Questions — OCR (4907 questions)

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OCR D2 Q1
8 marks Standard +0.3
  1. The payoff matrix for player \(A\) in a two-person zero-sum game with value \(V\) is shown below.
\cline { 3 - 5 } \multicolumn{2}{c|}{}\(B\)
\cline { 2 - 5 } \multicolumn{2}{c|}{}IIIIII
\multirow{3}{*}{\(A\)}I6- 4- 1
\cline { 2 - 5 }II- 253
\cline { 2 - 5 }III51- 3
Formulate this information as a linear programming problem, the solution to which will give the optimal strategy for player \(B\).
  1. Rewrite the matrix as necessary and state the new value of the game, \(v\), in terms of \(V\).
  2. Define your decision variables.
  3. Write down the objective function in terms of your decision variables.
  4. Write down the constraints.
OCR D2 Q2
8 marks Moderate -0.5
2. The owner of a small plane is planning a journey from her local airport, \(A\) to the airport nearest her parents, \(K\). On the journey she will make three refuelling stops, the first at \(B , C\) or \(D\), the second at \(E , F\) or \(G\) and the third at \(H , I\) or \(J\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{34728928-2a21-463d-982e-c46ab2dc05c8-2_723_1303_427_356} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows all the possible flights that can be made on the journey with the number by each arc indicating the distance of each flight in hundreds of miles. The owner of the plane wishes to choose a route that minimises the total distance that she flies. Use dynamic programming to find the route that she should use and state the total length of this route.
OCR D2 Q3
10 marks Moderate -0.3
  1. Four sales representatives ( \(R _ { 1 } , R _ { 2 } , R _ { 3 }\) and \(R _ { 4 }\) ) are to be sent to four areas ( \(A _ { 1 } , A _ { 2 } , A _ { 3 }\) and \(A _ { 4 }\) ) such that each representative visits one area. The estimated profit, in tens of pounds, that each representative will make in each area is shown in the table below.
\cline { 2 - 5 } \multicolumn{1}{c|}{}\(A _ { 1 }\)\(A _ { 2 }\)\(A _ { 3 }\)\(A _ { 4 }\)
\(R _ { 1 }\)37294451
\(R _ { 2 }\)45304341
\(R _ { 3 }\)32273950
\(R _ { 4 }\)43255155
Use the Hungarian method to obtain an allocation which will maximise the total profit made from the visits. Show the state of the table after each stage in the algorithm.
OCR D2 Q4
11 marks Moderate -0.3
4.
\$ FMMUMITI7 IP HIZ3 UFHGHQFHIT
ா\$ மோங்கோ
ா\%\%mmum \includegraphics[max width=\textwidth, alt={}, center]{34728928-2a21-463d-982e-c46ab2dc05c8-4_49_268_424_301} \includegraphics[max width=\textwidth, alt={}, center]{34728928-2a21-463d-982e-c46ab2dc05c8-4_46_465_482_301} \includegraphics[max width=\textwidth, alt={}, center]{34728928-2a21-463d-982e-c46ab2dc05c8-4_49_533_539_301} \includegraphics[max width=\textwidth, alt={}, center]{34728928-2a21-463d-982e-c46ab2dc05c8-4_49_472_593_303} \includegraphics[max width=\textwidth, alt={}, center]{34728928-2a21-463d-982e-c46ab2dc05c8-4_49_497_648_303} \includegraphics[max width=\textwidth, alt={}, center]{34728928-2a21-463d-982e-c46ab2dc05c8-4_54_501_703_306} \includegraphics[max width=\textwidth, alt={}, center]{34728928-2a21-463d-982e-c46ab2dc05c8-4_45_467_762_303} \includegraphics[max width=\textwidth, alt={}, center]{34728928-2a21-463d-982e-c46ab2dc05c8-4_49_463_813_303} \includegraphics[max width=\textwidth, alt={}, center]{34728928-2a21-463d-982e-c46ab2dc05c8-4_47_460_872_303} \(\square\) \(\square\) Fig. 2
Construct an activity network to model the work involved in laying the foundations and putting in services for an industrial complex.
