Interval complex neutrosophic soft set: some generalizations and applications

Abstract

Molodtsov originated the notion of the soft set (SS), which can be utilized as a generic mathematical tool for modeling complex systems involving uncertain or unclearly defined objects. Many researchers have studied and investigated the theory of SSs and formed some hybrid models to deal with the uncertainty that exists in some real-life problems, such as decision-making and medical diagnosis problems. The results of this thesis are split into two parts. In the first part, we introduce the concept of interval complex neutrosophic soft set (ICNSS) by incorporating the two concepts of complex neutrosophic set (CNS) and SS in interval form or by extending the interval neutrosophic soft range of three neutrosophic membership functions from [0,1] to a unit circle in the complex plane. The ICNSS is a hybrid structure that collects the properties of both the interval complex neutrosophic set (ICNS) and the SS, thus giving it a high efficiency in solving decision-making problems that involve two dimensions of data where the extra terms (phase terms) of the complex numbers play an important role in determining the final decision. Next, we discuss the basic set operations of ICNSS, namely subset, equality, complement, intersection, union, AND, and OR, and for these operators, we derive some properties and illustrative examples. Then, we introduce a new relation structure called interval complex neutrosophic soft relation (ICNSR) as an ICNS-subset of the Cartesian product between two ICNSSs. Related ideas like the equivalent ICNSR, composition, partition, and function, along with some properties of this new relation structure, are defined and proved. Then, we present the reflexive closure, symmetric closure, and transitive closure relations on ICNSS, followed by some numerical examples to illustrate these relations. In addition, in the concept of ICNSS, the parameters have the same degree of importance, which is considered as one. As a result, we develop the concept of fuzzy parameterized interval complex neutrosophic soft set (FP-ICNSS) based on the idea that each parameter set element has an important degree. For both ICNSS and FP-ICNSS, we discuss the relevant mappings and analyze their properties. On the other hand, all of the aforementioned models that are developed from the SS can only deal with one expert. This makes decisions hard for the user, especially those who use questionnaires at work. To overcome this drawback, in the second part of this thesis, we establish the notion of interval neutrosophic soft expert set (INSES) in real space by combining the two concepts of interval neutrosophic set (INS) and soft expert set (SES). Then we extend this notion from real space to complex space by establishing the notion of interval complex neutrosophic soft expert set (ICNSES). The novelty of ICNSES lies in its ability to represent the neutrosophic information using a time factor and show the opinions of all experts in one model under the interval form based. The fundamental theoretical operations, such as subset, equality, complement, intersection, union, AND, and OR, are obtained for both INSES and ICNSES. Finally, we employ all the aforementioned proposed models to solve some real-life problems, such as decision-making and medical diagnosis problems, by developing algorithms that work to convert these models from complex space to real space to help the user make the right decision without any complicated operations.