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RESOURCE MANAGEMENT TECHNIQUES
QUESTION BANK
UNIT – I
PART - A
1. What is operations research?
2. List some applications of OR.
3. What are the various types of models in OR?
4. What are main characteristics of OR?
5. Name some characteristics of good model.
6. What are the different phases of OR?
7. What is an artificial variable in linear programming? (May 2012)
8. What are the limitations of OR?
9. What is linear programming?
10. What is the mathematical model of linear programming? (May 2012)
11. What are the characteristic of LPP?
12. What are the characteristic of standard form of LPP?
13. What are the characteristics of canonical form of LPP?
14. A firm manufactures two types of products A and B and sells them at profit of Rs 2 on type
A and Rs 3 on type B. Each product is processed on two machines M1 and M2.Type A requires
1 minute of processing time on M1 and 2 minutes on M2 Type B requires 1 minute of processing
time on M1 and 1 minute on M2. Machine M1 is available for not more than 6 hours 40 minutes
while machine M2 is available for 10 hours during any working day. Formulate the problem as a
LPP so as to maximize the profit.
15. A company sells two different products A and B , making a profit of Rs.40 and Rs. 30 per
unit on them, respectively. They are produced in a common production process and are sold in
two different markets, the production process has a total capacity of 30,000 man-hours. It takes
three hours to produce a unit of A and one hour to produce a unit of B. The market has been
surveyed and company official feel that the maximum number of units of A that can be sold is
8,000 units and that of B is 12,000 units. Subject to these limitations, products can be sold in any
combination. Formulate the problem as a LPP so as to maximize the profit.
16. What is feasibility region? Is it necessary that it should always be a convex set?
17. Define solution, feasible solution and optimal solution of LPP
18. State the applications of linear programming
19. State the Limitations of LP.
20. Define Unbounded solution and Multiple Optimal solution?
21. What is slack variable?
22. What are surplus variables?
23. Define Basic solution?
24. Define non Degenerate Basic feasible solution and degenerate basic solution?
25. From the optimum simplex table how do you identify that the LPP has no solution?
26. Write the standard form of LPP in the matrix notation?
27. Define basic variable and non-basic variable in linear programming.
28. Define unrestricted variable and artificial variable.
29. What is the use of artificial variables in Linear Programming Problem?
UNIT II
PART A
1. Define transportation problem.
2. Define the following: Feasible solution & basic feasible solution
3. Define optimal solution in transportation problem
4. What are the methods used in transportation problem to obtain the initial basic feasible
solution.
5. Write down the basic steps involved in solving a transportation problem.
6. What do you understand by degeneracy in a transportation problem?
7. What is balanced transportation problem & unbalanced transportation problem?
8. How do you convert an unbalanced transportation problem into a balanced one?
9. Explain how the profit maximization transportation problem can be converted to an equivalent
cost minimization transportation problem.
10. Define transshipment problems?
11. What is the difference between Transportation problem & Transshipment Problem?
12. What is assignment problem?
13. Explain the difference between transportation and assignment problems?
14. Define unbounded assignment problem and describe the steps involved in solving it?
15. Explain how a maximization problem is solved using assignment model?
16. What do you understand by restricted assignment? Explain how you should overcome it?
17. How do you identify alternative solution in assignment problem?
18. What is a traveling salesman problem?
19. How do you convert the unbalanced assignment problem into a balanced one?
20. What are the methods for finding initial basic feasible solution to transportation problem?
21. Give the mathematical formulation of assignment model. (May 2012)
UNIT III
PART A
1. What is Integer Programming Model? (May 2012)
2. Explain the importance of Integer programming problem?
3. List out some of the applications of IPP?
4. List the various types of integer programming?
5. What is Zero-one problem?
6. What is the difference between Pure integer programming & mixed integer integer
programming.
7. Explain the importance of Integer Programming?
8. Why not round off the optimum values in stead of resorting to IP?
9. What are methods for IPP?
10.What is cutting method?
11. What is search method?
12. What is Branch and Bound Technique? (May 2012)
13. Give the general format of IPP?
14. Write an algorithm for Gomory’s Fractional Cut algorithm?
15. What is the purpose of Fractional cut constraints?
16.A manufacturer of baby dolls makes two types of dolls, doll X and doll Y. Processing of these
dolls is done on two machines A and B. Doll X requires 2 hours on machine A and 6 hours on
Machine B. Doll Y requires 5 hours on machine A and 5 hours on Machine B. There are 16
hours of time per day available on machine A and 30 hours on machine B. The profit is gained
on both the dolls is same. Format this as IPP?
17. Explain Gomory’s Mixed Integer Method?
18. What is the geometrical meaning of portioned or branched the original problem?
19. What is standard discrete programming problem?
20. How can you improve the efficiency of portioned method?
UNIT IV & V
PART A
1. What do you mean by project?
2. What are the three main phases of project?
3. What are the two basic planning and controlling techniques in a network analysis?
4. What are the advantages of CPM and PERT techniques?
5. Differentiate between PERT and CPM? (May 2012)
6. What is network?
7. What is Event in a network diagram?
8. Define activity & Critical Activities?
9. Define non critical activities?
10. Define Dummy Activities & duration?
11. Define total project time & Critical path?
12. Define float or slack?
13. Define total float, free float, Independent float?
14. Define Interfering float?
15. Define Optimistic?
16. Define Pessimistic?
17. Define most likely time estimation?
18. What is a parallel critical path?
19. What is standard deviation and variance in PERT network?
20. Give the difference between direct cost and indirect cost?
21. What is meant by resource analysis? (May 2012)
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PART – B
UNIT – I
1. A firm has available 240,370 and 180 Kgs of wood, plastic and steel respectively. The
firm produces two products A and B. Each until of A require 1, 3 and 3 kg of wood,
plastic and steel respectively. The corresponding requirement for each unit of B are 3,4,1
respectively. If A sells for Rs. 4 and B for Rs. 6. Determine how many units of A and B
are should be produced in order to obtain the maximum gross income. Use the simplex
method.
