Question 5 - A-Level Maths

OCR D2 2008 January — Question 5 15 marks

Exam Board	OCR
Module	D2 (Decision Mathematics 2)
Year	2008
Session	January
Marks	15
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Critical Path Analysis
Type	Draw resource histogram
Difficulty	Moderate -0.8 This is a standard Critical Path Analysis question with routine procedures: completing a precedence table, performing forward/backward passes (algorithmic), drawing a resource histogram (mechanical plotting), and basic resource leveling. All techniques are textbook exercises requiring careful execution but no problem-solving insight or novel reasoning.
Spec	7.05b Forward and backward pass: earliest/latest times, critical activities 7.05c Total float: calculation and interpretation 7.05d Latest start and earliest finish: independent and interfering float

5 Answer this question on the insert provided. The diagram shows an activity network for a project. The figures in brackets show the durations of the activities in days. \includegraphics[max width=\textwidth, alt={}, center]{95fbb09b-0301-4fc1-b694-838b8d0b64a6-06_956_921_495_612}

Complete the table in the insert to show the precedences for the activities.
Use the boxes on the diagram in the insert to carry out a forward pass and a backward pass. Find the minimum project duration and list the critical activities. The number of people required for each activity is shown in the table below. The workers are all equally skilled at all of the activities.
Activity \(A\) \(B\) \(C\) \(D\) \(E\) \(F\) \(G\) \(H\) \(I\) \(J\)
Number of workers 4 1 2 2 3 2 3 3 1 2
On graph paper, draw a resource histogram for the project with each activity starting at its earliest possible time.
Describe how the project can be completed in 21 days using just six workers.

Show mark scheme Show mark scheme source

Question 5:

Part (i):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
Precedences correct for \(A, B, C, D\)	B1
Precedences correct for \(E, F, G\)	B1
Precedences correct for \(H, I, J\)	B1

Part (ii):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
Forward pass, no more than one independent error	M1
Forward pass correct (cao)	A1
Backward pass, no more than one independent error	M1
Backward pass correct (cao)	A1
Minimum project duration \(= 17\) days	B1	17, cao
Critical activities \(= A\ D\ H\)	B1	\(ADH\), cao

Part (iii):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
A plausible histogram with no holes or overhanging blocks	M1	Answered on graph paper
Correct shape	A1

Part (iv):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
Start \(A\) and \(B\) as before but delay \(C\) to day 6; start \(D\) and \(F\) as before but delay \(E\) to day 11; start \(G\) on day 12, \(H\) on day 13, \(I\) and \(J\) on day 16	B1	Dealing with \(A\), \(B\) and \(C\); precedences not violated, durations correct
B1	Dealing with \(D\), \(E\) and \(F\)
M1	Dealing with \(G\), \(H\ I\) and \(J\)
A valid solution using 6 workers for 21 days	A1

# Question 5:

## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Precedences correct for $A, B, C, D$ | B1 | |
| Precedences correct for $E, F, G$ | B1 | |
| Precedences correct for $H, I, J$ | B1 | |

## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Forward pass, no more than one independent error | M1 | |
| Forward pass correct (cao) | A1 | |
| Backward pass, no more than one independent error | M1 | |
| Backward pass correct (cao) | A1 | |
| Minimum project duration $= 17$ days | B1 | 17, cao |
| Critical activities $= A\ D\ H$ | B1 | $ADH$, cao |

## Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| A plausible histogram with no holes or overhanging blocks | M1 | Answered on graph paper |
| Correct shape | A1 | |

## Part (iv):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Start $A$ and $B$ as before but delay $C$ to day 6; start $D$ and $F$ as before but delay $E$ to day 11; start $G$ on day 12, $H$ on day 13, $I$ and $J$ on day 16 | B1 | Dealing with $A$, $B$ and $C$; precedences not violated, durations correct |
| | B1 | Dealing with $D$, $E$ and $F$ |
| | M1 | Dealing with $G$, $H\ I$ and $J$ |
| A valid solution using 6 workers for 21 days | A1 | |

Show LaTeX source

5 Answer this question on the insert provided.
The diagram shows an activity network for a project. The figures in brackets show the durations of the activities in days.\\
\includegraphics[max width=\textwidth, alt={}, center]{95fbb09b-0301-4fc1-b694-838b8d0b64a6-06_956_921_495_612}\\
(i) Complete the table in the insert to show the precedences for the activities.\\
(ii) Use the boxes on the diagram in the insert to carry out a forward pass and a backward pass. Find the minimum project duration and list the critical activities.

The number of people required for each activity is shown in the table below. The workers are all equally skilled at all of the activities.

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | }
\hline
Activity & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ & $I$ & $J$ \\
\hline
Number of workers & 4 & 1 & 2 & 2 & 3 & 2 & 3 & 3 & 1 & 2 \\
\hline
\end{tabular}
\end{center}

(iii) On graph paper, draw a resource histogram for the project with each activity starting at its earliest possible time.\\
(iv) Describe how the project can be completed in 21 days using just six workers.

\hfill \mbox{\textit{OCR D2 2008 Q5 [15]}}

This paper (5 questions)

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Q1 14 Q2 17 Q3 12 Q4 14 Q5 15

Activity	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
Number of workers	4	1	2	2	3	2	3	3	1	2