---
title: "CPM network, 12 activities (A=10 to L=8)"
subject: "Project Engineering"
chapter: 3
key: "semester_7/project_engineering:cpm-network-twelve-act-as67"
source: https://www.bctengineer.com/q/semester_7/project_engineering/cpm-network-twelve-act-as67
site: BCT Engineering
---

# CPM network, 12 activities (A=10 to L=8)

Project Engineering · Chapter 3: Project Planning and Scheduling

- Asked in: 2067 Asar (old course), 4+12 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-twelve-act-as67
- Whole chapter: https://www.bctengineer.com/md/semester_7/project_engineering/chapter-3

## Question

Define forward and backward pass in the network analysis. Draw the network diagram and compute EST, EFT, LST, LFT, TF, FF, IF and interfering floats for each activity of the project having the precedence relationship shown below. Also find out the critical path.

| Activity | Duration (days) | Predecessor |
|---|---|---|
| A | 10 | - |
| B | 9 | - |
| C | 7 | A |
| D | 9 | A |
| E | 8 | B |
| F | 5 | B |
| G | 11 | D, E |
| H | 6 | C, G |
| I | 9 | H |
| J | 12 | G |
| K | 10 | G, F |
| L | 8 | K |

## Short answer

Forward pass computes EST/EFT from start to end; backward pass computes LFT/LST from end to start. Duration 48 days; critical path A-D-G-K-L. EST/EFT/LST/LFT/TF/FF: A 0,10,0,10,0,0; B 0,9,2,11,2,0; C 10,17,26,33,16,13; D 10,19,10,19,0,0; E 9,17,11,19,2,2; F 9,14,25,30,16,16; G 19,30,19,30,0,0; H 30,36,33,39,3,0; I 36,45,39,48,3,3; J 30,42,36,48,6,6; K 30,40,30,40,0,0; L 40,48,40,48,0,0.

## Answer

#### Forward pass and backward pass

- **Forward pass:** the calculation from the start event to the end event to find EST and EFT of each activity ($EST$ = largest EFT of the predecessors, $EFT = EST + D$). The largest EFT is the project duration.
- **Backward pass:** the calculation from the end event back to the start event to find LFT and LST ($LFT$ = smallest LST of the successors, $LST = LFT - D$), taking the project duration as LFT of the last activity.

