Lecturer: R.RajaramanSubject: Applied Operation Research Code: BA1603

UNIT I: Introduction to Linear Programming

PART-A01. Define LPP02. Explain the terminologies of linear programming model03. What are the major assumptions of linear programming ?04. What are the special cases of LPP?05. Define the following;

a) Basic solutionb) Non-degenerate solutionc) Degenerate solution

06. What is an unbounded solution in LP?07. Write a note on Sensitive Analysis08. Graphical solution is not possible for LPP with more than two constraints. True or False? Justify your answer.09. What is the use of Artificial variable in LP solution?10. Define slack variable and surplus variable11. Define Optimal solution12. What is the difference between feasible solution and

basic feasible solution?13. Define Unbounded solution14. What do you mean by standard form of LPP?15. What do your mean by canonical form of LPP?16. What is key column and how is it selected?17. What is key row and how is it selected?18. What is the difference between regular simplex method

and dual simplex method?19. What is a redundant constraint?20. solve the following LPP graphically: Maximize Z = 20X1 + 80X2

Subject to: 4X1 + 6X2 90 8X1 + 6X2 100

5X1 + 4X2 80 X1, X2 0

PART-B

01. ABC manufacturing company can make two products P1 and P2 . Each of the product require time on a cutting machine and a finishing machine relevant data are

ProductP1 P2

Cutting hrs(per unit) 2 1

Finishing hrs (per unit) 3 3

Profit (per unit) Rs.6 Rs.4

Max. sales (per week) -- --

The number of cutting hours available per week is 390 and the number of finishing hours available per week is 810. How much of each product should be produce in order to maximize the profit?

02. 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 check at the 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 per hour. The company has available for the inspection job eight Grade 1 inspectors, and ten Grade 2 inspectors, which will minimize the total cost of the inspection.

03. A farmer has a 100 acre farm. He can sell all tomatoes, lettuce and radishes and can raise the price to obtain Re1.00 per kg. for tomatoes. Rs.0.75 ahead for lettuce and Rs.2.00 per kg for radishes. The average yield per acre is 2000 kg. of tomatoes, 3000 heads of lettuce and 1000 kg of radishes. Fertilizers are available at Rs.0.50 per kg and the amount required per acre is 100 kgs each for tomatoes and lettuce and 50 kgs for radishes. Labour required for sowing, cultivating and harvesting per acre is 5 man-days or tomatoes and radishes and 6 man-days for lettuce. A total of 400 man-days of labour are available at Rs.20.000 per man-day. Formulate this problem as a linear programming model to maximize the farmer’s total profit.

04. A company produces two types of leather belts A and B. A is of superior quality and B is of inferior quality and B is of inferior quality. The respective profits are Rs.10 and Rs.5 per belt. The supply of raw material is sufficient for making 850 belts per day. For belt A, a special type of buckle is required and 500 are available per day. There are 700 buckles available for belt B per day. Belt A needs twice as much time as that required for belt B and the company can produce 500 belts if all of them were of the type A. Formulate a LP model for the above problem.

05. Egg contains 6 units of vitamin A per gram and 7 units of vitamin B per gram and cost 12 paise per gram. Milk contains 8 units of vitamin A per gram and 12 units of vitamin B per gram and costs 20 paise per gram. The daily minimum requirement of vitamin A and vitamin B are 100 units and 120 units respectively. Find the optimal product mix.

06. Solve the following LPP using dual simplex method:Min Z = X1 + X2

Subject to:2X1 + X2 2-X1 – X21X1, X20.

07. Use penalty method to Max Z = 3X1 +2X2

Subject to2X1 + X2 23X1 + 4X2 12X1, X2 0

08. Use two-phase method to solve the following LPPMax Z = 5X1-2X2+3x3

Subject to 2X1 + 2X2 – X323X1 – 4X23X2 + 3X35X1, X2, X3 0

09. Solve the following LPPMax Z = 2X1 + X2

Subject to4X1 + 3X2 124X1 + X2 84X1 – X2 8X1, X2 0

10. Solve the following LPP using simplex methodMin Z = X2 – 3X3 + 2X5

Subject to3X2 – X3 + 2X5 7-2X2 + 4X3 12-4X2 + 3X3 + 8X5 10

X2, X3, X5 0

UNIT – II Linear Programming Extensions

PART-A

01. What is unbalanced transportation problem?02. What are the two conditions to be satisfied to perform

optimality text?03. Give a note on traveling salesmen problem.04. State any two applications of zero-one programming.05. Where do you apply assignment model? Give an

example.06. Briefly explain transshipment problem.07. Define non-degenerate solution of a T.P08. List any three approaches used with T.P for determining

the starting solution.09. Define the optimal solution to a T.P10. What is the purpose of the MODI Method?11. What do you mean by degeneracy in a T.P?12. What is an assignment problem? Give two applications.13. State the difference between the T.P and A.P 14. What is the objective of the traveling salesmen problem?15. How do you convert the maximization assignment

problem into a minimization one?16. What is the name of the method used in getting the

optimum assignment?

