Questions — Edexcel D2 (231 questions)

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Edexcel D2 2009 June Q6
6. The table below shows the cost, in pounds, of transporting one unit of stock from each of three supply points, \(\mathrm { X } , \mathrm { Y }\) and Z to three demand points, \(\mathrm { A } , \mathrm { B }\) and C . It also shows the stock held at each supply point and the stock required at each demand point.
\(\mathbf { A }\)\(\mathbf { B }\)\(\mathbf { C }\)Supply
\(\mathbf { X }\)178722
\(\mathbf { Y }\)16121517
\(\mathbf { Z }\)610915
Demand161523
  1. This is a balanced problem. Explain what this means.
  2. Use the north west corner method to obtain a possible solution.
  3. Taking ZA as the entering cell, use the stepping-stone method to find an improved solution. Make your route clear and state your exiting cell.
  4. Perform one more iteration of the stepping-stone method to find a further improved solution. You must make your shadow costs, improvement indices, entering cell, exiting cell and route clear.
  5. State the cost of the solution you found in part (d).
Edexcel D2 2009 June Q7
7. Minty has \(\pounds 250000\) to allocate to three investment schemes. She will allocate the money to these schemes in units of \(\pounds 50000\). The net income generated by each scheme, in \(\pounds 1000\) s, is given in the table below.
\(\mathbf { \pounds 0 }\)\(\mathbf { \pounds 5 0 0 0 0 }\)\(\mathbf { \pounds 1 0 0 0 0 0 }\)\(\mathbf { \pounds 1 5 0 0 0 0 }\)\(\mathbf { \pounds 2 0 0 0 0 0 }\)\(\mathbf { \pounds 2 5 0 0 0 0 }\)
Scheme1060120180240300
Scheme 2065125190235280
Scheme 3055110170230300
Minty wishes to maximise the net income. She decides to use dynamic programming to determine the optimal allocation, and starts the table shown in your answer book.
  1. Complete the table in the answer book to determine the amount Minty should allocate to each scheme in order to maximise the income. State the maximum income and the amount that should be allocated to each scheme.
  2. For this problem give the meaning of the table headings
    1. Stage,
    2. State,
    3. Action.
Edexcel D2 2009 June Q8
8. Laura (L) and Sam (S) play a two-person zero-sum game which is represented by the following pay-off matrix for Laura.
S plays 1S plays 2S plays 3
L plays 1- 28- 1
L plays 274- 3
L plays 31- 54
Formulate the game as a linear programming problem for Laura, writing the constraints as inequalities. Define your variables clearly.
Edexcel D2 2010 June Q1
  1. The table below shows the least costs, in pounds, of travelling between six cities, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F .
ABCDEF
A-3618282422
B36-54222027
C1854-422724
D282242-2030
E24202720-13
F2227243013-
Vicky must visit each city at least once. She will start and finish at A and wishes to minimise the total cost.
  1. Use Prim's algorithm, starting at A , to find a minimum spanning tree for this network.
    (2)
  2. Use your answer to part (a) to help you calculate an initial upper bound for the length of Vicky's route.
    (1)
  3. Show that there are two nearest neighbour routes that start from A . You must make your routes and their lengths clear.
  4. State the best upper bound from your answers to (b) and (c).
  5. Starting by deleting A , and all of its arcs, find a lower bound for the route length.
Edexcel D2 2010 June Q2
2. A team of four workers, Harry, Jess, Louis and Saul, are to be assigned to four tasks, 1, 2, 3 and 4. Each worker must be assigned to one task and each task must be done by just one worker. Jess cannot be assigned to task 4.
The amount, in pounds, that each person would earn while assigned to each task is shown in the table below.
1234
Harry18242217
Jess202519-
Louis25242722
Saul19262314
  1. Reducing rows first, use the Hungarian algorithm to obtain an allocation that maximises the total amount earned by the team. You must make your method clear and show the table after each stage.
  2. State who should be assigned to each task and the total amount earned by the team.
Edexcel D2 2010 June Q3
3. The table below shows the cost of transporting one block of staging from each of two supply points, X and Y , to each of four concert venues, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D . It also shows the number of blocks held at each supply point and the number of blocks required at each concert venue. A minimal cost solution is required.
ABCDSupply
X2820191653
Y1512141747
Demand18312229
  1. Use the north-west corner method to obtain a possible solution.
    (1)
  2. Taking the most negative improvement index to indicate the entering square, use the stepping stone method twice to obtain an improved solution. You must make your method clear by stating your shadow costs, improvement indices, routes, entering cells and exiting cells.
