1. technology and computing
2. software
3. desktop publishing

The Grange School
Maths Department
Decision 2
OCR Past Papers
Jan 2007
1
2
Four friends have rented a house and need to decide who will have which bedroom. The table below
shows how each friend rated each room, so the higher the rating the more the room was liked.
Phil
Rob
Sam
Tim
Attic
room
Back
room
Downstairs
room
Front
room
5
1
4
3
1
6
2
5
0
1
2
0
4
2
3
0
The Hungarian algorithm is to be used to find the matching with the greatest total. Before the
Hungarian algorithm can be used, each rating is subtracted from 6.
(i) Explain why the ratings could not be used as given in the table.
[1]
(ii) Apply the Hungarian algorithm, reducing rows first, to match the friends to the rooms. You
must show your working and say how each matrix was formed.
[7]
© OCR 2007
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Jan 2007
2
3
The table shows the activities involved in a project, their durations, precedences and the number of
workers needed for each activity. The graph gives a schedule with each activity starting at its earliest
possible time.
Activity
A
B
C
D
E
F
G
Duration (hours)
3
5
3
3
3
5
3
Immediate predecessors
−
A
A
B
C
D, E
B, E
Number of workers
3
2
2
1
3
2
3
(i) Using the graph, find the minimum completion time for the project and state which activities are
critical.
[2]
(ii) Draw a resource histogram, using graph paper, assuming that there are no delays and that every
activity starts at its earliest possible time.
[2]
Assume that only four workers are available but that they are equally skilled at all tasks. Assume also
that once an activity has been started it continues until it is finished.
(iii) The critical activities are to start at their earliest possible times. List the start times for the
non-critical activities for completion of the project in the minimum possible time. What is this
minimum completion time?
[4]
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Jan 2007
3
Rebecca and Claire repeatedly play a zero-sum game in which they each have a choice of three
strategies, X , Y and Z .
The table shows the number of points Rebecca scores for each pair of strategies.
Claire
X
Rebecca
Y
Z
X
5
−3
1
Y
3
2
−2
Z
−1
1
3
(i) If both players choose strategy X , how many points will Claire score?
[1]
(ii) Show that row X does not dominate row Y and that column Y does not dominate column Z . [2]
(iii) Find the play-safe strategies. State which strategy is best for Claire if she knows that Rebecca
will play safe.
[3]
(iv) Explain why decreasing the value ‘5’ when both players choose strategy X cannot alter the playsafe strategies.
[2]
4
The table gives the pay-off matrix for a zero-sum game between two players, Rowan and Colin.
Colin
Strategy X
Rowan
Strategy Y
Strategy Z
Strategy P
5
−3
−2
Strategy Q
−4
3
1
Rowan makes a random choice between strategies P and Q, choosing strategy P with probability p
and strategy Q with probability 1 − p.
(i) Write down and simplify an expression for the expected pay-off for Rowan when Colin chooses
strategy X .
[2]
(ii) Using graph paper, draw a graph to show Rowan’s expected pay-off against p for each of Colin’s
choices of strategy.
[4]
(iii) Using your graph, find the optimal value of p for Rowan.
[2]
(iv) Rowan plays using the optimal value of p. Explain why, in the long run, Colin cannot expect to
win more than 0.25 per game.
[2]
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5
Jan 2007
5
Answer this question on the insert provided.
The diagram represents a system of pipes through which fluid can flow from a source, S, to a sink, T .
The arrows are labelled to show excess capacities and potential backflows (how much more and how
much less could flow in each pipe). The excess capacities and potential backflows are measured in
litres per second. Currently no fluid is flowing through the system.
(i) Calculate the capacity of the cut X = {S, B}, Y = {A, C, D, E, F , G, T }.
[2]
(ii) Update the first diagram in the insert to show the changes to the labelling when 3 litres per second
flow along SADT , 2 litres per second flow along SBET and 2 litres per second flow along SCFT .
[3]
(iii) Write down one more flow augmenting route, but do not change the labelling on the diagram.
How much can flow through your route?
[2]
(iv) What is the maximum flow through the network? Write down a cut that has capacity equal to the
maximum flow.
[2]
(v) Complete the second diagram in the insert to show a maximum feasible flow through the network.
You will need to mark the direction and amount of flow in each arc.
[3]
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Jan 2007
6
Answer this question on the insert provided.
The table shows a partially completed dynamic programming tabulation for solving a maximin
problem.
Stage
State
Action
Working
Maximin
1
0
1
0
0
0
1
0
1
4
3
min(6, 4) = 4
min(2, 3) = 2
min(2, 4) =
min(4, 3) =
4
3
0
1
min(2,
min(3,
0
min(5,
1
2
min(5,
min(2,
0
2
1
2
3
0
(i) Complete the last two columns of the table in the insert.
[8]
(ii) State the maximin value and write down the maximin route.
[4]
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7
Jan 2007
Annie (A), Brigid (B), Carla (C) and Diane (D) are hanging wallpaper in a stairwell. They have broken
the job down into four tasks: measuring and cutting the paper (M ), pasting the paper (P), hanging and
then trimming the top end of the paper (H ) and smoothing out the air bubbles and then trimming the
lower end of the paper (S). They will each do one of these tasks.
•
•
•
•
Annie does not like climbing ladders but she is prepared to do tasks M , P or S
Brigid gets into a mess with paste so she is only able to do tasks M or S
Carla enjoys hanging the paper so she wants to do task H or task S
Diane wants to do task H
Initially Annie chooses task M , Brigid task S and Carla task H .
(i) Draw a bipartite graph to show the available pairings between the people and the tasks. Write
down an alternating path to improve the initial matching and write down the complete matching
from your alternating path.
[3]
Hanging the wallpaper is part of a bigger decorating project. The table lists the activities involved,
their durations and precedences.
A
B
C
D
E
F
G
H
Activity
Strip the old wallpaper
Fill in plaster
Undercoat woodwork
Paint woodwork with gloss
Paint ceiling
Hang new wallpaper
Prepare floor
Lay new flooring
Duration (hours)
10
2
1
2
2
4
3
2
Immediate predecessors
−
A
B
C
A
B
A
C, E, G
(ii) Draw an activity network, using activity on arc, to represent this information. You will need to
label the activities and use some dummy activities. Calculate the early event time and late event
time for each event and show these clearly at the vertices of your network.
[6]
(iii) State the minimum completion time for the whole decorating project, assuming that there are
enough workers and there are no delays. Write down the critical activities.
[2]
(iv) Construct a cascade chart, showing each activity starting at its earliest possible time.
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[3]
Jan 2007 Insert
5
2
(i) Capacity = .....................................................................................................................................
(ii)
(iii) Route: ......................................................................................
Flow = .....................................
(iv) Maximum flow = .........................................................
Cut: X = {
}
(v)
© OCR 2007
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Y={
}
3
Jan 2007 Insert
6
(i)
Stage
State
Action
Working
Maximin
1
0
1
0
0
0
1
0
1
0
1
4
3
min(6, 4) = 4
min(2, 3) = 2
min(2, 4) =
min(4, 3) =
min(2,
min(3,
4
3
0
min(5,
1
2
min(5,
min(2,
0
2
1
2
3
0
(ii) Maximin value = ...................................................................
Route = ..........................................................................................................................................
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June 2007
1
Daniel needs to clean four houses but only has one day in which to do it. He estimates that each house
will take one day and so he has asked three professional cleaning companies to give him a quotation
for cleaning each house. He intends to hire the three companies to clean one house each and he will
clean the fourth house himself. The table below shows the quotations that Daniel was given by the
three companies.
Allclean
Brightenupp
Clean4U
House 1
House 2
House 3
House 4
£500
£300
£500
£400
£200
£300
£700
£400
£750
£600
£350
£680
(i) Copy the table and add a dummy row to represent Daniel.
[2]
(ii) Apply the Hungarian algorithm, reducing rows first, to find a minimum cost matching. You
must show your working and say how each matrix was formed.
[9]
(iii) Which house should Daniel ask each company to clean? Find the total cost of hiring the three
companies.
[2]
2
The table gives the pay-off matrix for a zero-sum game between two players, Amy and Bea. The
values in the table show the pay-offs for Amy.
Bea
Strategy X
Amy
Strategy P
Strategy Q
4
−1
Strategy Y
−2
5
Strategy Z
0
4
Amy makes a random choice between strategies P and Q, choosing strategy P with probability p and
strategy Q with probability 1 − p.
(i) Write down and simplify an expression for the expected pay-off for Amy when Bea chooses
strategy X . Write down similar expressions for the cases when Bea chooses strategy Y and when
[4]
she chooses strategy Z .
(ii) Using graph paper, draw a graph to show Amy’s expected pay-off against p for each of Bea’s
[5]
choices of strategy. Using your graph, find the optimal value of p for Amy.
Amy and Bea play the game many times. Amy chooses randomly between her strategies using the
optimal value for p.
(iii) Showing your working, calculate Amy’s minimum expected pay-off per game. Why might Amy
gain more points than this, on average?
[3]
(iv) What is Bea’s minimum expected loss per game? How should Bea play to minimise her expected
loss?
[3]
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June 2007
3
The table shows the activities involved in a project, their durations and precedences, and the number
of workers needed for each activity.
Duration (days)
Immediate predecessors
B
4
A
2
C
5
A
2
D
2
B, C
1
E
3
C
3
F
4
D
2
G
2
D, E
2
Activity
A
−
3
Number of workers
3
(i) Draw an activity network to represent the project, using activity on arc. You are advised to make
the diagram quite large. The activity network requires two dummy activities; explain why each
of these is needed.
[4]
(ii) Carry out a forward pass to find the early times for the events. Record these at the vertices on your
network. Also calculate and record the late times for the events. Find the minimum completion
time for the project and list the critical activities.
[6]
The number of workers required for each activity is shown in the table. Assume that each worker is
able to do any of the activities. Once an activity has been started, it must run for its duration.
(iii) Using graph paper, draw a resource histogram with each activity starting at its earliest possible
start time.
[3]
(iv) Explain why if only four workers are available, the project cannot be completed in the minimum
project completion time. Show how the project can be completed in one day more than the
minimum project completion time when there are only four workers.
[2]
© OCR 2007
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4
June 2007
4
Answer this question on the insert provided.
The table shows a partially completed dynamic programming tabulation for solving a minimax
problem.
Stage
1
State
Action
Working
Minimax
0
0
4
4
1
0
3
3
2
0
2
2
max(2, 3) = 3
3
0
0
1
2
0
2
1
1
2
0
2
3
0
max(6, 4) = 6
max(3, 2) = 3
max(2, 4) =
max(4, 3) =
max(5, 2) =
max(2,
1
max(3,
2
max(4,
0
max(5,
1
max(5,
2
max(2,
(i) On the insert, complete the last two columns of the table.
[9]
(ii) State the minimax value and write down the minimax route.
[4]
(iii) Complete the diagram on the insert to show the network that is represented by the table.
[3]
© OCR 2007
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5
June 2007
5
Answer this question on the insert provided.
The network represents a system of pipes through which fluid can flow from a source, S, to a sink, T .
The arrows are labelled to show excess capacities and potential backflows (how much more and how
much less could flow in each pipe). The excess capacities and potential backflows are measured
in litres per second. Currently the flow is 6 litres per second, all flowing along a single route through
the system.
(i) Write down the route of the 6 litres per second that is flowing from S to T .
[1]
(ii) What is the capacity of the pipe AG and in which direction can fluid flow along this pipe?
[2]
(iii) Calculate the capacity of the cut X = {S, A, B, C, D, E }, Y = {F , G, H , I , T }.
[3]
(iv) Describe how a further 7 litres per second can flow from S to T and update the labels on the
arrows to show your flow. Explain how you know that this is the maximum flow.
[7]
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June 2007 Insert
4
(i)
Stage
1
State
Action
Working
Minimax
0
0
4
4
1
0
3
3
2
0
2
2
max(2, 3) = 3
3
0
0
1
2
0
2
1
1
2
0
2
3
0
max(6, 4) = 6
max(3, 2) = 3
max(2, 4) =
max(4, 3) =
max(5, 2) =
max(2,
1
max(3,
2
max(4,
0
max(5,
1
max(5,
2
max(2,
(ii) Minimax value = ............................
Route = ..........................................................................................................................................
(iii)
© OCR 2007
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June 2007 Insert
5
3
(i) ........................................................................................................................................................
(ii) ........................................................................................................................................................
(iii) ........................................................................................................................................................
(iv) ........................................................................................................................................................
........................................................................................................................................................
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Jan 2008
1
Arnie (A), Brigitte (B), Charles (C), Diana (D), Edward (E) and Faye (F ) are moving into a home for
retired Hollywood stars. They all still expect to be treated as stars and each has particular requirements.
Arnie wants a room that can be seen from the road, but does not want a ground floor room; Brigitte
wants a room that looks out onto the garden; Charles wants a ground floor room; Diana wants a room
with a balcony; Edward wants a second floor room; Faye wants a room, with a balcony, that can be
seen from the road.
The table below shows the features of each of the six rooms available.
Room
Floor
Can be seen from road
1
Ground
2
Ground
3
First
4
First
5
Second
6
Second
Looks out onto garden
Has balcony
(i) Draw a bipartite graph to show the possible pairings between the stars (A, B, C, D, E and F ) and
the rooms (1, 2, 3, 4, 5 and 6).
[2]
Originally Arnie was given room 4, Brigitte was given room 3, Charles was given room 2, Diana was
given room 5, Edward was given room 6 and Faye was given room 1.
(ii) Identify the star that has not been given a room that satisfies their requirements. Draw a second
bipartite graph to show the incomplete matching that results when this star is not given a room.
[2]
(iii) Construct the shortest possible alternating path, starting from the star without a room and ending
at the room that was not used, and hence find a complete matching between the stars and the
rooms. Write a list showing which star should be given which room.
[2]
When the stars view the rooms they decide that some are much nicer than others. Each star gives each
room a value from 1 to 6, where 1 is the room they would most like and 6 is the room they would least
like. The results are shown below.
Room
1
2
3
4
5
6
Arnie (A)
3
6
4
1
5
2
Brigitte (B)
5
3
2
4
1
6
Charles (C)
2
1
3
4
5
6
Diana (D)
5
4
1
3
2
6
Edward (E )
5
6
4
3
2
1
Faye (F )
5
6
4
1
3
2
(iv) Apply the Hungarian algorithm to this table, reducing rows first, to find a minimum ‘cost’
allocation between the stars and the rooms. Write a list showing which star should be given
which room according to this allocation. Write down the name of any star whose original
requirements are not satisfied.
[8]
© OCR 2008
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3
Jan 2008
2
As part of a team-building exercise the reprographics technicians (Team R) and the computer network
support staff (Team C) take part in a paintballing game. The game ends when a total of 10 ‘hits’ have
been scored.
Each team has to choose a player to be its captain. The number of ‘hits’ expected by Team R for each
pair of captains is shown below.
Team C
Team R
Liam
Mike
Nicola
Philip
4
5
6
Sanjiv
3
2
4
Tina
6
5
3
The teams are each trying to maximise their number of hits.
(i) If Team R chooses Philip and Team C chooses Liam, how many hits will Team C expect?
[1]
(ii) Explain why subtracting 5 from each of the values in the table will convert the game into a
zero-sum game.
[1]
(iii) Find the play-safe strategies for the zero-sum game. Explain how you know that the game is not
stable. State which choice of captain is best for Team C if they know that Team R will play safe.
[5]
(iv) Explain carefully why increasing the expected number of hits for Team R when Philip and Liam
are chosen as the captains will not change the play-safe strategies.
[2]
(v) Explain why Team R should never choose Sanjiv as the captain.
[1]
Team R chooses its captain by using random numbers to choose between Philip and Tina, where the
probability of choosing Philip is p and the probability of choosing Tina is 1 − p.
(vi) Show that the expected number of hits for Team R when Team C choose Liam is given by 6 − 2p
and find the corresponding expressions when Mike is chosen and when Nicola is chosen.
[3]
(vii) Use a graphical method to find the optimal value of p for Team R and how many hits Team R can
expect when this value is used.
[4]
© OCR 2008
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4
Jan 2008
3
Answer this question on the insert provided.
The table shows a partially completed dynamic programming tabulation for solving a minimax
problem.
Stage
1
State
Action
Working
0
0
1
1
0
3
2
0
2
0
2
1
2
3
0
0
(4,
1
(2,
1
(3,
2
(5,
0
(2,
2
(4,
0
(5,
1
(3,
2
(1,
Minimax
(i) Complete the last two columns of the table in the insert.
[6]
(ii) State the minimax value and write down the minimax route.
