This series is about decision analysis: evaluating decision alternatives under uncertainty. Decision tree analysis and Monte Carlo simulation are the most popular calculation techniques. They were presented in preceding installments in this series.
There are other quantitative techniques useful in dealing with uncertainty in specific situations. One method employs what might be called pseudo-probability distributions. Others provide alternate ways to combine probability distributions.
We'll start with a summary of methods. Table 1 on the following page lists techniques and their key strengths and weaknesses. Afterward, the following methods will be profiled:
- Dynamic project modeling with Monte Carlo simulation
- Parameter method and moments method
- Fuzzy logic
- Fast calculus integration
The last two of these methods are newly emerging technologies that the author believes will become important in project management.
This article discusses four of the methods in Table 1. Here are the remaining six:
CPM (Critical Path Method) [1] and PERT (Program Evaluation and Review Technique) [2] are the classic modeling techniques for controlling project schedule. Both identify the deterministic critical path. PERT is CPM with the substitution of probability distributions for activity completion times. Slack times (CPM) or distributions of slack times (PERT) are calculated for activities that lie off the critical path. The principal drawbacks of PERT are that it does not recognize that: (1) activities outside the deterministic critical path can be late and delay the project; and (2), real-world projects are often inadequately represented by a simple network of independent activities.
Scenario analysis is a planning technique nique focusing on management responses to plausible alternative futures. Possible events are anticipated, and plans are made to take advantage of opportunities and to protect against threats. The emphasis is not on a forecast but on developing insights about what could impact the project.
Sensitivity analysis is primarily useful in determining which variables in an evaluation should receive the most attention. The next installment in this series, “Project Modeling,” will discuss various techniques.
Design of experiments and Taguchi methods provide an efficient approach to sensitivity analysis for many-decision-variable systems. This provides an effective way to prioritize alternatives for improving value or for reducing variance. Interested persons should consult the recent references in PMI publications [3].
Multi-criteria decision approaches generally use a value scoring function. Persons most frequently use a simple combination of attribute ratings multiplied by weighting factors. Often, risk is merely one of the attributes. The Analytic Hierarchy process is a clever way to establish the scores and weights through pair-wise comparisons.
Influence diagrams are a variant of decision trees. The graphical form is more compact and better in representing relationships (influences) between variables. The author feels that, presently, the benefit of the added formalism is not worth the burden of the method's theoretical complexity.
DYNAMIC MODELING WITH MONTE CARLO
Addressed in the fifth article in this series (see the January 1994 PMNET-work), Monte Carlo simulation is the method most often used to avoid the problems in simple PERT or in CPM within simulation. All the important project details, such as load-leveling, expediting, learning curve effect, and rework, can be incorporated into the model.
One of the valuable outputs of a Monte Carlo solution is a list of project activities with a probability for each that it lies on the critical path. Acceleration and other alternatives can be evaluated rigorously. Monte Carlo is also excellent for representing project contingencies (e.g., bad weather) and detailed cost relationships.
Monte Carlo can easily handle correlations between activities, The author prefers to decompose the system model so that common drivers are represented distinctly. For example, shared resources (e.g., a crane) can be represented, individually, as part of a project system model.
Table 1. Techniques of Uncertainty
| METHOD | KEY STRENGTHS | KEY WEAKNESSES |
| Discussed in Previous Articles | ||
| Decision Tree Analysis | Graphical expected value calculation | Must convert continuous into discrete distributions |
| Evacuating alternatives with sequential decisions (e.g., value of information) | Must limit number of decision alternatives and chance event outcomes | |
| Requires value function | ||
| Monte Carlo Simulation (revisited in this article) | Can accommodate complexity easily, such as dynamic behavior under dynamic behavior under | Time vs. accuracy trade-off Solution can be computationally time-consuming |
| Very generally applicable | Solution is approximate and sometimes difficult to duplicate | |
| Discussed in This Article | ||
| Parameter Method; Method of Moments | Medium complexity; Fast; Reproducible solutions | Provides only statistics about the shape of the solution distribution |
| Difficult to deal with subsequent decision points, correlations and non-linearities | ||
| Monte Carlo Simulation (revisited) | (see MCS above in table) | (see MCS above in table) |
| Fuzzy Logic | Low-medium complexity Fast; | Only approximates probabilistic reasoning |
| Reproducible solutions | Potential developments needed to improve accuracy | |
| Approximate integration | Fast; Repeatable solution |
Little recognition in practice and literature; Emerging technique |
| Identified in This Article | ||
| CPM and PERT | Simple | Simplistic project network model may be inadequate |
| Only one critical path is recognized | ||
| Scenario Analysis | Simple | Seldom quantifies risks and uncertainties |
| Sensitivity Analysis | Simple | Does not recognize risk vs. value trade-offs |
| Design of Experiments; Taguchi Methods | Value optimizing or variance reduction with efficient handling of many decision (controllable) variables | Limited representation of uncertainty, “noise,” e.g., using “Low” and “High” for each chance event |
| Multi-Criteria Approaches; Analytic Hierarchy Process | Simple if non-probabilistic | Risk or uncertainty is merely one of several attributes; problems with consistency. |
| Probabilities can be used if a value function is devised. | ||
| Influence Diagram | Similar to decision trees | More difficult theory and calculations |
| Better represents relationships between variables | ||
Merge Bias
Often, several activities, e.g., {AB,C}, must be completed before a successor activity, D, can be started. In a conventional, deterministic project analysis, we assume that the latest completion rime, Max(A,B,C), determines when D can begin. The notation Max(A,B,C) is read “maximum of A, B or C.”
