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The Dual Simplex Algorithm Operational Research-Level4. Prepared by T.M.J.A.Cooray Department of Mathemtics. Introduction .

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the dual simplex algorithm operational research level4

The Dual Simplex AlgorithmOperational Research-Level4

Prepared by T.M.J.A.Cooray

Department of Mathemtics

MA(4020) Operational Research,Dual simplex method

introduction
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.

MA(4020) Operational Research,Dual simplex method

slide3
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.

MA(4020) Operational Research,Dual simplex method

method
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.

MA(4020) Operational Research,Dual simplex method

method5
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 < 0.
  • if all the denominators are 0 or +ve , the problem has no feasible solution. (Can not get rid of infeasibility.)

MA(4020) Operational Research,Dual simplex method

slide6
Once we have identified the leaving and the entering variables , we perform the normal pivot operation to move to the next dictionary.

MA(4020) Operational Research,Dual simplex method

slide7
Min Zy=60Y1+40Y2

Subject to :5Y1+4Y2 6,

10Y1+4Y2 8

  • Y1,Y2 0

-5Y1-4Y2+s1 =- 6,

-10Y1-4Y2+s2=- 8

Ratio : -6 -10

Smallest absolute value

MA(4020) Operational Research,Dual simplex method

slide8

-8 -12

Smallest absolute value

MA(4020) Operational Research,Dual simplex method

the optimal solution
The optimal solution

This is a feasible solution and still optimal . Stop the procedure.

MA(4020) Operational Research,Dual simplex method

exercise
Exercise

MA(4020) Operational Research,Dual simplex method