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. 2024 Feb 29;14(1):5032.
doi: 10.1038/s41598-024-54910-3.

Hippopotamus optimization algorithm: a novel nature-inspired optimization algorithm

Mohammad Hussein Amiri  1 , Nastaran Mehrabi Hashjin  2 , Mohsen Montazeri  1 , Seyedali Mirjalili  3   4 , Nima Khodadadi  5
Affiliations

Hippopotamus optimization algorithm: a novel nature-inspired optimization algorithm

Mohammad Hussein Amiri et al. Sci Rep. .

Abstract

The novelty of this article lies in introducing a novel stochastic technique named the Hippopotamus Optimization (HO) algorithm. The HO is conceived by drawing inspiration from the inherent behaviors observed in hippopotamuses, showcasing an innovative approach in metaheuristic methodology. The HO is conceptually defined using a trinary-phase model that incorporates their position updating in rivers or ponds, defensive strategies against predators, and evasion methods, which are mathematically formulated. It attained the top rank in 115 out of 161 benchmark functions in finding optimal value, encompassing unimodal and high-dimensional multimodal functions, fixed-dimensional multimodal functions, as well as the CEC 2019 test suite and CEC 2014 test suite dimensions of 10, 30, 50, and 100 and Zigzag Pattern benchmark functions, this suggests that the HO demonstrates a noteworthy proficiency in both exploitation and exploration. Moreover, it effectively balances exploration and exploitation, supporting the search process. In light of the results from addressing four distinct engineering design challenges, the HO has effectively achieved the most efficient resolution while concurrently upholding adherence to the designated constraints. The performance evaluation of the HO algorithm encompasses various aspects, including a comparison with WOA, GWO, SSA, PSO, SCA, FA, GOA, TLBO, MFO, and IWO recognized as the most extensively researched metaheuristics, AOA as recently developed algorithms, and CMA-ES as high-performance optimizers acknowledged for their success in the IEEE CEC competition. According to the statistical post hoc analysis, the HO algorithm is determined to be significantly superior to the investigated algorithms. The source codes of the HO algorithm are publicly available at https://www.mathworks.com/matlabcentral/fileexchange/160088-hippopotamus-optimization-algorithm-ho .

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Conflict of interest statement

The authors declare no competing interests.

Figures

Figure 1
Figure 1
(a–d) shows the defensive behavior of the hippopotamus against the predator.
Figure 2
Figure 2
Graphic representation of the phase 2.
Figure 3
Figure 3
Drawing a Hippopotamus Escaping from the Predator.
Figure 4
Figure 4
HO's flowchart.
Algorithm 1
Algorithm 1
Pseudo-code of HO.
Figure 5
Figure 5
Boxplot illustrating the performance of the HO in comparison to competing algorithms for optimizing BFs (F1-F23).
Figure 5
Figure 5
Boxplot illustrating the performance of the HO in comparison to competing algorithms for optimizing BFs (F1-F23).
Figure 5
Figure 5
Boxplot illustrating the performance of the HO in comparison to competing algorithms for optimizing BFs (F1-F23).
Figure 6
Figure 6
Convergence curves of the top three algorithms in each benchmark functions (F1- F23).
Figure 6
Figure 6
Convergence curves of the top three algorithms in each benchmark functions (F1- F23).
Figure 6
Figure 6
Convergence curves of the top three algorithms in each benchmark functions (F1- F23).
Figure 7
Figure 7
Boxplot illustrating the performance of the HO in comparison to competing algorithms for ZP.
Figure 8
Figure 8
Convergence curves of the top three algorithms in each function in ZP.
Figure 9
Figure 9
Boxplot illustrating the performance of the HO in comparison to competing algorithms for optimizing CEC 2019.
Figure 9
Figure 9
Boxplot illustrating the performance of the HO in comparison to competing algorithms for optimizing CEC 2019.
Figure 10
Figure 10
Convergence curves of the top three algorithms in each function in CEC 2019.
Figure 10
Figure 10
Convergence curves of the top three algorithms in each function in CEC 2019.
Figure 11
Figure 11
Nemenyi test for top ten algorithms in each group with α= 0.05.
Figure 12
Figure 12
The convergence curves of HO during the investigation of sensitivity analysis regarding parameter N.
Figure 12
Figure 12
The convergence curves of HO during the investigation of sensitivity analysis regarding parameter N.
Figure 12
Figure 12
The convergence curves of HO during the investigation of sensitivity analysis regarding parameter N.
Figure 13
Figure 13
The convergence curves of HO during the investigation of sensitivity analysis regarding parameter T.
Figure 13
Figure 13
The convergence curves of HO during the investigation of sensitivity analysis regarding parameter T.
Figure 13
Figure 13
The convergence curves of HO during the investigation of sensitivity analysis regarding parameter T.
Figure 14
Figure 14
TCS.
Figure 15
Figure 15
WB.
Figure 16
Figure 16
PV.
Figure 17
Figure 17
WFLO with HO.
Figure 18
Figure 18
Boxplot illustrating the performance of the HO in comparison to twelve algorithms for optimizing TCS, WB, PV and WFLO.
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