Questions — OCR MEI D2 (54 questions)

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OCR MEI D2 2011 June Q1
1
  1. Heard in Parliament: "Will the minister not now discontinue her proposal to ban the protest?"
    The minister replied "Yes I will."
    To what had the minister committed herself logically, and why might that not have been her intention?
  2. In a cricket tournament an umpire might be required to decide whether or not a batsman is out 'lbw', ie 'leg before wicket'. The lbw law for the tournament refers to parts of the cricket pitch as shown in the diagram (assuming a right-handed batsman):
    \includegraphics[max width=\textwidth, alt={}, center]{52b8153f-e655-4852-a0f8-6f1c1e9c9170-2_254_1045_717_507} The umpire has to make a number of judgements:
    A Would the ball have hit the wicket?
    B Did the ball hit the batsman, or part of his equipment other than the bat, without hitting the bat?
    C Did the ball hit the batsman, or part of his equipment other than the bat, before hitting the bat?
    D Was the part of the batsman or his equipment which was hit by the ball, between the wickets when it was hit? E Was the part of the batsman or his equipment which was hit by the ball, outside of the wicket on the off side when it was hit? F Was the batsman attempting to play a stroke?
    The law can be interpreted as saying that the batsman is out lbw if \([ ( \mathrm { A } \wedge \mathrm { B } ) \vee ( \mathrm { A } \wedge \mathrm { C } ) ] \wedge [ \mathrm { D } \vee ( \mathrm { E } \wedge \sim \mathrm { F } ) ]\).
    The tournament's umpiring manual, in attempting to simplify the law, states that the batsman is out lbw if \(\mathrm { A } \wedge ( \mathrm { B } \vee \mathrm { C } ) \wedge ( \mathrm { D } \vee \mathrm { E } ) \wedge ( \mathrm { D } \vee \sim \mathrm { F } )\). For an lbw decision this requires 4 conditions each to be true.
    1. Use the rules of Boolean algebra to show that the manual's rule is logically equivalent to the law as stated above, naming the rules used at each step. A trainee umpire, using the manual, considers each condition in turn and judges that the following are true: A; B; E; D.
    2. What is her decision and why?
    3. What is odd about her judgement, and does this make the logic invalid?
OCR MEI D2 2011 June Q2
2 A government has just created a new ministry, the Ministry of Administrative Affairs. The ministry is to have four departments:
the Administration
the Bureaucracy
the Certification Service
the Duplication Section.
Each of these departments is to be established in a separate office on one of four existing sites. The diagram shows the direct journey times in minutes between these four sites.
\includegraphics[max width=\textwidth, alt={}, center]{52b8153f-e655-4852-a0f8-6f1c1e9c9170-3_342_403_721_829}
  1. Use Floyd's algorithm to find the shortest journey times between the four office sites.
  2. Draw a network showing your shortest times.
  3. Use appropriate algorithms to find upper and lower bounds for the optimum solution to the Travelling Salesperson Problem in the original network, briefly explaining the steps taken.
  4. A van is to be organised to deliver bundles of paperwork between the departments. Why might the optimum solution to the TSP be useful in planning this, and why might it not be?
  5. Journeys to locations 2 and 3, from locations 1 and 4, are subject to a congestion charge which is equivalent in costing terms to 15 minutes of journey time. What sort of network would be needed to model this?
OCR MEI D2 2011 June Q3
3 Magnus has been researching career possibilities. He has just completed his GCSEs, and could leave school and get a good job. He estimates, discounted at today's values and given a 49 year working life, that there is a \(50 \%\) chance of such a job giving him lifetime earnings of \(\pounds 1.5 \mathrm {~m}\), a \(30 \%\) chance of \(\pounds 1.75 \mathrm {~m}\), and a \(20 \%\) chance of \(\pounds 2 \mathrm {~m}\). Alternatively Magnus can stay on at school and take A levels. He estimates that, if he does so, there is a 75\% chance that he will achieve good results. If he does not achieve good results then he will still be able to take the same job as earlier, but he will have lost two years of his lifetime earnings. This will give a \(50 \%\) chance of lifetime earnings of \(\pounds 1.42 \mathrm {~m}\), a \(30 \%\) chance of \(\pounds 1.67 \mathrm {~m}\) and a \(20 \%\) chance of \(\pounds 1.92 \mathrm {~m}\). If Magnus achieves good A level results then he could take a better job, which should give him discounted lifetime earnings of \(\pounds 1.6 \mathrm {~m}\) with \(50 \%\) probability or \(\pounds 2 \mathrm {~m}\) with \(50 \%\) probability. Alternatively he could go to university. This would cost Magnus another 3 years of lifetime earnings and would not guarantee him a well-paid career, since graduates sometimes choose to follow less well-paid vocations. His research shows him that graduates can expect discounted lifetime earnings of \(\pounds 1 \mathrm {~m}\) with \(20 \%\) probability, \(\pounds 1.5 \mathrm {~m}\) with \(30 \%\) probability, \(\pounds 2 \mathrm {~m}\) with \(30 \%\) probability, and \(\pounds 3 \mathrm {~m}\) with \(20 \%\) probability.
  1. Draw up a decision tree showing Magnus's options.
  2. Using the EMV criterion, find Magnus's best course of action, and give its value. Magnus has read that money isn't everything, and that one way to reflect this is to use a utility function and then compare expected utilities. He decides to investigate the outcome of using a function in which utility is defined to be the square root of value.
