DHCRWOA: adaptive whale optimization algorithm with Cauchy-Rayleigh distribution for numerical and engineering design optimization

This paper proposes DHCRWOA, a stage-specific extension of the Whale Optimization Algorithm (WOA) designed to improve its exploration–exploitation transition and reduce stagnation around suboptimal leaders. Five modules are introduced, each targeting a single WOA search phase: Good Nodes Set (GNS) initialization, adaptive parameter control, statistical-guided exploration, Cauchy-based local exploitation, and Rayleigh-weighted spiral updating. On the CEC2005 benchmark suite at 30 dimensions, DHCRWOA achieves the best average rank (2.13) among ten algorithms and the best or tied-best mean on 22 of 23 functions, with no baseline outperforming it on more than one function. At 50 and 100 dimensions, DHCRWOA maintains the top Friedman rank (1.69 and 1.62, respectively) at average runtimes of 0.52 s and 1.03 s per run. A five-variant ablation study confirms that weakening the statistical-guided exploration or Cauchy-based exploitation produces the largest rank degradation (from 2.35 to 4.09 and 4.70, respectively), while all five modules contribute to overall performance. Five constrained engineering design problems further confirm competitive results on practical optimization tasks. Theoretical analysis establishes the relationship between DHCRWOA and standard WOA, including a perturbation bound on the adaptive parameter (), boundedness of the population trajectory, monotonicity of the best-so-far process, and a global reachability theorem.