A Tight Formulation for the Dial-a-Ride Problem

TL;DR

This work presents 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 and demonstrates the theoretical and computational superiority of the new model.

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.

Figures (5)

• Figure 1: Classification of formulations according to the information given by a binary variable with value 1
• Figure 2: Overview of developed methods
• Figure 3: Lifting for infeasible paths
• Figure 4: Network transformation
• Figure 5: Example for LB formulation

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• Theorem 3
• Example 4