The dual program makes it possible to check two other measures of sensitivities: new variable and new constraint.
Case study 1: introduction of a new variable
Here is the linear program:
We want to add a new activity:
We want to determine whether the new activity has an interest or not. In other words, if it does not have an interest in doing so, the optimal solution of the linear program above gives x3 = 0. This is equivalent to the fact that the third constraint of the dual is not satisfied: y1 + 3 y2 + 2y3 ≥ C3.
The solution of the dual is (200/3, 0, 100/3), one replaces in the third constraint of the dual:
- If C3< 200/3 then the new variable is zero in the primal, there is no interest in producing the new activity
- If C3> 200/3 then there is interest in producing the new activity, moreover the solution of the primal / dual will change
- If C3= 200/3 then the production of the new activity will not change the value of the optimal solution.
Case study 2: introduction of a new constraint
If the optimal solution satisfies the new constraint, the problem will admit the same solution. Otherwise the introduction of this constraint will generate a new optimal solution.