An Intelligent Multiattribute Decision-Support Framework Based on Parameterization of Neutrosophic Hypersoft Set

<abstract><p>The study on N-soft sets (NSSs) has been significantly developed recently. Hybrid models such as fuzzy N-soft sets, Intuitionistic fuzzy N-soft sets, and hesitant fuzzy N-soft sets were introduced to combine fuzzy sets, intuitionistic fuzzy sets and hesitant fuzzy sets with NSSs. Related to the hybrid models, it was also constructed some complements, operations and related properties. This article aims to construct a new hybrid model called hesitant intuitionistic fuzzy N-soft sets (HIFNSSs) to combine intuitionistic fuzzy N-soft sets and hesitant fuzzy N-soft sets. Moreover, we generalise HIFNSSs to generalized hesitant intuitionistic fuzzy N-soft sets (GHIFNSSs) as a hybrid model between generalized hesitant intuitionistic fuzzy sets and N-soft sets. It was also defined some complements of GHIFNSSs, intersection and union operations between GHIFNSSs, and proved that the operations between some particular complements hold De Morgan Law. In applying a GHIFNSS, we provide an algorithm for decision-making problems and its numerical illustration.</p></abstract>

<abstract><p>Soft set has limitation for the consideration of disjoint attribute-valued sets corresponding to distinct attributes whereas hypersoft set, an extension of soft set, fully addresses this scarcity by replacing the approximate function of soft sets with multi-argument approximate function. Some structures (i.e., possibility fuzzy soft set, possibility intuitionistic fuzzy soft set) exist in literature in which a possibility of each element in the universe is attached with the parameterization of fuzzy sets and intuitionistic fuzzy sets while defining fuzzy soft set and intuitionistic fuzzy soft set respectively. This study aims to generalize the existing structure (i.e., possibility intuitionistic fuzzy soft set) and to make it adequate for multi-argument approximate function. Therefore, firstly, the elementary notion of possibility intuitionistic fuzzy hypersoft set is developed and some of its elementary properties i.e., subset, null set, absolute set and complement, are discussed with numerical examples. Secondly, its set-theoretic operations i.e., union, intersection, AND, OR and relevant laws are investigated with the help of numerical examples, matrix and graphical representations. Moreover, algorithms based on AND/OR operations are proposed and are elaborated with illustrative examples. Lastly, similarity measure between two possibility intuitionistic fuzzy hypersoft sets is characterized with the help of example. This concept of similarity measure is successfully applied in decision making to judge the eligibility of a candidate for an appropriate job. The proposed similarity formulation is compared with the relevant existing models and validity of the generalization of the proposed structure is discussed.</p></abstract>

In this paper we introduce the concentration and dilation operators on interval valued intuitionistic fuzzy soft sets of root type which is a generalization of fuzzy set, intuitionistic fuzzy set, interval valued intuitionistic fuzzy set and intuitionistic fuzzy set of root type and establish some properties of these operators. We define Hamming distance between two interval valued intuitionistic fuzzy soft sets of root type and it is shown that it is a metric. A similarity measure based on this Hamming distance is defined and some properties are established. We also develop a decision making method based on the similarity measure of Hamming distance between interval valued intuitionistic fuzzy soft sets of root type. We develop an algorithm for the decision making problem and illustrate its working by means of examples.

This paper researched and constructed residuation implication operator of intuitionistic fuzzy T-norms.Based on this,the paper deducted the intuitionistic fuzzy rough sets model,and proved that the Pawlak rough sets,intuitionistic fuzzy sets,fuzzy rough sets,rough fuzzy sets and fuzzy T-norms rough sets were special cases of intuitionistic fuzzy rough sets.At last,some properties of intuitionistic fuzzy sets were presented and proved.

