Asymptotically Optimal Algorithms for Pickup and Delivery Problems with Application to Large-Scale Transportation Systems

The Stacker Crane Problem is NP-Hard and the best known approximation algorithm only provides a 9/5 approximation ratio. The objective of this paper is threefold. First, by embedding the problem within a stochastic framework, we present a novel algorithm for the SCP that: (i) is asymptotically optimal, i.e., it produces, almost surely, a solution approaching the optimal one as the number of pickups/deliveries goes to infinity; and (ii) has computational complexity O(n^{2+\eps}), where $n$ is the number of pickup/delivery pairs and \eps is an arbitrarily small positive constant. Second, we asymptotically characterize the length of the optimal SCP tour. Finally, we study a dynamic version of the SCP, whereby pickup and delivery requests arrive according to a Poisson process, and which serves as a model for large-scale demand-responsive transport (DRT) systems. For such a dynamic counterpart of the SCP, we derive a necessary and sufficient condition for the existence of stable vehicle routing policies, which depends only on the workspace geometry, the stochastic distributions of pickup and delivery points, the arrival rate of requests, and the number of vehicles. Our results leverage a novel connection between the Euclidean Bipartite Matching Problem and the theory of random permutations, and, for the dynamic setting, exhibit novel features that are absent in traditional spatially-distributed queueing systems.Comment: 27 pages, plus Appendix, 7 figures, extended version of paper being submitted to IEEE Transactions of Automatic Contro

CONFLICT RESOLUTION AND TRAFFIC COMPLEXITY OF MULTIPLE INTERSECTING FLOWS OF AIRCRAFT

This paper proposes a general framework to study conflict resolution for multiple intersecting flows of aircraft in a planar airspace.The conflict resolution problem is decomposed into a sequence of sub-problems each involving only two intersecting flows of aircraft.The strategy for achieving the decomposition is to displace the aircraft flows so that they intersect in pairs, instead of all at once, and so that the resulting conflict zones have no overlap.A conflict zone is defined as a circular area centered at the intersection of a pair of flows which allows aircraft approaching the intersection to resolve conflict completely within the conflict zone, without straying outside.An optimization problem is then formulated to displace the aircraft flows in a way that keeps airspace demand as low as possible.Although this optimization problem is difficult to solve in general due to its non-convex nature, a closed-form solution can be obtained for three intersecting flows.The metric used for the airspace demand is the radius of the smallest circular region (control space) encompassing all of the non-overlapping conflict zones.This radius can also be used as an indication of traffic complexity for multiple intersecting flows of aircraft.It is shown that the growth of the demand for control-space radius is of the fourth order against the number of intersecting flows of aircraft in a symmetric configuration

An asymptotically optimal algorithm for pickup and delivery problems

Pickup and delivery problems (PDPs), in which objects or people have to be transported between specific locations, are among the most common combinatorial problems in real-world operations. One particular PDP is the Stacker Crane problem (SCP), where each commodity/customer is associated with a pickup location and a delivery location, and the objective is to find a minimum-length tour visiting all locations with the constraint that each pickup location and its associated delivery location are visited in consecutive order. The SCP is a route optimization problem behind several transportation systems, e.g., Transportation-On-Demand (TOD) systems. The SCP is NP-Hard and the best know approximation algorithm only provides a 9/5 approximation ratio. We present an algorithm for the stochastic SCP which: (i) is asymptotically optimal, i.e., it produces a solution approaching the optimal one as the number of pickups/deliveries goes to infinity; and (ii) has computational complexity O(n[superscript 2+ϵ]), where n is the number of pickup/delivery pairs and ϵ is an arbitrarily small positive constant. Our results leverage a novel connection between the Euclidean Bipartite Matching Problem and the theory of random permutations.Singapore-MIT Alliance for Research and Technology Cente

Cost Bounds for Pickup and Delivery Problems with Application to Large-Scale Transportation Systems

Demand-responsive transport (DRT) systems, where users generate requests for transportation from a pickup point to a delivery point, are expected to increase in usage dramatically as the inconvenience of privately-owned cars in metropolitan areas becomes excessive. However, despite the increasing role of DRT systems, there are very few rigorous results characterizing achievable performance (in terms, e.g., of stability conditions). In this paper, our aim is to bridge this gap for a rather general model of DRT systems, which takes the form of a generalized Dynamic Pickup and Delivery Problem. The key strategy is to develop analytical bounds for the optimal cost of the Euclidean Stacker Crane Problem (ESCP), which represents a general static model for DRT systems. By leveraging such bounds, we characterize a necessary and sufficient condition for the stability of DRT systems; the condition depends only on the workspace geometry, the stochastic distributions of pickup and delivery points, customers' arrival rate, and the number of vehicles. Our results exhibit some surprising features that are absent in traditional spatially-distributed queueing systems.Singapore-MIT Alliance for Research and Technology (Future Urban Mobility project)Singapore. National Research Foundatio

