This paper addressed the computational challenges of solving large-scale fuzzy 0-1 Integer Linear Programming (ILP) problems by proposing a Fuzzy-Hybrid Quantum-Classical Optimization Algorithm (F-HQCOA).The novel approach employed α-cut based transformations to convert fuzzy ILP models into a family of crisp Quadratic Unconstrained Binary Optimization (QUBO) problems, which were then solved using hybrid quantumclassical techniques.Unlike multi-objective formulations, the proposed framework performed a parametric analysis over α-levels to generate a spectrum of solutions that reflected trade-offs between objective optimality and feasibility satisfaction under uncertainty.The degree of satisfaction (µ) was adopted as an evaluation metric to assess solution quality rather than as an independent optimization objective.The method was evaluated on fuzzy Multi-Dimensional Knapsack and Set Covering problems.Results showed that the proposed approach could efficiently handle larger instances and provided a diverse set of solutions across α-levels.While quantum-assisted methods demonstrated reduced Time-to-Solution (TTS) in certain settings, comparisons were presented with careful consideration of differences in computational paradigms.The proposed framework offers a practical decision-support tool for optimization under uncertainty and establishes a foundation for future extensions toward true multi-objective fuzzy quantum optimization.
This paper addressed the computational challenges of solving large-scale fuzzy 0-1 Integer Linear Programming (ILP) problems by proposing a Fuzzy-Hybrid Quantum-Classical Optimization Algorithm (F-HQCOA).The novel approach employed α-cut based transformations to convert fuzzy ILP models into a family of crisp Quadratic Unconstrained Binary Optimization (QUBO) problems, which were then solved using hybrid quantumclassical techniques.Unlike multi-objective formulations, the proposed framework performed a parametric analysis over α-levels to generate a spectrum of solutions that reflected trade-offs between objective optimality and feasibility satisfaction under uncertainty.The degree of satisfaction (µ) was adopted as an evaluation metric to assess solution quality rather than as an independent optimization objective.The method was evaluated on fuzzy Multi-Dimensional Knapsack and Set Covering problems.Results showed that the proposed approach could efficiently handle larger instances and provided a diverse set of solutions across α-levels.While quantum-assisted methods demonstrated reduced Time-to-Solution (TTS) in certain settings, comparisons were presented with careful consideration of differences in computational paradigms.The proposed framework offers a practical decision-support tool for optimization under uncertainty and establishes a foundation for future extensions toward true multi-objective fuzzy quantum optimization.
Quantum optimization provides a promising approach for solving large-scale combinatorial problems through quadratic unconstrained binary optimization (QUBO) formulations. However, integrating QUBO-based solvers into structured optimization frameworks while preserving solution guarantees remains a fundamental challenge. This paper develops a hybrid quantum-classical column-and-constraint generation (QCCG) framework for solving two-stage adaptive robust optimization problems with binary first-stage decisions and linear recourse under polyhedral uncertainty. The proposed approach reformulates the restricted master problem as a QUBO and solves it approximately using a quantum optimizer, while retaining a classical adversarial subproblem to compute worst-case recourse and certify solution quality. We construct a constraint-preserving QUBO encoding for inequality-constrained master problems using slack variables and penalty terms, enabling general mixed-integer structures to be mapped to quantum-compatible representations. To address inexactness arising from discretization, penalty modeling, and quantum optimization, we introduce a bound-adjustment mechanism that yields valid lower and upper bounds and provides a certified stopping criterion. We show that the proposed framework generalizes classical column-and-constraint generation and retains its convergence properties when the master problem is solved exactly. Numerical experiments on two-stage robust location-transportation problems demonstrate that the proposed hybrid approach achieves solution quality comparable to classical methods while reducing the computational burden associated with solving mixed-integer master problems, highlighting the potential of hybrid quantum-classical optimization for scalable decision-making under uncertainty.