  1. Execute a forward scan to find the minimum time in which the project can be completed.
  2. Execute a backward scan to determine which activities lie on the critical path. The contractor is committed to completing the project in this minimum time and faces a penalty of \(\pounds 50000\) for each day that the project is late. Unfortunately, before any work has begun, flooding means that activity \(E\) will take 3 days longer than the 7 days allocated.
  3. Activity \(K\) could be completed in 1 day at an extra cost of \(\pounds 90000\). Explain why doing this is not economical.
    (2 marks)
  4. If the time taken to complete any one activity, other than \(E\), could be reduced by 2 days at an extra cost of \(\pounds 80000\), for which activities on their own would this be profitable. Explain your reasoning.
    (3 marks)
    11 marks
OCR D2 Q5
11 marks Moderate -0.3
  1. A sheet is provided for use in answering this question.
A town has adopted a one-way system to cope with recent problems associated with congestion in one area. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{34728928-2a21-463d-982e-c46ab2dc05c8-5_684_1320_454_316} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 models the one-way system as a capacitated directed network. The numbers on the arcs are proportional to the number of vehicles that can pass along each road in a given period of time.
  1. Find the capacity of the cut which passes through the \(\operatorname { arcs } A E , B F , B G\) and \(C D\).
    (1 mark) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{34728928-2a21-463d-982e-c46ab2dc05c8-6_714_1280_171_333} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure} Figure 4 shows a feasible flow of 17 through the same network. For convenience, a supersource, \(S\), and a supersink, \(T\), have been used.
    1. Use the labelling procedure to find the maximum flow through this network listing each flow-augmenting route you use together with its flow.
    2. Show your maximum flow pattern and state its value.
  3. Prove that your flow is the maximum possible through the network.
  4. It is suggested that the maximum flow through the network could be increased by making road \(E F\) undirected, so that it has a capacity of 8 in either direction. Using the maximum flow-minimum cut theorem, find the increase in maximum flow this change would allow.
OCR D2 Q1
8 marks Moderate -0.8
  1. Whilst Clive is in hospital, four of his friends decide to redecorate his lounge as a welcomehome surprise. They divide the work to be done into four jobs which must be completed in the following order:
  • strip the wallpaper,
  • paint the woodwork and ceiling,
  • hang the new wallpaper,
  • replace the fittings and tidy up.
The table below shows the time, in hours, that each of the friends is likely to take to complete each job.
AliceBhavinCarlDieter
Strip wallpaper5354
Paint7564
Hang wallpaper8476
Replace fittings5323
As they do not know how long Clive will be in hospital his friends wish to complete the redecoration in the shortest possible total time.
  1. Use the Hungarian method to obtain the optimal allocation of the jobs, showing the state of the table after each stage in the algorithm.
  2. Hence find the minimum time in which the friends can redecorate the lounge.
OCR D2 Q2
9 marks Moderate -0.3
2. The payoff matrix for player \(A\) in a two-person zero-sum game is shown below.
\cline { 3 - 5 } \multicolumn{2}{c|}{}\(B\)
\cline { 2 - 5 } \multicolumn{2}{c|}{}IIIIII
\multirow{3}{*}{\(A\)}I35- 2
\cline { 2 - 5 }II7\({ } ^ { - } 4\)- 1
\cline { 2 - 5 }III9\({ } ^ { - } 4\)8
  1. Applying the dominance rule, explain, with justification, which strategy can be ignored by
    1. player \(A\),
    2. player \(B\).
  2. For the reduced table, find the optimal strategy for
    1. player \(A\),
    2. player \(B\).
OCR D2 Q3
9 marks Standard +0.3
  1. A sheet is provided for use in answering this question.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{df7b056f-1446-43f1-a2fd-c0d56533550e-3_588_1285_287_296} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a capacitated, directed network. The numbers on each arc indicate the minimum and maximum capacity of that arc. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{df7b056f-1446-43f1-a2fd-c0d56533550e-3_648_1288_1155_296} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows a feasible flow through the same network.
  1. Using this as your initial flow pattern, use the labelling procedure to find a maximal flow. You should list each flow-augmenting route you use together with its flow and draw the maximal flow pattern.
