M.E (OR) QB

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

First Semester

Computer Science and Engineering

MA9219 - OPERATION RESEARCH

(Common to M.E-Network Engineering, M.E-Software Engineering and M.Tech- IT)

(Regulation 2009)

Time: Three hours Maximum: 100 Marks

Answer all the 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 - Khinctchine 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 Quadratic programming

problem.

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

Part B – (5*16=80 Marks)

11. (a) (i) Explain system and solve it under steady state condition. (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 bump a train) distribution is also exponential with an

average 36 minutes. Calculate the following

(1) The average no 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 in case of first come first serve basis. Give a suitable

illustration. (8)

(ii) An automobile inspection in which there are three inspections 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 no 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 having a 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 no of customers

waiting for a haircut.

(2) The percentage of customers who have to wait prior getting into the Barber’s

chair.

(3) The percent of time an arrival can walk without having to wait. (9)

(Or)

(b) (i) Explain briefly open and closed networks models in a queue system. (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 & 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 hours at a cost of Rs.3 per truck unloaded.

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 & solutions 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

& 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 the computation of an

optimum solution to a linear programming problem? (4)

(ii) Explain the meaning of basic feasible solution and degenerate solution in a linear

programming problem. (4)

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

Subject to the constraints

. (8)

(Or)

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

when demand and supply are not equal? (8)

(ii) A company has factories at , and which supply to warehouses at ,

and . Weekly factory capacities are 200,180,120 and 150 units respectively.

Unit shipping costs (in rupees) are as follows.

Factor

Warehouse

Supply

16 20 12 200

14 8 18 160

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 quadratic programming 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)

Subject to the constraints

.

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

First Semester

Computer Science and Engineering

MA9219- OPERATION RESEARCH

(Common to M.E-Network Engineering, M.E-Software Engineering and M.Tech- IT)

(Regulation 2009)

Time: Three hours Maximum: 100 Marks

Answer all the questions

Part A – (10*2=20 Marks)

1. Define queue discipline.

2. What do you mean by (a) steady state and (b) transient state in a queuing system?

3. Write down Pollaczek Khintchine formulae.

4. What is meant by closed queuing network?

5. Mention the types of simulation.

6. Specify any two advantages of simulation.

7. Define Basic feasible solution of a LPP.

8. Write down the mathematical formulation of a transportation problem.

9. What are the Kuhn-Tucker conditions for solving a non-linear programming problem?

10. What is Quadratic programming?

Part B – (5*16=80 Marks)

11. (a) Arrivals at a telephone booth are considered to be Poisson, with an average time of

10 minutes between one arrival and the next. The length of the phone call is assumed to

be distributed exponentially.

(1) What is the probability that a person arriving at the booth will have to wait?

(2) What is the average length of the queues that form from time to time?

(3) The telephone department will install a second booth when convinced that an arrival

would expect to have a wait atleast three minutes for the phone. By how much must

the flow of arrivals be increased in order to justify a second booth? (16)

(Or)

(b) A super market has two salesmen 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.

(1) Calculate the probability that an arrival will have to wait for service.

(2) Find the expected percentage of idle time for each salesman.

(3) If a customer has to wait, find the expected length of his waiting time. (16)

12. (a) An automatic car wash facility operates with only one bay. Cars arriving according to

Poisson distribution with a mean of 4 cars per hour may wait in the facility’s parking lot

if the bay is busy. The time of washing and cleaning a car is exponential, with a mean of

10 minutes. Cars that cannot park in the lot can wait in the street bordering the wash

facility. The manager of the facility wants to determine the size of the parking lot.

Suppose that a new system is installed so that the service time for all cars is constant

and equal to 10 minutes, how does the new system affect the operation facility. (16)

(Or)

(b) 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. Inter - arrival

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

and server 2 can serve at an exponential rate of 30 customers per hour. After completing

service at serve 1, half the customers leave the system and half go to server 2. After

completing server 2, of the customers complete service and returns to server 1.