2. A firm plans to purchase atleast 200 quindals of scrap containing high quality metal X
and low quality metal Y. It decides that the scrap to be purchased must contain atleast
100 quintals of X metal and has no more than 35 quintals of Y metal. The firm can
purchase the scrap from two suppliers ( A and B) in unlimited quantities. The percentage
of x and Y metals interms of weight in the scrap supplied by A and B is given below:
Metals Supplier A Supplier B
X 25% 75%
Y 10% 20%
The price of A’s scrap is Rs. 200 per quintal and that of B’s Rs. 400 per quintal.
Determine the quantities that it should buy from the two suppliers so that total cost is
minimized. (Use graphical method)
3. Use graphical method to Maximize Z = 6x1 + 4x2 Subject to
-2x1 + x2 ≤ 2
x1 – x2 ≤ 2
3x1 + 2x2 ≤ 9
x1, x2 ≥ 0
4. Using Artificial variable technique to solve the following LP problems.
(i) Min Z = 2x1–3x2+6 x3
subject to 3x1 – 4x2 - 6x3 ≤ 2
2 x1 + x2 + 2x3 ≥ 11
x1 + 3 x2 - 2x3 ≤ 5
x1, x2, x3 ≥ 0
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(ii) Max Z = x1 + 2x2 + 3x3 – x4
subject to x1 + 2x2 + 3x3 = 15
2x1 + x2 + 5x3 = 20
x1 + 2x2 + x3 + x4 = 10
x1, x2, x3, x4 ≥ 0
5. Solve the following linear programming problems (i) using the simplex method (ii)
Using Graphical method
Maximize z = 5x1 + 7x2 subject to
4x1 + 7x2 ≤ 240
4x1 + 6x2 ≤ 300
2x1 +3x2 ≥ 0
x1, x2 ≥ 0
6. Show that the following LPP has a feasible solution but no finite optimal solution
of Maximizes z = 3x1 + 3x2 subject to
x1 - x2 ≤ 1
x1 + x2 ≥ 3
x1, x2 ≥ 0
7. Use simplex method to solve the LPP Maximizes z = 4 x1 + 10x2 subject to
2x1 + x2 ≤ 50
2x1 + 5x2 ≤ 100
2x1 + 3x2 ≤ 90
x1, x2 ≥ 0
8. Find the non-negative values of x1, x2 and x3 which maximize z = 3 x1+2 x2+5x3
subject to
x1 + 4x2 ≤ 420
3x1 + 2x2 ≤ 460
x1 + 2x2 + x3 ≤ 430
x1, x2, x3 ≥ 0
9. Solve the following Maximize 15x1 + 6x2 + 9x3 + 2x4 subject to
2x1 + x2 + 5x3 + 6x4 ≤ 20
3x1 + x2+ 3x3 + 25x4 ≤ 24
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7x3 + x4 ≤ 70
x1, x2, x3, x4 ≥ 0
10. Use simplex method to Min Z = x2 – 3x3 + 2x3 subject to
3x2 – x3 + 2x5 ≤ 7
-2x2 + 4x3 ≤ 12
-4x2 +3x3 + 8x5 ≤ 10
and x2, x3, x5 ≥ 0
11. (i) State and explain the general linear programming problem and define (a) feasible
solution and (b) basic feasible solution
(ii) Describe simplex method of solving linear programming problem
UNIT – II
1. A company has three plants A, B, C and three warehouse X,Y,Z. The number if
units available at the plants is 60, 70, 80 and the demand at X, Y, Z are 50, 80, 80
respectively. The unit cost of transportation is given by the following table
X Y Z
A 8 7 3
B 3 8 9
C 11 3 5
2. Solve the transportation problems with transportation cost in rupees, requirements
and availability as given below.
Distribution centre
D1 D2 D3 D4
F1 10 15 12 12 200
F2 8 10 13 10 150
F3 11 12 13 10 120
140 120 80 220
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3. A product is manufactured by four factories A, B, C and D. The unit production costs
in them are Rs. 2, Rs. 3, Rs. 1 and Rs. 5 respectively. Their production capacities are 50,
70, 30 and 50 units respectively. These factories supply the product to four stores,
demands of which are 25, 35, 105 and 20 units respectively. Unit transportation cost in
rupees from each factory to each store in given in the table below:
Stores
Factories
- 1 2 3 4
A 2 4 6 11
B 10 8 7 5
C 13 3 9 12
D 4 6 8 3
Determine the extent of deliveries from each of the factories to each of the stores so that
the total production and transportation cost in minimum.
4. Solve the assignment problem for maximizes for the profit matrix
P Q R S
A 51 53 54 50
B 47 50 48 50
C 49 50 60 61
D 63 64 60 60
5. Solve the following assignment problem
A B C D E
M1 4 6 10 5 6
M2 7 4 - 5 4
M3 - 6 9 6 2
M4 9 3 7 2 3
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6. A city corporation has decided to carry out road repairs on main four arteries of the
city. The government has agreed to make a special grant of Rs. 50 lakh towards the cost
with a condition that the repairs be done at the lowest cost and quickest time. If the
conditions warrant than a supplementary token grant will also be considered favourably.
The corporation has floated tenders and five contractors have sent in their bids. In order
to expedite work. One road will be awarded to only one contractor.
Cost of Repairs (Rs. Lakh)
Contractors/Road
- R1 R2 R3 R4
C1 9 14 19 15
C2 7 17 20 19
C3 9 18 21 18
C4 10 12 18 19
C5 10 15 21 16
(i) Find the best way of assigning the repair work to the contractors and the costs.
(ii) Which if the five contractors will be unsuccessful in his bid?
7. Travelling salesman has to visit 5 cities. He wishes to start from a particular city, visit
each city once and then returns to his starting point. The traveling cost from each city
from a particular point is given below.