PART – B

01. A company has three factories and 4 distribution centers. The following table given the unit transportation cost, Supply and demand details.

DistributionFactories Madras Madurai Trichy Salem Supply

A 3 2 7 6 5000

B 7 5 2 3 6000

C 2 5 4 5 2500

Requirement 6000 4000 2000 1500

Suggest optimum transportation schedule and find the corresponding cost.

02. Optimize the cost of the transportation problem which is Given below.

Supply9 12 9 6 9 1

06

7

3 7 7 5 5 2

6 5 9 11

3 11

5

6 8 11 2 2 10

9

Demand 5 4 5 4 2 2

03. Solve the following T.P

P Q R S Supply A 21 16 25 13 11

B 17 18 14 23 13 C 32 17 18 41 19

Demand

6 10 12 15 43

04. Solve the following T.P whose cost matrix is given below.

A B C D Supply 1 1 5 3 3 34

2 3 3 1 2 15 3 0 2 2 3 12 4 2 7 2 4 19Demand

21 25 17 17 80

05. A company has three plants A, B and C, 3 houses X, Y, Z. The number of units available at the plants is 60, 70

And 80 and the demand at X, Y, Z are 50, 80, 80 Respectively. The cost of the transportation is given

In the following table

X Y Z A 8 7 3 B 3 8 9 C 11 3 5

Find the allocation so that the total transportation cost is minimum.

06. Solve the following T.P to maximize the profit.

A B C D Supply 1 15 1 42 33 23

2 80 42 26 81 44 3 90 40 66 60 33Demand

23 31 16 30 100

07. Using the following cost matrix determine (a) optimal Job assignment (b) the cost of assignment.

1 2 3 4 5

A 10 3 3 2 8

B 9 7 8 2 7

Mechanic C 7 5 6 2 4

D 3 5 8 2 4

E 9 10 9 6 10

08. A Company has 4 machines to do 3 jobs. Each job can be assigned to one and only one machine. The cost of each job on each machine is given below. Determine the job assignments which will minimize the total cost.

MachineW X Y Z

A 18 24 28 32

Job B 8 13 17 18

C 10 15 19 22

09.A traveling salesman has to visit 5 cities. He wishes to start from a particular city, visit each city once and then return to his starting point. Cost of going from one city to another is shown below. Your are required to find the least cost route.

To City

A B C D E

A 4 10 14 2

B 12 6 10 4

From City C 16 14 8 14

D 24 8 12 10

E 2 6 4 16

10.A Salesman has to visit five cities A, B, C, D and E. The distances (in hundred miles) between the five cities are as follows.

To

A B C D E

A - 7 6 8 4

B 7 - 8 5 6

From C 6 8 - 9 7D 8 5 9 - 8

E 4 6 7 8 -

UNIT – III Integer Linear Programming and Game Theory

PART – A

01. What do you mean by integer programming problem?02. Give some application of IPP.03. Differentiate between pure and mixed IPP.04. What are the methods used in solving IPP?05. Where is Branch and Bound method used?06. Define mixed integer programming problem.07. Define a game.08. Define strategy.09. What are the classifications of strategy?10. Define a saddle point.11. When do players apply mixed strategies?12. Define two-person zero sum game.13. Define payoff.14. What types of games are solved graphically?15. What is meant by minimax, maximin?16. When is a game fair?17. What do you mean by zero-sum game?18. What do you mean by a pure strategy game?

PART – B

01. Find the optimum integer solution to the following LPP

Max Z = X2 + X3

Subject to3X1 + 2X2 5 X2 2X1, X2 0 and are integers.

02. Find the optimum integer solution of the following integer programming problem.

Min Z = -2X1 -3X2

Subject to2X1 + 2X2 7

X1 2X2 2

X1 ,X2 0 and are integers.

03. Solve the following mixed integer programming problem using Go Mary’s cutting plane method.

Max Z = X1 + X2

Subject to3X1 + 2X2 5

X2 2X1 + X2 0 and X1 is an integer.

04. Use Branch and Bound technique to solve the following;

Max Z = X1 + 4X2

Subject to2X1 + 4X2 75X1 + 3X2 15X1 , X2 0 and are integers.

05. Solve the game whose pay off matrix is given by

Player BB1 B2 B3

A1 1 3 1

Player A A2 0 -4 -3

A3 1 5 -1

06. Determine the optimal minimax strategies for each player in the following game.

B1 B2 B3 B4

A1 -5 2 0 7

A2 5 6 4 8

A3 4 0 2 -3

07. Solve the following payoff matrix, determine the optimal strategies and the value of game.

B

5 1 A 3 4

08. Solve the following 2x3 game graphically.

Player B

1 3 11

Player A 8 5 2

09. Is the following two person zero sum game stable? Solve the game.

Player B

5 -10 9 0

6 7 8 1

Player A 8 7 15 1

3 4 -1 4

10. Using the principle of dominance, solve the following game.

Player B

3 -2 4

Player A -1 4 2

2 2 6

UNIT – IV Dynamic Programming, Simulation and Decision Theory

PART – A

01. What is the Dynamic Programming?02. What are the advantages of dynamic programming?03. State Bellmans principle of optimality.04. Define random number.05. Define Pseudo-random number.06. Explain Monte-carlo technique.07. What are the advantages of simulation?08. What are the limitations of simulation?09. What are the uses of simulation?10. What are the classifications of decision?11. What are the types of decision making situations?12. What is Expected Monetary value (EMV)?13. What is Expected Opportunity Loss (EOL)?14. What is Expected value of Perfect Information (EVPI)?15. Describe some methods which are useful for decision

making under uncertainty?

