Basic theory of LP

1. LINEAR
PROGRAMMING
Presented By –
Meenakshi Tripathi

2. Linear Programming
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3. BASIC and BASIC FEASIBLE SOLUTION
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x1
x2
x3

4. Geometric Solution
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5. Definitions
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x* - 3 constraints active

6. SIMPLEX ALGORITHM
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7. SIMPLEX ALGORITHM: Basis notation
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8. SIMPLEX
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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
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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