Questions — Edexcel D2 (231 questions)

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Edexcel D2 2013 June Q1
  1. Four workers, Chris (C), James (J), Katie (K) and Nicky (N), are to be allocated to four tasks, 1, 2, 3 and 4. Each worker is to be allocated to one task and each task must be allocated to one worker.
The profit, in pounds, resulting from allocating each worker to each task, is shown in the table below. The profit is to be maximised.
1234
Chris127116111113
James225208205208
Katie130113112114
Nicky228212203210
  1. Reducing rows first, use the Hungarian algorithm to obtain an allocation that maximises the total profit. You must make your method clear and show the table after each stage.
  2. State which worker should be allocated to each task and the resulting total profit made.
Edexcel D2 2013 June Q2
2. The table shows the least distances, in km, between six towns, A, B, C, D, E and F.
ABCDEF
A-12221713710982
B122-110130128204
C217110-204238135
D137130204-98211
E10912823898-113
F82204135211113-
Liz must visit each town at least once. She will start and finish at A and wishes to minimise the total distance she will travel.
  1. Starting with the minimum spanning tree given in your answer book, use the shortcut method to find an upper bound below 810 km for Liz's route. You must state the shortcut(s) you use and the length of your upper bound.
    (2)
  2. Use the nearest neighbour algorithm, starting at A , to find another upper bound for the length of Liz's route.
  3. Starting by deleting F , and all of its arcs, find a lower bound for the length of Liz's route.
  4. Use your results to write down the smallest interval which you are confident contains the optimal length of the route.
Edexcel D2 2013 June Q3
3. Table 1 below shows the cost, in pounds, of transporting one unit of stock from each of four supply points, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , to four demand points \(1,2,3\) and 4 . It also shows the stock held at each supply point and the stock required at each demand point. A minimum cost solution is required. \begin{table}[h]
1234Supply
A2236193735
B2935303615
C2432254120
D2330233830
Demand30203020
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} 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]
1234
A305
B150
C20
D1020
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table} \begin{table}[h]
1234
Axx
Bxx
C82x1
D92xx
\captionsetup{labelformat=empty} \caption{Table 3}
\end{table}
  1. Explain why a zero has been placed in cell B3 in Table 2.
    (1)
  2. Calculate the shadow costs and the missing improvement indices and enter them into Table 3 in your answer book.
  3. Taking the most negative improvement index to indicate the entering cell, state the steppingstone route that should be used to obtain the next solution. You must state your entering cell and exiting cell.
Edexcel D2 2013 June Q4
4. Robin (R) and Steve (S) play a two-person zero-sum game which is represented by the following pay-off matrix for Robin.
S plays 1S plays 2S plays 3
R plays 1213
R plays 21- 12
R plays 3- 13- 3
Find the best strategy for Robin and the value of the game to him.
Edexcel D2 2013 June Q5
5. 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
\(r\)\(\frac { 1 } { 2 }\)\(- \frac { 1 } { 2 }\)010\(- \frac { 1 } { 2 }\)10
\(s\)\(1 \frac { 1 } { 2 }\)\(2 \frac { 1 } { 2 }\)001\(- \frac { 1 } { 2 }\)5
\(z\)\(\frac { 1 } { 2 }\)\(\frac { 1 } { 2 }\)100\(\frac { 1 } { 2 }\)5
\(P\)-5-1000020220
  1. Starting by increasing \(y\), perform one complete iteration of the Simplex algorithm, to obtain a new tableau, T. State the row operations you use.
  2. Write down the profit equation given by T .
  3. Use the profit equation from part (b) to explain why T is optimal.
Edexcel D2 2013 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0af7fd3e-68af-41fc-883b-3bc2589035bb-7_816_1138_178_459} \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.
  2. On Diagram 1 and Diagram 2 in the answer book, add a supersource S and a supersink T . On Diagram 1, show the minimum capacities of the arcs you have added.
