Decision analysis in projects

We will continue the discussion about the gold mine development project. A wastewater plant is needed to capture heavy metals from the mine's water discharge. In the last article, as shown in Figure 1 (see next page) and Table 1, four alternatives were considered for acquiring the wastewater treatment facility.

The chief uncertainties recognized in this analysis are (1) the time to implement the plant choice and (2) the longest completion time of all other mine development activities. Fixed assumptions include items such as mine life, plant operating costs, salvage value, and delay cost (per month impact on mine value of delaying or accelerating startup).

This article presents an alternative calculation technique that offers, in some cases, important advantages over decision tree analysis.

DIFFICULTIES IN EXTENDING THE TREE ANALYSIS

Decision tree analysis provides a logical, credible basis for the wastewater plant decision. So, why introduce another technique? Because some problems are very cumbersome to solve as decision trees.

First, consider the three base plant type alternatives. There are 27 possible outcomes:

•    3 decision alternatives
• x 3 time plant operational outcomes
• x 3 time other mine development activities complete outcomes.

The paths in the Evaluate option part of the tree add another 54 outcomes. It is easy to see how adding a few more uncertainties leads to thousands of decision tree paths to be evaluated.

Second, uncertainties must be represented as discrete probability distributions in decision tree analysis. For example, the Time Other Mine Activities Complete was characterized as being exactly three, four, or five months. However, for this variable, we considered only three completion time scenarios. In reality, the time could be measured in weeks, days or possibly hours. A continuous distribution would better represent our view of this variable, but discrete outcomes were used as a simplification. Additional outcome branches on the chance nodes would provide finer resolution, but this worsens the combinatorial explosion. A decision tree can easily become a bushy mess.

MONTE CARLO SIMULATION

Monte Carlo simulation does not suffer the two difficulties mentioned above. While not a panacea, it has advantages in certain situations. It allows a richer, more detailed representation, which can sometimes be important. For example, small differences are important in competitive bidding.

Conceptually, either decision tree analysis or simulation can solve any decision problem. However, there are advantages and disadvantages for either technique. The choice depends upon the problem and tools at hand.

Monte Carlo simulation has been around at least as long as computers.

“Monte Carlo” was the code name of a secret project during WWII. Legendary mathematician John von Neumann is credited with inventing the technique while designing atomic bombs. He saw that a simple, yet elegant, sampling technique solves certain mathematical problems that are otherwise impossible. It solves for expected value, i.e., the mean or probability-weighted average of a probability distribution. A valuable side-benefit is that outcome probability distributions are also obtained easily. Hereafter, I'll refer to Monte Carlo simulation as simply simulation. It is an alternative way to solve for expected value.

Simulation is based upon two essential elements: (1) a model that is used to measure outcome value and (2) a technique that samples input probability distribution values.

Inputs as Distributions

Probability distributions are used to represent judgments about uncertainty. For example, someone's forecast about time to complete an activity is better represented by a distribution than by a single-value estimate. This represents the expert's opinion about the range of possible outcomes and the relative likelihood of outcome values within that range.

Figure 1. Decision Tree Model

Figure 2. Probabilistic Model

Figure 3. Probability Density Distribution for One Input

Figure 4. Cumulative Probability Distribution

The foundation for simulation is sampling probability distributions at random. We generate—sample—many possible project scenarios and then examine the outcome distributions. These trials, in sufficient number, preserve the shape and other characteristics of the original probability distributions.

Every trial pass through the model generates a plausible scenario. We can examine extreme cases to see what conditions could give rise to these outlier results. The process is appealing because it is believable; we do not face a black box solution. A simulation model is a straightforward extension to the customary, single -valued deterministic model (so called because every input is singly determined). The evaluation model is relatively simple and understandable.

Figure 2 shows, conceptually, how a conventional, deterministic model is extended for a simulation analysis. The deterministic model often need not be changed. We must ensure that the model remains valid over the ranges and combinations of possible input values.

If some of the inputs are probability distributions, then the outputs are, quite naturally, going to be probability distributions. Instead of a single value measure, such as present value (PV), a distribution for value is obtained. Time spread variables, such as net cash flow (NCF), can be graphed as expected value projections and confidence curves.

