You are tasked with evaluating three possible foundation design alternatives for a new bridge considering extreme flood limit state, possible damage, and repair cost, and a $75-$year design service life. The occurrence of an Extreme Flood Event is best estimated by the Poisson distribution and the AASHTO standards require that foundations are designed for an extreme flood event with a $500-$year recurrence $($Hint: annual occurrence is $\mathrm{.}002/$year$).$ Associated costs and conditional probabilities are listed in the following table.

Find: Using Decision$-$Tree Analysis Method provide the following:

a$.)$ Construct the decision tree model representing each alternative decision and all possible outcomes $(10)$

b$.)$ Determine expected value $($cost$)$ for each foundation type and the preferred decision based on the minimum expected cost criteria. $(10)$

c$.)$ Determine the expected value of perfect information $($EVPI$).(5)$

d$.)$ Develop a cumulative risk profile for each foundation decision. $(10)$

e$.)$ Which strategy is preferred based on total risk? Explain $(5)$

f$.)$ Assume that a flood study can be performed at the proposed bridge location. The firm you plan to hire has conducted several flood studies in that region, and $15\%$ of those studies resulted in reports that forecasted significant flood events. Further, the actual occurrence of significant flood events in the region was $95\%$ accurate compared to the firm$$s reports. What is the expected value of this firm$$s flood study $($EVSI$)?$ Should you hire them? Explain? $(15$