  3. Complete the initialisation of the labelling procedure on Diagram 2 in the answer book by entering values on the arcs to S and T and on \(\operatorname { arcs } \mathrm { CD }\), DE , DG , FG, FI and GI.
  4. Find the maximum flow through the network. You must list each flow-augmenting route you use, together with its flow.
  5. Show your maximum flow on Diagram 3 in the answer book.
  6. Prove that your flow is maximal.
Edexcel D2 2013 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 11- 32
A plays 2- 23- 1
A plays 35- 10
Formulate the game as a linear programming problem for player A. Write the constraints as inequalities. Define your variables clearly.
(Total 7 marks)
Edexcel D2 2013 June Q8
8. A factory can process up to five units of carrots each month. Each unit can be sold fresh or frozen or canned.
The profits, in \(\pounds 100\) s, for the number of units sold, are shown in the table.
The total monthly profit is to be maximised.
Number of units012345
Fresh04585120150175
Frozen04570100120130
Canned03575125155195
Use dynamic programming to determine how many of the five units should be sold fresh, frozen and canned in order to maximise the monthly profit. State the maximum monthly profit.
(Total 12 marks)
Edexcel D2 2013 June Q1
1.
ABCDE
A-15192520
B15-151525
C1915-2211
D251522-18
E20251118-
The table shows the least distances, in km, between five hiding places, A, B, C, D and E.
Agent Goodie has to leave a secret message in each of the hiding places. He will start and finish at A , and wishes to minimise the total distance travelled.
  1. Use Prim's algorithm to find a minimum spanning tree for this network. Make your order of arc selection clear.
  2. Use your answer to part (a) to determine an initial upper bound for the length of Agent Goodie's route.
  3. Show that there are two nearest neighbour routes which start from A . State these routes and their lengths.
  4. State the better upper bound from your answers to (b) and (c).
  5. Starting by deleting B, and all of its arcs, find a lower bound for the length of Agent Goodie's route.
  6. Consider your answers to (d) and (e) and hence state an optimal route.
Edexcel D2 2013 June Q2
2. The table shows the cost, in pounds, of transporting one unit of stock from each of four supply points, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , to each of three demand points, 1, 2 and 3 . It also shows the stock held at each supply point and the stock required at each demand point. A minimum cost solution is required.
123Supply
A10112018
B1571314
C24151221
D9211812
Demand271820
  1. Use the north-west corner method to obtain an initial solution.
    (1)
  2. Taking D1 as the entering cell, use the stepping stone method to find an improved solution. Make your route clear.
    (2)
  3. Perform one further iteration of the stepping stone method to obtain an improved solution. You must make your method clear by stating your shadow costs, improvement indices, route, entering cell and exiting cell.
  4. Determine whether your current solution is optimal, giving a reason for your answer.
Edexcel D2 2013 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f3feef8a-32ba-4234-a5fa-cdd26ef6967d-4_778_1420_262_360} \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.
  2. State the capacity of cut \(\mathrm { C } _ { 1 }\). The labelling procedure has been used and the result drawn on Diagram 1 in the answer book.
  3. Use Diagram 1 to find the maximum flow through the network. You must list each flow-augmenting route you use, together with its flow.
    (4)
  4. Draw a maximum flow pattern on Diagram 2 in your answer book.
    (2)
  5. Prove that the flow shown in (d) is maximal.
    (2)
Edexcel D2 2013 June Q4
4. 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 154- 6
A plays 2- 1- 23
A plays 31- 12
  1. Reduce the game so that player B has only two possible actions.
  2. Write down the reduced pay-off matrix for player B.
  3. Find the best strategy for player B and the value of the game to him.
Edexcel D2 2013 June Q5
5. In solving a three-variable maximising linear programming problem, the following tableau was obtained after the first iteration.
Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
\(r\)- 1201018
\(s\)- 13001122
\(z\)- 21100111
\(P\)2- 5000\(\frac { 1 } { 2 }\)15
  1. State which variable was increased first, giving a reason for your answer.
  2. 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 the final values of each variable.
