Questions — Edexcel D2 (231 questions)

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Edexcel D2 2006 January Q1
  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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
Edexcel D2 2002 June Q10
10. While solving a maximizing linear programming problem, the following tableau was obtained.
Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
\(r\)00\(1 \frac { 2 } { 3 }\)10\(- \frac { 1 } { 6 }\)\(\frac { 2 } { 3 }\)
\(y\)01\(3 \frac { 1 } { 3 }\)01\(- \frac { 1 } { 3 }\)\(\frac { 1 } { 3 }\)
\(x\)10-30-1\(\frac { 1 } { 2 }\)1
P00101111
  1. Explain why this is an optimal tableau.
  2. Write down the optimal solution of this problem, stating the value of every variable.
  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 \(P\).
Edexcel D2 2002 June Q11
11. 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. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{c4c64221-0373-4be9-abe3-5ff281922cdb-10_723_1172_476_337}
\end{figure}
  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.
    1. State the maximum flow along \(S F _ { 1 } A B R\) and \(S F _ { 3 } C R\).
    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.
    2. Prove that your final flow is maximal.
Edexcel D2 2002 June Q12
12. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{c4c64221-0373-4be9-abe3-5ff281922cdb-11_618_1211_253_253}
\end{figure} 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 \(W\), 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.
  2. State the maximum flow along
    1. \(W \quad W _ { 1 } \quad A \quad R _ { 1 } \quad R\),
    2. \(W W _ { 3 } \quad C \quad R _ { 2 } \quad R\).
  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 flowaugmenting route you use, together with its flow.
  4. From your final flow pattern, determine the number of van loads passing through \(B\) each day.
Edexcel D2 2003 June Q1
  1. A two person zero-sum game is represented by the following pay-off matrix for player \(A\).
\cline { 2 - 4 } \multicolumn{1}{c|}{}\(B\) plays I\(B\) plays II\(B\) plays III
\(A\) plays I- 325
\(A\) plays II4- 1- 4
  1. Write down the pay off matrix for player \(B\).
  2. Formulate the game as a linear programming problem for player \(B\), writing the constraints as equalities and stating your variables clearly.
Edexcel D2 2003 June Q2
2. (a) Explain the difference between the classical and practical travelling salesman problems.
\includegraphics[max width=\textwidth, alt={}, center]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-1_691_1297_1014_383} The network in the diagram above shows the distances, in kilometres, between eight McBurger restaurants. An inspector from head office wishes to visit each restaurant. His route should start and finish at \(A\), visit each restaurant at least once and cover a minimum distance.
(b) Obtain a minimum spanning tree for the network using Kruskal's algorithm. You should draw your tree and state the order in which the arcs were added.
(c) Use your answer to part (b) to determine an initial upper bound for the length of the route.
(d) Starting from your initial upper bound and using an appropriate method, find an upper bound which is less than 135 km . State your tour.
Edexcel D2 2003 June Q3
3. Talkalot College holds an induction meeting for new students. The meeting consists of four talks: I (Welcome), II (Options and Facilities), III (Study Tips) and IV (Planning for Success). The four department heads, Clive, Julie, Nicky and Steve, deliver one of these talks each. The talks are delivered consecutively and there are no breaks between talks. The meeting starts at 10 a.m. and ends when all four talks have been delivered. The time, in minutes, each department head takes to deliver each talk is given in the table below.
\cline { 2 - 5 } \multicolumn{1}{c|}{}Talk ITalk IITalk IIITalk IV
Clive12342816
Julie13323612
Nicky15323214
Steve11333610
  1. Use the Hungarian algorithm to find the earliest time that the meeting could end. You must make your method clear and show
    1. the state of the table after each stage in the algorithm,
    2. the final allocation.
  2. Modify the table so it could be used to find the latest time that the meeting could end.
Edexcel D2 2003 June Q4
4. A two person zero-sum game is represented by the following pay-off matrix for player \(A\).
\cline { 2 - 4 } \multicolumn{1}{c|}{}\(B\) plays I\(B\) plays II\(B\) plays III
\(A\) plays I2- 13
\(A\) plays II130
\(A\) plays III01- 3
  1. Identify the play safe strategies for each player.
  2. Verify that there is no stable solution to this game.
  3. Explain why the pay-off matrix above may be reduced to
    \cline { 2 - 4 } \multicolumn{1}{c|}{}\(B\) plays I\(B\) plays II\(B\) plays III
    \(A\) plays I2- 13
    \(A\) plays II130
  4. Find the best strategy for player \(A\), and the value of the game.
Edexcel D2 2003 June Q5
5. The manager of a car hire firm has to arrange to move cars from three garages \(A , B\) and \(C\) to three airports \(D , E\) and \(F\) so that customers can collect them. The table below shows the transportation cost of moving one car from each garage to each airport. It also shows the number of cars available in each garage and the number of cars required at each airport. The total number of cars available is equal to the total number required.
Airport \(D\)Airport \(E\)Airport \(F\)Cars available
Garage \(A\)£20£40£106
Garage \(B\)£20£30£405
Garage C£10£20£308
Cars required694
  1. Use the North-West corner rule to obtain a possible pattern of distribution and find its cost.
    (3)
  2. Calculate shadow costs for this pattern and hence obtain improvement indices for each route.
  3. Use the stepping-stone method to obtain an optimal solution and state its cost.
Edexcel D2 2003 June Q6
6. Kris produces custom made racing cycles. She can produce up to four cycles each month, but if she wishes to produce more than three in any one month she has to hire additional help at a cost of \(\pounds 350\) for that month. In any month when cycles are produced, the overhead costs are \(\pounds 200\). A maximum of 3 cycles can be held in stock in any one month, at a cost of \(\pounds 40\) per cycle per month. Cycles must be delivered at the end of the month. The order book for cycles is
MonthAugustSeptemberOctoberNovember
Number of cycles required3352
Disregarding the cost of parts and Kris’ time,
  1. determine the total cost of storing 2 cycles and producing 4 cycles in a given month, making your calculations clear. There is no stock at the beginning of August and Kris plans to have no stock after the November delivery.
  2. Use dynamic programming to determine the production schedule which minimises the costs, showing your working in the table below.
    StageDemandStateActionDestinationValue
    \multirow[t]{3}{*}{1 (Nov)}\multirow[t]{3}{*}{2}0 (in stock)(make) 20200
    1 (in stock)(make) 10240
    2 (in stock)(make) 0080
    \multirow[t]{2}{*}{2 (Oct)}\multirow[t]{2}{*}{5}140\(590 + 200 = 790\)
    230
    The fixed cost of parts is \(\pounds 600\) per cycle and of Kris’ time is \(\pounds 500\) per month. She sells the cycles for \(\pounds 2000\) each.
  3. Determine her total profit for the four month period.
    (3)
    (Total 18 marks)