Pearson Edexcel GCE In AS Further Mathematics (8FM0) Paper 28
Decision Mathematics 2 QP & MS june 2024
, Paper
Afternoon
■ ■
Further Mathematics
Advanced Subsidiary
Further Mathematics options
28: Decision Mathematics 2
(Part of option K only)
Try to answer every questio
Turn over
P75678A
©2024 Pearson Education Ltd.
F:1/1/1/
,1.
C C
1
2
E
A 26
24 24
0 H
24
21 15
42 15 19
39 B 18 18
16 48
F 6
12 12 48
S 37 19
24 6
19 27 T
D 17 J
35 6
35 17
21 36
18 11 10 40
16 18 14
19 10
C
19
C1 G 26 26 K
C2
Figure 1
Figure 1 shows a capacitated, directed network of pipes. The number on each arc
represents the capacity of the corresponding pipe. The numbers in circles represent a
feasible flow from S to T.
(a) State the value of this flow.
(1)
(b) Explain why arcs CD and CG cannot both be saturated.
(1)
(c) Find the capacity of
(i) cut C1
(ii) cut C2
(2)
(d) Write down a flow augmenting route of weight 6 which saturates BF.
(1)
The flow augmenting route in part (d) is applied to give an increased flow.
(e) Prove that this increased flow is maximal.
(3)
(Total for Question 1 is 8 marks)
2 P75678A
, 2. A team of 5 players, A, B, C, D and E, competes in a quiz. Each player must answer one
of 5 rounds, P, Q, R, S and T.
Each player must be assigned to exactly one round, and each round must be answered
by exactly one player.
Player B cannot answer round Q, player D cannot answer round T, and player E cannot
answer round R.
The number of points that each player is expected to earn in each round is shown in
the table.
P Q R S T
A 32 40 35 41 37
B 38 – 40 27 33
C 41 28 37 36 35
D 35 33 38 36 –
E 40 38 – 39 34
The team wants to maximise its total expected score.
The Hungarian algorithm is to be used to find the maximum total expected score that
can be earned by the 5 players.
(a) Explain how the table should be modified.
(2)
(b) (i) Reducing rows first, use the Hungarian algorithm to obtain an allocation which
maximises the total expected score.
(ii) Calculate the maximum total expected score.
(6)
(Total for Question 2 is 8 marks)
P75678A 3
Decision Mathematics 2 QP & MS june 2024
, Paper
Afternoon
■ ■
Further Mathematics
Advanced Subsidiary
Further Mathematics options
28: Decision Mathematics 2
(Part of option K only)
Try to answer every questio
Turn over
P75678A
©2024 Pearson Education Ltd.
F:1/1/1/
,1.
C C
1
2
E
A 26
24 24
0 H
24
21 15
42 15 19
39 B 18 18
16 48
F 6
12 12 48
S 37 19
24 6
19 27 T
D 17 J
35 6
35 17
21 36
18 11 10 40
16 18 14
19 10
C
19
C1 G 26 26 K
C2
Figure 1
Figure 1 shows a capacitated, directed network of pipes. The number on each arc
represents the capacity of the corresponding pipe. The numbers in circles represent a
feasible flow from S to T.
(a) State the value of this flow.
(1)
(b) Explain why arcs CD and CG cannot both be saturated.
(1)
(c) Find the capacity of
(i) cut C1
(ii) cut C2
(2)
(d) Write down a flow augmenting route of weight 6 which saturates BF.
(1)
The flow augmenting route in part (d) is applied to give an increased flow.
(e) Prove that this increased flow is maximal.
(3)
(Total for Question 1 is 8 marks)
2 P75678A
, 2. A team of 5 players, A, B, C, D and E, competes in a quiz. Each player must answer one
of 5 rounds, P, Q, R, S and T.
Each player must be assigned to exactly one round, and each round must be answered
by exactly one player.
Player B cannot answer round Q, player D cannot answer round T, and player E cannot
answer round R.
The number of points that each player is expected to earn in each round is shown in
the table.
P Q R S T
A 32 40 35 41 37
B 38 – 40 27 33
C 41 28 37 36 35
D 35 33 38 36 –
E 40 38 – 39 34
The team wants to maximise its total expected score.
The Hungarian algorithm is to be used to find the maximum total expected score that
can be earned by the 5 players.
(a) Explain how the table should be modified.
(2)
(b) (i) Reducing rows first, use the Hungarian algorithm to obtain an allocation which
maximises the total expected score.
(ii) Calculate the maximum total expected score.
(6)
(Total for Question 2 is 8 marks)
P75678A 3
