QUESTION BANK ON OPERATIONS RESEARCH UNIT-1: Basics of operations research PDF

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QUESTION BANK ON OPERATIONS RESEARCH UNIT-1: Basics of operations research Q1. Discuss the origin and development of OR. Q2. How computer has helped in popularizing OR? Q3. What are the limitations of OR? Q4. Describe the various objectives of OR. Q5. What are the main characteristics of OR? Explain...

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QUESTION BANK ON OPERATIONS RESEARCH UNIT-1: Basics of operations research Q1. Discuss the origin and development of OR. Q2. How computer has helped in popularizing OR? Q3. What are the limitations of OR? Q4. Describe the various objectives of OR. Q5. What are the main characteristics of OR? Explain with suitable examples. Q6Give features of OR. Briefly discuss technique and tools of OR. Q7. What is the role of decision making in OR. Explain its scope. Q8. “OR is the application of scientific methods, technique and tool to problems involving the operation of a system so as to provide those in control of the system with optimum solution to the problems.” Q9. Discuss the significance and scope of OR in modern management. Q10. “Mathematics of OR is mathematics of optimization.” Discuss. Q11. “OR is an aid for the executive in making his decision by providing him with the needed quantitative information, based on the scientific method analysis.” Discuss the statement in detail, illustrating it with OR methods that you know. Q12. Discuss in brief the role of OR model in decision making.

UNIT-2: Linear Programming Q1. What are the essential characteristics of a linear programming model? Q2.What is linear programming? Discuss the application of linear programming to managerial decision making. Q3 Discuss the assumption of proportionality, additivity, continuity, certainty and finite choices in the context of linear programming problems. Q4. Explain the meaning of linear programming problem stating its uses and give its limitations. Q5. Write at least five application areas of linear programming. Q6. A small manufacturer employs 5 skilled men and 10 semi skilled men and makes an article in two qualities, a deluxe model and an ordinary model the making of deluxe model requires 2 hours work by a skilled man and 2 hours work by a semi-skilled man. The ordinary model requires 1 hour work by a skilled man and 3 hour work by a semiskilled man. By union rules no man can work more than 8 hours per day. The manufacturer’s clear profit of the deluxe model is Rs.10 and of ordinary model Rs.8. Formulate the model of the problem. Q7. Old hens can be bought for Rs.2 each but young one cost Rs.5 each. The old hens lay 3 eggs per week, and young one 5 eggs per week, each egg being worth 30 paise. A hen cost Re.1 per week to feed. If a person has only Rs.80 to spend on hens, how many of each kind should he buy to get a profit of more than Rs.6 per week assuming that he can’t house more than 20 hens? Q8. A firm manufactures three products A, B, and C. The profits are Rs.3, 2 and 4 respectively. The firm has two machines and required processing time in minutes for each machine on each product is given below: Product A B C Machine X 4 3 5 Y 2 2 4 Machine X and Y have 2000 and 1500 machine minutes respectively. The firm must manufacture 100 A’s, 200B;s and 50C’s but no more than 150 A’s. set up an Lp model to maximize the profit. Q9. The manager of an oil refinery has to decide upon the optimal mix of two possible blending processes, of which the input and output per production run are as follows:

Process 1 2

Input Crude A 5 4

Crude B 3 5

Gasoline X 5 4

output Gasoline Y 8 4

The maximum amount available of crude A and B is 200 units and 150 units respectively. Market requirement show that at least 100 units of gasoline X and 80 units of gasoline y must be produced. The profit per production run from process 1 and process 2 are Rs.3 and 4 respectively. Formulate the problem as a linear programming problem.

