DEPARTMENT OF MECHANICAL ENGINEERING

BM7002-OPERATIONS RESEARCH
QUESTION BANK
UNIT 1 – Linear Models
PART – A (2 Marks)

1. Define Operations Research?

2. What is linear programming?
3. What are slack and surplus variables?
4. Find the dual of the following LPP.
Max Z=x1+2x2+x3
Subject to 2x1+x2-x3≤2
-2x1+x2-5x3 ≥ -6
4x1+x2+x3 ≤ 6
X1, x2, x3 ≥ 0
5. List out the methods used to obtain initial basic feasible solution in Transportation
Problem.
6. Define an assignment Problem.
7. What are the phases of an operations research study?
8. Define Duality in LPP.
9. Define Decision variable.
10. What is the difference between feasible solution and basic feasible solution?
11.What do you mean by standard form of LPP?
12. What do you mean by canonical form of LPP?
13. . What do you mean by degeneracy in a Transportation Problem?
14. State the difference between the Transportation Problem and Assignment.Problem.
15. What is Two phase method?
16. What do you mean by an unbalanced Transportation Problem?
17. With an example, describe how to convert the minimization problem into maximization
problem in Simplex method?
18. What are the applications of O.R?
19. What is optimality test in transportation problem?
20. Define Artificial Variable.

Part-B (6 Marks)

1.Minimize Z=3x1+2x2 solve by graphically.

Subject to 5x1+x2≥10
x1+x2≥6
x1+4x2≥12
X1,x2,≥0

2. Write the steps involved in solving LPP Using Graphical method? And also write the
applications of Operations Research.

3. A Manufacturer produces two types of models M1 and M2. Each model of the type M1 requires
4 hours of grinding and 2 hours of polishing, whereas each model of the type M2 requires 2 hours
of grinding and 5 hours of polishing. The manufacturer has 2 grinders and 3 polishers. Each
grinder works 40 hours a week and each polisher works for 60 hours a week. Profit on M1 model
is Rs. 3.00 and on model M2 is Rs.4.00. whatever is produced in a week is sold in the market.
How should the manufacturer allocate his production capacity to the two types of models, so that
he may make the maximum profit in a week? Write a suitable LPP for the above question.
4. A company produces 2 types of hats. Every hat A require twice as much labour time as the
second hat be. If the company produces only hat B then it can produce a total of 500 hats a day.
The market limits daily sales of the hat A and hat B to 150 and 250 hats. The profits on hat A and
B are Rs.8 and Rs.5 respectively. Solve graphically to get the optimal solution.

5. Use Graphical method to solve the following LP problem

Maximize Z= 15x1+10x2
Subject to the constraints: 4x1+6x2≤360
3x1+0x2≤180
0x1+5x2≤200;
x1, x2≥0

6. Explain the scope of OR.

7. List the phases of OR and explain them.

8. A company manufactures two products A and B. Each unit of B takes twice as long to produce
as one unit of A and if the company were to produce only A it would have time to produce 2000
units per day. The availability of the raw material is sufficient to produce 1500 units per day of
both a and B combined. Product B recurring a special Ingredient only 600units can be
made per day. If A fetches a profit of Rs.2 per unit and B a profit of Rs.4 per unit, Formulate the
optimum product min.

9. Write down the mathematical formulation for transportation problem.

10. Use simplex method to solve the following LP problem

Maximize Z= x1+x2+3x3
Subject to 3x1+2x2+x3≤3
2x1+x2+2x3≤2
x1, x2≥0

11. Explain MODI method?

12.a)Obtain the dual of the following primal problem

Minimize z=3x1-2x2+ x3
Subject to : 2x1-3x2+ x3 ≤5
4x1-2x2 ≥9
-8x1+4x2+3x3=8
x1,x2 ≥0, x3 is unrestricted.

b) And also write difference between Primal and Dual in LPP.

13. Write down the steps involved in solving Assignment problem using Hungarian Method.

14. A person requires at least 10 and 12 units of chemicals A and B respectively, for his garden. A
liquid product contains 5 and 2 units of A and B respectively per bottle. A dry product contains 1
and 4 units of A and B respectively per box. The liquid products are sold for Rs. 30 per bottle, dry
products are sold for Rs. 40 per box. How many of each should be purchased in order to minimize
the cost and meet the requirements? Formulate the L.P.P.

