Swanson's Bakery is well known for producing the best fresh bread in the city, so the sales are very substantial. The daily demand for its fresh bread has a uniform distribution between 300 and 600 loaves. The bread is baked in the early morning, before the bakery opens for business, at a cost of $2 per loaf. It then is sold that day for $3 per loaf. Any bread not sold on the day it is baked is relabeled as day-old bread and sold subsequently at a discount price of $1.50 per loaf.

(a) Apply the stochastic single-period model for perishable products to determine the optimal service level.

(b) Apply this model graphically to determine the optimal number of loaves to bake each morning.

(c) With such a wide range of possible values in the demand distribution, it is difficult to draw the graph in part (b) carefully enough to determine the exact value of the optimal number of loaves. Use algebra to calculate this exact value.

(d) Given your answer in part (a), what is the probability of incurring a shortage of fresh bread on any given day?

(e) Because the bakery's bread is so popular, its customers are quite disappointed when a shortage occurs. The owner of the bakery, Ken Swanson, places high priority on keeping his customers satisfied, so he doesn't like having shortages. He feels that the analysis also should consider the loss of customer goodwill due to shortages. Since this loss of goodwill can have a negative effect on future sales, he estimates that a cost of $1.50 per loaf should be assessed each time a customer cannot purchase fresh bread because of a shortage. Determine the new optimal number of loaves to bake each day with this change. What is the new probability of incurring a shortage of fresh bread on any given day?