Questions - AQA D2 - A-Level Maths

Activity	Immediate Predecessors	Duration (days)	Number of Workers Required
A	-	2	3
B	A	4	2
C	A	6	1
D	$B , C$	8	3
E	C	3	2
F	D	2	2
G	D, E	4	2
H	D, E	6	1
I	$F , G , H$	2	3

Complete the activity network for the project on Figure 1.
Find the earliest start time for each activity.
Find the latest finish time for each activity.
Find the critical path and state the minimum time for completion.
State the float time for each non-critical activity.
Given that each activity starts as early as possible, draw a resource histogram for the project on Figure 2.
There are only 3 workers available at any time. Use resource levelling to explain why the project will overrun and state the minimum extra time required.

AQA D2 2006 January Q4

4 [Figures 3, 4 and 5, printed on the insert, are provided for use in this question.]
The network shows a system of pipes, with the lower and upper capacities for each pipe in litres per second.
\includegraphics[max width=\textwidth, alt={}, center]{30a88efe-fe9e-4384-a3e3-da2a05326797-04_547_1214_555_404}

Figure 3, on the insert, shows a partially completed diagram for a feasible flow of 10 litres per second from $S$ to $T$. Indicate, on Figure 3, the flows along the edges $M N , P Q , N P$ and $N T$.
1. Taking your answer from part (a) as an initial flow, use flow augmentation on Figure 4 to find the maximum flow from $S$ to $T$.
2. State the value of the maximum flow and illustrate this flow on Figure 5.
Find a cut with capacity equal to that of the maximum flow.

AQA D2 2006 January Q5

Display the following linear programming problem in a Simplex tableau. $$\begin{array} { l c } \text { Maximise } & P = 3 x + 2 y + 4 z
\text { subject to } & x + 4 y + 2 z \leqslant 8
& 2 x + 7 y + 3 z \leqslant 21
& x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
Use the Simplex method to perform one iteration of your tableau for part (a), choosing a value in the $z$-column as pivot.
1. Perform one further iteration.
2. State whether or not this is the optimal solution, and give a reason for your answer.

AQA D2 2006 January Q6

6 Sam is playing a computer game in which he is trying to drive a car in different road conditions. He chooses a car and the computer decides the road conditions. The points scored by Sam are shown in the table.

		Road Conditions
\cline { 2 - 5 }		$\boldsymbol { C } _ { \mathbf { 1 } }$	$\boldsymbol { C } _ { \mathbf { 2 } }$	$\boldsymbol { C } _ { \mathbf { 3 } }$
\cline { 2 - 5 }	$\boldsymbol { S } _ { \mathbf { 1 } }$	- 2	2	4
\cline { 2 - 5 } Sam's Car	$\boldsymbol { S } _ { \mathbf { 2 } }$	2	4	5
\cline { 2 - 5 }	$\boldsymbol { S } _ { \mathbf { 3 } }$	5	1	2
\cline { 2 - 5 }
\cline { 2 - 5 }

Sam is trying to maximise his total points and the computer is trying to stop him.

Explain why Sam should never choose $S _ { 1 }$ and why the computer should not choose $C _ { 3 }$.
Find the play-safe strategies for the reduced 2 by 2 game for Sam and the computer, and hence show that this game does not have a stable solution.
Sam uses random numbers to choose $S _ { 2 }$ with probability $p$ and $S _ { 3 }$ with probability $1 - p$.
1. Find expressions for the expected gain for Sam when the computer chooses each of its two remaining strategies.
2. Calculate the value of $p$ for Sam to maximise his total points.
3. Hence find the expected points gain for Sam.
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AQA D2 2007 January Q1

1 [Figure 1, printed on the insert, is provided for use in this question.]
A building project is to be undertaken. The table shows the activities involved.

Activity	Immediate Predecessors	Duration (weeks)
A	-	2
B	-	1
C	A	3
D	A, B	2
E	B	4
F	C	1
G	$C , D , E$	3
H	E	5
I	$F , G$	2
J	$H , I$	3

Complete an activity network for the project on Figure 1.
Find the earliest start time for each activity.
Find the latest finish time for each activity.
State the minimum completion time for the building project and identify the critical paths.

AQA D2 2007 January Q2

2 Five successful applicants received the following scores when matched against suitability criteria for five jobs in a company.

	Job 1	Job 2	Job 3	Job 4	Job 5
Alex	13	11	9	10	13
Bill	15	12	12	11	12
Cath	12	10	8	14	14
Don	11	12	13	14	10
Ed	12	14	14	13	14

It is intended to allocate each applicant to a different job so as to maximise the total score of the five applicants.

