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AQA D2 2008 June Q6
13 marks Standard +0.3
6 [Figures 4, 5 and 6, printed on the insert, are provided for use in this question.]
The network shows a system of pipes with the lower and upper capacities for each pipe in litres per second. \includegraphics[max width=\textwidth, alt={}, center]{f98d4434-458a-4118-92ed-309510d7975a-06_796_1337_518_338}
    1. Find the value of the cut \(C\).
    2. Hence state what can be deduced about the maximum flow from \(S\) to \(T\).
  2. Figure 4, printed on the insert, shows a partially completed diagram for a feasible flow of 32 litres per second from \(S\) to \(T\). Indicate, on Figure 4, the flows along the edges \(P Q , U Q\) and \(U T\).
    1. Taking your feasible flow from part (b) as an initial flow, indicate potential increases and decreases of the flow along each edge on Figure 5.
    2. Use flow augmentation on Figure 5 to find the maximum flow from \(S\) to \(T\). You should indicate any flow augmenting paths in the table and modify the potential increases and decreases of the flow on the network.
    3. Illustrate the maximum flow on Figure 6.
AQA D2 2009 June Q1
12 marks Moderate -0.8
1 [Figure 1, printed on the insert, is provided for use in this question.]
A decorating project is to be undertaken. The table shows the activities involved.
ActivityImmediate PredecessorsDuration (days)
A-5
B-3
C-2
DA, \(B\)4
E\(B , C\)1
\(F\)D2
GE9
H\(F , G\)1
I\(H\)6
\(J\)\(H\)5
\(K\)\(I , J\)2
  1. Complete an activity network for the project on Figure 1.
  2. On Figure 1, indicate:
    1. the earliest start time for each activity;
    2. the latest finish time for each activity.
  3. State the minimum completion time for the decorating project and identify the critical path.
  4. Activity \(F\) takes 4 days longer than first expected.
    1. Determine the new earliest start time for activities \(H\) and \(I\).
    2. State the minimum delay in completing the project.
AQA D2 2009 June Q2
11 marks Moderate -0.3
2 Two people, Rowena and Colin, play a zero-sum game.
The game is represented by the following pay-off matrix for Rowena.
\multirow{5}{*}{Rowena}Colin
Strategy\(\mathrm { C } _ { 1 }\)\(\mathbf { C } _ { \mathbf { 2 } }\)\(\mathrm { C } _ { 3 }\)
\(\mathbf { R } _ { \mathbf { 1 } }\)-454
\(\mathbf { R } _ { \mathbf { 2 } }\)2-3-1
\(\mathbf { R } _ { \mathbf { 3 } }\)-543
  1. Explain what is meant by the term 'zero-sum game'.
  2. Determine the play-safe strategy for Colin, giving a reason for your answer.
  3. Explain why Rowena should never play strategy \(R _ { 3 }\).
  4. Find the optimal mixed strategy for Rowena.
AQA D2 2009 June Q3
13 marks Moderate -0.3
3 Five lecturers were given the following scores when matched against criteria for teaching five courses in a college.
Course 1Course 2Course 3Course 4Course 5
Ron131391013
Sam1314121715
Tom161081414
Una1114121610
Viv1214141315
Each lecturer is to be allocated to exactly one of the courses so as to maximise the total score of the five lecturers.
  1. Explain why the Hungarian algorithm may be used if each number, \(x\), in the table is replaced by \(17 - x\).
  2. Form a new table by subtracting each number in the table above from 17. Hence show that, by reducing rows first and then columns, the resulting table of values is as below.
    00330
    43402
    06722
    52306
    31020
  3. Show that the zeros in the table in part (b) can be covered with two horizontal and two vertical lines. Hence use the Hungarian algorithm to reduce the table to a form where five lines are needed to cover the zeros.
  4. Hence find the possible allocations of courses to the five lecturers so that the total score is maximised.
  5. State the value of the maximum total score.
AQA D2 2009 June Q4
14 marks Standard +0.8
4 A linear programming problem involving variables \(x , y\) and \(z\) is to be solved. The objective function to be maximised is \(P = 4 x + y + k z\), where \(k\) is a constant. The initial Simplex tableau is given below.
