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AOR802S - APPLIED OPERATIONS RESEARCH - 1ST OPP - NOV 2022 |
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nAmlBIA UnlVERSITY
OF SCIEnCE Ano TECHnOLOGY
FACULTY OF HEALTH AND APPLIED SCIENCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATION CODE: 08BSHM
LEVEL: 8
COURSE CODE: AOR802S
COURSE NAME: APPLIED OPERATIONS RESEARCH
SESSION: NOVEMBER 2022
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 240 (To be Converted to 100%)
EXAMINER
MODERATOR:
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
PROF. S. A. REJU
PROF. 0. D. MAKINDE
INSTRUCTIONS
l. Attempt ALL the questions.
2. All written work must be done in blue or black ink and sketches must be
done in pencil.
3. Use of COMMA is not allowed as a DECIMAL POINT.
PERMISSIBLEMATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 4 PAGES (including this front page)
Page 1 of 4
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QUESTION 1 [SO MARKS]
(a) Discuss Game Reduction by Dominance procedure.
(4 Marks)
(b) Simplify by using reduction by dominance the game defined by the following payoff
matrix, showing progressively the reduced pay-off matrix:
(18 Marks)
B
ab C
A
-1
AB r42 -4
-;51
C 3 -3 -8
D 2 -5 -4
(c) Distinguish between game players' pure and mixed strategies, with clear characteristics
of the latter.
(6 Marks)
(d) Ezra has a 270-gallon capacity home heating oil tank, presently empty, meant to store
oil against the next winter. Consider the following winter heating oil quantity needed
and the oil prices during probable four levels of winter severity:
(22 Marks)
WINTER SEVERITY
Mild Winter (MW)
Average Winter (AW)
Severe Winter (SW)
Prolonged Winter (PW)
OIL STORAGENEEDED
125 Gallons
180 Gallons
240 Gallons
270 Gallons
OIL PRICESPERGALLON
N$1.00
N$1.85
N$2.00
N$3.00
Formulate a game model and employ the Minimax criterion technique to determine the
gallons of oil Ezra should stockpile at the current price of N$1 per gallon to avoid oil wastage
and to maximise his saving.
QUESTION 2 [53 MARKS]
(a) Two suspects A and B have been apprehended for a crime and are in cells in Tsumeb
police station, with no means of communicating with each other. The prosecutor has
separately told them the following:
If you confess and agree to testify against the other suspect, who does not confess, the
charges against you will be dropped and you will go scot-free. If you do not confess but
the other suspect does, you will be convicted and the prosecution will seek the maximum
sentence of three years. If both of you confess, you will both be sentenced to two years in
prison. If neither of you confesses, you will both be charged with misdemeanours and will
be sentenced to one year in prison.
Selecting Suspect A as the row player in a 2-person game, construct the game payoff matrix
and each suspect's payoff matrix.
(12 Marks)
Page 2 of 4
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(b) (i) Determine what the two suspects should do and discuss fully why.
(6 Marks)
(ii) Discuss the implication of the dominant strategy for each player (or prisoner).
(5 Marks)
(c) Consider a construction firm that is deciding to specialise in building High School blocks
or Elementary School blocks or a combination of both. The construction company must
submit a bid proposal, which costs money to prepare, and there are no guarantees that
it will be awarded the contract. If the company bids on the high school, it has a 35%
chance of getting the contract, and it expects to make $162,000 net profit. However, if
the company does not get the contract, it loses $11,500. If the company bids on the
elementary school, there is a 25% chance of getting the contract, and it would net
$140,000 in profit. However, if the company does not get the contract, it will lose $5,750.
(i) What should the construction company do?
(14 Marks)
(ii) How sensitive to the estimate of the probability of the award of a contract is the
decision (i):
• in either to build the High School or the Elementary School blocks? (6.5 Marks)
• to the net profit for each case, if awarded the contract?
(9.5 Marks)
QUESTION 3 [53 MARKS]
(a) Provide a comprehensive definition of a Decision tree and hence diagrammatically
show its basic characteristic components.
(14 Marks)
(b) Using the problem in Question 2(c) above, provide the Fold-Back method tree for its
solution.
(14 Marks)
(c)
(i) What is the Kendall's classification of Queuing Systems?
