Questions D2 (553 questions)

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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}
Edexcel D2 Q1
8 marks Moderate -0.8
\includegraphics{figure_1} Figure 1 shows a network of roads connecting six villages \(A\), \(B\), \(C\), \(D\), \(E\) and \(F\). The lengths of the roads are given in km.
  1. Complete the table in the answer booklet, in which the entries are the shortest distances between pairs of villages. You should do this by inspection. [2] The table can now be taken to represent a complete network.
  2. Use the nearest-neighbour algorithm, starting at \(A\), on your completed table in part (a). Obtain an upper bound to the length of a tour in this complete network, which starts and finishes at \(A\) and visits every village exactly once. [3]
  3. Interpret your answer in part (b) in terms of the original network of roads connecting the six villages. [1]
  4. By choosing a different vertex as your starting point, use the nearest-neighbour algorithm to obtain a shorter tour than that found in part (b). State the tour and its length. [2]
Edexcel D2 Q2
8 marks Moderate -0.8
A two-person zero-sum game is represented by the following pay-off matrix for player \(A\).
IIIIIIIV
I\(-4\)\(-5\)\(-2\)4
II\(-1\)1\(-1\)2
III05\(-2\)\(-4\)
IV\(-1\)3\(-1\)1
  1. Determine the play-safe strategy for each player. [4]
  2. Verify that there is a stable solution and determine the saddle points. [3]
  3. State the value of the game to \(B\). [1]
Edexcel D2 Q3
10 marks Standard +0.3
\includegraphics{figure_2} The network in Fig. 2 shows possible routes that an aircraft can take from \(S\) to \(T\). The numbers on the directed arcs give the amount of fuel used on that part of the route, in appropriate units. The airline wishes to choose the route for which the maximum amount of fuel used on any part of the route is as small as possible. This is the minimax route.
  1. Complete the table in the answer booklet. [8]
  2. Hence obtain the minimax route from \(S\) to \(T\) and state the maximum amount of fuel used on any part of this route. [2]
Edexcel D2 Q4
8 marks Moderate -0.3
Andrew (\(A\)) and Barbara (\(B\)) play a zero-sum game. This game is represented by the following pay-off matrix for Andrew. $$A \begin{pmatrix} 3 & 5 & 4 \\ 1 & 4 & 2 \\ 6 & 3 & 7 \end{pmatrix}$$
  1. Explain why this matrix may be reduced to $$\begin{pmatrix} 3 & 5 \\ 6 & 3 \end{pmatrix}$$ [8]
  2. Hence find the best strategy for each player and the value of the game.
Edexcel D2 Q5
11 marks Moderate -0.5
An engineering company has 4 machines available and 4 jobs to be completed. Each machine is to be assigned to one job. The time, in hours, required by each machine to complete each job is shown in the table below.
Job 1Job 2Job 3Job 4
Machine 114587
Machine 221265
Machine 37839
Machine 424610
Use the Hungarian algorithm, \emph{reducing rows first}, to obtain the allocation of machines to jobs which minimises the total time required. State this minimum time. [11]
Edexcel D2 Q6
14 marks Moderate -0.3
The table below shows the distances, in km, between six towns \(A\), \(B\), \(C\), \(D\), \(E\) and \(F\).
ABCDEF
A\(-\)85110175108100
B85\(-\)3817516093
C11038\(-\)14815673
D175175148\(-\)11084
E108160156110\(-\)92
F10093738492\(-\)
  1. Starting from \(A\), use Prim's algorithm to find a minimum connector and draw the minimum spanning tree. You must make your method clear by stating the order in which the arcs are selected. [4]
    1. Using your answer to part (a) obtain an initial upper bound for the solution of the travelling salesman problem. [2]
    2. Use a short cut to reduce the upper bound to a value less than 680. [4]
  3. Starting by deleting \(F\), find a lower bound for the solution of the travelling salesman problem. [4]
Edexcel D2 Q7
14 marks Standard +0.3
A steel manufacturer has 3 factories \(F_1\), \(F_2\) and \(F_3\) which can produce 35, 25 and 15 kilotomnes of steel per year, respectively. Three businesses \(B_1\), \(B_2\) and \(B_3\) have annual requirements of 20, 25 and 30 kilotomnes respectively. The table below shows the cost \(C_{ij}\) in appropriate units, of transporting one kilotome of steel from factory \(F_i\) to business \(B_j\).
