valliammai engineering college semester/mc7401-resource...apply graphical method to solve the...

VALLIAMMAI ENGINEERING COLLEGE

SRM Nagar, Kattankulathur – 603 203

DEPARTMENT OF COMPUTER APPLICATIONS

QUESTION BANK

IV SEMESTER

MC 7401 – RESOURCE MANAGMENT TECHNIQUES

Regulation – 2013

Academic Year 2017 – 18

Prepared by

Mr. S. K. Saravanan, Assistant Professor/MCA

VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur – 603 203.


QUESTION BANK

SUBJECT: MC 7401 – RESOURCE MANAGMENT TECHNIQUES

SEM / YEAR: IV / II

UNIT I - LINEAR PROGRAMMING MODELS

SYLLABUS : Mathematical Formulation - Graphical Solution of linear programming models – Simplex

method – Artificial variable Techniques- Variants of Simplex method

PART – A (2 Marks)

Q.No Questions BT Level Competence

1. What is infeasible solution? BTL1 Knowledge

2. Show some applications of Linear Programming Problem. BTL3 Application

3. What is Operation Research and Analyze its characteristics? BTL4 Analysis

4. What are the components of LPP? BTL4 Analysis

5. Define Convex Polygon. BTL1 Knowledge

6. What are the limitations of LPP? BTL2 Comprehension

7. What is linear programming problem? Describe its characteristics. BTL1 Knowledge

8. What is the mathematical model of linear programming? BTL1 Knowledge

9. Compare standard and canonical forms of LPP? BTL2 Comprehension

10. Discuss the characteristics of standard and canonical forms of LPP? BTL2 Comprehension

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 Application

12. What is feasibility region? Is it necessary that it should always be a convex

set?

BTL4 Analysis

13. Define solution, feasible solution and optimal solution of LPP. BTL1 Knowledge

14. State the applications of linear programming. BTL3 Application

15. Devise the meaning of unbounded solution and Multiple Optimal solution. BTL5 Synthesis

16. Propose the meaning of slack and surplus variables. BTL5 Synthesis

17. How to resolve degeneracy in a LPP? BTL6 Evaluation

18. What is Simplex method? BTL1 Knowledge

19. Explain basic variable and non-basic variable in LP . BTL2 Comprehension

20. What do you think about unrestricted variable and artificial variable? BTL6 Evaluation

PART – B (13Marks)

1. Use Two – Phase simplex method to solve the following LPP.

Maximize Z = 3X1 + 2 X2

Subject to the constraints

2X1 + X2 ≤ 2

3X1 + 4X2 ≥ 12

X1, X2 ≥ 0

BTL3 Application

2. Use Big-M method to solve the following LPP.

Maximize Z = 3X1 - X2


2X1 + X2 ≥ 2

X1 + 3X2 ≤ 3

X1, X2 ≥ 0

BTL3 Application

3. Solve by Big-M method.

Minimize Z = 2X1 + X2


3X1 + X2 = 3

4X1 + 3X2 ≥ 6

X1 + 2X2 ≤ 3

X1, X2 ≥ 0

BTL3 Application

4. Solve the following LPP using Big-M method.

Maximize Z = 2X1 +3X2 + 4X3


3X1 + X2 + 4X3 ≤ 600

2X1 + 4X2 + 2X3 ≥ 480

2X1 + 3X2 + 3X3 = 540

X1, X2, X3 ≥ 0

BTL3 Application

5. Apply Simplex method to solve the LPP

Maximize Z = 3X1 + 5X2


3X1 + 2X2 ≤ 18

0 ≤ X1 ≤ 4, 0 ≤ X2 ≤ 6

BTL5 Synthesis

6. Use simplex method to solve the LPP.



2X1 + X2 ≤ 50

2X1 + 5X2 ≤ 100

2X1 + 3X2 ≤ 90 and

X1, X2 ≥ 0.

BTL3 Application

7. i. Apply graphical method to solve the following LPP (10)



X1 + X2 ≤ 450

2X1 + X2 ≤ 600

X1, X2 ≥ 0.

ii. Write the steps of graphical method. (3)

BTL5 Synthesis

8. Solve the LPP by graphical Method



2X1 - X2 ≥ -2

2X1 + 3X2 ≥ 12

X1 ≤ 4

X2 ≥ 2 and X1, X2 ≥ 0.

