The University of New Mexico



Chapter TitleAuthor1, Author2, and Author3Department of Mathematics, Gaziantep University, Gaziantep27310-TurkeyE-mail: adilkilic@gantep.edu.tr, vulucay27@, mesahin@gantep.edu.tr, h.deniz@Abstract In recent times, refined neutrosophic sets introduced by Deli [6] has been one of the most powerful and flexible approaches for dealing with complex and uncertain situations of real world. In particular, the decision making methods between refined neutrosophic sets are important since it has applications in various areas such as image segmentation, decision making, medical diagnosis, pattern recognition and many more. The aim of this chapter is to introduce a new distance-based similarity measure for refined neutrosophic sets. The properties of the proposed new distance-based similarity measure have been studied and the findings are applied in medical diagnosis of some diseases with a common set of symptoms.Keywords: Neutrosophic sets, Refined neutrosophic sets, Similarity measure, Decision making, Medical diagnosis.IntroductionThe vagueness or uncertainty representation of imperfect knowledge becomes a crucial issue in the areas of computer science and artificial intelligence. To deal with the uncertainty, the fuzzy set proposed by Zadeh [39] allows the uncertainty of a set with a membership degree between 0 and 1. Then, Atanassov [1] introduced an intuitionistic Fuzzy set (IFS) as a generalization of the Fuzzy set. The IFS represents the uncertainty with respect to both membership and non-membership. However, it can only handle incomplete information but not the indeterminate and inconsistent information which exists commonly in real situations. Therefore, Smarandache [25] proposed a neutrosophic set. It can independently express truth-membership degree, indeterminacy-membership degree, and false membership degree and deal with incomplete, indeterminate, and inconsistent information. Also, several generalization of the set theories made such as fuzzy multi-set theory [28, 29], intuitionstic fuzzy multi-set theory [20-24] and refined neutrosophic set theory [5, 6, 8-11, 13, 26, 31]. Many research treating imprecision and uncertainty have been developed and studied. Since then, it is applied to various areas, such as decision making problems [2-4, 7, 12, 14-19, 27, 30, 33-37]. Another generalization of above theories that is relevant for our work is single valued neutrosophic refined (multi) set theory by introduced Ye [32, 38] which contain a few different values. A single valued neutrosophic multi set theory have truth-membership sequenceμA1u,μA2u,…μApu, indeterminacy-membership sequence vA1u,vA2u,…vApu and falsity-membership sequence wA1u,wA2u,…wApuof the element u∈U.The chapter is organized as follows: In section 2, introduces some concepts and basic operations are reviewed. In section 3, presents a new distance-based similarity measure for refined neutrosophic sets and investigates their properties. In section 4, the similarity measures are applied to medicine diagnosis. Finally, Conclusions and further research are contained.BackgroundDefinition 1. [25]Let U be a universe. A neutrosophic