Lecture on ‘Linear Programming problem’

1. Lecture on ‘Linear Programming problem’ Professor M. TajUddin Department of Statistics Shahjalal University of Science and Technology, Sylhet Cell: 01716 348194 Email: mtajstat@yahoo.com

2. Linear programming problem (LPP): • With the limited available resources (such as raw materials manpower, capitals, power and technical appliance etc.) the main objective of an industry is to produce different products in such a way that the maximum profit may be earned by selling them at market price. • Similarly, the main objective of a house wife is to buy food grains, vegetables, fruits and other food materials at a minimum cost which will satisfy the minimum need (regardless food value, calories, proteins, vitamins etc.) of the member of her family. Department of statistics, SUST, Sylhet

3. LPP • All these can be done mathematically by formulating a problem which is known as programming problem. Some restrictions or constraints are to be adopted to formulate the problem. • The function which is to be optimized (such as profit, cost etc. either maximized or minimized) is known as objective function. • Almost in all types of problem the objective function and the constraints are of linear type and these problems are known as the linear programming problem. Department of statistics, SUST, Sylhet

4. Formulation of LPP: Department of statistics, SUST, Sylhet

5. LPP Department of statistics, SUST, Sylhet

6. LPP Department of statistics, SUST, Sylhet

7. LPP Department of statistics, SUST, Sylhet

8. LPP Department of statistics, SUST, Sylhet

9. Assumptions of LPP: Department of statistics, SUST, Sylhet

10. Uses of LP: • Linear programming has found numerous application in the i) military ii) the Government, iii) industry, iv) Urban engineering. • Linear programming is also frequently used as a part of general computation schemes for solving non linear programming problem, discrete programs, combinatorial problems, problem of optimal controlled programming under uncertainty. Department of statistics, SUST, Sylhet

11. Inequalities and equation: Department of statistics, SUST, Sylhet

12. Standard form and Canonical form of LPP: • A given linear program can be put in different equivalent form by suitable manipulation. Two forms in particular will be useful. These are the standard and canonical form. A linear program is said to be in standard format if all restrictions are equalities and all variables are non negative. The simplex method is designed to be applied only after the problem is put in standard form. • The canonical format is also useful especially in exploiting duality relationship. A minimization problem is in canonical form if all variables are non negative and all constraints are of the type. A maximization problem is in canonical form if all variables are non negative and all constraints are of the type. The standard and canonical forms are summarized in a tabular form Department of statistics, SUST, Sylhet

13. Table of standard form and Canonical form Department of statistics, SUST, Sylhet

14. Example of Formulation of LPP • Ex-1: Suppose that the four metals Iron, Copper, Zinc and Manganese are required to produce three commodities A, B and C. To produce one unit of A, 40kg Iron, 30kg Copper, 7kg Zinc and 4kg Manganese are needed. • Similarly to produce one unit of B, 70kg Iron, 14kg Copper and 9kg manganese are needed and for producing one unit of C, 50kg Iron, 18kg Copper and 8kg zinc are required. • The total available quantities of metals are 1 metric tons of Iron, 5quintals of Copper, 2 quintals of Zinc and Manganese each. The profits are Tk. 300, Tk. 200, Tk. 100 in selling per unit of A, B and C respectively. Formulate the above problem mathematically. Department of statistics, SUST, Sylhet

15. Formulation of LPP Department of statistics, SUST, Sylhet

16. Formulation of LPP Department of statistics, SUST, Sylhet

17. Formulation of LPP Department of statistics, SUST, Sylhet

18. Many thanks for your attention!!! Department of statistics, SUST, Sylhet