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.
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Figure 2 is available at the schedule pages of the Utah Transit Authority. http://www.rideuta.com/mc/?page=Bus-BusHome-RouteF94. The shaded area with slashes represents auxiliary transit services available only in several rides, and our analysis does not consider their influence on the feasibility of the Route F94 flex-route feeder transit system.
Planners usually need to fix the maximum number of deviation stops in operation. It is hard to obtain the accurate service capacity using theoretical modeling, which overestimates the service capacity due to the variation of curb-to-curb demand (Fu 2002). Since the estimation of service capacity is not the scope of this paper, here the service capacity is defined when at least 85 % of curb-to-curb requests can be accepted in simulations. In reality, this definition has no direct influence on the discussion about Fig. 5.
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Acknowledgments
We would like to thank the anonymous reviewers for their valuable comments on the previous drafts. The research reported in this paper is supported by the National Key Basic Research Program of China (No. 2012CB725402), the National Nature Science Foundation of China (No. 51208099), and the Program for Postgraduates Research in University of Jiangsu Province (No.CXZZ12_0111).
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Appendix A
Appendix A
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(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 \( c_{c} \).
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 \( c_{c - 1} \).
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:
$$ A_{III - dr}^{c} = \varphi T_{c} \left[ {\frac{1}{2} - \frac{1}{{4\left( {S - 1} \right)}}\sum\limits_{i = 1}^{C - 1} {\frac{1}{{\left( {C - i} \right)}}} } \right] $$
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Qiu, F., Li, W. & Haghani, A. An exploration of the demand limit for flex-route as feeder transit services: a case study in Salt Lake City. Public Transp 7, 259–276 (2015). https://doi.org/10.1007/s12469-014-0097-9
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DOI: https://doi.org/10.1007/s12469-014-0097-9