CPG Bagels sta CFG Bagels starts the day with a large production run of bagels.
ID: 407053 • Letter: C
Question
CPG Bagels sta CFG Bagels starts the day with a large production run of bagels. Through out 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 S0.20 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 ot 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 of 54 and standard deviation of 21. 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)? 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? Suppose the store manager has 101 bagels at 3 p.m. How many bagels should the store manager expect to have at the end of the day?Explanation / Answer
a)this quantity to maximize the profit can be calculated by the formula z={(Q-mean)/std dev.}
For this we need to find z, Hence,
Underage cost: U=0.6-0.2 = $0.4 i.e. the gross margin lost if demand is not met
Overage cost : If more than required bagels are produced, only 2/3rd of day old bagels are sold at $0.165. Rest are thrown away. Hence, the actual money that is earned by selling the day old bagels is (2/3)*0.165 = $ 0.11.
Hence overage cost, O=0.20-0.11= $ 0.09 i.e. the actual loss incurred if the bagels is not sold and has to be thrown away.
Hence, critical ratio C= U/(O+U) = 0.4 / (0.09 + 0.4) = 0.8163
From the standard normal table, z (0.91) = 0.8186 (z(0.90)=0.8159 hence we are taking z = 0.91)
From the above formula, z={(Q-mean)/std dev.},
Q= mean + (z*std dev.) = 54 + (0.91 *21) = 73.11 = 73 (round off to the closes whole numebr)
Hence, tore should have 73 bagels at 3 pm to maximize the store's expected profit.
b) The new underage cost will be $5. Hence,
C= 5 / ( 0.09 + 5) = 5/5.09= 0.9823
z=2.11 ( for C= 0.9826)
Hence, Q = 54 +2.11*21 = 98.31, Hence, 98 bagels are required to maximize the profit.
c) if underage cost = $ 0.4 , 101-73 = 28 bagels may be left by the end of the day
if underage cost is $5 , 101- 98 = 3 bagels may be left by the end of the day.