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: .

Mon Aug 24 14:57:03 EDT 1998