Formulation from word problem

2 Rahul is decorating a room. He needs to decorate at least \(30 \mathrm {~m} ^ { 2 }\) of the walls using paint, panelling or wallpaper. The cost ( \(\pounds\) ) and the time required (hours) to decorate \(1 \mathrm {~m} ^ { 2 }\) of wall are shown in the table. Rahul wants to complete decorating the walls in no more than 8 hours and wants to minimise the cost.
Set up an LP formulation for Rahul's problem, defining your variables. You are not required to solve this LP problem.

6 Kai mixes hot drinks using coffee and steamed milk.
The amounts ( ml ) needed and profit ( \(\pounds\) ) for a standard sized cup of four different drinks are given in the table. The table also shows the amount of the ingredients available. Kai makes the equivalent of \(w\) standard sized americanos, \(x\) standard sized cappuccinos, \(y\) standard sized flat whites and \(z\) standard sized lattes. He can make different sized drinks so \(w , x , y , z\) need not be integers. Kai wants to find the maximum profit that he can make, assuming that the customers want to buy the drinks he has made.

1. What is the minimum value of X for it to be worthwhile for Kai to make cappuccinos? Kai makes no cappuccinos.
2. Use the simplex algorithm to solve Kai's problem. The grids in the Printed Answer Booklet should have at least enough rows and columns and there should be at least enough grids to show all the iterations needed. Only record the output from each iteration, not any intermediate stages.
   Interpret the solution and state the maximum profit that Kai can make.

6 Jack is making pizzas for a party. He can make three types of pizza:

• There is enough dough to make 30 pizzas.
• Type Z pizzas use vegan cheese. Jack only has enough vegan cheese to make 2 type Z pizzas.
• At least half the pizzas made must be suitable for vegetarians.
• Jack has enough onions to make 50 type X pizzas or 20 type Y pizzas or 20 type Z pizzas or some mixture of the three types.

Suppose that Jack makes \(x\) type X pizzas, \(y\) type Y pizzas and \(z\) type Z pizzas.

1. Formulate the constraints above in terms of the non-negative, integer valued variables \(x , y\) and \(z\), together with non-negative slack variables \(s , t , u\) and \(v\). Jack wants to find out the maximum total number of pizzas that he can make.
2. 1. Set up an initial simplex tableau for Jack's problem.
   2. Carry out one iteration of the simplex algorithm, choosing your pivot so that \(x\) becomes a basic variable. When Jack carries out the simplex algorithm his final tableau is:

      \(P\)	\(x\)	\(y\)	\(z\)	\(s\)	\(t\)	\(u\)	\(v\)	RHS
      1	0	0	0	0	0	\(\frac { 3 } { 7 }\)	\(\frac { 2 } { 7 }\)	\(28 \frac { 4 } { 7 }\)
      0	0	0	0	1	0	\(- \frac { 3 } { 7 }\)	\(- \frac { 2 } { 7 }\)	\(1 \frac { 3 } { 7 }\)
      0	0	0	1	0	1	0	0	2
      0	1	0	0	0	0	\(\frac { 5 } { 7 }\)	\(\frac { 1 } { 7 }\)	\(14 \frac { 2 } { 7 }\)
      0	0	1	1	0	0	\(- \frac { 2 } { 7 }\)	\(\frac { 1 } { 7 }\)	\(14 \frac { 2 } { 7 }\)

3. Use this final tableau to deduce how many pizzas of each type Jack should make. Jack knows that some of the guests are vegans. He decides to make 2 pizzas of type \(Z\).
4. 1. Plot the feasible region for \(x\) and \(y\).
   2. Complete the branch-and-bound formulation in the Printed Answer Booklet to find the number of pizzas of each type that Jack should make.
      You should branch on \(x\) first. \section*{END OF QUESTION PAPER}

3 Manon makes apple cakes and banana cakes. Each apple cake is made with 3 eggs and 100 grams of flour. Each banana cake is made with 2 eggs and 150 grams of flour. Manon has 36 eggs and 1500 grams of flour.
Manon wants to make as many cakes as possible.
Formulate Manon's situation as a linear programming problem, clearly defining any variables you introduce.

