Use multi-objective versions of NSGA-II or SPEA-II. You can download source code for many algorithms from following link:

http://delta.cs.cinvestav.mx/~ccoello/EMOO/EMOOsoftware.html

There is no best method to solve multi-objective optimization problems (MOP), all depends on the problem if it is multimodal, or if it has continuous variables, the shape of the front, etc.

You have to test which algorithm gets the best results for your problem. Also, there are some frameworks as jMetal which includes diverse multiobjetive algorithms (NSGA-II, MOEA/D, SPEA, PAES, etc.).

There is a list of other frameworks.  

http://jmetal.sourceforge.net

http://moeaframework.org

http://opt4j.sourceforge.net [Includes a list of more frameworks]

Just try some of them and choose whichever you feel more comfortable.

I recommend you to start with the simple/easy algorithms as MOEA/DE, NSGA-II, and Multi-objective simulated annealing. This algorithms are simple to implement by your own.

Hello

Sorry for the delay

The best method is simulated annealing

MOEA/D and SPGA II.

The links below are some papers about solving multi objective problems by using population based algorithms.

http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=5586458&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D5586458

https://ore.exeter.ac.uk/repository/handle/10871/11690

http://link.springer.com/article/10.1007/s11721-008-0022-4

genetic algorithms

Two_Arch2: An Improved Two-Archive Algorithm for Many-Objective Optimization,

NSGA-III

Perhaps I'm missing something but I considered the following reformulation (based on normalization of the objectives and maximization of their minima)

max z

subject to

1-(0.10/x1+0.64/x2+2.63/x1x2)/3.27>=z

1-(1.26/x1+1.11/x2+5.84/x1x2)/8.21>=z

1-(0.37/x1+0.81/x2+1.62/x1x2)/2.8>=z

(20000-(33*x1+10*x2+20*x1*x2))/(20000-63)>=z

x1, x2  - integer,

x1>=1, x2>=1 to avoid dision by zero

The solution is x1=6, x2=9

The optimal values of the initial objective functions are

0.13462963, 0.441481481, 0.181666667, 1368

 respectively.

Will this help in anything?

In case if the multiple objectives are strongly conflicting, then only we can search for MOEA like, NSGA, NSPSO and more existing in the literature, otherwise convert these multiple objectives in to single objective using weighted summation approach. 

You can try MILP (Mixed Integer Linear Programming). It is simple and useful for your case.

And MILP is just what I presented in my previous post.

Hello,

In my own research, I found that using NSGA-III or PESA-II work well with 3 objectives, and NSGA-III and SPEA2 work well with 4 objectives.

As for a toolbox, the MOEA Framework is fairly robust and has many of these algorithms already implemented. It also hooks into another framework called jMetal which makes even more algorithms available. If you have experience using Java, I would recommend these. Unfortunately I don't have experience using any non-java frameworks, so I can't provide any suggestions on that front. Best of luck!

Regards,

Ben

Dear Cassie

TOMLAB Package offer some methods to solve the multiobjective optimization problems

for more details please see the following link:

http://tomopt.com/tomlab/

You can use hybrid algorithm based on global methods such as MOGA and localal algorithm such as search method in all direction.

Dear Cassie Suh,

To my best knowledge, multi-objective cuckoo search is a powerful computational tool to do so because of it's efficient exploration capability.

Article Multiobjective cuckoo search for design optimization. Comput...

conic scalarization method can be used for these problems. This method can be applied  for  convex and concove functions

MOEA/D may be a good choice.

http://www.cs.bham.ac.uk/~likw/publications.html