  3. Using the expected utility criterion, find Magnus's best course of action, and give its utility.
  4. The possibility of high earnings ( \(\pounds 3 \mathrm {~m}\) ) swings Magnus's decision towards a university education. Find what value instead of \(\pounds 3 \mathrm {~m}\) would make him indifferent to choosing a university education under the EMV criterion. (Do not change the probabilities.)
OCR MEI D2 2011 June Q4
4 A small alpine hotel is planned. Permission has been obtained for no more than 60 beds, and these can be accommodated in rooms containing one, two or four beds. The total floor areas needed are \(15 \mathrm {~m} ^ { 2 }\) for a one-bed room, \(25 \mathrm {~m} ^ { 2 }\) for a two-bed room and \(40 \mathrm {~m} ^ { 2 }\) for a four-bed room. The total floor area of the bedrooms must not exceed \(700 \mathrm {~m} ^ { 2 }\). Marginal profit contributions per annum, in thousands of euros, are estimated to be 5 for a one-bed room, 9 for a two-bed room and 15 for a four-bed room.
  1. Formulate a linear programming problem to find the mix of rooms which will maximise the profit contribution within the two constraints.
  2. Use the simplex algorithm to solve the problem, and interpret your solution. It is decided that, for marketing reasons, at least 5 one-bed rooms must be provided.
  3. Solve this modified problem using either the two-stage simplex method or the big-M method. You may wish to adapt your final tableau from part (ii) to produce an initial tableau, but you are not required to do so.
  4. The simplex solution to the revised problem is to provide 5 one-bed rooms, 15 two-bed rooms and 6.25 four-bed rooms, giving a profit contribution of \(€ 253750\). Interpret this solution in terms of the real world problem.
  5. Compare the following solution to your answer to part (iv): 8 one-bed rooms, 12 two-bed rooms and 7 four-bed rooms. Explain your findings.
OCR MEI D2 2012 June Q1
1
  1. When marking coursework, a teacher has to complete a form which includes the following:
    □
    In your opinion is this the original work of the pupil? (tick as appropriate)
    I have no reason to believe that it is not □
    I cannot confirm that it is □
    1. The teacher suspects that a pupil has copied work from the internet. For each box, state whether the teacher should tick the box or not.
    2. The teacher has no suspicions about the work of another pupil, and has no information about how the work was produced. Which boxes should she tick?
    3. Explain why the teacher must always tick at least one box.
  2. Angus, the ski instructor, says that the class will have to have lunch in Italy tomorrow if it is foggy or if the top ski lift is not working. On the next morning Chloe, one of Angus's students, says that it is not foggy, so they can have lunch in Switzerland. Produce a line of a truth table which shows that Chloe's deduction is incorrect. You may produce a complete truth table if you wish, but you must indicate a row which shows that Chloe's deduction is incorrect.
  3. You are given that the following two statements are true. $$\begin{aligned} & ( \mathrm { X } \vee \sim \mathrm { Y } ) \Rightarrow \mathrm { Z }
    & \sim \mathrm { Z } \end{aligned}$$ Use Boolean algebra to show that Y is true.
OCR MEI D2 2012 June Q2
2 Adrian is considering selling his house and renting a flat.
Adrian still owes \(\pounds 150000\) on his house. He has a mortgage for this, for which he has to pay \(\pounds 4800\) annual interest. If he sells he will pay off the \(\pounds 150000\) and invest the remainder of the proceeds at an interest rate of \(2.5 \%\) per annum. He will use the interest to help to pay his rent. His estate agent estimates that there is a \(30 \%\) chance that the house will sell for \(\pounds 225000\), a \(50 \%\) chance that it will sell for \(\pounds 250000\), and a \(20 \%\) chance that it will sell for \(\pounds 275000\). A flat will cost him \(\pounds 7500\) per annum to rent.
  1. Draw a decision tree to help Adrian to decide whether to keep his house, or to sell it and rent a flat. Compare the EMVs of Adrian's annual outgoings, and ignore the costs of selling.
  2. Would the analysis point to a different course of action if Adrian were to use a square root utility function, instead of EMVs? Adrian's circumstances change so that he has to decide now whether to sell or not in one year's time. Economic conditions might then be less favourable for the housing market, the same, or more favourable, these occurring with probabilities \(0.3,0.3\) and 0.4 respectively. The possible selling prices and their probabilities are shown in the table.
    Economic conditions and probabilitiesSelling prices ( £) and probabilities
    less favourable0.32000000.22250000.32500000.5
    unchanged0.32250000.32500000.52750000.2
    more favourable0.42500000.33000000.53500000.2
  3. Draw a decision tree to help Adrian to decide what to do. Compare the EMVs of Adrian's annual outgoings. Assume that he will still owe \(\pounds 150000\) in one year's time, and that the cost of renting and interest rates do not change.
OCR MEI D2 2012 June Q3
3 The weights on the network represent distances.
\includegraphics[max width=\textwidth, alt={}, center]{eb4e9c34-7d8f-4118-b7ec-edcd9567077f-4_451_544_324_740}
  1. The answer book shows the initial tables and the results of iterations \(1,2,3\) and 5 when Floyd's algorithm is applied to the network.
    (A) Complete the two tables for iteration 4.
    (B) Use the final route table to give the shortest route from vertex \(\mathbf { 3 }\) to vertex \(\mathbf { 5 }\).
    (C) Use the final distance table to produce a complete network with weights representing the shortest distances between vertices.
