An architect has been awarded a contract to prepare plans for an urban renewal project. The job consists of the following activities and their estimated times:
Activity
Immediate Time Predecessors (days)
A Prepare preliminary sketches
B Outline specifications
2
1
C Prepare drawings
A
3
D Write specifications
A, B
2
E Run off prints
C, D
1
F Have specification
B, D
3
G Assemble bid packages
E, F
1

To solve this problem, we need to:

1. Construct a project schedule using a network diagram based on the activities and their immediate predecessors.
2. Identify the critical path, which is the longest path of dependent activities through the project. This path determines the minimum project duration.

### Step-by-Step Breakdown:

| Activity | Description              | Time (days) | Immediate Predecessors |
|----------|--------------------------|-------------|------------------------|
| A        | Prepare preliminary sketches | 2           | None                   |
| B        | Outline specifications      | 1           | None                   |
| C        | Prepare drawings            | 3           | A                      |
| D        | Write specifications        | 2           | A, B                   |
| E        | Run off prints              | 1           | C, D                   |
| F        | Have specifications         | 3           | B, D                   |
| G        | Assemble bid packages       | 1           | E, F                   |

### Step 1: Network Diagram
- **A** and **B** can start at the same time since they have no predecessors.
- **C** depends on **A**.
- **D** depends on both **A** and **B**.
- **E** depends on both **C** and **D**.
- **F** depends on both **B** and **D**.
- **G** depends on both **E** and **F**.

### Step 2: Calculate Earliest Start (ES), Earliest Finish (EF), Latest Start (LS), and Latest Finish (LF)

Let’s analyze the project using the forward and backward pass techniques to calculate the earliest and latest start and finish times for each activity. This will help us determine the critical path.

### Forward Pass (Earliest Start/Finish)

1. **A**: Earliest Start (ES) = 0, Earliest Finish (EF) = 2 (ES + Duration)
2. **B**: ES = 0, EF = 1 (ES + Duration)
3. **C**: ES = EF of A = 2, EF = 5 (ES + 3)
4. **D**: ES = Max(EF of A, EF of B) = Max(2, 1) = 2, EF = 4 (ES + 2)
5. **E**: ES = Max(EF of C, EF of D) = Max(5, 4) = 5, EF = 6 (ES + 1)
6. **F**: ES = Max(EF of B, EF of D) = Max(1, 4) = 4, EF = 7 (ES + 3)
7. **G**: ES = Max(EF of E, EF of F) = Max(6, 7) = 7, EF = 8 (ES + 1)

So, the minimum project duration (EF of G) is **8 days**.

### Step 3: Backward Pass (Latest Start/Finish)

We now perform the backward pass, starting from the last activity, to find the latest start and finish times.

1. **G**: LF = 8 (since it’s the last activity), LS = LF - Duration = 8 - 1 = 7
2. **E**: LF = LS of G = 7, LS = LF - Duration = 7 - 1 = 6
3. **F**: LF = LS of G = 7, LS = LF - Duration = 7 - 3 = 4
4. **C**: LF = LS of E = 6, LS = LF - Duration = 6 - 3 = 3
5. **D**: LF = Min(LS of E, LS of F) = Min(6, 4) = 4, LS = LF - Duration = 4 - 2 = 2
6. **B**: LF = Min(LS of D, LS of F) = Min(2, 4) = 2, LS = LF - Duration = 2 - 1 = 1
7. **A**: LF = Min(LS of C, LS of D) = Min(3, 2) = 2, LS = LF - Duration = 2 - 2 = 0

### Step 4: Determine the Critical Path
The critical path is the path where all activities have the same earliest and latest start times (i.e., zero slack). Activities on the critical path are:

- A → D → F → G

So, the critical path is **A → D → F → G**, and the project duration is **8 days**.

### Summary:
- **Critical Path**: A → D → F → G
- **Minimum Project Duration**: 8 days

Do you want further details or have any questions?

Here are some related questions you could explore:
1. What is the slack time for non-critical activities?
2. Can you calculate the total float for the non-critical activities?
3. How would adding 2 more days to Activity A affect the critical path?
4. What happens to the project duration if Activity D takes an additional day?
5. How would you adjust the project schedule to reduce the total project time?

**Tip:** Monitoring critical path activities is key to keeping a project on schedule.

Explore a detailed example of how to use the Critical Path Method (CPM) in project scheduling for an urban renewal project. Learn how to calculate the earliest and latest start and finish times, determine the critical path, and estimate the minimum project duration. Suitable for undergraduate project management students.

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