  In this paper, we develop a zonal-based flexible bus services (ZBFBS) by
considering both passenger demands spatial (origin-destination or OD) and
volume stochastic variations. Service requests are grouped by zonal OD pairs
and number of passengers per request, and aggregated into demand categories
which follow certain probability distributions. A two-stage stochastic program
is formulated to minimize the expected operating cost of ZBFBS, in which the
zonal visit sequences of vehicles are determined in Stage-1, whereas in
Stage-2, service requests are assigned to either regular routes determined in
Stage-1 or ad hoc services that incur additional costs. Demand volume
reliability and detour time reliability are introduced to ensure quality of the
services and separate the problem into two phases for efficient solutions. In
phase-1, given the reliability requirements, we minimize the cost of operating
the regular services. In phase-2, we optimize the passenger assignment to
vehicles to minimize the expected ad hoc service cost. The reliabilities are
then optimized by a gradient-based approach to minimize the sum of the regular
service operating cost and expected ad hoc service cost. We conduct numerical
studies on vehicle capacity, detour time limit and demand volume to demonstrate
the potential of ZBFBS, and apply the model to Chengdu, China, based on real
data to illustrate its applicability.
