Operations Research  Jun 2015
Mechanical Engg. (Semester 7)
TOTAL MARKS: 100
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks.
1 (a) List and briefly explain the various phases of O.R. study and state the limitations of O.R. models.(10 marks)
1 (b) The XYZ company has been a producer of electronic circuits for Television sets and certain printed circuit boards for Radios. The company has decided to expand into full scale production and marketing of AM and AMFM radios. It has built a new plant that can operate 48 hours per week. Production of an Am radio in the new plant will require 2 hours and production of AM. FM radio will require 3 hours. Each AM radio will contribute Rs.40 to profit, while an AMFM radio will contribute Rs 80 to profits. The marketing department, after extensive research, has determined that a maximum of 15 AM radio, and 10AMFM radios can be sold each week, formulate a L.P. model to determine the optimal production mix of AM and AMFM radios that will maximize profits and solve the problem using Graphical method.(10 marks)
2 (a) Obtain the Dual problem of the following Primal problem
Min Z=2x_{1}5x_{2}2x_{3}
Subject to
3x_{1}1x_{2}+2x_{3}=9
2x_{1}4x_{2}≥14
4x_{1}+3x_{2}+8x_{3}=12
x_{1}x_{2}≥0 and x_{3} is unrestricted.(4 marks)
2 (b) Use BIGM method to solve the following LPP.
Min Z=2x_{1}=x_{2}
Subjected to
3x_{1}+x_{2}=3
4x_{1}+3x_{2}≤6
x_{1}+2x_{2}≤3
x_{1}, x_{2} ≥0.(16 marks)
3 (a) The owner of a machine shop has four machines available to assign the jobs for the day. Five jobs are offered with the expected profit in ₹ for each machine on each job is as follows, Find the assignment of the machines to the jobs that will result in a maximum profit, which job to be declined.
A  B  C  D  E  
1  62  78  50  101  82 
2  71  84  61  73  59 
3  87  92  111  71  81 
4  48  64  87  77  80 
Max. Z=x_{1}+x_{2}
Subjected to
3x_{1}+2x_{2}≤12
x_{2}≤2
x_{1}, x_{2} ≥ 0 and integers.(14 marks) 5 (a) A project consists of the following activities with their duration in days and the precedence relationship.
i) Draw the network for the above information
ii) Identify the critical path and duration
iii) Calculate EST, EFT, LST, LFT, TF.
Activity  A  B  C  D  E  F  G  H  I 
Precedence    A  A  B,C  A  D,E  C  F,G  H 
Duration (days)  10  12  5  7  9  10  8  10  9 
i) Draw a project work, identify the critical path and its expected duration and variance.
ii) What is the probability of completing the project in 30 days schedule time?
iii) What due data has 90% chance of being met?
Activity  12  23  24  35  45  46  57  67  78  79  810  910 
t_{m}  2  2  3  4  3  5  5  7  4  6  2  5 
t_{o}  1  1  1  3  2  3  4  6  2  4  1  3 
t_{p}  3  3  5  5  4  7  6  8  6  8  3  7 
i) Expected fraction of the day that the phone will be in use
ii) Expected number of units in the queue
iii) What is the probability that an arrival will have to wait more than 6min in queue for service? iv) What is the probability that more than 5 units are in the system?(10 marks) 7 (a) Define and briefly explain the following terms with respect to GAME theory.
i) PURE STRATEGY ii) SADDLE POINT iii) VALUE OF GAME iv) TWO PERSON ZERO SUM GAME v) PAYOFF.(10 marks) 7 (b) Solve the following TWO PERSON ZERO SUM GAME by Graphical Method.
B  
A 

Machine  
Job 


Job  A  B  C 
1  8  3  8 
2  7  4  3 
3  6  5  7 
4  9  2  2 
5  10  1  5 
6  9  6  1 