# How can I formulate this multi-objective optimization problem?

Now, for each system $$X$$ $$(X=A,B,C,E)$$, my objective is $$\max\min\frac{s_{x_u}}{d_{x_u}}$$ here, $$x=a$$ for system A, $$x=b$$ for system B and follows...

and for the whole system, my objective is $$\max\min\frac{s_{X}}{d_{X}}$$

Here, $$s_{x_u}$$ is a function of optimization variable $$a$$, i.e.,$$s_{x_u}=f(a),$$ while $$s_{X}$$ is a function of optimization variable $$z$$ and of course $$s_{x_u}$$, i.e., $$s_{X}=f(z,s_{x_u}).$$

How can I formulate this optimization problem?

I mean how to have a single objective function.

## Ideas

For each system, I can rewrite the objective as $$\max t_x$$ by adding the following constraint $$s_{x_u}\ge t_x d_{x_u}$$. So, I now have $$\max t_a$$, $$\max t_b$$, $$\max t_c$$ and $$\max t_e$$.

Likewise, for the overall system I can reformulate it as $$\max t_X$$ and adding the following constraints $$s_X\ge t_X d_X$$, $$X=\{A,B,C,E\}$$.

So, now the objective function I have is

$$\max t_a\hspace{1mm} \& \max t_b \hspace{1mm} \& \max t_c \hspace{1mm} \& \max t_e \hspace{1mm} \& \max t_X.$$

Is it okay to write $$\max t_a+t_b+t_c+t_e+t_X$$?

• It’s not clear to me what you mean by “how to have a single objective function.” Do you mean you want to get rid of the max-min so you have only a max or a min? Or do you mean how to deal with the fact that you have an objective for each $x_u$ as well as for $s_X$? Or something else? – LarrySnyder610 Sep 1 '19 at 3:27
• @LarrySnyder610, Please see my edit. As you see I have multiple objective functions (four for four systems and one for the whole system). How can I have a single objective function? – dipak narayanan Sep 1 '19 at 13:33
• @LarrySnyder610, I mean 'how to deal with the fact that you have an objective for each xu as well as for sX' – dipak narayanan Sep 3 '19 at 7:56
• Are you familiar with the basics of multi-objective optimization (MOO)? If not, then that is the direction you should explore. And if you are already familiar with MOO but the standard setup for MOO doesn't do what you need it to do, please explain why not. As it stands, I can't really see any fundamental difference between what you're trying to do and what MOO usually does. – LarrySnyder610 Sep 4 '19 at 2:51