\(\boldsymbol { P }\)\(\boldsymbol { x }\)\(\boldsymbol { y }\)\(\boldsymbol { z }\)\(s\)\(\boldsymbol { t }\)value
1-4-1\(- k\)000
0123107
02140110
  1. In addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), write down two inequalities involving \(x , y\) and \(z\) for this problem.
    1. The first pivot is chosen from the \(\boldsymbol { x }\)-column. Identify the pivot and perform one iteration of the Simplex method.
    2. Given that the optimal value of \(P\) has not been reached after this first iteration, find the possible values of \(k\).
  3. Given that \(k = 10\) :
    1. perform one further iteration of the Simplex method;
    2. interpret the final tableau.
AQA D2 2009 June Q5
9 marks Standard +0.3
5 [Figure 2, printed on the insert, is provided for use in this question.]
A company has a number of stores. The following network shows the possible actions and profits over the next five years. The number on each edge is the expected profit, in millions of pounds. A negative number indicates a loss due to investment in new stores. \includegraphics[max width=\textwidth, alt={}, center]{1bf0d8b7-9f91-437a-bc18-3bfe5ca12223-06_1006_1583_591_223}
  1. Working backwards from \(\boldsymbol { T }\), use dynamic programming to maximise the expected profits over the five years. You may wish to complete the table on Figure 2 as your solution.
  2. State the maximum expected profit and the sequence of vertices from \(S\) to \(T\) in order to achieve this.
    (2 marks)
AQA D2 2009 June Q6
16 marks Standard +0.3
6 [Figures 3, 4 and 5, printed on the insert, are provided for use in this question.]
The network shows a system of pipes with the lower and upper capacities for each pipe in litres per second. \includegraphics[max width=\textwidth, alt={}, center]{1bf0d8b7-9f91-437a-bc18-3bfe5ca12223-07_849_1363_518_326}
  1. Find the value of the cut \(C\).
  2. Figure 3, on the insert, shows a partially completed diagram for a feasible flow of 40 litres per second from \(S\) to \(T\). Indicate, on Figure 3, the flows along the edges \(A E , E F\) and \(F G\).
    1. Taking your answer 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 \(S\) to \(T\). You should indicate any flow augmenting paths in the table and modify the potential increases and decreases of the flow on the network.
  4. Illustrate the maximum flow on Figure 5.
  5. Find a cut with value equal to that of the maximum flow.
AQA D2 2012 June Q1
14 marks Moderate -0.5
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 \(x\).
    1. \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}
    2. 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}
    3. \(\_\_\_\_\)
AQA D2 2012 June Q2
10 marks Standard +0.3
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.
AQA D2 2012 June Q3
14 marks Standard +0.8
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.
    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}
      1. \(\boldsymbol { P }\)\(\boldsymbol { x }\)\(\boldsymbol { y }\)\(\boldsymbol { Z }\)\(s\)\(\boldsymbol { t }\)value
        1\(- k\)-6-5000
        0
        0
      2. \(\boldsymbol { P }\)\(\boldsymbol { x }\)\(\boldsymbol { y }\)\(\boldsymbol { Z }\)\(\boldsymbol { s }\)\(\boldsymbol { t }\)value
        \section*{Answer space for question 3}
        1. \(\_\_\_\_\)
AQA D2 2012 June Q4
11 marks Standard +0.3
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.
AQA D2 2012 June Q5
10 marks Moderate -0.8
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)
YearAlready renovatedHouse renovatedCalculationValue
3\(A\) and \(B\)C
\(A\) and \(C\)B
\(B\) and \(C\)A
2AB
C
BA
C
CA
B
1
Optimum order \(\_\_\_\_\)
AQA D2 2012 June Q6
16 marks Moderate -0.5
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.
      CutValue
      \(\{ S \}\)\(\{ A , B , C , T \}\)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}
AQA D2 2014 June Q1
9 marks Moderate -0.8
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}
\includegraphics[max width=\textwidth, alt={}]{c2b62fee-d320-4701-a5bb-b2e4b8cc0952-03_424_410_296_685}
AQA D2 2014 June Q2
5 marks Easy -1.2
2 Alex and Roberto play a zero-sum game. The game is represented by the following pay-off matrix for Alex. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Roberto}
\multirow{5}{*}{Alex Strategy}DEFG
A5- 4- 11
B4301
C- 30- 5- 2
\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.
AQA D2 2014 June Q3
9 marks Moderate -0.5
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.
