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Nizar optimization algorithm: a novel metaheuristic algorithm for global optimization and engineering applications

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Abstract

This paper presents a novel and powerful population-based metaheuristic algorithm called Nizar Optimization Algorithm (NOA). This algorithm is based on two techniques. The first technique is to use effective mappings, which are divided into two types: mixing and transformation mappings. The mixing mappings return a mixed vector by replacing or shuffling the elements of two selected vectors, and the transformation mappings return the translation, dilation, or transfer of a selected vector. The second technique is to determine the effective points by using the effective mappings and individuals of the population, and then, these points are used in the learning process of NOA. To validate and demonstrate the performance of the proposed algorithm and its ability to balance exploration and exploitation, NOA is tested on 60 unconstrained benchmark functions and four classic constrained real-world engineering problems. The experimental results are verified by a comparative study with over 20 well-known and recently developed optimization algorithms. These results show that the proposed algorithm outperforms all other algorithms in terms of solution accuracy, convergence curve speed, statistical measurements, and computational expenses.

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Author notes

  1. Saif Eddine Khouni and Tidjani Menacer have contributed equally to this work.

Authors and Affiliations

  1. Laboratory of Applied Mathematics, Mohamed Khider University, P.O. Box 145 RP, 07000, Biskra, Algeria

    Saif Eddine Khouni & Tidjani Menacer

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  1. Saif Eddine Khouni
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  2. Tidjani Menacer
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A Appendix

A Appendix

1.1 A.1 Mathematical expression of benchmark functions

Tables 13 and 14.

Table 13 Mathematical expression of unimodal benchmark functions
Full size table
Table 14 Mathematical expression of multimodal benchmark functions
Full size table

1.2 A.2 Mathematical model WBD

Minimize \(f(x)=\,1.10471x_{1}^{2}x_{2}+0.04811x_{3}x_{4}(14+x_{2})\).

Subject to:

$$\begin{aligned} g_{1}(x)\,=\,& {} \tau (x)-\tau _{\max }\underline{<}0. \\ g_{2}(x)\,=\, & {} \sigma (x)-\sigma _{\max }\underline{<}0. \\ g_{3}(x)\,=\, & {} x_{1}-x_{4}\underline{<}0. \\ g_{4}(x)\,=\, & {} 0.10471x_{1}^{2}+0.04811x_{3}x_{4}(14+x_{2})-5\underline{<}0. \\ g_{5}(x)\,=\, & {} 0.125-x_{1}\underline{<}0. \\ g_{6}(x)\,=\, & {} \delta (x)-\delta _{\max }\underline{<}0. \\ g_{7}(x)\,=\, & {} P-P_{c}(x)\underline{<}0. \\ \end{aligned}$$

Where:

$$\begin{gathered} \tau (x) = \sqrt {(\tau ^{{\prime 2}} + 2\tau ^{\prime } \tau ^{{\prime \prime }} \frac{{x_{2} }}{{2R}} + (\tau ^{{\prime \prime }} )^{2} } ,\quad \tau ^{\prime } = \frac{P}{{\sqrt 2 x_{1} x_{2} }},\quad \tau ^{{\prime \prime }} = \frac{{MR}}{J} \hfill \\ J = 2\left( {\sqrt 2 x_{1} x_{2} \left( {\frac{{x_{2}^{2} }}{{12}} + \left( {\frac{{x_{1} + x_{3} }}{2}} \right)^{2} } \right)} \right),\quad R = \sqrt {\left( {\frac{{x_{2} }}{2}} \right)^{2} + \left( {\frac{{x_{1} + x_{3} }}{2}} \right)^{2} } ,\quad M = P\left( {L + \frac{{x_{2} }}{2}} \right), \hfill \\ P_{c} (x) = \frac{{4.013E\sqrt {\frac{{x_{3}^{2} x_{4}^{6} }}{{36}}} }}{{L^{2} }}\left( {1 - \frac{{x_{3} }}{{2L}}\sqrt {\frac{E}{{4G}}} } \right),\quad P = 6000lb,\;L = 14{\text{in}},\quad E = 30 \times 10^{6} {\text{psi}}, \hfill \\ G = 12 \times 10^{6} {\text{psi}},\delta _{{\max }} = 0.25{\text{in}},\quad \tau _{{\max }} = 13,600{\text{psi}},\quad \sigma _{{\max }} = 3 \times 10^{4} {\text{psi}}. \hfill \\ \end{gathered}$$

And

$$\begin{aligned}{} & {} 0.1\underline{<}x_{i}\underline{<}2,{ }i=1,4. \\{} & {} 0.1\underline{<}x_{i}\underline{<}10,{ }i=2,3. \end{aligned}$$

1.3 A.3 Mathematical model SRD

Minimize \(f(x)=0.7854x_{1}x_{2}^{2}(3.3333x_{3}^{2}+14.9334x_{3}-43.0934)-1.508x_{1}(x_{6}^{2}+x_{7}^{2})+7.4777(x_{6}^{3}+x_{7}^{3})+0.7854(x_{4}x_{6}^{2}+x_{5}x_{7}^{2}).\)

