---
title: "CPM network, 9 activities (old course)"
subject: "Project Engineering"
chapter: 3
key: "semester_7/project_engineering:cpm-network-nine-act-c70o"
source: https://www.bctengineer.com/q/semester_7/project_engineering/cpm-network-nine-act-c70o
site: BCT Engineering
---

# CPM network, 9 activities (old course)

Project Engineering · Chapter 3: Project Planning and Scheduling

- Asked in: 2070 Chaitra (old course), 4+6 marks
- Video: [How to draw a CPM network diagram](https://www.youtube.com/watch?v=gEBMPP0SOnU) — Engineer4Free
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/project_engineering/cpm-network-nine-act-c70o
- Whole chapter: https://www.bctengineer.com/md/semester_7/project_engineering/chapter-3

## Question

Draw a CPM network. Find EST, EFT, LST, LFT, TF, FF.

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

## Short answer

Duration 15 time units; critical path B-E-H-I. EST/EFT/LST/LFT/TF/FF: A 0,2,1,3,1,0; B 0,4,0,4,0,0; C 2,7,3,8,1,0; D 4,9,5,10,1,0; E 4,7,4,7,0,0; F 7,11,8,12,1,1; G 9,11,10,12,1,1; H 7,12,7,12,0,0; I 12,15,12,15,0,0.

## Answer

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 (time units). Events are circled numbers; dashed arrows are dummy activities (zero duration) that show dependencies only.

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

The network has 8 events and 1 dummy activity. 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 + 2 = 2
- B: EST = 0; EFT = 0 + 4 = 4
- C: EST = EFT of A = 2; EFT = 2 + 5 = 7
- D: EST = EFT of B = 4; EFT = 4 + 5 = 9
- E: EST = EFT of B = 4; EFT = 4 + 3 = 7
- F: EST = max(B:4, C:7) = 7; EFT = 7 + 4 = 11
- H: EST = EFT of E = 7; EFT = 7 + 5 = 12
- G: EST = EFT of D = 9; EFT = 9 + 2 = 11
- I: EST = max(F:11, G:11, H:12) = 12; EFT = 12 + 3 = 15

Project duration = largest EFT = **15 time units**.

#### 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 = 15; LST = 15 - 3 = 12
- G: LFT = LST of I = 12; LST = 12 - 2 = 10
- H: LFT = LST of I = 12; LST = 12 - 5 = 7
- F: LFT = LST of I = 12; LST = 12 - 4 = 8
- E: LFT = LST of H = 7; LST = 7 - 3 = 4
- D: LFT = LST of G = 10; LST = 10 - 5 = 5
- C: LFT = LST of F = 8; LST = 8 - 5 = 3
- B: LFT = min(D:5, E:4, F:8) = 4; LST = 4 - 4 = 0
- A: LFT = LST of C = 3; LST = 3 - 2 = 1

#### 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 | 2 | 0 | 2 | 1 | 3 | 1 | 0 | 1 | 0 |
| B | 4 | 0 | 4 | 0 | 4 | 0 | 0 | 0 | 0 |
| C | 5 | 2 | 7 | 3 | 8 | 1 | 0 | 1 | 0 |
| D | 5 | 4 | 9 | 5 | 10 | 1 | 0 | 1 | 0 |
| E | 3 | 4 | 7 | 4 | 7 | 0 | 0 | 0 | 0 |
| F | 4 | 7 | 11 | 8 | 12 | 1 | 1 | 0 | 0 |
| G | 2 | 9 | 11 | 10 | 12 | 1 | 1 | 0 | 0 |
| H | 5 | 7 | 12 | 7 | 12 | 0 | 0 | 0 | 0 |
| I | 3 | 12 | 15 | 12 | 15 | 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 | 2 | 3 | 1 |
| 3 | 4 | 4 | 0 |
| 4 | 9 | 10 | 1 |
| 5 | 7 | 7 | 0 |
| 6 | 7 | 8 | 1 |
| 7 | 12 | 12 | 0 |
| 8 | 15 | 15 | 0 |

#### Critical path and project duration

- Critical activities (TF = 0): B, E, H, I
- Critical path: B - E - H - I (= 4+3+5+3 = 15 time units)

**Answer: project duration = 15 time units; critical path = B - E - H - I**

## Check yourself

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

1. What is the project duration?
   A. 16
   B. 15
   C. 12
   D. 14
   Answer: B. I starts at max(F:11, G:11, H:12) = 12 and ends at 15, the largest EFT. (Key point: Project duration)

2. Which path is critical?
   A. B-D-G-I
   B. B-E-H-I
   C. A-C-F-I and B-D-G-I
   D. A-C-F-I
   Answer: B. B, E, H and I have TF = 0, and 4+3+5+3 = 15 days. (Key point: Critical path)

3. Why is a dummy activity needed from event 3 to event 6?
   A. To connect B to D
   B. Because H is the last activity
   C. F depends on both B and C, while C depends only on A
   D. To add time to the project
   Answer: C. F has predecessors B and C, so a zero-duration dummy shows that F must wait for B as well as C. (Key point: Need for a dummy)

4. F has EFT 11 and its successor I has EST 12. What is FF of F?
   A. 1
   B. 12
   C. 0
   D. 3
   Answer: A. FF = min EST of successors - EFT = 12 - 11 = 1. (Key point: Free float of F)

## Close variants set in other exams

- [CPM network, 9 activities with successors](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-nine-act-s65) — Project Engineering, chapter 3, asked 2×

## Related in Project Engineering

- [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 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×
- [CPM network, 10 activities (A=4 to J=9)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-ten-act-as70) — Project Engineering, chapter 3
- [CPM network, 10 activities with successors](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-ten-act-a74) — Project Engineering, chapter 3