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**Anna University Question Paper**

**M.E./M.Tech. DEGREE EXAMINATION, JANUARY 2012.**

**First semester**

**Computer Science and Engineering**

**MA 9219—OPERATIONS RESEARCH**

**(Common to M.E Software Engineering, M.E Network Engineering, and M.Tech Information Technology and also common to MA 9329- Operations Research for M.E Computer**

**Networks and M.E Computer Networking and Engineering)**

**(Regulation 2009)**

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**PART A-(10 × 2 = 20 marks)**

1. What is meant by Queue discipline? Name some common queue disciplines.

2. The number of glasses of juice ordered per hour at a hotel follows a Poisson distribution. With an average of 30 glasses per hour being ordered. Find the probability that exactly 60 glasses are ordered between 2 P.M and 4 P.M.

3. What are the characteristics of Kendall-Lee Notation for a Queueing system?

4. State the assumptions of Birth-Death Processes.

5. Define discrete and continuous systems with an example for each.

6. Draw the flowchart for breakdown and maintenance in Stochastic Simulation.

7. Find the graphical solution for the following LPP.

Maximize z=4x1+3x2

Subjext to x1+x2≤60;

2x1+x2≤60;

x1,x2≥0.

8. Illustrate how the following inequality constraints are converted into equality constraints.

Maximize z=50 x1+20 x2+30x3+80x4

Subject to 400x1+200x2+150x3+500x4≥500

3x1+2x2≥6

2x1+2x2+4x3+4x4≥10

2x1+4x2+x3+5x4≥8;

xi≥0(i=1,2,3,4).

9. State the Kuhm-Tucker conditions for an NLP with maximization.

10. Name two different algorithms to solve constrained NLP.

**PART B-(5 × 16 = 80 marks)**

11. (a) (i) An average of 10 cars per hour arrive at a single-server drive-in teller. Assume that

the average service time for each customer is 4 minutes, and both interarrival times and times

and service times are exponential.

(1) What is the probability that the teller is idle?

(2) What is the average number of cars waiting in line for the teller? (A car that is being

served is not considered to be waiting in line.)

(3) What is the average amount of time a drive-in customer spends in the bank parking lot

(including time in service)?

(4) On the average, how many customers per hour will be served by the teller? (8)

(ii) A one-man barber shop has a total of 10 seats. Interarrival times are exponentially

distributed, and an average of 20 prospective customers arrives each hour at the shop. Those

customers who find the shop full do not enter. The barber takes an average of 12 minutes to

cut each customer’s hair. Haircut times are exponentially distributed.

(1) On the average, how many haircuts per hour will the barber complete?

(2) On the average, how many times will be spent in the shop by a customer whoenters? (8)

Or

(b) Explain Machine Interference Model and solve the following problem. The Town Police

Department has 5 patrol cars. A patrol car breaks down and requires service once in every 30

days. The police department has two repair workers; each of them takes an average of 3 days

to repair a car. Breakdown times and repair times are exponential.

(i) Determine the average number of police cars in good condition.

(ii) Find the average down time for a police car that needs repairs.

(iii) Find the fraction of the time a particular repair worker is idle.

12. (a) Consider an M/M/I/GD/∞/∞ system with Î» = 5 customers per hour and Î¼=8 customers

per hour. Use the results of Pollaczek and Khinchin to analyze the efficiency of

M/M/1/GD/∞/∞ queueing system with M/M/I/GD/∞/∞ queueing system with

M/M/I/GD/∞/∞ queueing system.

Or

(b) (i) Consider two servers. An average of 8 customers per hour arrive from outside at server

1, and an average of 17 customers per hour arrive from outside at server 2. Interarrival times

are exponential. Server 1 can serve at an exponential rate of 20 customers per hour, and

server2 can serve at an exponential rate of 30 customers per hour. After completing service at

server 1, half of the customers leave the system, and half go to server 2. After completing

service at server 2 , ¾ of the customers complete service, and ¼ return to server 1.

(1) What fraction of the time is service 1 idle?

(2) Find the expected number of customers at each server.

(3) Find the average time a customer spends in the system.

(4) How would the answers to parts (1)—(3) change if the server 2 could serve only

an average of 20 customers per hour? (8)

(ii) The last two things that are done to a car before its manufacture is complete are installing

the engine and putting on the tyres. An average of 54 cars per hour arrives requiring these

two tasks. One worker is available to install the engine and can service an average of 60 cars

per hour. After the engine is installed, the car goes to the tyre station and waits for its tyres to

be attached. Three workers serve at the tyre station. Each works on one car at a time and can

put tyres on a car in an average of 3 minutes. Both interarrival times and service times are

exponential.

(1) Determine the mean queue length at each work station.

(2) Determine the total expected time that a car spends waiting for service. (8)

13. (a) Explain and draw the flowchart for simulation Model for Single-Server

QueueingSystem.

Or

(b) A Bakery bakes and sells French bread. Each morning, the bakery satisfies the demand

for the day using freshly baked bread. It can bake the bread only in batches of a dozen loaves

each. Each loaf costs Rs. 25 to make. Assume that the total daily demand for bread occurs in

multiples of 12. Past data have shown thatthis demand ranges from 36 to 96 loaves per day. A

loaf sells for Rs. 40, and any bread left over at the end of the day is sold to a charitable

kitchen for a salvage price of Rs. 10/loaf. If demand exceeds supply, assume that there is a

lost-profit cost of Rs. 15/loaf (because of loss of goodwill, loss of customers to competitors,

and so on). The bakery records show that the daily demand can be categorized into three

types: high, average, and low. These demands occur with probabilities of .30, .45, and .25,

respectively. The distribution of the demand by categories is given in the optional number of

loaves to bake each day to maximize profit(revenues + salvage revenues – cost of bread –

cost of lost profits). Table : demand probability distribution

Demand High Average low

36 0.05 0.10 0.15

48 0.10 0.20 0.25

60 0.25 0.30 0.35

72 0.30 0.25 0.15

84 0.20 0.10 0.05

86 0.10 0.05 0.05

14. (a) Solve the transportation problem to find the optional solution

8 6 10 9 35

9 12 13 7 50

14 9 16 5 40

45 20 30 30

Or

(b) Solve the given LPP using Big-M method:

Minimize z=2x1+3x2

Subject to ½ x1 +1/4 x2≤4

x1+3x2≥20

x1+x2=10

and x1,x2≥0

15. (a) A company is planning to spend $10,000 on advertising. It costs $3,000 per minute to

advertise on television and $1,000 per minute to advertise on radio. If the firm buys x minutes

of television advertising and y minutes of radio advertising, then its revenue in thousands of

dollars is given by f(x,y)=-2x2-y2-xy-8x-3y. How can the firm maximize its revenue?

Or

(b) Minimize f=x12+2x22+3x2

3 subject to the constraints

g1=x1-x2-2x3≤12

g2=x1+2x2-33x3≤8 using Kuhn-Tucker conditions.

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