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

Abstract

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.Keywords: Vehicle routing problem with pickup and delivery with time windows; Lagrangian relaxation; Timedependent least-cost path problem; Forward dynamic programming; Ride-sharing service optimization. 1 IntroductionAs population and personal travel activities continue to increase, traffic congestion has remained as one of the major concerns for transportation system agencies with tight resource constraints. The next generation of transportation system initiatives aims to integrate various demand management strategies and traffic control measures to actively achieve mobility, environment, and sustainability goals. A number of approaches hold promises of reducing the undesirable effects of traffic congestion due to driving-alone trips, to name a few, demand-responsive transit services, dynamic ride-sharing, and intermodal traffic corridor management.The optimized and coordinated ride-sharing services provided by transportation network companies (TNC) can efficiently utilize limited vehicle and driver resources while satisfying time-sensitive origin-to-destination transportation service requests. In a city with numerous travelers with different purposes, each traveler has his own traveling schedule. Instead of using his own car, the traveler can (by the aid of ride-sharing) bid and call a car just a few minutes before leaving his origin, or preschedule a car a day prior to his departure. The on-demand transportation system provides a traveler with a short waiting time even if he resides in a high-demand area. Currently, several realtime ride-sharing or, more precisely, app-based transportation network and taxi companies, such as Uber and Lyft are serving passengers in many metropolitan areas. In the ...

Mobility as a Service: A Critical Review of Definitions, Assessments of Schemes, and Key Challenges

“…Heuristic search algorithms have been proven to provide optimal solutions for these problems. To control and plan the requests with the desired level of service in the pick-up and drop-off times, the supplier uses the time window approach, as studied, for example, in (Mahmoudi & Zhou, 2016). As an extension of the DARP, the Integrated Dial-aride Problem (IDARP) includes the integration of demand responsive services with fixed route services.…”

Open-source VRPLite Package for Vehicle Routing with Pickup and Delivery: A Path Finding Engine for Scheduled Transportation Systems

“…A simple example with two travel requests and one vehicle in our previous study [20] is used to demonstrate key modeling features of space-time-state network representation. Consider a general physical transportation network containing six nodes.…”

“…Recently, the shortest path problem in three-dimensional networks or hyper-networks has attracted significant interests in transportation optimization problems. Mahmoudi and Zhou [20] developed a timedependent forward dynamic programming algorithm to reformulate the VRPPD as a special version of the shortest path problem with time-dependent and state-dependent arc costs where the state represents the number of passengers on the vehicle. Liu et al [21] further considered an extended version of household activity scheduling problem by using a cumulative space-time-state representation to reduce the number of complex constraints.…”

Multi-Objective Analysis of Ridesharing in Automated Mobility-on-Demand

“…Yet, the existing exact VRP methods focus on instances with tens of vehicles and requests [13,9] and as such they are not applicable to management or analysis of large-scale fleets that often consist of thousands of vehicles and requests.…”