Practical Issues

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