A company produces two products, A and B$.$ The unit revenues are $$2$ and $$3$

respectively. Two raw materials, M$1$ and M$2,$ 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 M$1$ and $3$ units of M$2,$ and $1$ unit of B uses $2$ units of M$1$ and $6$ units of

M$2.$

A$)$ Determine the dual prices of M$1$ and M$2$ and their feasibility ranges.

B$)$ Suppose that $4$ additional units of M$1$ 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 M$2?$

D$)$ If M$2$ availability is increased by $5$ units, determine the associated optimal

revenue.