---
title: "CPM components (AOA), activities A to I"
subject: "Project Engineering"
chapter: 3
key: "semester_7/project_engineering:cpm-aoa-nine-act-c76"
source: https://www.bctengineer.com/q/semester_7/project_engineering/cpm-aoa-nine-act-c76
site: BCT Engineering
---

# CPM components (AOA), activities A to I

Project Engineering · Chapter 3: Project Planning and Scheduling

- Asked in: 2076 Chaitra, 13 marks
- Video: [Finding the Critical Path, duration and Project Duration, Critical Path Method, float, EST, EFT, LST](https://www.youtube.com/watch?v=OKKcjeJt4cI) — Civil Engineering Exam
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/project_engineering/cpm-aoa-nine-act-c76
- Whole chapter: https://www.bctengineer.com/md/semester_7/project_engineering/chapter-3

## Question

Find all the components of CPM from the following information. Use the AOA method.

| Activity | Duration (week) | Predecessor |
|---|---|---|
| A | 1 | - |
| B | 3 | - |
| C | 2 | A, B |
| D | 5 | B |
| E | 3 | B |
| F | 1 | C, D |
| G | 3 | D |
| H | 4 | D, E |
| I | 5 | F, G, H |

## Short answer

Using the AOA method, all CPM components are found: network with 8 events and 3 dummies, event times, EST/EFT/LST/LFT and all floats tabulated. Project duration is 17 weeks. Critical path is B-D-H-I (3+5+4+5 weeks).

## Answer

The components of CPM are: the network, event times (TE, TL), activity times (EST, EFT, LST, LFT), floats (TF, FF, Int.F, Ind.F), the critical path and the project duration. All are found below with the activity-on-arrow (AOA) method.

Notation: EST/EFT = earliest start/finish, LST/LFT = latest start/finish, TF = total float, FF = free float, Int.F = interfering float, Ind.F = independent float.

#### Network diagram (activity-on-arrow)

Each activity is an arrow with its duration in brackets (weeks). Events are circled numbers; dashed arrows are dummy activities (zero duration) that show dependencies only.

```text
(1) --B(3)--> (2)
(1) --A(1)--> (4)
(2) --D(5)--> (3)
(2) - - dummy - -> (4)
(2) --E(3)--> (5)
(3) - - dummy - -> (5)
(3) - - dummy - -> (6)
(3) --G(3)--> (7)
(4) --C(2)--> (6)
(5) --H(4)--> (7)
(6) --F(1)--> (7)
(7) --I(5)--> (8)
```

The network has 8 events and 3 dummy activities. Start event is (1) and the end event is (8).

#### Forward pass (EST and EFT)

Rule: EST = largest EFT of the preceding activities (0 for start activities); EFT = EST + D.

- A: EST = 0; EFT = 0 + 1 = 1
- B: EST = 0; EFT = 0 + 3 = 3
- C: EST = max(A:1, B:3) = 3; EFT = 3 + 2 = 5
- D: EST = EFT of B = 3; EFT = 3 + 5 = 8
- E: EST = EFT of B = 3; EFT = 3 + 3 = 6
- F: EST = max(C:5, D:8) = 8; EFT = 8 + 1 = 9
- G: EST = EFT of D = 8; EFT = 8 + 3 = 11
- H: EST = max(D:8, E:6) = 8; EFT = 8 + 4 = 12
- I: EST = max(F:9, G:11, H:12) = 12; EFT = 12 + 5 = 17

Project duration = largest EFT = **17 weeks**.

#### Backward pass (LFT and LST)

Rule: LFT = smallest LST of the succeeding activities (project duration for end activities); LST = LFT - D.

- I: LFT = project duration = 17; LST = 17 - 5 = 12
- H: LFT = LST of I = 12; LST = 12 - 4 = 8
- G: LFT = LST of I = 12; LST = 12 - 3 = 9
- F: LFT = LST of I = 12; LST = 12 - 1 = 11
- E: LFT = LST of H = 8; LST = 8 - 3 = 5
- D: LFT = min(F:11, G:9, H:8) = 8; LST = 8 - 5 = 3
- C: LFT = LST of F = 11; LST = 11 - 2 = 9
- B: LFT = min(C:9, D:3, E:5) = 3; LST = 3 - 3 = 0
- A: LFT = LST of C = 9; LST = 9 - 1 = 8

#### Floats

- **Total float** $TF = LST - EST = LFT - EFT$
- **Free float** $FF = \min(\text{EST of successors}) - EFT$
- **Interfering float** $IntF = TF - FF$
- **Independent float** $IndF = \max\left(0,\ \min(\text{EST of successors}) - \max(\text{LFT of predecessors}) - D\right)$