A B C D E
A - 4 6 3 4
B 4 - 5 3 4
C 7 6 - 7 5
D 3 3 7 - 7
E 4 4 5 7 -
8. Determine the basic feasible solution to the following transportation problem
using North – West Corner Rule:
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Sink
origin
A B C D E Supply
P 2 11 10 3 7 4
Q 1 4 7 2 1 8
R 3 9 4 8 12 9
Demand 3 3 4 5 6
9. Solve the transportation problem with unit transportation costs, demands and supplies
as given below:
Destination
source
D1 D2 D3 D4 Supply
S1 6 1 9 3 70
S2 11 5 2 8 55
S3 10 12 4 7 70
Demand 85 35 50 45
10. Solve the following transportation problem to maximize profit
Profit (Rs) / Unit
Destination
source
A B C D Supply
1 40 25 22 33 100
2 44 35 30 30 30
3 38 38 28 30 70
Demand 40 20 60 30
11. Explain the concept of degeneracy in transportation problems. Obtain an optimum
basic feasible solution to the following transportation problem
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To
To
From
A B C Supply
F1 7 3 4 2
F2 2 1 3 3
F3 3 4 6 3
Demand 4 1 3 8
12. An automobile dealer is faced with the problem of determining the minimum cost
policy for supplying dealers with the desired number of automobiles. The relevant data
are given below. Obtain the minimum total cost of transportation
Dealers
Plant
1 2 3 4 5 Supply
A 1.2 1.7 1.6 1.8 2.4 300
B 1.8 1.5 2.2 1.2 1.6 400
C 1.5 1.4 1.2 1.5 1.0 100
Requirements 100 50 300 150 200
The cost unit is in 100 rupees.
UNIT – III
1. Write a Brief Note on Cutting Plane Method
2. Describe Gomory’s Entire Integer Programming Problem Method and Its Algorithm.
3. Describe Any One of the Method of Solving Mixed Integer Programming Problem.
4. Explain The Importance Of Integer Programming Problems And Their Applications.
5. Explain Branch And Bound Method In Integer Programming Problem.
6. Write Down The Branch And Bound Algorithm For Solving A Mixed Integer
Programming Problem.
RMT
7. Using Gomory’s Cutting Plane Method Maximize Z = 2x1 + 2x2 Subject To
5x1 +4 x2 ≤8
2 x1 + 4 x2 ≤8
x1, x2 ≥ 0 and are all Integers.
8. Solve The Following Integer Mixed Integer Programming Problem Maximize
Z = x1 + x2 subject to
2 x1 +5 x2 ≤16
6 x1 + 5 x2 ≤30
x2 ≥ 0, x1 , non-Negative Integer, and are all Integers.
9. Use Branch and Bound Method to solve the following Maximize Z = x1 + 4x2 subject
to
2 x1 +4 x2 ≤7
5 x1 + 3 x2 ≤15
x1, x2≥ 0and are all integers.
10. Solve the following Integer Mixed Integer Programming Problem by Gomory’s
Cutting Plane Method Maximize z = x1 + x2 subject to
3 x1 +2 x2 ≤5
x2 ≤ 2
x1, x2≥ 0 and x1 an integers.
UNIT – IV
1. A particular item has a demand of 9,000 units / year. The cost of one
procuirement is Rs. 100 and the holding cost per unit is Rs. 2.40 per year. The
replacement is instantaneous and no shortages allowed. Determine
(i) the economic lot size
(ii) the number of orders per year
(iii) the time between orders
(iv) the total cost per year if the cost of one unit Rs. 1
2. A manufacturer has to supply 12,000 units of a product per year to his customer.
The demand is fixes and known and the shortage cost is assumed to be infinite. The
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inventory holding cost is Rs. 0.20 per unit per month and the setup cost per run is Rs.
350. Determine
(i) the optimum run size
(ii) optimum scheduling period
(iii) minimum total expected yearly cost.
3. An item is produced at the rate of 50 items per day. The demand occurs at the rate
of 25 items per day. If a setup cost is assumed to be infinite. The inventory 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 cycle and minimum total cost for one
run.
4. A commodity is to be supplied at the constant rate of 200 units per day supplies of
any amount can be had at any required time but each ordering costs Rs. 50. Cost of
holding the commodity in inventory Rs. 2 per unit per day while the delay in the supply
of the item induces a penalty of Rs.10 per unit per delay of 1 day. Find the optimal policy
(q, t) where t is the reorder cycle period and q is the inventory level after reorder. What
would be the best policy if the penalty cost becomes infinity?
5. The average demand for an item is 120 units / year, the lead time is 1 month and
the demand during lead time follows normal distribution with average of 10 units and
standard deviation of 2 units. If the item is ordered once in 4 months and the policy of the
company is that there should not be more than 1 stock out every two years, determine the
reorder level.
6. 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 4 1 1 1 6 5 4 8 1 2 5 7
Construct PERT networks and find the critical path.
7. Draw the network and determine the critical path for the given data:
Jobs 1-2 1-3 2-4 3-4 3-5 4-5 4-6 5-6
Duration 6 5 10 3 4 6 2 9
Find the total float, Free float and Independent Float of each activity.
8. (i) Explain the three rime estimates used in PERT
(ii) What are the main assumptions under in PERT computations
(iii) Distinguish between PERT and CPM
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9. Explain PERT procedure to determine (a) the expected critical path (b) expected
variance of the project length and (c) the probability of completing the project with in a
specified time.
10. Determine the least cost schedule for the following project. Overhead cist is Rs. 60
per day.
Activity Normal duration Crash duration Cost of crashing / Day
(days) (days) (Rs.)
1-2 5 3 30
2-3 4 2 20
3-4 6 3 40
3-5 4 1 30
4-5 3 2 60
11. 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 the expected time?
12. (i) Write the rules of drawing network and define critical activities.
(ii) A project has the following details
Activity 1-2 1-3 2-6 3-4 3-5 3-6 4-5 5-6 5-7 6-7
Expected
Duration
8 6.83 9 4 8.17 13.5 4.17 5.17 9 4.5
Variance 1/9 1/4 1 0 1/4 9/4 1/4 ¼ 4/9 25/26
What should be the scheduled completion time for the probability of completion of
project to be 90%.
13. Listed in the table are the activities and sequencing requirements necessary for the
completion of research report.
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Activity: A B C D E F G H I J K L M
Duration(Weeks) 4 2 1 12 14 2 3 2 4 3 4 2 2
Immediate
Predecessors:
E A B K - E F F F I,L C,G,H D I,L
(i) Find the critical path
(ii) Find the total float and the free float for each non-critical activity.
