or question bank

UNIT I

LINEAR PROGRAMMING PROBLEM

1. Discuss the importance of Operations Research in decision making process.

2. Explain briefly the various applications of Operations Research. 3. With reference to the solution of LPP by simplex method / table, when one can conclude as the problem has i) unbounded solution ii) no feasible solution

4. What are the limitations of Operational Research

5. Write at least 5 application areas of linear programming.

6. Discuss in detail the role of linear programming in managerial decision making.

7. Define the term operations research and discuss its scope.

8. What are the steps / phases involved in operations research? Explain in brief. 05 Marks

9 Explain the applications of OR in Industry.

10. Mention the limitations of graphical method

MATHEMATICAL FORMULATION OF LPP

1. A farmer has 1000 acres of land on which he can grow corn, wheat or soybeans. Each acre of corn costs rs.100 for preparation, requires 7 man days of work and Yield a profit of Rs.30 .An acre of wheat cost Rs.120 to prepare, requires 7 man Days of work and Yields a profit of Rs.20.if the farmer has Rs 1, 00,000 for Preparation and can count on 8,000 man- days of work, how many acres should be Allocated to each crop to maximize profit? Write the mathematical formulation of the problem.

2. A company manufactures fm radio and calculators. The radios contribute Rs.10/unit and calculation Rs.15/unit has profit. Each radio requires 4 diodes and 4 resistors, while each calculator requires 10 diode and 2 resistors. A radio takes 12 min and calculators takes 9.6 mins of time on the company electronic testing machine and product manager. Estimate that 160hour of test time available. The company has 8000 diode and 3000 resistors in the stock. Formulate the problem has linear programming problem.

3. A company manufactures laptops’ and desktops that fetches profit of Rs.700 and 500 each unit of laptops takes4 hours of assembly time and 2 hours of testing time while each desktops takes 3hours of assembly and one hour f testing time. In the given month the total of hour for assembly is 210hour and for inspections is 90hour.formulate LLP in such way that total project is maximum.

4. A company wish to schedule the production of kitchen appliances that requires 2 sources labour and .The company in considering 3 different model and its productions engineering

1

department has furnished following data.

Model

A B C

Labour (hr/unit) 7 3 6

Material( per/unit) 4 4 5

Profit($ per/unit) 4 2 3

x1=A,x2=B,x3=c. The supply of raw material is restricted zmax = 4x1+2x2+3x3 to 200 pounds/day. The daily available of labour is 150hour. Formulate linear programming model to determine the daily production rate various model in order to maximize the profit.

5. A company can produce 3 types of cloth say A, B, C. 3 kinds of wool have required for it green, red, blue wool one unit length of type. A cloth needs 2m of red and 3m of blue. 1 unit length of B cloth needs 3m of red and 2m of green and 2m of blue. One unit length of C cloth need 5m green, 4m of blue. The company has only stock of 8m of red wool 10m of green wool and 15m of blue. It is assumed that the income obtain from one unit length A is Rs.3, B is Rs.5, C is Rs.4.Determine how the form should use the company. The company should use available material as to move income from finish cloth.

Wool type Cloth

A B C Red 2 3 0 8 Blue 0 2 4 15 Green 3 2 5 10 profit 3 5 4

6. A company has 2 grades of inspections 1st & 2nd who are to be assigned for quality control inspections that is required that at least 1800 pieces be inspected 8hrs/day grade1 inspector can check pieces at the rate of 25/1hr.with an accuracy of 98%grade 2 inspector check at the rate of 15/hr with an accuracy of 95%the wage rate of 1 inspector is $4.00per hour. grade 2 inspector is $3.00/hr each time error made by the inspector of the cost to the company $2.00.the company is available for the inspection job 8 grade 1 &10 grade 2 The company wants to determine the optimum assignment of inspector which minimizes the total cost of inspection.

7. The farmer has the plan to plant 2 kinds of trees 'A' and 'B' in a land and of 4400m2 tree 'A' requires at least 25m2 and 'B' requires 40m2 land. The annual water requirements for ‘A’ are 30 units and 'B' is 15 units per tree while 3300 unit of water is available. If the return of tree 'A' is expected to be 11/2 times as much as 'B' formulate the problem.

2

Trees A B

Land 25m2 40m2

Water 30units 15units

8. A farmer has 100 acres land he can sell all the tomatoes, potatoes and radishes he can grow. The price obtain is Rs.10/kg for tomato Rs.7/kg for potato and Rs.10/kg for radishes. The average yield per acres is 2000 kg of radishes. The labor requires for harvesting per acres 5 man days for tomato and radishes.6 man days for potatoes. A total of 400 man days are available at Rs.50 man days. Formulate LPP to maximize farmers’ total profit.

9. The handy-dandy company wishes to schedule the production of a kitchen appliance that requires two resources-labor and material. The company is considering three different models and its production engineering department has furnished the following data:

The supply of raw materials is restricted to 200 pounds per day. The daily availability of labor is 150 hours. Formulate a linear programming model to determine the daily production rate of various models in order to maximize the total profit.

