# Decision analysis in projects

## decision trees

Concerns of Project Managers

PM Tutorial

Editor's Note: This is the third in a series of articles describing decision analysis in projects. The first two articles laid the groundwork for decision making under uncertainty based upon value. This third installment begins tying together those concepts using an example decision tree analysis.

### PREFACE

The first article in this series discussed the expected value concept and how probability distributions can express judgments about uncertainty. Expected value (EV), the mean or probability weighted average, was shown to be the objective (unbiased) outcome predictor. The second article recommended maximizing expected monetary value (EMV) as an optimal decision policy suited to most large business organizations. That article included brief illustrations of the EMV decision rule using a payoff table and a decision tree.

In this and the next installments, a more comprehensive decision tree model illustrates the practice of decision analysis in project management.

### DECISION TREE CONSTRUCTION

A decision tree is graphical representation of EV calculations. The tree consists of decision, chance and terminal modes connected by branches. The diagram acts as a blackboard to document our understanding of a situation. This facilitates team collaboration, communication and instruction.

Theoretically, any decision, no matter how complex, can be analyzed using a decision tree analysis. Other alternatives, especially Monte Carlo simulation, have advantages and disadvantages for some problems. Decision tree analysis is especially suited to quick-and-dirty everyday problems where one simply wants to pick the best alternative.

There are three types of nodes in a decision tree:

• Decision nodes represented by squares: These are variables or actions the decision maker controls.
• Chance event nodes represented by circles: These variables or events cannot be controlled by the decision maker.
• Terminal nodes represented diagrammatically by unconnected branches. Usually an outcome value is attached.

By convention, the tree is drawn chronologically from left to right. Analysis lore includes many biological analogies. The starting decision node is called the root, and the radial lines are called branches (and, sometimes, twigs). The terminal nodes are sometimes called leaves. An overly-complex tree bears the label, a bushy mess.

A decision tree diagram is usually annotated with these numbers:

• Outcome values, representing present values of the resulting cash flow stream, discounted to the date of the root decision node. Sometimes, benefits and costs are placed along branches as they are realized. More commonly, all or most of the outcome value is assigned to terminal nodes.
• Node values, calculated when solving the tree.
• Probabilities for each outcome branch emanating from chance nodes.

Solving the decision tree is a simple back-calculation process. Starting from the terminal nodes at the right, each node is labeled with its EV (expected value). Usually, these EVs are EMVs (expected monetary value) when we measure value in dollars.

Here are the three simple rules for solving a tree:

• A chance node's value is the EMV of its branches.
• A decision node's value is the value of the best alternative, using the EMV decision rule.
• A present value cost or benefit value that was placed along a branch is subtracted (costs) or added (benefits) when traversed.

### EXAMPLE PROJECT DECISION

You are managing a new gold mine development. The mine will be turned over to someone else to manage for production. The “project,” for your purposes, is complete when the mine commences operation.

This week you learned that certain contaminant levels are above those seen in the original survey samples. The regulatory agencies have convinced you (mandated) that an improved treatment facility is required to recover heavy metals.

The mill and other mine development activities are nearly complete. This extended pollutant recovery capability was not designed into the nearly-completed mill facility.

The best general solution is to construct or bring in a wastewater module, referred to as the “Plant.”

In addition to the investment and operating costs of the Plant, adding this capability has the potential to delay mine startup.

### Data Assumptions

There are uncertainties affecting the development project's completion time. For all “Other” activities excepting the wastewater system, your engineers judge the completion time uncertainty in three scenarios as shown in Table 1. (For convenience, decimal notation will be used for probabilities, throughout, instead of percents.)

For simplicity, we'll assume that extending construction activities to meet a later startup date will not affect their costs. Further, their completion times are independent of the wastewater treatment system.

You are considering three Plant alternatives from suitable vendors:

Used       Buy a used, skid-mounted wastewater plant

Skid        Buy a new, skid-mounted plant

Fixed   Buy a new, fixed plant

These labels will be used for convenience in the discussion. Figure 1 shows how the root decision node will appear, adding a branch for another alternative to be evaluated later.

Table 1. Completion Time for All Activities Except Wastewater Plant

 Probability Other Activities Complete .30 in 3 months .40 in 4 months .30 in 5 months 1.00

Figure 1. Decision node

Figure 2. Chance Node Example

Table 2. Plant Cost Information. All costs are pre-tax \$M (\$000's).

