INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIESBANGALORE

QUANTITATIVE TECHNIQUES AND OPERATIONS RESEARCH

QUEUEING THEORY1. A TV repairman finds that the time spent on his jobs has an exponential distribution with mean

30 minutes. If he repairs sets in the order in which they come in. If the arrival of sets is approximately Poisson with an average rate of 10 per 8 hour day, what is the repairman’s expected idle time each day? How many jobs are ahead of the average set just brought in ?

2. At what average rate must a clerk at a super market work in order to insure a probability of 0.90 that the customers will not have to wait longer than 12 minutes ? It is assumed that there is only one counter to which customers arrive in a Poisson fashion at an average rate of 15 per hour. The length of service by the clerk has an exponential distribution.

3. Arrivals of telephone booth are considered to be Poisson Distribution with an average time of 10 minutes between one arrival and the next. The length of a phone call is assumed to be exponentially with mean 3 minutes. (a) What is the probability that a person arriving at the booth will have to wait ?(b) What is the average length of the queue that forms from time to time ?(c) The telephone department will install a second booth when convinced that an arrival would

expect to have to wait at least three minutes for the phone. By how much must the flow of arrivals be increased in order to justify second booth ?

4. Customers arrive at a one window drive in bank according to Poisson Distribution with mean 10 per hour. Service time per customer is exponential with mean 5 minutes. The space in front of the window including that for the serviced car can accommodate a maximum of 3 cars. Others can wait outside this space.(a) What is the probability that an arriving customer can drive directly to the space in front of

the window ?(b) What is the probability that an arriving customer will have to wait outside the indicated space

?(c) How long is an arriving customer expected to wait before starting service ?

5. In a Super market, the average arrival rate of customer is 10 every 30 minutes following Poisson process. The average time taken by a cashier to list and calculate the customer’s purchase is 2.5 minutes following exponential distribution. What is the probability that the queue length exceeds 6 ? What is the expected time spent by the customer in the system ?

6. On an average, 96 patients per 24 hour day require the service of an emergency clinic. Also, on an average, a patient requires 10 minutes of active attention. Assume that the facility can handle only one emergency at a time. Suppose that it cost the clinic Rs. 100 per patient treated to obtain an average servicing time of 10 minutes and thus each minute of decrease in this average time would cost Rs. 10 per patient treated. How much would have to be budgeted by the clinic to decrease the average size of the queue from 1 1/3 patients to ½ patients ?

7. In a public telephone booth, the arrivals are on the average 15 per hour. A call on the average takes 3 minutes. If there is just one phone, find (i) the expected number of callers in the booth at just one phone, and (ii) the proportion of the time the booth is expected to be idle ?

8. In a railway marshalling yard, goods train arrive at a rate of 30 trains per day. Assuming that inter arrival time and the service time distribution follows an exponential distribution with an average of 30 minutes. Calculate the following (a) The mean queue size(b) The probability that queue size exceeds 10(c) If the input of the train increases to an average of 33 per day, what will be the changes in (a)

and (b) ?

9. In a railway marshalling yard, Goods train arrive at the rate of 30 trains per day. Assume that the inter arrival time follows an exponential distribution and the service time is also to be assumed as exponential with mean of 36 minutes. Calculate (a) the probability that the yard is empty (b) the average queue length assuming that the line capacity of the yard is 9 trains.

10. A car park contains 5 cars. The arrival of cars is Poisson at a mean rate of 10 per hour. The length of time each car spends in the car park is exponential distribution with mean of 5 hours. How many cars are in the car park on an average ?

11. Customers arrive at a booking office window, being manned by a single individual at a rate of 25 per hour. Time required to serve a customer has exponential distribution with a mean of 120 seconds. Find the mean waiting time of a customer in the queue. A belt snapping for conveyors in an open cast mine occur at the rate of 2 per shift. There is only one hot plate available for vulcanizing, and it can vulcanize on an average 5 belts snap per shift. (a) What is the probability that when a belt snaps, the hot plate is readily available ?(b) What is the average number of belts in the system ?(c) What is the waiting time of an arrival ?(d) What is the average waiting time plus vulcanizing time ?

12. Telephone authorities have studied a locality where public telephone booth installed has the following data : Arrivals (Mean inter arrival time) = 10 minutes The length of phone call on an average = 3 minutesArrival, follow Poisson Distribution and the telephone call follows negative exponential distribution.

