Book Series, Vol. 20, 2018 Florentin Smarandache and Surapati Pramanik ISBN 978-1-59973-560-3 ISBN 978-1-59973-560-3 Neutrosophic Sets and Systems An International Book Series in Information Science and Engineering Copyright Notice Copyright @ Neutrosophics Sets and Systems All rights reserved. The authors of the articles do hereby ademic or individual use can be made by any user without grant Neutrosophic Sets and Systems non-exclusive, permission or charge. The authors of the articles published worldwide, royalty-free license to publish and distribute in Neutrosophic Sets and Systems retain their rights to use the articles in accordance with the Budapest Open Initia- this book as a whole or any part of it in any other publi- tive: this means that electronic copying, distribution and cations and in any way they see fit. 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Neutrosophic Sets and Systems, 20/2018 Neutrosophic Sets and Systems An International Book Series in Information Science and Engineering (cid:40)(cid:71)(cid:76)(cid:87)(cid:82)(cid:85)s(cid:16)(cid:76)(cid:81)(cid:16)(cid:38)(cid:75)(cid:76)(cid:72)(cid:73)(cid:29)(cid:3) (cid:40)(cid:71)(cid:76)(cid:87)(cid:82)(cid:85)(cid:86)(cid:29)(cid:3) Prof. Florentin Smarandache, PhD, Postdoc, W. B. Vasantha Kandasamy, Indian Institute of Technology, Chennai, Tamil Nadu, India. Math Department, University of New Mexico, A. A. Salama, Faculty of Science, Port Said University, Egypt. Gallup, NM 87301, USA. Yanhui Guo, University of Illinois at Springfield, One Univ. Plaza, Springfield, IL 62703, USA. Young Bae Jun, Gyeongsang National University, South Korea. Dr. Surapati Pramanik, Assistant Professor, Francisco Gallego Lupianez, Universidad Complutense, Madrid, Spain. Department of Mathematics, Nandalal Peide Liu, Shandong University of Finance and Economics, China. Ghosh B.T. College, Panpur, Narayanpur, Pabitra Kumar Maji, Math Department, K. N. University, India. Dist-North 24 Parganas, West Bengal, S. A. Albolwi, King Abdulaziz Univ., Jeddah, Saudi Arabia. India-743126 Jun Ye, Shaoxing University, China. Madad Khan, Comsats Institute of Information Technology, Abbottabad, Pakistan. (cid:36)(cid:86)(cid:86)(cid:82)(cid:70)(cid:76)(cid:68)(cid:87)(cid:72)(cid:3)(cid:40)(cid:71)(cid:76)(cid:87)(cid:82)(cid:85)(cid:86)(cid:29)(cid:3) Stefan Vladutescu, University of Craiova, Romania. Said Broumi, University of Hassan II, Valeri Kroumov, Okayama University of Science, Japan. Dmitri Rabounski and Larissa Borissova, independent researchers. Casablanca, Morocco. Selcuk Topal, Mathematics Department, Bitlis Eren University, Turkey. Mohamed Abdel-Baset, Faculty of Computers Luige Vladareanu, Romanian Academy, Bucharest, Romania. and Informatics, Zagazig University, Egypt. Ibrahim El-henawy, Faculty of Computers and Informatics, Zagazig University, Egypt. A. A. A. 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(cid:36)(cid:71)(cid:71)(cid:85)(cid:72)(cid:86)(cid:86)(cid:29)(cid:3) Victor Christianto, Malang Institute of Agriculture (IPM), Malang, Indonesia. (cid:179)(cid:49)(cid:72)(cid:88)(cid:87)(cid:85)(cid:82)(cid:86)(cid:82)(cid:83)(cid:75)(cid:76)(cid:70)(cid:3)(cid:54)(cid:72)(cid:87)(cid:86)(cid:3)(cid:68)(cid:81)(cid:71)(cid:3)(cid:54)(cid:92)(cid:86)(cid:87)(cid:72)(cid:80)(cid:86)(cid:180)(cid:3) Ganeshsree Selvachandran, UCSI University, Jalan Menara Gading, Kuala Lumpur, Malaysia. (cid:11)(cid:36)(cid:81)(cid:3) (cid:44)(cid:81)(cid:87)(cid:72)(cid:85)(cid:81)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:68)(cid:79)(cid:3) (cid:45)(cid:82)(cid:88)(cid:85)(cid:81)(cid:68)(cid:79)(cid:3) (cid:76)(cid:81)(cid:3) (cid:44)(cid:81)(cid:73)(cid:82)(cid:85)(cid:80)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3) Saeid