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Edexcel D2 Q3
9 marks Easy -3.0
3. 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.
Edexcel D2 Q4
10 marks Moderate -0.5
4. This question should be answered on the sheet provided. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4e50371b-0c1c-4b4e-b21d-60858ae160df-3_771_1479_1178_237} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A salesman is planning a four-day trip beginning at 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 2 shows the expected net profit, in tens of pounds, that he will gain on each day according to the route he chooses. Use dynamic programming to find the route which should maximise the salesman's net profit. State the expected profit from using this route.
(10 marks)
Edexcel D2 Q5
13 marks Moderate -0.3
5. A construction company has three teams of workers available, each of which is to be assigned to one of four jobs at a site. The following table shows the estimated cost, in tens of pounds, of each team doing each job:
WindowsConservatoryDoorsGreenhouse
Team A2780881
Team B2860571
Team C3090773
Use the Hungarian algorithm to find an allocation of jobs which will minimise the total cost. Show the state of the table after each stage in the algorithm and state the cost of the final assignment.
(13 marks)
Edexcel D2 Q6
14 marks Standard +0.3
6. This question should be answered on the sheet provided. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4e50371b-0c1c-4b4e-b21d-60858ae160df-5_664_1029_335_440} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} The network in Figure 3 shows the distances, in miles, between a newspaper distributor based at area \(A\), and five areas, \(B , C , D , E\), and \(F\), to which the distributor must deliver newspapers. Each morning a delivery van has to set out from \(A\) and visit each of these areas before again returning to \(A\), and the driver wishes to keep the total mileage to a minimum.
  1. Draw a complete network showing the shortest distances between the six areas.
    (3 marks)
  2. Obtain a minimum spanning tree for the complete network and hence find an upper bound for the length of the driver's route.
    (5 marks)
  3. Improve this upper bound to find an upper bound of less than 55 miles.
  4. By deleting \(A\), find a lower bound for the total length of the route.
Edexcel D2 Q7
16 marks Standard +0.3
7. Mrs. Hartley organises the tennis fixtures for her school. On one day she has to send a team of 10 players to a match against school \(A\) and a team of 6 players to a match against school \(B\). She has to select the two teams from a squad that includes 7 players who live in village \(C\), 5 players who live in village \(D\) and 8 players who live in village \(E\). Having a small budget, Mrs. Hartley wishes to minimise the total amount spent on travel. The table below shows the cost, in pounds, for one player to travel from each village to each of the schools they are competing against.
\cline { 2 - 3 } \multicolumn{1}{c|}{}\(A\)\(B\)
\(C\)23
\(D\)25
\(E\)76
  1. Use the north-west corner rule to find an initial solution to this problem.
  2. Obtain improvement indices for this initial solution.
  3. Use the stepping-stone method to obtain an optimal solution and state the pattern of transportation that this represents. \section*{Please hand this sheet in for marking}
    StageStateAction
    \multirow[t]{2}{*}{1}GGI
    HHI
    \multirow[t]{3}{*}{2}D
    DG
    DH
    E
    EG
    \(E H\)
    F
    FG
    FH
    \multirow[t]{3}{*}{3}A
    AD
    \(A E\)
    \(A F\)
    B
    BD
    BE
    \(B F\)
    C
    CD
    CE
    CF
    4Home
    Home-A
    Home-B
    Home-C
    \section*{Please hand this sheet in for marking}
  4. \includegraphics[max width=\textwidth, alt={}, center]{4e50371b-0c1c-4b4e-b21d-60858ae160df-8_662_1025_529_440}
  5. Sheet for answering question 6 (cont.)
Edexcel D2 Q1
6 marks Moderate -0.8
  1. This question should be answered on the sheet provided.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e892e87c-1c2d-4f97-ac23-41e38663d0f0-02_485_995_285_477} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} The network in Figure 1 shows the distances, in miles, between the five villages in which Sarah is planning to enquire about holiday work, with village \(A\) being Sarah's home village.
  1. Illustrate this situation as a complete network showing the shortest distances.
    (2 marks)
  2. Use the nearest neighbour algorithm, starting with \(A\), to find an upper bound to the length of a tour beginning and ending at \(A\).
    (2 marks)
  3. Interpret the tour found in part (b) in terms of the original network.
    (2 marks)
Edexcel D2 Q2
8 marks Standard +0.8
2. 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}{*}{\(A\)}I- 14- 3
\cline { 2 - 5 }II- 371
\cline { 2 - 5 }III5- 2- 1
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.
