  Optimization of on-demand transportation systems and ride-sharing services
involves solving a class of complex vehicle routing problems with pickup and
delivery with time windows (VRPPDTW). This paper first proposes a new
time-discretized multi-commodity network flow model for the VRPPDTW based on
the integration of vehicles carrying states within space-time transportation
networks, so as to allow a joint optimization of passenger-to-vehicle
assignment and turn-by-turn routing in congested transportation networks. Our
three-dimensional state-space-time network construct is able to comprehensively
enumerate possible transportation states at any given time along vehicle
space-time paths, and further allows a forward dynamic programming solution
algorithm to solve the single vehicle VRPPDTW problem. By utilizing a
Lagrangian relaxation approach, the primal multi-vehicle routing problem is
decomposed to a sequence of single vehicle routing sub-problems, with
Lagrangian multipliers for individual passengers requests being updated by
sub-gradient-based algorithms. We further discuss a number of search space
reduction strategies and test our algorithms, implemented through a specialized
program in C++, on medium-scale and large-scale transportation networks, namely
the Chicago sketch and Phoenix regional networks.