Jafari, College of Vestsjaelland South, Slagelse, Denmark. (cid:54)(cid:70)(cid:76)(cid:72)(cid:81)(cid:70)(cid:72)(cid:3)(cid:68)(cid:81)(cid:71)(cid:3)(cid:40)(cid:81)(cid:74)(cid:76)(cid:81)(cid:72)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:12)(cid:3) Paul Wang, Pratt School of Engineering, Duke University, USA. (cid:39)(cid:72)(cid:83)(cid:68)(cid:85)(cid:87)(cid:80)(cid:72)(cid:81)(cid:87)(cid:3)(cid:82)(cid:73)(cid:3)(cid:48)(cid:68)(cid:87)(cid:75)(cid:72)(cid:80)(cid:68)(cid:87)(cid:76)(cid:70)(cid:86)(cid:3)(cid:68)(cid:81)(cid:71)(cid:3)(cid:54)(cid:70)(cid:76)(cid:72)(cid:81)(cid:70)(cid:72)(cid:3) Arun Kumar Sangaiah, VIT University, Vellore, India. (cid:56)(cid:81)(cid:76)(cid:89)(cid:72)(cid:85)(cid:86)(cid:76)(cid:87)(cid:92)(cid:3)(cid:82)(cid:73)(cid:3)(cid:49)(cid:72)(cid:90)(cid:3)(cid:48)(cid:72)(cid:91)(cid:76)(cid:70)(cid:82)(cid:3) Kul Hur, Wonkwang University, Iksan, Jeollabukdo, South Korea. (cid:26)(cid:19)(cid:24)(cid:3)(cid:42)(cid:88)(cid:85)(cid:79)(cid:72)(cid:92)(cid:3)(cid:36)(cid:89)(cid:72)(cid:81)(cid:88)(cid:72)(cid:3) Darjan Karabasevic, University Business Academy, Novi Sad, Serbia. (cid:42)(cid:68)(cid:79)(cid:79)(cid:88)(cid:83)(cid:15)(cid:3)(cid:49)(cid:48)(cid:3)(cid:27)(cid:26)(cid:22)(cid:19)(cid:20)(cid:15)(cid:3)(cid:56)(cid:54)(cid:36)(cid:3) Dragisa Stanujkic, John Naisbitt University, Belgrade, Serbia. E. K. Zavadskas, Vilnius Gediminas Technical University, Vilnius, Lithuania. (cid:40)(cid:16)(cid:80)(cid:68)(cid:76)(cid:79)(cid:29)(cid:3)(cid:86)(cid:80)(cid:68)(cid:85)(cid:68)(cid:81)(cid:71)(cid:35)(cid:88)(cid:81)(cid:80)(cid:17)(cid:72)(cid:71)(cid:88)(cid:3) M. Ganster, Graz University of Technology, Graz, Austria. (cid:43)(cid:82)(cid:80)(cid:72)(cid:3)(cid:83)(cid:68)(cid:74)(cid:72)(cid:29)(cid:3)(cid:75)(cid:87)(cid:87)(cid:83)(cid:29)(cid:18)(cid:18)(cid:73)(cid:86)(cid:17)(cid:74)(cid:68)(cid:79)(cid:79)(cid:88)(cid:83)(cid:17)(cid:88)(cid:81)(cid:80)(cid:17)(cid:72)(cid:71)(cid:88)(cid:18)(cid:49)(cid:54)(cid:54)(cid:3)(cid:3) Willem K. M. Brauers, Faculty of Applied Economics, University of Antwerp, Antwerp, Belgium. (cid:57)(cid:82)(cid:79)(cid:88)(cid:80)(cid:72)(cid:3)20 (cid:21)(cid:19)(cid:20)8 (cid:38)(cid:82)(cid:81)(cid:87)(cid:72)(cid:81)(cid:87)(cid:86)(cid:3) Kalyan Mondal, Surapati Pramanik, Bibhas C. Giri. Sin- gle Valued Neutrosophic Hyperbolic Sine Similarity Tuhin Bera, Nirmal K. Mahapatra. On Neutrosophic Measure Based MADM Strategy ………………..…....… 3 Soft Prime Ideal …...…………………............................. 54 Kalyan Mondal, Surapati Pramanik, Bibhas C. Giri. Hy- Emad Marei. Single Valued Neutrosophic Soft brid Binary Logarithm Similarity Measure for MAGDM Approach to Rough Sets, Theory and Application …… 76 Problems under SVNS Assessments ………….....……... 12 M. Lellis Thivagar, Saeid Jafari, V. Sutha Devi, V. An- Seon Jeong Kim, Seok-Zun Song, Young Bae Jun. Gen- tonysamy. A novel approach to nano topology via neu- eralizations of Neutrosophic Subalgebras in BCK/BCI- trosophic sets ………....................................................… 86 Algebras Based on Neutrosophic Points ……………..… 26 S. Pramanik, Shyamal Dalapati, Shariful Alam, T. K. G. Muhiuddin, Hashem Bordbar, F. Smarandache, Roy: NC-VIKOR Based MAGDM Strategy under Neu- Young Bae Jun. Further results on (ԑ, ԑ)-neutrosophic trosophic Cubic Set Environment....…………………….. 95 subalgebras and ideals in BCK/BCI-algebras ……...…… 36 Surapati Pramanik, Rama Mallick, Anindita Dasgupta: Young Bae Jun, F. Smarandache, Mehmat Ali Ozturk. Contributions of Selected Indian Researchers to Multi- Commutative falling neutrosophic ideals in BCK- Attribute Decision Making in Neutrosophic Environ- algebras …………………………………….....