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Fuzzy Behavior Organization and Fusion for Mobile Robot Reactive Navigation
Published in Takushi Tanaka, Setsuo Ohsuga, Moonis Ali, Industrial and Engineering Applications of Artificial Intelligence and Expert Systems, 2022
Jiancheng Qiu, Michael Walters
The basic fuzzy set operations defined by [Zadeh65] are: intersection, A∧B=min(μA(x)μB(y)); union, A∨B=max(μA(x),μB(y)) and complement~A=1−μA(x).
Fuzzy Logic–Based Control Section
Published in Bogdan M. Wilamowski, J. David Irwin, Intelligent Systems, 2018
The combination of membership functions requires some form of set operations. Three basic operations of conventional set theory are intersection, union, and complement. The fuzzy logic counterparts to these operations are similar to those of conventional set theory. Fuzzy set operations such as union, intersection, and complement are defined in terms of the membership functions. Let A and B be two fuzzy sets with membership functions μA and μB, respectively, defined for all u ∈ U. The TALL membership function has been defined in Equation 4.6, and a FAST membership function is defined as (Figure 4.4) () FAST=μFAST(height)=0.56+0.86.2+0.96.4+16+0.756.8+0.67.
Fuzzy Linear Programming
Published in Anindya Ghosh, Prithwiraj Mal, Abhijit Majumdar, Advanced Optimization and Decision-Making Techniques in Textile Manufacturing, 2019
Anindya Ghosh, Prithwiraj Mal, Abhijit Majumdar
In fuzzy set operations, the membership function of the “union” of two fuzzy sets is the maximum of the two individual membership functions and is known as the maximum criterion. Again, the membership function of the “intersection” of two fuzzy sets is the minimum of the two individual membership functions and is known as the minimum criterion. Further, the membership function of the complement of a fuzzy set is the negation of the specified membership function and is known as the negation criterion. Symbolically, for the set D˜=A˜ U B˜ (“union” of fuzzy sets A˜ and B˜), the membership function is defined as μD˜(x)=μA˜(x)∨μB˜(x)=max{μA˜(x),μB˜(x)},x∈X where the symbol ∨ stands for the maximum operator.
Integrating fuzzy set function into floating catchment area measures: a determination of spatial accessibility of service points
Published in Annals of GIS, 2022
Frank Edward Mahuve, Beatrice Christopher Tarimo
To generate heterogeneous accessibility indices within a sub-zone that gradually change across neighbouring subzones, the grades to which a travel impedance belongs to the three sub-zones are combined using fuzzy set operations. These fuzzy set operations include union (OR), intersection (AND), algebraic product, algebraic sum, arithmetic mean and weighted mean (Rafati and Nikeghbal 2017). The arithmetic and weighted mean are often preferred among these six operations because they allow high grade on one fuzzy set to compensate for low grade on another fuzzy set. The resulting grade becomes higher than the lowest grade and lower than the highest grade (Grabisch, Orlovski, and Yager 1998). However, arithmetic mean operation, in some cases, generates grades of the same value for different travel distances within different sub-zones. For example, assuming population unit 1 in Figure 1 is at 10 m instead of 95 m from service point ‘a’ with grades of 0.8 to inner and 0.2 to central sub-zones, while population unit 2 is at 190 m instead of 105 m from the same service point ‘a’ with grades of 0.2 to central and 0.8 to outer sub-zones, the distances ( and ) from the two population units 1 and 2 to service point ‘a’ are equally weighed with a value of 0.5 obtained by averaging the respective grades. The resulting constant weight of 0.5 for the two different travel distances contradicts with Tobbler’s first law of Geography. Hence, the arithmetic mean operation is not recommended for calculating weights of travel distances based on the grades to which travel distances belong to different sub-zones. On the contrary, the weighted mean fuzzy operation, given by Equation (6), addresses the shortcoming of the arithmetic mean fuzzy operation of assigning uniform influence (i.e. weights) to different travel distances from different population units at different sub-zones (Burrough 1989; Grabisch, Orlovski, and Yager 1998).
Strait fuzzy sets, strait fuzzy rough sets and their similarity measures-based decision making systems
Published in International Journal of Systems Science, 2023
Akın Osman Atagün, Hüseyin Kamacı
In information science, information is sometimes imprecise or vaguely organised and has a level of granularity. In the twentieth century, fuzzy set theory (Zadeh, 1965), rough set theory (Pawlak, 1982), soft set theory (Molodtsov, 1999) were developed to address these issues of the knowledge system. Fuzzy logic is a form of multi-valued logic in which the true value of variables can be any real number (exact value) between 0 and 1, both being inclusive. Fuzzy systems as a subject was developed for modelling the uncertainty, ambiguity and vagueness present in the human thought process. Fuzzy set is a set in which each element is associated with value, which is between 0 to 1 based on the certainty. Basically it allows partial membership which means that it contain elements that have varying degrees of membership in the set. The use of fuzzy set theory, which extends the original set theory and gradually expands its application areas, to solve a certain complex problem has attracted the attention of many researchers in world-wide. Relatedly, the fuzzy set operations (Dubois & Prade, 1978; Mizumoto & Tanaka, 1981; Wu, 2020), fuzzy aggregation operators (Mardani et al., 2018; Sasikala & Petrou, 2001), fuzzy graph structures (Gani et al., 2016; Sitara et al., 2019) and fuzzy matrix forms (Kamacı et al., 2018a, 2018b; Pal, 2016; Petchimuthu et al., 2020; Petchimuthu & Kamacı, 2022) are studied in detail. But there are situations when assigning an exact number to an expert/decision maker opinion is too restrictive, and the assignment of an interval of values is more realistic. Therefore, there is an extensive theory of fuzzy sets, called an interval valued fuzzy set (Sambuc, 1975). In many real-life scenarios, these sets can model imperfect information better than fuzzy sets. Akram and Dudek (2011), Gehrke et al. (2001), Kamacı (2019), Peng and Garg (2016) developed some interval valued fuzzy structures and then studied some desirable properties of these structures. On the other hand, the rough set theory, introduced by Pawlak in the year 1982, is used as a mathematical tool for a perusal of uncertain, unsettled, and vagueness of imprecise data in information systems (Alisantoso & Khoo, 2009; Pawlak, 2002; Pawlak & Skowron, 2007). The rough set theory was applied in several fields including medical informatics, image processing, pattern recognition, data mining, expert systems and knowledge discovery. In the current literature, several research works were combined the rough set theory with other artificial intelligence methods such as neural networks (Ji et al., 2021; Pourpanah et al., 2021), fuzzy logic (Riaz & Hashmi, 2020; Zeng et al., 2019), additionally to other methods resulting in some good results (Kamacı, 2021a, 2021b; Karaaslan & Çagman, 2018). Mafarja and Abdullah (2015) and Shidpour et al. (2016) focussed on the combined form of rough set and fuzzy set. Mardani et al. (2017) and Yuan et al. (2021) systematically reviewed many studies that proposed or developed fuzzy rough sets theory. Besides, there exists quite a body of research work on applications of these set theories.