Questions D2 - A-Level Maths

Edexcel D2 2019 June Q7

7. A company has purchased a plot of land and has decided to build four holiday homes, A, B, C and D, on the land at the rate of one home per year. The company expects that the construction costs each year will vary, depending on which houses have already been constructed and which house is currently under construction. The expected construction costs, in thousands of pounds, are shown in the table below.

\cline { 2 - 7 } \multicolumn{1}{c\|}{}	A	B	C	D	E	F
A	-	53	47	39	35	40
B	53	-	32	46	41	43
C	47	32	-	51	47	37
D	39	46	51	-	36	49
E	35	41	47	36	-	42
F	40	43	37	49	42	-

\begin{table}[h]

	1	2	3	4	Supply
A	17	20	23	14	25
B	16	15	19	22	29
C	19	14	11	15	32
Demand	28	17	23	18

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

\end{table} 2. You may not need to use all of these tables \begin{table}[h]

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

	1	2	3	4	Supply
A					25
B					29
C					32
Demand	28	17	23	18

\end{table}

	1	2	3	4	Supply
A					25
B					29
C					32
Demand	28	17	23	18

	1	2	3	4	Supply
A					25
B					29
C					32
Demand	28	17	23	18

	1	2	3	4	Supply
A					25
B					29
C					32
Demand	28	17	23	18

	1	2	3	4	Supply
A					25
B					29
C					32
Demand	28	17	23	18

	1	2	3	4	Supply
A					25
B					29
C					32
Demand	28	17	23	18

	1	2	3	4	Supply
A					25
B					29
C					32
Demand	28	17	23	18

	1	2	3	4	Supply
A					25
B					29
C					32
Demand	28	17	23	18

	1	2	3	4	Supply
A					25
B					29
C					32
Demand	28	17	23	18

3.

	A	B	C	D	E
Frank	5	0	7	3	4
Gill	5	3	8	10	1
Harry	4	3	7	9	0
Imogen	6	3	6	5	4
Jiao	0	2	7	3	2

You may not need to use all of these tables

	$A$	$B$	$C$	$D$	$E$
$F$
$G$
$H$
$I$
$J$

	A	B	C	D	E
F
G
H
I
J

	$A$	$B$	$C$	$D$	$E$
$F$
$G$
$H$
$I$
$J$

	$A$	$B$	$C$	$D$	$E$
$F$
$G$
$H$
$I$
$J$

	$A$	$B$	$C$	$D$	$E$
$F$
$G$
$H$
$I$
$J$

	$A$	$B$	$C$	$D$	$E$
$F$
$G$
$H$
$I$
$J$

	$A$	$B$	$C$	$D$	$E$
$F$
$G$
$H$
$I$
$J$

	Stephen plays 1	Stephen plays 2	Stephen plays 3
Eugene plays 1	4	5	0
Eugene plays 2	-2	1	1
Eugene plays 3	-3	-4	3

4. 5. (a)

b.v.	$x$	$y$	$z$	$r$	$s$	$t$	Value
							30
							60
							80
							0

You may not need to use all of these tableaux

b.v.	$x$	$y$	$z$	$r$	$s$	$t$	Value	Row Ops



$P$

b.v.	$x$	$y$	$z$	$r$	$s$	$t$	Value	Row Ops



$P$

b.v.	$x$	$y$	$z$	$r$	$s$	$t$	Value	Row Ops



$P$

6. (a) Value of initial flow
(b) and (c)
\includegraphics[max width=\textwidth, alt={}, center]{e0c66144-9e34-42fc-9f40-a87a49331483-20_725_1251_404_349} \section*{Diagram 1}

\includegraphics[max width=\textwidth, alt={}]{e0c66144-9e34-42fc-9f40-a87a49331483-20_1070_1264_1322_349}

\section*{Diagram 2} (d)
(e)
\includegraphics[max width=\textwidth, alt={}, center]{e0c66144-9e34-42fc-9f40-a87a49331483-21_714_1385_1306_283} \section*{Diagram 3} (f)
7. (a)

Stage	State	Action	Dest.	Value
1	ABC	D	ABCD	65*

Stage	State	Action	Dest.	Value

\includegraphics[max width=\textwidth, alt={}]{e0c66144-9e34-42fc-9f40-a87a49331483-24_2642_1833_118_118}

OCR D2 2006 June Q1

1 The network represents a system of pipes along which fluid can flow from $S$ to $T$. The values on the arcs are lower and upper capacities in litres per second.
\includegraphics[max width=\textwidth, alt={}, center]{e879b1f5-edc7-4819-80be-2a90dbf3d451-02_696_1292_376_424}

Calculate the capacity of the cut with $\mathrm { X } = \{ S , A , B , C \} , \mathrm { Y } = \{ D , E , F , G , H , I , T \}$.
Show that the capacity of the cut $\alpha$, shown on the diagram, is 12 litres per second and calculate the minimum flow across the cut $\alpha$, from $S$ to $T$, (without regard to the remainder of the diagram).
Explain why the arc SC must have at least 5 litres per second flowing through it. By considering the flow through $A$, explain why $A D$ cannot be full to capacity.
Show that it is possible for 11 litres per second to flow through the system.
From your previous answers, what can be deduced about the maximum flow through the system?

OCR D2 2006 June Q2

2 A delivery company needs to transport heavy loads from its warehouse to a ferry port. Each of the roads that it can use has a bridge with a maximum weight limit. The directed network below represents the roads that can be used to get from the warehouse to the ferry port. Road junctions are labelled with (stage; state) labels. The weights on the arcs represent weight limits in tonnes.
\includegraphics[max width=\textwidth, alt={}, center]{e879b1f5-edc7-4819-80be-2a90dbf3d451-03_896_1561_468_292}

Explain what a maximin route is.
Set up a dynamic programming tabulation, working backwards from stage 1, to find the two maximin routes through the network. What is the maximum load that can be transported in one journey from the warehouse to the ferry port?
A new road is opened connecting ( $2 ; 0$ ) and ( $2 ; 1$ ). There is no bridge on this road so it does not restrict the weight of the load that can be carried. Using the new road, what is the maximum load that can be transported in one journey from the warehouse to the ferry port? State the route that the delivery company should use. (Do not attempt to apply dynamic programming in this part.)

