I am going to first layout the problem, then I'll get to the thing that is troubling me. I am enrolled in a course called "Optimization I", and this exercise is from a chapter called "Sensitivity analysis".
Here is the problem/exercise:
Assume that the right-hand-side is changed in the following constraints. State if these changes are a relaxation or a restriction to the problem, if the optimal objective function value will increase or decrease, and finally if the change will be larger or smaller if the right-hand-side is changed further. Assume that the problem is an LP problem, all variables are non-negative, and that the constraint is fulfilled.
a) maximization problem, 8x+8y>=12, increase the right-hand-side
b) minimization problem, 7x-3y<=19, decrease the right-hand-side
c) minimization problem, 4x-3y>=77, decrease the right-hand-side
d) maximization problem, 6x+4y<=12, increase the right-hand-side
e) maximization problem, 3x+4y>=2, decrease the right-hand-side
a) this is clearly a restriction, and since it is a maximization problem, the optimal objective function value will decrease
b) this is also a restriction, but since it is a minimization problem, the optimal objective function value will increase
c) on the other hand this is a relaxation, and since it is a minimization problem, the optimal objective function value will decrease
d) this is a relaxation as well, but since it is a maximization problem the optimal objective function value will increase as opposed to (c) where the it is a minimization problem.
e) also a relaxation, the optimal objective function value will increase for the same reason as in (d)
But, I am having trouble understanding what they mean by "will the change be larger or smaller if the right-hand-side is changed further". I have read the entire chapter a few times this past two days but I still don't know what they mean by that or even how to solve it.
And these are the correct answers (from solution manual) from (a) through (e)
a) larger
b) larger
c) smaller
d) smaller
e) smaller
The only thing I could do (even though I don't understand what they mean by larger or smaller change) was try to see a pattern, by comparing all five and my conclusion is that if you have a restriction then it is a larger change, and if it is a relaxation then you get a smaller change, but unfortunately this conclusion (even if it is "correct") doesn't help me with the next exercise that comes after this one, since you can't tell if you have a restriction or a relaxation. Now, can someone please explain this to me and how I am supposed to solve this problem? Thank you!
Edit:
After some thinking, this was my conclusion; If the change of RHS, leads to a "worse" objective function value, then the change is larger, and if the change leads to a "better/improved" objective function value, then the change is smaller. (a) and (b) were worse, thus larger change, while the rest were better, and therefore smaller change. I asked the professor and he said that is indeed the case, if you use the definition. Another way to think about, he said, was with convexity (just like @borisd stated below). But I think I'll use the definition, since the teacher himself does that, as it is faster because you can check if the change is a restriction/relaxation, if the optimal objective function value will increase or decrease, then after determining those two factors it is quite easy.
But if we're changing the objective function coefficients and not the constraints like above, then it's the "opposite". If the change leads to an improvement then the change is larger, while if the change leads to worsening/deterioration then the change is smaller.
Best Answer
I think that "the change will be larger or smaller if the right-hand-side is changed further" refers to the objective function difference when the change in RHS is more important.
More precisely, assume that when you change your RHS from $b$ to $b+\delta$, and that the objective function increases by $\Delta$. When you change your RHS from $b$ to $b+\delta'$ with $\delta'>\delta$, the objective function increases by $\Delta'$. The question is whether $\frac{\Delta}{\delta} < \frac{\Delta'}{\delta'}$? This may be easier to see what it means by taking $\delta=1$ and $\delta'=2$.
You can get the correct answer by recalling that the optimum value of an LP in minimization with $\ge$ constraints is a convex function of its RHS. Thus, the more the RHS increases, the more the marginal cost increase is important.