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AQA D1 2012 June Q7

11 marks Moderate -0.8

7 Rupta, a sales representative, has to visit six shops, $A , B , C , D , E$ and $F$. Rupta starts at shop $A$ and travels to each of the other shops once, before returning to shop $A$. Rupta wishes to keep her travelling time to a minimum. The table shows the travelling times, in minutes, between the shops.

AQA D1 2012 June Q8

8 marks Easy -1.2

8 The following algorithm finds an estimate of the value of the number represented by the symbol e:

Line 10	Let $A = 1 , B = 1 , C = 1$
Line 20	Let $D = A$
Line 30	Let $C = C \times B$
Line 40	Let $D = D + ( 1 / C )$
Line 50	If $B = 4$ then go to Line 80
Line 60	Let $B = B + 1$
Line 70	Go to Line 30
Line 80	Print 'An estimate of e is', $D$
Line 90	End

Trace the algorithm.
A student miscopied Line 70 . His line was
Line 70 Go to Line 10
Explain what would happen if his algorithm were traced.

AQA D1 2012 June Q9

14 marks Moderate -0.3

9 Ollyin is buying new pillows for his hotel. He buys three types of pillow: soft, medium and firm. He must buy at least 100 soft pillows and at least 200 medium pillows.
He must buy at least 400 pillows in total.
Soft pillows cost $\pounds 4$ each. Medium pillows cost $\pounds 3$ each. Firm pillows cost $\pounds 4$ each.
He wishes to spend no more than $\pounds 1800$ on new pillows.
At least $40 \%$ of the new pillows must be medium pillows.
Ollyin buys $x$ soft pillows, $y$ medium pillows and $z$ firm pillows.

In addition to $x \geqslant 0 , y \geqslant 0$ and $z \geqslant 0$, find five inequalities in $x , y$ and $z$ that model the above constraints.
Ollyin decides to buy twice as many soft pillows as firm pillows.
1. Show that three of your answers in part (a) become $$\begin{aligned} 3 x + 2 y & \geqslant 800 \\ 2 x + y & \leqslant 600 \\ y & \geqslant x \end{aligned}$$
2. On the grid opposite, draw a suitable diagram to represent Ollyin's situation, indicating the feasible region.
3. Use your diagram to find the maximum total number of pillows that Ollyin can buy.
4. Find the number of each type of pillow that Ollyin can buy that corresponds to your answer to part (b)(iii).
  \includegraphics[max width=\textwidth, alt={}]{1258a6d3-558a-46dc-a916-d71f71b175ff-20_2256_1707_221_153}

AQA D1 2013 June Q1

7 marks Moderate -0.8

1 Six people, Andy, Bob, Colin, Dev, Eric and Faisal, are to be allocated to six tasks, $1,2,3,4,5$ and 6 . The following table shows the tasks that each person is able to undertake.

Person	Task
Andy	1,3
Bob	1,4
Colin	2,3
Dev	$4,5,6$
Eric	$2,5,6$
Faisal	1,3

Represent this information on a bipartite graph.
Initially, Bob is allocated to task 1, Colin to task 3, Dev to task 5 and Eric to task 2. Demonstrate, by using an alternating path algorithm from this initial matching, how each person can be allocated to a different task.

AQA D1 2013 June Q2

5 marks Easy -1.8

Use the quicksort algorithm to rearrange the following numbers into ascending order, showing the new arrangement after each pass. You must indicate the pivot(s) being used on each pass. $$2 , \quad 12 , \quad 17 , \quad 18 , \quad 5 , \quad 13$$
For the first pass, write down the number of comparisons.

AQA D1 2013 June Q3

9 marks Moderate -0.5

3 The following network shows the lengths, in miles, of roads connecting ten villages, $A , B , C , \ldots , J$. \includegraphics[max width=\textwidth, alt={}, center]{77c4efd4-a905-48f3-a6f7-36b0e47dbc6d-06_899_1458_397_285}

1. Use Kruskal's algorithm, showing the order in which you select the edges, to find a minimum spanning tree for the network.
2. Find the length of your minimum spanning tree.
3. Draw your minimum spanning tree.
Prim's algorithm from different starting points produces the same minimum spanning tree. State the final edge that would be added to complete the minimum spanning tree if the starting point were:
1. $A$;
2. $F$.

