Reconsider Prob. 19.6-4. The bakery owner, Ken Swanson, now wants you to conduct a financial analysis of various inventory policies. You are to begin with the policy obtained in the first four parts of Prob. 19.6-4 (ignoring any cost for the loss of customer goodwill). As given with the answers in the back of the book, this policy is to bake 500 loaves of bread each morning, which gives a probability of incurring a shortage of  .

(a) For any day that a shortage does occur, calculate the revenue from selling fresh bread.

(b) For those days where shortages do not occur, use the probability distribution of demand to determine the expected number of loaves of fresh bread sold. Use this number to calculate the expected daily revenue from selling fresh bread on those days.

(c) Combine your results from parts (a) and (b) to calculate the expected daily revenue from selling fresh bread when considering all days

(d) Calculate the expected daily revenue from selling day-old bread.

(e) Use the results in parts (c) and (d) to calculate the expected total daily revenue and then the expected daily profit (excluding overhead).

(f) Now consider the inventory policy of baking 600 loaves each morning, so that shortages never occur. Calculate the expected daily profit (excluding overhead) from this policy.

(g) Consider the inventory policy found in part (e) of Prob. 19.6-4. As implied by the answers in the back of the book, this policy is to bake 550 loaves each morning, which gives a probability of incurring a shortage of   . Since this policy is midway between the policy considered here in parts (a) to (e) and the one considered in part (), its expected daily profit (excluding overhead and the cost of the loss of customer goodwill) also is midway between the expected daily profit for those two policies. Use this fact to determine its expected daily profit.

(h) Now consider the cost of the loss of customer goodwill for the inventory policy analyzed in part (g). Calculate the expected daily cost of the loss of customer goodwill and then the expected daily profit when considering this cost.

(i) Repeat part (h) for the inventory policy considered in parts (a) to (e).

Prob. 19.6-4

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?