Fuzzy average

XII. THE FUZZY AVERAGE OF CRISP SETS, THE FUZZY AVERAGE OF FUZZY SETS, THE CRISP AVERAGE OF CRISP SETS,... 

In all other cases, the fuzzy average Ff, of a family F of crisp sets is a genuine fuzzy set, since the average in definition (199) is over the necessarily binary membership function values for crisp sets, and if not all of the equations in Eq. (200) hold, for example, if... 

The crisp average F av (or simply F ) of a family F of crisp or fuzzy or mixed sets F, F2,...,F is defined in terms of the crisp a-cut Dp (a) of the fuzzy average /av of the family F, where the unique a value is selected so that the mass of Dp (a) is equal that of the fuzzy average... 

Similarly, the fuzzy scaling-nesting similarity measures serve as the basis of measures of approximate symmetry, with respect to the fuzzy average. 

The fuzzy folded set is the fuzzy average of thi defined by the fuzzy membership function (x) as follows ... 

As follows from its construction, the fuzzy average of crisp or fuzzy sets A,RA,R A,...,R" A does have the corresponding fuzzy R symmetry. 

A typical fuzzy continuum of chemical importance is an electron density cloud of a molecule. The fuzzy average of crisp or fuzzy subsets Fi,p2,..., and of a set X can... 

This is easily done. Rather than averaging the values of all consequents that predict membership of the same fuzzy set, we include separately the membership values for every rule in the calculation of equation (8.6). As Figure 8.18 shows, there are now three areas to consider, so the numerator in equation (8.6) becomes ... 

The quantity pBX can be calculated from vex, c / x, where vex. x is the excluded volume between one hindering fuzzy cylinder and the laminar region of unit surface area, and is approximately equal to Le[l + fde/Le)]. Furthermore, the average length b L is proportional to Le for the longer sides of ribbon-like obstacles and to de for the shorter sides of the obstacles. 

From a structural point of view, mechanism in a single crystal can be much closer to a set of identical atomic trajectories than to the kind of fuzzy statistical average with which one must be content in solution. It is not surprising that with this kind of structural uniformity the site problems that plague kinetic studies in rigid glasses disappear. Adherence to first-order rate laws can be as close in single crystals as it is in fluids, and equally valid activation parameters can be obtained for thermal unimolecular reactions of reaction intermediates [12]. 

In such a case, a unique HEN structure with satisfactory levels in nominal or average utility consumption and operational flexibility as well as unit numbers will be obtained. A two-phase fuzzy optimization method is proposed to find a best compromised solution for the multi-criteria HEN synthesis problem, as discussed in the next section. The basic number of constraints and variables for the multi-objective MTT.P formulation are summarized in the following. 

Fuzzy complements, intersections, and unions have been characterized and studied on axiomatic grounds. Efficient procedures are now available by which various classes of functions can be generated, each of which covers the whole recognized semantic range of the respective operation. In addition, averaging operations for fuzzy sets, which have no counterparts for crisp sets, have also been investigated in this way. This rather theoretical subject, which is beyond the scope of this overview, is thoroughly covered in ref. 18. 

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