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University of Leeds School of Mathematics MATH 1331 Linear algebra with applications The Simplex Method for linear programming problems in standard form. A linear programming problem is said to be in standard form if • the objective function is to be maximised ; • each variable has to be > 0; • all other inequalities are of the form (linear combination of variables) 6 (positive constant). A simplex tableau is an augmented m × (n + 1) matrix, with n > m, in which m of the first n columns are the columns of the identity matrix Im (in some order). These m columns (and the associated variables) are said to be of Type II; the remainder are of Type I. Step 1. Introduce slack variables and restate the problem in terms of a system of linear equations. Step 2. Construct the simplex tableau corresponding to the system, with the bottom row of the matrix corresponding to the objective equation. Step 3. If there are no negative entries in the bottom row of the unaugmented matrix, then the solution corresponding to the tableau (obtained by setting all the Type I variables to 0) yields a maximum and the problem is solved. Step 4. If there are negative entries, construct a new simplex tableau as follows. (a) Choose the pivot column to be the one containing the most negative element on the bottom row of the matrix. (b) Choose the pivot element by computing ratios associated with the positive entries in the pivot column. The ratio is the element in the augmented column divided by the corresponding element in the pivot column. The pivot element is the one corresponding to the smallest positive ratio. (c) Construct the new simplex tableau by pivoting about the selected element. Step 5. Repeat Steps 3 and 4 as many times as necessary to find a maximum. ECL/11–10–2004 How to deal with problems in nonstandard form. A. If the objective function is to be minimised, replace it by its negative and maximise that (not forgetting to change the sign back again when the solution has been found). B. If the inequalities are not of the form (linear combination of variables) 6 (positive constant), proceed as follows. Step 1. Convert all inequalities (except those saying that the variables are positive) into the form (linear combination of variables) 6 (constant) and construct the simplex tableau for the problem. Step 2. If a negative number appears in the upper part of the last column of the simplex tableau, remove it by pivoting: (a) Select a negative entry in its row. The column containing that entry will be the pivot column. (b) Select the pivot element by determining the least of the positive ratios associated with entries in the pivot column (except the bottom entry). (c) Pivot. Step 3. Repeat Step 2 until there are no negative entries in the upper part of the right hand column of the simplex tableau. Step 4. Proceed to apply the simplex method for a tableau in standard form. ECL/11–10–2004