Probabilistic on-line transportation problems with carrying-capacity constraints

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2014.Cataloged from PDF version of thesis.Includes bibliographical references (pages 175-184).This thesis presents new insights and techniques for the analysis and design of autonomous or technology-assisted ("intelligent") transportation systems. The focus is on cooperative, on-line planning and control, of a fleet of transport vehicles with limited carrying capacity, where new transportation demands enter the system in real time. The study extends an existing probabilistic framework which has provided numerous insights about vehicle scheduling and routing problems since its inception. Additionally, the thesis provides algorithms and new probabilistic cost bounds, for optimal bipartite matchings between large sets of random points and optimal stacker crane tours through large sets of random demands. A recurrent theme of the thesis is that capacity-constrained vehicles must drive passenger-less, inescapably, for some positive fraction of time (in almost any practical setting). Moreover, under probabilistic modelling for the uncertainty of demand, one can predict the aforementioned fraction precisely, using strong Laws of Large Numbers arguments; it relates to a quantity known as the Earth Mover's distance (EMD), described by a fundamental problem in transportation theory. Since the existence of an unavoidable extra cost term has significant implications, e.g., for operational budgets of shared-vehicle systems, the results illuminate a phenomenon whose neglect could prove an unfortunate oversight. To the author's knowledge, this connection of the EMD to on-line vehicle routing is novel. The thesis also provides a new study of the practical considerations imposed by the "street rules" ubiquitous among ground-based transport problems. A new efficient algorithm for the Bipartite Matching problem for points on a roadmap is given. Also given is a new explicit formulation of the EMD on road networks; very few explicit formulas for EMDs have been known previously.by Kyle Treleaven.Ph. D

Models and efficient algorithms for pickup and delivery problems on roadmaps

One of the most common combinatorial problems in logistics and transportation-after the Traveling Salesman Problem-is the Stacker Crane Problem (SCP), where commodities or customers are associated each with a pickup location and a delivery location, and the objective is to find a minimum-length tour `picking up' and `delivering' all items, while ensuring the number of items on-board never exceeds a given capacity. While vastly many SCPs encountered in practice are embedded in road or road-like networks, very few studies explicitly consider such environments. In this paper, first, we formulate an environment model capturing the essential features of a “small-neighborhood” road network, along with models for omni-directional vehicles and directed vehicles. Then, we formulate a stochastic version of the unit-capacity SCP, on our road network model, where pickup/delivery sites are random points along segments of the network. Our main contribution is a polynomial-time algorithm for the problem that is asymptotically constant-factor; i.e., it produces a solution no worse than κ+o(1) times the length of the optimal one, where o(1) goes to zero as the number of items grows large, almost surely. The constant κ is at most 3, and for omni-directional vehicles it is provably 1, i.e., optimal. Simulations show that with a number of pickup/delivery pairs as low as 50, the proposed algorithm delivers a solution whose cost is consistently within 10% of that of an optimal solution.Singapore-MIT Alliance for Research and Technology Cente

Fundamental performance limits and efficient polices for Transportation-On-Demand systems

Transportation-On-Demand (TOD) systems, where users generate requests for transportation from a pick-up point to a delivery point, are already very popular and are expected to increase in usage dramatically as the inconvenience of privately-owned cars in metropolitan areas becomes excessive. Routing service vehicles through customers is usually accomplished with heuristic algorithms. In this paper we study TOD systems in a formal setting that allows us to characterize fundamental performance limits and devise dynamic routing policies with provable performance guarantees. Specifically, we study TOD systems in the form of a unit-capacity, multiple-vehicle dynamic pick-up and delivery problem, whereby pick-up requests arrive according to a Poisson process and are randomly located according to a general probability density. Corresponding delivery locations are also randomly distributed according to a general probability density, and a number of unit-capacity vehicles must transport demands from their pick-up locations to their delivery locations. We derive insightful fundamental bounds on the steady-state waiting times for the demands, and we devise constant-factor optimal dynamic routing policies. Simulation results are presented and discussed.Singapore-MIT Alliance for Research and Technology CenterSingapore. National Research Foundatio

NSC148286

The objective of this work is to provide analytical guidelines and financial justification for the design of shared-vehicle mobility-on-demand systems. Specifically, we consider the fundamental issue of determining the appropriate number of vehicles to field in the fleet, and estimate the financial benefits of several models of car sharing. As a case study, we consider replacing all modes of ion in a city such as Singapore with a fleet of shared automated vehicles, able to drive themselves, e.g., to move to a customer’s location. Using actual transportation data, our analysis suggests a shared-vehicle mobility solution can meet the personal mobility needs of the entire population with a fleet whose size is approximately 1/3 of the total number of passenger vehicles currently in operation.Singapore. National Research FoundationSingapore-MIT Alliance for Research and Technology Center (Future Urban Mobility SMART IRG program