Suppose there's a model with 2 constraints originally. Now that another constraint is added. Which of the following are possible:
1) feasible region shrinks
2) feasible region stays the same
3) feasible region expands
and
Which of the following are possible:
1) optimal value in target cell decreases
2) optimal value in target cell stays the same
3) optimal value in target cell increases
I think the answer is 1) for both questions, but I want to ask do they hold in general? Or is there some exceptions?
For the first question, when an additional constraint is added to a model with 2 constraints, there can be different outcomes:
1) The feasible region shrinks: This is possible if the new constraint further restricts the allowable solution space. It means that the intersection of all the constraints becomes smaller, resulting in a smaller feasible region.
2) The feasible region stays the same: This is possible if the new constraint does not significantly affect the existing feasible region. It means that the new constraint does not impose any additional restrictions beyond the existing constraints.
3) The feasible region expands: This is less likely to occur when an additional constraint is added since the new constraint usually imposes additional restrictions on the solution space. Generally, it is more likely for the feasible region to either shrink or stay the same.
Regarding the second question, when a new constraint is added, it does not directly impact the optimal value in the target cell (usually representing the objective function). However, the addition of a new constraint can indirectly affect the optimal value through changes in the feasible region.
Therefore, the possibilities for the optimal value in the target cell are the same as those for the feasible region:
1) The optimal value in the target cell can decrease if the addition of the new constraint results in a smaller feasible region and the new optimal solution yields a lower value for the objective function.
2) The optimal value in the target cell can stay the same if the new constraint does not significantly change the feasible region or if the new optimal solution has the same value for the objective function as the previous optimal solution.
3) The optimal value in the target cell can increase, but this is less likely to occur since the addition of a new constraint usually leads to a smaller feasible region and a potential decrease in the optimal value.
It's important to note that the specific outcome will depend on the nature of the problem, the constraints, and the objective function. There can be exceptions, but these possibilities generally hold true for linear programming and optimization problems.