162 PROBABILISTIC EVALUATION: BUILDING AND PRUNING
THE TREE
the time required to evaluate the tree. We consider each of these options in turn.
What is a reasonable number of branches at a node? Our experience
indicates that chance nodes are usually well approximated by three branches.
Going to four or more branches seldom perceptibly affects the overall profit
distribution. While we can use two branches for uncertainties that deterministic
sensitivity analysis shows to be less important, there is no central branch
in a two-branch node to trace the effect of the base case. With decision nodes,
we can use the preliminary evaluation to eliminate the inferior alternatives
and narrow down to the three or four really distinct and most promising
alternatives.
What is a reasonable number of nodes in a tree? A good number to aim
for is five or six nodes. Normally, there are too many nodes in the initial version
of the tree, and the tree has to be pruned before it can be evaluated. In pruning
the tree, four types of nodes compete for a place on the final tree: decision
nodes, chance nodes with effects common to all alternatives, chance nodes
with effects that distinguish between alternatives, and chance (or decision)
nodes that are not really important to the tree but should be included for
political reasons.
By using the results of the deterministic sensitivity analysis, we can
usually manage to discuss and eliminate the politically important nodes from
consideration. Indeed, this may be one of the important insights from the
analysis. If necessary, chance nodes with effects common to all alternatives
can be combined into one or two nodes whose branches represent scenarios
(combinations of events). If there are still too many nodes, we must be creative
in combining variables, restructuring the tree, and using further sensitivity
analysis to narrow down the number of important uncertainties even more.
In cases of real necessity, we may have to run a separate tree for each
alternative. If we do this, we will have to do some calculations by hand to
reorder the tree (i.e., to obtain value of information), but we can still have
software do most of the work for us.
Creating asymmetric trees is another way to reduce tree size. For
instance, when uncertainty is important for one alternative but not for
another, we can have the chance node for this uncertainty follow only the
alternative for which it is important. Similarly, an uncertainty may be
important only for certain branches of a chance node, in which case you can
have the node follow only those branches.
Finally, we can simplify the model to shorten its running time and, thus,
the time required to evaluate the tree. Models can be simplified by eliminating
calculations of quantities that were initially of interest but that are not
necessary for calculating the net present value of cash flow. Models can
similarly be simplified by replacing complicated calculations with simplified
ones that produce approximately the same results.
In general, beginning with a small version of the tree is better, even if you
are using a software program to evaluate the tree. Nothing is as discouraging as starting off by running up against multiple errors from hardware/software