points, $40$ minutes$)$ Consider the following resource allocation problem and the accompanying optimal tableau $(x_4,x_5,$ and $x_6$ are slack variables$).$

maximize $z=4x_1+6x_2+5x_3$

$3x_1+x_2+2x_{3}64(Resource1)$

$x_1+x_2+x_{3}20(Resource2)$

$x_1+2x_2+3x_{3}30(Resource3)$

$x_1,x_2,x_{3}0$

$($Profit $$)$

subject to

$($Resource $1)$

$($Resource $3)$

a$)$ Determine the range on the unit profit of Product. $1(c_1)$ for which the current basis remains optimal.

b$)$ What would be the new optimal solution if $c_1$ was $8?$ Comment on the new solution.

c$)$ A new product is proposed. It is estimated that each unit of this new product consumes $4$ units of Resource $1,2$ units of Resource $2,$ and $1$ unit of Resource $3.$ What

should be the unit profit of this new product for it to be profitable to manufacture?

d$)$ Suppose that the production requirements for Product $3$ are modified such that each unit of product $3$ now requires $2$ units of resource $1,1$ unit of Resource $2,$ and

$1$ unit of resource $3.$ Find the new optimal solution.