Math 482 (Lecture 7): The two phase simplex method
This lecture covers Section 2.8 of the textbook.
I have also attached the following
notes.
Lectures 5 and 6 show that assuming we start with a basic feasible solution, then the
simplex method will complete, and we will decide
A. The problem has a finite minimum, and a bfs that attains this min.
B. The problem is unbounded.
However, it can happen that
C. The problem is infeasible.
D. The problem is feasible, but you aren't able to immediately detect a bfs.
(With regards to D., we've been focusing on the very good case that the problem has an
"obvious" basis and the coefficient vector b is nonnegative.)
The
first phase
of the
two phase simplex method
determines if C happens, or otherwise
finds us a bfs. (The second phase is what was covered in lectures 5 and 6.)
Example:
See here.
Phase 1 summary:
* Negate if necessary all constraints so that the coefficients are all positive.
* Add slack variables to as many equations as necessary so as to obtain an "obvious"
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 Spring '08
 Staff
 Math, Optimization, basic feasible solution, max z=2x1+3x2+x3

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