# Question 1:

## Part (a)

| Answer | Marks | Guidance |
|--------|-------|----------|
| Earliest start times: $A=0, B=0, C=0, D=0$ (given) | | |
| $E: \max(0+5, 0+5) = 5$ | B1 | For correct EST for E and H |
| $F: 0+5=5$ | | |
| $G: 0+7=7$ | | |
| $H: 5+3=8$ | | |
| $I: 5+4=9$ | | |
| $J: 0+6=6$ | | |
| $K: \max(7+6, 8+2)=13$ | B1 | For correct ESTs for K, L, M |
| $L: \max(8+2, 9+5)=14$ | | |
| $M: \max(9+5, 6+2)=14$ | | |
| $N: \max(13+5, 14+4, 14+3)=18$ | B1 | For correct EST for N |
| Latest finish times working backwards: $N=20$ | | |
| $K: 20-2=18$, $L: 20-2=18$, $M: 20-2=18$ | B1 | For correct LFTs |
| Total: | **4** | |

## Part (b)

| Answer | Marks | Guidance |
|--------|-------|----------|
| Critical path: $C \to F \to I \to M \to N$ | M1 | For identifying critical activities |
| Also: $D \to J \to M \to N$ | A1 | Both paths required |
| Minimum completion time = $\mathbf{20}$ days | A1 | |
| Total: | **3** | |

## Part (c)

| Answer | Marks | Guidance |
|--------|-------|----------|
| Gantt chart drawn with activities starting at EST | B1 | Scale correct, activities labelled |
| Critical activities $C, F, I, M, N$ and $D, J$ shown correctly | B2 | One mark per error, ft from (a)/(b) |
| Non-critical activities shown with float correctly indicated | B2 | |
| Total: | **5** | |

## Part (d)

| Answer | Marks | Guidance |
|--------|-------|----------|
| $J$ starts at day 6, so $6 + x \leq 18$ (must finish before $M$ can finish at 20) | M1 | Correct inequality formed |
| $J$ must complete in time: $6 + x \leq 18$ giving $x \leq 12$ | A1 | Accept $x < 13$ |
| Total: | **2** | |

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