# Multi objective optimization into single objective.

I read that it is possible to convert a multi-objective optimization problem into single objective by using weighted sum method. I wanted to know if it is a good idea to convert a two objective optimization problem into single objective by multiplication. for example:

Goal: minimize $f_1(x)$, maximize $f_2(x)$. So is it a good idea to calculate $f_2(x)/f_1(x)$, and use it as an optimal value.

PS: Any related link to academic literature would be helpful as well. Thanks.

• First thing that comes to my mind is that multiplying or dividing the objective functions you may end up with non-linear objective. Is that relevant to you to have a linear objective function? – Libra Oct 2 '13 at 6:11
• No. I don't mind having a linear or a non-linear objective function . – A. K. Oct 2 '13 at 11:57
• You may have a look at this: theory.stanford.edu/~megiddo/pdf/rational.pdf – Libra Oct 2 '13 at 12:49

## 1 Answer

In a multi-objective optimization, the objectives to be optimized are conflict. For the convenience of the description, supposing all the objectives are to be minimized, because the maximizing problem can be transformed to the minimizing problems by multiplying $-1$.

The "conflict" means that this is no single solution can simultaneously satisfy all objectives, but a set of solutions. These solutions form the Pareto-front in the objective space, and also these solutions are called Pareto-optimal solutions, and form the Pareto Set in the decision space. In addition, these solutions should evenly distribute on the Pareto-front.

In solving multi-objective optimization, our goal is to obtain these Pareto-optima solutions. Weight-sum method can not result in the optimal solutions that evenly distribute on the Pareto-front, therefore, this method cannot be used in this regard.

Typically, there are few good algorithms that convert a multi-objective optimization problem to several single-objective optimization problems. such as MOEA/D. In addition, NSGA-II performs very well in solving multi-objective optimization.