Business
\(B_1\)\(B_2\)\(B_3\)
\(F_1\)10411
Factory \(F_2\)1258
\(F_3\)967
The manufacturer wishes to transport the steel to the businesses at minimum total cost.
  1. Write down the transportation pattern obtained by using the North-West corner rule. [2]
  2. Calculate all of the improvement indices \(I_{ij}\) and hence show that this pattern is not optimal. [5]
  3. Use the stepping-stone method to obtain an improved solution. [3]
  4. Show that the transportation pattern obtained in part (c) is optimal and find its cost. [4]
Edexcel D2 Q8
14 marks Moderate -0.8
\includegraphics{figure_4} The network in Fig. 4 models a drainage system. The number on each arc indicates the capacity of that arc, in litres per second.
  1. Write down the source vertices. [2]
\includegraphics{figure_5} Figure 5 shows a feasible flow through the same network.
  1. State the value of the feasible flow shown in Fig. 5. [1]
Taking the flow in Fig. 5 as your initial flow pattern,
  1. use the labelling procedure on Diagram 1 to find a maximum flow through this network. You should list each flow-augmenting route you use, together with its flow. [6]
  2. Show the maximal flow on Diagram 2 and state its value. [3]
  3. Prove that your flow is maximal. [2]
Edexcel D2 Q9
17 marks Moderate -0.3
T42 Co. Ltd produces three different blends of tea, Morning, Afternoon and Evening. The teas must be processed, blended and then packed for distribution. The table below shows the time taken, in hours, for each stage of the production of a tonne of tea. It also shows the profit, in hundreds of pounds, on each tonne.
ProcessingBlendingPackingProfit (£100)
Morning blend3134
Afternoon blend2345
Evening blend4233
The total times available each week for processing, blending and packing are 35, 20 and 24 hours respectively. T42 Co. Ltd wishes to maximise the weekly profit. Let \(x\), \(y\) and \(z\) be the number of tonnes of Morning, Afternoon and Evening blend produced each week.
  1. Formulate the above situation as a linear programming problem, listing clearly the objective function, and the constraints as inequalities. [4]
An initial Simplex tableau for the above situation is
Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
\(r\)32410035
\(s\)13201020
\(t\)24300124
\(P\)\(-4\)\(-5\)\(-3\)0000
  1. Solve this linear programming problem using the Simplex algorithm. Take the most negative number in the profit row to indicate the pivot column at each stage. [11]
T42 Co. Ltd wishes to increase its profit further and is prepared to increase the time available for processing or blending or packing or any two of these three.
  1. Use your answer to part (b) to advise the company as to which stage(s) it should increase the time available. [2]
Edexcel D2 Q10
6 marks Moderate -0.3
While solving a maximizing linear programming problem, the following tableau was obtained.
Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
\(r\)00\(1\frac{1}{3}\)10\(-\frac{1}{3}\)\(\frac{5}{3}\)
\(y\)01\(3\frac{1}{3}\)01\(-\frac{1}{3}\)\(\frac{1}{3}\)
\(x\)10\(-3\)0\(-1\)\(\frac{1}{3}\)1
\(P\)00101111
  1. Explain why this is an optimal tableau. [1]
  2. Write down the optimal solution of this problem, stating the value of every variable. [3]
  3. Write down the profit equation from the tableau. Use it to explain why changing the value of any of the non-basic variables will decrease the value of \(P\). [2]
Edexcel D2 Q11
11 marks Standard +0.3
A company wishes to transport its products from 3 factories \(F_1\), \(F_2\) and \(F_3\) to a single retail outlet \(R\). The capacities of the possible routes, in van loads per day, are shown in Fig. 5. \includegraphics{figure_5}
  1. On Diagram 1 in the answer booklet add a supersource \(S\) to obtain a capacitated network with a single source and a single sink. State the minimum capacity of each arc you have added. [2]
    1. State the maximum flow along \(SF_1ABR\) and \(SF_2CR\). [2]
    2. Show these maximum flows on Diagram 2 in the answer booklet, using numbers in circles.
    Taking your answer to part (b)(ii) as the initial flow pattern,
    1. use the labelling procedure to find a maximum flow from \(S\) to \(R\). Your working should be shown on Diagram 3. List each flow-augmenting route you find together with its flow. [7]
    2. Prove that your final flow is maximal.
Edexcel D2 Q12
11 marks Standard +0.3
\includegraphics{figure_2} A company has 3 warehouses \(W_1\), \(W_2\) and \(W_3\). It needs to transport the goods stored there to 2 retail outlets \(R_1\) and \(R_2\). The capacities of the possible routes, in van loads per day, are shown in Fig. 2. Warehouses \(W_1\), \(W_2\) and \(W_3\) have 14, 12 and 14 van loads respectively available per day and retail outlets \(R_1\) and \(R_2\) can accept 6 and 25 van loads respectively per day.
  1. On Diagram 1 on the answer sheet add a supersource \(W\) and a supersink \(R\) and the appropriate directed arcs to obtain a single-source, single-sink capacitated network. State the minimum capacity of each arc you have added. [3]
  2. State the maximum flow along
    1. \(W_1W_1R_1R\),
    2. \(W_2CR_2R\).
    [2]
  3. Taking your answers to part (b) as the initial flow pattern, use the labelling procedure to obtain a maximum flow through the network from \(W\) to \(R\). Show your working on Diagram 2. List each flow-augmenting route you find together with its flow. [5]
  4. From your final flow pattern, determine the number of van loads passing through \(B\) each day. [1]
Edexcel D2 2004 June Q1
4 marks Easy -2.0
In game theory explain what is meant by
  1. zero-sum game, [2]
  2. saddle point. [2]
(Total 4 marks)
Edexcel D2 2004 June Q2
9 marks Moderate -0.8
In a quiz there are four individual rounds, Art, Literature, Music and Science. A team consists of four people, Donna, Hannah, Kerwin and Thomas. Each of four rounds must be answered by a different team member. The table shows the number of points that each team member is likely to get on each individual round.
ArtLiteratureMusicScience
Donna31243235
Hannah16101922
Kerwin19142021
Thomas18152123
Use the Hungarian algorithm, reducing rows first, to obtain an allocation which maximises the total points likely to be scored in the four rounds. You must make your method clear and show the table after each stage. [9] (Total 9 marks)
Edexcel D2 2004 June Q3
12 marks Moderate -0.3
The table shows the least distances, in km, between five towns, \(A\), \(B\), \(C\), \(D\) and \(E\).
\(A\)\(B\)\(C\)\(D\)\(E\)
\(A\)\(-\)15398124115
\(B\)153\(-\)74131149
\(C\)9874\(-\)82103
\(D\)12413182\(-\)134
\(E\)115149103134\(-\)
Nassim wishes to find an interval which contains the solution to the travelling salesman problem for this network.