………… 44 ment: An Overview ……………………......................… 109 Copyright © Neutrosophic Sets and Systems Neutrosophic Sets and Systems, Vol. 20, 2018 3 University of New Mexico Single Valued Neutrosophic Hyperbolic Sine Similarity Measure Based MADM Strategy Kalyan Mondal1, Surapati Pramanik2, and Bibhas C. Giri3 1Department of Mathematics, Jadavpur University, Kolkata: 700032, West Bengal, India. E mail:kalyanmathematic@gmail.com ²Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P O - Narayanpur, and District: North 24 Parganas, Pin Code: 743126, West Bengal, India. Email: sura_pati@yahoo.co.in, 3Department of Mathematics, Jadavpur University, Kolkata: 700032, West Bengal, India. Email: bibhasc.giri@jadavpuruniversity.in Abstract: In this paper, we introduce new type of similarity making strategy for single valued neutrosophic set based on the measures for single valued neutrosophic sets based on hyperbolic proposed weighted similarity measure. We present a numerical sine function. The new similarity measures are namely, single example to verify the practicability of the proposed strategy. valued neutrosophic hyperbolic sine similarity measure and Finally, we present a comparison of the proposed strategy with weighted single valued neutrosophic hyperbolic sine similarity the existing strategies to exhibit the effectiveness and practicality measure. We prove the basic properties of the proposed of the proposed strategy. similarity measures. We also develop a multi-attribute decision- Keywords: Single valued neutrosophic set, Hyperbolic sine function, Similarity measure, MADM, Compromise function 1 Introduction sets studied by Xu [28], Papakostas et al. [29], Biswas and Smarandache [1] introduced the concept of neutrosophic Pramanik [30], Mondal and Pramanik [31], etc. However, set (NS) to deal with imprecise and indeterminate data. In these strategies are not capable of dealing with the similari- the concept of NS, truth-membership, indeterminacy- membership, and falsity-membership are independent. In- ty measures involving indeterminacy. SVNS can handle determinacy plays an important role in many real world this situation. In the literature, few studies have addressed decision-making problems. NS generalizes the Cantor set similarity measures for neutrosophic sets and single valued discovered by Smith [2] in 1874 and introduced by neutrosophic sets [32, 33, 34, 35]. German mathematician Cantor [3] in 1883, fuzzy set Ye [36] proposed an MADM method with completely introduced by Zadeh [4], intuitionistic fuzzy set proposed unknown weights based on similarity measures under by Atanassov [5]. Wang et al. [6] introduced the concept SVNS environment. Ye [37] proposed vector similarity of single valued neutrosophic set (SVNS) that is the sub- measures of simplified neutrosophic sets and applied it in class of a neutrosophic set. SVNS is capable to represent multi-criteria decision making problems. Ye [38] imprecise, incomplete, and inconsistent information that developed improved cosine similarity measures of manifest the real world. simplified neutrosophic sets for medical diagnosis. Ye [39] Neutrosophic sets and its various extensions have been also proposed exponential similarity measure of studied and applied in different fields such as medical neutrosophic numbers for fault diagnoses of steam turbine. diagnosis [7, 8, 9], decision making problems [10, 11, 12, Ye [40] developed clustering algorithms based on 13, 14], social problems [15, 16], educational problem [17, similarity measures for SVNSs. Ye and Ye [41] proposed 18], conflict resolution [19], image processing [ 20, 21, Dice