5.3 Monte Carlo Simulation & Estimating

Monte Carlo Simulation

Monte Carlo simulation is the engine of quantitative analysis. It runs the cost or schedule model thousands of times (iterations), and on each pass it randomly samples a value from a probability distribution for every uncertain input. The collected results form a distribution of total project cost or finish dates.

It is the standard way to model overall project risk because it captures how many small uncertainties and dependent schedule paths combine — something a single deterministic estimate cannot.

A crucial schedule insight Monte Carlo reveals is criticality: as the simulation runs, paths that were not on the original critical path can become critical in some iterations. The simulation reports how often each path drives the finish date, exposing near-critical paths that deterministic critical-path method analysis hides. This is why a 50/50 estimate of the planned end date almost always overstates confidence — merge points and parallel paths conspire to push the realistic finish later.

Distributions and the S-Curve

Inputs are assigned probability distributions — commonly triangular (from three-point estimates), beta, normal, or uniform. The simulation's output is plotted as a cumulative probability curve, the S-curve.

Read the S-curve at confidence levels:

• P50 — 50% probability of finishing at or below this cost/date (the median)
• P80 — 80% probability of finishing at or below this value; a common target for setting contingency reserve
• P90 — used where the organization is risk-averse

The gap between the deterministic plan and, say, the P80 value is the reserve needed to reach that confidence. P-values are reserve-sizing tools, and the exam expects you to read them correctly.

The direction of the P-value trips people up. Higher P means more conservative for cost and schedule: P90 cost is larger than P50 cost, because you are buying more certainty of staying under budget. Choosing a confidence level is a governance decision tied to risk appetite — a risk-averse organization on a safety-critical program may fund to P90, while a routine internal project might accept P50 and run leaner. Always tie the chosen P-value back to the organization's stated appetite.

Three-Point Estimating and PERT

Single-point estimates hide uncertainty. Three-point estimating captures it with:

• O — Optimistic (best case)
• M — Most Likely
• P — Pessimistic (worst case)

Two ways to combine them:

PERT weights the most likely value four times, so it is less swayed by an extreme optimistic or pessimistic guess than the simple triangular average.

Three-point estimates are also the raw material for Monte Carlo: each activity's O/M/P range defines a triangular distribution the simulation samples from. So three-point estimating, PERT, and Monte Carlo form a chain — gather the ranges, weight them with PERT for a point estimate if you need one, or feed the ranges into a simulation for a full distribution. The standard deviation of a PERT estimate is (P − O) / 6, a measure of the estimate's uncertainty that the exam occasionally references.

Worked PERT Example

An activity is estimated at O = 8 days, M = 10 days, P = 18 days.

• PERT mean = (8 + 4×10 + 18) / 6 = (8 + 40 + 18) / 6 = 66 / 6 = 11 days
• Triangular mean = (8 + 10 + 18) / 3 = 36 / 3 = 12 days

Both land above the most likely 10 days because the pessimistic tail (18) is far from the optimistic value (8). PERT's 11 sits closer to M than the triangular 12 does, because PERT gives M four times the weight. When a question lists O, M, and P, default to the PERT formula unless it explicitly says triangular.

Sensitivity Analysis and the Tornado Diagram

Sensitivity analysis determines which individual risks or variables have the greatest potential effect on project outcomes. Its signature output is the tornado diagram: a horizontal bar chart with the most influential variable at the top and bars shrinking downward, giving the funnel shape.

Use it to focus response effort where it pays off:

• Long bars = high-leverage risks worth detailed response planning
• Short bars = low-leverage risks that can be monitored lightly

Monte Carlo gives the range of outcomes; the tornado diagram tells you which inputs drive that range. Together they convert raw simulation output into an action priority list.

A worked read: if a tornado diagram for a $4M project shows the "steel price" bar swinging the total by ±$700K, the "labor productivity" bar by ±$300K, and "permit timing" by ±$80K, the team concentrates response planning on steel price first — perhaps a fixed-price contract to transfer that exposure — and barely touches permit timing. The diagram converts a long risk register into a short, ranked action list, which is exactly the prioritization the PMI-RMP framework expects before committing response budget.