| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2020 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Calculate lower bound for workers |
| Difficulty | Standard +0.3 This is a standard critical path analysis question requiring routine application of well-defined algorithms: finding early/late times, calculating lower bound using total activity time divided by project duration, and scheduling activities. Part (d) requires identifying which activity is on the critical path, but this follows directly from part (a). All techniques are textbook procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 7.05a Critical path analysis: activity on arc networks7.05b Forward and backward pass: earliest/latest times, critical activities7.05c Total float: calculation and interpretation7.05e Cascade charts: scheduling and effect of delays |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| (a) [Table with forward and backward pass values, working values shown] | M1 A1 M1 A1 (4) | a1M1: All top boxes complete, values generally increasing in the direction of the arrows ('left to right'), condone one rogue. a1A1: CAO (top boxes). a2M1: All bottom boxes complete, values generally decreasing in the opposite direction of the arrows ('right to left'), condone one rogue. Condone missing 0 and/or their 33 (at the end event) for the M mark only. a2A1: CAO (bottom boxes). |
| (b) Lower bound is \(\frac{87}{33} = 2.6363… = 3\) | M1 A1 (2) | b1M1: Attempt to find lower bound: (a value in the interval [73 – 101] / their finish time) or (sum of the activities / their finish time) or (as a minimum) an awrt 2.6. b1A1: CSO – requires both a correct calculation or awrt 2.6 seen and 3. An answer of 3 with no working scores no marks. |
| (c) [Gantt chart showing activities with correct time intervals and worker allocation] | M1 A1 A1 A1 (4) | c1M1: Not a cascade chart. 4 workers used at most, at least 10 activities placed. c1A1: 3 workers. All 15 activities present (just once). Condone two errors. An activity can give rise to at most three errors; one on duration, one on time interval and only one on IPA. c2A1: 3 workers. All 15 activities present (just once). Condone one error. An activity can give rise to at most three errors; one on duration, one on time interval and only one on IPA. |
| (d) G is not a critical activity (it has a total float of 16 – 5 – 3 = 8 days) and so there is no benefit from reducing the duration of G by one day. Activities D and P are both critical activities. However, D appears in both critical paths therefore reducing P would not reduce the minimum completion time (as there is still a critical path \(A – D – J – N\) of length 33) and so activity D should be shortened by one day | M1 A1 (2) 12 marks | d1M1: D and P stated as being critical or activity G is not critical. d1A1: Correct answer of D with fully correct reason (G not critical, D and P are both critical but D appears in both/all critical paths or P appears in only one critical path). |
| Answer | Marks | Guidance |
|--------|-------|----------|
| (a) [Table with forward and backward pass values, working values shown] | M1 A1 M1 A1 (4) | a1M1: All top boxes complete, values generally increasing in the direction of the arrows ('left to right'), condone one rogue. a1A1: CAO (top boxes). a2M1: All bottom boxes complete, values generally decreasing in the opposite direction of the arrows ('right to left'), condone one rogue. Condone missing 0 and/or their 33 (at the end event) for the M mark only. a2A1: CAO (bottom boxes). |
| (b) Lower bound is $\frac{87}{33} = 2.6363… = 3$ | M1 A1 (2) | b1M1: Attempt to find lower bound: (a value in the interval [73 – 101] / their finish time) or (sum of the activities / their finish time) or (as a minimum) an awrt 2.6. b1A1: CSO – requires both a **correct calculation or awrt 2.6** seen and 3. An answer of 3 with no working scores no marks. |
| (c) [Gantt chart showing activities with correct time intervals and worker allocation] | M1 A1 A1 A1 (4) | c1M1: Not a cascade chart. 4 workers used at most, at least 10 activities placed. c1A1: 3 workers. All 15 activities present (just once). Condone **two errors**. An activity can give rise to at most three errors; one on duration, one on time interval and only one on IPA. c2A1: 3 workers. All 15 activities present (just once). Condone **one error**. An activity can give rise to at most three errors; one on duration, one on time interval and only one on IPA. |
| (d) G is not a critical activity (it has a total float of 16 – 5 – 3 = 8 days) and so there is no benefit from reducing the duration of G by one day. Activities D and P are both critical activities. However, D appears in both critical paths therefore reducing P would not reduce the minimum completion time (as there is still a critical path $A – D – J – N$ of length 33) and so activity D should be shortened by one day | M1 A1 (2) 12 marks | d1M1: D and P stated as being critical or activity G is not critical. d1A1: Correct answer of D with fully correct reason (G not critical, D and P are both critical but D appears in both/all critical paths or P appears in only one critical path). |
---5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3aa30e8f-7d55-4c3b-8b2c-55c3e822c8a0-06_501_1328_242_374}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
A project is modelled by the activity network shown in Figure 1. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete the corresponding activity. Each activity requires exactly one worker. The project is to be completed in the shortest possible time.
\begin{enumerate}[label=(\alph*)]
\item Complete Diagram 1 in the answer book to show the early event times and the late event times.
\item Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working.
\item Schedule the activities on Grid 1 in the answer book using the minimum number of workers so that the project is completed in the minimum time.
Additional resources become available, which can shorten the duration of one of activities D, G or P by one day.
\item Determine which of these three activities should be shortened to allow the project to be completed in the minimum time. You must give reasons for your answer.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2020 Q5 [12]}}