OCR D2 2006 June Q3

3 Rose and Colin repeatedly play a zero-sum game. The pay-off matrix shows the number of points won by Rose for each combination of strategies.

\multirow{6}{*}{Rose's strategy}	Colin's strategy
		$W$	$X$	$Y$	$Z$
	$A$	-1	4	-3	2
	$B$	5	-2	5	6
	C	3	-4	-1	0
	$D$	-5	6	-4	-2

What is the greatest number of points that Colin can win when Rose plays strategy $A$ and which strategy does Colin need to play to achieve this?
Show that strategy $B$ dominates strategy $C$ and also that strategy $Y$ dominates strategy $Z$. Hence reduce the game to a $3 \times 3$ pay-off matrix.
Find the play-safe strategy for each player on the reduced game. Is the game stable? Rose makes a random choice between the strategies, choosing strategy $A$ with probability $p _ { 1 }$, strategy $B$ with probability $p _ { 2 }$ and strategy $D$ with probability $p _ { 3 }$. She formulates the following LP problem to be solved using the Simplex algorithm: $$\begin{array} { l l } \text { maximise } & M = m - 5 ,
\text { subject to } & m \leqslant 4 p _ { 1 } + 10 p _ { 2 } ,
& m \leqslant 9 p _ { 1 } + 3 p _ { 2 } + 11 p _ { 3 } ,
& m \leqslant 2 p _ { 1 } + 10 p _ { 2 } + p _ { 3 } ,
& p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1 ,
\text { and } & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , m \geqslant 0 . \end{array}$$ (You are not required to solve this problem.)
Explain how $9 p _ { 1 } + 3 p _ { 2 } + 11 p _ { 3 }$ was obtained. A computer gives the solution to the LP problem as $p _ { 1 } = \frac { 7 } { 48 } , p _ { 2 } = \frac { 27 } { 48 } , p _ { 3 } = \frac { 14 } { 48 }$.
Calculate the value of $M$ at this solution.

OCR D2 2006 June Q4

4 Answer this question on the insert provided. The diagram shows an activity network for a project. The table lists the durations of the activities (in hours).
\includegraphics[max width=\textwidth, alt={}, center]{e879b1f5-edc7-4819-80be-2a90dbf3d451-05_680_1125_424_244} (ii) Key: \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e879b1f5-edc7-4819-80be-2a90dbf3d451-10_154_225_1119_1509} \captionsetup{labelformat=empty} \caption{Early event Late event time time}

\end{figure}

\includegraphics[max width=\textwidth, alt={}]{e879b1f5-edc7-4819-80be-2a90dbf3d451-10_762_1371_1409_427}

Minimum completion time = $\_\_\_\_$ hours Critical activities: $\_\_\_\_$
(iii) $\_\_\_\_$
(iv)
\includegraphics[max width=\textwidth, alt={}, center]{e879b1f5-edc7-4819-80be-2a90dbf3d451-11_513_1189_543_520} Number of workers required = $\_\_\_\_$

(i)	$A \bullet$
$B \bullet$	$\bullet J$
$C \bullet$	$\bullet K$
$D \bullet$	$\bullet L$
$E \bullet$	$\bullet M$
$F \bullet$	$\bullet N$

(ii) $\_\_\_\_$
(iii)

	$J$	$K$	$L$	$M$	$N$	$O$
$A$	2	5	2	2	5	2
$B$	2	5	2	0	5	5
$C$	5	0	5	5	2	2
$D$
$E$
$F$

Answer part (iv) in your answer booklet.

OCR D2 2010 June Q1

1 The famous fictional detective Agatha Parrot is investigating a murder. She has identified six suspects: Mrs Lemon $( L )$, Prof Mulberry $( M )$, Mr Nutmeg $( N )$, Miss Olive $( O )$, Capt Peach $( P )$ and Rev Quince $( Q )$. The table shows the weapons that could have been used by each suspect.

Suspect
		$L$	M	$N$	$O$	$P$	$Q$
Axe handle	A		✓	✓			✓
Broomstick	$B$		✓		✓
Drainpipe	D	✓		✓
Fence post	$F$			✓	✓
Golf club	$G$	✓				✓	✓
Hammer	$H$	✓			✓	✓

Draw a bipartite graph to represent this information. Put the weapons on the left-hand side and the suspects on the right-hand side. Agatha Parrot is convinced that all six suspects were involved, and that each used a different weapon. She originally thinks that the axe handle was used by Prof Mulberry, the broomstick by Miss Olive, the drainpipe by Mrs Lemon, the fence post by Mr Nutmeg and the golf club by Capt Peach. However, this would leave the hammer for Rev Quince, which is not a possible pairing.
Draw a second bipartite graph to show this incomplete matching.
Construct the shortest possible alternating path from $H$ to $Q$ and hence find a complete matching. Write down which suspect used each weapon.
Find a different complete matching in which none of the suspects used the same weapon as in the matching from part (iii).

OCR D2 2010 June Q2

2 In an investigation into a burglary, Agatha has five suspects who were all known to have been near the scene of the crime, each at a different time of the day. She collects evidence from witnesses and draws up a table showing the number of witnesses claiming sight of each suspect near the scene of the crime at each possible time. Suspect \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Time}

		1 pm	2 pm	3 pm	4 pm	5 pm
Mrs Rowan	$R$	3	4	2	7	1
Dr Silverbirch	$S$	5	10	6	6	6
Mr Thorn	$T$	4	7	3	5	3
Ms Willow	$W$	6	8	4	8	3
Sgt Yew	$Y$	8	8	7	4	3

\end{table}

Use the Hungarian algorithm on a suitably modified table, reducing rows first, to find the matchings for which the total number of claimed sightings is maximised. Show your working clearly. Write down the resulting matchings between the suspects and the times. Further enquiries show that the burglary took place at 5 pm , and that Dr Silverbirch was not the burglar.
Who should Agatha suspect?