14. A maintenance foreman has given the following estimate of times and cost of jobs in
a maintenance project.
Job Predecessor Normal time Normal Cost Crash time Crash cost
A - 8 80 6 100
B
UNIT – V
1. A bank has two tellers working on savings account. The first teller handles withdrawal
only. The second teller handles deposits only. It has been found the service time
distributions for both deposits and withdrawals are exponential with mean service time of
3 minutes and 4 seconds per customer. Depositors are found to arrive in a passion fashion
throughput the day with mean arrival rate of 16 per hour. Withdrawers also arrive in a
passion fashion with mean arrival rate of 14 per hour. What would be the effect on the
average waiting time for the customers if each teller could handle both withdrawals and
deposits?
2. In heavy machine shop, the overhead crane is 75% utilized. Time study observations
gave the average slinging time as 10.5 minutes with a standard deviation of 8.8 minutes.
What is the average calling rate for the services of the crane and what is the average
delay is getting service? If the average service time is cut to 8.0 minutes, with a standard
deviation of 6.0 minutes, how much reduction will occur, on average, in the delay of
getting served?
3. Explain the model (M/M/S): (œ / FIFO) (i.e) multiple server with infinite capscity.
Derive the average number of customers in the queue (i.e average queue length: LØ )
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4. Automatic car wash facility operates with only one bay. Cars arrive according to a
Possion distribution, with a mean of 4 cars per hour and may in the facility’s parking lot
if the bay is busy. If the service time for all cars is constant and equal to 10 minutes
determine Ls, Lf, Ws and Wf.
4. A two person barber shop has 5 chairs to accommodate waiting customers. Potential
customers who arrive when all chairs are full, leaving without entering barber shop.
Customers arrive at the average rate of 4 per hour and spend an average of 12 minutes in
the barber’s chair. Compute ρ0, ρ7 and average number of customers in the queue.
6. In a super market, the average arrival rate of customer is 10 in every 30 minutes
following passion process. The average time taken by the cashier to list and calculate the
customer’s purchases is 2.5 minutes, following exponential distribution. What is the
probability that the queue length exceeds 6? What is the expected time spent by a
customer in the system?
5. People arrive at a Theatre ticket booth in Possion distributed arrival rate of 25 per hour.
Service time is constant at 2 minutes. Calculate
(a) The mean number in the waiting line.
(b) The mean waiting time
(c) The utilization factor
6. A petrol station has two pumps. The service time follows the exponential distribution
with mean 4 minutes and car arrives for service in a Poisson process at the rate of ten cars
per hour. Find the probability that a customer has to wait for service. What proportion of
time the pump remains idel?
7. A customer owning a Maruti car right now has got the option to switch over to maruti.
Ambassador or Fiat next time with the probability (0.20 0.5 and 0.30) given the transition
matrix
0.40 0.30 0.30
P = 0.20 0.50 0.30
0.25 0.25 0.50
Find the probabilities with his fourth purchase?
8. State and explain about the Pure Birth Process in detail.
(i) Write the steady-state equations for the model (M/M/C) : (FIFO/ ∞/ ∞)
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(ii) Obtain the expected waiting time of a customer in the queue of the above model.
(iii) In the above model λ = 10/ hr, µ = 3/ hr, C = 4, what is the probability that a
customer has to wait before he gets service.
9. (i) For the steady state queueing model (M/M/1) : (FCFS/∞), with usual notations
prove that pn = (λ/µ)n po and P(n>=k) = (λ/µ)
k
(ii) A super market has two girls serving at the counters. The customers arrive in a
Poisson fashion at the rate of 12 per hour. The service time for each customer is
exponential with mean 6 minutes. Find the probability that an arriving customer has to
wait for service.
10. (i) For the queueing model (M/M/S) ; (∞/FIFO), derive the formula for average time
of a customer in the system.
(ii) Patients arrive at a clinic according to Poisson distribution at a rate of 60 patients per
hour. The waiting room does not accommodate more than 14 patients. Investigation time
per patient is exponential with mean rate of 40 per hour.
(a) What is the probability that an arriving patient will not wait?
(b) What is the expected waiting time until a patient is discharged from the clinic?
11. (i) Write the steady state equations for the model (M/M/C)(FIFO/∞/∞)
(ii) Obtain the expected waiting time of a customer in the queue of the above model.
(iii) In the above model λ=10/hour, µ=3/hour C=4, what is the probability that a
customer has to wait before he gets service?
12. Arrivals of a telephone booth are considered to be poisson with an average time of 10
minutes between one arrival and the next. The length of phone call is assumed to be
distributed exponentially, with mean 3 minutes.
(i) What is the probability that a person arriving at the booth will have to wait?
(ii) The telephone department will install a second booth when convinced that an
arrival would expect waiting for atleast 3 minutes for a phone call. By how much should
the flow of arrivals increase in order to justify a second booth?
(iii) What is the average length of the queue that form from time to time?
(iv) What is the probability that it will take him more than 10 minutes altogether to
wait for the phone and complete his call?
VALLIAMMAI ENGINEERING COLLEGE S.R.M. Nagar, Kattankulathur - 603 203.
