AQA D2 (Decision Mathematics 2) 2016 June

1
Figure 1 below shows an activity diagram for a project. Each activity requires one worker. The duration required for each activity is given in hours.

1. Find the earliest start time and the latest finish time for each activity and insert these values on Figure 1.
2. 1. Find the critical path.
   2. Find the float time of activity \(F\).
3. Using Figure 2 on page 3, draw a resource histogram to illustrate how the project can be completed in the minimum time, assuming that each activity is to start as early as possible.
4. 1. Given that there are two workers available for the project, find the minimum completion time for the project.
   2. Write down an allocation of tasks to the two workers that corresponds to your answer in part (d)(i). \section*{Answer space for question 1} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-02_687_1655_1941_189}
      \end{figure} \section*{Answer space for question 1} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-03_1115_1575_434_283}
      \end{figure}
      \includegraphics[max width=\textwidth, alt={}]{34de3f03-a275-44fb-88b2-b88038bcec97-03_1024_1593_1683_267}

2 Alan, Beth, Callum, Diane and Ethan work for a restaurant chain. The costs, in pounds, for the five people to travel to each of five different restaurants are recorded in the table below. Alan cannot travel to restaurant 1 and Beth cannot travel to restaurants 3 and 5, as indicated by the asterisks in the table.

3
Maximise \(\quad P = 2 x - 3 y + 4 z\)
subject to \(\quad x + 2 y + z \leqslant 20\)
\(x - y + 3 z \leqslant 24\)
\(3 x - 2 y + 2 z \leqslant 30\)
and \(\quad x \geqslant 0 , y \geqslant 0 , z \geqslant 0\).

1. Display the linear programming problem in a Simplex tableau.
2. 1. The first pivot to be chosen is from the \(z\)-column. Identify the pivot and explain why this particular value is chosen.
   2. Perform one iteration of the Simplex method.
   3. Perform one further iteration.
3. Interpret your final tableau and state the values of your slack variables.
   [0pt] [3 marks]

4 Monica and Vladimir play a zero-sum game. The game is represented by the following pay-off matrix for Monica.

5 Robert is planning to renovate four houses, \(A , B , C\) and \(D\), at the rate of one per month. The houses can be renovated in any order but the costs will vary because some of the materials left over from renovating one house can be used for the next one. The expected profits, in hundreds of pounds, are given in the table below.

6 The network shows a system of pipes with lower and upper capacities for each pipe in litres per second.
\includegraphics[max width=\textwidth, alt={}, center]{34de3f03-a275-44fb-88b2-b88038bcec97-22_817_744_397_648}

1. 1. Find the value of the cut \(X\).
   2. Hence state what can be deduced about the maximum flow from \(A\) to \(H\).
2. Figure 3 shows a partially completed diagram for a feasible flow of 28 litres per second from \(A\) to \(H\). Indicate, on Figure 3, the flows along the edges \(B D , B E\) and \(C D\).
3. 1. Using your feasible flow from part (b) as an initial flow, indicate potential increases and decreases of the flow along each edge on Figure 4.
   2. Use flow augmentation on Figure 4 to find the maximum flow from \(A\) to \(H\). You should indicate any flow augmenting paths in the table and modify the potential increases and decreases of the flow on the network.
   3. State the maximum flow and indicate a maximum flow on Figure 5. \section*{Answer space for question 6} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-23_682_689_312_397}
      \end{figure} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-23_935_1477_1037_365}
      \end{figure} Figure 5
      \includegraphics[max width=\textwidth, alt={}]{34de3f03-a275-44fb-88b2-b88038bcec97-24_2032_1707_219_153}