15. a) Difference between Transportation and Assignment Problem

b) Difference between Optimal solution and feasible solution

Part-C (10 Marks)

1. Solve by using simplex method,

Maximize Z= Max Z=4x1+10x2
Subject to 2x1+x2≤10
2x1+5x2≤20
2x1+3x2≤18
X1,x2≥0

2. A paper mill produces 2 grades of paper namely x and y. Because of raw material
restrictions, it cannot produce more than 400 tonnes of grade x and 300 tonnes of
grade y in a week. There are 160 production hours in a week. It requires 0.2 hours and
1.4 hours to produce a tone of product x and y respectively, with corresponding
profits of Rs.200 and Rs.500 per ton. Formulate the above LPP to maximize the profit
using the graphical method.

3. Solve by Using VAM Method

Origin/Destination D1 D2 D3 D4 Supply
O1 11 13 17 14 250
O2 16 18 14 10 300
O3 21 24 13 10 400
Demand 200 225 275 250 950

4. The Processing time in hours for the jobs when allocated to different machines is indicated
below. Assign the machines for the jobs so that the total processing time is minimum?

5. Solve using Vogel’s Approximation Method and perform optimality Test using
MODI method
D1 D2 D3 D4 Supply
O1 2 3 11 7 6
O2 1 0 6 1 1
O3 5 8 15 9 10
Demand 7 5 3 2 17

6.Use penalty method or Big M method to solve Linear Programming Problem

Minimize Z=4x1+x2
Subject to 3x1+x2=3
4x1+3x2≥6
x1+2x2≤3
X1,x2≥0
 DEPARTMENT OF MECHANICAL ENGINEERING
BM7002-OPERATIONS RESEARCH
QUESTION BANK
UNIT 2 – Network Models
PART – A (2 Marks)

1. Define Critical Path?

2. Define Critical Activity?
3. What is dummy activity?
4. What is network scheduling?
5. What is a network?
5. What is spanning tree?
6. What are the three types of float?
7. What is slack?
8. What is float?
9. What is merge event?
10. What is PERT method?
11. What is shortest route problem?
12. Define the expected variance of project length.
13. What is total float?
14. What are the common errors in construction of a network?
15. Define Maximum flow?
16. Define minimum spanning tree?
17. What is the difference between an event and an activity?
18. What are the three phases of project?
19. Difference between preceding and succeeding activity.
20. What is Time Analysis in a network model?

Part-B (6 Marks)
1. a) Distinguish between PERT and CPM,
b) Distinguish between Free float and Independent Float.
2. Explain the shortest route problem with an example
3. Write short notes on maximum flow models.
4. Explain the minimum spanning tree with an example.
5. Write the steps involved in forward pass and backward pass calculation.
6. Write down the steps used in solving Network Model using Fulkerson’s Rule.
7. Listed in the table are the activities and sequencing requirements necessary for
completing the research project. Find the critical path.

Activity A B C D E F G H I J K L M
Duration 4 2 1 12 14 2 3 2 4 3 4 2 2
Immediate E A B K - E F F F I,L C,G,H D I,L
Predecessor

8. Explain in detail about various phases of project management.

9. Calculate the earliest start, earliest finish, latest start and latest finish of each
activity of the project given below:
 Activity 1-2 1-3 1-5 2-3 2-4 3-4 3-5 3-6 4-6 5-6
Duration 8 7 12 4 10 3 5 10 7 4
(weeks)

10. Write short notes on Prim’s Algorithm in solving Minimum spanning tree with an
example.
11. A project schedule has the following characteristics. Draw the network diagram
and find the critical path.

Activity 1-2 1-3 2-4 3-4 3-5 4-5 4-6 5-6

Time(days) 6 5 10 3 4 6 2 9

12. A project schedule has the following characteristics. Draw the network diagram
and find the critical path.

Activity 1-2 1-3 2-3 2-4 3-4 4-5

Time(days) 20 25 10 12 6 10

13. Write notes on augmenting path algorithm.

14. A small project consists of seven activities for which the relevant data are given
below:
Activity Preceding Activity Duration(in days)
A - 4
B A 7
C - 6
D C 5
E B 7
F D,E 6
G F 5

Draw the network and find the project completion time.