Q10. A firm can produce three types of clothes say, A,B and C. Three kinds of wool are required for it say, red, green and blue. One unit length of type A cloth needs 2 yard of red wool and 3 yards of blue; one unit length of type B cloth needs 3 yards of red wool, 2 yards of green and 2 yards of blue; and one unit length of type C needs 5 yards of green and 4 yards of blue wool. The firm has a stock of only 8 yards of red wool, 10 of green 15 of blue. It is assumed that the income obtained from one unit length of type A is Rs.3, of type B cloth is Rs.5 and of type C cloth is Rs.4. Formulate LPP. Q11. Why do some problems have multiple optimal feasible solutions? How such information is useful for decision making? Use graphical method to solve the following problem: Q12. Maximize subject to

Z=5x1+3x2, x1+x2≤6, 0≤x1≥3, 0≤x2≥3, 2x1+3x2≥3

Q13. Minimize subject to

Z=20x1+10x2, x1+2x2≤40, 3x1+x2≥30 4x1+3x2≥60, x1, x2≥0

Q14. Minimize subject to

Z=3x1+2x2, 8x1+x2≥8, 2x1+x2≥6, x1+3x2≥6, x1+6x2≥8, x1, x2≥0

Q15. Maximize subject to

Z= -150x1-100 x2,+280000, 20≤ x1≤60, 70≤x2≤140, 120≤ x1+x2≤140, x1, x2≥0

Use simplex method to solve the following problem:

Q16. Maximize subject to

Z= 3x1+2 x2+ 5x3, x1+ x2+ x3≤9, 2x1+ 3x2+ 5x3≤30, 2x1-x2-x3≤8, x1, x2, x3≥0

Q17. Maximize subject to

Z= 2x1+4x2+ x3+ x4, x1+ 3x2+ x4≤4, 2x1+ x2≤3, x2+4x3+x4≤3, x1, x2, x3, x4≥0

Q18. Maximize subject to

Z= 2x1+3x2+ x3+7 x4, 8x1+ 3x2+ 4x3+ x4≤6, 2x1+6x2+ x3+5x4≤3, x1+4x2+5x3+2x4≤7, x1, x2, x3, x4≥0

Q19. Maximize subject to

Z= 6x1+7x2+ 9x3, 3x1+ 7x2+ 6x3 ≤245, 5x1+8x2+ 9x3≤424, 11x1+6x2+8x3≤235, x1, x2, x3, ≥0

Q20. Maximize subject to

Z= 2x1+3x2+ 4x3+x4, -x1- 5x2- 9x3+ 6x4≤2, 3x1-x2+ x3+3x4≤10, 2x1+3x2-7x3+8x4≤0, x1, x2, x3, x4≥0

Q21. Minimize subject to

Z= 4x1-3x2+ 7x3-x4, 7x1+ 3x2≤400, 5x1 + 4x3≥250, x1+x4=43, x1,x2,x3,and x4 are non negative and none is below 20.

Solve by Big M-method Q22. Maximize subject to

Z= x1+2x2+ 3x3-x4, x1+2x2+3x3=15, 2x1+x2+ 5x3=20, x1+2x2+x3+x4=10, x1, x2, x3, x4≥0

Q23. Maximize subject to

Z= x1+2x2+ 3x3, x1-x2+x3≥4, x1+x2+ 2x3≤8, x1-x3 ≥2, x1, x2, x3,≥0

Q24. A manufacturer produces three products A, B, and C. Each product requires processing on two machines I & II. The time required to produce one unit of each product on a machine is: Time to produce one unit (hrs.) Product Machine I Machine II A 0.5 0.6 B 0.7 0.8 C 0.9 1.05 There are 850 hrs are available on each machine. The operating cost is Rs.5/hr. for machine I and Rs.4/hr. for machine II. The market requirements are at least 90 units of A, at least 80 units of B and at least 60 units of C. The manufacturer wishes to meet the requirement at minimum cost. Solve the problem by simplex method.

Q25. A factory has decided to diversify its activities. The data collected by sales and production department is summarized below. Potential demand exist for three products A,B and C. market can take any amount of A and C. whereas the share of B for this organization is expected to be not more than 400 units a month.

For every 3 units of C produced, there will be one unit of by-product which sells at a contribution of Rs.3 a unit and only 100 units of this by-product can be sold per month. Contribution per unit of products A,B &C is expected to be Rs. 6, 8 ,4 respectively. These products require three different processes and the time required per unit product is given in the table below:

Process I Ii III

Product A 2 3

Hours/unit Product B 3 1 2

Product C 1 2 2

Determine the optimum product mix for maximizing the contribution.

