Questions - Edexcel D2 - A-Level Maths

Add a supersource and a supersink, and arcs of appropriate capacity, to Diagram 1 below. \section*{Diagram 1} \includegraphics[max width=\textwidth, alt={}, center]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-4_684_1531_1816_267}
Find the values of \(x\) and \(y\), explaining your method briefly.
Find the value of cuts \(C _ { 1 }\) and \(C _ { 2 }\). Starting with the given feasible flow of 68,
use the labelling procedure on Diagram 2 to find a maximal flow through this network. List each flow-augmenting route you use, together with its flow. \section*{Diagram 2} \includegraphics[max width=\textwidth, alt={}, center]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-5_647_1506_612_283}
Show your maximal flow on Diagram 3 and state its value. \section*{Diagram 3} \includegraphics[max width=\textwidth, alt={}, center]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-5_654_1511_1567_278}
Prove that your flow is maximal.

Edexcel D2 2003 June Q8

8. The tableau below is the initial tableau for a maximising linear programming problem.

Basic

For this problem \(x \geq 0 , y \geq 0 , z \geq 0\). Write down the other two inequalities and the objective function.
Solve this linear programming problem. You may not need to use all of these tableaux.
b.v. \(x\) \(y\) \(z\) \(r\) \(s\) Value
\(P\)

b.v. \(x\) \(y\) \(z\) \(r\) \(s\) Value
\(P\)

b.v. \(x\) \(y\) \(z\) \(r\) \(s\) Value
\(P\)

b.v. \(x\) \(y\) \(z\) \(r\) \(s\) Value
\(P\)
State the final value of \(P\), the objective function, and of each of the variables.

Edexcel D2 2004 June Q1

In game theory explain what is meant by
1. zero-sum game,
2. saddle point.
  (Total 4 marks)
3. In a quiz there are four individual rounds, Art, Literature, Music and Science. A team consists of four people, Donna, Hannah, Kerwin and Thomas. Each of four rounds must be answered by a different team member. The table shows the number of points that each team member is likely to get on each individual round.
\cline { 2 - 5 } \multicolumn{1}{c|}{} Art Literature Music Science
Donna 31 24 32 35
Hannah 16 10 19 22
Kerwin 19 14 20 21
Thomas 18 15 21 23
Use the Hungarian algorithm, reducing rows first, to obtain an allocation which maximises the total points likely to be scored in the four rounds. You must make your method clear and show the table after each stage.
(Total 9 marks)

Edexcel D2 2004 June Q3

3. The table shows the least distances, in km, between five towns, \(A , B , C , D\) and \(E\). Nassim wishes to find an interval which contains the solution to the travelling salesman problem for this network.

Making your method clear, find an initial upper bound starting at \(A\) and using
1. the minimum spanning tree method,
2. the nearest neighbour algorithm.
  \(A\) \(B\) \(C\) \(D\) \(E\)
  \(A\) - 153 98 124 115
  \(B\) 153 - 74 131 149
  \(C\) 98 74 - 82 103
  \(D\) 124 131 82 - 134
  \(E\) 115 149 103 134 -
By deleting \(E\), find a lower bound.
Using your answers to parts (a) and (b), state the smallest interval that Nassim could correctly write down.
(Total 12 marks)

Edexcel D2 2004 June Q4

4. Emma and Freddie play a zero-sum game. This game is represented by the following pay-off matrix for Emma. \(\left( \begin{array} { r r r } - 4 & - 1 & 3
2 & 1 & - 2 \end{array} \right)\)

Show that there is no stable solution.
Find the best strategy for Emma and the value of the game to her.
Write down the value of the game to Freddie and his pay-off matrix.

Edexcel D2 2004 June Q5

5. (a) Describe a practical problem that could be solved using the transportation algorithm. A problem is to be solved using the transportation problem. The costs are shown in the table. The supply is from \(A , B\) and \(C\) and the demand is at \(d\) and \(e\).

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	\(d\)	\(e\)	Supply
\(A\)	5	3	45
\(B\)	4	6	35
\(C\)	2	4	40
Demand	50	60

(b) Explain why it is necessary to add a third demand \(f\).
(c) Use the north-west corner rule to obtain a possible pattern of distribution and find its cost.

