$1.$ Claims Processing Specialists $($CPS$)$ processes insurance claims for health care providers. CPS has capacity for up to $45$ employees. They pay $$400$ per day for permanent claims processors and are considering the possibility of also using temporary help for $$240$ per day. Past history shows that an experienced, permanent processor can complete $15$ claims per day, but a temporary employee typically completes only $10$ per day. A permanent operator averages $0\mathrm{.}5$ claims with errors each day, whereas a temporary operator averages about $1\mathrm{.}4$ defective claims per day. The company wants to process at least $500$ claims each day and wants to limit claims with errors to less than $5\%(25$ per day$).$ How many permanent and temporary processors does CPS need on staff each day if they want to minimize costs?

Please note that fractional answers can be interpreted as hiring part time employees.

a$.$ Formulate a linear programming model for this problem. Provide a complete algebraic formulation including a careful definition of decision variables and write a sentence stating the solution in the context of the problem.