A warehouse manager must set up inventory ordering systems for two new distribution items, X and Y$.$ Item X can be ordered at any time, but item Y can only be ordered once every three weeks. The company operates $50$ weeks a year and the weekly demand for both items is normally distributed. The manager has gathered the following information about the items:

Item X

Item Y

Average weekly demand

$150$ Units

$100$ Units

Standard deviation

$30$ units per week

$15$ units per week

Unit cost

$$200$

$$120$

Annual holding cost

$30\%$

$30\%$

Ordering cost

$$250$

$$250$

Lead time

$5$ weeks

$4$ weeks

Acceptable stockout risk

$2\%$

$5\%$

a$.$ When should the manager reorder each item?

b$.$ Compute the order quantity for Item X$.$

c$.$ Compute the order quantity for Item Y if $500$ units are on hand at the time the order is placed.

d$.$ Compute the optimal order interval for Item Y $($i$.$e$.,$ the current three$-$week order interval may not be optimal$).$

e$.$ Compute the savings in average inventory carrying costs if Item Y can be ordered at any time $($rather than once every three weeks$).$