KUMARAGURU COLLEGE OF TECHNOLOGY, COIMBATORE – 06DEPARTMENT OF MATHEMATICS

DECISION MODELS FOR MANAGEMENTLINEAR PROGRAMMING FORMULATION AND GRAPHICAL METHOD

PART-A

1. What do you mean by a general LPP?

2. Define feasible region.

3. Define redundant constraint.

4. When we say that an LPP have (i) unique solution (ii) An infinite number of solutions

(iii) An unbounded solution (iv) No solution

5. What do you understand by the term ‘constraint’? Explain with examples.

6. When can we use the graphical method for solving a LPP?

7. Solve the following linear programming problem graphically.

8. State the limitations of the graphical method of solving a L.P.P.

9. Linear programming is the most widely used of problem solving,. Why?

10. What are the assumptions underlying linear programming?

11. What does the non-negativity restriction mean?

12. Name any two major requirements of LPP.

13. Distinguish between a resource and a constraint

14. What are the applications of linear programming?

15. Solve the following LPP graphically:

16. What are decision variables in the construction of operation research problems?

17. List the limitations of LPP.

18. What are the characteristics of linear programming?

19. State any three salient features of a L.P.P.

20. What are the decision variables in the construction of operations research problem?

*********************

KUMARAGURU COLLEGE OF TECHNOLOGY, COIMBATORE – 06

DEPARTMENT OF MATHEMATICS

DECISION MODELS FOR MANAGEMENTLINEAR PROGRAMMING FORMULATION AND GRAPHICAL METHOD

PART-B1. Production allocation problem

A firm produces three products. These products are processed on three different

machines. The time required to manufacture one unit of each of the three products and

the daily capacity of the three machines are given in the table below:

Machine Time Per unit(minutes) Machine capacity(minutes/day)Product 1 Product 2 Product 3

2 3 2 4404 - 3 4702 5 - 430

It is required to determine the number of units to be manufactured for each product daily.

The profit per unit for product 1, 2 and 3 is Rs.4, Rs.3 and Rs.6 respectively. It is

produced are consumed in the market, formulate the mathematical model for the problem.

2. Blending problemA firm produces an alloy having the following specifications:

(i) Specific gravity

(ii) Chromium

(iii) Melting point

Raw materials A, B and C having the properties shown in the table can be used to make

the alloy.

Property RAW MATERIALSA B C

Specific gravity 0.92 0.97 1.04

Chromium 7% 13% 16%

Melting point Cost of the various raw materials per unit tons are: Rs.90 for A, Rs.280 for B and Rs.40 for C. Find the proportions which A, B and C be used to obtain an alloy of desired Properties while the coast of raw materials is minimum.

3. Diet problem

A person wants to decide the constituents of a diet which will fulfill his daily

requirements of proteins, fats and carbohydrates at the minimum cost. The choice is to

be made from four different types of foods. The yields per unit of these foods are given

in the following table.

Food type Yield/unit Cost/unitProteins Fats Carbohydrates

1 3 2 6 452 4 2 4 403 8 7 7 854 6 5 4 65

Minimum requirement

800 200 700

4. A farmer has 100 acre farm. He can sell all tomamtoes, lettuce, or radishes he can raise. The price he can obtain is Rs. 1.00 per kg for tomatoes, RS 0.75 a head for lettuce and Rs. 2.00 per kg for radishes. The average yield per-acre is 2,000 kgs of tomatoes, 3000 heads of Lettuce, And 1000 Kgs of radishes. Fertilizer is available at Rs.0.50 per kg and the amount required per acre is 100 kgs each for tomatoes and lettuce and 50 kg of radishes.Labour required sowing, cultivating and harvesting per acre is 5 man-days for tomatoes and radishes, and 6 man-days for lettuce. A total of 400 man-days of labour are available at Rs 20.00 per man-day. Formulae this problem as LP model to maximize the farmer’s total profit.

5. Solve the following L.P.P by the graphical method

Subject to

(b) Solve the following L.P.P by the graphical method.

6. A pineapple firm produces two products canned pineapple and canned juice. The Specific amounts of material, labour and equipment required to produce each product and the availability of each of these resources are shown in the table given below:

Canned juice Canned pineapple Available resourcesLabour

(man/hours)3 2.0 12.0

Equipment (Mc/hours)

1 2.3 6.9

Material (unit)

1 1.4 4.9

Assuming one unit of canned juice and canned pineapple has profit margins Rs.2 and Rs.1 respectively. Formulate this as a L.P.P. and solve it graphically also.

7. A company manufacturers 2 types of printed circuits. The requirements of transistors, resistors and capacitors for each type of printed circuits along with other data are given below:

Circuit Stock available

A BTransistor 15 10 180Resistor 10 20 200Capacitor 15 20 210Profit Rs.5 Rs.8

How many circuits of each type should the company produce form the stock to earn maximum profit.