The model is surrounded by an iterative loop that controls generating—usually several hundred—trial solutions. Here is a summary of the simulation process:

1. Trial values are sampled from probability distributions representing several random, or stochastic, variables.
2. The values are put into the deterministic model, and the outcome values are calculated.
3. Selected outcome values for this trial, e.g., time and cost to complete, are stored in a file.
4. Return to Step 1 and repeat until the number of trials is sufficient to provide the required level of precision.
5. Analyze the stored results.

When the trials are complete, the stored values are analyzed. We can compute expected values, probability distributions, mean time spread projections, and time spread confidence bands. For example, for the value distribution outcome in Figure 2:

• Averaging the PV (present value) approximates expected monetary value (EMV).
• Aggregating PVs into groups by size permits displaying them as a histogram providing the approximate shape of the probability distribution.
• Sorting the PVs by magnitude and displaying PV as a function of rank yields the approximate shape of the cumulative probability distribution.

The precision of the approximation improves as the number of simulation trials increases.

How Sampling Works

Sample, trial values for random variables are obtained by a simple process. Figures 3 and 4 show how this works. The best judgment about the Time to Complete Other Activities is the probability distribution of Figure 3. This is a normal distribution with a mean of 4 and .775 standard deviation. This is our designated expert's forecast for this variable.

In the decision tree analysis, we abstracted this into the 3-value discrete distribution shown in Table 2.

With simulation, this abstraction is not needed, and we may use the full sweep of possible outcomes for this uncertainty.

Simulation software often allows directly entering the probability density distribution as an input assumption. However, conceptually, the sampling process uses the cumulative form of the distribution.

The curve of Figure 3 is converted into the cumulative curve of Figure 4. The conversion is straightforward. The probability density distribution has a normalization requirement that the area under the curve equal 1. The cumulative distribution is obtained by simply adding the area (i.e., integrating) as the curve is swept from left to right.

As in Figure 3, Figure 4 represents the uncertain completion time of all other mine development activities. For any completion time, t, on the x-axis, the curve intercept is the probability that the outcome will be less than t.

Let's see how this works by sampling the curve in Figure 4. For a single pass through the simulation model, we want to determine a trial value for this activity completion time. A random number generator function is used to randomly obtain a sampling parameter between 0 and 1, i.e., a random number. All values in this range are equally likely. The sampling parameter can be mapped to a cumulative probability. This is done by finding the random number value on the y-axis, moving right to the cumulative curve, and then moving down to the corresponding value on the x-axis.

Suppose we obtain a sampling parameter of, say, 0.685 from the random number generator. This corresponds to 4.4 weeks on the x-axis. This trial value is substituted into the model. When random variables for each distribution have been sampled and substituted into the model, a trial is generated. This is one particular outcome scenario in the simulation run.

If we sample an input distribution many times and place the values into a histogram, the shape obtained approximates the original probability distribution. The key to simulation is that this random sampling process preserves the character of the original distribution. The match improves with more trials and finer histogram divisions.

If we sample a distribution many times and average the results, then we obtain an approximation for the expected value. For the mathematically inclined, random sampling solves for:

where x is the outcome value, and f(x) is the probability density function. The very simple yet elegant simulation process performs the integration for us. In most evaluation problems of interest, the integration of x f(x) is very complex and defies direct mathematical solution.

Figure 5. Decision Model

Table 4. Expected Value Costs

Figure 6. Three Original Alternatives

WASTEWATER PLANT DECISION

Figure 5 shows a revised decision model for the gold mine development project. Continuous probability distributions, rather than discrete distributions, are used to represent activity completion times. Being able to use continuous distributions is an important advantage of simulation over decision trees.

The “Evaluate Used” alternative involves inspecting and testing the Used Unit. In the decision tree analysis, we considered only that the evaluation outcome was either “Favorable” or “Unfavorable.” These outcomes were correlated to the Time to Complete the Used Unit through a joint probability table. On the basis of the evaluation outcome, we were able to revise the probabilities for the different Time to Complete outcomes by using Bayes' rule.

With simulation, one can provide a richer representation. While Bayes-like dependencies can be modeled easily, we have used a more real-to-life description for illustration. The evaluation results in an estimation of Time to Complete for the Used Unit. The uncertainty of this imperfect information is represented by a separate Evaluation Error distribution. Here, we are modeling the estimation process as a function:

Simulation allows a rich representation of risks and uncertainties. Virtually any number of chance events can be represented in the model. This is unlike decision trees, which can quickly become too large to solve.