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	\(d\)	\(e\)	\(f\)	Supply
\(A\)	5	3		45
\(B\)	4	6		35
\(C\)	2	4		40
Demand	50	60

(d) Calculate shadow costs and improvement indices for this pattern.
(e) Use the stepping-stone method once to obtain an improved solution and its cost.
(Total 16 marks)

Edexcel D2 2004 June Q6

6. Joan sells ice cream. She needs to decide which three shows to visit over a three-week period in the summer. She starts the three-week period at home and finishes at home. She will spend one week at each of the three shows she chooses travelling directly from one show to the next. Table 1 gives the week in which each show is held. Table 2 gives the expected profits from visiting each show. Table 3 gives the cost of travel between shows. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Table 1}

Week	1	2	3
Shows	A, B, C	D, E	F, G, H

\end{table} Table 2 \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Table 1}

Show	A	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
Expected Profit \(( \pounds )\)	900	800	1000	1500	1300	500	700	600

\end{table} \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Table 3}

Travel costs (£)	A	B	C	D	E	F	G	H
Home	70	80	150			80	90	70
A				180	150
B				140	120
C				200	210
D						200	160	120
E						170	100	110

\end{table} It is decided to use dynamic programming to find a schedule that maximises the total expected profit, taking into account the travel costs.

Define suitable stage, state and action variables.
Determine the schedule that maximises the total profit. Show your working in a table.
Advise Joan on the shows that she should visit and state her total expected profit.
(3) (Total 18 marks)

Edexcel D2 2004 June Q7

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{343fdcef-660e-4e8c-bd9c-a7f929dc668e-3_526_903_1455_575}

\end{figure} Figure 1 shows a capacitated directed network. The number on each arc is its capacity. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{343fdcef-660e-4e8c-bd9c-a7f929dc668e-3_499_967_2231_548}

\end{figure} Figure 2 shows a feasible initial flow through the same network.

Write down the values of the flow \(x\) and the flow \(y\).
Obtain the value of the initial flow through the network, and explain how you know it is not maximal.
Use this initial flow and the labelling procedure on Diagram 1 below to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow. \section*{Diagram 1} \includegraphics[max width=\textwidth, alt={}, center]{343fdcef-660e-4e8c-bd9c-a7f929dc668e-4_1920_1175_742_450}
d) Show your maximal flow pattern on Diagram 2. \section*{Diagram 2} \includegraphics[max width=\textwidth, alt={}, center]{343fdcef-660e-4e8c-bd9c-a7f929dc668e-5_679_1102_315_479}
(2)
Prove that your flow is maximal.
(2)
(Total 13 marks)

Edexcel D2 2004 June Q8

8. 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 was obtained.

Basic variable	\(x\)	\(y\)	\(Z\)	\(r\)	\(s\)	\(t\)	Value
S	3	0	2	0	1	\(- \frac { 2 } { 3 }\)	\(\frac { 2 } { 3 }\)
\(r\)	4	0	\(\frac { 7 } { 2 }\)	1	0	8	\(\frac { 9 } { 2 }\)
\(y\)	5	1	7	0	0	3	7
P	3	0	2	0	0	8	63

State, giving your reason, whether this tableau represents the optimal solution.
State the values of every variable.
Calculate the profit made on each unit of \(y\).

Edexcel D2 2004 June Q9

9.
\includegraphics[max width=\textwidth, alt={}, center]{343fdcef-660e-4e8c-bd9c-a7f929dc668e-6_1088_1509_219_285} The diagram above shows a network of roads represented by arcs. The capacity of the road represented by that arc is shown on each arc. The numbers in circles represent a possible flow of 26 from \(B\) to \(L\). Three cuts \(\mathrm { C } _ { 1 } , \mathrm { C } _ { 2 }\) and \(\mathrm { C } _ { 3 }\) are shown on The diagram above.

