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AQA D2 2011 January Q1
14 marks Moderate -0.5
1
A group of workers is involved in a decorating project. The table shows the activities involved. Each worker can perform any of the given activities.
ActivityA\(B\)CD\(E\)\(F\)GHI\(J\)\(K\)\(L\)
Duration (days)256794323231
Number of workers required635252445324
The activity network for the project is given in Figure 1 below.
  1. Find the earliest start time and the latest finish time for each activity, inserting their values on Figure 1.
  2. Hence find:
    1. the critical path;
    2. the float time for activity \(D\).
      1. \begin{figure}[h]
        \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{172c5c92-4254-4593-b741-1caa83a1e833-02_647_1657_1640_180}
        \end{figure}
        1. The critical path is \(\_\_\_\_\)
        2. The float time for activity \(D\) is \(\_\_\_\_\)
    3. Given that each activity starts as early as possible and assuming that there is no limit to the number of workers available, draw a resource histogram for the project on Figure 2 below, indicating clearly which activities are taking place at any given time.
    4. It is later discovered that there are only 8 workers available at any time. Use resource levelling to construct a new resource histogram on Figure 3 below, showing how the project can be completed with the minimum extra time. State the minimum extra time required.
    5. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{172c5c92-4254-4593-b741-1caa83a1e833-03_586_1708_922_150}
      \end{figure}
    6. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{172c5c92-4254-4593-b741-1caa83a1e833-03_496_1705_1672_153}
      \end{figure} The minimum extra time required is \(\_\_\_\_\)
AQA D2 2011 January Q2
13 marks Moderate -0.8
2 A farmer has five fields. He intends to grow a different crop in each of four fields and to leave one of the fields unused. The farmer tests the soil in each field and calculates a score for growing each of the four crops. The scores are given in the table below.
Field AField BField CField DField E
Crop 1161281814
Crop 2201581612
Crop 3910121712
Crop 41811171519
The farmer's aim is to maximise the total score for the four crops.
    1. Modify the table of values by first subtracting each value in the table above from 20 and then adding an extra row of equal values.
    2. Explain why the Hungarian algorithm can now be applied to the new table of values to maximise the total score for the four crops.
    1. By reducing rows first, show that the table from part (a)(i) becomes
      26100\(p\)
      051248
      8750\(q\)
      18240
      00000
      State the values of the constants \(p\) and \(q\).
    2. Show that the zeros in the table from part (b)(i) can be covered by one horizontal and three vertical lines, and use the Hungarian algorithm to decide how the four crops should be allocated to the fields.
    3. Hence find the maximum possible total score for the four crops.
AQA D2 2011 January Q3
13 marks Easy -1.8
3 Two people, Rhona and Colleen, play a zero-sum game. The game is represented by the following pay-off matrix for Rhona.
\cline { 2 - 5 }Colleen
\cline { 2 - 5 } Strategy\(\mathbf { C } _ { \mathbf { 1 } }\)\(\mathbf { C } _ { \mathbf { 2 } }\)\(\mathbf { C } _ { \mathbf { 3 } }\)
\cline { 2 - 5 } Rhona\(\mathbf { R } _ { \mathbf { 1 } }\)264
\cline { 2 - 5 }\(\mathbf { R } _ { \mathbf { 2 } }\)3- 3- 1
\cline { 2 - 5 }\(\mathbf { R } _ { \mathbf { 3 } }\)\(x\)\(x + 3\)3
\cline { 2 - 5 }
\cline { 2 - 5 }
It is given that \(x < 2\).
    1. Write down the three row minima.
    2. Show that there is no stable solution.
  2. Explain why Rhona should never play strategy \(R _ { 3 }\).
    1. Find the optimal mixed strategy for Rhona.
    2. Find the value of the game.
AQA D2 2011 January Q4
15 marks Standard +0.8
4 The Simplex method is to be used to maximise \(P = 3 x + 2 y + z\) subject to the constraints $$\begin{aligned} - x + y + z & \leqslant 4 \\ 2 x + y + 4 z & \leqslant 10 \\ 4 x + 2 y + 3 z & \leqslant 21 \end{aligned}$$ The initial Simplex tableau is given below.
