--- title: "CPM network, 13 activities (A to M)" subject: "Project Engineering" chapter: 3 key: "semester_7/project_engineering:cpm-network-thirteen-act-k82" source: https://www.bctengineer.com/q/semester_7/project_engineering/cpm-network-thirteen-act-k82 site: BCT Engineering --- # CPM network, 13 activities (A to M) Project Engineering · Chapter 3: Project Planning and Scheduling - Asked in: 2082 Baisakh, 14 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-thirteen-act-k82 - Whole chapter: https://www.bctengineer.com/md/semester_7/project_engineering/chapter-3 ## Question Draw the CPM network diagram and compute EST, EFT, LST, LFT, TF, FF, Int.F and Ind.F from the information given below. Also compute the project duration and mark the critical path. | Activity | Duration (days) | Predecessor | |---|---|---| | A | 9 | - | | B | 8 | - | | C | 7 | - | | D | 9 | - | | E | 5 | A | | F | 4 | A | | G | 3 | D | | H | 6 | D | | I | 7 | B, F | | J | 8 | C, G | | K | 4 | B, E, F | | L | 5 | I, J | | M | 3 | C, G, H | ## Short answer Forward/backward pass gives project duration 25 days. Critical activities with zero total float: A, D, F, G, I, J, L. Critical paths: A-F-I-L and D-G-J-L (each 25 days). ## 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 (days). Events are circled numbers; dashed arrows are dummy activities (zero duration) that show dependencies only. ```text (1) --A(9)--> (2) (1) --D(9)--> (3) (1) --B(8)--> (4) (1) --C(7)--> (5) (2) --F(4)--> (4) (2) --E(5)--> (6) (3) --G(3)--> (5) (3) --H(6)--> (8) (4) - - dummy - -> (6) (4) --I(7)--> (7) (5) --J(8)--> (7) (5) - - dummy - -> (8) (6) --K(4)--> (9) (7) --L(5)--> (9) (8) --M(3)--> (9) ``` The network has 9 events and 2 dummy activities. Start event is (1) and the end event is (9). #### 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 + 9 = 9 - B: EST = 0; EFT = 0 + 8 = 8 - C: EST = 0; EFT = 0 + 7 = 7 - D: EST = 0; EFT = 0 + 9 = 9 - E: EST = EFT of A = 9; EFT = 9 + 5 = 14 - F: EST = EFT of A = 9; EFT = 9 + 4 = 13 - G: EST = EFT of D = 9; EFT = 9 + 3 = 12 - H: EST = EFT of D = 9; EFT = 9 + 6 = 15 - J: EST = max(C:7, G:12) = 12; EFT = 12 + 8 = 20 - I: EST = max(B:8, F:13) = 13; EFT = 13 + 7 = 20 - K: EST = max(B:8, E:14, F:13) = 14; EFT = 14 + 4 = 18 - M: EST = max(C:7, G:12, H:15) = 15; EFT = 15 + 3 = 18 - L: EST = max(I:20, J:20) = 20; EFT = 20 + 5 = 25 Project duration = largest EFT = **25 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 = 25; LST = 25 - 5 = 20 - M: LFT = project duration = 25; LST = 25 - 3 = 22 - K: LFT = project duration = 25; LST = 25 - 4 = 21 - I: LFT = LST of L = 20; LST = 20 - 7 = 13 - J: LFT = LST of L = 20; LST = 20 - 8 = 12 - H: LFT = LST of M = 22; LST = 22 - 6 = 16 - G: LFT = min(J:12, M:22) = 12; LST = 12 - 3 = 9 - F: LFT = min(I:13, K:21) = 13; LST = 13 - 4 = 9 - E: LFT = LST of K = 21; LST = 21 - 5 = 16 - D: LFT = min(G:9, H:16) = 9; LST = 9 - 9 = 0 - C: LFT = min(J:12, M:22) = 12; LST = 12 - 7 = 5 - B: LFT = min(I:13, K:21) = 13; LST = 13 - 8 = 5 - A: LFT = min(E:16, F:9) = 9; LST = 9 - 9 = 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 | 9 | 0 | 9 | 0 | 9 | 0 | 0 | 0 | 0 | | B | 8 | 0 | 8 | 5 | 13 | 5 | 5 | 0 | 5 | | C | 7 | 0 | 7 | 5 | 12 | 5 | 5 | 0 | 5 | | D | 9 | 0 | 9 | 0 | 9 | 0 | 0 | 0 | 0 | | E | 5 | 9 | 14 | 16 | 21 | 7 | 0 | 7 | 0 | | F | 4 | 9 | 13 | 9 | 13 | 0 | 0 | 0 | 0 | | G | 3 | 9 | 12 | 9 | 12 | 0 | 0 | 0 | 0 | | H | 6 | 9 | 15 | 16 | 22 | 7 | 0 | 7 | 0 | | I | 7 | 13 | 20 | 13 | 20 | 0 | 0 | 0 | 0 | | J | 8 | 12 | 20 | 12 | 20 | 0 | 0 | 0 | 0 | | K | 4 | 14 | 18 | 21 | 25 | 7 | 7 | 0 | 0 | | L | 5 | 20 | 25 | 20 | 25 | 0 | 0 | 0 | 0 | | M | 3 | 15 | 18 | 22 | 25 | 7 | 7 | 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 | 9 | 9 | 0 | | 3 | 9 | 9 | 0 | | 4 | 13 | 13 | 0 | | 5 | 12 | 12 | 0 | | 6 | 14 | 21 | 7 | | 7 | 20 | 20 | 0 | | 8 | 15 | 22 | 7 | | 9 | 25 | 25 | 0 | #### Critical path and project duration - Critical activities (TF = 0): A, D, F, G, I, J, L - Critical path: A - F - I - L (= 9+4+7+5 = 25 days) - Critical path: D - G - J - L (= 9+3+8+5 = 25 days) - There are two critical paths of the same length, so a delay on either one delays the project. **Answer: project duration = 25 days; critical paths = A - F - I - L and D - G - J - L** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Activity I follows B (EFT 8) and F (EFT 13). What is its EST? A. 8 B. 9 C. 20 D. 13 Answer: D. EST = max(8, 13) = 13, so EFT = 13 + 7 = 20. (Key point: Forward pass rule) 2. What is the project duration? A. 25 days B. 20 days C. 27 days D. 18 days Answer: A. L starts at 20 (after I and J, both finishing at 20) and takes 5 days, so the duration is 25 days. (Key point: Project duration) 3. Which statement about the critical paths is correct? A. Only A - F - I - L is critical B. Two paths, A-F-I-L and D-G-J-L, each of 25 days are critical C. Only D - G - J - L is critical D. B - I - L is critical Answer: B. Both paths total 25 days with zero float, so a delay on either delays the project. (Key point: Multiple critical paths) 4. Activity E has TF = 7 and FF = 0. What type of float is this? A. Independent float of 7 B. Free float of 7 C. Interfering float of 7 D. No float Answer: C. Int.F = TF - FF = 7 - 0 = 7; using it would delay the successor K. (Key point: Float of activity E) ## Close variants set in other exams - [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, 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× ## Related in Project Engineering - [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 - [CPM network, 12 activities (A=10 to L=8)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acpm-network-twelve-act-as67) — 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 - [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, 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 ## Related in other subjects - [CPM schedule, critical path and float](https://www.bctengineer.com/question.md?key=semester_6%2Findustrial_engineering_and_management%3Acpm-network-critical-path-float) — Industrial Engineering and Management, chapter 3