To further embellish the wastewater plant example, another chance event, Delay Cost, was added. This is the amount of penalty (or benefit) arising from delayed (or accelerated) project mine startup.

Table 3 shows the various distributions used in the simulation model. Note that the discrete distributions used before are now represented by either Normal or Beta distributions with specific values for each parameter involved. The normal distribution has two parameters (mean, standard deviation). Beta distributions are simple functions that can be tailored with two shape exponents to resemble many shapes ranging from normal distribution to highly skewed (either right or left) distributions. Here, they are used to provide bell-shaped distributions with positive skewness.

Table 4 shows the results of a 500-trial simulation run. The numbers, while close to those obtained from the decision tree analysis, are somewhat different due to the increased precision of the continuous distributions and the robustness of the model. Note that the Used Later cost is lower than the Used Now cost because, in this model, the testing and inspecting cost ($30M) is expensed rather than capitalized. This cost, however, reduces the investment required if this alternative is chosen after inspection.

Without the Evaluate option, the decision would be made today by comparing the distributions for the three base plant alternatives. Figure 6 shows the cumulative probability curves l for these options. The Skid alternative has a slight advantage in having the lowest expected value cost. The Skid plant also is the least risky (risk being indicated by the width of the distribution). The “stairstep” nature of the curves is caused by using a cash flow projection model that used monthly time periods.

As in the previous article, we have an option to inspect the Used plant before committing to a plant choice. After taking one month to evaluate the Used plant, the alternatives would be compared. Fixed and Skid alternatives would be delayed one month, thus slightly increasing their expected value costs. The Used alternative is also reassessed, based upon the evaluation information. The decision policy would be to choose the alternative having the lowest expected value cost. Figure 7 compares the Evaluate alternative with the Skid alternative. Pursuing the information alternative (inspection) adds $96M of value to the project².

Simulation is excellent for determining the distribution curves for various outcome parameters of interest. Figure 8 shows a histogram of simulation results for the improvement offered by the Evaluate alternative. Superimposed on the figure is the cumulative distribution. Random sampling and the accumulation buckets typically result in an irregular distribution shape. Although the histogram appears lumpy, this has little impact on the expected value calculation. The cumulative curve appears smoother because it avoids the bucketing effect and dilutes any random sampling errors. Cumulative curves are used, in general, to compare the risk-versus-value profiles of the different alternatives. The simulation model used for this analysis uses discrete time periods. Due to computer programming and running time considerations to do otherwise, monthly periods were used. This contributed to the discontinuities in the resulting distributions. Without too much effort, the model could be modified to use weeks, days, hours, etc., to meet the precision requirements of the study. To minimize computer running time while increasing precision, there are sampling techniques generally described as “design of experiments.” For a recent application, see the tutorial series “Design of Experiments for Project Managers” (Parts I, II, and III) by J.C. Warner in PMNETwork May (pp. 36-40), July (pp. 34-38, 40) and August (pp. 69-74), and “Optimization of Project Coordination Scheduling Through Application of Taguchi Methods,” M.P. Santell, J.R. Jung, and J.C. Warner, Project Management Journal., September 1992 (pp. 5.16).

Figure 7. Evaluation Alternative

Figure 8. Incremental Improvement

SUMMARY

Monte Carlo simulation is a complementary calculation alternative to decision tree analysis. Each technique has its advantages and disadvantages. The nature of the problem at hand and personal preference usuallydetermines the choice of method. Most importantly, simulation is usually preferred for situations:

• Having many important uncertainties and contingencies;
• Involving a portfolio of problems (e.g., decisions involving a collection of projects);
• Where outcome probability distributions are desired, providing additional insights and for comparing risk-versus-value profiles; or
• Involving multiple decision criteria (in situations where it is difficult or undesirable to condense multiple criteria to a single value (objective) function; a forthcoming installment will discuss multi-criteria decisions).

In contrast, decision trees are preferred for situations:

• Involving a sequence of decisions (e.g., value of information and control problems);
• Where correlations between chance events will be represented by Bayesian calculations (i.e., revising prior probabilities based upon new information); and
• Where a “quick and dirty” solution will suffice (often, decision analysts start with a decision tree model and then move to simulation when the model becomes complex).

Simulation and decision trees are the principal decision analysis techniques where alternatives are being valued with respect to a monetary-value-maximizing decision policy. The next article will survey additional probabilistic techniques for characterizing uncertainties in schedule and cost.