AQA D1 2013 June Q4

12 marks Moderate -0.8

4 Sarah is a mobile hairdresser based at $A$. Her day's appointments are at five places: $B , C , D , E$ and $F$. She can arrange the appointments in any order. She intends to travel from one place to the next until she has visited all of the places, starting and finishing at $A$. The following table shows the times, in minutes, that it takes to travel between the six places.

\cline { 2 - 7 } \multicolumn{1}{c\|}{}	$\boldsymbol { A }$	$\boldsymbol { B }$	$\boldsymbol { C }$	$\boldsymbol { D }$	$\boldsymbol { E }$	$\boldsymbol { F }$
$\boldsymbol { A }$	-	15	11	14	27	12
$\boldsymbol { B }$	15	-	13	19	24	15
$\boldsymbol { C }$	11	13	-	10	19	12
$\boldsymbol { D }$	14	19	10	-	26	15
$\boldsymbol { E }$	27	24	19	26	-	27
$\boldsymbol { F }$	12	15	12	15	27	-

Sarah decides to visit the places in the order $A B C D E F A$. Find the travelling time of this tour.
Explain why this answer can be considered as being an upper bound for the minimum travelling time of Sarah's tour.
Use the nearest neighbour algorithm, starting from $A$, to find another upper bound for the minimum travelling time of Sarah's tour.
By deleting $A$, find a lower bound for the minimum travelling time of Sarah's tour.
Sarah thinks that she can reduce her travelling time to 75 minutes. Explain why she is wrong.

AQA D1 2013 June Q5

17 marks Standard +0.3

5 The network on the page opposite shows the times, in minutes, taken by police cars to drive along roads connecting 12 places, $A , B , \ldots , L$. On a particular day, there are three police cars in the area at $A , E$ and $J$. There is an emergency at $G$ and all three police cars drive to $G$.

1. Use Dijkstra's algorithm on the network, starting from $\boldsymbol { G }$, to find the minimum driving time for each of the police cars to arrive at $G$.
2. For each of the police cars, write down the route corresponding to the minimum driving time in your answer to part (a)(i).
Each day, a police car has to drive along each road at least once, starting and finishing at $A$. For an optimal Chinese postman route:
1. find the driving time for the police car;
2. state the number of times that the vertex $B$ would appear.
  \includegraphics[max width=\textwidth, alt={}]{77c4efd4-a905-48f3-a6f7-36b0e47dbc6d-13_2486_1709_221_153}

AQA D1 2013 June Q6

9 marks Easy -1.2

6 A student is tracing the following algorithm. The function INT gives the integer part of any number, eg $\operatorname { INT } ( 2.3 ) = 2$ and $\operatorname { INT } ( 6.7 ) = 6$. Line 10 Input $A , B$ Line $20 \quad$ Let $C = \operatorname { INT } ( A \div B )$ Line 30 Let $D = B \times C$ Line $40 \quad$ Let $E = A - D$ Line 50 If $E = 0$ then go to Line 90
Line 60 Let $A = B$ Line $70 \quad$ Let $B = E$ Line 80 Go to Line 20
Line 90 Print $B$ Line 100 Stop

Trace the algorithm when the input values are:
1. $A = 36$ and $B = 16$;
2. $A = 11$ and $B = 7$.
State the purpose of the algorithm.

AQA D1 2013 June Q7

16 marks Moderate -0.3

7 Paul is a florist. Every day, he makes three types of floral bouquet: gold, silver and bronze. Each gold bouquet has 6 roses, 6 carnations and 6 dahlias.
Each silver bouquet has 4 roses, 6 carnations and 4 dahlias.
Each bronze bouquet has 3 roses, 4 carnations and 4 dahlias.
Each day, Paul must use at least 420 roses and at least 480 carnations, but he can use at most 720 dahlias. Each day, Paul makes $x$ gold bouquets, $y$ silver bouquets and $z$ bronze bouquets.

