Flow Shop Scheduling Problems (FSP) represent a challenging class of combinatorial optimization tasks with significant relevance across diverse domains ranging from manufacturing and logistics to service operations. Recent advances in quantum annealing and quantum-inspired approaches have opened avenues for tackling such problems via Quadratic Unconstrained Binary Optimization models. However, their performance depends strongly on how the problem is encoded as a QUBO. In this study, we perform a comparative evaluation of manually formulated QUBOs versus those automatically produced by the AutoQUBO framework. We test both formulations on D-Wave's hybrid and qbsolv solvers and the InfinityQ solver, exploring the effect of QUBO-solver-specific behaviors and sensitivity, and fine-tuning strategies on the quality of the solution. While carefully crafted manual QUBOs typically yield stronger solution fidelity, they demand greater domain expertise and computational overhead. In contrast, automated approaches greatly expedite modeling at the cost of certain performance trade-offs. We further observe that solver fine-tuning and hardware selection substantially shape outcomes, with certain solvers favoring domain-tailored encodings. These insights demonstrate the necessity of solver-aware QUBO design, offering practical guidance on QUBO construction and solver choice to solve the FSP using quantum and quantum-inspired hardware.
Flow Shop Scheduling Problems (FSP) represent a challenging class of combinatorial optimization tasks with significant relevance across diverse domains ranging from manufacturing and logistics to service operations. Recent advances in quantum annealing and quantum-inspired approaches have opened avenues for tackling such problems via Quadratic Unconstrained Binary Optimization models. However, their performance depends strongly on how the problem is encoded as a QUBO. In this study, we perform a comparative evaluation of manually formulated QUBOs versus those automatically produced by the AutoQUBO framework. We test both formulations on D-Wave's hybrid and qbsolv solvers and the InfinityQ solver, exploring the effect of QUBO-solver-specific behaviors and sensitivity, and fine-tuning strategies on the quality of the solution. While carefully crafted manual QUBOs typically yield stronger solution fidelity, they demand greater domain expertise and computational overhead. In contrast, automated approaches greatly expedite modeling at the cost of certain performance trade-offs. We further observe that solver fine-tuning and hardware selection substantially shape outcomes, with certain solvers favoring domain-tailored encodings. These insights demonstrate the necessity of solver-aware QUBO design, offering practical guidance on QUBO construction and solver choice to solve the FSP using quantum and quantum-inspired hardware.
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.
Large-scale combinatorial scheduling problems -- job-shop scheduling, flow-shop scheduling, and vehicle routing among them -- remain NP-hard at practically relevant scale, and quantum computing has entered this space along two related tracks: genuinely quantum solvers, principally quantum annealers and gate-based devices running the quantum approximate optimization algorithm (QAOA), and quantum-inspired classical solvers, such as coherent Ising machines and digital annealers, that borrow the same Ising/QUBO problem structure without requiring quantum hardware. This article synthesizes both tracks as applied specifically to scheduling. We trace the theoretical foundations from quantum and simulated annealing through the QUBO formulation techniques that make scheduling problems solver-ready, then review direct applications to job-shop, flow-shop, and vehicle-routing problems, reporting the concrete, verified numbers the underlying studies provide. The consistent finding is that quantum hardware remains constrained by qubit connectivity, embedding overhead, and shallow achievable circuit depth to small benchmark instances, while quantum-inspired classical solvers, unconstrained by qubit count, already reach scheduling-relevant problem sizes with reported performance competitive with or exceeding classical heuristics. We close by identifying benchmarking rigor, hardware connectivity, and the near-total absence of work on dynamic, online scheduling as the field's clearest open problems.
Large-scale combinatorial scheduling problems -- job-shop scheduling, flow-shop scheduling, and vehicle routing among them-- remain NP-hard at practically relevant scale, and quantum computing has entered this space along two related tracks:genuinely quantum solvers, principally quantum annealers and gate-based devices running the quantum approximateoptimization algorithm (QAOA), and quantum-inspired classical solvers, such as coherent Ising machines and digitalannealers, that borrow the same Ising/QUBO problem structure without requiring quantum hardware. This article synthesizesboth tracks as applied specifically to scheduling. We trace the theoretical foundations from quantum and simulated annealingthrough the QUBO formulation techniques that make scheduling problems solver-ready, then review direct applications tojob-shop, flow-shop, and vehicle-routing problems, reporting the concrete, verified numbers the underlying studies provide. Theconsistent finding is that quantum hardware remains constrained by qubit connectivity, embedding overhead, and shallowachievable circuit depth to small benchmark instances, while quantum-inspired classical solvers, unconstrained by qubitcount, already reach scheduling-relevant problem sizes with reported performance competitive with or exceeding classicalheuristics. We close by identifying benchmarking rigor, hardware connectivity, and the near-total absence of work on dynamic,online scheduling as the field's clearest open problems.
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.
Despite quantum computing is revealing an increasingly promising technology that has the potential to introduce a significant speed-up in many areas of computation, the number of problems that it can represent and solve is currently rather limited. Therefore, one of the current challenges faced by the quantum computing community is to broaden the class of problems that can be tackled. Among these problems, scheduling problems are a class of particularly interesting and hard combinatorial problems; in this paper, we present a novel solution for representing and solving the Flexible Open Shop Scheduling Problem (FOSSP) to optimality by minimizing the makespan. We firstly present a compact formulation of this problem as a Quadratic unconstrained binary optimization (QUBO), which can be used to solve this problem with a quantum annealer. Then, we proceed to the Quantum Approximate Optimization Algorithm (QAOA) problem formulation, thus producing both the cost and mix Hamiltonians related to the problem. From the Hamiltonians, we provide the complete description of the quantum circuit that can be used to tackle the FOSSP within the QAOA framework. This second approach can be used to solve the optimization problem with a general-purpose quantum gate-based hardware.
Real-world optimization problems must undergo a series of transformations before becoming solvable on current quantum hardware. Even for a fixed problem, the number of possible transformation paths-from industry-relevant formulations through binary constrained linear programs (BILPs), to quadratic unconstrained binary optimization (QUBO), and finally to a hardware-executable representation-is remarkably large. Each step introduces free parameters, such as Lagrange multipliers, encoding strategies, slack variables, rounding schemes or algorithmic choices-making brute-force exploration of all paths intractable. In this work, we benchmark a representative subset of these transformation paths using a realworld industrial production planning problem with industry data: the optimization of work allocation in a press shop producing vehicle parts. We focus on QUBO reformulations and algorithmic parameters for both quantum annealing (QA) and the Linear Ramp Quantum Approximate Optimization Algorithm (LR-QAOA). Our goal is to identify a reduced set of effective configurations applicable to similar industrial settings. Our results show that QA on D-Wave hardware consistently produces near-optimal solutions, whereas LR-QAOA on IBM quantum devices struggles to reach comparable performance. Hence, the choice of hardware and solver strategy significantly impacts performance. The problem formulation and especially the penalization strategy determine the solution quality. Most importantly, mathematically-defined penalization strategies are equally successful as hand-picked penalty factors, paving the way for automated QUBO formulation. Moreover, we observe a strong correlation between simulated and quantum annealing performance metrics, offering a scalable proxy for predicting QA behavior on larger problem instances.