Linear programming ppt
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This document provides an overview of linear programming. It discusses basic and basic feasible solutions, the geometric solution, definitions used in linear programming, and the simplex algorithm. It provides an example problem that is solved over multiple iterations using the simplex algorithm to find the optimal solution. Finally, it briefly discusses the primal dual relationship between linear programming problems.
Linear programming ppt
- 1. LINEAR PROGRAMMING Presented By – Meenakshi Tripathi
- 3. BASIC and BASIC FEASIBLE SOLUTION • x1 x2 x3
- 5. Definitions • x* - 3 constraints active
- 7. SIMPLEX ALGORITHM: Basis notation •
- 8. SIMPLEX •
- 9. Example • Z=30x1+20x2 s.t. 2x1+x2+s1=100 • X1+x2+s2=80 • X1+s3=40 • X1,x2,s1,s2,s3≥0 Iteration 1: column = minimum negative entry = -30; Row : min{ 100/2, 80/1,40/1}=40 => s3 leaving, x1 entering BV z x1 x2 s1 s2 s3 B Z 1 -30 -20 0 0 0 0 S1 0 2 1 1 0 0 100 S2 0 1 1 0 1 0 80 S3 0 1 0 0 0 1 40 BV z x1 x2 s1 s2 s3 B Z 1 0 -20 0 0 30 1200 S1 0 0 1 1 0 -2 20 S2 0 0 1 0 1 -1 40 x1 0 1 0 0 0 1 40
- 10. Example BV z x1 x2 s1 s2 s3 B Z 1 0 0 20 0 -10 1600 x2 0 0 1 1 0 -2 20 S2 0 0 0 -1 1 1 20 x1 0 1 0 0 0 1 40 Iteration 2: column = minimum negative entry = -20; Row : min{ 20/1, 40/1,-}=20 => s1 leaving, x2 entering Iteration 3: column = minimum negative entry = -10; Row : min{-, 20/1, 40/1,-}=20 => s2 leaving, s3 entering BV z x1 x2 s1 s2 s3 B Z 1 0 0 10 10 0 1800 x2 0 0 1 -1 2 0 60 S3 0 0 0 -1 1 1 20 x1 0 1 0 0 -1 1 20 All nonbasic variables with non-negative coefficients in row 0, Optimal solution : x1=20, x2=60 & Z=1800
- 11. PRIMAL-DUAL •
- 12. Primal Dual • Problem 1: Maximize Z Z=3x1+4x2 4x1+2x2 80 3x1+5x2 180 • Dual of Problem1: Minimize Z Z=80y1+180y2 4y1+3y2≥3 2y1+5y2 ≥4
- 13. Reference • Linear Programming and network flows, M.S. Bazaraa, J.J.Jarvis & H.D.Sherali • Coursera lectures,”Linear Optimization” • Wikipedia