Schmears Inc. is a catalog retailer of men's shirts. Daily demand for a particular SKU (style and size) is Poisson with mean 1.5. It takes three days for a replen- ishment order to arrive from Schmears' supplier and orders are placed daily. Schmears uses the order-up-to model to manage its inventory of this shirt.
a. Suppose Schmears uses an order-up-to level of 9. What is the average number of shirts on order?
b. Now suppose Schmears uses an order-up-to level of 8. What is the probability during any given day that Schmears does not have sufficient inventory to meet the demand from all customers?
c. Suppose Schmears wants to ensure that 90 percent of customer demand is satisfied immediately from stock. What order-up-to level should they use?
d. Schmears is considering a switch from a "service-based" stocking policy to a "cost- minimization" stocking policy. They estimate their holding cost per shirt per day is $0.01. Forty-five percent of customers order more than one item at a time, so they estimate their stockout cost on this shirt is $6 per shirt. What order-up-to level minimizes the sum of their holding and back-order costs?