Read Mini$-$Case Study.

Answer the question: Using the decision Tree, which alternative should the fleet manager for the Southern Company choose?

For alternative $4,$ do we add the cost of oil analysis to the costs related to each probability?

Southern Company

Fleet managers have a large pool of cars and trucks to maintain. ${?}^{16}$ One approach to the vehicle maintenance is to use periodic oil

analysis: the oil from the engine and transmission are subjected periodically to a test. These tests can sometimes signal an

impending failure $($for example, iron particles in the oil$),$ and preventive maintenance is then performed $($at a relatively low cost$),$

eliminating the risk of failure $($failure would result in a relatively high cost$).$ However, oil analysis costs money and it is not perfect$-$

it can indicate that a unit is defective when in fact it is not, and it can indicate that a unit is non$-$defective when in fact it is$.$ As a

possible substitute for oil analysis, the company could simply change the oil periodically, thereby reducing the probability of failure.

The fleet manager for the Southern Company, an electrical utility based in Atlanta $($parent of Georgia Power$),$ has four alternatives:

$(1)$ do nothing, $(2)$ use oil analysis only, $(3)$ replace oil only, or $(4)$ replace oil and do oil analysis. For option $(1)$ the probability of

failure is $0\mathrm{.}1,$ and the cost of failure is $$1,200.$ For option $(2),$ the probability of failure remains at $0\mathrm{.}1.$ If the unit is about to fail, the

oil analysis will indicate this with probability $0\mathrm{.}7$ ; if the unit is not about to fail, the oil analysis will indicate this with probability

$0\mathrm{.}8.$ The oil analysis itself costs $$20,$ and if it indicates that failure is about to occur, the oil will be changed at the cost of $$14\mathrm{.}80$

and preventive maintenance will be performed. The cost of preventive maintenance to restore a unit that is about to fail is $$500,$

whereas the cost of maintenance for a unit that is not about to fail is $$250.$ For options $(3)$ and $(4),$ probability of failure decreases

from $0\mathrm{.}1$ to $0\mathrm{.}04.$ Analyze this decision problem.

Question

Using a decision tree, which alternative should the fleet manager for the Southern Company choose?