CHAPTER 6 THE COMPLEXITY OF REAL-WO

CHAPTER 6 THE COMPLEXITY OF REAL-WORLD PROBLEMS 161
the swing is obtained by evaluating the model for the 10 percent
and 90 percent points for each variable and see Figure 10–17.)
4. Thus, the square of the swing for a variable determines how
important it is to the overall uncertainty.
Accordingly, one easy way to compare the importance of uncertainty in
the variables is to square the swing and express it as a percentage of the sum
of squares of all the swings. The final column in Figures 6–10 and 6–12 show
this swing. (Technically, the variation for Technical Success for the DiagStatic
New alternative is not a 10/50/90 variation and distorts the overall
uncertainty figures shown in the final column.)
Thus, we see that, for the DiagRF Plus alternative, the top four variables
(peak market share, initial price, RF fraction of market in 2010, and average
selling price) account for 95% of the uncertainty. In most problems with more
variables and complex models, we find that the top three or four variables
capture 80 to 90 percent of the effects of uncertainty.
Remember, though, that for this approximation to be valid, we need to
use probabilistically independent variables. With dependent variables, use
the joint sensitivity as illustrated above. This joint sensitivity must then be
probabilistically independent from the other variables listed.

Probabilistic Evaluation: Building and Pruning the Tree
Even after deterministic sensitivity analysis, it is often difficult to limit the
tree to a reasonable size. If there are, for instance, n nodes in a tree and each
node has three branches, a symmetric tree will have 3n paths. For example,
for seven nodes, we would have 2,187 paths. If the model takes 1 second to
calculate an answer (not untypical of a large spreadsheet model on a personal
computer), the evaluation command will take a little over half an hour to
execute. If we add several more nodes, change a node from three to four
branches, or construct a more elaborate model, we will have a tree that is
impractical to evaluate on a personal computer.
How large a tree is reasonable? This depends greatly on the type of
problem (and on the opinion of the facilitator). However, we feel that for most
problems, 50 to 200 paths per alternative is sufficient. This number of paths
allows us to include the three generic uncertainties that often affect an
alternative: uncertainty about the growth of the market, uncertainty in
competitive action or reaction, and uncertainty in how well we will fare. At
three branches a node, we have 27 paths per alternative, leaving room for
several other nodes if called for. After the full-scale analysis is complete, we
will probably find that around 20 paths per alternative are enough to draw
all the conclusions. Reducing the tree to this size is often important for
clarifying the results and drawing the tree for the final presentation to the
decision-maker.
How can you make your tree small enough to evaluate? By reducing the
number of branches at each node, by reducing the number of nodes, and by
creating asymmetric trees. Simplifying the model can also help by reducing