Available hours 900 600 1200

UNIT-3a: Transportation Model

Q1. Explain the following in the context of transportation problem. a) Degenerate transportation problem b) Modified distribution method. Q2. What is degeneracy in transportation problem? How it can be resolved? Q3. What are the conditions for the application of optimality test in case of transportation problem? Briefly explain as to why these conditions should be satisfied? Q4. Find the feasible solution of the following transportation problem using North West corner method. W1

W2

W3

W4

Supply

Factory F1

14

25

45

5

6

F2

65

25

35

55

8

F3

35

3

65

15

16

7

6

13

30 (Total)

Requirement 4

Q5. Find the initial basic feasible solution of the following transportation problem using Vogel’s approximation method. W1

W2

W3

W4

Capacity

Factory F1

19

30

50

10

7

F2

70

30

40

60

9

F3

40

8

70

20

18

8

7

14

34 (Total)

Requirement 5

Q6. Find the initial basic feasible solution of the following transportation problem using North West corner method and Vogel’s approximation method.

A1

B1

C1

D1

E1

Supply

Origin A

2

11

10

3

7

4

B

1

4

7

2

1

8

C

3

9

4

8

12

9

Requirement 3

3

4

5

6

Q7. Solve the transportation problem for which the cost, origin, availability and destinations requirements are given below: D1

D2

D3

D4

D5

D6

ai

O1

1

2

1

4

5

2

30

O2

3

3

2

1

4

3

50

O3

4

2

5

9

6

2

75

O4

3

1

7

3

4

6

20

bj

20

40

30

10

50

25

175 (Total)

Q8. A departmental store wishes to purchase the following quantities of ladies’ dresses: Dress type Quantity

A 150

B 100

C 75

D 250

Tenders are submitted by three different manufacturers who undertake to supply not more than the quantities below: Manufacturer Total Quantity

W 350

X 250

Y 150

The store estimates that its profit per dress will vary with the manufacturer as shown in the matrix below. How should orders be placed? Dress A B C D Manufacturer 2.75

3.50

4.25

2.25

W X

3.00

3.25

4.50

1.75

Y

2.50

3.50

4.75

2.00

Q9. A fertilizer company has three plants A, B and C which supply to six major distribution centres 1,2,3,4,5,6. the table below gives the transportation cost per case, the plant annual capacities and predicted annual demand at different centres in terms of thousands of cases. The variable production cost per case are Rs.8.50, 9.40, 7.20 respectively at plants A,B and C. Determine the minimum cost production and transportation allocation. 1

2

3

4

5

6

Annual production in thousands of cases

Plant A

2.50

3.50

5.50

4.50

1.50

4.00

220

B

4.60

3.60

2.60

5.10

3.10

4.10

3400

C

5.30

4.30

4.80

2.30

3.30

2.80

1800

Annual production 850 in thousands of cases

750

420

580

1020

920

Prove that if variable production cost are same at every plant , one can obtain an optimal allocation by using transportation costs only. Q10. A production control superintendent finds the following information on his desk: In department A,B,C the number of surplus pallet is 18,27,21 resp. In department G,H,I and J the no. of pallets required is 14,12,23, 17 resp. The time in minutes to move a pallet from one department to another is given below:

To From A B C

G

H

I

J

13 18 23

25 23 15

12 14 12

21 9 16

What is the optimal distribution plan to minimize the moving cost? Q11. Solve the following transportation problem: To From 1 2 3 Demand

A

B

C

D

E

Supply

20 15 18 30

19 15 20 40

14

21 19 20 40

16 16

40 60 90

18 70

60

Q12. A manufacturing company has three factories F1, F2, F3 with monthly manufacturing capacities of 7000,4000,10000 units of a product. The product is to be supplied to seven stores. The manufacturing costs of these factories are slightly different but the important factor is the shipping cost from each factory to a particular store. Following table represent the factory capacities, store requirement and unit cost in rupees of shipping from each factory to each store and slack. Here slack is difference between total factory capacity and total store requirement. S1

S2

S3

S4

S5

S6

S7

Slack

Factory capacity

Factory F1

5

6

4

3

7

5

4

0

7000

F2

9

4

3

4

3

2

1

0

4000

F3

8

4

2

5

4

8

3

0

10000

2000

4500

4000

2000

3500 3000

Store 1000 requirement

1000

Work out a transportation plan so as to minimize the transportation cost. Q13. General electrode is a big electrode manufacturing company. It has two factories and three main distribution centers in three cities. The supply and demand conditions for units of electrode are given below along with unit cost of production. How should the trips be scheduled so that cost of production is minimized?

