a Simplex
EXAMPLE OF SIMPLEX PROCEDURE
FOR A NON-STANDARD
LINEAR PROGRAMMING PROBLEM

 Below is original Problem
 1st step "NS-1"
 Red symbols below were changed by step NS-1
 2nd step "NS-2"
 Red variables sn below are called SLACK VARIABLES

 There is a tie for most negative right-hand number. Either "-12" is OK
 The pivot in the 1st Simplex Tableau. is circled below. Missing Z-COLUMN?? highlighted is the "ISM"
 In the matrix below, the pivot has become "1"
 Named below are the 4 row operations needed to pivot on the number "-2" encircled in red

 Row operations are named above; their results are below Highlighted is the new ISM
 Since one right-hand number (-6) remains negative, we remain in Phase I
 In the matrix below, the pivot has become "1"
 Named below are the 4 row operations needed to pivot on the number "-3/2" encircled in red

 Row operations are named above; their results are at the right. The new ISM is highlighted. All right-most numbers above the bottom (objective) row are now zero or bigger; Phase I is now ended. Phase II now begins, presented as "Step 8" in Rolf (Pg 323)

In many non-standard problems, we would now find a negative indicator in the bottom row. However, in our last tableau above, a nice coincidence finds all indicators (0, 0, 0, 4/3, 1/3) are zero or bigger; "-20" is not an indicator. Hence, Phase II is completed at it's start, because the above tableau is a final tableau, and the row operations of SIMPLEX are done!

 To obtain the final basic solution to our problem, 1st set equal to 0 each variable NOT associated with the highlighted ISM: variable tags are placed above each column in the final tableau. 2nd convert each row of the final tableau back into it's equation format, as at the right, to find the values of the remaining variables (shown in red).