Edexcel D2 2013 June Q6
6. Three workers, Harriet, Jason and Katherine, are to be assigned to three tasks, 1, 2 and 3. Each worker must be assigned to just one task and each task must be done by just one worker. The amount each person would earn, in pounds, while assigned to each task is shown in the table below.
Task 1Task 2Task 3
Harriet251243257
Jason244247255
Katherine249252246
The total income is to be maximised.
  1. Modify the table so it can be used to find the maximum income.
  2. Formulate the above situation as a linear programming problem. You must define your decision variables and make your objective function and constraints clear.
Edexcel D2 2013 June Q7
7. Nigel has a business renting out his fleet of bicycles to tourists. At the start of each year Nigel must decide on one of two actions:
  • Keep his fleet of bicycles, incurring maintenance costs.
  • Replace his fleet of bicycles.
The cost of keeping the fleet of bicycles, the cost of replacing the fleet of bicycles and the annual income are dependent on the age of the fleet of bicycles.
Table 1 shows these amounts, in \(\pounds 1000\) s. \begin{table}[h]
Age of fleet of bicyclesnew1 year old2 years old3 years old4 years old
Cost of keeping (£1000s)01238
Cost of replacing (£1000s)-78910
Income (£1000s)118520
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} Nigel has a new fleet of bicycles now and wishes to maximise his total profit over the next four years. He is planning to sell his business at the end of the fourth year.
The amount Nigel will receive will depend on the age of his fleet of bicycles.
These amounts, in £1000s, are shown in Table 2. \begin{table}[h]
Age of fleet of bicycles
at end of 4th year
1 year
old
2 years
old
3 years
old
4 years
old
Amount received at end
of 4th year \(( \pounds 1000 \mathrm {~s} )\)
6421
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table} Complete the table in the answer book to determine Nigel's best strategy to maximise his total profit over the next four years. You must state the action he should take each year (keep or replace) and his total profit.
(Total 13 marks)
Edexcel D2 2014 June Q1
  1. Four bakeries, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , supply bread to four supermarkets, \(\mathrm { P } , \mathrm { Q } , \mathrm { R }\) and S . The table gives the cost, in pounds, of transporting one lorry load of bread from each bakery to each supermarket. It also shows the number of lorry loads of bread at each bakery and the number of lorry loads of bread required at each supermarket. The total cost of transportation is to be minimised.
PQRSSupply
A2832332713
B312926314
C3026293212
D2530283411
Demand1110118
  1. Use the north-west corner method to obtain a possible solution. A partly completed table of improvement indices is given in Table 1 in the answer book.
  2. Complete Table 1.
  3. Taking the most negative improvement index to indicate the entering cell, use the steppingstone method once to obtain an improved solution. You must make your route clear and state your entering cell and exiting cell.
  4. State the cost of your improved solution.
Edexcel D2 2014 June Q2
2. (a) Explain the difference between the classical and the practical travelling salesperson problem.
ABCDEF
A-6548153040
B65-50513526
C4850-372034
D155137-1725
E30352017-14
F4026342514-
The table above shows the least distances, in km, between six towns, A, B, C, D, E and F. Keith needs to visit each town, starting and finishing at A , and wishes to minimise the total distance he will travel.
(b) Starting at A, use the nearest neighbour algorithm to obtain an upper bound. You must state your route and its length.
(c) Starting by deleting A, and all of its arcs, find a lower bound for the route length.
(d) Use your results to write down the smallest interval which you are confident contains the optimal length of the route.
Edexcel D2 2014 June Q3
3. 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- 22- 3
A plays 211- 1
A plays 32- 11
  1. Starting by reducing player B's options, find the best strategy for player B.
  2. State the value of the game to player B.
Edexcel D2 2014 June Q4
4. The tableau below is the initial tableau for a three-variable 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
\(r\)43\(\frac { 5 } { 2 }\)10050
\(s\)12101030
\(t\)05100180
\(P\)- 25- 40- 350000
  1. Taking the most negative number in the profit row to indicate the pivot column at each stage, perform two complete iterations of the simplex algorithm to obtain tableau T. Make your method clear by stating the row operations you use.
