Fuzzy operation

In order to prepare the final multi-class predictor map, the input weighted layers are fused using various Fuzzy operators (Fig. 5). Figure 6 is a reclassified final Fuzzy map, predicting the high potential areas for further drilling at East-Kahang. To validate the accuracy of the Fuzzy model, the projected Cu values of the completed drill holes are overlain on the final predictive map. The results show... 

In this chapter, it is sufficient to introduce only the most common fuzzy operations, which are the only fuzzy operations that are cutworthy (and also strongly cutworthy). These operations, usually called the standard fuzzy operations, are defined for all xElX by the following formulas ... 

Fuzzy decision from different LVs using fuzzy operators (D y)... 

Important is the monotonicity of the membership function. The special form of the membership function has only a weak influence on the result of fuzzy operations. The parabola function in Eq. (8.37) can be approached, therefore, by a triangular function... 

The overall security situation of the plant is safer above level. During calculating it has been found that the proportion of environmental factor is only 0.1. But the fuzzy operation results show that the security situation in the general level. Therefore, environmental factors are relatively weak, especially the electrostatic influence. And it is necessary to strengthen the supervision. 

Fuzzy sets and fuzzy operators are the subjects and verbs of fuzzy logic. The processing core of... 

Fuzzy rule sets usually have several antecedents that are combined using fuzzy operators. The combination is called a premise, and it generates a single truth value that determines the rule s outcome. In general, one rule by itself is not sufficient, but two or more rules that can play off one another are needed. The output of each rule is a fuzzy set, but in general, the output of an entire collection of rules should be a single number. 

If a given rule has several parts, once the inputs have been fuzzified, the fuzzy operators are applied to resolve the antecedents to a single number between 0 and 1. 

Fuzzy intersection Fuzzy intersection is the fuzzy operation for creating the intersection of fuzzy sets A and B on the universe of discourse X, which can be obtained as ... 

Fuzzy reasoning involves two parts evaluating the rule antecedent (the IF part of the rule) and applying the result to the consequent (the THEN part of the rule). Like the rules in expert systems, a fuzzy rule can have multiple antecedents joined by fuzzy operators AND or OR, or multiple consequents joined by fuzzy operator AND. For example ... 

After input data are fuzzified and their membership values obtained, the next step involves application of them to the antecedents of fuzzy rules. If a given fuzzy rule has multiple antecedents, a fuzzy operator (AND or OR) is used to obtain a single number that represents the result of antecedent evaluation. This number is then applied to a consequent membership function. 

AND is used to evaluate the conjunction of rule antecedents. Typically, fuzzy logic systems utilize the classical fuzzy operation intersection to implement this operation. Consider fuzzy rule 1 ... 

Similarly, OR is used to evaluate the disjunction of rule antecedents, which is implemented by the classical fuzzy operation union in fuzzy logic systems. Consider fuzzy rule 2 ... 

Running the fuzzy operations presented in Section 2 gives us the fhzzy probabiUty associated with operator error. This fiizzy probabiUty is presented as a triangular fuzzy munber (7.22 x 10 , 7.04 x 10 °, 4.09 X 10 ° ), which is shown by the red line in Fig. 4. On the other hand, the probability of success... 

Balopoulos, V., Hatzimichailidis, A. G. Papadoupoulos, B. K. (2007). Distance and similarity measures for fuzzy operators. Science Direct-Information Sciences (177) 2336-2348. 

It can be verified that the relations above reduce to their usual counterparts when applied to binary logic. The standard Lukasiewicz logic Li is isomorphic to fuzzy set theory based on the standard fuzzy operations in the same way the two-valued logic is isomorphic to the crisp set theory. The membership degree A(x) for x e X may be interpreted as the truth value of the proposition x is a member of the set A . The reciprocal is also valid. 

One realistic way to analyse a system with unavailable data is to employ subjective assessment using the combination of fuzzy logic and Evidential Reasoning (ER). Compared to the traditional fuzzy inference mechanism (i.e. max-min fuzzy operations), an ER approach has the advantage of avoiding the loss of useful information in its inference processes hence, it can be suitable for modelling complex systems. 

Table 3.1. Comparison between the Boolean and Fuzzy operations [Source Reference 6]...

A fuzzy system maps an input spaee to an output spaee by means linguistic rules, which is based on human reasoning. The linguistie representation presents an intuitive, natural description of a system allowing for relatively easy algorithm development compared to numerical systems. A fuzzy linguistic mle consists of an IF-THEN statement. A fuzzy mle is evaluated by means of fuzzy operators such as fuzzy AND , fuzzy OR etc. For example, in the case of two inputs (A and I2) and single output (O) fuzzy system, it can be expressed as shown below ... 

Similar fuzzy sets could be generated for describing the safety of other failure modes, which could be aggregated using conventional fuzzy operations to generate safety descriptions for the components, the subsystems and the whole system of the assessment hierarchy. However, this process may lead to information loss. 

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