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(10)--> (2)
(1) --B(9)--> (3)
(2) --D(9)--> (4)
(2) --C(7)--> (6)
(3) --E(8)--> (4)
(3) --F(5)--> (7)
(4) --G(11)--> (5)
(5) - - dummy - -> (6)
(5) - - dummy - -> (7)
(5) --J(12)--> (10)
(6) --H(6)--> (8)
(7) --K(10)--> (9)
(8) --I(9)--> (10)
(9) --L(8)--> (10)
```

The network has 10 events and 2 dummy activities. Start event is (1) and the end event is (10).

#### 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 + 10 = 10
- B: EST = 0; EFT = 0 + 9 = 9
- E: EST = EFT of B = 9; EFT = 9 + 8 = 17
- F: EST = EFT of B = 9; EFT = 9 + 5 = 14
- C: EST = EFT of A = 10; EFT = 10 + 7 = 17
- D: EST = EFT of A = 10; EFT = 10 + 9 = 19
- G: EST = max(D:19, E:17) = 19; EFT = 19 + 11 = 30
- H: EST = max(C:17, G:30) = 30; EFT = 30 + 6 = 36
- J: EST = EFT of G = 30; EFT = 30 + 12 = 42
- K: EST = max(G:30, F:14) = 30; EFT = 30 + 10 = 40
- I: EST = EFT of H = 36; EFT = 36 + 9 = 45
- L: EST = EFT of K = 40; EFT = 40 + 8 = 48

Project duration = largest EFT = **48 days**.

#### Backward pass (LFT and LST)

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

- L: LFT = project duration = 48; LST = 48 - 8 = 40
- I: LFT = project duration = 48; LST = 48 - 9 = 39
- K: LFT = LST of L = 40; LST = 40 - 10 = 30
- J: LFT = project duration = 48; LST = 48 - 12 = 36
- H: LFT = LST of I = 39; LST = 39 - 6 = 33
- G: LFT = min(H:33, J:36, K:30) = 30; LST = 30 - 11 = 19
- D: LFT = LST of G = 19; LST = 19 - 9 = 10
- C: LFT = LST of H = 33; LST = 33 - 7 = 26
- F: LFT = LST of K = 30; LST = 30 - 5 = 25
- E: LFT = LST of G = 19; LST = 19 - 8 = 11
- B: LFT = min(E:11, F:25) = 11; LST = 11 - 9 = 2
- A: LFT = min(C:26, D:10) = 10; LST = 10 - 10 = 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 | 10 | 0 | 10 | 0 | 10 | 0 | 0 | 0 | 0 |
| B | 9 | 0 | 9 | 2 | 11 | 2 | 0 | 2 | 0 |
| C | 7 | 10 | 17 | 26 | 33 | 16 | 13 | 3 | 13 |
| D | 9 | 10 | 19 | 10 | 19 | 0 | 0 | 0 | 0 |
| E | 8 | 9 | 17 | 11 | 19 | 2 | 2 | 0 | 0 |
| F | 5 | 9 | 14 | 25 | 30 | 16 | 16 | 0 | 14 |
| G | 11 | 19 | 30 | 19 | 30 | 0 | 0 | 0 | 0 |
| H | 6 | 30 | 36 | 33 | 39 | 3 | 0 | 3 | 0 |
| I | 9 | 36 | 45 | 39 | 48 | 3 | 3 | 0 | 0 |
| J | 12 | 30 | 42 | 36 | 48 | 6 | 6 | 0 | 6 |
| K | 10 | 30 | 40 | 30 | 40 | 0 | 0 | 0 | 0 |
| L | 8 | 40 | 48 | 40 | 48 | 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 | 10 | 10 | 0 |
| 3 | 9 | 11 | 2 |
| 4 | 19 | 19 | 0 |
| 5 | 30 | 30 | 0 |
| 6 | 30 | 33 | 3 |
| 7 | 30 | 30 | 0 |
| 8 | 36 | 39 | 3 |
| 9 | 40 | 40 | 0 |
| 10 | 48 | 48 | 0 |

#### Critical path and project duration

- Critical activities (TF = 0): A, D, G, K, L
- Critical path: A - D - G - K - L (= 10+9+11+10+8 = 48 days)

**Answer: project duration = 48 days; critical path = A - D - G - K - L**

## Check yourself

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

1. What is the forward pass?
   A. Drawing the network from end to start
   B. Start to end, giving EST and EFT; largest EFT is the project duration
   C. End to start, giving LST and LFT
   D. Finding only the float
   Answer: B. The forward pass moves from start to end, EST = largest predecessor EFT, EFT = EST + D. (Key point: Forward pass definition)

2. In the backward pass, how is LFT of an activity found?
   A. Largest LST of its successors
   B. EFT + float
   C. Smallest LST of its successors, with project duration for the last activity
   D. EST + D
   Answer: C. LFT is the smallest LST among successors, starting from the project duration; LST = LFT - D. (Key point: Backward pass definition)

3. What is the critical path and duration?
   A. A-C-H-I, 45 days
   B. A-D-G-J, 42 days
   C. B-E-G-J, 42 days
   D. A-D-G-K-L, 48 days
   Answer: D. L ends at 48 and A, D, G, K, L have TF = 0: 10+9+11+10+8 = 48. (Key point: Critical path)

4. Why is EST of G 19?
   A. It is the project duration
   B. EFT of D (19) is larger than EFT of E (17)
   C. It follows B
   D. EFT of E is larger
   Answer: B. G follows D and E; max(19, 17) = 19. (Key point: EST of G)

5. C has TF = 16 and FF = 13. What is its interfering float?
   A. 29
   B. 16
   C. 3
   D. 13
   Answer: C. Interfering float = TF - FF = 16 - 13 = 3. (Key point: Floats of C)

## Related in Project Engineering

- [CPM network, 10 activities (A=10 to J=4)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-ten-act-k79) — Project Engineering, chapter 3
- [CPM network, 12 activities (A=8 to L=3)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-twelve-act-b66) — Project Engineering, chapter 3
- [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 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, building project (A to K)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-eleven-act-s73) — 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