Asymptotically Optimal Algorithms for One-to-One Pickup and Delivery Problems With Applications to Transportation Systems

Abstract

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 logistical operations. A widely-encountered type of 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 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+ε ), where n is the number of pickup/delivery pairs and ε 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.

Autonomous Mobility-on-Demand Systems for Future Urban Mobility

“…The results presented in this chapter build upon a number of previous works by the author and his collaborators, namely [13] for the lumped approach, [14,15,16,17] for the spatial queueing-theoretical framework and the distributed approach, and [13,18] for the case studies. The rest of this chapter is structured as follows.…”

“…The key idea behind the distributed approach [14,15,16,17] is that the number of stations represents a continuum (i.e., N [20,21,22,23]. In other words, customers arrive at any point in a given bounded environment [15,16], or at any point along the segments of a road map [15].…”

“…In other words, customers arrive at any point in a given bounded environment [15,16], or at any point along the segments of a road map [15]. In the simplest scenario, a dynamical process generates spatially-localized origin-destination pairs (representing the transportation requests) in a geographical region Q R 2 .…”

Simulation Framework for Rebalancing of Autonomous Mobility on Demand Systems

“…In each specific DVR scenario the author adopts the methods from queuing theory, combinatorial optimization, and stochastic geometry for the automatic planning of optimal multivehicle routes to provide service to demands. Similarly, [13] and [14] consider routing of shared-use autonomous vehicles. The authors design a routing policy that minimizes the average steady-state time delay between the generation of an origin-destination pair and the time the trip is completed.…”

“…The authors design a routing policy that minimizes the average steady-state time delay between the generation of an origin-destination pair and the time the trip is completed. [2] reviews modeling, control and evaluation of AMOD systems presented in [5,[11][12][13][14].…”