maths operational research

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Reg. No. :

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

First Semester

Computer Science and Engineering

(Common to Information Technology, Software Engineering and

Network Engineering)

281110 OPERATIONS RESEARCH

(Regulation 2010)

Time : Three hours Maximum : 100 marks

Answer ALL questions.

PART A (10 2 = 20 marks)

1. What are the different parts of solution of queuing models?2. Write down the Littles formulae in queuing model.3. Write down the Pollaczek-Khinchin (P K) transform in non Poisson

queuing system.

4. Explain queues in Tandem and classify the above.5. Explain the elements of a simulation model.6. Explain the different types of simulation.7. Determine the initial basic feasible solution to the following transportation

problem using north west corner rule.

To

5 2 4 3 22

From 4 8 1 6 15 Availability

4 6 7 5 8

7 12 17 9

Demand

Question Paper Code : 20142

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8. Explain degenerate solution in LPP.9. Obtain the necessary conditions for the non-linear programming problem.

Max z2

3

2

2

2

1 53 xxx ++=

Subject to the constraints

.0,,,525,23 321321321 =++=++ xxxxxxxxx

10. State sufficient condition of Kuhn - Tucker conditions.PART B (5 16 = 80 marks)

11. (a) At a railway station, only one train is handled at a time. The railwayyard is sufficient only for two trains to wait while other is given signal to

leave the station. Trains arrive at the station at an average rate of 6 per

hour and the railway station can handle them on an average of 12 per

hour, Assuming Poisson arrivals and exponential service distribution,

find the steady state probabilities for the number of trains in the system.

Also find the average waiting time of a new train coming into the yard. If

the handling rate is reduced to half, what is the effect of the above

results?

Or

(b) At a port there are 6 unloading berths and 4 unloading crews. When all

the berths are full, arriving ships are diverted to an over flow facility

20 kms down the river. Tankers arrive according to a Poisson process

with a mean of 1 for every 2 hours. It takes for an unloading crew, on theaverage, 10 hours to unload a tanker; the unloading time follows an

exponential distribution. Determine

(i) How many tankers are at the port on the average?

(ii) How long does a tanker spend at the port on the average?

(iii) What is the average arrival rate at the overflow facility?

12. (a) A roofing company installs single roofs on new and old residences in apart of city. Prospective customers request the service randomly at the

rate of nine jobs per 30 day month and are placed on the waiting list to be

processed on a FCFS basis. Homes sizes vary, but it is fairly reasonable

to assume that the roof areas are uniformly distributed between 150 and

300 squares. The work crews can usually complex 75 squares a day.

Determine the following.

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(i) Companys average backlog of roofing jobs.

(ii) The average time a customer waits until a roofing job is completed.

(iii) If the work crew is increased to the point where they can complete

150 squares a day, how would this affect the average time until a

job is completed?

Or

(b) A barber shop has space to accommodate only 10 customers. He can

server only one person at a time, If a customer comes to his shops and

finds it full he goes to the next shop. Customers randomly arrive at an

average rate at d = 10 per hour and the barbers service time is negative

exponential with an average of 5/1 = minutes per customer.

(i) Write recurrence relations for the steady state queuing system

(FIFO) for above.

(ii) Determine oP and nP probability of having O and n customersrespectively in the shop.

13. (a) Use MonteCarlo technique to estimate the area of a circle whoseequation is given by 25)2()1( 22 =+ yx .

Or

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

inter-arrival and service times are constants 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 thepercentage idle time of the facility?

[Assume that the system starts at t = 0).

14. (a) Solve the following LPP by using simplex method.Max 4321 534 xxxxz +++=


.0,,,

202338

10423

204564

4321

4321

4321

4321

++

++

+++

xxxx

xxxx

xxxx

xxxx

Or

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(b) Solve the following LPP by using Two-Phase method.

Max z 3212 xxx ++=


,8364 321 ++ xxx ,1463 321 xxx

,4532 321 + xxx 0,, 321 xxx .

15. (a) Solve the following NLPP by using Kuhn-Tucker conditions.Max z 21

2

3

2

2

2

1 64 xxxxx ++=


,221 + xx ,1232 21 + xx .0, 21 xx

Or

(b) Solve the following QPP.

Max z 212

2

2

121 4102510 xxxxxx +=


,102 21 + xx ,921 + xx 0, 21 xx .

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maths operational research

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