(5 Marks)
Discuss specifically the M/M/1 queuing system and the process N(t) describing its state at
time t as a birth-death process. Provide its state independent parameter equations and define
its Traffic Intensity.
(3 Marks)
(ii) Consider a drive-in banking service modelled as an M/M/1 queuing system with customer
arrival rate of 2 per minute. It is desired to have fewer than 5 customers line up 99% of the
time. How fast should the service rate be?
(6 Marks)
(iii) Trucks arrive at garage for a stop-over service according to a Poisson process at a rate of
one per every 13 minutes, and the garage service time is an exponential rate variable with
mean 9 minutes.
(iiia) Find the average number L of trucks, the average time W a truck spends in the garage,
and the average time Wq a truck spends in waiting for service.
(5 Marks)
(iiib) Due to increased traffic, suppose that the arrival rate of the trucks increases by 5%.
Find the corresponding changes in L, W, and Wq.
(5 Marks)
(iiic) Discuss your observations.
(1 Mark)
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QUESTION 4 [84 MARKS]
(a) Consider a winning bid of $5.4 million to construct a new plant for a major manufacturer
and the manufacturer needs the plant to go into operation within 40 weeks. Below is the
list of the various project activities. The third column provides important additional
information for coordinating the scheduling of the project crews.
;=-,z;;;m-;)=..,.,,
~T-:::sP'C
-=-
..
==er-J
Activity
Activity Description
Immediate
Predecessors
Estimated
Duration
A
Excavate
-
2 weeks
B
Lay lhe foundation
A
4 weeks
C
rut up the rough wall
B
10 weeks
D
Put up the roof
C
6 weeks
E
Install the exterior plumbing
C
4 weeks
F
Install the interior plumbing
E
5 weeks
C
Put up the exterior siding
[)
7 'Necks
H
Do the exterior painting
F, G
9 weeks
I
Do the electrical work
C
7 weeks
J
Put up the wallbozird
K
Install the flooring
L
Do the interior painting
f, I
8 ,,veeks
I
4 weeks
I
5 weeks
flit
Install the exterior fixtures
H
2 weeks
N
Install the interior fixtures
K, L
.
6 weeks
(i) Define Critical Path Method (CPM) and Project Evaluation and Review Technique (PERT).
{3.5 Marks)
(ii) Sketch the project network diagram for the above project.
(16 Marks)
(iii) Distinguish between crashing a project and a project activity. Hence obtain the crash
costs per week saved for each activity from the following investigative time-cost trade-off
data.
{37 Marks)
Activity
,\\
8
C
D
[
r
G
Time
Normal
Crash
2 wtek.s
4 weeks
10 weeks
6 week.s
4 wcclo
5 weeks
7 weeks
I ,.,,1:t:k
2 week>
7 wt.'ek,
4 weeks
3 weeks
3 Wl'eks
•I week5
Co,t
Normal
1180,000
B20,000
1620,000
.1260,000
HI0,000
S 180,000
1900,000
Cruh
s 280,000
s 420,000
s 860,000
s 3•10,000
s 570,000
s 260,000
il,020,000
Activity
.~
,L•,.
N
Time
Normal
Cruh
7 wr-ck\\
8 \\'/i'('k.i
4 wcckl
5 weeks
2 We~l<S
6 weeks
5 wrek,
6 wC(·ks
3 weeks
3 we-ek..\\
I w~ek
3 weeks
Co,1
Normal
Cruh
1210,000
1430,000
I 160,000
1250,000
I 100,000
1330,000
270,000
,190,000
200,000
350,000
200,000
510,000
(iv) Discuss your observations.
(4 Marks)
(b) Consider a flow network with a directed graph with three vertices and three arcs,
described as follows: The first arc from vertex(l) to vertex(2) has capacity 3 and the
cost 1; the second arc from vertex(l) to vertex(3) has capacity 5 and the cost 4, and
the third arc from vertex(2) to vertex(3) has capacity 4 and the cost 2.
(i) Sketch the flow network.
(12 Marks)
(ii) State the matrices for the arc capacities, the arc costs and the demand function for
the vertices.
(11.5 Marks)
END OF EXAMINATION
TOTAL MARKS:240 CONVERTTO 100%
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