An exploration of the demand limit for flex-route as feeder transit services: a case study in Salt Lake City

Abstract

As an innovative combination of fixed-route transit and demand responsive service, a flex-route operating policy had been introduced into feeder transit services. In this paper, a system cost function, combining vehicle operation cost and transit customer cost, is constructed as the performance measure to explore the feasibility of replacing the fixed-route policy with a flex-route policy in feeder transit systems, without disturbing the existing coordination between the major transit and feeder service. The Route F94 flex-route feeder system in Salt Lake City which connects with UTA TRAX Blue Rail Line is chosen for the analysis. The upper bound of demand for implementing the flex-route policy in this feeder service is derived. The results indicate that the Route F94 flex-route feeder system is still likely to have a distinct system advantage in operating environments with occasional request rejections, in comparison with the fixed-route service. As a result of our findings, it is possible to substantially expand the application of the flex-route policy in the feeder transit market.

Appendix A

• (a) The derivation of Eq. (12)

  The distribution of type I demand in major transit customers is displayed in Fig. 6, which is uniformly distributed among checkpoints between $c_{2}$ and $c_{c}$.

  Fig. 6

  

  The distribution of type I demand in major transit customers in drop-off ride

  For travel demand from $c_{1}$ to $c_{2}$, the expected riding time is ${{T_{r} } \mathord{\left/ {\vphantom {{T_{r} } {\left( {C - 1} \right)}}} \right. \kern-0pt} {\left( {C - 1} \right)}}$;

  For travel demand from $c_{1}$ to $c_{3}$, the expected riding time is ${{2T_{r} } \mathord{\left/ {\vphantom {{2T_{r} } {\left( {C - 1} \right)}}} \right. \kern-0pt} {\left( {C - 1} \right)}}$;

  For travel demand from $c_{1}$ to $c_{c}$, the expected riding time is ${{\left( {C - 1} \right)T_{r} } \mathord{\left/ {\vphantom {{\left( {C - 1} \right)T_{r} } {\left( {C - 1} \right)}}} \right. \kern-0pt} {\left( {C - 1} \right)}}$;

  Thus the expected riding time of type I passengers in major transit customers in drop-off ride can be derived as follows:

  $$ R_{I - dr}^{m} = \frac{1}{{\left( {C - 1} \right)}}\frac{{T_{r} }}{{\left( {C - 1} \right)}} + \frac{1}{{\left( {C - 1} \right)}}\frac{{2T_{r} }}{{\left( {C - 1} \right)}} + \cdots + \frac{1}{{\left( {C - 1} \right)}}\frac{{\left( {C - 1} \right)T_{r} }}{{\left( {C - 1} \right)}} = \frac{{T_{r} }}{{\left( {C - 1} \right)^{2} }}\sum\limits_{i = 2}^{C} {\left( {i - 1} \right)} $$

• (b) The derivation of Eq. (18)

  The distribution of type II demand in community riders is shown in Fig. 7, which is uniformly distributed among checkpoints between $c_{2}$ and $c_{c - 1}$.

  Fig. 7

  

  The distribution of type II and III demand in community riders in drop-off ride

  For travel demand whose starting point is $c_{2}$, the probability of having no waiting time is

  $$ \frac{{\left[ {{L \mathord{\left/ {\vphantom {L {2\left( {S - 1} \right)}}} \right. \kern-0pt} {2\left( {S - 1} \right)}}} \right]}}{{\left[ {{L \mathord{\left/ {\vphantom {L {\left( {C - 1} \right)}}} \right. \kern-0pt} {\left( {C - 1} \right)}}} \right]\left( {C - 2} \right)}} $$

  For travel demand whose starting point is $c_{3}$, the probability of having no waiting time is

  $$ \frac{{\left[ {{L \mathord{\left/ {\vphantom {L {2\left( {S - 1} \right)}}} \right. \kern-0pt} {2\left( {S - 1} \right)}}} \right]}}{{\left[ {{L \mathord{\left/ {\vphantom {L {\left( {C - 1} \right)}}} \right. \kern-0pt} {\left( {C - 1} \right)}}} \right]\left( {C - 3} \right)}} $$

