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AQA D1 2016 June Q3

7 marks Moderate -0.3

3 The network below shows vertices $A , B , C , D$ and $E$. The number on each edge shows the distance between vertices. \includegraphics[max width=\textwidth, alt={}, center]{fb95068f-f76d-492a-b385-bce17b26ae30-06_563_736_402_651}

1. In the case where $x = 8$, use Kruskal's algorithm to find a minimum spanning tree for the network. Write down the order in which you add edges to your minimum spanning tree.
2. Draw your minimum spanning tree.
3. Write down the length of your minimum spanning tree.
Alice draws the same network but changes the value of $x$. She correctly uses Kruskal's algorithm and edge $C D$ is included in her minimum spanning tree.
1. Explain why $x$ cannot be equal to 7 .
2. Write down an inequality for $x$.

AQA D1 2016 June Q4

11 marks Moderate -0.3

4 Amal delivers free advertiser magazines to all the houses in his village. The network shows the roads in his village. The number on each road shows the time, in minutes, that Amal takes to walk along that road. \includegraphics[max width=\textwidth, alt={}, center]{fb95068f-f76d-492a-b385-bce17b26ae30-08_846_1264_445_388}

Amal starts his delivery round from his house, at vertex $A$. He must walk along each road at least once.
1. Find the length of an optimal Chinese postman route around the village, starting and finishing at Amal's house.
2. State the number of times that Amal passes his friend Dipak's house, at vertex $D$.
Dipak offers to deliver the magazines while Amal is away on holiday. Dipak must walk along each road at least once. Assume that Dipak takes the same length of time as Amal to walk along each road.
1. Dipak can start his journey at any vertex and finish his journey at any vertex. Find the length of time for an optimal route for Dipak.
2. State the vertices at which Dipak could finish, in order to achieve his optimal route.
1. Find the length of time for an optimal route for Dipak, if, instead, he wants to finish at his house, at vertex $D$, and can start his journey at any other vertex.
2. State the start vertex.

AQA D1 2016 June Q5

8 marks Standard +0.8

5 A fair comes to town one year and sets up its rides in two large fields that are separated by a river. The diagram shows the ten main rides, at $A , B , C , \ldots , J$. The numbers on the edges represent the times, in minutes, it takes to walk between pairs of rides. A footbridge connects the rides at $D$ and $F$.

1. Use Dijkstra's algorithm on the diagram below to find the minimum time to walk from $A$ to each of the other main rides.
2. Write down the route corresponding to the minimum time to walk from $A$ to $G$.
The following year, the fair returns. In addition to the information shown on the diagram, another footbridge has been built to connect the rides at $E$ and $G$. This reduces the time taken to travel from $A$ to $G$, but the time taken to travel from $A$ to $J$ is not reduced. The time to walk across the footbridge from $E$ to $G$ is $x$ minutes, where $x$ is an integer. Find two inequalities for $x$ and hence state the value of $x$. \section*{Answer space for question 5}
1. (i) \includegraphics[max width=\textwidth, alt={}, center]{fb95068f-f76d-492a-b385-bce17b26ae30-12_659_1591_1692_223}

AQA D1 2016 June Q6

7 marks Moderate -0.5

6 A connected graph is semi-Eulerian if exactly two of its vertices are of odd degree.

A graph is drawn with 4 vertices and 7 edges. What is the sum of the degrees of the vertices?
Draw a simple semi-Eulerian graph with exactly 5 vertices and 5 edges, in which exactly one of the vertices has degree 4 .
Draw a simple semi-Eulerian graph with exactly 5 vertices that is also a tree.
A simple graph has 6 vertices. The graph has two vertices of degree 5 . Explain why the graph can have no vertex of degree 1.

AQA D1 2016 June Q7

17 marks Moderate -0.5

7 A company operates a steam railway between six stations. The minimum cost (in euros) of travelling between pairs of stations is shown in the table below.