OCR D2 2010 June Q3

3

Set up a dynamic programming tabulation to find the minimum weight route from ( $0 ; 0$ ) to ( $4 ; 0$ ) on the following directed network.
\includegraphics[max width=\textwidth, alt={}, center]{406831f5-74a3-415e-8849-2c381bfe47f4-03_707_1342_1594_443} Give the route and its total weight.
Explain carefully how the route is obtained directly from the values in the table, without referring to the network.

OCR D2 2010 June Q4

4 Euan and Wai Mai play a zero-sum game. Each is trying to maximise the total number of points that they score in many repeats of the game. The table shows the number of points that Euan scores for each combination of strategies.

	Wai Mai
\cline { 2 - 5 }	$X$	$Y$	$Z$
$A$	2	- 5	3
\cline { 2 - 5 } $E u a n$	- 1	- 3	4
\cline { 1 - 5 } $C$	3	- 5	2
$D$	3	- 2	- 1

Explain what the term 'zero-sum game' means.
How many points does Wai Mai score if she chooses $X$ and Euan chooses $A$ ?
Why should Wai Mai never choose strategy $Z$ ?
Delete the column for $Z$ and find the play-safe strategy for Euan and the play-safe strategy for Wai Mai on the table that remains. Is the resulting game stable or not? State how you know. The value 3 in the cell corresponding to Euan choosing $D$ and Wai Mai choosing $X$ is changed to - 5 ; otherwise the table is unchanged. Wai Mai decides that she will choose her strategy by making a random choice between $X$ and $Y$, choosing $X$ with probability $p$ and $Y$ with probability $1 - p$.
Write down and simplify an expression for the expected score for Wai Mai when Euan chooses each of his four strategies.
Using graph paper, draw a graph showing Wai Mai's expected score against $p$ for each of Euan's four strategies and hence calculate the optimum value of $p$.

OCR D2 2010 June Q5

5 Answer this question on the insert provided. The network represents a system of irrigation channels along which water can flow. The weights on the arcs represent the maximum flow in litres per second.
\includegraphics[max width=\textwidth, alt={}, center]{406831f5-74a3-415e-8849-2c381bfe47f4-05_597_1553_479_296}

Calculate the capacity of the cut that separates $\{ S , B , C , E \}$ from $\{ A , D , F , G , H , T \}$.
Explain why neither arc $S C$ nor arc $B C$ can be full to capacity. Explain why the arcs $E F$ and $E H$ cannot both be full to capacity. Hence find the maximum flow along arc $H T$. When arc $H T$ carries its maximum flow, what is the flow along arc $H G$ ?
Show a flow of 58 litres per second on the diagram in the insert, and find a cut of capacity 58. The direction of flow in $H G$ is reversed.
Use the diagram in the insert to show the excess capacities and potential backflows for your flow from part (iii) in this case.
Without augmenting the labels from part (iv), write down flow augmenting routes to enable an additional 2 litres per second to flow from $S$ to $T$.
Show your augmented flow on the diagram in the insert. Explain how you know that this flow is maximal.

OCR D2 2010 June Q6

6 Answer parts (i), (ii) and (iii) of this question on the insert provided. The activity network for a project is shown below. The durations are in minutes. The events are numbered 1, 2, 3, etc. for reference.
\includegraphics[max width=\textwidth, alt={}, center]{406831f5-74a3-415e-8849-2c381bfe47f4-06_747_1249_482_447}

Complete the table in the insert to show the immediate predecessors for each activity.
Explain why the dummy activity is needed between event 2 and event 3, and why the dummy activity is needed between event 4 and event 5 .
Carry out a forward pass to find the early event times and a backward pass to find the late event times. Record your early event times and late event times in the table in the insert. Write down the minimum project completion time and the critical activities. Suppose that the duration of activity $K$ changes to $x$ minutes.
Find, in terms of $x$, expressions for the early event time and the late event time for event 9 .
Find the maximum duration of activity $K$ that will not affect the minimum project completion time found in part (iii). \section*{ADVANCED GCE
MATHEMATICS} Decision Mathematics 2
INSERT for Questions 5 and 6
Dummy activity is needed between event 2 and event 3 because $\_\_\_\_$
Dummy activity is needed between event 4 and event 5 because $\_\_\_\_$
Event 1 2 3 4 5 6 7 8 9 10
Early event time
Late event time
Minimum project completion time = $\_\_\_\_$ minutes Critical activities: $\_\_\_\_$ \section*{Answer part (iv) and part (v) in your answer booklet.} OCR
RECOGNISING ACHIEVEMENT

OCR MEI D2 Q1

1 The switching circuit in Fig. 1.1 shows switches, $s$ for a car's sidelights, $h$ for its dipped headlights and f for its high-intensity rear foglights. It also shows the three sets of lights. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{83d00f1f-25e5-4a26-bac5-dc2baf965438-02_284_917_404_580} \captionsetup{labelformat=empty} \caption{Fig. 1.1}

\end{figure} (Note: $s$ and $h$ are each "ganged" switches. A ganged switch consists of two connected switches sharing a single switch control, so that both are either on or off together.)

1. Describe in words the conditions under which the foglights will come on. Fig. 1.2 shows a combinatorial circuit. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{83d00f1f-25e5-4a26-bac5-dc2baf965438-02_367_1235_1183_431} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
  \end{figure}
2. Write the output in terms of a Boolean expression involving $s , h$ and $f$.
3. Use a truth table to prove that $\mathrm { s } \wedge \mathrm { h } \wedge \mathrm { f } = \sim ( \sim \mathrm { s } \vee \sim \mathrm { h } ) \wedge \mathrm { f }$.
A car's first gear can be engaged ( g ) if either both the road speed is low ( r ) and the clutch is depressed ( d ), or if both the road speed is low ( r ) and the engine speed is the correct multiple of the road speed (m).
1. Draw a switching circuit to represent the conditions under which first gear can be engaged. Use two ganged switches to represent $r$, and single switches to represent each of $\mathrm { d } , \mathrm { m }$ and g .
2. Draw a combinatorial circuit to represent the Boolean expression $\mathrm { r } \wedge ( \mathrm { d } \vee \mathrm { m } ) \wedge \mathrm { g }$.
3. Use Boolean algebra to prove that $\mathrm { r } \wedge ( \mathrm { d } \vee \mathrm { m } ) \wedge \mathrm { g } = ( ( \mathrm { r } \wedge \mathrm { d } ) \vee ( \mathrm { r } \wedge \mathrm { m } ) ) \wedge \mathrm { g }$.
4. Draw another switching circuit to represent the conditions under which first gear can be selected, but without using a ganged switch.

