$3\mathrm{.}18($EOQ with Batch Demands$)$ Consider an inventory system in which each order is for $Q$ units. Instead of the demand occurring continuously over time $($as in the EOQ model$),$ the customer purchases exactly half of the inventory exactly halfway through the order cycle and the remaining half exactly at the end of the order cycle. At that point, a new order is placed, and it arrives instantly. $($Therefore$,$ there is no time at which the inventory level equals $0.)$ The total demand per year is $,$ just as in the EOQ model, which means that each order cycle has the same length as in the EOQ model.

a$.$ Write an expression for the average annual total cost.

b$.$ What is the optimal order quantity, $Q^{**}?$