The present cost of transportation is around Rs.3100/month. What can be the maximum saving by proper scheduling? Centers Requirement Cost per trip from X plant Cost per trip from Y plant Capacity of plant X Capacity of plant Y

A 50 25

B 50 35

C 150 10

20

5

80

150 units of electrodes 100 units of electrodes

Q14. A company has three plants at location A,B,C which supply warehouses located at D,E,F,G,H.. Monthly plant capacities are 800,500,900 units resp. Monthly warehouse requirements are 400,350, 250,900. the unit transportation cost in rupees given below: To From A B C

D

E

F

G

H

8 5 8

8 8 9

9 5 7

4 11 3

3 6 3

Determine an optimum distribution for the company in order to minimize the total transportation cost. How much is the cost? Q15. Priyanshu enterprise has three auditors. Each auditor can work up to 160 hours during the next month, during which time 3 projects must be completed. Project I will take 130 hours, project II will take 140 hours and III will take 160 hours. The amount in rupee per hour that can be billed for assigning each auditor to each project is given below:

Auditor 1 2 3

Project 1 1200 1400 1600

Project 2 1500 1300 1400

Project 3 1900 1200 1500

Find the optimal solution. Also find the maximum total billing during the next month. Q16. Four suppliers have submitted sealed bids that quote the price per case of hairnets delivered to four regional stores of the army. The bids are summarized in the following table. The regional stores requirements as well as supplying capacities of suppliers are also shown. Supplier 4 has quoted for only region 1. Because of previous contractual obligations, region 3 will have to get a minimum of 200 cases from supplier 2.

R1

R2

R3

R4

Supplier S1

30

25

40

35

S2 S3 S4 Requirement (cases)

35 28 25 1000

32 30 800

38 35 1200

40 38 750

Max. Supply (cases) 800 1000 1500 600

a) Formulate this problem as a transportation model including all constraints. b) Find the initial basic solution using V.A.M. c) Use Modi method to establish whether the above solution obtained is optimal or not. Q17. A company has factories at location A, B, C which supply warehouses at D, E, F, G. Monthly factory capacities are 250,300,400 units resp. for regular production. If overtime production is utilized, factories A and B can produce 50 & 75 additional units respectively at overtime incremental cost of Rs.4 and 5 resp. The current warehouse requirements are 200,225, 300 units resp. Unit transportation cost in rupees from factories to warehouse are as follows: To From A B C

D

E

F

G

11 16 21

13 18 24

17 14 13

14 10 10

Determine the optimum distribution for this company to minimize cost.

Q18. A company produces a small component for an industrial product and distributes it to five wholesalers at a fixed delivered price of Rs. 250 per unit. Sales forecasts indicate that monthly deliveries will be 300,300,100,500,400 units to wholesalers 1, 2,3,4,5 resp. The direct costs of production of each unit are Rs.100, 90, 80 at plants 1, 2, 3 resp. The transportation cost of shipping a unit from plants to wholesaler are given below:

Plant 1 2 3

Wholesaler 1 5 8 10

2 7 6 9

3 10 9 8

4 15 12 10

5 15 14 15

Find how many components each plant must supply to each wholesaler to maximize the profit? What is the maximum total profit? Take the monthly production capacities of plant 1, 2 and 3 as 500,100, 1250units resp.

UNIT-3b: Assignment Model

Q19. Explain the following in the context of assignment problem: a) Balanced assignment problem b) The Hungarian method c) An infeasible assignment Q20. Show that the assignment model is a special case of the transportation model. Q21. Six machines M1, M2, M3, M4, M5, M6 are to be located in six places P1, P2, P3, P4, P5, P6. Cij the cost of locating machine Mi at place Pj is given in the matrix below:

M1 M2 M3 M4 M5 M6

P1 20 50 60 6 18 9

P2 23 20 30 7 19 10

P3 18 17 40 10 28 20

P4 10 16 55 20 17 30

P5 16 15 8 25 60 40

P6 20 11 7 9 70 55

Formulate an Lp model to determine an optimal assignment. Write the objective function a...