Gantt (Cascade) Charts (Edexcel A-Level Further Mathematics): Revision Notes

Example 1: Calculate Total Float and Draw a Gantt Chart

Step 1: Given Information

Consider the following activities, their durations, and the earliest and latest times:

Step 2: Calculate Total Float

Using $\text{TF} = L - E - d$:

• Activity $A$: $TF = 6 - 0 - 4 = :success[2]$
• Activity $B$: $TF = 9 - 4 - 3 = :success[2]$
• Activity $C$: $TF = 9 - 7 - 2 = :highlight[0]$ (critical activity)
• Activity $D$: $TF = 10 - 0 - 5 = :success[5]$

Step 3: Draw the Gantt Chart

• Activity A: Starts at time $0$, duration $4$, and has $2$ units of float.
• Activity B: Starts at time $4$, duration $3$, and has $2$ units of float.
• Activity C: Starts at time $7$, duration $2$, and no float (critical path).
• Activity D: Starts at time $0$, duration $5$, and has $5$ units of float.

Visualising the Gantt chart:

Time:       0   1   2   3   4   5   6   7   8   9   10
Activity A: ████----
Activity B:     ███----
Activity C:             ██
Activity D: █████----

Key:

• █: Task duration.
• - : Float.

Example 2: Construct a Gantt Chart for Dependencies

Step 1: Given Information

The project has the following dependencies:

Step 2: Calculate Total Float

• Activity $A$: $TF = 3 - 0 - 2 = :success[1]$
• Activity $B$: $TF = 6 - 2 - 3 = :success[1]$
• Activity $C$: $TF = 6 - 2 - 4 = :highlight[0]$ (critical activity)
• Activity $D$: $TF = 12 - 6 - 5 = :success[1]$

Step 3: Draw the Gantt Chart

Considering the dependencies:

• $A$ starts at time $0$, duration $2$, and has $1$ unit of float.
• $B$ and $C$ depend on $A$, so they cannot start before time $2$.
• $D$ depends on $B$ and $C$, so it cannot start before time $6$.

Visualising the Gantt chart:

Time:       0   1   2   3   4   5   6   7   8   9   10  11  12
Activity A: ██-
Activity B:     ███-
Activity C:     ████
Activity D:             █████

Common Mistakes

1. Incorrect float calculation: Forgetting to subtract the duration ($d$) in $TF = L - E - d$
2. Misinterpreting dependencies: Starting an activity before all its predecessors are complete.
3. Omitting float on the Gantt chart: Ensure that float is shown for non-critical activities.
4. Overlapping activities requiring one worker: Assign only one worker per activity to avoid conflicts.
5. Failing to identify the critical path: Recognise activities with zero float as critical.

Key Formulas

1. Total float:

2. Earliest Start Time (EST): The earliest time an activity can start, based on its predecessors.
3. Latest Finish Time (LFT): The latest time an activity can finish without delaying the project.
4. Gantt chart bars:

• Bar length = duration.
• Extend bar for float if present.

5. Dependencies:

• Activities must start only after their predecessors are completed.