Mapping the Work with PERT, Explained with Examples

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Mapping the Work with PERT

Quantitative Analysis · Chapter 7: Project Planning with PERT and GanttSection 7.1 · lesson 31 of 5510 min · free with a free account

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In this lesson

Network diagrams, the forward and backward pass, float, and the critical path.

  • Read a precedence table and build an activity-on-node network
  • Run the forward and backward pass to find ES, EF, LS and LF
  • Compute total float and free float for every activity
  • Identify the critical path and the minimum project duration

Transcript

Everything said in the video, section by section (1,627 words). Read along, or search it.

Project Scheduling · Section 7.1

Mapping the Work with PERT

Every large project is really a web of tasks that depend on one another. Some of those tasks can slip a little, and nobody ever notices. Others cannot slip at all. PERT is the technique that tells you which is which. It stands for Program Evaluation and Review Technique. In this section we build a project network from a table of dependencies. Then we sweep forward through it, to find the earliest each task can happen. Then we sweep backward, to find the latest each task can happen without hurting the deadline. The gap between those two answers is called float. Activities with zero float form the critical path. And the length of that path is the shortest time the project can possibly take.

Why PERT Exists

A Network View of a Large Project

PERT is a network-based way of planning, scheduling and controlling a project. The word network is doing real work there. The method treats the project as a connected structure, not as a list of jobs. That is what lets it answer questions a list simply cannot. Which tasks are genuinely critical? How short can this project be? The history explains the ambition. PERT was developed in nineteen fifty-seven by the United States Navy. They worked with Booz Allen Hamilton on the Polaris Fleet Ballistic Missile programme. That project involved thousands of contractors and sub-contractors, far too many to coordinate by intuition. PERT helped compress the original six-year schedule down to four years. For project management, that result was a landmark.

The Vocabulary

Activities, Precedence and the Network

Four words carry most of the machinery, so it is worth getting them exact. An activity is a well-defined unit of work, with a start, an end, and resources. Crucially, it consumes time. Time is what we will be adding up. A precedence relationship is a logical dependency constraint. It says an activity cannot begin until its predecessors are fully complete. Not mostly complete. Fully. The network diagram is a directed, acyclic graph of the whole project. Directed, because dependencies point one way. Acyclic, because a task cannot depend on itself. Activities can be drawn as arrows, or as boxes. An event, or node, is a milestone, where preceding work ends and succeeding work begins. In a node diagram, every node carries the four timing numbers we are about to compute.

Two Notations

Activity-on-Arrow vs Activity-on-Node

There are two drawing conventions, and they are easy to confuse. In activity-on-arrow, the arrows are the activities and the circles are events. In activity-on-node, the boxes are the activities, and events are implicit. Both encode exactly the same logic. The practical difference is the dummy activity. Activity-on-arrow sometimes needs a fake, zero-duration arrow, purely to express a dependency. That is an artefact of the notation, not a feature of the project. Activity-on-node never needs one. On a complex network, that makes the node form far easier to read. Which is why activity-on-node is the more common convention today. Everything we compute from here uses the node form.

Two Traversals

The Forward Pass and the Backward Pass

Every timing number in PERT comes from just two sweeps through the network. The forward pass runs left to right, from project start to finish. It asks a simple question. How soon can this happen? Earliest start is the maximum earliest finish among all immediate predecessors. Maximum, not minimum, because you have to wait for the slowest one. That single word is the most common mistake in the whole method. Earliest finish is then earliest start plus the duration, and start activities begin at time zero. The backward pass runs the other way, from the finish back to the start. It asks how late a task can happen without moving the deadline. Latest finish is the minimum latest start among all immediate successors. Minimum this time, because you must not hold up any of them. And latest start is simply latest finish minus the duration.

Slack

Total Float and Free Float

Now put the two passes side by side. The gap between them is float. Total float is latest start minus earliest start. Equivalently, latest finish minus earliest finish. The two always agree. It is the delay an activity can absorb before the project itself is delayed. Free float is a different, narrower idea. It is the earliest start of the successor, minus this activity's earliest finish. That is the delay you can take before any successor has to move at all. Here is the part that textbooks tend to bury. Total float is a property of a path, not of a single activity. If two activities share a path, spending float on one takes it away from the other. Free float, by contrast, belongs entirely to the individual activity. An activity whose total float is zero is called critical.

The Longest Path

What Makes a Path Critical

The critical path is the longest path from start to finish. Longest, measured in total duration. That sounds backwards until you see why. The project is not finished until every single path is finished. So the longest path sets the minimum possible project duration. Every activity on it has a total float of zero. Which means one day of delay anywhere on it is one day of delay for everything. A project may have more than one critical path, all of equal length. That makes management harder, not easier, because now there are two chains to protect. To shorten a project, managers crash it, adding resources to critical activities first. Crashing anything off the critical path just buys slack you already had.

