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204 Applying simulation to decision problems
you have obtained your $10 000, however, your relative preference for
money received now may decline, and you may then only be prepared
to give up the promise of $1 for each $0.91 you can receive now.
Clearly, if either of the NPV assumptions is seriously violated then
the NPV will not accurately represent the decision maker’s preferences
between sums of money arriving at different points in time. In this
case, converting the NPVs to utilities might not lead to a ranking of
investment options, which reflects the decision maker’s true preferences.
It is therefore worth questioning whether the implicit assumptions of
NPV are reasonable before applying the method.
Modeling dependence relationships
So far we have assumed, for simplicity, that all the probability distributions
in our models are independent. In reality, it is possible that
the value of some variables will depend upon the value of others. For
example, in the Alpha and Beta machine problem it is possible that
the maintenance costs will be related to sales revenue, since high sales
revenue implies high production levels and hence more wear and tear
on machinery. Similarly, the year 2 sales revenue may be closely related
to that achieved in year 1 since, for example, high sales in year 1 may
signify that the product is popular and hence increase the probability of
high sales in subsequent years.
Where these types of relationships exist, a number of methods have
been proposed for simulating them. One approach (see Hertz 8 ) is referred
to as conditional sampling. This involves the elicitation of a series of
probability distributions for one of the variables (the dependent variable)
with each distribution being elicited on the assumption that a
particular value of the other variable (the independent variable) has
occurred. For example, we might ask the decision maker to estimate
a probability distribution for annual delivery costs given that a certain
sales level will be achieved. We would then repeat the process
for other possible sales levels. When the simulation generated a sales
level it would then also automatically generate delivery costs from
the appropriate distribution. Obviously, this approach becomes impractical
if there is a very large number of possible sales levels. Other
approaches have therefore been developed (e.g. Eilon and Fawkes 9 and