Table 2 Benchmark functions used to find the effectiveness of MGOA, GOA and PSO algorithms.

Booth (F1)

\(f_{Booth} (x) = \left( {x_{1} + 2x_{2} - 7} \right)^{2} + \left( {2x_{1} + x_{2} - 5} \right)^{2}\)

2

[− 10, 10]

Branin (F2)

\(f_{Branin} (x) = a(x_{2} - bx_{1}^{2} + cx_{1} - r)^{2} + s.(1 - t).\cos (x_{1} ) + s\)

2

[− 5, 0], [0–15]

Griewank (F3)

\(f_{Griewank} (x) = \sum\limits_{i = 1}^{d} {\frac{{x_{i}^{2} }}{4000}} - \mathop \prod \limits_{i = 1}^{d} \cos \left( {\frac{{x_{i} }}{\sqrt i }} \right) + 1\)

2

[− 600, 600]

Matyas (F4)

\(f_{Matyas} (x) = 0.26.\left( {x_{1}^{2} + x_{2}^{2} } \right) - 0.48x_{1} .x_{2}\)

2

[− 10, 10]

Powell (F5)

\(f_{Powell} (x) = \left[ \begin{gathered} \left( {x_{4i - 3} + 10.x_{4i - 2} } \right)^{2} + 5.\left( {x_{4i - 1} - x_{4i} } \right)^{2} \hfill \\ + \left( {x_{4i - 2} - 2.x_{4i - 1} } \right)^{4} + 10.\left( {x_{4i - 3} - x_{4i} } \right)^{4} \hfill \\ \end{gathered} \right]\)

10

[− 4, 5]

Rotated hyper-ellipsoid (F6)

\(f_{Rothyp} (x) = \sum\limits_{i = 1}^{d} {\sum\limits_{j = 1}^{i} {x_{j}^{2} } }\)

30

[− 65.536, 65.536]