  2. Using the complete network of shortest distances, find a lower bound for the solution to the Travelling Salesperson Problem by deleting vertex 5 and its arcs, and by finding the length of a minimum connector for the remainder. (You may find the minimum connector by inspection.)
  3. Use the nearest neighbour algorithm, starting at vertex \(\mathbf { 1 }\), to produce a Hamilton cycle in the complete network. Give the length of your cycle.
  4. Interpret your Hamilton cycle in part (iii) in terms of the original network.
  5. Give a walk of minimum length which traverses every arc on the original network at least once, and which returns to the start. Give the length of your walk.
OCR MEI D2 2012 June Q4
4 A publisher is considering producing three books over the next week: a mathematics book, a novel and a biography. The mathematics book will sell at \(\pounds 10\) and costs \(\pounds 4\) to produce. The novel will sell at \(\pounds 5\) and costs \(\pounds 2\) to produce. The biography will sell at \(\pounds 12\) and costs \(\pounds 5\) to produce. The publisher wants to maximise profit, and is confident that all books will be sold. There are constraints on production. Each copy of the mathematics book needs 2 minutes of printing time, 1 minute of packing time, and \(300 \mathrm {~cm} ^ { 3 }\) of temporary storage space. Each copy of the novel needs 1.5 minutes of printing time, 0.5 minutes of packing time, and \(200 \mathrm {~cm} ^ { 3 }\) of temporary storage space. Each copy of the biography needs 2.5 minutes of printing time, 1.5 minutes of packing time, and \(400 \mathrm {~cm} ^ { 3 }\) of temporary storage space. There are 10000 minutes of printing time available on several printing presses, 7500 minutes of packing time, and \(2 \mathrm {~m} ^ { 3 }\) of temporary storage space.
  1. Explain how the following initial feasible tableau models this problem.
    P\(x\)\(y\)\(z\)\(s 1\)\(s 2\)\(s 3\)RHS
    1- 6- 3- 70000
    021.52.510010000
    010.51.50107500
    03002004000012000000
  2. Use the simplex algorithm to solve your LP, and interpret your solution.
  3. The optimal solution involves producing just one of the three books. By how much would the price of each of the other books have to be increased to make them worth producing? There is a marketing requirement to provide at least 1000 copies of the novel.
  4. Show how to incorporate this constraint into the initial tableau ready for an application of the two-stage simplex method. Briefly describe how to use the modified tableau to solve the problem. You are NOT required to perform the iterations.
OCR MEI D2 2013 June Q1
1
  1. A graph is simple if it contains neither loops nor multiple arcs, ie none of the following:
    \includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-2_81_134_219_1683}
    or
    \includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-2_79_589_301_328} In an examination question, students were asked to describe in words when a graph is simple. Mark the following responses as right or wrong, giving reasons for your decisions if you mark them wrong.
    1. A graph is simple if there are no loops and if two nodes are connected by a single arc.
    2. A graph is simple if there are no loops and no two nodes are connected by more than one arc.
    3. A graph is simple if there are no loops and two arcs do not have the same ends.
    4. A graph is simple if there are no loops and there is at most one route from one node to another.
  2. The following picture represents a two-way switch
    \includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-2_149_138_932_1119} It can either be in the up state
    \includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-2_104_104_1128_790}
    or in the down state • or in the down state . Two switches can be used to construct a circuit in which changing the state of either switch changes the state of a lamp.
    \includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-2_309_543_1448_762} Georgios tries to connect together three two-way switches so that changing the state of any switch changes the state of the lamp. His circuit is shown below. The switches have been labelled 1,2 and 3.
    \includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-2_496_547_1946_760}
    1. List the possible combination of switch states and determine whether the lamp is on or off for each of them.
    2. Say whether or not Georgios has achieved his objective, justifying your answer.
  3. Use a truth table to show that \(( \mathrm { A } \wedge ( \mathrm { B } \vee \mathrm { C } ) ) \vee \sim ( \sim \mathrm { A } \vee ( \mathrm { B } \wedge \mathrm { C } ) ) \Leftrightarrow \mathrm { A }\).
OCR MEI D2 2013 June Q2
2 Graham skis each year in an Italian resort which shares a ski area with a Swiss resort. He can buy an Italian lift pass, or an international lift pass which gives him access to Switzerland as well as to Italy. For his 6-day holiday the Italian pass costs \(€ 200\) and the international pass costs \(€ 250\). If he buys an Italian pass then he can still visit Switzerland by purchasing day supplements at \(€ 30\) per day. If the weather is good during his holiday, then Graham visits Switzerland three times. If the weather is moderate he goes twice. If poor he goes once. If the weather is windy then the lifts are closed, and he is not able to go at all. In his years of skiing at the resort he has had good weather on \(30 \%\) of his visits, moderate weather on \(40 \%\), poor weather on \(20 \%\) and windy weather on \(10 \%\) of his visits.
  1. Draw a decision tree to help Graham decide whether to buy an Italian lift pass or an international lift pass. Give the action he should take to minimize the EMV of his costs. When he arrives at the resort, and before he buys his lift pass, he finds that he has internet access to a local weather forecast, and to records of the past performance of the forecast. The 6-day forecast is limited to "good"/"not good", and the records show the actual weather proportions following those forecasts. It also shows that \(60 \%\) of historical forecasts have been "good" and \(40 \%\) "not good".