[3]
(iii) Draw the network represented by the table.
[3]
© OCR 2008
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5
Jan 2008
4
Answer this question on the insert provided.
The diagram represents a system of pipes through which fluid can flow from two sources, S1 and S2 ,
to two sinks, T1 and T2 . It also shows a cut α . The weights on the arcs show the capacities of the
pipes in gallons per hour.
(i) Add a supersource and a supersink to the network in the insert. Give appropriate weightings and
directions to the connecting arcs.
[2]
(ii) Explain why the arcs AE and BE cannot both be full to capacity.
[2]
(iii) Calculate the capacity of the cut α .
[1]
(iv) Calculate the capacity of the cut X = {S1 , S2 , B, D}, Y = {A, C, E, F , G, T1 , T2 }.
[2]
(v) On the diagram in the insert, show a flow through the network of 200 gallons per hour. Show
that this flow is maximal by finding a cut of 200 gallons per hour.
[3]
(vi) Vertex C now has a vertex restriction applied that means that no more than 20 gallons per hour
can flow through it. Amend the diagram in the insert to show this restriction. Find the value of
the maximum flow with the restriction.
[4]
© OCR 2008
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6
Jan 2008
5
Answer this question on the insert provided.
The diagram shows an activity network for a project. The figures in brackets show the durations of
the activities in days.
(i) Complete the table in the insert to show the precedences for the activities.
[3]
(ii) Use the boxes on the diagram in the insert to carry out a forward pass and a backward pass. Find
the minimum project duration and list the critical activities.
[6]
The number of people required for each activity is shown in the table below. The workers are all
equally skilled at all of the activities.
Activity
A
B
C
D
E
F
G
H
I
J
Number of workers
4
1
2
2
3
2
3
3
1
2
(iii) On graph paper, draw a resource histogram for the project with each activity starting at its earliest
possible time.
[2]
(iv) Describe how the project can be completed in 21 days using just six workers.
© OCR 2008
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[4]
Jan 2008 Insert
3
2
(i)
Stage
1
State
Action
Working
0
0
1
1
0
3
2
0
2
0
2
1
2
3
0
0
(4,
1
(2,
1
(3,
2
(5,
0
(2,
2
(4,
0
(5,
1
(3,
2
(1,
Minimax
(ii) Minimax value = ............................
Minimax route = ..........................................................................................................
(iii)
© OCR 2008
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Jan 2008 Insert
4
3
(i)
(ii) ........................................................................................................................................................
........................................................................................................................................................
(iii) ...................................................................................... = .................................. gallons per hour
(iv) ...................................................................................... = .................................. gallons per hour
(v)
(vi)
Maximum flow = ............. gallons per hour
© OCR 2008
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4
Jan 2008 Insert
5
(i)
Activity
Duration (days)
A
B
C
D
E
F
G
H
I
J
8
6
4
4
2
3
4
5
3
5
Immediate predecessors
(ii)
Minimum project completion time = ........ days;
critical activities = ........................................
(iii) To be answered on graph paper.
(iv) ........................................................................................................................................................
........................................................................................................................................................
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be
pleased to make amends at the earliest possible opportunity.
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.
© OCR 2008
4737/01 Ins Jan08
June 2008
1
2
(a) Five student teachers have been asked to each shadow an experienced teacher for a day. The
student teachers are Amy, Ben, Emily, Frank and Gina, and the experienced teachers are
Miss Patel, Mrs Quinn, Mr Roberts, Mr Thomas and Mrs Unwin.
Because of timetabling restrictions, Amy must shadow either Miss Patel or Mr Thomas; Ben
must shadow Miss Patel, Mr Roberts or Mr Thomas; Emily must shadow either Mrs Quinn or
Mrs Unwin; Frank must shadow either Mr Roberts or Mrs Unwin; and Gina can only shadow
Mrs Quinn.
(i) Draw a bipartite graph to represent this information. Put the student teachers (A, B, E , F and
G) on the left-hand side and the experienced teachers (P, Q, R, T and U ) on the right-hand
side.
[1]
Initially Amy is asked to shadow Mr Thomas, Ben to shadow Mr Roberts, Emily to shadow
Mrs Unwin and Gina to shadow Mrs Quinn.
(ii) Draw a second bipartite graph to show this incomplete matching.
[1]
(iii) Construct the shortest possible alternating path from F to P and hence find a complete
matching between the student teachers and the experienced teachers.
[2]
(iv) Amy would prefer to shadow one of the women (P, Q or U ). Find a complete matching that
will also satisfy this additional restriction.
[1]
[This question continues on the next page.]
© OCR 2008
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June 2008
3
(b) Mr Roberts teaches media studies. Some of his class are making a documentary about the five
student teachers in the school. He needs to choose a pupil to operate the camera (C), another to
be the director (D), a third pupil to be in charge of the lighting (L) and a fourth to be in charge of
the sound (S).
Five pupils have volunteered to do these tasks: Harry (H ), Iannos (I ), Jack (J ), Kerry (K ) and
Nadia (N ). Mr Roberts has assessed each pupil for their suitability for each task and has given
them a score out of 10. He has then subtracted each score from 10 to give a table on which the
Hungarian algorithm can be applied to find the best matching of pupils to tasks.
This table, showing ‘10 minus score’, is given below.
Task
Pupil
C
D
L
S
X
H
1
2
4
4
10
I
2
4
7
6
10
J
4
6
5
9
10
K
3
8
7
7
10
N
3
7
7
5
10
(i) Explain why the scores needed to be subtracted from 10, and explain the purpose of column X .
[2]
(ii) Apply the Hungarian algorithm, reducing columns first, to find a minimum cost matching.
You must show your working clearly. Which pupil should be given which task, and what is
[7]
the total score resulting from this allocation?
Harry says that he would rather take part in the documentary. This leaves just four pupils (I , J ,
K and N ) to be allocated to the four tasks (C, D, L and S).
(iii) Write out the 4 × 4 matrix showing ‘10 minus score’ for the pupils I , J , K and N and the
tasks (C, D, L and S). Apply the Hungarian algorithm to this reduced matrix to allocate the
pupils to the tasks.
[4]
© OCR 2008
4737/01 Jun08
[Turn over
4
June 2008
2
Rowena and Collette repeatedly play a zero-sum game in which Rowena has a choice of two strategies,
P and Q, and Collette has a choice of four strategies, W , X , Y and Z .
The table shows the number of points Rowena scores for each pair of strategies.
Collette
Rowena
W
X
Y
Z
P
2
−3
1
3
Q
1
2
−1
−4
(i) If Rowena chooses strategy P and Collette chooses strategy W , how many points will Collette
score?
[1]
(ii) Show that column W is dominated by one of the other columns, and state which column this is.
[2]
(iii) Find the play-safe strategy for Rowena and the play-safe strategy for Collette.
[3]
Rowena makes a random choice between strategies P and Q, choosing strategy P with probability p
and strategy Q with probability 1 − p.
(iv) Write down and simplify an expression for the expected pay-off for Rowena when Collette
chooses strategy X . Write down and simplify the corresponding expressions for when Collette
[2]
chooses strategy Y and for when she chooses strategy Z .
(v) Using graph paper, draw a graph to show Rowena’s expected pay-off against p for each of
Collette’s choices of strategy. Using your graph, find the optimal value of p for Rowena.
[4]
Calculate Rowena’s minimum expected pay-off if she plays using this value of p.
[This question continues on the next page.]
© OCR 2008
4737/01 Jun08
5
June 2008
In a variation of the game, Rowena has a choice of three strategies and Collette has a choice of just
two strategies. The table shows the number of points Rowena scores for each pair of strategies.
Collette
Y
Rowena
Z
P
0
3
Q
−1
−4
R
2
−2
Rowena intends to make a random choice between strategies P, Q and R, choosing strategy P with
probability p1 , strategy Q with probability p2 and strategy R with probability p3 .
She formulates the following linear programming problem so that she can find the optimal values of
p1 , p2 and p3 using the Simplex algorithm.
Maximise
M = m,
subject to
m ≤ 4p1 + 3p2 + 6p3 ,
m ≤ 7p 1
+ 2p3 ,
p1 + p2 + p3 ≤ 1,
and
p1 ≥ 0, p2 ≥ 0, p3 ≥ 0, m ≥ 0.
(vi) Show how Rowena obtained the expressions 4p1 + 3p2 + 6p3 and 7p1 + 2p3 . Also explain why
m cannot exceed either of these expressions.
[3]
(vii) Explain why the constraint p1 + p2 + p3 ≤ 1 is needed.
[1]
Rowena uses the Simplex algorithm to find the optimal values of the probabilities. She finds that the
optimal value of p1 is 47 and the optimal value of p2 is 0.
(viii) Calculate the optimal value of p3 and the corresponding minimum expected pay-off for Rowena.
[2]
© OCR 2008
4737/01 Jun08
[Turn over
6
June 2008
3
Answer this question on the insert provided.
The network below represents a system of pipes through which fluid can flow from a source, S, to a
sink, T . The weights on the arcs represent pipe capacities (maximum flow rates) in litres per minute.
(i) Calculate the capacity of the cut that separates {S, A, B, D, G} from {C, E, F , T }.
[2]
(ii) Explain why the arc GE cannot be full to capacity.
[1]
(iii) What is the maximum possible rate of flow through the vertex E? Show such a flow on the
diagram in the insert.
[3]
The diagram in the insert shows the graph on which the network was formed with arrows for use in
the labelling procedure. For each arc, the arrow pointing in the original direction of possible flow
is to show how much more could flow (excess capacity) and the arrow pointing against the original
direction of flow is to show how much less could flow (potential backflow).
(iv) Label the arrows to show a flow of 4 litres per minute along SACFT , a flow of 1 litre per minute
[3]
along SBET and a flow of 2 litres per minute along SDGT .
(v) Apply the labelling procedure to augment the flow using the route SBDET . State the amount
[3]
that flows along this route. Do not obliterate your values from part (iv).
(vi) Further augment the flow by 2 litres per minute using just one route. Leave your values from
[3]
part (iv) and part (v) clearly visible. Write down the route that you have used.
(vii) Show your resulting flow on the directed graph in the insert.
[1]
(viii) Show that your flow is maximal by finding a cut with capacity equal to the flow. Describe your
cut by stating the arcs that it crosses.
[2]
© OCR 2008
4737/01 Jun08
7
June 2008
4
Answer part (a) of this question on the insert provided.
(a) The table shows a partially completed dynamic programming tabulation for solving a longest
path (maximum path) problem.
Stage
2
State
Action
Working
0
0
5
1
0
4
2
0
4
0
1
1
2
0
0
0
3+
1
4+
1
2+
2
4+
1
6+
2
5+
0
4+
1
5+
2
2+
Suboptimal
maximum
On the insert, complete the last two columns of the table. State the length of the longest path and
write down its route.
[8]
(b) The table below shows the activities involved in a project, their durations and precedences.
Activity
A
B
C
D
E
F
G
H
I
J
K
L
Duration (days)
4
5
2
3
4
2
4
6
5
5
4
4
Immediate predecessors
−
−
−
A
A
B
B
C
C
D
E, F, H
G, I
(i) Draw an activity network to represent the project, using activity on arc. You are advised to
make your diagram as large as possible.
[2]
(ii) Carry out a forward pass to find the early times for the events. Record these at the vertices
on your network. Also calculate and record the late times for the events. Find the minimum
completion time for the project and list the critical activities.
[6]
It is now realised that activity K must follow activity G, as well as E , F and H .
(iii) Draw that part of the activity network that changes. This will mean showing the connections
[2]
between E, F , G, H , I and K and using two dummy activities.
© OCR 2008
4737/01 Jun08
June 2008 Insert
2
3
(i) ........................................................................................................................................................
Capacity of cut that separates {S, A, B, D, G} from {C, E , F , T } = ................ litres per minute
(ii) GE cannot be full to capacity since ..............................................................................................
........................................................................................................................................................
(iii) Maximum possible rate of flow through the vertex E = ..................................... litres per minute
© OCR 2008
4737/01 Ins Jun08
June 2008 Insert
(iv), (v) and (vi)
3
(v) Amount that flows along SBDET = ............................ litres per minute
(vi) Route used = ................................................................................................
(vii)
(viii) Cut through arcs ............................................................................................................................
........................................................................................................................................................
© OCR 2008
4737/01 Ins Jun08
[Turn over
4
June 2008 Insert
4
(a)
Stage
2
State
Action
Working
0
0
5
1
0
4
2
0
4
0
1
1
2
0
0
0
3+
1
4+
1
2+
2
4+
1
6+
2
5+
0
4+
1
5+
2
2+
Suboptimal
maximum
Length of longest path = ...............................................
Route = ..........................................................................................................................................
Answer part (b) in your answer booklet.
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be
pleased to make amends at the earliest possible opportunity.
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.
© OCR 2008
4737/01 Ins Jun08
2
Jan 2009
1
Answer this question on the insert provided.
The table shows a partially completed dynamic programming tabulation for solving a maximin
problem.
Stage
1
State
Action
Working
0
0
10
1
0
11
2
0
14
3
0
15
0
1
2
2
3
3
0
0
(12,
)=
2
(10,
)=
0
(13,
)=
1
(10,
)=
2
(11,
)=
1
( 9,
)=
2
(10,
)=
3
( 7,
)=
1
( 8,
)=
3
(12,
)=
0
(15,
)=
1
(14,
)=
2
(16,
)=
3
(13,
)=
Maximin
(i) Complete the last two columns of the table in the insert.
[6]
(ii) State the maximin value and write down the maximin route.
[3]
© OCR 2009
4737 Jan09
3
Jan 2009
2
Answer this question on the insert provided.
The diagram shows an activity network for a project. The figures in brackets show the durations of
the activities in days.
D (1)
A (8)
H (7)
B (10)
E (3)
J (5)
F (4)
C (12)
I (4)
G (3)
(i) Complete the table in the insert to show the precedences for the activities.
[3]
(ii) Use the boxes on the diagram in the insert to carry out a forward pass and a backward pass. Show
that the minimum project completion time is 28 days and list the critical activities.
[4]
The resource histogram below shows the number of workers required each day when the activities
each begin at their earliest possible start time. Once an activity has been started it runs for its duration
without a break.
Number of
workers
6
5
4
3
2
1
0
5
10
15
20
25
Day
(iii) By considering which activities are happening each day, complete the table in the insert to show
the number of workers required for each activity. You are advised to start at day 28 and work
back through the days towards day 1.
[6]
Only five workers are actually available, but they are all equally skilled at each of the activities. The
project can still be completed in 28 days by delaying the start of activity E .
(iv) Find the minimum possible delay and the maximum possible delay on activity E in this case.
[2]
© OCR 2009
4737 Jan09
Turn over
4
Jan 2009
3
Answer this question on the insert provided.
A
D
(0, 4)
F
(2, 3)
(4, 8)
(3, 3)
(2, 4)
E
(1, 3)
S
(0, 3)
(4, 7)
(8, 10)
(2, 8)
B
T
G
(10, 11)
(2, 2)
(4, 8)
(5, 7)
a
(1, 5)
C
H
Fig. 1
Fig. 1 represents a system of pipes through which fluid can flow from a source, S, to a sink, T . It also
shows a cut α . The weights on the arcs show the lower and upper capacities of the pipes in litres
per second.
(i) Calculate the capacity of the cut α .
[2]
(ii) By considering vertex B, explain why arc SB must be at its lower capacity. Then by considering
vertex E , explain why arc CE must be at its upper capacity, and hence explain why arc HT must
be at its lower capacity.
[4]
(iii) On the diagram in the insert, show a flow through the network of 15 litres per second. Write down
one flow augmenting route that allows another 1 litre per second to flow through the network.
Show that the maximum flow is 16 litres per second by finding a cut of 16 litres per second. [4]
A
D
(0, 4)
(2, 3)
F
(4, 8)
(3, 3)
(2, 4)
S
(0, 3)
E
(1, 3)
B
(4, 7)
(2, 8)
(8, 10)
T
G
(10, 11)
(2, 2)
(4, 8)
C
(5, 7)
(1, 5)
H
Fig. 2
Fig. 2 represents the same system, but with pipe EB installed the wrong way round.
(iv) Explain why there can be no feasible flow through this network.
© OCR 2009
4737 Jan09
[2]
5
4
Jan 2009
Anya (A), Ben (B), Connie (C), Derek (D) and Emma (E ) work for a local newspaper. The editor
wants them each to write a regular weekly article for the paper. The items needed are: problem page
(P), restaurant review (R), sports news (S), theatre review (T ) and weather report (W ).
Anya wants to write either the problem page or the restaurant review. She is given the problem page.