Find the capacity of each of the three cuts.
Verify that the flow of 26 is maximal. The government aims to maximise the possible flow from \(B\) to \(L\) by using one of two options.
Option 1: Build a new road from \(E\) to \(J\) with capacity 5.
or
Option 2: Build a new road from \(F\) to \(H\) with capacity 3.
By considering both options, explain which one meets the government's aim

Edexcel D2 2004 June Q10

10. Flatland UK Ltd makes three types of carpet, the Lincoln, the Norfolk and the Suffolk. The carpets all require units of black, green and red wool. For each roll of carpet, the Lincoln requires 1 unit of black, 1 of green and 3 of red, the Norfolk requires 1 unit of black, 2 of green and 2 of red, and the Suffolk requires 2 units of black, 1 of green and 1 of red. There are up to 30 units of black, 40 units of green and 50 units of red available each day.
Profits of \(\pounds 50 , \pounds 80\) and \(\pounds 60\) are made on each roll of Lincoln, Norfolk and Suffolk respectively. Flatland UK Ltd wishes to maximise its profit. Let the number of rolls of the Lincoln, Norfolk and Suffolk made daily be \(x , y\) and \(z\) respectively.

Formulate the above situation as a linear programming problem, listing clearly the constraints as inequalities in their simplest form, and stating the objective function. This problem is to be solved using the Simplex algorithm. The most negative number in the profit row is taken to indicate the pivot column at each stage.

Stating your row operations, show that after one complete iteration the tableau becomes

Basic variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value
\(r\)	\(\frac { 1 } { 2 }\)	0	\(1 \frac { 1 } { 2 }\)	1	\(- \frac { 1 } { 2 }\)	0	10
\(y\)	\(\frac { 1 } { 2 }\)	1	\(\frac { 1 } { 2 }\)	0	\(\frac { 1 } { 2 }\)	0	20
\(t\)	2	0	0	0	-1	1	10
\(P\)	-10	0	-20	0	40	0	1600

You may not need to use all of the tableaux.

Basic variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value	Row operations
\(r\)
\(S\)
\(t\)
\(P\)

Basic variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value	Row operations

Explain the practical meaning of the value 10 in the top row.
1. Perform one further complete iteration of the Simplex algorithm.
  Basic variable \(x\) \(y\) \(z\) \(r\) \(s\) \(t\) Value Row operations
  
  Basic variable \(x\) \(y\) \(z\) \(r\) \(s\) \(t\) Value Row operations
2. State whether your current answer to part (d)(i) is optimal. Give a reason for your answer.
3. Interpret your current tableau, giving the value of each variable.
  (8)

Edexcel D2 2005 June Q1

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 2005 June Q2

2.
\includegraphics[max width=\textwidth, alt={}, center]{be329a47-a709-4719-abe6-41d388a6c631-1_613_1269_1318_392} The network in the diagram 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 better 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.
(4) (Total 11 marks)

Edexcel D2 2005 June Q3

3. 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.
(Total 7 marks)

Edexcel D2 2005 June Q4

4. (a) Explain what is meant by a maximin route in dynamic programming, and give an example of a situation that would require a maximin solution.
(3)
\includegraphics[max width=\textwidth, alt={}, center]{be329a47-a709-4719-abe6-41d388a6c631-2_700_1392_1069_338} A maximin route is to be found through the network shown in the diagram.
(b) Complete the table in the answer book, and hence find a maximin route.
(9)
(c) List all other maximin routes through the network.
(Total 14 marks)

Edexcel D2 2005 June Q5

5. Four salesperson \(A , B , C\) 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.
(Total 15 marks)

Edexcel D2 2005 June Q6

6.
\includegraphics[max width=\textwidth, alt={}, center]{be329a47-a709-4719-abe6-41d388a6c631-3_696_1319_1292_374} This figure shows a capacitated directed network. The number on each arc is its capacity. The numbers in circles show a feasible flow through the network. Take this as the initial flow.

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. Diagram 2 in the answer book shows the first stage of the labelling procedure for the given initial flow.
Complete the initial labelling procedure in Diagram 2.
Find the maximum flow through the network. You must list each flow-augmenting route you use together with its flow, and state the maximal flow.
Show a maximal flow pattern on Diagram 3.
Prove that your flow is maximal.
Describe briefly a situation for which this network could be a suitable model.
(Total 16 marks)
\includegraphics[max width=\textwidth, alt={}, center]{be329a47-a709-4719-abe6-41d388a6c631-4_1486_1963_568_50}

Edexcel D2 2005 June Q7

7. (a) Explain briefly what is meant by a zero-sum game. 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