  2. Write down the profit equation given by T .
  3. Use your answer to (b) to determine whether T is optimal, justifying your answer.
Edexcel D2 2014 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d708ce08-4ea3-4a13-a39c-00efcde32c57-5_707_969_237_523} \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. Add a supersource, S , and a supersink, T , and corresponding arcs to Diagrams 1 and 2, in the answer book.
    2. Enter the flow value and appropriate capacity on each of the arcs you have added to Diagram 1.
  2. Complete the initialisation of the labelling procedure on Diagram 2 by entering values along the new arcs from \(S\) and \(T\), and along \(A B , A D\) and \(D _ { 2 }\).
  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. Draw a maximal flow pattern on Diagram 3 in the answer book.
  5. Prove that your flow is maximal.
Edexcel D2 2014 June Q6
6. Four workers, A, B, C and D, are to be assigned to four tasks, 1, 2, 3 and 4. Each worker must be assigned to just one task and each task must be done by just one worker. Worker C cannot do task 4 and worker D cannot do task 1. The cost of assigning each worker to each task is shown in the table below. The total cost is to be minimised.
1234
A29153230
B34264032
C282735-
D-213331
Formulate the above situation as a linear programming problem. You must define your decision variables and make the objective function and constraints clear.
Edexcel D2 2014 June Q7
7. Susie has hired a team of four workers who can make three types of toy. The total number of toys the team can produce will depend on which toys they make, and on how many workers are assigned to make each type of toy. The table shows how many of each toy would be made if different numbers of workers were assigned to make them. Each worker is to be assigned to make just one type of toy and all four workers are to be assigned. Susie wishes to maximise the total number of toys produced.
\cline { 3 - 7 } \multicolumn{2}{c|}{}Number of workers
\cline { 3 - 7 } \multicolumn{2}{c|}{}01234
\multirow{2}{*}{
T
O
Y
S
}		Bicycle080170260350
\cline { 2 - 7 }Dolls House095165245335
\cline { 2 - 7 }Train Set0100180260340
  1. Use dynamic programming to determine the allocation of workers which maximises the total number of toys made. You should show your working in the table provided in the answer book.
    (12)
  2. State the maximum total number of toys produced by this team.
    (1)
    (Total 13 marks)
Edexcel D2 2014 June Q1
  1. Four workers, A, B, C and D, are to be assigned to four tasks, 1, 2, 3 and 4. Each worker must be assigned to just one task and each task must be done by just one worker.
Worker A cannot do task 4 and worker B cannot do task 2. The amount, in pounds, that each worker would earn if assigned to the tasks, is shown in the table below.
1234
A191623-
B24-3023
C18172518
D24242624
Reducing rows first, use the Hungarian algorithm to obtain an allocation that maximises the total earnings. You must make your method clear and show the table after each stage.
Edexcel D2 2014 June Q2
2. The table shows the least times, in seconds, that it takes a robot to travel between six points in an automated warehouse. These six points are an entrance, A , and five storage bins, \(\mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F . The robot will start at A , visit each bin, and return to A . The total time taken for the robot's route is to be minimised.
ABCDEF
A-901308535125
B90-801008388
C13080-108106105
D85100108-11088
E3583106110-75
F125881058875-
  1. Show that there are two nearest neighbour routes that start from A . You must make the routes and their lengths clear.
  2. Starting by deleting F , and all of its arcs, find a lower bound for the time taken for the robot's route.
  3. Use your results to write down the smallest interval which you are confident contains the optimal time for the robot's route.
Edexcel D2 2014 June Q3
3. The tableau below is the initial tableau for a three-variable 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
\(r\)53\(- \frac { 1 } { 2 }\)1002500
\(s\)3210101650
\(t\)\(\frac { 1 } { 2 }\)- 12001800
\(P\)- 40- 50- 350000
  1. 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.
  2. State the final values of the objective function and each variable.