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<p>The Strengthened Hippopotamus Optimization (SHO) algorithm is an enhanced metaheuristic developed to overcome the limitations of the original Hippopotamus Optimization (HO) method. It incorporates three mechanisms: randomized fractional-order chaotic opposition-based learning to improve global exploration, quadratic interpolation to enhance local exploitation, and hybrid horizontal–vertical crossover to maintain population diversity. SHO was tested on 51 benchmark functions from the CEC 2017, 2019, and 2022 suites, seven engineering design problems, and eight system identification tasks. Experimental results show that SHO ranked within the top two in 34 benchmark functions (66.7%) and achieved first place in several of them, whereas HO achieved such rankings in only three cases (5.9%), indicating an approximately eleven-fold improvement. Statistical analyses (Friedman and Holm tests, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p&lt;0.05\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo /> <mn>0.05</mn> </mrow> </math></EquationSource> </InlineEquation>) confirmed that SHO significantly outperformed ten other metaheuristic algorithms. In real-world applications, SHO consistently produced optimal or near-optimal solutions, demonstrating strong convergence, robustness, and stability.</p>

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Strengthened hippopotamus optimization algorithm enhanced with quadratic interpolation, chaotic opposition-based learning, and horizontal–vertical crossover for continuous optimization

  • 期刊论文

  • 发表日期：

    2026年6月30日

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  • Shaghayegh Saghafi,
  • Nastaran Mehrabi Hashjin,
  • Mohammad Hussein Amiri,
  • Asma Alanazy

摘要

The Strengthened Hippopotamus Optimization (SHO) algorithm is an enhanced metaheuristic developed to overcome the limitations of the original Hippopotamus Optimization (HO) method. It incorporates three mechanisms: randomized fractional-order chaotic opposition-based learning to improve global exploration, quadratic interpolation to enhance local exploitation, and hybrid horizontal–vertical crossover to maintain population diversity. SHO was tested on 51 benchmark functions from the CEC 2017, 2019, and 2022 suites, seven engineering design problems, and eight system identification tasks. Experimental results show that SHO ranked within the top two in 34 benchmark functions (66.7%) and achieved first place in several of them, whereas HO achieved such rankings in only three cases (5.9%), indicating an approximately eleven-fold improvement. Statistical analyses (Friedman and Holm tests, \(p<0.05\) p 0.05 ) confirmed that SHO significantly outperformed ten other metaheuristic algorithms. In real-world applications, SHO consistently produced optimal or near-optimal solutions, demonstrating strong convergence, robustness, and stability.