15. Explain the steps in PERT method and also write the formula in calculating
project variance and estimated time.

Part-C (10 Marks)

1. The following table shows the jobs of a network along with their time estimates.
Job 1-2 1-6 2-3 2-4 3-5 4-5 6-7 5-8 7-8
a(days) 1 2 2 2 7 5 5 3 8
m(days) 7 5 14 5 10 5 8 3 17
b(days) 13 14 26 8 19 17 29 9 32
Draw the project network and find the probability of project completion in 40 days

2. A project schedule has the following characteristics.

Activity 1-2 1-3 2-4 3-4 3-5 4-9 5-6 5-7 6-8 7-8 8-10 9-10
Time(days) 4 1 1 1 6 5 4 8 1 2 5 7

From the above information, you are required to

1. Construct a network diagram.
2. Compute the earliest and latest event time
3. Determine the critical path and total project duration.
4. Compute the total and free float for each activity.

3. A project has the following activities and other characteristics:

Estimated Duration (in weeks)
Activity(i-j) Optimistic Most likely Pessimistic
1-2 1 1 7
1-3 1 4 7
1-4 2 2 8
2-5 1 1 1
3-5 2 5 14
4-6 2 5 8
5-6 3 6 15

(i) What is the expected project length? (ii) What is the probability that the project
will be completed no more than 4 weeks later than expected time?

4. A small maintenance project consists of the following jobs whose precedence

relationships are given below.
Job 1-2 1-3 2-3 2-5 3-4 3-6 4-5 4-6 5-6 6-7

Duration 15 15 3 5 8 12 1 14 3 14
(days)

(i) Draw an arrow diagram representing the project (ii) Find the total float for each
activity (iii) Find the critical path and the total project duration.

5. Explain Kruskal’s Algorithm to solve Minimum spanning tree with an example.

6. The Stagecoach Shipping Company transports oranges by six trucks from Los
Angeles to six cities in the West and Midwest. The different routes between Los
Angeles and the destination cities and the length of time, in hours, required by a truck
to travel each route are shown in figure. Find the shortest travel time to each this
destination.
 DEPARTMENT OF MECHANICAL ENGINEERING
BM7002-OPERATIONS RESEARCH
QUESTION BANK
UNIT 3 – Inventory Models
PART – A (2 Marks)

1. What is meant by inventory?

2. Mention the various types of inventory.
3. What are the different costs that are involved in the inventory
problem?
4. Define holding cost and setup cost
5. Briefly explain probabilistic inventory model.
6. Distinguish between deterministic model and probabilistic
model.
7. Define buffer stock or safety stock.
8. Define Lead time and reorder point.
9. Summarize the causes of poor inventory control.
10. Define shortage cost.
11. What is Economic order quantity?
12. Write the EOQ formula under purchasing model with
shortages and without shortages.
13. Write the EOQ formula under manufacturing model with
shortages and without shortages.
14. Write the formula for finding total cost with price breaks.
15. What is demand?
16. What are controlled variables in inventory problem?
17. What are uncontrolled variables in inventory problem?
18. What is order cycle? On what basis order cycle is created?
19. Write the four types of deterministic inventory models.
20. What is lot size inventory?

Part-B (6 Marks)
1. The demand for an item is 18000 units per year. The holding cost is
Rs1.20 per unit time and the cost of shortage is Rs.5.00. The production
cost is Rs.400.00. Assuming that replacement rate is instantaneous
determine the optimum order quantity.
2. The demand for an item is 12000 per year and the shortage is
allowed. If the unit cost is Rs.15 and the holding cost is Rs.20 per year
per unit determine the optimum total yearly cost. The cost of placing
one order is Rs.6000 and the cost of one shortage is Rs.100 per year.
3. Discuss briefly about the different types of inventory and various
costs involved in inventory problems.
4. A company has a demand of 12,000 units/year for an item and it can
produce 2000 such items per month. The cost of one setup is Rs.400
and the holding cost/unit/month is Rs. 0.15. Find the optimum lot size
and max inventory.
 5. A newspaper boy buys papers for 30 paise each and sells them for 70
paise. He cannot return unsold news papers. Daily demand has the
following distribution.

No. of 23 24 25 26 27 28 29 30 31 32
customers
Probability 0.01 0.03 0.06 0.10 0.20 0.25 0.15 0.10 0.05 0.05

If each day’s demand is independent of the previous day’s, how many

papers should he order each day?

