The multi-object optimization problem is the most sever problems in the optimization area. The severity comes from how to find the proper weights for the parts of the composite objective function. Considering the weights as extra variables complicates the problem. Evaluation of the local minima is possible under precautions of using several trials for the weight values and the initial values of the variables and how you change the variables.
Prof. S. EL-Rabaie
multiobject optimization requires you now a parto point , so you can convert the simple particle swarm optimization algorithms to deal withm mutltiobject like that.
1- First convert the two object function F1 and F2 to single obtimization problem called F
F=a*F1 + b*F2
2 -- (a+b=1) and thy called weights. take intial value for a and linear decreasing until reseached agiven iteration
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There are many methods to find the best answers for multi-objective models. The answer of these kind of problems will be a set of solutions. You can find the sets using a Pareto diagram. I suggest using NSGA algorithm for your model.
If you read the chapter given in the attachment, you have a idea which multi objective func. is proper for you
For the MOOP, finding Pareto optimal front is the best way. For this NSGA II by Kalyanmoy Deb is one of the best metaheuristic. One should read the MOO book authored by him.
The multi-objective problem can be converted into a single-objective by searching for one Pareto-optimal solution at a time. The inconvenient is the computational time and the hope to find different solutions each time.
Most of Multi Objective problem are formulated in Pareto format. there are various ways to solve them based on heuristic or meta-heuristic algorithms like: ant colony, PSO, cultural algorithms
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