However, consider when A, B, and C have probability distributions representing their completion dates. The expected value start date for D is often later than the latest expected value completion date for A, B, and C. This is called merge bias. This “joining” effect is one reason why many projects are late.
Let's assume that A, B and C activities have expected value (mean) completion times of 10, 11 and 9 days, respectively. There are uncertainties in these estimates as shown in Figure 1. The judged activity completion times are skewed (lognormal) probability distributions. The mean, or expected value, is the first value in parentheises. The width or uncertainty of each distribution is characterized by its standard deviation. These are the second values in the parentheses.
Figure 1. Completion Times and Earliest Start
With a conventional CPM model, we would assume that Task D can start in 11 days. This is the latest of the best single-value estimates for A, B and C. This assumption is often incorrect. Task D's expected start date would be 11 days if the completion times of A, B, and C are known precisely or at least within narrow ranges. This would also be true if distributions for A, B and C have no probability of exceeding 11 days. This is not the case shown in the figure. Any of the precedent activities could cause D to be delayed past the greatest single expected value completion date.
A simple Monte Carlo simulation demonstrates the effect. With the assumed lognormal distributions as shown, the expected start date for Task D turns out to be 13.05 days. This 2.05-day increase beyond the largest expected value completion date is the merge bias. This concept is analogous to the theory of system reliability [4].
If your project's schedule is lengthened, say 19 percent, what do you think the effect will be on costs? As always, risk and uncertainty should be explicitly ir-corporate into the analysis.
PARAMETER AND MOMENTS METHODS
Simple probability distributions can be combined mathematically. The parameter method and the method of moments are among the names given these evaluation techniques. “Parameter” refers to statistics about the shape of probability distributions. “Moment” refers to the calculation of the statistics. Usually, only the first two moments are solved.
The expected value, or mean, is the first moment about the origin. Outcome (x-axis) values are weighted with their probability of occurrence (y-axis). A mechanical analogy is moment arms in calculating torque. The “center” of a probability distribution is the value about which the torque is zero. Another engineering analogy for this balance point is the probability graph's center of gravity. The variance is the second moment about the mean.
The fundamental equations are the additivity of means and variances. When adding distributions, means (μ) add:
The total is normal distribution if A, B and C are normally distributed.
If there are many additive terms of similar magnitude, the sum will be approximately normally distributed regardless of the shape of the individual distributions (from the central limit theorem).
When normal and independent distributions are added, the variances (σ2, standard deviations squared) also add:
The variances of two added, non-independent, normal distributions can be calculated using:
where ρ AB (Greek letter rho) is the correlation coefficient between elements A and B. Systems of variable correlations can be modeled as a correlation matrix.
Most often, correlations are positive, meaning that changes in one variable are associated (i.e., correlated) with changes in another. This is frequently exhibited in project model. The fuzzy results propagate through the model in all calculations. The result is a fast, reproducible solution.
The drawbacks that the calculations do not have the logic of probability theory. The author has obtained some strange outcomes from certain seemingly innocent combinations of variables. However, the mathematical operations are somewhat arbitrary and may be improved eventually We can expect further applications of fuzzy logic projections as the techniques become better understood and refined.
APPROXIMATE INTEGRATION
Although its solution technique is very general, Monte Carlo simulation gives only an approximate solution. The result is slow to converge with additional trials. Further, repeating the same solution is often impractical, thus making it more difficult to value incremental improvements.
What we desire is a fast, repeatable solution to the expected value integral. Ideally we could assault the problem directly using symbolic mathematics. However, this is usually impossible with most decision problems.
Numeric integration holds promise. The results are reproducible and fast. The method is known by several names, Variously as convolution, quadrature, or cubic spline integration. (Splines were flexible strips of wood used by draftsmen [5, p. 295]. Another term is fast calculus integration.
Approximate integration provides suitably accurate results for simple probabilistic models involving addition and multiplication. One obtains the solution probability distribution. Persons working with signal processing have used these calculation techniques for over 25 years. This is emerging as a possible method for decision analysis. It will be interesting to learn more about the applications and limitations of this approach to solving decision problems under uncertainty.
SUMMARY
Traditionally project managers have attempted to optimize cost, schedule and performance. If completion time is your focus, then PERT and other activity-network models can be used. Completion time and its uncertainty can easily be projected. (lien, costs can be loaded easily to add this dimension.
Desiring more holistic decisions, a value-based approach is needed. Composite value, including the effects of project cost and completion time, should drive our selection of alternatives. Simpler methods are often inadequate. Inter-dependencies between activities, time value, and complex performance and cost relationships often require an expanded project model.
In the author's opinion, most significant projects need Monte Carlo simulation for at least some decisions. Monte Carlo, while not without its drawbacks, is superior in its ability to accommodate detail about risks and uncertainties. The top decision is usually the feasibility study. However, once the drivers of value are well understood, day-to-day project decisions often can be treated more simply.
For example, the value impact of completion delay is often linear and can be represented as a daily penalty cost. This allows using decision trees, PERT, and other simpler project modeling techniques.
Often, multiple analysis approaches will be employed on the same project. For example, decision trees are well suited to situations involving a sequence of decisions. This includes the important class of problems involving valuing information.
The next article in this series will discuss modeling techniques to value project outcomes.