Neutrosophic sets and systems : Book series, vol. 23, 2018 / : an international book series in information science and engineering. PDF
Preview Neutrosophic sets and systems : Book series, vol. 23, 2018 / : an international book series in information science and engineering.
Book Series, Vol. 23, 2018 Editors: Florentin Smarandache and Mohamed Abdel-Basset ISBN 978-1-59973-597-9 Neutrosophic Science International Association (NSIA) ISBN 978-1-59973-597-9 Neutrosophic Sets and Systems An International Book Series in Information Science and Engineering University of New Mexico ISBN 978-1-59973-597-9 University of New Mexico 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 grant Neutrosophic Sets and Systems non-exclusive, worldwide, royalty-free license to publish and distribute the articles in accordance with the Budapest Open Initi- ative: this means that electronic copying, distribution and printing of both full-size version of the journal and the individual papers published therein for non-commercial, academic or individual use can be made by any user without permission or charge. The authors of the articles published in Neutrosophic Sets and Systems retain their rights to use this journal as a whole or any part of it in any other publications and in any way they see fit. Any part of Neutrosophic Sets and Systems howsoever used in other publications must include an appropriate citation of this journal. Information for Authors and Subscribers “Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc. The submitted papers should be professional, in good English, containing a brief review of a problem and obtained results. Neutrosophy is a new branch of philosophy that studies the origin, nature, and scope of neutralities, as well as their inter- actions with different ideational spectra. This theory considers every notion or idea <A> together with its opposite or negation <antiA> and with their spectrum of neutralities <neutA> in between them (i.e. notions or ideas supporting neither <A> nor <antiA>). The <neutA> and <antiA> ideas together are referred to as <nonA>. Neutrosophy is a generalization of Hegel's dialectics (the last one is based on <A> and <antiA> only). According to this theory every idea <A> tends to be neutralized and balanced by <antiA> and <nonA> ideas - as a state of equilibrium. In a classical way <A>, <neutA>, <antiA> are disjoint two by two. But, since in many cases the borders between notions are vague, imprecise, Sorites, it is possible that <A>, <neutA>, <antiA> (and <nonA> of course) have common parts two by two, or even all three of them as well. Neutrosophic Set and Neutrosophic Logic are generalizations of the fuzzy set and respectively fuzzy logic (especially of intuitionistic fuzzy set and respectively intuitionistic fuzzy logic). In neutrosophic logic a proposition has a degree of truth (T), a degree of indeterminacy (I), and a degree of falsity (F), where T, I, F are standard or non-standard subsets of ]-0, 1+[. Neutrosophic Probability is a generalization of the classical probability and imprecise probability. Neutrosophic Statistics is a generalization of the classical statistics. What distinguishes the neutrosophics from other fields is the <neutA>, which means neither <A> nor <antiA>. <neutA>, which of course depends on <A>, can be indeterminacy, neutrality, tie game, unknown, contradiction, igno- rance, imprecision, etc. All submissions should be designed in MS Word format using our template file: http://fs.unm.edu/NSS/NSS-paper-template.doc. A variety of scientific books in many languages can be downloaded freely from the Digital Library of Science: http://fs.unm.edu/ScienceLibrary.htm. To submit a paper, mail the file to the Editor-in-Chief. To order printed issues, contact the Editor-in-Chief. This journal is non-commercial, academic edition. It is printed from private donations. Home page: http://fs.unm.edu/NSS The Educational Publisher Inc. 1313 Chesapeake Ave. Columbus, Ohio 43212, USA Copyright © Neutrosophic Sets and Systems, 2018 ISBN 978-1-59973-597-9 University of New Mexico Contents Ranjan Kumar, S A Edaltpanah, Sripati Jha, Said Broumi, Arindam Dey. Neutrosophic Shortest Path Problem ………………..……………………..………………..…………………………………..…… 5 Haitham Elwahsh, Mona Gamal, A. A. Salama, I.M. El-Henawy. Intrusion Detection System and Neutrosophic theory for MANETs: A Comparative study ………………………………….…….….. 16 R. A. Cruz, F. N. Irudhayam. Neutrosophic Soft Cubic Set in Topological Spaces …………………. 23 Noel Batista Hernández, Norberto Valcárcel Izquierdo, Maikel Leyva-Vázquez, Florentin Smarandache. Validation of the pedagogical strategy for the formation of the competence entrepreneurship in high education through the use of neutrosophic logic and Iadov technique .………………………… 45 Tuhin Bera, Nirmal Kumar Mahapatra. Neutrosophic Soft Normed Linear Space ….……..………….. 52 M. Mohseni Takallo, R.A. Borzooei, Young Bae Jun. MBJ-neutrosophic structures and its applica- tions in BCK/BCI-algebras ……………………………………………………………………..………... 72 Vasantha W.B., Ilanthenral Kandasamy, Florentin Smarandache. Algebraic. Structure of Neutrosoph- ic Duplets in Neutrosophic Rings <Z ∪ I>, Q ∪ I, and <R ∪ I> ...…………………………………….. 85 Riad K. Al-Hamido, T. Gharibah, S. Jafari, F. Smarandache. On Neutrosophic Crisp Topology via N- Topology ……………………………………………………………………………..…………………… 96 Serpil Altınırmak, Yavuz Gül, Basil Oluch Okoth, Çağlar Karamaşa. Performance Evaluation of Mu- tual Funds Via Single Valued Neutrosophic Set (SVNS) Perspective: A Case Study in Turkey …… 110 Rajab Ali Borzooei, M. Mohseni Takallo, Florentin Smarandache, Young Bae Jun. Positive implicative BMBJ-neutrosophic ideals in BCK-algebras …………………………………………………..……... 126 Vakkas Ulucay, Adil Kilic, Ismet Yildiz, Mehmet Sahin. A new approach for multi-attribute deci- sion-making problems in bipolar neutrosophic sets …...……………………………………..………… 142 Rodolfo González Ortega, Maikel Leyva Vázquez, João Alcione Sganderla Figueiredo, Alfonso Guijar- ro-Rodríguez. Sinos river basin social-environmental prospective assessment of water quality management using fuzzy cognitive maps and neutrosophic AHP-TOPSIS …………..……………..... 160 Address: “Neutrosophic Sets and Systems” (An International Book Series in Information Science and Engineering ) Department of Mathematics and Science University of New Mexico 705 Gurley Avenue Gallup, NM 87301, USA E-mail: [email protected] Home page: http://fs.unm.edu/NSS Copyright © Neutrosophic Sets and Systems, 2018 Editorial Board ISBN 978-1-59973-596-9 University of New Mexico Editors-in-Chief Prof. Florentin Smarandache, PhD, Postdoc, Mathematics Department, University of New Mexico, Gallup, NM 87301, USA, Email: [email protected]. Dr. Mohamed Abdel-Baset, Faculty of Computers and Informatics, Zagazig University, Egypt, Email: [email protected]. Associate Editors Dr. Said Broumi, University of Hassan II, Casablanca, Morocco, Email: [email protected]. Prof. Le Hoang Son, VNU Univ. of Science, Vietnam National Univ. Hanoi, Vietnam, Email: [email protected]. Dr. Huda E. Khalid, University of Telafer, College of Basic Education, Telafer - Mosul, Iraq, Email: [email protected]. Prof. Xiaohong Zhang, Department of Mathematics, Shaanxi University of Science &Technology, Xian 710021, China, Email: [email protected]. Dr. Harish