Effect of stochastic vehicle arrival and passenger demand on semi-flexible transit design

Abstract

Semi-flexible transit (SFT) is commonly discussed as a cost-effective alternative to serving public transportation users in low-demand conditions. Despite its considerable potential, the implementation of SFT is limited due to two primary operating challenges: (a) fluctuating travel demand and (b) service unreliability. Most researchers recently are rigorously involved in developing complex algorithms and heuristics to handle operational planning issues, while very few focus on the optimization of variables for SFT operations involving tactical decision making. Moreover, the optimization of decision variables is largely based on a single dimension of stochasticity, demand only. The present study proposes a methodology to optimize two decision variables, service headway and the proportion of requests accepted for curb-to-curb service per trip while operating SFT following a route-deviation operating policy. Implementing stochasticity in both demand and vehicle arrival, we perform multi-objective optimization with two conflicting objectives as minimization of operator cost and user cost. Pertaining to vehicle delays and demand variability linked to values of decision variables in the Pareto set, we define the risks associated with selecting each value for attaining Pareto optimality. The risk is proportionate to the occurrence of a decision variable value in the Pareto set. The study methodology can be adopted as a decision support tool to establish planning policies to optimize SFT operation while considering the interests of both the operator and the user.

Appendix 1: Properties of request arrival process/travel demand following a negative binomial distribution

The probability density function (PDF) of a negative binomial distribution (NBD) is characterized by a variance higher than the mean. The PDF of the NBD for a random variable X denoting the number of request arrivals (boarding/alighting) is expressed in Eq. (21). Further, the observed mean and variance are derived using Eqs. (22) and (23), respectively.

$$P\left(Y=y\right)=\left(\genfrac{}{}{0pt}{}{r+y-1}{y}\right){a}^{r}{b}^{y} \quad (y=0, 1, 2,\dots \dots ..)$$

(21)

$$\mu =\frac{rb}{a}$$

(22)

$${s}^{2}=\frac{rb}{{a}^{2}},$$

(23)

where X = number of failures preceding the r^th success (i.e., number of request arrivals), r = number of zero arrivals, p = probability of success representing the probability of arrival, a = probability of a zero-request arrival at a stop in the interval ‘h’, b = probability of a non-zero request arrival at a stop in the interval ‘h’

The parameters a and r are a function of μ = sample mean (i.e., mean arrival rate), and s² = sample variance (i.e., variance in request arrival pattern), as given in Eqs. (24) and (25), respectively.

$$a=\frac{\mu }{{s}^{2}}$$

(24)

$$r=\frac{{\mu }^{2}}{{s}^{2}-\mu }$$

(25)

Subsequently, considering a passenger request arrival pattern follows a NBD, the probability that a vehicle stops at a stop location is given in Eq. (26). Thus, for a low-demand route with n stops, the average number of times a transit stops for performing boarding and alighting operation E(s) along a route is as given in Eq. (27).

$$1-P\left(0\right)=1-{a}^{r}$$

(26)

$$E\left(s\right)=n\left[1-{a}^{r}\right]$$

(27)