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Rough Fuzzy Image Analysis Essay

1. Introduction

There are many real-life problems that are beyond a single expert. These are because of the need to involve a wide domain of knowledge. As a generalization of the intuitionistic fuzzy set, a neutrosophic set (NS) has been introduced by Smarandache [1]. This is a useful tool to deal with uncertainty in several social and natural aspects. Neutrosophy provides a foundation for a whole family of new mathematical theories with the generalization of both classical and fuzzy counterparts. In a NS, an element has three associated defining functions, the truth membership function (), the indeterminate membership function () and the false membership function (), defined on a universe of discourse X. These three functions are completely independent. To apply NSs in real-life problems more conveniently, Smarandache [2] and Wang et al. [3] defined single-valued NSs. Ye [4] studied the correlation coefficient and improved the correlation coefficient of NSs, as well as determined that in NSs the cosine similarity measure is a special case of the correlation coefficient.

Rough set theory was proposed by Pawlak [5] in 1982. Rough set theory is useful to study the intelligence systems containing incomplete, uncertain or inexact information. The lower and upper approximation operators of rough sets are used for managing hidden information in a system. Therefore, many hybrid models have been built, such as soft rough sets, rough fuzzy sets, fuzzy rough sets, soft fuzzy rough sets, neutrosophic rough sets, and rough NSs for handling uncertainty and incomplete information effectively. Dubois and Prade [6] introduced the notions of rough fuzzy sets and fuzzy rough sets. Liu and Chen [7] have studied different decision-making methods. Broumi et al. [8] introduced the concept of rough NSs. Yang et al. [9] proposed single-valued neutrosophic rough sets by combining single-valued NSs and rough sets, and they established an algorithm for decision-making problems based on single-valued neutrosophic rough sets on two universes. Mordeson and Peng [10] presented operations on fuzzy graphs. Akram et al. [11,12,13,14] considered several new concepts of neutrosophic graphs with applications. Zafer and Akram [15] dealt with rough fuzzy digraphs with applications. Recently, Sayed et al. [16] considered rough neutrosophic digraphs. They discussed some fundamental properties of rough neutrosophic digraphs. In this research paper, we investigate further new operations, including lexicographic products, strong products, rejection and tensor products on rough neutrosophic digraphs. We investigate some of their properties. We also consider an application of a rough neutrosophic digraph in decision-making.

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