| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Calculate lower bound for workers |
| Difficulty | Moderate -0.8 This is a standard Critical Path Analysis question covering routine D1 techniques: forward/backward pass calculations, identifying critical paths, calculating lower bounds using a simple formula (total time ÷ project duration), and basic scheduling. All parts follow textbook procedures with no novel problem-solving required, making it easier than average A-level maths questions. |
| 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 |
|---|---|
| B3, 2, 1, 0 (5) |
| Answer | Marks |
|---|---|
| B2, 1, 0 (a) |
| Answer | Marks |
|---|---|
| B1 / B1 (2) |
| Answer | Marks |
|---|---|
| m1 / A1 (2) |
| Answer | Marks |
|---|---|
| B2, 1, 0 (e) |
| Answer | Marks |
|---|---|
| B1 (1) |
| Answer | Marks |
|---|---|
| m1 / A1 / A1 (3) |
## (a)
[Network diagram with labeled vertices and edges]
| B3, 2, 1, 0 (5) |
## (b)
$A \in HK$
$AEL$
| B2, 1, 0 (a) |
## (c)
Idea of 'critical' - zero slack, no delay, immediate
- if late project will finish late etc.
Idea of 'path' - from start to end every time/sequence
- the event timing end up as critical
- from the start of the next
- sequence or slack or late
- S.c. 'longest path' gets pt 01 only
| B1 / B1 (2) |
## (d)
$\frac{11.0}{3.0} (= 3.7/s.f.)$ ✓ works
| m1 / A1 (2) |
## (e)
D, H, I, J, L
| B2, 1, 0 (e) |
## (f)
It will not be possible to find a solution with 5 workers to complete the project on the minimum time. 5 workers will be needed. Accept 'an extra worker is required'
| B1 (1) |
## (g)
[Gantt chart diagram]
$A < D$, $C < G$, $G < H$, $F < J$, $D < K$
$A \times \frac{D}{E}$, $Q \times F$, $\frac{E}{C}$, $\frac{G}{I}$, $E \times L$, $D < K$
| m1 / A1 / A1 (3) |
---\includegraphics{figure_5}
The network in Figure 5 shows the activities that need to be undertaken to complete a project. Each activity is represented by an arc. The number in brackets is the duration of the activity in days. The early and late event times are to be shown at each vertex and some have been completed for you.
\begin{enumerate}[label=(\alph*)]
\item Calculate the missing early and late times and hence complete Diagram 2 in your answer book. [3]
\item List the two critical paths for this network. [2]
\item Explain what is meant by a critical path. [2]
\end{enumerate}
The sum of all the activity times is 110 days and each activity requires just one worker. The project must be completed in the minimum time.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working. [2]
\item List the activities that must be happening on day 20. [2]
\item Comment on your answer to part (e) with regard to the lower bound you found in part (d). [1]
\item Schedule the activities, using the minimum number of workers, so that the project is completed in 30 days. [3]
\end{enumerate}
(Total 15 marks)
\hfill \mbox{\textit{Edexcel D1 2007 Q6 [15]}}