Ben wants the restaurant review, the sports news or the theatre review. The editor gives him the
restaurant review.
Connie wants either the theatre review or the weather report. The editor gives her the theatre review.
Derek wants the problem page, the sports news or the weather report. He is given the weather report.
Emma is only interested in writing the problem page but this has already been given to Anya.
(i) Draw a bipartite graph to show the possible pairings between the writers (A, B, C , D and E) and
the articles (P, R, S, T and W ). On your bipartite graph, show who has been given which article
by the editor.
[2]
(ii) Construct the shortest possible alternating path, starting from Emma, to find a complete matching
between the writers and the articles. Write a list showing which article each writer is given with
this complete matching.
[3]
When the writers send in their articles the editor assigns a sub-editor to each one to check it. The
sub-editors can check at most one article each.
The table shows how long, in minutes, each sub-editor would typically take to check each article.
Article
Sub-editor
P
R
S
T
W
Jeremy (J )
56
56
51
57
58
Kath (K )
53
52
53
54
54
Laura (L)
57
55
52
58
60
Mohammed (M )
59
55
53
59
57
Natalie (N )
57
57
53
59
60
Ollie (O)
58
56
51
56
57
The editor wants to find the allocation for which the total time spent checking the articles is as short
as possible.
(iii) Apply the Hungarian algorithm to the table, reducing rows first, to find an optimal allocation
between the sub-editors and the articles. Explain how each table is formed and write a list
showing which sub-editor should be assigned to which article. If each minute of sub-editor time
costs £0.25, calculate the total cost of checking the articles each week.
[11]
[Question 5 is printed overleaf.]
© OCR 2009
4737 Jan09
Turn over
6
5
Jan 2009
The local rugby club has challenged the local cricket club to a chess match to raise money for charity.
Each of the top three chess players from the rugby club has played 10 chess games against each of the
top three chess players from the cricket club. There were no drawn games. The table shows, for each
pairing, the number of games won by the player from the rugby club minus the number of games won
by the player from the cricket club. This will be called the score; the scores make a zero-sum game.
Cricket club
Doug
Sanjeev
Rugby club
Tom
Ursula
Euan
Fiona
0
4
−2
−4
2
−4
2
−6
0
(i) How many of the 10 games between Sanjeev and Doug did Sanjeev win? How many of the 10
games between Sanjeev and Euan did Euan win?
[3]
Each club must choose one person to play. They want to choose the person who will optimise the
score.
(ii) Find the play-safe choice for each club, showing your working. Explain how you know that the
game is not stable.
[5]
(iii) Which person should the cricket club choose if they know that the rugby club will play-safe and
which person should the rugby club choose if they know that the cricket club will play-safe? [2]
(iv) Explain why the rugby club should not choose Tom. Which player should the cricket club not
choose, and why?
[3]
The rugby club chooses its player by using random numbers to choose between Sanjeev and Ursula,
where the probability of choosing Sanjeev is p and the probability of choosing Ursula is 1 − p.
(v) Write down an expression for the expected score for the rugby club for each of the two remaining
choices that can be made by the cricket club. Calculate the optimal value for p.
[2]
Doug is studying AS Mathematics. He removes the row representing Tom and then models the cricket
club’s problem as the following LP.
maximise
M =m−4
subject to
m ≤ 4x
+ 6ß
m ≤ 2x + 10y + 4ß
x+y+ß≤1
and
m ≥ 0, x ≥ 0, y ≥ 0, ß ≥ 0
(vi) Show how Doug used the values in the table to get the constraints m ≤ 4x + 6ß and
m ≤ 2x + 10y + 4ß.
[3]
Doug uses the Simplex algorithm to solve the LP problem. His solution has x = 0 and y = 16 .
(vii) Calculate the optimal value of M .
© OCR 2009
[2]
4737 Jan09
2
Jan 2009 Insert
1
(i)
Stage
1
State
Action
Working
0
0
10
1
0
11
2
0
14
3
0
15
0
1
2
2
3
3
0
0
(12,
)=
2
(10,
)=
0
(13,
)=
1
(10,
)=
2
(11,
)=
1
( 9,
)=
2
(10,
)=
3
( 7,
)=
1
( 8,
)=
3
(12,
)=
0
(15,
)=
1
(14,
)=
2
(16,
)=
3
(13,
)=
Maximin
(ii) Maximin value = ............................
Maximin route = ............................................................................................................................
© OCR 2009
4737 Ins Jan09
3
Jan 2009 Insert
2
(i)
Activity
Duration (days)
A
8
B
10
C
12
D
1
E
3
F
4
G
3
H
7
I
4
J
5
Immediate predecessors
(ii)
Key:
Late event time
Early event time
D (1)
A (8)
H (7)
B (10)
E (3)
J (5)
F (4)
C (12)
I (4)
G (3)
Critical activities ...........................................................................................................................
(iii)
Activity
A
B
C
D
E
F
G
H
I
J
Number of workers
(iv) Minimum delay .................. days
Maximum delay .................. days
© OCR 2009
4737 Ins Jan09
Turn over
4
Jan 2009 Insert
3
A
(0, 4)
D
(2, 3)
F
(4, 8)
(3, 3)
(2, 4)
E
(1, 3)
S
(0, 3)
(4, 7)
(2, 8)
B
(8, 10)
T
G
(10, 11)
(2, 2)
(4, 8)
(5, 7)
a
(1, 5)
C
H
(i) Capacity of cut α = ........................... litres per second
(ii) ........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
(iii)
A
D
F
E
S
T
B
G
C
H
Flow augmenting route: ................................................................................................................
Cut: ...............................................................................................................................................
(iv) ........................................................................................................................................................
........................................................................................................................................................
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be
pleased to make amends at the earliest possible opportunity.
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.
© OCR 2009
4737 Ins Jan09
2
June 2009
1
(a) A caf´e sells five different types of filled roll. Mr King buys one of each to take home to his
family. The family consists of Mr King’s daughter Fiona (F ), his mother Gwen (G), his wife
Helen (H ), his son Jack (J ) and Mr King (K ).
The table shows who likes which rolls.
F
Avocado and bacon
(A)
Beef and horseradish
(B)
Chicken and stuffing
(C)
Duck and plum sauce
(D)
Egg and tomato
(E )
G
H
J
K
(i) Draw a bipartite graph to represent this information. Put the fillings (A, B, C, D and E) on
the left-hand side and the members of the family (F , G, H , J and K ) on the right-hand side.
[1]
Fiona takes the avocado roll; Gwen takes the beef roll; Helen takes the duck roll and Jack takes
the chicken roll.
(ii) Draw a second bipartite graph to show this incomplete matching.
[1]
(iii) Construct the shortest possible alternating path from E to K and hence find a complete
matching. State which roll each family member has with this complete matching.
[2]
(iv) Find a different complete matching.
[1]
(b) Mr King decides that the family should eat more fruit. Each family member gives a score out of
10 to five fruits. These scores are subtracted from 10 to give the values below.
Family member
F
G
H
J
K
L
8
8
8
8
1
Mandarin
M
4
8
6
4
2
Nectarine
N
9
9
9
7
1
Orange
O
4
6
5
4
3
Peach
P
6
9
7
5
0
Lemon
The smaller entries in each column correspond to fruits that the family members liked most.
Mr King buys one of each of these five fruits. Each family member is to be given a fruit.
Apply the Hungarian algorithm, reducing rows first, to find a minimum cost matching. You
must show your working clearly. Which family member should be given which fruit?
[8]
© OCR 2009
4737 Jun09
3
June 2009
2
(i) Set up a dynamic programming tabulation to find the maximum weight route from (0; 0) to (3; 0)
on the following directed network.
(1; 0)
5
(2; 0)
6
8
7
4
(0; 0)
9
(1; 1)
(2; 1)
5
6
(3; 0)
6
10
7
8
9
6
(1; 2)
(2; 2)
Give the route and its total weight.
[11]
(ii) The actions now represent the activities in a project and the weights represent their durations.
This information is shown in the table below.
Activity
Duration
Immediate predecessors
A
B
C
D
E
F
G
H
I
J
K
L
M
N
8
9
7
5
6
4
5
6
10
9
6
7
6
8
−
−
−
A
A
B
B
B
C
C
C
D, F , I
E , G, J
H, K
Make a large copy of the network with the activities A to N labelled appropriately. Carry out a
forward pass to find the early event times and a backward pass to find the late event times. Find
the minimum completion time for the project and list the critical activities.
[7]
(iii) Compare the solutions to parts (i) and (ii).
© OCR 2009
4737 Jun09
[2]
Turn over
4
June 2009
3
The ‘Rovers’ and the ‘Collies’ are two teams of dog owners who compete in weekly dog shows. The
top three dogs owned by members of the Rovers are Prince, Queenie and Rex. The top four dogs
owned by the Collies are Woof, Xena, Yappie and Zulu.
In a show the Rovers choose one of their dogs to compete against one of the dogs owned by the Collies.
There are 10 points available in total. Each of the 10 points is awarded either to the dog owned by the
Rovers or to the dog owned by the Collies. There are no tied points. At the end of the competition,
5 points are subtracted from the number of points won by each dog to give the score for that dog.
The table shows the score for the dog owned by the Rovers for each combination of dogs.
Collies
Rovers
W
X
Y
Z
P
1
2
−1
3
Q
−2
1
−3
−1
R
2
−4
1
0
(i) Explain why calculating the score by subtracting 5 from the number of points for each dog makes
this a zero-sum game.
[2]
(ii) If the Rovers choose Prince and the Collies choose Woof, what score does Woof get, and how
many points do Prince and Woof each get in the competition?
[2]
(iii) Show that column W is dominated by one of the other columns, and state which column this is.
[2]
(iv) Delete the column for W and find the play-safe strategy for the Rovers and the play-safe strategy
for the Collies on the table that remains.
[3]
Queenie is ill one week, so the Rovers make a random choice between Prince and Rex, choosing
Prince with probability p and Rex with probability 1 − p.
(v) Write down and simplify an expression for the expected score for the Rovers when the Collies
choose Xena. Write down and simplify the corresponding expressions for when the Collies
choose Yappie and for when they choose Zulu.
[2]
(vi) Using graph paper, draw a graph to show the expected score for the Rovers against p for each
of the choices that the Collies can make. Using your graph, find the optimal value of p for the
Rovers.
[3]
[This question continues on the next page.]
© OCR 2009
4737 Jun09
5
June 2009
If Queenie had not been ill, the Rovers would have made a random choice between Prince, Queenie
and Rex, choosing Prince with probability p1 , Queenie with probability p2 and Rex with probability
p3 .
The problem of choosing the optimal values of p1 , p2 and p3 can be formulated as the following linear
programming problem:
maximise
M = m−4
subject to
m ≤ 6p1 + 5p2 ,
m ≤ 3p1 + p2 + 5p3 ,
m ≤ 7p1 + 3p2 + 4p3 ,
p1 + p2 + p3 ≤ 1
and
p1 ≥ 0, p2 ≥ 0, p3 ≥ 0, m ≥ 0.
(vii) Explain how the expressions 6p1 + 5p2 , 3p1 + p2 + 5p3 and 7p1 + 3p2 + 4p3 were obtained. Also
explain how the linear programming formulation tells you that M is a maximin solution.
[3]
The Simplex algorithm is used to find the optimal values of the probabilities. The optimal value of p1
is 58 and the optimal value of p2 is 0.
(viii) Calculate the optimal value of p3 and the corresponding value of M .
© OCR 2009
4737 Jun09
[2]
Turn over
6
June 2009
4
The network represents a system of pipes through which fluid can flow from a source, S, to a sink, T .
The weights on the arcs represent pipe capacities in gallons per minute.
B
8
C
5
S
4
6
A
E
6
T
5
4
3
6
7
5
D
7
6
4
F
(i) Calculate the capacity of the cut that separates {S, A, C, D} from {B, E , F , T }.
[2]
(ii) Explain why the arcs AC and AD cannot both be full to capacity and why the arcs DF and EF
cannot both be full to capacity.
[2]
(iii) Draw a diagram to show a flow in which as much as possible flows through vertex E but none
flows through vertex A and none flows through vertex D. State the maximum flow through
vertex E.
[4]
An engineer wants to find a flow augmenting route to improve the flow from part (iii).
(iv) (a) Explain why there can be no flow augmenting route that passes through vertex A but not
through vertex D.
[1]
(b) Write down a flow augmenting route that passes through vertex D but not through vertex A.
State the maximum by which the flow can be augmented.
[2]
(v) Prove that the augmented flow in part (iv)(b) is the maximum flow.
[4]
(vi) A vertex restriction means that the flow through E can no longer be at its maximum rate. By
how much can the flow through E be reduced without reducing the maximum flow from S to T ?
Explain your reasoning.
[3]
The pipe represented by the arc BE becomes blocked and cannot be used.
(vii) Draw a diagram to show that, even when the flow through E is reduced as in part (vi), the same
[2]
maximum flow from S to T is still possible.
© OCR 2009
4737 Jun09
Jan 2010
1
2
Andy (A), Beth (B), Chelsey (C ), Dean (D) and Elly (E) have formed a quiz team. They have
entered a quiz in which, as well as team questions, each of them must answer individual questions
on a specialist topic. The specialist topics could be any of: food (F ), geography (G), history (H ),
politics (P), science (S) and television (T ). The team members do not know which five specialist
topics will arise in the quiz.
Andy wants to answer questions on either food or television; Beth wants to answer questions on
geography, history or science; Chelsey wants to answer questions on geography or television; Dean
wants to answer questions on politics or television; and Elly wants to answer questions on history or
television.
(i) Draw a bipartite graph to show the possible pairings between the team members and the specialist
topics.
[1]
In the quiz, the first specialist topic is food, and Andy is chosen to answer the questions. The second
specialist topic is geography, and Beth is chosen. The next specialist topic is history, and Elly is
chosen. The fourth specialist topic is science. Beth has already answered questions so Dean offers to
try this round. The final specialist topic is television, and Chelsey answers these questions.
(ii) Draw a second bipartite graph to show these pairings, apart from Dean answering the science
questions. Write down an alternating path starting from Dean to show that there would have
been a better way to choose who answered the questions had the topics been known in advance.
Write down which team member would have been chosen for each specialist topic in this case.
[4]
(iii) In a practice, although the other team members were able to choose topics that they wanted, Beth
had to answer the questions on television. Write down which topic each team member answered
questions on, and which topic did not arise.
[2]
2
Dudley has three daughters who are all planning to get married next year. The girls are named April,
May and June, after the months in which they were born. Each girl wants to get married on her own
birthday.
Dudley has already obtained costings from four different hotels. From past experience, Dudley knows
that when his family get together they are likely to end up with everyone fighting one another, so he
cannot use the same hotel twice.
The table shows the costs, in £100, for each hotel to host each daughter’s wedding.
Hotel
Daughter
Palace
Regent
Sunnyside
Tall Trees
April
30
28
32
25
May
32
34
32
35
June
40
40
39
38
Dudley wants to choose the three hotels to minimise the total cost.
Add a dummy row and then apply the Hungarian algorithm to the table, reducing rows first, to find
an optimal allocation between the hotels and Dudley’s daughters. State how each table is formed and
write out the final solution and its cost to Dudley.
[7]
© OCR 2010
4737 Jan10
Jan 2010
3
3
The table lists the duration (in hours), immediate predecessors and number of workers required for
each activity in a project.
Activity
Duration
Immediate
predecessors
Number of
workers
A
B
C
D
E
F
G
H
6
5
4
1
2
1
2
3
−
−
−
A, B
B
B, C
D, E
D, E, F
2
4
1
3
2
2
4
3
(i) Draw an activity network, using activity on arc, to represent the project. You should make your
diagram quite large so that there is room for working.
[2]
(ii) Carry out a forward pass and a backward pass through the activity network, showing the early
and late event times clearly at the vertices of your network.
State the minimum project completion time and list the critical activities.
[6]
(iii) Using graph paper, draw a resource histogram to show the number of workers required each
hour. Each activity begins at its earliest possible start time. Once an activity has started it runs
for its duration without a break.
[2]
A delay from the supplier means that the start of activity F is delayed.
(iv) By how much could the start of activity F be delayed without affecting the minimum project
completion time?
[1]
Suppose that only six workers are available after the first four hours of the project.
(v) Explain carefully what delay this will cause on the completion of the project. What is the
maximum possible delay on the start of activity F, compared with its earliest possible start time
in part (iii), without affecting the new minimum project completion time? Justify your answer.
[4]
© OCR 2010
4737 Jan10
Turn over
Jan 2010
4
4
The diagram represents a map of an army truck-driving course that includes several bridges. The start
and a ‘safe point’ just after each bridge have been given (stage; state) labels. The number below each
bridge shows its weight limit, in tonnes.
