Table 1 Constrained test problems used in the evaluation (all objectives are to be minimized)

SCH

1

[- 10³,10³]

f ₁(x)=x ²,

x ∈ [ 0,2]

f ₂(x)=(x−2)²

ZDT1

30

[0, 1]

f ₁(x)=x ₁,

x∈ [ 0,1],

\(f_{2}(x)=g(x)[1-\sqrt {\frac {x_{1}}{g(x)}}],\)

x _i =0,

\(g(x)=1+9\frac {\sum _{i=2}^{n} x_{i}}{(n-1)}\)

i=2,…,n

ZDT2

30

[0, 1]

f ₁(x)=x ₁,

x∈ [ 0,1],

\(f_{2}(x)=g(x)[1-(\frac {x_{1}}{g(x)})^{2}],\)

x _i =0,

\(g(x)=1+9\frac {\sum _{i=2}^{n} x_{i}}{(n-1)}\)

i=2,…,n

ZDT3

30

[0, 1]

f ₁(x)=x ₁,

x∈ [ 0,1],

\(f_{2}(x)=g(x)[1-\sqrt {\frac {x_{1}}{g(x)}}-\frac {x_{1}}{g(x)}\sin {(10\pi x_{1})}],\)

x _i =0,

\(g(x)=1+9\frac {\sum _{i=2}^{n} x_{i}}{(n-1)},\)

i=2,…,n

ZDT6

10

[0, 1]

f ₁(x)=1−exp(−4x ₁)sin6(6π x ₁),

x∈ [ 0,1],

\(f_{2}(x)=g(x)[1-(\frac {f_{1}(x)}{g(x)})^{2}],\)

x _i =0,

\(g(x)=1+9[\frac {\sum _{i=2}^{n} x_{i}}{(n-1)}]^{0.25}\)

i=2,…,n