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Bat algorithm for constrained optimization tasks

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Abstract

In this study, we use a new metaheuristic optimization algorithm, called bat algorithm (BA), to solve constraint optimization tasks. BA is verified using several classical benchmark constraint problems. For further validation, BA is applied to three benchmark constraint engineering problems reported in the specialized literature. The performance of the bat algorithm is compared with various existing algorithms. The optimal solutions obtained by BA are found to be better than the best solutions provided by the existing methods. Finally, the unique search features used in BA are analyzed, and their implications for future research are discussed in detail.

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Acknowledgments

The authors gratefully acknowledge the work and help of Engineer Parvin Arjmandi (The University of Akron).

Author information

Authors and Affiliations

  1. Young Researchers Club, Central Tehran Branch, Islamic Azad University, Tehran, Iran

    Amir Hossein Gandomi

  2. Mathematics and Scientific Computing, National Physical Laboratory, Teddington, TW11 0LW, UK

    Xin-She Yang

  3. Young Researchers Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran

    Amir Hossein Alavi

  4. Marand Faculty of Engineering, University of Tabriz, Tabriz, Iran

    Siamak Talatahari

Authors
  1. Amir Hossein Gandomi
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  2. Xin-She Yang
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  3. Amir Hossein Alavi
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  4. Siamak Talatahari
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Corresponding author

Correspondence to Amir Hossein Gandomi.

Appendix

Appendix

The 13 benchmarked problems

1.1 Problem G 1

$$ \begin{gathered} \mathop {\min }\limits_{x} f(x) = 5\sum\limits_{i = 1}^{4} {x_{i} - 5} \sum\limits_{i = 1}^{4} {x_{i}^{2} } - \sum\limits_{i = 5}^{13} {x_{i} } \hfill \\ s.t.\quad g_{1} (x) = 2x_{1} + 2x_{2} + x_{10} + x_{11} - 10 \le 0, \hfill \\ \quad \quad g_{2} (x) = 2x_{1} + 2x_{3} + x_{10} + x_{12} - 10 \le 0, \hfill \\ \quad \quad g_{3} (x) = 2x_{2} + 2x_{3} + x_{11} + x_{12} - 10 \le 0, \hfill \\ \quad \quad g_{4} (x) = - 8x_{1} + x_{10} \le 0, \hfill \\ \quad \quad g_{5} (x) = - 8x_{2} + x_{11} \le 0, \hfill \\ \quad \quad g_{6} (x) = - 8x_{3} + x_{12} \le 0, \hfill \\ \quad \quad g_{7} (x) = - 2x_{4} - x_{5} + x_{10} \le 0, \hfill \\ \quad \quad g_{8} (x) = - 2x_{6} - x_{7} + x_{11} \le 0, \hfill \\ \quad \quad g_{9} (x) = - 2x_{8} - x_{9} + x_{12} \le 0, \hfill \\ \quad \quad x_{i} \ge 0\quad i = 1, \ldots ,13, \hfill \\ \quad \quad x_{i} \le 1\quad i = 1, \ldots ,9,13. \hfill \\ \end{gathered} $$

The bound constraints:

$$ U = \left( {1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,100,\,100,\,100,\,1} \right)\,{\text{and}}\,L = \left( {0,\, \ldots ,\,0} \right). $$

Global minimum:

$$ x^{ * } = \left( {1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,1,\,3,\,3,\,3,\,1} \right),\,f(x^{ * } ) = - 15. $$

1.2 Problem G 2

$$ \begin{gathered} \mathop {\max }\limits_{x} f(x) = \left| {\frac{{\sum\nolimits_{i = 1}^{n} {\cos^{4} \left( {x_{i} } \right) - 2\prod\nolimits_{i = 1}^{n} {\cos^{2} \left( {x_{i} } \right)} } }}{{\sqrt {\sum\nolimits_{i = 1}^{n} {ix_{i}^{2} } } }}} \right| \hfill \\ s.t.\,\quad g_{1} (x) = - \prod\limits_{i = 1}^{n} {x_{i} } + 0.75 \le 0, \hfill \\ \quad \quad g_{2} (x) = \sum\limits_{i = 1}^{n} {x_{i} } - 7.5n \le 0. \hfill \\ \end{gathered} $$