(1; 0)
9
(2; 0)
6
(2; 1)
START
(0; 0)
7
(1; 1)
7
FINISH
10
(3; 0)
(2; 2)
8
An army cadet needs to drive a truck through the course from start to finish, crossing exactly three
bridges.
(i) Draw a network, using the (stage; state) labels given, to represent the routes through the course.
The weights on the arcs should show the weight limits for the bridges.
[3]
The cadet wants to find out the weight of the heaviest truck she can use.
(ii) Which network problem does she need to solve?
[1]
(iii) Set up a dynamic programming tabulation to solve the cadet’s problem. Write down the weight
of the heaviest truck she can use and write down the (stage; state) labels for the route she should
take.
[9]
© OCR 2010
4737 Jan10
Jan 2010
5
5
Robbie received a new computer game for Christmas. He has already worked through several levels
of the game but is now stuck at one of the levels in which he is playing against a character called
Conan. Robbie has played this particular level several times.
Each time Robbie encounters Conan he can choose to be helped by a dwarf, an elf or a fairy.
Conan chooses between being helped by a goblin, a hag or an imp. The players make their choices
simultaneously, without knowing what the other has chosen.
Robbie starts the level with ten gold coins. The table shows the number of gold coins that Conan must
give Robbie in each encounter for each combination of helpers (a negative entry means that Robbie
gives gold coins to Conan). If Robbie’s total reaches twenty gold coins then he completes the level,
but if it reaches zero the game ends. This means that each attempt can be regarded as a zero-sum
game.
Conan
Goblin
Hag
−1
−4
2
Elf
3
1
−4
Fairy
1
−1
1
Dwarf
Robbie
Imp
(i) Find the play-safe choice for each player, showing your working. Which helper should Robbie
choose if he thinks that Conan will play-safe?
[5]
(ii) How many gold coins can Robbie expect to win, with each choice of helper, if he thinks that
[2]
Conan will choose randomly between his three choices (so that each has probability 31 )?
Robbie decides to choose his helper by using random numbers to choose between the elf and the fairy,
where the probability of choosing the elf is p and the probability of choosing the fairy is 1 − p.
(iii) Write down an expression for the expected number of gold coins won at each encounter by
Robbie for each of Conan’s choices. Calculate the optimal value of p.
[4]
Robbie’s girlfriend thinks that he should have included the possibility of choosing the dwarf. She
denotes the probability with which Robbie should choose the dwarf, the elf and the fairy as x, y and ß
respectively. She then models the problem of choosing between the three helpers as the following LP.
Maximise
M = m − 4,
subject to
m ≤ 3x + 7y + 5ß,
m ≤ 5y + 3ß,
m ≤ 6x + 5ß,
x + y + ß ≤ 1,
m ≥ 0, x ≥ 0, y ≥ 0, ß ≥ 0.
and
(iv) Explain how the expression 3x + 7y + 5ß was formed.
[2]
Robbie’s girlfriend uses the Simplex algorithm to solve the LP problem. Her solution has x = 0 and
y = 27 .
(v) Calculate the optimal value of M .
© OCR 2010
[3]
4737 Jan10
Turn over
Jan 2010
6
6
The diagram represents a system of pipes through which fluid can flow from a source, S, to a sink, T .
It also shows two cuts, α and β . The weights on the arcs show the lower and upper capacities of the
pipes in litres per second.
(1, 4)
A
F
(3, 6)
(1, 4)
(4, 8)
(3, 4)
(2, 3)
C
(2, 4)
S
(2, 5)
(0, 5)
T
(2, 8)
(3, 8)
(3, 4)
G
E
(2, 5)
(0, 3)
a
B
(2, 5)
D
(2, 5)
b
(i) Calculate the capacities of the cuts α and β .
[2]
(ii) Explain why the arcs AC and AF cannot both be at their lower capacities.
[1]
(iii) Explain why the arcs BC , BD, DE and DT must all be at their lower capacities.
[2]
(iv) Show that a flow of 10 litres per second is impossible. Deduce the minimum and maximum
feasible flows, showing your working.
[6]
Vertex E becomes blocked so that no fluid can flow through it.
(v) Draw a copy of the network with this vertex restriction. You are advised to make your diagram
quite large. Show a flow of 9 litres per second on your diagram.
[3]
© OCR 2010
4737 Jan10
June 2010
1
2
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
Axe handle
A
Broomstick
B
Drainpipe
D
Fence post
F
Golf club
G
Hammer
H
P
M
P
P
P
P
N
P
P
P
O
P
P
P
P
P
P
Q
P
P
(i) Draw a bipartite graph to represent this information. Put the weapons on the left-hand side and
the suspects on the right-hand side.
[1]
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.
(ii) Draw a second bipartite graph to show this incomplete matching.
[1]
(iii) Construct the shortest possible alternating path from H to Q and hence find a complete matching.
Write down which suspect used each weapon.
[2]
(iv) Find a different complete matching in which none of the suspects used the same weapon as in
the matching from part (iii).
[2]
© OCR 2010
4737 Jun10
June 2010
2
3
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.
Time
Suspect
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
Y
8
8
7
4
3
Sgt Yew
(i) 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.
[9]
Further enquiries show that the burglary took place at 5 pm, and that Dr Silverbirch was not the
burglar.
(ii) Who should Agatha suspect?
3
[1]
(i) Set up a dynamic programming tabulation to find the minimum weight route from (0; 0) to (4; 0)
on the following directed network.
(1; 0)
2
(2; 0)
5
(3; 0)
6
6
3
5
3
8
(0; 0)
2
(2; 1)
(1; 1)
3
5
3
(1; 2)
8
4
(3; 1)
3
(2; 2)
Give the route and its total weight.
(4; 0)
6
2
(3; 2)
[9]
(ii) Explain carefully how the route is obtained directly from the values in the table, without referring
to the network.
[2]
© OCR 2010
4737 Jun10
Turn over
June 2010
4
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
X
Euan
Y
Z
A
2
−5
3
B
−1
−3
4
C
3
−5
2
D
3
−2
−1
(i) Explain what the term ‘zero-sum game’ means.
[1]
(ii) How many points does Wai Mai score if she chooses X and Euan chooses A?
[1]
(iii) Why should Wai Mai never choose strategy Z ?
[2]
(iv) 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. [4]
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.
(v) Write down and simplify an expression for the expected score for Wai Mai when Euan chooses
each of his four strategies.
[3]
(vi) 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.
[4]
© OCR 2010
4737 Jun10
June 2010
5
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.
A
D
20
G
28
21
25
31
23
S
16
B
22
T
F
36
29
20
7
18
C
17
E
16
18
H
48
(i) Calculate the capacity of the cut that separates {S, B, C, E} from {A, D, F, G, H , T }.
[2]
(ii) Explain why neither arc SC nor arc BC can be full to capacity. Explain why the arcs EF and
EH cannot both be full to capacity. Hence find the maximum flow along arc HT . When arc HT
carries its maximum flow, what is the flow along arc HG?
[4]
(iii) Show a flow of 58 litres per second on the diagram in the insert, and find a cut of capacity 58.
[3]
The direction of flow in HG is reversed.
(iv) Use the diagram in the insert to show the excess capacities and potential backflows for your flow
from part (iii) in this case.
[2]
(v) 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 .
[2]
(vi) Show your augmented flow on the diagram in the insert. Explain how you know that this flow is
maximal.
[2]
© OCR 2010
4737 Jun10
Turn over
June 2010
6
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.
2
7
D(9)
A(6)
1
B(5)
G(2)
E(4)
3
6
F(2)
8
I(5)
M(12)
L(4)
J(6)
9
N(6)
10
H(3)
C(3)
K(10)
4
5
(i) Complete the table in the insert to show the immediate predecessors for each activity.
[3]
(ii) 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 .
[2]
(iii) 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.
[5]
Suppose that the duration of activity K changes to x minutes.
(iv) Find, in terms of x, expressions for the early event time and the late event time for event 9 . [4]
(v) Find the maximum duration of activity K that will not affect the minimum project completion
time found in part (iii).
[1]
© OCR 2010
4737 Jun10
June 2010
5
2
(i) Capacity of the cut that separates {S, B, C , E} from {A, D, F , G, H , T } =
................... litres per second
(ii) Neither arc SC nor arc BC can be full to capacity since ...............................................................
........................................................................................................................................................
Arcs EF and EH cannot both be full to capacity since ................................................................
........................................................................................................................................................
Maximum flow along arc HT is .................. litres per second
When arc HT carries its maximum flow, the flow along arc HG is .................. litres per second
(iii) Flow of 58 litres per second
A
D
20
G
28
21
25
31
23
S
16
B
22
T
F
36
29
20
7
18
C
17
E
16
18
H
48
Cut of capacity 58 litres per second ..............................................................................................
© OCR 2010
4737 Ins Jun10
June 2010
3
(iv)
A
S
G
D
B
T
F
C
E
H
(v) Augment flow in route ..................................................................... by ............ litres per second
Augment flow in route ..................................................................... by ............ litres per second
(vi)
A
D
20
G
28
21
31
23
S
B
20
7
18
17
T
F
36
C
29
16
22
25
E
16
18
H
48
Flow is maximal because ..............................................................................................................
........................................................................................................................................................
........................................................................................................................................................
© OCR 2010
4737 Ins Jun10
Turn over
4
June 2010
6
(i)
Activity
Duration
A
B
C
D
E
F
G
H
I
J
K
L
M
N
6
5
3
9
4
2
2
3
5
6
10
4
12
6
Immediate predecessors
(ii) Dummy activity is needed between event 2 and event 3 because ..............................................
........................................................................................................................................................
Dummy activity is needed between event 4 and event 5 because ..............................................
........................................................................................................................................................
(iii)
Event
1
2
3
4
5
6
7
8
9
10
Early event time
Late event time
Minimum project completion time = .............................................................................. minutes
Critical activities: ..........................................................................................................................
Answer part (iv) and part (v) in your answer booklet.
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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
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For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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of the University of Cambridge.
© OCR 2010
4737 Ins Jun10
2
Jan 2011
1
Four friends, Amir (A), Bex (B), Cerys (C) and Duncan (D), are visiting a bird sanctuary. They have
decided that they each will sponsor a different bird. The sanctuary is looking for sponsors for a kite
(K ), a lark (L), a moorhen (M ), a nightjar (N ), and an owl (O).
Amir wants to sponsor the kite, the nightjar or the owl; Bex wants to sponsor the lark, the moorhen
or the owl; Cerys wants to sponsor the kite, the lark or the owl; and Duncan wants to sponsor either
the lark or the owl.
(i) Draw a bipartite graph to show which friend wants to sponsor which birds.
[1]
Amir chooses to sponsor the kite and Bex chooses the lark. Cerys then chooses the owl and Duncan
is left with no bird that he wants.
(ii) Write down the shortest possible alternating path starting from the nightjar, and hence write
down one way in which all four friends could have chosen birds that they wanted to sponsor.
[2]
(iii) List a way in which all four friends could have chosen birds they wanted to sponsor, with the
owl not being chosen.
[1]
2
Amir, Bex, Cerys and Duncan all have birthdays in January. To save money they have decided that
they will each buy a present for just one of the others, so that each person buys one present and receives
one present. Four slips of paper with their names on are put into a hat and each person chooses one
of them. They do not tell the others whose name they have chosen and, fortunately, nobody chooses
their own name.
The table shows the cost, in £, of the present that each person would buy for each of the others.
To
From
Amir
Bex
Cerys
Duncan
Amir
−
15
21
19
Bex
20
−
16
14
Cerys
25
12
−
16
Duncan
24
10
18
−
As it happens, the names are chosen in such a way that the total cost of the presents is minimised.
Assign the cost £25 to each of the missing entries in the table and then apply the Hungarian algorithm,
reducing rows first, to find which name each person chose.
[7]
© OCR 2011
4737 Jan11
3
Jan 2011
3
The table lists the duration, immediate predecessors and number of workers required for each activity
in a project.
Activity
A
B
C
D
E
F
G
H
I
Duration
(hours)
3
2
2
3
3
3
2
5
4
Immediate
predecessors
−
−
A
A, B
C
C, D
D
E, F
F, G
Number of
workers
1
1
2
2
3
3
3
1
2
(i) Represent the project by an activity network, using activity on arc. You should make your
diagram quite large so that there is room for working.
[3]
(ii) Carry out a forward pass and a backward pass through the activity network, showing the early
event times and late event times clearly at the vertices of your network.
State the minimum project completion time and list the critical activities.
[5]
(iii) Draw a resource histogram to show the number of workers required each hour when each activity
begins at its earliest possible start time.
[2]
(iv) Show how it is possible for the project to be completed in the minimum project completion time
when only six workers are available.
[2]
© OCR 2011
4737 Jan11
Turn over
Jan 2011
4
4
Answer parts (v) and (vi) of this question on the insert provided.
The diagram represents a system of pipes through which fluid can flow. The weights on the arcs show
the lower and upper capacities of the pipes in litres per second.
A
(1, 4)
B
(3, 8)
G
(3, 6)
(1, 2)
(3, 5)
(3, 3)
D
(2, 3)
(1, 4)
E
(2, 4)
(0, 5)
(3, 6)
(2, 5)
H
(2, 4)
(0, 3)
C
I
(2, 5)
F
(2, 5)
(i) Which vertex is the source and which vertex is the sink?
[1]
(ii) Cut α partitions the vertices into the sets {A, B, C}, {D, E, F , G, H , I }. Calculate the capacity
of cut α .
[2]
(iii) Explain why partitioning the vertices into sets {A, D, G}, {B, C, E, F , H , I } does not give a
cut.
[1]
(iv) (a) How many litres per second must flow along arc DG?
[1]
(b) Explain why the arc AD must be at its upper capacity. Hence find the flow in arc BA.
[2]
(c) Explain why at least 7 litres per second must flow along arc BC.
[2]
(v) Use the diagrams in the insert to show a minimum feasible flow and a maximum feasible flow.
[4]
The upper capacity of BC is now increased from 8 to 18.
(vi) (a) Use the diagram in the insert to show a flow of 19 litres per second.
[1]
(b) List the saturated arcs when 19 litres per second flows through the network. Hence, or
otherwise, find a cut of capacity 19.
[2]
(vii) Explain how your answers to part (vi) show that 19 litres per second is the maximum flow. [2]
© OCR 2011
4737 Jan11
Jan 2011
5
5
A card game between two players consists of several rounds. In each round the players both choose
a card from those in their hand; they then show these cards to each other and exchange tokens. The
number of tokens that the second player gives to the first player depends on the colour of the first
player’s card and the design on the second player’s card.
The table shows the number of tokens that the first player receives for each combination of colour
and design. A negative entry means that the first player gives tokens to the second, zero means that
no tokens are exchanged.
Second player
Square
Circle
2
−1
1
Yellow
−2
0
−3
Blue
−5
1
3
Red
First player
Triangle
(i) Explain how you know that the game is zero-sum. Describe what zero-sum means for the way
in which the players play the game.
[2]
(ii) Find the play-safe choice for each player, showing your working. Explain how you know whether
the game is stable or unstable. Describe what ‘stable’ and ‘unstable’ mean for the way in which
the players play the game.
[6]
The first player decides not to risk playing a blue card.
(iii) Show that in this reduced game the circle strategy dominates the square strategy, and explain
what this means for the way in which the second player plays the game.
[2]
The first player uses random numbers to choose between the other two colours, where the probability
of choosing a red card is p and the probability of choosing a yellow card is 1 − p.
(iv) Write down an expression for the expected number of tokens that the first player is given in each
round for each choice of design. Calculate the optimal value of p, showing your working. [3]
The entries in the row for ‘Blue’ in the original table are now all multiplied by −1. So, for example,
when the first player chooses blue and the second chooses square, instead of the first player giving
the second player 5 tokens, the second player now gives the first player 5 tokens.
The first player now uses random numbers to choose between the three colours, letting x, y and ß
denote the probabilities of choosing red, yellow and blue respectively.
The problem of choosing between the three colours is modelled as the following LP.
Maximise
M = m − 3,
subject to
m ≤ 5x + y + 8ß,
m ≤ 2x + 3y + 2ß,
m ≤ 4x,
x + y + ß ≤ 1,
m ≥ 0, x ≥ 0, y ≥ 0, ß ≥ 0.
and
(v) Explain how the expression 5x + y + 8ß was formed.
[2]
The Simplex algorithm is used to solve the LP problem. The solution has x = 0.6, y = 0.4 and ß = 0.
