# How to normalize the objective functions of multi-objective optimization into uniform form?

In my bi-objective model, the range of solution value for the first objective is large than the second objective. I decide to obtain a single solution by the weighted sum approach and solve it using metaheuristics. How can I do that?

## 1 Answer

You should scale the objective functions so that they have similar size values and then weight them. (The scaling factors effectively become part of the weights.) If your metaheuristics are fast, it would be wise to try different weightings and get a few Pareto efficient solutions.

There is another approach which is similar but might produce a different solution. It involves first optimizing each objective separately, again scaling them as needed to make their magnitudes comparable. If $$z_1^*$$ and $$z_2^*$$ are the optimal values of the two scaled objectives, we refer to the point $$z^* = (z_1^*, z_2^*)\in \mathbb{R}^2$$ as the "utopia point". The objective function for the metaheuristics is then to minimize the norm of the distance of the scaled objective values $$(z_1, z_2)$$ from the utopia point, using your favorite norm (1-norm, 2-norm, infinity-norm).