$*3-69.$ Consider the TOYCO model.

$($a$)$ Suppose that any additional time for operation $1$ beyond its current capacity of $430$ mins per day must be done on an overtime basis at $$50$ an hour. The hourly cost includes both labor and the operation of the machine. Is it economically advantageous to use overtime with operation $1?$

$($b$)$ Suppose that the operator of operation $2$ has agreed to work $2$ hrs of overtime daily at $$45$ an hour. Additionally, the cost of the operation itself is $$10$ an hour. What is the net effect of this activity on the daily revenue?

$($c$)$ Is overtime needed for operation $3?$

$($d$)$ Suppose that the daily availability of operation $1$ is increased to $440$mins. Any overtime used beyond the current maximum capacity will cost $$40$ an hour. Determine the new optimum solution, including the associated net revenue.

$($e$)$ Suppose that the availability of operation $2$ is decreased by $15$mins a day and that the hourly cost of the operation during regular time is $$30.$ Is it advantageous to decrease the availability of operation $2?$