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Molodtsov (Comput Math Appl 37:19–31, 1999 [6]) initiated a novel concept called soft sets, a new mathematical tool for dealing with uncertainties. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts (Aktas and Cagman in Inf Sci 1(77):2726–2735, 2007 [1]). Research works on soft sets are very active and progressing rapidly in these years. In 2001, Maji et al. (J Fuzzy Math 9(3):589–602, 2001 [5]) proposed the idea of intuitionistic fuzzy soft set theory and established some results on them. Based on an equivalence relation on the universe of discourse, Dubois and Prade (Int J Gen Syst 17:191–209, 1990 [3]) introduced the lower and upper approximation of fuzzy sets in a Pawlak approximation space and obtained a new notion called rough fuzzy sets. Feng et al. (Soft Compt 14:899–911, 2009 [4]) introduced lower and upper soft rough approximation of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy sets which is the extension of Dubois and Prade’s rough fuzzy sets. The aim of this chapter is to consider lower and upper soft rough intuitionistic fuzzy approximation of intuitionistic fuzzy sets in intuitionistic fuzzy soft approximation space (IF soft approximation space) and obtain a new hybrid model called soft rough intuitionistic fuzzy sets which can be seen as extension of both the previous work by Dubois and Prade and Feng et al.

Lattices, soft sets, fuzzy sets and their generalizations have always been important for Mathematicians and the researchers working on uncertaities. In this paper our aim is to introduce the concept of lattice ordered intuitionistic fuzzy soft sets. After introducing extended union, extended intersection, AND-product, OR-product, basic union, basic intersection of intuitionistic fuzzy soft sets, in this paper the affects of lattice ordered intuitionistic fuzzy soft sets and anti-lattice ordered intuitionistic fuzzy soft sets on restricted union, restricted intersection, extended union, extended intersection,AND-product, OR-product, basic union, basic intersection of intuitionistic fuzzy sets are discussed. Further a decision making problem is solved by using these concepts.

The extensions of ordinary fuzzy sets such as intuitionistic fuzzy sets (IFS), Pythagorean fuzzy sets (PFS), and neutrosophic sets (NS), whose membership functions are based on three dimensions, aim at collecting experts’ judgments more informatively and explicitly. In the literature, generalized three-dimensional spherical fuzzy sets have been developed by Kutlu Gündoğdu and Kahraman [1], including their arithmetic operations, aggregation operators, and defuzzfication operations. Spherical Fuzzy Sets (SFS) are a new extension of Intuitionistic, Pythagorean and Neutrosophic Fuzzy sets. In this paper, our aim is to employ SF-VIKOR method to waste management problems. We handle a waste disposal site selection problem with five alternatives and four criteria in order to demonstrate the performance of the proposed SF-VIKOR method.

The basic notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation. This paper proposes an intuitionistic fuzzy rough set (IFRS) model which is a combination of intuitionistic fuzzy set (IFS) and rough set. We approximate an IFS by using soft binary relations instead of fixed binary relations. By using this technique, we get two pairs of intuitionistic fuzzy (IF) soft sets, called the upper approximation and lower approximation with respect to foresets and aftersets. Properties of newly defined rough set model (IFRS) are studied. Similarity relations between IFS with respect to this rough set model (IFRS) are also studied. Finally, an algorithm is constructed depending on these approximations of IFSs and score function for decision-making problems, although a method of decision-making algorithm has been introduced for fuzzy sets already. But, this new IFRS model is more accurate to solve the problem because IFS has degree of nonmembership and degree of hesitant.

Neutrosophic sets (NSs) handle uncertain information while fuzzy sets (FSs) and intuitionistic fuzzy sets (IFs) fail to handle indeterminate information. Soft set theory, neutrosophic set theory, and rough set theory are different mathematical models for handling uncertainties and they are mutually related. The neutrosophic soft rough set (NSRS) model is a hybrid model by combining neutrosophic soft sets with rough sets. We apply neutrosophic soft rough sets to graphs. In this research paper, we introduce the idea of neutrosophic soft rough graphs (NSRGs) and describe different methods of their construction. We consider the application of NSRG in decision-making problems. In particular, we develop efficient algorithms to solve decision-making problems.

