New types of Neutrosophic Set/Logic/Probability, Neutrosophic Over-/Under-/ Off-Set, Neutrosophic Refined Set, and their Extension to Plithogenic Set/Logic/ Probability, with Applications Printed Edition of the Special Issue Published in Symmetry www.mdpi.com/journal/symmetry Florentin Smarandache Edited by New types of Neutrosophic Set/Logic/Probability, Neutrosophic Over-/Under-/Off-Set, Neutrosophic Refined Set, and their Extension to Plithogenic Set/Logic/Probability, with Applications New types of Neutrosophic Set/Logic/Probability, Neutrosophic Over-/Under-/Off-Set, Neutrosophic Refined Set, and their Extension to Plithogenic Set/Logic/Probability, with Applications Special Issue Editor Florentin Smarandache MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Special Issue Editor Florentin Smarandache University of New Mexico USA Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Symmetry (ISSN 2073-8994) in 2019 (available at: https://www.mdpi.com/journal/symmetry/special issues/ Neutrosophic Set Logic Probability). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year , Article Number , Page Range. ISBN 978-3-03921-938-4 (Pbk) ISBN 978-3-03921-939-1 (PDF) c © 2019 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Special Issue Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Preface to ”New types of Neutrosophic Set/Logic/Probability, Neutrosophic Over-/Under-/Off-Set, Neutrosophic Refined Set, and their Extension to Plithogenic Set/Logic/Probability, with Applications” . . . . . . . . . . . . . . . . . . . . . . . . xi Yingcang Ma, Xiaohong Zhang, Florentin Smarandache, Juanjuan Zhang The Structure of Idempotents in Neutrosophic Rings and Neutrosophic Quadruple Rings Reprinted from: Symmetry 2019 , 11 , 1254, doi:10.3390/sym11101254 . . . . . . . . . . . . . . . . . 1 Xiaohong Zhang, Florentin Smarandache and Yingcang Ma Symmetry in Hyperstructure: Neutrosophic Extended Triplet Semihypergroups and Regular Hypergroups Reprinted from: Symmetry 2019 , 11 , 1217, doi:10.3390/sym11101217 . . . . . . . . . . . . . . . . . 16 Erick Gonz ́ alez Caballero, Florentin Smarandache and Maikel Leyva V ́ azquez On Neutrosophic Offuninorms Reprinted from: Symmetry 2019 , 11 , 1136, doi:10.3390/sym11091136 . . . . . . . . . . . . . . . . . 34 Junhui Kim, Florentin Smarandache, Jeong Gon Lee and Kul Hur Ordinary Single Valued Neutrosophic Topological Spaces Reprinted from: Symmetry 2019 , 11 , 1075, doi:10.3390/sym11091075 . . . . . . . . . . . . . . . . . 60 Jingqian Wang, Xiaohong Zhang A New Type of Single Valued Neutrosophic Covering Rough Set Model Reprinted from: Symmetry 2019 , 11 , 1074, doi:10.3390/sym11091074 . . . . . . . . . . . . . . . . . 86 Muhammad Akram, Sumera Naz and Florentin Smarandache Generalization of Maximizing Deviation and TOPSIS Method for MADM in Simplified Neutrosophic Hesitant Fuzzy Environment Reprinted from: Symmetry 2019 , 11 , 1058, doi:10.3390/sym11081058 . . . . . . . . . . . . . . . . . 109 Marcel-Ioan Bolos, Ioana-Alexandra Bradea and Camelia Delcea Modeling the Performance Indicators of Financial Assets with Neutrosophic Fuzzy Numbers Reprinted from: Symmetry 2019 , 11 , 1021, doi:10.3390/sym11081021 . . . . . . . . . . . . . . . . . 135 Qiaoyan Li, Yingcang Ma, Xiaohong Zhang and Juanjuan Zhang Study on the Algebraic Structure of Refined Neutrosophic Numbers Reprinted from: Symmetry 2019 , 11 , 954, doi:10.3390/sym11080954 . . . . . . . . . . . . . . . . . 158 Chunxin Bo, Xiaohong Zhang and Songtao Shao Non-Dual Multi-Granulation Neutrosophic Rough Set with Applications Reprinted from: Symmetry 2019 , 11 , 910, doi:10.3390/sym11070910 . . . . . . . . . . . . . . . . . 171 Mohamed Abdel-Basset, Rehab Mohamed, Abd El-Nasser H. Zaied and Florentin Smarandache A Hybrid Plithogenic Decision-Making Approach with Quality Function Deployment for Selecting Supply Chain Sustainability Metrics Reprinted from: Symmetry 2019 , 11 , 903, doi:10.3390/sym11070903 . . . . . . . . . . . . . . . . . 