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Multi-attribute utility 131
0.5
0.5
Figure 5.22
Overrun Costs
1 week
$80000
3 weeks $50000
(a)
0.5
0.5
Overrun Costs
6 weeks $50000
1 week $120000
(b)
0.5
0.5
Overrun Costs
0 weeks $120000
6 weeks $120000
option has performed well and another has performed badly. If the
decision maker does not feel that the explanations are consistent with his
preferences then the analysis may need to be repeated. In fact, it is likely
that several iterations will be necessary before a consistent representation
is achieved and, as the decision maker gains a greater understanding of
his problem, he may wish to revise his earlier responses.
Another way of checking consistency is to offer the decision maker a
new set of lotteries and to ask him to rank them in order of preference.
For example, we could offer the project manager the three lotteries
shown in Figure 5.22. The expected utilities of these lotteries are A:
0.726, B: 0.888 and C: 0.620, so if he is consistent then he should rank
them in the order B, A, C. We should also carry out sensitivity analysis
on the probabilities and utilities by, for example, examining the effect of
changes in the values of k1 and k2.
Interpreting multi-attribute utilities
In the analysis above we derived an expected utility of 0.872 for the ‘hire
extra labor ...’ option, but what does this figure actually represent? We
demonstrated earlier that we could use the concept of utility to convert a
decision problem to a simple choice between lotteries which the decision
maker regarded as being equivalent to the original outcomes. Each of
these lotteries would result in either the best or worst possible outcome,
but with different probabilities. The same is true for multi-attribute
utility. This time the lotteries will result in either the best outcome on
both attributes (i.e. the best/best outcome) or the worst possible outcome
on both attributes (i.e. worst/worst). Thus the expected utility of 0.872
for the ‘hire extra labor ...’ option implies that the decision maker
regards this option as being equivalent to a lottery offering a 0.872
chance of the best/best outcome (and a complementary probability of
the worst/worst outcome). It therefore seems reasonable that he should
prefer this option to the ‘work normally’ alternative, which is regarded
(c)