(vi) Calculate the value of each of the expressions 5x + y + 8ß, 2x + 3y + 2ß and 4x. Hence write
[3]
down the optimal value of M .
© OCR 2011
4737 Jan11
Turn over
6
Jan 2011
6
Answer this question on the insert provided.
Four friends have decided to sponsor four birds at a bird sanctuary. They want to construct a route
through the bird sanctuary, starting and ending at the entrance/exit, that enables them to visit the four
birds in the shortest possible time. The table below shows the times, in minutes, that it takes to get
between the different birds and the entrance/exit. The friends will spend the same amount of time
with each bird, so this does not need to be included in the calculation.
Entrance/exit
Kite
Lark
Moorhen
Nightjar
Entrance/exit
−
10
14
12
17
Kite
10
−
3
2
6
Lark
14
3
−
2
4
Moorhen
12
2
2
−
3
Nightjar
17
6
4
3
−
Let the stages be 0, 1, 2, 3, 4, 5. Stage 0 represents arriving at the sanctuary entrance. Stage 1
represents visiting the first bird, stage 2 the second bird, and so on, with stage 5 representing leaving
the sanctuary. Let the states be 0, 1, 2, 3, 4 representing the entrance/exit, kite, lark, moorhen and
nightjar respectively.
(i) Calculate how many minutes it takes to travel the route
(0; 0) − (1; 1) − (2; 2) − (3; 3) − (4; 4) − (5; 0).
[1]
The friends then realise that if they try to find the quickest route using dynamic programming with
this (stage; state) formulation, they will get the route (0; 0) − (1; 1) − (2; 2) − (3; 3) − (4; 1) − (5; 0),
or this in reverse, taking 27 minutes.
(ii) Explain why the route (0; 0) − (1; 1) − (2; 2) − (3; 3) − (4; 1) − (5; 0) is not a solution to the
friends’ problem.
[1]
Instead, the friends set up a dynamic programming tabulation with stages and states as described
above, except that now the states also show, in brackets, any birds that have already been visited. So,
for example, state 1(234) means that they are currently visiting the kite and have already visited the
other three birds in some order. The partially completed dynamic programming tabulation is shown
opposite.
(iii) For the last completed row, i.e. stage 2, state 1(3), action 4(13), explain where the value 18 and
the value 6 in the working column come from.
[2]
(iv) Complete the table in the insert and hence find the order in which the birds should be visited to
give a quickest route and find the corresponding minimum journey time.
[9]
© OCR 2011
4737 Jan11
Jan 2011
7
Stage
4
3
State
1(234)
2(134)
3(124)
4(123)
1(23)
1(24)
1(34)
2(13)
2(14)
2(34)
3(12)
3(14)
3(24)
4(12)
4(13)
4(23)
1(2)
1(3)
Action
0
0
0
0
4(123)
3(124)
2(134)
4(123)
3(124)
1(234)
4(123)
2(134)
1(234)
3(124)
2(134)
1(234)
3(12)
4(12)
2(13)
4(13)
Working
10
14
12
17
17 + 6 = 23
12 + 2 = 14
14 + 3 = 17
17 + 4 = 21
12 + 2 = 14
10 + 3 = 13
17 + 3 = 20
14 + 2 = 16
10 + 2 = 12
12 + 3 = 15
14 + 4 = 18
10 + 6 = 16
20 + 2 = 22
15 + 6 = 21
21 + 3 = 24
18 + 6 = 24
1(4)
2(1)
2(3)
2(4)
2
3(1)
3(2)
3(4)
4(1)
4(2)
4(3)
1
2
1
3
4
0
© OCR 2011
0
1
2
3
4
4737 Jan11
Suboptimal minimum
10
14
12
17
23
14
17
21
14
13
20
16
12
15
18
16
21
24
2
Jan 2011
4
(v)
Minimum flow
D
A
G
E
H
B
C
I
F
Maximum flow
D
A
G
E
H
B
C
I
F
© OCR 2011
4737 Ins Jan11
Jan 2011
3
(vi) (a)
D
A
G
E
B
H
C
I
F
(b) ................................................................................................................................................
................................................................................................................................................
................................................................................................................................................
................................................................................................................................................
[Answer Question 6 overleaf.]
© OCR 2011
4737 Ins Jan11
Turn over
Jan 2011
6
4
(i) ........................................................................................................................................................
........................................................................................................................................................
(ii) ........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
(iii) ........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
........................................................................................................................................................
(iv) The table for this part of the question is on the opposite page.
Quickest route ...............................................................................................................................
Minimum journey time ..................................... minutes
© OCR 2011
4737 Ins Jan11
June 2011
1
2
Adam, Barbara and their children Charlie, Donna, Edward and Fiona all want cereal for breakfast.
The only cereal in the house is a pack of six individual portions of different cereals.
The table shows which family members like each of the cereals in the pack.
Family member
A
Cereal
Cornflakes
(1)
Rice pips
(2)
Wheat biscs
(3)
Oatie bits
(4)
Choco pips
(5)
Honey footballs
(6)
B
C
D
(i) Draw a bipartite graph to represent this information.
E
F
[1]
Adam gives the cornflakes to Fiona, the oatie bits to Edward, the rice pips to Donna, the choco pips
to Charlie and the wheat biscs to Barbara. However, this leaves the honey footballs for Adam, which
is not a possible pairing.
(ii) Draw a second bipartite graph to show this incomplete matching.
[1]
(iii) Construct the shortest possible alternating path from 6 to A and hence find a complete matching
between the cereals and the family members. Write down which family member is given each
cereal with this complete matching.
[2]
(iv) Adam decides that he wants cornflakes. Construct an alternating path starting at A, based on
your answer to part (iii) but with Adam being matched to the cornflakes, to find another complete
matching. Write down which family member is given each cereal with this matching.
[2]
© OCR 2011
4737 Jun11
3
June 2011
2
Granny is on holiday in Amsterdam and has bought some postcards. She wants to send one card to
each member of her family. She has given each card a score to show how suitable it is for each family
member. The higher the score the more suitable the card is.
Family member
Postcard
Adam
Barbara
Charlie
Donna
Edward
Fiona
Painted barges
P
2
4
2
6
0
4
Quaint houses
Q
3
5
3
5
3
4
Reichsmuseum
R
6
7
6
6
6
8
Scenic view
S
4
6
4
4
0
4
Tulips
T
1
0
1
4
0
5
University
U
3
4
4
4
3
3
View from air
V
7
5
7
6
7
5
Windmills
W
4
6
5
4
5
5
Granny adds two dummy columns, G and H , both with score 0 for each postcard. She then modifies
the resulting table so that she can use the Hungarian algorithm to find the matching for which the total
score is maximised.
(i) Explain why the dummy columns were needed, why they should not have positive scores and
how the resulting table was modified.
[3]
(ii) Show that, after reducing rows and columns, Granny gets this reduced cost matrix.
A
B
C
D
E
F
G
H
P
4
2
4
0
6
2
2
2
Q
2
0
2
0
2
1
1
1
R
2
1
2
2
2
0
4
4
S
2
0
2
2
6
2
2
2
T
4
5
4
1
5
0
1
1
U
1
0
0
0
1
1
0
0
V
0
2
0
1
0
2
3
3
W
2
0
1
2
1
1
2
2
[3]
(iii) Complete the application of the Hungarian algorithm, showing your working clearly. Write down
which family member is sent each postcard, and which postcards are not used, to maximise the
score.
[6]
© OCR 2011
4737 Jun11
Turn over
June 2011
3
4
Basil runs a luxury hotel. He advertises summer breaks at the hotel in several different magazines.
Last summer he won the opportunity to place a full-page colour advertisement in one of four magazines
for the price of the usual smaller advertisement. The table shows the expected additional weekly
income, in £, for each of the magazines for each possible type of weather. Basil wanted to maximise
the additional income.
Weather
Magazine
Rainy
Sunny
Activity holidays
4000
5000
British beaches
1000
7000
Country retreats
3000
6000
Dining experiences
5000
3000
(i) Explain carefully why no magazine choice can be rejected using a dominance argument.
[2]
(ii) Treating the choice of strategies as being a zero-sum game, find Basil’s play-safe strategy and
show that the game is unstable.
[4]
(iii) Calculate the expected additional weekly income for each magazine choice if the weather is
rainy with probability 0.4 and sunny with probability 0.6.
[2]
Suppose that the weather is rainy with probability p and sunny with probability 1 − p.
(iv) Which magazine should Basil choose if the weather is certain to be sunny (p = 0), and which
should he choose if the weather is certain to be rainy (p = 1)?
[1]
(v) Graph the expected additional weekly income against p. Hence advise Basil on which magazine
he should choose for the different possible ranges of values of p.
[3]
© OCR 2011
4737 Jun11
5
June 2011
4
Jamil is building a summerhouse in his garden. The activities involved, the duration, immediate
predecessors and number of workers required for each activity are listed in the table.
Activity
A:
B:
C:
D:
E:
F:
G:
H:
I:
J:
Choose summerhouse
Buy slabs for base
Take goods home
Level ground
Lay slabs
Treat wood
Install floor, walls and roof
Fit windows and door
Fit patio rail
Fit shelving
Duration
(hours)
2
1
2
3
2
3
4
2
1
1
Immediate
predecessors
−
−
A, B
−
C, D
C
E, F
G
G
G
Number of
workers
2
2
2
1
2
1
2
1
1
1
(i) Represent the project by an activity network, using activity on arc. You should make your
diagram quite large so that there is room for working.
[3]
(ii) Carry out a forward pass and a backward pass through the activity network, showing the early
event times and late event times at the vertices of your network. State the minimum project
completion time and list the critical activities.
[5]
(iii) Draw a resource histogram to show the number of workers required each hour when each activity
begins at its earliest possible start time.
[2]
(iv) Describe how it is possible for the project to be completed in the minimum project completion
time when only four workers are available.
[1]
(v) Describe how two workers can complete the project as quickly as possible. Find the minimum
time in which two workers can complete the project.
[3]
© OCR 2011
4737 Jun11
Turn over
6
June 2011
5
The network represents a simplified map of a town centre. On certain days, large numbers of visitors
need to travel through the town centre, from S to T . The arcs represent roads and the weights show
the maximum number of visitors per hour who can use each road. To find the maximum rate at
which visitors can travel through the town centre without any of them being delayed, the problem is
modelled as a maximum flow problem.
D
1000
S
800
B
3000
E
2000
2000
1000
1000
T
3500
500
C
A
500
F
1800
1200
600
G
800
(i) Calculate the capacity of the cut that separates {S, A, C, G} from {B, D, E, F , T }.
[2]
(ii) Explain why neither arc SA nor arc ET can be full to capacity. Also explain why the arcs AC
and BC cannot simultaneously be full to capacity.
[3]
(iii) Show a flow of 3300 people per hour, and find a cut of capacity 3300.
[3]
The direction of flow in BC is reversed.
(iv) Show the excess capacities and potential backflows when there is no flow.
[2]
(v) Without obscuring your answer to part (iv), augment the labels to show a flow of 2000 people
per hour along SBET .
[2]
(vi) Write down further flow augmenting routes and augment the labels, without obscuring your
previous answers, to find the maximum flow from S to T .
[4]
(vii) Show the maximum flow and explain how you know that this flow is maximal.
© OCR 2011
4737 Jun11
[3]
7
June 2011
6
Set up a dynamic programming tabulation to find the maximin route from (0; 0) to (3; 0) on the
following directed network.
[9]
(1; 0)
7
(0; 0)
4
(2; 0)
5
5
(1; 1)
2
6
(2; 1)
6
4
3
3
(1; 2)
© OCR 2011
4
4737 Jun11
(2; 2)
(3; 0)
Jan 2012
1
2
Five film studies students need to review five different films for an assignment, but only have one evening
left before the assignment is due in. They decide that they will share the work out so that each of them
reviews just one film.
Jack (J ) wants to review a horror movie; Karen (K ) wants to review an animated film; Lee (L) wants to
review a film that is suitable for family viewing; Mark (M ) wants to review an action adventure film and
Nikki (N ) wants to review anything that is in 3D.
The film “Somewhere” (S ) has been classified as a horror movie and is being shown in 3D; “Tornado
Terror” (T ) has been classified as an action adventure film that is suitable for family viewing; “Underwater”
(U ) is an animated action adventure film; “Vampires” (V ) is an animated horror movie that is suitable for
family viewing and “World” (W ) is an animated film.
(i) Draw a bipartite graph to show which student (J, K, L, M, N ) wants to review which films (S, T, U,
V, W).
[1]
Initially Jack says that he will review “Somewhere”, Karen then chooses “Underwater” and Lee chooses
“Tornado Terror”, but this would leave both Mark and Nikki with films that they do not want.
(ii) Write down the shortest possible alternating path starting from Nikki, and hence write down an
improved, but still incomplete, matching.
[2]
(iii) From this incomplete matching, write down the shortest possible alternating path starting from
“World”, and hence write down a complete matching between the students and the films.
[2]
(iv) Show that this is the only possible complete matching between the students and the films.
© OCR 2012
4737 Jan12
[1]
3
Jan 2012
2
The table lists the durations (in minutes), immediate predecessors and number of workers required for each
activity in a project to decorate a room.
Activity
Duration
(minutes)
Immediate
predecessors
Number of
workers
A
Cover furniture with dust sheets
20
–
1
B
Repair any cracks in the plaster
100
A
1
C
Hang wallpaper
60
B
1
D
Paint feature wall
90
B
1
E
Paint woodwork
120
C, D
1
F
Put up shelves
30
C
2
G
Paint ceiling
60
A
1
H
Clean paintbrushes
10
E, G
1
I
Tidy room
20
F, H
2
(i) Draw an activity network, using activity on arc, to represent the project. Your network will require a
dummy activity.
[2]
(ii) Carry out a forward pass and a backward pass through the activity network, showing the early event
time and the late event time at each vertex of your network.
State the minimum project completion time and list the critical activities.
[5]
(iii) Draw a resource histogram to show the number of workers required at each time when each activity
begins at its earliest possible start time.
[2]
Suppose that there is only one worker available at the start of the project, but another two workers are
available later.
(iv) Find the latest possible time for the other workers to start and still have the project completed on time.
Which activities could happen at the same time as painting the ceiling if the other two workers arrive at
this latest possible time?
[Do not change your resource histogram from part (iii).]
[2]
© OCR 2012
4737 Jan12
Turn over
4
Jan 2012
3
The famous fictional detective Agatha Parrot has been called in to investigate the theft of some jewels. Each
thief is known to have taken just one item of jewellery.
Agatha has invented a scoring system based on motive, opportunity and past experience. The table shows
the score for each of four suspects with each of three items of jewellery. The higher the score the more likely
the suspect is to have stolen that item of jewellery.
Suspect
Pearl necklace
Ruby ring
Sapphire
bracelet
Butler
80
100
20
Cook
40
35
60
Gardener
60
45
30
Handyman
20
100
80
(i) Assuming that three of these four suspects are the thieves, find who is most likely to have stolen each
item of jewellery for the total score to be maximised. State how each table of working was calculated.
Write down the two possible solutions for who should be suspected of stealing each item of jewellery
and who should be thought to be innocent.
[9]
Further evidence shows that the butler stole the sapphire bracelet.
(ii) Using this additional information, find out which suspect should be thought to be innocent. Explain
your reasoning.
[2]
© OCR 2012
4737 Jan12
5
Jan 2012
4
The diagram represents a system of roads through which traffic flows from a source, S, to a sink, T. The
weights on the arcs show the capacities of the roads in cars per minute.
S
80
A
20
10
40
B
25
30
C
25
45
25
D
T
20
E
30
65
F
(i) (a)
The cut α partitions the vertices into the sets {S, A, B, C}, {D, E, F, T}. Calculate the capacity of
cut α.
[1]
(b)
The cut β partitions the vertices into the sets {S, A, B, D}, {C, E, F, T}. Calculate the capacity of
cut β.
[1]
(c)
Using only the capacities of cuts α and β, what can you deduce about the maximum possible flow
through the system?
[1]
(ii) Explain why partitioning the vertices into sets {S, A, D, T}, {B, C, E, F} does not give a cut.
[1]
(iii) What do the double arcs between D and E and between E and F represent?
[1]
(iv) Explain why the maximum possible flow along CF must be less than 45 cars per minute.
[1]
(v) (a)
Show how a flow of 60 cars per minute along FT can be achieved.
[2]
Show that 60 cars per minute is the maximum possible flow through the system.
[2]
(b)
An extra road is opened linking S to C. Let the capacity of this road be x cars per minute.
(vi) Find the maximum possible flow through the new system, in terms of x where necessary, for the
different possible values of x.
