Duality, Operation Research

For every LP formulation there exists another unique linear programming formulation called the 'Dual' (the original formulation is called the 'Primal'). Same data can be used for both 'Dual' and 'Primal' formulation.  Both can be solved in a similar manner as the Dual is also an LP formulation.

The Dual can be considered as the 'inverse' of the Primal in every respect. The column coefficients in the Primal constraints become the row co-efficients in the Dual constraints. The coefficients in the Primal objective function become the right-hand-side constraints in the Dual constraints. The column of constants on the right hand side of the Primal constraints becomes the row of coefficients of the dual objective function. The direction of the inequalities are reversed. If the primal objective function is a 'Maximization' function then the dual objective function is a 'Minimization' function and vice versa.


Consider the following 'Primal' LP formulation.

Maximize   12x1 + 10x2

subject to         2x1 +   3x2  <  18

                       2x1 +    x2   <  14

                             x1, x >    0

The 'Dual' formulation for this problem would be

Minimize      18y1 + 14y2

subject to   2y1 +   2y2    > 12

                   3y1 + y2    > 10

                   y1 > 0,  y2  >  0

Note the following:

  1. The column coefficient in the Primal constraint namely (2,2) and (3,1) have become the row coefficient in the Dual constraints.

  2. The coefficient of the Primal objective function namely, 12 and 10 have become the constants in the right-hand-side of the Dual constraints.

  3. The constants of the Primal constraints, namely 18 and 14, have become the coefficient in the Dual objective function.

  4. The direction of the inequalities have been reversed. The Primal constraints have the inequalities of < while the Dual constraints have the inequalities of >.

  5. While the Primal is a 'Maximization' problem the Dual is a 'Minimization' problem and vice versa.              

Posted Date: 9/13/2012 9:05:19 AM | Location : United States

Related Discussions:- Duality, Assignment Help, Ask Question on Duality, Get Answer, Expert's Help, Duality Discussions

Write discussion on Duality
Your posts are moderated
Related Questions
In your own words, describe the special cases of integer programming and binary programming: what makes these problems different? Give an example of each, pointing out why they mus

A farmer in the Midwest has 1,000 acres of land on which she intends to plant corn, wheat, and soybeans. Each acre of corn costs $100 for preparation, requires 7 worker-days of la

Area under Standard Normal Curve The graph  of the  standard  normal  curve y = p(z) with  different percentages areas is  shown here. Since p(z) is  a probability  distributi

Problems based on solution of a given LPP when it has multiple optimal solution: 1.    Find the maximum and minimum values of 5x+2y, subject to the constraints -2x-3y ≤ -6

meaning,significance of operations research in bussiness management?

Write a research paper in relation to a Software Design related topic. Diagrams and drawing of attention to key points through highlighting, bulleting etc is encouraged. Questi

A paper mill produces two grades of paper viz X and Y. Because of raw material restrictions, it cannot produce more than 400 tons of grade X paper and 300 tons of grade Y paper in

Background: The use of "Tablet Personal Computers" such as HP's TouchPad and Apple's iPad has been increasing over the past several years. Task: Using a web browser and quality,

Graphic Methods a. Scatter Diagram: Scatter  diagram  is a special  type of dot chart.  Under  this methods  the given data are  plotted in  a graph  paper  in the form of  d