Synthetic realization approach to fuzzy global optimization via gamma algorithm

A new approach is proposed for global optimization problems with fuzzy cost functions and fuzzy box and equality constraints. It allows one to avoid complex operations with fuzzy sets and the use of various subjective indices of choice. To resolve the contradiction between economically better solutions with low possibility of realization and a little poorer solution with higher possibility of realization, the synthetic realization is defined as certain fixed α-level cut for all membership functions. Consideration of such realizations guarantees a level of credibility not less than given α ∈ (0, 1] for all globally optimal solutions. Then, so defined α-cuts are rectified to cut off realizations with possibility less than α and to retain higher possibility realizations which are assigned credibility μ = 1 for the whole interval of possible realizations. This construction results in a set-valued band of credibility not less than α for a given fuzzy cost function f̃(x) which band has crisp Lipschitz continuous lower- and upper-value functions f*(x), f*(x) such that f*(x) ≤ f̃(x) ≤ f*(x) for all x ∈ X̃ ⊂ Rn. Then, the gamma algorithm is applied to obtain the interval global optimal solution f̄0(x) = [f0*(x), f*0(x)]. To further simplify the computations, the fuzziness in the feasible set X̃ is transferred to the function value space transforming X̃ into the crisp unit cube in Rn+ common for all fuzzy optimization problems in Rn with box and equality constraints.