    \backslashbox{Forecast}{Actual}goodmoderatepoorwindyproportion of forecasts
    good0.40.50.10.00.6
    not good0.150.250.350.250.4
  2. Draw a decision tree to help Graham decide the worth of consulting the forecast before buying his lift pass. Give the actions he should take to minimize the EMV of his costs.
OCR MEI D2 2013 June Q3
3 Five towns, 1, 2, 3, 4 and 5, are connected by direct routes as shown. The arc weights represent distances.
\includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-4_632_540_312_744}
  1. The printed answer book shows the initial tables and the results of iterations \(1,2,3\) and 5 when Floyd's algorithm is applied to the network.
    (A) Complete the two tables for iteration 4.
    (B) Use the final route table to give the shortest route from vertex 5 to vertex 2.
    (C) Use the final distance table to produce a complete network with weights representing the shortest distances between vertices.
  2. Use the nearest neighbour algorithm, starting at vertex \(\mathbf { 4 }\), to produce a Hamilton cycle in the complete network. Give the length of your cycle.
  3. Interpret your Hamilton cycle from part (ii) in terms of towns actually visited.
  4. Find an improved Hamilton cycle by applying the nearest neighbour algorithm starting from one of the other vertices.
  5. Using the complete network of shortest distances (excluding loops), find a lower bound for the solution to the Travelling Salesperson Problem by deleting vertex 4 and its arcs, and by finding the length of a minimum connector for the remainder. (You may find the minimum connector by inspection.)
  6. Given that the sum of the road lengths in the original network is 43 , give a walk of minimum length which traverses every arc on the original network at least once, and which returns to the start. Show your methodology. Give the length of your walk.
OCR MEI D2 2013 June Q4
4 Colin has a hobby from which he makes a small income. He makes bowls, candle holders and key fobs.
The materials he uses include wood, metal parts, polish and sandpaper. They cost, on average, \(\pounds 15\) per bowl, \(\pounds 6\) per candle holder and \(\pounds 2\) per key fob. Colin has a monthly budget of \(\pounds 100\) for materials. Colin spends no more than 30 hours per month on manufacturing these objects. Each bowl takes 4 hours, each candle holder takes 2 hours and each key fob takes half an hour.
  1. Let \(b\) be the number of bowls Colin makes in a month, \(c\) the number of candle holders and \(f\) the number of key fobs. Write out, in terms of these variables, two constraints corresponding to the limit on monthly expenditure on materials, and to the limit on Colin's time. Colin sells the objects at craft fairs. He charges \(\pounds 30\) for a bowl, \(\pounds 15\) for a candle holder and \(\pounds 3\) for a key fob.
  2. Set up an initial simplex tableau for the problem of maximising Colin's monthly income subject to your constraints from part (i), assuming that he sells all that he produces.
  3. Use the simplex algorithm to solve your LP, and interpret the solution from the simplex algorithm. Over a spell of several months Colin finds it difficult to sell bowls so he stops making them.
  4. Modify and solve your LP, using simplex, to find how many candle holders and how many key fobs he should make, and interpret your solution. At the next craft fair Colin takes an order for 4 bowls. He promises to make exactly 4 bowls in the next month.
  5. Set up this modified problem either as an application of two-stage simplex, or as an application of the big-M method. You are not required to solve the problem. The solution now is for Colin to produce 4 bowls, \(6 \frac { 2 } { 3 }\) candle holders and no key fobs.
  6. What is Colin's best integer solution to the problem?
  7. Your answer to part (vi) is not necessarily the integer solution giving the maximum profit for Colin. Explain why.
OCR MEI D2 2014 June Q1
1 marks
1 Keith is wondering whether or not to insure the value of his house against destruction. His friend Georgia has told him that it is a waste of money. Georgia argues that the insurance company sets its premiums (how much it charges for insurance) to take account of the probability of destruction, plus an extra fee for its profit. Georgia argues that house-owners are, on average, simply paying fees to the insurance company. Keith's house is valued at \(\pounds 400000\). The annual premium for insuring its value against destruction is \(\pounds 100\). Past statistics show that the probability of destruction in any one year is 0.0002 .
  1. Draw a decision tree to model Keith's decision and the possible outcomes.
  2. Compute Keith's EMV and give the course of action which corresponds to that EMV.
  3. What would be the insurance premium if there were no fee for the insurance company? For the remainder of the question the insurance premium is still \(\pounds 100\).
    Suppose that, instead of EMV, Keith uses the utility function utility \(= ( \text { money } ) ^ { 0.5 }\).
  4. Compute Keith's utility and give his corresponding course of action. Keith suspects that it may be the case that he lives in an area in which the probability of destruction in a given year, \(p\), is not 0.0002 .
  5. Draw a decision tree, using the EMV criterion, to model Keith's decision in terms of \(p\), the probability of destruction in the area in which Keith lives.
  6. Find the value of \(p\) which would make it worthwhile for Keith to insure his house using the EMV criterion.
  7. Explain why Keith may wish to insure even if \(p\) is less than the value which you found in part (vi). [1]
    (a) A national Sunday newspaper runs a "You are the umpire" series, in which questions are posed about whether a batsman in cricket is given "out", and why, or "not out". One Sunday the readers were told that a ball had either hit the bat and then the pad, or had missed the bat and hit the pad; the umpire could not be sure which. The ball had then flown directly to a fielder, who had caught it. The LBW (leg before wicket) rule is complicated. The readers were told that this batsman should be given out (LBW) if the ball had not hit the bat. On the other hand, if the ball had hit the bat, then he should be given out (caught). Readers were asked what the decision should be. The answer given in the newspaper was that this batsman should be given not out because the umpire could not be sure that the batsman was out (LBW), and could not be sure that he was out (caught).