similarity measure between single valued 22], etc. neutrosophic multisets. Ye et al. [42] proposed distance- The concept of similarity is very important in studying based similarity measures of single valued neutrosophic almost every scientific field. Many strategies have been multisets for medical diagnosis. Ye and Fu [43] developed proposed for measuring the degree of similarity between a single valued neutrosophic similarity measure based on tangent function for multi-period medical diagnosis. fuzzy sets studied by Chen [23], Chen et al. [24], Hyung et In hybrid environment Pramanik and Mondal [44] al. [25], Pappis and Karacapilidis [26], Pramanik and Roy proposed cosine similarity measure of rough neutrosophic [27], etc. Several strategies have been proposed for meas- sets and provided its application in medical diagnosis. uring the degree of similarity between intuitionistic fuzzy Pramanik and Mondal [45] also proposed cotangent Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Single Valued Neutrosophic Hyperbolic Sine Similarity Measure based MADM Strategy 4 Neutrosophic Sets and Systems, Vol. 20, 2018 2 Neutrosophic preliminaries similarity measure of rough neutrosophic sets and its 2.1 Neutrosophic set (NS) application to medical diagnosis. Definition 2.1 [1] Let U be a universe of discourse. Then Research gap: MADM strategy using similarity measure the neutrosophic set P can be presented of the form: based on hyperbolic sine function under single valued P = {< x:T (x ), I (x ), F (x)> | x U}, where the P P P neutrosophic environment is yet to appear. functions T, I, F: U→ ]−0,1+[ define respectively the degree of membership, the degree of indeterminacy, and Research questions: the degree of non-membership of the element xU to the  Is it possible to define a new similarity measure set P satisfying the following the condition. between single valued neutrosophic sets using hyper- −0 ≤ supT (x) + supI ( x) + supF (x) ≤ 3+ P P P bolic sine function?  Is it possible to develop a new MADM strategy based 2.2 Single valued neutrosophic set (SVNS) on the proposed similarity measures in single valued Definition 2.2 [6] Let X be a space of points with generic neutrosophic environment? elements in X denoted by x. A SVNS P in X is characterized by a truth-membership function T (x), an Having motivated from the above researches on P indeterminacy-membership function I (x), and a falsity neutrosophic similarity measures, we have introduced the P membership function F (x), for each point x in X. concept of hyperbolic sine similarity measure for SVNS P T (x), I (x), F (x)[0, 1]. When X is continuous, a environment. The new similarity measures called single P P P SVNS P can be written as follows: valued neutrosophic hyperbolic sine similarity measure (SVNHSSM) and single valued neutrosophic weighted P TP(x),IP(x),FP(x):xX hyperbolic sine similarity measure (SVNWHSSM). The X x properties of hyperbolic sine similarity are established. We When X is discrete, a SVNS P can be written as have developed a MADM model using the proposed follows: SVNWHSSM. The proposed hyperbolic sine similarity T (x ),I (x ),F (x ) Pn P i P i P i :x X measure is applied to multi-attribute decision making. i1 x i i The objectives of the paper: For two SVNSs,  To define hyperbolic sine similarity measures for P = { | x X} and SVNS P P P SVNS environment and prove some of it’s basic Q = { | xX } the two relations SVNS Q Q Q properties. are defined as follows:  To define conpromise function for determining (1) PSVNSQSVNS if and only if TP(x)TQ(x), unknown weight of attributes. IP(x)IQ(x), FP(x)FQ(x)  To develop a multi-attribute decision making model (2) PSVNS = QSVNS if and only if TP(x) = TQ(x), IP(x) = based on proposed similarity measures. IQ(x), FP(x) = FQ(x) for any xX .  