  3. Is your current solution optimal? Give a reason for your answer.
    (1)
Edexcel D2 2010 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a91f96a3-a9b4-4bf5-bfdf-93c34912e54a-5_774_1623_264_221} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 represents the maintenance choices a council can make and their costs, in \(\pounds 1000\) s, over the next four years. The council wishes to minimise the greatest annual cost of maintenance.
  1. Use dynamic programming to find a minimax route from S to T .
    (9)
  2. State your route and the greatest annual cost incurred by the council.
    (2)
  3. Calculate the average annual cost to the council.
    (2)
Edexcel D2 2010 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a91f96a3-a9b4-4bf5-bfdf-93c34912e54a-6_1012_1696_233_182} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a capacitated, directed network. The number on each arc represents the capacity of that arc. The numbers in circles represent an initial flow.
  1. State the value of the initial flow.
    (1)
  2. State the capacities of cuts \(\mathrm { C } _ { 1 }\) and \(\mathrm { C } _ { 2 }\).
    (2)
  3. By entering values along DH, FH, FI and IT, complete the labelling procedure on Diagram 1 in the answer book.
    (2)
  4. Using Diagram 1, increase the flow by a further 4 units. You must list each flow-augmenting route you use, together with its flow.
    (3)
  5. Prove that the flow is now maximal.
    (2)
Edexcel D2 2010 June Q6
6. The tableau below is the initial tableau for a linear programming problem in \(x , y\) and \(z\). The objective is to maximise the profit, \(P\).
Basic Variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
\(r\)01210024
\(s\)21401028
\(t\)-1\(\frac { 1 } { 2 }\)300122
\(P\)-1-2-60000
  1. Write down the profit equation represented in the initial tableau.
    (1)
  2. Taking the most negative number in the profit row to indicate the pivot column at each stage, solve this linear programming problem. Make your method clear by stating the row operations you use.
  3. State the final value of the objective function and of each variable.
    (3)
Edexcel D2 2010 June Q7
7. A two person zero-sum game is represented by the following pay-off matrix for player A.
B plays 1B plays 2B plays 3
A plays 1- 451
A plays 23- 1- 2
A plays 3- 302
Formulate the game as a linear programming problem for player A. Write the constraints as inequalities and define your variables.
Edexcel D2 2011 June Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d28d78c1-052d-4350-a7e3-284380e3bbab-2_703_958_351_552} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The network in Figure 1 shows the distances, in km, between six towns, A, B, C, D, E and F. Mabintou needs to visit each town. She will start and finish at A and wishes to minimise the total distance travelled.
  1. By inspection, complete the two copies of the table of least distances in your answer book.
  2. Starting at A, use the nearest neighbour algorithm to find an upper bound for the length of Mabintou's route. Write down the route which gives this upper bound.
  3. Starting by deleting A, and all of its arcs, find a lower bound for the route length.
Edexcel D2 2011 June Q2
2. The table below shows the cost of transporting one unit of stock from each of four supply points, 1 , 2, 3 and 4, to each of three demand points, \(\mathrm { A } , \mathrm { B }\) and C . It also shows the stock held at each supply point and the stock required at each demand point. A minimal cost solution is required.
ABCSupply
131293220
222332722
325273220
423263838
Demand352530
  1. Add a dummy demand point and appropriate values to Table 1 in the answer book. Table 2 shows an initial solution given by the north-west corner method.
    Table 3 shows some of the improvement indices for this solution. \begin{table}[h]
    ABCD
    120
    2157
    3182
    42810
    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table} \begin{table}[h]
    ABCD
    1- 13- 9
    2- 11
    3
    41- 7
    \captionsetup{labelformat=empty} \caption{Table 3}
    \end{table}
  2. Calculate the shadow costs and the missing improvement indices and enter them into Table 3 in the answer book.
  3. Taking the most negative improvement index to indicate the entering square, use the steppingstone method once to obtain an improved solution. You must make your route clear and state your entering cell and exiting cell.
Edexcel D2 2011 June Q3
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 is obtained.
Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
\(r\)\(- \frac { 1 } { 2 }\)021\(- \frac { 1 } { 2 }\)010
\(y\)\(\frac { 1 } { 2 }\)1\(\frac { 3 } { 4 }\)0\(\frac { 1 } { 4 }\)05
\(t\)\(\frac { 1 } { 2 }\)010\(- \frac { 1 } { 4 }\)14
\(P\)- 701040320
  1. Write down the profit equation represented in the tableau.
  2. Taking the most negative number in the profit row to indicate the pivot column at each stage, solve this linear programming problem. Make your method clear by stating the row operations you use.
  3. State the value of the objective function and of each variable.
Edexcel D2 2011 June Q4
4. Laura and Sam play a zero-sum game. This game is represented by the following pay-off matrix for Laura.
S plays 1S plays 2S plays 3
L plays 1- 4- 11
L plays 23- 1- 2
L plays 3- 302
Find the best strategy for Laura and the value of the game to her.
Edexcel D2 2011 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d28d78c1-052d-4350-a7e3-284380e3bbab-6_663_1363_242_351} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a capacitated directed network. The number on each arc is its capacity. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d28d78c1-052d-4350-a7e3-284380e3bbab-6_665_1363_1117_351} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows an initial flow through the same network.
  1. State the values of flows \(a , b\) and \(c\), and the value of the initial flow.
  2. By entering values along HG, HT and FG, complete the labelling procedure on Diagram 1 in the answer book.
  3. Find the maximum flow through the network. You must list each flow-augmenting route you use, together with its flow.
  4. State the value of the maximum flow through the network.
  5. Show your maximum flow on Diagram 2 in the answer book.
  6. Prove that your flow is maximal.
Edexcel D2 2011 June Q6
6. Three workers, \(\mathrm { P } , \mathrm { Q }\) and R , are to be assigned to three tasks, A, B and C. Each worker must be assigned to just one task and each task must be assigned to just one worker.
Table 1 shows the cost of using each worker for each task. The total cost is to be minimised. \begin{table}[h]
Task ATask BTask C
Worker P273125
Worker Q263034
Worker R352932
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
  1. Formulate the above situation as a linear programming problem. You must define your decision variables and make the objective and constraints clear.
    You are not required to solve the problem. Table 2 shows the profit gained by using each worker for each task. The total profit is to be maximised. \begin{table}[h]
    Task ATask BTask C
    Worker P333731
    Worker Q323640
    Worker R413538
    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table}
  2. Modify Table 2 in the answer book so that the Hungarian Algorithm could be used to find the maximum total profit. You are not required to solve the problem.
    (2)
    (Total 9 marks)
Edexcel D2 2011 June Q7
7. Patrick is to take orders for his company's products. He will visit four countries over the next four weeks.
He will visit just one country each week.
He will leave from his office in London and will only return there after visiting the four countries.
He will travel directly from one country to the next.
He wishes to determine a schedule of four countries to visit. Table 1 shows the countries he could visit in each week. \begin{table}[h]
WeekWeek 1Week 2Week 3Week 4
Possible countriesA or BC, D or EF or GH or I
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} Table 2 shows the value of the orders, in \(\pounds 100\) s, he expects to take in each country. \begin{table}[h]
CountryABCDEFGHI
Value of expected orders in \(\pounds 100\) s221742413929273638
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table} Table 3 shows the cost, in \(\pounds 100\) s, of travelling between the various countries. \begin{table}[h]
Travel costs in £100sABCDEFGHI
London5354
A542
B443
C65
D63
E44
F67
G56
\captionsetup{labelformat=empty} \caption{Table 3}
\end{table} The expected income is the value of the expected orders minus the cost of travel. It is decided to use dynamic programming to find a schedule that maximises the total expected income for these four weeks.
  1. Complete the table in the answer book to determine the optimal expected income.
  2. State Patrick's two optimal schedules.
Edexcel D2 2012 June Q1
  1. Five workers, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E , are to be assigned to five tasks, \(1,2,3,4\) and 5 . Each worker is to be assigned to one task and each task must be assigned to one worker.
The cost, in pounds, of assigning each person to each task is shown in the table below. The cost is to be minimised.
12345
A129127122134135
B127125123131132
C142131121140139
D127127122131136
E141134129144143
  1. Reducing rows first, use the Hungarian algorithm to obtain an allocation that minimises the cost. You must make your method clear and show the table after each stage.
  2. Find the minimum cost.
Edexcel D2 2012 June Q2
2. The table shows the least distances, in km, between six towns, A, B, C, D, E and F.
ABCDEF
A-1625211215
B16-24222112
C2524-183027
D212218-1512
E12213015-18
F1512271218-
Toby must visit each town at least once. He will start and finish at A and wishes to minimise the total distance.