\(\boldsymbol { P }\)\(\boldsymbol { x }\)\(\boldsymbol { y }\)\(\boldsymbol { z }\)\(s\)\(t\)\(\boldsymbol { u }\)value
1-3-2-10000
0-1111004
021401010
042300121
    1. The first pivot is to be chosen from the \(x\)-column. Identify the pivot and explain why this particular value is chosen.
    2. Perform one iteration of the Simplex method and explain how you know that the optimal value has not been reached.
    1. Perform one further iteration.
    2. Interpret the final tableau and write down the initial inequality that still has slack.
      \includegraphics[max width=\textwidth, alt={}]{172c5c92-4254-4593-b741-1caa83a1e833-11_2486_1714_221_153}
AQA D2 2011 January Q5
9 marks Moderate -0.8
5 Each path from \(S\) to \(T\) in the network below represents a possible way of using the internet to buy a ticket for a particular event. The number on each edge represents a charge, in pounds, with a negative value representing a discount. For example, the path SAEIT represents a ticket costing \(23 + 5 - 4 - 7 = 17\) pounds. \includegraphics[max width=\textwidth, alt={}, center]{172c5c92-4254-4593-b741-1caa83a1e833-12_1023_1330_540_350}
  1. By working backwards from \(\boldsymbol { T }\) and completing the table on Figure 4, use dynamic programming to find the minimum weight of all paths from \(S\) to \(T\).
  2. State the minimum cost of a ticket for the event and the paths corresponding to this minimum cost.
    (3 marks)
    1. \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Figure 4}
      StageStateFromValue
      1I\(T\)-7
      \(J\)\(T\)-6
      \(K\)\(T\)-5
      2\(E\)\(I\)\(- 7 - 4 = - 11\)
      FI
      \(J\)
      GI
      \(J\)
      \(K\)
      \(H\)\(K\)
      3
      \end{table}
AQA D2 2011 January Q6
11 marks Moderate -0.5
6 A retail company has warehouses at \(P , Q\) and \(R\), and goods are to be transported to retail outlets at \(Y\) and \(Z\). There are also retaining depots at \(U , V , W\) and \(X\). The possible routes with the capacities along each edge, in van loads per week, are shown in the following diagram. \includegraphics[max width=\textwidth, alt={}, center]{172c5c92-4254-4593-b741-1caa83a1e833-14_673_1193_577_429}
  1. On Figure 5 opposite, add a super-source, \(S\), and a super-sink, \(T\), and appropriate edges so as to produce a directed network with a single source and a single sink. Indicate the capacity of each edge that you have added.
  2. On Figure 6, write down the maximum flows along the routes SPUYT and SRVWZT.
    1. On Figure 7, add the vertices \(S\) and \(T\) and the edges connecting \(S\) and \(T\) to the network. Using the maximum flows along the routes SPUYT and SRVWZT found in part (b) as the initial flow, indicate the potential increases and decreases of the flow on each edge of these routes.
    2. Use flow augmentation to find the maximum flow from \(S\) to \(T\). You should indicate any flow-augmenting routes on Figure 6 and modify the potential increases and decreases of the flow on Figure 7.
  4. Find a cut with value equal to the maximum flow. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{172c5c92-4254-4593-b741-1caa83a1e833-15_629_1100_342_477}
    \end{figure} \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Figure 6}
    RouteFlow
    SPUYT
    SRVWZT
    \end{table} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 7} \includegraphics[alt={},max width=\textwidth]{172c5c92-4254-4593-b741-1caa83a1e833-15_634_1109_1838_470}
    \end{figure}
AQA D2 2012 January Q1
14 marks Standard +0.3
1 The diagram shows the activity network and the duration, in days, of each activity for a particular project. Some of the earliest start times and latest finish times are shown on the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-02_830_1447_678_301}
  1. Find the values of the constants \(x , y\) and \(z\).
  2. Find the critical paths.
  3. Find the activity with the largest float and state the value of this float.
  4. The number of workers required for each activity is shown in the table.
    Activity\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
    Number of workers required4234243356
    Given that each activity starts as early as possible and assuming that there is no limit to the number of workers available, draw a resource histogram for the project on Figure 1 below, indicating clearly which activities are taking place at any given time.
