Towards Asymptotically Optimal One-to-One PDP Algorithms for Capacity 2+ Vehicles

TL;DR

This paper introduces a polynomial-time algorithm for the one-to-one Pickup and Delivery Problem in Euclidean space, achieving asymptotic optimality for vehicles with capacity growing slower than a certain rate, applicable to both multi-vehicle and single-vehicle scenarios.

Contribution

It presents the first asymptotically optimal polynomial-time algorithms for capacity 2+ PDP in Euclidean space, extending to LIFO constraints and single-vehicle cases.

Findings

Algorithm is asymptotically optimal for capacity o(n^{1/2d})

Extends to LIFO loading/unloading constraints

Applicable to both multi-vehicle and single-vehicle PDP

Abstract

We consider the one-to-one Pickup and Delivery Problem (PDP) in Euclidean Space with arbitrary dimension $d$ where $n$ transportation requests are picked i.i.d. with a separate origin-destination pair for each object to be moved. First, we consider the problem from the customer perspective where the objective is to compute a plan for transporting the objects such that the Euclidean distance traveled by the vehicles when carrying objects is minimized. We develop a polynomial time asymptotically optimal algorithm for vehicles with capacity $o(\sqrt[2d]{n})$ for this case. This result also holds imposing LIFO constraints for loading and unloading objects. Secondly, we extend our algorithm to the classical single-vehicle PDP where the objective is to minimize the total distance traveled by the vehicle and present results indicating that the extended algorithm is asymptotically optimal for a fixed vehicle capacity if the origins and destinations are picked i.i.d. using the same distribution.