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Edexcel D1 Q6
15 marks Standard +0.3
6. This question should be answered on the sheet provided. 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]{e1fd42f7-c97c-4bf2-92d3-69afc8bb6e29-06_670_1301_459_331} \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 \(C D\).
    (2 marks) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e1fd42f7-c97c-4bf2-92d3-69afc8bb6e29-07_659_1278_196_335} \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. You must list 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.
    (2 marks)
  5. An alternative suggestion is to widen a single road in order to increase its capacity. Which road, on its own, could lead to the biggest improvement, and what would be the largest maximum flow this could achieve.
    (2 marks)
Edexcel D1 Q7
16 marks Moderate -0.8
7. A project involves six tasks, some of which cannot be started until others have been completed. This is shown in the table below. \includegraphics[max width=\textwidth, alt={}, center]{e1fd42f7-c97c-4bf2-92d3-69afc8bb6e29-09_2036_1555_349_248}
  1. \(\_\_\_\_\)
  2. \(\_\_\_\_\) \section*{Sheet for answering question 5} NAME \section*{Please hand this sheet in for marking} Sheet for answering question 6
    NAME \section*{Please hand this sheet in for marking}
    1. \includegraphics[max width=\textwidth, alt={}, center]{e1fd42f7-c97c-4bf2-92d3-69afc8bb6e29-11_666_1280_461_374}
    2. \includegraphics[max width=\textwidth, alt={}, center]{e1fd42f7-c97c-4bf2-92d3-69afc8bb6e29-11_657_1276_1356_376} Maximum Flow = \(\_\_\_\_\)
  5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
  6. \(\_\_\_\_\)
  7. \(\_\_\_\_\)
Edexcel D1 Q1
7 marks Easy -1.8
  1. (a) Use the binary search algorithm to locate the name PENICUIK in the following list.
\begin{displayquote} ANKERDINE CULROSS DUNOON ELGIN FORFAR FORT WILLIAM HADDINGTON KINCARDINE LARGS MALLAIG MONTROSE PENICUIK ST. ANDREWS THURSO
(b) Use the same algorithm to attempt to locate PENDINE.
(c) Explain the purpose of the mid-point in dividing up the ordered list when using this algorithm. \end{displayquote}
Edexcel D1 Q2
7 marks Moderate -0.8
  1. The following lengths of cloth (in metres) are to be cut from standard 24 metre rolls.
$$\begin{array} { l l l l l l l l l l l l } 6 & 6 & 4 & 6 & 8 & 8 & 4 & 12 & 14 & 6 & 14 & 8 \end{array}$$
  1. Considering only the total amount, what is the least number of rolls that are needed?
  2. Using the first-fit decreasing algorithm, show that the lengths could be cut from 5 rolls.
  3. Using "full-bin" combinations show that it is possible to cut these lengths from the number of rolls found in part (a).
  4. Comment on this result.
Edexcel D1 Q3
11 marks Moderate -0.8
3. This question should be answered on the sheet provided. A firm of auditors is to place one trainee accountant at each of its five offices. These are situated in Dundee, Edinburgh, Glasgow and London. There are two offices in London, one of which is the company's Head Office. The table summarises the trainees' preferences.
TraineePreferences
\(P\)Dundee, London (either)
\(Q\)Dundee, Edinburgh, Glasgow
\(R\)Glasgow, London (Head Office only)
SEdinburgh
\(T\)Edinburgh
  1. Draw a bipartite graph to model this situation.
  2. Explain why no complete matching is possible. Trainee \(T\) is persuaded that either office in London would be just as good for her personal development as the Edinburgh office.
  3. Draw a second bipartite graph to model the changed situation. In an initial matching, trainee \(P\) is placed in the Dundee office, trainee \(R\) in the Glasgow office, and trainee \(S\) in the Edinburgh office.
  4. Draw this initial matching.
  5. Starting from this initial matching, use the maximum matching algorithm to find a complete matching. You must indicate how the algorithm has been applied, stating clearly the alternating paths you use and the final matching.
    (7 marks) Turn over
Edexcel D1 Q4
11 marks Standard +0.3
4. The following matrix gives the capacities of the pipes in a system.
To FromS\(T\)A\(B\)\(C\)D
S--1626--
T------
A----135
B-16---11
C-11----
D-11----
  1. Represent this information as a digraph.
  2. Find the minimum cut, expressing it in the form \(\{ \} \mid \{ \}\), and state its value.
  3. Starting from having no flow in the system, use the labelling procedure to find a maximal flow through the system. You should list each flow-augmenting route you use, together with its flow.