6. The demand of an item is uniform , at a rate of 25 unit per month.

The fixed cost is Rs.15 each time a production run is made. The
production cost is Rs.1 per item and the inventory carrying cost is
Rs.0.30 per item per month. If the shortage cost is Rs.1.50 per item per
month, determine the frequency and size of the production run that is to
be made.
7. The annual consumption of an item is 2000 units. The ordering cost
is Rs.100 per order. The carrying cost is Rs.0.80 per unit, per year.
Assuming working days as 200, lead time as 20 days, and safety stock
as 100 units, calculate i) EOQ, ii) The number of orders per year.

8. For a fixed order quantity, determine i)EOQ, ii) Optimum buffer

stock.
Annual consumption, R=10,000 units, cost of one unit = Rs.1, C3=
Rs.12 per production run, C1=0.24 per unit. Maximum lead time =30
days and normal lead time =15 days.

9.A particular item has an annual demand of 9000 units. The carrying
cost is Rs.2 per unit, per year. The ordering cost is Rs.90. Find i)EOQ
ii) Determine the number of orders to be placed per annum.

10. Calculate EOQ and buffer stock from the following data. Annual
consumption is 12000 units at the cost of Rs.7.50 per unit. Set up cost
is Rs.6 and the average inventory holding cost is Rs.0.12 per unit.
Normal lead time is 15 days and maximum lead time is 20 days.

11. Find the optimum order quantity for a product, the price break for
which is as follows.
Quantity Unit cost(Rs.)
0≤q1≤500 10
500≤q2 9.25

The monthly demand for the product is 200 units, the cost of storage is 2
percent of the unit cost and the cost of ordering is Rs.350.

12.The annual demand for the product is 10,000 units, Each unit costs
Rs.100 for orders placed in quantities below 200 units but for orders of 200
or above the price is Rs.95. The annual inventory holding cost is 10 percent
of the value of the item and the ordering cost is Rs.5 per order. Find the
economic lot size. The cost of storage is 2 percent of the unit cost and the
cost of ordering is Rs.500 per order. Find the economic lot size?
13. An item is produced at the rate of 50 units per day. The demand occurs
at the rate of 25 items per day. If the set up cost is Rs.100 per run and the
holding cost is Rs.0.01 per unit of item, per day, find the economic lot size
for one run, assuming that shortages are not permitted. Also find the time of
the cycle and minimum cost for one run.

14. Discuss the assumptions made in purchase model with shortages and
without shortages and also write the formula involved in their calculations.

15. Discuss the assumptions made in manufacturing model with shortages

and without shortages and also write the formula involved in their
calculations.

Part-C (10 Marks)

1. A manufacturer has to supply his customer with 600 units of his
products per year. Shortages are not allowed and storage cost
amounts to 60 paise per unit per year. The set up cost is Rs 80.00 find
i) EOQ ii) The minimum average yearly cost. iii) The optimum
number of orders per year. iv) The optimum period of supply per
optimum order.

2. The following table gives the annual demand and unit price of four
items.
Item A B C D
Annual demand(Units) 800 400 392 13800
Unit Price(Rs.) 0.02 1.00 8.00 0.20

Order cost is Rs.5 per order and holding cost is 10 percent of the
price.
i) Determine the EOQ in units.
ii)Calculate total variable cost.
Iii)Calculate the EOQ in year of supply
iv)Determine the number of orders per year?

3. The annual requirement for a product is 3000 units. The ordering

cost is Rs. 100 per order. The cost per unit is Rs.10. The carrying cost
per unit , per year is 30 percent of the unit cost. i)Find the EOQ. By
using better organizational methods, the ordering cost per order can
be brought down to Rs. 80 per order, but the same quantity as
determined above ha to be ordered. Ii) If a new EOQ I found by
using the ordering cost as Rs. 80, what would be the further saving in
cost?

4.Find the optimum order quantity for a product or which the price
breaks are as follows.
Quantity Unit Cost(Rs.)
0≤q1≤100 Rs.20 per unit
100≤q2≤200 Rs.18 per unit
200≤q3 Rs.16 per unit
The monthly demand for the product is 400 units. The storage cost is
20% of the unit cost of the product and the cost of ordering is Rs.25.
5.The demand for an item is deterministic and constant over tme and
is equal to 600 units per year. The per unit cost is Rs.50, while the
cost of placing an order is Rs.5. The inventory carrying cost is 20
percent of the cost of inventory per annum and the cost of shortage is
Rs.1 per unit, per month. Find the optimal quantity when stock outs
are permitted. If stock outs are not permitted, what would be the loss
to the company?