In addition to $x \geqslant 0 , y \geqslant 0$ and $z \geqslant 0$, find three inequalities in $x , y$ and $z$ that model the above constraints.
On a particular day, Paul makes the same number of silver bouquets as bronze bouquets.
1. Show that $x$ and $y$ must satisfy the following inequalities. $$\begin{aligned} & 6 x + 7 y \geqslant 420 \\ & 3 x + 5 y \geqslant 240 \\ & 3 x + 4 y \leqslant 360 \end{aligned}$$
2. Paul makes a profit of $\pounds 4$ on each gold bouquet sold, a profit of $\pounds 2.50$ on each silver bouquet sold and a profit of $\pounds 2.50$ on each bronze bouquet sold. Each day, Paul sells all the bouquets he makes. Paul wishes to maximise his daily profit, $\pounds P$. Draw a suitable diagram, on the grid opposite, to enable this problem to be solved graphically, indicating the feasible region and the direction of the objective line.
  (6 marks)
3. Use your diagram to find Paul's maximum daily profit and the number of each type of bouquet he must make to achieve this maximum.
On another day, Paul again makes the same number of silver bouquets as bronze bouquets, but he makes a profit of $\pounds 2$ on each gold bouquet sold, a profit of $\pounds 6$ on each silver bouquet sold and a profit of $\pounds 6$ on each bronze bouquet sold. Find Paul's maximum daily profit, and the number of each type of bouquet he must make to achieve this maximum.
(3 marks) Turn over -

AQA D2 2010 January Q1

13 marks Moderate -0.8

1 [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 float for activity $G$.
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 2 2 3 2 3 2 1 3 5 2
Given that each activity starts as late 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.

AQA D2 2010 January Q2

11 marks Standard +0.3

2 The following table shows the times taken, in minutes, by five people, Ron, Sam, Tim, Vic and Zac, to carry out the tasks $1,2,3$ and 4 . Sam takes $x$ minutes, where $8 \leqslant x \leqslant 12$, to do task 2.

	Ron	Sam	Tim	Vic	Zac
Task 1	8	7	9	10	8
Task 2	9	$x$	8	7	11
Task 3	12	10	9	9	10
Task 4	11	9	8	11	11

Each of the four tasks is to be given to a different one of the five people so that the total 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.
1. Use the Hungarian algorithm, reducing columns first and then rows, to reduce the matrix to a form, in terms of $x$, from which the optimum matching can be made.
2. Hence find the possible way of allocating the four tasks so that the total time is minimised.
3. Find the minimum total time.
After special training, Sam is able to complete task 2 in 7 minutes and is assigned to task 2. Determine the possible ways of allocating the other three tasks so that the total time is minimised.

AQA D2 2010 January Q3

12 marks Easy -1.3

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

\multirow{5}{*}{Ann}	Bill
	Strategy	$\mathbf { B } _ { \mathbf { 1 } }$	$\mathbf { B } _ { \mathbf { 2 } }$	$\mathbf { B } _ { \mathbf { 3 } }$
	$\mathbf { A } _ { \mathbf { 1 } }$	-1	0	-2
	$\mathbf { A } _ { \mathbf { 2 } }$	4	-2	-3
	$\mathbf { A } _ { \mathbf { 3 } }$	-4	-5	-3

Show that this game has a stable solution and state the play-safe strategies for Ann and Bill.

Russ and Carlos play a different zero-sum game, which does not have a stable solution. The game is represented by the following pay-off matrix for Russ.

	Carlos
\cline { 2 - 5 }	Strategy	$\mathbf { C } _ { \mathbf { 1 } }$	$\mathbf { C } _ { \mathbf { 2 } }$	$\mathbf { C } _ { \mathbf { 3 } }$
\cline { 2 - 5 } Russ	$\mathbf { R } _ { \mathbf { 1 } }$	- 4	7	- 3
\cline { 2 - 5 }	$\mathbf { R } _ { \mathbf { 2 } }$	2	- 1	1

Find the optimal mixed strategy for Russ.
Find the value of the game.