(b) Verify that there is no stable solution to this game.
(c) Find the best strategy for player \(A\) and the value of the game to her.
(d) Formulate the game as a linear programming problem for player \(B\). Write the constraints as inequalities and define your variables clearly.
(Total 17 marks)

Edexcel D2 2005 June Q8

8. Polly has a bird food stall at the local market. Each week she makes and sells three types of packs \(A , B\) and C. Pack \(A\) contains 4 kg of bird seed, 2 suet blocks and 1 kg of peanuts.
Pack \(B\) contains 5 kg of bird seed, 1 suet block and 2 kg of peanuts.
Pack \(C\) contains 10 kg of bird seed, 4 suet blocks and 3 kg of peanuts.
Each week Polly has 140 kg of bird seed, 60 suet blocks and 60 kg of peanuts available for the packs.
The profit made on each pack of \(A , B\) and \(C\) sold is \(\pounds 3.50 , \pounds 3.50\) and \(\pounds 6.50\) respectively. Polly sells every pack on her stall and wishes to maximise her profit, \(P\) pence. Let \(x , y\) and \(z\) be the numbers of packs \(A , B\) and \(C\) sold each week.
An initial Simplex tableau for the above situation is

Basic variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value
\(r\)	4	5	1	1	0	0	140
\(s\)	2	1	4	0	1	0	60
\(t\)	1	2	3	0	0	1	60
\(P\)	- 350	- 350	- 650	0	0	0	0

Explain the meaning of the variables \(r , s\) and \(t\) in the context of this question.
Perform one complete iteration of the Simplex algorithm to form a new tableau \(T\). Take the most negative number in the profit row to indicate the pivotal column.
State the value of every variable as given by tableau \(T\).
Write down the profit equation given by tableau \(T\).
Use your profit equation to explain why tableau \(T\) is not optimal. Taking the most negative number in the profit row to indicate the pivotal column,
identify clearly the location of the next pivotal element.

Edexcel D2 2005 June Q9

9.
\includegraphics[max width=\textwidth, alt={}, center]{be329a47-a709-4719-abe6-41d388a6c631-6_540_1291_203_411} This diagram shows a capacitated directed network. The number on each arc is its capacity.

State the maximum flow along
1. SADT,
2. SCET,
3. \(S B F T\).
Show these maximum flows on Diagram 1 below. \section*{Diagram 1}
\includegraphics[max width=\textwidth, alt={}]{be329a47-a709-4719-abe6-41d388a6c631-6_561_1187_1283_721}
Take your answer to part (b) as the initial flow pattern.
1. Use the labelling procedure to find a maximum flow from \(S\) to \(T\). Your working should be shown on Diagram 2 below. List each flow-augmenting route you use, together with its flow. \section*{Diagram 2} \includegraphics[max width=\textwidth, alt={}, center]{be329a47-a709-4719-abe6-41d388a6c631-7_718_1525_205_269}
2. Draw your final flow pattern on Diagram 3 below.
  \includegraphics[max width=\textwidth, alt={}, center]{be329a47-a709-4719-abe6-41d388a6c631-7_611_1196_1082_717}
3. Prove that your flow is maximal.
Give an example of a practical situation that could have been modelled by the original network.

Edexcel D2 2006 June Q1

(a) State Bellman's principle of optimality.
(b) Explain what is meant by a minimax route.
(c) Describe a practical problem that would require a minimax route as its solution.
(Total 4 marks)
Three workers, \(P , Q\) and \(R\), are to be assigned to three tasks, 1,2 and 3 . Each worker is to be assigned to one task and each task must be assigned to one worker. The cost, in hundreds of pounds, of using each worker for each task is given in the table below. The cost is to be minimised.