sets A over U is defined by A=?u,(μAu,vAu,wAu?:u∈U}where, μAu, vAu and wAu are called truth-membership function, indeterminacy-membership function and falsity- membership function, respectively. They are respectively defined byμA:U→]-0,1+[ , vA:U→]-0,1+[ , wA: U→]-0,1+[ such that 0-≤μAu+vAu+wAu≤3+.Definition 2 [31] Let U be a universe. An single valued neutrosophic set (SVN-set) over U is a neutrosophic set over U, but the truth-membership function, indeterminacy-membership function and falsity- membership function are respectively defined byμA:U→[0,1], vA:U→[0,1] , wA: U→[0,1] Such that 0≤μAu+vAu+wAu≤3.Definition 3. [32] Let U be a universe. A neutrosophic multiset set (Nms) A on U can be defined as follows:A={?u,μA1u,μA2u,…μApu,vA1u,vA2u,…vApu,wA1u,wA2u,…wApu?:u∈U}where,μA1u,μA2u,…μApu:U→0,1,vA1u,vA2u,…vApu:U→0,1,andwA1u,wA2u,…wApu:U→0,1such that0≤supμAiu+supvAiu+supwAiu≤3(i=1,2,…,P)andμA1u,μA2u,…,μApu,vA1u,vA2u,…,vApuand wA1u,wA2u,…,wApuIs the truth-membership sequence, indeterminacy-membership sequence and falsity- membership sequence of the element u, respectively. Also, P is called the dimension (cardinality) of Nms A, denoted d(A). We arrange the truth- membership sequence in decreasing order but the corresponding indeterminacy- membership and falsity-membership sequence may not be in decreasing or increasing order.The set of all Neutrosophic multisets on U is denoted by NMSU.Definition 4. [5,32,38] Let A,B∈NMS( U). Then,A is said to be Nm-subset of B is denoted by A?B if μAiu≤μBiu, vAiu≥vBiu, wAiu≥wBiu, ? u∈U and i=1,2,…P. (2) A is said to be neutrosophic equal of B is denoted by A=B if μAiu=μBiu,vAiu=vBiu, wAiu=wBiu, ? u∈U and i=1,2,…P.(3) The complement of A denoted by Ac and is defined by Ac=?u,wA1u,wA2u,…,wApu,vA1u,vA2u,…vApu,μA1u,μA2u,…μApu?:u∈U}(4) If μAiu=0 and vAiu= wAiu=1 for all u∈U and i=1,2,…P, thenA is called null ns-set and denoted by Φ.(5) If μAiu=1 and vAiu= wAiu=0 for all u∈U and i=1,2,…P, thenA is called universal ns-set and denoted by U.(6) The union of A and B is denoted by A∪B=C and is defined by C={?u,μC1u,μC2u,…μCpu,vC1u,vC2u,…vCpu,wC1u,wC2u,…wCpu?:u∈U}Where μCi=μAiu∨μBiu, vCi=vAiu∧vBiu, wCi=wAiu∧wBiu, ? u∈U and i=1,2,…P. (7) The intersection of A and B is denoted by A∩B=D and is defined byD={?u,μD1u,μD2u,…μDpu,vD1u,vD2u,…vDpu,wD1u,wD2u,…wDpu?:u∈U}where μDi=μAiu∨μBiu, vDi=vAiu∧vBiu, wDi=wAiu∧wBiu, ? u∈U and i=1,2,…P.(8) The addition of A and B is denoted by A+B=U1 and is defined byU1={?u,μU11u,μU12u,…μU1pu,vU11u,vU12u,…vU1pu,wU11u,wU12u,…wU1pu?:u∈U}where μU1i=μAiu+μBiu-μAiu.μBiu, vU1i=vAiu.vBiu, wU1i=wAiu.wBiu? u∈U and i=1,2,…P.(9) The multiplication of A and B is denoted by AxB=U2 and is defined byU2={?u,μU21u,μU22u,…μU2pu,vU21u,vU22u,…vU2pu,wU21u,wU22u,…wU2pu?:u∈U}where μU2i=μAiu.