  4. Explain how you know that this flow is maximal.
Edexcel D1 Q5
11 marks Moderate -0.5
5. This question should be answered on the sheet provided. An algorithm is described by the flow chart shown in Figure 1 below. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1e518ab0-9852-4d1d-a4c9-344a5edf9547-05_1337_937_388_404} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Complete the table on the answer sheet recording the results of each instruction as the algorithm is applied and state the final output.
  2. Explain what the algorithm achieves.
  3. Attempt to apply the algorithm again, with the initial value of \(a\), as specified in Box 2, changed to 5 . Explain what happens.
    (2 mark)
  4. Find the set of positive initial values of \(a\) for which the algorithm will work.
    (2 marks)
Edexcel D1 Q6
14 marks Moderate -0.5
6. The manager of a new leisure complex needs to maximise the Revenue \(( \pounds R )\) from providing the following two weekend programmes.
\(\frac { \text { Participants } } { \text { Children } }\)7 hours windsurfing, 2 hours sailing\(\frac { \text { Revenue } } { \pounds 50 }\)
Adults5 hours windsurfing, 6 hours sailing\(\pounds 100\)
The following restrictions apply to each weekend.
No more than 90 participants can be accommodated.
There must be at most 40 adults.
A maximum of 600 person-hours of windsurfing can be offered.
A maximum of 300 person-hours of sailing can be offered.
  1. Formulate the above information as a linear programming problem, listing the constraints as inequalities and stating the objective function \(R\).
  2. On graph paper, illustrate the constraints, indicating clearly the feasible region.
  3. Solve the problem graphically, stating how many adults and how many children should be accepted each weekend and what the revenue will be. The manager is considering buying more windsurfing equipment at a cost of \(\pounds 2000\). This would increase windsurfing provision by \(10 \%\).
  4. State, with a reason, whether such a purchase would be cost effective.
Edexcel D1 Q7
14 marks Standard +0.3
7. This question should be answered on the sheet provided. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1e518ab0-9852-4d1d-a4c9-344a5edf9547-07_576_1360_331_278} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows an activity network modelling the tasks involved in widening a bridge over the B451. The arcs represent the tasks and the numbers in brackets gives the time, in days, to complete each task.
  1. Find the early and late times for each event.
  2. Determine those activities which lie on the critical path and list them in order.
  3. State the minimum length of time needed to widen the bridge. Each task needs a single worker.
  4. Show that two men would not be sufficient to widen the bridge in the shortest time.
    (2 marks)
  5. Draw up a schedule showing how 3 men could complete the project in the shortest time. \section*{Please hand this sheet in for marking}
  6. Complete matching:
    \(P\)\(\bullet\)\(\bullet\)\(D\)
    \(Q\)\(\bullet\)\(\bullet\)\(G\)
    \(R\)\(\bullet\)\(\bullet\)\(E\)
    \(S\)\(\bullet\)\(\bullet\)\(L ( H )\)
    \(T\)\(\bullet\)\(\bullet\)\(L\)
    \section*{Please hand this sheet in for marking}
  7. \(x\)\(a\)\(b\)\(( a - b ) < 0.01\) ?
    1005026No
    -2614.923No
    Final output
  8. \(\_\_\_\_\)
  9. \(x\)\(a\)\(b\)\(( a - b ) < 0.01 ?\)
    100
  10. \(\_\_\_\_\) \section*{Please hand this sheet in for marking}
  11. \includegraphics[max width=\textwidth, alt={}, center]{1e518ab0-9852-4d1d-a4c9-344a5edf9547-11_768_1689_427_221}
  12. \(\_\_\_\_\)
  13. \(\_\_\_\_\)
  14. 051015202530354045505560
    Worker 1
    Worker 2
  15. 051015202530354045505560
    Worker 1
    Worker 2
    Worker 3
Edexcel D2 2006 January Q1
13 marks Moderate -0.8
  1. A theme park has four sites, A, B, C and D, on which to put kiosks. Each kiosk will sell a different type of refreshment. The income from each kiosk depends upon what it sells and where it is located. The table below shows the expected daily income, in pounds, from each kiosk at each site.
Hot dogs and beef burgers (H)Ice cream (I)Popcorn, candyfloss and drinks (P)Snacks and hot drinks (S)
Site A267272276261
Site B264271278263
Site C267273275263
Site D261269274257
Reducing rows first, use the Hungarian algorithm to determine a site for each kiosk in order to maximise the total income. State the site for each kiosk and the total expected income. You must make your method clear and show the table after each stage.
(Total 13 marks)
Edexcel D2 2006 January Q2
12 marks Standard +0.8
2. An engineering firm makes motors. They can make up to five in any one month, but if they make more than four they have to hire additional premises at a cost of \(\pounds 500\) per month. They can store up to two motors for \(\pounds 100\) per motor per month. The overhead costs are \(\pounds 200\) in any month in which work is done.