Garg, School of Mathematics, Thapar Institute of Engineering & Technology, Patiala 147004, Punjab, India, Email: [email protected]. Editors W. B. Vasantha Kandasamy, School of Computer [email protected]. Science and Engineering, VIT, Vellore 632014, India, Maikel Leyva-Vazquez, Universidad de Guayaquil, Email: [email protected] Ecuador, Email: [email protected]. A. A. Salama, Faculty of Science, Port Said University, Muhammad Akram, University of the Punjab, New Egypt, Email: [email protected]. Campus, Lahore, Pakistan, Young Bae Jun, Gyeongsang National University, Email: [email protected]. South Korea, Email: [email protected]. Irfan Deli, Muallim Rifat Faculty of Education, Kilis 7 Vakkas Ulucay, Gaziantep University, Gaziantep, Aralik University, Turkey, Turkey, Email: [email protected]. Email: [email protected]. Peide Liu, Shandong University of Finance and Ridvan Sahin, Department of Mathematics, Faculty of Economics, China, Email: [email protected]. Science, Ataturk University, Erzurum 25240, Turkey, Mehmet Şahin, Department of Mathematics, Gaziantep Email: [email protected]. University, Gaziantep 27310, Turkey, Abduallah Gamal, Faculty of Computers and Email: [email protected]. Informatics, Zagazig University, Egypt, Email: Mohammed Alshumrani & Cenap Ozel, King [email protected]. Abdulaziz Univ., Jeddah, Saudi Arabia, Emails: Ibrahim M. Hezam, Department of computer, Faculty [email protected], [email protected]. of Education, Ibb University, Ibb City, Yemen, Email: Jun Ye, Shaoxing University, China, Email: [email protected]. [email protected]. Pingping Chi, China-Asean International College, Madad Khan, Comsats Institute of Information Dhurakij Pundit University, Bangkok 10210, Thailand, Technology, Abbottabad, Pakistan, Email: Email: [email protected]. [email protected]. Ameirys Betancourt-Vázquez, 1 Instituto Superior Dmitri Rabounski and Larissa Borissova, independent Politécnico de Tecnologias e Ciências (ISPTEC), researchers, Email: [email protected], Luanda, Angola, E-mail: [email protected]. Email: [email protected] Karina Pérez-Teruel, Universidad Abierta para Adultos Selcuk Topal, Mathematics Department, Bitlis Eren (UAPA), Santiago de los Caballeros, República University, Turkey, Email: [email protected]. Dominicana, E-mail: [email protected]. Ibrahim El-henawy, Faculty of Computers and Neilys González Benítez, Centro Meteorológico Pinar Informatics, Zagazig University, Egypt, Email: del Río, Cuba, E-mail: [email protected]. [email protected]. Jesus Estupinan Ricardo, Centro de Estudios para la A. A. A. Agboola, Federal University of Agriculture, Calidad Educativa y la Investigation Cinetifica, Toluca, Abeokuta, Nigeria, Email: [email protected]. Mexico, Email: [email protected]. Luu Quoc Dat, Univ. of Economics and Business, B. Davvaz, Department of Mathematics, Yazd Vietnam National Univ., Hanoi, Vietnam, Email: University, Iran, Email: [email protected]. Copyright © Neutrosophic Sets and Systems, 2018 Editorial Board ISBN 978-1-59973-596-9 University of New Mexico Fernando A. F. Ferreira, ISCTE Business School, BRU- Victor Christianto, Malang Institute of Agriculture IUL, University Institute of Lisbon, Avenida das Forças (IPM), Malang, Indonesia, Email: Armadas, 1649-026 Lisbon, Portugal, [email protected]. Email: [email protected] Wadei Al-Omeri, Department of Mathematics, Al- Julio J. Valdés, National Research Council Canada, M- Balqa Applied University, Salt 19117, Jordan, Email: 50, 1200 Montreal Road, Ottawa, Ontario K1A 0R6, [email protected]. Canada, Email: [email protected] Ganeshsree Selvachandran, UCSI University, Jalan Tieta Putri, College of Engineering