OCR MEI D2 Q4

$\mathbf { 4 }$ & 9 & 7 & 6 & 12
\hline \end{tabular} \end{center} Route Matrix

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	$\mathbf { 1 }$	$\mathbf { 2 }$	$\mathbf { 3 }$	$\mathbf { 4 }$
$\mathbf { 1 }$	3	2	3	3
$\mathbf { 2 }$	1	1	3	3
$\mathbf { 3 }$	1	2	1	4
$\mathbf { 4 }$	3	3	3	3

Draw the complete network of shortest distances.
(ii) Starting at vertex 1, apply the nearest neighbour algorithm to the complete network of shortest distances to find a Hamilton cycle. Give the length of your cycle and interpret it in the original network.
(iii) By temporarily deleting vertex $\mathbf { 1 }$ and its connecting arcs from the complete network of shortest distances, find a lower bound for the solution to the Travelling Salesperson's Problem in that network. Say what this implies in the original network.
(b) Solve the route inspection problem in the original network, and say how you can be sure that your solution is optimal. 4 A factory's output includes three products. To manufacture a tonne of product $\mathrm { A } , 3$ tonnes of water are needed. Product B needs 2 tonnes of water per tonne produced, and product C needs 5 tonnes of water per tonne produced. Product A uses 5 hours of labour per tonne produced, product B uses 6 hours and product C uses 2 hours. There are 60 tonnes of water and 50 hours of labour available for tomorrow's production.
(i) Formulate a linear programming problem to maximise production within the given constraints.
(ii) Use the simplex algorithm to solve your LP, pivoting first on your column relating to product C.
(iii) An extra constraint is imposed by a contract to supply at least 8 tonnes of A . Use either two stage simplex or the big M method to solve this revised problem. 4772
Decision Mathematics 2 \section*{Candidates answer on the Answer Booklet} \section*{OCR Supplied Materials:}

Answer Booklet (8 pages)
Graph paper
MEI Examination Formulae and Tables (MF2)

\section*{Other Materials Required:} None \section*{Wednesday 17 June 2009
Morning} Duration: 1 hour 30 minutes
|||||||||||||||||||||||

Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces provided on the Answer Booklet.
Use black ink. Pencil may be used for graphs and diagrams only.
Read each question carefully and make sure that you know what you have to do before starting your answer.
Answer all the questions.
You are permitted to use a graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.
Do not write in the bar codes.

The number of marks is given in brackets [ ] at the end of each question or part question.
You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
The total number of marks for this paper is 72.
This document consists of $\mathbf { 4 }$ pages. Any blank pages are indicated.

1 (a) The following was said in a charity appeal on Radio 4 in October 2006.
"It is hard to underestimate the effect that your contribution will make."
Rewrite the comment more simply in your own words without changing its meaning.
(b) A machine has three components, A, B and C, each of which is either active or inactive.

The machine is active if A and B are both active.
The machine is active if A is inactive and C is active.
The machine is active if B is inactive and C is active.
Otherwise the machine is inactive.

The states (active or inactive) of the components and the machine are to be modelled by a combinatorial circuit in which "active" is represented by "true" and "inactive" is represented by "false". Draw such a circuit.
(c) Construct a truth table to show the following. $$[ ( ( \mathrm { a } \wedge \mathrm {~b} ) \vee ( ( \sim \mathrm { a } ) \wedge \mathrm { c } ) ) \vee ( ( \sim \mathrm { b } ) \wedge \mathrm { c } ) ] \Leftrightarrow \sim [ ( ( \sim \mathrm { a } ) \wedge ( \sim \mathrm { c } ) ) \vee ( ( \sim \mathrm { b } ) \wedge ( \sim \mathrm { c } ) ) ]$$ 2 Zoe is preparing for a Decision Maths test on two topics, Decision Analysis (D) and Simplex (S). She has to decide whether to devote her final revision session to D or to S . There will be two questions in the test, one on D and one on S . One will be worth 60 marks and the other will be worth 40 marks. Historically there is a $50 \%$ chance of each possibility. Zoe is better at $D$ than at $S$. If her final revision session is on $D$ then she would expect to score $80 \%$ of the $D$ marks and $50 \%$ of the $S$ marks. If her final session is on $S$ then she would expect to score $70 \%$ of the S marks and $60 \%$ of the D marks.
(i) Compute Zoe's expected mark under each of the four possible circumstances, i.e. Zoe revising $D$ and the D question being worth 60 marks, etc.
(ii) Draw a decision tree for Zoe. Michael claims some expertise in forecasting which question will be worth 60 marks. When he forecasts that it will be the D question which is worth 60 , then there is a $70 \%$ chance that the D question will be worth 60 . Similarly, when he forecasts that it will be the S question which is worth 60 , then there is a $70 \%$ chance that the S question will be worth 60 . He is equally likely to forecast that the D or the S question will be worth 60.
(iii) Draw a decision tree to find the worth to Zoe of Michael's advice. 3 A farmer has 40 acres of land. Four crops, A, B, C and D are available.
Crop A will return a profit of $\pounds 50$ per acre. Crop B will return a profit of $\pounds 40$ per acre.
Crop C will return a profit of $\pounds 40$ per acre. Crop D will return a profit of $\pounds 30$ per acre.
The total number of acres used for crops A and B must not be greater than the total number used for crops C and D. The farmer formulates this problem as:
Maximise $\quad 50 a + 40 b + 40 c + 30 d$,
subject to $\quad a + b \leqslant 20$,
$a + b + c + d \leqslant 40$.
(i) Explain what the variables $a , b , c$ and $d$ represent. Explain how the first inequality was obtained.
Explain why expressing the constraint on the total area of land as an inequality will lead to a solution in which all of the land is used.
(ii) Solve the problem using the simplex algorithm. Suppose now that the farmer had formulated the problem as:
Maximise $\quad 50 a + 40 b + 40 c + 30 d$,
subject to $\quad a + b \leqslant 20$,
$a + b + c + d = 40$.
(iii) Show how to adapt this problem for solution either by the two-stage simplex method or the big-M method. In either case you should show the initial tableau and describe what has to be done next. You should not attempt to solve the problem. 4 The diagram shows routes connecting five cities. Lengths are in km .
\includegraphics[max width=\textwidth, alt={}, center]{83d00f1f-25e5-4a26-bac5-dc2baf965438-21_442_995_319_534}
(i) Produce the initial matrices for an application of Floyd's algorithm to find the complete network of shortest distances between the five cities. The following are the distance and route matrices after the third iteration of Floyd's algorithm. \begin{center} \begin{tabular}{ | c | c | c | c | c | c | } \cline { 2 - 6 } \multicolumn{1}{c|}{} & $\mathbf { 1 }$ & $\mathbf { 2 }$ & $\mathbf { 3 }$ & $\mathbf { 4 }$ & $\mathbf { 5 }$
\hline $\mathbf { 1 }$ & 44 & 22 & 42 & 15 & 15
\hline $\mathbf { 2 }$ & 22 & 44 & 20 & 5 & 23
\hline $\mathbf { 3 }$ & 42 & 20 & 40 & 25 & 43
\hline $\mathbf { 4 }$ & 15 & 5 & 25 & 10 & 16
\hline

