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, five days per week, 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.

$\backslash$table$[[,$Gadget,Widget$],[$Average demand,$6,000$ units per year,$50$ units per week$],[$Standard deviation,$50$ units per week,$5$ units per day$],[$Unit cost $($C$),$$$18,$$$200],[$Production rate,$-,100$ units per day$],[$Ordering cost$/$Setup cost$,$$$550,$$$2,650],[$Annual holding cost,$10\%,15\%],[$Lead time,$3$ weeks,$1$ week$],[$Service level,$99\mathrm{.}5\%,97\%]]$

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?