  5. It is later discovered that there are only 9 workers available at any time. Use resource levelling to find the new earliest start time for activity \(J\) so that the project can be completed with the minimum extra time. State the minimum extra time required. (d) Number of workers \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-03_803_1330_1224_468}
    \end{figure}
AQA D2 2012 January Q2
12 marks Moderate -0.3
2 A team with five members is training to take part in a quiz. The team members, Abby, Bob, Cait, Drew and Ellie, attempted sample questions on each of the five topics and their scores are given in the table.
Topic 1Topic 2Topic 3Topic 4Topic 5
Abby2729253531
Bob3322172929
Cait2329253321
Drew2229292731
Ellie2727192127
For the actual quiz, each topic must be allocated to exactly one of the team members. The maximum total score for the sample questions is to be used to allocate the different topics to the team members.
  1. Explain why the Hungarian algorithm may be used if each number, \(x\), in the table is replaced by \(35 - x\).
  2. Form a new table by subtracting each number in the table above from 35 . Hence show that, by reducing rows first then columns, the resulting table of values is as below, stating the values of the constants \(p\) and \(q\).
    86804
    011\(p\)44
    1046012
    \(q\)2040
    00660
  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.
    1. Hence find the possible allocations of topics to the five team members so that the total score for the sample questions is maximised.
    2. State the value of this maximum total score.
AQA D2 2012 January Q3
13 marks Easy -2.5
3 Two people, Roz and Colum, play a zero-sum game. The game is represented by the following pay-off matrix for Roz.
Colum
\cline { 2 - 5 }Strategy\(\mathbf { C } _ { \mathbf { 1 } }\)\(\mathbf { C } _ { \mathbf { 2 } }\)\(\mathbf { C } _ { \mathbf { 3 } }\)
\multirow{3}{*}{\(\operatorname { Roz }\)}\(\mathbf { R } _ { \mathbf { 1 } }\)- 2- 6- 1
\cline { 2 - 5 }\(\mathbf { R } _ { \mathbf { 2 } }\)- 52- 6
\cline { 2 - 5 }\(\mathbf { R } _ { \mathbf { 3 } }\)- 33- 4
  1. Explain what is meant by the term 'zero-sum game'.
  2. Determine the play-safe strategy for Colum, giving a reason for your answer.
    1. Show that the matrix can be reduced to a 2 by 3 matrix, giving the reason for deleting one of the rows.
    2. Hence find the optimal mixed strategy for Roz.
AQA D2 2012 January Q4
13 marks Standard +0.3
4 A linear programming problem consists of maximising an objective function \(P\) involving three variables, \(x , y\) and \(z\), subject to constraints given by three inequalities other than \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\). Slack variables \(s , t\) and \(u\) are introduced and the Simplex method is used to solve the problem. One iteration of the method leads to the following tableau.
\(\boldsymbol { P }\)\(\boldsymbol { x }\)\(\boldsymbol { y }\)\(\boldsymbol { Z }\)\(\boldsymbol { s }\)\(\boldsymbol { t }\)\(\boldsymbol { u }\)value
1-21103006
02311002
06-300-6103
0-1-90-3014
    1. State the column from which the pivot for the next iteration should be chosen. Identify this pivot and explain the reason for your choice.
    2. Perform the next iteration of the Simplex method.
    1. Explain why you know that the maximum value of \(P\) has been achieved.
    2. State how many of the three original inequalities still have slack.
    1. State the maximum value of \(P\) and the values of \(x , y\) and \(z\) that produce this maximum value.
    2. The objective function for this problem is \(P = k x - 2 y + 3 z\), where \(k\) is a constant. Find the value of \(k\).
