It is capable of representing dubious, incomplete, and fragmentary information and can additionally incorporate objective, fluctuating, and data-based information in the fuzziness description. Requirements regarding special properties of the information do generally not exist. That is, for example, environmental conditions can be unknown or arbitrarily fluctuating. With respect to the regularity of information within the uncertainty, the uncertainty model fuzziness is less rigorous in comparison with probabilistic models. It specifies a lower information content and thus possesses the advantage of requiring less information for an adequate uncertainty quantification.

There is no general algorithm for fuzzification, that is, for quantifying uncertainty with the model fuzziness. In order to specify a fuzzy set on the fundamental set X it is necessary to formulate a criterion that is more or less satisfactorily fulfilled by the elements x ∈ X. This criterion may either represent an uncertain proposition or an event. In order to specify the membership function μ_A(x), all x ∈ X are gradually assessed in relation to the formulated criterion. Generally, it is appropriate to choose simple functional formulations such as linear or polygonal types for the μ_A(x). In view of engineering applications, the uncertainty is usually bounded, e.g., by physical restrictions or by reasonable limits, so that the specified fuzzy sets are also bounded, which is advantageous for their numerical processing in a fuzzy analysis with α-level optimization.

Primarily, fuzzification is a subjective assessment, which depends on the available information. In this context, four types of information are distignuished to formulate at least course guidelines for fuzzification. If the information consists of various types, different fuzzification methods may be combined.

information type I – sample of very small size

The membership function is specified on the basis of existing data comprising elements of a sample. The assessment criterion for the elements x is directly related to numerical values derived from X. An initial draft for a membership function may be generated with the aid of simple interpolation algorithms applied to the objective information, e.g., represented by a histogram. This is subsequently adapted, corrected, or modified by means of subjective aspects.

information type II: – linguistic assessment

The assessment criterion for the elements x of X may be expressed using linguistic variables and associated terms, such as "low" or "high". As numerical values are required for a fuzzy analysis, it is necessary to transform the linguistic variables to a numerical scale. By combining the terms of a linguistic variable with modifiers, such as "very" or "reasonably", a wide spectrum is available for the purpose of assessment.

information type III: – single uncertain measured value

If only a single numerical value from X is available as an uncertain result of measurement _m, the assessment criterion for the elements x may be derived from the uncertainty of the measurement, which is quantified on the assigned numerical scale.  The uncertainty of the measurement is obtained as a "gray zone" comprising more or less trustworthy values. This can be induced, e.g., by the imprecision of a measurement device or by a not clearly specifiable measuring point.

The experimenter evaluates the uncertain observation for different membership levels. For the level μ_A(x) = 1 a single measurement or a measurement interval is specified in such a way that the observation may be considered to be “as crisp as possible”. For the level of the support, μ_A(x) = 0, a measurement interval is determined that contains all possible measurements within the scope of the observation. An assessment of the uncertain measurements for intermediate levels is left up to the experimenter. The membership function is generated by interpolation or by connecting the determined points (x, μ_A(x)).



                     a) imprecision of the measuring device                                                                           b) imprecise measuring point

information type IV: – knowledge based on experience

The specification of a membership function generally requires the consideration of opinions of experts or expert groups, of experience gained from comparable problems, and of additional information where necessary. Also, possible errors in measurement, and other inaccuracies attached to the fuzzification process may be accounted for. These subjective aspects generally supplement the initial draft of a membership function. If neither reliable data nor linguistic assessments are available, fuzzification depends entirely on estimates by experts. As an example, consider a single measurement carried out under dubious conditions, which only yields some plausible value range.

In those cases, a crisp set may initially be specified as a kernel set of the fuzzy set. The boundary regions of this crisp kernel set are finally “smeared” by assigned membership values μ_A(x) < 1 to elements close to the boundary and leading the branches of μ_A(x) beyond the boundaries of the crisp kernel set monotonically to μ_A(x) = 0. By this means elements that do not belong to the crisp kernel set, but are located “in the neighborhood” of the latter, are also assessed with membership values of μ_A(x) > 0. This approach may be extended by selecting several crisp kernel sets for different membership levels (α-level sets) and by specifying the μ_A(x) in level increments.