) optimize BRT frequencies by minimizing total cost without transfers. Chen et al. (2012) also minimize the total cost and design the limited-stop bus service for a single bus route for a given travel demand. Chiraphadhanakul and Barnhart (2013) solve the service generation and frequency optimization problems simultaneously. Zhang et al. (2016) optimize frequencies for limited-stop and short turning services by minimizing total costs for given trip matrix. Martínez et al. (2017) simultaneously solve route design and frequency setting problems for a BRT corridor.

All the works mentioned above optimize service frequencies (or some other design elements) while directly imposing capacity constraints to a total cost minimization problem, meaning that by construction they will overestimate the social benefits. It is crucial, then, to be able to quantify the size of the gap between this approximate, naïve solution, and the costs that a more realistic passenger assignment model would predict. If this gap is small enough, we could argue that this kind of approach is a good approximation and simplification of the original problem, and its utilization could even be encouraged.

The objective of this work is to measure the size of this error and to understand how it misrepresents passenger flows and bus occupation rates. To the best of our knowledge, we are the first ones who have identified this issue (Larrain and Muñoz, 2016, Leiva et al., 2010, Soto et al., 2017), and this is the first work that aims to quantify this effect. This work also provides a methodology to validate the results of the naïve approach. It will help the transportation planners or researchers who would like to implement their solutions based on the naïve, simplified approach. We perform an experiment over a set of instances of the problem for a simple bus corridor. First, we design the services for this corridor using a naïve LSDP model. Then, we estimate a benchmark passenger assignment by taking the resulting service design from the naïve LSDP, and estimating its “intended” passenger assignment, i.e., the passenger assignment that would result if bus capacity were not binding. From now on, we will call it the benchmark assignment. We perform our experiments on two versions of the LSDP model, based on different types of passenger assignment models, explained in detail in the following sections: itinerary-based, and route-based. The benchmark passenger assignment is obtained using the same type of behavioral model behind the LSDP variant under consideration.

We acknowledge that assuming unlimited bus capacity for this comparison is a debatable decision. When the demand for a service approaches or exceeds its capacity, congestion appears in the form of increasing waiting times and discomfort. A better prediction of passenger assignment on these scenarios would involve dealing with user equilibrium or some other tool that considers congestion. There are different approaches for dealing with this equilibrium problem. One way to do this is to assume that waiting times are flow dependent (De Cea & Fernández, 1993), which leads to an asymmetric equilibrium problem which is not trivial to solve. Another possible approach to estimate passenger assignment on a congested network would be to use a transit simulation tool (Binder et al., 2017, Cats and Hartl, 2016).

There are three main reasons we opted to use the benchmark passenger assignment (i.e., how passengers would assign to a given corridor design if capacity was not binding) to compare against the naïve approach. The first one is that it allows us to understand the magnitude of the problem, i.e., how many users are assigned to a route that they would not take under the route choices provided in the network. Second, this benchmark assignment gives a lower bound for the user costs (recall that it is yielded by a total cost minimizing model) in contrast of a more realistic equilibrium assignment, so it is a conservative estimate of the impacts of this simplification. The third reason is more practical: obtaining the benchmark passenger assignment is trivial, which allows us to study more instances, and to cover more cases to perform a deeper analysis. In other words, this simplification makes the problem tractable, while still giving a good and objective measure of how distant from reality the passenger assignment from the naïve solution can be. Further research on this subject will look into the differences between the altruistic assignment from the naïve approach and the more realistic assignment that an equilibrium or simulation model would predict.

In order to compare the passenger assignment from the naïve approach against the benchmark assignment for the same network, we define two indicators. The first one is the level of passenger diversion (i.e., the proportion of total passenger travel time that differs between assignments), and the capacity deficit in the service design (i.e., the proportion of additional capacity of the system required to cater for the benchmark passenger assignment).

The rest of the paper is structured as follows. Section 2 presents the state-of-the-art review pertaining to the LSDP, and the different approaches authors have considered for optimizing frequencies, modelling passenger assignment, and dealing with capacity in their formulations. Section 3 presents the mathematical formulation for the LSDP using a naïve approach, considering both route-based and itinerary-based user behavior. Section 4 details the solution methodology and performance indicators introduced in this work. Section 5 presents the results of different scenarios for a 5-stop bus corridor with three services. The last section summarizes our findings and recommendations.