μBiu, vU2i=vAiu+vBiu-vAiu.vBiu, wU2i=wAiu+wBiu-wAiu.wBiu ? u∈U and i=1,2,…P.Here ∨, ∧,+,.,- denotes maximum, minimum, addition, multiplication, subtraction of real numbers respectively.Definition 5. [6] LetA={?u,μA1u,μA2u,…μApu,vA1u,vA2u,…vApu,wA1u,wA2u,…wApu?:u∈U}andB={?u,μB1u,μB2u,…μBpu,vB1u,vB2u,…vBpu,wA1u,wA2u,…wApu?:u∈U}And be two NMSs, then the normalized hamming distance between A and B can be defined as follows:dpA,B =13i=1pωiμAui-μBuip+vAui-vBuip+wAui-wBuip1pwhere A,B are two SVNSs p>0, wii=1,2,…,p are the weight of the element xii=1,2,…,p with wi≥0 and i=1nwi=1.In the next section, we will define a new Hybrid Distance-Based Similarity Measures for Refined Neutrosophic Sets (RNSs).Hybrid Distance-Based Similarity Measures for Refined Neutrosophic SetsDefinition 6 For two refined neutrosophic sets A and B in a universe of discourse which are denoted by A={?u,μA1u,μA2u,…μApu,vA1u,vA2u,…vApu,wA1u,wA2u,…wApu?:u∈U}andB={?u,μB1u,μB2u,…μBpu,vB1u,vB2u,…vBpu,wB1u,wB2u,…wBpu?:u∈U}μAiu,μBiu,vAiu,vBiu,wAiu,wBiu∈[0,1] for every ,i=1,2,…P. Let us consider the weight ωi i=1,2,…P with ωi≥0,i=1,2,…P and i=1Pωi=1.The, we define the refined generalized neutrosophic weighted distance measure:dPA,B =13i=1Pωi[μAiui-μBiuiP+vAiui-vBiuiP+wAiui-wBiuiP]1P, (1)where P>0.As the Hamming distance and Euclidean distance, which are two typical distance measures, are usually used in practical applications [11 ye ] when P=1,2. we can obtain the refined neutrosophic weighted Hamming distance and the refined neutrosophic weighted Euclidean distance, respectively, as follows: dA,B =13i=1Pωi[μAiui-μBiui+vAiui-vBiui+wAiui-wBiui, (2)d2A,B =13i=1PωiμAiui-μBiui2+vAiui-vBiui2+wAiui-wBiui212 (3)Therefore, Eqs. (2) and (3) are the special cases of Eq. (1). Then, for the distance measure, we have the following proposition.Proposition 7 The distance measure dpA,B for p>0 satisfies the following properties:(H1) 0≤dpA,B ≤1;(H2)dpA,B =0 if and only if A=B ;(H3)dpA,B =dpB,A ;(H4)If ?B ?C , A3 is a refined neutrosophic in U, then dp(A,B )≤dp(A,C) and dp(B,C)≤dp(A,C).Proof: It is easy to see that dpA,B satisfies the properties (H1)-(H3). Therefore, we only prove (H4). Let A?B ?C, thenμAiu≤μBiu≤μCiu , vAiu≥vBiu≥μCiu, wAiu≥wBiu≥μCiu, ? ui∈U and i=1,2,…P we obtain following relations:μAiui-μBiuiP≤μAiui-μCiuiP;μBiui-μCiuiP≤μAiui-μCiuiP,vAiui-vBiuiP≤vAiui-vCiuiP;vBiui-vCiuiP≤vAiui-vCiuiP ,wAiui-wBiuiP≤wAiui-wCiuiP;wBiui-wCiuiP≤wAiui-wCiuiP,Hence,μAiui-μBiuiP+vAiui-vBiuiP+wAiui-wBiuiP≤μAiui-μCiuiP+vAiui-vCiuiP+wAiui-wCiuiP,μBiui-μCiuiP+vBiui-vCiuiP+wBiui-wCiuiP≤μAiui-μCiuiP+vAiui-vCiuiP+wAiui-wCiuiPdλ(A,C) ≥dλ(A,B ) and dλ(A,C) ≥dλ(B,C) for λ>0. Example 8: Assume that we have the following three refined neutrosophic weighted Hamming distance and the refined neutrosophic weighted Euclidean distance, in a universe of discourse u∈U: let ω1=0.2, ω2=0.4, ω3=0.4. A=u,0.8,0.5,0.6,0.3,0.1,0.5,(0.2,0.3,0.4)B=u,0.5,0.7,0.6,0.2,0.3,0.4,(0.1,0.3,0.2)dA,B =13i=1PωiμAiui-μBiui+vAiui-vBiui+wAiui-wBiui =13ω1μA1u1-μB1u1+vA1u1-vB1u1+wA1u1-wB1u1 +ω2μA2u2-μB2u2+vA2u2-vB2u2+wA2u2-wB2u2+ω3μA3u3-μB3u3+vA3u3-vB3u3+wA3u3-wB3u3 =13[0.20.8-0.5+0.3-0.2+0.2-0.1+0.40.5-0.7+0.1-0.3+0.3-0.3+0.4(0.6-0.6+0.5-0.4+0.4-0.2)] =130.20.3+0.1+0.1+0.40.2+0.2+0.40.1+0.2dA,B =0,127d2A,B =13i=1Pωi[μAiui-μBiui2+vAiui-vBiui2+wAiui-wBiui2]12 =13[ω1μA1u1-μB1u12+vA1u1-vB1u12+wA1u1-wB1u12+ω2μA2u2-μB2u22+vA2u2-vB2u22+wA2u2-wB2u22+ω3(μA3u3-μB3u32+vA3u3-vB3u32+wA3u3-wB3u32)]12 =13[0.20.8-0.52+0.3-0.22+0.2-0.12+0.40.5-0.72+0.1-0.32+0.3-0.32+0.4(0.6-0.62+0.5-0.42+0.4-0.22)]12 =130,20,09+0,01+0,01+0,40,04+0,04+0,40,01+0,0412d2A,B =0,157Definition 9 LetA={?u,μA1u,μA2u,…μApu,vA1u,vA2u,…vApu,wA1u,wA2u,…wApu?:u∈U}andB={?u,μB1u,μB2u,…μBpu,vB1u,vB2u,…vBpu,wB1u,wB2u,…wBpu?:u∈U}be two refined neutrosophic sets. Then, hybrid similarity measure between refined neutrosophic sets A and B, denoted HybdA,B =φ13i=1PωiμAiui-μBiui+vAiui+vBiui+wAiui-wBiui+1-φ13i=1PωiμAiui-μBiui2+vAiui+vBiui2+wAiui-wBiui212, i=1,2,…,P Note that similarity and distance (dissimilarity) measures are complementary: when the first increases, the second decreases. Normalized distance measure and similarity measure below are dual concepts. Thus,δA,B=1-HybdA,B and vice versa. The properties of distance measures below are complementary to those of similarity measures.Proposition 10 The similarity measure δpA,B for p>0 satisfies the following properties;(HD1) 0≤δpA,B≤1;(HD2) δpA,B=1 if and only if A=B;(HD3) δpA,B=δpB,A;(HD4) If ?B ?C , C is a refined neutrosophic in U, then δpA,C≤δpA,B and δpA,C≤δpB,C.Assume that there are two refined neutrosophic A={?u,μA1u,μA2u,…μApu,vA1u,vA2u,…vApu,wA1u,wA2u,…wApu?:u∈U}andB={?u,μB1u,μB2u,…μBpu,vB1u,vB2u,…vBpu,wB1u,wB2u,…wBpu?:u∈U}in a universe of distance ? ui∈U. Thus, according to the relationship between the distance and the similarity measure, we can obtain the following refined neutrosophic similarity measure:δA,B=1-HybdA,B =1-φ13i=1PωiμAiui-μBiui+vAiui+vBiui+wAiui-wBiui+ 1-φ13i=1PωiμAiui-μBiui2+vAiui+vBiui2+wAiui-wBiui212 (4)Obviously, we can easily prove that δ1A,B satisfied the properties (HD1)-(HD4) in proposition 10 by the relationship between the distance and the similarity measure and the proof of proposition 7, which is omitted here. Furthermore, we can also propose another refined neutrosophic similarity measure:δ2A,B=1-HybdA,B 1+HybdA,B =1-φ13i=1PωiμAiui-μBiui+vAiui+vBiui+wAiui-wBiui+1-φ13i=1PωiμAiui-μBiui2+vAiui+vBiui2+wAiui-wBiui2121+φ13i=1PωiμAiui-μBiui+vAiui+vBiui+wAiui-wBiui+1-φ13i=1PωiμAiui-μBiui2+vAiui+vBiui2+wAiui-wBiui212 (5)Then, the similarity measure δ2A,B also satisfied the properties (HD1)-(HD4) in Proposition 2.Proof: It is easy to see that δ2A,B satisfies the properties (HD1)-(HD3). Therefore, we only property (HD4).As we obtain δpA,B≤δpA,C and δpB,C≤δpA,C for p>0 from the property (H4) in proposition 1, there are 1-δpA,B≥1-δpA,C, 1-δpB,C≥1-δpA,C 1+δpA,B≤1+δpA,C and 1+δpB,C≤1+δpA,C. Then, there are the following inequalities:1-dp(A,B )1+dp(A,B )≥1-dpA,C 1+dpA,C and 1-dpB, C1+dpB,C ≥1-dpA,C 1+dpA,C .Then, there are δpA,C≤δpA,B and δpA,C≤δpB,C. Hence, the property (HD4) is satisfied.Example 11: Assume that we have the following two refined neutrosophic hybrid similarity measure in a universe of discourse u∈U; let ω1=0.2, ω2=0.3, ω3=0.5.A=u,0.5,0.5,0.3,0.8,0.1,0.2,(0.2,0.8,0.9)B=u,0.5,0.1,0.6,0.2,0.3,0.9,(0.6,0.3,0.2)HybdA,B =φ13i=1PωiμAiui-μBiui+vAiui+vBiui+wAiui-wBiui+1-φ13i=1PωiμAiui-μBiui2+vAiui+vBiui2+wAiui-wBiui212=φ13ω1μA1u1-μB1u1+vA1u1-vB1u1+wA1u1-wB1u1 +ω2μA2u2-μB2u2+vA2u2-vB2u2+wA2u2-wB2u2+ω3μA3u3-μB3u3+vA3u3-vB3u3+wA3u3-wB3u3+(1-φ)13[ω1μA1u1-μB1u12+vA1u1-vB1u12+wA1u1-wB1u12+ω2μA2u2-μB2u22+vA2u2-vB2u22+wA2u2-wB2u22+ω3(μA3u3-μB3u32+vA3u3-vB3u32+wA3u3-wB3u32)]12 =0.3130.20.5-0.5+0.8-0.2+0.2-0.6 +0.30.5-0.1+0.1-0.3+0.8-0.3+0.50.3-0.6+0.2-0.9+0.9-0.2+0.713[0.20.5-0.52+0.8-0.22+0.2-0.62+0.30.5-0.12+0.1-0.32+0.8-0.32+0.5(0.3-0.62+0.2-0.92+0.9-0.22)]12=0.3130,20,6+0,4+0,30,4+0,2+0,5+0,50,3+0,7+0,7 +0.7130,20,36+0,16+0,30,16+0,04+0,25+0,50,09+0,49+0,4912HybdA,B =0.4935δA,B=1-HybdA,B δA,B=0.5065Medical diagnosis using the Hybrid similarity measureWe consider a medical diagnosis problem from practical point of view for illustration of the proposed approach. Medical diagnosis comprises of uncertainties and increased volume of information available to physicians from new medical technologies. The process of classifying different set of symptoms under a single name of a disease is very difficult task. In some practical situations, there exists possibility of each element within a lower and an upper approximation of refined neutrosophic sets. It can deal with the medical diagnosis involving more indeterminacy. Actually this approach is more flexible and easy to use. The proposed similarity measure among the patients versus symptoms and symptoms versus diseases will provide the proper medical diagnosis. The main feature of this proposed approach is that it considers truth membership, indeterminate and false membership of each element between two approximations of refined neutrosophic sets by taking one time inspection for diagnosis.Now, an example of a medical diagnosis is presented. Let P={Erol,Harun,Deniz} be a set of patients, D={Viral Fever, Tuberculosis, Throat disease} be a set of diseases and S={Throat pain, headache, body pain} be a set of symptoms. Our solution is to examine the patient at different time intervals (three times a day), which in turn give arise to different truth membership, indeterminate and false membership function for each patient. Let ω1=0.2, ω2=0.5, ω3=0.3Table 1: Q (the relation Between Patient and Symptoms)QThroat painHeadacheBody PainErol0.1,0.3,0.6,0.3,0.8,0.3,(0.3,0.3,0.4)0.8,0.2,0.9,0.7,0.4,0.4,(0.2,0.7,0.5)0.9,0.5,0.6,0.3,0.3,0.5,(0.2,0.3,0.4)0.1,0.5,0.6,0.3,0.5,0.8,(0.9,0.3,0.7)0.5,0.5,0.5,0.6,0.3,0.5,(0.4,0.3,0.4)0.6,0.7,0.8,0.5,0.3,0.5,(0.5,0.4,0.1)0.7,0.2,0.1,0.5,0.6,0.3,(0.2,0.3,0.4)0.5,0.5,0.3,0.3,0.4,0.5,(0.3,0.2,0.2)0.1,0.3,0.1,0.6,0.9,0.3,(0.2,0.4,0.4)Harun0.6,0.5,0.5,0.5,0.4,0.5,(0.4,0.3,0.4)0.3,0.5,0.6,0.6,0.9,0.4,(0.5,0.3,0.9)0.8,0.7,0.1,0.8,0.7,0.5,(0.3,0.7,0.4)0.1,0.5,0.6,0.6,0.7,0.5,(0.3,0.3,0.4)0.3,0.3,0.6,0.4,0.4,0.7,(0.3,0.3,0.9)0.8,0.5,0.6,0.5,0.8,0.5,(0.8,0.5,0.4)0.8,0.6,0.6,0.5,0.6,0.4,(0.4,0.3,0.5)0.5,0.2,0.3,0.9,0.6,0.5,(0.9,0.3,0.4)0.8,0.5,0.7,0.6,0.5,0.4,(0.5,0.3,0.4)Deniz0.8,0.6,0.6,0.6,0.6,0.6,(0.6,0.7,0.5)0.1,0.3,0.6,0.8,0.8,0.7,(0.8,0.5,0.9)0.8,0.5,0.6,0.6,0.7,0.9,(0.6,0.9,0.5)0.1,0.5,0.6,0.9,0.6,0.8,(0.9,0.5,0.7)0.7,0.4,0.2,0.9,0.8,0.8,(0.6,0.8,0.5)0.9,0.5,0.7,0.6,0.8,0.7,(0.5,0.8,0.6)0.4,0.5,0.6,0.7,0.6,0.6,(0.7,0.8,0.6)0.5,0.2,0.3,0.8,0.9,0.7,(0.9,0.5,0.5)0.7,0.2,0.1,0.8,0.6,0.9,(0.5,0.9,0.9)Let the samples be taken at three different timings in a day (in 07:00, 15:00 and 23:00)Table 2: R (the relation among Symptoms and Diseases)RViral FeverTuberculosisTyphoidThroat