PREFACE
Your project involves developing a new gold mine. A wastewater plant is needed to capture heavy metals from the mine's water discharge. In the last article, we evaluated three alternatives for acquiring the wastewater treatment facility:
| Used | Rent a used, skid-mounted wastewater plant |
| Skid | Buy a new, skid-mounted plant |
| Fixed Buy a new, fixed plant | |
These alternatives were evaluated with decision tree analysis and found to be remarkably similar. Expected costs range from $1120M to $1136M (M=000's). Figure 1 shows the initial decision model. There are two chance events affecting the cost outcome: (1) the time until the wastewater plant becomes operational and (2) the time until all other mine development activities are complete. The longer outcome controls when the mine can begin production.
The recommended decision policy is to choose the alternative having the lowest expected cost. Schedule and performance criteria are explicitly incorporated into the analysis as cost equivalents, based upon their impact on mine value.
VALUE OF INFORMATION
Creating good decision alternatives is an important project management function. Often, there are opportunities to do additional analysis or to otherwise get more information before making a resource commitment.
Figure 1. Current Decision Model
Usually the information, although imperfect, is useful in revising prior judgments. Here are example sources of additional information:
- Further analysis, e.g., modeling in greater detail
- Prototyping, market studies and similar directed research
- Obtaining and analyzing more data
- Spending additional effort in eliciting judgments about risks and uncertainties.
Figure 2. Expanded Decision Model
Time permitting, it might always appear that additional analysis is desirable. However, we should weigh the cost, any delay, and other possible impacts against the value added.
Often, in project management, we need to assess the value of additional information. This is calculated by inserting an additional chance node in the tree to represent the outcome of future information-gathering. The new information is used to revise certain assessments, usually prior probabilities.
Here is the equation to determine the value of additional information:
(value of additional information)= (value of new decision tree) - (value of old decision tree)
In this calculation we temporarily ignore the added cost, if any, of acquiring the additional information.
A similar concept is the value of control. Here, “control” means being able to influence, or partially influence, the outcome of chance events. Examples of controlling or affecting chance events include:
- Acquiring services or products through turnkey contracts
- Limiting losses with insurance
- Protecting commodity prices by forward-selling production
- Using higher-grade materials to reduce chance of failure
- Using conservative operating practices
- Investing in redundancy and flexibility.
Valuing additional control is done similarly to valuing additional information: Compare the new control alternative with the previous best alternative.
The value of information and control concepts often lead people to search for other, new alternatives with desirable characteristics.
Bayesian Process
For persons who are interested in how prior probability assessments are revised based upon new information, here is a brief explanation. The process is credited to Rev. Thomas Bayes, an 18th century British clergyman. Bayes’ theorem is most often stated as:
| where | ei | = | outcome of the chance event of interest |
| A | = | attribute (additional information) and P (A I ei) is read as “the probability of A given ei” |
While the formula is more ominous than difficult, a more convenient solution technique is to construct and inspect a joint probability table. The two nodes in Figure 3 can be expressed in the following table.
The values inside the table are the joint probabilities, i.e., probabilities for the compound events comprised of combinations of Evaluation Outcome and Months to Complete Plant as shown in Figure 3.
The probabilities for the inverted tree, shown below, can be read from the table by inspection. For example, let's assume we have a “Favorable” Evaluation Outcome. There is a .44 chance of this occurring. When it does, we are restricted to the left column of the table, which has three possibilities for Months to Complete Plant: {3, 4, or 12]. The probabilities of the Months to Complete Plant Outcomes are in proportion to { .25, .14, or .05]. However, these values do not sum to 1. They must be normalized in order to total 1. This is accomplished by dividing each number by the column total. The normalized numbers then represent the revised probabilities conditional on the “Favorable” Evaluation Outcome.
Bayesian revision usually requires a more detailed explanation plus considerable practice before the user is comfortable with the process.
NEW ALTERNATIVE FOR THE WASTEWATER PLANT
Let's now consider an information alternative for the wastewater plant. This is to “Evaluate Used Plant”: Spend one month testing and evaluating the used rental plant to determine when it could be operational. The evaluation will not change the lead time, but would provide us with information to reassess the lead time probabilities.
This testing and evaluation would be a necessary step in the lead time for this alternative, anyway. However, lead times for other alternatives would be the same from the date of commitment. For example, if you spend one month evaluating the used plant and then decide to purchase a new one, then the new wastewater plant acquisition would be delayed one month.