The bound constraints:

$$ U = \left( {10,\, \ldots ,\,0} \right)\,{\text{and}}\,L = \left( {0,\, \ldots ,\,0} \right). $$

Best-known value:

$$ f(x^{ * } ) = 0.803619,\,{\text{for}}\,n = 20. $$

1.3 Problem G 3

$$ \begin{gathered} \mathop {\max }\limits_{x} f(x) = \left( {\sqrt n } \right)^{n} \prod\limits_{i = 1}^{n} {x_{i} } \hfill \\ s.t.\,\quad h_{1} (x) = \sum\limits_{i = 1}^{n} {x_{i}^{2} - 1 = 0} . \hfill \\ \end{gathered} $$

The bound constraints:

$$ U = \left( {1,\, \ldots ,\,1} \right)\,{\text{and}}\,L = \left( {0,\, \ldots ,\,0} \right). $$

Global maximum:

$$ x^{ * } = \left( {\frac{1}{\sqrt n },\; \ldots ,\;\frac{1}{\sqrt n }} \right),\,f(x^{ * } )\; = 1. $$

1.4 Problem G 4

$$ \begin{gathered} \mathop {\min }\limits_{x} f(x) = 5.3578547x_{2}^{3} + 0.8356891x_{1} x_{5} + 37.293239x_{1} - 40792.141 \hfill \\ s.t.\quad g_{1} (x) = u\left( x \right) - 92 \le 0, \hfill \\ \quad \quad g_{2} (x) = - u\left( x \right) \le 0, \hfill \\ \quad \quad g_{3} (x) = v\left( x \right) - 110 \le 0, \hfill \\ \quad \quad g_{4} (x) = - v\left( x \right) + 90 \le 0, \hfill \\ \quad \quad g_{5} (x) = w\left( x \right) - 25 \le 0, \hfill \\ \quad \quad g_{6} (x) = - w\left( x \right) + 20 \le 0, \hfill \\ \end{gathered} $$

where

$$ \begin{gathered} u\left( x \right) = 85.334407 + 0.0056858x_{2} x_{5} + 0.0006262x_{1} x_{4} - 0.0022053x_{3} x_{5} , \hfill \\ v\left( x \right) = 80.51249 + 0.0071317x_{2} x_{5} + 0.0029955x_{1} x_{2} + 0.0021813x_{3}^{2} , \hfill \\ w\left( x \right) = 9.300961 + 0.0047026x_{3} x_{5} + 0.0012547x_{1} x_{3} - 0.0019085x_{3} x_{4} . \hfill \\ \end{gathered} $$

The bound constraints:

$$ U = \left( {102,\,45,\,45,\,45,\,45} \right)\,{\text{and}}\;L = \left( {78,\,33,\,27,\,27,\,27} \right). $$

Global minimum:

$$ x^{ * } = \left( {78,33,29.995256025682,45,36.775812905788,33} \right),\,f(x^{ * } ) = - 30665.539. $$

1.5 Problem G 5

$$ \begin{gathered} \mathop {\min }\limits_{x} f(x) = 3x_{1} + 10^{ - 6} x_{1}^{3} + 2x_{2} + \frac{2}{3} \times 10^{ - 6} x_{2}^{3} \hfill \\ s.t.\quad g_{1} (x) = x_{3} - x_{4} - 0.55 \le 0, \hfill \\ \quad \quad g_{2} (x) = x_{4} - x_{3} - 0.55 \le 0, \hfill \\ \quad \quad h_{1} (x) = 1000\left[ {\sin \left( { - x_{3} - 0.25} \right) + \sin \left( { - x_{4} - 0.25} \right)} \right] + 894.8 - x_{1} = 0, \hfill \\ \quad \quad h_{2} (x) = 1000\left[ {\sin \left( {x_{3} - 0.25} \right) + \sin \left( {x_{3} - x_{4} - 0.25} \right)} \right] + 894.8 - x_{2} = 0, \hfill \\ \quad \quad h_{3} (x) = 1000\left[ {\sin \left( {x_{4} - 0.25} \right) + \sin \left( {x_{4} - x_{3} - 0.25} \right)} \right] + 1294.8 = 0. \hfill \\ \end{gathered} $$