Subject to:

$$\begin{aligned} g_{1}(x)\,= \, & {} \frac{27}{x_{1}x_{2^{\chi }3}^{2}}-1\leqslant 0. \\ g_{2}(x)\,= \,& {} \frac{397.5}{x_{1}x_{2}^{2}x_{3}}-1\leqslant 0. \\ g_{3}(x)=\, & {} \frac{1.93x_{4}^{3}}{x_{2}x_{6}^{4}x_{3}}-1\le 0. \\ g_{4}(x)\,= \, & {} \frac{1.93x_{5}^{3}}{x_{2}x_{7}^{4}x_{3}}-1\le 0. \\ g_{5}(x)\,= \, & {} \frac{\sqrt{\left( \left( \frac{745x_{4}}{x_{2}x_{3}}\right) ^{2}+16.9\times 10^{6}\right) }}{110x_{6}^{3}}-1\le 0. \\ g_{6}(x)\,= \, & {} \frac{\sqrt{\left( \left( \frac{745x_{5}}{x_{2}x_{3}}\right) ^{2}+157.5\times 10^{6}\right) }}{85x_{7}^{3}}-1\le 0. \\ g_{7}(x)= & {} \frac{x_{2}x_{3}}{40}-1\le 0. \\ g_{8}(x)\,= \, & {} \frac{5x_{2}}{x_{1}}-1\le 0. \\ g_{9}(x)\,= \, & {} \frac{x_{1}}{12x_{2}}-1\le 0. \\ g_{10}(x)\,= \, & {} -\frac{1.5x_{6}+1.9}{x_{4}}-1\leqslant 0. \\ g_{11}(x)\,= \, & {} \frac{1.1x_{7}+1.9}{x_{5}}-1\leqslant 0. \end{aligned}$$

Where:

$$\begin{aligned}{} & {} 2.6\le x_{1}\le 3.6, \quad 0.7\le x_{2}\le 0.8, \quad 17\le x_{3}\le 28, \quad 7.3\le x_{4}\le 8.3, \\{} & {} 7.3\le x_{5}\le 8.3, \quad 2.9\le x_{6}\le 3.9, \quad 5.0\le x_{7}\le 5.5. \end{aligned}$$

1.4 A.4 Mathematical model PVD

Minimize \(f(x)=0.6224x_{1}x_{3}x_{4}+1.7781x_{2}x_{3}^{2}+3.1661x_{1}^{2}x_{4}+19.84x_{1}^{2}x_{3}\).

Subject to:

$$\begin{aligned} g_{1}(x)= & {} -x_{1}+0.0193x_{3}\underline{<}0. \\ g_{2}(x)= & {} -x_{2}+0.00954x_{3}\underline{<}0. \\ g_{3}(x)= & {} -\pi x_{3}^{2}x_{4}-\frac{4}{3}\pi x_{3}^{3}+1296,000\underline{<}0. \\ g_{4}(x)= \,& {} x_{4}-240\underline{<}0. \end{aligned}$$

Where:

$$\begin{aligned}{} & {} 0\underline{<}x_{i}\underline{<}100,{ }i=1,2. \\{} & {} 10\underline{<}x_{i}\underline{<}200,{ }i=3,4. \end{aligned}$$

1.5 A.5 Mathematical model TCSD

Minimize \(f(x)=(x_{3}+2)x_{2}x_{1}^{2}.\)

Subject to:

$$\begin{aligned} g_{1}(x)\,=\, & {} 1-\frac{x_{2}^{3}x_{3}}{71,785x_{1}^{4}}\underline{<}0. \\ g_{2}(x)\,=\, & {} 4x_{2}^{2}-\frac{x_{1^{\chi }2}}{12.566(x_{2}x_{1}^{3}-x_{1}^{4})}+ \frac{1}{5108x_{1}^{2}}-1\underline{<}0. \\ g_{3}(x)\,=\, & {} 1-\frac{140.45x_{1}}{x_{2}^{2}x_{3}}\underline{<}0. \\ g_{4}(x)\,=\, & {} x_{2}+\frac{x_{1}}{1.5}-1\underline{<}0. \end{aligned}$$

Where:

$$\begin{aligned}{} & {} 0.05\underline{<}x_{1}\underline{<}2. \\{} & {} 0.25\underline{<}x_{2}\underline{<}1.30. \\{} & {} 2.00\underline{<}x_{3}\underline{<}15.00. \end{aligned}$$

1.6 A.6 Optimal solutions of benchmark engineering problems

Table 15 Comparison of the optimal solutions obtained by NOA and the compared algorithms on WBD problem
Full size table
Table 16 Comparison of the optimal solutions obtained by NOA and the compared algorithms on SRD problem
Full size table
Table 17 Comparison of the optimal solutions obtained by NOA and the compared algorithms on PVD problem
Full size table
Table 18 Comparison of the optimal solutions obtained by NOA and the compared algorithms on TCSD problem
Full size table

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Khouni, S.E., Menacer, T. Nizar optimization algorithm: a novel metaheuristic algorithm for global optimization and engineering applications. J Supercomput 80, 3229–3281 (2024). https://doi.org/10.1007/s11227-023-05579-4

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