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Bacteria phototaxis optimizer

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Abstract

This paper introduces a new metaheuristic algorithm called bacteria phototaxis optimizer (BPO). It is designed to solving optimization issues. Inspired by the bacteria phototaxis under the control of photosensory proteins in nature, and based on the basic law of bacterial colony growth and evolution, we have designed the photosensory protein concentration, phototaxis motion and growth operators. These three operators exhibit a highly adaptive and information interaction mechanism. The goal is to simulate the phototaxis process of bacteria and form a complete model of BPO. At the same time, BPO is compared with eight most representative as well as newly generated metaheuristics. Its performance is verified by using 23 well-known benchmark functions with three different types. Additionally, we have conducted several evaluation processes, such as qualitative and quantitative analysis as well as parametric and nonparametric tests. Finally, five classical engineering design problems are used to further test the effectiveness of the algorithm in solving constrained problems. The aforementioned experimental results show that compared with other algorithms, BPO has better accuracy, convergence, and robustness and shows strong competitiveness and optimization performance.

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Funding

This paper was supported by the National Natural Science Foundation Project of China [grant number 62073330] and the Natural Science Foundation Project of Hunan Province of China [grant number 2019JJ20021].

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  1. College of Systems Engineering, National University of Defense Technology, Changsha, 410073, China

    Qingtao Pan, Jun Tang, Jianjun Zhan & Hao Li

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Appendix

Appendix

1.1 Process synthesis problem

This problem incorporates a nonlinear constraint and its mathematical formulation is described as follows [150].

Minimize:

$$f\left(\overline{x}\right)={x}_{2}+2{x}_{1}$$

Subject to:

$${g}_{1}\left(\overline{x}\right)=-{x}_{1}^{2}-{x}_{2}+1.25\le 0$$
$${g}_{2}\left(\overline{x}\right)={x}_{1}+{x}_{2}\le 1.6$$

Variable ranges:

$$0\le {x}_{1}\le 1.6$$
$${x}_{2}\in \left\{\mathrm{0,1}\right\}$$

1.2 Process flow sheeting problem

This problem can be formulated as a nonconvex constrained optimization problem, which is expressed as follows [151].

Minimize:

$$f\left(\overline{x}\right)=-0.7{x}_{3}+0.8+5{\left(0.5-{x}_{1}\right)}^{2}$$

Subject to:

$${g}_{1}\left(\overline{x}\right)=-\mathit{exp}\left({x}_{1}-0.2\right)-{x}_{2}\le 0$$
$${g}_{2}\left(\overline{x}\right)={x}_{2}+1.1{x}_{3}\le -1.0$$
$${g}_{3}\left(\overline{x}\right)={x}_{1}-{x}_{3}\le 0.2$$

Variable ranges:

$$0.2\le {x}_{1}\le 1$$
$$-2.22554\le {x}_{2}\le -1$$
$${x}_{3}\in \left\{\mathrm{0,1}\right\}$$

1.3 Tension/compression spring design

This problem needs to optimize the weight of a tension or compression spring and contains four constraints and three variables: the diameter of the wire (x1), the mean of the diameter of coil (x2), and the number of active coils (x3). It is defined specifically as follows [152].

Minimize:

$$f\left(\overline{x}\right)={x}_{1}^{2}{x}_{2}\left(2+{x}_{3}\right)$$

Subject to:

$${g}_{1}\left(\overline{x}\right)=1-\frac{{x}_{2}^{3}{x}_{3}}{71785{x}_{1}^{4}}\le 0$$
$${g}_{2}\left(\overline{x}\right)=\frac{4{x}_{2}^{2}-{x}_{1}{x}_{2}}{12566\left({x}_{2}{x}_{1}^{3}-{x}_{1}^{4}\right)}+\frac{1}{5108{x}_{1}^{2}}-1\le 0$$
$${g}_{3}\left(\overline{x}\right)=1-\frac{140.45{x}_{1}}{{x}_{2}^{2}{x}_{3}}\le 0$$
$${g}_{4}\left(\overline{x}\right)=\frac{{x}_{1}+{x}_{2}}{1.5}-1\le 0$$

Variable ranges:

$$0.05\le {x}_{1}\le 2.00$$
$$0.25\le {x}_{2}\le 1.30$$
$$2.00\le {x}_{3}\le 15.0$$

1.4 Three-bar truss design problem

This problem with an accidented constrained space is taken from civil engineering. Its main objective is to minimize the weight of the bar structures. The stress constraints of each bar are considered, which finally constitute three nonlinear constraints of this problem. The mathematical description is given below [153].

Minimize:

$$f\left(\overline{x}\right)=l\left({x}_{2}+2\sqrt{2}{x}_{1}\right)$$

Subject to:

$${g}_{1}\left(\overline{x}\right)=\frac{{x}_{2}}{2{x}_{2}{x}_{1}+\sqrt{2}{x}_{1}^{2}}P-\sigma \le 0$$
$${g}_{2}\left(\overline{x}\right)=\frac{{x}_{2}+\sqrt{2}{x}_{1}}{2{x}_{2}{x}_{1}+\sqrt{2}{x}_{1}^{2}}P-\sigma \le 0$$
$${g}_{3}\left(\overline{x}\right)=\frac{1}{{x}_{1}+\sqrt{2}{x}_{2}}P-\sigma \le 0$$
$$l=100, P=2,\text{ and }\sigma =2.$$

Variable ranges:

$$0\le {x}_{1},{x}_{2}\le 1$$

1.5 Himmelblau’s function

This problem is not only used to simulate the process design problems, but also as a common benchmark to analyze non-linear constrained optimization algorithms. It contains six nonlinear constraints and five variables, which are shown below [154].

Minimize:

$$f\left(\overline{x}\right)=5.3578547{x}_{3}^{2}+0.8356891{x}_{1}{x}_{5}+37.293239{x}_{1}-40792.141$$

Subject to:

$${g}_{1}\left(\overline{x}\right)=-G1\le 0$$
$${g}_{2}\left(\overline{x}\right)=G1-92\le 0$$
$${g}_{3}\left(\overline{x}\right)=90-G2\le 0$$
$${g}_{4}\left(\overline{x}\right)=G2-110\le 0$$
$${g}_{5}\left(\overline{x}\right)=20-G3\le 0$$
$${g}_{6}\left(\overline{x}\right)=G3-25\le 0$$
$$G1=85.334407+0.0056858{x}_{2}{x}_{5}+0.0006262{x}_{1}{x}_{4}-0.0022053{x}_{3}{x}_{5}$$
$$G2=80.51249+0.0071317{x}_{2}{x}_{5}+0.0029955{x}_{1}{x}_{2}+0.0021813{x}_{3}^{2}$$
$$G3=9.300961+0.0047026{x}_{3}{x}_{5}+0.00125447{x}_{1}{x}_{3}+0.0019085{x}_{3}{x}_{4}$$

Variable ranges:

$$78\le {x}_{1}\le 102$$
$$33\le {x}_{2}\le 45$$
$$27\le {x}_{3},{x}_{4},{x}_{5}\le 45$$

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Pan, Q., Tang, J., Zhan, J. et al. Bacteria phototaxis optimizer. Neural Comput & Applic 35, 13433–13464 (2023). https://doi.org/10.1007/s00521-023-08391-6

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