Automatically compute the critical path for AON networks with up to 12 activities. Calculate on-time probability with PERT three-point estimation. Real-time Gantt chart rendering.
Project Settings
Number of Activities
Target Deadline T (days)
Predecessor input example: enter predecessor activity numbers separated by commas (e.g., 1,2). Leave blank if none.
$$Z = \frac{T - \sum_{CP} t_e}{\sqrt{\sum_{CP} \sigma^2}},\quad P = \Phi(Z)$$
What is PERT/CPM Scheduling?
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What exactly is the "critical path" in a project? I see it highlighted in red in the simulator.
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Basically, it's the longest chain of dependent activities in your project network. In practice, it determines the shortest possible project duration. A one-day delay on any activity along this path delays the entire project. Try adding more activities in the simulator—you'll see how the red path can shift, changing which tasks are most critical.
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Wait, really? So the critical path can change? And what are those three time estimates (a, m, b) for each activity?
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Yes, if a non-critical task gets delayed enough, it can become part of a new critical path! The three estimates account for uncertainty: a (optimistic), m (most likely), and b (pessimistic) duration. The simulator uses them to calculate a weighted average expected time. For instance, in software development, "coding a module" might have a=3, m=5, b=10 days.
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That makes sense. So what does the "On-Time Probability" slider do? It asks for a Target Deadline T.
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Great question! That's the power of PERT. Once we know the expected duration and variance of the critical path, we can estimate the probability of finishing by a chosen deadline. Slide the "Target Deadline T" control. If you set it far beyond the expected finish, the probability nears 100%. Set it tighter, and the probability drops, telling you the risk of being late.
Physical Model & Key Equations
The core of PERT is calculating a probabilistic expected time for each activity from three-point estimates, reducing uncertainty to a single value and a variance.
Here, $t_e$ is the expected activity duration, $\sigma^2$ is its variance, and $a$, $m$, $b$ are the optimistic, most likely, and pessimistic times. The formula heavily weights the most likely estimate.
The project's overall duration is the sum of $t_e$ along the critical path (CP). To find the probability of meeting a deadline $T$, we treat the CP duration as a normal random variable (Central Limit Theorem) and calculate a Z-score.
$$Z = \frac{T - \sum_{CP} t_e}{\sqrt{\sum_{CP} \sigma^2}},\quad P = \Phi(Z)$$
$Z$ is the number of standard deviations the deadline $T$ is from the expected finish. $P = \Phi(Z)$ is the probability from the standard normal CDF, which the simulator calculates automatically.
Frequently Asked Questions
This tool supports up to 12 activities. If you need 13 or more, please divide the project into multiple subnetworks and connect them with milestones, or use a larger-scale project management software.
Set the optimistic value as the shortest duration when everything goes smoothly, the most likely value as the most probable duration under normal conditions, and the pessimistic value as the longest duration when major problems occur. If historical data from similar tasks is available, use it as a reference; otherwise, determine realistic values based on the experience of the person in charge.
In this tool, when multiple critical paths exist, the variances of each path are summed, and the probability of meeting the deadline is calculated based on the path with the largest variance (the path with the greatest schedule fluctuation). In actual projects, it is recommended to monitor multiple paths.
After entering activities, be sure to click the 'Run Calculation' button. Also, if predecessor relationships (dependencies) are not set correctly, the Gantt chart will not render properly. Check that all activities are connected to a start node and an end node.
Real-World Applications
CAE Simulation Workflow Management: In Computer-Aided Engineering, a single analysis (e.g., crash test) involves sequential steps: geometry cleanup, meshing, solver setup, computation, and post-processing. PERT/CPM is used to schedule these resource-intensive steps, identifying the critical path—often the solver run time—to optimize cluster resource booking and meet design review deadlines.
Aerospace & Automotive Development: Managing the timeline for complex design reviews (DR1, DR2, etc.) is crucial. Each review depends on hundreds of pre-requisite analysis tasks. This method helps program managers concentrate mitigation efforts (like adding engineers) on the critical path activities to prevent cascading delays in the product launch.
New Product Introduction (NPI): Launching a new consumer electronics product or machine involves parallel tracks for design, procurement, manufacturing line setup, and marketing. CPM networks synchronize these tracks, clearly showing how a delay in tooling procurement (a critical path activity) directly impacts the final launch date.
Construction & Plant Turnarounds: A common case is scheduling a refinery shutdown for maintenance. Thousands of interdependent tasks (inspection, repair, replacement) must be completed within a tight window to restart production. The critical path analysis determines the minimum shutdown duration and highlights high-risk tasks for extra oversight.
Common Misconceptions and Points to Note
When you start using this tool, there are several points where mistakes are easy to make, especially with PERT three-point estimation. First, understand that the "optimistic estimate a" and the "pessimistic estimate b" are not just best-case and worst-case guesses. These are values assuming a statistically meaningful range, like "there's a 99% probability the task will finish within this period." For example, even if you think mesh generation will "absolutely take 3 days," if historical data shows it rarely took 5 days, then b=5 is correct. Relying on gut feeling can make the variance too small, leading to painful surprises later.
Next, remember that the critical path is not fixed. Changing the estimate for a task or its dependencies can cause the critical path to shift to a different route. For instance, if you increase the pessimistic value b too much for a non-critical task A (te=5 days), its variance and expected duration increase, potentially reducing its slack to zero and creating a new critical path. Don't just calculate once and forget; get into the habit of recalculating whenever the situation changes.
Finally, don't forget that the calculated probability of meeting the deadline is an "approximation". It's based on the Central Limit Theorem, assuming the critical path duration follows a normal distribution. However, if the number of activities is small (up to 12 in this tool) or if individual task distributions deviate significantly from a Beta distribution, the probability figure is only a guideline. Don't get complacent with a 90% probability or despair at 30%; the practical tip is to use it as "an indicator of risk level."