A novel chaotic optimal foraging algorithm for unconstrained and constrained problems and its application in white blood cell segmentation

Abstract

Meta-heuristic is defined as an iterative generation process with some random characters, which aims to generate a sufficiently good solution(s) for the global optimization problems. The process of solving a global optimization problem with near-optimal solutions is based on combining in some intelligent way different methodologies for exploiting and exploring the search space and building a structure information through different learning strategies. Optimal foraging algorithm (OFA) is a robust meta-heuristic algorithm inspired by following the animal foraging behavior. Recently, chaos and meta-heuristics have been combined in different algorithms with the aim of overcoming the limitations of meta-heuristics. In this paper, a novel algorithm for combining chaos with OFA is presented. The presented chaotic optimal foraging algorithm (COFA) is introduced with a set of unconstrained and constrained optimization problems using different chaotic maps. The results show that the proposed COFA outperforms the standard OFA for these benchmarks regarding exploitation, exploration, the trajectory of foraging individuals, search history, fitness improvement of the population and convergence rate. The performance of COFA has also compared with other most recent and popular meta-heuristic algorithms proofing its superior. An application of white blood cell segmentation in microscopic images has been chosen, and the proposed chaotic optimal foraging algorithm has been applied to see its ability and accuracy to identify and segment the white blood cell for further diagnosis. According to the statistical analysis of objective values, COFA algorithm is more accurate and robust than original OFA algorithm. COFA proves its ability to converge to the optimal multiple thresholds level-based segmentation more accurate than OFA. The experimental results also show that the proposed COFA proves its high degree of stability compared with original OFA in finding optimal multiple threshold values.

Appendices

Appendix A: List of benchmark global optimization problems

See Tables 15, 16, 17 and 18.

Table 15 Mathematical formula of unimodal benchmark functions

Table 16 The mathematical formula of multi-modal benchmark functions

Table 17 Properties of unimodal benchmark functions, opt denotes optimum value, lb denotes lower bound, dim denotes dimensions, and ub denotes upper bound

Table 18 Properties of multi-modal benchmark functions, opt denotes optimum value, lb denotes lower bound, dim denotes dimensions, and ub denotes upper bound

Appendix B: List of engineering design problems

1.1 The three-bar truss design problem

The three-bar truss design problem is a minimization problem. The mathematical definition of this problem as follows:

$$\begin{aligned} \begin{aligned}&{\text {Minimize}}\ {\text {the}}\ {\text {function}} \\&f(\mathbf {z})=(2\sqrt{2} z_1+z_2)\times l,\\&{\text {Subject}}\,{\text {to}},\\&h_1 (\mathbf {z})=\frac{\sqrt{2} z_1+z_2}{\sqrt{2} z_1^2+2z_1 z_2 } p-\alpha \le 0,\\&h_2 (\mathbf {z})= \frac{z_2}{\sqrt{2} z_1^2+2z_1 z_2 } p-\alpha \le 0,\\&h_3 (\mathbf {z})= \frac{ 1}{\sqrt{2} z_2+z_1 } p-\alpha \le 0,\\&{\text {where,}}\\&0\le z_i \le 1,i=1,2,\\&l= 100\,{\text {cm}},\ p=2\,{\text {kN/cm}}^2\ \alpha =2\,{\text {kN/cm}}^2. \end{aligned} \end{aligned}$$

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1.2 Compression spring design problem

Compression spring design problem is a minimization problem. It has three main variables. These variables are wire diameter (WD), coil diameter (CD), and the number of active coils (NAC). The mathematical definition of this problem as follows:

