Single-valued neutrosophic set (SVNS) can be viewed as an instance of the neutrosophic set (NS) proposed by Smarandache in 2005 to solve real decision-making problems that contain imperfect knowledge of the domain. Embedding the idea of SVNS with the soft set (SS), a new mathematical structure known as a single-valued neutrosophic soft set (SVNSS) is introduced to model the vague concept parametrically. In this chapter, the authors introduce a new soft model, namely inverse single-valued neutrosophic soft set (ISVNSS), that can be perceived as another mathematical model which is used to solve the purposes that are just opposite in the context of SVNSS. Also, they define different properties and operators associated with ISVNSSs. Finally, by using the ISVNSSs, an algorithm has been introduced that will serve the purpose of decision-makers to solve various real decision-making problems, and it will enable to unveil another type of uncertainty domain hidden in nature. This new approach surely provides more information to the decision-makers before they make any decisions on various attributes.
Top1. Introduction
Presently, researchers are facing such kinds of problems in their real-life that cannot be answered by using classical mathematics. In a classical or ordinary or crisp sense, a decision-makers decision in the cognitive domain is restricted within 0 or 1. So, it will unable to answer those questions whose value lies between 0 and 1 as it is the demand of the decision-makers. For example, it is difficult to explain the qualitative term “excellent” by using two-valued logic(0 0r 1) as there are different categories to measure excellent such as very excellent, average excellent, less excellent, and so on. We give another example, in the classical sense, the quality of a student can be determined by using the linguistic term ‘good’ or ‘bad. We assign 1 for good and 0 for bad. But, what value we assign for ‘very good’, ‘relatively good’, ‘average’, and ‘satisfactory’ etc. Like this, researchers give many instances that they are experienced in our day- to- day activity. To eradicate such issues to some extent Zadeh(Zadeh,1965) introduced the fuzzy set (FS) in 1965 and since then it can attract the attention of many researchers and mathematicians over the decades due to its novelty. It is a mathematical technique to model uncertainty. FS can be viewed as an extension of the classical or crisp sets. In FS, every object in the universe has a membership value that belongs to the interval [0,1]. For instance, suppose an individual having an IQ within 85 and 115 can be considered to be an intelligent person. So, here we consider “intelligent” as a fuzzy word. Therefore, we can assign membership values corresponding to some IQs of a group of individuals as (85,0), (100,0.5), (115,1) etc. Membership values decide the degree of intelligence of the group of individuals. As we know every theory has its inherent limitation like this FS cannot solve the purpose of incomplete information in data. For incomplete information, a decision-maker needs to assign a non-membership value along with a membership value to every member of a universe. To remove such an issue Atanassov(Atanassov,1986) introduced an intuitionistic fuzzy set(IFS) in 1986. An IFS is a direct extension of FS with the restriction that the sum of the membership and the non-membership degree does not exceed the unity where both the membership degrees belong to the interval [0,1]. If there is a situation where the non-membership value becomes zero then the IFS reduces to FS. To deal various types of uncertainties present in different fields of knowledge, some other extensions and applications of FSs are given in the literature proposed in the following: In 1989, the notion of interval-valued intuitionistic fuzzy set was given by Atanassov(Atanassov,1989). Bustince et al.(Bustince & Burillo, 1996) defined vague sets are intuitionistic fuzzy sets. In 2010, interval-valued fuzzy sets in soft computing was introduced by Bustice(Bustince, 2010). An application of interval-valued fuzzy soft was introduced by Chetia et al. (Chetia & Das, 2010). Cuong et al.(Cuong & Kreinovich, 2013) presented the picture fuzzy set for computational intelligence problems. Robust decision-making using intuitionistic fuzzy numbers were given by Das et al. (Das, Kar & Pal, 2017). In 2016, Garg(Garg, 2016) defined correlation coefficients between Pythagorean fuzzy sets. The linguistic Pythagorean fuzzy sets and its applications in multi-attribute decision‐making process was initiated by Garg(Garg,2018). In 2014, Rodriguez et al. (Rodriguez, Martínez, Torra, Xu & Herrera, 2014) used hesitant fuzzy sets in future directions. The idea of the distance and similarity measures for dual hesitant fuzzy sets and their applications in pattern recognition was given by Su et al.(Su, Xu, Liu & Liu, 2015). The dual hesitant fuzzy aggregation operator in multiple attribute decision-making was introduced by Wang et al. (Wang, Zhao & Wei, 2014). In 2015, Zadeh(Zadeh, 2015) used fuzzy logic as a personal perspective. Fuzzy set theory and its various applications were given by Zimmermann(Zimmermann, 2010). Zhu et al.(Zhu, Xu & Xia, 2012) gave the notion of dual hesitant fuzzy sets.