AQA D2 2010 January Q4

14 marks Standard +0.3

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

$\boldsymbol { P }$	$\boldsymbol { x }$	$\boldsymbol { y }$	$\boldsymbol { Z }$	$s$	$t$	$\boldsymbol { u }$	value
1	-2	-4	-3	0	0	0	0
0	2	2	1	1	0	0	14
0	-1	1	2	0	1	0	6
0	4	4	3	0	0	1	29

1. What name is given to the variables $s , t$ and $u$ ?
2. Write down an equation involving $x , y , z$ and $s$ for this problem.
1. By choosing the first pivot from the $\boldsymbol { y }$-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, stating the values of $P , x , y$ and $z$.

AQA D2 2010 January Q5

10 marks Moderate -0.5

5 [Figure 3, printed on the insert, is provided for use in this question.]
A landscape gardener has three projects, $A , B$ and $C$, to be completed over a period of 4 months: May, June, July and August. The gardener must allocate one of these months to each project and the other month is to be taken as a holiday. Various factors, such as availability of materials and transport, mean that the costs for completing the projects in different months will vary. The costs, in thousands of pounds, are given in the table.

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	May	June	July	August
Project $\boldsymbol { A }$	17	16	18	16
Project $\boldsymbol { B }$	14	13	12	10
Project $\boldsymbol { C }$	14	17	15	14

By completing the table of values on Figure 3, or otherwise, use dynamic programming, working backwards from August, to find the project schedule that minimises total costs. State clearly which month should be taken as a holiday and which project should be undertaken in which month.

AQA D2 2010 January Q6

15 marks Moderate -0.5

6 [Figures 4, 5, 6 and 7, printed on the insert, are provided for use in this question.]

The network shows a flow from $S$ to $T$ along a system of pipes, with the capacity, in litres per minute, indicated on each edge. \includegraphics[max width=\textwidth, alt={}, center]{3ac580ff-f9c8-4e47-b4ca-97d186b0936c-6_350_878_532_591}
1. Show that the value of the cut shown on the diagram is 97 .
2. The cut shown on the diagram can be represented as $\{ S , C \} , \{ A , B , T \}$. Complete the table on Figure 4, giving the value of each of the 8 possible cuts.
3. State the value of the maximum flow through the network, giving a reason for your answer.
4. Indicate on Figure 5 a possible flow along each edge corresponding to this maximum flow.
Extra pipes, $B D , C D$ and $D T$, are added to form a new system, with the capacity, in litres per minute, indicated on each edge of the network below. \includegraphics[max width=\textwidth, alt={}, center]{3ac580ff-f9c8-4e47-b4ca-97d186b0936c-6_483_977_1724_520}
1. Taking your values from Figure 5 as the initial flow, use the labelling procedure on Figure 6 to find the new 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 new maximum flow, and, on Figure 7, indicate a possible flow along each edge corresponding to this maximum flow.

AQA D2 2011 January Q1

14 marks Moderate -0.5

1
A group of workers is involved in a decorating project. The table shows the activities involved. Each worker can perform any of the given activities.

Activity	A	$B$	C	D	$E$	$F$	G	H	I	$J$	$K$	$L$
Duration (days)	2	5	6	7	9	4	3	2	3	2	3	1
Number of workers required	6	3	5	2	5	2	4	4	5	3	2	4

The activity network for the project is given in Figure 1 below.

Find the earliest start time and the latest finish time for each activity, inserting their values on Figure 1.
Hence find:
1. the critical path;
2. the float time for activity $D$.
  1. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{172c5c92-4254-4593-b741-1caa83a1e833-02_647_1657_1640_180}
    \end{figure}
  2. 1. The critical path is $\_\_\_\_$
    2. The float time for activity $D$ is $\_\_\_\_$
3. 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 below, indicating clearly which activities are taking place at any given time.
4. It is later discovered that there are only 8 workers available at any time. Use resource levelling to construct a new resource histogram on Figure 3 below, showing how the project can be completed with the minimum extra time. State the minimum extra time required.
5. \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{172c5c92-4254-4593-b741-1caa83a1e833-03_586_1708_922_150}
  \end{figure}
6. \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{172c5c92-4254-4593-b741-1caa83a1e833-03_496_1705_1672_153}
  \end{figure} The minimum extra time required is $\_\_\_\_$

AQA D2 2011 January Q2

13 marks Moderate -0.8

2 A farmer has five fields. He intends to grow a different crop in each of four fields and to leave one of the fields unused. The farmer tests the soil in each field and calculates a score for growing each of the four crops. The scores are given in the table below.

