Strong Duality Theory

$$ Optimal solution to $($P$)$: $(12,0,0),($s$1=4,$s$2=0)$

$$ Using the complementarity conditions

$$ y$1(8$x$1+6$x$2+4$x$3-100)=0,$ y$1=0$;

$$ x$1(8$y$1+5$y$2-80)=0,$ y$2=16.$

$$ The optimal solution to $($D$)$ is y$1=0,$ y$2=16.$