GATE 2022 ME (ME2) – Question 59
A project consists of five activities (A, B, C, D and E). The duration of each activity follows beta distribution. The three time estimates (in weeks) of each activity and immediate predecessor(s) are listed in the table. The expected time of the project completion is ______ weeks (in integer).
| Activity | Optimistic | Most likely | Pessimistic | Predecessors |
|---|---|---|---|---|
| A | 4 | 5 | 6 | None |
| B | 1 | 3 | 5 | A |
| C | 1 | 2 | 3 | A |
| D | 2 | 4 | 6 | C |
| E | 3 | 4 | 5 | B, D |
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Correct answer: 15
Explanation
**Step 1: PERT expected durations**, $t_e=\dfrac{a+4m+b}6$.
| Activity | $t_e$ |
|---|---|
| A | $(4+20+6)/6=5$ |
| B | $(1+12+5)/6=3$ |
| C | $(1+8+3)/6=2$ |
| D | $(2+16+6)/6=4$ |
| E | $(3+16+5)/6=4$ |
**Step 2: network.** A first; then B and C in parallel (both after A); D follows C; E follows both B and D.
**Step 3: paths.**
- A → B → E: $5+3+4=12$ weeks
- A → C → D → E: $5+2+4+4=15$ weeks
**Step 4: project time** is the longest path, the critical path A–C–D–E:
$$T_E=\mathbf{15\text{ weeks}}.$$