Worked Example

A Software Deployment Project

Let us work a full example, a software deployment project with nine activities, labelled A through I. Each one has a duration in days, and a list of immediate predecessors. A is requirements gathering, four days, and it depends on nothing. B is system architecture design, three days, after A. C is database schema design, two days, also after A. So B and C can run in parallel, once A is done. D is backend development, six days, and it follows B. E is frontend development, five days, and it waits for both B and C. F is A P I integration, three days, after D. G is unit and integration testing, four days, after D and E. H is user acceptance testing, three days, after F and G. And I is deployment and go-live, two days, after H.

Forward Pass

Earliest Start and Earliest Finish

Now the forward pass. Start at time zero and move left to right. A has no predecessor, so its earliest start is zero and its earliest finish is four. B and C both follow A, so both of them start at four. B takes three days and finishes at seven. C takes two and finishes at six. D follows B alone. It starts at seven and finishes at thirteen. E is the first merge point, and merge points are where errors creep in. E waits for B and for C. B finishes at seven, C finishes at six. Take the maximum, which is seven. E runs five days and finishes at twelve. F follows D, running from thirteen to sixteen. G merges D at thirteen and E at twelve. Take thirteen, add four, finish at seventeen. H merges F at sixteen and G at seventeen. Take seventeen, add three, finish at twenty. I follows H, running from twenty to twenty-two. The earliest finish of the last activity is twenty-two, so the project takes twenty-two days.

Backward Pass

Latest Times and the Float They Reveal

Now turn around. Set the latest finish of the final activity to twenty-two. I takes two days, so its latest start is twenty. H must finish by twenty, so it has to start by seventeen. F and G both feed H, so both have a latest finish of seventeen. F takes three days, giving a latest start of fourteen. G takes four, giving thirteen. D feeds both F and G, so take the minimum of fourteen and thirteen. D gets a latest finish of thirteen, and a latest start of seven. Now the floats. Total float is latest start minus earliest start. A, B, D, G, H and I all come out at zero, while C gets two days, and E and F one day each. Watch C closely. Its total float is two, but its free float is only one. Delay C by two days and the project still lands on time. But E can no longer start as early as it could, and E's own slack absorbs the difference.

The Answer

Critical Path and Project Duration

So which chain is the critical one? A to B to D to G to H to I. Six activities, every one of them carrying zero total float. Their durations are four, three, six, four, three and two. Add them up and you get twenty-two days, exactly the project duration. That is not a coincidence. That is the definition. Off the path sit C, E and F. C has two days of total float, E has one, and F has one. They can be rescheduled inside those windows without touching the deadline. But do not read float as free time. Spend all of C's float, and E is left with no room at all. Float is a shared budget, and the network decides who is allowed to spend it.

Key Formulas

The Whole Procedure in One Place

Let us gather the formulas in one place. Earliest finish is earliest start plus duration. Earliest start is the maximum earliest finish across all predecessors. Latest start is latest finish minus duration. Latest finish is the minimum latest start across all successors. Total float is latest start minus earliest start, which also equals latest finish minus earliest finish. Free float is the earliest start of the earliest successor, minus this activity's earliest finish. And project duration is the earliest finish of the final activity. The procedure itself never changes. Tabulate, draw the network, forward pass, backward pass, float, then read off the critical path. And if you must shorten the project, crash on the critical path. Anywhere else, the finish date will not move.

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FAQ

Questions students ask us

About Mapping the Work with PERT

What is PERT in project management?
PERT, the Program Evaluation and Review Technique, is a network-based method for planning, scheduling and controlling a project. It treats the project as a connected web of activities linked by precedence relationships rather than a simple task list, which reveals which tasks can slip and which cannot. It was developed in the late 1950s by the U.S. Navy with Booz Allen Hamilton for the Polaris missile program, which involved thousands of contractors.
How do you do a forward pass and a backward pass in a project network?
The forward pass moves from start to finish: an activity's earliest start is the maximum earliest finish of its immediate predecessors, and earliest finish is earliest start plus duration, with start activities at time zero. The backward pass moves from the end back: latest finish is the minimum latest start of the immediate successors, and latest start is latest finish minus duration. Using the minimum on the forward pass is a classic mistake.
What is the difference between total float and free float?
Total float is how long an activity can be delayed without delaying the project, calculated as latest start minus earliest start. Free float is how long it can slip before any immediate successor must start later, calculated as the successor's earliest start minus the activity's earliest finish. Total float is shared along a path; in one software project, activity C has two days of total float but only one of free float, because using both squeezes its successor.
What is the difference between activity-on-arrow and activity-on-node diagrams?
In an activity-on-arrow network, arrows represent activities and circles represent events; in an activity-on-node network, boxes represent activities and events are implicit. Both express the same precedence logic, but activity-on-arrow sometimes needs dummy activities, zero-duration arrows added purely to show a dependency. Activity-on-node does not need dummies, which makes complex networks easier to read and is why it is the more common convention today.

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