Objective Function

Although the standard LP model can be either the maximization or the minimization type, it is sometimes useful to convert one form to the other.

The maximization of a function is equivalent to the minimization of the negative of the same function, and vice versa.

For example:  Max. Z   = 5X1 + 2X2 + 3X3 is mathematically equivalent to
                    Min.  (-Z) = -5X1 - 2X2 - 3X3

Equivalence means that for the same set of constraints the optimum values of X₁, X₂ and X₃ are the same in both cases. The only difference is that of the values of the objective functions, although equal numerically, will appear with opposite signs. Example: Write the following LP model in the standard form:

Minimum:        Z = 2X1 + 3X2       Minimum:    Z = X1' - 2X1'' + 3X2

Subject to:       X1 + X2 = 10        Subject to:    X'1 - 2X''1 + 3X2 = 10
                     -2X1' + 3X2≤ -5                        2X'1 - 2X''1 - 3X2 - S2 = 5
                      7X1 - 4X2 ≤ 6                          7X'1 - 7X''1 - 4X2 + S3 = 6
                      X1 Unrestricted                         X1', X1'', X2, S2, S3 ≥ 0
                      X2 ≥ 0