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128 Decision making under uncertainty
independent then it can be shown that the multi-attribute utility function
will have the following form:
where
u(x1, x2) = k1u(x1) + k2u(x2) + k3u(x1)u(x2)
x1 = the level of attribute 1,
x2 = the level of attribute 2,
u(x1, x2) = the multi-attribute utility if attribute 1 has a level x1
and attribute 2 has a level x2,
u(x1) = the single-attribute utility if attribute 1 has a level x1,
u(x2) = the single-attribute utility if attribute 2 has a level x2
and k1, k2, andk3 are numbers which are used to ‘weight’ the singleattribute
utilities.
In stage 1 we derived u(x1) and u(x2), so we now need to find the
values of k1, k2 and k3.Wenotethatk1 is the weight attached to the utility
for overrun time. In order to find its value we offer the project manager
a choice between the following alternatives:
A: A project where overrun is certain to be at its best level (i.e. 0 weeks),
but where the cost is certain to be at its worst level (i.e. $140 000); or
B: A lottery which offers a probability of k1 that both cost and overrun
will be at their best levels (i.e. 0 weeks and $50 000) and a 1 − k1
probability that they will both be at their worst levels (i.e. 6 weeks
and $140 000, respectively).
These options are shown in Figure 5.19. Note that because we are
finding k1 it is attribute 1 (i.e. overrun) which appears at its best level in
the certain outcome.
The decision maker is now asked what value the probability k1 must
have to make him indifferent between the certain outcome and the
A:
1.0
Figure 5.19
Overrun Cost
Best = 0 weeks Worst = $140000
B:
k 1
1 − k 1
Overrun Costs
Best = 0 weeks Best = $50000
Worst = 6 weeks Worst = $140000