Explain why the Hungarian algorithm may be used if each number, $x$, in the table is replaced by $15 - x$.
Form a new table by subtracting each number in the table from 15. Use the Hungarian algorithm to allocate the jobs to the applicants so that the total score is maximised.
It is later discovered that Bill has already been allocated to Job 4. Decide how to alter the allocation of the other jobs so as to maximise the score now possible.

AQA D2 2007 January Q3

Display the following linear programming problem in a Simplex tableau. $$\begin{array} { l l } \text { Maximise } & P = 5 x + 8 y + 7 z
\text { subject to } & 3 x + 2 y + z \leqslant 12
& 2 x + 4 y + 5 z \leqslant 16
& x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
The Simplex method is to be used by initially choosing a value in the $y$-column as a pivot.
1. Explain why the initial pivot is 4 .
2. Perform two iterations of your tableau from part (a) using the Simplex method.
3. State the values of $P , x , y$ and $z$ after your second iteration.
4. State, giving a reason, whether the maximum value of $P$ has been achieved.

AQA D2 2007 January Q4

Two people, Ros and Col, play a zero-sum game. The game is represented by the following pay-off matrix for Ros.
\multirow{2}{*}{} \multirow[b]{2}{*}{Strategy} Col
X Y Z
\multirow{3}{*}{Ros} I -4 -3 0
II 5 -2 2
III 1 -1 3
1. Show that this game has a stable solution.
2. Find the play-safe strategy for each player and state the value of the game.

Ros and Col play a different zero-sum game for which there is no stable solution. The game is represented by the following pay-off matrix for Ros.

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	Col
\cline { 2 - 5 } \multicolumn{1}{c\|}{}	Strategy	$\mathbf { C } _ { \mathbf { 1 } }$	$\mathbf { C } _ { \mathbf { 2 } }$	$\mathbf { C } _ { \mathbf { 3 } }$
\multirow{2}{*}{Ros}	$\mathbf { R } _ { \mathbf { 1 } }$	3	2	1
\cline { 2 - 5 }	$\mathbf { R } _ { \mathbf { 2 } }$	- 2	- 1	2

Find the optimal mixed strategy for Ros.
Calculate the value of the game.

AQA D2 2007 January Q5

5 A three-day journey is to be made from $S$ to $T$, with overnight stops at the end of the first day at either $A$ or $B$ and at the end of the second day at one of the locations $C , D$ or $E$. The network shows the number of hours of sunshine forecast for each day of the journey.
\includegraphics[max width=\textwidth, alt={}, center]{be283950-ef4c-482f-94cb-bdb3def9ff6d-05_753_1284_479_386} The optimal route, known as the maximin route, is that for which the least number of hours of sunshine during a day's journey is as large as possible.

Explain why the three-day route $S A E T$ is better than $S A C T$.
Use dynamic programming to find the optimal (maximin) three-day route from $S$ to $T$. (8 marks)

AQA D2 2007 January Q6

6 [Figures 2 and 3, printed on the insert, are provided for use in this question.]
The diagram shows a network of pipelines through which oil can travel. The oil field is at $S$, the refinery is at $T$ and the other vertices are intermediate stations. The weights on the edges show the capacities in millions of barrels per hour that can flow through each pipeline.
\includegraphics[max width=\textwidth, alt={}, center]{be283950-ef4c-482f-94cb-bdb3def9ff6d-06_956_1470_593_283}

1. Find the value of the cut marked $C$ on the diagram.
2. Hence make a deduction about the maximum flow of oil through the network.
State the maximum possible flows along the routes $S A B T , S D E T$ and $S F T$.
1. Taking your answer to part (b) as the initial flow, use a labelling procedure on Figure 2 to find the maximum flow from $S$ to $T$. Record your routes and flows in the table provided and show the augmented flows on the network diagram. (6 marks)
2. State the value of the maximum flow, and, on Figure 3, illustrate a possible flow along each edge corresponding to this maximum flow.
3. Prove that your flow in part (c)(ii) is a maximum.
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AQA D2 2008 January Q1

1 [Figures 1 and 2, printed on the insert, are provided for use in this question.]
A group of workers is involved in a building project. The table shows the activities involved. Each worker can perform any of the given activities.