OCR MEI D2 Q5

$\mathbf { 5 }$ & 15 & 23 & 43 & 16 & 30
\hline \end{tabular} \end{center}

\cline { 2 - 6 } \multicolumn{1}{c\|}{}	$\mathbf { 1 }$	$\mathbf { 2 }$	$\mathbf { 3 }$	$\mathbf { 4 }$	$\mathbf { 5 }$
$\mathbf { 1 }$	2	2	2	4	5
$\mathbf { 2 }$	1	1	3	4	5
$\mathbf { 3 }$	2	2	2	2	2
$\mathbf { 4 }$	1	2	2	2	5
$\mathbf { 5 }$	1	2	2	4	1

(ii) Perform the fourth iteration. There are no changes on the fifth iteration, so your answer to part (ii) should give the complete network of shortest distances.
(iii) Use your matrices to find the shortest distance and route from vertex $\mathbf { 3 }$ to vertex $\mathbf { 1 }$, and explain how you do it.
(iv) Draw the complete network of shortest distances, not including the loops.
(v) Apply the nearest neighbour algorithm to your network in part (iv), starting at vertex 2. Give the length of the Hamilton cycle that is produced. Interpret the Hamilton cycle in terms of the original diagram and state what the algorithm has achieved. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations, is given to all schools that receive assessment material and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1PB.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. \section*{ADVANCED GCE
MATHEMATICS (MEI)} Decision Mathematics 2 \section*{Candidates answer on the Answer Booklet} \section*{OCR Supplied Materials:}

8 page Answer Booklet
MEI Examination Formulae and Tables (MF2)
Graph paper

\section*{Other Materials Required:}

Scientific or graphical calculator

\section*{Tuesday 22 June 2010 Afternoon} Duration: 1 hour 30 minutes $\| \| \| \| \| \| \| \| \| \| \| \| \|$

Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces provided on the Answer Booklet.
Use black ink. Pencil may be used for graphs and diagrams only.
Read each question carefully and make sure that you know what you have to do before starting your answer.
Answer all the questions.
Do not write in the bar codes.
You are permitted to use a graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.

The number of marks is given in brackets [ ] at the end of each question or part question.
You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
The total number of marks for this paper is $\mathbf { 7 2 }$.
This document consists of $\mathbf { 8 }$ pages. Any blank pages are indicated.

1

Mickey ate the last of the cheese. Minnie was put out at this. Mickey's defence was "There wasn't enough left not to eat it all". Let "c" represent "there is enough cheese for two" and "e" represent "one person can eat all of the cheese".
1. Which of the following best captures Mickey's argument? $$\mathrm { c } \Rightarrow \mathrm { e } \quad \mathrm { c } \Rightarrow \sim \mathrm { e } \quad \sim _ { \mathrm { c } } \Rightarrow \mathrm { e } \quad \sim \mathrm { c } \Rightarrow \sim \mathrm { e }$$ In the ensuing argument Minnie concedes that if there's a lot left then one should not eat it all, but argues that this is not an excuse for Mickey having eaten it all when there was not a lot left.
2. Prove that Minnie is right by writing down a line of a truth table which shows that $$( c \Rightarrow \sim e ) \Leftrightarrow ( \sim c \Rightarrow e )$$ is false.
  (You may produce the whole table if you wish, but you need to indicate a specific line of the table.)
1. Show that the following combinatorial circuit is modelling an implication statement. Say what that statement is, and prove that the circuit and the statement are equivalent.
  \includegraphics[max width=\textwidth, alt={}, center]{83d00f1f-25e5-4a26-bac5-dc2baf965438-23_188_533_1272_767}
2. Express the following combinatorial circuit as an equivalent implication statement.
  \includegraphics[max width=\textwidth, alt={}, center]{83d00f1f-25e5-4a26-bac5-dc2baf965438-23_314_835_1599_616}
3. Explain why $( \sim \mathrm { p } \wedge \sim \mathrm { q } ) \Rightarrow \mathrm { r }$, together with $\sim \mathrm { r }$ and $\sim \mathrm { q }$, give p . 2 The network is a representation of a show garden. The weights on the arcs give the times in minutes to walk between the six features represented by the vertices, where paths exist.
  \includegraphics[max width=\textwidth, alt={}, center]{83d00f1f-25e5-4a26-bac5-dc2baf965438-24_483_985_342_539}
4. Why might it be that the time taken to walk from vertex $\mathbf { 2 }$ to vertex $\mathbf { 3 }$ via vertex $\mathbf { 4 }$ is less than the time taken by the direct route, i.e. the route from $\mathbf { 2 }$ to $\mathbf { 3 }$ which does not pass through any other vertices? The matrices shown below are the results of the first iteration of Floyd's algorithm when applied to the network. \begin{center} \begin{tabular}{ | c | c | c | c | c | c | c | } \cline { 2 - 7 } \multicolumn{1}{c|}{} & $\mathbf { 1 }$ & $\mathbf { 2 }$ & $\mathbf { 3 }$ & $\mathbf { 4 }$ & $\mathbf { 5 }$ & $\mathbf { 6 }$
  \hline $\mathbf { 1 }$ & $\infty$ & 15 & $\infty$ & $\infty$ & 7 & 8
  \hline $\mathbf { 2 }$ & 15 & 30 & 6 & 2 & 6 & 23
  \hline $\mathbf { 3 }$ & $\infty$ & 6 & $\infty$ & 3 & $\infty$ & $\infty$
  \hline $\mathbf { 4 }$ & $\infty$ & 2 & 3 & $\infty$ & 10 & 17
  \hline $\mathbf { 5 }$ & 7 & 6 & $\infty$ & 10 & 14 & 8
  \hline