DEPARTMENT OF COMPUTER APPLICATIONS
MC7401– RESOURCE MANAGEMENT TECHNIQUES IV SEM/MCA / R-2013
Question Bank
UNIT – I PART – A 1. What is operations research? BTL1 2. List some applications of OR. BTL1 3. What are main characteristics of OR? BTL4 4. What are the different phases of OR? BTL4 5. What is an artificial variable in linear programming? BTL1 6. What are the limitations of OR? BTL2 7. What is linear programming? BTL1 8. What is the mathematical model of linear programming? BTL1 9. What are the characteristics of LPP? BTL2 10. What are the characteristic of standard form of LPP? BTL2 11. A firm manufactures two types of products A and B and sells them at profit of Rs 2 on type A and Rs
3 on type B. Each product is processed on two machines M1 and M2.Type A requires 1 minute of processing time on M1 and 2 minutes on M2 Type B requires 1 minute of processing time on M1 and 1 minute on M2. Machine M1 is available for not more than 6 hours 40 minutes while machine M2 is available for 10 hours during any working day. Formulate the problem as a LPP so as to maximize the profit. BTL3
12. What is feasibility region? Is it necessary that it should always be a convex set? BTL4 13. Define solution, feasible solution and optimal solution of LPP. BTL1 14. State the applications of linear programming. BTL3 15. Devise the meaning of unbounded solution and Multiple Optimal solution. BTL5 16. Devise the meaning of slack variable? BTL5 17. What are surplus variables? BTL2 18. State the standard form of LPP in the matrix notation. BTL3 19. What do you think about basic variable and non-basic variable in linear programming? BTL6 20. What do you think about unrestricted variable and artificial variable? BTL6 PART – B 1. Use Two – Phase simplex method to solve the following LPP. (16) BTL3 Maximize Z = 3X1 + 2 X2 Subject to the constraints 2 X1 + X2 ≤ 2 3 X1 + 4 X2 ≥ 12 X1, X2 ≥ 0.
2. Use Big-M method to solve the following LPP. (16) BTL3 Maximize Z = 3X1 - X2
Subject to the constraints 2 X1 + X2 ≥ 2 X1 + 3 X2 ≤ 3 X2 ≤ 4 X1, X2 ≥ 0. 3. Solve by Big-M method. (16) BTL3
Minimize Z = 2X1 + X2 Subject to the constraints 3X1 + X2 = 3 4X1 + 3X2 ≥ 6 X1 + 2X2 ≤ 3 X1, X2 ≥ 0. 4. Solve the following LPP using Big-M method. (16) BTL3
Maximize Z = 2X1 +3 X2 + 4X3 Subject to the constraints 3X1 + X2 + 4X3 ≤ 600 2X1 + 4X2 + 2X3 ≥ 480
2X1 + 3X2 + 3X3 = 540 X1, X2, X3 ≥ 0. 5. Apply Simplex method to solve the LPP. (16) BTL5
Maximize Z = 3X1 + 5X2 Subject to the constraints 3X1 + 2X2 ≤ 18 0 ≤ X1 ≤ 4, 0 ≤ X2 ≤ 6 6. Use simplex method to solve the LPP. (16) BTL3
Maximize Z = 4X1 + 10X2 Subject to the constraints 2X1 + X2 ≤ 50 2X1 + 5X2 ≤ 100 2X1 + 3X2 ≤ 90 and X1, X2 ≥ 0. 7. (a) Apply graphical method to solve the following LPP. (8) BTL5
Maximize Z = 3X1 + 4X2 Subject to the constraints X1 + X2 ≤ 450 2X1 + X2 ≤ 600 X1, X2 ≥ 0.
(b). Write the Procedure for solving Linear Programming Problem using Charne’s Big M method. (8) BTL 4
8. Solve the LPP by graphical Method. (16) BTL3
Maximize Z = 3X1 + 5X2 Subject to the constraints -3X1 +4 X2 ≤ 12 2X1 - X2 ≥ -2 2X1 + 3X2 ≥ 12 X1 ≤ 4
X2 ≥ 2 and X1, X2 ≥ 0.
9. (a). A company produces refrigerator in Unit I and heater in Unit II. The two products are produced and sold on a weekly basis. The weekly production cannot exceed 25 in unit I and 36 in Unit II, due to constraints 60 workers are employed. A refrigerator requires 2 man week of labour, while a heater requires 1 man week of labour, the profit available is Rs. 600 per refrigerator and Rs. 400 per heater. Formulate the LPP problem. (8) BTL4
(b). Solve the following LPP Graphically:
Minimize Z = 4X1 + 3X2 Subject to the constraints X1 + 3X2 ≥ 9 2X1 + 3X2 ≥ 12 X1 + X2 ≥ 5 X1, X2 ≥ 0. (8) BTL 3 10. (a). A firm manufactures two types of products A and B and sells them at profit of Rs 2 on
type A and Rs 3 on type B. Each product is processed on two machines M1 and M2.Type A requires 1 minute of processing time on M1 and 2 minutes on M2 Type B requires 1 minute of processing time on M1 and 1 minute on M2. Machine M1 is available for not more than 6 hours 40 minutes while machine M2 is available for 10 hours during any working day. Formulate the problem as a LPP so as to maximize the profit. (8) BTL4
(b). Write the Procedure for solving Linear Programming Problem using Simplex Method. (8).
BTL4.
UNIT-II PART - A 1. Define transportation problem. BTL1 2. Define Feasible solution & basic feasible solution BTL1 3. Provide a definition for optimal solution in transportation problem. BTL2 4. Write the methods used in transportation problem to obtain the initial basic feasible solution. BTL4 5. Can you provide the basic steps involved in solving a transportation problem? BTL2 6. What do you understand by degeneracy in a transportation problem? BTL4 7. What is balanced transportation problem & unbalanced transportation problem? BTL1 8. How do you convert an unbalanced transportation problem into a balanced one? BTL5
9. Explain how the profit maximization transportation problem can be converted to an equivalent cost minimization transportation problem. BTL3
10. Define transshipment problems? BTL1 11. What differences exist between Transportation problem & Transshipment Problem? BTL2 12. What is assignment problem? BTL1 13. Define unbounded assignment problem and describe the steps involved in solving it. BTL1 14. Describe how a maximization problem is solved using assignment model. BTL4 15. What do you understand by restricted assignment? How do you overcome it? BTL6 16. How do you identify alternative solution in assignment problem? BTL6 17. Illustrate the traveling salesman problem. BTL3 18. Can you say how a minimization problem is solved using assignment model? BTL2 19. Can you propose which method is best for finding initial basic feasible solution in transportation problem? BTL5 20. Show the mathematical formulation of assignment model. BTL3
********** PART – B 1.(a). Write the steps involved for solving Transportation problem using Vogel’s Approximation
method. (8). BTL4 (b). Obtain the Initial Basic Feasible Solution for the following TP using North West Corner
method. (8) BTL 5
I II III SUPPLY
S1 8 5 6 120 S2 15 10 12 80 S3 3 9 10 150 DEMAND 150 80 50
2. Solve the following assignments problems (16) BTL3
I II III IV V
A 10 5 9 18 11 B 13 19 6 12 14 C 3 2 4 4 5 D 18 9 12 17 15 E 11 6 14 19 10
3. Solve the TP where cell entries are unit costs. Use vogel’s approximate method to find the initial basic solution (16) BTL1
D1 D2 D3 D4 D5 AVAILABLE O1 68 35 4 74 15 18 O2 57 88 91 3 8 17 O3 91 60 75 45 60 19 O4 52 53 24 7 82 13 O5 51 18 82 13 7 15 Required 16 18 20 14 14
4.A small garments making units has five tailors stitching five different types of garments all the
five tailors are capable of stiching all the five types of garments .the output per day per tailor and the profit(Rs.)for each type of garments are given below.