On Figure 1 below, use Prim's algorithm, starting from $P$, to find a minimum spanning tree for the graph connecting $P , Q , R , S , T$ and $U$. State clearly the order in which you select the vertices and draw your minimum spanning tree. \section*{Question 7 continues on page 20} \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Figure 1}

	$\boldsymbol { P }$	$\boldsymbol { Q }$	$\boldsymbol { R }$	$\boldsymbol { S }$	$\boldsymbol { T }$	$\boldsymbol { U }$
$\boldsymbol { P }$	-	14	7	11	6	12
$\boldsymbol { Q }$	14	-	8	10	9	10
$\boldsymbol { R }$	7	8	-	12	13	15
$\boldsymbol { S }$	11	10	12	-	5	11
$\boldsymbol { T }$	6	9	13	5	-	10
$\boldsymbol { U }$	12	10	15	11	10	-

\end{table}

Another station, $V$, is opened. The minimum costs (in euros) of travelling to and from $V$ to each of the other stations are added to the table in part (a), as shown in Figure 2(i) below. Further copies of this table are shown in Figure 2(ii). Arjen is on holiday and he plans to visit each station. He intends to board a train at $V$ and visit all the other stations, once only, before returning to $V$.

By first removing $V$, obtain a lower bound for the minimum travelling cost of Arjen's tour. (You may use Figure 2(i) for your working.)
Use the nearest neighbour algorithm twice, starting each time from $V$, to find two different upper bounds for the minimum cost of Arjen's tour. State, with a reason, which of your two answers gives the better upper bound. (You may use Figure 2(ii) for your working.)

Hence find an optimal tour of the seven stations. Explain how you know that it is optimal. Answer space for question 7(b) \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Figure 2(ii)}

	$\boldsymbol { P }$	$Q$	$\boldsymbol { R }$	$\boldsymbol { S }$	$T$	$\boldsymbol { U }$	$V$
$\boldsymbol { P }$	-	14	7	11	6	12	15
$Q$	14	-	8	10	9	10	18
$\boldsymbol { R }$	7	8	-	12	13	15	14
$\boldsymbol { S }$	11	10	12	-	5	11	14
$T$	6	9	13	5	-	10	17
$\boldsymbol { U }$	12	10	15	11	10	-	12
$V$	15	18	14	14	17	12	-

\end{table}

AQA D1 2016 June Q8

13 marks Easy -1.2

8 Nerys runs a cake shop. In November and December she sells Christmas hampers. She makes up the hampers herself, in two sizes: Luxury and Special. Each day, Nerys prepares $x$ Luxury hampers and $y$ Special hampers.
It takes Nerys 10 minutes to prepare a Luxury hamper and 15 minutes to prepare a Special hamper. She has 6 hours available, each day, for preparing hampers. From past experience, Nerys knows that each day:

she will need to prepare at least 5 hampers of each size
she will prepare at most a total of 32 hampers
she will prepare at least twice as many Luxury hampers as Special hampers.

Each Luxury hamper that Nerys prepares makes her a profit of $\pounds 15$; each Special hamper makes a profit of $\pounds 20$. Nerys wishes to maximise her daily profit, $\pounds P$.

Show that $x$ and $y$ must satisfy the inequality $2 x + 3 y \leqslant 72$.
In addition to $x \geqslant 5$ and $y \geqslant 5$, write down two more inequalities that model the constraints above.
On the grid opposite draw a suitable diagram to enable this problem to be solved graphically. Indicate a feasible region and the direction of an objective line.
1. Use your diagram to find the number of each type of hamper that Nerys should prepare each day to achieve a maximum profit.
2. Calculate this profit.
  \includegraphics[max width=\textwidth, alt={}]{fb95068f-f76d-492a-b385-bce17b26ae30-27_2490_1719_217_150}
  \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}

AQA D2 Q4

Standard +0.3

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]{c18db720-6fe8-4e6c-bd0c-dc51cc341b47-005_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 Q7

Moderate -0.3

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]{c18db720-6fe8-4e6c-bd0c-dc51cc341b47-144_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.
1. Stage State From Value
  1 G I
  H I
  2

AQA D2 Q8

Moderate -0.3

8 The network below represents a system of pipes. The capacity of each pipe, in litres per second, is indicated on the corresponding edge. \includegraphics[max width=\textwidth, alt={}, center]{c18db720-6fe8-4e6c-bd0c-dc51cc341b47-146_743_977_404_536}

Find the maximum flow along each of the routes $A B E H , A C F H$ and $A D G H$ and enter their values in the table on Figure 4 opposite.
1. Taking your answers to part (a) 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 routes 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 opposite, illustrate a possible flow along each edge corresponding to this maximum flow.
Confirm that you have a maximum flow by finding a cut of the same value. List the edges of your cut. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Figure 4}
Route Flow
$A B E H$
$A C F H$
$A D G H$
\end{table} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{c18db720-6fe8-4e6c-bd0c-dc51cc341b47-147_746_972_397_845}
\end{figure} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{c18db720-6fe8-4e6c-bd0c-dc51cc341b47-147_739_971_1311_539}
\end{figure}

AQA D2 2006 January Q1

9 marks Moderate -0.5

1 Five trainers, Ali, Bo, Chas, Dee and Eve, held an initial training session with each of four teams over an assault course. The completion times in minutes are recorded below.