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

(2) Find the expected no 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 server 2 could serve only an

average of 20 customers per hour. (16)

13. (a) The occurrence of rain in a city on a day depends upon whether or not it rained on the

previous day. If it has rained on the previous day, the rain distribution is

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 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

rainfall during the period. Use the random numbers 67 63 39 55 29 78 70 06 78 76

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

day before. (16)

(Or)

(b) Records of 100 truckloads of finished jobs arriving in a department’s check out area

show the following: Checking out takes 5 minutes and checker takes care of only one truck at

a time. The data is summarized in the following table:

Truck Inter Arrival Time 1 2 3 4 5 6 7 8 9 10

Frequency 1 4 7 17 31 23 7 5 3 2 (Total = 100)

As soon as the trucks are checked out, the truck drivers take them to the next

departments. Using Monte – Carlo simulations determine:

(1) What is the average waiting time before service?

(2) What is likely to be the largest? (16)

14. (a) Use two phase simplex method to solve the problem.

Subject to the constraints

. (16)

(Or)

(b) A company has 5 jobs to be done. The following matrix shows the return in rupees on

assigning machine to the . Assign the five

jobs to the five machines so as to maximize the total expected profit. (16)

Machine

A B C D E

1 5 11 10 12 4

2 2 4 6 3 5

3 3 12 5 14 6

4 6 14 4 11 7

5 7 9 8 12 5

15. (a) Find the dimensions of a rectangular parallelepiped with largest volume whose sides are

parallel to the coordinate planes, to be inserted in the ellipsoid,

. (16)

(Or)

(b) Apply Wolfe’s Method for solving the quadratic programming problem

Subject to the constraints

. (16)

M.E / M.Tech DEGREE EXAMINATION, NOV / DEC 2010

First Semester

Computer Science and Engineering

MA9219- OPERATION RESEARCH

(Common to M.E-Network Engineering, M.E-Software Engineering and M.Tech- IT)

(Regulation 2009)

Time: Three hours Maximum: 100 Marks

Answer all the questions

Part A – (10*2=20 Marks)

1. Solve graphically the following LLP:

S.t.c .

2. How is maximization converted into minimization in assignment problem?

3. What are the customer behaviors in queuing system?

4. Define an absorbing state in Markov chain.

5. Define Monte – Carlo method of simulation.

6. What are the tests used to ensure the uniformity and independence of random numbers?

7. Write the Pollaczek – Khinchine formula for mean time delay in queue.

8. Consider the problem : Subject to . Show that for the Lagrangian has

a stationary point at . Show also that does not maximize the

Lagrangian function .

9. Find all the local maxima and minima (if any) of the following functions and determine

whether each local extremism in a global extremis .

10. Find the stationary points of the following function using the method of constrained

variation optimize subject to .

Part B – (5*16=80 Marks)

11. (a) A petrol station has two pumps. The service time follows the exponential distribution

with mean 4 minutes and cars arrive for service in a Poisson process at the rate of 10 cars

per hour. Find the probability that a customer has to wait for service. What proportion of

time the pump remain idle? (16)

(Or)

(b) Cars arrive at a petrol pump, having one petrol unit, in Poisson fashion with an

average of 10 cars per hour. The service time in distributed exponentially with a mean of

3 minutes. Find

(1) Average number of cars in the system.

(2) Average waiting time in the queue

(3) Average queue length

(4) The probability that the number of cars in the system is 2. (16)

12. (a) A barber shop has two barbers and three chairs for customers. Assume that the customers

arrive in Poisson fashion at a rate of 5 per hour and that each barber services customers

according to an exponential distribution with mean of 15 minutes. Further if a customer

arrives and there are no empty chairs in the shop, he will leave. What is the probability

that the shop is empty? What is the expected number of customers in the shop? (16)

(Or)

(b) An order picking process in a warehouse gets calls for service at an average rate

of 8.5 per hour. The average time to fill the order in 0.1 hours. For analysis purpose

assumes both times are exponentially distributed. Analyzing the system as an

queue, the average time in the queue is 0.5667 hours. An opportunity arises to reduce the

variability of the process for filling orders. The inventory manager wonders if the change

is worth the cost. Analyze the problem using Non – Morkovian method. (16)

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

service time are constants and given by 1.5 and 4 minutes respectively. Simulate the

system by hand computations for 14 minutes.