The bound constraints:

$$ U = \left( {1200,\,1200,0.55,\,0.55} \right)\,{\text{and}}\;L = \left( {0,\,0,\, - 0.55,\; - 0.55} \right). $$

Best-known solution:

$$ x^{ * } = \left( {679.9453,\,1026,\,0.118876,\, - 0.3962336} \right),\,f(x^{ * } ) = 5126.4981. $$

1.6 Problem G 6

$$ \begin{gathered} \mathop {\min }\limits_{x} f(x) = \left( {x_{1} - 10} \right)^{3} + \left( {x_{2} - 20} \right)^{3} \hfill \\ s.t.\quad g_{1} (x) = - \left( {x_{1} - 5} \right)^{2} - \left( {x_{2} - 5} \right)^{2} + 100 \le 0, \hfill \\ \quad \quad g_{2} (x) = \left( {x_{1} - 5} \right)^{2} + \left( {x_{2} - 5} \right)^{2} - 828 \le 0, \hfill \\ \end{gathered} $$

The bound constraints:

$$ U = \left( {100,\,100} \right)\,{\text{and}}\;L = \left( {13,\,0} \right). $$

Global minimum:

$$ x^{ * } = \left( {14.095,\,0.84296} \right),\,f(x^{ * } ) = 6961.81388. $$

1.7 Problem G 7

$$ \begin{gathered} \mathop {\min }\limits_{x} f(x) = x_{1}^{2} + x_{2}^{2} + x_{1} x_{2} - 14x_{1} - 16x_{2} + \left( {x_{3} - 10} \right)^{2} + 4\left( {x_{4} - 5} \right)^{2} + \left( {x_{5} - 3} \right)^{2} \hfill \\ \quad \quad \quad \quad \quad + 2\left( {x_{6} - 1} \right)^{2} + 5x_{7}^{2} + 7\left( {x_{8} - 11} \right)^{2} + 2\left( {x_{9} - 10} \right)^{2} + \left( {x_{10} - 7} \right)^{2} + 45 \hfill \\ s.t.\quad g_{1} (x) = 4x_{1} + 5x_{2} - 3x_{7} + 9x_{8} - 105 \le 0, \hfill \\ \quad \quad g_{2} (x) = 10x_{1} - 8x_{2} - 17x_{7} + 2x_{8} \le 0, \hfill \\ \quad \quad g_{3} (x) = - 8x_{1} + 2x_{2} + 5x_{9} - 2x_{10} - 12 \le 0, \hfill \\ \quad \quad g_{4} (x) = 3\left( {x_{1} - 2} \right)^{2} + 4\left( {x_{2} - 3} \right)^{2} + 2x_{2}^{3} - 7x_{4} - 120 \le 0, \hfill \\ \quad \quad g_{5} (x) = 5x_{1}^{2} + 8x_{2} + \left( {x_{3} - 6} \right)^{2} - 2x_{4} - 40 \le 0, \hfill \\ \quad \quad g_{6} (x) = - 0.5\left( {x_{1} - 8} \right)^{2} + 2\left( {x_{2} - 4} \right)^{2} + 3x_{5}^{2} - x_{6} - 30 \le 0, \hfill \\ \quad \quad g_{7} (x) = x_{1}^{2} + 2\left( {x_{2} - 2} \right)^{2} - 2x_{1} x_{2} + 14x_{5} - 6x_{6} \le 0, \hfill \\ \quad \quad g_{8} (x) = - 3x_{1} + 6x_{2} + 12\left( {x_{9} - 8} \right)^{2} - 7x_{10} \le 0. \hfill \\ \end{gathered} $$

