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lecture+3.3

# lecture+3.3 - Analytical Solutions to Linear Programming(LP...

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Artificial Variables Analytical Solutions to Linear Programming (LP) Problems

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Plastic Design of a Frame Structure Example: Simplex Solution - Artificial Variables 20’ 15’ 15’ 2 kips 1 kip A B C D E
Model: Augmented Form subject to: and non-negativity conditions: 1 2 3 4 5 6 4 30 15 10 5 2 25 2 25 B B C B C C B C B C P P P P P P P P P P M S M M S M M S M S M M S M M S - = + - = + - = - = + - = + - = Min 30 40 B C P P Z M M = + 4 30 15 10 5 2 25 2 25 B B C B C C B C B C P P P P P P P P P P M M M M M M M M M M + + + + , 0 B C P P M M 1 2 3 4 5 6 Min 30 40 0 0 0 0 0 0 B C P P Z M M S S S S S S = + + + + + + + 1 2 3 4 5 6 , , , , , , , 0 B C P P M M S S S S S S

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Model: Fully Augmented Form Artificial Variable “Large” Coefficient Not a valid initial feasible solution 1 2 3 4 5 6 4 30 15 10 5 2 25 2 25 B B C B C C B C B C P P P P P P P P P P M S M M S M M S M S M M S M M S - = + - = + - = - = + - = + - = 1 2 3 4 5 6 Min 30 40 0 0 0 0 0 0 B C P P Z M M S S S S S S = + + + + + + + 1 2 3 4 5 6 , , , , , , , 0 B C P P M M S S S S S S 1 2 3 4 5 6 Min 30 40 0 0 0 0 0 0 100 B C P P Z M M S S S S S S A = + + + + + + + + 1 2 3 4 5 6 4 30 15 10 5 2 25 2 25 B B C B C C B C B C P P P P P P P P P P M S A M M S A M M S A M S A M M S A M M S A - + = + - + = + - + = - + = + - + = + - + = 1 2 3 4 5 6 , , , , , , , , 0 B C P P M M S S S S S S A 1 2 3 4 5 6 30 15 10 5 25 25 S S S S S S = - = - = - = - = - = - , 0 B C P P M M Big Problem: setting
Model: Fully Augmented Form [ ] 1 2 3 4 5 6 Min 30 40 0 0 0 0 0 0 100 B C P P M M S S Z S S S S A =

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Model: Fully Augmented Form subject to: 1 2 3 4 5 6 4 0 1 0 0 0 0 0 1 30 1 1 0 1 0 0 0 0 1 15 1 1 0 0 1 0 0 0 1 10 0 1 0 0 0 1 0 0 1 5 2 1 0 0 0 0 1 0 1 25 1 2 0 0 0 0 0 1 1 25 B C P P M M S S S S S S A - - - = - - -
Model: Fully Augmented Form and, non-negativity constraints 1 2 3 4 5 6 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 B C P P M M S S S S S S A                              

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Model: Fully Augmented Form 1 2 3 4 5 6 1 2 3 4 5 6 30 40 0 0 0 0 0 0 100 0 0 0 0 0 0 4 30 15 10 5 2 25 0 2 25 B C B B C B C C B C B C P P P P P P P P P P P P Z M M S S S S S S A M S A M M S A M M S A M S A M M S A M M Z Z Z Z Z Z S A - - - - - - - - - = + - + = + + - + = + + - + = + - + = + + - + = + + - + = 1 2 3 4 5 6 , , , , , , , , 0 B C P P M M S S S S S S A Min Z
Model: Fully Augmented Form 1 2 3 4 5 6 1 2 3 4 5 6 30 40 0 0 0 0 0 0 100 0 0 0 0 0 0 4 30 15 10 5 2 25 0 2 25 B C B B C B C C B C B C P P P P P P P P P P P P Z M M S S S S S S A M S A M M S A M M S A M S A M M S A M M Z Z Z Z Z Z S A - - - - - - - - - = + - + = + + - + = + + - + = + - + = + + - + = + + - + = 1 2 3 4 5 6 1 30 40 0 0 0 0 0 0 100 0 0 4 0 1 0 0 0 0 0 1 30 0 1 1 0 1 0 0 0 0 1 15 0 1 1 0 0 1 0 0 0 1 10 0 0 1 0 0 0 1 0 0 1 5 0 2 1 0 0 0 0 1 0 1 25 0 1 2 0 0 0 0 0 1 1 25 B C P P Z M M S S S S S S A

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