	Field A	Field B	Field C	Field D	Field E
Crop 1	16	12	8	18	14
Crop 2	20	15	8	16	12
Crop 3	9	10	12	17	12
Crop 4	18	11	17	15	19

The farmer's aim is to maximise the total score for the four crops.

1. Modify the table of values by first subtracting each value in the table above from 20 and then adding an extra row of equal values.
2. Explain why the Hungarian algorithm can now be applied to the new table of values to maximise the total score for the four crops.
1. By reducing rows first, show that the table from part (a)(i) becomes
  2 6 10 0 $p$
  0 5 12 4 8
  8 7 5 0 $q$
  1 8 2 4 0
  0 0 0 0 0
  State the values of the constants $p$ and $q$.
2. Show that the zeros in the table from part (b)(i) can be covered by one horizontal and three vertical lines, and use the Hungarian algorithm to decide how the four crops should be allocated to the fields.
3. Hence find the maximum possible total score for the four crops.

AQA D2 2011 January Q3

13 marks Easy -1.8

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


\cline { 2 - 5 }	Colleen
\cline { 2 - 5 } Strategy	$\mathbf { C } _ { \mathbf { 1 } }$	$\mathbf { C } _ { \mathbf { 2 } }$	$\mathbf { C } _ { \mathbf { 3 } }$
\cline { 2 - 5 } Rhona	$\mathbf { R } _ { \mathbf { 1 } }$	2	6	4
\cline { 2 - 5 }	$\mathbf { R } _ { \mathbf { 2 } }$	3	- 3	- 1
\cline { 2 - 5 }	$\mathbf { R } _ { \mathbf { 3 } }$	$x$	$x + 3$	3
\cline { 2 - 5 }
\cline { 2 - 5 }

It is given that $x < 2$.

1. Write down the three row minima.
2. Show that there is no stable solution.
Explain why Rhona should never play strategy $R _ { 3 }$.
1. Find the optimal mixed strategy for Rhona.
2. Find the value of the game.

AQA D2 2011 January Q4

15 marks Standard +0.8

4 The Simplex method is to be used to maximise $P = 3 x + 2 y + z$ subject to the constraints $$\begin{aligned} - x + y + z & \leqslant 4 \\ 2 x + y + 4 z & \leqslant 10 \\ 4 x + 2 y + 3 z & \leqslant 21 \end{aligned}$$ The initial Simplex tableau is given below.

$\boldsymbol { P }$	$\boldsymbol { x }$	$\boldsymbol { y }$	$\boldsymbol { z }$	$s$	$t$	$\boldsymbol { u }$	value
1	-3	-2	-1	0	0	0	0
0	-1	1	1	1	0	0	4
0	2	1	4	0	1	0	10
0	4	2	3	0	0	1	21

1. The first pivot is to be chosen from the $x$-column. Identify the pivot and explain why this particular value is chosen.
2. Perform one iteration of the Simplex method and explain how you know that the optimal value has not been reached.
1. Perform one further iteration.
2. Interpret the final tableau and write down the initial inequality that still has slack.
  \includegraphics[max width=\textwidth, alt={}]{172c5c92-4254-4593-b741-1caa83a1e833-11_2486_1714_221_153}

AQA D2 2011 January Q5

9 marks Moderate -0.8

5 Each path from $S$ to $T$ in the network below represents a possible way of using the internet to buy a ticket for a particular event. The number on each edge represents a charge, in pounds, with a negative value representing a discount. For example, the path SAEIT represents a ticket costing $23 + 5 - 4 - 7 = 17$ pounds. \includegraphics[max width=\textwidth, alt={}, center]{172c5c92-4254-4593-b741-1caa83a1e833-12_1023_1330_540_350}

