Moonbucks roasts 2 types of coffee: Guatemala Gold and SumatraSilver. Each month, the demand for each coffee type is uncertain.For Guatemala Gold, the mean demand is 20,000 pounds and thestandard deviation is 5,000 pounds. For Sumatra Silver, the meandemand is 10,000 pounds and the standard deviation is 5,000 pounds.The demand for Guatemala Gold and Sumatra Silver is negativelycorrelated with a correlation of ?0.4, since some customers tend tobuy whichever coffee is
on sale that month. It takes time to roast each type of coffee, andboth coffees are roasted on the Clover Roasting Machine. The CloverMachine can process 125 pounds of Guatemala Gold per hour, but only50 pounds of Sumatra Silver per hour. Although the Clover Machinecan only roast one of the two coffees at any given moment, it issimple to switch between roasting Guatemala Gold and SumatraSilver, so there is no setup time required in addition to theroasting times mentioned above.
a. What is the covariance of the demand for Guatemala Gold and thedemand for Sumatra Silver?
b. First express T (total roasting time) in terms of G (demand forGuatemala Gold) and S (demand for Sumatra Silver). T = a constanttimes G plus another constant times S. You need to determine
these constants.
c. What is the expected value of the total roasting time needed tohandle the total demand for
Guatemala Gold and Sumatra Silver in one month?
d. What is the variance of the total roasting time needed to handlethe total demand for
Guatemala Gold and Sumatra Silver in one month?
e. Moonbuck's operations manager has reserved 640 hours on theClover Machine to process next
month’s demand. Assuming that total roasting time is normallydistributed, do you think this will suffice? What is theprobability that 640 hours will be enough?