#### Result table

| Activity | D | EST | EFT | LST | LFT | TF | FF | Int.F | Ind.F |
|---|---|---|---|---|---|---|---|---|---|
| A | 1 | 0 | 1 | 8 | 9 | 8 | 2 | 6 | 2 |
| B | 3 | 0 | 3 | 0 | 3 | 0 | 0 | 0 | 0 |
| C | 2 | 3 | 5 | 9 | 11 | 6 | 3 | 3 | 0 |
| D | 5 | 3 | 8 | 3 | 8 | 0 | 0 | 0 | 0 |
| E | 3 | 3 | 6 | 5 | 8 | 2 | 2 | 0 | 2 |
| F | 1 | 8 | 9 | 11 | 12 | 3 | 3 | 0 | 0 |
| G | 3 | 8 | 11 | 9 | 12 | 1 | 1 | 0 | 1 |
| H | 4 | 8 | 12 | 8 | 12 | 0 | 0 | 0 | 0 |
| I | 5 | 12 | 17 | 12 | 17 | 0 | 0 | 0 | 0 |

Event times (TE from the forward pass, TL from the backward pass):

| Event | TE (earliest) | TL (latest) | Slack |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 3 | 3 | 0 |
| 3 | 8 | 8 | 0 |
| 4 | 3 | 9 | 6 |
| 5 | 8 | 8 | 0 |
| 6 | 8 | 11 | 3 |
| 7 | 12 | 12 | 0 |
| 8 | 17 | 17 | 0 |

#### Critical path and project duration

- Critical activities (TF = 0): B, D, H, I
- Critical path: B - D - H - I (= 3+5+4+5 = 17 weeks)

**Answer: project duration = 17 weeks; critical path = B - D - H - I**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. H has predecessors D (EFT 8) and E (EFT 6). What is the EST and EFT of H (duration 4)?
   A. EST 6, EFT 10
   B. EST 8, EFT 12
   C. EST 14, EFT 18
   D. EST 3, EFT 7
   Answer: B. EST = largest predecessor EFT = 8; EFT = 8 + 4 = 12. (Key point: Forward pass for H)

2. D is followed by F (LST 11), G (LST 9) and H (LST 8). What is LFT and LST of D (duration 5)?
   A. LFT 8, LST 3
   B. LFT 11, LST 6
   C. LFT 9, LST 4
   D. LFT 8, LST 8
   Answer: A. LFT is the smallest successor LST = 8 (from H), so LST = 8 - 5 = 3, equal to EST 3, so D is critical. (Key point: Backward pass for D)

3. Which activities form the critical path and what is the project duration?
   A. B-D-G-I, 16 weeks
   B. B-D-H-I, 17 weeks
   C. A-C-F-I, 9 weeks
   D. B-E-H-I, 15 weeks
   Answer: B. B+D+H+I = 3+5+4+5 = 17 weeks, the longest path, with zero float on each. (Key point: Critical path)

4. A has EFT 1, TF 8 and its successor C has EST 3. What are FF and interfering float of A?
   A. FF 8, IntF 0
   B. FF 3, IntF 5
   C. FF 2, IntF 6
   D. FF 0, IntF 8
   Answer: C. FF = EST of C - EFT of A = 3 - 1 = 2. Interfering float = TF - FF = 8 - 2 = 6. (Key point: Float of A)

## Close variants set in other exams

- [CPM components (AOA), 9 activities](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-aoa-nine-act-b78) — Project Engineering, chapter 3, asked 4×
- [CPM components (AOA), 10 activities](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-aoa-ten-act-b80) — Project Engineering, chapter 3, asked 5×
- [CPM components (AOA), 13 activities (months)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-aoa-thirteen-act-c75) — Project Engineering, chapter 3, asked 5×

## Related in Project Engineering

- [CPM components, 9 activities with successors](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-components-nine-act-c70) — Project Engineering, chapter 3, asked 3×
- [CPM network, 9 activities (A=3 to I=1)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-nine-act-b82) — Project Engineering, chapter 3, asked 5×
- [CPM network, 9 activities (A=5 to I=2)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-nine-act-a75) — Project Engineering, chapter 3, asked 5×
- [CPM precedence diagram, 9 activities](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-precedence-k72) — Project Engineering, chapter 3
- [CPM network, 13 activities (A to M)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-thirteen-act-k82) — Project Engineering, chapter 3, asked 5×
- [CPM network, 13 activities (A=2 to M=7)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-thirteen-act-b81) — Project Engineering, chapter 3, asked 5×

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