Schedule risk and simulation

01The problem

The sponsor asked: "Will we open by 30 September?" The plan says 11 September with most likely durations, and later with PERT. Both are single dates. Neither answers the real question, which is about chance: how likely is it that everything that matters goes well enough?

02The concept

Monte Carlo simulation

1. For each task, draw a random duration from its range (here, a triangular distribution between optimistic and pessimistic, peaking at most likely).
2. Schedule the project with those durations and record the end date.
3. Repeat thousands of times.
4. The results form a distribution of end dates: you can read off the probability of meeting any date.

Merge bias

When several paths join (training waits for hiring and system testing), the task starts when the slowest finishes. Each path alone might be on time, but the chance that all of them are is lower. That's why simulation usually shows later dates than adding up one path's estimates.

Using the results

• Quote dates with confidence: "P80" is the date you have an 80% chance of meeting.
• A schedule buffer is the gap between the plan's date and the date you commit to.

03Example

Simulate the project 10,000 times. The random generator has a fixed seed so your numbers match these:

import numpy as np
import pandas as pd

base = "https://academy.cloudtechanalytics.com/datasets/project/"
tasks = pd.read_csv(base + "tasks.csv").fillna({"predecessors": ""})
START = np.datetime64("2026-06-01")
DEADLINE_DAYS = int(np.busday_count(START, np.datetime64("2026-09-30")) + 1)   # working days to 30 September

rng = np.random.default_rng(42)
N = 10_000
durations = {t.task_id: rng.triangular(t.optimistic_days, t.likely_days, t.pessimistic_days, N) for t in tasks.itertuples()}

finish = {}
for t in tasks.itertuples():
    preds = [p for p in t.predecessors.split(";") if p]
    start = np.max([finish[p] for p in preds], axis=0) if preds else np.zeros(N)
    finish[t.task_id] = start + durations[t.task_id]
end = finish["F3"]

print("Deadline is working day", DEADLINE_DAYS)
print("Chance of opening by 30 September:", f"{(end <= DEADLINE_DAYS).mean():.0%}")
print("Chance of opening by 11 September (the most likely plan):", f"{(end <= 75).mean():.0%}")
for p in [50, 80, 90]:
    day = int(np.ceil(np.percentile(end, p)))
    print(f"P{p}: day {day}, {np.busday_offset(START, day - 1)}")

Deadline is working day 88
Chance of opening by 30 September: 64%
Chance of opening by 11 September (the most likely plan): 3%
P50: day 86, 2026-09-28
P80: day 92, 2026-10-06
P90: day 95, 2026-10-09

The most likely plan's date had almost no chance of being met. The 30 September deadline is roughly a two-in-three chance before anything has even gone wrong. Which tasks drive the risk? Count how often each task was on the longest path:

critical = pd.Series(0.0, index=tasks["task_id"])
latest = {}
for t in tasks.itertuples():
    preds = [p for p in t.predecessors.split(";") if p]
    if preds:
        stacked = np.vstack([finish[p] for p in preds])
        latest[t.task_id] = np.array(preds)[stacked.argmax(axis=0)]
for i in range(N):
    task = "F3"
    while True:
        critical[task] += 1
        if task not in latest:
            break
        task = latest[task][i]
print((critical / N).sort_values(ascending=False).head(10).round(2).to_string())

task_id
A1    1.00
A2    1.00
F3    1.00
F2    1.00
A3    0.99
B2    0.99
C5    0.99
C3    0.99
D3    0.85
E2    0.15

The planned critical path is the longest path in almost every run. The interesting split is at the end: staff training (D3) is on the longest path about 85% of the time, but in the other runs it's stocking the opening inventory (E2) that the trial run waits for. Racking and hiring almost never decide the date. So the project manager's attention belongs on permits, fit-out, network and system testing, and both training and stocking.

04Walkthrough

1. Run the cells. What's the P80 date? Would you tell the sponsor that, or 30 September?
2. Make the permits certain (O = M = P = 15) and rerun. How much does the chance improve?
3. What would cutting the fit-out's pessimistic estimate to 30 days do?
4. Write the answer to the sponsor (the task below).

05Practice

06Check your understanding