BTL3 Application

9. i. 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)

ii. Solve the following LPP Graphically:

Minimize Z = 4X1 + 3X2


X1 + 3X2 ≥ 9

2X1 + 3X2 ≥ 12

X1 + X2 ≥ 5

X1, X2 ≥ 0.

BTL3 Application

10.

i. 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.

(5)

ii. Solve the following LPP using Graphical Method

Max Z = 3x1 + 4 x 2


5x1 + 4x2 ≤ 200, 3x1 + 5x2 ≤ 150

5x1 + 4x2 ≥ 100, 8x1 + 4x2 ≥ 80 and

x1, x2 ≥ 0 (8)

BTL3 Application

11. Solve the following LPP.



3X1 + X2 ≥ 27

X1 + X2 ≥ 21

X1 + 2X2 ≥ 30

X1, X2 ≥ 0.

BTL3 Application

12. Apply graphical method to solve the LPP.

Maximize Z = X1 - 2X2


-X1 + X2 ≤ 1

6X1 +4X2 ≥ 24

0 ≤ X1 ≤ 5, 2 ≤ X2 ≤ 4

BTL5 Synthesis

13. Solve the following LPP by simplex method.

Minimize Z = 8 X1 - 2X2


-4X1 + 2X2 ≤ 1

5X1 - 4X2≤ 3

and X1, X2 ≥ 0.

BTL3 Application

14. Solve the following LPP.

Maximize Z = 5X1 + 6X2 + X3


9X1 + 3X2 - 2X3≤ 5

4X1 + 2X2 - X3 ≤ 2

X1 - 4X2 + X3 ≤ 3

X1, X2, X3 ≥ 0.

BTL5 Synthesis

PART – C (15Marks)

1. i. Use the graphical method to solve the following LPP. (10)

Minimize Z = 20 X1 + 10X2


X1 + 2X2 ≤ 40

3X1 + X2 ≥ 30 and

X1, X2 ≥ 0.

ii. Draw the flowchart for the Simplex method(Minimization case). (5)

BTL5 Synthesis

2. i. Use Simplex method to solve the LPP. (10)

Minimize Z = X1 - 3X2 + 2X3


3X1 - X2 + 2X3 ≤ 7

-2X1 + 4X2 ≤ 12

-4X1 + 3X2 + 8X3 ≤ 10

X1, X2, X3 ≥ 0.

ii. Write the Simplex Algorithm (Maximization case). (5)

ii. Draw the flowchart for the Simplex method(Minimization case).

BTL3 Application

3. i. Use Big-M method to solve the following LPP.



2X1 + X2 ≤ 18

3X1 + 2X2 ≥ 30

X1 + 2X2 = 26

X1, X2 ≥ 0

ii. Write the procedure of Big-M method.

(5)

(10) BTL3 Application

4. i. Use two-phase Simplex method to solve.

Maximize Z = 5X1 - 4X2 + 3X3


2 X1 + X2 - 6X3 = 20

6 X1 + 5X2 + 10X3 ≤ 76

8 X1 - 3X2 + 6X3 ≤ 50

X1, X2, X3 ≥ 0

ii. Write the procedure of two-phase method.

(5)

(10) BTL3 Application



QUESTION BANK

SUBJECT : MC 7401 – RESOURCE MANAGMENT TECHNICS

SEM / YEAR: IV / II

UNIT II - TRANSPORTATION AND ASSIGNMENT MODELS

SYLLABUS : Mathematical formulation of transportation problem- Methods for finding initial basic feasible

solution – optimum solution - degeneracy – Mathematical formulation of assignment models – Hungarian

Algorithm – Variants of the Assignment problem



1. Define transportation problem and its basic purpose. BTL1 Knowledge

2. Define feasible solution & basic feasible solution in transportation problem. BTL1 Knowledge

3. Provide a definition for optimal solution and mathematical formation of

transportation problem.

BTL2 Comprehension

4. Obtain the initial solution for the transportation problem using least cost

method.

A B C SUPPLY

1 5

2 8

3 7

4 14

DEMAND 34

BTL4 Analysis

5. Can you provide the basic steps involved in solving a transportation

problem?

BTL2 Comprehension

6. What do you understand by degeneracy in a transportation problem? BTL4 Analysis

7. What are balanced & unbalanced transportation problems? BTL1 Knowledge

8. How do you convert an unbalanced transportation problem into a balanced BTL5 Synthesis

9. Describe how the profit maximization transportation problem can be

converted to an equivalent cost minimization transportation problem.

