Question $1(40$ points$)$ Consider the following linear programming problem.

maximize $Z=5*x_1+7*x_2$

Subject $to$:

Time Limit: $2*x_1+4*x_{2}28$

Labour Limit: $y_2{x}_1+2*x_{2}20$

Input $1$ Limit: $y_3{x}_1=10$

Input $2$ Limit: $y_4{x}_{2}22$

Input $3$ Limit: $y_{5}2*x_1+x_{2}10$

Capacity Limit: $y_6{x}_1+x_{2}40$

Non$-$Negativity: $x_{1}0,x_{2}0$

a$)$ Construct the complete dual problem.

b$)$ If you had to choose, conceptually, which problem would lead to the lowest computational effort,

the primal or the dual? Explain your answer.

c$)$ If the primal problem has an optimal basic feasible solution, what can be said about the solution

of the dual problem?