Consider the following LP:

MAXZ $=151+102$

subject $to$

$41+62360$

$31+22180$

$x1+52200$

and $x1,x20$

$($a$)$ Convert the LP given above to the standard form. Determine all the basic feasible solutions $($bfs$)$ of the problem. Give the values of both basic and non$-$basic variables in each bfs$.$

$($b$)$ Identify the adjacent basic feasible solutions of each extreme point of the feasible region.

$($c$)$ Using the graphical solution technique, solve the problem.

$($d$)$ Which constraint$($s$)$ is$($are$)$ active $($binding$)$ in the optimal solution? Which constraint$($s$)$ is$($are$)$ redundant?

$($e$)$ Using the big$-$M simplex method, find optimal solution.

$($f$)$ Using the method of transformation technique, write the dual of the problem.

$($g$)$ Write directly the dual of the original problem, without using the method of transformation.