Department of Menara Gading, Kuala Lumpur, Malaysia, Email: Computer Science and Software Engineering, [email protected]. University of Canterbury, Christchurch, New Zeeland. Ilanthenral Kandasamy, School of Computer Science M. Al Tahan, Department of Mathematics, Lebanese and Engineering (SCOPE), Vellore Institute of International University, Bekaa, Lebanon, Email: Technology (VIT), Vellore 632014, Tamil Nadu, India, [email protected] Sudan Jha, Pokhara Email: [email protected] University, Kathmandu, Nepal, Kul Hur, Wonkwang University, Iksan, Jeollabukdo, Email: [email protected] South Korea, Willem K. M. Brauers, Faculty of Applied Economics, Email: [email protected]. University of Antwerp, Antwerp, Belgium, Email: Kemale Veliyeva & Sadi Bayramov, Department of [email protected]. Algebra and Geometry, Baku State University, 23 Z. M. Ganster, Graz University of Technology, Graz, Khalilov Str., AZ1148, Baku, Azerbaijan, Email: Austria, Email: [email protected]. [email protected], Email: [email protected]. Umberto Rivieccio, Department of Philosophy, Inayatur Rehman, College of Arts and Applied University of Genoa, Italy, Email: Sciences, Dhofar University Salalah, Oman, Email: [email protected]. [email protected]. F. Gallego Lupiaňez, Universidad Complutense, Riad K. Al-Hamido, Math Departent, College of Madrid, Spain, Email: [email protected]. Science, Al-Baath University, Homs, Syria, Email: Francisco Chiclana, School of Computer Science and [email protected]. Informatics, De Montfort University, The Gateway, Faruk Karaaslan, Çankırı Karatekin University, Leicester, LE1 9BH, United Kingdom, E-mail: Çankırı, Turkey, [email protected]. E-mail: [email protected]. Yanhui Guo, University of Illinois at Springfield, One Suriana Alias, Universiti Teknologi MARA (UiTM) University Plaza, Springfield, IL 62703, United States, Kelantan, Campus Machang, 18500 Machang, Email: [email protected] Kelantan, Malaysia, Email: [email protected]. Angelo de Oliveira, Ciencia da Computacao, Universidade Federal de Rondonia, Porto Velho - Rondonia, Brazil, Email: [email protected]. Valeri Kroumov, Okayama University of Science, Japan, Email: [email protected]. E. K. Zavadskas, Vilnius Gediminas Technical University, Vilnius, Lithuania, Email: [email protected]. Darjan Karabasevic, University Business Academy, Novi Sad, Serbia, Email: [email protected]. Dragisa Stanujkic, Technical Faculty in Bor, University of Belgrade, Bor, Serbia, Email: [email protected]. Luige Vladareanu, Romanian Academy, Bucharest, Romania, Email: [email protected]. Stefan Vladutescu, University of Craiova, Romania, Email: [email protected]. Philippe Schweizer, Independant Researcher, Av. de Lonay 11, 1110 Morges, Switzerland, Email: [email protected]. Saeid Jafari, College of Vestsjaelland South, Slagelse, Denmark, Email: [email protected]. Copyright © Neutrosophic Sets and Systems, 2018 Neutrosophic Sets and Systems, Vol. 23, 2018 5 University of New Mexico Neutrosophic Shortest Path Problem Ranjan Kumar1, S A Edaltpanah2, Sripati Jha1, Said Broumi3 and Arindam Dey4 1 Department of Mathematics, National Institute of Technology, Adityapur, Jamshedpur, 831014, India. 