A company has two machines for manufacturing a product, Machine $1$ makes $2$ units per hour, while machine $2$ makes $3$ units per hour. The company has an order of $80$ units. Energy restrictions dictate that only one machine can operate at one time. The company has $40$ hours of regular machine time, but overtime is available. It costs $$4$ to run machine for $1$ hour, while machine $2$ costs $$5$ per hour. As for the goals:

$1.$ Meet the demand of $80$ units.

$2.$ Limit machine overtime to $10$ hours.

$3.$ Use the $40$ hours of normal machining time.

$4.$ Minimize costs.