    (6 marks)
  2. Find a cut of the same value as your maximum flow and explain why this proves it gives the maximim possible flow.
OCR D2 Q4
11 marks Moderate -0.3
4.
ActivityTimePrecedence
A12
B5
C10
D8A
E5A, B , C
F9C
G11D, E
H6G, F
I6H
J2H
K3I
Construct an activity network to show the tasks involved in widening a bridge over the B451.
  1. Find those tasks which lie on the critical path and list them in order.
  2. State the minimum length of time needed to widen the bridge.
  3. Represent the tasks on a Gantt diagram. Tasks \(F\) and \(J\) each require 3 workers, tasks \(B\), \(D\) and \(I\) each require 2 workers and the remaining tasks each require one worker.
  4. Draw a resource histogram showing how it is possible for a team of 4 workers to complete the project in the minimum possible time.
OCR D2 Q5
11 marks Standard +0.8
  1. A company wishes to plan its production of a particular item over the coming four months based on its current orders. In each month the company can manufacture up to three of the item with the costs according to how many it makes being as follows:
No. of Items Produced0123
Cost in Pounds05500970013100
There are no items in stock at the start of the period and the company wishes to meet all its orders on time and have no stock left at the end of the 4-month period. If any items are not to be supplied in the month they are made there is also a storage cost incurred of \(\pounds 400\) per item per month. The orders for each of the four months being considered are as follows:
MonthMarchAprilMayJune
No. of Orders1241
Use dynamic programming to find how many of the item the company should make in each of these four months in order to minimise the total cost for this period. \section*{Please hand this sheet in for marking} \includegraphics[max width=\textwidth, alt={}, center]{df7b056f-1446-43f1-a2fd-c0d56533550e-6_588_1285_504_276} \includegraphics[max width=\textwidth, alt={}, center]{df7b056f-1446-43f1-a2fd-c0d56533550e-6_588_1280_1361_276}
OCR D2 Q1
8 marks Moderate -0.8
  1. A linear programming problem is defined as follows:
$$\begin{array} { l l } \text { Maximise } & P = 3 x + 3 y + 4 z \\ \text { subject to } & x + 2 y + z \leq 30 \\ & 5 x + y + 3 z \leq 60 \\ \text { and } & x \geq 0 , y \geq 0 , z \geq 0 . \end{array}$$
  1. Display the problem in a Simplex Tableau.
  2. Starting with a pivot chosen from the \(z\)-column, perform one iteration of your tableau.
  3. Write down the resulting values of \(x , y , z\) and \(P\) and state with a reason whether or not these values give an optimal solution.
OCR D2 Q2
8 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d88fdf1e-7547-434d-ba87-7f816e4386ba-1_627_1116_1190_388} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a capacitated, directed network. The numbers on each arc indicate the maximum capacity of that arc. In addition to the restrictions on flow through the arcs a maximum flow of 6 units is allowed to pass through vertex \(C\).
  1. Redraw the network to take into account this restriction.
  2. Starting with an initial flow of 6 units along SADT and 6 units along SBT use the labelling procedure to find a maximal flow. You should list each flow-augmenting route you use together with its flow and draw the maximal flow pattern.
OCR D2 Q3
10 marks Standard +0.3
3. A travel company offers a touring holiday which stops at four locations, \(A , B , C\) and \(D\). The tour may be taken with the locations appearing in any order, but the number of days spent in each location is dependent on its position in the tour, as shown in the table below.
\multirow{2}{*}{}Stage
1234
A7856
\(B\)6965
C9857
\(D\)7766
Showing the state of the table at each stage, use the Hungarian algorithm to find the order in which to complete the tour so as to maximise the total number of days. State the maximum total number of days that can be spent in the four locations.
OCR D2 Q4
10 marks Moderate -0.8
4. A rally consisting of four stages is being planned. The first stage will begin at \(A\) and the last stage will end at \(L\). Various routes are being considered, with the end of one stage being the start of the next. The organisers want the total length of the route chosen to be as small as possible. The table below shows the length, in miles, of each of the possible stages.
