Lizhi Wang lzwangiastateedu IE 534 Linear Programming 22 30

Lizhi wang lzwangiastateedu ie 534 linear programming

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Lizhi Wang ([email protected]) IE 534 Linear Programming September 24, 2012 22 / 30
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Outline 1 Review 2 How to check the optimality of a fbp? 3 How to check the unboundedness of an LP? 4 How to find a better fbp? Lizhi Wang ([email protected]) IE 534 Linear Programming September 24, 2012 23 / 30
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The Simplex diagram Lizhi Wang ([email protected]) IE 534 Linear Programming September 24, 2012 24 / 30
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Improving a fbp Suppose the current fbp is ( B , N ) . The way we improve this fbp is by selecting an j * from N and a i * from B and then updating the fbp as ( B 1 = B\{ i * } ∪ { j * } , N 1 = N\{ j * } ∪ { i * } ) . The variable x j * is called the entering variable , because it will enter the basis. Similarly, x i * is called the leaving variable . Geometrically, such an update means a move from a fbs to an adjacent one. Lizhi Wang ([email protected]) IE 534 Linear Programming September 24, 2012 25 / 30
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Bland’s rule Suppose the current fbp is ( B , N ) . The rule for selecting the entering and leaving variables is called a pivoting rule . Bland’s rule is a commonly used pivoting rule. The Bland’s rule j * = min { j ∈ N : ( c > N - c > B A - 1 B A N ) j > 0 } . i * = min ( argmin i ∈B ( ( A - 1 B b ) i ( A - 1 B A N ) i,j * : ( A - 1 B A N ) i,j * > 0 )) . Lizhi Wang ([email protected]) IE 534 Linear Programming September 24, 2012 26 / 30
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Bland’s rule According to Bland’s rule The entering variable has the smallest index with a positive reduced cost. The leaving variable is the winner of the ratio test : min ( ( A - 1 B b ) i ( A - 1 B A N ) i,j * ) for all i such that ( A - 1 B A N ) i,j * > 0 . In case of a tie, choose the smallest index as the winner of the ratio test. Lizhi Wang ([email protected]) IE 534 Linear Programming September 24, 2012 27 / 30
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Example 5 Consider the following dictionary. ζ = 0 + 5 4 3 x 1 x 2 x 3 > w 4 w 5 w 6 = 5 11 8 - 2 3 1 4 1 2 3 4 2 x 1 x 2 x 3 . The current fbp is ( B 0 = { 4 , 5 , 6 } , N 0 = { 1 , 2 , 3 } ) . The entering variable is x 1 . The ratio test for w 4 w 5 w 6 is 5 / 2 11 / 4 8 / 3 = 2 . 5 2 . 75 2 . 67 . The leaving variable is w 4 , because it’s the winner of the ratio test. The updated fbp is ( B 1 = { 1 , 5 , 6 } , N 1 = { 4 , 2 , 3 } ) . Lizhi Wang ([email protected]) IE 534 Linear Programming September 24, 2012 28 / 30
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Exercise Solve the following LP using the Simplex algorithm and the Matlab programs dictionaryL1.m and dictionaryL0.m , which can be downloaded from the course web. max x ζ = 13 x 1 + 7 x 2 - 12 x 3 s . t . 2 x 1 + 3 x 2 - x 3 5 - 4 x 1 - 7 x 2 + 2 x 3 ≤ - 11 3 x 1 - 4 x 2 - 2 x 3 ≤ - 8 x 1 , x 2 , x 3 0 . Lizhi Wang ([email protected]) IE 534 Linear Programming September 24, 2012 29 / 30
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Exercise solution A=[2 3 -1; -4 -7 2; 3 -4 -2]; b=[5; -11; -8]; c=[13; 7; -12]; Phase I I dictionaryL1(A,b,c,[4 7 8],[1 2 3 5 6]); I dictionaryL1(A,b,c,[1 7 8],[4 2 3 5 6]); I dictionaryL1(A,b,c,[1 2 8],[4 7 3 5 6]); I dictionaryL1(A,b,c,[1 2 3],[4 7 8 5 6]); Phase II I
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