The initial cost of the “Evaluate Used Plant” alternative is a non-refundable $30M (pre-tax) deposit. This would apply to the installation expense if you decide to proceed with the used unit.
Figure 2 shows the expanded decision model. Here, decision tree shorthand is used to illustrate the structure in compact form: A copy of every following node is implicitly attached to the end of each branch of the preceding node. In this series, the nodes are numbered to help the reader correlate the figures.
REVISING PROBABILITIES
The evaluation and testing provide information useful for revising the prior probability assessments for the “Time Plant Operational” nodes. Note that the information node in this section of the tree is before the investment decision. This is necessary for the new information to add value.
Figure 3. Used Plant Prior Probabilities (Note: This is not the same as the decision tree used in the example.)
For convenience and simplicity, we'll classify the evaluation results into two outcomes: “Favorable” and “Unfavorable.” Your engineers judge the quality of this imperfect information as shown in Figure 3. These nodes are not numbered because they relate the information in the sequence the probabilities are assessed. The sequence of these two nodes is reversed in the decision model. The probabilities for the Evaluation Outcomes are assessed conditional upon the Time Plant Operational event outcome. Representing conditional probabilities is a natural and powerful feature of decision trees.
The representation in Figure 3 is the natural way to express the assessments and relationships between the two events. However, the sequence is not in the order needed to solve our real world problem. The actual sequence would be (1) get the evaluation result, (2) make the plant decision, and (3) observe how long it takes to complete the plant. We need to reverse or invert the nodes. This is done using a straightforward technique called Bayesian revision (using Bayes’ theorem). We revise the prior probabilities about Time Plant Operational based upon the Evaluation Outcome. The original and revised probabilities for each “Time Plant Operational” outcome are shown in Table 1 and in Figure 4.
EVALUATING THE NEW ALTERNATIVE
Figure 4 shows the partial decision model with the tree extended for the “Evaluate Used Plant” alternative. The Used plant evaluations show the revised probabilities and expected cost calculations. The values along the right side of the diagram are terminal values. These are the present value costs for the respective paths through the tree. These costs include an implicit penalty for value lost for a delayed mine opening.
Figure 5 shows the calculations for the expected cost of the delayed skid alternative given the “Examine Used Plant” choice. This value changed from $1120M to $1134M because of the added evaluation cost (about $18M after-tax) less the present value timing adjustment (about $4M) of delayed investment.
The new information adds value only if a decision is potentially changed. In our example, the project manager will choose to install the Used rental plant, instead of the skid plant, if the evaluation result is favorable.
The Evaluate Used Plant option is found to be the best alternative. Spending $30M pre-tax on inspection adds a net $62M (l120-1058) of after-tax value to the project.
The decision tree was back-solved to compute the expected cost for each alternative. Starting at the right, each chance node is replaced (annotated) with its expected cost; each decision node is replaced (annotated) with its best alternative. Here are some example calculations, starting at the upper-right of Figure 4 and working backward:
The previous article provided another, more detailed example of a decision tree evaluation.
Table 1. Probabilities for Used Plant
Figure 4. Additional Information Alternative
Figure 5. Delayed Skid Alternative
CLOSING REMARKS
Many projects have a sequence of decision points. Analyzing these options are an extremely important part of an evaluation. Decision trees are especially useful for analyzing such situations.
This article extends an example decision tree analysis to value an additional information alternative. Many projects have such opportunities to improve value. In most cases, it is convenient to use the information to revise prior probabilities. Alternatively, one can revise chance assessments for event outcomes. It is sometimes appropriate to revise both event probabilities and values.
We have seen how increasingly complex problems can be handled in an orderly way using decision tree analysis. For some problems, the tree can become exceedingly large. In these situations, other ways may be more convenient for calculating expected values. In the next article in this series we'll use the wastewater plant example to demonstrate another popular technique, Monte Carlo simulation. For some decision problems, this method provides an elegant and more flexible way to calculate expected value (or costs).
John R. Schuyler, PE, CMA, is principal of Decision PrecisionSM, an Aurora, Colorado, firm providing training and assistance in risk and economic decision analysis. Mr. Schuyler teaches Petroleum Risks and Decision Analysis worldwide in association with Oil & Gas Consultants International. His services focus on modeling capital investments, acquisitions, and other corporate planning decisions. He received B.S. and M.S. degrees in engineering from Colorado School of Mines and an M.B.A. from the University of Colorado. His prior experience includes vice president and evaluation engineer with the nation ‘s fifth largest bank, planning and evaluation analyst for a major oil company, and senior management consultant with a national CPA firm.