A project to open a coffee shop has the following activities, durations, and dependencies:

A (2 days) → Start
B (3 days) → after A
C (4 days) → after A
D (2 days) → after B
E (5 days) → after C
F (3 days) → after D and E

What is the project duration using the Critical Path Method (CPM)?

Question

A project to open a coffee shop has the following activities, durations, and dependencies:

A (2 days) → Start
B (3 days) → after A
C (4 days) → after A
D (2 days) → after B
E (5 days) → after C
F (3 days) → after D and E

What is the project duration using the Critical Path Method (CPM)?

BlackTom AI · Accepted Answer

Critical path equals the longest-duration path through the network, so we must compute earliest and latest times and identify the path with zero total float. Relevant concepts: activity-on-node sequencing, earliest start/finish (ES/EF), latest start/finish (LS/LF), total float = LS − ES, and critical path = sequence of activities with zero float whose durations sum to the project length.

Step-by-step reasoning and calculations:

1. Forward pass (compute ES and EF):
- A (duration 2): ES = 0, EF = ES + 2 = 2.
- B (3) after A: ES = EF_A = 2, EF = 2 + 3 = 5.
- C (4) after A: ES = EF_A = 2, EF = 2 + 4 = 6.
- D (2) after B: ES = EF_B = 5, EF = 5 + 2 = 7.
- E (5) after C: ES = EF_C = 6, EF = 6 + 5 = 11.
- F (3) after D and E: ES = max(EF_D, EF_E) = max(7, 11) = 11, EF = 11 + 3 = 14.

The forward pass shows project earliest completion = EF_F = 14 days.

2. Backward pass (compute LF and LS), start with LF_F = project duration = 14:
- F: LF = 14, LS = LF − 3 = 11.
- D: successor F has LS = 11, so LF_D = 11, LS_D = 11 − 2 = 9.
- E: successor F has LS = 11, so LF_E = 11, LS_E = 11 − 5 = 6.
- B: successor D has LS = 9, so LF_B = 9, LS_B = 9 − 3 = 6.
- C: successor E has LS = 6, so LF_C = 6, LS_C = 6 − 4 = 2.
- A: successors B (LS=6) and C (LS=2) → choose minimum LF = 2, so LF_A = 2, LS_A = 2 − 2 = 0.

3. Compute total float = LS − ES:
- A: 0 − 0 = 0 (critical)
- B: 6 − 2 = 4
- C: 2 − 2 = 0 (critical)
- D: 9 − 5 = 4
- E: 6 − 6 = 0 (critical)
- F: 11 − 11 = 0 (critical)

Why the correct answer is correct:
The critical path is A → C → E → F (all zero float), with duration 2 + 4 + 5 + 3 = 14 days. The project duration equals the length of this longest/critical path.

Why other options are incorrect:
- 10 days corresponds to A → B → D → F (2+3+2+3=10), which is shorter and has positive float, so it does not determine project duration.
- 12 and 16 days do not match any correct sum of constrained longest paths; 12 undercounts and 16 overcounts relative to computed earliest finish (14).

Conclusion: The project duration using CPM is 14 days, matching the provided answer.

类似问题

What is the latest finish time (LFT) of Activity D?

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