Question 6 of 7: Construction Project — CPM Network and a Duration Change

Question 6: Construction Project — CPM Network and a Duration Change (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Ten activities with precedence and durations (days), late completion penalty $\$5,000$/day.

Find. (a) The project network and critical path; (b) earliest/latest start times (and slack) of every activity; (c) the effect on finish date if D becomes 35 days, plus at least two recovery strategies.

Approach. Run a forward pass (earliest start/finish) then a backward pass (latest start/finish) through the precedence network to get every activity's slack; the critical path is the chain of zero-slack activities, and its total duration is the project length.

1. Forward pass (earliest times). Starting A at day 0 and working through the precedence chain ($ES_i=\max$ of predecessors' $EF$; $EF_i=ES_i+d_i$): $$EF_A=15,\ EF_B=27,\ EF_C=21,\ EF_D=20,\ EF_E=\max(27,21)+3=30,\ EF_G=21+8=29,$$ $$EF_H=20+9=29,\ EF_F=\max(30,29)+8=38,\ EF_J=\max(29,29)+14=43,\ EF_I=38+7=45.$$ The project duration is the later of the two end activities: $\boxed{T=\max(EF_I,EF_J)=\max(45,43)=45\ \text{days}}$.
2. Backward pass and slack (part b). Setting $LF_I=LF_J=45$ (project length) and working backward ($LF_i=\min$ of successors' $LS$; $LS_i=LF_i-d_i$; slack $=LS_i-ES_i$) gives every activity's latest start and slack, tabulated below. Zero-slack activities are critical.
3. Critical path (part a). The chain of zero-slack activities is $\boxed{A\to B\to E\to F\to I}$, with $15+12+3+8+7=45$ days, matching the project duration — this is the path shown in red in Figure 1. All other activities (C, D, G, H, J) carry positive slack (1–2 days) and are not critical.
4. Effect of D → 35 days (part c). D is not on the original critical path (2 days of slack), so the first 2 days of its 20-day increase are absorbed harmlessly, but the remaining increase pushes its successor chain past the old critical path. Re-running the forward pass with $d_D=35$: $EF_D=15+35=50\Rightarrow EF_H=50+9=59\Rightarrow EF_J=\max(29,59)+14=73$, while the F/I side is unaffected ($EF_I=45$ still). The new project duration is $$\boxed{T'=\max(45,73)=73\ \text{days}},$$ a delay of $73-45=\boxed{28\ \text{days}}$ versus the original plan (D's activity duration increases from 5 to 35 days, a 30-day increase, of which 2 days of slack absorb part, giving a net delay of $30-2=28$ days, confirmed by the recomputation above). The critical path shifts entirely to $A\to D\to H\to J$ (new duration $15+35+9+14=73$ days). At $\$5{,}000$/day, an unmitigated 28-day delay costs $\boxed{28\times\$5{,}000=\$140{,}000}$ in late-completion penalties.

Recovery strategies (part c, continued). Two genuinely different approaches to pull the finish date back toward 45 days: (1) Crash the new critical path (A–D–H–J). Since D itself is now the bottleneck, negotiate a second subcontractor crew to run in parallel with the strike-affected one (splitting D's scope so two crews attack it concurrently), or authorize overtime/expedited crews on H and J once D finally releases them — buying back days at an added cost that should be compared against the $\$5,000$/day penalty (any crashing move that costs less than $\$5,000$ to save a day is worth it). (2) Fast-track by overlapping activities instead of paying for extra resources. Rather than waiting for D to fully finish before starting H, begin the portions of H that do not depend on D's still-incomplete scope as soon as they are technically able to proceed (and similarly overlap H into J where feasible) — this trades some rework risk for schedule compression at little or no direct cost, and can be combined with strategy (1) rather than used alone.