Vector Similarity Measures For Simplied Neutrosophic hesitant fuzzy Set And Their Applications

In this article we present three similarity measures between simpli…ed neutrosophic hesitant fuzzy sets, which contain the concept of single valued neutrosophic hesitant fuzzy sets and interval valued neutrosophic hesitant fuzzy sets, based on the extension of Jaccard similarity measure, Dice similarity measure and Cosine similarity in the vector space. Then based on these three de…ned similarity measures we present a muliple attribute decision making method to solve the simpli…ed neutrosophic hesitant fuzzy multiple criteria decision making problem, in which the evaluated values of the alternative with respect to the criteria is represented by simpli…ed neutrosophic hesitant fuzzy elements. Further we applied the proposed similarity measures to pattern recognition. At the end a numerical examples are discussed to show the e¤ectiveness of the proposed similarity measures. Keywords: Vector similarity measure, Jaccard similarity measure, Dice similarity measure, Cosine similarity measure, hesitant fuzzy sets, interval valued hesitant fuzzy sets, Single valued neutrosophic hesitant fuzzy sets, interval neutrosophic hesitant fuzzzy sets, multi-criteria decision making, pattern recognition. 1 Introduction The evidence provided by people to examine issues is frequently fuzzy and crisp, and the fuzzy set (g FS) has been veri…ed to be very important in multi-criteria decision making. g FS theory was …rst presented by L. A. Zadeh in 1965[46] seeing degrees of membership to progressively asses the membership of elements 1 in a set. After the introduction of g FS, atanassov[1] presented the concept of intuitionistic fuzzy set (g IFS) by considering membership and non-membership degrees of an element to the set. Atanassov et al. extended g IFS to interval valued intuitionistic fuzzy set ( ^ IV IFS)[2]. After the introduction of g IFS alot of similarity measures are proposed by many researchers as the extension of the similarity measures for g FS, as a similarity measure is an important tool for determining similarity between objects. Li and cheng[6] presented similarity measure for g IFS which is the …rst to applied them to pattern recognition. S.K De et al. gave some application of g IFS to medical daignosis[7]. Ye[12] presented cosine similarity measure for g IFS and applied it to medical diagnosis and pattern recognition. Ye[13, 14] also presented cosine similarity measure and vector similarity measures for ^ IV IFS and trapezoidal ] IFN and applied them to multiple criteria decision making problem. Neutrosophic set(g NS) was …rst presented by F. Samarandache[29, 30] which simpli…es the concept of crisp set,g FS, g IFS, ^ IV IFS, and so on. Ing NS its indeterminacy is computed unambiguously and its truth-membership, indeterminacymembership and falsity-membership are signi…ed freely. g NS simpli…es all the above de…ned sets from philosophical point of view. Several similarity measures for g NS has been de…ned by Broumi. S. et al. [31]. Majumdar et al.[26] presented similarity and entropy for g NS: g NS is di¢ cult to apply in real life and engineering problem. To overcome this di¢ culty wang et. al. [36, 37]. introduced the concept of single valued neutrosophic sets / ( ^ SV NSs) and interval neutrosophic sets ( ÎNSs) and provided basic operation and properties of ^ SV NSs and ÎNSs. The ŜV NSs and ÎNSs is a subclass of g NS. Sahin R. et al.[28] de…ned subsethood for ^ SV NSs. Zhang et al[42] also de…ned some operation for ] INS. Ye[15, 16, 17, 18] de…ned correlation coe¢ cient, improved correlation,entropy and similarity measure for ŜV NS and ] INS and gave their application in multiple attribute decision making problems (M̂CDM). Broumi. S. et al. [32, 33] presented correlation and cosine similarity measure for ] INS. Ye[19] presented the concept of simpli…ed neutrosophic sets ( ŜNSs) which contained the concept of ŜV NS and ] INS and de…ned some basic operation for ŜNSs and applied them to multi-criteria decision making problem. Ye [20, 21] also de…ned vector similarity measures for ŜNSs and improved cosine similarity measures for ŜNSs. Peng et al[11] …nd some drawbacks of the operation for ] SNS de…ned by Ye and presented new operations for ] SNS. Hesitant fuzzy set ( ĤFS) is another generalization of g FS proposed by Torra and Narukawa[34, 35]. The advantege of ĤFS is that it authorizations the membership degree of an element to a given set with a limited di¤erent values, which can ascend in a group decision making problem. Recently a ĤFS has acknowledged more and more consideration since its presence. Xia and Xu [38, 39, 40, 41] presented hesitant fuzzy information, similarity measures