OCR MEI D2 Q6

$\mathbf { 6 }$ & 8 & 23 & $\infty$ & 17 & 8 & 16
\hline \end{tabular} \end{center}

The teacher suspects that a pupil has copied work from the internet. For each box, state whether the teacher should tick the box or not.
The teacher has no suspicions about the work of another pupil, and has no information about how the work was produced. Which boxes should she tick?
Explain why the teacher must always tick at least one box.
(b) Angus, the ski instructor, says that the class will have to have lunch in Italy tomorrow if it is foggy or if the top ski lift is not working. On the next morning Chloe, one of Angus's students, says that it is not foggy, so they can have lunch in Switzerland. Produce a line of a truth table which shows that Chloe's deduction is incorrect. You may produce a complete truth table if you wish, but you must indicate a row which shows that Chloe's deduction is incorrect.
(c) You are given that the following two statements are true. $$\begin{aligned} & ( \mathrm { X } \vee \sim \mathrm { Y } ) \Rightarrow \mathrm { Z }
& \sim \mathrm { Z } \end{aligned}$$ Use Boolean algebra to show that Y is true. 2 Adrian is considering selling his house and renting a flat.
Adrian still owes $\pounds 150000$ on his house. He has a mortgage for this, for which he has to pay $\pounds 4800$ annual interest. If he sells he will pay off the $\pounds 150000$ and invest the remainder of the proceeds at an interest rate of $2.5 \%$ per annum. He will use the interest to help to pay his rent. His estate agent estimates that there is a $30 \%$ chance that the house will sell for $\pounds 225000$, a $50 \%$ chance that it will sell for $\pounds 250000$, and a $20 \%$ chance that it will sell for $\pounds 275000$. A flat will cost him $\pounds 7500$ per annum to rent.
Draw a decision tree to help Adrian to decide whether to keep his house, or to sell it and rent a flat. Compare the EMVs of Adrian's annual outgoings, and ignore the costs of selling.

Would the analysis point to a different course of action if Adrian were to use a square root utility function, instead of EMVs? Adrian's circumstances change so that he has to decide now whether to sell or not in one year's time. Economic conditions might then be less favourable for the housing market, the same, or more favourable, these occurring with probabilities $0.3,0.3$ and 0.4 respectively. The possible selling prices and their probabilities are shown in the table.

Economic conditions and probabilities		Selling prices ( £) and probabilities
less favourable	0.3	200000	0.2	225000	0.3	250000	0.5
unchanged	0.3	225000	0.3	250000	0.5	275000	0.2
more favourable	0.4	250000	0.3	300000	0.5	350000	0.2

Draw a decision tree to help Adrian to decide what to do. Compare the EMVs of Adrian's annual outgoings. Assume that he will still owe $\pounds 150000$ in one year's time, and that the cost of renting and interest rates do not change. 3 The weights on the network represent distances.
\includegraphics[max width=\textwidth, alt={}, center]{83d00f1f-25e5-4a26-bac5-dc2baf965438-36_451_544_324_740}
The answer book shows the initial tables and the results of iterations $1,2,3$ and 5 when Floyd's algorithm is applied to the network.
(A) Complete the two tables for iteration 4.
(B) Use the final route table to give the shortest route from vertex $\mathbf { 3 }$ to vertex $\mathbf { 5 }$.
(C) Use the final distance table to produce a complete network with weights representing the shortest distances between vertices.
Using the complete network of shortest distances, find a lower bound for the solution to the Travelling Salesperson Problem by deleting vertex 5 and its arcs, and by finding the length of a minimum connector for the remainder. (You may find the minimum connector by inspection.)
Use the nearest neighbour algorithm, starting at vertex $\mathbf { 1 }$, to produce a Hamilton cycle in the complete network. Give the length of your cycle.
Interpret your Hamilton cycle in part (iii) in terms of the original network.
Give a walk of minimum length which traverses every arc on the original network at least once, and which returns to the start. Give the length of your walk. 4 A publisher is considering producing three books over the next week: a mathematics book, a novel and a biography. The mathematics book will sell at $\pounds 10$ and costs $\pounds 4$ to produce. The novel will sell at $\pounds 5$ and costs $\pounds 2$ to produce. The biography will sell at $\pounds 12$ and costs $\pounds 5$ to produce. The publisher wants to maximise profit, and is confident that all books will be sold. There are constraints on production. Each copy of the mathematics book needs 2 minutes of printing time, 1 minute of packing time, and $300 \mathrm {~cm} ^ { 3 }$ of temporary storage space. Each copy of the novel needs 1.5 minutes of printing time, 0.5 minutes of packing time, and $200 \mathrm {~cm} ^ { 3 }$ of temporary storage space. Each copy of the biography needs 2.5 minutes of printing time, 1.5 minutes of packing time, and $400 \mathrm {~cm} ^ { 3 }$ of temporary storage space. There are 10000 minutes of printing time available on several printing presses, 7500 minutes of packing time, and $2 \mathrm {~m} ^ { 3 }$ of temporary storage space.
Explain how the following initial feasible tableau models this problem.
P $x$ $y$ $z$ $s 1$ $s 2$ $s 3$ RHS
1 - 6 - 3 - 7 0 0 0 0
0 2 1.5 2.5 1 0 0 10000
0 1 0.5 1.5 0 1 0 7500
0 300 200 400 0 0 1 2000000
Use the simplex algorithm to solve your LP, and interpret your solution.
The optimal solution involves producing just one of the three books. By how much would the price of each of the other books have to be increased to make them worth producing? There is a marketing requirement to provide at least 1000 copies of the novel.
Show how to incorporate this constraint into the initial tableau ready for an application of the two-stage simplex method. Briefly describe how to use the modified tableau to solve the problem. You are NOT required to perform the iterations.