pain0.2,0.5,0.6,0.3,0.1,0.6,(0.2,0.8,0.4)0.6,0.5,0.7,0.2,0.1,0.5,(0.4,0.3,0.4)0.1,0.5,0.6,0.3,0.1,0.2,(0.5,0.3,0.7)Headache0.4,0.5,0.6,0.2,0.7,0.5,(0.7,0.4,0.1)0.7,0.2,0.1,0.5,0.7,0.3,(0.3,0.2,0.4)0.3,0.2,0.3,0.3,0.1,0.8,(0.9,0.3,0.4)Body Pain0.9,0.5,0.7,0.6,0.9,0.1,(0.5,0.9,0.4)0.7,0.5,0.6,0.3,0.1,0.5,(0.5,0.3,0.9)0.8,0.5,0.6,0.3,0.1,0.5,(0.2,0.3,0.4)Table 3: The Hamming weighted distance refined neutrosophic sets Q and RHamming weightedViral FeverTuberculosisTyphoidErol0,2311110,2222220,323333Harun0,1744440,2222220,342222Deniz0,3166670,2111110,364444Optimal-Erol(Tuberculosis); Harun( Viral Fever); Deniz( Tuberculosis)Table 4: The weighted Euclidean distance refined neutrosophic sets Q and RWeighted Euclidean distanceViral FeverTuberculosisTyphoidErol0,0109570,015990,027775Harun0,0031310,0067280,027159Deniz0,0233680,0080540,032489Optimal-Erol(Viral Fever); Harun( Viral Fever); Deniz( Tuberculosis)Table 5: The Weighted hybrid distance refined neutrosophic sets Q and R. let φ1+=0.43Weighted hybridViral FeverTuberculosisTyphoidErol0,1056230,1335420,196243Harun0,100780,129560,206745Deniz0,1905480,1237970,221704Optimal-Erol(Viral Fever); Harun( Viral Fever); Deniz( Tuberculosis)Table 6: The Weighted similarity Measure hybrid distance refined neutrosophic sets Q and R. let φ1+=0.43.Weighted similarity Measure hybridViral FeverTuberculosisTyphoidErol0,8943770,8664580,803757Harun0,899220,870440,793255Deniz0,8094520,8762030,778296Optimal-Erol(Viral Fever); Harun( Viral Fever); Deniz( Tuberculosis)Table 7: The Weighted similarity Measure hybrid distance refined neutrosophic sets Q and R. let φ2+=0.5.Weighted similarity Measure hybridViral FeverTuberculosisTyphoidErol0,8789660,8808940,824446Harun0,9112120,8855250,815309Deniz0,8299830,8904170,801533Optimal-Erol(Tuberculosis); Harun( Viral Fever); Deniz( Tuberculosis)Table 8: The Weighted similarity Measure hybrid distance refined neutrosophic sets Q and R. let φ2+=0.67Weighted similarity Measure hybridViral FeverTuberculosisTyphoidErol0,841540,9159530,874691Harun0,9403350,9221590,86887Deniz0,8798430,9249370,857965Optimal-Erol(Tuberculosis); Harun( Viral Fever); Deniz( Tuberculosis)ConclusionsIn this paper, a new hybrid similarity measure and a weighted hybrid similarity measure for refined neutrosophic sets are presented and some of its basic properties are discussed. The proposed hybrid similarity measure enriches the theories and techniques for measuring the degree of hybrid similarity between refined neutrosophic sets. This measure greatly reduces the influence of imprecise measures and provides an extremely intuitive quantification. The effectiveness of the proposed hybrid similarity measure is demonstrated in a numerical example with the help of measure of performance and measure of error. Moreover, medical diagnosis problems have been exhibited through a hypothetical case study by using this proposed hybrid similarity measure. The authors hope that the proposed concept can be applied in solving realistic multi-criteria decision making problems.Future Research Directions …………………………………………………………………..………………………………………………………………………………References[1] Atanassov, K. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20 87-96.