Fuzzy Optimality and Evolutionary Multiobjective Optimization

Pareto optimality is someway ineffective for optimization problems with several (more than three) objectives. In fact the Pareto optimal set tends to become a wide portion of the whole design domain search space with the increasing of the numbers of objectives. Consequently, little or no help is given to the human decision maker. Here we use fuzzy logic to give two new definitions of optimality that extend the notion of Pareto optimality. Our aim is to identify, inside the set of Pareto optimal solutions, different “degrees of optimality” such that only a few solutions have the highest degree of optimality; even in problems with a big number of objectives. Then we demonstrate (on simple analytical test cases) the coherence of these definitions and their reduction to Pareto optimality in some special subcases. At last we introduce a first extension of (1+1)ES mutation operator able to approximate the set of solutions with a given degree of optimality, and test it on analytical test cases.