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Edexcel D2 2016 June Q7
15 marks Standard +0.8
7. Remy builds canoes. He can build up to five canoes each month, but if he wishes to build more than three canoes in any one month he has to hire an additional worker at a cost of \(\pounds 400\) for that month. In any month when canoes are built, the overhead costs are \(\pounds 150\) A maximum of three canoes can be held in stock in any one month, at a cost of \(\pounds 25\) per canoe per month. Canoes must be delivered at the end of the month. The order book for canoes is
MonthJanuaryFebruaryMarchAprilMay
Number ordered22564
There is no stock at the beginning of January and Remy plans to have no stock after the May delivery.
  1. Use dynamic programming to determine the production schedule that minimises the costs given above. Show your working in the table provided in the answer book and state the minimum cost.
    (13) The cost of materials is \(\pounds 200\) per canoe and the cost of Remy's time is \(\pounds 450\) per month. Remy sells the canoes for \(\pounds 700\) each.
  2. Determine Remy's total profit for the five-month period.
    (2)
    (Total 15 marks)
Edexcel D2 Q1
9 marks Moderate -0.5
  1. A company has five machines \(A , B , C , D\) and \(E\), which it can assign to four tasks \(1,2,3\) and 4 . Each task must be assigned to just one machine and each machine may only be assigned to just one task.
The profit, in \(\pounds 100\) s, of using each machine to do each task is given in the table below.
1234
\(A\)14121117
\(B\)14131516
\(C\)17161012
\(D\)16141312
\(E\)13151315
  1. Explain why it is necessary to add a dummy column to the table.
    (2)
  2. Use the Hungarian algorithm to allocate machines to tasks in order to maximise the total profit. You must make your method clear and show the state of the table after each iteration.
    (7)
    (Total 9 marks)
Edexcel D2 Q2
9 marks Moderate -0.3
2. The following transportation problem is to be solved.
\(P\)\(Q\)\(R\)Supply
\(A\)75712
\(B\)5657
\(C\)1412911
Demand10911
A possible north-west corner solution is:
\(P\)\(Q\)\(R\)
\(A\)102
\(B\)70
\(C\)11
  1. Use the stepping-stone method once to obtain an improved solution. You must make your shadow costs, improvement indices, entering cell, exiting cell and stepping-stone route clear.
  2. Demonstrate that your solution is optimal.
    (3)
Edexcel D2 Q3
7 marks Moderate -0.5
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- 243
A plays 24- 12
Find the best strategy for player A and the value of the game.
(Total 7 marks)
Edexcel D2 Q4
8 marks Moderate -0.8
4. The table shows the least distances, in km, between six towns \(A , B , C , D , E\) and \(F\).
\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)
\(A\)-98123689671
\(B\)98-7412947120
\(C\)12374-10211163
\(D\)68129102-8559
\(E\)964711185-115
\(F\)711206359115-
  1. Starting at \(A\), and making your method clear, find an upper bound for the travelling salesman problem using the nearest neighbour algorithm.
  2. By deleting \(A\), and all of its arcs, find a lower bound for the travelling salesman problem.
  3. Write down an inequality about the length of the optimal route.
Edexcel D2 Q5
14 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e80fcab6-7c7d-4a0c-84e0-c23f5a969a75-4_924_1646_221_207} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a capacitated, directed, network. The capacity of each arc is shown on that arc and the numbers in circles represent an initial flow from S to T . Two cuts, \(\mathrm { C } _ { 1 }\) and \(\mathrm { C } _ { 2 }\) are shown on Figure 1.
  1. Find the capacity of each of the two cuts and the value of the initial flow.
    (3)
  2. Complete the initialisation of the labelling procedure on Figure 1 in the answer book, by entering values along \(\mathrm { SB } , \mathrm { AB } , \mathrm { BE }\) and BG .
    (2)
  3. Hence use the labelling procedure to find a maximum flow of 85 through the network. You must list each flow-augmenting path you use, together with its flow.
    (5)
  4. Show your flow pattern on Figure 2.
    (2)
  5. Prove that your flow is maximal.
    (2)
Edexcel D2 Q6
14 marks Standard +0.3
6. The tableau below is the initial tableau for a maximising linear programming problem.
Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
\(r\)1624100350
\(s\)18- 26010480
\(t\)505001360
\(P\)- 18- 7- 200000
  1. Write down the four equations represented in the initial tableau.
  2. 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. State the row operations that you use.
