Linear Programs in Standard Form

Before we start discussing the simplex method, we point out that every linear program can be converted into ``standard'' form

where the objective is maximized, the constraints are equalities and the variables are all nonnegative.

This is done as follows:

• If the problem is min z, convert it to max -z.
• If a constraint is , convert it into an equality constraint by adding a nonnegative slack variable . The resulting constraint is , where .
• If a constraint is , convert it into an equality constraint by subtracting a nonnegative surplus variable . The resulting constraint is , where .
• If some variable is unrestricted in sign, replace it everywhere in the formulation by , where and .

Let us first turn the objective into a max and the constraints into equalities.

The last step is to convert the unrestricted variable into two nonnegative variables: .