☞ Comprehensive Study Notes: Critical Path Analysis (D1)
Welcome to Critical Path Analysis!
Hello there! If you’re tackling Decision Mathematics 1, Critical Path Analysis (CPA) is one of the most practical and interesting topics. Don't worry if graph theory seems a little abstract at first—CPA is essentially a powerful tool for planning and managing projects efficiently.
What are we learning? We will learn how to model a complex project (like building a house or launching a product) as a network. We then use simple arithmetic to find the shortest possible time to complete the whole project and identify the most important tasks that cannot be delayed. This minimum time is called the Critical Path.
Section 1: Modeling a Project – Activity Networks
1.1 Activities, Duration, and Precedence
Before we draw anything, we need to understand the components of a project.
- Activity: A specific task that needs to be performed (e.g., 'Lay the foundation'). Activities take time.
- Duration: The time (in hours, days, or weeks) required to complete that activity.
- Precedence (or Dependency): Which activities must be finished before a new activity can start. This is the core information used to build our network.
Example: If Activity B requires Activity A to be completed first, we say A is the immediate predecessor of B.
1.2 Drawing the Activity Network (AOA)
In Decision Maths, we typically use the Activity on Arc (AOA) format.
- Arcs (Arrows): Represent the Activities themselves. The activity name and its duration are written on the arrow.
- Nodes (Circles/Vertices): Represent the Events—moments in time when a set of activities has been completed, and new ones can start.
The Rule of Direction: Arrows must always flow from an earlier event (node) to a later event.
Step-by-Step Guide to Drawing the Network
- Start with a single node (Node 1) for the project start.
- Draw arrows from Node 1 for all activities that have no predecessors.
- Continue adding nodes and arcs, ensuring that an activity only starts after all its predecessors arrive at the preceding node.
- End the network at a single final node.
⚠ Accessibility Checkpoint: Think of the nodes like train stations and the activities like train journeys. You can only leave a station after the previous train has successfully arrived!
1.3 The Essential Concept: Dummy Activities
This is often the trickiest part, but it's vital for drawing a correct network. A Dummy Activity is a conceptual activity that takes zero time (duration = 0). It is represented by a dashed arrow.
When Must We Use a Dummy Activity?
We use dummies for two main reasons to ensure the network accurately reflects the dependencies:
- Uniqueness Rule Violation: If two or more activities share the exact same start node AND end node. (This is illegal in AOA, as it makes identifying the activities impossible.)
-
Complex Dependency Rule (The 'Overlap' Rule): This is the most common reason. Look at the dependencies:
- Activity C depends on A only.
- Activity D depends on A and B.
💡 Memory Aid for Dummies: Dummies are like a mathematical necessity, not a real task. They allow us to obey the rules: One Activity = One Unique Arc.
Key Takeaway for Section 1: A well-drawn network correctly shows all tasks (arcs) and dependencies using nodes and, critically, zero-duration dashed arrows (dummies) where needed.
---Section 2: The Critical Path Algorithm
Now that the network is drawn, we determine the timing. We fill in the nodes with two important times: the Earliest Start Time (EST) and the Latest Start Time (LST).
We use a standard box notation at each node, usually formatted like this:
\[
\begin{array}{|c|c|}
\hline
\text{EST} & \text{Node Number} \\
\hline
\text{LST} & \\
\hline
\end{array}
\]
2.1 The Forward Pass (Calculating EST)
The Earliest Start Time (EST) at any node is the earliest time that event can possibly be achieved. This is calculated by working forward through the network, from the start node to the end node.
Forward Pass Step-by-Step
- Start Node: The EST of the first node (Node 1) is always 0.
- Calculation: For any subsequent node, find the EST of its preceding node, add the duration of the connecting activity (arc). $$ \text{EST}_{\text{next node}} = \text{EST}_{\text{current node}} + \text{Duration} $$
- The Maximum Rule: If a node has multiple incoming activities (arcs), you must calculate the arrival time for every single path leading to it. The EST of that node is the LARGEST of these arrival times. This ensures all preceding activities are fully completed before the next phase can begin.
💭 Analogy: Imagine running several races at once. The overall project can only start the next round of activities when the slowest runner (the maximum time) crosses the finish line.
Final Result: The EST of the final node gives us the minimum completion time for the entire project.
2.2 The Backward Pass (Calculating LST)
The Latest Start Time (LST) at any node is the latest time that event can occur without delaying the overall minimum completion time of the project. This is calculated by working backward through the network, from the end node to the start node.