  3. State whether or not your last tableau is optimal. Give a reason for your answer.
Edexcel D2 Q7
14 marks Challenging +1.8
7. D2 make industrial robots. They can make up to four in any one month, but if they make more than three they need to hire additional labour at a cost of \(\pounds 300\) per month. They can store up to three robots at a cost of \(\pounds 100\) per robot per month. The overhead costs are \(\pounds 500\) in any month in which work is done. The 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 at the end of May. The order book for January to May is:
MonthJanuaryFebruaryMarchAprilMay
Number of robots required32254
Use dynamic programming to determine the production schedule that minimises the costs, showing your working in the table provided in the answer book. State the minimum cost.
(Total 14 marks)
Edexcel D2 Specimen Q1
5 marks Easy -1.2
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{899a26d1-7599-4051-b1cf-596542624997-2_730_1534_285_264} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a directed, capacitated network where the number on each arc is its capacity. A possible flow is shown from \(S\) to \(T\) and the value in brackets on each arc is the flow in that arc.
  1. Find the values of \(x , y\), and \(z\).
    (3)
  2. Find, by inspection, the maximal flow from \(S\) to \(T\) and verify that it is maximal.
    (2)
Edexcel D2 Specimen Q2
7 marks Moderate -0.8
2. 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 initial tableau was obtained.
Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)Value
\(r\)2041080
\(s\)14201160
\(P\)- 2- 8- 20000
  1. Taking the most negative number in the profit row to indicate the pivot column, perform one complete iteration of the simplex algorithm, to obtain tableau \(T\). State the row operations that you use.
  2. Write down the profit equation shown in tableau \(T\).
  3. State whether tableau \(T\) is optimal. Give a reason for your answer.
Edexcel D2 Specimen Q3
11 marks Moderate -0.5
3. Freezy Co. has three factories \(A , B\) and \(C\). It supplies freezers to three shops \(D , E\) and \(F\). The table shows the transportation cost in pounds of moving one freezer from each factory to each outlet. It also shows the number of freezers available for delivery at each factory and the number of freezers required at each shop. The total number of freezers required is equal to the total number of freezers available.
\cline { 2 - 5 } \multicolumn{1}{c|}{}\(D\)\(E\)\(F\)Available
\(A\)21241624
\(B\)18231732
\(C\)15192514
Required203020
\cline { 1 - 4 }
\cline { 1 - 4 }
  1. Use the north-west corner rule to find an initial solution.
  2. Obtain improvement indices for each unused route.
  3. Use the stepping-stone method once to obtain a better solution and state its cost.
Edexcel D2 Specimen Q4
11 marks Moderate -0.5
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{899a26d1-7599-4051-b1cf-596542624997-5_602_1255_196_406} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The network in Figure 2 shows the distances, in km , of the cables between seven electricity relay stations \(A , B , C , D , E , F\) and \(G\). An inspector needs to visit each relay station. He wishes to travel a minimum distance, and his route must start and finish at the same station. By deleting C, a lower bound for the length of the route is found to be 129 km .
  1. Find another lower bound for the length of the route by deleting \(F\). State which is the best lower bound of the two.
  2. By inspection, complete the table of least distances. The table can now be taken to represent a complete network.
  3. Using the nearest-neighbour algorithm, starting at \(F\), obtain an upper bound to the length of the route. State your route.
Edexcel D2 Specimen Q5
6 marks Moderate -0.5
5. Three warehouses \(W , X\) and \(Y\) supply televisions to three supermarkets \(J , K\) and \(L\). The table gives the cost, in pounds, of transporting a television from each warehouse to each supermarket. The warehouses have stocks of 34,57 and 25 televisions respectively, and the supermarkets require 20, 56 and 40 televisions respectively. The total cost of transporting the televisions is to be minimised.
\(J\)\(K\)\(L\)
\(W\)363
\(X\)584
\(Y\)257
Formulate this transportation problem as a linear programming problem. Make clear your decision variables, objective function and constraints.
Edexcel D2 Specimen Q6
9 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{899a26d1-7599-4051-b1cf-596542624997-6_705_1424_1034_338} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A maximin route is to be found through the network shown in Figure 3.
Complete the table in the answer book, and hence find a maximin route.
Edexcel D2 Specimen Q7
15 marks Moderate -0.5
7. Four salespersons \(A , B , C\) and \(D\) are to be sent to visit four companies 1,2,3 and 4. Each salesperson will visit exactly one company, and all companies will be visited.