To present a numerical example for the efficiency 3.Hyperbolic sine similarity measures for SVNSs and effectiveness of the proposed strategy. Rest of the paper is structured as follows. Section 2 pre- Let A = and B = be two SVNSs. Now hyperbolic sine similarity neutrosophic sets. Section 3 is devoted to introduce hyper- function which measures the similarity between two bolic sine similarity measure for SVNSs and some of its SVNSs can be presented as follows (see Eqn. 1): properties. Section 4 presents a method to determine un- SVNHSSM(A,B)  known attribute weights. Section 5 presents a novel decision making strategy based on proposed neutrosophic hy-  T (x)T (x)  I (x)I (x) perbolic sine similarity measure. Section 6 presents an il- sinh A i B i A i B i  lustrative example for the application of the proposed 11 n  FA(xi)FB(xi)  (1) method. Section 7 presents a comparison analysis for the n  11  i1 applicability of the proposed strategy. Section 8 presents     the main contributions of the proposed strategy. Finally,   section 9 presents concluding remarks and scope of future Theorem 1. The defined hyperbolic sine similarity research. measure SVNHSSM(A, B) between SVNSs A and B satisfies the following properties: Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Single Valued Neutrosophic Hyperbolic Sine Similarity Measure Based MADM Strategy Neutrosophic Sets and Systems, Vol. 20, 2018 5 1. 0 SVNHSSM(A, B)1 FA(x)FB(x)  FA(x)FR(x) , 2. SVNHSSM(A, B) = 1 if and only if A = B 3. SVNHSSM (A, B) = SVNHSSM(B, A) FB(x)FR(x)  FA(x)FR(x) . 4. If R is a SVNS in X and ABR then Thus, SVNHSSM(A, R)  SVNHSSM(A, B) and SVNHSSM(A, R)  SVNHSSM(A, B) and SVNHSSM(A, R)  SVNHSSM(B, R). SVNHSSM(A, R)  SVNHSSM(B, R). 3.1 Weighted hyperbolic sine similarity measures Proofs: for SVNSs 1.For two neutrosophic sets A and B, Let A = and B = be two SVNSs. Now weighted hyperbolic A i A i A i B i B i B i B B 0T (x )T (x ) I (x )I (x ) sine similarity function which measures the similarity A i B i A i B i between two SVNSs can be presented as follows (see Eqn.  F (x )F (x ) 3 A i B i 2):  T (x )T (x ) I (x )I (x ) sinh A i B i A i B i  SVNWHSSM(A,B) 0  FA(xi)FB(xi) 1  T (x)T (x)  I (x)I (x)  11  sinh A i B i A i B i    1n wi  FA(xi)FB(xi)  (2)  11  i1 Hence 0 SVNHSSM(A, B) 1     2.For any two SVNSs A and B, if A = B,   TA(x) = TB(x), IA(x) = IB(x), FA(x) = FB(x) Here, 0 w 1, n w1. i i i1  T (x)T (x) 0 , I (x)I (x) 0, A B A B Theorem 2. The defined weighted hyperbolic sine F (x)F (x) 0 similarity measure SVNWHSSM(A, B) between SVNSs A A B and B satisfies the following properties: Hence SVNHSSM(A, B) = 1. 1. 0 SVNWHSSM(A, B)1 Conversely, 2. SVNWHSSM (A, B) = 1 if and only if A = B 3. SVNWHSSM (A, B) = SVNWHSSM (B, A) SVNHSSM(A, B) = 1 4. If R is a SVNS in X and ABR then  T (x)T (x) 0, I (x)I (x) 0, A B A B SVNWHSSM (A, R)  SVNWHSSM(A, B) and FA(x)FB(x) 0. SVNWHSSM (A, R)  SVNWHSSM (B, R). This implies, T (x)= T (x) , I (x)= I (x), F (x) = F (x). A B A B A B Proofs: Hence A = B. 1.For two neutrosophic sets A and B, 3.Since, 0T (x),I (x),F (x),T (x),I (x),F (x)1 A i A i A i B i B i B i TA(x)TB(x) TB(x)TA(x) , 0TA(xi)TB(xi) IA(xi)IB(xi) IA(x)IB(x)  IB(x)IA(x) ,  FA(xi)FB(xi) 3 F (x)F (x)  F (x)F (x) . A B B A  T (x )T (x ) I (x )I (x ) We can write, SVNHSSM(A, B) = SVNHSSM(B, A). sinh A i B i A i B i  4.ABR 0  FA(xi)FB(xi) 1 TA(x)TB(x)TR(x), IA(x)IB(x)IR(x),  11  FA(x)FB(x)FR(x) for xX.   