      \includegraphics[max width=\textwidth, alt={}]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-11_2486_1714_221_153}
AQA D2 2012 January Q5
9 marks Moderate -0.5
5 A firm is considering various strategies for development over the next few years. In the network, the number on each edge is the expected profit, in millions of pounds, moving from one year to the next. A negative number indicates a loss because of building costs or other expenses. Each path from \(S\) to \(T\) represents a complete strategy. \includegraphics[max width=\textwidth, alt={}, center]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-12_748_1575_559_228}
  1. By completing the table on the page opposite, or otherwise, use dynamic programming working backwards from \(\boldsymbol { T }\) to find the maximum weight of all paths from \(S\) to \(T\).
  2. State the overall maximum profit and the paths from \(S\) to \(T\) corresponding to this maximum profit.
    1. StageStateFromCalculationValue
      1G\(T\)
      H\(T\)
      I\(T\)
      2DG
      \(H\)
      EG
      \(H\)
      I
      \(F\)\(H\)
      I
      3
    2. Maximum profit is £ \(\_\_\_\_\) million Corresponding paths from \(S\) to \(T\) \(\_\_\_\_\)
AQA D2 2012 January Q6
14 marks Standard +0.3
6 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]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-14_807_1472_429_276}
  1. Find the value of the cut \(Q\).
  2. Figure 2 shows most of the values of a feasible flow of 34 litres per second from \(S\) to \(T\).
    1. Insert the missing values of the flows along \(D E\) and \(F G\) on Figure 2.
    2. Using this feasible flow as the initial flow, indicate potential increases and decreases of the flow along each edge on Figure 3.
    3. Use flow augmentation on Figure 3 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.
    1. State the value of the maximum flow.
    2. Illustrate your maximum flow on Figure 4.
  4. Find a cut with capacity equal to that of the maximum flow. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-15_1646_1463_280_381}
    \end{figure} Figure 3 \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-15_668_1230_1998_404}
    \end{figure}
AQA D2 2013 January Q1
13 marks Moderate -0.5
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 their values on Figure 1.
  2. On Figure 2 opposite, complete the precedence table.
  3. Find the critical path.
  4. Find the float time of activity \(E\).
  5. Using Figure 3 on page 5, 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.
  6. Given that there are two workers available for the project, find the minimum completion time for the project.
  7. Given that there is only one worker available for the project, find the minimum completion time for the project. Figure 1 \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{(a)} \includegraphics[alt={},max width=\textwidth]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-02_629_1550_1818_292}
    \end{figure} (b) \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2}
    ActivityImmediate predecessor(s)
    A
    B
    C
    D
    E
    \(F\)
    G
    H
    I
    J
    \(K\)
    \end{table}
    \includegraphics[max width=\textwidth, alt={}]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-05_2486_1717_221_150}
AQA D2 2013 January Q2
5 marks Easy -2.5
2 Harry and Will play a zero-sum game. The game is represented by the following pay-off matrix for Harry.
Will
\cline { 2 - 6 }Strategy\(\boldsymbol { D }\)\(\boldsymbol { E }\)\(\boldsymbol { F }\)\(\boldsymbol { G }\)
Harry\(\boldsymbol { A }\)- 123
\cline { 2 - 6 }\(\boldsymbol { B }\)4637
\cline { 2 - 6 }\(\boldsymbol { C }\)13- 24
  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 2013 January Q3
9 marks Moderate -0.5
3 Four pupils, Wendy, Xiong, Yasmin and Zaira, are each to be allocated a different memory coach from five available coaches: Asif, Bill, Connie, Deidre and Eric. Each pupil has an initial training session with each coach, and a test which scores their improvement in memory-recall produces the following results.
AQA D2 2013 January Q4
6 marks Moderate -0.5
4
  1. When investigating three network flow problems, a student finds:
    1. a flow of 50 and a cut with capacity 50 ;
    2. a flow of 35 and a cut with capacity 50 ;
    3. a flow of 50 and a cut with capacity 35 . In each case, write down what the student can deduce about the maximum flow.
