Decision Trees: Exercise Sheet 1
1 An entertainment company is organising a pop concert in London. The company has
to decide how much it should spend on publicising the event, and three options have been
identified:
Option 1: Advertise only in the music press;
Option 2: As option 1, but also advertise in the national press;
Option 3: As options 1 and 2, but also advertise on commercial radio.
For simplicity, the demand for tickets is categorised as low, medium or high. The payoff
table below shows how the profit (in thousands of pounds) that the company will earn for
each option depends upon the level of demand:
Option Low Medium High
1 -20 -20 100
2 -60 -20 60
3 -100 -60 20
It is estimated that if option 1 is adopted the probabilities of low, medium and high
demand are 0.4, 0.5 and 0.1 respectively. For option 2 the respective probabilities are
0.1, 0.3 and 0.6, while for option 3 they are 0.05, 0.15 and 0.8. Determine the option
that would lead to the highest expected profit. Would you have any reservations about
recommending this option to the company.
2 A team of scientists is due to spend 6 months in Antarctica carrying out research. One
major piece of equipment they will be taking is subject to breakdowns caused by the
sudden failure of a particular component. Because a failed component cannot be repaired,
the team intend to carry a stock of spare units of the component, but it will cost them
roughly £3000 each for each spare unit they take with them. However, if the equipment
breaks down and a spare is not available, a new unit will have to be specially flown in and
the team will incur a total cost of £4000 for each unit that is delivered in this way. An
engineer who will be travelling with the team has estimated that the number of spares
that will be required during the 6 months follows the probability distribution shown below
Number of spares 0 1 2 3
Probability 0.2 0.3 0.4 0.1
Determine the number of spares that the team should carry if their objective is to minimise
expected costs.
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3 A large machine in a factory has broken down and the company that owns the factory
will incur costs of £3200 for each day that the machine is out of action. The factory’s
engineer has three immediate options.
Option 1: He can return the machine to the supplier who has agreed to collect, repair and
return it free of charge, but not to compensate the company for any losses they might incur
while the repair is being carried out. The supplier will not agree to repair the machine
if any other person has previously attempted to repair it. If the machine is returned, the
supplier will guarantee to return it in working order in 10 days time.
Option 2: He can call in a specialist local engineering company. They will charge £20000
to carry out the repair, and they estimate that there is a 30% chance that they will be
able to return the machine to working order in 2 days. However, there is a 70% chance
that repairs will take 4 days.
Option 3: He can attempt to carry out the repair work himself, and he estimates that
there is a 50% chance that he could mend the machine in 5 days. However, if at the end of
5 days the attempted repair has not been successful, he will have to decide whether to call
in the local engineering company or to make a second attempt at repair by investigating
a different part of the mechanism. This would take two further days, and he estimates
that there is a 25% chance that this second attempt would be successful. If he fails in the
second attempt, he will have to call in the local engineering company. It can be assumed
that the probability distribution for the local engineering company’s repair time will be
unaffected by any work that the factory engineer carries out.
Assuming that the engineer’s objective is to minimise expected costs, what course of action
should he take?
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Decision Trees: Exercise Sheet 2
1 The Bonsante Drug Company is aiming to develop a new drug that will alleviate the
symptoms of arthritis with few side effects. The earlier the company can develop, test and
market the drug, the greater will be the returns it will earn in a market that is thought
to be worth billions of pounds.
Tow alternative technologies of developing the drug are being considered, and given the
resources available to the company, only one of these approaches can be pursued at a given
time. the first approach is based upon a substance called HMP acid, and it is estimated
that there is a 0.4 probability that this approach would lead to development of the drug
in five years, with a 0.6 probability that the development would take seven years.
There is more uncertainty about the development time of the second approach, which
is based on a derivative of the chemical zylogen. It is estimated that the use of this
chemical has a 0.3 probability of leading to completion of development in three years.
If development has not ben completed in this period, then a decision would have to be
made between switching to the HMP acid technology or attempting to modify the zylogen
approach. It is thought that the modification has a 0.8 probability of leading to completion
after a further two years.
If this has not occurred,a decision would then have to be made between switching to the
HMP acid approach or persevering with zylogen for a further seven years, by which time
it is assumed that successful development is certain to have been achieved.
Assuming that the objective of Bonsante’s directors is to minimise the expected develop-
ment time of the drug, determine their optimum policy.
2 In January, a sales manager estimates that there is only a “30% chance” that the sales
of a new product will exceed a million units in the coming year. However, she is then
handed the results of a sales forecast. This suggests that the sales will exceed a million
units. The probability that this indication will be given when sales will exceed a million
units is 0.8. However, the probability that the forecast will give this indication when sales
will not exceed a million units is 0.4. Revise the sale manager’s estimate in light of the
new forecast.
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3 A company has just received some “state of the art” electronic equipment from an over-
seas supplier. The packaging has been damaged during delivery, and the company must
decide whether to accept the equipment. If the equipment itself has not been damaged,
it could be sold for a profit of £10000 . However, if the batch is accepted and it turns
out to be damaged, a loss of £5000 will be made. Rejection of the equipment will lead
to no change in the company’s profit. after a cursory inspection, the company’s engineer
estimates that there is a 60% chance that the equipment has not been damaged. The
company has another option. The equipment could be tested by a local specialist com-
pany. Their test, however, is not perfectly reliable and has only an 80% chance of giving
a correct indication.
How much would it be worth paying for the information from the test (assuming that the
company’s objective is to maximise expected profit)?
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