BTL1 Knowledge

10. Solve the assignment problem BTL3 Application

A B C D

I 20 25 22 28

II 15 18 23 17

III 19 17 21 24

IV 25 23 24 24

11. What differences exist between Transportation & Assignment Problem? BTL2 Comprehension

12. What are assignment problem and unbalanced assignment problem? BTL1 Knowledge

13. Define unbounded assignment problem and describe the steps involved in

solving it.

BTL1 Knowledge

14. Describe how a maximization problem is solved using assignment model. BTL4 Analysis

2 7 4

3 3 1

5 4 7

1 6 2

7 9 18

15. What do you understand by restricted assignment? How do you overcome

it?

BTL6 Evaluation

16. How do you identify alternative solution in assignment problem? BTL6 Evaluation

17. Illustrate the travelling salesman problem. BTL3 Application

18. Can you say how a minimization problem is solved using assignment model? BTL2 Comprehension

19. Can you propose which method is best for finding initial basic feasible

solution in transportation problem?

BTL5 Synthesis

20. Show the mathematical formulation of assignment model. BTL3 Application


1. Good has to be transported from sources S1, S2 and S3 destinations D1, D2

and D3. The transportation cost per unit, capacities of the sources and

requirements of the destinations are given in the following table.

I II III SUPPLY

S1

S2

S3

DEMAND

Determine a transportation schedule so that cost is Minimum.

BTL5 Synthesis

2. Solve the following assignments problems

I II III IV V

A

B

C

D

E

BTL3 Application

3. Solve the TP where cell entries are unit costs. Use Vogel’s approximate

method to find the initial basic solution.

D1 D2 D3 D4 D5 AVAILABLE

O1 18

O2 17

O3 19

O4 13

O5 15

Required 16 18 20 14 14

BTL1 Knowledge

8 5 6 120

15 10 12 80

3 9 10 150

150 80 50

10 5 9 18 11

13 19 6 12 14

3 2 4 4 5

18 9 12 17 15

11 6 14 19 10

68 35 4 74 15

57 88 91 3 8

91 60 75 45 60

52 53 24 7 82

51 18 82 13 7

4. A small garments making units has five tailors stitching five different types

of garments all the five tailors are capable of stitching 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

B

C

D

E

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.

BTL4 Analysis

5. Solve the following TP to maximize profit

A B C D SUPPLY

1 100

2 30

3 70

DEMANDS 40 20 60 30

BTL3 Application

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

W2

W3

W4

W5

Determine whether the new machine can be accepted and also determine

optimal assignments and the associated saving in cost

BTL6 Evaluation

7. Solve the following Transportation Problem

A B C D SUPPLY

30

I

40

II

30

III

20 20 25 35 100

DEMAND

BTL3 Application

7 9 4 8 6

4 9 5 7 8

8 5 2 9 8

6 5 8 10 10

7 8 10 9 9

40 25 22 33

44 35 30 30

38 38 28 30

12 3 6 - 5 8

4 11 - 5 - 3

8 2 10 9 7 5

- 7 8 6 12 10

5 8 9 4 6 -

15 18 22 16

15 19 20 14

13 16 23 17

8. Determine the IFBS using North-West Corner Rule and VAM

1 2 3 4 5

A

B

C

Demand

Supply

BTL3 Application

9. Solve the following assignment problem.

M1

J1

J2

J3

M2

M3

M4

BTL3 Application

10. Solve the following Transportation Problem. BTL3 Application

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

11. Solve the following travelling salesman problem so as to minimize the cost

per cycle.

A B C D E

A

B

C

D

E

BTL3 Application

12. A Company has one surplus truck in each of the cities A,B,C,E&E and one

deficit truck in each of the cities 1,2,3,4,5&6. The distance between the

cities (in km) is shown in below matrix.

To Cities (Deficit)

1 2 3 4 5 6

From cities A 12 10 15 22 18 8 (Surplus)

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

Find the assignment of trucks from cities in surplus to cities in deficit so that

the distance covered these vehicles is minimum.

