Pickup and Delivery Problem with Two Dimensional Loading/Unloading Constraints

Abstract

This article addresses the Pickup and Delivery Problem with Two Dimensional Loading/Unloading Constraints. In this problem, we are given a weighted complete graph with 2n + 1 vertices, a collection ${\mathcal S} = \{S_1, \ldots, S_n\}$ of sets of rectangular items and a bin B of width W and height H. Vertex 0 is a depot and the remaining vertices are grouped into n pairs (p _i ,d _i ), 1 ≤ i ≤ n representing a pickup and a delivery point for each costumer. Each set S _i corresponds to items that have to be transported from p _i to d _i . As in the well-known Pickup and Delivery Problem, we have to find the shortest Hamiltonian cycle (route) such that each vertex p _i is visited before d _i 1 ≤ i ≤ n, but here we have the additional constraint that there must exist a feasible packing of all items from ${\mathcal S}$ into B satisfying loading and unloading constraints. These constraints ensure that exists a free way to insert/remove items into/from B along the route. To the best of our knowledge this is the first work to address this problem, which considers practical constraints of loading and unloading items along a delivery route. Our main contribution is the proposal of two exact algorithms and a GRASP heuristic for the problem, providing an extensive computational experiment with these algorithms showing that they can be effectively used to solve the problem.