**Example: Product
Planning**

The actual replacement of x_{3} by x_{2}
is accomplished making use of two steps. The goal is to create in the x_{2}
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 x_{2} column and the leaving x_{3}
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 x_{3}
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
x_{2} with the c_{2} 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 x_{2} column
to zero. In our example, the only other value in the x_{2} column
is a 1 in the x_{4} row. This can be accomplished by multiplying
the entire new x_{2} row by the value in the x_{2} column,
x_{4} row-that is, the value – 1;and subtracting the result from
the old x_{4} row see Table 2. In this case we multiply the new
pivot line by 1 – no change.

Pivot