2 Department of Industrial Engineering, Ayandegan Institute of Higher Education, Iran 3 Laboratory of Information Processing, Faculty of Science, Ben M’sik, University Hassan II, B.P. 7955, Casablanca, Morocco 4 Department of computer Science and Engineering, Saroj Mohan Institute of Technology, West Bengal, India 1 Corresponding Author: Ranjan Kumar, [email protected] Abstract. Neutrosophic set theory provides a new tool to handle the uncertainties in shortest path problem (SPP). This paper intro- duces the SPP from a source node to a destination node on a neutrosophic graph in which a positive neutrosophic number is as- signed to each edge as its edge cost. We define this problem as neutrosophic shortest path problem (NSSPP). A simple algorithm is also introduced to solve the NSSPP. The proposed algorithm finds the neutrosophic shortest path (NSSP) and its corresponding neutrosophic shortest path length (NSSPL) between source node and destination node. Our proposed algorithm is also capable to find crisp shortest path length (CrSPL) of the corresponding neutrosophic shortest path length (NSSPL) which helps the decision maker to choose the shortest path easily. We also compare our proposed algorithm with some existing methods to show efficiency of our proposed algorithm. Finally, some numerical experiments are given to show the effectiveness and robustness of the new model. Numerical and graphical results demonstrate that the novel methods are superior to the existing method. Keywords: Trapezoidal neutrosophic fuzzy numbers; scoring, accuracy and certainty index, shortest path problem 1 Introduction Let G = (V, E) be a graph, where V is a set of all the nodes (or vertices) and E is a set of all the edges (or arcs). The aim of the shortest path problem (SPP) is to find a path between two nodes and optimizing the weight of the path. The SPP is known as one of the well-studied fields in the area operations research and mathematical optimization and it is commonly encountered in wide array of practical applications including road network [1], flow shop scheduling [2], routing problems [3], transportation planning [4], geographical information system(GIS) field [5], optimal path[6-7] and so on. There are several methods for solving traditional SPP such as Dijkstra [8] algorithm or the label-correcting Bellman [9] algorithm. Due to uncertain factors in real-world problems, such as efficiency, expense, and path capacity variation, we must consider SPP with imprecise information. Under some circumstances, an approx- imate method applies fuzzy numbers to solve SPP, called Fuzzy-SPP (FSPP). Many researchers have focused on FSPP and intuitionistic FSPP (IFSPP) formulations and solution approaches. Dubois and Prade [10] first intro- duced FSPP. Later, different approaches were presented by various researchers/scientists to evaluate the FSPP. Some of them are as follows; Keshavarz and Khorram [11] used the highest reliability, Deng et al. [12] sug- gested extended Dijkstra Principle technique, Hassanzadeh et al. [13], and Syarif et al. [14] proposed a genetic algorithm model, Ebrahimnejad et al. [15] using the artificial bee colony model, Li et al.[16]; Zhong and Zhou [17] used neural networks for finding FSPP. Moreover, Motameni and Ebrahimnejad [18] considered constraint SPP, Mukherjee [19], Geetharamani and Jayagowri [20] and Biswas et al. [21] considered the IFSPP. In recent years, research on this subject has increased and that is of continuing interest such as Kristianto et al.[22], Zhang et al.[23], Mukherjee [24], Huang and Wang [25], Dey et al. [26], Niroomand et al. [27], Rashmanlou et al. [28], Mali and Gautam [29], Wang et al. [30], Yen and Cheng [31] and so on. Recently, neutrosophic set (NS) theory is proposed by Smarandache [32-33], and this is generalised from the fuzzy set [34] and intuitionistic fuzzy set [35]. NS deals with uncertain, indeterminate and incongruous data where the indeterminacy is quantified explicitly. Moreover, falsity, indeterminacy and truth membership are completely independent. It overcomes some limitations of the existing methods in depicting uncertain