Dual And Primal

Introduction:

The fact that the value of the objective function in the optimal program equals to the imputed value of the available resources has been called the FUNDAMENTAL THEOREM OF LINEAR PROGRAMMING. By changing the rows of the primal problem (dual problem) into columns we get the dual problem (primal problem) and vice versa.

 Characteristics:

A.      If in the primal, the objective function is to be maximized, then in the dual it is to be minimized. Conversely, if in the primal the objective function is to be minimized, then in the dual it is to be maximized.

B.      The objective function coefficients of the prima appear as right-hand side numbers in the dual and vice versa.

C.      The right hand side elements of the primal appear as objective function coefficients in the dual and vice versa.

D.      The input - output coefficient matrix of the dual is the transpose of the input – output coefficient matrix of the primal and vice versa.

E.       If the inequalities in the primal are of the “less than or equal to” type then in the dual they are of the “greater than or equal to” type. Conversely, if the inequalities in the primal are of the “greater than or equal to” type; then in the dual they are of the “less than or equal to” type.

F.       The necessary and sufficient condition for any linear programming problem and its dual to have optimum solution is that both have feasible solution. Moreover if one of them has a finite optimum solution, the other also has a finite optimum solution. The solution of the other (dual or primal) can be read from the net evaluation row (elements under slack/surplus variable column in net evaluation row). Then the values of dual variables are called shadow prices.

G.     If the primal (both) problem has an unbound solution, and then the dual has no solution.

H.     If the i ith dual constraints are multiplied by –1, then i th primal variable computed from net evaluation row of the dual problem must be multiplied by –1.

I.        If the dual has no feasible solution, then the primal also admits no feasible solution.

J.        If k th constraint of the primal is equality, then the k th dual variable is unrestricted in sign.

K.      If p th variable of the primal is unrestricted in sign, then the p th constraint of the dual is a strict equality.

Difference between primal and dual:

Note:

1. Primal of a Prima is Primal

2. Dual of a Dual is Primal.

3. Primal of a Dual is Primal.

4. Dual of a Primal is Dual.

5. Dual of a Dual of a Dual is Primal.