$($a$)$ A certain chain of sports stores is considering to change its current inventory control

system to manage its inventory of bicycle and sports helmets. The information regarding

the helmets are as follows:

Average demand: $100$ units$/$week

Lead time $=5$ weeks

Order cost $=$ $$65/$order

Unit cost $=$ $$90$

Carrying rate $=10\%$

Standard deviation of weekly demand $=30$

Desired service level $=95\%($z$-$value$=1\mathrm{.}645)$

Number of weeks per year $=52$

Inventory position $($IP$)=300$ units

The inventory manager is planning to use a fixed$-$period inventory system model. Calculate

the optimal time interval $($T$)$ and replenishment level $($M$)$ for a fixed$-$period inventory

model with and without safety stock.

Comment briefly on your findings and how this system operates.