Minimum Completion Time (HSC SSCE Mathematics Standard): Revision Notes

Worked Example 11: Simple Project Analysis

Part a: What is the minimum completion time?

Solution:

The minimum completion time is found by looking at the EST value in the finish node of the network diagram.

From the diagram, we can see that the EST at the finish node is $13$.

Therefore, the minimum completion time for this project is 13 hours.

Part b: Which activities would not reduce completion time if crashed?

Solution:

To answer this question, we need to identify which activities are NOT on the critical path.

Looking at the network diagram, the critical path (shown in red) runs through activities D and E.

Activities A, B, and C are on an alternative path and are not part of the critical path.

Therefore, if activities A, B, or C were reduced in duration individually, this would not result in an earlier completion of the project.

These activities have float time, meaning they could take longer without delaying the project. Conversely, making them shorter doesn't speed up the project overall.

Part c: Recalculating after crashing activity D

Now suppose activity D was reduced in time from $6$ hours to $4$ hours. What would be the new minimum completion time?

Solution:

When an activity is crashed, we need to update the network diagram and recalculate the EST and LST values for all activities.

Step 1: Update activity D's duration from $6$ to $4$ hours

Step 2: Recalculate the EST values by working forward through the network

Step 3: Recalculate the LST values by working backward through the network

Step 4: Find the new EST at the finish node

After recalculating, we can see that the EST at the finish node is now $11$.

Therefore, the new minimum completion time is 11 hours.

Notice that crashing activity D reduced the project time by $2$ hours (from $13$ to $11$ hours). This worked because D was on the critical path. The critical path remains the same: D → E.

Worked Example 12: Complex Project with Multiple Activities

Part a: Finding the critical path and minimum completion time

Solution:

To find the critical path, we look for activities that have zero float time. Float time is calculated as:

$\text{Float time} = \text{LST} - \text{EST}$

Activities with zero float are on the critical path because any delay to these activities will delay the entire project.

Examining the network diagram:

The critical path is: B → C → F → H → I

The minimum completion time is the EST value at the finish node.

The minimum completion time is 19 weeks.

Part b: Which activities would not reduce completion time?

The builders can potentially speed up activities A, C, F, and G at additional cost. However, not all of these would be worth crashing.

Solution:

We need to identify which of these four activities are NOT on the critical path.

From part a, we know the critical path is B → C → F → H → I.

Examining the given activities:

• Activity A is not on the critical path
• Activity C is on the critical path
• Activity F is on the critical path
• Activity G is not on the critical path

Additionally, activities D and E are also not on the critical path.

Therefore, activities A and G (from the given list) would not result in an earlier completion of the project if reduced individually.

We should only consider crashing activities C and F, as these are on the critical path.

Part c: Finding minimum time after crashing multiple activities

The project owner will pay to reduce activities on the critical path. Activities C and F can each be reduced by a maximum of $2$ weeks. What is the new minimum completion time?

Solution:

Step 1: Reduce activity C from $3$ weeks to $1$ week (reduction of $2$ weeks)

Step 2: Reduce activity F from $4$ weeks to $2$ weeks (reduction of $2$ weeks)

Step 3: Update the network diagram with these new durations

Step 4: Recalculate all EST and LST values

Step 5: Identify the new critical path

After recalculating, we can observe that:

The critical path has changed! The new critical path is: B → E → H → I

This is an important lesson: crashing activities can change which path is critical. Once we reduced C and F significantly, path B → E → H → I became longer than the original critical path.

The new minimum completion time is 16 weeks.

This represents a reduction of $3$ weeks from the original $19$ weeks. Even though we reduced C and F by $2$ weeks each (total $4$ weeks), the project was only shortened by $3$ weeks because the alternative path through E became the limiting factor.