Motors are delivered to buyers at the end of each month. There are no motors in stock at the beginning of May and there should be none in stock after the September delivery. The order book for motors is:
MonthMayJuneJulyAugustSeptember
Number of motors33754
Use dynamic programming to determine the production schedule that minimises the costs, showing your working in the table provided below.
Stage (month)State (Number in store at start of month)Action (Number made in month)Destinatio n (Number in store at end of month)Value (cost)
\section*{Production schedule}
MonthMayJuneJulyAugustSeptember
Number to be
made
Total cost: \(\_\_\_\_\)
Edexcel D2 2006 January Q3
8 marks Moderate -0.5
3. Three depots, F, G and H, supply petrol to three service stations, S, T and U. The table gives the cost, in pounds, of transporting 1000 litres of petrol from each depot to each service station. F, G and H have stocks of 540000,789000 and 673000 litres respectively.
S, T and U require 257000,348000 and 412000 litres respectively. The total cost of transporting the petrol is to be minimised.
STU
F233146
G353851
H415063
Formulate this problem as a linear programming problem. Make clear your decision variables, objective function and constraints.
Edexcel D2 2006 January Q4
16 marks Moderate -0.8
4. The following minimising transportation problem is to be solved.
JKSupply
A12159
B81713
C4912
Demand911
  1. Complete the table below.
    JKLSupply
    A12159
    B81713
    C4912
    Demand91134
  2. Explain why an extra demand column was added to the table above. A possible north-west corner solution is:
    JKL
    A90
    B112
    C12
  3. Explain why it was necessary to place a zero in the first row of the second column. After three iterations of the stepping-stone method the table becomes:
    JKL
    A81
    B13
    C93
  4. Taking the most negative improvement index as the entering square for the stepping stone method, solve the transportation problem. You must make your shadow costs and improvement indices clear and demonstrate that your solution is optimal.
Edexcel D2 2006 January Q5
13 marks Moderate -0.5
5. A two-person zero-sum game is represented by the following pay-off matrix for player A.
B plays 1B plays 2B plays 3B plays 4
A plays 1- 213- 1
A plays 2- 1321
A plays 3- 420- 1
A plays 41- 2- 13
  1. Verify that there is no stable solution to this game.
  2. Explain why the \(4 \times 4\) game above may be reduced to the following \(3 \times 3\) game.
  3. Formulate the \(3 \times 3\) game as a linear programming problem for player A. Write the
    - 213
    - 132
    1- 2- 1
    constraints as inequalities. Define your variables clearly.
Edexcel D2 2006 January Q6
13 marks Moderate -0.5
6. The network in the figure above, shows the distances in km , along the roads between eight towns, A, B, C, D, E, F, G and H. Keith has a shop in each town and needs to visit each one. He wishes to travel a minimum distance and his route should start and finish at A . By deleting D, a lower bound for the length of the route was found to be 586 km .
By deleting F, a lower bound for the length of the route was found to be 590 km .
  1. By deleting C, find another lower bound for the length of the route. State which is the best lower bound of the three, giving a reason for your answer.
  2. By inspection complete the table of least distances. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{(8)
    (8)
    (Total 13 marks)} \includegraphics[alt={},max width=\textwidth]{a5d69a77-c196-483c-a550-1a55363555af-3_780_889_1069_1078}
    \end{figure} (4) The table can now be taken to represent a complete network. The nearest neighbour algorithm was used to obtain upper bounds for the length of the route: Starting at D, an upper bound for the length of the route was found to be 838 km .
    Starting at F, an upper bound for the length of the route was found to be 707 km .
  3. Starting at C , use the nearest neighbour algorithm to obtain another upper bound for the length of the route. State which is the best upper bound of the
    ABCDEFGH
    A-848513817314952
    B84-13077126213222136
    C85130-53888392
    D1387753-49190
    E1731268849-100180215
    F21383100-163115
    G14922292180163-97
    H5213619021511597-
    three, giving a reason for your answer.
    (4) (Total 13 marks)
Edexcel D2 2006 January Q7
11 marks Moderate -0.5
7.
  1. Define the terms
    1. cut,
    2. minimum cut, as applied to a directed network flow. \includegraphics[max width=\textwidth, alt={}, center]{a5d69a77-c196-483c-a550-1a55363555af-4_844_1465_338_299} The figure above shows a capacitated directed network and two cuts \(C _ { 1 }\) and \(C _ { 2 }\). The number on each arc is its capacity.
  2. State the values of the cuts \(C _ { 1 }\) and \(C _ { 2 }\). Given that one of these two cuts is a minimum cut,
  3. find a maximum flow pattern by inspection, and show it on the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{a5d69a77-c196-483c-a550-1a55363555af-4_597_1470_1656_296}
  4. Find a second minimum cut for this network. In order to increase the flow through the network it is decided to add an arc of capacity 100 joining \(D\) either to \(E\) or to \(G\).