10. A company has two grades of inspectors, 1 and 2, who are to be assigned for a quality control inspection. It is required that at least 1800 pieces be inspected per 8-hour day. Grade 1 inspectors can check pieces at the rate of 25 per hour, with an accuracy of 98%. Grade 2 inspectors can check at rate of 15 pieces per hour, with an accuracy of 95%.

The wage rate of a grade 1 inspector is $4.00 per hour, while that of a grade 2 inspector is $3.00 per hour. Each time an error is made by an inspector, the cost to the company is $2.00. The company has available for the inspection job eight grade 1 inspectors, and ten grade 2 inspectors. The company wants determine the optional assignment of inspectors, which will minimize the total cost of the inspection.

11 A machine shop has one drill press and five milling machines, which are to be used to produce an assembly considering of two parts, 1 and 2. the productivity of each machine for two parts is given below:

A B CLabor(hours per unit) 7 3 6Material(pounds per unit) 4 4 5Profit($ per unit) 4 2 3

3

It is designed to maintain a balanced loading on all machines such that no machine runs more than 30 minutes per day longer than any other machine (assume that the milling load is split evenly among all five machines). Divide the work time of each machine to obtain the maximum number of completed assemblies assuming an 8-hour working day.

12. A furniture maker has 6 units of wood and 28 h of free time , in which he will make decorative screens. Two models have sold well in the past, so he will restrict himself to those two. He estimates that model 1 requires two units of wood and 7 h of time, while model 2 requires 1 unit of wood and 8 h of time . The prices of the models are $120 and $80, respectively. How many screens of each model should the furniture maker assemble if he wishes to maximize his sales revenue?

13. A plastic manufacturer has 1200 boxes of transparent wrap in stock at one factory and another 1000 boxes at its second factory. The manufacturer has orders for this product from three different retailers, in quantities of 1000, 700 and 500 boxes, respectively. The unit shipping costs (in cents per box) from the factories to the retailers are as follows:

14. A village butcher shop traditionally makes it meat loaf from a combination of lean round beef and ground pork. The ground beef contains 80% meat and 20%fat, and costs the shop 80$ per pound: the ground pork contains 68%meat and 32% fat and costs 60$ per pound. How much of each kind of meat should the shop use in each pound of meat loaf if it wants to minimize its costs and to keep the fat content of the meat loaf to no more than 25%?

15. A firm manufactures 3 products A, B and C. The profits are 3, 2 and 4 respectively. The firm has 2 machines and below is the required processing time in minutes for each machine and each product. Machine G and H have 2,000 and 2,500 machine- minutes respectively. The firm must manufacture 100 A’s, 200 B’s and 50 C’s, but no more than 150A’s.

Product

Machine

Setup an LP problem to maximize the profit.

Production time in minutes per piecePart Drill Mill

1 3 202 5 15

Retailer1 Retailer2 Retailer3Factory1 14 13 11Factory2 13 13 12

A B CG 4 3 5H 2 2 4

4

16. A 24 hour supermarket has the following minimal requirements for cashiers:

Period 1 2 3 4 5 6Time of day(24-h clock)

03-07 07-11 11-15 15-19 19-23 23-03

Minimum No 7 20 14 20 10 5

Period 1 follows immediately after period 6. A cashier works eight consecutive hours, starting at the beginning of one of the six periods. Determine a daily employee worksheet which satisfies the requirements with the least number of personnel.

GRAPHICAL SOLUTION FOR LPP

1. Solve the following L P problem by graphically. Maximize Z=3x1+5x2

Subjected to x1+2x2<2000 x1+x2<1500 x2<600

X1,x2>02. Solve the following L P problem by graphically.

Maximize Z=x1+x2

Subjected to x1+2x2<2000 X1+x1<1500

x2<600 X1,x2>0

3. Solve the following L P problem by graphically. Minimize Z=40x1+36x2

Subjected to x1<8 X2<10

5X1+3X2<2X1,x2>0

4. Solve the following L P problem by graphically. Maximize Z=8000x1+7000x2

Subjected to 3x1+x2<66 x1<20 X2<40 X1+X2<45

X1,x2>05. Solve the following LP model using graphical method: Max Z = 40X1 +100X2, S.T. 12X1+6X2≥3000, 4X1+10X2≤ 2000, 2X1+3X2≤900, X1, X2≥0

5

UNIT II

SIMPLEX METHOD

1. Define the following terms in connection with LPP: i. Slack variable ii. surplus variable iii. Basic feasible solution iv. optimal solution2. Write a note on artificial variable.3. What is degeneracy in LPP and how do you resolve it?