Now we have the following inequalities: n TA(x)TB(x)  TA(x)TR(x) , Again, 0 wi1, wi1. i1 TB(x)TR(x)  TA(x)TR(x) ; Hence 0SVNWHSSM(A, B) 1 IA(x)IB(x)  IA(x)IR(x) , 2.For any two SVNSs A and B, if A = B, IB(x)IR(x)  IA(x)IR(x) ; Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Single Valued Neutrosophic Hyperbolic Sine Similarity Measure Based MADM Strategy 6 Neutrosophic Sets and Systems, Vol. 20, 2018 TA(x) = TB(x), IA(x) = IB(x), FA(x) = FB(x) The weight of j-th attribute is defined as follows (see Eqn.  T (x)T (x) 0 , I (x)I (x) 0, 4). A B A B FA(x)FB(x) 0 w  Cj(A) (4) j n C (A) j1 j Hence SVNWHSSM(A, B) = 1. Conversely, n Here, w 1. j SVNWHSSM(A, B) = 1 j1  TA(x)TB(x) 0, IA(x)IB(x) 0, Theorem 3. The compromise function Cj(A) satisfies the F (x)F (x) 0. following properties: A B This implies, TA(x) = TB(x) , IA(x) = IB(x), FA(x) = FB(x). P1. Cj(A)1, if Tij1,FijIij0. Hence A = B. P2. C (A)0, if T ,I ,F  0,1,1 . j ij ij ij 3.Since, P3. Cj(A)Ej(B),ifTiAj TiBj and IiAjFiAj IiBjFiBj . T (x)T (x) T (x)T (x) , A B B A Proofs. I (x)I (x)  I (x)I (x) , A B B A F (x)F (x)  F (x)F (x) . P1. Tij1,FijIij0 A B B A 1 m 1  C (A) 33 .m1 We can write, SVNWHSSM(A, B) = SVNWHSSM(B, A). j m m i1 4.ABR P2. Tij,Iij,Fij  0,1,1 . TA(x)TB(x)TR(x), IA(x)IB(x)  IR(x), 1 m FA(x)  FB(x)FR(x) for xX.  Cj(A)m0 30 i1 Now we have the following inequalities: P3. C (A)C (B) j j TA(x)TB(x)  TA(x)TR(x) ,  1 m2TAIAFA 3 1 m2TBIBFB 30 TB(x)TR(x)  TA(x)TR(x) ; mi1 ij ij ij mi1 ij ij ij  IA(x)IB(x)  IA(x)IR(x) ,  Cj(A)Cj(B)0, Since, TiAj TiBj andIiAjFiAjIiBjFiBj. IB(x)IR(x)  IA(x)IR(x) ; Hence,C (A)C (B). j j FA(x)FB(x)  FA(x)FR(x ) , FB(x)FR(x)  FA(x)FR(x) . 5.Decision making procedure Let A , A , ..., A be a discrete set of alternatives, C , C , Thus SVNWHSSM(A, R)  SVNWHSSM(A, B) and ..., C1 be 2the set mof attributes of each alternative. The1 val2- SVNWHSSM(A, R)  SVNWHSSM(B, R). ues anssociated with the alternatives A (i = 1, 2,..., m) i against the attribute C (j = 1, 2, ..., n) for MADM problem j 4.Determination of unknown attribute weights is presented in a SVNS based decision matrix. When attribute weights are completely unknown to The steps of decision-making (see Figure 2) based on decision makers, the entropy measure [46] can be used to single valued neutrosophic weighted hyperbolic sine simi- calculate attribute weights. Biswas et al. [47] employed larity measure (SVNWHSSM) are presented using the fol- entropy measure for MADM problems to determine lowing steps. completely unknown attribute weights of SVNSs. Step 1: Determination of the relation between alternatives and attributes 4.1 Compromise function The relation between alternatives A (i = 1, 2, ..., m) i The compromise function of a SVNS A = TiAj,IiAj,FiAj and the attribute Cj (j = 1, 2, ..., n) is presented in the Eqn. (i = 1, 2, ..., m; j = 1, 2, ..., n) is defined as follows (see (5). Eqn. 3): m  C (A)  2TAIAFA 3 (3) j ij ij ij i1 Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Single Valued Neutrosophic Hyperbolic Sine Similarity Measure Based MADM Strategy Neutrosophic Sets and Systems, Vol. 20, 2018 7 D[A|C] nection. Therefore, it is necessary to select suitable SIM  C C  C  card for his/her mobile connection. After initial screening,  1 2 n  there are four possible alternatives (SIM cards) for mobile A T ,I ,F T ,I ,F  T ,I ,F   1 11 11 11 12 12 12 1n 1n 1n  (5) connection. The alternatives (SIM cards) are presented as A T ,I ,F T ,I ,F  T ,I ,F  2 21 21 21 22 22 22 2n 2n 2n  follows:          Am Tm1,Im1,F1m1 Tm2,Im2,Fm2  Tmn,Imn,Fmn   A1: Airtel  A : Vodafone 2 Here T ,I ,F (i = 1, 2, ..., m; j = 1, 2, ..., n) be SVNS  A : BSNL ij ij ij 3 assessment value.  A4: Reliance Jio The person must take a decision based on the Step 2: Determine the weights of attributes following five attributes of SIM cards:  C : Service quality Using the Eqn. (3) and (4), decision-maker calculates the 1  C : Cost weight of the attribute C (j = 1, 2, …, n). 