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Basic theory of LP

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- 1. LINEAR PROGRAMMING Presented By – Meenakshi Tripathi
- 2. Linear Programming •
- 3. BASIC and BASIC FEASIBLE SOLUTION • x1 x2 x3
- 4. Geometric Solution •
- 5. Definitions • x* - 3 constraints active
- 6. SIMPLEX ALGORITHM •
- 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

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