Welcome to Critical Path Analysis!
Hello future project managers! This chapter, Critical Path Analysis (CPA), is one of the most practical and engaging topics in Decision Mathematics. It’s all about planning complex projects efficiently.
Don't worry if networks seem intimidating at first. We will break down project planning into simple steps: mapping the tasks, calculating the earliest and latest times, and finding the essential bottleneck activities. By the end of these notes, you’ll know how to find the quickest way to complete any project!
What is Critical Path Analysis?
- CPA is a powerful mathematical tool used for planning, scheduling, and controlling large projects (like building a skyscraper, launching a satellite, or even planning a major concert).
- Its main goal is to determine the minimum time required to complete the entire project and identify which activities absolutely must finish on time.
Section 1: Foundations – Activities and Precedence
1.1 Activities and Precedence Tables
A project is made up of individual activities (tasks) that take a certain amount of time (duration) and must be completed in a specific order.
The order of tasks is called precedence. For example, you can't paint the walls (Activity B) until you have built the walls (Activity A).
We use a Precedence Table to list all activities, their durations, and which activities they depend on.
Key Terminology:
- Activity: A task that requires time and resources (e.g., Dig foundations).
- Duration: The time taken to complete the activity (e.g., 5 days).
- Immediate Predecessor: The activity (or activities) that must be completed immediately before this activity can start.
Analogy: Think about getting ready for school. You can't put your shoes on (Activity C) until you put your socks on (Activity B). Activity B is the immediate predecessor to Activity C.
1.2 Activity Networks (AOA Diagrams)
To perform CPA calculations, we convert the precedence table into a diagram called an Activity Network, often using the Activity on Arc (AOA) format in D1.
In an AOA diagram:
- The tasks (Activities) are represented by the arcs (arrows). The duration is written next to the arc.
- The start and end points of the arcs are called nodes (or events). Nodes represent a point in time when one or more activities have been completed and others can begin.
Rule 1: Starting and Finishing
Every project starts at a single source node (Node 1) and ends at a single sink node.
Rule 2: Unique Nodes
Every activity must have a unique start node and a unique end node. An activity cannot start and finish at the same two nodes as another activity.
Key Takeaway (Section 1)
The network diagram is the map. Activities are the journeys (arcs), and nodes are the major milestones (events). We must follow the rules of precedence exactly when drawing the map.
Section 2: The Dreaded Dummy Activities
This is often the trickiest part of drawing the network, but if you remember the two main reasons they exist, you will be fine!
2.1 What is a Dummy Activity?
A dummy activity is represented by a dashed arc on the network.
- It has a duration of zero. It represents a logical dependency, not a task that takes time.
- Dummies ensure the network drawing rules are followed correctly.
2.2 When to Use a Dummy
There are two main reasons we introduce a dummy activity:
Reason 1: Maintaining Unique Nodes (Rule 2)
If Activity A and Activity B both start at Node 1 and end at Node 2, this violates Rule 2 (two activities share the same start and end nodes).
The Fix: We must introduce a dummy to separate them logically.
Reason 2: Showing Complex Dependencies
This is the most common use. Dummies are needed when a set of activities depends on some, but not all, of the completed previous activities.
Example:
| Activity | Immediate Predecessors |
|---|---|
| A | - |
| B | - |
| C | A |
| D | A and B |
Look at this table:
- Activity C only depends on A.
- Activity D depends on both A and B.
If A and B start at Node 1 and meet at Node 2, then both C and D would depend on both A and B finishing at Node 2. This is wrong for C!
The Fix: We need to split the dependency. A dummy is drawn from the finish of A to the finish of B (or vice versa), allowing D to depend on both end nodes, while C only depends on A's finish node.
Memory Aid: If you ever draw two activities starting/ending at the same pair of nodes, or if the dependencies don't match the required flow, think dummy.
Common Mistake to Avoid!
Don't confuse a dummy (logical link, dashed line, duration 0) with a standard activity (actual task, solid line, duration > 0).
Section 3: Time Calculations – The Forward and Backward Pass
Once the network is drawn correctly, we calculate the earliest and latest times for every node (event) using a special box at each node.
The box usually looks like this:
\[ \begin{array}{|c|c|} \hline \text{EST} & \text{LST} \\ \hline \end{array} \]
- EST: Earliest Start Time for activities leaving the node.
- LST: Latest Start Time for activities leaving the node (or Latest Finish Time for activities entering the node).
