Questions - Edexcel D2 - A-Level Maths

Basic variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	Value
\(r\)	2	0	4	1	0	80
\(s\)	1	4	2	0	1	160
\(P\)	- 2	- 8	- 20	0	0	0

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.
Write down the profit equation shown in tableau \(T\).
State whether tableau \(T\) is optimal. Give a reason for your answer.

Edexcel D2 Specimen Q3

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
\(A\)	21	24	16	24
\(B\)	18	23	17	32
\(C\)	15	19	25	14
Required	20	30	20
\cline { 1 - 4 }
\cline { 1 - 4 }

Use the north-west corner rule to find an initial solution.
Obtain improvement indices for each unused route.
Use the stepping-stone method once to obtain a better solution and state its cost.

Edexcel D2 Specimen Q4

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 .

Find another lower bound for the length of the route by deleting \(F\). State which is the best lower bound of the two.
By inspection, complete the table of least distances. The table can now be taken to represent a complete network.
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

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.

	\(J\)	\(K\)	\(L\)
\(W\)	3	6	3
\(X\)	5	8	4
\(Y\)	2	5	7

Formulate this transportation problem as a linear programming problem. Make clear your decision variables, objective function and constraints.

Edexcel D2 Specimen Q6

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

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\|}{}	1	2	3	4
Ann	26	30	30	30
Brenda	30	23	26	29
Connor	30	25	27	24
Dave	30	27	25	21

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.
State the value of the maximum sales.
Show that there is a second allocation that maximises the sales.

Edexcel D2 Specimen Q8

8. A two person zero-sum game is represented by the following pay-off matrix for player \(A\).

	I	II	III
I	5	2	3
II	3	5	4

Verify that there is no stable solution to this game.
Find the best strategy for player \(A\) and the value of the game to her.
(Total 11 marks)

Edexcel D2 Q1

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\|}{}	I	II	III
\multirow{3}{*}{\(A\)}	I	- 3	4	0
\cline { 2 - 5 }	II	2	2	1
\cline { 2 - 5 }	III	3	- 2	- 1

Find the optimal strategy for each player and the value of the game.

Edexcel D2 Q2

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\|}{}	\(D\)	\(E\)	\(F\)
\(A\)	19	22	13
\(B\)	18	14	26
\(C\)	27	16	19

Formulate this information as a linear programming problem.

State your decision variables.
Write down the objective function in terms of your decision variables.
Write down the constraints, explaining what each one represents.

Edexcel D2 Q3

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

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.

	A	B	\(C\)	D	\(E\)	\(F\)	G	\(H\)
A	-	85	59	31	47	52	74	41
B	85	-	104	73	51	68	43	55
C	59	104	-	54	62	88	61	45
D	31	73	54	-	40	59	65	78
E	47	51	62	40	-	56	71	68
\(F\)	52	68	88	59	56	-	53	49
G	74	43	61	65	71	53	-	63
H	41	55	45	78	68	49	63	-

Showing your method clearly, use

the nearest neighbour algorithm, beginning with \(A\),
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

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}{*}{\(X\)}	\(X _ { 1 }\)	10	4	3
\cline { 2 - 5 }	\(X _ { 2 }\)	\({ } ^ { - } 4\)	\({ } ^ { - } 1\)	9

Using a graphical method, find the optimal strategy for player \(X\).
Find the optimal strategy for player \(Y\).
Find the value of the game.

Edexcel D2 Q6

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\|}{}	\(A _ { 1 }\)	\(A _ { 2 }\)	\(A _ { 3 }\)	\(A _ { 4 }\)
\(R _ { 1 }\)	37	29	44	51
\(R _ { 2 }\)	45	30	43	41
\(R _ { 3 }\)	32	27	39	50
\(R _ { 4 }\)	43	25	51	55

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

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
\(W _ { 1 }\)	7	8	10	10
\(W _ { 2 }\)	9	6	5	8
\(W _ { 3 }\)	11	5	7	7
Required	5	12	8

Write down the initial solution given by the north-west corner rule.
Obtain improvement indices for the unused routes.
Use the stepping-stone method to find an improved solution and state why it is degenerate.
Placing a zero in cell \(( 2,2 )\), show that the improved solution is optimal and state the transportation pattern.

