Starting Solutions and Convergence

1
Starting Solutions and Convergence

• CONTENTS
• The Initial Basic Feasible Solutions
• The Two-Phase Method
• The Big-M Method
• Degeneracy, Cycling, and Stalling
• Reference Chapter 4 in BJS book.

2
Starting Solutions

• The simplex method assumes the existence of a
  basic feasible solution
• When a basic feasible solution is not readily
  available, then we need to create one such
  solution by adding slack, surplus, or artificial
  variable

3
Starting Solutions (contd)

• Starting basis

4
Starting Solutions (contd)

• In both cases, the constraint matrix does not
  contain an identity matrix.

5
Artificial Variables

• In order to obtain identity in the constraint
  matrix, sometimes we must add artificial
  variables.
• The use of artificial variables changes the
  solution space hence we must guarantee that
  these variables will eventually drop to zero

6
Artificial Variables (contd)

• Let P1
• and P2
• Result 1 If P1 has a feasible solution,
• them P2 has a feasible solution with xa0.
• Result 2 If P2 has a feasible solution with
  xa0,
• then P1 has a feasible solution.
• Theorem There is a one-to-one correspondence
  between
• feasible solution of P1 and feasible solutions
  of P2 with xa0.

7
Two Phase Method

• Phase I
• If at optimality , then stop the
  original problem has no feasible solutions.
• If at optimality , then the original
  problem has a feasible solution (xB) and we go to
  phase 2.

8
Two Phase Method (contd)

• Phase II Solve the following LP

9
Optimization of the Simplex Tableau
Phase I Objective
Phase II Objective
10
Big M Method

• Solve the following LP
• where M is a very large number.
• The term can be interpreted as a
  penalty to be paid by any solution with
  .

11
Nondegenerate Linear Program
z-value of the new BFS z-value of the
current BFS (value of the
entering variable)(zj cj) for the
entering variable We know that the
reduced cost coeff. of the entering variable is
positive. RESULT 1 If value of the entering
variable gt 0, then z-value of the new BFS is
strictly less than the BFS of the current
BFS. RESULT 2 If value of the entering variable
0, then z-value of the new BFS is the same as
that for the BFS of the current BFS.
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Nondegenerate Linear Programs (contd.)
Nondegenerate LP We call a LP to be
nondegenerate if in each BFS of the LP all the
basic variables are positive. RESULT 3 For a
nondegenerate LP, the value of the entering
variable is always positive. RESULT 4 For a
nondegenerate LP, the simplex algorithm never
repeats a BFS and terminates within nCm
iterations.
13
Degenerate Linear Programs
Degenerate LP We call a LP to be degenerate if
it has at least one BFS in which a basic variable
is equal to zero. The following LP is
degenerate max z 5x1 2x2 max z 5x1
2x2 s. t. x1 x2 ? 6 s. t.
s. t. x1 x2 s1 6 x1
x2 ? 0 x1 x2
s2 0 x1, x2 ? 0
x1, x2 , s1, s2 ? 0
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Degenerate Linear Programs (contd.)
Starting Solution
Solution after the first iteration. It is a
degenerate iteration.
15
Degenerate Linear Programs (contd.)
Solution after the first iteration
Solution after the second iteration. It is a
nondegenerate iteration.
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Degenerate Linear Programs (contd.)
BFS and Extreme Points
B
6
5
4
D
3
2
1
A
1
2
3
4
5
6
C
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Degenerate Linear Programs (contd.)

• In a degenerate iteration, the basis changes, but
  the BFS solution remains unchanged.
• In the presence of degeneracy, the objective
  function value may not increase from one
  iteration to next.
• For very large LPs, degeneracy is a real problem
  and over 90 of the pivot operations are
  degenerate iterations.
• For degenerate LPs, cycling can occur (that is,
  simplex algorithm can perform an infinite
  sequence of iterations without any improvement)
  and the algorithm may not obtain an optimal
  solution.
• Cycling can theoretically occur and but has never
  occurred in practice.