[3]
© OCR 2012
4737 Jan12
Turn over
6
Jan 2012
5
Henry is doing a sponsored cycle ride for charity. He needs to finish at noon on Sunday. He can ride up to 50
miles each day, except Sunday when he can ride at most 20 miles if he is to finish on time.
The total length of the ride is 95 miles so Henry has allowed 3 days for the ride. Henry will start his ride at A
and travel through B, C, D and E, in that order, and finish on Sunday at F. He will stay overnight on Friday
and Saturday at two of the places B, C, D and E.
The distances between the places along the route are:
A–B = 30 miles, B–C = 15 miles, C–D = 35 miles,
D–E = 12 miles,
E–F = 3 miles.
To reach F on Sunday he must have reached at least D by Saturday night (since the distance from D to F is
less than 20 miles but C to F is more than 20 miles.)
Henry wants to use dynamic programming to minimise the maximum distance that he cycles on any day.
The stages will be the days. The places where Henry stays overnight will be the states. Henry starts on
Friday morning at A which has the (stage; state) label (0; 0). On Friday night he can either stay at B (1; 0) or
at C (1; 1). Depending on where he stays on Friday night, he can spend Saturday night at D (2; 0) or E (2; 1).
On Sunday he arrives at F (3; 0).
(i) Use this information and the table below to draw a network, labelled with stages and states, to show
the possible transitions between states. The arc weights should be the distances between the states. [2]
Henry uses dynamic programming, working backwards from stage 3, to find where he should stay overnight
to give the route for which the maximum on any day is a minimum. His tabulation is shown below.
Stage
State
Action
Working
Suboptimal minimax
2
0
0
15
15
1
0
3
3
0
0
max(50, 15)
50
1
0
max(35, 15)
35
1
max(47, 3)
0
max(30, 50)
1
max(45, 35)
1
0
(ii) (a)
(ii)(b)
0
45
In the last row of the table, the action value is 1. Explain what this tells you.
[1]
In the last row of the table, the working column is max(45, 35). Explain where each of the values
45 and 35 comes from and how they relate to the (stage; state) values for this row and for a row
from the next stage.
[2]
(iii) Use the table to deduce where Henry should make his overnight stops to minimise the maximum
distance that he cycles on any day. Explain how you obtained this solution from the table.
[2]
© OCR 2012
4737 Jan12
Jan 2012
7
Henry is so pleased with his ride that he decides to do a longer ride. Again he will cycle up to 50 miles each
day, except the last day when he will cycle at most 20 miles. He wants to complete the ride in five days, and
he wants to minimise the maximum distance that he rides on any one day.
He will start at A and travel through B, C, D, E, F, G, H, I, J and K, in that order, and finish at L.
He will stay overnight on Wednesday, Thursday, Friday and Saturday at four of B, C, D, E, F, G, H, I, J
and K.
The distances between the places along the route are:
A–B = 30 miles, B–C = 15 miles, C–D = 35 miles,
E–F = 3 miles,
F–G = 30 miles, G–H = 10 miles,
I–J = 10 miles,
J–K = 10 miles, K–L = 5 miles.
D–E = 12 miles,
H–I = 25 miles,
(iv) (a)
Which is the furthest place from L that Henry must reach by Saturday night if he is to finish on
time?
[1]
(b)
Work backwards to deduce the furthest place from L that Henry must reach by Friday night,
Thursday night and Wednesday night.
[2]
(v) Find out where Henry could stay each night, and hence define appropriate states for each of stages 1, 2,
3 and 4. (Note that not every place need correspond to a (stage; state) label.)
[3]
(vi) Set up a dynamic programming tabulation, working backwards from stage 5, to minimise the maximum
distance that Henry must ride on any one day. Where should he make his overnight stops?
[5]
[Question 6 is printed overleaf.]
© OCR 2012
4737 Jan12
Turn over
8
Jan 2012
6
Rowena and Colin play a game in which each chooses a letter. The table shows how many points Rowena
wins for each combination of letters. In each case the number of points that Colin wins is the negative of the
entry in the table. Both Rowena and Colin are trying to win as many points as possible.
Colin’s letter
N
P
Q
T
4
1
−1
−2
1
3
1
−1
5
1
2
−1
0
1
1
−1
W
X
Y
Z
Rowena’s
letter
(i) Write down Colin’s play-safe strategy, showing your working. What is the maximum number of points
that Colin can win if he plays safe?
[3]
(ii) Explain why Rowena would never choose the letter W.
[1]
Rowena uses random numbers to choose between her three remaining options, so that she chooses X, Y and
Z with probabilities x, y and z, respectively. Rowena then models the problem of which letter she should
choose as the following LP.
Maximise
M=m−1
subject to
m 2x + 6y + z,
m 4x + 2y,
m 3y + 2z,
m 2x + 2z,
x+y+z 1
and
m
0, x
0, y
0, z
0
(iii) Show how the expression 2x + 6y + z was formed.
[2]
The Simplex algorithm is used to solve the LP problem. The solution has x = 0.3, y = 0.2 and z = 0.5.
(iv) Show that the optimal value of M is 0.6.
[2]
Colin then models the problem of which letter he should choose in a similar way. When Rowena plays using
her optimal solution, from above, Colin finds that he should never choose the letter N. Letting p, q and t
denote the probabilities that he chooses P, Q and T, respectively, Colin obtains the following equations.
−3p + q – t = −0.6
−p – 2q + t = −0.6
p − q − t = −0.6
p+q+t=1
(v) Explain how the equation −3p + q – t = −0.6 is obtained.
[3]
(vi) Use the third and fourth equations to find the value of p. Hence find the values of q and t.
[2]
Copyright Information
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 and is freely available to download from our public website (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 1GE.
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.
© OCR 2012
4737 Jan12
Jan 2012
1
2
Five film studies students need to review five different films for an assignment, but only have one evening
left before the assignment is due in. They decide that they will share the work out so that each of them
reviews just one film.
Jack (J ) wants to review a horror movie; Karen (K ) wants to review an animated film; Lee (L) wants to
review a film that is suitable for family viewing; Mark (M ) wants to review an action adventure film and
Nikki (N ) wants to review anything that is in 3D.
The film “Somewhere” (S ) has been classified as a horror movie and is being shown in 3D; “Tornado
Terror” (T ) has been classified as an action adventure film that is suitable for family viewing; “Underwater”
(U ) is an animated action adventure film; “Vampires” (V ) is an animated horror movie that is suitable for
family viewing and “World” (W ) is an animated film.
(i) Draw a bipartite graph to show which student (J, K, L, M, N ) wants to review which films (S, T, U,
V, W).
[1]
Initially Jack says that he will review “Somewhere”, Karen then chooses “Underwater” and Lee chooses
“Tornado Terror”, but this would leave both Mark and Nikki with films that they do not want.
(ii) Write down the shortest possible alternating path starting from Nikki, and hence write down an
improved, but still incomplete, matching.
[2]
(iii) From this incomplete matching, write down the shortest possible alternating path starting from
“World”, and hence write down a complete matching between the students and the films.
[2]
(iv) Show that this is the only possible complete matching between the students and the films.
© OCR 2012
4737 Jan12
[1]
Jan 2012
2
3
The table lists the durations (in minutes), immediate predecessors and number of workers required for each
activity in a project to decorate a room.
Activity
Duration
(minutes)
Immediate
predecessors
Number of
workers
A
Cover furniture with dust sheets
20
–
1
B
Repair any cracks in the plaster
100
A
1
C
Hang wallpaper
60
B
1
D
Paint feature wall
90
B
1
E
Paint woodwork
120
C, D
1
F
Put up shelves
30
C
2
G
Paint ceiling
60
A
1
H
Clean paintbrushes
10
E, G
1
I
Tidy room
20
F, H
2
(i) Draw an activity network, using activity on arc, to represent the project. Your network will require a
dummy activity.
[2]
(ii) Carry out a forward pass and a backward pass through the activity network, showing the early event
time and the late event time at each vertex of your network.
State the minimum project completion time and list the critical activities.
[5]
(iii) Draw a resource histogram to show the number of workers required at each time when each activity
begins at its earliest possible start time.
[2]
Suppose that there is only one worker available at the start of the project, but another two workers are
available later.
(iv) Find the latest possible time for the other workers to start and still have the project completed on time.
Which activities could happen at the same time as painting the ceiling if the other two workers arrive at
this latest possible time?
[Do not change your resource histogram from part (iii).]
[2]
© OCR 2012
4737 Jan12
Turn over
Jan 2012
3
4
The famous fictional detective Agatha Parrot has been called in to investigate the theft of some jewels. Each
thief is known to have taken just one item of jewellery.
Agatha has invented a scoring system based on motive, opportunity and past experience. The table shows
the score for each of four suspects with each of three items of jewellery. The higher the score the more likely
the suspect is to have stolen that item of jewellery.
Suspect
Pearl necklace
Ruby ring
Sapphire
bracelet
Butler
80
100
20
Cook
40
35
60
Gardener
60
45
30
Handyman
20
100
80
(i) Assuming that three of these four suspects are the thieves, find who is most likely to have stolen each
item of jewellery for the total score to be maximised. State how each table of working was calculated.
Write down the two possible solutions for who should be suspected of stealing each item of jewellery
and who should be thought to be innocent.
[9]
Further evidence shows that the butler stole the sapphire bracelet.
(ii) Using this additional information, find out which suspect should be thought to be innocent. Explain
your reasoning.
[2]
© OCR 2012
4737 Jan12
Jan 2012
5
4
The diagram represents a system of roads through which traffic flows from a source, S, to a sink, T. The
weights on the arcs show the capacities of the roads in cars per minute.
S
80
A
20
10
40
B
25
30
C
25
45
25
D
T
20
E
30
65
F
(i) (a)
The cut α partitions the vertices into the sets {S, A, B, C}, {D, E, F, T}. Calculate the capacity of
cut α.
[1]
(b)
The cut β partitions the vertices into the sets {S, A, B, D}, {C, E, F, T}. Calculate the capacity of
cut β.
[1]
(c)
Using only the capacities of cuts α and β, what can you deduce about the maximum possible flow
through the system?
[1]
(ii) Explain why partitioning the vertices into sets {S, A, D, T}, {B, C, E, F} does not give a cut.
[1]
(iii) What do the double arcs between D and E and between E and F represent?
[1]
(iv) Explain why the maximum possible flow along CF must be less than 45 cars per minute.
[1]
(v) (a)
Show how a flow of 60 cars per minute along FT can be achieved.
[2]
Show that 60 cars per minute is the maximum possible flow through the system.
[2]
(b)
An extra road is opened linking S to C. Let the capacity of this road be x cars per minute.
(vi) Find the maximum possible flow through the new system, in terms of x where necessary, for the
different possible values of x.
[3]
© OCR 2012
4737 Jan12
Turn over
6
Jan 2012
5
Henry is doing a sponsored cycle ride for charity. He needs to finish at noon on Sunday. He can ride up to 50
miles each day, except Sunday when he can ride at most 20 miles if he is to finish on time.
The total length of the ride is 95 miles so Henry has allowed 3 days for the ride. Henry will start his ride at A
and travel through B, C, D and E, in that order, and finish on Sunday at F. He will stay overnight on Friday
and Saturday at two of the places B, C, D and E.
The distances between the places along the route are:
A–B = 30 miles, B–C = 15 miles, C–D = 35 miles,
D–E = 12 miles,
E–F = 3 miles.
To reach F on Sunday he must have reached at least D by Saturday night (since the distance from D to F is
less than 20 miles but C to F is more than 20 miles.)
Henry wants to use dynamic programming to minimise the maximum distance that he cycles on any day.
The stages will be the days. The places where Henry stays overnight will be the states. Henry starts on
Friday morning at A which has the (stage; state) label (0; 0). On Friday night he can either stay at B (1; 0) or
at C (1; 1). Depending on where he stays on Friday night, he can spend Saturday night at D (2; 0) or E (2; 1).
On Sunday he arrives at F (3; 0).
(i) Use this information and the table below to draw a network, labelled with stages and states, to show
the possible transitions between states. The arc weights should be the distances between the states. [2]
Henry uses dynamic programming, working backwards from stage 3, to find where he should stay overnight
to give the route for which the maximum on any day is a minimum. His tabulation is shown below.
Stage
State
Action
Working
Suboptimal minimax
2
0
0
15
15
1
0
3
3
0
0
max(50, 15)
50
1
0
max(35, 15)
35
1
max(47, 3)
0
max(30, 50)
1
max(45, 35)
1
0
(ii) (a)
(ii)(b)
0
45
In the last row of the table, the action value is 1. Explain what this tells you.
[1]
In the last row of the table, the working column is max(45, 35). Explain where each of the values
45 and 35 comes from and how they relate to the (stage; state) values for this row and for a row
from the next stage.
[2]
(iii) Use the table to deduce where Henry should make his overnight stops to minimise the maximum
distance that he cycles on any day. Explain how you obtained this solution from the table.
[2]
© OCR 2012
4737 Jan12
Jan 2012
7
Henry is so pleased with his ride that he decides to do a longer ride. Again he will cycle up to 50 miles each
day, except the last day when he will cycle at most 20 miles. He wants to complete the ride in five days, and
he wants to minimise the maximum distance that he rides on any one day.
He will start at A and travel through B, C, D, E, F, G, H, I, J and K, in that order, and finish at L.
He will stay overnight on Wednesday, Thursday, Friday and Saturday at four of B, C, D, E, F, G, H, I, J
and K.
The distances between the places along the route are:
A–B = 30 miles, B–C = 15 miles, C–D = 35 miles,
E–F = 3 miles,
F–G = 30 miles, G–H = 10 miles,
I–J = 10 miles,
J–K = 10 miles, K–L = 5 miles.
D–E = 12 miles,
H–I = 25 miles,
(iv) (a)
Which is the furthest place from L that Henry must reach by Saturday night if he is to finish on
time?
[1]
(b)
Work backwards to deduce the furthest place from L that Henry must reach by Friday night,
Thursday night and Wednesday night.
[2]
(v) Find out where Henry could stay each night, and hence define appropriate states for each of stages 1, 2,
3 and 4. (Note that not every place need correspond to a (stage; state) label.)
[3]
(vi) Set up a dynamic programming tabulation, working backwards from stage 5, to minimise the maximum
distance that Henry must ride on any one day. Where should he make his overnight stops?
[5]
[Question 6 is printed overleaf.]
© OCR 2012
4737 Jan12
Turn over
Jan 2012
6
8
Rowena and Colin play a game in which each chooses a letter. The table shows how many points Rowena
wins for each combination of letters. In each case the number of points that Colin wins is the negative of the
entry in the table. Both Rowena and Colin are trying to win as many points as possible.
Colin’s letter
N
P
Q
T
4
1
−1
−2
1
3
1
−1
5
1
2
−1
0
1
1
−1
W
X
Y
Z
Rowena’s
letter
(i) Write down Colin’s play-safe strategy, showing your working. What is the maximum number of points
that Colin can win if he plays safe?
[3]
(ii) Explain why Rowena would never choose the letter W.
[1]
Rowena uses random numbers to choose between her three remaining options, so that she chooses X, Y and
Z with probabilities x, y and z, respectively. Rowena then models the problem of which letter she should
choose as the following LP.
Maximise
M=m−1
subject to
m 2x + 6y + z,
m 4x + 2y,
m 3y + 2z,
m 2x + 2z,
x+y+z 1
and
m
0, x
0, y
0, z
0
(iii) Show how the expression 2x + 6y + z was formed.
[2]
The Simplex algorithm is used to solve the LP problem. The solution has x = 0.3, y = 0.2 and z = 0.5.
(iv) Show that the optimal value of M is 0.6.
[2]
Colin then models the problem of which letter he should choose in a similar way. When Rowena plays using
her optimal solution, from above, Colin finds that he should never choose the letter N. Letting p, q and t
denote the probabilities that he chooses P, Q and T, respectively, Colin obtains the following equations.
−3p + q – t = −0.6
−p – 2q + t = −0.6
p − q − t = −0.6
p+q+t=1
(v) Explain how the equation −3p + q – t = −0.6 is obtained.
[3]
(vi) Use the third and fourth equations to find the value of p. Hence find the values of q and t.
[2]
Copyright Information
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 and is freely available to download from our public website (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 1GE.
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.
© OCR 2012
4737 Jan12
2
June 2012
1
The six cadets in Red Team have been told to guard a building through the night, starting at 2200 hours and
finishing at 0800 hours the next day. Each will be on duty for either one hour or three hours and will then
hand over to the next cadet.
The table shows which duty each cadet has offered to take.
