note3 - Note 3: LP Duality If the primal problem (P) in the...

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Note 3: LP Duality If the primal problem (P) in the canonical form is min Z = n j =1 c j x j s.t. n j =1 a ij x j b i i = 1 , 2 ,...,m x j 0 j = 1 , 2 ,...,n, (1) then the dual problem (D) in the canonical form is max W = m i =1 b i y i s.t. m i =1 a ij y i c j j = 1 , 2 ,...,n y i 0 i = 1 , 2 ,...,m. (2) We may think of j as a food type, i as a nutrition type, c j as the per-unit price of food type j , b i as the required quantity of nutrition type i , a ij as the quantity of nutrition type i contained in each unit of food type j , x j as the quantity of food type j to purchase, and y i as the per-unit price to be charged for nutrition type i . The primal problem may be considered as finding the least costing quantities of foods to buy to satisfy nutritional needs, and the dual problem can be considered as finding the most profitable pricing scheme for pure nutritions that is competitive in the face of existing prices of food types. In essence, (P)’s variable corresponds to (D)’s constraint, and (D)’s variable corresponds to (P)’s constraint; If (P) is a minimization (maximization) problem, then (D) is a maximization (minimization) problem; (P)’s free variable corresponds to (D)’s = constraint and its nonnegative variable corresponds to (D)’s max / constraint (meaning that (D) is a maximization problem and the constraint is of the “ ” type) or min / constraint; on the other hand, (D)’s free variable corresponds to (P)’s = constraint, its nonnegative variable corresponds to (P)’s max / or min / constraint. The dual of dual is the primal itself. In lieu of the above, if the primal is our standard form min Z = n j =1 c j x j s.t. n j =1 a ij x j = b i i = 1 , 2 ,...,m x j 0 j = 1 , 2 ,...,n, 1
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with n m , then its dual is max W = m i =1 b i y i s.t. m i =1 a ij y i c i y i is free i = 1 , 2 ,...,m. By introducing the slack variables
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note3 - Note 3: LP Duality If the primal problem (P) in the...

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