3.3 Fuzzy convolution scheme

Zadeh [14] put the theory of fuzzy sets forward with explicit reference to the vagueness of natural language, when describing quantitative or qualitative goals of the system. Here we assume that local criteria as well as different constraints in the ill-structured situation can be represented by fuzzy sets. A final decision is defined by the Bellman and Zadeh model [17] as the intersection of all fuzzy criteria and constraints and is represented by its membership function μ(Х) as follows:

$$
\mu(X) = \mu\_K(X)\,\mu\_C(X), \quad \mathbf{X} \in \mathbf{X}\_P. \tag{12}
$$

This problem is reduced to the standard nonlinear programming problems: to find the values Х and λ that maximize λ that is subject to

$$\begin{aligned} \lambda \le \mu\_{\text{Ki}}(X), \quad i = 1, 2, \dots, n; \\\lambda \le \mu\_{\text{Cj}}(X), \quad j = 1, 2, \dots, m \end{aligned} \tag{13}$$

Here, by way of illustration, the new approach was introduced to estimate the Redlich-Kwong EoS parameters for simultaneous description of the phase equilibria and critical line data in binary mixtures, thermodynamically consistent description of the inhomogeneous data, and other inconsistency problems arising in the modeling of phase equilibria.