6.Find the optimum order quantity for a product, the price breaks of
which are as follows.
Quantity Unit cost(Rs.)
0≤q1≤800 Rs.1
800≤q2 Rs.0.98

The yearly demand for the product is 1,600 units, cost of placing an
order is Rs.5 and the cost of storage is 10 percent per year.
 DEPARTMENT OF MECHANICAL ENGINEERING
BM7002-OPERATIONS RESEARCH
QUESTION BANK
UNIT 4– Replacement Models

PART – A (2 Marks)
1. What is replacement?

2. When should the replacement be done?

3. What are the categories into which the replacements of items are
classified?

4. When do we replace a machine considering the time t as a discrete

variable and ignoring changes in the value of money?

5. Describe briefly some of the replacement policies?

6. Define group replacement.

7. Define individual replacement.

8. Differentiate between individual and group replacement.

9. Define discount factor.

10. Write is salvage value?

11. Write the formula for optimum replacement when salvage is

considered.

12. Write the formula for optimum replacement when salvage value is
negligible whose money value changes with time.

13. What is present worth factor?

14. State the conditions under which group replacement is superior to

individual replacement.

15. Define Sequencing.

16. Define processing order and processing time.

17. What is the principle assumption made in sequencing problems?

18. When can we apply Johnson’s algorithm in finding the optimal

ordering of n jobs through 3 machines?

19. Find the present worth factor of the money to be spent in a year, if
the money is worth 5 percent per year.
20. Write the expression for weighted average cost.

Part-B(6Marks)
1. A machine owner finds from his past records that the costs per year
of maintaining a machine, whose purchase price is Rs.6000, are as
given below.
Yr. 1 2 3 4 5 6 7 8
Maintenance 1000 1200 1400 1800 2300 2800 3400 4000
cost
Resale price 3000 1500 750 375 200 200 200 200
Determine at what age a replacement is due.

2. The cost of a new machine is Rs. 5000. The maintenance cost of

the nth year is given by Cn=500(n-1), n=1, 2…..
Suppose money is worth 5 percent per year, after how many years
will it be economical to replace the machine?

3. A machine owner finds from his past records that the costs per year
of maintaining a machine, whose purchase price is Rs.8000, are as
given below.
Yr. 1 2 3 4 5 6 7 8
Maintenance 1000 1300 1700 2200 2900 3800 4800 6000
cost
Resale price 4000 2000 1200 600 500 400 400 400
Determine the time at which it is profitable to replace the truck.

4. The cost pattern for two machines A and B, when money value is
not considered, is given in the table below.
Year Cost at the beginning of year
Machine A Machine B
1 900 1400
2 600 100
3 700 700

Find the cost pattern for each machine when money is worth 10
percent per year and hence, find which machine is less costly.

5. Explain the replacement of items that deteriorate with time under

the value of money doesn’t change with time and change with time.

6. The following mortality rates have been observed for a certain type
of light bulbs.
Week 1 2 3 4 5
Percent failing by the 10 25 50 80 100
end of week

There are 1000 bulbs in use and it costs Rs.2 to replace an individual
bulb, which has burnt out. If all the bulbs were replaced simultaneously,
it would cost 50 paise per bulb. Find the average cost of group
replacement policy.
7. The following table gives the running costs per year and resale
price of certain equipment, whose purchase price is Rs.5000.
Year 1 2 3 4 5 6 7 8
Running 1500 1600 1800 2100 2500 2900 3400 4400
cost
Resale 3500 2500 1700 1200 800 500 500 500
value
In what year is the replacement due?
8. The following table gives the running costs per year and resale
price of certain machine, whose purchase price is Rs.50,000.
Year 1 2 3 4 5 6 7 8
Running cost 15 16 18 21 25 29 43 40
(in 1000)
Resale value 35 25 17 12 8 5 5 5
(in 1000)
In what year is the replacement due?

