A fraction of all health care claims requires special processing. On any given day at a particular third$-$party processing company, $500$ claims need to be analyzed and processed. There are two types of employees that handle the claims processing; permanent employees that can each process $25$ claims a day, and on average make only $2$ mistakes per day and temporary employees that can process $20$ claims per day but make on average $4$ errors. Each permanent employee costs the processing company $$80/$day while a temporary employee only costs $$50.$

The company has set a quality goal of no more than $50$ errors day. The company has $40$ workstations available; each employee needs one.

$($a$)$ What is the optimal mix of claims processors needed at the firm?

In order to meet the new quality goal, the processing company can hire specially trained checkers, that each catch $10$ errors a day, for the cost of

$$45/$day$.($and will require a workstation each$)$

$($b$)$ Should the processing company hire any checkers? If so$,$ how many?