simplex2b.html - The Simplex Method pivoting to find the...

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The Simplex Method : pivoting to find the optimum Simplex tableau Recall that: The Simplex method uses (invariant) row operations on matrices in Linear Algebra to find the optimal solution of an LP Review: invariant row operations in Linear Algebra Consider the following system of equations: 2 x 1 + 4 x 2 = 16 3 x 1 + 2 x 2 = 12 The equality (=) means that the values in the left hand side and right hand side of "=" are in "balance" . The systems of equations above can be represented by this figure: 2 x 1 + 4 x 2 = 16 3 x 1 + 2 x 2 = 12 Invariant row operation 1: You can multiply (or divide ) both sides of an equation by the same factor , and the resulting equality will hold (= true) . Example: divide the first equation by 2 (= multiply first equation by 0.5) 2 x 1 + 4 x 2 = 16 (equation 1)
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3 x 1 + 2 x 2 = 12 (equation 2) Manipulation: 0.5 × (2 x 1 + 4 x 2 ) = 0.5 × (16) ==> x 1 + 2 x 2 = 8 Result: x 1 + x 2 = 8 (equation 1) 3 x 1 + 2 x 2 = 12 (equation 2) Invariant row operation2: You can add (or subtract ) a multiple of one equation (row) to another equation (row) and the resulting equality will hold (= true) Example: subtract 3 × the first equation from the second equation x 1 + 2 x 2 = 8 (row 1) 3 x 1 + 2 x 2 = 12 (row 2) (row 2) ===> 3 x 1 + 2 x 2 = 12 3 x (row 1) ===> 3 x 1 + 6 x 2 = 24 - (subtract) ------------------------------- 0 x 1 - 4 x 2 = -12 Pivot operation: transform one basic solution into another (better) basic solution Pivot operating: (a.k.a.: pivoting )
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Pivot operation = to apply invariant row operations to Simplex Tableau such that: A non-basic variable will become a basic variable and A (some other) basic variable will become a non-basic variable In other words: A pivot operation will change the basic solution !!! Note: Before we perform the pivot operation , we will make sure that the objective function will increase when we perform the pivot operation !!! We will learn how to select a pivot later .... Pivot or pivot element Pivot = an element in the Simplex Tableau that forms the central element in the pivot operation The pivot operation is apply using the pivot element as its parameter The steps of the pivot operation : 1. Divide the row containing the pivot element by the value of the pivot This step will make the pivot element = 1 2. Use row operations to make the other elements of the column vector containg the pivot equal to 0 This step will make the column vector containg the pivot
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equal to a
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