Authors:
Saman M. Almufti, Amira Bibo Sallow

Addresses:
Department of Information Technology, Technical College of Informatics-Akre, Akre University for Applied Sciences, Akre, Nineveh Governorate, Iraq. Department of Information Technology, Technical College of Duhok, Duhok Polytechnic University, Duhok, Iraq. 

Abstract:

The compound gear-train benchmark is a discrete engineering optimisation problem in which four integer tooth counts must approximate a prescribed transmission ratio. Although the mathematical model is compact, the attainable objective values are highly discontinuous, and several recent reports have produced artificially smaller values by treating tooth counts as continuous variables. This study develops an integer-preserving Scuba Diver Optimisation Algorithm (SDOA) for the benchmark and evaluates ten parameter configurations through 30 independent runs per configuration. Every movement operator is followed by rounding and bound repair, and the best-known solution is verified by exhaustive enumeration of the 49^4 admissible designs. The strongest setting, S10, used 150 divers and 600 iterations. It obtained a mean objective of 3.935622e-10, a median of 2.307816e-11, and the lowest mean rank of 3.617. It reached f <= 1e-9 in 86.67% of runs, f <= 1e-10 in 60.00%, and the exact discrete optimum in 13.33%. The best design [19, 16, 43, 49] is symmetric to [16, 19, 43, 49] and gives f = 2.700857148886513e-12. A Friedman test detected a configuration effect (chi-square (9) = 120.972634, p = 8.447286e-22, Kendall W = 0.448047). Holm-adjusted Wilcoxon tests confirmed that S10 outperformed the three lowest-budget settings, while several higher-budget alternatives remained statistically competitive. Convergence and search-stage records show that the selected configuration allocated 30.90% of diver-iterations to fine-tuning and limited resets to 13.76%. The results establish a reproducible discrete SDOA implementation and show that parameter selection, integer repair, and exact-optimum verification materially affect conclusions for this benchmark.

Keywords: Discrete Optimisation; Engineering Design; Gear Train; Integer Repair and Metaheuristics; Parameter Sensitivity; Scuba Diver Optimisation Algorithm; Nonparametric Statistics.

Received on: 07/05/2025, Revised on: 22/08/2025, Accepted on: 13/09/2025, Published on: 05/06/2026

DOI: 10.69888/FTSSM.2026.000753

FMDB Transactions on Sustainable Structures and Materials, 2026 Vol. 2 No. 1, Pages: 1-16

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