Harvey$$s Specialty Shop is a popular spot that specializes in international gourmet foods. One of the items that Harvey sells is a popular mustard that he purchases from an English company. The mustard costs $$25$ a jar and requires a $3-$month lead time for replenishment of stock. The replenishment time is constant. Harvey uses a $20\%$ annual interest rate to compute holding costs. Bookkeeping expenses for placing an order amount to about $$60.$ During the $3-$month supply time, Harvey estimates that he sells an average of $150$ jars but there is substantial variation. He estimates the standard deviation of demand for each $3-$month period is $25.$ Assume that demand is described by a normal distribution.

a$)$ What is the optimal order quantity?

b$)$ What will be the average time between transactions?

c$)$ How much safety stock should be maintained for $96\%$ cycle service level?

d$)$ What should be the reorder point for $96\%$ fill rate?

e$)$ Now suppose that the English company is ready to send the mustard by a faster ship, which will reduce the replenishment lead time to $1-$month and the standard deviation of demand during that period to $14\mathrm{.}4,$ but increase the cost to $$30$ per jar. What will be the new reorder point for $96\%$ fill rate service level?

f$)$ In order to achieve $96\%$ fill rate service, what will be the optimal total inventory cost $($Ordering $+$ Holding $+$ Purchase$)$ before and after the decrease of lead time $($and resultant price increase$)?$ What managerial insights can you get from this?