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Edexcel D2 2011 June Q7
13 marks Moderate -0.5
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
9 marks Moderate -0.8
  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
7 marks Easy -1.2
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
12 marks Moderate -0.5
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
8 marks Moderate -0.5
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
9 marks Moderate -0.3
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
11 marks Moderate -0.3
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 marks Moderate -0.8
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
12 marks Standard +0.3
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)
Edexcel D2 2013 June Q1
10 marks Moderate -0.8
  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
8 marks Standard +0.3
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
8 marks Moderate -0.5
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
9 marks Standard +0.8
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
8 marks Moderate -0.5
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
13 marks Standard +0.8
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 marks Challenging +1.2
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
12 marks Standard +0.3
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
12 marks Moderate -0.3
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
10 marks Moderate -0.3
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
10 marks Standard +0.3
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
11 marks Moderate -0.5
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
11 marks Standard +0.8
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
8 marks Moderate -0.8
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
13 marks Standard +0.8
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
10 marks Standard +0.3
  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.