 Expenditure Used Skid Fixed Acquisition + Installation \$650 \$1480 \$1150 Salvage Value \$0 \$600 \$200 Operating Costs Per Month \$8.0 \$6.3 \$5.0

Although costing more initially, a skid-mounted plant costs less and is faster to install. It would also have a greater salvage value at mine abandonment.

Times for the various acquisition options are judged using three representative scenarios for each. Between the vendors and your engineers, three-level discrete distributions are assessed as follows:

Time to Acquire and Install Wastewater Plant

Used  {.3, 3; .35,4; .35, 12 months}

Skid       {.2, 4; .6,5; .2,6 months}

Fixed  {.2, 5; .5,6; .3,8 months]

These pairs of points represent outcomes values (second in each pair) and their associated probabilities (first value, a decimal, in each pair). This notation is a popular way for expressing a discrete chance event. Figure 2 illustrates what the “Time Plant Operational” chance node for the “Skid” option looks like as a chance node.

An important consideration is the cost (actual or opportunity) of delaying the mine startup. From the project feasibility model, we determine that changes to mine startup date are valued at \$150M per month (pre-tax). The customary financial abbreviation “M” will be used for thousands. The mine is projected to operate for six years. Other information about the options are shown in Table 2. We'll assume these values are known sufficiently well that single-value estimates can be used. The model details will be described in later installments. The cash flow projection assumptions include such items as inflation, cost of capital, depreciation and taxes.

If you have a great deal of experience in problems of this sort, perhaps you can judge the best alternative. Try estimating EV costs for each option. How much better is the first choice over the next best alternative?

For most of us, even this small problem is much too complex to internalize in our heads. The Plant choice is not obvious because each alternative has advantages:

Used    Lowest initial cost.

Skid     Fastest and lowest-risk acquisition time, medium operating costs, highest salvage value.

Fixed    Lowest operating costs, medium investment.

We need some type of analytic tool to have any confidence in our selection.

### Evaluating Options

Let's use the appraisal approach of decision analysis to evaluate the three plant acquisition alternatives. There are two uncertainties affecting value: the time the Plant becomes operational and the time all Other development activities are complete.

Figure 3 shows the complete decision tree model. Terminal nodes (rightmost values) show outcomes realized by the respective path traversed in the tree. The outcome value is present value costs. Since all monetary values are costs, it is cleaner, i.e., no minus signs, to show EV costs instead of EMVs. Minimizing EV cost is exactly equivalent to maximizing EMV.

The root node is the Type Plant decision. There are two chance events affecting outcome value: (1) time for the wastewater plant to become operational and (2) time for all other mine development activities to be complete. Whichever event is later controls when the mine development project is complete. These chance event outcomes are annotated with labels (months to complete) and the probability of the respective outcome. Each chance and decision node is annotated with its EV cost.

The three alternatives are evaluated by back-solving the decision tree. Table 3 shows the sequence of node EV calculations.

Note that there are fewer calculations at each column of nodes as we progress from right to left. For large trees, the progressive thinning of branches provides considerable calculation efficiency.

Acquiring a skid-mounted plant (\$1120M) appears the best of the three alternatives, although its superiority is slight (\$16M). The “cut” marks indicate alternatives that have been “pruned.”

Later in this series, we will examine a fourth, new alternative which is clearly superior.

Figure 3. Evaluating Plant Options

### CLOSING REMARKS

This article describes decision tree analysis as a convenient way to analyze project decisions having one or several important uncertainties. Probability distributions are used to encode judgments about risk and uncertainty. For all but the simplest of problems, decision models greatly bolster our intuition.

A single value measure, measuring goodness or progress toward the organization's objective, is a central idea in decision analysis. Money is a convenient measurement scale for most purposes. Notice how this example combined cost, schedule and performance criteria into a single monetary value measure, EV costs:

• Schedule was translated into cost equivalents by recognizing the value lost due to project delay.
• Performance was recognized by different plant operating costs during the mine life.

Many practitioners believe that insights gained into decision problems are more important than the numerical results. The largest part of a decision analysis effort is usually constructing the project cash flow model. These ideas and this example will be continued into the next articles in this series.

John R. Schuyler, PE, CMA, is principal of Decision Precision, 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.

JULY 1993

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