  This paper considers the design of On-Demand Multimodal Transit Systems
(ODMTS) that combine fixed bus/rail routes between transit hubs with on-demand
shuttles to serve the first/last miles to/from the hubs. The design problem
aims at finding a network design for the fixed routes to allow a set of riders
to travel from their origins to their destinations, while minimizing the sum of
the travel costs, the bus operating costs, and rider travel times. The paper
addresses two gaps in existing tools for designing ODMTS. First, it generalizes
prior work by including ridesharing in the shuttle rides. Second, it proposes
novel fleet-sizing algorithms for determining the number of shuttles needed to
meet the performance metrics of the ODMTS design. Both contributions are based
on Mixed-Integer Programs (MIP). For the ODMTS design, the MIP reasons about
pickup and dropoff routes in order to capture ridesharing, grouping riders who
travel to/from the same hub. The fleet-sizing optimization is modeled as a
minimum flow problem with covering constraints. The natural formulation leads
to a dense graph and computational issues, and is refined in an optimized model
working on a sparse graph. The methodological contributions are evaluated on a
real case study: the public transit system of the broader Ann Arbor and
Ypsilanti region in Michigan. The results demonstrate the substantial potential
of ridesharing for ODMTS, as costs are reduced by about 26% with minimal
increase in transit times. Compared to the existing system, the designed ODMTS
also cuts down costs by 35% and reduces transit times by 38%.
