A manager must set up inventory ordering systems for two new items, Gadget and Widget. Gadget is ordered from a supplier and is delivered at once, while Widget is produced in$-$house in batches of fixed size. The company operates $52$ weeks a year and demand for each item is normally distributed. The company uses a continuous inventory system. The manager has gathered the following information about the items.

$$

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Gadget

Widget

Average demand

$7,000$ units per year

$50$ units per week

Standard deviation

$80$ units per week

$15$ units per day

Unit cost $($C$)$

$$28$

$$350$

Production rate $($pt$)$

$-$

$100$ units per day

Ordering cost$/$Setup cost $($Co$)$

$$750$

$$4,650$

Annual carrying cost $($Cc$)$

$18\%$

$25\%$

Lead time $($Lt$)$

$3$ weeks

$1$ week

Service level $($$)$

$99\mathrm{.}5\%$

$92\%$

$$

a$.$ Compute the optimal order quantity for item Gadget.

b$.$ How many orders of item Gadget will be placed per year?

c$.$ What should the safety stock be for item Gadget?

d$.$ Compute the optimal order quantity for Widget.

e$.$ When should the manager reorder item Widget?

f$.$ Determine the length of each production run of item Widget, in days.

g$.$ What is the time between two consecutive orders of Widget, in weeks?