---
title: "CPM schedule, critical path and float"
subject: "Industrial Engineering and Management"
chapter: 3
key: "semester_6/industrial_engineering_and_management:cpm-network-critical-path-float"
source: https://www.bctengineer.com/q/semester_6/industrial_engineering_and_management/cpm-network-critical-path-float
site: BCT Engineering
---

# CPM schedule, critical path and float

Industrial Engineering and Management · Chapter 3: Loading and Scheduling Technique

- Asked in: year not given, 8 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_6/industrial_engineering_and_management/cpm-network-critical-path-float
- Whole chapter: https://www.bctengineer.com/md/semester_6/industrial_engineering_and_management/chapter-3

## Question

A project has the following activities (times in days):

| Activity | A | B | C | D | E | F | G | H |
|---|---|---|---|---|---|---|---|---|
| Duration | 4 | 3 | 5 | 6 | 4 | 3 | 5 | 2 |
| Predecessors | - | - | A | A, B | B | C, D | D, E | F, G |

(a) Draw the network (describe it), (b) find the earliest and latest start and finish times of every activity, (c) find the total float, the critical path and the project duration.

## Short answer

By forward and backward pass, project duration is 17 days. Critical path is A-D-G-H. Floats: B 1 day, C 3 days, E 3 days, F 2 days.

## Answer

#### (a) Network (activity-on-node)
```text
 A(4) --+----------> C(5) ---------+
        |                          v
        +----------> D(6) --+---> F(3) ----+
                     ^      |              v
 B(3) --+------------+      +---> G(5) --> H(2)
        |                         ^
        +----------> E(4) --------+
```
A and B start the project. C follows A; D needs both A and B; E follows B; F needs C and D; G needs D and E; H needs F and G.

#### (b) Forward and backward pass
Forward pass: $EF = ES + t$, where ES is the largest EF of the predecessors. Backward pass: $LS = LF - t$, where LF is the smallest LS of the successors. Project duration = largest EF = 17.

| Activity | $t$ | ES | EF | LS | LF | Float (LS - ES) |
|---|---|---|---|---|---|---|
| A | 4 | 0 | 4 | 0 | 4 | 0 |
| B | 3 | 0 | 3 | 1 | 4 | 1 |
| C | 5 | 4 | 9 | 7 | 12 | 3 |
| D | 6 | 4 | 10 | 4 | 10 | 0 |
| E | 4 | 3 | 7 | 6 | 10 | 3 |
| F | 3 | 10 | 13 | 12 | 15 | 2 |
| G | 5 | 10 | 15 | 10 | 15 | 0 |
| H | 2 | 15 | 17 | 15 | 17 | 0 |

Sample: ES of D = max(EF of A, EF of B) = max(4, 3) = 4. ES of F = max(9, 10) = 10. LF of D = min(LS of F, LS of G) = min(12, 10) = 10.

#### (c) Critical path
Activities with zero total float are critical: A, D, G, H.

Check: $4 + 6 + 5 + 2 = 17$ days. The other paths are shorter (e.g. B-D-G-H = 16, A-D-F-H = 15, A-C-F-H = 14).

**Answer: Critical path A-D-G-H; project duration 17 days; floats: B = 1, C = 3, E = 3, F = 2 days.**

## Check yourself

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

1. How is ES of an activity found in the forward pass?
   A. Smallest EF of predecessors
   B. Largest EF of predecessors
   C. LF minus t
   D. Sum of all durations
   Answer: B. ES is the largest EF of predecessors, so D starts at max(4, 3) = 4. (Key point: Forward pass)

2. How is LF found in the backward pass?
   A. Largest LS of successors
   B. ES + t
   C. Smallest LS of successors
   D. Largest EF
   Answer: C. LF = smallest LS of successors; LF of D = min(12, 10) = 10. (Key point: Backward pass)

3. Which is the critical path and project duration?
   A. A-D-G-H, 17 days
   B. A-C-F-H, 14 days
   C. B-D-G-H, 16 days
   D. B-E-G-H, 14 days
   Answer: A. A, D, G, H have zero float and 4+6+5+2 = 17 days. (Key point: Critical path)

4. What is the total float of activity C?
   A. 0 days
   B. 1 day
   C. 2 days
   D. 3 days
   Answer: D. Float = LS - ES = 7 - 4 = 3 days. (Key point: Float)

## Related in Industrial Engineering and Management

- [PERT expected time and completion probability](https://www.bctengineer.com/question.md?key=semester_6%2Findustrial_engineering_and_management%3Apert-probability-of-completion) — Industrial Engineering and Management, chapter 3
- [Difference between CPM and PERT](https://www.bctengineer.com/question.md?key=semester_6%2Findustrial_engineering_and_management%3Adifference-between-cpm-and-pert) — Industrial Engineering and Management, chapter 3
- [EOQ, total cost and reorder point numerical](https://www.bctengineer.com/question.md?key=semester_6%2Findustrial_engineering_and_management%3Aeoq-numerical-reorder-point) — Industrial Engineering and Management, chapter 4

## Related in other subjects

- [CPM network, road project (days)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-road-project-b79) — Project Engineering, chapter 3
- [Dummy activity; use of critical path in CPM](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Adummy-critical-path-use) — 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 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