[2] Athar, K. (2014). A neutrosophic multi-criteria decision making method. New Mathematics and Natural Computation, 10(02), 143-162.[3] Broumi S. and F. Smarandache, (2013). Several similarity measures of neutrosophic sets, Neutrosophic Sets and Systems 1(1) 54-62.[4] Chatterjee, R. P. Majumdar, S. K. Samanta, Single valued neutrosophic multisets, Annals of FuzzyMathematics and Informatics, x/x (2015) xx–xx.[5] Deli, I., Broumi, S., Ali, M. (2014). Neutrosophic Soft Multi-Set Theory and Its Decision Making. Neutrosophic Sets and Systems, 5, 65-76.[6] Deli, I. (2016). Refined Neutrosophic Sets and Refined Neutrosophic Soft Sets: Theory and Applications. Handbook of Research on Generalized and Hybrid Set Structures and Applications for Soft Computing, 321-343.[7] Deli, I., Broumi S (2015) Neutrosophic Soft Matrices and NSM-decision Making. Journal of Intelligent and Fuzzy Systems, 28: 2233-2241.[8] Deli,I., S. Broumi, F. Smarandache, On neutrosophic refined sets and their applications in medical diagnosis, Journal of New Theory, 6 (2015) 88-98.[9] Deli, I., Eraslan, S., & ?a?man, N. (2016). ivnpiv-Neutrosophic soft sets and their decision making based on similarity measure.?Neural Computing and Applications, 1-17.[10] Fua, J., & Yeb, J. Simplified neutrosophic exponential similarity measures for the initial evaluation/diagnosis of benign prostatic hyperplasia symptom.[11] Fathi, S., ElGhawalby, H., & Salama, A. A. (2016). A Neutrosophic Graph Similarity Measures.?New Trends in Neutrosophic Theories and Applications,?1.[12] Karaaslan, F. (2016). Correlation Coefficient between Possibility Neutrosophic Soft Sets. Math. Sci. Lett. 5/1, 71-74.[13] Karaaslan, F. (2016). Correlation coefficients of single-valued neutrosophic refined soft sets and their applications in clustering analysis. Neural Computing and Applications, 1-13.[14] Liu, P., & Wang, Y. (2014). Multiple attribute decision-making method based on single-valued neutrosophic normalized weighted Bonferroni mean.?Neural Computing and Applications,?25(7-8), 2001-2010.[15] Mondal, K., Pramanik, S. (2015). Neutrosophic tangent similarity measure and its application to multiple attribute decision making. Neutrosophic Sets and Systems, 9, 85-92.[16] P?tra?cu, V. (2016). Refined Neutrosophic Information Based on Truth, Falsity, Ignorance, Contradiction and Hesitation.?Neutrosophic Sets and Systems, 11:57-65. [17] Pramanik, S., & Mondal, K. (2015). Cotangent similarity measure of rough neutrosophic sets and its application to medical diagnosis.?Journal of New Theory,?4, 90-102.[18] Pramanik, S., Biswas, P., Giri, B. C. (2015). Hybrid vector similarity measures and their applications to multi-attribute decision making under neutrosophic environment. Neural computing and Applications, 1-14.[19] Pramanik, S., Mondal, K. (2015). Cosine similarity measure of rough neutrosophic sets and its application in medical diagnosis. Global Journal of Advanced Research, 2(1), 212-220.[20] P. Rajarajeswari, N. Uma, Correlation Measure For Intuitionistic Fuzzy Multi Sets, International Journalof Research in Engineering and Technology, 3(1) (2014) 611-617.[21] Rajarajeswari P. and N. Uma, On Distance and Similarity Measures of Intuitionistic Fuzzy Multi Set, IOSR Journal of Mathematics, 5(4) (2013) 19–23.