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RISK ANALYSIS AND SENSITIVITY ANALYSIS 553

Computer software of nature and learn even more about how changes in the probabilities affect the
packages for decision recommended decision alternative. The drawback to this approach is the numerous
analysis make it easy to
calculate these revised calculations required to evaluate the effect of several possible changes in the state-
scenarios. of-nature probabilities.
For the special case of two states of nature, a graphical procedure can be used to
determine how changes for the probabilities of the states of nature affect the
recommended decision alternative. To demonstrate this procedure, we let p denote
the probability of state of nature s 1 ; that is, P(s 1 ) ¼ p. With only two states of nature
in the PDC problem, the probability of state of nature s 2 is:
Pðs 2 Þ¼ 1 Pðs 1 Þ¼ 1 P
Using Equation (13.4) and the payoff values in Table 13.1, we determine the
expected value for decision alternative d 1 as follows:

EVðd 1 Þ¼ Pðs 1 Þð8Þþ Pðs 2 Þð7Þ
¼ pð8Þþð1 pÞð7Þ
¼ 8p þ 7 7p ¼ p þ 7 (13:6)

Repeating the expected value computations for decision alternatives d 2 and d 3 ,we
obtain expressions for the expected value of each decision alternative as a function of p:

EVðd 2 Þ¼ 9p þ 5 (13:7)

EVðd 3 Þ¼ 29p 9 (13:8)

Thus, we have developed three equations that show the expected value of the three
decision alternatives as a function of the probability of state of nature s 1 .
We continue by developing a graph with values of p on the horizontal axis and the
associated EVs on the vertical axis. Because Equations (13.6), (13.7) and (13.8) are
linear equations, the graph of each equation is a straight line. For each equation, then,
we can obtain the line by identifying two points that satisfy the equation and drawing a
line through the points. For instance, if we let p ¼ 0 in Equation (13.6), EV(d 1 ) ¼ 7.
Then, letting p ¼ 1, EV(d 1 ) ¼ 8. Connecting these two points, (0,7) and (1,8), provides
the line labelled EV(d 1 ) in Figure 13.5. Similarly, we obtain the lines labelled EV(d 2 )
and EV(d 3 ); these lines are the graphs of Equations (13.7) and (13.8), respectively.
Figure 13.5 shows how the recommended decision changes as p, the probability of
the strong demand state of nature (s 1 ), changes. Note that for small values of p,
decision alternative d 1 (small complex) provides the largest expected value and is
thus the recommended decision. When the value of p increases to a certain point,
decision alternative d 2 (medium complex) provides the largest expected value and is
the recommended decision. Finally, for large values of p, decision alternative d 3
(large complex) becomes the recommended decision.
The value of p for which the expected values of d 1 and d 2 are equal is the value of
p corresponding to the intersection of the EV(d 1 ) and the EV(d 2 ) lines. To deter-
mine this value, we set EV(d 1 ) ¼ EV(d 2 ) and solve for the value of p:

p þ 7 ¼ 9p þ 5
8p ¼ 2
2
p ¼ ¼ 0:25
8

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