- A diesel truck supercharger can cause a lot of damage if it fails critically.

Question

- A diesel truck supercharger can cause a lot of damage if it fails critically. The chief mechanic of the trucking fleet can choose to replace the old supercharger with a new one or leave it until the next repair cycle. The state of the supercharger can be categorized as “good”, acceptable”, or ‘poor”. The probability of a critical failure per maintenance cycle (maintenance every quarter year) depends on the state of the supercharger: Pr(failire | good) = 0.0001, Pr(failure | acceptable) = 0.001, Pr(failure | poor) = 0.01 The engineering department made a model of the degradation of the supercharger. In the model, it is assumed that if the state is “good” or “acceptable”, then the end of the maintenance period, it stays with a probability of 95% in the same condition or degrades with a probability of 5% (that is, “good” becomes “acceptable”, and “acceptable” becomes “poor”. If the state is “poor”, it stays as “poor” (a) Determine the probability that a supercharger that is “good”, “acceptable”, and “poor” in the next three maintenance periods, and does not fail. The cost of repairing a failure is $150,000 and the cost of a new supercharger is $1000. (b) Considerasituationwhereanoldsuperchargeriscategorizedas“acceptable”;determinethe optimal decision using expected monetary loss (risk). What would the decision be if the old supercharger was “good” with a 10% probability and “acceptable” with a 90% probability?

Explanation / Answer

P(failure|good)=0.0001

P(failure|acceptable)=0.001

P(failure|poor)=0.01

P(good staying good)=P(acceptable staying acceptable)=0.95

P(good becoming acceptable)=P(acceptable becoming poor)=0.05

a)Probability of good,acceptable,poor,non failure= P(nonfailure|good)*P(nonfailure|acceptable)*P(nonfailure|poor)*P(good becoming acceptable)*P(acceptable becoming poor) =0.9999*0.999*0.99*0.05*0.05=2.47x10^-3

b)Expected monitory loss is calculated for an year given condition is acceptable=P(failure|aceptable)*(cost if failed) =0.001*(150000+1000)=$151

c)Given 10% good and 90% acceptable

Expected Loss=0.1*0.0001*151000+0.9*0.001*151000=$137.41

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