2 j  C : Initial talk time 3 Step 3: Determine ideal solution  C : Call rate per second 4  C : Internet and other facilities 5 Generally, the evaluation attribute can be categorized into The decision-making strategy is presented using the fol- two types: benefit type attribute and cost type attribute. In lowing steps. the proposed decision-making method, an ideal alternative can be identified by using a maximum operator for the Step 1: Determine the relation between alternatives benefit type attributes and a minimum operator for the cost and attributes type attributes to determine the best value of each attribute The relation between alternatives A , A , A , and A 1 2 3 4 among all the alternatives. Therefore, we define an ideal and the attributes C , C , C , C , C is presented in the Eqn. 1 2 3 4 5 alternative as follows: (8). 𝐴* = {C *, C *, … , C *}. 1 2 m D[A|C ,C ,C ,C ,C ] 1 2 3 4 5 Here, benefit attributeC*jcan be presented as follows:  C1 C2 C3 C4 C5  A .7,.3,.3 .6,.4,.3 .8,.1,.1 .5,.4,.4 .5,.3,.2  C*maxT (Ai),minI (Ai),minF (Ai) (6)  1  (8) j  i Cj i Cj i Cj  A2 .5,.3,.1 .7,.1,.3 .7,.3,.1 .6,.1,.1 .5,.2,.3  for j = 1, 2, ..., n. A3 .8,.2,.2 .6,.4,.3 .6,0,.1 .7,.3,0 .5,.3,.4    Similarly, the cost attribute C*j can be presented as A4 .6,.1,.3 .5,.1,.2 .6,.3,.1 .5,.1,.2 .9,.1,.1  follows: Step 2: Determine the weights of attributes C*minT (Ai),maxI (Ai),maxF (Ai) (7) j  i Cj i Cj i Cj  Using the Eq. (3) and (4), we calculate the weight of the for j = 1, 2, ..., n attributes C1, C2, C3, C4, C5 as follows: Step 4: Determine the similarity values [w1, w2, w3, w4, w5] = Using Eqns. (2) and (5), calculate SVNWHSSM values [0.2023, 0.1917, 0.2078, 0.2009, 0.1973] for each alternative between positive (or negative) ideal so- Step 3: Determine ideal solution lutions and corresponding single valued neutrosophic from decision matrix D[A|C]. In this problem, attributes C , C , C , C are benefit type 1 3 4 5 attributes and , C is the cost type attribute. Step 5: Ranking the alternatives 2 𝐴* = {(0.8, 0.1, 0.1), (0.5, 0.4, 0.3), (0.8, 0.0, 0.1), (0.7, Ranking the alternatives is prepared based on the de- 0.1, 0.0), (0.9, 0.1, 0.1)}. scending order of similarity measures. Highest value indicates the best alternative. Step 4: Determine the weighted similarity values Step 6: End Using Eq. (2) and Eq. (8), we calculate similarity measure values for each alternative as follows. 6.Numerical example SVNWHSSM(A*, A )=0.92422 1 In this section, we illustrate a numerical example as an ap- SVNWHSSM(A*, A )=0.95629 plication of the proposed approach. We consider a deci- 2 sion-making problem stated as follows. Suppose a person SVNWHSSM(A*, A )=0.97866 who wants to purchase a SIM card for his/her mobile con- 3 Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Single Valued Neutrosophic Hyperbolic Sine Similarity Measure Based MADM Strategy 8 Neutrosophic Sets and Systems, Vol. 20, 2018 SVNWHSSM(A*, A )=0.96795 2) We have proposed ‘compromise function’ for cal- 4 culating unknown weights structure of attributes in Step 5: Ranking the alternatives SVNS environment. Ranking the alternatives is prepared based on the de- 3) We develop a decision making strategy based on scending order of similarity measures (see Figure 1). Now the proposed weighted similarity measure the final ranking order will be as follows. (SVNWHSSM). A3 A4 A2 A1 4) Steps and calculations of the proposed strategy are Highest value indicates the best alternative. easy to use. Step 6: End 5) We have solved a numerical example to show the feasibility, applicability, and effectiveness of the proposed strategy. 