Question

Moonbucks roasts 2 types of coffee: Guatemala Gold and Sumatra Silver. Each month, the demand for each coffee type is uncertain. For Guatemala Gold, the mean demand is 20,000 pounds and the standard deviation is 5,000 pounds. For Sumatra Silver, the mean demand is 10,000 pounds and the standard deviation is 5,000 pounds. The demand for Guatemala Gold and Sumatra Silver is negatively correlated with a correlation of −0.4, since some customers tend to buy whichever coffee is
on sale that month. It takes time to roast each type of coffee, and both coffees are roasted on the Clover Roasting Machine. The Clover Machine can process 125 pounds of Guatemala Gold per hour, but only 50 pounds of Sumatra Silver per hour. Although the Clover Machine can only roast one of the two coffees at any given moment, it is simple to switch between roasting Guatemala Gold and Sumatra Silver, so there is no setup time required in addition to the roasting times mentioned above.
a. What is the covariance of the demand for Guatemala Gold and the demand for Sumatra Silver?
b. First express T (total roasting time) in terms of G (demand for Guatemala Gold) and S (demand for Sumatra Silver). T = a constant times G plus another constant times S. You need to determine
these constants.
c. What is the expected value of the total roasting time needed to handle the total demand for
Guatemala Gold and Sumatra Silver in one month?
d. What is the variance of the total roasting time needed to handle the total demand for
Guatemala Gold and Sumatra Silver in one month?
e. Moonbuck's operations manager has reserved 640 hours on the Clover Machine to process next
month’s demand. Assuming that total roasting time is normally distributed, do you think this will suffice? What is the probability that 640 hours will be enough?

Answer 1

First we define the following random variables

Let G =demand for Guatemala Gold (in pounds) and S=demand for Sumatra Silver (in pounds)

We know the following

For Guatemala Gold, the mean demand is 20,000 pounds

and the standard deviation is 5,000 pounds.

For Sumatra Silver, the mean demand is 10,000 pounds

and the standard deviation is 5,000 pounds

The demand for Guatemala Gold and Sumatra Silver is negatively correlated with a correlation of −0.4

a. What is the covariance of the demand for Guatemala Gold and the demand for Sumatra Silver?
The covariance is

ans: the covariance of the demand for Guatemala Gold and the demand for Sumatra Silver is -10,000,000

b. First express T (total roasting time) in terms of G (demand for Guatemala Gold) and S (demand for Sumatra Silver). T = a constant times G plus another constant times S.

The Clover Machine can process 125 pounds of Guatemala Gold per hour,

Total roasting time for G pounds of Guatemala Gold is

hours

but only 50 pounds of Sumatra Silver per hour

Total roasting time for S pounds of Sumatra Silver is

hours

Since there is no setup time involved, switching from roasting one type to another takes no time.

The total time to roast G pounds of Guatemala Gold and S pounds of Sumatra Silver is

ans:

c. What is the expected value of the total roasting time needed to handle the total demand for Guatemala Gold and Sumatra Silver in one month?

The expected value of T is

ans: the expected value of the total roasting time needed to handle the total demand for Guatemala Gold and Sumatra Silver in one month is 360 hours

d. What is the variance of the total roasting time needed to handle the total demand for Guatemala Gold and Sumatra Silver in one month?

The variance of T is

ans: the variance of the total roasting time needed to handle the total demand for Guatemala Gold and Sumatra Silver in one month is 8400

e. Moonbuck's operations manager has reserved 640 hours on the Clover Machine to process next month’s demand. Assuming that total roasting time is normally distributed, do you think this will suffice? What is the probability that 640 hours will be enough?

T is the total roasting time and it is normally distributed with

mean and standard deviation

the probability that the total roasting time is less than 640 hours is

ans: the probability that 640 hours will be enough is 0.9989