Pivoting: Simplex Algorithm choosing a pivot column for the second time

Question

I found online a set of notes from a university. I am studying an example of the Simplex algorithm and I am somewhat confused:

• We defined the pivot column to be the leftmost column with a negative operator in the last row

• So in the first run of the algorithm we have:

Now what I don't get is what's coming next:

Question: $x, v, w$ are exactly the ones having non-negative values in the last row, so how come they become pivot columns? Isn't the other way around ? $y, z$ are possible pivot columns, aren't they?

Answer 1

As pointed out in the comments by Brian, the author of this textbook is denoting the basic variable columns as "pivot columns", and the non-basic variable columns as "free variables". The terminology is confusing, however it's probably more rooted from the traditional terms used in Linear Algebra than from most modern Linear-Programming terminology.

Here are more examples of the terminology the author is using to help provide clarity:

• what do free variable and leading variables mean?
• Row Reduction and Echelon Forms (PDF, Page 8)
• Basic Variables, Free Variables, and Parameters (PDF)

Thus, the next pivoting column, using the author's definition of pivoting, would be the $y$ "free" variable as it is the left-most negative column.