  1. Use the nearest neighbour algorithm, starting at A , to find an upper bound for the length of Toby's route.
  2. Starting by deleting A, and all of its arcs, find a lower bound for the route length.
Edexcel D2 2012 June Q3
3. The table below shows the cost, in pounds, of transporting one tonne of concrete from each of three supply depots, \(\mathrm { A } , \mathrm { B }\) and C , to each of four building sites, \(\mathrm { D } , \mathrm { E } , \mathrm { F }\) and G . It also shows the number of tonnes that can be supplied from each depot and the number of tonnes required at each building site. A minimum cost solution is required.
DEFGSupply
A1719212018
B2120192223
C1817162129
Demand15241813
The north-west corner method gives the following possible solution.
DEFGSupply
A15318
B21223
C161329
Demand15241813
Taking AG as the first entering cell,
  1. use the stepping stone method twice to obtain an improved solution. You must make your method clear by stating your shadow costs, improvement indices, routes, entering cells and exiting cells.
  2. Determine whether your current solution is optimal. Justify your answer.
Edexcel D2 2012 June Q4
4. The tableau below is the initial tableau for a maximising linear programming problem in \(x , y\) and \(z\) which is to be solved.
Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
\(r\)5\(\frac { 1 } { 2 }\)01005
\(s\)1-240103
\(t\)8460016
\(P\)-5-7-40000
  1. Starting by increasing \(y\), perform one complete iteration of the simplex algorithm, to obtain tableau T. State the row operations you use.
  2. Write down the profit equation given by tableau T .
  3. Use the profit equation from part (b) to explain why tableau T is optimal.
Edexcel D2 2012 June Q5
5. Agent Goodie is planning to break into Evil Doctor Fiendish's secret base. He uses game theory to determine whether to approach the base from air, sea or land.
Evil Doctor Fiendish decides each day which of three possible plans he should use to protect his base. Agent Goodie evaluates the situation. He assigns numbers, negative indicating he fails in his mission, positive indicating success, to create a pay-off matrix. The numbers range from - 3 (he fails in his mission and is captured) to 5 (he successfully achieves his mission and escapes uninjured) and the pay-off matrix is shown below.
Fiendish uses plan 1Fiendish uses plan 2Fiendish uses plan 3
Air045
Sea2-31
Land-23-2
  1. Reduce the game so that Agent Goodie has only two choices, explaining your reasoning.
  2. Use game theory to determine Agent Goodie's best strategy.
  3. Find the value of the game to Agent Goodie.
Edexcel D2 2012 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d0bede44-05dc-4edd-8436-cf5eea710d1a-7_1120_1691_260_187} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a capacitated, directed network. The number on each arc represents the capacity of that arc. The numbers in circles represent an initial flow.
  1. State the value of the initial flow.
    (1)
  2. Complete the initialisation of the labelling procedure on Diagram 1 in the answer book by entering values along \(\mathrm { SB } , \mathrm { BD } , \mathrm { CF }\) and FT .
    (2)
  3. Hence use the labelling procedure to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow.
    (4)
  4. Draw a maximal flow pattern on Diagram 2 in your answer book.
    (2)
  5. Prove that your flow is maximal.
    (2)
Edexcel D2 2012 June Q7
7. Four workers, A, B, C and D, are to be assigned to four tasks, P, Q, R and S. Each worker is to be assigned to exactly one task and each task must be assigned to just one worker. The cost, in pounds, of using each worker for each task is given in the table below. The total cost is to be minimised.
PQRS
A23413444
B21453342
C26433140
D20473546
Formulate the above situation as a linear programming problem. You must define your decision variables and make the objective function and constraints clear.
(Total 7 marks)
Edexcel D2 2012 June Q8
8. A company makes industrial robots. They can make up to four robots in any one month, but if they make more than three they will have to hire additional labour at a cost of \(\pounds 400\) per month.
They can store up to two robots at a cost of \(\pounds 150\) per robot per month.
The overhead costs are \(\pounds 300\) in any month in which work is done.
Robots are delivered to buyers at the end of each month. There are no robots in stock at the beginning of January and there should be none in stock after the April delivery. The order book for robots is
MonthJanuaryFebruaryMarchApril
Number of robots required2234
Use dynamic programming to determine the production schedule which minimises the costs, showing your working in the table provided in the answer book.
(Total 12 marks)