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Multi-attribute utility 129
lottery. After some thought, he indicates that this probability is 0.8, so
k1 = 0.8. This suggests that the ‘swing’ from the worst to the best overrun
time is seen by the project manager to be significant relative to project
cost. If he hardly cared whether the overrun was 0 or 6 weeks, it would
have taken only a small value of k1 to have made him indifferent to a
gamble where overrun time might turn out to be at its worst level.
To obtain k2, the weight for project cost, we offer the project manager
a similar pair of options. However, in the certain outcome, project cost
is now at its best level and the other attribute at its worst level. The
probability of the best outcome in the lottery is now k2. These two options
are shown in Figure 5.20.
We now ask the project manager what value k2 would need to be to
make him indifferent between the two options. He judges this probability
to be 0.6, so k2 = 0.6. The fact that k2 is less than k1 suggests that the
project manager sees the swing from the worst to the best cost as being
less significant than the swing from the worst to the best overrun time.
Having been offered a project which is certain to incur the lowest cost,
he requires a smaller probability to tempt him to the lottery, where he
might gain a project where overrun is also at its best level but where
there is also a risk of a project with costs at their worst level.
Finally, we need to find k3. This is a simple calculation and it can be
shown that:
k1 + k2 + k3 = 1, so k3 = 1 − k1 − k2
Thus, in our case k3 = 1 − 0.8 − 0.6 =−0.4. The project manager’s multiattribute
utility function is therefore:
u(x1, x2) = 0.8 u(x1) + 0.6 u(x2) − 0.4 u(x1)u(x2)
We can now use the multi-attribute utility function to determine the
utilities of the different outcomes in the decision tree. For example, to
find the utility of a project which overruns by 3 weeks and costs $60 000
we proceed as follows. From the single-attribute functions we know that
Figure 5.20
Overrun Cost
1.0
Worst = 6 weeks Best: $50000
k 2
1 − k 2
Overrun Costs
Best = 0 weeks Best: $50000
Worst = 6 weeks Worst: $140000