A Tight Formulation for the Dial-a-Ride Problem

TL;DR

This paper introduces a new, tight mixed-integer linear programming formulation for the dial-a-ride problem, improving solution efficiency and optimality in routing and passenger assignment for ridepooling services.

Contribution

The paper presents a novel location-augmented-event-based MILP formulation for DARP, combining existing models with new inequalities and bounding techniques for enhanced performance.

Findings

Formulation is tight, yielding integer solutions when time windows are single points.

Computational times are reduced by approximately 50% compared to existing methods.

Numerical experiments confirm the model's theoretical and computational advantages.

Abstract

Ridepooling services play an increasingly important role in modern transportation systems. With soaring demand and growing fleet sizes, the underlying route planning problems become increasingly challenging. In this context, we consider the dial-a-ride problem (DARP): Given a set of transportation requests with pick-up and delivery locations, passenger numbers, time windows, and maximum ride times, an optimal routing for a fleet of vehicles, including an optimized passenger assignment, needs to be determined. We present tight mixed-integer linear programming (MILP) formulations for the DARP by combining two state-of-the-art models into novel location-augmented-event-based formulations. Strong valid inequalities and lower and upper bounding techniques are derived to further improve the formulations. We then demonstrate the theoretical and computational superiority of the new model: First, the formulation is tight in the sense that, if time windows shrink to a single point in time, the linear programming relaxation yields integer (and hence optimal) solutions. Second, extensive numerical experiments on benchmark instances show that computational times are on average reduced by 49.7% compared to state-of-the-art event-based approaches.