Two-objective method for crisp and fuzzy interval comparison in optimization

In real optimization we always meet two main groups of criteria: requirements of useful outcomes increasing or expenses decreasing and demands of lower uncertainty or, in other words, risk minimization. Therefore, it seems advisable to formulate optimization problem under conditions of uncertainty, at least, two-objective on the basis of local criteria of outcomes increasing or expenses reduction and risk minimization. Generally, risk may be treated as the uncertainty of obtained result. In the considered situation, the degree of risk (uncertainty) may be defined in a natural way through the width of final interval objective function at the point of optimum achieved. To solve the given problem, the two-objective interval comparison technique has been developed taking into account the probability of supremacy of one interval over the other one and relation of compared widths of intervals. To illustrate the efficiency of the proposed method, the simple examples of minimization of interval double-extreme discontinuous cost function and fuzzy extension of Rosenbrockʹs test function are presented.