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Operation Research Ii Question Paper
Operation Research Ii
Course:Bachelor Of Commerce
Institution: Kca University question papers
Exam Year:2008
UNIVERSITY EXAMINATIONS: 2008/2009
THIRD YEAR STAGE 1 EXAMINATION FOR THE DEGREE OF
BACHELOR OF COMMERCE
CMS 300: OPERATION RESEARCH II
DATE: DECEMBER 2008 TIME: 2 Hours
INSTRUCTIONS: Answer question ONE and any other TWO questions
QUESTION ONE (30 MARKS)
(a) Explain the following terms
(i) Network [2 Marks]
(ii) Shortest route problem [2 Marks]
(iii) Production problems [2 Marks]
(iv) Stochastic process [2 Marks]
(v) Absorbing states [2 Marks]
(b) A researcher for population studies found out by its survey that the mobility of the
population (in percent) of a state to a village, town and city in the following
percentages;
Village Town City
Village 50 30 20
Town 10 70 20
City 10 40 50
2
Calculate
i) the proportion of population in the village, Town, and City after two years, given
that the present population has proportions 0.7, 0.2 and 0.1 in the village, Town and
City, respectively. [4 Marks]
ii) The long run probabilities [6 Marks]
(c) Using examples, briefly discuss three types of replacement situations.
[3 Marks]
(d) Using step-by step illustrations identify a path from source A to sink G that can
accommodate positive flow from the diagram below. [7 Marks]
AB=5, AC=AD=0, AE=3, AF=1, BA=2, BC=BE=BG=0, CA=CE=1, CB=3,
CD=2, CG=0, DA=DF=4, DG=1, DC=0, EA=1, EB=7, EC=2, EG=0,
FA=FG=0, FD=4, GB=GC=2, GF=5, GD=3, GE=5.
QUESTION TWO (20 MARKS)
(a) In the context of theory of replacement models, define with an example the
following terms:
(i) Individual replacement policy [3 Marks]
(ii) Group replacement policy [3 Marks]
(b) A computer has a large number of electronic tubes. They are subject to mortality as
given below
A
B
D
F
E
C
G
3
Period Age of failure (Hours) Probability of failure
1 0-200 0.10
2 201-400 0.26
3 401-600 0.35
4 601-800 0.22
5 801-1000 0.07
If the tubes are group replaced, the cost of replacement is Kenya pounds 15 per tube. Group
replacement can be done at fixed intervals in the night shift when the computer is not
normally used. Replacement of individual tubes which fail in service costs Kenya pounds
60 per tube. Determine
i) Expected number of failures (Replacements). [5 Marks]
ii) The cost of individual replacements per period [3 Marks]
iii) Group average cost per period due to group replacement at fixed intervals, hence
comment on your results [6 Marks]
QUESTION THREE (20 MARKS)
(a) The number of units of an item that are withdrawn from inventory requirements on
a day-to-day basis is a Markov chain process in which requirements for tomorrow
depend on today’s requirements. A one-day transition matrix is given below:
Tomorrow
5 10 12
5 0.6 0.30 0.0
Today 10 0.3 0.30 0.4
12 0.1 0.30 0.6
i) Construct a tree diagram showing inventory requirements on two consecutive
days. [9 Marks]
4
ii) Develop a two-day transition matrix. [3 Marks]
iii) Comment on how a two-day transition matrix might be helpful to a manager who
is responsible for inventory management. [3 Marks]
(b) Differentiate between the following terms as used in Operation Research.
i) Gradual failure and sudden failure [2 Marks]
ii) LINDO and LINGO [3 Marks]
QUESTION FOUR (20 MARKS)
(a) The national Park service plans to develop a wilderness area for tourism. Four
locations in the area are designated for automobile access. These sites, and their
distances (in Kilometres) between them, are listed below.
Park
Entrance
Wild falls Majestic
Rock
Sunset
Point
Hippo
point
Park Entrance … 7.1 19.5 19.1 25.7
Wild falls 7.1 … 8.3 16.2 13.2
Majestic Rock 19.5 8.3 … 18.1 5.2
Sunset Point 19.1 16.2 18.1 … 17.2
Hippo point 25.7 13.2 5.2 17.2 …
To inflict the least harm on the environment, the park service wants to minimize the miles
of roadway required to provide the desired accessibility.
(i) Draw the network diagram to represent this problem [3 Marks]
(ii) Determine how roads should be built to achieve this objective [4Marks]
(b) A manufacturing concern has been awarded a contract to produce casings. The
contract is for four (4) years and is not expected to be renewed. The production process
requires a specialized machine, which the concern does not have. The concern can buy
the machine, maintain it for 4 years of the contract, and then sell it for scrap value; or it
can replace the machine at the end of any given year by a new model. New models
require less maintenance than older ones. Estimated net operating cost (Purchase price
plus maintenance minus trade-in) for buying a machine in the beginning of year i and
5
trading it in at the beginning of year j is given in the table below, with all figures
expressed in thousand dollar units
j
i
1
2
3
4
5
1 … 12 19 33 49
2 … … 14 23 38
3 … … 16 26
4 … … … 13
(i) Determine the optimal path [9 Marks]
(ii) Determine a replacement policy that will minimize the total operating cost for the
machine over the life of the contract. [4 Marks]
QUESTION FIVE (20 MARKS)
(a) Define a general non-linear programming problem [5 Marks]
(b) The Operations Research (OR) team of ABC Company has come up with the
mathematical data needed for two products which the firm manufacture. It has
determined that this is a non linear programming problem, having linear constraints and
objective function which is the sum of a linear and a quadratic form. The pertinent data
gathered by the OR team is:
Max(contribution) 2
2 2
2
1 1 Z = 8x - x + 8x - x subject to the constraints
12 1 2 x + x =
4 1 2 x - x =
, 0 1 2 x x =
where 1 x and 2 x represent the number of units of the two products.
Use the graphical method to find the maximum contribution and number of units that
can be expected from these products which are a part of the firm''s total output.
[10 Marks]
(c) Explain two differences between dynamic programming and linear programming
[5 Marks]
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