$($Decision Tree, $40\%).$ A water resources engineer is considering the economic feasibility of a

centralized well field source for supplying a small community in an arid region. Surface water supplies

are available, but at such a distance that they would be expensive to secure$.$ The engineer must decide

whether to construct $($act $a_1)$ or not to construct $($act $a_2)$ the well field. It is uncertain as to whether the

well field will provide $(1)$ only an insignificant amount of ground water $($state ${}_1),(2)$ an amount

sufficient supply part of the city's needs $($state ${}_2),$ or $(3)$ an amount sufficient to completely supply the

community needs $($state ${}_3).$ Since there is no prior evidence about the likelihoods of the three states, it

is believed that the three states have equal likelihood of $\frac{1}{3}.$

If a well water supply is found, part or all of the cost of the surface water source can be avoided, but the

cost of the well field must be considered. The net returns $($savings$)$ under the different states of nature

are calculated as the table below. The cost for installing the well field is $$700000$ and the cost of the

remote reservoir source is $$2700000.$

Table $1$: Payoff table of the decision options

In addition, at a cost of $$100000,$ the engineer could drill a series of small test wells to determine

whether they produced a low yield $z_1,$ a medium yield $z_2,$ or a high yield $z_3.$ Based upon historical

records and experience, the engineer estimates the reliability of the testing wells for the true state of

nature, which is described as the conditional probabilities of the yield state of test wells with a known

true state of the nature, orPr$(z|).$ The estimation results are shown in the table below.

Table $2$: Reliability of Testing wells, $Pr(z|)$

The engineer wants to know $(1)$ the optimal action if experimentation $($test wells$)$ is not conducted, $(2)$

the optimal strategy for experimentation and action, $(3)$ the best overall strategy, and $(4)$ the expected

monetary value of information in this case.