BTL5 Synthesis

2 11 10 3 7 4

1 4 7 2 1 8

3 9 4 8 12 9

3 3 4 5 6 21

18 24 28 32

8 13 17 18

10 15 19 22

- 3 6 2 3

3 - 5 2 3

6 5 - 6 4

2 3 6 - 6

3 3 4 6 -

13. Find the optimal solution of the following transportation problem.

Destination X Y Z SUPPLY

P 30

Origin Q 35

R 35

DEMAND 30 40 30

BTL3 Application

14. A Transport corporation has three vehicles in three cities. Each of

vehicles can be assigned to any of the four other cities. The distance

from one city to another as under : W X Y Z

A

B

C

You are required :

i. to assign a vehicle to a city in such a way that the total distance

travelled is minimized. (10)

ii. to build a mathematical model (3)

BTL5 Synthesis


1. i. Solve the following Travelling Salesman Problem. (10)

To

P Q R S

A

From B

C

D

ii. Discuss the various degeneracy occur in Transportation problem and how to resolve it. (5)

BTL3 Application

2. i. Find an optimal solution to the following Transportation Problem. A B C SUPPLY

1 50

2 70

3 80

4 140

DEMAND 340 (10)

ii. Write the procedure for Modified Distribution Method (MODI) to

solve the Transportation problem. (5)

BTL3 Application

1 2 0

2 3 4

1 5 6

33 40 43 32

45 28 31 23

42 29 36 29

∞ 46 16 40

41 ∞ 50 40

82 32 ∞ 60

40 40 36 ∞

2 7 4

3 3 7

5 4 1

1 6 2

70 90 180

3. i. Consider the following profit matrix.

SUPPLY

1 150

2 200

3 125

DEMAND

Minimize the profit. (10)

ii. Write the procedure of Vogel’s Approximation method. (5)

BTL3 Application

4. i. Four jobs are to be done on four different machines. Assign the jobs

so as to maximize the total profit. (10)

ii. Draw the flowchart of Hungarian method. (5)

BTL3 Application

A B C D E

19 21 16 15 15

9 13 11 19 11

18 19 20 24 14

80 100 75 45 125

M1 M2 M3 M4

J1 15 11 13 15

J2 17 12 12 13

J3 14 15 10 14

J4 16 13 11 17



QUESTION BANK


SEM / YEAR: IV / II

UNIT III - INTEGER PROGRAMMING MODELS

SYLLABUS : Formulation – Gomory’s IPP method – Gomory’s mixed integer method – Branch

and bound technique.



1. State the Integer Programming Model. BTL1 Knowledge

2. Explain the characteristics of Integer programming problem. BTL4 Analysis

3. List some of the applications of Integer Programming. BTL3 Application

4. Define Mixed Integer Programming Problem. BTL2 Comprehension

5. What is Zero-one problem? BTL1 Knowledge

6. Write the differences between Pure & mixed integer programming. BTL2 Comprehension

7. Explain the importance of Integer Programming. BTL4 Analysis

8. Why not round off the optimum values instead of resorting to IP? BTL5 Synthesis

9. Write the difference between LPP and IPP. BTL2 Comprehension

10. What is cutting Plane method? BTL1 Knowledge

11. What do you think about search method? BTL6 Evaluation

12. Write about Branch and Bound Technique. BTL5 Synthesis

13. Construct the general format of IPP. BTL3 Application

14. Write the Gomory’s IPP method. BTL2 Comprehension

15. What is the purpose of Fractional cut constraints? BTL1 Knowledge

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

Application

17. Explain Gomory’s Mixed Integer Method. BTL4 Analysis

18. What the geometrical meaning is of partitioned or branched the original

problem? BTL1 Knowledge

19. What is standard discrete programming problem? BTL1 Knowledge

20. How can you improve the efficiency of partitioned method? BTL6 Evaluation


1. Solve the following mixed integer linear programming problem.

Maximize Z = X1 + X2

Subject to

2X1 + 5X2 ≤ 16

6X1 + 5X2 ≤ 30

X1, X2 ≥ 0 and X1 is integer.

BTL3 Application

2. Solve the following ILPP.



-X1 + 2X2 ≤ 4

5X1 + 2X2 ≤ 16

2X1 - X2 ≤ 4

X1, X2 ≥ 0 and are non negative integers.

BTL3 Application

3. Solve the integer programming problem.

Maximize Z = 2X1 + 20X2 - 10X3


2X1 + 20X2 + 4X3 ≤ 15

6X1 + 20X2 + 4X3 = 20

X1, X2, X3 ≥ 0 and are integers.

BTL3 Application

4. Explain the geometrical interpretation of Branch and Bound method by

solving the following Integer Linear Programming.



3X1 + 2X2 ≤ 12

X2 ≤ 2

X1, X2 ≥ 0 and are integers.

