Questions - AQA D2 - A-Level Maths

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

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

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

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

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

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

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 $D$.
\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}
1. The critical path is $\_\_\_\_$
2. The float time for activity $D$ is $\_\_\_\_$
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.
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.
\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}
\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

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

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

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

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)
\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

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

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.
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

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

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.

AQA D2 2012 January Q4

4 A linear programming problem consists of maximising an objective function $P$ involving three variables, $x , y$ and $z$, subject to constraints given by three inequalities other than $x \geqslant 0 , y \geqslant 0$ and $z \geqslant 0$. Slack variables $s , t$ and $u$ are introduced and the Simplex method is used to solve the problem. One iteration of the method leads to the following tableau.

$\boldsymbol { P }$	$\boldsymbol { x }$	$\boldsymbol { y }$	$\boldsymbol { Z }$	$\boldsymbol { s }$	$\boldsymbol { t }$	$\boldsymbol { u }$	value
1	-2	11	0	3	0	0	6
0	2	3	1	1	0	0	2
0	6	-30	0	-6	1	0	3
0	-1	-9	0	-3	0	1	4

1. State the column from which the pivot for the next iteration should be chosen. Identify this pivot and explain the reason for your choice.
2. Perform the next iteration of the Simplex method.
1. Explain why you know that the maximum value of $P$ has been achieved.
2. State how many of the three original inequalities still have slack.
1. State the maximum value of $P$ and the values of $x , y$ and $z$ that produce this maximum value.
2. The objective function for this problem is $P = k x - 2 y + 3 z$, where $k$ is a constant. Find the value of $k$.
  
  \includegraphics[max width=\textwidth, alt={}]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-11_2486_1714_221_153}

AQA D2 2012 January Q5

5 A firm is considering various strategies for development over the next few years. In the network, the number on each edge is the expected profit, in millions of pounds, moving from one year to the next. A negative number indicates a loss because of building costs or other expenses. Each path from $S$ to $T$ represents a complete strategy.
\includegraphics[max width=\textwidth, alt={}, center]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-12_748_1575_559_228}

By completing the table on the page opposite, or otherwise, use dynamic programming working backwards from $\boldsymbol { T }$ to find the maximum weight of all paths from $S$ to $T$.
State the overall maximum profit and the paths from $S$ to $T$ corresponding to this maximum profit.
Stage State From Calculation Value
1 G $T$
H $T$
I $T$
2 D G
$H$
E G
$H$
I
$F$ $H$
I
3
Maximum profit is £ $\_\_\_\_$ million Corresponding paths from $S$ to $T$ $\_\_\_\_$

AQA D2 2012 January Q6

6 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]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-14_807_1472_429_276}

Find the value of the cut $Q$.
Figure 2 shows most of the values of a feasible flow of 34 litres per second from $S$ to $T$.
1. Insert the missing values of the flows along $D E$ and $F G$ on Figure 2.
2. Using this feasible flow as the initial flow, indicate potential increases and decreases of the flow along each edge on Figure 3.
3. Use flow augmentation on Figure 3 to find the maximum flow from $S$ to $T$. You should indicate any flow-augmenting paths in the table and modify the potential increases and decreases of the flow on the network.
1. State the value of the maximum flow.
2. Illustrate your maximum flow on Figure 4.
Find a cut with capacity equal to that of the maximum flow. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-15_1646_1463_280_381}
\end{figure} Figure 3 \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-15_668_1230_1998_404}
\end{figure}

AQA D2 2013 January Q1

1
Figure 1 below shows an activity diagram for a project. Each activity requires one worker. The duration required for each activity is given in hours.

