Example: Product Planning

The actual replacement of x₃ by x₂ is accomplished making use of two steps. The goal is to create in the x₂ column vector with 1 instead of the pivot and other elements.

The first step is conversion of the pivot line. The element in the entering x₂ column and the leaving x₃ row is designated the pivot element. (In our example, the pivot element is circled in The Leaving Variable and has a value of 3.) In order to obtain a new solution containing X2, this element must be converted to a + 1.To convert the pivot element to + 1 requires that we divide every element in the present x₃ row by the value of the pivot element (i.e., by 3). Recall that each row is in reality a constraint equation, and one may divide a whole equation by a constant.

The calculations are:

Thus, the new top row will be see Table1 labelled x₂ with the c₂ objective function coefficient in the very left column.

The second part of the procedure is aimed at converting all the elements (except the pivot element) in the x₂ column to zero. In our example, the only other value in the x₂ column is a 1 in the x₄ row. This can be accomplished by multiplying the entire new x₂ row by the value in the x₂ column, x₄ row-that is, the value – 1;and subtracting the result from the old x₄ row see Table 2. In this case we multiply the new pivot line by 1 – no change.

Pivot

So the whole second simlex tableau (Table 3) includes new  row and new  with circled new pivot.