The bound constraints:

$$ U = \left( {10,\, \ldots ,\,10} \right)\,{\text{and}}\;L = \left( { - 10,\, \ldots ,\, - 10} \right). $$

Global minimum:

$$ \begin{gathered} x^{ * } = \left( {2.171996,\,2.363683,\,8.773926,\,5.095984,\,0.9906548,\,1.430574,\,1.321644,\,} \right. \hfill \\ 9.828726,\,8.280092,\,8.375927,\,f(x^{ * } ) = 24.3062091. \hfill \\ \end{gathered} $$

1.8 Problem G 8

$$ \begin{gathered} \mathop {\max }\limits_{x} f(x) = \frac{{\sin \left( {2\pi x_{1} } \right)\sin \left( {2\pi x_{2} } \right)}}{{x_{1}^{3} \left( {x_{1} + x_{2} } \right)}} \hfill \\ s.t.\quad g_{1} (x) = x_{1}^{2} - x_{2} + 1 \le 0, \hfill \\ \quad \quad g_{2} (x) = 1 - x_{1} + \left( {x_{2} - 4} \right)^{2} \le 0, \hfill \\ \end{gathered} $$

The bound constraints:

$$ U = \left( {10,\,10} \right)\,{\text{and}}\,L = \left( {0,\,0} \right). $$

Global maximum:

$$ x^{ * } = \left( {1.2279713,\,4.2453733} \right),\,f(x^{ * } ) = 0.095825. $$

1.9 Problem G 9

$$ \begin{gathered} \mathop {\min }\limits_{x} f(x) = \left( {x_{1} - 10} \right)^{2} + 5\left( {x_{2} - 12} \right)^{2} + x_{3}^{4} + 3\left( {x_{4} - 11} \right)^{2} + 10x_{5}^{6} + 7x_{6}^{2} + x_{7}^{4} - 4x_{6} x_{7} \hfill \\ \quad \quad \quad - 10x_{6} - 8x_{7} \quad \quad \hfill \\ s.t.\quad g_{1} (x) = 2x_{1}^{2} + 3x_{2}^{4} + x_{3} + 4x_{4}^{2} + 5x_{5} - 127 \le 0, \hfill \\ \quad \quad g_{2} (x) = 7x_{1} + 3x_{2} + 10x_{3}^{2} + x_{4} - x_{5} - 282 \le 0, \hfill \\ \quad \quad g_{3} (x) = 23x_{1} + x_{2}^{2} + 6x_{6}^{2} - 8x_{7} - 196 \le 0, \hfill \\ \quad \quad g_{4} (x) = 4x_{1}^{2} + x_{2}^{2} - 3x_{1} x_{2} + 2x_{3}^{2} + 5x_{6} - 11x_{7} \le 0, \hfill \\ \end{gathered} $$

The bound constraints:

$$ U = \left( {10,\, \ldots ,\,10} \right)\,{\text{and}}\;L = \left( { - 10,\, \ldots ,\, - 10} \right). $$

Global minimum:

$$ \begin{gathered} x^{ * } = \left( {2.330499,\,1.951372, - 0.4775414,4.365726,\, - 0.6244870,1.038131,\,} \right.\quad \hfill \\ 1.59422\left. 7 \right),\,f(x^{ * } ) = 680.6300573. \hfill \\ \end{gathered} $$

1.10 Problem G 10

$$ \begin{gathered} \mathop {\min }\limits_{x} f(x) = x_{1} + x_{2} + x_{3} \quad \quad \hfill \\ s.t.\quad g_{1} (x) = - 1 + 0.0025\left( {x_{4} + x_{6} } \right) \le 0, \hfill \\ \quad \quad g_{2} (x) = - 1 + 0.0025\left( { - x_{4} + x_{5} + x_{7} } \right) \le 0, \hfill \\ \quad \quad g_{3} (x) = - 1 + 0.01\left( { - x_{5} + x_{8} } \right) \le 0, \hfill \\ \quad \quad g_{4} (x) = 100x_{1} - x_{1} x_{6} + 833.33252x_{4} - 83333.333 \le 0, \hfill \\ \quad \quad g_{5} (x) = x_{2} x_{4} - x_{2} x_{7} - 1250x_{4} + 1250x_{5} \le 0, \hfill \\ \quad \quad g_{6} (x) = x_{3} x_{5} - x_{3} x_{8} - 2500x_{5} + 1250000 \le 0, \hfill \\ \end{gathered} $$

