Table 2 Multi-objective test functions.

\({F}_{8}={\sum }_{j=1}^{m}-{y}_{j}sin\left(\sqrt{\left|{y}_{j}\right|}\right)\)

60

[− 100,100]

228.289 × m

\({\text{F}}_{9}={\sum }_{j=1}^{m}\left[{y}_{j}^{3}-15cos\left(3\pi {y}_{j}\right)+15\right]\)

60

[− 4.12,5.12]

0.012

\(\begin{aligned} {\text{F}}_{12} = & \frac{\pi }{m}\left\{ {15sin\left( {\pi x_{2} } \right) + \mathop \sum \limits_{i = 1}^{m - 2} \left( {x_{i} - 1} \right)^{3} \left[ {1 + 15sin^{3} \left( {\pi x_{i + 1} } \right) + \left( {x_{m} - 1} \right)^{3} } \right]} \right\} \\ & + \mathop \sum \limits_{j = 1}^{m} u\left( {y_{i} ,15,50,5} \right) \\ \end{aligned}\)

60

[− 30,30]

0.012

\({\text{F}}_{13}=0.2\left\{{sin}^{3}\left(4\pi {y}_{1}\right)+{\sum }_{i=1}^{m}{\left({y}_{i}-1\right)}^{3}\left[2+{sin}^{3}\left(4\pi {y}_{i}+1\right)\right]+{\left({y}_{m}-1\right)}^{3}\left[1+{sin}^{4}\left(3\pi {y}_{m}\right)\right]\right\}\)

60

[− 40,40]

0.012