| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Calculate early and late times |
| Difficulty | Moderate -0.3 This is a standard Critical Path Analysis question requiring forward/backward scans and identifying the critical path—core D1 techniques practiced extensively. Parts (d) and (e) add some application of float/slack concepts with cost-benefit reasoning, but these are routine extensions. The question is methodical rather than conceptually challenging, 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 interpretation7.05d Latest start and earliest finish: independent and interfering float |
| A | B | C | D | E | \(F\) | |
| A | - | 130 | 190 | 155 | 140 | 125 |
| B | 130 | - | 215 | 200 | 190 | 170 |
| C | 190 | 215 | - | 110 | 180 | 100 |
| D | 155 | 200 | 110 | - | 70 | 45 |
| E | 140 | 190 | 180 | 70 | - | 75 |
| \(F\) | 125 | 170 | 100 | 45 | 75 | - |
| \(n\) | \(x _ { n }\) | \(a\) | Any more data? | \(x _ { n + 1 }\) | \(b\) | \(( b - a ) > 0\) ? | \(a\) |
| 1 | 8 | 8 | Yes | 2 | 2 | No | 2 |
| 2 | - | - |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | Network diagram with activities labeled and durations shown; activities with float values in parentheses shown | M3 A3 |
| (b) | \(B, C, D, F, I, N, P\) | M1 A1 |
| (c) | 48 days | A1 |
| (d) | \(F\) on critical path \(\therefore\) £150 000 penalty; if reduce \(N\) by more than 1 day it is no longer on critical path \(\therefore\) only reduces penalty by £50 000 at cost of £90 000 | B3 |
| (e) | \(B, D\) and \(P\): reducing any of these by 2 days reduces minimum time by 2 days; this reduces penalty by £100 000 at cost of £80 000 \(\therefore\) profitable | B3 |
**(a)** | Network diagram with activities labeled and durations shown; activities with float values in parentheses shown | M3 A3 | |
**(b)** | $B, C, D, F, I, N, P$ | M1 A1 | |
**(c)** | 48 days | A1 | |
**(d)** | $F$ on critical path $\therefore$ £150 000 penalty; if reduce $N$ by more than 1 day it is no longer on critical path $\therefore$ only reduces penalty by £50 000 at cost of £90 000 | B3 | |
**(e)** | $B, D$ and $P$: reducing any of these by 2 days reduces minimum time by 2 days; this reduces penalty by £100 000 at cost of £80 000 $\therefore$ profitable | B3 | (15) |7. This question should be answered on the sheet provided.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{acc09687-11a3-4392-af17-3d4d331d5ab4-08_586_1372_333_303}
\captionsetup{labelformat=empty}
\caption{Fig. 5}
\end{center}
\end{figure}
The activity network in Figure 5 models the work involved in laying the foundations and putting in services for an industrial complex. The activities are represented by the arcs and the numbers in brackets give the time, in days, to complete each activity. Activity $C$ is a dummy.
\begin{enumerate}[label=(\alph*)]
\item Execute a forward scan to calculate the early times and a backward scan to calculate the late times, for each event.
\item Determine which activities lie on the critical path and list them in order.
\item State the minimum length of time needed to complete the project.
The contractor is committed to completing the project in this minimum time and faces a penalty of $\pounds 50000$ for each day that the project is late. Unfortunately, before any work has begun, flooding means that activity $F$ will take 3 days longer than the 7 days allocated.
\item Activity $N$ could be completed in 1 day at an extra cost of $\pounds 90000$. Explain why doing this is not economical.\\
(3 marks)
\item If the time taken to complete any one activity, other than $F$, could be reduced by 2 days at an extra cost of $\pounds 80000$, for which activities on their own would this be profitable. Explain your reasoning.\\
(3 marks)
END
\section*{Please hand this sheet in for marking}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
& A & B & C & D & E & $F$ \\
\hline
A & - & 130 & 190 & 155 & 140 & 125 \\
\hline
B & 130 & - & 215 & 200 & 190 & 170 \\
\hline
C & 190 & 215 & - & 110 & 180 & 100 \\
\hline
D & 155 & 200 & 110 & - & 70 & 45 \\
\hline
E & 140 & 190 & 180 & 70 & - & 75 \\
\hline
$F$ & 125 & 170 & 100 & 45 & 75 & - \\
\hline
\end{tabular}
\end{center}
\section*{Please hand this sheet in for marking}
(a)
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
$n$ & $x _ { n }$ & $a$ & Any more data? & $x _ { n + 1 }$ & $b$ & $( b - a ) > 0$ ? & $a$ \\
\hline
1 & 8 & 8 & Yes & 2 & 2 & No & 2 \\
\hline
2 & - & - & & & & & \\
\hline
\end{tabular}
\end{center}
Final output\\
(b) $\_\_\_\_$\\
Sheet for answering question 3\\
NAME
\section*{Please hand this sheet in for marking}
(a)\\
(b) (i)\\
\includegraphics[max width=\textwidth, alt={}, center]{acc09687-11a3-4392-af17-3d4d331d5ab4-11_716_1218_502_331}\\
(ii)\\
\includegraphics[max width=\textwidth, alt={}, center]{acc09687-11a3-4392-af17-3d4d331d5ab4-11_709_1214_1498_333}
Maximum flow =\\
(c) (i) $\_\_\_\_$\\
(ii) $\_\_\_\_$\\
\section*{Please hand this sheet in for marking}
(a)\\
\includegraphics[max width=\textwidth, alt={}, center]{acc09687-11a3-4392-af17-3d4d331d5ab4-12_764_1612_402_255}\\
(b) $\_\_\_\_$\\
(c) $\_\_\_\_$\\
(d) $\_\_\_\_$\\
(e) $\_\_\_\_$
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 Q7 [15]}}