Previous sales figures show that each salesperson will make sales of different values, depending on the company that they visit. These values (in \(\pounds 10000\) s) are shown in the table below.
\cline { 2 - 5 } \multicolumn{1}{c|}{}1234
Ann26303030
Brenda30232629
Connor30252724
Dave30272521
  1. Use the Hungarian algorithm to obtain an allocation that maximises the sales. You must make your method clear and show the table after each stage.
  2. State the value of the maximum sales.
  3. Show that there is a second allocation that maximises the sales.
Edexcel D2 Specimen Q8
11 marks Standard +0.8
8. A two person zero-sum game is represented by the following pay-off matrix for player \(A\).
IIIIII
I523
II354
  1. Verify that there is no stable solution to this game.
  2. Find the best strategy for player \(A\) and the value of the game to her.
    (Total 11 marks)
Edexcel D2 Q1
5 marks Easy -1.8
  1. The payoff matrix for player \(A\) in a two-person zero-sum game is shown below.
\cline { 3 - 5 } \multicolumn{2}{c|}{}\(B\)
\cline { 3 - 5 } \multicolumn{2}{c|}{}IIIIII
\multirow{3}{*}{\(A\)}I- 340
\cline { 2 - 5 }II221
\cline { 2 - 5 }III3- 2- 1
Find the optimal strategy for each player and the value of the game.
Edexcel D2 Q2
6 marks Moderate -0.5
2. A supplier has three warehouses, \(A , B\) and \(C\), at which there are 42,26 and 32 crates of a particular cereal respectively. Three supermarkets, \(D , E\) and \(F\), require 29, 47 and 24 crates of the cereal respectively. The supplier wishes to minimise the cost in meeting the requirements of the supermarkets. The cost, in pounds, of supplying one crate of the cereal from each warehouse to each supermarket is given in the table below.
\cline { 2 - 4 } \multicolumn{1}{c|}{}\(D\)\(E\)\(F\)
\(A\)192213
\(B\)181426
\(C\)271619
Formulate this information as a linear programming problem.
  1. State your decision variables.
  2. Write down the objective function in terms of your decision variables.
  3. Write down the constraints, explaining what each one represents.
Edexcel D2 Q3
9 marks Easy -1.2
3. This question should be answered on the sheet provided. A couple are making the arrangements for their wedding. They are deciding whether to have the ceremony at their church, a local castle or a nearby registry office. The reception will then be held in a marquee, at the castle or at a local hotel. Both the castle and hotel offer catering services but the couple are also considering using Deluxe Catering or Cuisine, who can both provide the food at any venue. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f662b4da-12c1-4f30-ab5d-fb132f19e643-3_944_1504_605_258} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} The network in Figure 1 shows the costs incurred (including transport), in hundreds of pounds, according to the choice the couple make for each stage of the day. Use dynamic programming to find how the couple can minimise the total cost of their wedding and state the total cost of this arrangement.
(9 marks)
Edexcel D2 Q4
11 marks Moderate -0.3
4. This question should be answered on the sheet provided. A travelling salesman problem relates to the network represented by the following table of distances in kilometres. You may assume that the network satisfies the triangle inequality.
AB\(C\)D\(E\)\(F\)G\(H\)
A-85593147527441
B85-1047351684355
C59104-5462886145
D317354-40596578
E47516240-567168
\(F\)5268885956-5349
G744361657153-63
H41554578684963-
Showing your method clearly, use
  1. the nearest neighbour algorithm, beginning with \(A\),
  2. Prim's algorithm with \(H\) deleted,
    to show that the minimum distance travelled, \(d \mathrm {~km}\), satisfies the inequality \(357 \leq d \leq 371\).
