As we discussed in Chapter 2, in each iteration of the simplex method, we first choose an entering variable looking at the objective row of the current tableau, and then identify a leaving variable by comparing the ratios of the numbers on the right-hand side and the column for the entering variable. Once these two variables are identified we update the tableau. Clearly, the most time-consuming job among these steps of the method is the tableau update. If we can save some time on this bottleneck step then we can make the simplex method much faster. The revised simplex method is a variant of the simplex method developed with precisely that intention.

The crucial question here is whether it is necessary to update the whole tableau in every iteration. To answer this question, let us try to identify what parts of the tableau are absolutely necessary to run the simplex algorithm. As we mentioned before, the first task in each iteration is to find an entering variable. Let us recall how we do that. In a maximization problem, we look for a nonbasic variable with a positive rate of improvement. In terms of the tableau notation, this translates into having a negative coefficient in the objective row, where Z is the basic variable.

To facilitate the discussion below let us represent a simplex tableau in an algebraic form, using the notation from Section 2.4.1.