1. Solve the following problems by simplex method1. Max Z=5X1+3X2, subjected to

3X1+5X2≤155X1+2X2≤10X1 ,X2≥0

2. Max Z=7X1+5X2, subjected to

-X1-2 X2≥-6 4X1+3X2≤12 X1, X2≥0

3. Max Z=5X1+7X2, subjected to X1+ X2≤4 3X1-8X2≤24 10X1+7X2≤35 X1, X2≥0

4. Max Z=3X1+2X2, subjected to 2X1+ X2≤40 X1+X2≤24 2X1+3X2≤60 X1, X2≥0

5. Max Z=3X1+2X2, subjected to 2X1+1 X2≤5 X1+X2≤3 X1, X2≥0

6. Max Z=2X1+4X2, subjected to 2X1+3X2≤48 X1+3X2≤42 X1+X2≤21 X1, X2≥0

7. Max Z=3X1+4X2, subjected to X1- X2≤1 -X1+X2≤2 X1, X2≥0

6

8. Max Z=3X1+2X2, subjected to 2X1+X2≤10 X1+3X2≤6 X1, X2>0

9. Max Z=2X1+5X2, subjected to X1+3X2≤3 3X1+2X2≤6 X1, X2≥0

10. Max Z=3X1+5X2, subjected to 3X1+ 2X2≤18 X1≤4 X2≤6 X1, X2≥0

11. Max Z=2X1+X2, subjected to X1+ 2X2≤10 X1+X2≤6 X1-X2≤2 X1-2X2≤1 X1, X2≥0

12. Max Z=2X+5Y, subjected to X+Y≤600 0≤X≤400 0≤Y≤300

13. Max Z=X1-X2+3X3, subjected to X1+X2+X3≤10 2X1- X3≤2 2X1-2X2+3X3≤0 X1, X2, X3≥0

14. Max Z=X1+X2+X3, subjected to 4X1+5X2+3X3≤15 10X1+7X2+ X3≤12 X1, X2, X3≥0

15. Max Z=8X1+19X2+7X3, subjected to 3X1+4X2+X3≤25 X1+3X2+3X3≤50 X1, X2, X3≥0

16. Max Z=X1+X2+3X3, subjected to 3X1+2X2+X3≤3 2X1+X2+2X3≤2 X1, X2, X3≥0

7

17. Max Z=4X1+3X2+4X3 +6X4, subjected to X1+2X2+2X3+4X4≤80 2X1+2X3 +X4≤60 3X1+3X2+X3+X4≤80 X1, X2, X3, X4≥0

18. Max Z=4X1+5X2+9X3 +11X4, subjected to X1+X2+X3+X4≤15 7X1+5X2+3X3 +2X4≤120 3X1+5X2+10X3+15X4≤100 X1, X2, X3, X4≥0

19. Max Z=2X1+4X2+X3 +X4, subjected to 2X1+X2+2X3+3X4≤12 3X1+2X3 +2X4≤20 2X1+X2+4X3≤16 X1, X2, X3, X4≥0

20. Max Z=5X1+3X2, subjected to X1+ X2≤2 5X1+2X2≤10 3X1+8X2≤12 X1, X2≥0

21. Max Z=8X1+11X2, subjected to 3X1+ X2≤7 X1+3X2≤8 X1, X2≥0

22. Max Z=10X1+X2+2X3, subjected to X1+2X2-3X3≤10 4X1+X2+X3≤20 X1, X2, X3≥0

23. Max Z=2X1+4X2+X3 +X4, subjected to X1+3X2+X4≤4 2X1+X2≤3 X2+4X3+X4≤3 X1, X2, X3, X4≥0

24. Max Z=10X1+6X2, subjected to X1+ X2≤2 2X1+X2≤4 3X1+8X2≤12 X1, X2≥0

25. Max Z=102X1+X2+2X3 , subjected to 14X1+X2-6X3+3X4=7

8

16X1+1/2X2 -6X3≤5 3X1-X2-X3≤0 X1, X2, X3, X4≥0

26 Max Z=3X1+2X2-2X3, subjected to X1+2X2+2X3≤10 2X1+4X2+ 3X3≤15 X1, X2 <X3≥0

27. Max Z=30X1+23X2+29X3, subjected to 6X1+5X2+3X3≤52 6X1+2X2+ 5X3≤14 X1, X2, X3≥0

28. Max Z=7X1+X2+2X3, subjected to X1 allocated +X2-2X3≤10 4X1+X2+ X3≤20 X1, X2, X3≥0

29. Max R=2X-3Y+Z, subjected to 3X+6Y+Z≤6 4X+2Y+ Z≤4 X-Y+Z≤3 X, Y, Z≥0

. 30. Max R=2X+4Y+3Z, subjected to

3X+4Y+2Z≤60 2X+Y+2Z≤40 X+3Y+2Z≤80 X, Y, Z≥0

31. Max Z=X1-X2+X3 +X4+X5-X6, subjected to X1+X4+6X6=9 3X1+X2-4X3 +2X6=2 X1+2X3+X5+2X6=6 Xi≥0, i=1, 2, 3,4,5,6

32. Solve the following L P problem by simplex method. Maximize Z=x1+3x2

Subjected to x1+2x2<100<x1<50<x2<4

33. Solve the following L P problem by simplex method. Maximize Z=3x1+2x2

Subjected to x1+x2<40X1-X2<20

9

X1,x2>0


Subjected to x1+x2<4X1-X2<2X1,x2>0

35. Solve the following L P problem by simplex method. Maximize Z=x1-3x2+2x3

Subjected to 3x1-x2+2x3<7-2x1+4x3<12-4x1+3x2+8x3<10

36. Solve the following L P problem by simplex method. Minimize Z=2x1+3x2+x3

Subjected to 3x1+2x2+x3<3 2x1+x2+x3<2 x1,x2,x3>0


Subjected to 4x1+3x2<124x1+x2<8x1,x2>0

38. Solve the following L P problem by simplex method. Maximize Z=2x1+4x2+3x3