    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} (c) \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 = \(\_\_\_\_\)
AQA D2 2014 June Q4
11 marks Standard +0.3
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.
AQA D2 2014 June Q5
8 marks Standard +0.3
5 Mark and Owen play a zero-sum game. The game is represented by the following pay-off matrix for Mark.
Owen
\cline { 2 - 5 }\cline { 2 - 5 }StrategyDEF
A41- 1
\cline { 2 - 5 } MarkB3- 2- 2
\cline { 2 - 5 }C- 203
  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]
AQA D2 2014 June Q6
12 marks Standard +0.3
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}
    StageStateFromCalculationValue
    1IK
    \(J\)K
AQA D2 2014 June Q7
11 marks Standard +0.3
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 \(4 < x < 11\).
AQA D2 2014 June Q8
10 marks Moderate -0.8
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.
AQA D2 2015 June Q1
14 marks Moderate -0.5
1 Figure 2, on the page opposite, shows an activity diagram for a project. Each activity requires one worker. The duration required for each activity is given in hours.
  1. On Figure 1 below, complete the precedence table.
  2. Find the earliest start time and the latest finish time for each activity and insert their values on Figure 2.
  3. List the critical paths.
  4. Find the float time of activity \(E\).
  5. Using Figure 3 opposite, draw a Gantt diagram to illustrate how the project can be completed in the minimum time, assuming that each activity is to start as early as possible.
  6. Given that there is only one worker available for the project, find the minimum completion time for the project.
  7. Given that there are two workers available for the project, find the minimum completion time for the project. Show a suitable allocation of tasks to the two workers.
    [0pt] [2 marks] \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1}
    ActivityImmediate predecessor(s)
    A
    B
    C
    D
    E
    \(F\)
    G
    \(H\)
    I
    J
    \end{table} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{b0f9523e-51dd-495f-99ec-4724243b5619-03_1071_1561_376_278}
    \end{figure} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{b0f9523e-51dd-495f-99ec-4724243b5619-03_801_1301_1644_420}
    \end{figure}
AQA D2 2015 June Q2
8 marks Moderate -0.8
2 Stan and Christine play a zero-sum game. The game is represented by the following pay-off matrix for Stan. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Christine}
\multirow{5}{*}{Stan}StrategyDEF
A3- 3- 1
B- 1- 42
C10- 3
\cline { 2 - 5 }- 2
\end{table}
  1. Find the play-safe strategy for each player.
  2. Show that there is no stable solution.
  3. Explain why a suitable pay-off matrix for Christine is given by
AQA D2 2015 June Q3
11 marks Moderate -0.3
3 In the London 2012 Olympics, the Jamaican \(4 \times 100\) metres relay team set a world record time of 36.84 seconds. Athletes take different times to run each of the four legs.
The coach of a national athletics team has five athletes available for a major championship. The lowest times that the five athletes take to cover each of the four legs is given in the table below. The coach is to allocate a different athlete from the five available athletes, \(A , B , C , D\) and \(E\), to each of the four legs to produce the lowest total time.
Leg 1Leg 2Leg 3Leg 4
Athlete \(\boldsymbol { A }\)9.848.918.988.70
Athlete \(\boldsymbol { B }\)10.289.069.249.05
Athlete \(\boldsymbol { C }\)10.319.119.229.18
Athlete \(\boldsymbol { D }\)10.049.079.199.01
Athlete \(\boldsymbol { E }\)9.918.959.098.74
Use the Hungarian algorithm, by reducing the columns first, to assign an athlete to each leg so that the total time of the four athletes is minimised. State the allocation of the athletes to the four legs and the total time.
[0pt] [11 marks]
\includegraphics[max width=\textwidth, alt={}]{b0f9523e-51dd-495f-99ec-4724243b5619-08_1200_1705_1507_155}
AQA D2 2015 June Q4
13 marks Standard +0.8
4
  1. Display the following linear programming problem in a Simplex tableau. $$\begin{array} { l r } \text { Maximise } & P = 2 x + 3 y + 4 z \\ \text { subject to } & x + y + 2 z \leqslant 20 \\ & 3 x + 2 y + z \leqslant 30 \\ & 2 x + 3 y + z \leqslant 40 \\ \text { and } & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
    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.
    1. Perform one further iteration.
    2. Interpret your final tableau and state the values of your slack variables.