# Duality

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## Definition: Duality

The linear programming problem addresses resource allocation and also resource valuation. While the resource allocation is solved using the primal problem, the value of each resource is identified using the dual problem.

Every linear programming problem which has a primal problem can also be converted into a dual problem. The rules for converting a primal into a dual problem are as follows:

 Primal Dual Decision variable: x Decision variable:y Max Z=cx Min Z=by Such that Ax<=b yA>=c

For any primal and dual problem all relationships between them must be symmetric.

 Primal Dual Obj function: Max Min If constraint is Variable is <= >=0 >= <=0 = If variable is Constraint is >=0 >= <=0 <= URS =

Duality theorem:

Weak Duality

– If a constraint is inactive in an optimal solution of the primal problem then corresponding dual variable equal to zero .

– Or an inactive constraint in an dual problem will have its corresponding variable equal to zero in primal solution

Strong Duality:

If constraint is active then corresponding dual variable is unequal to zero. Conversely an active constraint of the dual problem will have its corresponding variable unequal to zero.

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