9. The cost of a machine is Rs. 61,000 and its scrap value is Rs.1000.
The maintenance costs found from the past experiences are as
follows.
Year 1 2 3 4 5 6 7 8
Maintenance 1000 2500 4000 6000 9000 12000 16000 20000
cost in Rs.
When should the machine be replaced?
10. There are five jobs, each of which must go through the two
machines A and B in the order AB. Processing times are given
below.
Job 1 2 3 4 5
Machine A 5 1 9 3 10
Machine B 2 6 7 8 4
Determine a sequence for the five jobs that will minimize the total
elapsed time.
11. A company has six jobs, A to F. All the jobs have to go through two
machines M1 and M2. The time required for the jobs on each
machine in hours is given below. Find the optimum sequence that
minimizes the total elapsed time.
Job A B C D E F
M/c1 1 4 6 3 5 2
M/c2 3 6 8 8 1 5

12. The failure rates of 1000 street bulbs in a colony are summarized in
table.
End of month 1 2 3 4 5 6
Probability of 0.05 0.20 0.40 0.65 0.85 1.00
failure to date

The cost of replacing an individual bulb is Rs. 60. If all the bulbs are
replaced simultaneously it would cost Rs. 25 per bulb. Any one of the
following two options can be followed to replace the bulbs. Replace
the bulbs individually when they fail (individual replacement policy).
13. A truck is priced at Rs.60,000 and running costs are estimated at Rs.
6000 for each of the first four years, increasing by Rs.2000 per year in the
fifth and subsequent years. If the money is worth 10 percent per year, when
the truck should be replaced. Assume that the truck will eventually be sold
for scrap at a negligible price.

14. There are 1000 bulbs in the system. Survival rate is given below.
Week 1 2 3 4 5
Percent failing by the 10 20 40 60 100
end of week

Find the optimal costs under individual replacement policy if the cost of
replacement is Rs.5 per bulb.

15. There are 500 bulbs in the system. Survival rate is given below.
Week 1 2 3 4 5
Percent failing by the 5 25 50 75 100
end of week

Find the optimal costs under group replacement policy if the cost of
replacement is Rs.5 per bulb. If all the bulbs were replaced simultaneously
it could cost Rs. 2 per bulb.

Part-C (10 Marks)

1. A manufacturer is offered two machines A and B. A is priced at
Rs.50,000 and running costs are estimated at Rs.8000 for each of the
first five years, increasing by 2000 per year in the sixth and
subsequent years. Machine B of the same capacity costs Rs.
Rs.25,000 but will have running costs of Rs.12000 per year for six
years increasing by Rs.2000 per year thereafter. If money is worth
10% per year, which machine should be purchased?

2. The data on the running costs per year and the resale price of an
equipment A whose purchase price is Rs.2 lakhs are as follows.
i)What is the average period of replacement. ii) When A is 2years
old, an equipment B which is a newly available model. The optimum
period of replacement is 4 years with average costs of Rs.72, 000.
Should A be changed with B?
year 1 2 3 4 5 6 7
Running 30,000 38,000 46,000 58,000 72,000 90,000 1,10,000
cost
Resale 100,000 50,000 25,000 12,000 8,000 8,000 8,000
value

3. The probability Pn of failure just before age n is shown below. If

individual replacement costs Rs.12.50 and group replacement costs Rs.3
per item. Find the optimal replacement policy.
n 1 2 3 4 5
Pn 0.1 0.2 0.25 0.3 0.15
4. A machine costs Rs.6,000. The running cost and the salvage value at the
end of the year is given in the table below.
year 1 2 3 4 5 6 7
Running 1200 1400 1600 1800 2000 2400 3000
cost
Salvage 4000 2666 2000 1500 1000 600 600
value

If the interest rate is 10 percent per year, when should the machine be
replaced?
5. Find the sequence that minimizes the total elapsed time(in hours)
required to complete the following tasks on two machines.
Task A B C D E F G H I
M/c1 2 5 4 9 6 8 7 5 4
M/c2 6 8 7 4 3 9 3 8 11

6. Four jobs 1,2,3 and 4 are to be processed on each of the five machines
A,B,C,D and E in the order ABCDE. Find the total minimum elapsed time
if no passing of jobs is permitted. Also find the idle time for each machine.
Machines Jobs
1 2 3 4
A 7 6 5 8
B 5 6 4 3
C 2 4 5 3
D 3 5 6 2
E 9 10 8 6
 DEPARTMENT OF MECHANICAL ENGINEERING
BM7002-OPERATIONS RESEARCH
QUESTION BANK
UNIT 5 – Queuing Theory
PART – A (2 Marks)

1. Define a queue
2. What are the basic characteristics of a queuing system?
3. Define transient and steady state.
4. Explain Kendall’s notation.
5. Write Little’ formula
6. Define the following (1) Balking (2) Reneging (3) Jockeying
7. List the characteristic of a queueing system
8. Explain the queue discipline and its various forms:
9. Difference between Transient and steady states.
10. Classify Queuing models.
11. Define utilization factor.
12. Write Little’s formula.
13. Define a customer.
14. What is the distribution for service time and inter arrival time?
15. Define priority in customer’s behaviour.
16. What is efficiency of M/M/S model?
17. Write the meaning of (M/M/1):(∞/FCFS).
18. Write the meaning of (M/M/1):(N/FCFS).
19. Write the formula for finding expected waiting line the queue and
queue length Lq of model1.
20. Find the traffic intensity for the mean arrival rate of the customer is
30 per day and the service rate of the server is 48 per day.