Activity	Immediate predecessors	Duration (days)	Number of workers required
A	-	3	5
B	A	8	2
C	A	7	3
$D$	$B , C$	8	4
E	C	10	2
$F$	C	3	3
$G$	D, E	3	4
H	$F$	6	1
I	$G , H$	2	3

Complete the activity network for the project on Figure 1.
Find the earliest start time and the latest finish time for each activity, inserting their values on Figure 1.
Find the critical path and state the minimum time for completion.
The number of workers required for each activity is given in the table above. Given that each activity starts as early as possible and assuming there is no limit to the number of workers available, draw a resource histogram for the project on Figure 2, indicating clearly which activities take place at any given time.
It is later discovered that there are only 7 workers available at any time. Use resource levelling to explain why the project will overrun and indicate which activities need to be delayed so that the project can be completed with the minimum extra time. State the minimum extra time required.

AQA D2 2008 January Q2

2 The following table shows the times taken, in minutes, by five people, Ash, Bob, Col, Dan and Emma, to carry out the tasks 1, 2, 3 and 4 . Dan cannot do task 3.

	Ash	Bob	Col	Dan	Emma
Task 1	14	10	12	12	14
Task 2	11	13	10	12	12
Task 3	13	11	12	**	12
Task 4	13	10	12	13	15

Each of the four tasks is to be given to a different one of the five people so that the overall time for the four tasks is minimised.

Modify the table of values by adding an extra row of non-zero values so that the Hungarian algorithm can be applied.
Use the Hungarian algorithm, reducing columns first then rows, to decide which four people should be allocated to which task. State the minimum total time for the four tasks using this matching.
After special training, Dan is able to complete task 3 in 12 minutes. Determine, giving a reason, whether the minimum total time found in part (b) could be improved.
(2 marks)

AQA D2 2008 January Q3

3 Two people, Rob and Con, play a zero-sum game. The game is represented by the following pay-off matrix for Rob.

\multirow{5}{*}{Rob}	Con
	Strategy	$\mathrm { C } _ { 1 }$	$\mathbf { C } _ { \mathbf { 2 } }$	$\mathrm { C } _ { 3 }$
	$\mathbf { R } _ { \mathbf { 1 } }$	-2	5	3
	$\mathbf { R } _ { \mathbf { 2 } }$	3	-3	-1
	$\mathbf { R } _ { \mathbf { 3 } }$	-3	3	2

Explain what is meant by the term 'zero-sum game'.
Show that this game has no stable solution.
Explain why Rob should never play strategy $R _ { 3 }$.
1. Find the optimal mixed strategy for Rob.
2. Find the value of the game.

AQA D2 2008 January Q4

4 A linear programming problem involving the variables $x , y$ and $z$ is to be solved. The objective function to be maximised is $P = 2 x + 3 y + 5 z$. The initial Simplex tableau is given below.

$\boldsymbol { P }$	$\boldsymbol { x }$	$\boldsymbol { y }$	$\boldsymbol { z }$	$s$	$\boldsymbol { t }$	$\boldsymbol { u }$	value
1	-2	-3	-5	0	0	0	0
0	1	0	1	1	0	0	9
0	2	1	4	0	1	0	40
0	4	2	3	0	0	1	33

In addition to $x \geqslant 0 , y \geqslant 0 , z \geqslant 0$, write down three inequalities involving $x , y$ and $z$ for this problem.
1. By choosing the first pivot from the $z$-column, perform one iteration of the Simplex method.
2. Explain how you know that the optimal value has not been reached.
1. Perform one further iteration.
2. Interpret the final tableau and state the values of the slack variables.

AQA D2 2008 January Q5

5 [Figure 3, printed on the insert, is provided for use in this question.]
The following network shows 11 vertices. The number on each edge is the cost of travelling between the corresponding vertices. A negative number indicates a reduction by the amount shown.
\includegraphics[max width=\textwidth, alt={}, center]{68ff5b50-6aff-4b28-8fad-d16a8bb4779e-05_824_1504_559_278}

Working backwards from $\boldsymbol { T }$, use dynamic programming to find the minimum cost of travelling from $S$ to $T$. You may wish to complete the table on Figure 3 as your solution.
State the minimum cost and the routes corresponding to this minimum cost.