PART – B

01. Use dynamic programming t solve

Maximum Z = Y1. Y2. Y3

Subject to Y1 + Y2 + Y3 = 5

And Y1, Y2, Y3 0

02. A student has to take examination in three courses X, Y, Z. He has three days available for study. He feels it

would be best to devote a whole day to study the same course, so that he may study a course for one day , two days or three days or not at all. His estimates of grades he may get by studying are as follows.

Study days X Y Z 0 1 2 1 1 2 2 2 2 2 4 4 3 4 5 4

How should he plan to study so that he maximizes the sum of his grades?

03. Use dynamic programming to solve the LPP

Max Z = X1 + 9X2

Subject to

2X1 + X2 25 X2 11

X1, X2 0

04. A manufacturing company keeps stock of a special product. Previous experience indicates the daily demand as given below.

Daily demand 5 10 15 20 25 30Probability 0.01 0.20 0.15 0.5

00.12

0.02

Simulate the demand for the next 10 days. Also find the daily average demand for that product on the

basis of simulated data.

05.Suppose that the sales of a particular item per day is poisson with mean 5, then generate 20 days of sales by Monte-Carlo method.

07. A small ink manufacturer produces a certain type of ink at a total average cost of Rs.3 per bottle and sells at a price of Rs.5 per bottle. The ink is produced over the week-end and is sold during the following week. According to the past experience the weekly demand has never been less than 78 or greater than 80 bottles in his place.

You are required to formulate pay off table.

08. The research department of consumer products division has recommended to the marketing department to launch soap with three different perfumes. The marketing manager has to decide the type of perfume to launch under the following estimated pay off for the various levels of sales.

Estimated level of sales (units)Types of perfume 20,000 10,000 2,000

I 250 15 10

II 40 20 5

III 60 25 3

Examine which type can be chosen under maximax, minimax, maximin, laplace and Hurwicz Alpha criteria.

Unit – V Queuing Theory And Replacement Models.

Part – A

01. Define a customer02. What are the basic characteristics of a queuing system?03. Define transient and steady state.04. Explain Kendall’s notation.05. Write Little’s formula.06. When is the replacement to be done?07. What are the categories into which the replacements of

items are classified?08. Describe briefly some of the replacement policies?09. Define group replacement.10. Define replacement model for items that fail

completely.11. Define discount factor.12. What is present worth factor?13. Name the three categories of replacement items which

follow sudden failure mechanism.14. What is meant by running cost?

Part – B

01. A machine owner finds from his past records that the costs per year of maintaining a machine whose

purchase rice is Rs.6000 are as given below.

Year 1 2 3 4 5 6 7 8Maintenance cost

1000

1200

1400

1800

2300

2800

3400

4000

Resale price 3000

1500

750 375 200 200 200 200

Determine at what age a replacement is due.

02.A machine costs Rs.15000. The running cost for the different years are given below:

Year 1 2 3 4 5 6 7Running 2500 3000 4000 5000 6500 8000 10000

Find the optimum replacement period if the capital is worth 10% and has no salvage value.

03. The probability Pn of failure just before age n is shown below. If individual replacement costs Rs.12.50 and group replacement costs Rs.3.00 per item. Find the optimal replacement policy.

n 1 2 3 4 5Pn 0.1 0.2 0.25 0.3 0.15

04. Customers arrive at a one window drive in bank according

to Poisson distribution with mean 10 per hour. Service time per customer is exponential with mean 5

minutes. The space in front of the window including that for the serviced car can accommodate a maximum of 3 cars. Others can wait outside this space.

(i) What is the probability that an arriving customer can drive directly to the space in front of the

window?(ii) What is the probability that an arriving customer will have to wait outside the indicated space?(iii) How long is an arriving customer expected to

wait before starting service.

05. In a supermarket, the average arrival rate of customer is 10 every 30 minutes following Poisson process. The average time taken by a cashier to list and calculate the customer’s purchase is 2.5 minutes following

exponential distribution. What is the probability that the queue length exceeds 6? What is the expected time spent by a customer in the system?

06. In a public telephone booth the arrivals are on the average 15 per hour. A call on the average takes 3 minutes. If there is just one phone, find (i) the expected number of callers in the booth at any time (ii) the proportion of the time the booth is expected to be idle?

07. A petrol station has two pumps. The service time follows the exponential distribution with mean 4 minutes and

cars arrive for service in a Poisson process at the rate of 10 cars per hour. Find the probability that a customer has to wait for service. What proportion of time the pumps remain idle?