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Reg. No. :

M.E./M.Tech. DEGREE EXAMINATION, JANUARY 2010

First Semester

Computer Science and Engineering

MA 9219 OPERATIONS RESEARCH

(Common to M.E. Network Engineering, M.E. Software Engineering and

M.Tech. Information Technology)

(Regulation 2009)

Time: Three hours Maximum: 100 Marks

Answer ALL Questions

PART A (10 2 = 20 Marks)

1. Explain the main characteristics of the queuing system.2. State the steady-state measures of performance in a queuing system.3. State Pollaczek-Khintchine formula for non-Markovian queuing system.4. Mention the different types of queuing models in series.5. What is Monte Carlo simulation? Mention its advantages.6. Give one application area in which stochastic simulation can be used in

practice.

7. Define slack and surplus variable in a linear programming problem.8. Mention the different methods to obtain an initial basic feasible solution of a

transportation problem.

9. State the Kuhn-Tucker conditions for an optimal solution to a Quadraticprogramming problem.

10. Define Non-linear programming problem. Mention its uses.

Question Paper Code:W7671

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PART B (5 16 = 80 Marks)

11. (a) (i) Explain M/M/1 (N/FCFS) system and solve it under steady-statecondition. (8)

(ii) In a railway marshalling yard, goods trains arrive at a rate of 30

trains per day. Assuming that the inter-arrival time follows an

exponential distribution and the service time (the time taken to

hump a train) distribution is also exponential with an average 36

minutes. Calculate the following

(1) The average number of trains in the queue

(2) The probability that the queue size exceeds 10. If the input of

trains increases to an average 33 per day, what will be

changed in (1) and (2)? (8)

Or

(b) (i) Explain the model M/M/S in case of first come first served basis.

Give a suitable illustration. (8)(ii) An automobile inspection in which there are three inspection stalls.

Assume that cars wait in such a way that when stall becomes

vacant, the car at the head of the line pulls up to it. The station can

accommodate almost four cars waiting (Seven in station) at one

time. The arrival pattern is Poisson with a mean of one car every

minute during the peak hours. The service time is exponential with

mean of 6 minutes. Find the average number of customers in the

system during peak hours, the average waiting time and the

average number per hour that cannot enter the station because of

full capacity. (8)

12.

(a) (i) Discuss the queuing model which applied to queuing system havinga single service channel, Poisson input, exponential service,

assuming that there is no limit on the system capacity while the

customers are served on a first in first out basis. (7)

(ii) At a one-man barber shop, customers arrive according to Poisson

distribution with a mean arrival rate of 5 per hour and his hair

cutting time was exponentially distributed with an average hair cut

taking 10 minutes. It is assumed that because of his excellence,

reputation customers were always willing to wait. Calculate the

following

(1) Average number of customers in the shop and the average

number of customers waiting for a hair cut.

(2) The percentage of customers who have to wait prior of getting

into the Barbers chair.

(3) The percent of time an arrival can walk without having to

wait. (9)

Or

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(b) (i) Explain briefly open and closed networks model in a queuingsystem. (8)

(ii) Truck drivers who arrive to unload plastic materials for recycling

currently wait an average of 15 minutes before unloading. The cost

of driver and truck time wasted while in queue is valued Rs. 100

per hour. A new device is installed to process truck loads at a

constant rate of 10 trucks per hour at a cost of Rs. 3 per truckunloaded. Trucks arrive according to a Poisson distribution at an

average rate of 8 per hour. Suggest whether the device should be

put to use or not. (8)

13. (a) (i) Distinguish between solutions derived from simulation models andsolutions derived from analytical models. (6)

(ii) Describe the kind of problems for which Monte Carlo will be an

appropriate method of solution. (5)

(iii) Explain what factors must be considered when designing a

simulation experiment. (5)

Or

(b) (i) Discuss stochastic simulation method of solving a problem. What

are the advantages and limitations of stochastic simulation? (9)

(ii) What are random numbers? Why are random numbers useful in

simulation models and solutions derived from analytical models? (7)

14. (a) (i) Explain various steps of the simplex method involved in thecomputation of an optimum solution to a linear programming

problem. (4)

(ii) Explain the meaning of basic feasible solution and degeneratesolution in a linear programming problem. (4)

(iii) Solve the following LPP using the simplex method. (8)

Max 21 35 xxz +=

subject to

0,

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2

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21

21

+

+

+

xx

xx

xx

xx

Or

(b) (i) What is degeneracy in transportation problem? How is

transportation problem solved when demand and supply are not

equal? (8)

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(ii) A company has factories at 21 , FF and 3F which supply towarehouses at 21,WW and 3W . Weekly factory capacities are 200,

160 and 90 units respectively. Weekly warehouse requirement are

180, 120 and 150 units respectively. Unit shipping costs (in rupees)

are as follows.

WarehouseW1 W2 W3 Supply

F1 16 20 12 200

Factory F2 14 8 18 160

F3 26 24 16 90

Demand 180 120 150 450

Determine the optimal distribution for this company to minimize total

shipping cost. (8)

15. (a) (i) What is meant by quadratic programming? How does a quadraticprogramming differ from a linear programming problem? (8)

(ii) Explain briefly the various methods of solving a quadratic

programming problem. (8)

Or

(b) (i) Explain the role of Lagrange multipliers in a non-linear

programming problem. (6)

(ii) Solve the following quadratic programming problem (10)

Maximize 21212 xxxz +=

Subject to the constraints

0,

and

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+

+

xx

xx

xx

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