AQA D2 2008 January Q6

6 [Figures 4, 5 and 6, printed on the insert, are provided for use in this question.]
Water has to be transferred from two mountain lakes $P$ and $Q$ to three urban reservoirs $U$, $V$ and $W$. There are pumping stations at $X , Y$ and $Z$. The possible routes with the capacities along each edge, in millions of litres per hour, are shown in the following diagram.
\includegraphics[max width=\textwidth, alt={}, center]{68ff5b50-6aff-4b28-8fad-d16a8bb4779e-06_862_1316_662_338}

On Figure 4, add a super-source, $S$, and a super-sink, $T$, and appropriate edges so as to produce a directed network with a single source and a single sink. Indicate the capacity of each of the edges you have added.
1. Find the value of the cut $C$.
2. State what can be deduced about the maximum flow from $S$ to $T$.
On Figure 5, write down the maximum flows along the routes $S Q Z W T$ and $S P Y X Z V T$.
1. On Figure 6, add the vertices $S$ and $T$ and the edges connecting $S$ and $T$ to the network. Using the maximum flows along the routes SQZWT and SPYXZVT found in part (c) as the initial flow, indicate the potential increases and decreases of flow on each edge.
2. Use flow augmentation to find the maximum flow from $S$ to $T$. You should indicate any flow augmenting paths on Figure 5 and modify the potential increases and decreases of the flow on Figure 6.
State the value of the flow from $Y$ to $X$ in millions of litres per hour when the maximum flow is achieved.

AQA D2 2009 January Q1

1 The times taken in minutes for five people, $\mathrm { P } , \mathrm { Q } , \mathrm { R } , \mathrm { S }$ and T , to complete each of five different tasks are recorded in the table.

	P	Q	R	S	T
Task 1	17	20	19	17	17
Task 2	19	18	18	18	15
Task 3	13	16	16	14	12
Task 4	13	13	15	13	13
Task 5	10	11	12	14	13

Using the Hungarian algorithm, each of the five people is to be allocated to a different task so that the total time for completing the five tasks is minimised.

By reducing the columns first and then the rows, show that the new table of values is as follows.
3 5 3 0 1
6 4 3 2 0
3 5 4 1 0
3 2 3 0 1
0 0 0 1 1
Show that the zeros in the table in part (a) can be covered with three lines, and use adjustments to produce a table where five lines are required to cover the zeros.
Hence find the two possible ways of allocating the five people to the five tasks so that the total completion time is minimised.
Find the minimum total time for completing the five tasks.

AQA D2 2009 January Q2

2 [Figures 1 and 2, printed on the insert, are provided for use in this question.]
Figure 1 shows the activity network and the duration in days of each activity for a particular project.

On Figure 1:
1. find the earliest start time for each activity;
2. find the latest finish time for each activity.
Find the critical paths and state the minimum time for completion.
The number of workers required for each activity is shown in the table.
Activity $A$ $B$ $C$ $D$ $E$ $F$ $G$ $H$ $I$ $J$
Number of
workers required
3 3 4 2 3 4 1 2 2 5
1. Given that each activity starts as early as possible and assuming that there is no limit to the number of workers available, draw a resource histogram for the project on Figure 2, indicating clearly which activities take place at any given time.
2. It is later discovered that there are only 6 workers available at any time. Explain why the project will overrun, and use resource levelling to indicate which activities need to be delayed so that the project can be completed with the minimum extra time. State the minimum extra time required.

AQA D2 2009 January Q3

Display the following linear programming problem in a Simplex tableau. $$\begin{array} { l l } \text { Maximise } & P = 4 x - 5 y + 6 z
\text { subject to } & 6 x + 7 y - 4 z \leqslant 30
& 2 x + 4 y - 5 z \leqslant 8
& x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
The Simplex method is to be used to solve this problem.
1. Explain why it is not possible to choose a pivot from the $z$-column initially.
2. Identify the initial pivot and explain why this particular element should be chosen.
3. Perform one iteration using your initial tableau from part (a).
4. State the values of $x , y$ and $z$ after this first iteration.
5. Without performing any further iterations, explain why $P$ has no finite maximum value.
Using the same inequalities as in part (a), the problem is modified to $$\text { Maximise } \quad Q = 4 x - 5 y - 20 z$$
1. Write down a modified initial tableau and the revised tableau after one iteration.
2. Hence find the maximum value of $Q$.

AQA D2 2009 January Q4

Two people, Raj and Cal, play a zero-sum game. The game is represented by the following pay-off matrix for Raj.
Cal
\cline { 2 - 5 } Strategy X Y Z
Raj I - 7 8 - 5
\cline { 2 - 5 } II 6 2 - 1
\cline { 2 - 5 } III - 2 4 - 3
\cline { 2 - 5 }
\cline { 2 - 5 }
Show that this game has a stable solution and state the play-safe strategy for each player.

