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Cms 300: Operation Research Ii (Evening Class) Question Paper
Cms 300: Operation Research Ii (Evening Class)
Course:Bachelor Of Commerce
Institution: Kca University question papers
Exam Year:2009
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UNIVERSITY EXAMINATIONS: 2009/2010
THIRD YEAR STAGE I EXAMINATION FOR THE DEGREE OF
BACHELOR OF COMMERCE
CMS 300: OPERATION RESEARCH II (EVENING CLASS)
DATE: DECEMBER 2009 TIME: 2 HOURS
INSTRUCTIONS: Answer question ONE and any other TWO questions
QUESTION ONE (30 MARKS)
(a)Explain the following terms as used in Operation Research
(i) Network [1 Mark]
(ii) A stochastic process [2 Mark]
(iii) Maximum flow problem [2 Marks]
(b) Explain the Four Characteristics of Markov Chains [3Marks]
(c) Represent the following transition matrix using a transition diagram;
Succeeding state
S1 S2 S3
S1 0 P22 0
Initial state S2 0 P22 P23
S3 p31 0 p33
[5 Marks]
(d) A housewife buys three kinds of cereals; A, B, and C. She never buys the same cereals on
successive weeks. If she buys cereals A, then the next week she buys B. However, if she buys
either B or C, then the next week she is three times as likely to buy A as the other brands. In the
long run, how often does she buy each of the three brands? [6 Marks]
(e) Using examples explain two Categories of replacement of items [4 Marks]
(f) A fleet owner finds from his past records that the costs per year of running a truck whose
purchase price is US$ 6,000 are given below:
Year: 1 2 3 4 5 6 7
Running cost (US$): 1000 1200 1400 1800 2300 2800 3400
Resale value (US$): 3000 1500 750 375 200 200 200
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Determine when the equipment should be replaced [7 Marks]
QUESTION TWO (20 MARKS)
(a) Differentiate between a minimum span problem a shortest route problem [2 Marks]
(b) An individual who lives in Thika would like to travel to Westlands using intermediate points.
The distances between these towns are given below. Any empty cell means that there is no
route connecting the two intermediate towns.
Thika Juja Kasarani Kiambu Forest
road
Westlands
Thika 8 32
Juja 8 15 28
Kasarani 15 13 32
Kiambu 32 28 13 10 13
Forest road 10 5
Westlands 32 13 5
By using the first two letters to denote the various towns, determine the minimum time this individual
takes to travel from Thika to Westlands [7 Marks]
(c) The following failure rates have been observed for a certain type of light bulb:
End of week 1 2 3 4 5 6 7 8
Probability of
failure to date
0.05 0.13 0.25 0.43 0.68 0.88 0.96 1.00
Given that the number of light bulbs in the beginning is 1000, and the cost of replacing an individual
failed bulb is Kshs 40. The decision is made to replace all bulbs simultaneously at fixed intervals, and
also to replace individual bulbs as they fail in services. If the cost of group replacement is Kshs 35 per
bulb;
(i) find the best interval between group replacements. [8 Marks]
(ii) find at what group replacement price per bulb would a policy of strictly individual
replacement become preferable to the adopted policy. [2 Marks]
QUESTION THREE (20 MARKS)
(a) Differentiate between an ergodic state and a transient state in Markov chains [2 Marks]
(b) Explain steps in constructing matrix of transition probabilities [3 Marks]
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(c) Explain four application areas of Markov chains [4 Marks]
(d) A market survey is made on three brands of breakfast foods X, Y and Z. Every time the customer
purchases a new package, he may buy the same brand or switch to another brand. The following
estimates are obtained, expressed in decimal fractions
Brand just purchased
X Y Z
X 0.6 0.1 0.3
Y 0.3 0.4 0.3
Present Brand Z 0.4 0.1 0.5
At this time it is estimated that 20 percent of the people buy brand X, 30 percent brand Y and 50
percent brand Z. What will the distribution of customers be
(i) in two time periods late [4 Marks]
(ii) at equilibrium [7 Marks]
QUESTION FOUR (20 MARKS)
(a) Define a dynamic programming problem [2 Marks]
(b) Explain a general algorithm for solving a shortest route problem using dynamic problem
[5 Marks]
(c) A salesman located in a city A decided to travel to city J. He knew the distances of alternative
route from city A to J, The distances between various towns are shown in the table below.
TOWNS
TOWNS
A B C D E F G H I J
A 4 6 3
B 4 7 10 5
C 6 5 8 4
D 3 6 5 5
E 7 5 6 4 8
F 10 8 5 3 7
G 5 4 5 8 4
H 4 3 8 7
I 8 7 4 9
J 7 9
(i) Draw an oriented network for this problem [3 Marks]
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(ii) find the shortest route that covers all the selected cities fro city a to city j using dynamic
rogramming Method [10 Marks]
QUESTION FIVE (20 MARKS)
(a) Give two conditions to for a mathematical problem to be a non linear programming problem
[2 Marks]
(b) An Operation Research team of ABC Company has come up with a mathematical data for two
products which the firm manufactures. It has been determined to be a non linear programming
problem. The data gathered is as follows:
2
2 2
2
1 1 Z = 8X - X + 8X - X
Subject to the constraints
0
0
4
12
2
1
1 2
1 2
=
=
- =
+ =
X
X
X X
X X
Find the maximum contribution and the number of units that can be expected from this
products which are a part of the firms total output.
[8 Marks]
(c) Ndege Television (TV) Company Ltd manufactures two types of TV sets, the “Astro” and the
“Cosmo”. There are two production lines, one for each set. The Astro production line has a
capacity of 60 sets per day, where as the capacity for the Cosmo production line has a capacity is
only 50 sets per day. The labor requirements for the Astro sets are 1 person-hour, whereas the
Cosmo requires a full 2 person-hours of labor. Presently, there is a maximum of 120 man-hors of
labor per day that can be assigned to production of he two types of sets. If the profit contribution
are Kshs 2000 and 3,000 for each Astro and Cosmo set respectively. The company’s problem is to
find a daily mix of production quantities of each of the two types of TV sets that would maximize
their profits
(i) Formulate the above problem [5Marks]
(ii) Write a LINGO program that would solve the problem. [5 Marks]
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