OCR MEI D2 2014 June Q3
3 Three products, A, B and C are to be made.
Three supplements are included in each product. Product A has 10 g per kg of supplement \(\mathrm { X } , 5 \mathrm {~g}\) per kg of supplement Y and 5 g per kg of supplement Z . Product B has 5 g per kg of supplement \(\mathrm { X } , 5 \mathrm {~g}\) per kg of supplement Y and 3 g per kg of supplement Z .
Product C has 12 g per kg of supplement \(\mathrm { X } , 7 \mathrm {~g}\) per kg of supplement Y and 5 g per kg of supplement Z .
There are 12 kg of supplement X available, 12 kg of supplement Y , and 9 kg of supplement Z .
Product A will sell at \(\pounds 7\) per kg and costs \(\pounds 3\) per kg to produce. Product B will sell at \(\pounds 5\) per kg and costs \(\pounds 2\) per kg to produce. Product C will sell at \(\pounds 4\) per kg and costs \(\pounds 3\) per kg to produce. The profit is to be maximised.
  1. Explain how the initial feasible tableau shown in Fig. 3 models this problem. \begin{table}[h]
    Pabcs 1s 2s 3RHS
    1- 4- 3- 10000
    01051210012000
    055701012000
    05350019000
    \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{table}
  2. Use the simplex algorithm to solve this problem, and interpret the solution.
  3. In the solution, one of the basic variables appears at a value of 0 . Explain what this means. There is a contractual requirement to provide at least 500 kg of product A .
  4. Show how to incorporate this constraint into the initial tableau ready for an application of the two-stage simplex method. Briefly describe how the method works. You are not required to perform the iterations.
OCR MEI D2 2015 June Q1
1 A furniture manufacturer is planning a production run. He will be making wardrobes, drawer units and desks. All can be manufactured from the same wood. He has available \(200 \mathrm {~m} ^ { 2 }\) of wood for the production run. Allowing for wastage, a wardrobe requires \(5 \mathrm {~m} ^ { 2 }\), a drawer unit requires \(3 \mathrm {~m} ^ { 2 }\), and a desk requires \(2 \mathrm {~m} ^ { 2 }\). He has 200 hours available for the production run. A wardrobe requires 4.5 hours, a drawer unit requires 5.2 hours, and a desk requires 3.8 hours. The completed furniture will have to be stored at the factory for a short while before being shipped. The factory has \(50 \mathrm {~m} ^ { 3 }\) of storage space available. A wardrobe needs \(1 \mathrm {~m} ^ { 3 }\), a drawer unit needs \(0.75 \mathrm {~m} ^ { 3 }\), and a desk needs \(0.5 \mathrm {~m} ^ { 3 }\). The manufacturer needs to know what he should produce to maximise his income. He sells the wardrobes at \(\pounds 80\) each, the drawer units at \(\pounds 65\) each and the desks at \(\pounds 50\) each.
  1. Formulate the manufacturer's problem as an LP.
  2. Use the Simplex algorithm to solve the LP problem.
  3. Interpret the results.
  4. An extra \(25 \mathrm {~m} ^ { 2 }\) of wood is found and is to be used. The new optimal solution is to make 44 wardrobes, no drawer units and no desks. However, this leaves some of each resource (wood, hours and space) left over. Explain how this can be possible.
  5. Given that \(x\) and \(y\) are propositions, draw a 4-line truth table for \(x \Rightarrow y\), allowing \(x\) and \(y\) to take all combinations of truth values. If \(x\) is false and \(x \Rightarrow y\) is true, what can be deduced about the truth value of \(y\) ? A story has it that, in a lecture on logic, the philosopher Bertrand Russell (1872-1970) mentioned that a false proposition implies any proposition. A student challenged this, saying "In that case, given that \(1 = 0\), prove that you are the Pope."
    Russell immediately replied, "Add 1 to both sides of the equation: then we have \(2 = 1\). The set containing just me and the Pope has 2 members. But \(2 = 1\), so the set has only 1 member; therefore, I am the Pope." Russell's string of statements is an example of a deductive sequence. Let \(a\) represent " \(1 = 0\) ", \(b\) represent " \(2 = 1\) ", \(c\) represent "Russell and the Pope are 2" and \(d\) represent "Russell and the Pope are 1". Then Russell's deductive sequence can be written as \(( a \wedge ( a \Rightarrow b ) \wedge c ) \Rightarrow d\).
  6. Assuming that \(a\) is false, \(b\) is false, \(a \Rightarrow b\) is true, \(c\) is true, and that \(d\) can take either truth value, draw a 2-line truth table for \(( a \wedge ( a \Rightarrow b ) \wedge c ) \Rightarrow d\).
  7. What does the table tell you about \(d\) with respect to the false proposition \(a\) ?
  8. Explain why Russell introduced propositions \(b\) and \(c\) into his argument.
  9. Russell could correctly have started a deductive sequence:
    \(a \wedge [ a \Rightarrow ( ( 0.5 = - 0.5 ) \Rightarrow ( 0.25 = 0.25 ) ) ]\).
    Had he have done so could he correctly have continued it to end at \(d\) ?
    Justify your answer.
  10. Draw a combinatorial circuit to represent \(( a \wedge ( a \Rightarrow b ) \wedge c ) \Rightarrow d\). 3 Floyd's algorithm is applied to the incomplete network on 4 nodes drawn below. The weights on the arcs represent journey times.