Subjected to 3x1+4x2+2x3<602x1+x2+2x3<40X1+3x2+2x3<80x1,x2,x3>0


Subjected to x1+x2<25x1+2x2<103x1+8x2<12

x1,x2>01 40. Solve the LPP by Simplex Method

Minimize Z= X2-3X3+2X5, subject to the constraints: 3X2-X3+2X5≤7 -2X2+4X3≤12 -4X2+3X3+8X5≤10 and X2, X3, X5≥0

USE PENALTY (BIG -M) METHOD

1. Solve the following L P problem by penalty method.

10

Maximize Z=6x1+4x2

Subjected to 2x1+3x2<303x1+2x2<24X1+x2>3X1 >0, x2>0

2. Solve the following L P problem by penalty method. Maximize Z=2x1+3x2

Subjected to x1+2x2<4X1+x2=3x 1, x2>0

3. Solve the following L P problem by penalty method. Minimize Z=600x1+500x2

Subjected to 2x1+x2>80X1+2x2>60x 1, x2 >0

4. Solve the following program using Big M method. Maximize: Z=-8X1+3X2-6X3

Subject to: X1-3X2+5X3=4 5X1+3X2-4X3≥6 with all variables nonnegative

TWO PHASE METHOD

1. Solve the following L P problem by 2 phase method. Maximize Z=5x1+3x2

Subjected to 2x1+x2<1X1+4x2>6X1,x2,>0

2. Solve the following L P problem by 2 phase method. Maximize Z=2x1+3x2+4x3

Subjected to 3x1+x2+6x3<6002x1+4x2+2x3>4802x1+3x2+3x3=540


Subjected to 2x1+x2+x3=603x1+3x2+5x3>120


Subjected to 2x1+x2+x3<23x1+4x2+2x3>8 X1,x2,x3>0

11

DUAL SIMPLEX METHOD

1. Solve the following L P problem by dual simplex method. Maximize Z=-2x1-3x2

Subjected to x1+x2>22x1+x2<10X1+x2>8

2. Solve the following L P problem by dual simplex method. Maximize Z=-3x1-2x2

Subjected to x1+x2>1X1+x2<7X1+x2>10x 2<3 x 1, x2 >0

3. Solve the following L P problem by dual simplex method. Maximize Z=2x1+x2+3x3

Subjected to x1-2x2+3x3>42x1+x2+x3<8X1-x2>0

4. Write the dual for the following primal: Minimize Z = 3x1 + x2 - 7x3 subject to x1 - 2x2 + 3x3≤ 10 3x1+5x2-x3≥ 9 -x1-4x2+x3 = 6, x1 is unrestricted; x2, x3 ≥0

5. Write the dual of the following LPP:

Max Z = X1 + 2X2 + X3 subject to 2X1 + X2 – X3 ≤ 2 -2X1 + X2 -5X3≥-6 4X1 + X2 + X3 ≤6 X1, X2, X3 ≥ 0

6. Write the dual of the following LPP.

Maximize Z=2x1+3x2+4x3

Subjected to 3x1+x2+6x3≤600 2x1+4x2+2x3≥480 2x1+3x2+3x3=540

X1, X2, X3≥0 05 Marks

7. Write the dual of the following LPP. Max Z=102X1+X2+2X3 , subjected to 14X1+X2-6X3+3X4=7 16X1+1/2X2 -6X3≤5 3X1-X2-X3≤0 X1, X2, X3, X4≥0

12

8.Write the dual of the following LPP.


Subjected to 3x1+4x2+2x3≤602x1+x2+2x3≤40X1+3x2+2x3≤80x1, x2, x3>0

05 Marks

9. Write the dual of the following LPP.


Subjected to 2x1+x2+x3<23x1+4x2+2x3>8 x1, x2, x3>0

05 Marks GAME THEORY

1. Give the formulation of a game.2. Explain two person zero-sum game with a suitable example.3. Explain the types of games.4. Explain Mini max & Maxi min principal.5. Define saddle point & value of game.6. Explain the concept of the dominance.7. List the applications of game theory.8. Explain the following with an example.

a. Saddle pointb. Mixed strategy

9. Explain the following terms.a. Play off matrix.b. Strategyc. Fair gamed. Zero sum game.

1. Two competitors A and B are competing for the same product. Their different strategies are given in the following payoff matrix:

Company B

Company A

I II III VIIIIIIIV

3 2 4 03 4 2 44 2 4 00 4 0 8

13

Use dominance principle to find the optimal strategies. Also calculate the value of the game to the player A.