OCR D2 Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{89e3545e-fa4b-47dd-8651-7c8f998df9e7-1_762_1475_205_239} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} A salesman is planning a four-day trip beginning at his home and ending at town $I$. He will spend the first night in town $A , B$ or $C$, the second night in town $D , E$ or $F$ and the third night in town $G$ or $H$. The network in Figure 1 shows the distances, in tens of miles, that he will drive each day according to the route he chooses. Use dynamic programming to find the shortest route the salesman can take and state the distance he will drive in total using this route.

OCR D2 Q2

2. Four athletes are put forward for selection for a mixed stage relay race at a local competition. They may each be selected for a maximum of one stage and only one athlete can be entered for each stage. The average time, in seconds, for each athlete to complete each stage is given below, based on past performances.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Stage
\cline { 2 - 4 } \multicolumn{1}{c\|}{}	$\mathbf { 1 }$	$\mathbf { 2 }$	$\mathbf { 3 }$
Alex	19	69	168
Darren	22	64	157
Leroy	20	72	166
Suraj	23	66	171

Use the Hungarian algorithm to find an optimal allocation which will minimise the team's total time. Your answer should show clearly how you have applied the algorithm.

OCR D2 Q3

3. A project consists of 11 activities, some of which are dependent on others having been completed. The following precedence table summarises the relevant information.

Activity	Depends on	Duration (hours)
A	-	5
B	A	4
C	A	2
D	B, C	11
E	C	4
$F$	D	3
G	D	8
$H$	D, E	2
I	$F$	1
J	$F , G , H$	7
$K$	$I , J$	2

Draw an activity network for the project.
Find the critical path and the minimum time in which the project can be completed. Activity $F$ can be carried out more cheaply if it is allocated more time.
Find the maximum time that can be allocated to activity $F$ without increasing the minimum time in which the project can be completed.

OCR D2 Q4

A sheet is provided for use in answering this question.

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{89e3545e-fa4b-47dd-8651-7c8f998df9e7-3_725_1303_274_340} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 above shows a capacitated, directed network. The number on each arc indicates the capacity of that arc.

Calculate the values of cuts $C _ { 1 }$ and $C _ { 2 }$.
Find the minimum cut and state its value. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{89e3545e-fa4b-47dd-8651-7c8f998df9e7-3_645_1316_1430_338} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows a feasible flow through the same network.
State the values of $x , y$ and $z$.
Using this as your initial flow pattern, use the labelling procedure to find a maximal flow. You should list each flow-augmenting route you use together with its flow. State how you know that you have found a maximal flow.

OCR D2 Q5

5. The payoff matrix for player $X$ in a two-person zero-sum game is shown below.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	$Y$
\cline { 2 - 5 } \multicolumn{2}{c\|}{}	$Y _ { 1 }$	$Y _ { 2 }$	$Y _ { 3 }$
\multirow{2}{*}{$X$}	$X _ { 1 }$	10	4	3
\cline { 2 - 5 }	$X _ { 2 }$	${ } ^ { - } 4$	${ } ^ { - } 1$	9

Using a graphical method, find the optimal strategy for player $X$.
Find the optimal strategy for player $Y$.
Find the value of the game.

OCR D2 Q1

The payoff matrix for player $A$ in a two-person zero-sum game with value $V$ is shown below.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	$B$
\cline { 2 - 5 } \multicolumn{2}{c\|}{}	I	II	III
\multirow{3}{*}{$A$}	I	6	- 4	- 1
\cline { 2 - 5 }	II	- 2	5	3
\cline { 2 - 5 }	III	5	1	- 3

Formulate this information as a linear programming problem, the solution to which will give the optimal strategy for player $B$.

Rewrite the matrix as necessary and state the new value of the game, $v$, in terms of $V$.
Define your decision variables.
Write down the objective function in terms of your decision variables.
Write down the constraints.

OCR D2 Q2

2. The owner of a small plane is planning a journey from her local airport, $A$ to the airport nearest her parents, $K$. On the journey she will make three refuelling stops, the first at $B , C$ or $D$, the second at $E , F$ or $G$ and the third at $H , I$ or $J$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{34728928-2a21-463d-982e-c46ab2dc05c8-2_723_1303_427_356} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Figure 1 shows all the possible flights that can be made on the journey with the number by each arc indicating the distance of each flight in hundreds of miles. The owner of the plane wishes to choose a route that minimises the total distance that she flies. Use dynamic programming to find the route that she should use and state the total length of this route.

OCR D2 Q3

Four sales representatives ( $R _ { 1 } , R _ { 2 } , R _ { 3 }$ and $R _ { 4 }$ ) are to be sent to four areas ( $A _ { 1 } , A _ { 2 } , A _ { 3 }$ and $A _ { 4 }$ ) such that each representative visits one area. The estimated profit, in tens of pounds, that each representative will make in each area is shown in the table below.

