[Paper Review] Aircraft Loading Optimization -- QUBO models under multiple constraints

Giovanni Pilon, Nicola Gugole+1 more|arXiv (Cornell University)|Feb 16, 2021

Quantum Computing Algorithms and Architecture1 references4 citations

TL;DR

This paper formulates the aircraft loading optimization problem from the Airbus Quantum Computing Challenge as a Quadratic Unconstrained Binary Optimization (QUBO) model with multiple physical constraints—payload limits, center of gravity bounds, and shear limits—using penalty functions to encode constraints. The authors demonstrate that the QUBO model is compatible with quantum annealers like D-Wave 2000Q and benchmark it against classical solvers, showing feasible solutions with moderate success rates and suboptimal average payload compared to classical methods.

ABSTRACT

In this submission we solve the Aircraft Loading Optimization problem of the Airbus Quantum Computing Challenge. Finding the optimal loading for a plane is a challenging task for classical algorithms, especially because the solution must respect several flight constraints. The contribution of this work is formulating this problem and its constraints in a model based on QUBO equations which are compatible with quantum annealers. We then benchmarked the model on different solvers to evaluate the performances and capabilities of current technologies.

Motivation & Objective

• To address the complex aircraft loading optimization problem under real-world flight constraints such as payload, center of gravity, and shear limits.
• To reformulate these constraints into a QUBO-compatible penalty function framework suitable for quantum annealers.
• To evaluate the performance of the QUBO model using both classical QUBO solvers and a D-Wave 2000Q quantum annealer.
• To assess the feasibility and scalability of quantum-inspired optimization for industrial logistics problems.
• To identify key challenges in penalty scaling and slack variable usage for future quantum advantage in this domain.

Proposed method

• The problem is modeled using binary position assignment variables $ p_{i,j} $ indicating whether container $ i $ occupies position $ j $.
• The objective function maximizes total payload by minimizing a negative linear function of container masses and their assignments.
• Constraints are encoded as quadratic penalty functions: no-overlap, no-duplicate, center of gravity, and shear limit constraints are converted into QUBO-compatible penalties.
• Slack variables are introduced to convert inequality constraints into equality forms for penalty formulation.
• Penalty coefficients $ P^{( ext{constraint})} $ are used to scale constraint violations relative to the objective function.
• The full QUBO model is tested on classical solvers (e.g., QBsolv) and the D-Wave 2000Q via API, with results compared across multiple runs.

Experimental results

Research questions

• RQ1 Can the aircraft loading optimization problem with multiple physical constraints be effectively modeled as a QUBO problem?
• RQ2 How do penalty functions for constraints like center of gravity and shear limits perform in a QUBO framework?
• RQ3 What is the performance gap between classical QUBO solvers and quantum annealers (D-Wave 2000Q) on this problem?
• RQ4 How does the inclusion of additional constraints affect solution quality and computational time?
• RQ5 What are the key bottlenecks in scaling the QUBO model to larger aircraft configurations on current quantum hardware?

Key findings

• The QUBO model successfully encodes all required constraints—payload limits, center of gravity bounds, and shear limits—into penalty functions compatible with quantum annealing.
• Classical solvers (e.g., QBsolv) achieved a 33% success rate in finding the optimal solution for the PL-only case, with consistent feasibility.
• The D-Wave 2000Q achieved feasible solutions in 80% of runs but loaded on average less weight than classical solvers, indicating higher noise and suboptimal performance.
• Adding center of gravity constraints improved the distribution of center of gravity around the target, though shear limit constraints showed minimal improvement due to potential conflicts with CG constraints.
• The model reached a practical hardware limit on D-Wave 2000Q with 48 qubits (24 slack variables), suggesting scalability challenges for larger instances.
• The study identifies penalty scaling and automatic slack variable selection as key areas for future improvement to enhance quantum advantage potential.