Part-B (6Marks)

1. Write the steady-state equation for the model (M/M/C):(FIFO/∞/∞).

2. Obtain the expected waiting time of a customer in the queue of the

model λ=10/hour, μ=3/hour C=4, what is the probability that a
customer has to wait before he gets service?

3. 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 any time (ii) the
proportion of the time the booth is expected to be idle?

4. 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 park on the average?

5. 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 15 minutes. Further, if a customer arrives and
 there are no empty chairs in the shop, he will leave. What is the
expected number of customers in the shop?
6. A T.V mechanic 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 eight day, which is the mechanic’s expected
idle time each day? How many jobs are ahead of the average set just
brought in?

7. 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 service by the clerk has an exponential distribution.

8. In a super market, the average arrival rate of customers is 10 every 30

minutes, following Poisson process. The average time taken by a
cashier to list and calculate the customer’s purchase is two and half
minutes following exponential distribution. What is the probability that
the queue length exceeds six? What is the expected time spent by a
customer in the system?
9. In a public telephone booth, the arrivals on an average are 15 per hour.
A call on an average takes three minutes. If there is just one phone, find
i) the expected number of callers in the booth at any time ii) the
proportion of the time, the booth is expected to be idle.

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

serve only one person at a time. If a customer comes to his shop and
finds it full, he goes to the next shop. Customers randomly arrive at an
average rate λ=10 per hour and the berbe’s service time is negative
exponential with an average of 1/µ = 5 minutes per customer. Find P0
and Pn.

11. People arrive at a theatre ticket centre in a Poisson distributed arrival

rate of 25 per hour. Serve time is constant at two minutes. Calculate,

i) The mean number in the waiting line.

ii) the mean waiting time.
iii) Utilization factor.

12. Assuming for a period of two hours in a day (8-10 am), trains arrive at
the yard every 20 minutes, then calculate for this period.
i)The probability that the yard is empty.
ii) Average queue length, assuming that the capacity of the yard is 4
trains only.

13. Four counters are being run on the frontier of a country to check the
passports and necessary papers of the tourists. The tourists choose any
counter at random. If the arrival at the frontier is Poisson at the rate λ
and the service time is exponential with parameter λ/2, what is the
steady state average queue at each counter?
14. At a public telephone booth in a post office, arrivals are considered to
be Poisson, with an average inter arrival time of 12 minutes. The length
of the phone call may be assumed to be distributed exponentially with
an average of four minutes. Calculate the following.
 i) What is the probability that a fresh arrival will not have to wait for the
phone?
ii) What is the average length of the queue that forms from time to time?

15. A two channel waiting line with Poisson arrivals has a mean arrival
rate of 50 per hour and exponential service with a mean service rate of
75 per hour for each channel.
i) The probability of an empty system.
ii) The probability that an arrival in the system will have to wait.

Part-C (10 Marks)

1. 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 five minutes. The space in front of the
window including that for the serviced car can accommodate a maximum
of three cars. Others can wait outside this space.
i) What is the probability that an arriving customer can drive directly to the
space in front of the window?
ii) What is the probability that an arriving customer will have to wait
outside the indicated space?
iii) How long is an arriving customer expected to wait before starting
service?

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

3.A super market has two girls ringing up sales at the counters. If the
service time for each customer is exponential with mean four minutes and if
people arrive in a Poison 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.

4.A petrol pump has two pumps. The service time follows the exponential
distribution with mean four 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 do the pumps remain idle?

5.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 costs 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 1⅓ patients to 1/2 patients?
6.In a railway marshalling yard, goods train 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, i) the probability that the yard is empty.
ii) the average queue length, assuming that the line capacity of the yard is
nine trains.
iii) And also find the average queue length, if the goods train arrive at the
rate of 40 trains per day.