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L11_Explanation of Simplex Tableau Entries

L11_Explanation of Simplex Tableau Entries - Explanation of...

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Explanation of the entries in any simplex tableau in terms of the entries of the starting tableau
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In this lecture we explain how the starting Simplex tableau (in matrix form) gets transformed after ‘some’ iterations. We also give the meaning of the entries in the new tableau in terms of the entries of the starting tableau. We illustrate how we can write the simplex tableau from the starting tableau by choosing a different set of basic variables.
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The Simplex tableau in Matrix form Consider the LPP in matrix form as: z = cX Subject to = AX b X 0, b 0 Maximize (or Minimize) The problem can be written equivalently as 1 0 z -   =     c 0 A X b
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Suppose B is a m x m non-singular sub-matrix of the coefficient matrix A. Let X B be the corresponding set of basic variables with c B as its associated objective vector. The corresponding solution (and the objective function value ) may be computed as follows: 1 1 0 0 1 z - - - - - = = = - 1 1 B B B 1 1 B c c B c B b X 0 B b b 0 B B b 1 0 z = - B B c X 0 B b Size m × 1 Hence
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The general simplex tableau in matrix form can be derived from the original equations as follows: 1 0 1 1 z - - - - -   =     1 1 B B 1 1 c c B c B 0 A X b 0 B 0 B Multiplying out the matrices, we get 1 z - - - - = - 1 1 B B 1 1 c B A C c B b X 0 B A B b
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Thus in the new simplex tableau the z column will be 1 0 . . 0         The column of x j (in the constraint equations) will be is the jth column of A . The solution column will be (Note: The text book uses the symbol P j for the jth column of A.) 1 , where j j A A - B 1 .
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