OPERATION RESEARCH QUESTION BANK PART-I

1. Formulate the dual of the following LPP:Maximise : z= 2x1 + 3x2

Subject to Constraints:-x1 + 2x2 ≤ 4 x1 + x2 ≤ 6 x1 + 3x2 ≤ 9x1, x2 0

2. From the following two persons zero sum game find the value of the game:

B 8 -3 7 A 6 -4 5 -2 2 -3

3. A project is characterized by activities A and G and the times required and costs for regular programme for each activity are as follows:

EventActivityEvent

Name ofActivity

Time (weeks)Regular

1123245

2344566

ABCDEFG

5234435

Draw a network for the problem using only time for regular programme activities. Trace the critical path and state how long will it take to complete the project. What would be the cost of completing this project in this fashion? 4 . Solve the following LPP by Simplex method: Maximize : Z = 4X1 + 3X2

Subject to :

2X1 + X2 ≤ 72

X1 + 2X2 ≤ 48

X1 0, X2 0

1

5. Solve the following Transportation Problem :

P Destination Q

R Availability (units)

Origins Cost (Rs.)

Requirement (units)A B C D

2 3 5 1 7 3 4 6

4 1 7 2

5 8 7 14

7

9

18

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6. An automobile dealer wishes to put 4 repairmen to 4 different jobs. The repairers have some different kinds of skills and exhibit different levels of efficiency from one job to another. The dealer had estimated the number of man hours that are required for each job man combination. This is given in the matrix form as follows:

Jobs A B C D 1 5 3 2 8

2 7 9 2 6

3 6 4 5 7

4 5 7 7 8

7. A company manufactures four products X1, X2, X3, X4 each of which requires resources I, II and III. The data available are as per table given below :

Resource

Products Total availability

(units)X1 X2 X3 X4

IIIIII

Contribution Per unit (Rs.)

1 1 1 16 4 2 12 4 9 10

3 4 6 10

129070

2

It is required to find the quantities of each product that would maximize the contribution

8. Solve the game whose payoff matrix is given by

Play B

1 3 1

Play A 0 -4 -3

1 5 -1

9. You are given the following information:

ProjectEvent

ActivitiesTo event

Name of the activity

PrecedingActivities

EstimatedTime

(months)

1112345

2345566

ABCDEFG

---ABC

D,E

912156302932

Construct a network diagram .

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ActivityPrecedingActivity

Time Estimates (in weeks )

OptimisticMost

LikelyPessimistic

ABCDEFGH

NoneNone

AAA

B,CD

E,F,G

2108107933

4129157.59

3.55

1226102011975

(i) Determine the critical path(ii) If a 30 weeks deadline is imposed, what is the probability that the project will

be finished within the time limit?(iii) If the project manager wants to be 99% sure that the project is completed on

the scheduled date, how many weeks before that date should be start the

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project work. (Area of normal curve between z = 0.41 is 0.1591 and the value of z which covers 49% of area is 2.33

11. Determine the optimum strategies and the value of game for the payoff matrix:

B

12. Assume that the time estimates for an activity in a PERT network t0, tp and. tm are 4, 12 and 5 weeks respectively. What is the expected time of the activity and is standard deviation? 13. Determine the value of the following game : Player B A1 Player A A2

14. Using the graphical technique, find the optimal mixed strategies for both players A and B and the value of the game for the following competitive situation.

Player B

B1 B2 B3 B4 B5

A1 Player A A2

15. Consider the following project :

ActivityPredecessor

ActivityActivity duration

(days )

ABCDEFG

--AA

B.CB.C

F

1512734281814

(i) Construct the PERT chart (ii) Determine the project Completion Time.

-2 -4

3 8

5 -1 11 -3 13

-3 9 -5 3 1

4

16. Consider the following LP Problem:

Maximize z = 4x1 + 6x2 + 8x3

Subject to constraints - 4x1 + 2x2 + 2x3 20 3x2 + 6x3 45

and x1, x2 0

Solve this problem using the Simplex algorithm.

