# Optimizing multiple objectives with different scales

I have multiple objectives, such as $$f(\mathbf{x})$$, $$g(\mathbf{x})$$, and $$h(\mathbf{x})$$.

I would like to find a set of $$x$$ that can $$\underset{\mathbf{x}}{argmin} [ f(\mathbf{x}) + g(\mathbf{x}) + h(\mathbf{x}) ]$$.

However, the three objectives have quite different scales. For instance the output of $$f(\mathbf{x})$$ is normally in [1000, 5000], while that of $$g(\mathbf{x})$$ is in [0, 1].

Does this negatively impact the oprimization? To compute the $$argmin$$, should I normalize them to an identical scale?

• It depends on what the various objectives mean. When there's no meaningful interpretation, it's common to add tunable parameters as coefficients: $f(\mathbf{x}) + \lambda g(\mathbf{x}) + \gamma h(\mathbf{x})$. By changing $\lambda$ and $\gamma$, you change the relative importance of $f$, $g$, and $h$ in the overall optimization. – tddevlin Apr 11 at 17:58

## 1 Answer

As mentioned in a comment by @tddevlin, it depends on what those objectives are. I give you an example. Suppose I'm solving a vehicle routing problem and I want to minimize both cost and number of vehicles. Then, my $$f(x)$$ is cost and $$g(x)$$ is the number of trucks, which both fit the profile that you mentioned. Now, if I prefer costs over the number of trucks, I don't even need to add any weight to my $$f$$ and $$g$$ functions. In fact, I can use them exactly as is and know that because of the small scale of $$g$$ compared to $$f$$, always $$f$$ will be preferred first, which is the desired behavior. That's why it matters what are your objectives.