Examination style paper 1
Answer all questions
1.
Time allowed 90 minutes
13
A
100 m
64
385
12
65
B
400 m
390
63
14
C
1500 m
375
Downhill School has been invited to enter three pupils for an interschool athletic competition. The competition consists of three track
events: 100 m, 400 m and 1500 m. A pupil can only enter one event
and the winning team is the one whose total time is a minimum.
Downhill School has held trials to find the times, in seconds, taken
by three pupils, A, B and C, in each of the events. These times are
given in the diagram above. The teacher's problem is to decide who
shall run which event to obtain the best result in the competition.
Formulate the teacher's problem as a linear programming
problem.
(7 marks)
2. An electrical shop has just received three new repair jobs: a
television (1), a microwave (2) and a vacuum cleaner (3). Four men
are available to do the repairs. The manager estimates what it will
cost, in wages, to assign each of the workers to each of the jobs.
These estimates, in £s, are given in the table below.
Jobs
Workers
1
2
3
Alf
14
16
11
Ben
13
15
12
Cyril
12
12
11
Dennis
16
18
16
D2
Examination style paper 1
Use the Hungarian algorithm to obtain the assignment of workers
to jobs that results in the minimum overall cost.
(9 marks)
3.
D
15
18
16
A
30
S
H
E
16
20
24
19
14
B
T
15
25
20
10
12
26
F
I
14
C
13
18
G
Members of a scout troup assemble at S and are told to make their
way to T by any of the routes shown in the above figure. The
weights on the edges give the time, in minutes, required to complete
that leg. At least one person will take each of the possible routes
from S to T.
Use dynamic programming to determine the maximum time that
any scout could take to travel from S to T. State the route used.
Show your calculations in tabular form.
(1 mark)
4. Allan and Barbara play a zero-sum game and each has three
possible strategies. The payoff matrix is shown below.
B
A
I
II
III
I
3
5
4
II
2
3
3
III
6
3
8
(a) Explain why Allan will never choose strategy II.
(2 marks)
(b) Explain why Barbara will never choose strategy III. (2 marks)
(c) Obtain the optimal strategies for both players and the value of
the game.
(10 marks)
121
122
Examination style paper 1
5.
Destination
Source
A
B
C
Available
1
25
20
26
30
2
15
20
20
100
3
30
18
12
80
Required
100
50
60
The above table shows the unit transportation costs, the source
availabilities and the destination requirements for a transportation
problem. Use the north-west corner rule to obtain an initial basic
feasible solution and then use the stepping-stone method to obtain
the optimal solution. State the transportation pattern and give its
cost.
(16 marks)