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Pivot Column - Assignment Example

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Although simplices are not used in the method, it has been suggested that is operates on simplicial cones, hence the name.
Pivot operations are a geometrical process…
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Pivot Column
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Running Head: THE SIMPLEX METHOD – PIVOTS The Simplex Method: What is a pivot column, pivot row and pivot element?
In the simplex method, how is a pivot column selected? A pivot row? A pivot element? Give examples of each.
The simplex method, or simplex algorithm as it is also known, is an algorithm used for linear programming. Although simplices are not used in the method, it has been suggested that is operates on simplicial cones, hence the name.
Pivot operations are a geometrical process that moves from a basic feasible solution to a second adjacent basic feasible solution. If you take a pivot element (non-zero) in a non-basic column, and then multiply the row that the pivot element is in by its reciprocal then you end up with a pivot element that is 1, and the other entries in the column become 0 (zero). The column (r) becomes an r-nth column.
In a tableau the pivot column is the column that has the most negative integer. If there are no negative indicators then the tableau is already considered final and the solution is already evident. The pivot row is the row with the smallest non-negative ratio (quotient). The pivot element is the number that sits at the intersection of the column with the most negative indictor (pivot column) and the row with the smallest quotient (pivot row). The example below is from “The Simplex Method” (n.d. web). This scenario was based on three different products ( s , t, u) and three different machines required to make those products. Note all calculations are done from the basic solution that x, y, and z, (number of units per product) are all zero. The original numbers used and the full calculation can be found in the reference section.
x
y
z
s
t
u
P
constant
1
.5
.5
.5
0
0
0
90
1
3
2
0
1
0
0
300
2
1
2
0
0
1
0
240
-6
-5
-4
0
0
0
1
0
In this first set of calculations the pivot column is in blue, the pivot row is in yellow and the pivot element is in green. The pivot element is found by first selecting the pivot column (the one with the most negative indicator) – in this example it is column 1 with -6. Then the constant on the right of the bar needs to be divided by the corresponding non-zero element in the pivot column and this will give you the smallest quotient (pivot row) and the intersection of these two (row and column) will give you the pivot element.
References
“The Simplex Method” Pellissippi State Community College. N.d. http://www.pstcc.edu/facstaff/jahrens/math1630/simplex.pdf September 2011 Read More
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