Punjab Technical University 2006 M.C.A OPTIMIZATION TECHNIQUES - Question Paper

M.C.A. DEGREE exam OPTIMIZATION TECHNIQUES ques. Paper


OPTIMIZATION TECHNIQUES

Time: 3 hours maximum: 100marks

PART A ans all ques. (8x5=40 marks)

1. (a) discuss how will you formulate a mathematical model to a provided linear programming
problem. Or (b) Use graphical method to solve the subsequent Max z=3x1 +4X2
Subject to 5X1 + 4X2 <= 200 3x1 + 5X2 <=150 5x1 + 4X2 >= 100 SX1 + 4X2>=8O X1X2
>=O.

2. (a) discuss the various categories of stochastic processes with simple examples. Or (b)
discuss the recurrent class of Markov chain and state the criteria for recurrence.

3. (a) discuss the main characteristics of the Queuing system. Or (b) establish the
probability distribution formula for pure-death process.

4. (a) What is queuing theory? discuss the basic elements of queues. Or (b) At a public
telephone booth in a post office arrivals are considered to Poisson with an avg.
inter-arrival time of 12 minutes. The length of the phone. Call may be presumed to be
distributed exponentially with an avg. of four minutes. compute the following: (i)
What is the probability that a fresh arrival will not have to wait for phone? (ii) What is
the probability that an arrival will have to wait more than 10 minutes before the phone
is free?

5. (a) provide a brief outline of the revised simplex method. Or (b) Write down the dual of
the subsequent LPP Min z = 4x1 + 3X2 -2x3 Subject to 3X1 + 6X2 + 4x3 >=6 seven X1 + X2 +
2x3 >= 5. 6x1 - 2X2 – X3<= nine 2x1 - X2 + 3x3 >= four 4x1 + 6X2 – X3 >= two X1, X2,
X3>=0.

6. (a) discuss a few of the practical applications of Integer programming issue. Or
(b) discuss how the assignment issue can be treated as a particular case of
transportation issue.


7. (a) elaborate the unbalanced assignment problem? How are they solved? Or(b) discuss
the nature of a travelling salesman issue and provide its mathematical formulation.


8. (a) discuss the mechanism of queuing process by considering a few illustration.Or (b) A
customer owning a Maruti car right now has got the choice to switch over to Maruti,
Ambassador or Fiat next time with the probability (0.20, 0.5 and 0.30) provided the
transition matrix.
0040 0.30 0.30
P = 0.20 10.50 0.30
0.25 0.25 0.50 obtain the probabilities with his 4th purchase?


PART B ans ALL ques.. (5 x 12 = 60 marks)


9. (a) Solve the subsequent LPP using simplex method Maximize z =3X1 + 5X2 + 4X3 . Subject
to 2X1 +3X2<=8 2X2 + 5x3 <=10 3x1 + 2X2 + 4X3 <=15 and Xl. X2 X3 >= 0 Or (b) Use revised
simplex method to solve the LPP. Minimize z= -4x1 + X2 + 2X3Subject to 2x1 - 3X2 + 2X3 <=12
-5x1 + 2X2+3X3 >=4 3x1-2x3=-1 and X1, X2, X3 >=0.


10. (a) Use penalty method to solve the subsequent LPP. Minimize Z=4X1+X2 Subject
to3X1+X2=3 4X1 + 3X2 >= six x1 + 2X2 <=3 and X1,X2 >=0.Or (b) Solve by the dual simplex
method the subsequent LPP Minimize z = 5x1 + 6X2Subject to X +X2 2:24xj + X2 2: four X2 2:0.
(b) A fair die is tossed repeatedly. If Xn denotes the maximum of the numbers occurring in the
first n tosses, obtain the transition probability matrix p of the Markov chain {XnJ. obtain also p2
and P(X2 = 6).


11. (a) A supermarket has 2 girls ringing up sales at the counters. If the service time for every
customer is exponential with mean four minutes and if the people arrive in a Poisson fashion at
the rate of 10 per hour (i) What is the probability of having to wait for service?(ii) What is
the expected percentage of idle time for every girl?(iii) If a customer has to wait, what is the
expected length of his waiting time. Or (b) Discus the fields of application for queuing.
discuss queue discipline and its different form.


12. (a) A travelling salesman has to visit five cities. He wishes to begin from a particular city visit
every city once and then return to his starting point cost of going from 1 city to a different is
shown beneath. You are needed to obtain the lowest cost route.
To city
A B C D E
A 00 four 10 14 2
B 12 00 six 10 4
From City C 16 14 00 eight 14
D 24 eight 12 00 10
E two six four 16 00
Or (b) obtain the optimum integer solution to the subsequent linear programming issue
Maximize Z =XI + 2X2 Subject to 2X257 xI + X2 572x] 51 and xI' X2 2: 0 and are integers.


13. (a) describe the Markov -'property for a discrete space continuous time process. Prove that a
process having independent and stationary increments is Markov

Earning:   Approval pending.