you can merge the two equations in one and multiply each function by a weighting factor between 0 and 1. The important function take higher weighting factor.

e.x the important takes 0.8 and the second 0.2 (0.7 and 0.

3 0.75 0.25,..........)and try other factors till reach the best solution for your problem

Mohamed I. Mossad Thanks for your answer! i would like know what is the difference between weighted GA (like the one you explained) and the multi-objective GA (Pareto graph)?

@ Harimurugan Devarajan multi-objective functions are combined into a single objective scalar function, which is the weighted sum of the them. In the single objective (combined ) function the optimization can not differentiate between the different terms of the function but consider it as a one combined function

Hi, if you do a weighted sum of the objectives, then you will only find a single solution to the optimisation problem for each run of the optimiser; a multi-objective GA will return many results that all are (hopefully) on the Pareto front. The Pareto front is the trade-off surface of the optimisation problem, so if one objective is more important than the other, the solutions that lie more to one end of the Pareto front will be the ones that interest you.

For many optimisation problems, the weighted sum approach is fine, however there are limitations; in particular if the shape of the Pareto front is concave (i.e. bends backwards on itself). Solutions in the concavity region are still optimal choices, but there are no combinations of weights for the weighted sum that will allow you to find them. Therefore if your ideal solution lies within a Pareto concavity, the weighted sum method will never find it, and the optimiser will report a result that is from the closest non-concave Pareto section it can reach.

If you have a situation where with one set of weights you find a solution, and then changing the weights slightly keeps giving you the same solution, up to a point when the optimiser output suddenly jumps to another solution; you have a concavity in that situation and should not be using a weighted sum.

Sometimes a good approach is to use a quick run (i.e. small population or a few generations only) of a multi-objective optimiser to give you an approximate Pareto front, allowing you to see the rough shape of the front (i.e. seeing if there is a concavity in the front shape), and also then allowing you to gauge what weightings may work to give you the solutions you need. You can then do a much more detailed run with a weighted sum optimiser (not necessarily even a GA if the problem allows it) to get a high precision single optimum.

Evan Hughes Thanks a lot for your clear explanation. In my problem, first objective function is to minimize the value of X below 1, where X is a function of two variables (a, b ) for which i am finding optimal values. Second objective function is to minimization of difference between the standard values (A, B) and the variables (a, b). In other words, i need to find the optimal values of a, b such that the value of X is less than 1 and difference (A-a)+(B-b) is minimum. In my problem, making the value of X less than 1 is major objective. Currently, i am using multi-objective GA, from the available Pareto front, i am selecting the value on the Y-axis (Objective-2), i.e, selecting the value for which fval of objective 1 is zero and fval of objective 2 is maximum. Is this fval of objective-2 is minimum for which i will get fval of objective-1 zero? Thanks for your time.

Yes once you have sample points on a Pareto front, moving along it from high values of X until you find the first value that gives X below 1 will correspond to the optimal solution you are looking for. Essentially you seem to be using the multi-objective process to solve a constrained problem where X has an inequality constraint applied. Applying constraints is an area that is notoriously difficult in evolutionary algorithms and treating the constraints as objectives is often a very practical way of doing it. It is always worth considering if any of the non-EA more classical optimisation approaches may be able to solve your problem, as there are classical approaches that can handle constraints very elegantly; although they can struggle if the primary problem being optimised is very non-linear.

Evan Hughes Thank you for your response sir. I Will try some non-EA approaches too.

Two of the best choices are to use either NSGA-II or MOEA/D.

Best regards

Many thanks for the question and replies. By the way, we have two Excel-based programs for MOO (one is based on NSGA-II and another based on multi-objective differential evolution) free of cost. Both programs can handle inequality constraints (but not equality constraints) and progress-based termination criteria (besides maximum number of generations). Readers interested in either of these are welcome to contact me via email to [email protected].

Dear Harimurugan Devarajan ,

In order to formulate multi objective problem, first you need to verify the selected objectives are conflicting with each other and if its not you can formulate it as a single objective problem. Multi objective optimization algorithms are mainly used to obtain a multiple trade off solutions in a single run. Region of interest in the objective space is very important. If you are very clear about the region of interest in the Pareto front, u can easily use the weighted sum with the known weights. If you don't know that, u can come for Pareto based or decomposition based multi objective algorithms for getting many optimal trade-off solutions in a single run.