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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

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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

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