Trade-offs between headway, fare, and real-time bus information under different weather conditions

Abstract

Given the increasing interest in real-time bus information, quantifying the value of such information from a user’s perspective is useful for transport modelers and service planners. Although a number of studies have investigated several other aspects of real-time bus information systems, there is a lack of studies that compare the disutility associated with the bus headway of a scheduled arrival information system and that of a real-time information system from a user’s perspective. In addition, no analyses in the literature examined the value of real-time information as affected by trip purpose and weather, which is important especially for the cities in which the weather remains below zero degrees during winter. The primary objectives of this research are to elucidate these issues. A stated preference survey describing the choice between scheduled and real-time information systems was conducted in Calgary, Canada. A total of 426 people participated in the survey, and each person was presented with three randomly selected choice situations. This data set was utilized to estimate the coefficients in different utility functions using a mixed logit model, which avoided several major limitations of a standard multinomial logit model. It was found that the disutility of the headway of a real-time information system was about half of the disutility of a scheduled information system. The analysis also showed that there was a nonlinear trend for the real-time information system, in which people found a higher disutility rate for a longer headway. Further, the value of real-time information availability was normally distributed in the population, with a mean of $0.50 and a standard deviation of $0.40. The results also revealed that the value of real-time information was significantly different when the weather was below and above 0 °C, those values were $0.59 and $0.41, respectively. Many of the findings obtained here are novel and have implications for both theory and practice. Particularly, they are important for transport modelers and service planners to design or adjust the headway for a desired level of service for a given (or a change in) bus arrival information type.

Appendix

The following six equations for mixed logit model are essentially quoted from Train (2003).

If a sampled individual (q = 1, 2,…, Q) faces a choice among J alternatives in each of T choice situations, the utility that individual q derives from choosing alternative j on choice occasion t is given by:

$$ U_{qjt} = \beta_{q} X_{qjt} + \varepsilon_{qjt} $$

(15)

where X_qjt is a vector of explanatory variables that are observed by the analyst relating to individual q and alternative j on choice occasion t; β_q is a vector of coefficients of the variables for individual q representing that person’s tastes; and ε_qjt is a random term assumed to be an independently and identically distributed extreme value (Revelt and Train 1998).

The density for β_q is described as $f(\beta_{q} |\varphi )$, where φ are the parameters of the distribution. Conditional on knowing β_q, the probability of respondent q choosing alternative i from J alternatives on choice occasion t is given by the following:

$$ L_{qit} (\beta_{q} ) = \frac{{{ \exp }(\beta_{q} X_{qit} )}}{{\sum\nolimits_{j = 1}^{J} {{ \exp }(\beta_{q} X_{qjt} )} }} $$

(16)

The probability of the observed sequence of choices conditional on knowing β_q is given by the following:

$$ S_{q} (\beta_{q} ) = \mathop \prod \limits_{t = 1}^{T} L_{{qi\left( {q,t} \right)t}} \beta_{q} $$

(17)

where i(q, t) denotes the alternative chosen by individual q on choice occasion t. The unconditional probability of the observed sequence of choices is the expected value of the logit probability over all of the possible values of β_q, i.e. integrated over these values weighted by the density of β_q:

$$ P_{q} (\varphi ) = \int {S_{q} (\beta_{q} )f(\beta_{q} |\varphi )d\beta_{q} } $$

(18)

Models of this form are called mixed logit because the choice probability P_q is a mixture of logits with f as the mixing distribution. The log likelihood (LL) for the model is as follows:

$$ LL(\varphi ) = \mathop \sum \limits_{q = 1}^{Q} lnP_{q} (\varphi ) $$

(19)

This expression cannot be solved analytically, as the integral in the previous equation does not have a closed form. It is, therefore, approximated using a simulation method (Train 2003). The simulated log likelihood (SLL) is given by the following:

$$ SLL(\varphi ) = \mathop \sum \limits_{q = 1}^{Q} ln\left\{ {\frac{1}{R}\mathop \sum \limits_{r = 1}^{R} S_{q} (\beta_{q}^{r} ) } \right\} $$

(20)

where R is the number of replications and $\beta_{q}^{r}$ is the rth draw from $f(\beta_{q} |\varphi )$.