(i) Find the probability that the telephone is available when you go to the booth. (ii) The P and T Department is prepared to install second booth provided the customer has to

wait at least 3 minutes. By how much the flow of arrivals to be increased to justify the same?

INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIESBANGALORE

QUANTITATIVE TECHNIQUES AND OPERATIONS RESEARCH

LINEAR PROGRAMMING PROBLEM1. The Manager of an oil company must decide on the optimal mix of 2 possible

blending processes of which the inputs and outputs per production run are as follows:

Process Input OutputI 6 3 6 9II 5 6 5 5The maximum availability of crude A & B are 250 units and 200 units respectively. The market requirements show that atleast 150 units of gasoline X and 130 units of gasoline Y respectively. Formulate the problem for maximizing the profit.

2. Write the duality in Linear Programming Problem.Minimise z = 2 x

INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIESBANGALORE

QUANTITATIVE TECHNIQUES AND OPERATIONS RESEARCH

THEORY QUESTIONSModule ISection A

1. Define Operations Research.2. What is an OR model ?3. What is the importance of OR in the business ?4. State the applications of Operations Research in Marketing Management.5. Define the concept of optimization in Operations Research.6. Write any four models in Operations Research.

Section B & C1. Explain the phases in the usages of Operations Research.2. Explain the scope of OR models.3. Explain the different types of OR models.4. Explain the different techniques of Operations Research.5. Give Problems of Operations Research in real life.

Module IISection A

1. Define a Linear Programming Problem in general terms.2. What is Linear Programming Problem ? Explain the term optimal solution to LPP ?3. Illustrate multicriteria optimization.4. Give an application of assignment technique.5. What is Sensitivity analysis ?6. Give an example of Transportation problem.7. What is meant by Dual in LPP ?8. What are the assumptions considered in solving assignment problems ?9. What is the application of travelling salesmen problem ?10. Explain the two phase method.11. What is Graphical method of problem solving ?12. What are the basic steps involved in Transportation method ?13. What is Unbalanced and Balanced Transportation model ?14. What are the steps involved in assignment model ?15. What is assignment model ?16. What are Unbalanced and Balanced assignment model ?17. What are surplus variables in LPP ?18. Explain and illustrate the concept of optimization

19. Specify the transportation problem. 20. Give an example of assignment problem.

Section B1. Explain the following statement with an example. “A Transportation problem is a special type

of linear programming problem”.2. What is meant by Duality in Linear Programming Problem ? Explain the steps.3. Give structure of a Linear Programming Problem. Give one illustration each for maximizing

problem and minimizing problem. Explain that a transportation problem is also a linear programming problem by giving an example.

4. Explain the scope and structure of Operations Research.

Module IIISection A

1. Explain queuing theory.2. What are the two objectives of queuing problem solving ?3. Explain Replacement Policy.4. Write the formula for probability of ‘n’ number in the system.

Section B1. Give a general structure of a queuing system. Give some examples of queuing situations.2. What are the assumptions for solving the scheduling models ?

Module IVSection A

1. State the objectives of Network analysis.2. What are the advantages of Network analysis ?3. What is PERT ?4. What is the “Distribution” PERT follows ? Write the formula for expected time calculation and

standard deviation for PERT.5. Define critical path.6. Define shortest path.

Section B1. Distinguish between PERT and CPM.2. What are the important phases in Project Management ?

Module VSection A

1. What is carrying cost of inventory ?2. What is ABC analysis ?3. What is VED analysis ?4. What is FSN analysis ?5. What is EOQ analysis ?

Section B1. Classify the inventory models.2. Explain Economic Order Quantity Model or Economic Lot Size model.

Module VI

Section A1. Define Game theory.2. State the types of games theory model.3. Explain Zero sum game.4. State any two methods of measuring seasonal variations in a time series.5. Mention the components of a time series.6. What is the property of random number and what is the test to certify the random numbers ?7. What is simulation ?8. State one merit and one limitation of simulation techniques.9. Explain simulation procedures.10. Distinguish between pure strategy and mixed strategy.

Section B1. Outline a forecasting method using time series analysis.2. What is game theory ? State the assumptions underlying it. Discuss its importance to business

decisions.3. What is simulation ? When do you use simulation ?4. What is Group replacement ? How do you calculate the mean age of a bulb for group

replacement of bulbs on a industry.5. What is simulation ? What are its advantages ? What is a random number table ? Explain

Monte Carlo simulation.6.