Question 4:
4 | 5 | 2 | 1 | 2
1 | 2 | 3 | 4
4 | 2 | 2 | 3 | 3
4
4 Noel is designing a hotel patio. It will consist of decking and paving.
Decking costs £4 per m2 and paving costs £2 per m2. He has a budget of £2500.
Noel prefers paving to decking, and he wants the area given to paving to be at least twice that given
to decking.
He wants to have as large a patio as possible.
Noel’s problem is formulated as the following LP.
Let x be the number of m2 of decking.
Let y be the number of m2 of paving.
Maximise P= x+y
subject to 2x+ y (cid:3)1250
2x- y (cid:3)0
x (cid:4)0
y (cid:4)0
(i) Use the simplex algorithm to solve this LP. Pivot first on the positive element in the ycolumn.
[6]
Noel would like to have at least 200m2 of decking.
(ii) Add a line corresponding to this constraint to your solution tableau from part (i), and modify
the resulting table either for two-stage simplex or the big-M method. Hence solve the
problem. [9]
Noel finally decides that he will minimise the annual cost of maintenance, which is given by
£(0.75x(cid:2)1.25y), subject to the additional constraint that there is at least 1000 m2 of patio.
(iii) Starting from your solution to part (ii), use simplex to solve this problem. [5]
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 2007 4772/01 June 07
INSTRUCTIONS TO CANDIDATES
(cid:127) Write your name in capital letters, your Centre Number and Candidate Number in the spaces
provided on the Answer Booklet.
(cid:127) Read each question carefully and make sure you know what you have to do before starting
your answer.
(cid:127) Answer all the questions.
(cid:127) You are permitted to use a graphical calculator in this paper.
(cid:127) Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
(cid:127) The number of marks is given in brackets [ ] at the end of each question or part question.
(cid:127) The total number of marks for this paper is 72.
(cid:127) You are advised that an answer may receive no marks unless you show sufficient detail of the
working to indicate that a correct method is being used.
This document consists of 4 printed pages.
SP (KN) T44381/4 © OCR 2008 [L/102/2660] OCR is an exempt Charity [Turn over
*CUP/T44381*
PMT
4772/01
ADVANCED GCE UNIT
MATHEMATICS (MEI)
Decision Mathematics 2
MONDAY 16 JUNE 2008 Afternoon
Time: 1 hour 30 minutes
Additional materials (enclosed): None
Additional materials (required):
Answer booklet (8 pages)
Graph paper
MEI Examination Formulae and Tables (MF2)
PMT
2
1 (a) The Plain English Society presents an annual “Foot in Mouth” award for “a truly baffling
comment”. In 2004 it was presented to Boris Johnson MP for a comment on the 12th December
2003 edition of “Have I Got News For You”:
“I could not fail to disagree with you less.”
(i) Explain why this can be rewritten as:
“I could succeed in agreeing with you more.” [2]
(ii) Rewrite the comment more simply in your own words without changing its meaning. [2]
(b) Two switches are to be wired between a mains electricity supply and a light so that when the state
of either switch is changed the state of the light changes (i.e. from off to on, or from on to off).
Draw a switching circuit to achieve this.
The switches are both 2-way switches, thus: [5]
(c) Construct a truth table to show the following.
[(a ∧ b) ∨ (((cid:1) a) ∧ ((cid:1) b))] ⇔ [(((cid:1) a) ∨ b) ∧ (a ∨ ((cid:1) b))] [7]
2 Jane has a house on a Mediterranean island. She spends eight weeks a year there, either visiting twice
for four weeks each trip or four times for two weeks each trip. Jane is wondering whether it is best for
her to fly out and rent a car, or to drive out.
Flights cost £500 return and car rental costs £150 per week.
Driving out costs £900 for ferries, road tolls, fuel and overnight expenses.
(i) Draw a decision tree to model this situation. Advise Jane on the cheapest option. [6]
As an alternative Jane considers buying a car to keep at the house. This is a long-term alternative, and
she decides to cost it over 10 years. She has to cost the purchase of the car and her flights, and compare
this with the other two options.
In her costing exercise she decides that she will not be tied to two trips per year nor to four trips per
year, but to model this as a random process in which she is equally likely to do either.
(ii) Draw a decision tree to model this situation. Advise Jane on how much she could spend on a car
using the EMV criterion. [8]
(iii) Explain what is meant by “the EMV criterion” and state an alternative approach. [2]
© OCR 2008 4772/01 Jun08
1 | 2 | 3 | 4
4 | 23 | 27 | 12 | 24
1 | 2 | 3 | 4
4 | 3 | 3 | 3 | 3
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.
*OCE/T67864*
PMT
ADVANCED GCE
4772
MATHEMATICS (MEI)
Decision Mathematics 2
Wednesday 17 June 2009
Candidates answer on the Answer Booklet
Morning
OCR Supplied Materials:
(cid:127) Answer Booklet (8 pages)
Duration: 1 hour 30 minutes
(cid:127) Graph paper
(cid:127) MEI Examination Formulae and Tables (MF2)
Other Materials Required:
None
* 4 7 7 2 *
INSTRUCTIONS TO CANDIDATES
(cid:127) Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces
provided on the Answer Booklet.
