Table 2.

Benchmark functions for multiobjective problem optimization.

Problem	Definition
Zitzler–Deb–Thiele 1	Minimize f1(x)=x1
	Minimize f2(x)=g(x) xh(f1(x), g(x))
	where g(x)=1+((9/(N − 1))∑i=2Nxi)
	$h\left(f_1\left(x\right),g\left(x\right)\right)=1-\sqrt{\left(\left(f_1\left(x\right)\right)/\left(g\left(x\right)\right)\right)},0\le \hspace{0.17em}\hspace{0.17em}{x}_i\le \mathrm{1,1}\le i\le 30$
Zitzler–Deb–Thiele 2	Minimize f1(x)=x1
	Minimize f2(x)=g(x) xh(f1(x), g(x))
	where g(x)=1+((9/(N − 1))∑i=2Nxi)
	h(f1(x), g(x))=1 − ((f1(x))/(g(x)))2, 0 ≤ xi  ≤ 1,1 ≤ i ≤ 30
Zitzler–Deb–Thiele 1 (with linear pareto front)	Minimize f1(x)=x1
	Minimize f2(x)=g(x) xh(f1(x), g(x))
	where g(x)=1+(9/(N − 1))∑i=2Nxi
	h(f1(x), g(x))=1 − ((f1(x))/(g(x))), 0 ≤ xi   ≤ 1,1 ≤ i ≤ 30
Zitzler–Deb–Thiele 3	Minimize f1(x)=x1
	Minimize f2(x)=g(x)  xh(f1(x), g(x))
	where g(x)=1+(9/29)∑i=2Nxi
	$h\left(f_1\left(x\right),g\left(x\right)\right)=1-\sqrt{\left(f_1\left(x\right)\right)/\left(g\left(x\right)\right)}-\left(\left(f_1\left(x\right)\right)/\left(g\left(x\right)\right)\right)\sin \left(10\pi f_1\left(x\right)\right),0\le x_i\le \mathrm{1,1}\le i\le 30$
Zitzler–Deb–Thiele 4	Minimize f1(x)=x1
	Minimize f2(x)=g(x) xh(f1(x), g(x))
	where g(x)=1+10(N − 1)+∑i=2N(xi2 − 10  sin(4πxi))
	$h\left(f_1\left(x\right),g\left(x\right)\right)=1-\sqrt{\left(f_1\left(x\right)\right)/\left(g\left(x\right)\right)},0\le x_i\le \mathrm{1,1}\le i\le 30$
Zitzler–Deb–Thiele 6	Minimize f1(x)=1 − exp(−4 ∗ x1) ∗ sin (6πx1)6
	Minimize f2(x)=g(x) xh(f1(x), g(x))
	where g(x)=1+  9((∑i=2Nxi)/(N − 1)0.25)
	h(f1(x), g(x))=1 − ((f1(x))/(g(x)))2, 0 ≤ xi   ≤ 1,1 ≤ i ≤ 30