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

Alessia Ciacco, Francesca Guerriero, Eneko Osaba

TL;DR

This paper introduces the STSP-TWPD, a novel routing problem combining Steiner connectivity, time windows, and pickup–delivery with capacity limits. It develops two MILP formulations, ABF and NBF, and analyzes two problem variants STSP-PD and STSP-TW to isolate the effects of time windows and precedence constraints. A preprocessing arc-filtering method (AFGR) drastically reduces model size and improves solver performance, enabling larger instances to be solved classically with Gurobi and explored via a quantum-hybrid platform (D-Wave LeapCQMHybrid). Comprehensive computational experiments demonstrate equivalence of ABF and NBF, quantify AFGR benefits (roughly 40-60% reductions in variables and constraints), and assess quantum experimentation, highlighting both feasibility and current hardware-driven limitations. The work provides reproducible instance generation, robust modeling frameworks, and insights into the potential of quantum–classical hybrids for complex logistics optimization in last-mile, reverse logistics, and time-constrained delivery contexts.

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.

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

TL;DR

This paper introduces the STSP-TWPD, a novel routing problem combining Steiner connectivity, time windows, and pickup–delivery with capacity limits. It develops two MILP formulations, ABF and NBF, and analyzes two problem variants STSP-PD and STSP-TW to isolate the effects of time windows and precedence constraints. A preprocessing arc-filtering method (AFGR) drastically reduces model size and improves solver performance, enabling larger instances to be solved classically with Gurobi and explored via a quantum-hybrid platform (D-Wave LeapCQMHybrid). Comprehensive computational experiments demonstrate equivalence of ABF and NBF, quantify AFGR benefits (roughly 40-60% reductions in variables and constraints), and assess quantum experimentation, highlighting both feasibility and current hardware-driven limitations. The work provides reproducible instance generation, robust modeling frameworks, and insights into the potential of quantum–classical hybrids for complex logistics optimization in last-mile, reverse logistics, and time-constrained delivery contexts.

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.

Paper Structure

This paper contains 49 sections, 40 equations, 9 figures, 14 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related works
  3. Classical approaches
  4. Works on the STSP
  5. Works on the VRP-TWPDP
  6. Quantum Approaches
  7. Problem definition
  8. Mathematical Formulation
  9. Arc-Based Formulation - ABF
  10. Decision Variables.
  11. Objective Function.
  12. Single service for customer.
  13. Time Constraints.
  14. Capacity Constraints.
  15. Variable Domains.
  16. ...and 34 more sections

Figures (9)

  • Figure 1: General structure of D-Wave’s hybrid solvers, adapted from osaba2025d. Each branch includes a CM and a QM that interact asynchronously to refine candidate solutions.
  • Figure 3: Comparison of optimal routes generated by the STSP-TWPD, STSP-PD and STSP-TW models for $V = 7$.
  • Figure 4: Comparison of optimal routes generated by the STSP-TWPD, STSP-PD, and STSP-TW models for $V = 8$.
  • Figure 5: Comparison of optimal routes generated by the STSP-TWPD, STSP-PD, and STSP-TW models for $V = 9$.
  • Figure 6: Comparison of optimal routes generated by the STSP-TWPD, STSP-PD, and STSP-TW models for $V = 10$.
  • ...and 4 more figures