The bound constraints:

$$ \begin{gathered} U = \left( {10000,10000\,,\,10000\,,\,1000\,,\,1000\,,\,1000\,,\,1000\,,\,1000} \right)\,{\text{and}}\, \hfill \\ L = \left( {100,\,1000,\,1000,\,10,\,10,\,10,\,10,\,10} \right). \hfill \\ \end{gathered} $$

Global minimum:

$$ \begin{gathered} x^{ * } = \left( {579.3167,\,1359.943,5110.071,182.0174,\,295.5985,217.9799,286.4162\,} \right.\quad \hfill \\ \left. {395.5979} \right),\,f(x^{ * } ) = 7049.3307. \hfill \\ \end{gathered} $$

1.11 Problem G 11

$$ \begin{gathered} \mathop {\min }\limits_{x} f(x) = x_{1}^{2} + \left( {x_{2} - 1} \right)^{2} \hfill \\ s.t.\quad h_{1} (x) = x_{2} - x_{1}^{2} = 0. \hfill \\ \end{gathered} $$

The bound constraints:

$$ U = \left( {1,\,1} \right)\,{\text{and}}\;L = \left( { - 1, - 1} \right). $$

Global maximum:

$$ x^{ * } = \left( { \pm \frac{1}{\sqrt 2 },\;\frac{1}{2}} \right)\,,\,\,f(x^{ * } )\; = 0.75. $$

1.12 Problem G 12

$$ \begin{gathered} \mathop {\min }\limits_{x} f(x) = 1 - 0.01\left[ {\left( {x_{1} - 5} \right)^{2} + \left( {x_{3} - 5} \right)^{2} } \right]\quad \quad \hfill \\ s.t.\quad g_{i,j,k} (x) = \left( {x_{1} - i} \right)^{2} + \left( {x_{2} - j} \right)^{2} + \left( {x_{3} - k} \right)^{2} - 0.0625 \le 0,\quad i,j,k = 1,\,2,\, \ldots ,\,9\;. \hfill \\ \end{gathered} $$

The bound constraints:

$$ U = \left( {10,10\,,\,10\,} \right)\,{\text{and}}\,L = \left( {0,\,0,\,0} \right). $$

Global minimum:

$$ x^{ * } = \left( {5,\;5,\;5} \right),\,f(x^{ * } ) = 1. $$

1.13 Problem G 13

$$ \begin{gathered} \mathop {\min }\limits_{x} f(x) = e^{{x_{1} x_{2} x_{3} x_{4} x_{5} }} \quad \hfill \\ s.t.\quad h_{1} (x) = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + x_{5}^{2} - 10 = 0, \hfill \\ \quad \quad h_{2} (x) = x_{2} x_{3} - 5x_{4} x_{5} = 0, \hfill \\ \quad \quad h_{3} (x) = x_{1}^{3} + x_{2}^{3} + 1 = 0, \hfill \\ \hfill \\ \end{gathered} $$

The bound constraints:

$$ U = \left( {2.3,\,2.3,\,2.3,\,2.3,\,2.3} \right)\,{\text{and}}\;L = \left( { - 2.3,\, - 2.3\,,\, - 2.3,\, - 2.3,\, - 2.3} \right). $$

Global minimum:

$$ x^{ * } = \left( { - 1.717143,\,1.595709,1.827247, - 0.7636413,\, - 0.763645} \right),\,f(x^{ * } ) = 0.0539498. $$

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Gandomi, A.H., Yang, XS., Alavi, A.H. et al. Bat algorithm for constrained optimization tasks. Neural Comput & Applic 22, 1239–1255 (2013). https://doi.org/10.1007/s00521-012-1028-9

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