Edexcel D2 Q3
9 marks Standard +0.3
3. This question should be answered on the sheet provided. Arthur is planning a bus journey from town \(A\) to town \(L\). There are various routes he can take but he will have to change buses three times - at \(B , C\) or \(D\), at \(E , F , G\) or \(H\) and at \(I , J\) or \(K\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e892e87c-1c2d-4f97-ac23-41e38663d0f0-03_764_1410_477_315} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows the bus routes that Arthur can use. The number on each arc shows the average waiting time, in minutes, for a bus to come on that route. As the forecast is for rain, Arthur wishes to plan his journey so that the maximum waiting time at any one stop is as small as possible. Use dynamic programming to find the route that Arthur should use.
(9 marks)
Edexcel D2 Q4
11 marks Moderate -0.5
4. A furniture manufacturer has three workshops, \(W _ { 1 } , W _ { 2 }\) and \(W _ { 3 }\). Orders for rolls of fabric are to be placed with three suppliers, \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\). The supply, demand and cost per roll in pounds, according to which supplier each workshop uses, are given in the table below.
\(W _ { 1 }\)\(W _ { 2 }\)\(W _ { 3 }\)Available
\(S _ { 1 }\)12111730
\(S _ { 2 }\)751025
\(S _ { 3 }\)56810
Required201530
Starting with the north-west corner method of finding an initial solution, find an optimal transportation pattern which minimises the total cost. State the final solution and its total cost.
(11 marks)
Edexcel D2 Q5
11 marks Standard +0.3
5. 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
D7766
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.
(11 marks)
Edexcel D2 Q6
13 marks Standard +0.3
6. 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- 48
  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 \(B\).
  3. Find the value of the game. Turn over
Edexcel D2 Q7
17 marks Moderate -0.5
7. This question should be answered on the sheet provided. A tinned food producer delivers goods to six supermarket warehouses, \(B , C , D , E , F\) and \(G\), from its base, \(A\). The distances, in kilometres, between each location are given in the table below. \section*{Please hand this sheet in for marking}
Edexcel D2 Q1
6 marks Moderate -0.5
  1. This question should be answered on the sheet provided.
The table below shows the distances in miles between five villages. Jane lives in village \(A\) and is about to take her daughter's friends home to villages \(B , C , D\) and \(E\). She will begin and end her journey at \(A\) and wishes to travel the minimum distance possible.
\(A\)\(B\)\(C\)\(D\)\(E\)
\(A\)-4782
\(B\)4-156
\(C\)71-27
\(D\)852-3
\(E\)2673-
  1. Obtain a minimum spanning tree for the network and hence find an upper bound for the length of Jane's journey.
  2. Using a shortcut, improve this upper bound to find an upper bound of less than 15 miles.
    (2 marks)
Edexcel D2 Q2
8 marks Standard +0.8
2. 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\({ } ^ { - } 2\)53
\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.
Edexcel D2 Q3
9 marks Challenging +1.2
3. This question should be answered on the sheet provided. The table below gives distances, in miles, for a network relating to a travelling salesman problem. Use dynamic programming to find the route which satisfies the wish of the organisers. State the length of the shortest stage on this route.
Edexcel D2 Q5
11 marks Easy -1.2
5. 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.
\multirow{2}{*}{}Stage
123
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.
Edexcel D2 Q6
13 marks Moderate -0.8
6. The payoff matrix for player \(X\) in a two-person zero-sum game is shown below.
\cline { 3 - 4 } \multicolumn{2}{c|}{}\(Y\)
\cline { 3 - 4 } \multicolumn{2}{c|}{}\(Y _ { 1 }\)\(Y _ { 2 }\)
\multirow{2}{*}{\(X\)}\(X _ { 1 }\)\({ } ^ { - } 2\)4
\cline { 2 - 4 }\(X _ { 2 }\)6\({ } ^ { - } 1\)
  1. Explain why the game does not have a saddle point.
  2. Find the optimal strategy for
    1. player \(X\),
    2. player \(Y\).