2. Solve the game given in the table by graphical method. B

A

Also calculate the value of the game to the player A.


A

Also calculate the value of the game to the player A. 10 Marks


A

Also calculate the value of the game to the player A. 10 Marks

5. Solve the following game graphically:

B 1 2 3 4

1 -6 0 6 -15A

2 7 -3 -8 2

6. Solve the following game by graphical method

Y1 Y2 Y3 Y4 X1X2X3X4

19 6 7 57 3 14 612 8 18 48 7 13 -1

I II IIII 1 3 11II 8 5 2

I II IIIIIIIV

2 42 33 2-2 6

14

B1 B2 B3 B4

A1 2 2 3 -1 A2 4 3 2 6 08 Marks

7. Solve the following game by using the dominance concept

Player B B1 B2 B3

A1Player A A2 A3

UNIT III

TRANSPORTATION & ASSIGNMENT PROBLEMS

Transportation problems

1. Give the mathematical formulation of transportation problems.2. What is an unbalanced transportation problem? How do you solve it.3. What is degeneracy in transportation problem? How do you resolve it.4. List & explain variations in transportation problems.5. Write the differences between transportation problem & assignment problem.6. Give the mathematical formulation of an assignment problem.7. Explain Hungarian method to solve an assignment problem.8. Write a note on solving procedure a maximization problem.9. List & explain variations in assignment problem.10. Explain clearly travelling sales man problem & give its formulation.

11. The solution to assignment problems are inherently degenerative. Explain 04 Marks

PROBLEMS:

1. Find the optimum solution to the following transportation problem in which the cells contain the transportation costs in rupees.

W1 W2 W3 W4 W5 Available F1 40 F2 30 F3 20

F4 10 Required 30 30 15 20 05

4 5 86 4 64 2 4

7 6 4 5 98 5 6 7 86 8 9 6 55 7 7 8 6

15

2. There are three factories A, B, and C supplying goods to four dealers D1, D2, D3 and D4. The production capacities of these factories are 1000, 700 and 900 units respectively. The requirements from the dealers are 900, 800, 500 and 400 units per month respectively. The per unit return (excluding transportation cost) are Rs 8, Rs 7and Rs9 at the three factories. The following table gives the unit transportation costs from the factories to dealers. Determine the optimum solution to maximize the total returns.

15 Marks

3. Find the optimal transportation plan.

1 2 3 4 5 Available A 80 B 60 C 40 D 20 Required 60 60 30 40 10

15 Marks

4.Solve the following transportation problem where cell entries are unit costs.

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

5.A product is produced by four factories A, B, C, and D. The unit production costs in them are Rs 2, Rs 3 Rs 1, and R5 respectively. Their production capacities are: factory A-50 units, B- 70 units, C – 30 units and D-50 units. These factories supply the product to four stores, demands of which are 25, 35, 105 and 20 units respectively. Unit transport cost in rupees from each factory to each store is given in the table below.

Stores 1 2 3 4

D1 D2 D3 D4A 2 2 2 4B 3 5 3 2C 4 3 2 1

4 3 1 2 65 2 3 4 53 5 6 3 22 4 4 5 3

16

A B C D

Determine the extent of deliveries from each of the factories to each of the stores so that the total production and transportation cost is minimum.

6. A firm produces a component and distribute them to 5 wholesalers at fixed priced of Rs 10 per unit. Sales forecast indicate that monthly demand will be 3000, 3000, 1000, 5000 and 4000 units at whole sale dealers a,b,c,d and e respectively. the monthly production capacities are 5000,1000 and 10000 at plants A,B and C respectively. the production cost per unit are Rs 2,1 and 3 at plants A,B and C respectively. The unit transportation cost in Rs between plants and whole salers are given in the following table:

Whole Sales a b c d e

A Plants B C Determine the transportation schedule between plants and wholesales in order to maximize the total profit per month. Use VAM to obtain the initial basic feasible solution.

7. Obtain an initial basic feasible solution to the transportation table shown below the elements of the matrix indicates cost in rupees.

Destination

Origin

8. A company is spending Rs.1,000 on transportation of its units from three plants to four distribution centers. The supply and demand units with unit cost of transportation are given as Distributes center

Plants D1 D2 D3 D4 CapacityP1 19 30 50 12 7P2 70 30 40 60 10P3 40 10 60 20 18Demand 5 8 7 15

2 4 6 1110 8 7 513 3 9 124 6 8 3

0.5 0.5 1.0 1.5 1.51.0 0.5 1.0 1.0 1.51.0 1.0 0.5 1.5 1.0

D1 D2 D3O1 2 7 4 5O2 3 3 1 8O3 5 4 7 7O4 1 6 8 14

8 8 18

17

What can be the maximum saving to the company by optimum scheduling?

