An improved multi-population whale optimization algorithm

Abstract

Clustering techniques and metaheuristic algorithms (MA) have demonstrated being efficient tools in their respective action fields. However, working together is an area marginally explored. One of the main disadvantages of MA is the lack of diversity in the solutions. Besides, most of them use only a single population to analyze the search space; this affects the capabilities to find the optimal solutions. This article proposes an approach called K-WOA that merges the benefits of two methods into a single algorithm. The K-means is a popular clustering technique based on centroids. Due to its simplicity and efficiency, combining it with a MA as the whale optimization algorithm (WOA) is ideal. This proposed K-WOA aims to increase the diversity of solutions in optimization problems by creating multiple groups of search agents operating cooperatively to explore the search space. To perform this task, the K-means is used in the initialization process to separate the population into different subgroups that the WOA independently evolves. In each sub-population, the best search agent is chosen to compare with the best agents of the other whale groups. By doing this, the algorithm can explore different regions of the search space simultaneously with more than one element. The K-WOA is proposed as an improved optimization algorithm that simultaneously searches for optimal solutions in multiple regions of the search space. The experimental results and comparisons with state-of-the-art approaches show that the proposed algorithm is competitive for solving complex optimization problems.

Appendices

Appendix A

The sets of benchmark test functions that are described in Tables 18, 19 and 20, representing the unimodal, multimodal and composite functions. In the column of minimum we have f which is the optimum value of the function and are the optimum positions, the column S is the continuous search space, every function has n dimensions, in this case we performed the experiments using 30, 50 and 100 dimensions.

Table 18 Unimodal test benchmark functions considered in the experiments

Table 19 Multimodal test benchmark functions considered in the experiments

Table 20 Composite test benchmark functions considered in the experiments

Appendix B

1.1 B.1: Gear train design problem

Consider	$\longrightarrow$	$\mathbf {x}=\left[ x_1\ x_2\ x_3\ x_4\right] =\left[ n_A\ n_B\ n_C\ n_D\right] ,$
Minimize	$\longrightarrow$	$f(\mathbf {x})=\left( \frac{1}{6.931}\ -\frac{x_3x_2}{x_1x_4}\right) ^2,$
Subject to	$\longrightarrow$	$12\le x_1,x_2,x_3,x_4\le 60$

1.2 B.2: Pressure vessel design problem

Consider	$\longrightarrow$	$\mathbf {x}=\left[ x_1\ x_2\ x_3\ x_4\right] =\left[ T_s\ T_h\ R\ L\right] ,$
Minimize	$\longrightarrow$	$f(\mathbf {x})=0.6224x_1x_3x_4+1.7781x_2x_3^2+3.1661x_1^2x_4+19.84x_1^2x_3$
Subject to	$\longrightarrow$	$g_1(\mathbf {x})=-x_1+0.0193x_3\le 0,$
		$g_2(\mathbf {x})=-x_3+0.00954x_3\le 0,$
		$g_3(\mathbf {x})=-\pi x_3^2x_4-\frac{4}{3}\pi x_3^3+1296000\le 0,$
		$g_4(\mathbf {x})=x_4-240\le 0$
Variable Range	$\longrightarrow$	$0\le x_1\le 99 ,$
		$0\le x_2\le 99 ,$
		$10\le x_3\le 200 ,$
		$10\le x_4\le 200$

1.3 B.3: Tension/compression spring design

Consider	$\longrightarrow$	$\mathbf {x}=\left[ x_1\ x_2\ x_3\right] =\left[ d\ D\ N\right] ,$
Minimize	$\longrightarrow$	$f(\mathbf {x})=(x_3+2)x_2x_1^2,$
Subject to	$\longrightarrow$	$g_1(\mathbf {x})=1-\frac{x_2^3x_3}{71785x_1^4}\le 0,$
		$g_2(\mathbf {x})=\frac{4x_2^2-x_1x_2}{12566\left( x_2x_1^3-x_1^4\right) }+\frac{1}{5108x_1^2}\le 0,$
		$g_3(\mathbf {x})=1-\frac{140.45x_1}{x_2^2x_3}\le 0,$
		$g_4(\mathbf {x})=\frac{x_1+x_2}{1.5}-1\le 0$
Variable Range	$\longrightarrow$	$0.05\le x_1\le 2.00 ,$
		$0.25\le x_2\le 1.30 ,$
		$2.00\le x_3\le 15.0$