QUESTION $1$

A manufacturer produces an item with a production capacity of $946$ units per $7-$hour shift. The annual demand of the item is $44220$ units. Preparation costs are $$55$ each time an order is

placed; the carrying cost rate is $$0\mathrm{.}2$ per dollar of inventory per year and the units cost $$3$ each. Assuming that the plant works one shift a day, $4$ days a week and $51$ weeks a year, what is the

optimal production order quantity? Write your answer to $1$ decimal place.