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176 Decision trees and influence diagrams
that it will decline to 15 000. The annual profits associated with these
passenger numbers are estimated to be $3 million and $1 million,
respectively.
If fares are reduced, but television advertising is not used, then
it is thought that there is a 0.6 probability that the mean number of
passengers carried will increase to 25 000 and a 0.4 probability that
it will increase to 22 000. The resulting profits generated by these
passenger numbers are estimated to be $2 million and $1.7 million,
respectively. Advertising the fare reduction on television would
increase the probability of an increase to a mean of 25 000 passengers
to 0.8 and reduce the probability that the mean will be 22 000 to 0.2.
However, it would reduce the profits associated with these mean
passenger numbers by $0.6 million. The railway’s objectives are to
maximize profit and to maximize passenger numbers (since this
brings environmental benefits such as reduced traffic congestion).
(a) Utility functions for the mean numbers of passengers carried
and profit have been elicited from the railway’s chief executive
and these are given below.
Mean number of passengers Utility Profit Utility
15 000 0.00 $1.0m 0.00
20 000 0.80 $1.1m 0.20
22 000 0.95 $1.4m 0.60
25 000 1.00 $1.7m 0.75
$2.0m 0.90
$3.0m 1.00
Plot these utility functions and interpret them.
(b) The elicitation session revealed that, for the chief executive,
mean number of passengers and profit are mutually utility
independent. You are reminded that, in this case, a two-attribute
utility function can be obtained from:
u(x1, x2) = k1u(x1) + k2u(x2) + k3u(x1)u(x2)
where k3 = 1 − k1 − k2.
The elicitation session also revealed that k1 = 0.9andk2 = 0.6,
where attribute number 1 is the mean number of passengers.
Determine the policy that the railway should pursue in the
light of the above utilities and comment on your answer.
(13) (a) Use an influence diagram to represent the following decision
problem, stating any assumptions you have made.