Probabilistic scheduling and schedule risk

A critical path schedule built with single durations gives a single finish date, and that date is almost certainly wrong. Not because the scheduler made a mistake, but because every duration is an estimate and every estimate is uncertain. Some activities will go faster than planned and some slower; weather, deliveries, inspections and productivity all vary. Probabilistic scheduling asks a more honest question: given the uncertainty in each activity, how likely is it that the project will finish by a given date, and which risks drive that likelihood? This lesson introduces the program evaluation and review technique (PERT), Monte Carlo schedule risk analysis, and the practical use of risk-adjusted schedules and contingency in construction.

Why single-point dates mislead

Suppose three activities in a row each have a most likely duration of 10 days, but each could take as little as 8 or as long as 16. The deterministic schedule shows 30 days. But the distributions are skewed: there is more room to run long than short, as anyone who has watched construction knows. The expected total is more than 30 days, and the chance of finishing in 30 days or less is well below one half.

A second effect makes things worse. When two or more paths merge into one activity, the merge activity can start only when the latest path finishes. Even if each path is equally likely to be early or late, the latest of several uncertain paths tends to be late. This is called merge bias, and it means that schedules with many parallel paths are more likely to overrun than their critical path alone suggests. A deterministic CPM calculation cannot show merge bias; probabilistic methods can.

Three-point estimates and the PERT formulas

PERT was developed in the late 1950s for the US Navy's Polaris missile program, where many activities had never been done before. It asks for three estimates for each activity: the optimistic duration \(a\), the most likely duration \(m\) and the pessimistic duration \(b\). Assuming the duration follows a beta distribution, PERT approximates the mean and standard deviation as follows.

For a path of activities in series, PERT adds the means to get the expected path duration and adds the variances (not the standard deviations) to get the path variance. By the central limit theorem, the total of several independent durations is approximately normally distributed, so the probability of finishing by a target date can be read from the standard normal distribution.

Limitations of PERT

PERT is simple and teaches the right instincts, but it has well-known weaknesses. It analyses only one path, usually the deterministic critical path, and so ignores merge bias: a near-critical path with high uncertainty can easily become the controlling path. It assumes the activities are independent, when in construction they are often correlated: a wet spring slows excavation, foundations and site work together, and a weak superintendent slows everything. And it relies on three estimates that people find hard to give honestly, especially the pessimistic one. For these reasons PERT is now used mainly for teaching and quick checks, and Monte Carlo simulation is the standard tool for schedule risk analysis on significant projects.

Monte Carlo schedule risk analysis

Monte Carlo simulation runs the whole schedule many times, often 1,000 to 10,000 iterations. In each iteration, the software picks a duration for every uncertain activity at random from its distribution, recalculates the critical path, and records the finish date and which activities were critical. The results are a probability distribution of the finish date, usually shown as a histogram with a cumulative curve, and a measure of how often each activity was critical, called the criticality index.

Because every iteration recalculates the whole network, Monte Carlo captures merge bias and shifting critical paths automatically. It can model correlation between activities, discrete risk events (such as a 30 % chance that a permit is delayed by 4 weeks), and different distributions, most commonly triangular, beta-PERT or uniform. AACE International's recommended practices describe several methods of schedule risk analysis, and many public owners require a probabilistic analysis on large projects.

Results are usually reported as percentile dates. The P50 date is the date with a 50 % chance of being met; the P80 or P90 date is often used for commitments and contingency. The spread between the deterministic date and the P80 date tells management how much schedule contingency is needed.

Risk registers and the schedule

Schedule risk analysis is most useful when it is tied to the project's risk register, the list of identified risks with their likelihood, impact, owners and responses. Risks that affect time are mapped to the activities they would delay. The analysis then shows which risks matter most to the finish date, and the team can compare the cost of a response with the reduction in schedule risk it buys.

Schedule contingency

Contingency is time held in reserve for risks. It can be held in several ways, and the choice matters for transparency and for float ownership.

• A visible contingency activity placed near the end of the schedule, before the contract completion milestone, so everyone can see it and its consumption can be tracked.
• Contingency buffers at the end of each phase or before key milestones, as in critical chain project management.
• Hidden padding in individual activity durations, which is generally poor practice because it is invisible, is consumed unconsciously and makes the critical path unreliable.

Contracts often regulate contingency. Some prohibit contractor contingency activities in the baseline because they look like float the contractor is claiming for itself; others allow or require them. Whatever the rule, the amount should be based on analysis, such as the gap between the deterministic and P80 dates, not on habit.

Making probabilistic thinking practical

Many construction schedules never receive a formal Monte Carlo analysis, and that is acceptable on simpler projects. The thinking still applies. A planner can identify the ten or twenty most uncertain activities, ask the people responsible for honest ranges, look for near-critical paths that merge before key milestones, and judge whether the stated completion date is realistic. Saying that the schedule shows an 80-day finish but the team believes there is only an even chance of meeting it is more useful to an owner than a confident date that everyone privately doubts.