Abstract
In this chapter, the concepts of bipolar q-rung picture fuzzy set and bipolar q-rung picture fuzzy number are introduced. Moreover, the fundamentals of bipolar q-rung picture fuzzy sets and bipolar q-rung picture fuzzy numbers are studied. Relatedly, some basic operations, aggregation operators, and relations on the bipolar q-rung picture fuzzy sets/numbers are derived. The intuitive definition of Tanimoto's similarity coefficient is proposed to measure the similarity between two bipolar q-rung picture fuzzy sets. Finally, it is shown that the proposed Tanimoto similarity measures can be used to deal with the problems of medical diagnosis and pattern recognition under the bipolar q-rung picture fuzzy environment.
TopIntroduction
Decision-making is a process in which decision maker(s) is invited to participate in the evaluation of some given alternatives, and then the most suitable to be selected or all of them are ranked according to the evaluation information. As the decision making problems become more complex, it is now very difficult for decision-maker(s) to evaluate alternatives considering real values. In 1965, Zadeh (Zadeh, 1965) initiated the theory of fuzzy set (FSs), which provides the decision maker(s) with an efficient way to model the fuzzy information. Nevertheless, the FSs are incapable of expressing the non-membership degree of each element in the universe of discourse belonging to an FS. To deal with this imperfection of FSs, Atanassov (Atanassov, 1986) proposed the concept of intuitionistic fuzzy sets (IFSs) in which each intuitionistic fuzzy number (IFN) consists of membership degree and non-membership degree, and the sum of them is less than or equal to 1. In the following years, by improving membership degree and non-membership degree in the structure of IFS, many researchers discussed new types of IFSs, such as interval-valued intuitionistic fuzzy sets (IVIFSs) (Atanassov, 1989; Broumi and Smarandache, 2014; Garg and Kumar 2019, 2020; Kamacı 2019), intuitionistic fuzzy multisets (IFMSs) (Ejegwa, 2015; Shinoj and John, 2012), Pythagorean fuzzy sets (PyFSs) (Yager, 2014; Garg, 2016, 2017), q-rung orthopair fuzzy sets (q-ROFSs) (Garg, 2020; Garg et al., 2020; Riaz et al., 2020a, 2020b, 2020c; Yager, 2017) and linear Diophantine fuzzy sets (LDFSs) (Riaz and Hashmi, 2019; Kamacı, 2021a). Moreover, the matrix representations of fuzzy sets and their extensions were also widely studied (Guleria and Bajaj, 2019; Kim and Roush, 1980; Padder and Murugadas, 2019; Kamacı et al., 2018a, 2018b; Petchimuthu et al., 2020; Petchimuthu and Kamacı, 2019, 2020). However, all these above theories failed to deal with abstinence, therefore, Cuong (Cuong, 2014) developed a new concept called picture fuzzy set (PFS) which generalizes fuzzy sets and intuitionistic fuzzy sets and so on. Later, some aspects of this type of fuzzy sets were developed (Kamacı, 2020, 2021b, 2021c, 2021d; Kamacı et al., 2021a, 2021b; Khalil et al., 2019). In 2019, Mahmood et al. (Mahmood et al., 2019) defined the concept of spherical fuzzy sets (SFSs) in which each spherical fuzzy number (SFN) consists of membership degree, abstinence degree, and non-membership degree whose square sum is less than or equal to 1. Li et al. (Li et al., 2018) described the q-rung picture fuzzy sets (q-RPFSs) extending the SFSs, where q≥1. Especially, Mahmood et al. (Mahmood et al., 2019) named these sets as T-spherical fuzzy sets (TSFSs) for 
.The lax constraint of q-RPFS is that the sum of qth power of the membership degree, abstinence degree, and non-membership degree is equal to or less than 1. He et al. (He et al., 2019) described the q-rung picture fuzzy Dombi Hamy mean operators and a new q-rung picture fuzzy MAGDM method. Akram and Habib (Akram and Habib, 2020) and Sitara et al. (Sitara et al., 2021) studied q-rung picture fuzzy graph structures and their applications. Recently, q-rung picture fuzzy models have been interested in many researchers.
Key Terms in this Chapter
Intuitionistic Fuzzy Set: Intuitionistic fuzzy sets are sets whose elements have membership grades and non-membership grades. The intuitionistic fuzzy set generalizes fuzzy set, since the indicator function of fuzzy set is a special case of the membership function and non-membership function of intuitionistic fuzzy set.
Picture Fuzzy Set: Like intuitionistic fuzzy sets, picture fuzzy sets have the previous two functions: one for membership and another for non-membership. The major difference is that picture fuzzy sets have one more function: for abstinence. This value indicates that the grade of abstinence. This concept of having abstinence value can be particularly useful when one cannot be very confident on the membership or non-membership values for alternative/object.
Information Fusion: Fusion can be described as a process of merging information from multiple sources to produce the most specific and comprehensive unified data about an activity, entity or event. Information fusion involves the combination of information into a new set of information towards reducing redundancy and uncertainty.
Fuzzy Set: In mathematics, fuzzy sets (a.k.a. uncertain sets) are somewhat like sets whose elements have membership grades. The fuzzy set generalizes classical set, since the characteristic function of classical set is a special case of the membership function of fuzzy set, if the latter only take values 0 or 1. These sets can be used in a wide range of domains in which information is imprecise or incomplete.
Bipolarity: Bipolarity refers to the tendency of the human psyche in reasoning and settling on choices based on positive and negative impacts. Where positive information reflects what is considered, satisfactory, permitted, possible, or desired as being acceptable. Otherwise, negative statements represent what is rejected, impossible or forbidden. Negative inclinations compare to restraints, since they indicate which esteems or articles must be dismissed, while positive inclinations relate to wishes, as they specify which items are more alluring than others without dismissing those that don’t meet the desires.
Medical Diagnosis: Medical diagnosis is the process of determining which disease explains a person's symptoms and signs.
Pattern Recognition: Pattern recognition is the automated recognition of patterns and regularities in data. It is a technology that helps to recognize patterns, which are continuous and repetitive shapes.
Information Measure: The mathematical theory of information measures information with several quantities of information. Information measure is a system of measurement of information based on the probabilities of the events that convey information. It is a system of related measures that facilitates the quantification of some particular characteristic.