By working backwards from $\boldsymbol { T }$ and completing the table on Figure 4, use dynamic programming to find the minimum weight of all paths from $S$ to $T$.
State the minimum cost of a ticket for the event and the paths corresponding to this minimum cost.
(3 marks)
1. \begin{table}[h]
  \captionsetup{labelformat=empty} \caption{Figure 4}
  Stage State From Value
  1 I $T$ -7
  $J$ $T$ -6
  $K$ $T$ -5
  2 $E$ $I$ $- 7 - 4 = - 11$
  F I
  $J$
  G I
  $J$
  $K$
  $H$ $K$
  3
  \end{table}

AQA D2 2011 January Q6

11 marks Moderate -0.5

6 A retail company has warehouses at $P , Q$ and $R$, and goods are to be transported to retail outlets at $Y$ and $Z$. There are also retaining depots at $U , V , W$ and $X$. The possible routes with the capacities along each edge, in van loads per week, are shown in the following diagram. \includegraphics[max width=\textwidth, alt={}, center]{172c5c92-4254-4593-b741-1caa83a1e833-14_673_1193_577_429}

On Figure 5 opposite, 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 edge that you have added.
On Figure 6, write down the maximum flows along the routes SPUYT and SRVWZT.
1. On Figure 7, add the vertices $S$ and $T$ and the edges connecting $S$ and $T$ to the network. Using the maximum flows along the routes SPUYT and SRVWZT found in part (b) as the initial flow, indicate the potential increases and decreases of the flow on each edge of these routes.
2. Use flow augmentation to find the maximum flow from $S$ to $T$. You should indicate any flow-augmenting routes on Figure 6 and modify the potential increases and decreases of the flow on Figure 7.
Find a cut with value equal to the maximum flow. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{172c5c92-4254-4593-b741-1caa83a1e833-15_629_1100_342_477}
\end{figure} \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Figure 6}
Route Flow
SPUYT
SRVWZT
\end{table} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 7} \includegraphics[alt={},max width=\textwidth]{172c5c92-4254-4593-b741-1caa83a1e833-15_634_1109_1838_470}
\end{figure}

AQA D2 2012 January Q1

14 marks Standard +0.3

1 The diagram shows the activity network and the duration, in days, of each activity for a particular project. Some of the earliest start times and latest finish times are shown on the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-02_830_1447_678_301}

Find the values of the constants $x , y$ and $z$.
Find the critical paths.
Find the activity with the largest float and state the value of this float.
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 4 2 3 4 2 4 3 3 5 6
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 1 below, indicating clearly which activities are taking place at any given time.
It is later discovered that there are only 9 workers available at any time. Use resource levelling to find the new earliest start time for activity $J$ so that the project can be completed with the minimum extra time. State the minimum extra time required. (d) Number of workers \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-03_803_1330_1224_468}
\end{figure}

AQA D2 2012 January Q2

12 marks Moderate -0.3

2 A team with five members is training to take part in a quiz. The team members, Abby, Bob, Cait, Drew and Ellie, attempted sample questions on each of the five topics and their scores are given in the table.

	Topic 1	Topic 2	Topic 3	Topic 4	Topic 5
Abby	27	29	25	35	31
Bob	33	22	17	29	29
Cait	23	29	25	33	21
Drew	22	29	29	27	31
Ellie	27	27	19	21	27

For the actual quiz, each topic must be allocated to exactly one of the team members. The maximum total score for the sample questions is to be used to allocate the different topics to the team members.

Explain why the Hungarian algorithm may be used if each number, $x$, in the table is replaced by $35 - x$.
Form a new table by subtracting each number in the table above from 35 . Hence show that, by reducing rows first then columns, the resulting table of values is as below, stating the values of the constants $p$ and $q$.
8 6 8 0 4
0 11 $p$ 4 4
10 4 6 0 12
$q$ 2 0 4 0
0 0 6 6 0
Show that the zeros in the table in part (b) can be covered with two horizontal and two vertical lines. Hence use the Hungarian algorithm to reduce the table to a form where five lines are needed to cover the zeros.
1. Hence find the possible allocations of topics to the five team members so that the total score for the sample questions is maximised.
2. State the value of this maximum total score.