8. A company has three car manufacturing factories located in cities C1,C2,C3 which can supply cars to four showrooms located in towns T1,T2,T3,T4? EachPlant can supply 6,1,and 10 truckload of cars daily respectively.The transportation costs per truckload of cars (in hundreds of rupees) from each factory to each show room are as follows.Find the optimum distribution schedule and cost.

2 3 11 71 0 6 15 8 15 9

9. Solve the following transportation problem in which cell entries represent the unitcosts (in lackhs of rupees)of transportation

10. Consider the transportation problem having the following parameter table destination.

Source(0)Dummy

1 2 3 4 51 8 6 3 7 5 202 5 M 8 4 7 303 6 3 9 6 8 304 0 0 0 0 0 20

25 25 20 10 20

After several iterations a basic feasible solution is obtained that has the following basic variables.

x13=20 x32=25x21=25 x34=5x24=5 x45=20Continue the transportation method for two more iterations and statewhether the solution is optimum or not(M indicates prohibited / restricted route).

11. A dairy firm has three plants located throughout a state. daily milk production at each plant is as follows: Plant 1 …. 6 million liters, Plant 2 …. 1 million liters, Plant 3 …. 10 million liters.

Each day the firm must fulfill the needs of its four distribution centre. Milk requirement at each centre as follows:

2 7 4 58714

3 3 15 4 71 6 27 9 18

18

Distribution centre 1 …. 7 million liters, Distribution centre 2 …. 5 million liters, Distribution centre 3 …. 3 million liters, Distribution centre 4 ….. 2 million liters,

Cost of shipping one million liters of milk from each plant to each distribution centre is given in the following table in hundreds of rupees:

Distribution centers1 2 3 4

1 Plants 2 3

The dairy firm wishes to determine as to how much should be the shipment from which milk plant to which distribution centers so that the total cost of shipment is the minimum.Determine the optimal transportation policy.

12. A company has four warehouses and six stores. the warehouses altogether have a surplus of 22 units of a given commodity, divided among them as follows:

Warehouse 1 2 3 4 Surplus 5 6 2 9

The six stores altogether need 22 units of the commodity. Individual requirements at stores 1, 2,3,4,5 and 6 are 4, 4,6,2,4 and 2 units respectively.Cost of shipping one unit of commodity from warehouse i to store j in rupees is given in the matrix below.

Stores 1 2 3 4 5 6 1 2 Warehouse 3 4

How should the products be shipped from the warehouses to the stores so that the transportation cost is minimum?

2 3 11 71 0 6 15 8 15 9

9 12 9 6 9 107 3 7 7 5 56 5 9 11 3 116 8 11 2 2 10

19

13. Find the optimum solution to the following transportation problem in which the cells contain the transportation cost in rupees. W1 W2 W3 W3 W4

F1 F2

F3 F4

14. Find the optimum solution to the following transportation problem in which the cells contain the transportation cost in rupees.(north-west corner rule) 1 2 3 4 5 a b c d

15. Solve the following transportation problem where cell entries are unit costs.

D1 D2 D3 D4 D5 O1 O2 O3 O4

16. A product is produced by four factories A, B, C, and D. the unit production costs in them are Rs.2 Rs.3. Re.1 and Rs.5 respectively. Their production capacities are: factory A-50 units. B-70 units. C-30 units. These factories supply the product to four stores, demands of which are 25, 35, 105, and 20 units respectively. Unit transport cost in rupees from each store is given in the table below.

7 6 4 5 9 40302010

8 5 6 7 86 8 9 6 55 7 7 8 6

30 30 15 20 05 100

4 3 1 2 6 80604020

5 2 3 4 53 5 6 3 22 4 4 5 3

60 60 30 40 10 200

68 35 4 74 15 1817601315

57 88 91 3 891 60 75 45 6052 53 24 7 8251 18 82 13 7

16 18 20 14 14 82

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Stores 1 2 3 4 AFactories B C D

ASSIGNMENT PROBLEMS

1. The captain of a cricket team has to allot five middle batting positions to five batsmen. The average runs scored by each batsman at these positions are as follows:

Find the assignment of batsmen to positions which would give the maximum number of runs.

2. Consider the problem of assigning five operators to five machines. The assignment costs are given below.

Operators I II III IV V

ABCDE

Assign the operators to different machines so that total cost is minimized.

3. Five wagons are available at stations 1, 2, 3, 4, and 5. These are required at five stations I, II, III, IV and V. The mileages between various stations are given by the table below. How should the wagons be transported so as to minimize the total mileage covered?

