Usually, ppl are interested to convert a multi-objective optimization problem in to a single objective optimization problem. However, working the other way around is not impossible. It can be done based on the property of the problem, for example if your objective function f(x1,x2,...,xk), for vectors xi possibly different dimensions, is separable as f=g1(x1)+g2(x2)+...+gk(xk) then you can have an optimization problem with k objective functions of gi(xi).

So one can deal with the objective function at hand in different ways.

Assume you have feasible domain X, feasible solution x0, sequence of objective functions f1(x), f2(x),...,fn(x) (F(x)=(f1(x), f2(x),...,fn(x)) , single objective algorithm A. To simplify the explanations, consider maximum. Use A to solve max{ f1(x), X1},

where X1={x from X: F(x)>=F(x0)}.

Find Arg max{x from X1 f1(x), X}=x1.

Consider Domain X2={x from X1: F(x)>=F(x1)}.

Again, using A, solve max{ f2(x), X2} with Arg max{ f2(x), X2}=x2,

where X2={x from X1: F(x)>=F(x1)}.

And so on...

The last in the sequence Arg max{ fn(x), Xn}=xn Is the efficient solution. If there is a problem max{ fk(x), Xk}, k

The use of weights can adapt a single objective algorithm for a multi objective problem. However this strategy is poor to adjust a pareto front.

There are robust approaches to fit pareto front, like SPEA, NSGA or PESA.

Using pre-defined set of weighs to form a single objective function, a number of optimal solutions can be found. Sequential quadratic programming, a gradient based solver can be used for subsequent minimization or maximization of the composite objective function. The choice of designer is important in weight selection !

As suggested by Bruno Prata, a better option would be to find the Pareto-set.

If your question is only about the modelling of a problem the previous answers are ok. But, I think your question is about the proposal of a new multi-objetive metaheuristic by using a previous single objective version. I think the easiest way to achieve this task is to add an external archive and put solutions on it based on dominance (or an weighted function as well). The choice of candidate solutions to the archive should be motivated by the original operators from the single objective version. This is a very simple approach, but it could be your first attempt in order to get a MOO version from a SO one.

Sorry, and apologies in case my answer may sound a bit too tough :

The idea "to convert a single objective optimization algorithm to a multi-objective one" only shows a lack of understanding of the essence of the multi-objective optimization. Indeed, it is just about as off the point, as trying, for instance, to convert in general partial differential equations into ordinary differential equations.

Of course, one can from the start simplify grossly and brutally the multi-objective nature of the given problem, and reduce it to one single objective.

However, the real and truly valuable practical issue is to avoid that, and instead, to keep the multi-objective situation alive all the time, that is, throughout the whole process of solving of the optimization problem.

Now, since the late 1970s, it is known that in a multi-objective context, the role of preference type information is fast diminishing with the increase in the number of objectives. Therefore, instead of preference type information, one is obliged to use other information, such as for instance, indifference information, that is, to what extent one is indifferent between two possible outcomes.

Details in this regard can be found, for instance, at arxiv:math/0506619

A multi-objective optimization problem will be converted into a single-objective one by using a fuzzy system. To achieve this purpose, at first, the value of each objective function, which is considered as an input in the (fuzzy inference system) FIS, is divided into several regions using fuzzy membership functions. Next, relations between the inputs and the final output, i.e. the final objective function that wants to optimize it, are made through the appropriate rules!

Sorry, but of course, there are any number of methods to ELIMINATE UP FRONT the multiplicity of objectives, and reduce from the start all of them to one single objective.

However, those who have significant PRACTICAL EXPERIENCE with genuine multi-objective optimization problems DO know very well how inadequate such approaches are in fact.

Yes, by far the MAIN issue in multi-objective optimization is NOT to eliminate the many objectives, and instead, to deal with them ALL the time, until the very last moment the solution is obtained.

What is not realized is that in the multi-objective context, the very CONCEPT OF SOLUTION IS NOT UNIQUE : there is indeed NO unique, natural, canonical concept of solution, and the choice of the concept of solution which one does eventually use may in fact be itself a multi-objective optimization.

Thus in fact, one must solve TWO multi-objective problems : first, to chose a concept of solution, and second, to implement that chosen concept.

Otherwise, one merely keeps publishing papers, an none of them would be accepted for application by any more sophisticated executive in charge of a real practical application ...

usually we want to find a global (single) objective function to either minimize of maximize. In the case of a multidisciplinary optimization, a certain trade off is made depending on the algorithm, but to work the other way around, to propose a multi-disciplinary problem from a single objective optimization problem is logically impossible. I do not understand the optimization problem you are trying to solve !!

If I understand your need, you want to cast of a single-objective optimization problem as a multiobjective one, for this, you need a transformation that can be achieved by the addition of supplementary objectives or by the decomposition of the original objective function.

the following links will be very useful

http://sys.ci.ritsumei.ac.jp/~sin/Paper/file/cec_camera_242_watanabe.pdf

https://www.macs.hw.ac.uk/~dwcorne/rlo.pdf

http://dbkgroup.org/handl/ppsn_decomp.pdf

Have a look on this paper:

Multi objective optimization algorithm based on sperm Fertilization procedure (MOSFP)

If the objective function has conflictic terms, you can separate them and used as different objective functions