A column generation and Combinatorial Benders Decomposition algorithm for the Selective Dial-A-Ride-Problem

In Pickup-and-Delivery Problems (PDP), one must design a set of vehicle routes that visit matching pickup and delivery locations. The Selective Dial-A-Ride Problem (SDARP) is a PDP which aims to serve as many pickup–delivery pairs as possible, where each pair has a time window and a maximum ride time. This paper introduces Extended Fragments to solve the SDARP. Extended Fragments stem from a subclass of vehicle routes which can be directly enumerated. An Extended Fragment formulation for the SDARP is proposed and solved using Combinatorial Benders Decomposition, augmented by a time discretisation and a novel variable-fixing technique. This algorithm, along with other recent ‘fragment’-based methods for PDP, is an a priori column generation method, combined with Combinatorial Benders Decomposition. The new method solves the existing benchmark for the SDARP, including 11 previously-unsolved instances.