    \includegraphics[max width=\textwidth, alt={}, center]{4b5bc097-1052-4e44-8623-a84ceaab0289-4_400_558_347_751} The final matrices are shown below. \begin{table}[h] \begin{center} \captionsetup{labelformat=empty} \caption{final time matrix} \begin{tabular}{ | l | r | r | r | r | } \cline { 2 - 5 } \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{\(\mathbf { 1 }\)} & \(\mathbf { 2 }\) & \(\mathbf { 3 }\) & \multicolumn{1}{c|}{\(\mathbf { 4 }\)}
    \hline \(\mathbf { 1 }\) & 6 & 5 & 3 & 10
    \hline
OCR MEI D2 2015 June Q3
3 Floyd's algorithm is applied to the incomplete network on 4 nodes drawn below. The weights on the arcs represent journey times.
\includegraphics[max width=\textwidth, alt={}, center]{4b5bc097-1052-4e44-8623-a84ceaab0289-4_400_558_347_751} The final matrices are shown below. \begin{table}[h] \begin{center} \captionsetup{labelformat=empty} \caption{final time matrix} \begin{tabular}{ | l | r | r | r | r | } \cline { 2 - 5 } \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{\(\mathbf { 1 }\)} & \(\mathbf { 2 }\) & \(\mathbf { 3 }\) & \multicolumn{1}{c|}{\(\mathbf { 4 }\)}
\hline \(\mathbf { 1 }\) & 6 & 5 & 3 & 10
\hline \(\mathbf { 2 }\) & 5 & 4 & 2 & 5
\hline \(\mathbf { 3 }\) & 3 & 2 & 4 & 7
\hline
OCR MEI D2 2015 June Q4
\(\mathbf { 4 }\) & 10 & 5 & 7 & 10
\hline \end{tabular} \end{center} \end{table} \begin{table}[h]
\captionsetup{labelformat=empty} \caption{final route matrix}
\cline { 2 - 5 } \multicolumn{1}{c|}{}\(\mathbf { 1 }\)\(\mathbf { 2 }\)\(\mathbf { 3 }\)\(\mathbf { 4 }\)
\(\mathbf { 1 }\)3333
\(\mathbf { 2 }\)3334
\(\mathbf { 3 }\)1222
\(\mathbf { 4 }\)2222
\end{table}
  1. Draw the complete network of shortest times.
  2. Explain how to use the final route matrix to find the quickest route from node \(\mathbf { 4 }\) to node \(\mathbf { 1 }\) in the original incomplete network. Give this quickest route. A new node, node 5, is added to the original incomplete network. The new journey times are shown in the table. \begin{center} \begin{tabular}{ | l | c | c | c | c | } \cline { 2 - 5 } \multicolumn{1}{c|}{} & \(\mathbf { 1 }\) & \(\mathbf { 2 }\) & \(\mathbf { 3 }\) & \(\mathbf { 4 }\)
    \hline
OCR MEI D2 2015 June Q5
\(\mathbf { 5 }\) & 4 & - & - & 2
\hline \end{tabular} \end{center} (iii) Draw the complete 5-node network of shortest times. (You are not required to use an algorithm to find the shortest times.)
(iv) Write down the final time matrix and the final route matrix for the complete 5 -node network. (You do not need to apply Floyd's algorithm.)
(v) (A) Apply the nearest neighbour algorithm to the complete 5-node network of shortest times, starting at node 1. Give the time for the cycle you produce.
(B) Starting at node 3, a possible cycle in the complete 5-node network of shortest times is \(\mathbf { 3 2 1 5 4 3 . }\) Give the actual cycle to which this corresponds in the incomplete 5-node network of journey times.
(vi) By deleting node 5 and its arcs from the complete 5 -node network of shortest times, and then using Kruskal's algorithm, produce a lower bound for the solution to the associated practical travelling salesperson problem. Show clearly your use of Kruskal's algorithm.
(vii) In the incomplete 5-node network of journey times, find a quickest route starting at node \(\mathbf { 5 }\) and using each of the 7 arcs at least once. Give the time of your route. 4 Helen has a meeting to go to in London. She has to travel from her home in G on the south coast to KC in London. She can drive from G to the west to A to catch a train, or she can drive to the east to W to catch a train on a different line. From both A and W she can travel to mainline terminuses V or LB in London. She will then travel by tube either from V to KC , or from LB to KC . The times for the various steps of her journey are shown in the tables. Both train services and tube services are subject to occasional delays, and these are modelled in the tables.
Driving timesto Ato W
From G20 min15 min
\multirow{2}{*}{Train journey}To VTo LB
normal timeprobability of delaydelaynormal timeprobability of delaydelay
From A1 hr 40 min0.0510 min1 hr 45 min0.0510 min
From W1 hr 30 min0.1020 min1 hr 35 min0.1020 min
\multirow{2}{*}{
Tube
journey
}		To KC
\cline { 2 - 4 }normal timeprobability of delaydelay
From V7 min0.202 min
From LB9 min0.102 min
Helen wants to choose the route which will give the shortest expected journey time.
(i) Draw a decision tree to model Helen's decisions and the possible outcomes.
(ii) Calculate Helen's shortest expected journey time and give the route which corresponds to that shortest expected journey time. As she gets into her car, Helen hears a travel bulletin on the radio warning of a broken escalator at V. This means that routes through V will take Helen 10 minutes longer.
