## The Basic Tree for Projects

Figure 4-3 shows the basic layout. It is customary, as described by John Schuyler in his book, Risk and Decision Analysis in Projects, Second Edition, [31 to show the tree laying on its side with the root to the left. Such an orientation facilitates adding to the tree by adding paper to the right. We use a somewhat standard notation: the square is the decision node; the decision node is labeled with the statement of the decision needed. Circles are summing nodes for quantitative values of alternatives or of different probabilistic outcomes. Diamonds are the starting point for random variables, and the triangle is the starting point for deterministic variables.

Summing nodes

In Figure 4-3, the decision maker is trying to decide between alternative "A" and alternative "B". There are several components to each decision as illustrated on the far right of the figure. Summing nodes combine the disparate inputs until there is a value for "A" and a value for "B". An example of a decision of this character is well known to project managers: "make or buy a particular deliverable."

Since our objective is to arrive at the decision node with a quantitative value for "A" and "B" so that the project manager can pick according to best advantage to the project, we apply values in the following way:

■ Fixed deterministic values, whether positive or negative, are usually shown on the connectors between summing nodes or as inputs to a summing node. We will show them as inputs to the summing node.

■ Random variable values are assigned a value and a probability, one such value-probability pair on each connector into the summing node. The summing node then sums the expected value (value * probability) for all its inputs. Naturally, the probabilities of all random variables leading to a summing node must sum to 1. Thus, the project manager must be cognizant of the 1-p space as the inputs are arrayed on the decision tree.

Figure 4-4 shows a simple example of how the summing node works. Alternative "A" is a risky proposition: it has an upside of \$5,000 with 0.8 probability, but it has a downside potential of -\$3,000 with a 0.2 probability. "A" also requires a fixed procurement of \$2,000 in order to complete the scope of "A". The expected value of the risky components of "A" is \$3,400. Combined with the \$2,000 fixed expenditure, "A" has an expected value of \$5,400 at this node. The most pessimistic outcome of "A" at this node is -\$1,000: \$2,000 - \$3,000; the most optimistic figure is \$7,000: \$2,000 + \$5,000. These figures provide the range of threat and opportunity that make up the risk characteristics of "A".

S2,000 + S4,000 - \$600 = S5.400 Figure 4-4: Summing Node Detail.

Now, let's add in the possibility of event "A4". The situation is shown in Figure 4-5. If the project team estimates the probability of occurrence of "A4" as 0.4, then the probability of the events on the other leg coming into the final summing node becomes equal to the "1-p" of "A4", or 0.6. Adding risk-weighted values, we come to a final conclusion that the expected value of "A" is \$4,840.

Summing node -0.0 * \$5,000 + 0.2 1 (-33,000} = 53,400

Summing node -0.0 * \$5,000 + 0.2 1 (-33,000} = 53,400

S4.000 = \$3,240 + S1.600 = 54,940 Figure 4-5: Summing Node A.

The most pessimistic outcome of "A" at this node remains -\$1,000 since if "A2" should occur, "A-i" and "A4" will not; the most optimistic figure remains \$7,000 since if "A-i" occurs, then the other two will not. If "A4" should occur, then "A1" and "A2" will not. However, "A3" is deterministic; "A3" always occurs. So the optimistic value with "A4" is \$6,000: \$2,000 + \$4,000. Obviously, \$6,000 is less than the outcome with "A1".

If the analysis of "B" done in similar manner to that of "A" should result in "B" having a value less than \$4,840, the decision would be to pick "A". Of course, the risk tolerance of the business must be accommodated. At the decision node there will be an expected value for "A" and another of "B". The project manager can follow the tree branches and determine the most pessimistic outcomes. If the most pessimistic outcomes fit within the risk tolerance of the business, then the outcome, "A" or "B", is decided on the basis of best advantage to the project and to the business. If the most pessimistic outcomes are not within the risk tolerance of the business, and if there is not a satisfactory plan for mitigating the risks to a tolerable level, then the choice of project defaults to the decision policy element of picking on the basis of the risk to the business. Risk managing the most pessimistic outcome is a subject unto itself and beyond the scope of this book.

By now you may have picked up on a couple of key points about decision trees. First, all the As and Bis must be identified in order to have a fair and complete input set to the methodology. The responsibility for assembling estimates for the As and Bis rests with the project manager and the project team. Second, there is a need to estimate the probability of an occurrence for each discrete input. Again, the project team is left with the task of coming up with these probabilities. The estimating task may not be straightforward. Delphi techniques t4] applied to bottom-up estimates or other estimating approaches may be required. Last, the judgment regarding risk tolerance is subjective. The concept of tolerance itself is subjective, and then the ability of the project team to adequately mitigate the risk is a judgment as well.