Page 302 - DCOM303_DMGT504_OPERATION_RESEARCH

Unit 14: Decision-making

If the probability distribution of X admits a probability density function f(x), then the expected Notes
value can be computed as

E(X)    xf(x)dx
It follows directly from the discrete case definition that if X is a constant random variable, i.e.
X = b for some fixed real number b, the expected value of X is also b.

The expected value of an arbitrary function of X, g(X), with respect to the probability density
function f(x) is given by the inner product of f and g:

E(g(X)) =    g(x)f(x) dx
Using representations as Riemann-Stieltjes integral and integration by parts the formula can be
restated as
E(g(X)) = a   g(x)dP(X  x) g(a)  a   g'(x)P(X  x)dx

if P(X  a) = 1
As a special case let  denote a positive real number, then

E(|X| ) = 0    1 P(|X| t)dt


t
In particular, for  = 1, this reduces to:

E(X) = 0  1 {  F(t)}dt ,
if P[X  0] = 1

14.6 EVPI

In probabilistic situation, there in no control over the occurrence of given state of nature.
However, what will happen, if decision maker had exact information about the occurrence of
particular state of nature.

The Expected Profit with Perfect Information (EPPI) is the maximum attainable Expected Monetary
Value (EMV) based on perfect information about the state of nature that will occur. The expected
profit with perfect information may be defined as the sum of the product of best state of nature
corresponding to each optimal course of action and its probability.
The Expected Value of Perfect Information (EVPI) may now be defined as the maximum amount
one would be willing to pay to obtain perfect information about the state of nature that would
occur. EMV* represents the maximum attainable expected monetary value given only the prior
outcome probabilities, with no information as to which state of nature will actually occur.
Therefore, perfect information would increase profit from EMV* up to the value of EPPI. This
increased amount is termed as Expected Value of Perfect Information (EVPI),
i.e., EVPI = EPPI – EMV

Example: A wholesaler of sports goods has an opportunity to buy 5,000 pairs of skiis, that
have been declared surplus by the government. The wholesaler will pay ` 50 per pair and can
obtain ` 100 a pair by selling skiis to retailers. The price is well established, but the wholesaler
is in doubt as to just, how many pairs he will be able to sell. Any skiis left over, he can sell to
discount outlets at ` 20 a pair. After a careful consideration of the historical data, the wholesaler
assigns probabilities to the demand as shown in Table 14.4.

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