# PERT Network Analyzer (Expected Duration & Variance)

Three-point PERT estimates: te and σ² per activity, critical path, expected project duration and variance, and normal completion probability for a target time.

> Canonical page: https://elysiatools.com/en/tools/pert-network-analyzer

- **Category:** Math & Numbers

- **Keywords:** pert, program evaluation and review technique, three-point estimate, expected duration, variance, completion probability, critical path, project management, beta distribution, operations research

## Overview

The PERT Network Analyzer performs Program Evaluation and Review Technique calculations using three-point duration estimates (optimistic, most likely, and pessimistic). It computes expected duration and variance for each activity, identifies the critical path across network dependencies, calculates total project expected duration and variance, and determines the probability of meeting an optional target completion deadline.

## Inputs

- **Activities (one per line: name, optimistic, most likely, pessimistic, predecessors)** (textarea): One activity per line: unique name, optimistic a ≤ most likely m ≤ pessimistic b, then predecessor names ('-' if none).
- **Target completion time (optional)** (number): Optional deadline T: the tool reports Z = (T − TE)/σ and the normal probability of finishing by T.
- **Decimal Places** (number)

## When to use

- Evaluating project schedule risk when task durations involve uncertainty and require three-point estimations.
- Identifying the critical path and total expected project duration from a set of dependent activities.
- Calculating the statistical probability (Z-score) of completing a project by a specific target deadline.

## How it works

- Enter activity details line-by-line with name, optimistic time (a), most likely time (m), pessimistic time (b), and predecessor names.
- The tool calculates the expected duration te = (a + 4m + b) / 6 and variance σ² = ((b - a) / 6)² for each activity.
- The network resolves dependencies to find the critical path, total expected duration (TE), and total critical path variance.
- If an optional target time (T) is provided, the tool calculates Z = (T - TE) / σ and outputs the normal completion probability.

## Use cases

- Project managers assessing schedule confidence intervals for complex multi-phase initiatives.
- Operations research analysts modeling project network dependencies and activity float.
- Engineering leads presenting statistical delivery timelines and contingency buffers to stakeholders.

## Frequently asked questions

### What format is required for the activity input list?

Enter one activity per line formatted as: name, optimistic (a), most likely (m), pessimistic (b), followed by predecessor names separated by spaces, or '-' if there are no predecessors.

### How are expected duration and variance calculated for each activity?

Expected duration uses the standard PERT formula te = (a + 4m + b) / 6, and variance is computed as σ² = ((b - a) / 6)².

### How is the total project variance calculated?

Total project variance is calculated by summing the variances (σ²) of only the activities located on the critical path.

### What happens if an activity violates the a ≤ m ≤ b rule?

The analyzer validates that the optimistic duration does not exceed the most likely duration, and the most likely does not exceed the pessimistic duration, rejecting invalid entries.

### What does the completion probability represent?

Using the standard normal cumulative distribution Φ(Z), it calculates the statistical probability of finishing the project within your specified target time T.

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