Ros and Carly play a different zero-sum game for which there is no stable solution. The game is represented by the following pay-off matrix for Ros, where $x$ is a constant.

	Carly
\cline { 2 - 4 }	Strategy		$\mathbf { C } _ { \mathbf { 1 } }$
\cline { 2 - 4 }\cline { 2 - 3 } $\operatorname { Ros }$	$\mathbf { R } _ { \mathbf { 1 } }$	5	$\mathbf { C } _ { \mathbf { 2 } }$
\cline { 2 - 4 }	$\mathbf { R } _ { \mathbf { 2 } }$	- 2	$x$
\cline { 2 - 4 }			4

Ros chooses strategy $\mathrm { R } _ { 1 }$ with probability $p$.

Find expressions for the expected gains for Ros when Carly chooses each of the strategies $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$.
Given that the value of the game is $\frac { 8 } { 3 }$, find the value of $p$ and the value of $x$.

AQA D2 2009 January Q5

5 [Figure 3, printed on the insert, is provided for use in this question.]
A truck has to transport stones from a quarry, $Q$, to a builders yard, $Y$. The network shows the possible roads from $Q$ to $Y$. Along each road there are bridges with weight restrictions. The value on each edge indicates the maximum load in tonnes that can be carried by the truck along that particular road.
\includegraphics[max width=\textwidth, alt={}, center]{6c407dbf-efe5-49e4-881f-91e7de5c46d9-6_723_1280_589_372} The truck is able to carry a load of up to 20 tonnes. The optimal route, known as the maximin route, is that for which the possible load that the truck can carry is as large as possible.

Explain why the route $Q A C Y$ is better than the route $Q B E Y$.
By completing the table on Figure 3, or otherwise, use dynamic programming, working backwards from $\boldsymbol { Y }$, to find the optimal (maximin) route from $Q$ to $Y$. Write down the maximin route and state the maximum possible load that the truck can carry from $Q$ to $Y$.

AQA D2 2009 January Q6

6 [Figures 4 and 5, printed on the insert, are provided for use in this question.]
The network shows the routes along corridors from two arrival gates to the passport control area, $P$, in a small airport. The number on each edge represents the maximum number of passengers that can travel along a particular corridor in one minute.
\includegraphics[max width=\textwidth, alt={}, center]{6c407dbf-efe5-49e4-881f-91e7de5c46d9-7_933_1063_552_486}

State the vertices that represent the arrival gates.
Find the value of the cut shown on the network.
State the maximum flow along each of the routes UTSP and RQVP.
1. Taking your answers to part (c) as the initial flow, use the labelling procedure on Figure 4 to find the maximum flow through the network. You should indicate any flow augmenting paths in the table and modify the potential increases and decreases of the flow on the network.
2. State the value of the maximum flow, and, on Figure 5, illustrate a possible flow along each edge corresponding to this maximum flow.
On a particular day, there is an obstruction allowing no more than 50 passengers per minute to pass through vertex $V$. State the maximum number of passengers that can move through the network per minute on this particular day.
(2 marks)

AQA D2 2006 June Q1

1 [Figures 1 and 2, printed on the insert, are provided for use in this question.] A construction project is to be undertaken. The table shows the activities involved.

Activity	Immediate Predecessors	Duration (days)
A	-	2
B	A	5
C	A	8
D	B	8
E	B	10
F	B	4
G	$C , F$	7
$H$	D, E	4
I	$G , H$	3

Complete the activity network for the project on Figure 1.
Find the earliest start time for each activity.
Find the latest finish time for each activity.
Find the critical path.
State the float time for each non-critical activity.
On Figure 2, draw a cascade diagram (Gantt chart) for the project, assuming each activity starts as late as possible.

AQA D2 2006 June Q2

2 Four of the five students Phil, Quin, Ros, Sue and Tim are to be chosen to make up a team for a mathematical relay race. The team will be asked four questions, one each on the topics A, B, C and D. A different member of the team will answer each question. Each member has to give the correct answer to the question before the next question is asked. The team with the least overall time wins.
The average times, in seconds, for each student in some practice questions are given below.

	Phil	Quin	Ros	Sue	Tim
Topic A	18	15	19	20	17
Topic B	23	24	22	25	23
Topic C	20	16	18	22	19
Topic D	21	17	18	23	20

Modify the table of values by adding an extra row of values so that the Hungarian algorithm can be applied.
Use the Hungarian algorithm, reducing columns first, then rows, to decide which four students should be chosen for the team. State which student should be allocated to each topic and state the total time for the four students on the practice questions using this matching.