    (11 marks)
Edexcel D2 Q5
13 marks Moderate -1.0
5. The payoff matrix for player \(X\) in a two-person zero-sum game is shown below.
\cline { 3 - 5 } \multicolumn{2}{c|}{}\(Y\)
\cline { 2 - 5 } \multicolumn{2}{c|}{}\(Y _ { 1 }\)\(Y _ { 2 }\)\(Y _ { 3 }\)
\multirow{2}{*}{\(X\)}\(X _ { 1 }\)1043
\cline { 2 - 5 }\(X _ { 2 }\)\({ } ^ { - } 4\)\({ } ^ { - } 1\)9
  1. Using a graphical method, find the optimal strategy for player \(X\).
  2. Find the optimal strategy for player \(Y\).
  3. Find the value of the game.
Edexcel D2 Q6
13 marks Moderate -0.3
6. Four sales representatives ( \(R _ { 1 } , R _ { 2 } , R _ { 3 }\) and \(R _ { 4 }\) ) are to be sent to four areas ( \(A _ { 1 } , A _ { 2 } , A _ { 3 }\) and \(A _ { 4 }\) ) such that each representative visits one area. The estimated profit, in tens of pounds, that each representative will make in each area is shown in the table below.
\cline { 2 - 5 } \multicolumn{1}{c|}{}\(A _ { 1 }\)\(A _ { 2 }\)\(A _ { 3 }\)\(A _ { 4 }\)
\(R _ { 1 }\)37294451
\(R _ { 2 }\)45304341
\(R _ { 3 }\)32273950
\(R _ { 4 }\)43255155
Use the Hungarian method to obtain an allocation which will maximise the total profit made from the visits. Show the state of the table after each stage in the algorithm.
(13 marks)
Edexcel D2 Q7
18 marks Moderate -0.5
7. A distributor has six warehouses. At one point the distributor needs to move 25 lorries from warehouses \(W _ { 1 } , W _ { 2 }\) and \(W _ { 3 }\) to warehouses \(W _ { \mathrm { A } } , W _ { \mathrm { B } }\) and \(W _ { \mathrm { C } }\) for the minimum possible cost. The transportation tableau below shows the unit cost, in tens of pounds, of moving a lorry between two warehouses, and the relevant figures regarding the number of lorries available or required at each warehouse.
\(W _ { \text {A } }\)\(W _ { \mathrm { B } }\)\(W _ { \mathrm { C } }\)Available
\(W _ { 1 }\)781010
\(W _ { 2 }\)9658
\(W _ { 3 }\)11577
Required5128
  1. Write down the initial solution given by the north-west corner rule.
  2. Obtain improvement indices for the unused routes.
  3. Use the stepping-stone method to find an improved solution and state why it is degenerate.
  4. Placing a zero in cell \(( 2,2 )\), show that the improved solution is optimal and state the transportation pattern.
  5. Find the total cost of the optimal solution. \section*{Please hand this sheet in for marking}
    StageStateDestinationCostTotal cost
    \multirow[t]{3}{*}{1}MarqueeDeluxe Cuisine
    CastleDeluxe Castle Cuisine
    HotelDeluxe Cuisine Hotel
    \multirow[t]{3}{*}{2}ChurchMarquee Castle Hotel
    CastleMarquee Castle
    Registry OfficeMarquee Castle Hotel
    3HomeCastle Church Registry
    \section*{Please hand this sheet in for marking}
    1. AB\(C\)D\(E\)\(F\)\(G\)\(H\)
      A-85593147527441
      B85-1047351684355
      C59104-5462886145
      D317354-40596578
      E47516240-567168
      \(F\)5268885956-5349
      \(G\)744361657153-63
      \(H\)41554578684963-
    2. A\(B\)\(C\)D\(E\)\(F\)\(G\)\(H\)
      A-85593147527441
      B85-1047351684355
      C59104-5462886145
      D317354-40596578
      E47516240-567168
      \(F\)5268885956-5349
      G744361657153-63
      \(H\)41554578684963-
Edexcel D2 Q1
6 marks Moderate -0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4e50371b-0c1c-4b4e-b21d-60858ae160df-2_659_986_203_479} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} The network in Figure 1 shows the shortest distance by road, in kilometres, between five villages. Find the best achievable upper bound for a tour of the network, of minimum length, using the nearest neighbour algorithm.
Edexcel D2 Q2
7 marks Easy -1.2
2. A school entrance examination consists of three papers - Mathematics, English and Verbal Reasoning. Three teams of markers are to mark one style of paper each. The table below shows the average time, in minutes, taken by each team to mark one script for each style of paper.
\cline { 2 - 4 } \multicolumn{1}{c|}{}MathsEnglishVerbal
Team 1392
Team 2471
Team 3583
It is desired that the scripts are marked as quickly as possible.
Formulate this information as a linear programming problem.
  1. State your decision variables.
  2. Write down the objective function in terms of your decision variables.
  3. Write down the constraints, explaining what each one represents.