We use it to justify the weights.

Of course multi-objective optimization (MOO) is more valuable in terms of information you can collect, but also more expensive than single objective optimization. Yet, once you have found a wide set of solutions belonging to the Pareto front, what to do with them? If, for instance, I am trying to optimize the functioning of some component I need to choose one among the many Pareto configurations and apply it. MCDM techniques respond to this question.

This does not mean MOO is useless and I would have rather chosen some single objective weighted based objective function. The insight you get from a MOO is still valuable, and most of all, if in the future I decide different "weights" between the objectives could be better for my component I don't need to run any optimization again.

There are many MCDM methods, weight-based methods are just a portion of them. I am not an expert in MCDM, yet I believe that non weight-based methods are generally more appropriate.

Last but not least, if possible, try not to decide a priori the weights and run a single objective optimization: you never know where you will end up. Choosing a posteriori you have already gained knowledge on your problem from the results of the optimization and have the chance to choose more wisely. After that, if you have time and resources, try to run a single objective optimization starting from the chosen configuration, in this way you could refine your search quickly in an area where otherwise a stochastic algorithm would converge very slowly.

Thank you Marco for your comprehensive response. It really helped me to justify this practice by researchers. I would summarize your comments as follows:

1) Actually, my concern was that the solution by MOP then MCDM exactly equals to MCDM then single objective run. However, it is not true for every cases. 

2) All MCDM methods are not weighted based.

3) Pareto set gives more valuable information about solutions.

4) Single-objective optimization may lead a solution which is a dominated solution according to the proposed Pareto set at hand.

5) By changing the weights by different decision makers (DMs), there is no need for solving several single objective optimization problems that are time consuming compared with MCDM methods.

Thank you Marco again!

Dear Behrang,

In my opinion, MOP finds only ranking of sets. Afterwards, MCDM methods are used to find rankings in each set. Both approaches involve the same criteria, but the main different is that MCDM methods take into account weights of criteria. 

Let's considere a simple example to explain it.

We have a group of students Si to assessment, where each student has two grades Si(Gi1, Gi2) (from 0 to 100). There we have two criteria G1 and G2. Our task is, please find the ranking of students from the best to the worst.

S1(45, 55); S2(50, 50); S3(95, 60); S4(70, 70)

If we use MPO, we can say that S3 and S4 is better than S1 and S2. We cannot obtain a relation between S3 and S4 (as in the case of S1 and S2). We find only ranking of sets. Therefore, we must use MCDM method to obtain ranking of alternatives. I hope that simple example explain difference between MCDM and MPO. Any comments are welcome.

Best Regards WS

Dear Behrang and Marco,

I was writing my comment and you added your comments in the meantime. I have one question to you. Could you give an example of MCDM method, which don't base on criteria weights?

Best Regards WS

Thank you Wojciech for your kind cooperation.

Dear Behrang,

I've just read with interest your questions/comments, and

comments/discussions added by colleagues.

I discuss and explain my students such questions/directions within MCDA course at my Univ each year/semester.

The field of using MCDA(M), including MODM- Multi-Objective Dec.Making - many/or infinite number of alternatives which are given, as a rule, imlicitly, and MADM (finite number of explicitly given alternatives)...are very wide, and the best way is to consider  several case studies   with...

These directions and their sprcific algorithms/merhods are supplemental, as were written above by WS and others.

We shouldn't try to substitute MODM for MADM..., and, certainly, vise versa.

U don't  find soft for MODM, where U can implememt effective weight- and value function- sensitivity analysis, etc (eg, see DecernsMCDA). 

And...we should not afraid subjectivity- weighting znd using Value Functions.... There is no Objective Approach to solving MC problems!-). Y, analyses say, AHP is the most popular method (:= no explicitly assigned weights and Value Functions). Y, they avoid using weights, VF, etc...They think, ratio scale in AHP is optimal. Y, they may use all the methods... There is no the Best MCDA meth.

I must confess, analyzing each year preferences of my students and colleagues, they don't think (70-90%, depending on the speciality), that AHP  is the most suitable/valuable/preferred method .. as they may use MANY METHODS (using Decerns s

ystem) and compare AHP (for them, in fact, a kind of black box, though without explicitly setted weights and VF) with other method, taking into account input info and possibility for extended analysis and presentation of the results for a transparent discussions..

Thus, I say my students, there is a wrong approach to find very eassy way to solve mcda problems without subjectivity and aporoaches to "weight" criteria and value of criterion ranges (which we use for weighting and assessing dufference of criterion values via diiferent meth, including MAVT/MAUT, PROMETHEE, and so on).

So let's use MODM abd MADM and different their meth and dont afraid weighting 

=Borisz

Dear: Bootaki

Maybe you interest with this paper (attached)

*Hengky

Thank you Mr Hengky.

My pleasure Mr Bootaki

Dear Behrang Bootaki

Criteria and objective are completely different. Criteria used for decision making and objective used for optimization. 

Use Linear Programming (LP) where you do not need to find any weights. There are some MOP applying LP such as Goal Programming.

If you are interested in multi objective MCDM models you can use Simus, which does not provide an optimal solution but a compromise solution as AHP and TOPSIS.do

In interested in Simus I can send you infotrmatyion as well as the software

MOO gives non dominated solutions like NSGA II algorithm. They are called pareto optimal solutions. If say for example we get 50 non dominated pareto solutions for a MOO problem. MCDM methods like TOPSIS help to choose the best compromise solution (BCS) among those 50 pareto solutions.

This is exactly the optimization case I use myself. First I find Pareto front by SPEA (multi-objective evolutionary algorithm), then apply TOPSIS or simple weighting factor as MCDM to sort the set according to user's preferences (weights). Why to do that?

Because:

1. I want to find a true (or approximated) Pareto front (which might be very much a time consuming process ...)

2. I do not want to a priori set the weights and convert the problem into single objective, because time consumption is not that lesser and the results are not general any more

3. I want to give user a chance to make up his mind on weights when the most time consuming part is ready. If the user changes his/her mind on weight values then I can re-apply MCDM very fast on a previously found Pareto set

Thank you Joanna for your comprehensive response.

can someone tell me how to incorporate the Topsis solution with a Multi-objective optimization problem?

It is the other way around. You solve your multi-objective problem first with some method e.g. multi-objective evolutionary algorithm. As a result you obtain a set of Pareto-optimal solutions. The set possibly can be numerous and quite often you might be interested in a single ("the best") solution. Thus, you can

1. apply a MCDM method, eg. TOPSIS to select one "the best" solution (according to Decision Maker's preferences given by weights) from the Pareto-optimal set.

2. another approach is to narrow the optimization exploration via including preferences already during multi-objective optimization.

An example of how to apply the approach 2. is provided here: Article Preference-based evolutionary multi-objective optimization i...

That is an interesting question. I would say the MOOP will give you a better tool to obtain different solution first and then using MCDM method you will select one out of them. Thought, weighted average optimization or having weights at the beginning, wont allow you to search in the domain and find other alternative solutions.