2 4 6 1110 8 7 513 3 9 124 6 8 3

Batsman Batting positionI II III IV V

P 40 40 35 25 50Q 42 30 16 25 27R 50 48 40 60 50S 20 19 20 18 25T 58 60 59 55 53

10 5 13 15 163 9 18 3 610 7 2 2 25 11 9 7 127 9 10 4 12

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I II III IV V1 10 5 9 18 112 13 9 6 12 143 3 2 4 4 54 18 9 12 17 155 11 6 14 19 10

4. A car company has one car at each of the 5 depots (A, B, C, D, and E). A customer requires a car in each town namely (P, Q, R, S and T). Distance between depots and towns (in km) are given in the following matrix. How the cars should be assigned to the customer to minimize the distance traveled.

5. Solve the following assignment problems:

J1 J2 J3 J4P1 11 3 5 8P2 9 9 8 4P3 10 3 5 10P4 4 13 12 11P5 8 9 10 4

The entries indicate the profits by assigning jobs to persons. Who will be idle person?

workers are available to work on four machines and the respective costs associated with each machine worker assignment is given below:

MACHINEM1 M2 M3 M4

w1 12 3 6 --w2 4 10 -- 5 w3 7 2 8 9w4 -- 7 8 6

The sign(--) indicates that the particular worker machine assignment is not permitted.i) Determine the optimum assignment

A B C D EP 160 130 175 190 200Q 135 130 130 160 175R 140 110 155 170 185S 50 50 180 80 110T 55 35 70 80 105

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ii) A fifth machine is available to replace one of the existing machines and the associated costs are w1= 4 Rs w2 = 3 Rs w3 = 3 Rs w4 = 2 Rs

Determine whether the new machine can be accepted and if so, which machine does it replace?

6. A medical representative has to visit five stations A, B, C, D & E. He does not want to visit any station twice before completing his tour of all the stations and wishes to return to the starting station. Costs of going from one station to another are given below. Determine the optimal route:

A B C D E

A B C D E

UNIT IVSEQUENCING

1. State the assumptions made in sequencing. 2. Explain how you sequence 2 jobs on m machines? Discuss 3. Use graphical method to minimize the time required to process the following jobs on the machines. For each machine specify the job which should be done first. Also calculate the total elapsed time to complete both jobs.

Machines Job 1 Sequence: A B C D E

Time(hr): 6 8 4 12 4

Job 2 Sequence: B C A D E Time(hr): 10 8 6 4 12

3. There are five jobs, each of which is to be processed through three machines A, B and C in the order ABC. Processing times in hours are

Job A B C1 3 4 72 8 5 93 7 1 54 5 2 65 4 3 10

5. Use graphical method to obtain the sequencing of jobs and machines and find the total elapsed time, idle times of the jobs.

∞ 2 4 7 15 ∞ 2 8 27 6 ∞ 4 610 3 5 ∞ 41 2 2 8 ∞

Job 1Sequenceand machine time

A B C D E3 4 2 6 2

Job 2Sequence and machine time

B C A D E5 4 3 2 6

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6. In a factory there are 6 jobs to perform, each of which should go through two machines A and B in the order A, B. The processing timings (in minutes)for the jobs are given below. Determine the sequence for performing the jobs that would minimize the total elapsed time also find the idle time of both the machines.

Job J1 J2 J3 J4 J5 J6Machine A 1 3 8 5 6 3Machine B 5 6 3 2 2 10

SIMULATION1. What is the need of simulation? How can you use Monte Carlo simulation for industrial problems? Give examples. 2. What is simulation? Explain the various steps involved in carrying out Monte Carlo simulation technique. 3. What are the advantages and limitations of Simulation technique? Explain 4. What is the need of simulation? Explain the various steps involved in carrying out Monte Carlo simulation technique: 4. Explain the application of simulation technique to the inventory problems. 6. A bakery keeps stock of a popular brand of cake. Daily demand based on past experience is given below: Daily demand 0 15 25 35 45 50 Probability 0.01 0.15 0.20 0.50 0.12 0.02 Consider the following sequence of random numbers: 48, 78, 09, 51, 56, 77, 15, 14, 68 and 09

i) Using the sequence, simulate the demand for the next 10 days.ii) Find the stock situation if the owner of the bakery decides to make 35 cakes every

day. Also estimate the daily average demand for the cakes on the basis of the simulated data.

UNIT V

QUEUING MODEL

1. State the basic elements of a queuing model. 2. Briefly explain the elements of a queuing system (structure of a queuing system) 3Write a note on the basic characteristics of a queuing phenomenon 4. The arrival of customers at a banking counter follows poisson distribution with mean of 45

per hour. The service rate of the counter clerk also follows poisson distribution with mean of 60 per hour.

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i. What is the probability of having zero customers in the system (Po)?ii. What is the probability of having 5 customers in the system? Probability

of having 10 customers in the system?iii. Determine the steady state performance statistics namely. Ls, Lq , Ws and

Wq . 5. The mean arrival rate to a service center is three per hour. The mean service time is ten minutes.