3.1 The Forward Pass (Calculating Earliest Times)
We start at Node 1 and move forward through the network. This calculates the earliest time each event can possibly occur.
Formula Used: \[ \text{EST}_{\text{next node}} = \text{EST}_{\text{current node}} + \text{Duration}_{\text{activity}} \]
Step-by-Step Forward Pass:
- Start Node (Node 1): The EST is always 0.
- Move Forward: For every activity leaving a node, calculate its finish time by adding its duration to the current node's EST.
-
The Maximum Rule: When an activity arrives at a node that has multiple activities entering it, the EST for that node is the maximum of all arriving times.
Why maximum? Because the event (node) can only occur when the last activity leading into it has finished. If three tasks finish at times 5, 8, and 10, the earliest you can start the next stage is 10.
- The EST of the final sink node gives you the Minimum Completion Time for the entire project.
3.2 The Backward Pass (Calculating Latest Times)
We start at the final node (sink node) and move backward through the network. This calculates the latest time each event can occur without delaying the overall project.
Formula Used: \[ \text{LST}_{\text{current node}} = \text{LST}_{\text{next node}} - \text{Duration}_{\text{activity}} \]
Step-by-Step Backward Pass:
- Start Node (Final Node): The LST is set equal to the EST (the minimum completion time).
- Move Backward: For every activity entering a node, calculate its latest start time by subtracting its duration from the next node’s LST.
-
The Minimum Rule: When calculating the LST for a node that has multiple activities leaving it, the LST is the minimum of all calculated latest start times.
Why minimum? The event (node) must occur by the latest time dictated by the earliest required start time of the activities leaving it. If activities leaving the node must start by times 12, 15, and 18, you must finish by time 12 to avoid delaying the first activity.
Quick Review Box: Calculations
Forward Pass: Start at 0, move forward, take the MAXIMUM at nodes.
Backward Pass: Start at project end time, move backward, take the MINIMUM at nodes.
Section 4: The Critical Path and Total Float
Now that we have the earliest and latest times for every node, we can identify the most important parts of the project.
4.1 The Critical Path
The Critical Path is the sequence of activities from the start node to the finish node that determines the minimum total completion time.
It is the longest path through the network.
How to Identify Critical Activities:
An activity is critical if, and only if, a delay in its completion will lead to a delay in the overall project.
On the network diagram, the Critical Path is where the EST and LST are the same at both the start and end nodes of the activity.
$$ \text{Critical Activity Condition:} \quad \text{EST}_{\text{Start Node}} = \text{LST}_{\text{Start Node}} $$
Tip: Critical paths are usually drawn with a double line or highlighted to make them clear in the final answer.
4.2 Total Float
Not all activities are critical. Non-critical activities have some spare time available – this spare time is called float.
Total Float is the maximum amount of time an activity can be delayed without affecting the overall minimum completion time of the project.
Calculating Total Float (\(F\))
The float is calculated for individual activities (arcs), not nodes.
Let \(D\) be the Duration of the activity.
Method 1: Using Finish Times (EFT and LFT)
$$
F = \text{Latest Finish Time (LFT)} - \text{Earliest Finish Time (EFT)}
$$
Where \(EFT = EST + D\).
Where \(LFT = \text{LST of the end node}\).
Method 2: Using Start Times (EST and LST)
$$
F = \text{Latest Start Time (LST)} - \text{Earliest Start Time (EST)}
$$
Where \(LST = \text{LST of the end node} - D\).
Where \(EST = \text{EST of the start node}\).
Method 3 (The most common calculation): $$ F = \text{LST}_{\text{End Node}} - \text{EST}_{\text{Start Node}} - \text{Duration} $$
Important: If an activity has a Total Float of 0, it means it is a critical activity. It has no spare time.
Did You Know?
Float is essential for resource levelling. If you have two non-critical activities competing for the same limited resource (e.g., one specialized engineer), you can use the float to schedule one activity later, making sure they don't clash.
4.3 Interpretation and Summary
- The number calculated at the final node (EST/LST) is the minimum project duration.
- The critical path identifies activities that must be monitored closely. Any delay here delays the whole project.
- Activities with large total float offer flexibility in scheduling and resource allocation.
Final Encouragement
CPA combines logic and arithmetic. Draw your network clearly, take your time with the forward and backward passes (especially watching the MINIMUM and MAXIMUM rules), and you will master this chapter! Practice drawing those tricky dummy situations until they become second nature. Good luck!