OCR MEI D2 (Decision Mathematics 2) 2012 June

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.

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

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.

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.

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	- 7	0	0	0	0
   0	2	1.5	2.5	1	0	0	10000
   0	1	0.5	1.5	0	1	0	7500
   0	300	200	400	0	0	1	2000000

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.