Problem 2: A certain industrial machine generates revenue for some factory at a

Question

Problem 2: A certain industrial machine generates revenue for some factory at a rate of 250 KD per hour of operation. However, the machine breaks down an average of 4 times per 20 hours of operation. Let Y denote the number of breakdowns of the machine in t hours of operation. The revenue generated by the machine is given by A(t)-250t-40%(t) (KD) a) What is the expected revenue after 25 hours of operation? b) Determine the optimal maintenance interval t* that will maximize the expected revenue between shutdowns? Hint: The number of breakdowns of the machine can be modeled as a Poisson random variable.

Explanation / Answer

E(R(t)) = E(250 t - 40 y^2)

= 250 t - 40 E(Y^2)

for t = 25

Y(t) follow poisson with lambda = 4/20 * 25=5

E(Y^2) = var(Y) + E(Y)^2

= 5 + 5^2 = 30

hence

E(R(t)) = 250 * 25 - 40 * 30

= 5050

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