Duty start time (24 hour clock time)
Cadet
Amir
(A)
Becca
(B)
Chris
(C)
Dan
(D)
2200
0100
✓
✓
✓
(F)
0300
0400
✓
✓
✓
✓
0500
✓
✓
Emma (E)
Finn
0200
✓
✓
✓
(i) Draw a bipartite graph to represent this information.
[1]
Amir suggests that he should take the 2200 duty, hand over to Becca at 0100, she can hand over to Chris
at 0200, and Dan can take the 0400 duty. However, this leaves Emma and Finn to cover the 0300 and 0500
duties, and neither of them wants either of these.
(ii) Write down the shortest possible alternating path starting at the 0500 duty and hence write down an
improved but still incomplete matching between the cadets and the duties.
[2]
(iii) Augment this second incomplete matching by writing down a shortest possible alternating path, this
time starting from one of the cadets, to form a complete matching between the cadets and the duties.
Write down which cadet should take which duty.
[2]
© OCR 2012
4737 Jun12
June 2012
2
3
The cadets in Blue Team have been set a task that requires them to get inside a guarded building. Every two
hours one of them will attempt to get inside the building. Each cadet will have one attempt.
The table shows a score for each cadet attempting to get inside the building at each time. The higher the
score the more likely the cadet is to succeed.
Time
Cadet
2330
0130
0330
0530
0730
Gary
G
8
0
7
1
1
Hilary
H
9
2
7
0
2
Ieuan
I
10
4
9
3
5
Jenni
J
7
2
6
1
2
Ken
K
10
8
9
6
7
(i) Explain how to modify the table so that the Hungarian algorithm can be used to find the matching for
which the total score is maximised.
[1]
(ii) Show that, after modifying the table and then reducing rows and then columns, the reduced cost matrix
becomes:
2330
0130
0330
0530
0730
G
0
6
0
3
4
H
0
5
1
5
4
I
0
4
0
3
2
J
0
3
0
2
2
K
0
0
0
0
0
[3]
(iii) Complete the application of the Hungarian algorithm, stating how each table was formed. Write down
the order in which the cadets should attempt to get into the building to maximise the total score. If the
cadets use this solution, which one is least likely to succeed?
[4]
© OCR 2012
4737 Jun12
Turn over
June 2012
4
3
Throughout this question all clock times are in Greenwich Mean Time (GMT).
An aeroplane needs to arrive at its destination at 3pm. The places it can pass through on its route are shown
in the network, together with the flying times, in hours, between them.
(1; 0)
A
2
3
5
(2; 0)
D
(3; 0)
4
3
G
4
4
3
2
(0; 0)
start
6
B
(1; 1)
7
C
(1; 2)
3
E
(2; 1)
4
2
4
2
(4; 0)
destination
H (3; 1)
F
(2; 2)
3
Use a dynamic programming tabulation, working backwards from 3pm at the destination, to find the latest
time that the aeroplane could set off. If the aeroplane takes off at its latest time, which places does it pass
through, and at what time does it reach each of these places?
[9]
© OCR 2012
4737 Jun12
June 2012
4
5
A group of rowers have challenged some cyclists to see which team is fitter. There will be several rounds to
the challenge. In each round, the rowers and the cyclists each choose a team member and these two compete
in a series of gym exercises. The time by which the winner finishes ahead of the loser is converted into
points. These points are added to the score for the winner’s team and taken off the score for the loser’s team.
The table shows the expected number of points added to the score for the rowers for each combination of
competitors.
Cyclists
Rowers
Chris
Jamie
Wendy
Andy
−3
2
−4
Kath
5
4
−6
Zac
1
−4
−5
(i) Regarding this as a zero-sum game, find the play-safe strategy for the rowers and the play-safe strategy
for the cyclists. Show that the game is stable.
[5]
Unfortunately, Wendy and Kath are needed by their coaches and cannot compete.
(ii) Show that the resulting reduced game is unstable.
[2]
(iii) Suppose that the cyclists are equally likely to choose Chris or Jamie. Calculate the expected number of
points added to the score for the rowers when they choose Andy and when they choose Zac.
[2]
Suppose that the cyclists use random numbers to choose between Chris and Jamie, so that Chris is chosen
with probability p and Jamie with probability 1− p.
(iv) Showing all your working, calculate the optimum value of p for the cyclists.
[3]
(v) The rowers use random numbers in a similar way to choose between Andy and Zac, so that Andy is
chosen with probability q and Zac with probability 1− q. Calculate the optimum value of q.
[3]
© OCR 2012
4737 Jun12
Turn over
June 2012
5
6
The network represents a system of pipes through which fluid can flow. The weights on the arcs show the
lower and upper capacities for the pipes, in litres per second.
(2, 5)
A
(4, 6)
D
(2, 2)
(3, 4)
(0, 5)
(1, 3)
(1, 3)
K
(3, 7)
(1, 4)
F
(1, 5)
(2, 4)
I
J
(1, 3)
H
(0, 6)
C
(2, 4)
E
B
(3, 5)
G
(0, 6)
(3, 5)
L
(i) Identify the source and explain how you know that the sink is at G.
[2]
(ii) Calculate the capacity of the cut that separates {A, B, C, D, E, F} from {G, H, I, J, K, L}.
[2]
(iii) Assuming that a feasible flow exists, explain why arc JG must be at its lower capacity. Write down the
flows in arcs HK and IL.
[3]
(iv) Assuming that a feasible flow exists, explain why arc HI must be at its upper capacity. Write down the
flows in arcs EH and CB.
[4]
(v) Show a flow of 10 litres per second through the system.
[2]
(vi) Using your flows from part (v), label the arrows on the diagram to show the excess capacities and the
potential backflows.
[2]
(vii) Write down a flow augmenting path from your diagram in part (vi), but do not update the excess
capacities and the potential backflows. Hence show a maximum flow through the system, and state
how you know that the flow is maximal.
[3]
© OCR 2012
4737 Jun12
7
June 2012
6
Tariq wants to advertise his gardening services. The activities involved, their durations (in hours) and
immediate predecessors are listed in the table.
Activity
Duration
(hours)
Immediate
predecessors
A
Choose a name for the gardening service
2
–
B
Think about what the text needs to say
3
–
C
Arrange a photo shoot
2
B
D
Visit a leaflet designer
3
A, C
E
Design website
5
A, C
F
Get business cards printed
3
D
G
Identify places to publicise services
2
A, C
H
Arrange to go on local radio
3
B
I
Distribute leaflets
4
D, G
J
Get name put on van
1
E
(i) Draw an activity network, using activity on arc, to represent the project.
[2]
(ii) Carry out a forward pass and a backward pass through the activity network, showing the early event
time and the late event time at each vertex of your network. State the minimum project completion
time and list the critical activities.
[5]
Tariq does not have time to complete all the activities on his own, so he gets some help from his friend Sally.
Sally can help Tariq with any of the activities apart from C, H and J. If Tariq and Sally share an activity, the
time it takes is reduced by 1 hour. Sally can also do any of F, G and I on her own.
(iii) Describe how Tariq and Sally should share the work so that activity D can start 5 hours after the start of
the project.
[2]
(iv) Show that, if Sally does as much of the work as she can, she will be busy for 18 hours. In this case, for
how many hours will Tariq be busy?
[3]
(v) Explain why, if Sally is busy for 18 hours, she will not be able to finish until more than 18 hours from
the start. How soon after the start can Sally finish when she is busy for 18 hours?
[2]
(vi) Describe how Tariq and Sally can complete the project together in 18 hours or less.
© OCR 2012
4737 Jun12
[3]
Jan 2013
2
1
A TV soap opera has five main characters, Alice (A), Bob (B), Charlie (C), Dylan (D) and Etty (E). A
different character is scheduled to play the lead in each of the next five episodes. Alice, Dylan and Etty are
all in the episode about the fire (F), but Bob and Charlie are not. Alice and Bob are the only main characters
in the episode about the gas leak (G). Alice, Charlie and Etty are the only main characters in the episode
about the house break-in (H). The episode about the icy path (I) stars Alice and Charlie only. The episode
about the jail break (J) does not star any of the main characters who were in the episodes about the fire or
the house break-in.
(i) Draw a bipartite graph to show which main characters (A, B, C, D, E) are in which of the next five
episodes (F, G, H, I, J).
[1]
The writer initially decides to make Alice play the lead in the episode about the fire, Bob in the episode
about the gas leak and Charlie in the episode about the house break-in.
(ii) Write down the shortest possible alternating path starting from Dylan. Hence draw the improved, but
still incomplete, matching that results.
[2]
(iii) From this incomplete matching, write down the shortest possible alternating path starting from the
character who still has no leading part allocated. Hence draw the complete matching that results. [2]
(iv) By starting with the episode about the jail break, explain how you know that this is the only possible
complete matching between the characters and the episodes.
[2]
© OCR 2013
4737 Jan13
3
Jan 2013
2
A project is represented by this activity network. The weights (in brackets) on the arcs represent activity
durations, in minutes.
D(6)
A(10)
B(8)
G(5)
E(4)
H(3)
C(9)
F(8)
(i) Complete the table in the answer book to show the immediate predecessors for each activity.
[2]
(ii) Carry out a forward pass and a backward pass through the activity network, showing the early event
time and the late event time at each vertex of your network. State the minimum project completion
time and list the critical activities.
[5]
Suppose that the start of one activity is delayed by 2 minutes.
(iii) List each activity which could be delayed by 2 minutes with no change to the minimum project
completion time.
[2]
(iv) Without altering your diagram from part (ii), state the effect that a delay of 2 minutes on activity A
would have on the minimum project completion time. Name another activity which could be delayed
by 2 minutes, instead of A, and have the same effect on the minimum project completion time.
[2]
(v) Without altering your diagram from part (ii), state what effect a delay of 2 minutes on activity C would
have on the minimum project completion time.
[1]
© OCR 2013
4737 Jan13
Turnover
Jan 2013
4
3
Agatha Parrot is in her garden and overhears her neighbours talking about four new people who have moved
into her village. Each of the new people has a different job, and Agatha’s neighbours are guessing who has
which job.
Using the information she has overheard, Agatha counts how many times she heard it guessed that each
person has each job.
Nurse
Police officer
Radiographer
Teacher
Jill Jenkins
7
8
8
8
Kevin Keast
8
4
5
7
Liz Lomax
5
1
0
4
Mike Mitchell
8
3
4
4
Agatha wants to find the allocation of people to jobs that maximises the total number of correct guesses. She
intends to use the Hungarian algorithm to do this. She starts by subtracting each value in the table from 10.
(i) Write down the table which Agatha gets after she has subtracted each value from 10. Explain why she
did a subtraction.
[2]
(ii) Apply the Hungarian algorithm, reducing rows first, to find which job Agatha concludes each person
has. State how each table of working was calculated from the previous one.
[8]
Agatha later meets Liz Lomax and is surprised to find out that she is the radiographer.
(iii) Using this additional information, but without formally using the Hungarian algorithm, find which job
Agatha should now conclude each person has. Explain how you know that there is no better solution in
which Liz is the radiographer.
[2]
© OCR 2013
4737 Jan13
5
Jan 2013
4
The diagram represents a system of pipes through which fluid can flow from two sources, S1 and S2, to a
sink, T. Most of the pipes have valves which restrict the flow to one direction only. However, the flow in arc
DE can be in either direction. The weights on the arcs show the lower capacities and the upper capacities of
the pipes in litres per second.
S1
A
(5, 12)
(3, 10)
D
(0, 10)
(8, 10)
(1, 8)
B
(5, 9)
(2, 3)
S2
(0, 9)
C
(2, 15)
T
(4, 5)
(10, 20)
E
(i) Add a supersource, S, to the copy of the diagram in the answer book, and weight the arcs attached to it
with appropriate lower and upper capacities.
[2]
(ii)The cut a partitions the vertices into the sets {S, S1, S2, A, C}, {B, D, E, T}. By considering the cut arcs
[3]
only, calculate the maximum and minimum capacities of cut a.
(iii) Show that the maximum capacity of the cut {S, S1, S2, A, E}, {B, C, D, T} is 47 litres per second.
A flow is set up in which the arcs S1A, S1B, S2C, AE, CE and DT are all at their lower capacities.
(iv) Show the flow in each arc on the diagram in the answer book, indicating the direction of the flow in
arc DE.
[2]
(v) What is the maximum amount, in litres per second, by which the flow can be augmented using the
[1]
routes S1ADT and S2CET ?
(vi) Find the maximum possible flow through the system, explaining how you know both that this is
feasible and that it cannot be exceeded.
[2]
© OCR 2013
4737 Jan13
[2]
Turnover
6
Jan 2013
5
Rose and Colin are playing a game in which they each have four cards. Each player chooses a card from
those in their hand, and simultaneously they show each other the cards they have chosen. The table below
shows how many points Rose wins for each combination of cards. In each case the number of points that
Colin wins is the negative of the entry in the table. Both Rose and Colin are trying to win as many points as
possible.
Colin’s card
–2
4
2
–6
Rose’s
card
3
–3
–5
5
–4
4
–2
–5
1
5
–1
–3
(i) What is the greatest number of points that Colin can win when Rose chooses ● and which card does
Colin need to choose to achieve this?
[1]
(ii) Explain why Rose should never choose ◆ and find the card that Colin should never choose. Hence
reduce the game to a 3 × 3 pay-off matrix.
[3]
(iii) Find the play-safe strategy for each player on the reduced game and show whether or not the game is
stable.
[4]
Rose makes a random choice between her cards, choosing ● with probability x, ■ with probability y, and ▲
with probability z. She formulates the following LP problem to be solved using the Simplex algorithm:
maximise
M = m – 6,
subject to
m G 4x + 10y,
m G 9x + 3y + 11z,
m G 2x + 10y + z,
x + y + z G 1,
x H 0, y H 0, z H 0, m H 0.
(You are not required to solve this problem.)
and
(iv) Explain how 9x + 3y + 11z was obtained.
The Simplex algorithm is used to solve the LP problem. The solution has x =
[2]
(v) Calculate the optimal value of M.
© OCR 2013
7
27
14
,y=
,z=
.
48
48
48
[2]
4737 Jan13
7
Jan 2013
6
Simon makes playhouses which he sells through an agent. Each Sunday the agent orders the number of
playhouses she will need Simon to deliver at the end of each day. The table below shows the order for the
coming week.
Day
Number of
playhouses
Monday
Tuesday
Wednesday
Thursday
Friday
2
3
2
2
4
Simon can make up to 3 houses each day, except for Wednesday when he can make at most 2 houses.
Because of limited storage space, Simon can store at most 2 houses overnight from one day to the next,
although the number in store does not restrict how many houses Simon can make the next day.
The process is modelled by letting the stages be the days and the states be the numbers of houses stored
overnight.
Simon starts the week, on Monday morning, with no houses in storage. This means that the start of Monday
morning has (stage; state) label (0; 0). Simon wants to end the week on Friday afternoon with no houses in
storage, so the start of Saturday morning will have (stage; state) label (5; 0).
(i) Explain why the (stage; state) label (4; 0) is not needed.
[2]
Simon wants to draw up a production plan showing how many houses he needs to make each day. He prefers
not to have to make several houses on the same day so he assigns a ‘cost’ that is the square of the number of
houses made that day, apart from Monday when the ‘cost’ is the cube of the number of houses made. So, for
example, if he makes 3 houses one day the cost is 9 units, unless it is Monday when the cost is 27 units.
(ii) Complete the diagram in the answer book to show all the possible production plans and weight the arcs
with the costs.
[6]
Simon wants to find a production plan that minimises the sum of the costs.
(iii) Set up a dynamic programming tabulation, working backwards from (5; 0), to find a production plan
that solves Simon’s problem.
[8]
(iv) Write down the number of houses that he should make each day with this plan.
© OCR 2013
4737 Jan13
[1]
2
June 2013
1
Six students, Alice (A), Beth (B), Correy (C), Drew (D), Edmund (E) and Fred (F) are moving into a shared
house together at university. They have viewed the house and now they are choosing their rooms.
Alice wants a room at the front of the house, so that she can keep a check on her car, but she does not want
a ground floor room; Beth wants a room at the back of the house; Correy wants a ground floor room; Drew
wants a first floor room, and he does not want a small room because he needs enough space for his drum kit;
Edmund wants a room that is next to the bathroom; Fred wants a large room on the first floor, because he
thinks that will be the best room in the house.
The table below shows the features of each of the six rooms available.