To date, research in quantum computation promises potential for outperforming classical heuristics in combinatorial optimization. However, when aiming at provable optimality, one has to rely on classical exact methods like integer programming. State-of-the-art integer programming algorithms can compute strong relaxation bounds even for hard instances, but may have to enumerate a large number of subproblems for determining an optimum solution. If the potential of quantum computing is realized, it can be expected that in particular finding high-quality solutions for hard problems can be done fast. Still, near-future quantum hardware considerably limits the size of treatable problems. In this work, we go one step into integrating the potentials of quantum and classical techniques for combinatorial optimization. We propose a hybrid heuristic for the weighted maximum-cut problem and for quadratic unconstrained binary optimization. The heuristic employs a linear programming relaxation, rendering it well-suited for integration into exact branch-and-cut algorithms. For large instances, we reduce the problem size according to a linear relaxation such that the reduced problem can be handled by quantum machines of limited size. Moreover, we improve the applicability of depth-1 QAOA, a parameterized quantum algorithm, by deriving a parameter estimate for arbitrary instances. We present numerous computational results from real quantum hardware.
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.
Abstract As interest in quantum and related optimization paradigms continues to grow, fairly and systematically comparing their performance across practical problem settings remains challenging. This challenge arises because these approaches often target different problem classes, use different mathematical representations, and are affected differently by modeling choices. As a result, observed performance depends not only on the solver itself, but also on how a problem is formulated, reformulated, and translated into a solver-specific input language. This work presents a publicly available and extensible benchmarking framework that unifies advances in quantum, quantum-hybrid, quantum-inspired, and classical optimization by enabling consistent comparison across use cases, solvers, and models. We address the diversity of solver “languages”—from mixed-integer programming (MIP) formulations for branch-and-bound methods over quadratic unconstrained binary optimization (QUBO) models for quantum annealers to constrained quadratic models (CQM) for hybrid approaches — as well as the influence of modeling choices, such as binary encodings and penalty formulations, on both performance and solution fidelity. By supporting systematic variation across the three dimensions of use case, solver, and model, the framework reveals trade-offs that single-dimension benchmarks overlook, clarifies when native formulations are preferable, and enables rigorous evaluation of practical combinatorial optimization problems. The two case studies, vehicle routing and maximum cut, illustrate this principle: their contrasting outcomes show that the effectiveness of a solver depends strongly on how well the chosen model representation matches the solver backend and the use case.
Mixed-integer linear programming problems are extensively used in industry for a wide range of optimization tasks. However, as they get larger, they present computational challenges for classical solvers within practical time limits. Quantum annealers can, in principle, accelerate the solution of problems formulated as quadratic unconstrained binary optimization instances, but their limited scale currently prevents achieving practical speedups. Quantum-classical algorithms have been proposed to take advantage of both paradigms and to allow current quantum computers to be used in larger problems. In this work, a hardware-agnostic Benders' decomposition algorithm and a series of enhancements with the goal of taking the most advantage of quantum computing are presented. The decomposition consists of a master problem with integer variables, which is reformulated as a quadratic unconstrained binary optimization problem and solved with a quantum annealer, and a linear subproblem solved by a classical computer. The enhancements consist, among others, of different embedding processes that substantially reduce the pre-processing time of the embedding computation without compromising solution quality, a conservative handling of cut constraints, and a stopping criterion that accounts for the limited size of current quantum computers and their heuristic nature. The proposed algorithm is benchmarked against classical approaches using a D-Wave quantum annealer for a scalable family of transmission network expansion planning problems.
Abstract We propose a new approach to utilize quantum computers for binary linear programming (BLP), which can be extended to general integer linear programs (ILP). Quantum optimization algorithms, hybrid or quantum-only, are currently general purpose, standalone solvers for ILP. However, to consider them practically useful, we expect them to overperform the current state of the art classical solvers. That expectation is unfair to quantum algorithms: in classical ILP solvers, after many decades of evolution, many different algorithms work together as a robust machine to get the best result. This is the approach we would like to follow now with our quantum ‘solver’ solutions. In this study we wrap any suitable quantum optimization algorithm into a quantum informed classical constraint generation framework. First we relax our problem by dropping all constraints and encode it into an Ising Hamiltonian for the quantum optimization subroutine. Then, by sampling from the solution state of the subroutine, we obtain information about constraint violations in the initial problem, from which we decide which coupling terms we need to introduce to the Hamiltonian. The coupling terms correspond to the constraints of the initial binary linear program. Then we optimize over the new Hamiltonian again, until we reach a feasible solution, or other stopping conditions hold. Since one can decide how many constraints they add to the Hamiltonian in a single step, our algorithm is at least as efficient as the (hybrid) quantum optimization algorithm it wraps. We support our claim with results on small scale minimum cost exact cover problem instances.