• First use of QAOA for Facility Location (FL) and Load Balancing (LB) in supply chains optimisation. • QAOA successfully solves FL and LB using compact unbalanced-penalisation QUBO models. • Hardware-calibrated QAOA retrieves optimal FL solutions even under realistic gate and readout errors. • Provides reproducible QUBO–to–Ising transformations and full QAOA implementations in PennyLane and Qiskit. Quantum computing is emerging as a promising paradigm for solving combinatorial optimisation problems that are intractable for classical methods. Among the most prominent quantum algorithms is the Quantum Approximate Optimisation Algorithm (QAOA), which has demonstrated potential in addressing NP-hard problems using shallow circuits compatible with Noisy Intermediate-Scale Quantum (NISQ) devices. While QAOA has been applied to logistics and scheduling tasks, foundational supply chain problems such as Facility Location (FL) and Load Balancing (LB) remain largely unexplored. This paper addresses this gap by formulating FL and LB as Quadratic Unconstrained Binary Optimisation (QUBO) models and solving them using QAOA under both noiseless and hardware-calibrated noisy conditions. Two encoding methods are compared: the conventional slack-variable approach and the unbalanced penalisation technique. Results show that unbalanced penalisation consistently outperforms the slack-variable method by achieving higher feasibility and optimality rates and by avoiding the qubit inflation that limits slack-based encodings. FL instances, modelled as minimum dominating sets, exhibit strong performance in both noiseless and noisy simulations, whereas LB instances suffer from depth-induced noise due to their denser constraints. Furthermore, we outline the QUBO-to-Ising transformation process, with open-source implementations in PennyLane and Qiskit. To the best of our knowledge, this work represents the first application of QAOA to FL and LB, bridging the gap between quantum algorithm design and practical supply chain optimisation.
Recent advances in quantum computing and the increasing availability of quantum hardware have substantially enhanced the practical relevance of quantum approaches to discrete optimization. Among these, the Quadratic Unconstrained Binary Optimization (QUBO) formulation provides a unifying modeling framework for a broad class of
NP-hard problems and is naturally suited to quantum computing and quantum-inspired algorithms. Location science, network design, and logistics represent core application domains of discrete optimization, combining high practical impact with substantial computational challenges. In this work, we develop QUBO formulations for several fundamental problems in these domains, including a nonlinear integer formulation of the Discrete Ordered Median Problem (DOMP). Beyond their modeling relevance, these QUBO formulations serve as representative benchmark problems for assessing quantum algorithms and quantum hardware. We further derive a tight bound for the penalty parameter ensuring equivalence between the QUBO formulation and its underlying integer program. Finally, we conduct a comprehensive computational study using QAOA, WS-QAOA, and classical heuristics for QUBO instances of the
p-Median Problem and the Fixed-Charge Facility Location Problem (FCFLP), and introduce two effective warm-start strategies for WS-QAOA based on its linear programming relaxation.