1 2 3 4 5
A 7 9 4 8 6 B 4 9 5 7 8 C 8 5 2 9 8 D 6 5 8 10 10 E 7 8 10 9 9
PROFIT 2 3 2 3 4
Which type of garments should be assigned to which tailor in order to maximize profit, assuming that there are no others constructs (16) BTL4 5. Solve the following TP to maximize profit (16) BTL3
A B C D SUPPLY
1 40 25 22 33 100
2 44 35 30 30 30
3 38 38 28 30 70
DEMANDS 40 20 60 30
6. Five workers are available to work with the machines and respective cost associated with each worker –machine assignments is given below. A sixth machine is available to replace one of the existing machines and the associated costs are also given below.
M1 M2 M3 M4 M5 M6
W1 12 3 6 - 5 8 W2 4 11 - 5 - 3 W3 8 2 10 9 7 5 W4 - 7 8 6 12 10 W5 5 8 9 4 6 -
Determine whether the new machine can be accept and also determine optimal assignments and the associated saving in cost (16) BTL6 7. Solve the following Transportation Problem (16) BTL3
A B C D SUPPLY I 15 18 22 16 30
II 15 19 20 14 40
III 13 16 23 17 30
DEMAND 20 20 25 35 100
8. Solve the assignment problem (16) BTL3
1 2 3 4 5 6 A 12 10 15 22 18 8 B 10 18 25 15 16 12 C 11 10 3 8 5 9 D 6 14 10 13 13 12 E 8 12 11 7 13 10
9. Solve the following assignment problems (16) BTL3
M1 M2 M3 M4 J1 18 24 28 32
J2 8 13 17 18
J3 10 15 19 22
10. Solve the following TP (16) BTL3
D1 D2 D3 D4 SUPPLY
S1 6 1 9 3 70 S2 11 5 2 8 55
S3 10 12 4 7 70
DEMANDS 85 35 50 45
UNIT-III PART- A 1. State the Integer Programming Model. BTL1 2. Explain the importance of Integer programming problem. BTL4 3. Illustrate some of the applications of IPP. BTL3 4. Can you provide various types of integer programming? BTL2 5. What is Zero-one problem? BTL1 6. Write the differences between Pure & mixed integer programming. BTL2 7. Explain the importance of Integer Programming. BTL4 8. Why not round off the optimum values instead of resorting to IP? BTL5 9. Write the methods for IPP. BTL2 10. What is cutting Plane method? BTL1 11. What do you think about search method? BTL6 12. Write about Branch and Bound Technique. BTL5 13. Construct the general format of IPP. BTL3 14. Write an algorithm for Gomory’s Fractional Cut algorithm. BTL2 15. What is the purpose of Fractional cut constraints? BTL1 16.A manufacturer of baby dolls makes two types of dolls, doll X and doll Y. Processing of these dolls is done on two machines A and B. Doll X requires 2 hours on machine A and 6 hours on Machine B. Doll Y requires 5 hours on machine A and 5 hours on Machine B. There are 16 hours of time per day available on machine A and 30 hours on machine B. The profit is gained on both the dolls is same. Format this as IPP? BTL3 17. Explain Gomory’s Mixed Integer Method. BTL4
18. What is the geometrical meaning of portioned or branched the original problem? BTL1 19. What is standard discrete programming problem? BTL1 20. How can you improve the efficiency of portioned method? BTL6
*********** PART - B
1. Suppose the owner of the readymade garments decide to make X1 Zee-shirts and X2 Button-
down shirts. Then the availability of time to tailors has the following restrictions: (16) BTL 2 2X1 + 4X2 ≤ 7
5X1 + 3X2 ≤ 15 and X1, X2 ≥ 0 and are integers. The problem of the owner is to find the values of X1 and X2 to maximize the profit Z = X1 + 4X2. 2. Solve the following ILPP using Gomary’s Cutting Plane Method. (16) BTL3
Maximize Z = X1 + 2X2 Subject to the constraints 2 X2 ≤ 7 X1 + X2 ≤ 7 2X2 ≤ 11 X1, X2 ≥ 0 and are integers. 3. Solve the following ILPP. (16) BTL3
Maximize Z = 11X1 + 4X2 Subject to the constraints -X1 + 2X2 ≤ 4 5X1 + 2X2 ≤ 16 2X1 - X2 ≤ 4 X1, X2 ≥ 0 and are non negative integers. 4. Solve the integer programming problem. (16) BTL3
Maximize Z = 2X1 + 20X2 - 10X3 Subject to the constraints 2X1 + 20X2 + 4X3 ≤ 15 6X1 + 20X2 + 4X3 ≤ 20 X1, X2, X3 ≥ 0 and are integers. 5. Solve the following mixed integer linear programming problem using Gomarian’s cutting plane method. (16) BTL3
Maximize Z = X1 + X2 Subject to the constraints 3 X1 + 2 X2 ≤ 5 X2 ≤ 2 X1, X2 ≥ 0 and X1 is an integer.
