This study explores the application of a hybrid quantum-classical algorithm for solving the parallel machine scheduling problem (PMSP) with sequence-dependent set-up times, a pivotal scheduling problem that has applications in multiple industries. Using a column generation-based approach, we propose a heuristic that combines a classical linear relaxation for the master problem with quantum annealing (QA) for solving the pricing subproblem (PSP). Whereas the PSP generates columns (i.e. a sequence of jobs that are assigned to a machine), the master problem selects which columns to use in order to minimize the makespan of the schedule. To generate columns, the PSP solves a travelling salesman problem (TSP) that is formulated as a quadratic unconstrained binary optimization (QUBO) problem. The big advantage thereof is that subtours can be eliminated by use of quadratic terms in the objective function. In addition, our approach also leverages the quantum annealer's capability to generate many high-quality solutions (i.e. columns) in a very short time. To assess the performance of our hybrid column generation-based heuristic, we perform a computational experiment. The results of this experiment demonstrate the synergy of hybrid methods for tackling complex decision-making problems, achieving competitive high-quality solutions and computational advantages when compared with classical solution methods. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.
This study explores the application of a hybrid quantum-classical algorithm for solving the parallel machine scheduling problem (PMSP) with sequence-dependent set-up times, a pivotal scheduling problem that has applications in multiple industries. Using a column generation-based approach, we propose a heuristic that combines a classical linear relaxation for the master problem with quantum annealing (QA) for solving the pricing subproblem (PSP). Whereas the PSP generates columns (i.e. a sequence of jobs that are assigned to a machine), the master problem selects which columns to use in order to minimize the makespan of the schedule. To generate columns, the PSP solves a travelling salesman problem (TSP) that is formulated as a quadratic unconstrained binary optimization (QUBO) problem. The big advantage thereof is that subtours can be eliminated by use of quadratic terms in the objective function. In addition, our approach also leverages the quantum annealer's capability to generate many high-quality solutions (i.e. columns) in a very short time. To assess the performance of our hybrid column generation-based heuristic, we perform a computational experiment. The results of this experiment demonstrate the synergy of hybrid methods for tackling complex decision-making problems, achieving competitive high-quality solutions and computational advantages when compared with classical solution methods. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.
Efficient production planning is essential in modern manufacturing to improve performance indicators such as lead time and to reduce reliance on human intuition. While mathematical optimization approaches, formulated as job shop scheduling problems, have been applied to automate this process, solving large-scale production planning problems remains computationally demanding. Moreover, many practical scenarios involve conflicting objectives, making traditional scalarization techniques ineffective in finding diverse and useful Pareto-optimal solutions. To address these challenges, we developed a quantum-classical hybrid algorithm that decomposes the problem into two subproblems: resource allocation and task scheduling. Resource allocation is formulated as a quadratic unconstrained binary optimization problem and solved using annealing-based methods that efficiently explore complex solutions. Task scheduling is modeled as a mixed-integer linear programming problem and solved using conventional solvers to satisfy detailed scheduling constraints. We validated the proposed method using benchmark instances based on foundry production scenarios. Experimental results demonstrate that our hybrid approach achieves superior solution quality and computational efficiency compared to traditional monolithic methods. This work offers a promising direction for high-speed, multi-objective scheduling in industrial applications.
This paper investigates several Quadratic Unconstrained Binary optimisation (QUBO) formulations for the parallel machine scheduling problem P m | | C max with the goal of assessing their suitability for quantum annealing and related heuristic optimisation methods. Three QUBO models are constructed, ranging from formulations derived from classical integer and one-hot encodings to a more compact load-balancing based representation. Extensive computational experiments using Simulated Annealing (SA) and Quantum Annealing (QA) reveal substantial differences in scalability and solution quality among the formulations. The alternative load-balanced model consistently yields near-optimal makespans, typically within 1%–3% of the theoretical lower bound, highlighting its stability and effectiveness compared to the more penalty sensitive formulations, whose performance deteriorates for larger instances. Sensitivity analyses further reveal that the appropriate selection of penalty weights and problem-specific parameter tuning significantly influences both solution accuracy and computational effort in QUBO formulations. While current quantum hardware limitations restrict experimentation to small problem sizes, the findings illustrate how compact QUBO designs can better exploit quantum optimisation frameworks. The insights gained in this study provide practical modelling guidelines for future quantum scheduling research and lay the groundwork for extending QUBO-based approaches to more complex scheduling environments. • Three QUBO formulations are compared for identical parallel machine scheduling. • A compact load-balancing QUBO avoids auxiliary variables. • Load balancing yields makespans on average within 1–3%. • Both simulated and quantum annealing produce near-optimal schedules. • Penalty selection and encoding strongly affect scalability and solution quality.