[22] Rajarajeswari P. and N. Uma, A Study of Normalized Geometric and Normalized Hamming Distance Measures in Intuitionistic Fuzzy Multi Sets, International Journal of Science and Research, Engineering and Technology, 2(11) (2013) 76–80.[23] Rajarajeswari, P. N. Uma, Intuitionistic Fuzzy Multi Relations, International Journal of Mathematical Archives, 4(10) (2013) 244-249.[24] Rajarajeswari P. and N. Uma, Zhang and Fu’s Similarity Measure on Intuitionistic Fuzzy Multi Sets, International Journal of Innovative Research in Science, Engineering and Technology, 3(5) (2014) 12309–12317.[25] Smarandache, F. (1998) A Unifying Field in Logics Neutrosophy: Neutrosophic Probability, Set and Logic. Rehoboth: American Research Press.[26] Smarandache, F. (2013) n-Valued Refined Neutrosophic Logic and Its Applications in Physics, Progress in Physics, 4; 143-146.[27] Smarandache, F. (2005) Neutrosophic set, a generalisation of the intuitionistic fuzzy sets. Int. J Pure Appl Math 24:287-297.[28] Shinoj T. K. and S. J. John, Intuitionistic fuzzy multisets and its application in medical diagnosis, World Academy of Science, Engineering and Technology, 6 (2012) 01–28.[29] Sebastian S. and T. V. Ramakrishnan, Multi-Fuzzy Sets, International Mathematical Forum, 5(50)(2010) 2471–2476.[30] Sahin,M., I. Deli, V. Ulucay, (2016). Jaccard Vector Similarity Measure of Bipolar Neutrosophic Set Based on Multi-Criteria Decision Making, International Conference on Natural Science and Engineering (ICNASE'16), March 19-20, Kilis. [31] Wang H, Smarandache FY, Q. Zhang Q, Sunderraman R (2010). Single valued neutrosophic sets. Multispace and Multistructure 4:410-413[32] Ye,S., and J. Ye, (2014). Dice Similarity Measure between Single Valued Neutrosophic Multisets anf Its Application in Medical Diagnosis, Neutrosophic Sets and Systems, 6, 49-54.[33] Ye, J. (2014). Vector similarity measures of simplified neutrosophic sets and their application in multicriteria decision making. International Journal of Fuzzy Systems, 16(2), 204-215.[34] Ye J (2014). Improved correlation coefficients of single valued neutrosophic sets and interval neutrosophic sets for multiple attribute decision making. J Intell Fuzzy Syst 27:2453-2462[35] Ye, J., Zhang, Q. S. (2014). Single valued neutrosophic similarity measures for multiple attribute decision making. Neutrosophic Sets and Systems, 2, 48-54.[36] Ye,J., and J. Fub, Multi-period medical diagnosis method using a single-valued neutrosophic similarity measure based on tangent function, computer methods and programs in biomedicine doi: 10.1016/j.cmpb.2015.10.002.[37] Ye,J., Single-valued neutrosophic similarity measures based on cotangent function and their application in the fault diagnosis of steam trbine, Soft Computing, DOI 10.1007/s00500-015- 1818-y.[38] Ye, S., Fu, J., Ye, J. (2015). Medical Diagnosis Using Distance-Based Similarity Measures of Single Valued Neutrosophic Multisets. Neutrosophic Sets and Systems, 7, 47-52.[39] L.A. Zadeh, (1965). Fuzzy Sets, Inform. and Control 8: 338-353. ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download