dual1 (1) - Dual Problem of an LPP Given a LPP(called the...

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Dual Problem of an LPP Given a LPP (called the primal problem), we shall associate another LPP called the dual problem of the original (primal) problem. We shall see that the Optimal values of the primal and dual are the same provided both have finite feasible solutions. This topic is further used to develop another method of solving LPPs and is also used in the sensitivity (or post-optimal) analysis.
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Definition of the dual problem Given the primal problem (in standard form) Maximize n n x c x c x c z + + + = ... 2 2 1 1 subject to 11 1 12 2 1 1 21 1 22 2 2 2 1 1 2 2 1 2 1 2 ... ... . . ... , ,..., 0, , ,..., 0 n n n n m m mn n m n m a x a x a x b a x a x a x b a x a x a x b x x x b b b + + + = + + + = + + + =
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the dual problem is the LPP Minimize m m y b y b y b w + + + = ... 2 2 1 1 subject to 11 1 21 2 1 1 12 1 22 2 2 2 1 1 2 2 1 2 ... ... . . ... , ,..., unrestricted in sign m m m m n n mn m n n a y a y a y c a y a y a y c a y a y a y c y y y + + + + + + + + +
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If the primal problem (in standard form) is Minimize n n x c x c x c z + + + = ... 2 2 1 1 subject to 11 1 12 2 1 1 21 1 22 2 2 2 1 1 2 2 1 2 1 2 ... ... . . ... , ,..., 0, , ,..., 0 n n n n m m mn n m n m a x a x a x b a x a x a x b a x a x a x b x x x b b b + + + = + + + = + + + =
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Then the dual problem is the LPP Maximize m m y b y b y b w + + + = ... 2 2 1 1 subject to 11 1 21 2 1 1 12 1 22 2 2 2 1 1 2 2 1 2 ... ... . . ... , ,..., unrestricted in sign m m m m n n mn m n n a y a y a y c a y a y a y c a y a y a y c y y y + + + + + + + + +
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1. In the dual, there are as many (decision) variables as there are constraints in the primal. We usually say y i is the dual variable associated with the ith constraint of the primal.
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