decision information. Some extensions of NSs, including interval NS [36-38] , bipolar NS[39], single-valued NS [40-44], Ranjan Kumar, S A Edaltpanah, Sripati Jha, Said Broumi and Arindam Dey. Neutrosophic shortest path problem 6 Neutrosophic Sets and Systems, Vol. 23, 2018 multi-valued NS [45-47], neutrosophic linguistic set [48-49], rough neutrosophic set [50-62], triangular fuzzy neutrosophic set [63], and neutrosophic trapezoidal set [64-67] have been proposed and applied to solve various problems. However, to the best of our knowledge, there are few methods which deal with NSSPP. Recently, Broumi et al.[68-71] proposed some models to solve SPP in the neutrosophic environment. Broumi et al. [68- 71], proposed a new method for the TrNSSPP and TNSSPP. However, the mentioned methods [68-71] have some shortcomings and are not valid. In this paper, for finding NSSPP, the shortcomings of the mentioned mod- els are pointed out, and a new method is proposed for the same. 2 Preliminaries Definition 2.1: [72]: Let 1 is a special NS on the real number set R, whose truth-MF (x),indeterminacy-MF a (x),and falsity-MF (x)are given as follows: a a T xa a T a xa , a a T I I T T a xa , (x) a I P (1) a T a x a S a xa , a a P S S P 0 otherwise. a xI xa I a T a xa , a a T I I T I a xa , (x) a I P (2) a xa I a x P a S a xa , a a P S S P 1 otherwise. a xF xa I a T a xa , a a T I I T F a xa , (x) a I p (3) a xa F a x P a S a xa , a a P S S P 1 otherwise. The graphical representation of the TrNS number a a ,a ,a ,a ,T ,I ,F is shown in Fig. 1, where, the T I P S a a a burgundy colour graph show truth-MF, the yellow colour graph shows indeterminacy-MF, and the red colour graph shows the falsity-MF. Blackline represent the truth value, the cyan line represents the indeterminacy value, and the blue line represents the falsity value ( here, we consider T I F ). a a a Additionally, when a 0, a a ,a ,a ,a ,T ,I ,F is called a positive TrNS number. Similarly, T T I P S a a a when a 0, a a ,a ,a ,a ,T ,I ,F becomes a negative TrNS number. When S T I P S a a a 0a a a a 1 and T ,I ,F [0,1], a is called a normalised TrNS number. When T I P S a a a I 1T F , the TrNS number is reduced to triangular intuitionistic fuzzy numbers (TrIFN). When a a a a a ,a a ,a ,a ,a ,T ,I ,F transforming into a TNS number. When I 0,F 0,a TrNS number I P T I P S a a a a a is reduced to generalised TrFN, a a ,a ,a ,a ,T . T I P S a Definition 2.2: [40] : Let X be a space point or objects, with a genetic element in X denoted by x. A single- valued NS, V in X is characterised by three independent parts, namely truth-MFT ,indeterminacy-MF I and V V falsity-MF F ,such that T :X [0,1],I :X [0,1], and F :X [0,1]. V V V V Now, V is denoted as V x,T (x),I (x),F (x) |xX, satisfying 0T (x)I (x)F (x)3. V V V V V V Definition 2.3: [72]: Let rˆN rˆ ,rˆ,rˆ ,rˆ ,T ,I ,F andsˆN sˆ ,sˆ ,sˆ ,sˆ ,T ,I ,F be two arbitrary T I M E rˆ rˆ rˆ T I M E sˆ sˆ sˆ TrNSNs, and0;then arithmetic operation on TrNS are as follows: Ranjan Kumar, S A Edaltpanah, Sripati Jha, Said Broumi and Arindam Dey. Neutrosophic shortest path problem Neutrosophic Sets and Systems, Vol. 23, 2018 7 rˆN sˆN rˆ sˆ ,rˆ sˆ ,rˆ sˆ ,rˆ sˆ ,T T TT ,I I ,FF T T I I M M E E rˆ sˆ rˆ sˆ rˆ sˆ rˆ sˆ rˆN sˆN rˆ sˆ ,rˆ sˆ ,rˆ sˆ ,rˆ sˆ ,T T ,I I I I ,F F FF T T I I M M E E rˆ sˆ rˆ sˆ rˆ sˆ rˆ sˆ rˆ sˆ rˆN rˆ ,rˆ,rˆ ,rˆ ,11T ,I ,F T I M E rˆ rˆ rˆ Figure 1. The graphical representation of the membership functions of TrNS. Definition 2.4: [73]: (Comparison of any two random TrNS numbers): Let rˆN rˆ ,rˆ,rˆ ,rˆ ,T ,I ,F be a T I M E rˆ rˆ rˆ TrNS number,and then the score and accuracy function is defined, as follow: 1 srˆN