BTL2 Comprehension

5. Solve the following mixed integer programming problem.

Maximize Z = 7X1 + 9 X2


-X1 + 3X2 ≤ 6

7X1 + X2 ≤ 35

and X1, X2, ≥ 0, X1 is an integer.

BTL3 Application


Maximize Z = 4X1 + 6X2 + 2X3


4X1 - 4X2 ≤ 5

-X1 + 6X2 ≤ 5

-X1 + X2 + X3 ≤ 5

and X1, X2, X3 ≥ 0, and X1 , X3 are integers.

BTL3 Application

7. Use Branch and bound algorithm to solve the following ILPP



7X1 + 16X2 ≤ 52

3X1 - 2X2 ≤ 18


BTL5 Synthesis

8. Apply Branch and bound algorithm to solve the following ILPP

Maximize Z = X1 + 4X2


2X1 + 4X2 ≤ 7

5X1 + 3X2 ≤ 15


BTL4 Analysis

9. Apply Branch and bound algorithm to solve the following ILPP



5X1 + 3X2 ≤ 8

X1 + 2X2 ≤ 4

X1, X2 ≥ 0 and are integers

BTL4 Analysis

10. Using Gomory’s cutting plane method to solve the problem.



5X1 + 3X2 ≤ 8

2X1 + 4X2 ≤ 8

X1, X2 ≥ 0 and are all integers.

BTL5 Synthesis

11. Describe the Branch and Bound Technique for Pure and Mixed IPP with

flowchart.

BTL1 Knowledge

12. i. Explain the Gomory’s Cutting Plane algorithms for Pure (All) IPP. (7)

ii. Explain the Gomory’s Cutting Plan algorithm for Mixed IPP. (6)

BTL2 Comprehension

13. Solve the following mixed integer problem.



X1 ≤ 8

X2 ≤ 10

5X1 + 3X2 ≥ 45

X1, X2 ≥ 0, and X1 integer.

BTL3 Application


Maximize Z = X1 - 3X2


X1 + X2 ≤ 5

-2X1 + 4X2 ≤ 11

X1, X2 ≥ 0 and and X2 is an integer.

BTL3 Application

PART – C (15 Marks)

1. Solve the following ILPP using Gomary’s Cutting Plane Method.



2 X1+5X2 ≤ 16

6X1 +5 X2 ≤ 30


BTL3 Application

2. Solve the following all-integer programming problem using the Branch and

Bound Method.

Miniimize Z = 3X1 + 2.5X2


X1 + 2X2 ≥ 20

3X1 + 2X2 ≥ 50


BTL3 Application

3. Solve the following mixed integer linear programming problem using

Gomarian’s cutting plane method.



3 X1 + 2 X2 ≤ 5

X2 ≤ 2

X1, X2 ≥ 0 and X1 is an integer.

BTL3 Application

4. Solve the following mixed integer problem.

Maximize Z = -3X1 + X2 + 3 X3

Subject to

- X1 + 2X2 + X3 ≤ 4

2X2 + 1.5X3 ≤ 1

X1 - 3X2 + 2X3 ≤ 3

X1, X2 ≥ 0 and X3 non-negative integer.

BTL3 Application



QUESTION BANK


SEM / YEAR: IV / II

UNIT IV - SCHEDULING BY PERT AND CPM

SYLLABUS : Network Construction – Critical Path Method – Project Evaluation and Review Technique – Resource Analysis in Network Scheduling



1. What do you mean by project? Give an example. BTL6 Evaluation

2. What are the assumptions made in PERT calculations? BTL4 Analysis

3. What are the two basic planning and controlling techniques in a network? BTL4 Analysis

4. Write down the advantages of CPM and PERT techniques? BTL2 Comprehension

5. Differentiate between PERT and CPM. BTL2 Comprehension

6. Draw the network for the project whose activities with predecessor

relationship are given below :

A,C,D can start simultaneously. E > B,C; F,G > D; H,I >E,F; J > I, G; K> H; B > A;