Multi-attribute group decision-making based on q-rung orthopair fuzzy Aczel–Alsina power aggregation operators

The presented work is an extensive study of the existing non-linear programming (NLP) methods in an uncertain environment of an interval-valued intuitionistic fuzzy soft set (IVIFSS) for solving multi-attribute decision-making (MADM) problems. IVIFSS is an intriguing extension of a fuzzy set (FS) involving both interval-valued intuitionistic fuzzy set (IVIFS) (which considers interval-value of both membership and non-membership elements of an intuitionistic fuzzy set (IFS)) and soft set (SS) (which gives importance to each parameter in an alternative). A comprehensive study projects that the existing NLP method in accordance with the technique for order preference by similarity to ideal solution (TOPSIS) for solving interval-valued intuitionistic fuzzy soft multi-attribute decision-making (IVIFSMADM) problems (decision-making (DM) problems in which assessment of rating of each alternative, over each character is rendered by an IVIFSS) is posing some limitations due to some mathematical incorrect assumptions and hence incorporating ambiguous results in real-life applications. Henceforth, an attempt has been made to properly understand the root cause of the posed shortcoming and suggested a new NLP method for solving IVIFSMADM problems, and also to validate this proposed NLP method a real-life problem is solved successfully.

In addition to the well-known fuzzy sets, a novel type of fuzzy sets called as spherical fuzzy set (SFS) is recently introduced in the literature. SFS is the generalized structure over existing structures of fuzzy sets (intuitionistic fuzzy sets, Pythagorean fuzzy sets, and neutrosophic sets) based on three dimensions (truth, falsehood, and indeterminacy) to provide a wider choice for decision makers (DMs). Although the SFS is introduced recently, the topic attracts the attention of academicians at a remarkable rate. Hence, a comprehensive literature review of recent and state-of-the-art papers is vital to draw a framework of the past, and to shed light on future directions. To do so, a systematic review methodology is followed and 43 academic papers include SFS are selected and reviewed. The papers are then analyzed and categorized based on the titles and abstracts to construct a useful foundation of past research. Finally, trends and gaps in the literature are identified to clarify and to suggest future research opportunities in fuzzy logic area.

Social responsibility is the most important thing to consider while working on a project. When deciding on a project or taking part in a bid, it is crucial to understand the nature and potential consequences of the risks involved. Attempting to implement projects with cutting-edge technologies appears to be necessary, necessitating up-to-date and continuous planning to implement the relevant matters in light of the ever-increasing growth of urban communities, the need to carry out tasks, and the rising standard of living. Due to the strong demand, these strategies try to improve quality while decreasing prices. In the beginning, Smarandache suggests using neutrosophic sets. These sets are an improvement above traditional fuzzy set theory in reflecting the uncertainty and fuzziness of real-world issues. Three decision-making states are considered: uncertainty, truthiness, and falseness. The fuzzy set degree in Zadeh's classic theory is merely the membership function. On the other hand, three membership functions are considered in a neutrosophic setting. An indeterminacy degree is considered, which is not the case with intuitionistic fuzzy sets. To express decision makers' perspectives on the truthiness (T), falsity (F), and indeterminacy (I) for a fuzzy set concurrently, this paper expands the usual neutrosophic TOPSIS approach to interval-valued neutrosophic. One example of how the suggested strategy might be put to use is to prioritize initiatives in the realm of corporate social responsibility using a combination of expert opinion and objective criteria.

We present a general method that can be used to unify the theoretical results for several new MV-valued fuzzy sets, such as intuitionistic fuzzy sets, neutrosophic fuzzy sets, fuzzy soft sets, or multivalued fuzzy sets and their mutual combinations. The results created in this way have properties identical to analogous results for classic MV-valued fuzzy sets, and it is not necessary to prove these results separately for each new type of fuzzy sets. To illustrate this general method, we will show how two related fuzzy Chang topologies can be derived in these new types of fuzzy sets from general approximation spaces.

Interval-valued intuitionistic fuzzy soft sets and their properties