Steiner Traveling Salesman Problem with Time Windows and Pickup-Delivery: integrating classical and quantum optimization

TL;DR

This paper introduces an advanced routing problem combining Steiner TSP with time windows, pickup-delivery, and capacity constraints, solved via classical and quantum optimization methods, including a preprocessing reduction for efficiency.

Contribution

It develops two novel mathematical formulations for this complex problem and demonstrates their solution using classical and quantum hybrid optimization platforms.

Findings

Preprocessing reduces problem size and improves computational performance.

Quantum hybrid approach effectively solves the complex routing problem.

Formulations show promising scalability and solution quality in experiments.

Abstract

We propose the Steiner Traveling Salesman Problem with Time Windows and Pickup and Delivery, an advanced and practical extension of classical routing models. This variant integrates the characteristics of the Steiner Traveling Salesman Problem with time-window constraints, pickup and delivery operations and vehicle capacity limitations. These features closely mirror the complexities of contemporary logistics challenges, including last-mile distribution, reverse logistics and on-demand service scenarios. To tackle the inherent computational difficulties of this NP-hard problem, we propose two specialized mathematical formulations: an arc-based model and a node-oriented model, each designed to capture distinct structural aspects of the problem. We further introduce a preprocessing reduction method that eliminates redundant arcs, significantly enhancing computational performance and scalability. Both formulations are implemented using classical and quantum optimization approaches. In particular, the classical models are solved with Gurobi, whereas the quantum implementation is carried out on D-Wave's LeapCQMHybrid platform, a hybrid quantum-classical environment that integrates quantum annealing with classical optimization techniques for constrained problem solving. Numerical experiments are conducted to validate the proposed formulations and the preprocessing reduction method. The analyses performed assess the structural properties of the two models, their computational behavior, and the impact of preprocessing on problem size and solution efficiency.