  2. The diagram below shows a network. The numbers on the arcs represent the minimum and maximum flow along each arc respectively. By considering the flow at an appropriate vertex, explain why a flow is not possible through this network. \includegraphics[max width=\textwidth, alt={}, center]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-10_1189_1559_1105_246}
    (2 marks)
AQA D2 2013 January Q5
13 marks Standard +0.3
5
  1. Display the following linear programming problem in a Simplex tableau.
    Maximise \(\quad P = x - 2 y + 3 z\) subject to $$\begin{array} { r } x + y + z \leqslant 16 \\ x - 2 y + 2 z \leqslant 17 \\ 2 x - y + 2 z \leqslant 19 \end{array}$$ and \(x \geqslant 0 , y \geqslant 0 , z \geqslant 0\).
    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 the tableau that you obtained in part (c)(i) and state the values of your slack variables.
AQA D2 2013 January Q6
12 marks Moderate -0.5
6 Kate and Pippa play a zero-sum game. The game is represented by the following pay-off matrix for Kate. \includegraphics[max width=\textwidth, alt={}, center]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-18_2482_1707_223_155}
AQA D2 2013 January Q7
8 marks Moderate -0.8
7 The network below shows a system of one-way roads. The number on each edge represents the number of bags for recycling that can be collected by driving along that road. A collector is to drive from \(A\) to \(I\). \includegraphics[max width=\textwidth, alt={}, center]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-20_867_1644_552_191}
  1. Working backwards from \(\boldsymbol { I }\), use dynamic programming to find the maximum number of bags that can be collected when driving from \(A\) to \(I\). You must complete the table opposite as your solution.
  2. State the route that the collector should take in order to collect the maximum number of bags.
    1. StageStateFromValue
      1GI
      HI
      2
AQA D2 2013 January Q8
9 marks Moderate -0.5
8 The network below represents a system of pipes. The capacity of each pipe, in litres per second, is indicated on the corresponding edge. \includegraphics[max width=\textwidth, alt={}, center]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-22_743_977_404_536}
  1. Find the maximum flow along each of the routes \(A B E H , A C F H\) and \(A D G H\) and enter their values in the table on Figure 4 opposite.
    1. Taking your answers to part (a) as the initial flow, use the labelling procedure on Figure 4 to find the maximum flow through the network. You should indicate any flow-augmenting routes in the table and modify the potential increases and decreases of the flow on the network.
    2. State the value of the maximum flow and, on Figure 5 opposite, illustrate a possible flow along each edge corresponding to this maximum flow.
  3. Confirm that you have a maximum flow by finding a cut of the same value. List the edges of your cut. \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Figure 4}
    RouteFlow
    \(A B E H\)
    \(A C F H\)
    \(A D G H\)
    \end{table} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-23_746_972_397_845}
    \end{figure} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-23_739_971_1311_539}
    \end{figure}
    \includegraphics[max width=\textwidth, alt={}]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-24_2253_1691_221_153}
AQA D2 2010 June Q1
13 marks Moderate -0.8
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 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. A delay in supplies means that Activity \(I\) takes 9 days instead of 2 .
    1. Determine the effect on the earliest possible starting times for activities \(K\) and \(L\).
    2. State the number of days by which the completion of the project is now delayed.
      (1 mark) \section*{Figure 1}
      1. \includegraphics[max width=\textwidth, alt={}, center]{c4dc61a7-47ee-4d5c-bf6d-30a5da2ee8dd-02_815_1337_1573_395}
      2. Critical paths are \(\_\_\_\_\) Minimum completion time is \(\_\_\_\_\) days. QUESTION PART REFERENCE
      3. \begin{figure}[h]
        \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{c4dc61a7-47ee-4d5c-bf6d-30a5da2ee8dd-03_978_1207_354_461}
        \end{figure}
AQA D2 2010 June Q2
10 marks Moderate -0.5
2 Five students attempted five different games, and penalty points were given for any mistakes that they made. The table shows the penalty points incurred by the students.
