The Dual Simplex Algorithm Operational Research-Level4

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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 AlgorithmOperational Research-Level4

Prepared by T.M.J.A.Cooray

Department of Mathemtics

MA(4020) Operational Research,Dual simplex method

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

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
• 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

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

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

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

-8 -12

Smallest absolute value

MA(4020) Operational Research,Dual simplex method

The optimal solution

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

MA(4020) Operational Research,Dual simplex method

Exercise

MA(4020) Operational Research,Dual simplex method