The Entering Variable

Example: Product Planning

To determine the variable that will become solution variable we have to calculate the values  for each column. zj 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 zj we have to multiply the objective function coefficients cj 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 zj value can be calculated for the "Solution values" column:

zj represents the profit of the current solution.

The values in the row  labelled. Subtract the total zj from cj (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 x2 is added to the solution (replacing some amounts of x3 and x4), $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)

According to the values of  the entering variable is  because the net profit added by introducing this variable is maximal. Since variable  will remain as outside variable equal to zero. The entering column  is marked with