AQA D2 (Decision Mathematics 2) 2014 June

1 A major project has been divided into a number of tasks, as shown in the table. The minimum time required to complete each task is also shown. \section*{Answer space for question 1}

2 Alex and Roberto play a zero-sum game. The game is represented by the following pay-off matrix for Alex. \begin{table}[h] \end{table}

1. Show that this game has a stable solution and state the play-safe strategy for each player.
2. List any saddle points.

3 The diagram below shows a network of pipes with source \(A\) and \(\operatorname { sink } J\). The capacity of each pipe is given by the number on each edge.
\includegraphics[max width=\textwidth, alt={}, center]{c2b62fee-d320-4701-a5bb-b2e4b8cc0952-08_816_1280_443_386}

1. Find the values of the cuts \(\mathrm { C } _ { 1 }\) and \(\mathrm { C } _ { 2 }\).
2. Find by inspection a flow of 60 units, with flows of 25,10 and 25 along \(H J , G J\) and \(I J\) respectively. Illustrate your answer on Figure 1.
3. 1. On a certain day the section \(E H\) is blocked, as shown on Figure 2. Find, by inspection or otherwise, the maximum flow on this day and illustrate your answer on Figure 2.
   2. Show that the flow obtained in part (c)(i) is maximal. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{c2b62fee-d320-4701-a5bb-b2e4b8cc0952-09_595_1065_376_475}
      \end{figure}
4. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{c2b62fee-d320-4701-a5bb-b2e4b8cc0952-09_617_1061_1142_477}
   \end{figure} Maximum flow = \(\_\_\_\_\)

4

1. Display the following linear programming problem in a Simplex tableau. $$\begin{array} { l c } \text { Maximise } & P = 3 x + 6 y + 2 z
   \text { subject to } & x + 3 y + 2 z \leqslant 11
   & 3 x + 4 y + 2 z \leqslant 21
   \text { and } & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 . \end{array}$$
2. The first pivot to be chosen is from the \(y\)-column. Perform one iteration of the Simplex method.
3. Perform one further iteration.
4. Interpret the tableau obtained in part (c) and state the values of your slack variables.

5 Mark and Owen play a zero-sum game. The game is represented by the following pay-off matrix for Mark.

1. Explain why Mark should never play strategy B.
2. It is given that the value of the game is 0.6 . Find the optimal strategy for Owen.
   (You are not required to find the optimal mixed strategy for Mark.)
   [0pt] [7 marks]

6 The network below has 11 vertices and 16 edges connecting some pairs of vertices. The numbers on the edges are their weights. The weight of the edge \(D G\) is given in terms of \(x\). There are three routes from \(A\) to \(K\) that have the same minimum total weight.
\includegraphics[max width=\textwidth, alt={}, center]{c2b62fee-d320-4701-a5bb-b2e4b8cc0952-16_863_1444_552_299} Working backwards from \(\boldsymbol { K }\), use dynamic programming, to find:

1. the minimum total weight from \(A\) to \(K\);
2. the value of \(x\);
3. the three routes corresponding to the minimum total weight. You must complete the table opposite as your solution.
   [0pt] [12 marks] \section*{Answer space for question 6}

   Stage	State	From	Calculation	Value
   1	I	K
   	\(J\)	K

7 The table shows the times taken, in minutes, by four people, \(A , B , C\) and \(D\), to carry out the tasks \(W , X , Y\) and \(Z\). Some of the times are subject to the same delay of \(x\) minutes, where \(4 < x < 11\).

8 An activity diagram for a project is shown below. The duration of each activity is given in weeks. The earliest start time and the latest finish time for each activity are shown on the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{c2b62fee-d320-4701-a5bb-b2e4b8cc0952-22_640_1626_475_209}

1. Find the values of \(x , y\) and \(z\).
2. State the critical path.
3. Some of the activities can be speeded up at an additional cost. The following table lists the activities that can be speeded up together with the minimum possible duration of these activities. The table also shows the additional cost of reducing the duration of each of these activities by one week.