**Example: Product
Planning**

To determine the variable that will become solution
variable we have to calculate the values
for each column. z_{j }total (where the j subscript refers to the
specific column being totalled) is the amount of profit given by replacing
some of the present solution mix with one unit of the item heading the
column. For obtaining z_{j} we have to multiply the objective function
coefficients c_{j }and the values in j-th column
(called substitution coefficients) and summarise the results of multiplication
for all rows:

The calculations are as follows:

Similarly, a z_{j} value can be calculated
for the "Solution values" column:

z_{j} represents the profit of the current
solution.

The values in the row
labelled. Subtract the total z_{j} from c_{j} (amount at
the very top of the column) the to find the net profit (in absolute numbers)
that is added by one unit of the product (if
is positive) or the amount of profit that will be lost (if
is negative). Thus, if one unit of x_{2} is added to the solution
(replacing some amounts of x_{3} and x_{4}), $7 of net
profit will be added.

The row is in reality an equation, like the other rows in the linear programming table, and expresses the profit at that stage of the computations in terms of the outside variables. Thus, the row is equivalent to: (6;7;0;0)