In the simplex method, how is a pivot column selected? A pivot row? A pivot ele

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I’m studying for my Mathematics class and don’t understand how to answer this. Can you help me study?

In the simplex method, how is a pivot column selected?  A pivot row? A pivot element?  Give examples of each.

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1) Pivot column corresponds to a negative value (normally the smallest, or one of them if there are several smallest values) in the index row if the problem is to maximize the object function, or to a positive value (normally the greatest) in the index row if the problem is to minimize (if all the values in the index row are non-negative/non-positive the maximum/minimum is already attained). In the example given these values are 0, 0, 2, -4, 0, 1, so when maximizing the pivot (leading) column should be #4 (corresponding to value -4); when minimizing it is #3 (value 2 - the greatest positive) as shown.

2) The pivot column being determined, consider all POSITIVE components in it (if there is no positive element in the pivot column, the object function is unbounded - infinite optimum). Compare the ratios of each free term and the corresponding positive element in the pivot column - the least of them determines the pivot (leading) row. If there is more than one least ratio, make an arbitrary choice between the corresponding rows. In the example the pivot column is #3, both elements 2 and 4 are positive, so we must compare 10/2 and 13/4 - the second one is the smallest, so the pivot row is #2. If the pivot column was #4, the only ratio is 13/3 in the 2nd row - it is again pivot row.

3) The common element of the pivot row and the pivot column is the pivot element (it is always positive in the standard simplex-method according the above). After the iteration the pivot column should become a unit vector with 1 instead of the pivot element.

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Benson K (358)
Boston College

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