Find the earliest start time and the latest finish time for each activity and insert their values on Figure 1.
On Figure 2 opposite, complete the precedence table.
Find the critical path.
Find the float time of activity $E$.
Using Figure 3 on page 5, draw a resource histogram to illustrate how the project can be completed in the minimum time, assuming that each activity is to start as early as possible.
Given that there are two workers available for the project, find the minimum completion time for the project.
Given that there is only one worker available for the project, find the minimum completion time for the project. Figure 1 \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{(a)} \includegraphics[alt={},max width=\textwidth]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-02_629_1550_1818_292}
\end{figure}
\begin{table}[h]
\captionsetup{labelformat=empty} \caption{Figure 2}
Activity Immediate predecessor(s)
A
B
C
D
E
$F$
G
H
I
J
$K$
\end{table}
\includegraphics[max width=\textwidth, alt={}]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-05_2486_1717_221_150}

AQA D2 2013 January Q2

2 Harry and Will play a zero-sum game. The game is represented by the following pay-off matrix for Harry.

	Will
\cline { 2 - 6 }	Strategy	$\boldsymbol { D }$	$\boldsymbol { E }$	$\boldsymbol { F }$	$\boldsymbol { G }$
Harry	$\boldsymbol { A }$	- 1	2	3
\cline { 2 - 6 }	$\boldsymbol { B }$	4	6	3	7
\cline { 2 - 6 }	$\boldsymbol { C }$	1	3	- 2	4

Show that this game has a stable solution and state the play-safe strategy for each player.
List any saddle points.

AQA D2 2013 January Q3

3 Four pupils, Wendy, Xiong, Yasmin and Zaira, are each to be allocated a different memory coach from five available coaches: Asif, Bill, Connie, Deidre and Eric. Each pupil has an initial training session with each coach, and a test which scores their improvement in memory-recall produces the following results.

AQA D2 2013 January Q4

When investigating three network flow problems, a student finds:
1. a flow of 50 and a cut with capacity 50 ;
2. a flow of 35 and a cut with capacity 50 ;
3. a flow of 50 and a cut with capacity 35 . In each case, write down what the student can deduce about the maximum flow.
The diagram below shows a network. The numbers on the arcs represent the minimum and maximum flow along each arc respectively. By considering the flow at an appropriate vertex, explain why a flow is not possible through this network.
\includegraphics[max width=\textwidth, alt={}, center]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-10_1189_1559_1105_246}
(2 marks)

AQA D2 2013 January Q5

Display the following linear programming problem in a Simplex tableau.
Maximise $\quad P = x - 2 y + 3 z$
subject to $$\begin{array} { r } x + y + z \leqslant 16
x - 2 y + 2 z \leqslant 17
2 x - y + 2 z \leqslant 19 \end{array}$$ and $x \geqslant 0 , y \geqslant 0 , z \geqslant 0$.
1. The first pivot to be chosen is from the $z$-column. Identify the pivot and explain why this particular value is chosen.
2. Perform one iteration of the Simplex method.
1. Perform one further iteration.
2. Interpret the tableau that you obtained in part (c)(i) and state the values of your slack variables.

AQA D2 2013 January Q6

6 Kate and Pippa play a zero-sum game. The game is represented by the following pay-off matrix for Kate. \includegraphics[max width=\textwidth, alt={}, center]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-18_2482_1707_223_155}

AQA D2 2013 January Q7

7 The network below shows a system of one-way roads. The number on each edge represents the number of bags for recycling that can be collected by driving along that road. A collector is to drive from $A$ to $I$.
\includegraphics[max width=\textwidth, alt={}, center]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-20_867_1644_552_191}

Working backwards from $\boldsymbol { I }$, use dynamic programming to find the maximum number of bags that can be collected when driving from $A$ to $I$. You must complete the table opposite as your solution.
State the route that the collector should take in order to collect the maximum number of bags.
Stage State From Value
1 G I
H I
2

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

Stage	State	From	Calculation	Value
1	G	\(T\)
	H	\(T\)
	I	\(T\)

2	D	G
		\(H\)
	E	G
		\(H\)
		I
	\(F\)	\(H\)
		I

3

	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

Questions — AQA D2 (121 questions)

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