AQA D2 (Decision Mathematics 2) 2012 June

1
Figure 1 below shows an activity diagram for a construction project. The time needed for each activity is given in days.

1. Find the earliest start time and the latest finish time for each activity and insert their values on Figure 1.
2. Find the critical paths and state the minimum time for completion of the project.
3. On Figure 2 opposite, draw a cascade diagram (Gantt chart) for the project, assuming that each activity starts as early as possible.
4. Activity \(J\) takes longer than expected so that its duration is \(x\) days, where \(x \geqslant 3\). Given that the minimum time for completion of the project is unchanged, find a further inequality relating to the maximum value of \(x\).
5. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d0902228-7041-4449-9ccb-770352ce6bef-02_910_1355_1414_411}
   \end{figure}
6. Critical paths are \(\_\_\_\_\)
   Minimum completion time is \(\_\_\_\_\) days. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d0902228-7041-4449-9ccb-770352ce6bef-03_940_1160_390_520}
   \end{figure}
7. \(\_\_\_\_\)

2 The times taken in minutes for five people, Ann, Baz, Cal, Di and Ez, to complete each of five different tasks are recorded in the table below. Neither Ann nor Di can do task 2, as indicated by the asterisks in the table.

3

1. Given that \(k\) is a constant, complete the Simplex tableau below for the following linear programming problem. Maximise $$P = k x + 6 y + 5 z$$ subject to $$\begin{gathered} 2 x + y + 4 z \leqslant 11
   x + 3 y + 6 z \leqslant 18
   x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{gathered}$$
2. Use the Simplex method to perform one iteration of your tableau for part (a), choosing a value in the \(\boldsymbol { y }\)-column as pivot.
3. 1. In the case when \(k = 1\), explain why the maximum value of \(P\) has now been reached and write down this maximum value of \(P\).
   2. In the case when \(k = 3\), perform one further iteration and interpret your new tableau. \section*{Answer space for question 3}

4. \(\boldsymbol { P }\)	\(\boldsymbol { x }\)	\(\boldsymbol { y }\)	\(\boldsymbol { Z }\)	\(s\)	\(\boldsymbol { t }\)	value
   1	\(- k\)	-6	-5	0	0	0
   0
   0

5. \(\boldsymbol { P }\)	\(\boldsymbol { x }\)	\(\boldsymbol { y }\)	\(\boldsymbol { Z }\)	\(\boldsymbol { s }\)	\(\boldsymbol { t }\)	value

   \section*{Answer space for question 3}
6. 1. \(\_\_\_\_\)

4

1. Two people, Adam and Bill, play a zero-sum game. The game is represented by the following pay-off matrix for Adam. 4
2. Roza plays a different zero-sum game against a computer. The game is represented by the following pay-off matrix for Roza.

5 Dave plans to renovate three houses, \(A , B\) and \(C\), at the rate of one per year. The order in which they are renovated is a matter of choice, but some costs vary over the three years. The expected costs, in thousands of pounds, are given in the table below. (b) Optimum order \(\_\_\_\_\)

6

1. The network shows a flow from \(S\) to \(T\) along a system of pipes, with the capacity in litres per second indicated on each edge.
   \includegraphics[max width=\textwidth, alt={}, center]{d0902228-7041-4449-9ccb-770352ce6bef-14_510_936_411_552}
   1. Show that the value of the cut shown on the diagram is 36 .
   2. The cut shown on the diagram can be represented as \(\{ S , B \} , \{ A , C , T \}\). Complete the table below to give the value of each of the 8 possible cuts.

      Cut		Value
      \(\{ S \}\)	\(\{ A , B , C , T \}\)	30
      \(\{ S , A \}\)	\(\{ B , C , T \}\)	29
      \(\{ S , B \}\)	\(\{ A , C , T \}\)	36
      \(\{ S , C \}\)	\(\{ A , B , T \}\)	33
      \(\{ S , A , B \}\)	\(\{ C , T \}\)
      \(\{ S , A , C \}\)	\(\{ B , T \}\)
      \(\{ S , B , C \}\)	\(\{ A , T \}\)
      \(\{ S , A , B , C \}\)	\(\{ T \}\)	30

   3. State the value of the maximum flow through the network, giving a reason for your answer. Maximum flow is \(\_\_\_\_\) because \(\_\_\_\_\)
   4. Indicate on the diagram below a possible flow along each edge corresponding to this maximum flow.
      \includegraphics[max width=\textwidth, alt={}, center]{d0902228-7041-4449-9ccb-770352ce6bef-15_469_933_406_550}
2. The capacities along \(S C\) and along \(A T\) are each increased by 4 litres per second.
   1. Using your values from part (a)(iv) as the initial flow, indicate potential increases and decreases on the diagram below and use the labelling procedure to find the new maximum flow through the network. You should indicate any flow augmenting paths in the table and modify the potential increases and decreases of the flow on the diagram.
      \includegraphics[max width=\textwidth, alt={}, center]{d0902228-7041-4449-9ccb-770352ce6bef-15_470_935_1315_260}

      Path	Additional Flow

   2. Use your results from part (b)(i) to illustrate the flow along each edge that gives this new maximum flow, and state the value of the new maximum flow. New maximum flow is \(\_\_\_\_\)
      \includegraphics[max width=\textwidth, alt={}, center]{d0902228-7041-4449-9ccb-770352ce6bef-15_474_933_2078_550}