rˆ rˆ rˆ rˆ 2T I F 12 T I M E rˆ rˆ rˆ 1 arˆN rˆ rˆ rˆ rˆ 2T I F 12 T I M E rˆ rˆ rˆ Let rˆN rˆ ,rˆ,rˆ ,rˆ ,T ,I ,F and sˆN sˆ ,sˆ ,sˆ ,sˆ ,T ,I ,F be two TrNS numbers, the ranking of T I M E rˆ rˆ rˆ T I M E sˆ sˆ sˆ rˆN and sˆNby score function is described as follows: 1. if srˆNssˆN then rˆN sˆN 2. if srˆNssˆN and if a. arˆNasˆN then rˆN sˆN b. arˆNasˆN then rˆN sˆN c. arˆNasˆN then rˆN sˆN Definition 2.5:[73]: (Comparison of any two random TNS numbers).Let rˆNS rˆ ,rˆ,rˆ ,T ,I ,F be a TNS T I P rˆ rˆ rˆ number, and then the score and accuracy functions are defined as follows: 1 srˆNS rˆ 2rˆ rˆ 2T I F 12 T I P rˆ rˆ rˆ 1 arˆNS rˆ 2rˆ rˆ 2T I F 12 T I P rˆ rˆ rˆ Let rˆNS rˆ ,rˆ,rˆ ,T ,I ,F and sˆNS sˆ ,sˆ ,sˆ ,T ,I ,F be two arbitrary TNSNs, the ranking T I P rˆ rˆ rˆ T I P sˆ sˆ sˆ ofrˆNSand sˆNSby score function is defined as follows: 1. if srˆNS ssˆNS then rˆNS sˆNS 2. if srˆNSssˆNS and if a. arˆNSasˆNS then rˆNS sˆNS b. arˆNSasˆNS then rˆNS sˆNS c. arˆNSasˆNS then rˆNS sˆNS Ranjan Kumar, S A Edaltpanah, Sripati Jha, Said Broumi and Arindam Dey. Neutrosophic shortest path problem 8 Neutrosophic Sets and Systems, Vol. 23, 2018 Definition 2.6: [72]: Let rˆrˆ ,rˆ,rˆ ,rˆ be a TrFN, and rˆ rˆ rˆ rˆ then the centre of gravity (COG) of T I M E T I M E rˆ can be defined as rˆ, if rˆ rˆ rˆ rˆ T I M E COG(rˆ)1 rˆ rˆ rˆrˆ 3rˆT rˆI rˆM rˆE rˆE ErˆMM rˆII T rˆT , otherwise Definition 2.7:[72]: ( Comparison of any two random TrNS numbers).Let sˆNS sˆ ,sˆ ,sˆ ,sˆ ,T ,I ,F be a T I M E sˆ sˆ sˆ TrNSNs,and then the score function, accuracy function, and certainty functions are defined as follows: 2T I F E(sˆN)COG(rˆ) sˆ sˆ sˆ , 3 A(sˆN)COG(rˆ)T F , sˆ sˆ C(sˆN)COG(rˆ)T sˆ LetrˆNS rˆ ,rˆ,rˆ ,T ,I ,F andsˆNS sˆ ,sˆ ,sˆ ,sˆ ,T ,I ,F be two arbitrary TrNSNs, the ranking of T I P rˆ rˆ rˆ T I M E sˆ sˆ sˆ rˆNS and sˆNS by score function is defined as follows: 1. if ErˆNSEsˆNS then rˆNS sˆNS 2. if ErˆNSEsˆNS and if ArˆNS AsˆNSthen rˆNS sˆNS 3. if ErˆNSEsˆNS and if ArˆNS AsˆNSthen rˆNS sˆNS 4. if ErˆNSEsˆNS and if ArˆNS AsˆNS and CrˆNSCsˆNSthen rˆNS sˆNS 5. if ErˆNSEsˆNS and if ArˆNS AsˆNS and CrˆNSCsˆNSthen rˆNS sˆNS 6. if ErˆNSEsˆNS and if ArˆNS AsˆNS and CrˆNSCsˆNSthen rˆNS sˆNS 2.1 List of Abbreviation used throughout this paper. SPP stands for “shortest path problem.” NSSPP stands for “neutrosophic shortest path problem.” NSSP stands for “neutrosophic shortest path.” NSSPL stands for “neutrosophic shortest path length.” CrSPL stands for “crisp shortest path length.” FSPP stands for “fuzzy shortest path problem.” IFSPP stands for “intuitionistic fuzzy shortest path problem.” NS stands for “neutrosophic set.” TrNSSPP stands for “trapezoidal neutrosophic shortest path.” TNSSPP stands for “triangular neutrosophic shortest path.” TrNS stands for “trapezoidal neutrosophic set.” TNS stands for “triangular neutrosophic set.” MF stands for “membership function.” TFN stands for “triangular fuzzy number.” TrFN stands for “trapezoidal fuzzy number.” 3 The Proposed model Before we start the main algorithm, we introduce a sub-section i.e., shortcoming and limitation of some of the existing models: 3.1 Discussion on shortcoming of some of the existing methods At first, we discussed the shortcoming and limitation of the existing methods under two different type of NS environment. Broumi et al. [68-69] first proposed a method to find the shortest path under TrNS environment. It is a very well known and propular paper in the field of neutrosophic set and system. However, the authors used some Ranjan Kumar, S A Edaltpanah, Sripati Jha, Said Broumi and Arindam Dey. Neutrosophic shortest path problem
The list of books you might like
