Enginnering Optimization using Decision Tree

An engineer must decide between spread footings or piles for the foundation to support each column of a

building. The soil strength is not known for certain, but could be low, medium, or high. Spread footings

for the building have a total cost of $$150,000,$ while piles cost $$250,000.$ The piles are considered "safe"

in that if the actual soil strength turns out to be low, no damage will result to the building. The spread

footings are less expensive but would result in some building damage if the soil strength is low or

medium, and they would have to be replaced with piles $($under difficult construction conditions$)$ if the soil

strength is low. The cost data for both design alternatives are given below.

The prior probabilities of the soil strength are $=0\mathrm{.}30,0\mathrm{.}40$ and $0\mathrm{.}30.$ For $$10,000,$ field studies and a

triaxial strength test can be conducted. Due to the test uncertainties, the test reliability expressed as a

conditional probability is shown below.

Using a decision tree, determine the optimal strategy of the foundation design conditional on the testing

result. Also determine if the test is worth doing.