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The notion of classical soft multisets is extended to neutrosophic vague soft multisets by applying the theory of soft multisets to neutrosophic vague sets to be more effective and useful. We also define its basic operations, namely complement, subset, union, intersection along with illustrative examples, and study some related properties with supporting proofs. Lastly, this notion is applied to a decision making problem and its effectiveness is demonstrated using an illustrative example. [ABSTRACT FROM AUTHOR]

Entropy measure is an important tool in measuring uncertain information and plays a vital role in solving Multi Criteria Decision Making (MCDM). At present, various entropy measures in literature are developed to measure the degree of fuzziness. However, they could not be used to deal with interval neutrosophic vague set (INVS) environment. In this study, two kinds of entropy measures are proposed as the extension of the entropy measure of single valued neutrosophic set (SVNS). First, we construct the axiomatic definition of INVS and propose a new formula for the entropy measure of INVS. Based on this measure, we develop two multi criteria decision making methods. Subsequently, an illustrative example of investment problems under INVS is given to demonstrate the proposed entropy measures. Finally, a comparative analysis is presented to illustrate the rationality and effectiveness of the proposed entropy measures. [ABSTRACT FROM AUTHOR]

The basic aim of soft computing is to trade precision for a tractableness and reduction in solution cost by pushing the limits of tolerance for imprecision and uncertainty. This paper introduces a novel soft computing technique called complex neutrosophic relation (CNR) to evaluate the degree of interaction between two complex neutrosophic sets (CNSs). CNSs are used to represent two-dimensional information that are imprecise, uncertain, incomplete and indeterminate. The Cartesian product of CNSs and subsequently the complex neutrosophic relation is formally defined. This relation is generalised from a conventional single valued neutrosophic relation (SVNR), based on CNSs, where the ranges of values of CNR are extended to the unit circle in complex plane for its membership functions instead of [0, 1] as in the conventional SVNR. A new algorithm is created using a comparison matrix of the SVNR after mapping the complex membership functions from complex space to the real space. This algorithm is then applied to scrutinise the impact of some teaching strategies on the student performance and the time frame(phase) of the interaction between these two variables. The notion of inverse, complement and composition of CNRs along with some related theorems and properties are introduced. The performance and utility of the composition concept in real-life situations is also demonstrated. Then, we define the concepts of projection and cylindric extension for CNRs along with illustrative examples. Some interesting properties are also obtained. Finally, a comparison between different existing relations and CNR to show the ascendancy of our proposed CNR is provided. [ABSTRACT FROM AUTHOR]