Consider the following LP problem :

Maxz$=3x_1+2x_2+5x_3$

subject to the constraints :

$x_1+2x_2+x_3{a}_1$

$3x_1+2x_3{a}_2$

$x_1+4x_2{a}_3$

where $a_1,a_2,a_3$ are constants $.$ For specific

values of $a_1,a_2,a_3$ the optimal solution is :

where $b_i\text{'}$s and $c_i\text{'}$s are constants $.$

Determine :

$($i$)$ The values of $a_1,a_2$ and $a_3$ that yield the

given optimal solution.

$($ii$)$ The values of $b_1,b_2$ and $b_3,8c_1,c_2,c_3$ in

the optimal tableau.

$($iii$)$The optimal dual solution.