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Optimization of on-demand transportation systems and ride-sharing servicesinvolves solving a class of complex vehicle routing problems with pickup anddelivery with time windows (VRPPDTW). This paper first proposes a newtime-discretized multi-commodity network flow model for the VRPPDTW based onthe integration of vehicles carrying states within space-time transportationnetworks, so as to allow a joint optimization of passenger-to-vehicleassignment and turn-by-turn routing in congested transportation networks. Ourthree-dimensional state-space-time network construct is able to comprehensivelyenumerate possible transportation states at any given time along vehiclespace-time paths, and further allows a forward dynamic programming solutionalgorithm to solve the single vehicle VRPPDTW problem. By utilizing aLagrangian relaxation approach, the primal multi-vehicle routing problem isdecomposed to a sequence of single vehicle routing sub-problems, withLagrangian multipliers for individual passengers requests being updated bysub-gradient-based algorithms. We further discuss a number of search spacereduction strategies and test our algorithms, implemented through a specializedprogram in C++, on medium-scale and large-scale transportation networks, namelythe Chicago sketch and Phoenix regional networks.

Finding optimal solutions for vehicle routing problem with pickup and  delivery services with time windows: A dynamic programming approach based on  state-space-time network representations