To determine the minimum cost of crashing the project by 4 days, we need to follow a step-by-step approach, focusing on which activities to crash in a way that minimizes the total cost.

Step 1: Calculate the Crash Cost per Day for Each Activity

For each activity, we'll calculate how much it costs to reduce its duration by 1 day, known as the crash cost per day. This is calculated by the formula:

$\text{Crash Cost per Day} = \frac{\text{Crash Cost} - \text{Normal Cost}}{\text{Normal Time} - \text{Crash Time}}$

Activity A (Design): $\frac{300 - 200}{9 - 7} = \frac{100}{2} = 50$

Activity B (Wiring): $\frac{875 - 800}{7 - 6} = \frac{75}{1} = 75$

Activity C (Chip Install): $\frac{675 - 600}{5 - 4} = \frac{75}{1} = 75$

Activity D (Software): $\frac{1,300 - 1,100}{6 - 4} = \frac{200}{2} = 100$

Activity E (Testing): $\frac{1,875 - 1,200}{9 - 6} = \frac{675}{3} = 225$

Step 2: Identify the Critical Path

The critical path is the sequence of activities that determines the minimum time to complete the project. Crashing a project involves reducing the time of activities on the critical path, since any delay or acceleration in non-critical activities will not impact the overall project time.

Immediate predecessors and activities provide insight into dependencies:

• Activity D (Software) depends on A (Design) and B (Wiring).
• Activity E (Testing) depends on C (Chip Install).

The overall project can be laid out as follows:

1. Path 1: A → D

   • Duration: $9 \text{ (A)} + 6 \text{ (D)} = 15$ days.

2. Path 2: B → D

   • Duration: $7 \text{ (B)} + 6 \text{ (D)} = 13$ days.

3. Path 3: C → E

   • Duration: $5 \text{ (C)} + 9 \text{ (E)} = 14$ days.

The longest path, A → D (15 days), is the critical path. To reduce the project time, we must crash activities on this path.

Step 3: Crash the Activities on the Critical Path

We need to reduce the project duration by 4 days. We will prioritize crashing activities with the lowest cost per day on the critical path.

The critical path is A → D. So, we can crash Activity A and Activity D.

1. Crash Activity D (Software):

   • Maximum crash time: 2 days (from 6 to 4 days).
   • Cost: $2 \times 100 = 200$.

   After crashing Activity D by 2 days, the new duration is $13$ days. We still need to reduce the project by 2 more days.

2. Crash Activity A (Design):

   • Maximum crash time: 2 days (from 9 to 7 days).
   • Cost: $2 \times 50 = 100$.

Now, the project duration is reduced by a total of 4 days, and the new project duration is $11$ days.

Step 4: Calculate the Total Cost of Crashing

The total cost of crashing is the sum of the costs to crash Activity A and Activity D:

$\text{Total Cost} = 200 \text{ (for D)} + 100 \text{ (for A)} = 300$

Conclusion:

The minimum cost to crash the project by 4 days is $300.

Do you want more details or have any questions?

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Related Questions:

1. How is the critical path determined in project management?
2. What are the limitations of crashing a project beyond the critical path?
3. Can activity E (Testing) impact the project timeline?
4. What happens if crashing costs exceed the expected project benefits?
5. How can you apply fast-tracking in project management to reduce time?

Tip: When crashing a project, always focus on reducing activities on the critical path for the most cost-effective time savings.