\multirow{2}{*}{}Finishing point
BCDE\(F\)G\(H\)\(I\)\(J\)\(K\)\(L\)
\multirow{11}{*}{Starting point}A54.51310
B8114
C510.5
D96
E12715
\(F\)522
G893
\(H\)1029
I5
J6
K10
Use dynamic programming to find the route which satisfies the wish of the organisers.
OCR D2 Q5
12 marks Moderate -0.8
  1. A project involves six tasks, some of which cannot be started until others have been completed. This is shown in the table below.
TaskDuration (minutes)Immediate predecessors
A18-
B23-
C13\(A , B\)
D9A
E28\(B , D\)
\(F\)23C
  1. Draw an activity network for this project.
  2. By labelling your network, find the critical path and the minimum duration of the project. An extra condition is now imposed. Task \(A\) may not begin until task \(B\) has been underway for at least 10 minutes.
  3. Draw a new network taking into account this restriction.
  4. Find a revised value for the minimum duration of the project and state the new critical path.
OCR D2 Q6
12 marks Standard +0.8
6. The payoff matrix for player \(A\) in a two-person zero-sum game is shown below.
\cline { 3 - 4 } \multicolumn{2}{c|}{}\(B\)
\cline { 3 - 4 }III
\multirow{2}{*}{\(A\)}I4\({ } ^ { - } 8\)
\cline { 2 - 4 }II2\({ } ^ { - } 4\)
\cline { 2 - 4 }III\({ } ^ { - } 8\)2
  1. Explain why the game does not have a saddle point.
  2. Using a graphical method, find the optimal strategy for player \(B\).
  3. Find the optimal strategy for player \(A\).
  4. Find the value of the game.
OCR D2 Q1
7 marks Moderate -0.5
  1. A couple are making the arrangements for their wedding. They are deciding whether to have the ceremony at their church, a local castle or a nearby registry office. The reception will then be held in a marquee, at the castle or at a local hotel. Both the castle and hotel offer catering services but the couple are also considering using Deluxe Catering or Cuisine, who can both provide the food at any venue.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b8eb80d5-5af5-4a8b-8335-6fae95f3aa73-1_938_1514_520_248} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} The network in Figure 1 shows the costs incurred (including transport), in hundreds of pounds, according to the choice the couple make for each stage of the day. Use dynamic programming to find how the couple can minimise the total cost of their wedding and state the total cost of this arrangement.
OCR D2 Q2
8 marks Standard +0.3
2. A two-person zero-sum game is represented by the payoff matrix for player \(A\) shown below.
\cline { 3 - 5 } \multicolumn{2}{c|}{}\(B\)
\cline { 3 - 5 } \multicolumn{2}{c|}{}IIIIII
\multirow{2}{*}{\(A\)}I1- 12
\cline { 2 - 5 }II35- 1
  1. Represent the expected payoffs to \(A\) against \(B\) 's strategies graphically and hence determine which strategy is not worth considering for player \(B\).
  2. Find the best strategy for player \(A\) and the value of the game.
OCR D2 Q3
9 marks Easy -1.2
3. Four people are contributing to the entertainment section of an email magazine. For one issue reviews are required for a film, a musical, a ballet and a concert such that each person reviews one show. The people in charge of the magazine will pay each person's expenses and the cost, in pounds, for each reviewer to attend each show are given below.
FilmMusicalBalletConcert
Andrew5201218
Betty6181516
Carlos421915
Davina5161113
Use the Hungarian algorithm to find an optimal assignment which minimises the total cost. State the total cost of this allocation.
OCR D2 Q4
9 marks Standard +0.3
  1. A sheet is provided for use in answering this question.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b8eb80d5-5af5-4a8b-8335-6fae95f3aa73-3_881_1310_319_315} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows a capacitated, directed network.
The numbers in bold denote the capacities of each arc.
The numbers in circles show a feasible flow of 48 through the network.
  1. Find the values of \(x\) and \(y\).
    1. Use the labelling procedure to find the maximum flow through this network, listing each flow-augmenting route you use together with its flow.
    2. Show your maximum flow pattern and state its value.
    1. Find a minimum cut, listing the arcs through which it passes.
    2. Explain why this proves that the flow found in part (b) is a maximum.
OCR D2 Q5
12 marks Moderate -0.3
5. A leisure company owns boats of each of the following types: 2-person boats which are 4 metres long and weigh 50 kg .
4-person boats which are 3 metres long and weigh 20 kg .
