What is the minimum cost of crashing the following project at Sawaya Robotics by 4​ days?
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            
ACTIVITY
NORMAL TIME​ (DAYS)
CRASH TIME​ (DAYS)
NORMAL COST
CRASH COST
IMMEDIATE​ PREDECESSOR(S)
Design​ (A)
9
7
200
300
​-
Wiring​ (B)
7
6
800
875
​-
Chip install​ (C)
5
4
600
675
​-
Software​ (D)
6
4
1 comma 100
1 comma 300
​A, B
Testing​ (E)
9
6
1 comma 200
1 comma 875
C
 
Part 2


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?

---

**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.

What is the minimum cost of crashing the following project at Sawaya Robotics by 4 days?

This solution explores the minimum cost to crash a project at Sawaya Robotics by 4 days using project management techniques like the Critical Path Method (CPM). The problem focuses on cost optimization while reducing project time, with a step-by-step breakdown of crash cost calculations for each activity. Ideal for undergraduate students studying project management.

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