Question

Find the dual of the following LP, using direct method. minz=4X1 +2X2 -X3

subject to

X1 +2X2 ≤6

X1 -X2 +2X3 =8

X1 ≥0,X2 ≥0,X3 urs

Answer 1

The LP is of the Minimization type. So, the dual shall be of maximization type.

Also, the first constraint is of type "<=", so it has to be first converted into ">=" type by multiplying it with -1.

There are 3 decision variables. So, there shall be 3 constraints in the dual.

There are 2 constraints. So, there shall be 2 decision variables in the dual

The decision variable X3 is unrestricted in sign. So, the third constraint in the dual shall be of "=" type.

The second constraint is of "=" type. So, the second decision variable in the dual shall be unrestricted in sign

Considering the decision variables in the dual as Y1, and Y2.

The original LP is as follows:

Minimize Z = 4X1 + 2X2 - X3

subject to

-X1 - 2X2 >= -6

X1 - X2 + 2X3 = 8

X1 >= 0, X2 >= 0, X3 urs

The dual is as follows:

Maximize Zd = -6Y1 + 8Y2

subject to:

-1Y1 + 1Y2 <= 4

-2Y1 - 1Y2 <= 2

0Y1 + 2Y2 = -1

Y1 >= 0, Y2 urs