Room
Floor
Front or back
Size of room
Next to
1
Ground
Front
Small
Lounge
2
Ground
Back
Large
Kitchen
3
First
Back
Medium
Stairs
4
First
Front
Large
Stairs
5
First
Back
Medium
Bathroom
6
First
Front
Small
Bathroom
(i) Draw a bipartite graph to show the possible pairings between the students (A, B, C, D, E and F) and the
rooms (1, 2, 3, 4, 5 and 6).
[2]
Initially Alice chooses room 4, Beth chooses room 3 and Correy chooses room 2.
(ii) Show how the remaining rooms can be allocated so that two of Drew, Edmund and Fred are happy
with the choices. State which student is not happy with this arrangement. Draw a second bipartite
graph to show the resulting incomplete matching between five of the students and five of the rooms.
[2]
(iii) Construct an alternating path, starting from the student without a room and ending at the room that was
not used, and hence find a complete matching between the students and the rooms. Write down a list
showing which student should be given which room.
[2]
© OCR 2013
4737/01 Jun13
3
June 2013
2
Alice (A), Beth (B), Correy (C), Drew (D), Edmund (E) and Fred (F) live in a shared student house. They
have decided that they will each do the cooking one day of the week, and on Saturdays they will have a
takeaway meal. They each give a score from 1 to 10 to the different days of the week. The higher the score
the more the student would like to cook on that day. The results are shown below.
Day
Student
Sun
Mon
Tue
Wed
Thur
Fri
Alice (A)
10
6
10
8
5
5
Beth (B)
6
2
10
1
4
6
Correy (C)
10
9
8
4
2
10
Drew (D)
9
4
9
5
5
8
Edmund (E)
9
8
9
8
7
8
Fred (F)
3
4
10
3
1
4
(i) Explain how to modify the table so that the Hungarian algorithm can be used to find the matching for
which the total score is maximised.
[1]
(ii) Show that modifying the table, and then reducing rows first and columns second gives the reduced cost
matrix below.
Day
Student
Sun
Mon
Tue
Wed
Thur
Fri
Alice (A)
0
3
0
1
3
5
Beth (B)
4
7
0
8
4
4
Correy (C)
0
0
2
5
6
0
Drew (D)
0
4
0
3
2
1
Edmund (E)
0
0
0
0
0
1
Fred (F)
7
5
0
6
7
6
[3]
(iii) Complete the application of the Hungarian algorithm, stating how each table was formed.
Use your final matrix to decide on which day Fred cooks.
[5]
Suppose that Alice cooks on Sunday.
(iv) Write a list showing on which day each student cooks according to the matching in which the total
score is maximised.
[2]
© OCR 2013
4737/01 Jun13
Turn over
4
June 2013
3
Molly is trying to book a holiday cottage for a short holiday. The activities involved, their durations (in
minutes) and immediate predecessors are listed in the table.
Activity
Duration (mins)
Immediate
predecessors
A
Choose a weekend for the
holiday
10
–
B
Decide on a region to visit for
the holiday
5
–
C
Look at maps and find suitable
locations
20
B
D
Go online and find out what
cottages are available
15
A, C
E
Find out what there is to do
near each location
25
C
F
Decide how much she wants
to spend on the cottage
5
D
G
Look up train services to the
nearest stations
15
A, C
H
Choose a cottage and book it
30
E, F, G
(i) Draw an activity network, using activity on arc, to represent the project.
[2]
(ii) Carry out a forward pass and a backward pass through the activity network, showing the early event
time and the late event time at each vertex of your network.
[3]
(iii) State the minimum project completion time, assuming that Molly can share the tasks out with her
friends, and list the critical activities.
[2]
Each activity requires one person and activities cannot be shared between people. Molly must do activities
A, B, F and H herself.
(iv) Draw up a schedule to show how Molly and two friends can complete the project in the minimum
project completion time.
[2]
(v) Draw up a schedule to show how Molly and just one friend can complete the project in just 5 minutes
more than the minimum project completion time from part (iii).
[2]
© OCR 2013
4737/01 Jun13
5
June 2013
4
The network represents a system of pipes through which oil can flow. The weights on the arcs show the
capacities for the pipes, in litres per second.
A
D
5
6
B
G
2
4
E
H
10
5
20
F
13
K
3
12
J
7
12
10
18
C
8
I
4
6
L
(i) Explain how you know that the source is at F and identify the sink.
[2]
(ii) Calculate the capacity of the cut that separates {A, B, C, D, E, F} from {G, H, I, J, K, L}.
[1]
(iii) Explain why in any feasible flow pipe BA cannot be at its full capacity.
Explain why the flow in pipe FC cannot exceed 15 litres per second.
[3]
(iv) Show a flow through the system in which the flows in pipes FC, IL and KJ are as large as possible and
nothing flows in pipes ED, FE and IH.
[2]
(v) Using your flows from part (iv), label the arrows on the diagram in the answer book to show the excess
capacities and the potential backflows. Write down a flow augmenting path but do not update the
excess capacities and potential backflows on your diagram. Hence state the maximum flow through the
system, in litres per second, and write down a cut that shows that the flow is maximal.
[5]
© OCR 2013
4737/01 Jun13
Turn over
June 2013
5
6
A delivery company needs to transport heavy loads from its base to a depot. Each of the roads which it can
use has a maximum weight limit. The directed network below represents the roads which can be used to get
from the base to the depot. Road junctions are labelled with (stage; state) labels. The weights on the arcs
represent weight limits in tonnes.
(1; 0)
18
16
20
22
(0; 0)
26
17
(3; 0)
20
16
15
(2; 1)
18
14
(3; 1)
15
(2; 2)
(i) Explain what a maximin route is.
(4; 0)
24
13
13
(1; 2)
12
17
14
(1; 1)
(2; 0)
11
(3; 2)
[1]
(ii) Set up a dynamic programming tabulation, working backwards from stage 4, to find the maximum
load which one truck can carry (in tonnes, including the weight of the truck) in one journey from the
base to the depot. Find all the routes for which this is the maximin.
[12]
(iii) The road connecting (2; 0) to (3; 1) is to be strengthened so that it can carry 20 tonnes. Find the
maximum load which one truck will be able to carry (in tonnes, including the weight of the truck),
explaining how you know that no greater load can be carried. (Do not use a dynamic programming
tabulation for this part.)
[2]
© OCR 2013
4737/01 Jun13
7
June 2013
6
A team from the Royal Hotel have challenged a team from the Carlton Hotel to a darts competition. In this
competition there are seven rounds, and the teams must choose a player for each round, although players
may play more than one round.
Each round is made up of five ‘legs’ and the team that wins the most legs (out of 35) wins the competition.
On the basis of their performances so far, the number of legs that each member of the Royal team can expect
to win (out of five) against each member of the Carlton team is shown below.
Carlton
Legs won
Royal
Jeff
Kathy
Leo
Greg
2
3
4
Hakkim
4
3
1
Iona
1
0
3
The teams want to choose which player to put in for each round to maximise the number of legs they expect
to win.
(i) If the Royal team chooses Greg and the Carlton team chooses Jeff, how many legs will the Carlton
team expect to win?
[1]
To convert the game into a zero-sum game, each value in the table is doubled and then 5 is subtracted.
(ii) Construct the resulting table for the zero-sum game.
[1]
(iii) Find the play-safe strategies for the zero-sum game, showing your working. Explain how you know
that the game is not stable. State which player is best for the Carlton team if they know that the Royal
team will play safe.
[5]
(iv) Use a dominance argument to explain why the Royal team should not choose Iona.
[1]
The Royal team chooses a player for the next round by using random numbers to choose between Greg and
Hakkim, where the probability of choosing Greg is p and the probability of choosing Hakkim is 1− p.
(v) Show that the expected number of legs that the Royal team win when the Carlton team chooses Jeff
is given by 4 − 2p and find the corresponding expressions for when Kathy is chosen and when Leo is
chosen.
[3]
(vi) Use a graphical method to find the optimal value of p for the Royal team, and calculate how many legs
the Royal team can expect when this value of p is used.
[4]
Suppose, instead, the team that wins the most rounds wins the competition. The winner of each round is the
team that wins the most legs (out of five) in that round and there are still seven rounds in the competition.
(vii) Give an example to show that it is possible to win the most legs without winning the most rounds.
[1]
© OCR 2013
4737/01 Jun13
2
Jan 2014
1
Six students are choosing their tokens for a board game. The bipartite graph in Fig. 1 shows which token
each student is prepared to have.
Adele
(A)
(B)
Battleship
Ezra
(E)
(F)
Flat iron
Jonah
(J)
(O)
Old boot
Lily
(L)
(R)
Racing car
Molly
(M)
(S)
Scottie dog
Noah
(N)
(T)
Top hat
Fig.1
Initially Ezra takes the flat iron, Jonah the old boot, Lily the racing car and Molly the scottie dog. This
leaves Adele and Noah with tokens that they do not want. This incomplete matching is shown in Fig. 2
below.
Adele
(A)
(B)
Battleship
Ezra
(E)
(F)
Flat iron
Jonah
(J)
(O)
Old boot
Lily
(L)
(R)
Racing car
Molly
(M)
(S)
Scottie dog
Noah
(N)
(T)
Top hat
Fig.2
(i) Write down the shortest possible alternating path that starts at A and finishes at either B or T. Hence
write down a matching that only excludes Noah and one of the tokens.
[2]
(ii) Working from the incomplete matching found in part (i), write down the shortest possible alternating
path that starts at N and finishes at whichever of B and T has still not been taken. Hence write down a
complete matching between the students and the tokens.
[2]
(iii) By starting at B on Fig. 1, show that there are exactly two complete matchings between the students
and the tokens.
[2]
© OCR 2014
4737/01 Jun14
3
Jan 2014
2
The network models a cooling system in a factory. Coolant starts at A, B and C and flows through the
system.
The arcs model components of the cooling system and the weights show the maximum amount of coolant
that can flow through each component of the system (measured in litres per second). The arrows show the
direction of flow.
A
8
D
12
G
6
7
B
5
E
4
3
C
H
6
F
5
(i) Add a supersource, S, and a supersink, T, to the copy of the network in your answer book. Connect S
and T to the network using appropriately weighted arcs.
[1]
(ii) (a) Find the capacity of the cut that separates A, B, C and E from D, F, G and H.
[1]
(b) Find the capacity of the cut that separates A, B, C, D, E and F from G and H.
[1]
(c) What can you deduce from this value about the maximum flow through the system?
[1]
(iii) Find the maximum possible flow through the system and prove that this is the maximum.
[3]
(iv) Describe what effect increasing the capacity of CE would have on the maximum flow.
[2]
© OCR 2014
4737/01 Jun14
Turnover
4
Jan 2014
3
Each of five jobs is to be allocated to one of five workers, and each worker will have one job. The table
shows the cost, in £, of using each worker on each job. It is required to find the allocation for which the total
cost is minimised.
Job
Worker
Plastering
Rewiring
Shelving
Tiling
Upholstery
Gill
25
50
34
40
25
Harry
36
42
48
44
45
Ivy
27
50
45
42
26
James
40
46
28
45
50
Kelly
34
48
34
50
40
(i) Construct a reduced cost matrix by first reducing rows and then reducing columns. Cross through the
0’s in your reduced cost matrix using the least possible number of horizontal or vertical lines. [Try to
ensure that the values in your table can still be read.]
[4]
(ii) Augment your reduced cost matrix and hence find a minimum cost allocation. Write a list showing
which job should be given to which worker for your minimum cost allocation, and calculate the total
cost in this case.
[4]
Gill decides that she does not like the job she has been allocated and increases her cost for this job by £100.
New minimum cost allocations can be found, each allocation costing just £1 more than the minimum cost
allocation found in part (ii).
(iii) Use the grid in your answer book to show the positions of the 0’s and 1’s in the augmented reduced
cost matrix from part (ii). Hence find three allocations, each costing just £1 more than the minimum
cost allocation found in part (ii) and with Gill having a different job to the one allocated in part (ii). [5]
© OCR 2014
4737/01 Jun14
Jan 2014
4
5
Ross and Collwen are playing a game in which each secretly chooses a magic spell. They then reveal their
choices, and work out their scores using the tables below. Ross and Collwen are both trying to get as large a
score as possible.
Collwen’s choice
Score for
Ross
Ross’s
choice
Fire
Ice
Gale
Fire
1
7
2
Ice
6
2
4
Gale
5
1
3
Collwen’s choice
Score for
Collwen
Fire
Ice
Gale
Fire
7
1
6
Ice
2
6
4
Gale
3
7
5
Ross’s
choice
(i) Explain how this can be rewritten as the following zero-sum game.
Collwen’s choice
Ross’s
choice
Fire
Ice
Gale
Fire
–3
3
–2
Ice
2
–2
0
Gale
1
–3
–1
[2]
(ii) If Ross chooses Ice what is Collwen’s best choice?
(iii) Find the play-safe strategy for Ross and the play-safe strategy for Collwen, showing your working.
Explain how you know that the game is unstable.
[5]
(iv) Show that none of Collwen’s strategies dominates any other.
(v) Explain why Ross would never choose Gale, hence reduce the game to a 2×3 zero-sum game, showing
the pay-offs for Ross.
[2]
Suppose that Ross uses random numbers to choose between Fire and Ice, choosing Fire with probability p
and Ice with probability 1 - p .
(vi) Use a graphical method to find the optimal value of p for Ross.
© OCR 2014
4737/01 Jun14
[1]
[3]
[3]
Turnover
6
Jan 2014
5
Following a promotion at work, Khalid needs to clear out his office to move to a different building. The
activities involved, their durations (in hours) and immediate predecessors are listed in the table below.
You may assume that some of Khalid’s friends will help him and that once an activity is started it will be
continued until it is completed.
Activity
Duration (hours)
Immediate
predecessors
A
Sort through cupboard and throw out rubbish
4
-
B
Get packing boxes
1
-
C
Sort out items from desk and throw out rubbish
3
-
D
Pack remaining items from cupboard in boxes
2
A, B
E
Put personal items from desk into briefcase
0.5
C
F
Pack remaining items from desk in boxes
1.5
B, C
G
Take certificates down and put into briefcase
1
-
H
Label boxes to be stored
0.5
D, F
(i) Represent this project using an activity network.
(ii) Carry out a forward pass and a backward pass through the activity network, showing the early event
time and late event time at each vertex of your network. State the minimum project completion time
and list the critical activities.
[5]
(iii) How much longer could be spent on sorting the items from the desk and throwing out the rubbish
(activity C ) without it affecting the overall completion time?
[1]
Khalid says that he needs to do activities A, C, E and G himself. These activities take a total of 8.5 hours.
(iv) By considering what happens if Khalid does A first, and what happens if he does C first, show that the
project will take more than 8.5 hours.
[2]
[4]
(v) Draw up a schedule to show how just two people, Khalid and his friend Mia, can complete the project
in 9 hours. Khalid must do A, C, E and G and activities cannot be shared between Khalid and Mia. [2]
© OCR 2014
4737/01 Jun14
7
Jan 2014
6
The table below shows an incomplete dynamic programming tabulation to solve a maximin problem. Do
not write your answer on this copy of the table.
Stage
State
Action
Working
Suboptimal
maximin
3
0
0
6
6
1
0
1
1
2
0
3
3
0
0
min(3, 6) = 3
0
min(1, 6) = 1
1
min(1, 1) = 1
2
min(2, 3) = 2
2
2
2
min(1, 3) = 1
1
0
0
min (3, ) =
1
min (4, ) =
1
1
min (3, ) =
2
1
min (3, ) =
2
min (1, ) =
0
min (5, ) =
1
min (3, ) =
2
min (4, ) =
2
1
1
0
0
3
(i) Complete the working and suboptimal maximin columns on the copy of the table in your answer book.
[4]
(ii)Use your answer to part (i) to write down the maximin value and the corresponding route. Give your
route using (stage; state) variables.
[3]
(iii) Draw the network that is represented in the table.
The network represents a system of pipes and the arc weights show the capacities of the pipes, in litres per
second.
(iv) What does the answer to part (ii) represent in this network?
[3]
[1]
Question6continuesonpage8.
© OCR 2014
4737/01 Jun14
Turnover
8
Jan 2014
The weights of the arcs in the maximin route are each reduced by the maximin value and then a maximin
is found for the resulting network. This is done until the maximin value is 0. At this point the network is as
shown below.
(1; 0)
0
1
3
0
(0; 0)
0
(1; 1)
3
0
0
(3; 0)
0
2
0
0
(2; 1)
(3; 1)
0
3
(1; 2)
(2; 0)
(2; 2)
0
(4; 0)
0
(3; 2)
(v) (a) Describe how this solves the maximum flow problem on the original network.
(b) Draw this maximum flow and draw a cut with value equal to the value of the flow.
[1]
[2]
ENDOFQUESTIONPAPER
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