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International Journal of Creative and Open Research in Engineering and Management

Formulations for Quantum Annealed Project Scheduling

U Kanade

Published Aug 22, 2026• 0 citations
Publisher

Abstract

Resource constrained project scheduling and related allocation tasks can be expressed as Quadratic Unconstrained Binary Optimization (QUBO) models and submitted to quantum annealers. This paper develops the QUBO formulation in detail: choice of binary variables, linear and quadratic objective terms, conversion of precedence and resource constraints into penalty functions, and calibration of penalty weights. The resulting Hamiltonian is embedded onto quantum hardware, annealed, and post-processed. Hybrid classical–quantum workflows that wrap the QUBO solver inside Monte Carlo risk analysis or a digital twin are examined. Supporting process flowcharts and energy-landscape figures illustrate every stage. The treatment remains grounded in construction and logistics applications while remaining transferable to other combinatorial domains such as ULD configuration. Keywords: QUBO formulation; quantum annealing; Ising model; penalty methods; resource-constrained project scheduling; hybrid quantum-classical; minor embedding; Monte Carlo; digital twin.

DOI
10.55041/ijcope.v2i8.065

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Resource constrained project scheduling and related allocation tasks can be expressed as Quadratic Unconstrained Binary Optimization (QUBO) models and submitted to quantum annealers. This paper develops the QUBO formulation in detail: choice of binary variables, linear and quadratic objective terms, conversion of precedence and resource constraints into penalty functions, and calibration of penalty weights. The resulting Hamiltonian is embedded onto quantum hardware, annealed, and post-processed. Hybrid classical–quantum workflows that wrap the QUBO solver inside Monte Carlo risk analysis or a digital twin are examined. Supporting process flowcharts and energy-landscape figures illustrate every stage. The treatment remains grounded in construction and logistics applications while remaining transferable to other combinatorial domains such as ULD configuration. Keywords: QUBO formulation; quantum annealing; Ising model; penalty methods; resource-constrained project scheduling; hybrid quantum-classical; minor embedding; Monte Carlo; digital twin.
Explored in this paper is the theory and practical implementation of quantum annealing for project scheduling and resource-allocation tasks that can be mapped to Ising or Quadratic Unconstrained Binary Optimization (QUBO) models. Quantum annealing is a meta-heuristic that exploits quantum tunnelling to search for the energy landscape of combinatorial optimization problems. The Quantum Annealing Process Flowchart is detailed stage by stage. Hybrid quantum-classical workflows that embed annealing inside a Monte Carlo or digital-twin loop are described, together with supporting figures of the energy landscape and the hybrid architecture. Cross-domain analogies with ULD configuration and related combinatorial problems illustrate transferability. The paper concludes with implementation considerations and the current maturity of the technology for construction and logistics applications. Keywords: quantum annealing implementation; Ising model; QUBO; combinatorial optimization; project scheduling; hybrid quantum-classical; energy landscape; digital twin; Monte Carlo; supply-chain resilience; ULD configuration
Presented in this paper is the theory and practical implementation of quantum annealing for project scheduling and resource-allocation tasks that can be mapped to Ising or Quadratic Unconstrained Binary Optimization (QUBO) models. Quantum annealing is a meta-heuristic that exploits quantum tunnelling to search the energy landscape of combinatorial optimization problems. The Quantum Annealing Process Flowchart is detailed stage by stage. Hybrid quantum-classical workflows that embed annealing inside a Monte Carlo or digital-twin loop are described, together with supporting figures of the energy landscape and the hybrid architecture. Cross-domain analogies with ULD configuration and related combinatorial problems illustrate transferability. The paper concludes with implementation considerations and the current maturity of the technology for construction and logistics applications. Keywords: quantum annealing implementation; Ising model; QUBO; combinatorial optimization; project scheduling; hybrid quantum-classical; energy landscape; digital twin; Monte Carlo; supply-chain resilience; ULD configuration
Quantum annealing has emerged as an effective meta-heuristic for solving combinatorial optimization problems by leveraging quantum tunnelling to navigate complex energy landscapes. This paper examines both the theoretical foundations and the practical implementation of quantum annealing for project scheduling and resource-allocation problems formulated as Ising or Quadratic Unconstrained Binary Optimization (QUBO) models. A comprehensive explanation of the quantum annealing process is provided through a stage-by-stage process flowchart. In addition, the study discusses hybrid quantum classical frameworks in which quantum annealing is integrated with Monte Carlo simulations or digital twin environments, supported by illustrations of the optimization energy landscape and the hybrid system architecture. The adaptability of the proposed approach is further demonstrated through analogies with unit load device (ULD) configuration and other related combinatorial optimization problems across multiple application domains. Finally, the paper outlines key implementation considerations and evaluates the current level of technological maturity of quantum annealing for applications in construction planning and logistics optimization. Keywords: quantum annealing implementation; Ising model; QUBO; combinatorial optimization; project scheduling; hybrid quantum-classical; energy landscape; digital twin; Monte Carlo; supply-chain resilience; ULD configuration
Quantum annealing (QA) offers a physics-inspired approach for such subproblems by mapping them onto an Ising Hamiltonian, which is algebraically equivalent to a Quadratic Unconstrained Binary Optimization (QUBO) model executable on quantum annealers or hybrid solvers. However, their practical employment in wireless systems critically depends on the specific problem formulation, on the particular variable encoding, constraint handling, solver selection, hardware embedding, and fair benchmarking. This paper develops wireless-oriented QUBO modeling guidelines for QA-assisted resource allocation. We first present a beginner-friendly toy example in which a terrestrial-aerial resource-allocation problem with a fractional signal-to-interference-plus-noise ratio (SINR) maximization objective is reformulated into a two-bit QUBO via Boolean interpolation, converted into an Ising representation, and decoded back into a wireless scheduling decision. The same scheduling is applicable to UAV, aircraft, satellite, and terrestrial links in SAGIN systems. We then review the QA foundations and solver ecosystems, including binary quadratic, discrete quadratic, constrained quadratic, and hybrid models from the perspective of NG wireless optimization. Furthermore, we discuss both binary and discrete encoding, penalty-based constraint conversion, embedding overhead, solution decoding, and feasibility verification. A QUBO Compatibility Index is introduced as a lightweight diagnostic metric for assessing the associated QUBO formulation readiness, rather than a predictor of runtime or solution quality. Finally, a case study on max-min uplink power control in cell-free massive MIMO illustrates how a fairness-oriented wireless problem can be discretized, constrained, converted into the concept of a QUBO, and evaluated against classical baselines. A host of key challenges and future directions are also discussed.
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.
Quantum annealing is a model for quantum computing that is aimed at solving hard optimization problems by representing them as quadratic unconstrained binary optimization (QUBO) problems. Although many NP-hard problems can easily be formulated as binary-variable problems with a quadratic objective function, such formulations typically also include constraints, which are not allowed in a QUBO. Hence, such constraints are usually incorporated in the objective function as additive penalty terms. While there is substantial previous work on implementing linear equality constraints, the case of inequality constraints has not much been studied. In this paper, we propose a new approach for formulating and embedding inequality constraints as penalties and describe early implementation results.