objective functionalthough the standard lp model

Question

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