BTL1 Knowledge

7. How does the PERT technique help a business manager in decision-making? BTL6 Evaluation

8. Can you write about activity & Critical Activities? BTL5 Synthesis

9. Define Dummy Activities and duration. BTL1 Knowledge

10. List three main managerial functions for any project. BTL4

11. What are float and slack? BTL1 Knowledge

12. Classify Crash time and Crash cost. BTL3 Application

13. Can you list the network construction steps? BTL3 Application

14. Define Optimistic time time estimate in PERT. BTL1 Knowledge

15. Show and define Pessimistic time estimate in PERT. BTL3 Application

16. How do you calculate most likely time estimation? BTL5 Synthesis

17. What is a parallel critical path? BTL1 Knowledge

18. Distinguish standard deviation and variance in PERT network? BTL2 Comprehension

19. Give the difference between direct cost and indirect cost. BTL2 Comprehension

20. What is meant by resource analysis? BTL1 Knowledge


1. A project schedule has the following characteristics.

i. Construct Network diagram (3)

ii. Compute Earliest time and latest time for each event. (5)

iii. Find the critical path. Also obtain the Total float, Free float and slack time

and Independent float. (5)

BTL1 Knowledge

2. A small project is composed of seven activities whose time estimates are

listed in the table as follows:

i. Draw the network and find the project completion time. (7)

ii. Calculate the three floats for each activity. (6)

BTL3 Application

3. Calculate the total float, free float and independent float for the project

whose activities are given below:

Find the critical path also.

BTL1 Knowledge

4. Draw the network for the following project and compute the earliest and

latest times for each event and also find the critical path.

BTL6 Evaluation

Activity 1 – 2 1 – 3 2 – 4 3 – 4 3 – 5 4 – 9

Time 4 1 1 1 6 5

Activity 5 – 6 5 – 7 6 – 8 7 - 8 8 - 10 9 - 10

Time 4 8 1 2 5 7

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

Activity 1 –2 1 – 3 1 – 5 2 – 3 2 – 4 3 – 4

Time 8 7 12 4 10 3

Activity 3 – 5 3 – 6 4 - 6 5 - 6

Time 5 10 7 4

Activity 1 – 2 1 – 3 2 – 4 3 – 4 4 – 5 4 – 6

Immediate

Predecessor --- --- 1 – 2 1 – 3 2 – 4

2 – 4 &

3 - 4

Time 5 4 6 2 1 7

Activity 5 – 7 6 – 7 7 - 8

Immediate

Predecessor 4 – 5 4 – 6

6 – 7 &

5 - 7

Time 8 4 3

5. The following table lists the jobs of a network with their time estimates:

i. Draw the project network. (4)

ii. Calculate the length and variance of the Critical Path. (3)

iii.What is the approximate probability that the jobs on the critical path will

be completed by the due date of 42 days? (3)

iv.What due date has about 90 % chance of being met? (3)

BTL5 Synthesis

6. 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.

i. Draw the project network and identify all the paths through it. (5)

ii. Find the expected duration and variance for each activity. What is the

expected project length? (3)

ii. Calculate the variance and standard deviation of the project length.What

is the probability that the project will be completed at least 4 weeks earlier

than expected time? (5)

BTL2 Comprehension

7. Draw the network and determine the critical path for the given data.

Find the Total Float, Free Float and Independent float for the given data

BTL3 Application

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

Job(i, j)

Duration


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

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

8. 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

BTL6 Evaluation

9. The following table lists the jobs of a network along with their time

estimates.

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

BTL3 Application

10. A project has the following activities and characteristics.

i. Find expected duration of each activity.

ii. Draw the project network and expected duration of the project.

iii. Find variances of activities on critical path and its standard deviation.

BTL4 Analysis

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

Job(I, j)

Duration


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

Activity

Estimated duration in days

Optimistic Most likely Pessimistic

1 – 2 2 5 8

1 – 3 4 10 16

1 – 4 1 7 13

2 – 5 5 8 11

3 – 5 2 8 14

4 – 6 6 9 12

5 – 6 4 7 10

11. Draw a network from the following activity and find a critical path and

duration of project.

BTL3 Application

Activity

1 –2 1 – 3 2 – 4

(Dummy)

2 – 7 3 – 4

(Dummy)

3 – 8 4 – 5

Time 3 8 0 1 0 2 4

12. The following information is available:

i. Draw the network and find the critical path.

ii. What is the peak requirement of Manpower? On which day(s) will this

occur?

iii. If the maximum labour available on any day is only 10, when can the

project be completed?

BTL4 Analysis

13. i. Explain the different phases of Network Analysis .

ii. Explain the Network components in detail with neat sketches.

(6)

(7)

BTL2 Comprehension

14. i. Describe the precedence relationship in detail with diagrams.

ii. Describe the different types of Floats and Slacks.

