Weighted-sum method for solving a bi-objective optimization problem

Hello all,

I would like to solve the problem that can be found attached below ('bi_objective.png') and then plot the pareto frontier solutions of it. I know that this can be done by gamultiobj , but in this case I need to export it and do it by hand. The decision vector of this problem is, let's say: x = [ y ;z], where y is the vector for the first problem and z is for the second.

After normalizing the two conditioned sub-problems, w1 and w2 should be applied (their sum equals to one) in order to form the weighted-sum problem formulation and search for the pareto frontier solutions.

Can someone help me how could this happen? If you need some further clarifications, please feel free to ask me. P.S.: the denominators and the theta variable is known to us...

Regards, Chris