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Concept of Primal-Dual relationship or duality in Linear Programming.

The original LPP as we have studied is called the Primal. For every LP problem there exists another related unique LP problem involving the same data which also describes the original problem. The original or primal programme can be solved by transposing or reversing the rows and columns of the statement of the problem. Reversing the rows and columns in this way gives us the dual program. Solution to dual program problem can be found out in a similar manner as we use for solving the primal problem. Each LP maximizing problem has its corresponding dual, a minimizing problem. Also, each LP minimizing problem has its corresponding dual, a maximizing problem. This duality is an extremely important and interesting feature of Linear Programming Problems (LPP). Important facts of this property are:-

a)     The optimal solution of the dual gives complete information about the optimal solution of the primal and vice versa.

b)    Sometimes converting the LPP into dual and then solving it gives many advantages, for example, if: the primal problem contains a large number of constraints in the form of rows and comparatively a lesser number of variables in the form of columns, the solution can be considerably simplified by converting the original problem into dual and then solving it.

c)     Duality can provide us economic information useful to management. Hence it has certain far reaching consequences of economic nature, since it helps managers in decision making.

d)    It provides us information as to how the optimal solution changes due to the results of the changes in coefficient and formulation of the problem. This can be used for sensitivity analysis after optimality tests are carried out.

e)     Duality indicates that there is a fairly close relationship between LP and Games theory as its shows each LPP is equivalent to a two-person zero-sum game.

f)      Dual of the dual is a primal.