$$\begin{aligned} \begin{aligned}&{\text {Minimize}}\ {\text {the}}\ {\text {function}}\\&f(\mathbf {z})=(z_3+2)z_2 z_1^2, \mathbf {z}=[z_1 z_2 z_3 ]=(CD)(WD)(NAC),\\&{\text {Subject}}\,{\text {to}},\\&h_1 (\mathbf {z})= 1-\frac{z_2^3 z_3}{717854z_1^4 }\le 0,\\&h_2 (\mathbf {z})= \frac{4z_2^2-z_1 z_2}{12566(z_2 z_1^3-z_1^4) } +\frac{1}{5108z_1^2 } -1 \le 0,\\&h_3 (\mathbf {z})= 1-\frac{140.45z_1}{z_2^2 z_3 }\le 0,\\&h_4 (\mathbf {z})=\frac{z_1+z_2}{1.5}-1\le 0 \\&{\text {where,}} \ \\&0.05\le z_1 \le 2,\\&0.25 \le z_2 \le 1.30,\\&2 \le z_3 \le 15\\ \end{aligned} \end{aligned}$$

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1.3 Welded Beam Design Problem

This problem is a minimization problem. It has four variables. These variables are weld thickness (h), the length of bar attached to the weld (l), the height of the bar (t), and the thickness of the bar. Furthermore, this problem has some constraints. These constraints are the bending stress in the beam $(\alpha )$, the beam deflection $(\beta )$, side and the end deflection of the beam $(\delta )$ constraints, and buckling load on the bar (BL). The mathematical definition of this problem is defined in the following:

$$\begin{aligned} \begin{aligned}&{\text {Minimize}}\ {\text {the}}\ {\text {function}}\ \\&f(\mathbf {z})=1.10471 z_1^2 z_2+0.04811z_3 z_4 (14.0+z_2 ), \mathbf {z}=[z_1 z_2 z_3 ]=[hltb],\\&{\text {Subject}}\,{\text {to}},\\&h_1 (\mathbf {z})= \delta (\mathbf {z})-\delta _{\max }\le 0,\\&h_2 (\mathbf {z})= \alpha (\mathbf {z})- \alpha _{\max }\le 0,\\&h_3 (\mathbf {z})= z_1-z_4\le 0,\\&h_4 (\mathbf {z})= 0.10471z_1^2+0.04811z_3 z_4 (14.0+z_2 )-5.0\le 0,\\&h_5 (\mathbf {z})= 0.125-z_1\le 0,\\&h_6 (\mathbf {z})=\delta (\mathbf {z})-\delta _{\max }\le 0,\\&h_7 (\mathbf {z})= B-BL(\mathbf {z})\le 0 \\&{\text {where}},\\&0.1\le z_1\le 2,\\&0.1\le z_2 \le 10,\\&0.1\le z_3 \le 10,\\&0.1\le z_4 \le 20\\&{\text {and}}\\&\delta (\mathbf {z})=\sqrt{\delta ^{\prime 2}+2\delta ^{\prime } \delta ^n \frac{z_2}{2R}+\delta ^{n2} }\\&\delta ^\prime =\frac{B}{\sqrt{2} z_1 z_2}\\&\delta ^n=\frac{MR}{J}\\&M=B(L+\frac{z_2}{2})\\&R=\sqrt{\frac{z_2^2}{4}+\left( \frac{z_1+z_3}{2}\right) ^2 }\\&J=2\left( \sqrt{2} z_1 z_2 \left[ \frac{z_2^2}{12} \left(\frac{z_1+z_3}{2}\right)^2\right] \right) \\&\alpha (\mathbf {z})=\frac{6PL}{z_4 z_3^2 },\beta (\mathbf {z})=\frac{4PL^3}{Ez_3^2 z+z_4 }\\&BL(\mathbf {z})=\frac{4.013E\sqrt{\frac{z_3^2 z_4^6}{36}}}{L^2} \left(1-\frac{z_3}{2L} \sqrt{\frac{E}{4G}}\right)\\&P=6000\,{\text {lb}},\ L=14\,{\text {in.}},\beta _{\max }=0,\ 25\,{\text {in.}},\ E=30 \times 10^6\, {\text {psi}},G=12\times 10^6\,{\text {psi}},\\&\delta _{\max }=13{,}600,{\text {psi}},\ \alpha _{\max }=30{,}000\,{\text {psi}} \end{aligned} \end{aligned}$$

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