Quaternion-based formulations for volume maximisation problems

This paper introduces a mathematical formulation for the problem of determining the optimal position for a three-dimensional item inside a convex container, where its scale can be increased the most and thus its volume maximised. Until now, no methods have been presented that guarantee optimal solutions to this volume maximisation problem while considering continuous free rotation of the item, with approaches relying on heuristics, approximations or enforcing a discrete number of rotations. We aim to find optimal solutions when considering continuous rotation, represented using quaternions. This enables modelling rotation through quadratic constraints. The resulting quadratically constrained problem can be solved to optimality by mathematical solvers. To keep the required computation time within reasonable limits, various improvements to the model such as symmetry breaking are introduced. Experiments show that the majority of our benchmark instances can be solved to optimality within minutes. The expansion to concave containers is also explored, but proves to be more challenging as the required number of quadratic constraints quickly becomes prohibitive.

The source code used to perform the experiments described in this paper can be found at https://github.com/JonasTollenaere/svmp-quaternion-qcp. All instance files can be found in our repository at https://gitlab.kuleuven.be/codes/datasets/3d-irregular-data-sets.

1. This repository can be found at https://gitlab.kuleuven.be/codes/datasets/3d-irregular-data-sets.

This contribution was supported by the KU Leuven Internal Funding C2E/23/013 project. Editorial feedback and review was provided by Luke Connolly.

Authors and Affiliations

1. CODeS/NUMA, Department of Computer Science, KU Leuven, Gebroeders de Smetstraat, 9000, Gent, Belgium

   Jonas Tollenaere & Tony Wauters

Contributions

J. Tollenaere: Conceptualization, Methodology, Software, Formal analysis and investigation, Writing. T. Wauters: Supervision, Funding acquisition, Writing—review.

Corresponding author

Correspondence to Jonas Tollenaere.

Conflict of interest

The authors have no competing interests to declare.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Datasets

1.1 Appendix A1: Experiments with convex containers

See Fig. 13.

The set of objects used as containers (gray) and items (locust) in the convex container experiments

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1.2 Appendix A2: Experiments with concave containers

See Fig. 14.

The set of objects used as containers (gray) and items (locust) in the concave container experiments

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All instances can be found in our repository at: https://gitlab.kuleuven.be/codes/datasets/3d-irregular-data-sets.

Appendix B: Detailed convex container results

Tables 11 and 12 provide a detailed overview of the computation times for the 80 instances in the convex container benchmark. Meanwhile, Tables 13 and 14 provide a detailed overview of the experiments where Gurobi and SCIP are compared.

The experiments were executed with Gurobi 11.0.0 and SCIP 9.1.0. To show the impact of a different number of available threads, the experiments with Gurobi were executed with 1, 12, and 64 threads. 12 threads is the default configuration for the M2 Pro CPU used in the experiments.

Appendix C: Convex container benchmarks compared to other approaches

This appendix provides detailed results comparing the solution quality and computation times against the heuristics introduced in Silva et al. (2021) and Tollenaere et al. (2024). This shows that the heuristic solution can be slightly worse than the optimal ones.

See Tables 15 and 16.

Appendix D: Detailed concave container results

Table 17 provides the individual results for the concave container benchmark. A scale of 0 and dash in the gap column indicates that no feasible solution was found by the solver in the 1 h time limit. The LA Rel. column shows the relative scale found by the late acceptance heuristic (Tollenaere et al. 2024) compared to the optimal scale.

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