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.

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