A Hybrid Quantum Classical Optimization Framework for Pickup and Delivery Problems with Parcel Lockers Using Quantum Graph Attention Networks

Aqdas Shehzad, Zhao Xu, Muhammad Aurangzeb, Wang Xiu-Xin, Muhammad Akbar, Tarek Salem Abdennaji, Aymen Flah

Published Aug 19, 2026• 0 citations

Abstract

Abstract The significant growth of e-commerce has created a vast demand for last-mile parcel delivery, in which case, parcel lockers will be the future of home delivery. But the Pickup and Delivery Problem with Lockers (PDPL) is computationally hard as it is an NP-hard problem with locker capacities, time windows, and stochastic customer availability. In this paper, we present Q-PDPL, a novel quantum-assisted hybrid approach for optimization that employs Variation Quantum Eigensolver (VQE), Quantum Approximate Optimization Algorithm (QAOA), and Quantum Graph Attention Networks (Q-GAT) in a Dantzig-Wolfe decomposition (DW) scheme. The framework recasts the pricing problem as a ground state of the energy optimization problem, which is quantum efficiently solvable by means of parameterized quantum circuits. In a stochastic setting, a Q-GAT-enhanced deep reinforcement learning module is employed that delivers online routing decisions. Tests with Solomon VRPTW and Li and Lim PDPTW datasets over the SQL demonstrate that Q-PDPL is 18.7% cheaper than classical Adaptive Large Neighborhood Search, 23.4% better than Branch-and-Price, and handles over 500 customers. The quantum pricing sub problem has O(√N⋅"poly" (logN)) complexity (classical O(N 2 )) and a solution quality within 2.1% of the optimal bound. These results hold out promise for quantum-classical hybrid algorithms for large-scale logistics.

DOI: 10.21203/rs.3.rs-10549098/v1

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