(iii) Find the value of the radio information, explaining your calculation.
(iv) Why might the shortest expected journey time not be the best criterion for choosing a route for Helen?
OCR MEI D2 2016 June Q1
1 Martin is considering paying for a vaccination against a disease. If he catches the disease he would not be able to work and would lose \(\pounds 900\) in income because he would have to stay at home recovering. The vaccination costs \(\pounds 20\). The vaccination would reduce his risk of catching the disease during the year from 0.02 to 0.001 .
  1. Draw a decision tree for Martin.
  2. Evaluate the EMV of Martin's loss at each node of your tree, and give the action that Martin should take to minimise the EMV of his loss. Martin can answer a medical questionnaire which will give an estimate of his susceptibility to the disease. If he is found to be susceptible, then his chance of catching the disease is 0.05 . Vaccination will reduce that to 0.0025 . If he is found not to be susceptible, then his chance of catching the disease is 0.01 and vaccination will reduce it to 0.0005 . Historically, \(25 \%\) of people are found to be susceptible.
  3. What is the EMV of this questionnaire? Martin decides not to answer the questionnaire. He also decides that there is more than just his EMV to be considered in deciding whether or not to have the vaccination. The vaccination itself is likely to have side effects, but catching the disease would be very unpleasant. Martin estimates that he would find the effects of the disease 1000 times more unpleasant than the effects of the vaccination.
  4. Analyse which course of action would minimise the unpleasantness for Martin.
OCR MEI D2 2016 June Q2
2
  1. Emelia: 'I won't go out for a walk if it's not dry or not warm.'
    Gemma: ‘It’s warm. Let’s go!’
    Will what Gemma has said convince Emelia, and if not, why not?
  2. If it is daytime and the car headlights are on, then it is raining. If the dashboard lights are dimmed then the car headlights are on.
    It is daytime.
    It is not raining.
    1. What can you deduce?
    2. Prove your deduction.
  3. In this part of the question the switch X is represented by
    \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_104_138_824_1226} The switch can be wired into a circuit so that current flows when
    the switch is up
    \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_103_177_1005_593}
    but does not flow when it is down
    \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_111_167_1000_1334} Or the switch can be wired so that current flows when
    the switch is down
    \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_109_172_1228_639}
    but does not flow when it is up
    \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_109_174_1228_1327}
    1. Explain how the following circuit models \(( \mathrm { A } \wedge \mathrm { B } ) \Rightarrow \mathrm { C }\).
      \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_365_682_1484_694} In the following circuit B1 and B2 represent 'ganged' switches. This means that the two switches are either both up or both down.
      \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_364_1278_2042_397}
    2. Given that A is down, C is up and current is flowing, what can you deduce?
OCR MEI D2 2016 June Q3
3 Neil is refurbishing a listed building. There are two types of paint that he can use for the inside walls. One costs \(\pounds 1.45\) per \(\mathrm { m } ^ { 2 }\) and the other costs \(\pounds 0.95\) per \(\mathrm { m } ^ { 2 }\). He must paint the lower half of each wall in the more expensive paint. He has \(350 \mathrm {~m} ^ { 2 }\) of wall to paint. He has a budget of \(\pounds 400\) for wall paint. The more expensive paint is easier to use, and so Neil wants to use as much of it as possible. Initially, the following LP is constructed to help Neil with his purchasing of paint.
Let \(x\) be the number of \(\mathrm { m } ^ { 2 }\) of wall painted with the expensive paint.
Let \(y\) be the number of \(\mathrm { m } ^ { 2 }\) of wall painted with the less expensive paint. $$\begin{array} { l l } \text { Maximise } & P = x + y
\text { subject to } & 1.45 x + 0.95 y \leqslant 400
& y - x \leqslant 0
& x \geqslant 0
& y \geqslant 0 \end{array}$$
  1. Explain the purpose of the inequality \(y - x \leqslant 0\).
  2. The formulation does not include the inequality \(x + y \geqslant 350\). State what this constraint models and why it has been omitted from the formulation.
  3. Use the simplex algorithm to solve the LP. Pivot first on the "1" in the \(y\) column. Interpret your solution. The solution shows that Neil needs to buy more paint. He negotiates an increase in his budget to \(\pounds 450\).
  4. Find the solution to the LP given by changing \(1.45 x + 0.95 y \leqslant 400\) to \(1.45 x + 0.95 y \leqslant 450\), and interpret your solution. Neil realises that although he now has a solution, that solution is not the best for his requirements.
  5. Explain why the revised solution is not optimal for Neil. In order to move to an optimal solution Neil needs to change the objective of the LP and add another constraint to it.
  6. Write down the new LP and the initial tableau for using two-stage simplex to solve it. Give a brief description of how to use two-stage simplex to solve it.
    \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-5_497_558_269_751}
    (a) Solve the route inspection problem in the network above, showing the methodology you used to ensure that your solution is optimal. Show your route.