8 Alli is planting garlic cloves and leek seedlings in a garden. The planting density is the number of plants that are planted per \(\mathrm { m } ^ { 2 }\) The planting densities and costs are shown in the table below.

3 Ben is packing eggs into boxes, labelled Town Box or Country Box. Each Town Box must contain 10 chicken eggs and 2 duck eggs. Each Country Box must contain 4 chicken eggs and 8 duck eggs. Ben has 253 chicken eggs and 151 duck eggs. Ben wants to pack as many boxes as possible. Formulate Ben's situation as a linear programming problem, defining any variables you introduce.

8 A sweet shop sells three different types of boxes of chocolate truffles. The cost of each type of box and the number of truffles of each variety in each type of box are given in the table below. \section*{2. Subject-specific Marking Instructions for AS Level Further Mathematics A} Annotations should be used whenever appropriate during your marking. The A, M and B annotations must be used on your standardisation scripts for responses that are not awarded either 0 or full marks. It is vital that you annotate standardisation scripts fully to show how the marks have been awarded. For subsequent marking you must make it clear how you have arrived at the mark you have awarded. An element of professional judgement is required in the marking of any written paper. Remember that the mark scheme is designed to assist in marking incorrect solutions. Correct solutions leading to correct answers are awarded full marks but work must not be judged on the answer alone, and answers that are given in the question, especially, must be validly obtained; key steps in the working must always be looked at and anything unfamiliar must be investigated thoroughly. Correct but unfamiliar or unexpected methods are often signalled by a correct result following an apparently incorrect method. Such work must be carefully assessed. When a candidate adopts a method which does not correspond to the mark scheme, escalate the question to your Team Leader who will decide on a course of action with the Principal Examiner.
If you are in any doubt whatsoever you should contact your Team Leader.
The following types of marks are available. \section*{M} A suitable method has been selected and applied in a manner which shows that the method is essentially understood. Method marks are not usually lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. In some cases the nature of the errors allowed for the award of an M mark may be specified. \section*{A} Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated Method mark is earned (or implied). Therefore M0 A1 cannot ever be awarded. \section*{B} Mark for a correct result or statement independent of Method marks. \section*{E} Mark for explaining a result or establishing a given result. This usually requires more working or explanation than the establishment of an unknown result.
Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored. Sometimes this is reinforced in the mark scheme by the abbreviation isw. However, this would not apply to a case where a candidate passes through the correct answer as part of a wrong argument.
d When a part of a question has two or more 'method' steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. (The notation 'dep*' is used to indicate that a particular mark is dependent on an earlier, asterisked, mark in the scheme.) Of course, in practice it may happen that when a candidate has once gone wrong in a part of a question, the work from there on is worthless so that no more marks can sensibly be given. On the other hand, when two or more steps are successfully run together by the candidate, the earlier marks are implied and full credit must be given.
e The abbreviation FT implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A and B marks are given for correct work only - differences in notation are of course permitted. A (accuracy) marks are not given for answers obtained from incorrect working. When A or B marks are awarded for work at an intermediate stage of a solution, there may be various alternatives that are equally acceptable. In such cases, what is acceptable will be detailed in the mark scheme. If this is not the case please, escalate the question to your Team Leader who will decide on a course of action with the Principal Examiner.
Sometimes the answer to one part of a question is used in a later part of the same question. In this case, A marks will often be 'follow through'. In such cases you must ensure that you refer back to the answer of the previous part question even if this is not shown within the image zone. You may find it easier to mark follow through questions candidate-by-candidate rather than question-by-question.
f Unless units are specifically requested, there is no penalty for wrong or missing units as long as the answer is numerically correct and expressed either in SI or in the units of the question (e.g. lengths will be assumed to be in metres unless in a particular question all the lengths are in km , when this would be assumed to be the unspecified unit.) We are usually quite flexible about the accuracy to which the final answer is expressed; over-specification is usually only penalised where the scheme explicitly says so. When a value is given in the paper only accept an answer correct to at least as many significant figures as the given value. This rule should be applied to each case. When a value is not given in the paper accept any answer that agrees with the correct value to 2 s .f. Follow through should be used so that only one mark is lost for each distinct accuracy error, except for errors due to premature approximation which should be penalised only once in the examination. There is no penalty for using a wrong value for \(g\). E marks will be lost except when results agree to the accuracy required in the question.
g Rules for replaced work: if a candidate attempts a question more than once, and indicates which attempt he/she wishes to be marked, then examiners should do as the candidate requests; if there are two or more attempts at a question which have not been crossed out, examiners should mark what appears to be the last (complete) attempt and ignore the others. NB Follow these maths-specific instructions rather than those in the assessor handbook.
h For a genuine misreading (of numbers or symbols) which is such that the object and the difficulty of the question remain unaltered, mark according to the scheme but following through from the candidate's data. A penalty is then applied; 1 mark is generally appropriate, though this may differ for some papers. This is achieved by withholding one A mark in the question. Marks designated as cao may be awarded as long as there are no other errors. E marks are lost unless, by chance, the given results are established by equivalent working. 'Fresh starts' will not affect an earlier decision about a misread. Note that a miscopy of the candidate's own working is not a misread but an accuracy error.
i If a calculator is used, some answers may be obtained with little or no working visible. Allow full marks for correct answers (provided, of course, that there is nothing in the wording of the question specifying that analytical methods are required). Where an answer is wrong but there is some evidence of method, allow appropriate method marks. Wrong answers with no supporting method score zero. If in doubt, consult your Team Leader.
j If in any case the scheme operates with considerable unfairness consult your Team Leader. \end{table} PS = Problem Solving
M = Modelling \section*{Accredited} \section*{AS Level Further Mathematics A} \section*{Unit Y534 Discrete Mathematics} \section*{Printed Answer Booklet} \section*{Date - Morning/Afternoon} \section*{Time allowed: 1 hour 15 minutes} OCR supplied materials:

• Printed Answer Booklet
• Formulae AS Level Further Mathematics A

You must have:

• Printed Answer Booklet
• Formulae AS Level Further Mathematics A
• Scientific or graphical calculator \includegraphics[max width=\textwidth, alt={}, center]{da50ab63-a6f5-4533-ba6d-f9941b71038f-25_321_1552_1320_246}

\section*{INSTRUCTIONS}

• Use black ink. HB pencil may be used for graphs and diagrams only.
• Complete the boxes provided on the Printed Answer Booklet with your name, centre number and candidate number.
• Answer all the questions.
• Write your answer to each question in the space provided in the Printed Answer Booklet.
• Additional paper may be used if necessary but you must clearly show your candidate number, centre number and question number(s).
• Do not write in the bar codes.
• You are permitted to use a scientific or graphical calculator in this paper.
• Final answers should be given to a degree of accuracy appropriate to the context.
• The acceleration due to gravity is denoted by \(g \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Unless otherwise instructed, when a numerical value is needed, use \(g = 9.8\).

\section*{INFORMATION}

• You are reminded of the need for clear presentation in your answers.
• The Printed Answer Booklet consists of \(\mathbf { 1 2 }\) pages. The Question Paper consists of \(\mathbf { 8 }\) pages.

5(iv) 6(i) \section*{PLEASE DO NOT WRITE ON THIS PAGE} \section*{Copyright Information:} }{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.
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A company produces two types of party bag, Infant and Junior. Both types of bag contain a balloon, a toy and a whistle. In addition the Infant bag contains 3 sweets and 3 stickers and the Junior bag contains 10 sweets and 2 stickers. The sweets and stickers are produced in the company's factory. The factory can produce up to 3000 sweets per hour and 1200 stickers per hour. The company buys a large supply of balloons, toys and whistles. Market research indicates that at least twice as many Infant bags as Junior bags should be produced. Both types of party bag are sold at a profit of 15p per bag. All the bags are sold. The company wishes to maximise its profit. Let \(x\) be the number of Infant bags produced and \(y\) be the number of Junior bags produced per hour.