  1. Making your method clear, find an initial upper bound starting at \(A\) and using
    1. the minimum spanning tree method, [4]
    2. the nearest neighbour algorithm. [3]
  2. By deleting \(E\), find a lower bound. [4]
  3. Using your answers to parts (a) and (b), state the smallest interval that Nassim could correctly write down. [1]
(Total 12 marks)
Edexcel D2 2004 June Q4
14 marks Standard +0.3
Emma and Freddie play a zero-sum game. This game is represented by the following pay-off matrix for Emma. \(\begin{pmatrix} -4 & -1 & 3 \\ 2 & 1 & -2 \end{pmatrix}\)
  1. Show that there is no stable solution. [3]
  2. Find the best strategy for Emma and the value of the game to her. [8]
  3. Write down the value of the game to Freddie and his pay-off matrix. [3]
(Total 14 marks)
Edexcel D2 2004 June Q5
18 marks Moderate -0.8
  1. Describe a practical problem that could be solved using the transportation algorithm. [2]
A problem is to be solved using the transportation problem. The costs are shown in the table. The supply is from \(A\), \(B\) and \(C\) and the demand is at \(d\) and \(e\).
\(d\)\(e\)Supply
\(A\)5345
\(B\)4635
\(C\)2440
Demand5060
  1. Explain why it is necessary to add a third demand \(f\). [1]
  2. Use the north-west corner rule to obtain a possible pattern of distribution and find its cost.
    \(d\)\(e\)\(f\)Supply
    \(A\)5345
    \(B\)4635
    \(C\)2440
    Demand5060
    [5]
  3. Calculate shadow costs and improvement indices for this pattern. [5]
  4. Use the stepping-stone method once to obtain an improved solution and its cost. [5]
(Total 16 marks)
Edexcel D2 2004 June Q6
18 marks Standard +0.3
Joan sells ice cream. She needs to decide which three shows to visit over a three-week period in the summer. She starts the three-week period at home and finishes at home. She will spend one week at each of the three shows she chooses travelling directly from one show to the next. Table 1 gives the week in which each show is held. Table 2 gives the expected profits from visiting each show. Table 3 gives the cost of travel between shows. Table 1
Week123
ShowsA, B, CD, EF, G, H
Table 2
Show\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Expected Profit (£)900800100015001300500700600
Table 3
Travel costs (£)\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Home7080150809070
\(A\)180150
\(B\)140120
\(C\)200210
\(D\)200160120
\(E\)170100110
It is decided to use dynamic programming to find a schedule that maximises the total expected profit, taking into account the travel costs.
  1. Define suitable stage, state and action variables. [3]
  2. Determine the schedule that maximises the total profit. Show your working in a table. [12]
  3. Advise Joan on the shows that she should visit and state her total expected profit. [3]
(Total 18 marks)
Edexcel D2 2004 June Q7
13 marks Moderate -0.5
\includegraphics{figure_1} Figure 1 shows a capacitated directed network. The number on each arc is its capacity. \includegraphics{figure_2} Figure 2 shows a feasible initial flow through the same network.
  1. Write down the values of the flow \(x\) and the flow \(y\). [2]
  2. Obtain the value of the initial flow through the network, and explain how you know it is not maximal. [2]
  3. Use this initial flow and the labelling procedure on Diagram 1 below to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow. Diagram 1 \includegraphics{figure_3} [5]
  4. Show your maximal flow pattern on Diagram 2. Diagram 2 \includegraphics{figure_4} [2]
  5. Prove that your flow is maximal. [2]
(Total 13 marks)
Edexcel D2 2004 June Q8
6 marks Moderate -0.3
A three-variable linear programming problem in \(x\), \(y\) and \(z\) is to be solved. The objective is to maximise the profit P. The following tableau was obtained.
Basic variable\(x\)\(y\)\(Z\)\(r\)\(s\)\(t\)Value
\(s\)30201\(-\frac{2}{3}\)\(\frac{2}{3}\)
\(r\)40\(\frac{7}{2}\)108\(\frac{9}{2}\)
\(y\)5170037
P30200863
  1. State, giving your reason, whether this tableau represents the optimal solution. [1]
  2. State the values of every variable. [3]
  3. Calculate the profit made on each unit of \(y\). [2]
(Total 6 marks)