	Ali	Bo	Chas	Dee	Eve
Team 1	16	19	18	25	24
Team 2	22	21	20	26	25
Team 3	21	22	23	21	24
Team 4	20	21	21	23	20

Each of the four teams is to be allocated a trainer and the overall time for the four teams is to be minimised. No trainer can train more than one team.

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 trainers should be allocated to which team. State the minimum total training time for the four teams using this matching.

AQA D2 2006 January Q2

9 marks Moderate -0.8

2 A manufacturing company is planning to build three new machines, $A , B$ and $C$, at the rate of one per month. The order in which they are built is a matter of choice, but the profits will vary according to the number of workers available and the suppliers' costs. The expected profits in thousands of pounds are given in the table.

\multirow[b]{2}{*}{Month}	\multirow[b]{2}{*}{Already built}	Profit (in units of £1000)
		$\boldsymbol { A }$	$\boldsymbol { B }$	$\boldsymbol { C }$
1	-	52	47	48
\multirow[t]{3}{*}{2}	A	-	58	54
	B	70	-	54
	$\boldsymbol { C }$	68	63	-
\multirow[t]{3}{*}{3}	$\boldsymbol { A }$ and $\boldsymbol { B }$	-	-	64
	$\boldsymbol { A }$ and $\boldsymbol { C }$	-	67	-
	$\boldsymbol { B }$ and $\boldsymbol { C }$	69	-	-

Draw a labelled network such that the most profitable order of manufacture corresponds to the longest path within that network.
Use dynamic programming to determine the order of manufacture that maximises the total profit, and state this maximum profit.

AQA D2 2006 January Q3

18 marks Moderate -0.3

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

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

14 marks Standard +0.3

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

13 marks Standard +0.3

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

11 marks Moderate -0.8

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.
  Surname Other Names
  Centre Number Candidate Number
  Candidate Signature
  \section*{General Certificate of Education January 2006
  Advanced Level Examination} \section*{MATHEMATICS
  Unit Decision 2} MD02 \section*{Insert} Wednesday 18 January 20061.30 pm to 3.00 pm Insert for use in Questions 3 and 4.
  Fill in the boxes at the top of this page.
  Fasten this insert securely to your answer book.

AQA D2 2007 January Q1

11 marks Easy -1.2

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

13 marks Moderate -0.5

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

13 marks Standard +0.8

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

13 marks Moderate -0.8

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

10 marks Standard +0.3

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

15 marks Moderate -0.5

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.
  Surname Other Names
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  \section*{General Certificate of Education
  January 2007
  Advanced Level Examination} \section*{MATHEMATICS
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AQA D2 2008 January Q1

15 marks Moderate -0.3

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

11 marks Standard +0.3

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

13 marks Standard +0.3

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

14 marks Standard +0.3

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.

Route	Flow
\(A B E H\)
\(A C F H\)
\(A D G H\)

\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

	\(\boldsymbol { P }\)	\(\boldsymbol { Q }\)	\(\boldsymbol { R }\)	\(\boldsymbol { S }\)	\(\boldsymbol { T }\)	\(\boldsymbol { U }\)
\(\boldsymbol { P }\)	-	14	7	11	6	12
\(\boldsymbol { Q }\)	14	-	8	10	9	10
\(\boldsymbol { R }\)	7	8	-	12	13	15
\(\boldsymbol { S }\)	11	10	12	-	5	11
\(\boldsymbol { T }\)	6	9	13	5	-	10
\(\boldsymbol { U }\)	12	10	15	11	10	-

	\(\boldsymbol { P }\)	\(Q\)	\(\boldsymbol { R }\)	\(\boldsymbol { S }\)	\(T\)	\(\boldsymbol { U }\)	\(V\)
\(\boldsymbol { P }\)	-	14	7	11	6	12	15
\(Q\)	14	-	8	10	9	10	18
\(\boldsymbol { R }\)	7	8	-	12	13	15	14
\(\boldsymbol { S }\)	11	10	12	-	5	11	14
\(T\)	6	9	13	5	-	10	17
\(\boldsymbol { U }\)	12	10	15	11	10	-	12
\(V\)	15	18	14	14	17	12	-

		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

Stage	State	From	Value
1	G	I
	H	I

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

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