(1) What is the waiting time per customer?

(2) What is the percentage idle time for the facility? (Assume that the system

starts at t = 0). (16)

(Or)

(b) Explain the components of Discrete - event Simulation and the Simulation engine logic?

Mention the application areas of Discrete – event simulation? (16)

14. (a) Prove using duality theory that the following linear program in feasible but has no

optimal solution

Subject to the constraints

. (16)

(Or)

(b) Goods have to be transported from sources , and to destinations , , ,

and respectively. The transportation cost per unit capacities the sources and

requirements of the destinations are given below. Determine the transportation schedule,

so that cost is minimized. (16)

Supply

4 1 2 6 9 100

6 4 3 5 7 120

5 2 6 4 8 120

Demand 40 50 70 90 90

15. (a) For each possible value of the constant a, solve the problem:

Subject to and . (16)

(Or)

(b) Consider the problem:

, Subject to for j = 1, …, m and for i = 1, …, n.

(i) Write down the Kuhn – Tucker conditions for this problem when it is written in the form:

Subject to for j = 1, …, m+n where for i = 1, …, n

(Write the derivative of the Lagrangian explicity in terms of the derivatives of f and

for j = 1, … , m using the notation for the derivatives of with respect to

at x. denote the Lagrange multiplier associated with the constraint for by

for j = 1, …, m and the multiplier associated with the constraint by

for I = 1, … , n).

(i) Write down the Kuhn – Tucker conditions tailored to problem with non – negativity

constraints.

(ii) Show that if satisfies the conditions in (i) the

satisfies the conditions in (ii) then there exists numbers such that

satisfies the conditions in (i). (16)

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

First Semester

Computer Science and Engineering

MA9219- OPERATION RESEARCH

(Common to M.E-Network Engineering, M.E-Software Engineering and M.Tech- IT)

(Regulation 2009)

Time: Three hours Maximum: 100 Marks

Answer all the questions

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.

Subject to

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

Subject to

.

9. State the Kuhn – 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 inter arrival 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. Inter arrival times are exponentially

distributed, and an average of 20 prospective customers arrive 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 much time will be spent in the shop by a customer

who enters? (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

whom takes an average of 3 days to repair a car. Breakdown times and repair times

are exponential.

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

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

(3) Find the fraction of the time a particular repair worker is idle. (16)

12. (a) Consider an system with customers per hour and

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

efficiency of queueing system with

queueing system. (16)

(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.

Inter arrival times are exponential. Server 1 can serve at an exponential rate of

20 customers per hour, and server 2 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, of the customers

complete service, and return to server 1.

(1) What fraction of the time is server 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 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 Queueing

System. (16)

(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 that this demand ranges from 30

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 following Table. Use Monte Carlo Simulation

to determine the optimal number of loaves to bake each day to maximize profit

(revenues + salvage revenues – cost of bread – cost of lost profits). (16)

TableDemand 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 optimal solution (16)

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

Subject to . (16)

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 . How can the firm

maximize its revenue? (16)

(Or)

(b) Minimize

subject to the constraints

Using Kuhn – Tucker conditions. (16)

M.E / M.Tech DEGREE EXAMINATION, JUNE 2012

First Semester

Computer Science and Engineering

MA9219- OPERATION RESEARCH

(Common to M.E-Network Engineering, M.E-Software Engineering and M.Tech- IT)

(Regulation 2009)

Time: Three hours Maximum: 100 Marks

Answer all the questions

Part A – (10*2=20 Marks)

1. Find the traffic intensity given per hour.

2. In a production company materials arrive at a rate of 30 bags per day. Service time

distribution is exponentially distributed with an average of 36 minutes. Find the probability

that the queue exceeds 10?

