Table 3.

Multi-objective optimization test problems.

Test Problem	Mathematical Formulation
BNH	$\begin{array}{ll}\underset{x}{min}\hfill & U=\{\begin{array}{l}{f}_1=4x_1^2+4x_2^2\hfill \\ f_2={(x_1-5)}^2+{(x_2-5)}^2\hfill \end{array}\hfill \\ s.t.\hfill & {(x_1-5)}^2+x_2^2\le 25\hfill \\ \hfill & {(x_1-8)}^2+{(x_2+3)}^2\ge 7.7\hfill \\ \hfill & x_1\in [0,5]\hfill \\ \hfill & x_2\in [0,3]\hfill \end{array}$
CONSTR	$\begin{array}{ll}\underset{x}{min}\hfill & U=\{\begin{array}{l}{f}_1=x_1\hfill \\ f_2=\frac{(1+x_2)}{x_1}\hfill \end{array}\hfill \\ s.t.\hfill & 9x_1+x_2-6\ge 0\hfill \\ \hfill & 9x_1-x_2-1\ge 0\hfill \\ \hfill & x_1\in [0.1,1]\hfill \\ \hfill & x_2\in [0,5]\hfill \end{array}$
Car-side Impact	$\begin{array}{ll}\underset{x}{min}\hfill & U=\{\begin{array}{l}{f}_1=1.98+4.9x_1+6.67x_2+6.98x_3+4.01x_4+1.78x_5+0.00001x_6+2.73x_7\hfill \\ f_2=F\hfill \\ f_3=0.5(V_{\mathit{MBP}}+V_{FD})\hfill \end{array}\hfill \\ s.t.\hfill & 1.16-0.3717x_2{x}_4-0.0092928x_3\le 1\hfill \\ \hfill & 0.261-0.0159x_1{x}_2-0.06486x_1-0.019x_2{x}_7+0.0144x_3{x}_5+0.0154464x_6\le 0.32\hfill \\ \hfill & 0.214+0.00817x_5-0.0587x_1+0.03099x_2{x}_6-0.018x_2{x}_7+0.0304x_3-0.00364x_5{x}_6-0.018x_2^2\le 0.32\hfill \\ \hfill & 0.74-0.61x_2-0.031296x_3-0.031872x_7+0.227x_2^2\le 0.32\hfill \\ \hfill & 28.98+3.818x_3-4.2x_1{x}_2+1.27296x_6-2.68065x_7\le 32\hfill \\ \hfill & 33.86+2.95x_3-5.057x_1{x}_2-3.795x_2-3.4431x_7+1.45728\le 32\hfill \\ \hfill & 46.36-9.9x_2-4.4505x_1-32\hfill \\ \hfill & F=4.72-0.5x_4-0.19x_2{x}_3\le 4\hfill \\ \hfill & V_{\mathit{MBP}}=10.58-0.674x_1{x}_2-0.67275x_2\le 9.9\hfill \\ \hfill & V_{FD}=16.45-0.489x_3{x}_7-0.843x_5{x}_6\le 15.7\hfill \\ \hfill & x_1\in [0.5,1.5]\hfill \\ \hfill & x_2\in [0.45,1.35]\hfill \\ \hfill & x_3\in [0.5,1.5]\hfill \\ \hfill & x_4\in [0.5,1.5]\hfill \\ \hfill & x_5\in [0.875,2.625]\hfill \\ \hfill & x_6\in [0.4,1.2]\hfill \\ \hfill & x_7\in [0.4,1.2]\hfill \end{array}$