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168 Decision trees and influence diagrams
and there would be an additional cost of $0.1 million for each foot in
the barrier’s height. For technical reasons, the height of this barrier
would be either 11 or 13 feet, and it is thought that there would be
no chance of the barrier being damaged if flooding did occur. The
probability of the river’s height exceeding the 13-foot barrier in any
one year is estimated to be only 0.004.
(a) Draw a decision tree to represent the River Authority’s problem.
(b) Determine the optimum policy for the Authority, assuming that
their objective is to minimize expected costs. (For simplicity,
you should ignore time preferences for money.)
(5) An engineering company is about to undertake a major overhaul of a
factory’s machinery for a customer. The overhaul will be carried out
on a Saturday and Sunday, but, if it is not completed by the Monday
morning, the factory will experience serious production losses. In
this event, the engineering company has agreed to compensate the
customer by paying a penalty of $20 000.
The manager of the engineering company has to decide how many
engineers to include in the overhaul team. Each engineer in the team
will be paid $480 for working over the weekend, but because of
the nature of the work, only teams of 10, 15 or 20 engineers can be
considered. The manager estimates that the chances of a 10-person
team completing the overhaul by the Monday morning is only 0.4.
A 15-person team has, he estimates, a 0.6 probability of meeting the
deadline, while a 20-person team has a 0.9 probability of completing
the work in time for Monday morning.
(a) Assuming that the manager wants to minimize expected costs,
how large a team should he choose?
(b) Having made a provisional decision about the size of the team,
the manager hears that a piece of specialized equipment will
be available for hire on the Sunday, at short notice. The cost of
hiring this equipment would be $4400, and it would require at
least 15 engineers to operate it. However, it is virtually certain
that the overhaul would be completed on time if the equipment
was used.
Before making a decision on whether to hire the equipment,
the manager will review the progress that has been made on
the Saturday evening. He reckons that there is a 0.5 probability
that a 15-person team would be behind schedule by Saturday
evening, while there is only a 0.2 probability that a 20-person
team will be in this position. He then calculates the probabilities
of the overhaul overrunning the deadline if the equipment