Alternative Processing of Verbal Data
by Milan Mare
The vague quantitative values which we call here verbal data are usually processed by means of the theory of fuzzy numbers or fuzzy quantities. This theory is well elaborated and the algebraical properties of fuzzy numbers are well known. It is worth mentioning that these properties are not completely identical with the properties of deterministic numbers ñ but they are well known and analytically managed.
The attempts to apply the theory of fuzzy quantities to the formal analysis of practical situations like, eg, the CPM (critical path method) optimization techniques show another serious discrepancy connected with this, today already classical, processing of fuzzy quantities based on so called extension principle. It is used for the summation and multiplication of fuzzy quantities and it has a very logical formal structure consisting in re-combination of possible values and their possibilities of the input fuzzy quantities. Anyhow, the extension principle enlarges the range of possible values (namely in the case of summation) in a significant measure and, namely if it is applied repeatedly in processing a large number of fuzzy quantities, the extent of possible values of the resulting fuzzy quantity becomes enormously (and unacceptable) large.
The everyday experience with practical manipulation with vague data (and many traditional techniques do manipulate with them without admitting that they process fuzzy quantities let us remember quantitative units like pinch, handful, step, two-days marching distance, proportional weight, acceptable profit, etc.) shows that the results of it are much less dispersed and their realistic values are concentrated near some periodic achieved expected values. As such practical vagueness is usually connected with verbal description of quantitative values, we use the term verbal data for their characterization. The above practical experience shows that there exists a contradiction between the fuzzy set theoretical model and real state regarding the processing of such verbal data. Its roots may be quite deep and worth for discussion, which was already done. Let us mention, at least briefly, its essence.
It appears to be the crucial point that the operations based on the extension principle do not reflect the specific semantic structure of the fuzzy quantities modelling the verbal data. Namely, these fuzzy quantities are, in fact, combinations of at least two components. Verbal datum like, eg, approximately 8 consists of a crisp core ñ its proper quantitative value (in our case 8) and a semantic shape mathematically describing the structure of vagueness connected with the verbal expression (in our example with approximately). Meanwhile the crisp cores are real numbers in the deterministic sense and they can be processed by means of classical deterministic operations, the shapes are mathematically real-valued functions mapping the real line into the unit interval with modal value 1 achieved for the argument 0. They can be interpreted as some kind of normalized membership functions of the fuzzy quantities modelling the verbal data. As the shapes are connected with the semantic component of the fuzzy datum, they would be rather processed by fuzzy logical methods using operators like maximum, minimum, or product, etc., even if the formula of the extension principle is also one of the applicable methods.
Each algebraical operation realized over such verbal data is to be decomposed, then, into operation with crisp cores leading to a deterministic real value, and logical or semantical operation over shapes leading to a resulting shape. This output crisp core and shape define the fuzzy quantity which reflects the result verbal datum. Due to the applied operation over shapes the possible values of the result are more or less closely concentrated near their crisp core (it is worth mentioning that usually much more closely than if the extension principle is applied).
The above method can be very easily completed also for the manipulation with verbal data having anonymous crisp core (like several, few, etc.) or combined structure of relation between vague and crisp component (eg something between 20 and 25, a bit more than 10, etc.). In all these cases it offers the possibility to reflect the logical relation between the processed verbal data much better than the traditional approaches limited to the extension principle.
Milan Mare CRCIM
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