  5. State, with a reason, which of these arcs should be added, and the value of the increased flow.
Edexcel D2 2002 June Q1
8 marks Easy -1.2
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{c4c64221-0373-4be9-abe3-5ff281922cdb-01_675_1052_378_485}
\end{figure} 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. 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 2002 June Q2
8 marks Easy -1.8
2. A two-person zero-sum game is represented by the following pay-off matrix for player \(A\).
\(B\)
IIIIIIIV
\multirow{3}{*}{\(A\)}I- 4- 5- 24
II- 11- 12
III05- 2- 4
IV- 13- 11
  1. Determine the play-safe strategy for each player.
  2. Verify that there is a stable solution and determine the saddle points.
  3. State the value of the game to \(B\).
Edexcel D2 2002 June Q3
10 marks Standard +0.3
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{c4c64221-0373-4be9-abe3-5ff281922cdb-03_764_1514_283_141}
\end{figure} 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 rninimax 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 2002 June Q4
8 marks Moderate -0.5
4. Andrew ( \(A\) ) and Barbara ( \(B\) ) play a zero-sum game. This game is represented by the following payoff matrix for Andrew. $$A \left( \begin{array} { c c c } & B & \\ 3 & 5 & 4 \\ 1 & 4 & 2 \\ 6 & 3 & 7 \end{array} \right)$$
  1. Explain why this matrix may be reduced to $$\left( \begin{array} { l l } 3 & 5 \\ 6 & 3 \end{array} \right)$$
  2. Hence find the best strategy for each player and the value of the game.
    (8)
Edexcel D2 2002 June Q5
11 marks Moderate -0.8
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.
\cline { 2 - 5 } \multicolumn{1}{c|}{}Job 1Job 2Job 3Job 4
Machine 114587
Machine 221265
Machine 37839
Machine 424610
Use the Hungarian algorithm, reducing rows first, to obtain the allocation of machines to jobs which minimises the total time required. State this minimum time.
Edexcel D2 2002 June Q6
12 marks Moderate -0.3
6. The table below shows the distances, in km, between six towns \(A , B , C , D , E\) and \(F\).
\cline { 2 - 7 } \multicolumn{1}{c|}{}\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)
\(A\)-85110175108100
\(B\)85-3817516093
\(C\)11038-14815673
\(D\)175175148-11084
\(E\)108160156110-92
\(F\)10093738492-
  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 hound for the solution of the travelling salesman problem.
    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 2002 June Q7
10 marks Moderate -0.3
7. A steel manufacturer has 3 factories \(F _ { 1 } , F _ { 2 }\) and \(F _ { 3 }\) which can produce 35,25 and 15 kilotonnes of steel per year, respectively. Three businesses \(B _ { 1 } , B _ { 2 }\) and \(B _ { 3 }\) have annual requirements of 20,25 and 30 kilotonnes respectively. The table below shows the cost \(C _ { i j }\) in appropriate units, of transporting one kilotonne of steel from factory \(F _ { i }\) to business \(B _ { j }\).
\cline { 3 - 5 } \multicolumn{2}{c|}{}Business
\cline { 3 - 5 } \multicolumn{2}{c|}{}\(B _ { 1 }\)\(B _ { 2 }\)\(B _ { 3 }\)
\multirow{3}{*}{Factory}\(F _ { 1 }\)10411
\cline { 2 - 5 }\(F _ { 2 }\)1258
\cline { 2 - 5 }\(F _ { 3 }\)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. Calculate all of the improvement indices \(I _ { i j }\), and hence show that this pattern is not optimal.
  3. Use the stepping-stone method to obtain an improved solution.
  4. Show that the transportation pattern obtained in part (c) is optimal and find its cost.
Edexcel D2 2002 June Q8
14 marks Moderate -0.3
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{c4c64221-0373-4be9-abe3-5ff281922cdb-07_521_1404_285_343}
\end{figure} 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. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{c4c64221-0373-4be9-abe3-5ff281922cdb-07_521_1402_1170_343}
    \end{figure} Figure 5 shows a feasible flow through the same network.
  2. State the value of the feasible flow shown in Fig. 5. Taking the flow in Fig. 5 as your initial flow pattern,
  3. 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.
  4. Show the maximal flow on Diagram 2 and state its value.
  5. Prove that your flow is maximal.
Edexcel D2 2002 June Q9
17 marks Moderate -0.5
9. 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.
\cline { 2 - 5 } \multicolumn{1}{c|}{}ProcessingBlendingPackingProfit ( \(\pounds 100\) )
Morning blend3124
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- 30000
  2. 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. 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.
  3. Use your answer to part (b) to advise the company as to which stage(s) it should increase the time available.