Duality in lpp
Download as PPTX, PDF1 like5,153 views
The document discusses how for every linear programming (LP) primal formulation, there exists a unique dual formulation that is derived from the same data. The dual LP formulation can be solved in the same way as the primal by interchanging various elements between the constraints and objective functions. Specifically, the column coefficients of the primal constraints become the row coefficients of the dual constraints, and the coefficients of the primal objective function become the right-hand side constants of the dual constraints. Additionally, the primal's maximization problem becomes a minimization problem in the dual, and vice versa.
Duality in lpp
- 1. Abu Bashar
- 2. For every LP formulation there exists another unique linear programming formulation called the 'Dual' (the original formulation is called the 'Primal'). The - Dual formulation can be derived from the same data from which the primal was formulated. The Dual formulated can be solved in the same manner in which the Primal is solved since the Dual is also a LP formulation.
- 3. The Dual can be considered as the 'inverse' of the Primal in every respect. The column coefficients in the Primal constrains become the row coefficients 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.
- 4. For each constraint in primal problem there is an associated variable in dual problem. The elements of right hand side of the constraints will be taken as the co-efficient of the objective function in the dual problem. If the primal problem is maximization, then its dual problem will be minimization and vice versa. The inequalities of constraints should be interchanged from >= to <= and vice versa and the variables in both the problems and non-negative. The row of primal problem are changed to columns in the dual problem. In other words the matrix A of the primal problem will be changed ti its transpose (A) for the dual problem. The co-efficient of the objective function will be taken the right hand side of the constraints of the dual problem.
- 6. The column coefficient in the Primal constraint namely (2,2) and (3,1) have become the row co- efficient in the Dual constraints. The co-efficient of the Primal objective function namely, 12 and 10 have become the constants in the right hand side of the Dual constraints. The constants of the Primal constraints, namely 18 and 14, have become the co-efficeint in the Dual obejctive function. The direction of the inequalities have been reserved. The Primal constraints have the in equalities of::; while the Dual constraints have the inequalities of ~ . While the Primal is a 'Maximization' problem the Dual is a 'Minimization' problem