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Reg. No. :

M.E./M.Tech. DEGREE EXAMINATION, JUNE 2011.

Common to M.E. – Computer Science and Engineering/M.Tech. – Information

Technology/M.E. – Software Engineering/M.E. – Network Engineering

First Semester

281110 — OPERATIONS RESEARCH

(Regulation 2010)

Time : Three hours Maximum : 100 marks

Answer ALL questions.

PART A — (10 × 2 = 20 marks)

1. Suppose that customers arrive at a Poisson rate of one per every 12 minutes,

and that the service time is exponential at a rate of one service per 8 minutes.

What is the average time of a customer spends in the system?

2. Define transient and steady state in a queuing system.

3. Define Non-Markovan queue with a suitable example.

4. State Pollaczek Khintchine formula.

5. What are the advantages of simulation?

6. Define Pseudo-random number.

7. What is the difference between slack variable and surplus variable?

8. Distinguish between transportation problem and assignment problem.

9. When necessary conditions become sufficient conditions for a maximum

(minimum) of the objective function for a general NLPP?

10. State Kuhn-Tucker conditions for the optimal solution of general NLPP.

Question Paper Code : 31215

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31215 2

PART B — (5 × 16 = 80 marks)

11. (a) 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 distribution is also exponential with an

average 36 minutes. Calculate the following :

(i) the mean queue size

(ii) the probability that the queue size exceeds 10 and

(iii) if the input of trains increases to an average 33 trains per day,

what will be the change in (i) and (ii).

Or

(b) A super market has two girls ringing up sales at the counters. If the

service time for each customer is exponential with mean 4 minutes and if

people arrive in a Poisson fashion at the counter at the rate of 10 per

hour, then calculate

(i) the probability of having to wait for service.

(ii) the expected percentage of idle time for each girl.

(iii) if a customer has to wait find the expected length of his waiting

time.

12. (a) In a heavy machine shop, the overhead crane is 75 percent utilized. Time

study observations gave the average slinging time as 10.5 minutes with

standard deviations of 8.8 minutes. What is the average calling rate for

the services of the crane, and what is the average delay in getting

service? If the average service time is cut to 8 minutes, with standard

deviation of 6 minutes, how much reduction will occur, on average, in the

delay of getting served?

Or

(b) For {(M/G/1):(∞ /FCFS)} queueing model, derive Pollaczek — Khintchine

formula for expected number of customers in the system.

13. (a) Customers arrive at a milk booth for the required service. Assume that

inter-arrival and service times are constant and given by 1.8 and 4 time

units respectively. Simulate the system by hand computations for 14 time

units. What is the average waiting time per customer? What is the

percentage idle time of the facility? (Assuming that the system starts at

time t = 0).

Or

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31215 3

(b) The occurrence of rain in a city on a day is dependent upon whether or

not it rained the previous day. If it rained on the previous day, the rain

distribution is as follows :

Event : No rain 1 cm rain 2 cm rain 3 cm rain 4 cm rain 5 cm rain

Probability : 0.5 0.25 0.15 0.05 0.03 0.02

If it did not rain on the previous day, the rain distribution is :

Event : No rain 1 cm rain 2 cm rain 3 cm rain

Probability : 0.75 0.15 0.06 0.04

Simulate the city’s weather for 10 days and determine by simulation the

total days without rain as welt as the total rainfall during the period. Use

the following random numbers :

67 63 39 55 29 78 70 6 78 76

for simulation. Assume that for the first day of the simulation it had not

rained the day before.

14. (a) An automobile manufacturer makes auto-mobiles and trucks in a factory

that is divided into two shops. Shop A, which performs the basic

assembly operation must work 5 man-days on each truck but only 2 man-

days on each automobile. Shop B, which performs finishing operation

must work 3 man-days for each truck or automobile that it produces.

Because of men and machine limitations shop A has 180 man-days per

week available while shop B has 135 man-day per week. If the

manufacturer makes a profit of Rs. 300/- on each truck and Rs. 200/- on

each automobile, how many of each should he produce to maximize his

profit? Use the Simplex method.

Or

(b) Solve the following transportation problem to minimize the total cost of

transportation :

Destination

D1 D2 D3 D4 Supply

O1 14 56 48 27 70

Origin O2 82 35 21 81 47

O3 99 31 71 63 93

Demand 70 35 45 60 210

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15. (a) Solve the following non-linear programming problems :

Maximize yxz 32 +=

Subject to the constraints :

20,8 22 ≤+≤ yxxy and

0, ≥yx .

Verify that the Kuhn- Tucker conditions hold for the maxima.

Or

(b) Use Wolfe’s method to solve the quadratic programming problem :

Maximize 2232 xyxz −+=

Subject to the constraints :

2,44 ≤+≤+ yxyx and 0, ≥yx .

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