1.0 measure values00..68 9In. Cthoe npcalpuesr,i owne have proposed hyperbolic sine similarity milarity 0.4 mfoer asSuVreN aSnsd awnde igphrtoevde dh ytpheeribro lbiacs sicin ep rsoimpeirltaireisty. Wmeea shuarvees Weighted si0.2 pwreoipgohstse do f ctohme partotrmibiusete sf uinn cStiVonN St oe ndveirtoenrmmiennet . uWnke nhoawvne developed a novel MADM strategy based on the proposed 0.0 A1 A2 A3 A4 weighted similarity measure to solve decision problems. Alternatives We have solved a numerical problem and compared the obtained result with other existing strategies to demon- FIGURE 1: Graphical representation of alternatives versus weighted similarity measures. strate the effectiveness of the proposed MADM strategy. The proposed MADM strategy can be applied in other decision-making problem such as supplier selection, pat- 7.Comparison analysis tern recognition, cluster analysis, medical diagnosis, weav- er selection [51-53], fault diagnosis [54], brick selection The ranking results calculated from proposed strategy and [55-56], data mining [57], logistic centre location selection different existing strategies [38, 48, 49, 50] are furnished in [58-60], teacher selection [61, 62], etc. Table 1. We observe that the ranking results obtained from proposed and existing strategies in the literature differ. The proposed strategy reflects that the optimal alternative is A . The ranking result obtained from Ye [38] is similar 3 to the proposed strategy. The ranking results obtained from Ye and Zhang [48] and Mondal and Pramanik [49] differ from the optimal result of the proposed strategy. In Ye [50], the ranking order differs but the best alternative is the same to the proposed strategy. Table 1 The ranking results of existing strategies Strategies Ranking results Ye and Zhang[48] A4A2A3A1 Mondal and Pramanik [49] A4A3A2A1 Ye [38] A3A4A2A1 Ye [50] A3A2A4A1 Proposed strategy A3A4A2A1 8.Contributions of the proposed strategy 1) SVNHSSM and SVNWHSSM in SVNS environment are firstly defined in the literature. We have also proved their basic properties. Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Single Valued Neutrosophic Hyperbolic Sine Similarity Measure Based MADM Strategy Neutrosophic Sets and Systems, Vol. 20, 2018 9 Multi attribute decision making problem Decision making analysis phase Determination of the relation between alternatives and attributes Step-1 Determine the weights of attributes Step- 2 D e t e r m i n e i d e a l s o l u t i o n Step- 3 Determine the similarity values Step-4 Ranking the alternatives Step-5 End Step- 6 FIGURE 2: Phase diagram of the proposed decision making strategy References [4] L. A. Zadeh. Fuzzy sets. Information and Control, 8(1965), 338–353. [1] F. Smarandache, A unifying field in logics, neutrosophy: [5] K. Atanassov. Intuitionistic fuzzy sets. Fuzzy Sets and neutrosophic probability, set and logic. Rehoboth, American Systems, 20(1986), 87–-96. Research Press, 1998. [6] H. Wang, F. Smarandache, Y. Q. Zhang, and R. [2] H. J. S. Smith. On the integration of discontinuous Sunderraman. Single valued neutrosophic sets. Multispace functions. Proceedings of the London Mathematical Society, and Multistructure, 4(2010), 410–413. Series, 1(6) (1874), 140–153. [7] S. Ye, and J. Ye. Dice similarity measure between single [3] G. Cantor. Über unendliche, lineare valued neutrosophic multisets and its application in medical Punktmannigfaltigkeiten V On infinite, linear point- diagnosis. Neutrosophic Sets and Systems, 6(2014). 49–54. manifolds (sets). Mathematische Annalen, 21 (1883), 545– 591. Kalyan Mondal, Surapati Pramanik, and Bibhas C. Giri. Single Valued Neutrosophic Hyperbolic Sine Similarity Measure Based MADM Strategy

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