8-person boats which are 14 metres long and weigh 100 kg .
The leisure company is willing to donate boats to a local sports club to accommodate up to 40 people at any one time. However, storage facilities mean that the combined length of the boats must not be more than 75 metres. Also, it must be possible to transport all the boats on a single trailer which has a maximum load capacity of 600 kg . The club intends to hire the boats out to help with the cost of maintaining them. It plans to charge \(\pounds 10 , \pounds 12\) and \(\pounds 8\) per day, for the 2 -, 4 - and 8 -person boats respectively and wishes to maximise its daily revenue ( \(\pounds R\) ). Let \(x , y\) and \(z\) represent the number of 2-, 4- and 8-person boats respectively given to the club.
  1. Model this as a linear programming problem. Using the Simplex algorithm the following initial tableau is obtained:
    \(R\)\(x\)\(y\)\(z\)\(s\)\(t\)\(u\)
    1\({ } ^ { - } 10\)\({ } ^ { - } 12\)\({ } ^ { - } 8\)0000
    012410020
    0431401075
    0521000160
  2. Explain the purpose of the variables \(s , t\) and \(u\).
  3. By increasing the value of \(y\) first, work out the next two complete tableaus.
  4. Explain how you know that your final tableau gives an optimal solution and state this solution in practical terms.
OCR D2 Q6
15 marks Moderate -0.3
  1. A project consists of the activities listed in the table below. For each activity the table shows how long it will take, which other activites must be completed before it can be done and the number of workers needed to complete it.
ActivityDuration (hours)Immediate Predecessor(s)No. of Workers
A3-9
B2A5
C5\(A\)6
D3C5
E6\(B , D\)2
\(F\)13D5
\(G\)4E6
\(H\)12E4
I3\(F\)4
J5H, I3
K7\(G , J\)8
  1. Draw an activity network for the project.
  2. Find the critical path and the minimum time in which the project can be completed.
  3. Represent all of the activities on a Gantt diagram.
  4. By drawing a resource histogram, find out the maximum number of workers required at any one time if each activity is begun as soon as possible.
  5. Draw another resource histogram to show how the project can be completed in the minimum time possible using a maximum of 10 workers at any one time. Sheet for answering question 4 \section*{Please hand this sheet in for marking} \includegraphics[max width=\textwidth, alt={}, center]{b8eb80d5-5af5-4a8b-8335-6fae95f3aa73-6_729_1227_482_338} \includegraphics[max width=\textwidth, alt={}, center]{b8eb80d5-5af5-4a8b-8335-6fae95f3aa73-6_723_1223_1466_338}
OCR H240/01 Q1
4 marks Moderate -0.8
1 Solve the simultaneous equations. $$\begin{array} { r } x ^ { 2 } + 8 x + y ^ { 2 } = 84 \\ x - y = 10 \end{array}$$
OCR H240/01 Q2
5 marks Moderate -0.8
2 The points \(A\), \(B\) and \(C\) have position vectors \(3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } , - \mathbf { i } + 6 \mathbf { k }\) and \(7 \mathbf { i } - 4 \mathbf { j } - 2 \mathbf { k }\) respectively. M is the midpoint of BC .
  1. Show that the magnitude of \(\overrightarrow { O M }\) is equal to \(\sqrt { 17 }\). Point D is such that \(\overrightarrow { B C } = \overrightarrow { A D }\).
  2. Show that position vector of the point D is \(11 \mathbf { i } - 8 \mathbf { j } - 6 \mathbf { k }\).
OCR H240/01 Q3
3 marks Moderate -0.8
3 The diagram below shows the graph of \(y = \mathrm { f } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{6c16d9e2-7698-48e4-a3ed-5aae3b6f041e-05_801_1483_413_251}
  1. On the diagram in the Printed Answer Booklet, draw the graph of \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\).
  2. On the diagram in the Printed Answer Booklet, draw the graph of \(y = \mathrm { f } ( x - 2 ) + 1\).