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 mean 54 and standard deviation of 21. Required:
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 the store manager has 96 bagels at 3 p. m. How many bagels should the store manager expect to have at the end of the day?
c. Suppose the manager would like to have a 0.92 in-stock probability on demand that occurs after 3 p. m. How many bagels should the store have at 3 p. m. to ensure that level of service?

A. 85.5

B. 42.1785

C. 85.5

Explanation:

a. Calculation to determine How many bagels should the store have to maximize the store's expected profit

First step is to calculate the Critical ratio

Using this formula

Critical ratio = Cu/(Co + Cu)

Where,

Cu = $0.60 - $0.20

Cu= $0.40

Co = $0.20 - $0.165

Co = $0.035

Let plug in the formula

Critical ratio= $0.40/($0.035 + $0.40)

Critical ratio= $0.40/0.435

Critical ratio= 0.91954023

Second step is to use Standard Normal Distribution Function to Find the above Critical ratio 0.91954023 in F(z) column which gives us 1.5 from the corresponding z value

Now let determine How many bagels should the store

Using this formula

Q = (mean) + (z x SD)

Let plug in the formula

Q= 54 + (1.5 x 21)

Q= 54 + 31.5

Q= 85.5

Therefore the numbers of bagels that the store should have at 3 p.m to maximize the store's expected profit is 85.5

(b) Calculation to determine How many bagels should the store manager expect to have at the end of the day

First step is to calculate the z using this formula

z = (Q - (mean))/SD

Let plug in the formula

z= (96- 54)/21

z= 42/21

z= 2

Second Second step is to use Standard Normal Distribution Function to Find 2 in z column which will gives us 2.0085 as the corresponding I(z)

Now let calculate the Expected Inventory using this formula

Expected Inventory = SD x I(z)

Let plug in the formula

Expected Inventory= 21 x 2.0085

Expected Inventory= 42.1785

Therefore The numbers of bagels that the store manager should expect to have at the end of the day is 42.1785

(c) Calculation to determine How many bagels should the store have at 3 p.m. to ensure that level of service

First step is to use the Standard Normal Distribution Function to Find 0.92 in f(z) column which will give us 1.5 as the corresponding z

Now let calculate Q using this formula

Q = (mean) + (z x SD)

Let plug in the formula

Q= 54 + (1.5x 21)

Q= 54 + 31.5

Q= 85.5

Therefore The number of bagels that the store should have at 3 p.m. to ensure that level of service is 85.5