• We propose a new hybrid quantum-classical algorithm for solving the UC problem. • The hybrid model successfully optimizes 24-hour UC systems from 10 to 1000 units. • The model outperforms classical metaheuristics (GA and SA) and classical MINLP. • This represents an improvement over previous QCUC methods, a step to real-world use. Resource scheduling is critical in many industries, especially in power systems where the Unit Commitment (UC) problem determines the on/off status and output levels of generators under physical and economic constraints. Traditional exact methods, such as Branch‑and‑Bound, Branch‑and‑Cut, dynamic programming and mixed‑integer linear programming (MILP), remain the backbone of UC solution techniques, but they often rely on linear approximations or exhaustive search, leading to high computational burdens as system size grows. Metaheuristic approaches, such as genetic algorithms, particle swarm optimization, and other evolutionary methods, have been explored to mitigate this complexity; however, they typically lack optimality guarantees, exhibit sensitivity to initial conditions, and can become prohibitively time‑consuming for large‑scale systems. In this paper, we introduce a quantum‑classical hybrid algorithm for UC, and other resource scheduling problems by extension, that leverages Benders decomposition to decouple binary commitment decisions from continuous economic dispatch. The binary “master problem” is formulated as a quadratic unconstrained binary optimization (QUBO) model and solved on a quantum annealer. The continuous “subproblem,” which minimizes generation costs, with Lagrangian cuts feeding back to the master until convergence. We evaluate our hybrid framework on systems scaled from 10 to 1,000 generation units. Compared against a classical mixed‑integer nonlinear programming (MINLP) baseline, the hybrid algorithm achieves a consistently lower computation‑time growth rate and maintains an absolute optimality gap below 1.63%. These results demonstrate that integrating quantum annealing within a hybrid quantum-classical Benders decomposition loop can significantly accelerate large‑scale resource scheduling without sacrificing solution quality, pointing toward a viable path for addressing the escalating complexity of modern power grids.
The optimization of front-end crude oil scheduling is a critical determinant of refinery profitability and operational stability. However, the coupling of discrete logistics events (e.g., vessel berthing) with continuous material flows (e.g., pipeline transfers) renders this problem an NP-hard Mixed-Integer Linear Programming (MILP) challenge, often intractable for classical solvers at industrial scales. This study proposes a novel hybrid quantum-classical framework to address these computational bottlenecks. We employ Benders Decomposition to decouple the monolithic model into a discrete Master Problem (MP) and a continuous Subproblem (SP). To exploit the search capabilities of quantum computing, the MP is reformulated as a Quadratic Unconstrained Binary Optimization (QUBO) model and solved via a hybrid quantum solver, while the SP enforces mass balance and quality constraints through iterative optimality and feasibility cuts. Extensive experiments on 15 multi-scale instances demonstrate that the proposed framework significantly outperforms traditional metaheuristics (e.g., Genetic Algorithms, Tabu Search), reducing total operating costs by approximately 73--80% and achieving computational speeds comparable to state-of-the-art commercial solvers (Gurobi). By effectively leveraging global optimality cuts, the method overcomes the tendency of heuristic approaches to trap in local optima, providing a robust and scalable solution for complex refinery logistics.
In recent years, the Flow shop Scheduling Problem (FSP) has gained significant attention, prompting researchers to develop various methods, from conventional to hybrid optimization algorithms. Driven by the rapid growth of quantum computing capabilities, quantum approaches applied to optimization problems have attracted an increasing research effort. This work attempts to solve the permutation flow shop scheduling problem using quantum annealing by formulating the problem as a Quadratic Unconstrained Binary Optimization model (QUBO). Two QUBO formulations tailored to address the FSP are used. The first formulation is based on the position-based modelling while the second is based on five approximations of FSP as a Traveling Salesman Problem (TSP). The QUBO formulations are tested, using D-Wave's quantum annealers, on the well-known Taillard FSP benchmark and compared against each other. Results show that the proposed position-based QUBO reaches better solutions than the TSP-based QUBOs. This work is an attempt to highlight the increasing effectiveness of quantum annealers in addressing complex optimization problems and their limitations compared to conventional classical methods.
Abstract Airline itinerary and scheduling represent some of the most difficult computational problems that exist in transportation. The problems require choosing the best flight routes while determining schedules and allocating resources through various constraints which include regulatory requirements and maintenance schedules and dynamic demand patterns. The solution of small and medium-scale instances has been achieved through mixed-integer programming and heuristic methods including simulated annealing and genetic algorithms and branch-and-price but these methods become impractical for larger real-world instances due to exponential scaling issues. Quantum computing has introduced fresh solutions which enable the resolution of these problems. Quantum mechanical phenomena particularly superposition and entanglement enable optimization algorithms to examine an exponentially expanded solution space simultaneously. Our research applies a basic airline scheduling problem to the Traveling Salesman Problem (TSP) before converting it into a Quadratic Unconstrained Binary Optimization (QUBO) problem. The Quantum Approximate Optimization Algorithm (QAOA) serves as a hybrid quantum-classical method to solve this formulation. The experimental evaluation of a 4-city scenario shows that quantum-based optimization generates performance levels comparable to traditional methods and indicates that improved quantum hardware development will provide computational benefits for complex airline scheduling problems. This research holds dual importance because it can minimize operational expenses and improve scheduling performance and it establishes fundamental principles for complex combinatorial problem optimization across the transportation industry. The research establishes fundamental principles for quantum-enabled applications that will revolutionize airline operations and other transportation networks by solving critical scalability and solution quality problems.