  3. Find the value of the game.
Edexcel D2 Q7
18 marks Moderate -0.5
7. A transportation problem has costs, in pounds, and supply and demand, in appropriate units, as given in the transportation tableau below. (c) \section*{Please hand this sheet in for marking}
StageStateAction
\multirow[t]{3}{*}{1}IIL
JJL
KKL
\multirow[t]{3}{*}{2}\(F\)FI FJ FK
GGI GJ GK
HHI HJ HK
\multirow[t]{4}{*}{3}B\(B F B G B H\)
CCF CH
DDF DH
EEF \(E G E H\)
4A\(A B A C\) AD \(A E\)
Edexcel D2 Q1
6 marks Moderate -0.8
  1. A glazing company runs a promotion for a special type of window. As a result of this the company receives orders for 30 of these windows from business \(B _ { 1 } , 18\) from business \(B _ { 2 }\) and 22 from business \(B _ { 3 }\). The company has stocks of 20 of these windows at factory \(F _ { 1 } , 35\) at factory \(F _ { 2 }\) and 15 at factory \(F _ { 3 }\). The table below shows the profit, in pounds, that the company will make for each window it sells according to which factory supplies each business.
\cline { 2 - 4 } \multicolumn{1}{c|}{}\(B _ { 1 }\)\(B _ { 2 }\)\(B _ { 3 }\)
\(F _ { 1 }\)201417
\(F _ { 2 }\)181919
\(F _ { 3 }\)151723
The glazing company wishes to supply the windows so that the total profit is a maximum.
Formulate this information as a linear programming problem.
  1. State your decision variables.
  2. Write down the objective function in terms of your decision variables.
  3. Write down the constraints and state what each one represents.
Edexcel D2 Q2
7 marks Moderate -0.8
2. This question should be answered on the sheet provided. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{726bca96-7f98-4ed5-b642-f5007a958c8b-03_492_862_301_502} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a network in which the nodes represent five major rides in a theme park and the arcs represent paths between these rides. The numbers on the arcs give the length, in metres, of the paths.
  1. By inspection, add additional arcs to make a complete network showing the shortest distances between the rides.
    (2 marks)
  2. Use the nearest neighbour algorithm, starting at \(A\), and your complete network to find an upper bound to the length of a tour visiting each ride exactly once.
  3. Interpret the tour found in part (b) in terms of the original network.
Edexcel D2 Q3
7 marks Moderate -0.3
3. 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.
    (6 marks)
  2. Hence, find the minimum time in which the friends can redecorate the lounge.
    (1 mark)
Edexcel D2 Q4
10 marks Standard +0.3
4. This question should be answered on the sheet provided. 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]{726bca96-7f98-4ed5-b642-f5007a958c8b-05_727_1303_523_356} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 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. As her plane does not have a large fuel tank, the owner wishes to choose a route that minimises the maximum distance of any one flight. Find the route that she should use and state the maximum distance of any one stage on this route.
Edexcel D2 Q5
10 marks Moderate -0.3
5. A car-hire firm has six branches in a region. Three of the branches, \(A , B\) and \(C\), have spare cars, whereas the other three, \(D , E\) and \(F\), require cars. The total number of cars required is equal to the number of cars available. The table below shows the cost in pounds of sending one car from each branch with spares to each branch needing more cars and the number of cars available or required by each branch.
\backslashbox{Branches with spare cars}{Branches needing cars}\(D\)\(E\)\(F\)Available
\(A\)6477
B8538
C4425
Required596
  1. Use the north-west corner method to obtain a possible pattern of moving cars and find its cost. The firm wishes to minimise the cost of redistributing the cars.
  2. Calculate shadow costs for the pattern found in part (a) and improvement indices for each unoccupied cell.
  3. State, with a reason, whether or not the pattern found in part (a) is optimal.
Edexcel D2 Q6
14 marks Moderate -0.3
6. This question should be answered on the sheet provided. A furniture company in Leeds is considering opening outlets in six other cities.
The table below shows the distances, in miles, between all seven cities.
LeedsLiverpoolManchesterNewcastleNottinghamOxfordYork
Leeds-7140967116528
Liverpool71-311559215593
Manchester4031-1366214167
Newcastle96155136-15625078
Nottingham719262156-9478
Oxford16515514125094-172
York2893677878172-
  1. Starting with Leeds, obtain and draw a minimum spanning tree for this network of cities showing your method clearly.
    (4 marks)
    A representative of the company is to visit each of the areas being considered. He wishes to plan a journey of minimum length starting and ending in Leeds and visiting each of the other cities in the table once. Assuming that the network satisfies the triangle inequality,
  2. find an initial upper bound for the length of his journey,
  3. improve this upper bound to find an upper bound of less than 575 miles.
  4. By deleting Leeds, find a lower bound for his journey. Turn over
Edexcel D2 Q7
21 marks Challenging +1.2
7. The payoff matrix for player \(A\) in a two-person zero-sum game is shown below.