---
title: "CPM network, 9 activities with successors"
subject: "Project Engineering"
chapter: 3
key: "semester_7/project_engineering:cpm-network-nine-act-s65"
source: https://www.bctengineer.com/q/semester_7/project_engineering/cpm-network-nine-act-s65
site: BCT Engineering
---

# CPM network, 9 activities with successors

Project Engineering · Chapter 3: Project Planning and Scheduling

- Asked in: 2065 Shrawan (old course), 11 marks
- Video: [Use forward and backward pass to determine project duration and critical path](https://www.youtube.com/watch?v=4oDLMs11Exs) — 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-s65
- Whole chapter: https://www.bctengineer.com/md/semester_7/project_engineering/chapter-3

## Question

Draw the network. Find EST, EFT, LST, LFT, TF, FF and IF.

| SN | Activity | Duration | Predecessor | Successor |
|---|---|---|---|---|
| 1 | A | 5 | - | B, C, D |
| 2 | B | 4 | A | E |
| 3 | C | 2 | A | F, H |
| 4 | D | 3 | A | G |
| 5 | E | 2 | B | H |
| 6 | F | 1 | C | I |
| 7 | G | 3 | D | I |
| 8 | H | 1 | C, E | - |
| 9 | I | 2 | F, G | - |

## Short answer

Draw the activity-on-arrow network with one dummy, then compute EST/EFT forward and LST/LFT backward. Total, free, interfering and independent floats follow from the event times. Project duration is 13 days; critical path is A-D-G-I.

## Answer

IF is taken as independent float.

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

```text
(1) --A(5)--> (2)
(2) --B(4)--> (3)
(2) --C(2)--> (4)
(2) --D(3)--> (5)
(3) --E(2)--> (6)
(4) - - dummy - -> (6)
(4) --F(1)--> (7)
(5) --G(3)--> (7)
(6) --H(1)--> (8)
(7) --I(2)--> (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 + 5 = 5
- B: EST = EFT of A = 5; EFT = 5 + 4 = 9
- C: EST = EFT of A = 5; EFT = 5 + 2 = 7
- D: EST = EFT of A = 5; EFT = 5 + 3 = 8
- F: EST = EFT of C = 7; EFT = 7 + 1 = 8
- G: EST = EFT of D = 8; EFT = 8 + 3 = 11
- E: EST = EFT of B = 9; EFT = 9 + 2 = 11
- H: EST = max(C:7, E:11) = 11; EFT = 11 + 1 = 12
- I: EST = max(F:8, G:11) = 11; EFT = 11 + 2 = 13

Project duration = largest EFT = **13 days**.

#### 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 = 13; LST = 13 - 2 = 11
- H: LFT = project duration = 13; LST = 13 - 1 = 12
- E: LFT = LST of H = 12; LST = 12 - 2 = 10
- G: LFT = LST of I = 11; LST = 11 - 3 = 8
- F: LFT = LST of I = 11; LST = 11 - 1 = 10
- D: LFT = LST of G = 8; LST = 8 - 3 = 5
- C: LFT = min(F:10, H:12) = 10; LST = 10 - 2 = 8
- B: LFT = LST of E = 10; LST = 10 - 4 = 6
- A: LFT = min(B:6, C:8, D:5) = 5; LST = 5 - 5 = 0

#### 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 | 5 | 0 | 5 | 0 | 5 | 0 | 0 | 0 | 0 |
| B | 4 | 5 | 9 | 6 | 10 | 1 | 0 | 1 | 0 |
| C | 2 | 5 | 7 | 8 | 10 | 3 | 0 | 3 | 0 |
| D | 3 | 5 | 8 | 5 | 8 | 0 | 0 | 0 | 0 |
| E | 2 | 9 | 11 | 10 | 12 | 1 | 0 | 1 | 0 |
| F | 1 | 7 | 8 | 10 | 11 | 3 | 3 | 0 | 0 |
| G | 3 | 8 | 11 | 8 | 11 | 0 | 0 | 0 | 0 |
| H | 1 | 11 | 12 | 12 | 13 | 1 | 1 | 0 | 0 |
| I | 2 | 11 | 13 | 11 | 13 | 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 | 5 | 5 | 0 |
| 3 | 9 | 10 | 1 |
| 4 | 7 | 10 | 3 |
| 5 | 8 | 8 | 0 |
| 6 | 11 | 12 | 1 |
| 7 | 11 | 11 | 0 |
| 8 | 13 | 13 | 0 |

#### Critical path and project duration

- Critical activities (TF = 0): A, D, G, I
- Critical path: A - D - G - I (= 5+3+3+2 = 13 days)

**Answer: project duration = 13 days; critical path = A - D - G - I**

## Check yourself

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

1. What are the project duration and critical path?
   A. 13 days, A-D-G-I
   B. 13 days, A-B-E-H
   C. 11 days, A-C-F-I
   D. 12 days, A-B-E-H
   Answer: A. I ends at 13; A, D, G, I have zero float: 5+3+3+2 = 13. (Key point: Critical path)

2. Why is EST of H = 11?
   A. EFT of A
   B. EFT of I
   C. EFT of C only
   D. Max of EFT of C (7) and EFT of E (11)
   Answer: D. H follows C and E; the larger EFT, 11 from E, controls. (Key point: EST of H)

3. Why is a dummy drawn from event 4 to event 6?
   A. To add time
   B. To show A is critical
   C. H depends on C as well as E, but F depends only on C
   D. To link D and G
   Answer: C. C leads to both F and H, so a dummy lets H wait for C without making F wait for E. (Key point: Role of the dummy)

4. LST of A is 0. How?
   A. LFT of A = min(LST of B = 6, C = 8, D = 5) = 5, so LST = 5 - 5
   B. LFT equals EFT of B
   C. LFT of A = max of successors' LST
   D. LFT of A = project duration
   Answer: A. The smallest successor LST is 5 (D), so LFT = 5 and LST = 0. (Key point: Backward pass for A)

## Close variants set in other exams

- [CPM network, 9 activities (old course)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-nine-act-c70o) — Project Engineering, chapter 3, asked 2×

## Related in Project Engineering

- [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
- [CPM network, 9 activities with successors](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-nine-act-a76) — Project Engineering, chapter 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 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×