AQA D2 2012 January Q3

13 marks Easy -2.5

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

Colum
\cline { 2 - 5 }	Strategy	$\mathbf { C } _ { \mathbf { 1 } }$	$\mathbf { C } _ { \mathbf { 2 } }$	$\mathbf { C } _ { \mathbf { 3 } }$
\multirow{3}{*}{$\operatorname { Roz }$}	$\mathbf { R } _ { \mathbf { 1 } }$	- 2	- 6	- 1
\cline { 2 - 5 }	$\mathbf { R } _ { \mathbf { 2 } }$	- 5	2	- 6
\cline { 2 - 5 }	$\mathbf { R } _ { \mathbf { 3 } }$	- 3	3	- 4

Explain what is meant by the term 'zero-sum game'.
Determine the play-safe strategy for Colum, giving a reason for your answer.
1. Show that the matrix can be reduced to a 2 by 3 matrix, giving the reason for deleting one of the rows.
2. Hence find the optimal mixed strategy for Roz.

Activity	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
Number of workers required	2	2	3	2	3	2	1	3	5	2

Activity	A	\(B\)	C	D	\(E\)	\(F\)	G	H	I	\(J\)	\(K\)	\(L\)
Duration (days)	2	5	6	7	9	4	3	2	3	2	3	1
Number of workers required	6	3	5	2	5	2	4	4	5	3	2	4

Stage	State	From	Value
1	I	\(T\)	-7
	\(J\)	\(T\)	-6
	\(K\)	\(T\)	-5

2	\(E\)	\(I\)	\(- 7 - 4 = - 11\)
	F	I
		\(J\)
	G	I
		\(J\)
		\(K\)
	\(H\)	\(K\)

3

Activity	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
Number of workers required	4	2	3	4	2	4	3	3	5	6

Line 10	Let \(A = 1 , B = 1 , C = 1\)
Line 20	Let \(D = A\)
Line 30	Let \(C = C \times B\)
Line 40	Let \(D = D + ( 1 / C )\)
Line 50	If \(B = 4\) then go to Line 80
Line 60	Let \(B = B + 1\)
Line 70	Go to Line 30
Line 80	Print 'An estimate of e is', \(D\)
Line 90	End

\cline { 2 - 7 } \multicolumn{1}{c\|}{}	\(\boldsymbol { A }\)	\(\boldsymbol { B }\)	\(\boldsymbol { C }\)	\(\boldsymbol { D }\)	\(\boldsymbol { E }\)	\(\boldsymbol { F }\)
\(\boldsymbol { A }\)	-	15	11	14	27	12
\(\boldsymbol { B }\)	15	-	13	19	24	15
\(\boldsymbol { C }\)	11	13	-	10	19	12
\(\boldsymbol { D }\)	14	19	10	-	26	15
\(\boldsymbol { E }\)	27	24	19	26	-	27
\(\boldsymbol { F }\)	12	15	12	15	27	-

	Ron	Sam	Tim	Vic	Zac
Task 1	8	7	9	10	8
Task 2	9	\(x\)	8	7	11
Task 3	12	10	9	9	10
Task 4	11	9	8	11	11

\(\boldsymbol { P }\)	\(\boldsymbol { x }\)	\(\boldsymbol { y }\)	\(\boldsymbol { Z }\)	\(s\)	\(t\)	\(\boldsymbol { u }\)	value
1	-2	-4	-3	0	0	0	0
0	2	2	1	1	0	0	14
0	-1	1	2	0	1	0	6
0	4	4	3	0	0	1	29

2	6	10	0	\(p\)
0	5	12	4	8
8	7	5	0	\(q\)
1	8	2	4	0
0	0	0	0	0

Route	Flow
SPUYT
SRVWZT

8	6	8	0	4
0	11	\(p\)	4	4
10	4	6	0	12
\(q\)	2	0	4	0
0	0	6	6	0

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