Cost (in \(\pounds 100\) s)	Task 1	Task 2	Task 3
Worker \(P\)	8	7	3
Worker \(Q\)	9	5	6
Worker \(R\)	10	4	4

Formulate the above situation as a linear programming problem, defining the decision variables and making the objective and constraints clear.
(Total 7 marks)

Edexcel D2 2006 June Q3

3. A college wants to offer five full-day activities with a different activity each day from Monday to Friday. The sports hall will only be used for these activities. Each evening the caretaker will prepare the hall by putting away the equipment from the previous activity and setting up the hall for the activity next day. On Friday evening he will put away the equipment used that day and set up the hall for the following Monday. The 5 activities offered are Badminton (B), Cricket nets (C), Dancing (D), Football coaching (F) and Tennis (T). Each will be on the same day from week to week. The college decides to offer the activities in the order that minimises the total time the caretaker has to spend preparing the hall each week. The hall is initially set up for Badminton on Monday.
The table below shows the time, in minutes, it will take the caretaker to put away the equipment from one activity and set up the hall for the next.

\multirow{3}{*}{}	To
	Time	\(B\)	C	D	\(F\)	\(T\)
	\(B\)	-	108	150	64	100
\multirow[t]{4}{*}{From}	C	108	-	54	104	60
	D	150	54	-	150	102
	\(F\)	64	104	150	-	68
	\(T\)	100	60	102	68	-

Explain why this problem is equivalent to the travelling salesman problem. A possible ordering of activities is
Monday Tuesday Wednesday Thursday Friday
\(B\) \(C\) \(D\) \(F\) \(T\)
Find the total time taken by the caretaker each week using this ordering.
Starting with Badminton on Monday, use a suitable algorithm to find an ordering that reduces the total time spent each week to less than 7 hours.
By deleting \(B\), use a suitable algorithm to find a lower bound for the time taken each week. Make your method clear.

Edexcel D2 2006 June Q4

4. During the school holidays four building tasks, rebuilding a wall ( \(W\) ), repairing the roof ( \(R\) ), repainting the hall \(( H )\) and relaying the playground \(( P )\), need to be carried out at a Junior School. Four builders, \(A , B , C\) and \(D\) will be hired for these tasks. Each builder must be assigned to one task. Builder \(B\) is not able to rebuild the wall and therefore cannot be assigned to this task. The cost, in thousands of pounds, of using each builder for each task is given in the table below.

Cost	\(H\)	\(P\)	\(R\)	\(W\)
\(A\)	3	5	11	9
\(B\)	3	7	8	-
\(C\)	2	5	10	7
\(D\)	8	3	7	6

Use the Hungarian algorithm, reducing rows first, to obtain an allocation that minimises the total cost. State the allocation and its total cost. You must make your method clear and show the table after each stage.
State, with a reason, whether this allocation is unique.

Edexcel D2 2006 June Q5

5. Victor owns some kiosks selling ice cream, hot dogs and soft drinks. The network below shows the choices of action and the profits, in thousands of pounds, they generate over the next four years. The negative numbers indicate losses due to the purchases of new kiosks.
\includegraphics[max width=\textwidth, alt={}, center]{83ddfbb1-d035-4fae-b39d-a643c877cfed-3_887_1342_340_365} Use a suitable algorithm to determine the sequence of actions so that the profit over the four years is maximised and state this maximum profit.
(Total 12 marks)

Edexcel D2 2006 June Q6

6. (a) Explain briefly the circumstances under which a degenerate feasible solution may occur to a transportation problem.
(b) Explain why a dummy location may be needed when solving a transportation problem. The table below shows the cost of transporting one unit of stock from each of three supply points \(A , B\) and \(C\) to each of two demand points 1 and 2 . It also shows the stock held at each supply point and the stock required at each demand point.

	1	2	Supply
\(A\)	62	47	15
\(B\)	61	48	12
\(C\)	68	58	17
Demand	16	11

	1	2	3
\(A\)
\(B\)			0
\(C\)

(d) Use the stepping-stone method to obtain an optimal solution and state its cost. You should make your method clear by stating shadow costs, improvement indices, stepping-stone route, and the entering and exiting squares at each stage.

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	Art	Literature	Music	Science
Donna	31	24	32	35
Hannah	16	10	19	22
Kerwin	19	14	20	21
Thomas	18	15	21	23

Monday	Tuesday	Wednesday	Thursday	Friday
\(B\)	\(C\)	\(D\)	\(F\)	\(T\)

	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
\(A\)	-	153	98	124	115
\(B\)	153	-	74	131	149
\(C\)	98	74	-	82	103
\(D\)	124	131	82	-	134
\(E\)	115	149	103	134	-

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