Assuming Poisson arrival rate and exponential service time, determine the following: i. Utilization factor ii. Probability of having two units in the system iii. Expected number of units in the system iv. Expected number of units in the queue and

iv. The expected time in minutes the customer has to wait in the system. 5. A drive in bank window has a mean service time of 2 minutes, while the customers arrive at a rate of 20 per hour. Assuming that customers arrive at position distribution and service time follows exponential distribution.

a. What percentage of time will the teller be idle?b. After driving up, how long will it take the average customer to wait in line and

served?c. What fraction of customers will have to wait in line?

6. The men’s department of a large store employs one tailor for customer fittings. The number of customers requiring fitting appear to follow a Poisson distribution with mean arrival rate 24 per hour. Customers are fitted on a first –come , first –served basis and they are always willing to wait for the tailor’s service because alterations are free. The time it takes to fit a customer appears to be exponentially distributed, with a mean of 2 minutes.

a. What is the average number of customer in the fitting room?b. How much time a customer is expected to spend in the fitting room?c. What percentage of the time is the tailor idle?

7. A repair shop attended by a single mechanic has an average of four customers an hour bring small appliances for repair. The mechanic inspects them for defects and quite often can fix them right away or render a diagnosis. This takes him six minutes on the average. Arrivals are poisson and service time has an exponential distribution. Find,

i. The probability of finding at least one customer in the shop.ii. The average time, spent by a customer in the shop.

8. The rate of arrival of customers at a public telephone booth follows poisson distribution , with an average time of 10 minutes between one customer and next. The duration of a call is assumed to follow exponential distribution with mean time of 3 minutes.

i. What is the probability that a person arriving at the booth will have to wait?

ii. What is the average length of the non-empty queues that form from time to time?

iii. The telephone department will install a second booth when it is convinced that the customers would expect waiting for atleast 3 minutes for their turn to make a call. By how much time should the flow of customers increase in order to justify a second booth?

iv. Estimste the fraction of a day that the telephone will be in use?

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PERT & CPM1. Describe crashing of project network2. Differentiate between i. PERT and CPM ii. Event and Activity 3. Define float. Explain its different types and their importance.4. What do you mean by slack? Define critical path in the light of the definition of slack.5. Explain the Fulkerson’s rule of numbering the events in network method.6. Write a note on the followings in network technique: i. Critical path ii. Dummy activity7. Bring out the differences between PERT & CPM.8. Explain the steps involved in PERT/CPM.9. Explain Fulkerson’s I – J rule for known numbering.10. Give the applications of PERT & CPM.11. Differences between an event & activity with an example.12. Explain the reasons for incorporating dummy activities in a network diagram. In what

way do these differ from the normal activities.13. Define the following

a. Earliest start timeb. Floatc. Estimated timed. Latest finish time

14. Describe crashing of network.

PROBLEMS

1. The following table shows the jobs of a network along with their time estimation in days.

Job 1-2 1-3 1-4 2-5 3-5 4-6 5-6to 1 1 2 1 2 24 3tm 1 4 2 1 5 5 6tp 7 7 8 1 14 8 15

1. Draw the project network2. Compute the expected duration and variance of each activity3. Identify the critical path4. Calculate the variance and standard deviation of the project5. What is the probability of completion of the project

a. 4 days earlier than expectedb. Not more than 4 days latter than expected

2. The details of a small project are given in the following table.

Activity ImmediatePredecessor(s)

Duration(Week)

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A - 2B - 6C - 4D B 3E A 6F A 8G B 3H C,D 7I C,D 2J E 5K F,G,H 4L F,G,H 3M IN J,K

i. Draw an activity on arrow diagram to represent the project.ii. Compute early start, early finish, late start, late finish, total float and

free float for each activity. Present the answer in tabular form.iii. What is the critical path and project duration?

3. Consider the following project

Time estimate in weeksActivity to tm tp Predecessor

A 3 6 9 -B 2 5 8 -C 2 4 6 AD 2 3 10 BE 1 3 11 BF 4 6 8 C,DG 1 5 15 E

Draw the project network and compute the probability of completion of the project in 18 weeks.4. A project has the following time schedule

Activity Time in weeks

1-2 41-3 12-4 13-4 13-5 64-9 55-6 45-7 86-8 17-8 28-9 1

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8-10 89-10 8

Construct a network diagram and compute

i. TE and TL for each event ii. Float for each activityiii. Critical path and its duration

5. The details of a small project are given in the following table

i) Draw an activity on arrow diagram to represent the project.ii) Compute early start, early finish, late start, late finish, total float and free float

for each activity. Present the answer in tabular form.iii) What is the critical path and project duration?

ActivityImmediate

Predecessor(s)Duration(Weeks)

ABCDEFGHIJKLMN

---BAAB

C,DC,D

EF,G,HF,G,H

IJ,K

264368372543

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or question bank

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