    (b) Floyd's algorithm is applied to the same network to find the complete network of shortest distances. After three iterations the distance and route matrices are as follows. \begin{center} \begin{tabular}{ | c | c | c | c | c | c | } \cline { 2 - 6 } \multicolumn{1}{c|}{} & \(\mathbf { 1 }\) & \(\mathbf { 2 }\) & \(\mathbf { 3 }\) & \(\mathbf { 4 }\) & \(\mathbf { 5 }\)
    \hline \(\mathbf { 1 }\) & 48 & 24 & 28 & 11 & 15
    \hline \(\mathbf { 2 }\) & 24 & 8 & 4 & 11 & 16
    \hline \(\mathbf { 3 }\) & 28 & 4 & 8 & 7 & 12
    \hline
OCR MEI D2 2016 June Q4
\(\mathbf { 4 }\) & 11 & 11 & 7 & 14 & 14
\hline
OCR MEI D2 2016 June Q5
\(\mathbf { 5 }\) & 15 & 16 & 12 & 14 & 24
\hline \end{tabular} \end{center}
\cline { 2 - 6 } \multicolumn{1}{c|}{}\(\mathbf { 1 }\)\(\mathbf { 2 }\)\(\mathbf { 3 }\)\(\mathbf { 4 }\)\(\mathbf { 5 }\)
\(\mathbf { 1 }\)22245
\(\mathbf { 2 }\)13333
\(\mathbf { 3 }\)22245
\(\mathbf { 4 }\)13335
\(\mathbf { 5 }\)13343
  1. Perform the fourth iteration of the algorithm, and show that there is no change to either matrix in the final iteration.
  2. Show how to use the matrices to give the shortest distance and the shortest route from vertex \(\mathbf { 1 }\) to vertex 2.
  3. Draw the complete network of shortest distances.
  4. Starting at vertex 1, apply the nearest neighbour algorithm to the complete network of shortest distances to find a Hamilton cycle. Give the length of your cycle and interpret it in the original network.
  5. By temporarily deleting vertex \(\mathbf { 1 }\) and its connecting arcs from the complete network of shortest distances, find a lower bound for the solution to the Travelling Salesperson's Problem in that network. Say what this implies in the original network.
OCR MEI D2 2014 June Q2
  1. Rachel thinks that the answer given in the newspaper article is not sensible. Give a verbal argument why Rachel might think that the batsman should be given out. Rachel tries to formalise her argument. She defines four simple propositions.
    o: "The batsman is given out."
    lb: "The batsman is given out (LBW)."
    c: "The batsman is given out (caught)."
    b: "The ball hit the bat."
  2. An implication of the batsman not being out (LBW) is that the ball has hit the bat. Write this down in terms of Rachel's propositions.
  3. Similarly, write down the implication of the batsman not being out (caught).
  4. Using your answers to parts (ii) and (iii) write down the implication of a batsman being not out, in terms of \(b\) and \(\sim b\).
    [0pt] [You may assume that if \(\mathrm { w } \Rightarrow \mathrm { y }\) and \(\mathrm { x } \Rightarrow \mathrm { z }\), then \(( \mathrm { w } \wedge \mathrm { x } ) \Rightarrow ( \mathrm { y } \wedge \mathrm { z } )\). ]
  5. By writing down the contrapositive of your implication from part (iv), produce an implication which supports Rachel's argument.
    (b) A classroom rule has been broken by either Anja, Bobby, Catherine or Dimitria, or by a subset of those four. The teacher knows that Dimitria could not have done it on her own. Let \(a\) be the proposition "Anja is guilty", and similarly for \(b , c\) and \(d\).
  6. Express the teacher's knowledge as a compound proposition. Evidence emerges that Bobby and Catherine were elsewhere at the time, so they cannot be guilty. This can be expressed as the compound proposition \(\sim ( b \vee c )\).
  7. Construct a truth table to show the truth values of the compound proposition given by the conjunction of the two compound propositions, one from part (i) and one given above.
  8. What does your truth table tell you about who is guilty? 3 Three products, A, B and C are to be made.
    Three supplements are included in each product. Product A has 10 g per kg of supplement \(\mathrm { X } , 5 \mathrm {~g}\) per kg of supplement Y and 5 g per kg of supplement Z . Product B has 5 g per kg of supplement \(\mathrm { X } , 5 \mathrm {~g}\) per kg of supplement Y and 3 g per kg of supplement Z .
    Product C has 12 g per kg of supplement \(\mathrm { X } , 7 \mathrm {~g}\) per kg of supplement Y and 5 g per kg of supplement Z .
    There are 12 kg of supplement X available, 12 kg of supplement Y , and 9 kg of supplement Z .
    Product A will sell at \(\pounds 7\) per kg and costs \(\pounds 3\) per kg to produce. Product B will sell at \(\pounds 5\) per kg and costs \(\pounds 2\) per kg to produce. Product C will sell at \(\pounds 4\) per kg and costs \(\pounds 3\) per kg to produce. The profit is to be maximised.
  9. Explain how the initial feasible tableau shown in Fig. 3 models this problem. \begin{table}[h]
    1(v)
    1(vi)
    1
  10. \begin{center} \begin{tabular}{|l|l|} \hline 2(a)(i) &
    \hline &
    \hline &
    \hline &
    \hline &
    \hline &
    \hline &
    \hline &
    \hline &
    \hline &
    \hline
OCR MEI D2 2006 June Q2
2 Answer this question on the insert provided. Fig. 2 shows a network in which the weights on the arcs represent distances. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9716cf3f-afa5-44a4-a8cd-f7511449d06b-2_405_497_1046_776} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Apply Floyd's algorithm on the insert provided to find the complete network of shortest distances.
  2. Show how to use your final matrices to find the shortest route from vertex \(\mathbf { 1 }\) to vertex 3, together with the length of that route.
  3. Use the nearest neighbour algorithm, starting at vertex 1, to find a Hamilton cycle in the complete network of shortest distances. Give the corresponding cycle in the original network, together with its length.