Find the total cost of the optimal solution. \section*{Please hand this sheet in for marking}

Stage	State	Destination	Cost	Total cost
\multirow[t]{3}{*}{1}	Marquee	Deluxe Cuisine
	Castle	Deluxe Castle Cuisine
	Hotel	Deluxe Cuisine Hotel
\multirow[t]{3}{*}{2}	Church	Marquee Castle Hotel
	Castle	Marquee Castle
	Registry Office	Marquee Castle Hotel
3	Home	Castle Church Registry

\section*{Please hand this sheet in for marking}

A B \(C\) D \(E\) \(F\) \(G\) \(H\)
A - 85 59 31 47 52 74 41
B 85 - 104 73 51 68 43 55
C 59 104 - 54 62 88 61 45
D 31 73 54 - 40 59 65 78
E 47 51 62 40 - 56 71 68
\(F\) 52 68 88 59 56 - 53 49
\(G\) 74 43 61 65 71 53 - 63
\(H\) 41 55 45 78 68 49 63 -
A \(B\) \(C\) D \(E\) \(F\) \(G\) \(H\)
A - 85 59 31 47 52 74 41
B 85 - 104 73 51 68 43 55
C 59 104 - 54 62 88 61 45
D 31 73 54 - 40 59 65 78
E 47 51 62 40 - 56 71 68
\(F\) 52 68 88 59 56 - 53 49
G 74 43 61 65 71 53 - 63
\(H\) 41 55 45 78 68 49 63 -

Edexcel D2 Q1

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

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\|}{}	Maths	English	Verbal
Team 1	3	9	2
Team 2	4	7	1
Team 3	5	8	3

It is desired that the scripts are marked as quickly as possible.
Formulate this information as a linear programming problem.

State your decision variables.
Write down the objective function in terms of your decision variables.
Write down the constraints, explaining what each one represents.

Edexcel D2 Q3

3. A two-person zero-sum game is represented by the payoff matrix for player \(A\) shown below.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	\(B\)
\cline { 3 - 5 } \multicolumn{2}{c\|}{}	I	II	III
\multirow{2}{*}{\(A\)}	I	1	- 1	2
\cline { 2 - 5 }	II	3	5	- 1

Represent the expected payoffs to \(A\) against \(B\) 's strategies graphically and hence determine which strategy is not worth considering for player \(B\).
Find the best strategy for player \(A\) and the value of the game.

Edexcel D2 Q4

4. This question should be answered on the sheet provided. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4e50371b-0c1c-4b4e-b21d-60858ae160df-3_771_1479_1178_237} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} A salesman is planning a four-day trip beginning at home and ending at town \(I\). He will spend the first night in town \(A , B\) or \(C\), the second night in town \(D , E\) or \(F\) and the third night in town \(G\) or \(H\). The network in Figure 2 shows the expected net profit, in tens of pounds, that he will gain on each day according to the route he chooses. Use dynamic programming to find the route which should maximise the salesman’s net profit. State the expected profit from using this route.
(10 marks)

Edexcel D2 Q5

5. A construction company has three teams of workers available, each of which is to be assigned to one of four jobs at a site. The following table shows the estimated cost, in tens of pounds, of each team doing each job:

	Windows	Conservatory	Doors	Greenhouse
Team A	27	80	8	81
Team B	28	60	5	71
Team C	30	90	7	73

Use the Hungarian algorithm to find an allocation of jobs which will minimise the total cost. Show the state of the table after each stage in the algorithm and state the cost of the final assignment.
(13 marks)

Edexcel D2 Q6

6. This question should be answered on the sheet provided. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4e50371b-0c1c-4b4e-b21d-60858ae160df-5_664_1029_335_440} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} The network in Figure 3 shows the distances, in miles, between a newspaper distributor based at area \(A\), and five areas, \(B , C , D , E\), and \(F\), to which the distributor must deliver newspapers. Each morning a delivery van has to set out from \(A\) and visit each of these areas before again returning to \(A\), and the driver wishes to keep the total mileage to a minimum.

Draw a complete network showing the shortest distances between the six areas.
(3 marks)
Obtain a minimum spanning tree for the complete network and hence find an upper bound for the length of the driver's route.
(5 marks)
Improve this upper bound to find an upper bound of less than 55 miles.
By deleting \(A\), find a lower bound for the total length of the route.

