A local supermarket sells a popular brand of shampee at a fairly steady rate of $380bo\frac{t}{c}g$ Per month. The cost of each bottle to the supermarket is $45$ cents, and the cost of $p{l}_{aci{n}_h}$ an order has been estimated at $$8\mathrm{.}5.$ Assume that holding costs are based on a $25$ pereent annual interest rate. Stock$-$outs of the shampoe are not allowed.

a$)$ Determine the optimal lot size the supermarket should order and the time between placements of onders for this product. $[2]$

b$)$ If the lead time is three months, find the reorder point based on the on$-$hand inventory.

$[1]$