Backward Pass Step-by-Step
- End Node: The LST of the final node is set equal to its EST (the minimum completion time).
- Calculation: For any node, find the LST of the subsequent node and subtract the duration of the connecting activity (arc). $$ \text{LST}_{\text{current node}} = \text{LST}_{\text{next node}} - \text{Duration} $$
- The Minimum Rule: If a node has multiple outgoing activities (arcs), you must calculate the required start time for every single outgoing path. The LST of that node is the SMALLEST of these times. This ensures we don't delay any subsequent tasks.
⚠ Common Mistake Alert: Students often confuse the rules!
- Forward Pass (EST): Look back, choose Maximum.
- Backward Pass (LST): Look forward, choose Minimum.
Key Takeaway for Section 2: The Forward Pass finds the absolute shortest time (EST). The Backward Pass finds the latest possible timeline (LST) without overrunning that shortest time.
---Section 3: Float and Identifying the Critical Path
The most important output of CPA is identifying which tasks are crucial (the critical path) and which have some flexibility (float).
3.1 Calculating Total Float
Total Float (often just called "Float") is the amount of time an activity can be delayed without affecting the overall minimum completion time of the project. It represents the "spare time" for that specific task.
To calculate the Total Float for Activity $X$, running from Node $i$ (start node) to Node $j$ (end node) with duration $D$:
$$ \text{Total Float} = \text{LST}_j - \text{EST}_i - D $$Where:
- \( \text{LST}_j \): Latest Start Time of the end node.
- \( \text{EST}_i \): Earliest Start Time of the start node.
- \( D \): Duration of the activity.
Example: If Activity A has an EST of 5, an LST of 15, and a duration of 8, the float is \( 15 - 5 - 8 = 2 \). This means Activity A can be delayed by 2 units of time without slowing down the project.
3.2 Defining the Critical Path
The Critical Path is the sequence of activities where the Total Float is zero (Float = 0).
- These activities are the "bottlenecks." If any critical activity is delayed, the entire project will be delayed by the same amount.
- In the network diagram, the critical path is usually marked by double lines or by drawing a solid line connecting the critical nodes.
How to Identify Critical Nodes: A node is critical if its EST equals its LST. $$ \text{EST} = \text{LST} $$ The Critical Path is the route connecting these critical nodes.
🗄 Accessibility Checkpoint: Think of the Critical Path as the longest, slowest route through the network. Since it is the longest, it determines the overall time. Any time saved on a non-critical activity won't help, but any time lost on a critical activity ruins the schedule!
3.3 Minimum Completion Time
The Minimum Completion Time is simply the EST of the final node. This is the shortest possible time in which the project can be completed, assuming all tasks are done efficiently.
🧩 Quick Review Box: Essential CPA Terms
EST: Earliest time a task can start (Forward Pass, Max rule).
LST: Latest time a task must start (Backward Pass, Min rule).
Float: Spare time (LST end - EST start - Duration).
Critical Path: Activities with zero float (cannot be delayed).
Section 4: Advanced Concepts and Common Pitfalls
4.1 Resource Allocation and Constraints (Brief Mention)
While the basic CPA determines the minimum time, in real life, we often face resource constraints (e.g., only two builders available, or only one machine).
If two non-critical activities are scheduled to happen simultaneously but both require the same limited resource, one of them must be delayed. This delay consumes its float. We must always ensure that the total float for an activity is never exceeded, otherwise, the minimum project time increases.
4.2 Common Examination Mistakes to Avoid
- Forgetting the Dummy: Always check your precedence table carefully. If the rules of uniqueness or complex dependency are violated, you MUST insert a dummy activity (Duration = 0).
- Mixing Up Min/Max: Remember the rules! EST = Maximum incoming time. LST = Minimum outgoing time.
- Incorrect Float Calculation: Ensure you are using the node times correctly: LST of the end node minus EST of the start node, minus the duration.
- Not Marking the Path: In exams, if you are asked to identify the critical path, you must draw double lines on the network diagram (or list the nodes/activities) in addition to stating the minimum time.
🎉 You Got This! CPA is methodical. Once you master the forward and backward passes, the rest is smooth sailing. Practice drawing networks until the placement of nodes and dummies becomes second nature.
Final Key Takeaway: CPA transforms a complex list of tasks into a robust schedule, clearly highlighting the activities that require the most careful monitoring to ensure the project finishes on time.