{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 5

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online