Interval Valued Trapezoidal Neutrosophic Set for Prioritization of Non-functional Requirements

Index Terms— Non-functional Requirements (NFRs), Multi Criteria Decision Making (MCDM), Multi Attribute Decision Making (MADM), Neutrosophic Set, Interval Valued Neutrosophic Set, Trapezoidal Neutrosophic Set , Interval Valued Trapezoidal Neutrosophic Set(IVTrNS), Interval Valued Trapezoidal Neutrosophic Number(IVTrNN), Interval Valued Trapezoidal Neutrosophic Weighted Arithmetic Averaging Operator(IVTrNWAA)

1. INTRODUCTION Zadeh developed the fuzzy set theory [1] to deal the impreciseness, incompleteness and uncertainty in the information. Later, Zadeh [2] in 1975 proposed the interval valued fuzzy sets(IVFS) if grade of membership is uncertain and cannot be expressed in terms of a crisp value. Atanassov [3] extended the fuzzy set theory and developed an intuitionistic fuzzy set(IFS) [3][4][5]. Various researchers have explored the use of IFSs in MCDM situations[6][7][8], stock market prediction [9] and medical diagnosis[10].
   Liu and Yuan [11] combined the concept of IFS and triangular fuzzy numbers (TFN), and introduced the triangular intuitionistic fuzzy sets (TIFS). Further, Atanassov and Gargov [12] combined the IFS and IVFS, and introduced the interval valued intuitionistic fuzzy set (IVIFS). Further, the use of IVIFS was demonstrated in MADM [13] and multi attribute group decision making(MAGDM) [14] situations. Wang [15] proposed the weighted geometric and hybrid geometric operators using triangular intuitionistic fuzzy sets. Further, he applied both the operators to handle MAGDM problems. Wei et al. [16] proposed an induced ordered weighted geometric operator on the basis of Fuzzy number intuitionistic fuzzy numbers and introduced an approach based on the proposed operator to solve group decision making problems. Ye [17] extended the TIFS and proposed the trapezoidal intuitionistic fuzzy set (TrIFS) for representing the membership and non-membership values in the form of a trapezoid. Smarandache [18] extended the concept of classic, fuzzy and IFS, and proposed the neutrosophic set(NS) to deal imprecise, incomplete and uncertain information. Later, A variation of a NS i.e. single-valued neutrosophic set(SVNS) is proposed which can be applied in real world scenarios [19]. Jun Ye [20] introduced the TrNS as an extension of trapezoidal fuzzy numbers (TrFN) and SVNS. He also introduced weighted arithmetic and geometric averaging operator based on the trapezoidal neutrosophic number. Further, using these operators, he introduced a method to handle MADM problems. As discussed, various methods have been proposed by the researchers based on IVIFS, TrIFS, and TrNS set to handle inconsistency, impreciseness, uncertainty, incompleteness and indeterminacy in the information where information is either (1) neutrosophic and can be represented in the form of a trapezoid (2) or the information is intuitionistic fuzzy and in the unit interval of real numbers and can be represented in the form of a triangle/trapezoid. But the proposed methodology handles the information which is neutrosophic in nature and in the unit interval of real numbers and can be represented in the form of a trapezoid or a triangle.

Interval Valued Trapezoidal Neutrosophic Set for Prioritization of Non-functional Requirements Kiran Khatter, Department of Computer Science, BML Munjal University

2 Thus the paper proposes an interval valued trapezoidal neutrosophic set (IVTrNS) based on the combination of IVIFN and TrNS. The paper also introduces the operational laws for IVTrNN. Further an interval valued trapezoidal neutrosophic weighted arithmetic averaging (IVTrNWAA) operator is introduced to combine the trapezoidal information which is neutrosophic and in the unit interval of real numbers. Finally, a method is developed to handle the problems in the MADM environment using IVTrNWAA operator followed by a numerical example of NFRs prioritization to illustrate the relevance of the developed method. Remaining sections of the paper are o

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