OCR MEI D2 (Decision Mathematics 2) 2006 June

1

1. Use a truth table to prove \(\sim ( \sim \mathrm { T } \Rightarrow \sim \mathrm { S } ) \Leftrightarrow ( \sim \mathrm { T } \wedge \mathrm { S } )\).
2. Prove that \(( \mathrm { A } \Rightarrow \mathrm { B } ) \Leftrightarrow ( \sim \mathrm { A } \vee \mathrm { B } )\) and hence use Boolean algebra to prove that $$\sim ( \sim \mathrm { T } \Rightarrow \sim \mathrm {~S} ) \Leftrightarrow ( \sim \mathrm { T } \wedge \mathrm {~S} ) .$$
3. A teacher wrote on a report "It is not the case that if Joanna doesn't try then she won't succeed." He meant to say that if Joanna were to try then she would have a chance of success. By letting T be "Joanna will try" and S be "Joanna will succeed", find the real meaning of what the teacher wrote.

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] \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.

3 Emma has won a holiday worth \(\pounds 1000\). She is wondering whether or not to take out an insurance policy which will pay out \(\pounds 1000\) if she should fall ill and be unable to go on the holiday. The insurance company tells her that this happens to 1 in 200 people. The insurance policy costs \(\pounds 10\). Thus Emma's monetary value if she buys the insurance and does not fall ill is \(\pounds 990\).

1. Draw a decision tree for Emma's problem. Use the EMV criterion in your calculations.
2. Interpret your tree and say what the maximum cost of the insurance would have to be for Emma to consider buying it if she uses the EMV criterion. Suppose that Emma's utility function is given by utility \(= \sqrt [ 3 ] { \text { monetary value } }\).
3. Using expected utility as the criterion, should Emma purchase the insurance? Under this criterion what is the cost at which she will be indifferent to buying or not buying it? Emma could pay for a blood pressure check to help her to make her decision. Statistics show that \(75 \%\) of checks are positive, and that when a check is positive the chance of missing a holiday through ill heath is 0.001 . However, when a check is negative the chance of cancellation through ill health is 0.017.
4. Draw a decision tree to help Emma decide whether or not to pay for the check. Use EMV, not expected utility, in your calculations and assume that the insurance policy costs \(\pounds 10\). What is the maximum amount that she should pay for the blood pressure check?

4 The "Cuddly Friends Company" produces soft toys. For one day's production run it has available \(11 \mathrm {~m} ^ { 2 }\) of furry material, \(24 \mathrm {~m} ^ { 2 }\) of woolly material and 30 glass eyes. It has three soft toys which it can produce: The "Cuddly Aardvark", each of which requires \(0.5 \mathrm {~m} ^ { 2 }\) of furry material, \(2 \mathrm {~m} ^ { 2 }\) of woolly material and two eyes. Each sells at a profit of \(\pounds 3\). The "Cuddly Bear", each of which requires \(1 \mathrm {~m} ^ { 2 }\) of furry material, \(1.5 \mathrm {~m} ^ { 2 }\) of woolly material and two eyes. Each sells at a profit of \(\pounds 5\). The "Cuddly Cat", each of which requires \(1 \mathrm {~m} ^ { 2 }\) of furry material, \(1 \mathrm {~m} ^ { 2 }\) of woolly material and two eyes. Each sells at a profit of \(\pounds 2\). An analyst formulates the following LP to find the production plan which maximises profit. $$\begin{array} { l l } \text { Maximise } & 3 a + 5 b + 2 c
\text { subject to } & 0.5 a + b + c \leqslant 11 ,
& 2 a + 1.5 b + c \leqslant 24 ,
& 2 a + 2 b + 2 c \leqslant 30 . \end{array}$$

1. Explain how this formulation models the problem, and say why the analyst has not simplified the last inequality to \(a + b + c \leqslant 15\).
2. The final constraint is different from the others in that the resource is integer valued. Explain why that does not impose an additional difficulty for this problem.
3. Solve this problem using the simplex algorithm. Interpret your solution and say what resources are left over. On a particular day an order is received for two Cuddly Cats, and the extra constraint \(c \geqslant 2\) is added to the formulation.
4. Set up an initial simplex tableau to deal with the modified problem using either the big-M approach or two-phase simplex. Do not perform any iterations on your tableau.
5. Show that the solution given by \(a = 8 , b = 2\) and \(c = 5\) uses all of the resources, but that \(a = 6 , b = 6\) and \(c = 2\) gives more profit. What resources are left over from the latter solution?