1. Formulate the above situation as a linear programming problem. [5]
2. Represent your inequalities graphically, indicating clearly the feasible region. [6]
3. Find the number of Infant bags and Junior bags that should be produced each hour and the maximum hourly profit. Make your method clear. [3]

In order to increase the profit further, the company decides to buy additional equipment. It can buy equipment to increase the production of either sweets or stickers, but not both.

1. Using your graph, explain which equipment should be bought, giving your reasoning. [2]

The manager of the company does not understand why the balloons, toys and whistles have not been considered in the above calculations.

1. Explain briefly why they do not need to be considered. [2]

\includegraphics{figure_6} The captain of the Malde Mare takes passengers on trips across the lake in her boat. The number of children is represented by \(x\) and the number of adults by \(y\). Two of the constraints limiting the number of people she can take on each trip are $$x < 10$$ and $$2 \leq y \leq 10$$ These are shown on the graph in Figure 6, where the rejected regions are shaded out.

1. Explain why the line \(x = 10\) is shown as a dotted line. (1)
2. Use the constraints to write down statements that describe the number of children and the number of adults that can be taken on each trip. (3)

For each trip she charges £2 per child and £3 per adult. She must take at least £24 per trip to cover costs. The number of children must not exceed twice the number of adults.

1. Use this information to write down two inequalities. (2)
2. Add two lines and shading to Diagram 1 in your answer book to represent these inequalities. Hence determine the feasible region and label it R. (4)
3. Use your graph to determine how many children and adults would be on the trip if the captain takes:
   1. the minimum number of passengers,
   2. the maximum number of passengers.
   (4)

(Total 14 marks)

A company produces two types of self-assembly wooden bedroom suites, the 'Oxford' and the 'York'. After the pieces of wood have been cut and finished, all the materials have to be packaged. The table below shows the time, in hours, needed to complete each stage of the process and the profit made, in pounds, on each type of suite.

	Oxford	York
Cutting	4	6
Finishing	3.5	4
Packaging	2	4
Profit (£)	300	500

The times available each week for cutting, finishing and packaging are 66, 56 and 40 hours respectively. The company wishes to maximise its profit. Let \(x\) be the number of Oxford, and \(y\) be the number of York suites made each week.

1. Write down the objective function. [1]
2. In addition to $$2x + 3y \leq 33,$$ $$x \geq 0,$$ $$y \geq 0,$$ find two further inequalities to model the company's situation. [2]
3. On the grid in the answer booklet, illustrate all the inequalities, indicating clearly the feasible region. [4]
4. Explain how you would locate the optimal point. [2]
5. Determine the number of Oxford and York suites that should be made each week and the maximum profit gained. [3]

It is noticed that when the optimal solution is adopted, the time needed for one of the three stages of the process is less than that available.

1. Identify this stage and state by how many hours the time may be reduced. [3]

Leone is designing her new garden. She wants to have at least 1000 m\(^2\), split between lawn and flower beds. Initial costs are £0.80 per m\(^2\) for lawn and £0.40 per m\(^2\) for flowerbeds. Leone's budget is £500. Leone prefers flower beds to lawn, and she wants the area for flower beds to be at least twice the area for lawn. However, she wants to have at least 200 m\(^2\) of lawn. Maintenance costs each year are £0.15 per m\(^2\) for lawn and £0.25 per m\(^2\) for flower beds. Leone wants to minimize the maintenance costs of her garden.

1. Formulate Leone's problem as a linear programming problem. [7]
2. Produce a graph to illustrate the inequalities. [6]
3. Solve Leone's problem. [2]
4. If Leone had more than £500 available initially, how much extra could she spend to minimize maintenance costs? [1]

A Young Enterprise Company decides to sell two types of cakes at a breakfast club. The two types of cakes are blueberry and chocolate. From its initial market research, the company knows that it will: • sell at most 200 cakes in total • sell at least twice as many blueberry cakes as they will chocolate cakes • make 20p profit on each blueberry cake they sell • make 15p profit on each chocolate cake they sell. The company's objective is to maximise its profit. Formulate the Young Enterprise Company's situation as a linear programming problem. [4 marks]