Question

Problem 13.35. A biotech manufacturing company can make test kits at a cost of $ 20.00. Each “kit” for which there is a glove demand the week it is manufactured can be sold for $ 100.00. However, due to the short life of its components, each “kit” that cannot be sold during that week has to be discarded at a cost of $ 5.00

The weekly demand for that product is a random variable with the following pmf:

Weekly demand (no. “kits”)	0	50	100	200
Probabilty of the demand	0.05	0.4	0.3	0.25

1) Calculate the expected value of the demand for “kits” during one week (5 pts)

2) Calculate the variance of demand for a week (5 pts)

3) The company could manufacture 50 “kits” per shift. If they decide to go into business, how many shifts must they work (1, 2, or 3) to maximize the expected profit?

Answer 1

Let X be the random variable denoting weakly demand for the product, then X has the following pmf,

x    0 50 100 200
P(X = x) 0.05 0.4 0.3 0.25

a) Hence, Expected weakly demad can be calculated as,

E(X) = 0*0.05 + 50*0.4 + 100*0.3 + 200*0.25
= 20 + 30 + 50
= 100

b) Var(X) = E(X - E(X))^2 = E(X^2) - (E(X))^2
Consider, E(X^2) = 0*0.05 + (50^2)*0.4 + (100^2)*0.3 + (200^2)*0.25
= 1000 + 3000 + 10000
= 14000
So, Var(X) = 14000 - 10000 = 4000

c) Let Y be the amount of kits being produced in a week. Then if X is the demand of the week, X kits are sold at a price of $100.00 and remaining Y - X kits are discarded at a cost of $5.00. Thus profit given demand X for Y kits in production is,
P(Y, X) = X*100 - (Y-X)*5 - Y*20 = X*105 - Y*25
Expected profit for Y kits in production is,
EP(Y) = E(P(Y,X)) = E(X)*105 - Y*205 = 10500 - Y*25

From the above expression, it is evident that is advisable to make as less kits as possible, that is to work only 1 shift to maximize the expected profit.