1. Practical Issues Finding an initial feasible solution Cycling
Two phase method
Fundamental Theorem
Efficiency of Simplex Method

2. Initialization What if the origin is not a feasible solution?
Create auxiliary problem as follows
Add a new variable x0
Subtract x0 from all constraints
Objective: minimize x0 (or maximize –x0)
Solve auxiliary problem to find feasible solution or prove problem is infeasible

3. Cycling Can the simplex method cycle?
Degenerate basic solutions lead can cause degenerate pivots and can lead to cycling
Example in text given for the rules
Max coefficient enters
Smallest subscript leaves
2 methods are given for eliminating cycling
Perturbation or lexicographic method
Smallest subscript method

4. Fundamental Theorem
Every LP in the standard form has the following three properties:
If it has no optimal solution, it is infeasible or unbounded
If it has a feasible solution, then it has a basic feasible solution
First phase of 2-phase simplex proves infeasible or gives basic feasible solution
If it has an optimal solution, then it has a basic optimal solution
2nd phase gives basic optimal solution or proves unbounded