# Step-by-Step Solution: Critical Path Method (CPM) Let's find the critical path, activity times, and slack times. --- ## **Step 1: List Activities, Durations, and Dependencies** | ID | Activity | Predecessor | Duration (Days) | |----|---------------------------|-------------|-----------------| | A | Order Review | None | 2 | | B | Order Standard Parts | A | 3 | | C | Produce Standard Parts | A | 10 | | D | Design Custom Parts | A | 13 | | E | Software Development | A | 18 | | F | Manufacture Custom Hardware | C, D | 15 | | G | Assemble | B, F | 10 | | H | Test | E, G | 5 | --- ## **Step 2: Draw the Network Diagram** Let's represent the paths based on dependencies: 1. **A → B → G** 2. **A → C → F → G** 3. **A → D → F → G** 4. **A → E** 5. **E → H** 6. **G → H** --- ## **Step 3: Forward Pass (Earliest Start and Finish Times)** ### **Activity A** - **ES = 0** - **EF = ES + Duration = 0 + 2 = 2** ### **Activity B, C, D, E (all depend on A)** - **ES = EF of A = 2** - **B: EF = 2 + 3 = 5** - **C: EF = 2 + 10 = 12** - **D: EF = 2 + 13 = 15** - **E: EF = 2 + 18 = 20** ### **Activity F (depends on C and D)** - **ES = max(EF of C, EF of D) = max(12, 15) = 15** - **EF = 15 + 15 = 30** ### **Activity G (depends on B and F)** - **ES = max(EF of B, EF of F) = max(5, 30) = 30** - **EF = 30 + 10 = 40** ### **Activity H (depends on E and G)** - **ES = max(EF of E, EF of G) = max(20, 40) = 40** - **EF = 40 + 5 = 45** --- ## **Step 4: Backward Pass (Latest Start and Finish Times)** ### **Activity H** - **LF = EF = 45** - **LS = LF - Duration = 45 - 5 = 40** ### **Activity G and E (predecessors to H)** - **G: LF = LS of H = 40, LS = 40 - 10 = 30** - **E: LF = LS of H = 40, LS = 40 - 18 = 22** ### **Activity F and B (predecessors to G)** - **F: LF = LS of G = 30, LS = 30 - 15 = 15** - **B: LF = LS of G = 30, LS = 30 - 3 = 27** ### **Activity C and D (predecessors to F)** - **F depends on both C and D, so:** - **C: LF = LS of F = 15, LS = 15 - 10 = 5** - **D: LF = LS of F = 15, LS = 15 - 13 = 2** ### **Activity A (predecessor to B, C, D, E)** - **A's LF = min(LS of B, C, D, E) = min(27, 5, 2, 22) = 2** - **LS = 2 - 2 = 0** --- ## **Step 5: Calculate Slack for Each Activity** - **Slack = LS - ES (or LF - EF)** | ID | ES | EF | LS | LF | Slack | |----|-----|-----|-----|-----|-------| | A | 0 | 2 | 0 | 2 | 0 | | B | 2 | 5 | 27 | 30 | 25 | | C | 2 | 12 | 5 | 15 | 3 | | D | 2 | 15 | 2 | 15 | 0 | | E | 2 | 20 | 22 | 40 | 20 | | F | 15 | 30 | 15 | 30 | 0 | | G | 30 | 40 | 30 | 40 | 0 | | H | 40 | 45 | 40 | 45 | 0 | --- ## **Step 6: Identify the Critical Path** The critical path is the sequence of activities with **zero slack**: **A → D → F → G → H** - Duration: \(2 + 13 + 15 + 10 + 5 = 45\) days --- ## **Final Answers** ### **Critical Path:** **A → D → F → G → H** ### **Project Duration:** **45 days** ### **Slack Times:** - **A, D, F, G, H:** 0 (critical path) - **B:** 25 - **C:** 3 - **E:** 20 --- ### **Summary Table** | Activity | ES | EF | LS | LF | Slack | On Critical Path? | |----------|----|----|----|----|-------|-------------------| | A | 0 | 2 | 0 | 2 | 0 | Yes | | B | 2 | 5 | 27 | 30 | 25 | No | | C | 2 | 12 | 5 | 15 | 3 | No | | D | 2 | 15 | 2 | 15 | 0 | Yes | | E | 2 | 20 | 22 | 40 | 20 | No | | F | 15 | 30 | 15 | 30 | 0 | Yes | | G | 30 | 40 | 30 | 40 | 0 | Yes | | H | 40 | 45 | 40 | 45 | 0 | Yes | --- **If you need a visual diagram, let me know!**