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Complex fuzzy sets and complex intuitionistic fuzzy sets cannot handle imprecise, indeterminate, inconsistent, and incomplete information of periodic nature. To overcome this difficulty, we introduce complex neutrosophic set. A complex neutrosophic set is a neutrosophic set whose complex-valued truth membership function, complex-valued indeterminacy membership function, and complex-valued falsehood membership functions are the combination of real-valued truth amplitude term in association with phase term, real-valued indeterminate amplitude term with phase term, and real-valued false amplitude term with phase term, respectively. Complex neutrosophic set is an extension of the neutrosophic set. Further set theoretic operations such as complement, union, intersection, complex neutrosophic product, Cartesian product, distance measure, and δ-equalities of complex neutrosophic sets are studied here. A possible application of complex neutrosophic set is presented in this paper. Drawbacks and failure of the current methods are shown, and we also give a comparison of complex neutrosophic set to all such methods in this paper. We also showed in this paper the dominancy of complex neutrosophic set to all current methods through the graph. [ABSTRACT FROM AUTHOR]

The most widely used technique for solving and optimizing a real-life problem is linear programming (LP), due to its simplicity and efficiency. However, in order to handle the impreciseness in the data, the neutrosophic set theory plays a vital role which makes a simulation of the decision-making process of humans by considering all aspects of decision (i.e., agree, not sure and disagree). By keeping the advantages of it, in the present work, we have introduced the neutrosophic LP models where their parameters are represented with a trapezoidal neutrosophic numbers and presented a technique for solving them. The presented approach has been illustrated with some numerical examples and shows their superiority with the state of the art by comparison. Finally, we conclude that proposed approach is simpler, efficient and capable of solving the LP models as compared to other methods. [ABSTRACT FROM AUTHOR]

This article introduces the structure of the (t,s)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(t,s)$$\end{document}-regulated interval-valued neutrosophic soft set (abbr. (t,s)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(t,s)$$\end{document}-INSS). The structure of (t,s)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(t,s)$$\end{document}-INSS is shown to be capable of handling the sheer heterogeneity and complexity of real-life situations, i.e. multiple inputs with various natures (hence neutrosophic), uncertainties over the input strength (hence interval-valued), the existence of different opinions (hence soft), and the perception at different strictness levels (hence (t,s)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(t,s)$$\end{document}-regulated). Besides, a novel distance measure for the (t,s)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(t,s)$$\end{document}-INSS model is proposed, which is truthful to the nature of each of the three membership (truth, indeterminacy, falsity) values present in a neutrosophic system. Finally, a Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) and a Viekriterijumsko Kompromisno Rangiranje (VIKOR) algorithm that works on the (t,s)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(t,s)$$\end{document}-INSS are introduced. The design of the proposed algorithms consists of TOPSIS and VIKOR frameworks that deploy a novel distance measure truthful to its intuitive meaning. The conventional method of TOPSIS and VIKOR will be generalized for the structure of (t,s)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(t,s)$$\end{document}-INSS. The parameters t\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$t$$\end{document} and s\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$s$$\end{document} in the (t,s)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(t,s)$$\end{document}-INSS model take the role of strictness in accepting a collection of data subject to the amount of mutually contradicting information present in that collection of data. The proposed algorithm will then be subjected to rigorous testing to justify its consistency with human intuition, using numerous examples which are specifically made to tally with the various human intuitions. Both the proposed algorithms are shown to be consistent with human intuitions through all the tests that were conducted. In comparison, all other works in the previous literature failed to comply with all the tests for consistency with human intuition. The (t,s)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(t,s)$$\end{document}-INSS model is designed to be a conclusive generalization of Pythagorean fuzzy sets, interval neutrosophic sets, and fuzzy soft sets. This combines the advantages of all the three previously established structures, as well as having user-customizable parameters t\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$t$$\end{document} and s\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$s$$\end{document}, thereby enabling the (t,s)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(t,s)$$\end{document}-INSS model to handle data of an unprecedentedly heterogeneous nature. The distance measure is a significant improvement over the current disputable distance measures, which handles the three types of membership values in a neutrosophic system as independent components, as if from a Euclidean vector. Lastly, the proposed algorithms were applied to data relevant to the ongoing COVID-19 pandemic which proves indispensable for the practical implementation of artificial intelligence. [ABSTRACT FROM AUTHOR]