  For all type II demand in community riders in drop-off ride, the probability of having no waiting time is

  $$ \frac{1}{C - 2}\frac{{\left[ {{L \mathord{\left/ {\vphantom {L {2\left( {S - 1} \right)}}} \right. \kern-0pt} {2\left( {S - 1} \right)}}} \right]}}{{\left[ {{L \mathord{\left/ {\vphantom {L {\left( {C - 1} \right)}}} \right. \kern-0pt} {\left( {C - 1} \right)}}} \right]\left( {C - 2} \right)}} + \frac{1}{C - 2}\frac{{\left[ {{L \mathord{\left/ {\vphantom {L {2\left( {S - 1} \right)}}} \right. \kern-0pt} {2\left( {S - 1} \right)}}} \right]}}{{\left[ {{L \mathord{\left/ {\vphantom {L {\left( {C - 1} \right)}}} \right. \kern-0pt} {\left( {C - 1} \right)}}} \right]\left( {C - 3} \right)}} + \cdots + \frac{1}{C - 2}\frac{{\left[ {{L \mathord{\left/ {\vphantom {L {2\left( {S - 1} \right)}}} \right. \kern-0pt} {2\left( {S - 1} \right)}}} \right]}}{{\left[ {{L \mathord{\left/ {\vphantom {L {\left( {C - 1} \right)}}} \right. \kern-0pt} {\left( {C - 1} \right)}}} \right]\left( {C - C + 1} \right)}} $$

  Thus, the expected waiting time of type II demand in community riders in drop-off ride can be expressed as follows:

  $$ A_{II - dr}^{c} = \varphi T_{c} \left[ {\frac{1}{2} - \frac{C - 1}{{4\left( {C - 2} \right)\left( {S - 1} \right)}}\sum\limits_{i = 2}^{C - 1} {\frac{1}{{\left( {C - i} \right)}}} } \right] $$

• (c) The derivation of Eq. (19)

  Similar to the derivation of Eq. (18), for type III demand in community riders in drop-off ride (see Fig. A2), the probability of having no waiting time is

  $$ \frac{1}{C - 1}\frac{{\left[ {{L \mathord{\left/ {\vphantom {L {2\left( {S - 1} \right)}}} \right. \kern-0pt} {2\left( {S - 1} \right)}}} \right]}}{{\left[ {{L \mathord{\left/ {\vphantom {L {\left( {C - 1} \right)}}} \right. \kern-0pt} {\left( {C - 1} \right)}}} \right]\left( {C - 1} \right)}} + \frac{1}{C - 1}\frac{{\left[ {{L \mathord{\left/ {\vphantom {L {2\left( {S - 1} \right)}}} \right. \kern-0pt} {2\left( {S - 1} \right)}}} \right]}}{{\left[ {{L \mathord{\left/ {\vphantom {L {\left( {C - 1} \right)}}} \right. \kern-0pt} {\left( {C - 1} \right)}}} \right]\left( {C - 2} \right)}} + \cdots + \frac{1}{C - 1}\frac{{\left[ {{L \mathord{\left/ {\vphantom {L {2\left( {S - 1} \right)}}} \right. \kern-0pt} {2\left( {S - 1} \right)}}} \right]}}{{\left[ {{L \mathord{\left/ {\vphantom {L {\left( {C - 1} \right)}}} \right. \kern-0pt} {\left( {C - 1} \right)}}} \right]\left( {C - C + 1} \right)}} $$

  Thus, the expected waiting time of type III demand in community riders in drop-off ride can be obtained:

  $$ A_{III - dr}^{c} = \varphi T_{c} \left[ {\frac{1}{2} - \frac{1}{{4\left( {S - 1} \right)}}\sum\limits_{i = 1}^{C - 1} {\frac{1}{{\left( {C - i} \right)}}} } \right] $$