(6)

(7)

BTL1 Knowledge


1. A project consists of a series or tasks labelled A, B,...H,I with the following

relationships (W < X , Y means D and Y cannot start until W is completed;

X,Y < W means W cannot start until both X and Y are completed). With this

notation, construct the network diagram having the following constraints:

A < D, E; B,D < F; C < G ; B < H; F, G < I

Find also the critical path of the project, when the time (in days) of each task

is as follows :

BTL1 Knowledge

5 – 6 6 – 7

(Dummy)

6 – 8

(Dummy)

7 - 9 8 - 9 9 -10 10 - 11

7 0 0 5 8 8 9

Activity No. of Days No. of men reqd. per day

A 1-2 4 2

B 1-3 2 3

C 1-4 8 5

D 2-6 6 3

E 3-5 4 2

F 5-6 1 3

G 4-6 1 8

Task A B C D E F G H I

Time 23 8 29 16 24 18 19 4 10

2. A project schedule ahs the following characteristics:

Construct a network and find critical path, total duration of the project and

various time estimates.

BTL3 Application

3. The following table gives the activities in a construction project and other

relevant information.

i. Draw a PERT diagram.

ii. Find the probability that the project will be completed in less than 60

days.

BTL6 Evaluation

4. The following project network and associated costs are given below:

i. Draw the network diagram

ii. What is the earliest the project can be completed? What is the lowest cost

for completing it in this time?

BTL4 Analysis

Activity 1 – 2 1 – 4 1 – 7 2 – 3 3 –6 4 – 5 4 – 8 5 – 6

Duration 3 2 1 3 2 4 6 5

Activity 6 – 9 7 – 8 8 – 9

Duration 4 4 5

Activity Optimistic Time

T1

Normal Time T2

Pessimistic Time

T3

1 – 2 30 44 54

1 – 3 8 12 16

2 – 3 1 2 3

2 – 4 2 3 5

3 – 4 8 10 12

4 – 5 14 22 25

Activity Predecess

or

Normal Crash

Tn (Days) Cn (Rs.) Tc(Days) Cc (Rs.)

A - 9 10 6 16

B - 8 9 5 18

C A 5 7 4 8

D A 8 9 6 19

E B 7 7 3 15

F C 5 5 5 5

G E,D 5 8 2 23



QUESTION BANK


SEM / YEAR: IV / II

UNIT V - QUEUEING MODELS

SYLLABUS : Characteristics of Queuing Models – Poisson Queues - (M / M / 1) : (FIFO / ∞ /∞), (M / M / 1)

: (FIFO/ N / ∞), (M / M / C) : (FIFO / ∞ / ∞), (M / M / C) : (FIFO / N / ∞) models.



1. Define Kendal’s notation for representing queuing models. BTL1 Knowledge

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

Synthesis

3. Explain Queue discipline and its various forms. BTL4 Analysis

4. Distinguish between transient and steady state queuing system. BTL4 Analysis

5. Define steady state. BTL1 Knowledge

6. Define Jockeying. BTL1 Knowledge

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 Synthesis

8. List the basic elements of queuing system. BTL2 Comprehensi

9. What do you meant by Balking and Reneging of customer behavior in a

queuing system? BTL2 Comprehension

10. What are the characteristics of queuing model? BTL2 Comprehension

11. Define Poisson process with its properties. BTL1 Knowledge

12. A two channel waiting line with Poisson arrival has a mean rate of 50 per

hour and exponential service with mean rate and 75 per hour for each

channel. Find the probability of empty system and the probability than an

arrival in the system will have to wait.

BTL4

Analysis

13. Customer arrives at a one-man barber shop according to a Poisson process

with a 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

Evaluation

14. Define pure birth process. BTL1 Knowledge

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 Knowledge

17. What is “traffic intensity”? BTL3 Application

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 Evaluation

19. Give the applications of Queuing Theory. BTL3 Application

20. State the operating characteristics of queuing system. BTL2 Comprehension


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? (2½)

ii. What is the average number of customers in the queuing system? (2½)

iii. What is the average time a customer spends in the system? (2½)

iv. What is the average number of customers in the queue? (3)

V. What is the average time a customer spends in the queue waiting for

service? (2½)

BTL1 Knowledge

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? (7)

ii. What would be the effect if this could only be accomplished by increasing

the service time to 3.5 minutes? (6)

BTL5 Synthesis

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. (2)

(ii). the probability that there are more than two customers at the counter. (2)

(iii). the probability that there is no customer to be served. (2)

(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 Evaluation

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? (7

ii. How much time, on the average, does a claimant spend in the branch

office?

