Do you have to normalize objectives when using the weighted sum approch?

Question

Do you have to normalize objectives when using the weighted sum approch when having multiple objectives?

Actually I thought that I should do it. But now I have run several experiments with different optimization models and the results reveal that there is no benefit when scaling the different objectives (by using the optimal value obtained from solving the single-objective problem). In contrary, I would even say that in my experiments, scaling the objective functions when using a weighted sum approach makes the results even worse, even when having objectives with different magnitudes.

Answer 1

It depends on what you want to achieve, but I would like to argue, contrary to the answer by @Merve Özer, that you do not need to (explicitly) normalize the objectives.

If you have, two objective functions $f_1$ and $f_2$ and you want to bring $f_2$ into the same magnitude as $f_1$, you would find a constant, say $\alpha$, and multiply $f_2$ by this constant and obtain a new normalized objective $\tilde{f}_2(x)=\alpha f_2(x)$. Next you would create a weighted sum of $f_1$ and $\tilde{f}_2$, with weights $\tilde{w}_1,\tilde{w}_2>0$, and optimize this: \begin{equation} \min \tilde{w}_1f_1(x)+\tilde{w}_2\tilde{f}_2(x) \end{equation} This is simply the same as optimizing the weighted sum of the original objective functions with weights $w_1=\tilde{w}_1$ and $w_2=\alpha \tilde{w}_2$. Hence, you should "just" find good weights for the original objectives.

A further thing to observe is that some times it is advised to scale both objectives such that the set of non-dominated outcome vectors are contained in a (unit) hypercube. This will in many cases lead to numerical problems if larger values are "squeezed" into a small box.

This answer relies on two assumptions:

1. You want a Pareto optimal solution to your multi objective optimisation problem, which is (only) guaranteed to be a supported efficient solution.

2. It is often very difficult, even after normalizing the objectives, to predict the characteristics of the resulting solution obtained after optimising the weighted sum scalarisation. For some problems, even small changes to the weights will lead to very different solutions. For others, large changes does not change the solution much.

Answer 2

Yes, you need to normalize. It is not about weighting. If you don't, one of them dominates the others. The reason that you got worse result may be because you use weight coefficients improperly or the problem arises from something else. Generally, when I use weight coefficients, their sum is equal to 1.