Edexcel D2 Q7

7. Mrs. Hartley organises the tennis fixtures for her school. On one day she has to send a team of 10 players to a match against school \(A\) and a team of 6 players to a match against school \(B\). She has to select the two teams from a squad that includes 7 players who live in village \(C\), 5 players who live in village \(D\) and 8 players who live in village \(E\). Having a small budget, Mrs. Hartley wishes to minimise the total amount spent on travel. The table below shows the cost, in pounds, for one player to travel from each village to each of the schools they are competing against.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	\(A\)	\(B\)
\(C\)	2	3
\(D\)	2	5
\(E\)	7	6

Use the north-west corner rule to find an initial solution to this problem.
Obtain improvement indices for this initial solution.
Use the stepping-stone method to obtain an optimal solution and state the pattern of transportation that this represents. \section*{Please hand this sheet in for marking}
Stage State Action
\multirow[t]{2}{*}{1} G GI
H HI
\multirow[t]{3}{*}{2} D
DG
DH
E
EG
\(E H\)
F
FG
FH
\multirow[t]{3}{*}{3} A
AD
\(A E\)
\(A F\)
B
BD
BE
\(B F\)
C
CD
CE
CF
4 Home
Home-A
Home-B
Home-C
\section*{Please hand this sheet in for marking}
\includegraphics[max width=\textwidth, alt={}, center]{4e50371b-0c1c-4b4e-b21d-60858ae160df-8_662_1025_529_440}
Sheet for answering question 6 (cont.)

Edexcel D2 Q1

This question should be answered on the sheet provided.

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e892e87c-1c2d-4f97-ac23-41e38663d0f0-02_485_995_285_477} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} The network in Figure 1 shows the distances, in miles, between the five villages in which Sarah is planning to enquire about holiday work, with village \(A\) being Sarah's home village.

Illustrate this situation as a complete network showing the shortest distances.
(2 marks)
Use the nearest neighbour algorithm, starting with \(A\), to find an upper bound to the length of a tour beginning and ending at \(A\).
(2 marks)
Interpret the tour found in part (b) in terms of the original network.
(2 marks)

Edexcel D2 Q2

2. The payoff matrix for player \(A\) in a two-person zero-sum game with value \(V\) is shown below.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	\(B\)
\cline { 2 - 5 } \multicolumn{2}{c\|}{}	I	II	III
\multirow{3}{*}{\(A\)}	I	- 1	4	- 3
\cline { 2 - 5 }	II	- 3	7	1
\cline { 2 - 5 }	III	5	- 2	- 1

Formulate this information as a linear programming problem, the solution to which will give the optimal strategy for player \(B\).

Rewrite the matrix as necessary and state the new value of the game, \(v\), in terms of \(V\).
Define your decision variables.
Write down the objective function in terms of your decision variables.
Write down the constraints.

Edexcel D2 Q3

3. This question should be answered on the sheet provided. Arthur is planning a bus journey from town \(A\) to town \(L\). There are various routes he can take but he will have to change buses three times - at \(B , C\) or \(D\), at \(E , F , G\) or \(H\) and at \(I , J\) or \(K\). \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e892e87c-1c2d-4f97-ac23-41e38663d0f0-03_764_1410_477_315} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 shows the bus routes that Arthur can use. The number on each arc shows the average waiting time, in minutes, for a bus to come on that route. As the forecast is for rain, Arthur wishes to plan his journey so that the maximum waiting time at any one stop is as small as possible. Use dynamic programming to find the route that Arthur should use.
(9 marks)

Edexcel D2 Q4

4. A furniture manufacturer has three workshops, \(W _ { 1 } , W _ { 2 }\) and \(W _ { 3 }\). Orders for rolls of fabric are to be placed with three suppliers, \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\). The supply, demand and cost per roll in pounds, according to which supplier each workshop uses, are given in the table below.

	\(W _ { 1 }\)	\(W _ { 2 }\)	\(W _ { 3 }\)	Available
\(S _ { 1 }\)	12	11	17	30
\(S _ { 2 }\)	7	5	10	25
\(S _ { 3 }\)	5	6	8	10
Required	20	15	30

Starting with the north-west corner method of finding an initial solution, find an optimal transportation pattern which minimises the total cost. State the final solution and its total cost.
(11 marks)

Questions — Edexcel D2 (231 questions)

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Edexcel D2 Specimen Q2

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Edexcel D2 Q4