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	$A _ { 1 }$	$A _ { 2 }$	$A _ { 3 }$	$A _ { 4 }$
$R _ { 1 }$	37	29	44	51
$R _ { 2 }$	45	30	43	41
$R _ { 3 }$	32	27	39	50
$R _ { 4 }$	43	25	51	55

Use the Hungarian method to obtain an allocation which will maximise the total profit made from the visits. Show the state of the table after each stage in the algorithm.

OCR D2 Q4

4.
\$ FMMUMITI7 IP HIZ3 UFHGHQFHIT
ா\$ மோங்கோ
ா\%\%mmum
\includegraphics[max width=\textwidth, alt={}, center]{34728928-2a21-463d-982e-c46ab2dc05c8-4_49_268_424_301}
\includegraphics[max width=\textwidth, alt={}, center]{34728928-2a21-463d-982e-c46ab2dc05c8-4_46_465_482_301}
\includegraphics[max width=\textwidth, alt={}, center]{34728928-2a21-463d-982e-c46ab2dc05c8-4_49_533_539_301}
\includegraphics[max width=\textwidth, alt={}, center]{34728928-2a21-463d-982e-c46ab2dc05c8-4_49_472_593_303}
\includegraphics[max width=\textwidth, alt={}, center]{34728928-2a21-463d-982e-c46ab2dc05c8-4_49_497_648_303}
\includegraphics[max width=\textwidth, alt={}, center]{34728928-2a21-463d-982e-c46ab2dc05c8-4_54_501_703_306}
\includegraphics[max width=\textwidth, alt={}, center]{34728928-2a21-463d-982e-c46ab2dc05c8-4_45_467_762_303}
\includegraphics[max width=\textwidth, alt={}, center]{34728928-2a21-463d-982e-c46ab2dc05c8-4_49_463_813_303}
\includegraphics[max width=\textwidth, alt={}, center]{34728928-2a21-463d-982e-c46ab2dc05c8-4_47_460_872_303}
$\square$
$\square$ Fig. 2
Construct an activity network to model the work involved in laying the foundations and putting in services for an industrial complex.

Execute a forward scan to find the minimum time in which the project can be completed.
Execute a backward scan to determine which activities lie on the critical path. The contractor is committed to completing the project in this minimum time and faces a penalty of $\pounds 50000$ for each day that the project is late. Unfortunately, before any work has begun, flooding means that activity $E$ will take 3 days longer than the 7 days allocated.
Activity $K$ could be completed in 1 day at an extra cost of $\pounds 90000$. Explain why doing this is not economical.
(2 marks)
If the time taken to complete any one activity, other than $E$, could be reduced by 2 days at an extra cost of $\pounds 80000$, for which activities on their own would this be profitable. Explain your reasoning.
(3 marks)
11 marks

OCR D2 Q5

A sheet is provided for use in answering this question.

A town has adopted a one-way system to cope with recent problems associated with congestion in one area. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{34728928-2a21-463d-982e-c46ab2dc05c8-5_684_1320_454_316} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} Figure 3 models the one-way system as a capacitated directed network. The numbers on the arcs are proportional to the number of vehicles that can pass along each road in a given period of time.

Find the capacity of the cut which passes through the $\operatorname { arcs } A E , B F , B G$ and $C D$.
(1 mark) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{34728928-2a21-463d-982e-c46ab2dc05c8-6_714_1280_171_333} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Figure 4 shows a feasible flow of 17 through the same network. For convenience, a supersource, $S$, and a supersink, $T$, have been used.
1. Use the labelling procedure to find the maximum flow through this network listing each flow-augmenting route you use together with its flow.
2. Show your maximum flow pattern and state its value.
Prove that your flow is the maximum possible through the network.
It is suggested that the maximum flow through the network could be increased by making road $E F$ undirected, so that it has a capacity of 8 in either direction. Using the maximum flow-minimum cut theorem, find the increase in maximum flow this change would allow.

	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
\(F\)
\(G\)
\(H\)
\(I\)
\(J\)

	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
\(F\)
\(G\)
\(H\)
\(I\)
\(J\)

	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
\(F\)
\(G\)
\(H\)
\(I\)
\(J\)

	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
\(F\)
\(G\)
\(H\)
\(I\)
\(J\)

	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
\(F\)
\(G\)
\(H\)
\(I\)
\(J\)

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\multirow{6}{*}{Rose's strategy}	Colin's strategy
		\(W\)	\(X\)	\(Y\)	\(Z\)
	\(A\)	-1	4	-3	2
	\(B\)	5	-2	5	6
	C	3	-4	-1	0
	\(D\)	-5	6	-4	-2

(i)	\(A \bullet\)
\(B \bullet\)	\(\bullet J\)
\(C \bullet\)	\(\bullet K\)
\(D \bullet\)	\(\bullet L\)
\(E \bullet\)	\(\bullet M\)
\(F \bullet\)	\(\bullet N\)

	\(J\)	\(K\)	\(L\)	\(M\)	\(N\)	\(O\)
\(A\)	2	5	2	2	5	2
\(B\)	2	5	2	0	5	5
\(C\)	5	0	5	5	2	2
\(D\)
\(E\)
\(F\)

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	\(\mathbf { 1 }\)	\(\mathbf { 2 }\)	\(\mathbf { 3 }\)	\(\mathbf { 4 }\)
\(\mathbf { 1 }\)	3	2	3	3
\(\mathbf { 2 }\)	1	1	3	3
\(\mathbf { 3 }\)	1	2	1	4
\(\mathbf { 4 }\)	3	3	3	3

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	\(A _ { 1 }\)	\(A _ { 2 }\)	\(A _ { 3 }\)	\(A _ { 4 }\)
\(R _ { 1 }\)	37	29	44	51
\(R _ { 2 }\)	45	30	43	41
\(R _ { 3 }\)	32	27	39	50
\(R _ { 4 }\)	43	25	51	55

b.v.	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value
							30
							60
							80
							0

b.v.	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value	Row Ops



\(P\)

b.v.	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value	Row Ops



\(P\)

b.v.	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value	Row Ops



\(P\)