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AOR802S - APPLIED OPERATIONS RESEARCH - 1ST OPP - NOVEMBER 2024 |
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n Am I BI A uni V ERs ITY
OF SCIEnCE AnDTECHnOLOGY
FacultyofHealth,Natural
ResourceasndApplied
Sciences
Schoolof Natural andApplied
Sciences
Department of Mathematics,
Statistics and Actuarial Science
p
13JacksonKaujeuaStreet
Private Bag 13388
Windhoek
NAMIBIA
T: •264 6120729i3
E: msas@nust.na
IN: www.nust.nil
QUALIFICATIONS: BACHELOR of SCIENCE HONOURS IN APPLIED MATHEMATICS
BACHELOR of SCIENCE HONOURS IN APPLIED STATISTICS
QUALIFICATION CODES: 08BSMH, 08BSSH
LEVEL:8
COURSE: APPLIED OPERATIONS RESEARCH
DATE: NOVEMBER 2024
DURATION: 3 HOURS
COURSECODE: AOR802S
SESSION: 1
MARKS: 140 (To be converted to 100%)
FIRST OPPORTUNITY: EXAMINATION QUESTION PAPER
EXAMINER:
MODERATOR:
Prof Sunday A. Reju
Prof Oluwo/e D. Makinde
INSTRUCTIONS
1. Answer all questions on the separate answer sheet.
2. Please write neatly and legibly.
3. Do not use the left side margin of the exam paper. This must be allowed for the
examiner.
4. No books, notes and other additional aids are allowed.
5. Mark all answers clearly with their respective question numbers.
6. Use of COMMA is NOT ALLOWED for a DECIMAL POINT.
PERMISSIBLME ATERIALS
1. Non-Programmable Calculator
ATTACHMENTS
NONE
This paper consists of 4 pages including this front page.
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QUESTION 1 [25 MARKS]
(a) Consider a competition between two companies, Coca-Cola and Pepsi, and assume the
former is thinking of cutting the price of its iconic soda. If it does so, Pepsi may have no
choice but to follow suit for its cola to retain its market share. This may result in a
significant drop in profits for both companies. Let's assume that the incremental profits
that accrue to Coca-Cola and Pepsi are as follows: If both keep prices high, profits for
each company increase by $500 million (because of normal growth in demand). If one
drops prices (i.e. defects) but the other does not (i.e. cooperates), profits increase by
$750 million for the former because of greater market share and are unchanged for the
latter. If both companies reduce prices, the increase in soft drink consumption offsets the
lower price, and profits for each company increase by $250 million.
(13 marks)
(i) Construct the payoff matrix for the game model, taking Coca-Cola as the row player
and each company's payoff matrix.
(ii) What should each company do?
(b) Suspects A and B have been apprehended for a crime and are in cells in Outapi police
station, with no means of communicating with each other. The prosecutor has separately
told them the following:
(12 marks)
"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".
(i) Selecting Suspect A as the row player in a 2-person game, construct the game payoff
matrix and each suspect's payoff matrix to determine what the two suspects should do
and discuss fully why.
(ii) Discuss the implication of the dominant strategy for each prisoner.
QUESTION 2 [30 MARKS]
(a) Discuss the Reduction by Dominance procedure and hence simplify by using reduction
by dominance the game defined by the following payoff matrix, showing progressively
the reduced pay-off matrix:
B
ab c
[1-1
AB 4 -4
A
C 3 -3
D 5 -5
(8 Marks)
(b) Distinguish between pure and mixed strategies.
(5 Marks)
Course Name (AOR802S)
1st Opportunity November 2024
2
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(c) Eugene has a 250-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:
WINTER SEVERITY
Mild Winter (MW)
Average Winter (AW)
Severe Winter (SW)
Prolonged Winter (PW)
OIL STORAGE NEEDED
110 Gallons
180 Gallons
230 Gallons
250 Gallons
OIL PRICES PER GALLON
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 Eugene should stockpile at the current price of N$1 per gallon to avoid wasted
unused oil and to maximise his saving.
(17 Marks)
QUESTION 3 [25 MARKS]
(a) A company licensed to mine in possession of a natural resource field has 0.25 chance for
diamond discovery. However, the company has the options to either mine the resource or to
sell it the field to another mining company wishing to buy the land for $90,000,000. The cost
of mining by the licensee is $100,000,000 with a revenue yield of $800,000,000 if diamond is
found. Formulate a game of strategy model to perform a mathematical decision analysis of
the problem and determine the decision of the licensee, stating appropriate assumptions for
your method.
(8 Marks)
(b) State the Maximum Likelihood Criterion and confirm the above decision obtained in (a)
with the criterion.
(4 Marks)
(c) Assuming the company feels that the true chances of discovering diamond are likely to be
between 15% and 35%, define sensitivity analysis and the decision crossover point, and hence
show that the decision is sensitive to these prior probabilities, providing an appropriate
sketch to substantiate your decision analysis and conclusion.
(13 Marks)
QUESTION 4 [60 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.
Course Name (AOR802S)
1'1 Opportunity November 2024
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Activity
A
B
C
D
E
F
C
H
/(
L
Iv/
N
-·~-~~
r.;:
Activity Description
Exc,watc
Lay lhe foundation
Pul up lhe rough willl
Put up the roof
Install the exlerior plumbing
lnst,1II the interior plumbing
f>ut up the exterior siding
Do the exterior painting
Do the electric,11work
Put up the 1,vallboard
Install the flooring
Do the interior painting
Install the exterior fixtures
Install the interior fixtures
:::'
lmmedii:ite
Predecessors
A
B
C
C
E
[)
F.,G
C
I
J
I
H
K, I.
Estimated
Duration
2 'Necks
4 weeks
10 weeks
6 .veeks
4 weeks
5 weeks
7 1.vceks
9 weeks
7 weeks
8 1NCckS
4 1Neeks
5 'Neek.s
2 weeks
6 weeks
(i) Define Critical Path Method (CPM) and Project Evaluation and Review Technique (PERT).
(3 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
A
B
C
D
[
r
G
Time
Co,1
Normal
Cr.uh
Norm.ti
Cr,uh
2 W('t'kS
4 wct·k\\
10 Wt'l'k\\
6 \\'ltt S
4 wt>el(\\
S wrrh
7 Wtl•k.S
I \\"It,'(:"
2. WC'Ck.l
7 •,Vt""i'k\\
.\\ W(•CkS
WC"('k)
J W('(:k\\
.1 ._.._.t:eks
1160,000
\\320,000
\\620,000
l2o0.000
\\410,000
l lX0,000
\\900,000
l 260,000
l 420,000
H60,000
340,000
570,000
260,000
l l .020,000
Activity
K
M
N
(iv) Discuss your observations.
Time
Normal
Cr.uh
7 we-eks.
8 we~~j
4 wt~do
s WC-<'k,
2 w(·t.·~
6 wccb
S w~k\\
6 •.'.'~t-1-.S
3 wtcb
3 ·...-~ek,
I -.·,•re~
3 week,
Co,t
Normal
Cr.uh
Ill 0,000
l-130,000
1160,000
1250,000
1100,000
IH0.000
210,000
,:90,000
200,000
350,000
200,000
l .510,000
(4 Marks)
END OF EXAMINATION
TOTAL MARKS= 140
Course Name (AOR8025)
1st Opportunity November 2024
4