  This paper reconsiders the ODMTS Design with Adoptions problem (ODMTS-DA) to
capture the latent demand in on-demand multimodal transit systems. The ODMTS-DA
is a bilevel optimization problem, for which Basciftci and Van Hentenryck
(2022) proposed an exact combinatorial Benders decomposition. Unfortunately,
their proposed algorithm only finds high-quality solutions for medium-sized
cities and is not practical for large metropolitan areas. The main contribution
of this paper is to propose a new path-based optimization model, called P-Path,
to address these computational difficulties. The key idea underlying P-Path is
to enumerate two specific sets of paths which capture the essence of the choice
model associated with the adoption behavior of riders. With the help of these
path sets, the ODMTS-DA can be formulated as a single-level mixed-integer
programming model. In addition, the paper presents preprocessing techniques
that can reduce the size of the model significantly. P-Path is evaluated on two
comprehensive case studies: the mid-size transit system of the Ann Arbor
Ypsilanti region in Michigan (which was studied by Basciftci and Van Hentenryck
(2022)) and the large-scale transit system for the city of Atlanta. The
experimental results show that P-Path solves the Michigan ODMTS-DA instances in
a few minutes, bringing more than two orders of magnitude improvements compared
to the existing approach. For Atlanta, the results show that P-Path can solve
large-scale ODMTS-DA instances (about 17 millions of variables and 37 millions
of constraints) optimally in a few hours or in a few days. These results show
the tremendous computational benefits of P-Path which provides a scalable
approach to the design of on-demand multimodal transit systems with latent
demand.