17. Assign the Typists to the Departments such that the total number of letters typed is maximum represents a situation where a given typist can be assigned to the particular department.

X Y Z A Typist B C 18. Consider the following projects activities :

ActivityPredecessor Activity

ABCDEFG

--AA

B,CB,C

F

Construct the PERT chart for the project.19. For the PERT chart Determine the project completion time

(i) What is the critical path?(ii) What are the critical activities?

D A 28 G 12 B E 5 20 10

5

1

2

5

3

6

4 7

14 C H F 16

20. Solve the game with the following pay-off matrix Player B

B1 B2 B3 B4 B5

A1

Player A A2 A3

A4

21. Consider the transportation problem shown in the tableau below:

To

From Destination

SupplierK L M N

Source

Demand

A

B

C

200

- 6 5 8

15 21 24 7

9 10 12 11

300 100 200

300

200

300

Solve this problem so that the total cost of transporting good from all Sources to destination is minimum.

22. Given the following payoff matrix, find the value of the game and strategies both the players can play. Player B

B1 B2 B3 B4

A2 3 2 4 0 A2 3 4 2 4

A3 4 2 4 0

A4 0 4 0 8

9 10 5 2 11

7 1 -3 3 5

13 6 -1 12 10

8 3 2 -3 4

6

23. Using North west Corner rule and MODI solve the transportation problem with unit transportation costs, availability and demand as given below:

Destinations D1 D2 D3 D4Availability

(in units)

Origin

Demand(in units)

01

02

03

6 2 8 9

12 6 16 12

4 6 18 4

40 50 130 60

80

80

120

280

24. The following table gives data on Normal time, Crash time, Normal cost and Crash cost of a project

ActivityNormalTime

(in weeks)

CrashTime

(in weeks)

NormalCost

(in Rs.)

CrashCost

(in Rs.)

1-22-32-42-53-44-65-6

3354433

2334121

300752001201009060

45075300120190130110

(i) Draw the network and find out the critical path and the normal project duration .

(ii) Find out the total float associated with each activity.(iii) If the indirect costs are Rs. 100 per week, find out the optimum duration by

crashing and the corresponding projects costs.(iv) With the crash durations indicated, what would be the minimum crash

duration possible, ignoring indirect costs?25. Solve the following Transportation problem by taking the initial basic feasible Solution by North West Corner Rule

From To 1 2 3 4 Supply 1

2

3

4

13 14 12 11

12 11 14 13

10 12 15 13

11 13 12 14

120

120

60

80

7

5

Demand

15 14 11 12

160 120 80 40

20

400

26. Obtain the optimal assignment from the following table: I II III IV

A

B

C

D

27. Determine the optimal strategies for both the players and the value of the game from the following payoff matrix. Player A a1 a2 a3 a4

b1

Player B b2

b3

8. The details of a small project are given below: Draw the network

Activity A B C D E F G

Dependence

Duration (in days)

Activity

Dependence

Duration (in days)

- - - B,C A C E

9 4 7 8 7 5 10

H I J K L

E D,F,H E I,J G

8 6 9 10 2

29. The owner of a Machine shop has four machinists available to assign to jobs for the day. Four jobs are offered with an expected profit in rupees for each machinist on each machinist on each job as follows: Jobs A B C D

1 62 78 50 101

Machinists 2 71 84 61 73

3 87 92 111 71

4 48 64 87 77 Determine the assignment of machinists to jobs that will result in a maximum

160 130 175 190

135 120 130 160

140 110 125 170

50 50 80 80

4 4 4 4

3 4 2 4

1 3 9 2

8

Profit.

