There are many MCDM techniques such as TOPSIS, ELECTRE, AHP, BWM and etc to use for your problem. You need to check which one is the most fitted to your problem. The only problem with techniques like TOPSIS, ELECTRE, AHP, BWM and etc is they usually provide a ranking for the identified alternative and they suffer from providing independent index for each alternative. For example, AHP/FAHP provides relative importance weight of each criteria/alternative based on pairwise comparison . TOPSIS provides a closeness index based on distance of each alternative from negative and positive ideal solutions. These values are relative/comparative and don't show the exact/independent alternative score/index. It means calculation of the final score for each alternative depends on the behaviour of the other alternatives.
Fuzzy Inference System doesn't suffer from this issue.
Again, it all depends on your problem charechtristics.
You are absolutely correct. All methods have pros and cons, no exception, however, perhaps one of the most important PROS is ther ability to model a problem with as reasonable resemblance or, as you say, the problem description