The Dual Simplex Algorithm

1
The Dual Simplex Algorithm

• Operational Research
• BY
• TMJA Cooray

2
Introduction

• The simplex method starts with a dictionary which
  is feasible but does not satisfy the optimality
  condition on the Z equation. It then performs
  successive pivot operations , preserving
  feasibility , to find a dictionary which is both
  feasible and optimal.?

3

• The dual simplex algorithm starts with a
  dictionary which satisfies the optimality
  condition on the z- equation, but is not
  feasible.
• It then performs successive pivot operations,
  which preserve optimality, to find a dictionary
  which is both feasible and optimal.

This Dual simplex method is very useful in
sensitivity analysis and also in solving Integer
programming problems.
4
Method

• Feasibility condition variable having the most
  negative value. (break ties arbitrarily)
• Optimality condition find the ratios of the
  coefficients of the objective row and the leaving
  variable row.

5
Method

• Leaving variable basic variable having the most
  negative value. (break ties arbitrarily)
• .
• Entering variable non basic variable with the
  smallest absolute ratio , that is min Zj/aij
  such that aij lt 0.
• if all the denominators are 0 or ve , the
  problem has no feasible solution. (Can not get
  rid of infeasibility.)

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• Once we have identified the leaving and the
  entering variables , we perform the normal pivot
  operation to move to the next dictionary.

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• Min Zy60Y140Y2
• Subject to 5Y14Y2? 6,
• 10Y14Y2? 8
• Y1,Y2? 0
• -5Y1-4Y2s1 - 6,
• -10Y1-4Y2s2- 8

Ratio -6 -10
Smallest absolute value
8
-8
-12
Smallest absolute value
9
The optimal solution
This is a feasible solution and still optimal
. Stop the procedure.
10
Exercise