6. Solve the following mixed integer programming problem. (16) BTL3 Maximize Z = 7X1 + 9 X2
Subject to the constraints -X1 + 3X2 ≤ 6 7X1 + X2 ≤ 35 and X1, X2, ≥ 0, X1 is an integer. 7. Solve the following mixed integer programming problem. (16) BTL3
Maximize Z = 4X1 + 6X2 + 2X3 Subject to the constraints 4X1 - 4X2 ≤ 5 -X1 + 6X2 ≤ 5
-X1 + X2 + X3 ≤ 5 and X1, X2, X3 ≥ 0, and X1 , X3 are integers. 8. Use Branch and bound algorithm to solve the following ILPP (16) BTL5
Maximize Z = 3X1 + 4X2 Subject to the constraints 7X1 + 16X2 ≤ 52 3X1 - 2X2 ≤ 18 X1, X2 ≥ 0 and are integers. 9. Apply Branch and bound algorithm to solve the following ILPP (16) BTL4
Maximize Z = X1 + 4X2 Subject to the constraints 2X1 + 4X2 ≤ 7 5X1 + 3X2 ≤ 15 X1, X2 ≥ 0 and are integers. 10. Apply Branch and bound algorithm to solve the following ILPP (16) BTL4
Maximize Z = 2X1 + 2X2 Subject to the constraints 5X1 + 3X2 ≤ 8 X1 + 2X2 ≤ 4 X1, X2 ≥ 0 and are integers.
********* UNIT-IV
PART - A 1. What do you mean by project? BTL6 2. What are the three main phases of project? BTL4 3. What are the two basic planning and controlling techniques in a network analysis? BTL4 4. Write down the advantages of CPM and PERT techniques. BTL2 5. Differentiate between PERT and CPM. BTL2 6. Define network. BTL1
7. What do you mean Event in a network diagram? BTL6 8. Can you write about activity & Critical Activities? BTL5 9. Define Dummy Activities and duration. BTL1 10. Explain Total Project Time & Critical path. BTL4 11. What is float or slack? BTL1 12. Classify Total float, Free float and Independent float? BTL3 13. Can you list the characteristics of Interfering Float? BTL3 14. Define Optimistic time. BTL1 15. Show and define Pessimistic time. BTL3 16. How do you calculate most likely time estimation? BTL5 17. What is a parallel critical path? BTL1 18. Distinguish standard deviation and variance in PERT network? BTL2 19. Give the difference between direct cost and indirect cost. BTL2 20. What is meant by resource analysis? BTL1
************ PART-B
1. A project schedule has the following characteristics
Activity 0 – 1 0 – 2 1 – 3 2 – 3 2 – 4 3 – 4 3 – 5 4 – 5 4 - 6 5 - 6 Time 2 3 2 3 3 0 8 7 8 6
(i). Construct Network diagram. (4) (ii). Compute Earliest time and latest time for each event. (6) (iii). Find the critical path. Also obtain the Total float, Free float and slack time and Independent float. (6) BTL1
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 4 1 1 1 6 5 4 8 1 2 5 7
(i). Construct Network diagram (4)
(ii). Compute Earliest time and latest time for each event. (6) (iii). Find the critical path. Also obtain the Total float, Free float and slack time and Independent float. (6) BTL1
3. A small project is composed of seven activities whose time estimates are listed in the table as follows:
Activity Preceding Activities Duration
A ---- 4 B ---- 7 C ---- 6
D A,B 5 E A,B 7 F C,D,E 6 G C,D,E 5
(i). Draw the network and find the project completion time. (8) (ii). Calculate the three floats for each activity. (8) BTL3
4. Calculate the total float, free float and independent float for the project whose activities are given below: (16)
Activity 1 –2 1 – 3 1 – 5 2 – 3 2 – 4 3 – 4 3 – 5 3 – 6 4 - 6 5 - 6 Key 8 7 12 4 10 3 5 10 7 4
Find the critical path also. BTL1
5. Draw the network for the following project and compute the earliest and latest times for each event and also find the critical path. (16) BTL6
Activity 1 – 2 1 – 3 2 – 4 3 – 4 4 – 5 4 – 6 5 – 7 6 – 7 7 - 8 Immediate Predecessor --- --- 1 – 2 1 – 3 2 – 4 2 – 4 &
3 - 4 4 – 5 4 – 6 6 – 7 & 5 - 7
Time 5 4 6 2 1 7 8 4 3 6. The following table lists the jobs of a network with their time estimates:
Job(i, j) Duration
Optimistic (to) Most likely(tm) Pessimistic (tp)
1 – 2 3 6 15 1 – 6 2 5 14 2 – 3 6 12 30 2 – 4 2 5 8 3 – 5 5 11 17 4 – 5 3 6 15 6 – 7 3 9 27 5 – 8 1 4 7 7 – 8 4 19 28
(i). Draw the project network. (4) (ii). Calculate the length and variance of the Critical Path. (4) (iii).What is the approximate probability that the jobs on the critical path will be completed by the due date of 42 days? (4) (iv).What due date has about 90 % chance of being met? (4) BTL5
7. A small project is composed of 7 activities, whose time estimates are listed in the table below. Activities are identified by their beginning (i) and (j) node numbers.
Job(i, j) Duration
Optimistic (to) Most likely(tm) Pessimistic (tp)
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). Draw the project network and identify all the paths through it. (6) BTL2 (ii). Find the expected duration and variance for each activity. What is the expected project length? (4) (iii). Calculate the variance and standard deviation of the project length. What is the probability that the project will be completed atleast 4 weeks earlier than expected time? (6)
8. Draw the network and determine the critical path for the given data. (16)
Jobs 1 – 2 1 – 3 2 – 4 3 – 4 3 – 5 4 – 5 4 – 6 5 – 6
Duration 6 5 10 3 4 6 2 9 Find the Total Float, Free Float and Independent float for the given data. BTL3
9. For the data given in the table below, draw the network, crash systematically the activities and determine the optimal project duration and cost. Indirect cot Rs. 70 per day. (16) BTL 4
Activity) Normal Time (days) Cost Crash Time
(days) Cost
1 – 2 8 100 6 200 1 – 3 4 150 2 350 2 – 4 2 50 1 90 2 – 5 10 100 5 400 3 – 4 5 100 1 200 4 – 5 3 80 1 100
