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.

 Example 

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
The Upjohn Company, based in Kalamazoo, Michigan, manufactures and markets pharmaceuticals and health-related products. With more than 19,000 employees and distribution in over 30

in a rectangular game pay off matrix of player a is as follows B1 B2 A1 5 7 A2 4 0 salve the game write down the pay off matrix of B and then solve the ga

Six Operators are to be assigned to five jobs with the cost of assignment in Rs. given in the matrix below. Determine the optimal assignment. Which operator will have no assignment

Maximize p = (3)x + 2y subject to 2x + y 3x + 4y >= 12

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 i

#what is the similarity and differences between transportation and linear programing models?

if one no. is repeated 2-3 times in machine-1 and machine-2 what is to be prefer first

a question was given but i cannot identify the alternatives and i do not know how to calculate the right amount of payoff.what are the right ways to calculate the payoff table.

1: explain vogel''s approximation method? 2: explain optimality test?

Case Study - Experimental Design Dental Clinic A leading dental clinic  with  three well qualified  dentists is seriously  considering  keeping  in touch  with  its  pat