3. Write the Pollaczek – Khinchine formula.

4. What is called Non – Markovian queue?

5. The inter arrival time of customers follows a probability distribution as shown below. Write

the general purpose simulation system block.

Interval Time 1 2 3 4 5 6

Probability 0.10 0.20 0.25 0.25 0.10 0.10

6. The arrival rate of customers at a banking counter follows poisson distribution with a mean of

30 per hour. The service rate of the counter clerk also follows poisson with a mean of 45 per

hour. What is the probability of “zero customer” in the system?

7. Solve the following LPP using graphical method

Maximize

Subject to .

8. Discuss any two similarities between Transportation problem and Assignment problem.

9. Write the procedure to solve quadratic programming problem.

10. What is Non Linear programming problem?

Part B – (5*16=80 Marks)

11. (a) (i) There are three clerks in the loan section of a bank to process the initial queries of

customers. The arrival rate of customers follow poisson distribution and it is 20 per

hour. The service rate also follows poisson distribution and it is 9 customers per

hour. Find

(1) Average waiting number of customers in the queue as well as in the system.

(2) Average waiting time per customers in the queue as well as in the system.

(8)

(ii) Explain the various models of Queueing theory. (8)

(Or)

(b) (i) For the with service rate customers per hour. How and

increases as the arrival rate increases from 5 to 8.64 by increments of 20% and

then to . (8)

(ii) Patients arrive at a clinic according to Poisson distribution at a rate of 30 patients per

hour. The waiting room does not accommodate more than 14 patients. Examination

time per patient is exponential with mean rate of 20 per hour.

(1) Find the effective rate at the clinic.

(2) What is the expected waiting time until a patient is discharged from the

clinic? (8)

12. (a) (i) Derive the Pollaczec – Khintchine formula for an queueing model. (8)

(ii) At a Driver’s Licence branch office drivers arrive at a rate of 50 per hour. All arrivals

must first check in with one of two clerks with the average check in time being 2

minutes. After checking in 15% of the drivers need to take a written test that lasts

approximately 20 minutes. All arrivals must wait to have their picture taken and their

license produced, this station can process about 60 drivers per hour. How to reduce

the customers delay? Whether adding a check in clerk (or) a new photo station?

(8)

(Or)

(b) (i) Write about the multiple server Poisson Queue model. (8)

(ii) A petrol pump station has 4 pumps. The service times follow the exponential

distribution with a mean of 6 minutes and cars arrive for service in a poisson process

at the rate of 30 cars per hour.

(1) What is the probability that an arrival would have to wait in line?

(2) For what % of time would a pump lie idle on an average? (8)

13. (a) (i) Write short note on Discrete event simulation. (8)

(ii) A toll gate in a highway consists of 5 lanes. The inter arrival time of the vehicles at

the toll gate follows uniform distribution with seconds. The service time also

follows uniform distribution with seconds. Draw a GPSS block diagram and

prepare a program to simulate the system for 10 hours. (8)

(Or)

(b) (i) Write short note on types of simulation. (8)

(ii) Explain the use of Transfer block with an example. (8)

14. (a) (i) Solve the following LPP using simplex method.

Subject to . (16)

(Or)

(b) (i) Find the optimal solution (16)

Plants

Warehouses

1 2 3 4 5

1 10 2 3 15 9 25

2 5 10 15 2 4 30

3 15 5 14 7 15 20

4 20 15 13 -- 8 30

20 20 30 10 25

15. (a) Solve the following Nonlinear programming problem using Lagrangian method.

Subject to . (16)

(Or)

(b) Solve the following quadratic programming problem

Subject to . (16)