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Hbc 2239: Operation Research Ii Question Paper
Hbc 2239: Operation Research Ii
Course:Bachelor Of Commerce
Institution: Dedan Kimathi University Of Technology question papers
Exam Year:2014
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DEDAN KIMATHI UNIVERSITY OF TECHNOLOGY
University Examination 2013/2014
THIRD YEAR FIRST SEMESTER EXAMINATIONS FOR THE DEGREE OF BACHELOR OF COMMERCE
HBC 2239: OPERATION RESEARCH II
DATE: 22nd April 2014 TIME: 2.00pm – 4.00pm
INSTRUCTIONS: Answer Question One (Compulsory) and any other Two Questions.
QUESTION ONE (COMPULSORY) (30 MARKS)
a) Define the following terms as used in network flows.
(i) A chain
(ii) A path
(iii)An arc (3 marks)
b) Define the following terms as used in Markov chains Analysis.
(i) Communicating states
(ii) Transient state
(iii)Periodic state
(iv) Closed set
(4 marks)
c) Suppose that a new vehicle cost $ 10000 and that the annual operational cost and resale value of the vehicle are as shown in the below;
Age of vehicle (yrs)
Resale value ($)
Operating cost ($)
1
7000
300(Yr 1)
2
6000
500(Yr 2)
3
4000
800(Yr 3)
4
3000
1200(Yr 4)
5
2000
1600(Yr 5)
6
1000
2200(Yr 6)
If you purchase a new vehicle now, determine a replacement policy that minimises the net cost of owning and operating a vehicle for the next six years. (8 marks)
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d) Consider the three-state marker chain having transition probability matrix
(i) Draw a transition graph of the chain. (2 marks)
(ii) Is the chain irreducible? Give reasons for your answer. (1 mark)
(iii)Determine the steady state distribution of the marker chain. (6 marks)
e) Determine the total output required to satisfy final demands of 150, 260 and 80 for three industries respectively given the following matrix of technical coefficients.
(6 marks)
QUESTION TWO (20 MARKS)
a) A company produces two types of Juices, type 1 and type 2. Given that if a person last purchased type 1, there is 90% chance that that her next purchase will be type 1. Given that if a person last purchased type 2, there is an 80% chance that her next purchase will be type 2.
(i) If a person is currently a type 2 purchaser, what is the probability that she will purchased type 2 two purchases from now? (4 marks)
(ii) If a person is currently a type 1 purchaser, what is the probability that she will purchase type 2 three purchases from now? (3 marks)
(iii)Find the steady-state probabilities for the company. (4 marks)
b) A manager has developed the following transition matrix for a firms account receivables
P
1
2
b
P
1
0
0
0
1
0.8
0
0.2
0
2
0.6
0
0
0.4
b
0
0
0
1
where
P – paid
1 – 1 to 30 days overdue
2 – 31 to 60 days overdue
b – bad debts
(i) Obtain the fundamental matrix (4 marks)
(ii) If there is currently Kshs. 20000 in account in the first category and kshs. 15000 in the second category determine the expected amount of bad debts. (5 marks)
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QUESTION THREE (20 MARKS)
a) A factory can produce four products denoted by . Each product must be processed in each of two workshops. The processing times (in hours per unit produced) are given in the following table.
P1
P2
P3
P4
Workshop 1
3
4
8
6
Workshop 2
6
2
5
8
400 hours of labour are available in each workshop. The profit margins are 4, 6, 10 and 9 dollars per unit of P1, P2, P3 and P4 produced respectively. Everything that is produced can be sold.
(i) Set up a linear programming problem and determine the optional solution.
(12 marks)
(ii) Assume that 20 units of P3 have been produced by mistake. What is the resulting decrease in profit? (2 marks)
(iii)In what range can the profit margin per unit of P1 vary without changing the optimal basis? (2 marks)
(iv) In what range can the profit margin of P2 vary without changing the optional basis
(4 marks)
QUESTION FOUR (20 MARKS)
a) Solve the following integer programming problem.
Maximise
Subject to
Subject to
(10 marks)
b) The following table shows the maintenance costs and trade-in prices for a car that has just been purchased for $ 12000
Age of car (yrs)
Annual maintenance cost($)
Car trade in prices ($)
0
2000
7000
1
4000
6000
2
5000
2000
3
9000
1000
4
12000
0
Find the minimum net cost that can be incurred in operating the car during the next five years. (10 marks)
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QUESTION FIVE (20 MARKS)
a) Show that where is the change in the final output vector, is the change in the final demand vector and A is the matrix of technical coefficient. (4marks)
b) Three sectors of economy consist of wood, chair and gum industries i.e W, C and G. The flow of inputs and outputs between the industries is represented in the table below.
INPUTS
Final demand
Outputs (tons)
W
C
G
W
10000
16250
18750
5000
R
15000
32500
18750
15000
G
20000
16250
6250
20000
(i) Obtain the technical coefficient matrix. (2 marks)
(ii) If the final demand for wood sector increases by 600 tons and that of the gum sector falls by 300 tons and the final demand for the chairs remain unchanged, what is the output level for each sector? (7 marks)
c) The following are capacities of water system network.
C(S, 1) = 2 C (2, T) = 3
C (S, 2) =6 C (3, 2)= 2
C (1, T) = 1 C (3, T) = T
C (1, 3) = 2
C (2, 3) = 2
Where S is the source and T is the sink. Sketch the network and find the maximum flow and the corresponding capacity of the minimum cut. (7 marks)
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