Question

CPG Bagels starts the day with a large production run of bagels. Throughout the morning, additional bagels are produced as needed. The last bake is completed at 3 p.m. and the store closes at 8 p.m. It costs approximately $0.20 in materials and labor to make a bagel. The price of a fresh bagel is $0.60. Bagels not sold by the end of the day are sold the next day as “day old” bagels in bags of six, for $0.99 a bag. About two-thirds of the day-old bagels are sold; the remainder are just thrown away. There are many bagel flavors, but for simplicity, concentrate just on the plain bagels. The store manager predicts that demand for plain bagels from 3 p.m. until closing is normally distributed with a mean of 66 and a standard deviation of 33.

a. How many bagels should the store have at 3 p.m. to maximize the store’s expected profit (from sales between 3 p.m. until closing)? (Hint: Assume day-old bagels are sold for $0.99/6 = $0.165 each; that is, don’t worry about the fact that day-old bagels are sold in bags of six.)

b. Suppose that the store manager is concerned that stockouts might cause a loss of future business. To explore this idea, the store manager feels that it is appropriate to assign a stockout cost of $5 per bagel that is demanded but not filled. (Customers frequently purchase more than one bagel at a time. This cost is per bagel demanded that is not satisfied rather than per customer that does not receive a complete order.) Given the additional stockout cost, how many bagels should the store have at 3 p.m. to maximize the store’s expected profit?

c. Suppose the store manager has 109 bagels at 3 p.m. How many bagels should the store manager expect to have at the end of the day?

Answer 1

Selling price of a fresh bagel = $ 0.60

Material cost of a fresh bagel = $ 0.20

Selling price of day old bagels = $ 0.99/6 = $ 0.165

Mean demand = 66

Standard deviation of demand = 33

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a)

Cost of shortage, Cu = selling price - cost = 0.6 - 0.2 = $ 0.40

Cost of overage, Co = cost - selling price of day old bagel = 0.20 - 0.165 = $ 0.035

Critical ratio = Cu/(Cu+Co) = 0.4/(0.4+0.035) = 0.9195

z = NORMSINV(0.9195) = 1.4017

Number of bagels store should have at 3 pm = mean demand + z * standard deviation of demand

= 66 + 1.4017*33

= 112  bagels

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b)

Cost of shortage, Cu = selling price - cost + stockout cost = 0.6 - 0.2 + 5 = $ 5.40

Cost of overage, Co = cost - selling price of day old bagel = 0.20 - 0.165 = $ 0.035

Critical ratio = Cu/(Cu+Co) = 5.4/(5.4+0.035) = 0.9936

z = NORMSINV(0.9936) = 2.4893

Number of bagels store should have at 3 pm = mean demand + z * standard deviation of demand

= 66 + 2.4893*33

= 148  bagels

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c)

z = (109-66)/33 = 1.3030

L(z) =NORMDIST(1.3030,0,1,FALSE)-1.3030*NORMSDIST(-1.3030)

= 0.0452

Expected shortage = 33*0.0452 = 1.5

Expected Sales = 66 - 1.5 = 64.5

Number of bagels the store manager should expect to have at the end of the day = 109 - 64.5

= 44.5 bagels