Questions D2 (553 questions)

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OCR D2 2010 June Q3
11 marks Standard +0.3
3
  1. Set up a dynamic programming tabulation to find the minimum weight route from ( \(0 ; 0\) ) to ( \(4 ; 0\) ) on the following directed network. \includegraphics[max width=\textwidth, alt={}, center]{406831f5-74a3-415e-8849-2c381bfe47f4-03_707_1342_1594_443} Give the route and its total weight.
  2. Explain carefully how the route is obtained directly from the values in the table, without referring to the network.
OCR D2 2010 June Q4
15 marks Moderate -0.3
4 Euan and Wai Mai play a zero-sum game. Each is trying to maximise the total number of points that they score in many repeats of the game. The table shows the number of points that Euan scores for each combination of strategies.
Wai Mai
\cline { 2 - 5 }\(X\)\(Y\)\(Z\)
\(A\)2- 53
\cline { 2 - 5 } \(E u a n\)- 1- 34
\cline { 1 - 5 } \(C\)3- 52
\(D\)3- 2- 1
  1. Explain what the term 'zero-sum game' means.
  2. How many points does Wai Mai score if she chooses \(X\) and Euan chooses \(A\) ?
  3. Why should Wai Mai never choose strategy \(Z\) ?
  4. Delete the column for \(Z\) and find the play-safe strategy for Euan and the play-safe strategy for Wai Mai on the table that remains. Is the resulting game stable or not? State how you know. The value 3 in the cell corresponding to Euan choosing \(D\) and Wai Mai choosing \(X\) is changed to - 5 ; otherwise the table is unchanged. Wai Mai decides that she will choose her strategy by making a random choice between \(X\) and \(Y\), choosing \(X\) with probability \(p\) and \(Y\) with probability \(1 - p\).
  5. Write down and simplify an expression for the expected score for Wai Mai when Euan chooses each of his four strategies.
  6. Using graph paper, draw a graph showing Wai Mai's expected score against \(p\) for each of Euan's four strategies and hence calculate the optimum value of \(p\).
OCR D2 2010 June Q5
15 marks Standard +0.3
5 Answer this question on the insert provided. The network represents a system of irrigation channels along which water can flow. The weights on the arcs represent the maximum flow in litres per second. \includegraphics[max width=\textwidth, alt={}, center]{406831f5-74a3-415e-8849-2c381bfe47f4-05_597_1553_479_296}
  1. Calculate the capacity of the cut that separates \(\{ S , B , C , E \}\) from \(\{ A , D , F , G , H , T \}\).
  2. Explain why neither arc \(S C\) nor arc \(B C\) can be full to capacity. Explain why the arcs \(E F\) and \(E H\) cannot both be full to capacity. Hence find the maximum flow along arc \(H T\). When arc \(H T\) carries its maximum flow, what is the flow along arc \(H G\) ?
  3. Show a flow of 58 litres per second on the diagram in the insert, and find a cut of capacity 58. The direction of flow in \(H G\) is reversed.
  4. Use the diagram in the insert to show the excess capacities and potential backflows for your flow from part (iii) in this case.
  5. Without augmenting the labels from part (iv), write down flow augmenting routes to enable an additional 2 litres per second to flow from \(S\) to \(T\).
  6. Show your augmented flow on the diagram in the insert. Explain how you know that this flow is maximal.
OCR D2 2010 June Q6
15 marks Standard +0.3
6 Answer parts (i), (ii) and (iii) of this question on the insert provided. The activity network for a project is shown below. The durations are in minutes. The events are numbered 1, 2, 3, etc. for reference. \includegraphics[max width=\textwidth, alt={}, center]{406831f5-74a3-415e-8849-2c381bfe47f4-06_747_1249_482_447}
  1. Complete the table in the insert to show the immediate predecessors for each activity.
  2. Explain why the dummy activity is needed between event 2 and event 3, and why the dummy activity is needed between event 4 and event 5 .
  3. Carry out a forward pass to find the early event times and a backward pass to find the late event times. Record your early event times and late event times in the table in the insert. Write down the minimum project completion time and the critical activities. Suppose that the duration of activity \(K\) changes to \(x\) minutes.
  4. Find, in terms of \(x\), expressions for the early event time and the late event time for event 9 .
  5. Find the maximum duration of activity \(K\) that will not affect the minimum project completion time found in part (iii). \section*{ADVANCED GCE
    MATHEMATICS} Decision Mathematics 2
    INSERT for Questions 5 and 6
  6. Dummy activity is needed between event 2 and event 3 because \(\_\_\_\_\) Dummy activity is needed between event 4 and event 5 because \(\_\_\_\_\)
  7. Event12345678910
    Early event time
    Late event time
    Minimum project completion time = \(\_\_\_\_\) minutes Critical activities: \(\_\_\_\_\) \section*{Answer part (iv) and part (v) in your answer booklet.} OCR
    RECOGNISING ACHIEVEMENT
OCR D2 Q1
8 marks Easy -1.2
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{89e3545e-fa4b-47dd-8651-7c8f998df9e7-1_762_1475_205_239} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A salesman is planning a four-day trip beginning at his home and ending at town \(I\). He will spend the first night in town \(A , B\) or \(C\), the second night in town \(D , E\) or \(F\) and the third night in town \(G\) or \(H\). The network in Figure 1 shows the distances, in tens of miles, that he will drive each day according to the route he chooses. Use dynamic programming to find the shortest route the salesman can take and state the distance he will drive in total using this route.
OCR D2 Q2
9 marks Moderate -0.8
2. Four athletes are put forward for selection for a mixed stage relay race at a local competition. They may each be selected for a maximum of one stage and only one athlete can be entered for each stage. The average time, in seconds, for each athlete to complete each stage is given below, based on past performances.
\cline { 2 - 4 } \multicolumn{1}{c|}{}Stage
\cline { 2 - 4 } \multicolumn{1}{c|}{}\(\mathbf { 1 }\)\(\mathbf { 2 }\)\(\mathbf { 3 }\)
Alex1969168
Darren2264157
Leroy2072166
Suraj2366171
Use the Hungarian algorithm to find an optimal allocation which will minimise the team's total time. Your answer should show clearly how you have applied the algorithm.
OCR D2 Q3
10 marks Standard +0.3
3. A project consists of 11 activities, some of which are dependent on others having been completed. The following precedence table summarises the relevant information.
ActivityDepends onDuration (hours)
A-5
BA4
CA2
DB, C11
EC4
\(F\)D3
GD8
\(H\)D, E2
I\(F\)1
J\(F , G , H\)7
\(K\)\(I , J\)2
  1. Draw an activity network for the project.
  2. Find the critical path and the minimum time in which the project can be completed. Activity \(F\) can be carried out more cheaply if it is allocated more time.
  3. Find the maximum time that can be allocated to activity \(F\) without increasing the minimum time in which the project can be completed.
OCR D2 Q4
11 marks Standard +0.3
  1. A sheet is provided for use in answering this question.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{89e3545e-fa4b-47dd-8651-7c8f998df9e7-3_725_1303_274_340} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 above shows a capacitated, directed network. The number on each arc indicates the capacity of that arc.
  1. Calculate the values of cuts \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Find the minimum cut and state its value. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{89e3545e-fa4b-47dd-8651-7c8f998df9e7-3_645_1316_1430_338} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure} Figure 3 shows a feasible flow through the same network.
  3. State the values of \(x , y\) and \(z\).
  4. 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. State how you know that you have found a maximal flow.
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.