8. Apply graphical method to solve the L.P.P: maximize

9. A company manufacturers two type of cloth, using three different colors of wool. One yard length of type A cloth require 4 OZ of red wool, 5 OZ of green wool and 3 OZ of yellow wool. One yard length of type B cloth requires 5 OZ of red wool, 2 OZ of green wool and 8 OZ of yellow wool. The wool available for manufacture is 1000 OZ of red wool, 1000 OZ of green wool and 1200 OZ of yellow wool. The manufacturer can take a profit of Rs.5 on one yard type A cloth and Rs. 3 on one yard of type B cloth. Find the best combination of the quantities of type A and type B cloth which gives him maximum Profit by solving the L.P.P graphically.

10. A company making cold drinks has two bottling plants at town each plant produces three drinks A,B and C and their production capacity per day is given below:

Cold drinks Plant At

A 6000 2000B 1000 2500C 3000 3000

The marketing department of the company forecast a demand of 80,000 bottles of A, 22,000 bottles of B and 40,000 bottles of C during the month of June. The operating costs per day of plants at and are Rs. 6000 and Rs. 4000 respectively. Find

graphically, the number of days for which each plant must be run in June so as to minimize the operating costs while meeting the market demand.

11. A firm manufactures two products A and B on which the profits earned per unit Rs.3 and Rs.4 respectively. Each product is processed on two machines product A requires one minute of processing time on and two minutes on while B requires one minute on and one minute on Machine is available for not more than 7 hours 30 minutes while machine is available for 10 hours during any working day. Find the number of units of products A and B to be manufactured to get maximum profit. Formulate the above as a L.P.P and solve by graphical method.

12. Solve the following L.P.P graphically

13. Using graphical method

14. Solve graphically the following L.P.P

PART-ASimplex methods-Linear programming

1. Define feasible solution.

2. Define optimal solution.

3. Define infeasible solution.

4. Define slack variables.

5. Define surplus variables.

6. Define unrestricted variables.

7. What do you mean by canonical form of a LPP?

8. What do you mean by standard form of a LPP?

9. Write the standard form of LPP in matrix form.

10. Is it meaning full to have a LPP for which the requirement vector b=0?

11. State the characteristics of the characteristics of the canonical form.

12. State the characteristics of the standard form.

13. State the characteristics of canonical form.

14. Express the following LPP in standard form:

15. Express the following LPP in standard form:

16. Express the following LPP in standard form:

17. Define basic solution to an LPP.

18. What do you mean by basic variable and non-basic variables?

19. Find all the basic solutions of the following LPP.

20. Write the general mathematical model of LPP in matrix form.

21. What is meant by optimally test?

22. Define (i) Feasible solution

(ii) Basic solution and

(iii) Basic feasible solution of a LPP.

23. Define (i) non-degenerate basic solution.

(ii) Degenerate basic solution.

24. What is meant by degeneracy of a LPP?

25. Explain the term feasible solution with an example.

26. What do you mean by an optimal solution of a LPP?

27. State the central problem of linear programming

28. What are slack and surplus variables?

29. What is the physical interpretation of a slack variable?

30. What is the difference between slack variables and surplus variable?

31. Which of the following solutions are feasible, in feasible, degenerate and

non- degenerate 32. Define a basic solution to a given system m simultaneous linear equations in n

unknown.

33. How many basic feasible solutions are there to a given system of 3 simultaneous

linear equations in 4 unknowns?

34. Find all the basic feasible solution of the simultaneous equations

35. State the necessary and sufficient condition for a basic feasible solution of a LPP to

be an optimum solution.

36. What is key column? How is it selected?

37. What is key row? How is it selected?

38. What is the test of optimality in the simplex method? And how is the solution read

from simplex tableau?

39. What information can be derived from the optimum simplex tableau?

40. State the condition for an un bounded solution of a LPP?

41. When does the simplex arithmetic indicate that the LPP has unbounded solution?

42. What is infeasible solution? How is it identified in the simplex tableau?

43. How can you find whether the solution to a LPP is unique or not?

44. How is the presence of more than one optimal solution found out from the optimal

simplex table of a LPP?

45. How will you find whether a LPP has got an alternate optimal solution or not from

the optimal simplex table?

46. Why the artificial variables are called so?

47. What do you mean by BIG-M method?

48. When does an LPP posses a pseudo optimal solution?

49. Define artificial variables.

50. What are the uses of artificial variables.

51. Write the solution of the maximization LPP from the following optimum simplex

table. Also state the nature of the solution.

(-1 2 0 0)

2

0

2

3

4 2 0 1 0

52. Write the nature of the solution of the maximization LPP from the following simplex table.

(2 1 0 0)

2

1

30

20

1 0 -1 1

0 1 -2 1

80 0 0 - 4 3

53. What are slack variables in the solution of LPP?54. While solving a LPP by simplex method, how will you find that the problem has got an unbounded solution?55. What is the difference between feasible solution and basic feasible solution?56. What the disadvantages of Big-M method over two phase method?

PART-B1. Using simplex method

2. Find the non-negative values of which

3. Solve the following:

4. Solve the following L.P.P. by simplex method

5. Using simplex method to

6. Solve the following LPP using simplex method

7. Solve simplex method:

8. Solve by simplex method:

9. Solve by simplex method:

BIG - M METHOD

10. Solve the following LPP

11. Solve the following LPP minimize

Subject to

12. Solve the following LPP: minimize

Subject to

13.

14.

15.

**************************