A company produces two products, A and $B.$ The unit revenues are $$2$ and $$3,$ respectively. Two raw materials, $M1$ and $M2,$ used in the manufacture of the two products have respective daily availabilities of $8$ and $18$ units. One unit of A uses $2$ units of $M1$ and $2$ units of $M2,$ and $1$ unit of $B$ uses $3$ units of $M1$ and $6$ units of $M2.$ Let $x_1$ be the number of product A produced daily, and $x_2$ be the number of product $B$ produced daily. Then, the LP model for this problem can be given as follows:

maxz$=2x_1+3x_2$

$s.t.2x_1+3x_{2}8$

$2x_1+6x_{2}18$

$x_1,x_{2}0$

$($a$)$ Determine the dual prices of $M1$ and $M2$ and their feasibility ranges.

$($b$)$ Suppose that $4$ additional units of $M1$ can be acquired at the cost of $30$ cents per unit. Would you recommend the additional purchase?

$($c$)$ What is the most the company should pay per unit of $M2?$

$($d$)$ If $\frac{M}{2}$ availability is increased by $5$ units, determine the assocaited optimum revenue.