30. To solve the following transportation problem. Obtain the initial solution by VAM method.

SourceDestination

1 2 3 4 Supply

A

B

C

7 3 8 5

5 2 6 11

3 6 5 2

160

180

100

Demand 40 100 120 180 440

31. A small project consisting of eight activities has the following characteristics :

Activity A B C D E F G H

Preceeding

ActivityTime aEstimates m(in weeks) b

- - A A A B,C D E,F G 2 13 10 8 7 9 3 5 6 4 15 12 9 8 10 4 5 812 23 26 10 9 17 5 5 10

a- most optimistic, m- most likely, b- most pessimistic. Draw PERT, find critical path and project completion time.

32. The following table gives the activities in a Project:

ActivityTime(days)

NormalTime

(Days)

CrashCost(Rs.)

Normal CrashCost(Rs.)

1-21-32-32-43-44-54-65-76-7

202510125105108

17258625353

600200300400300300600500400

720200440700420600900800700

Draw the activity network of the project.(i) Find the total float and free float for each activity.

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(ii) Using the above information crash the activity step by step until all the paths are critical.

33. Solve the following Transportation Problem:

Plants

P1 P2 P3 P4 P5 P6 Demand

Warehouses W1

W2

W3

W4

W5

5 7 13 10 9 8

8 6 14 13 22 15

12 10 9 11 14 26

5 18 12 14 20 7

6 3 5 4 14 9

300

200

400

600

700

Supply 500 300 600 200 400 200 2,200

34. Find the initial basic feasible solution by Vogel’s approximation method find the cost that allocation.

To

From

Distribution CentreSupply

D1 D2 D3 D4

P1

Plants P2

P3

Demand

2 3 5 1

7 3 4 6

4 1 6 2

5 8 7 15

7

10

18

35. Find the optimum assignment for the following problem.

M/C

A B C D

1

Job 2

3

7 4 8 5

4 6 10 6

9 3 7 2

3 6 9 6

10

4

36. Draw the Network for the following project.

Activity Immediate Predecessor activities

ABCDEFG

StartStartStart

ACA

B,D,E

37. Solve the following transportation problem and obtain optimum solution.

To Availability

X Y Z W

From A B C D

3 3 5 45 1 3 36 4 4 34 1 4 2

15351219

Demand 21 26 17 17

38. Draw the PERT NETWORK for the following case, Identify the critical path. Find the expected project completion time.

ActivityImmediatePrecedence

Requirement

Time Required in Days

OptimisticMost likely

Pessimistic

ABCDEFGHIJKL

-ABADAFG

C,E,HG

L,I,JA

342317101015210

477558701020715

51012139217010251220

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39. Solve the game given in the table below.

B

40. Four computer programming hobs have to be assigned to four programmers. Time taken by each programmer on each job is shown in the table below. Find an optimum assignment. Programmers

P1 P2 P3 P4 A Job B

C

D 41. Draw the network for the following activities and precedence relationship.

Activity Immediate predecessor activity

ABCDEFGH

---ABB

D,C,EF,G

42. Find the value of the following game:

Player B

Player A 2 1

-1 0

9 3 7 2

4 4 10 5

4 6 9 6

7 4 8 5

12

43. Determine the basic feasible solution and optimum solution for the following transportation problem: Destinations

D1 D2 D3 D4 D5 Supply

P1 12 9 10 9 6 50

P2 3 5 5 7 7 40

P3 5 3 11 9 11 10 P4 8 2 10 11 2 80 40 40 20 60 20 180

44. Draw the network consisting of the following activities. Find the critical path, project duration, variance of the critical path.

Duration (Days)

Activity t0 tm tp

1-21-42-32-53-64-54-75-66-87-8

4351325622

697474111255

28151571161730148

45. Apply North-West corner rule to the following TP To D1 D2 D3 Availability 01 2 3 4 10

From 02 1 2 3 18

03 5 4 3 12

Requirement 15 20 546. Draw an arrow diagram to represent the following project.

