Table 1 The ZDT functions of multi-objective optimization⁷⁹.

ZDT1

\(f_1=x_1\)

\(g=1+9\cdot \sum _{i=2}^m x_i/\left( m-1\right)\)

\(h=1-\sqrt{f_1/g}\)

\(m=30,\quad x_i \in \left[ 0,1\right]\)

\(0\le x_1\le 1\)

\(and \ x_i=0 \ for \ i=2,\dots ,n\)

ZDT2

\(f_1=x_1\)

\(g=1+9\cdot \sum _{i=2}^m x_i/\left( m-1\right)\)

\(h=1-\left( f_1/g\right) ^2\)

\(m=30,\quad x_i \in \left[ 0,1\right]\)

\(0\le x_1\le 1\)

\(and \ x_i=0 \ for \ i=2,\dots ,n\)

ZDT3

\(f_1=x_1\)

\(g=1+9\cdot \sum _{i=2}^m x_i/\left( m-1\right)\)

\(h=1-\sqrt{f_1/g}-\left( f_1/g\right) \sin \left( 10\pi f_1 \right)\)

\(m=30,\quad x_i \in \left[ 0,1\right]\)

\(0\le x_1\le 0.0830\)

\(or\ 0.1822\le x_1\le 0.2577\)

\(or\ 0.4093\le x_1\le 0.4538\)

\(or\ 0.6183 \le x_1\le 0.6525\)

\(or\ 0.8233\le x_1\le 0.8518\)

\(and\ x_i=0 \ for \ i=2,\dots ,n\)

ZDT4

\(f_1=x_1\)

\(g=1+10\left( m-1\right) +\sum _{i=2}^m\left( x_i^2-10 \cos \left( 4\pi x_i \right) \right)\)

\(h=1-\sqrt{f_1/g}-\left( f_1/g\right) \sin \left( 10\pi f_1 \right)\)

\(m=10,\quad x_1\in \left[ 0,1\right] \ and \ x_2,\dots ,x_m\epsilon \left[ -5,5\right]\)

\(0\le x_1\le 1\)

\(and \ x_i=0\ for\ i=2,\dots ,n\)