$1)$ Powertrain Design $(25$ Points$)$

As the lead systems engineer in an automotive company, you need to make a critical

decision: whether to $$outsource$$ the design and maintenance of your powertrain or to

keep that responsibility $$in$-$house$.$ While making that decision, there is uncertainty

associated with how often the powertrains will fail: either $$rare$$ or $$often$.$

$$ If you decide to outsource the design and maintenance, you pay a fixed contract

rate of $$20$M to an external company, which delivers the design and takes the

responsibility of the maintenance service under a warranty regardless of how

often the vehicle powertrain fails.

$$ If you decide to keep the design and maintenance $$in$-$house$,$ your company

must decide whether to use the $$existing design$$ with minimal changes or to

invest in a $$new design$.$

o The existing design is difficult to maintain and cheap to develop. If

failures occur often, it will cost $$25$M in the long run for the design and

maintenance, and if the failures are rare, it will cost $$15$M$.$ The new

design will be easier to maintain but costly to develop. If the failures are

often, it will cost $$23$M in the long run; and if the failures are rare, it will

cost $$16$M$.$

Based on your rough estimates, the failures will be often with a probability of $0\mathrm{.}4,$ and

it will be rare with a probability of $0\mathrm{.}6.$

a$)[5$pt$]$ Create a decision tree diagram based on the information above. Show

decision nodes, chance nodes, probabilities and the payoffs $($costs$).$

b$)[10$pt$]$ Find the expected payoffs of each decision and determine the most

rational decision out these two options $($outsource or in$-$house$).$ If you decide to

keep it in$-$house, specify whether to use the existing design or the new design.

Show your work.

c$)[10$pt$]$ Find the maximum amount you should pay to obtain perfect forecasts

about the failure rates in the future, i$.$e$.,$ find the expected value of perfect

information.