(cid:127) Use black ink. Pencil may be used for graphs and diagrams only.
(cid:127) Read each question carefully and make sure that you know what you have to do before starting your answer.
(cid:127) Answer all the questions.
(cid:127) You are permitted to use a graphical calculator in this paper.
(cid:127) Final answers should be given to a degree of accuracy appropriate to the context.
(cid:127) Do not write in the bar codes.
INFORMATION FOR CANDIDATES
(cid:127) The number of marks is given in brackets [ ] at the end of each question or part question.
(cid:127) You are advised that an answer may receive no marks unless you show sufficient detail of the working to
indicate that a correct method is being used.
(cid:127) The total number of marks for this paper is 72.
(cid:127) This document consists of 4 pages. Any blank pages are indicated.
© OCR 2009 [L/102/2660] OCR is an exempt Charity
SP (CW/CGW) T67864/7 Turn over
PMT
2
1 (a) The following was said in a charity appeal on Radio 4 in October 2006.
“It is hard to underestimate the effect that your contribution will make.”
Rewrite the comment more simply in your own words without changing its meaning. [2]
(b) A machine has three components, A, B and C, each of which is either active or inactive.
• The machine is active if A and B are both active.
• The machine is active if A is inactive and C is active.
• The machine is active if B is inactive and C is active.
• Otherwise the machine is inactive.
The states (active or inactive) of the components and the machine are to be modelled by a
combinatorial circuit in which “active” is represented by “true” and “inactive” is represented by
“false”.
Draw such a circuit. [7]
(c) Construct a truth table to show the following.
[(( ) (( ) )) (( ) )] [(( ) ( )) (( ) ( ))]
a ∧ b ∨ ∼ a ∧ c ∨ ∼ b ∧ c ⇔ ∼ ∼ a ∧ ∼ c ∨ ∼ b ∧ ∼ c [7]
2 Zoe is preparing for a Decision Maths test on two topics, Decision Analysis (D) and Simplex (S). She
has to decide whether to devote her final revision session to D or to S.
There will be two questions in the test, one on D and one on S. One will be worth 60 marks and the
other will be worth 40 marks. Historically there is a 50% chance of each possibility.
Zoe is better at D than at S. If her final revision session is on D then she would expect to score 80% of
the D marks and 50% of the S marks. If her final session is on S then she would expect to score 70% of
the S marks and 60% of the D marks.
(i) Compute Zoeʼs expected mark under each of the four possible circumstances, i.e. Zoe revising D
and the D question being worth 60 marks, etc. [5]
(ii) Draw a decision tree for Zoe. [5]
Michael claims some expertise in forecasting which question will be worth 60 marks. When he forecasts
that it will be the D question which is worth 60, then there is a 70% chance that the D question will be
worth 60. Similarly, when he forecasts that it will be the S question which is worth 60, then there is
a 70% chance that the S question will be worth 60. He is equally likely to forecast that the D or the S
question will be worth 60.
(iii) Draw a decision tree to find the worth to Zoe of Michaelʼs advice. [6]
© OCR 2009 4772 Jun09
PMT
3
3 A farmer has 40 acres of land. Four crops, A, B, C and D are available.
Crop A will return a profit of £50 per acre. Crop B will return a profit of £40 per acre.
Crop C will return a profit of £40 per acre. Crop D will return a profit of £30 per acre.
The total number of acres used for crops A and B must not be greater than the total number used for
crops C and D.
The farmer formulates this problem as:
Maximise 50a + 40b + 40c + 30d,
subject to a + b (cid:1) 20,
a + b + c + d (cid:1) 40.
(i) Explain what the variables a, b, c and d represent.
Explain how the first inequality was obtained.
Explain why expressing the constraint on the total area of land as an inequality will lead to a
solution in which all of the land is used. [3]
(ii) Solve the problem using the simplex algorithm. [10]
Suppose now that the farmer had formulated the problem as:
Maximise 50a + 40b + 40c + 30d,
subject to a + b (cid:1) 20,
a + b + c + d = 40.
(iii) Show how to adapt this problem for solution either by the two-stage simplex method or the big-M
method. In either case you should show the initial tableau and describe what has to be done next.
You should not attempt to solve the problem. [7]
Turn over
© OCR 2009 4772 Jun09
1 | 2 | 3 | 4 | 5
4 | 15 | 5 | 25 | 10 | 16
4 | 1 | 2 | 2 | 2 | 5
4 | ∞ | 2 | 3 | ∞ | 10 | 17
4 | 1 | 2 | 3 | 4 | 5 | 6
4 | 15 | 2 | 3 | 4 | 8 | 16
4 | 2 | 2 | 3 | 2 | 2 | 2