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
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