BTL5 Synthesis

5. A supermarket has two girls at the counters .The customers arrive in an

poisson fashion at the rate of 12 per hour .The service time for each

customer is exponential with mean 6 minutes. Find

i. The probability that an arriving customer has to wait for service

ii. The average number of customer in the system and

iii. The average time spent by a customer in the supermarket

BTL4 Analysis

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.

i.What percentage of the time is the paramedic busy? (7)

ii. How many patients on average are refused entry in a 24 hour day? (3)

iii. What is the average number of patients in the facility at any given time?

(3)

BTL4 Analysis

7. i. Explain the fundamental components of a queuing process. (9)

ii. Write the conditions for single channel queuing model. (4)

BTL2 Comprehension

8. Draw the basic structures of Queuing Models and explain the various

processes. (13)

BTL1 Knowledge

9. 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?

(5)

ii. The telephone department will install a second booth when convinced that

an arrival would expect waiting for at least 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 Knowledge

10. 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 present time an

arriving letter has to wait, average system time and average idle time cost of

the typewriter per day.

BTL6 Evaluation

11. 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 delays of ships waiting for berths is to be kept below 14 hours. How

many berths should be provided at the port?

BTL3 Application

12. Obtain the steady state difference equations for the queuing model

(M/M/1) : (FCFS/N/∞) with usual nota0ons and solve them for P0 and Pn.

Also find the average number of units in the system and average queue

length.

BTL1 Knowledge

13. A T.V 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 came in,and if they arrival of sets is approximately Poisson with the

average rate of 10per 8-hour day, what is repairman 's expected idle time

each day? How many jobs are ahead of the average set just brought in?

BTL3 Application

14. A factory has five machines. On an average there are two machine

breakdowns every 5 weeks .Assuming the repairing capacity is one machine

a week, the repairing time being exponential distributed ,determine

i. The probability that the service facility will be idle

ii. The probability that there shall be exactly 3 machines to be, and being

repaired

iii. The excepted number of the queue,

iv. The excepted time a machine waiting to be queue to be repaired ,and

v. The excepted time machine shall wait in the queue to be repaired, and

vi. The excepted time that a machine shall spend in the system-ie waiting for

and getting repaired

BTL3 Application


1. A tailor specializes in ladies dresses .The number if customers approaching

the tailor appear to be Poisson distributed with a mean of 6 customers per

the hour .The tailor attends the customers on the first come first served -

basis and the customers wait, it the need be. The tailor can attend the

customers at an average rate of 10 customers per hour with the service time

exponentially distributed.

i. Find the probability of the number of arrivals (0 through 5) during

(a) a 15 minutes interval and (b) a 30 minute interval

ii. The utilization parameter

iii. The probability that the queuing system is idle

iv. The average time that the tailor is free on an 10 hour working day

BTL1 Knowledge

2. Past records indicated that of the five machine that a factory owns,

breakdown occurs at random and the average time between the breakdown

is 2 days .Assuming that the repairing capacity of the workman is one

machine a day and the repairing is distributed exponentially ,determine the

following

i. The probability that the service facility will be idle

ii. The probability of various numbers of machine (0 through 5)to be and

being repaired

iii. The excepted length of the queue

iv. The Excepted number of machine waiting to be and being repaired

v. The excepted time that a machine shall wait in the queue to be repaired.

vi. The excepted time that a machine will be in the system

BTL3 Application

3. A tax consulting firm has three counters in its office to receive people who

have problems concerning their income, wealth and sales taxes .On the

average 48 persons arrive in an 8 hour in a day. Each tax advisor spends 15

minutes on an average on an arrival. If the arrivals are poissonly distributed

and service times are according to exponential distribution. Find

i. The average number if customers in the system

ii. Average number of customers waiting to be srved

iii. Average time a customer spends in the systems

BTL4 Analysis

4. Let there be an automobile inspection situation with three inspection stalls.

Assume that cars wait in such a way that when a stall becomes vacant. The

car at the of the line pulls up to it. The station can accommodate almost four

cars waiting (seven in station) at one time. The arrival pattern is poisson

with a mean of one car every minute during the peak hours. The service

time is exponential with a mean 6 minutes. Find the average number of

customers in the system during the peak hours, the average waiting time

and the average number per hour that cannot enter the station because of

full capacity.

BTL6 Evaluation

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