AQA D2 2006 June Q3

3 [Figure 3, printed on the insert, is provided for use in this question.]
The following network shows eight vertices. The number on each edge is the cost of travelling between the corresponding vertices. A negative number indicates a reduction by the amount shown.
\includegraphics[max width=\textwidth, alt={}, center]{587bccdf-abd7-4a08-a76e-61374f322e2e-03_595_1234_1866_386}

Use dynamic programming to find the minimum cost of travelling from $A$ to $H$. You may use Figure 3 for your working.
State the minimum cost and the possible routes corresponding to this minimum cost.

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	Col
\cline { 2 - 5 } \multicolumn{1}{c\|}{}	Strategy	\(\mathbf { C } _ { \mathbf { 1 } }\)	\(\mathbf { C } _ { \mathbf { 2 } }\)	\(\mathbf { C } _ { \mathbf { 3 } }\)
\multirow{2}{*}{Ros}	\(\mathbf { R } _ { \mathbf { 1 } }\)	3	2	1
\cline { 2 - 5 }	\(\mathbf { R } _ { \mathbf { 2 } }\)	- 2	- 1	2

\(\boldsymbol { P }\)	\(\boldsymbol { x }\)	\(\boldsymbol { y }\)	\(\boldsymbol { z }\)	\(s\)	\(\boldsymbol { t }\)	\(\boldsymbol { u }\)	value
1	-2	-3	-5	0	0	0	0
0	1	0	1	1	0	0	9
0	2	1	4	0	1	0	40
0	4	2	3	0	0	1	33

		Road Conditions
\cline { 2 - 5 }		\(\boldsymbol { C } _ { \mathbf { 1 } }\)	\(\boldsymbol { C } _ { \mathbf { 2 } }\)	\(\boldsymbol { C } _ { \mathbf { 3 } }\)
\cline { 2 - 5 }	\(\boldsymbol { S } _ { \mathbf { 1 } }\)	- 2	2	4
\cline { 2 - 5 } Sam's Car	\(\boldsymbol { S } _ { \mathbf { 2 } }\)	2	4	5
\cline { 2 - 5 }	\(\boldsymbol { S } _ { \mathbf { 3 } }\)	5	1	2
\cline { 2 - 5 }
\cline { 2 - 5 }

\multirow{2}{*}{}	\multirow[b]{2}{*}{Strategy}	Col
		X	Y	Z
\multirow{3}{*}{Ros}	I	-4	-3	0
	II	5	-2	2
	III	1	-1	3

\multirow{5}{*}{Rob}	Con
	Strategy	\(\mathrm { C } _ { 1 }\)	\(\mathbf { C } _ { \mathbf { 2 } }\)	\(\mathrm { C } _ { 3 }\)
	\(\mathbf { R } _ { \mathbf { 1 } }\)	-2	5	3
	\(\mathbf { R } _ { \mathbf { 2 } }\)	3	-3	-1
	\(\mathbf { R } _ { \mathbf { 3 } }\)	-3	3	2

Activity	Immediate Predecessors	Duration (weeks)
A	-	2
B	-	1
C	A	3
D	A, B	2
E	B	4
F	C	1
G	\(C , D , E\)	3
H	E	5
I	\(F , G\)	2
J	\(H , I\)	3

Activity	Immediate predecessors	Duration (days)	Number of workers required
A	-	3	5
B	A	8	2
C	A	7	3
\(D\)	\(B , C\)	8	4
E	C	10	2
\(F\)	C	3	3
\(G\)	D, E	3	4
H	\(F\)	6	1
I	\(G , H\)	2	3

Activity	Immediate Predecessors	Duration (days)
A	-	2
B	A	5
C	A	8
D	B	8
E	B	10
F	B	4
G	\(C , F\)	7
\(H\)	D, E	4
I	\(G , H\)	3

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AQA D2 2007 January Q4

AQA D2 2007 January Q5

AQA D2 2007 January Q6

AQA D2 2008 January Q1

AQA D2 2008 January Q2

AQA D2 2008 January Q3

AQA D2 2008 January Q4

AQA D2 2008 January Q5

AQA D2 2008 January Q6

AQA D2 2009 January Q1

AQA D2 2009 January Q2

AQA D2 2009 January Q3

AQA D2 2009 January Q4

AQA D2 2009 January Q5

AQA D2 2009 January Q6

AQA D2 2006 June Q1

AQA D2 2006 June Q2

AQA D2 2006 June Q3