10. The following table lists the jobs of a network along with their time estimates.
Job(I, j) Duration
Optimistic (to) Most likely(tm) Pessimistic (tp)
1 – 2 2 5 14 1 – 3 9 12 15 2 – 4 5 14 17 3 – 4 2 5 8 4 – 5 6 6 12 3 – 5 8 17 20
Draw the network. Calculate the length and variance of the critical path and find the probability that the project will be completed within 30 days. (16) BTL3
UNIT-V PART – A 1. Define Kendal’s notation for representing queuing models. BTL1 2. In a super market, the average arrival rate of customer is 5 in every 30 minutes following Poisson process. The average time is taken by the cashier to list and calculate the customer’s purchase is 4.5 minutes; following exponential distribution. What is the probability that the queue length exceeds 5? BTL5 3. Explain Queue discipline and its various forms. BTL4 4. Distinguish between transient and steady state queuing system. BTL4 5. Define steady state? BTL1 6. Write down the little formula? BTL1 7. If traffic intensity of M/M/I system is given to be 0.76, what percent of time the system would be idle? BTL5 8. List the basic elements of queuing system. BTL2 9. What do you meant by queue discipline? BTL2 10. What are the characteristics of queuing model? BTL2 11. Define Poisson process. BTL1 12. Identify the properties of Poisson process. BTL4 13. Customer arrives at a one-man barber shop according to a Poisson process with an mean inter arrival time of 12 minutes. Customers spend a average of 10 minutes in the barber’s chain.What is the expected no of customers in the barber shop and in the queue? BTL6 14. Define pure birth process. BTL1 15. Write down the postulates of birth and death process? BTL3 16. What is the formula for the problem for a customer to wait in the system under (m/m/1 : N/FCFS)? BTL1 17. What are the assumptions in m/m/1 model? BTL3
18. People arrive at a theatre ticket booth in Poisson distributed arrival rate of 25/hour. Service time is constant at 2 minutes. Calculate the mean? BTL6 19. Give any two applications of Queuing Theory. BTL3 20. State the assumptions made in queuing model. BTL2
************ PART – B
1. A departmental store has a single cashier. During the rush hours customers arrive at a rate of 20 customers per hour. The average number of customers that can be processed by the cashier is 24 per hour. Assume that the conditions for use of the single channel queuing model apply. (i). What is the probability that the cashier is idle? (3) (ii). What is the average number of customers in the queuing system? (3) (iii). What is the average time a customer spends in the system? (3) (iv). What is the average number of customers in the queue? (4) (v). What is the average time a customer spends in the queue waiting for service? (3) BTL1
2. A bank has two tellers working on savings accounts. The first teller handles withdrawals only.
The second teller handles depositors only. It has been found that the service time distributions of both depositors and withdrawals are exponential with a mean service time of 3 minutes per customer. Depositors and withdrawers are found to arrive in a Poisson fashion throughout the day with mean arrival rate of 16 and 14 per hour. (i). What would be the effect on the average waiting time for depositors and withdrawers if each teller could handle both withdrawals and deposits? (8) (ii). What would be the effect if this could only be accomplished by increasing the service time to 3.5 minutes. (8) BTL5
3. At a certain filling station, customers arrive in a Poisson process with an average time of 12 per
hour. The time intervals between services follow exponential distribution and as such the mean time taken to service a unit is 2 minutes. Evaluate: (i). the probability that there is no customer at the counter. (3) (ii). the probability that there are more than two customers at the counter. (3) (iii). the probability that there is no customer to be served. (3) (iv). the expected number of customers waiting in the system. (2) (v). the expected number of customers in the waiting line. (2) (vi). the expected time a customer spends in the system. (3) BTL6
4. An insurance company has three claims adjusters in its branch office. People with claims against
the company are found to arrive in a Poisson fashion, at an average rate of 20 per 8 hour day. The amount of time that an adjuster with a claimant is found to have an exponential distribution, with mean service time 40 minutes. Claimants are processed in the order of their appearance. (i). How many hours a week an adjuster expected to spend with claimants? (8) (ii). How much time, on the average, does a claimant spend in the branch office? (8) BTL5
5. A super market has two sales girls at the sales counter, if the service time for each customer is exponential with a mean of 4 minutes, and if the people arrive in a Poisson fashion at the rate of 10 an hour. (a). Calculate the probability that a customer has to wait for being server. (8) (b). Expected percentage of idle time for each sales girl? (4) ( c). If a customer has to wait, what is the expected length of his waiting time? (4) BTL4
6. An airport emergency medical facility has a single paramedic and room for a total of three
patients, including the one being treated. Patients arrive with an exponentially distributed inter arrival time with a mean of one hour. Service time is exponentially distribute with a mean of 30 minutes. BTL 4
(i). What percentage of the time is the paramedic busy? (8) (ii). How many patients on average are refused entry in a 24 hour day? (4) (iii). What is the average number of patients in the facility at any given time? (4)
7. Arrivals of a telephone booth are considered to be poisson with an average time of 10 minutes
between one arrival and the next. The length of phone call is assumed to be distributed exponentially, with mean 3 minutes.
(i). What is the probability that a person arriving at the booth will have to wait? (6) (ii). The telephone department will install a second booth when convinced that an arrival would
expect waiting for atleast 3 minutes for a phone call. By how much should the flow of arrivals increase in order to justify a second booth? (6)
(iii). What is the average length of the queue that forms from time to time? (4) BTL1 8. A branch of a national bank has only one typist. Since the typing work varies in length, the typing
rate is randomly distributed approximating Poisson distribution with mean rate of 8 letters per hour. The letter arrives at a rate of 5 per hour during the entire 8 hour work day. If the typewrite is valued at Rs.1.50 per hour. Determine equipment utilization, the percent time an arriving letter has to wait, average system time and average idle time cost of the typewriter per day. (16) BTL6
9. Ships arrive at a port at the rate of one in every 4 hours with exponential distribution of inter
arrival times. The time a ship occupies a berth for unloading has exponential distribution with an average of 10 hours. If the average delay of ships waiting for berths is to be kept below 14 hours. How many berths should be provided at the port? (16) BTL3
10. Obtain the steady state difference equations for the queuing model (M/M/1) : (FCFS/N/∞) with
usual notations and solve them for P0 and Pn. Also find the average number of units in the system and average queue length. (16) BTL1
