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borrowed from communication theory). The encoding activities occur at the transmitter while the decoding take place at the receiver. Figure 14.12 illustrates encoding and decoding with the use of the codebook A. The channel functions as follows. Any input information, whatever its nature, is encoded (represented) in terms of the elements of the codebook. In this internal format, encoded information is sent with or without processing across the channel. Using the same codebook, the output message is decoded at the receiver.

Figure 14.12. Fuzzy-communication channel with fuzzy encoding and decoding.

Fuzzy-set literature has traditionally used the terms fuzzification and defuzzification to denote encoding and decoding, respectively. These are, unfortunately, quite misleading and meaningless terms because they mask the very nature of the processing that takes place in fuzzy reasoning. They neither address any design criteria nor introduce any measures aimed at characterizing the quality of encoding and decoding information completed by the fuzzy channel.

The next two sections are examples of the application of fuzzy logic and fuzzy reasoning to decision-making processes, where the available data sets are ambiguous. These applications include multifactorial evaluation and extraction of fuzzy rules-based models from large numeric data sets.

14.5 MULTIFACTORIAL EVALUATION

Multifactorial evaluation is a good example of the application of the fuzzy-set theory to decision-making processes. Its purpose is to provide a synthetic evaluation of an object relative to an objective in a fuzzy-decision environment that has many factors. Let U = {u1, u2, … , un} be a set of objects for evaluation, let F = {f1, f2, … , fm} be the set of basic factors in the evaluation process, and let E = {e1, e2, … , ep} be a set of descriptive grades or qualitative classes used in the evaluation. For every object u ∈ U, there is a single-factor evaluation matrix R(u) with dimensions m × p, which is usually the result of a survey. This matrix may be interpreted and used as a 2-D MF for fuzzy relation F × E.

With the preceding three elements, F, E, and R, the evaluation result D(u) for a given object u ∈ U can be derived using the basic fuzzy-processing procedure: the product of fuzzy relations through max–min composition. This has been shown in Figure 14.13. An additional input to the process is the weight vector W(u) for evaluation factors, which can be viewed as a fuzzy set for a given input u. A detailed explanation of the computational steps in the multifactorial-evaluation process will be given through two examples.

Figure 14.13. Multifactorial-evaluation model.

14.5.1 A Cloth-Selection Problem

Assume that the basic factors of interest in the selection of cloth consist of f1 = style, f2 = quality, and f3 = price, that is, F = {f1, f2, f3}. The verbal grades used for the selection are e1 = best, e2 = good, e3 = fair, and e4 = poor, that is, E = {e1, e2, e3, e4}. For a particular piece of cloth u, the single-factor evaluation may be carried out by professionals or customers by a survey. For example, if the survey results of the “style” factor f1 are 60% for the best, 20% for the good, 10% for the fair, and 10% for the poor, then the single-factor evaluation vector R1(u) is

Similarly, we can obtain the following single-factor evaluation vectors for f2 and f3:

Based on single-factor evaluations, we can build the following evaluation matrix:

If a customer’s weight vector with respect to the three factors is

then it is possible to apply the multifactorial-evaluation model to compute the evaluation for a piece of cloth u. “Multiplication” of matrices W(u) and R(u) is based on the max–min composition of fuzzy relations, where the resulting evaluation is in the form of a fuzzy set D(u) = [d1, d2, d3, d4]:

where, for example, d1 is calculated through the following steps:

The values for d2, d3, and d4 are found similarly, where ∧ and ∨ represent the operations min and max, respectively. Because the largest components of D(u)