ActivityA B C D E F G H I J K

Immediate - A - B,C C G,H D B F G E,I,J

13

Predecessor :

47. Solve the following transportation problem.

To

W1 W2 W3 W4 Capacity

F1 5 7 3 8 300

From F2 4 6 9 5 500

F3 2 6 4 5 200

Requirement 200 300 400 100 1000

48. The following table gives the jobs of a network along with their time estimates.

Duration days

Jobi-j

OptimisticMost likely

Prssimistic

1-21-62-32-43-54-56-75-87-8

326253314

65251169419

1514308171527728

Draw the project network and find the critical path.

49. Use Vogel’s Approximation method to obtain an initial basic feasible solution.

Cost of Transport

D1 D2 D3 D4 Availability

01

02

03

5 8 3 6

4 5 7 4

6 2 4 6

30

50

20

Requirement 30 40 20 10 100

14

50. A project consists of 9 jobs the following precedence relations and estimates of time. Draw a project network.

Job A B C D E F G H I

Predecessor - - A,B A,B B D,E C,F D,E G,H

Time (day) 15 10 10 10 5 5 20 10 15

51. (i) Explain a two person zero-sum game with examples. (ii) What are the different steps in managerial decision making? Explain with an example.

52. A project consists of the following 12 jobs. Draw a project Network and determine the critical path.

Job 1-2 2-3 2-4 3-4 3-5 4-6 5-8 6-7 6-10 7-9 8-9 9-10Duration 2 7 3 3 5 3 5 8 4 4 1 7

53. The following table gives the activities of a construction project and duration (in days )

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

Duration 20 25 10 12 6 10

Draw the network diagram and find the critical path. 54. Solve the following transportation problem with unit transportation costs given as

under: Destinations

Origins D1 D2 D3 D4 Supply 01

02

03

40 25 22 33

44 35 30 30

38 38 28 30

100

30

70

Demand 40 20 60 30

55. A project has the following characteristics and time estimates. Optimistic time (a) Most likely time (m), Pessimistic time (b) Construct a PERT network. Find the critical path.

Activity 1-2 2-3 2-4 3-5 4-5 4-6 5-7 6-7 7-8 7-9 8-10 9-10

a 1 1 1 3 2 3 4 6 2 4 1 3

b 5 3 5 5 4 7 6 8 6 8 3 7

15

m 1.5 2 3 4 3 5 5 7 4 6 2 5

56. The following table gives the activities of a project and their duration in days.

Activity 1-2 1-6 2-3 2-4 3-5 4-5 6-7 5-8 7-8Duration 7 6 14 5 11 7 11 4 18

57. Solve the following transportation problem, with given cost coefficients: Destination

D1 D2 D3 D4 Availabilities

01

Origins 02

03

15 10 17 18

16 13 12 13

12 17 20 11

20

60

70 Requirements 30 30 40 50

58. (i) Explain the use of probability in Decision theory. (i) Explain, with suitable illustrative examples, decision making under uncertainty

59. The following table gives the activities of construction project and duration.Activity 1-2 1-3 1-4 2-4 2-5 3-4 4-5

Duration (days) 3 2 6 5 7 2 4 60. Solve the following transportation problem, with the following costs of transportation, demand and supply : Destination

D1 D2 D3 D4 D5 Supply

01

02

03

10 15 20 20 40

20 40 15 30 30

30 35 40 55 25

50

100

150Demand 25 115 60 30 70

61. A network diagram with PERT time estimates, is given below. Find the critical path and also determine the project completion time and is variance . 6-12-30 5-11-17

2-5-8 3-6-15 1-4-7

3-6-15 2-5-14 4-19 -28

16

1

2

6

3

4

5

7

8

3-9-27

62. Discuss the solution of the following game, with pay-off for player A

Player B

Player A

63. Solve the following transportation problem with the following information on costs, supply and demand: Destination

D1 D2 D3 D4 Supply01

Origins 02

03

2 2 2 1

10 8 5 4

7 6 6 8

3

7

5

Demand 4 3 4 4

Prepared byProf. L .M. Alphonse Ligori

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