Game 1Game 2Game 3Game 4Game 5
Ali57388
Beth86487
Cat612103
Di443107
Ell46479
Using the Hungarian algorithm, each of the five students is to be allocated to a different game so that the total number of penalty points is minimised.
  1. By reducing the rows first and then the columns, show that the new table of values is
    24023
    42011
    501\(k\)0
    11042
    02003
    and state the value of the constant \(k\).
  2. Show that the zeros in the table in part (a) can be covered with three lines, and use augmentation to produce a table where five lines are required to cover the zeros.
  3. Hence find the possible ways of allocating the five students to the five games with the minimum total of penalty points.
  4. Find the minimum possible total of penalty points.
    \includegraphics[max width=\textwidth, alt={}]{c4dc61a7-47ee-4d5c-bf6d-30a5da2ee8dd-05_2484_1709_223_153}
AQA D2 2010 June Q3
15 marks Standard +0.8
3
  1. Given that \(k\) is a constant, display the following linear programming problem in a Simplex tableau. $$\begin{array} { l l } \text { Maximise } & P = 6 x + 5 y + 3 z \\ \text { subject to } & x + 2 y + k z \leqslant 8 \\ & 2 x + 10 y + z \leqslant 17 \\ & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
    1. Use the Simplex method to perform one iteration of your tableau for part (a), choosing a value in the \(x\)-column as pivot.
    2. Given that the maximum value of \(P\) has not been achieved after this first iteration, find the range of possible values of \(k\).
  3. In the case where \(k = - 1\), perform one further iteration and interpret your final tableau.
    \includegraphics[max width=\textwidth, alt={}]{c4dc61a7-47ee-4d5c-bf6d-30a5da2ee8dd-07_2484_1707_223_155}
AQA D2 2010 June Q4
13 marks Moderate -0.5
4 Two people, Roger and Corrie, play a zero-sum game.
The game is represented by the following pay-off matrix for Roger.
Corrie
\cline { 2 - 5 }Strategy\(\mathbf { C } _ { \mathbf { 1 } }\)\(\mathbf { C } _ { \mathbf { 2 } }\)\(\mathbf { C } _ { \mathbf { 3 } }\)
\cline { 2 - 5 } Roger\(\mathbf { R } _ { \mathbf { 1 } }\)73- 5
\cline { 2 - 5 }\(\mathbf { R } _ { \mathbf { 2 } }\)- 2- 14
\cline { 2 - 5 }
\cline { 2 - 5 }
    1. Find the optimal mixed strategy for Roger.
    2. Show that the value of the game is \(\frac { 7 } { 13 }\).
  2. Given that the value of the game is \(\frac { 7 } { 13 }\), find the optimal mixed strategy for Corrie.
    \includegraphics[max width=\textwidth, alt={}]{c4dc61a7-47ee-4d5c-bf6d-30a5da2ee8dd-09_2484_1709_223_153}
AQA D2 2010 June Q5
10 marks Standard +0.3
5 A three-day journey is to be made from \(P\) to \(V\), with overnight stops at the end of the first day at one of the locations \(Q\) or \(R\), and at the end of the second day at \(S , T\) or \(U\). The network shows the journey times, in hours, for each day of the journey. \includegraphics[max width=\textwidth, alt={}, center]{c4dc61a7-47ee-4d5c-bf6d-30a5da2ee8dd-10_737_1280_447_388} The optimal route, known as the minimax route, is that in which the longest day's journey is as small as possible.
  1. Explain why the route \(P Q S V\) is better than the route \(P Q T V\).
  2. By completing the table opposite, or otherwise, use dynamic programming, working backwards from \(\boldsymbol { V }\), to find the optimal (minimax) route from \(P\) to \(V\). You should indicate the calculations as well as the values at stages 2 and 3.
    (8 marks)
    \(\ldots . . .\).\includegraphics[max width=\textwidth, alt={}]{c4dc61a7-47ee-4d5c-bf6d-30a5da2ee8dd-11_1000_114_1710_159}