187 v Keli Hu, Wei He, Jun Ye, Liping Zhao, Hua Peng and Jiatian Pi Online Visual Tracking of Weighted Multiple Instance Learning via Neutrosophic Similarity-Based Objectness Estimation Reprinted from: Symmetry 2019 , 11 , 832, doi:10.3390/sym11060832 . . . . . . . . . . . . . . . . . 208 Qingqing Hu, Xiaohong Zhang Neutrosophic Triangular Norms and Their Derived Residuated Lattices Reprinted from: Symmetry 2019 , 11 , 817, doi:10.3390/sym11060817 . . . . . . . . . . . . . . . . . 232 Mohammed A. Al Shumrani and Florentin Smarandache Introduction to Non-Standard Neutrosophic Topology Reprinted from: Symmetry 2019 , 11 , 706, doi:10.3390/sym11050706 . . . . . . . . . . . . . . . . . 254 Muhammad Jamil, Saleem Abdullah, Muhammad Yaqub Khan, Florentin Smarandache and Fazal Ghani Application of the Bipolar Neutrosophic Hamacher Averaging Aggregation Operators to Group Decision Making: An Illustrative Example Reprinted from: Symmetry 2019 , 11 , 698, doi:10.3390/sym11050698 . . . . . . . . . . . . . . . . . 268 Qiaoyan Li, Yingcang Ma, Xiaohong Zhang, Juanjuan Zhang Neutrosophic Extended Triplet Group Based on Neutrosophic Quadruple Numbers Reprinted from: Symmetry 2019 , 11 , 696, doi:10.3390/sym11050696 . . . . . . . . . . . . . . . . . 286 Shao Songtao, Zhang Xiaohong and Zhao Quan Multi-Attribute Decision Making Based on Probabilistic Neutrosophic Hesitant Fuzzy Choquet Aggregation Operators Reprinted from: Symmetry 2019 , 11 , 623, doi:10.3390/sym11050623 . . . . . . . . . . . . . . . . . 301 Mohamed Abdel-Basset, Mai Mohamed, Victor Chang and Florentin Smarandache IoT and Its Impact on the Electronics Market: A Powerful Decision Support System for Helping Customers in Choosing the Best Product Reprinted from: Symmetry 2019 , 11 , 611, doi:10.3390/sym11050611 . . . . . . . . . . . . . . . . . 316 Firoz Ahmad, Ahmad Yusuf Adhami and Florentin Smarandache Neutrosophic Optimization Model and Computational Algorithm for Optimal Shale Gas Water Management under Uncertainty Reprinted from: Symmetry 2019 , 11 , 544, doi:10.3390/sym11040544 . . . . . . . . . . . . . . . . . 337 Florentin Smarandache Extended Nonstandard Neutrosophic Logic, Set, and Probability Based on Extended Nonstandard Analysis Reprinted from: Symmetry 2019 , 11 , 515, doi:10.3390/sym11040515 . . . . . . . . . . . . . . . . . 371 Jihong Chen, Kai Xue, Jun Ye, Tiancun Huang, Yan Tian, Chengying Hua and Yuhua Zhu Simplified Neutrosophic Exponential Similarity Measures for Evaluation of Smart Port Development Reprinted from: Symmetry 2019 , 11 , 485, doi:10.3390/sym11040485 . . . . . . . . . . . . . . . . . 396 Mohamed Abdel-Basset, Victor Chang, Mai Mohamed and Florentin Smarandache A Refined Approach for Forecasting Based on Neutrosophic Time Series Reprinted from: Symmetry 2019 , 11 , 457, doi:10.3390/sym11040457 . . . . . . . . . . . . . . . . . 408 Ashraf Al-Quran, Nasruddin Hassan and Emad Marei A Novel Approach to Neutrosophic Soft Rough Set under Uncertainty Reprinted from: Symmetry 2019 , 11 , 384, doi:10.3390/sym11030384 . . . . . . . . . . . . . . . . . 431 vi Ashraf Al-Quran, Nasruddin Hassan and Shawkat Alkhazaleh Fuzzy Parameterized Complex Neutrosophic Soft Expert Set for Decision under Uncertainty Reprinted from: Symmetry 2019 , 11 , 382, doi:10.3390/sym11030382 . . . . . . . . . . . . . . . . . 447 Shahzaib Ashraf, Saleem Abdullah, Florentin Smarandache and Noor ul Amin Logarithmic Hybrid Aggregation Operators Based on Single Valued Neutrosophic Sets and Their Applications in Decision Support Systems Reprinted from: Symmetry 2019 , 11 , 364, doi:10.3390/sym11030364 . . . . . . . . . . . . . . . . . 466 Muhammad Aslam and Mohammed Albassam Application of Neutrosophic Logic to Evaluate Correlation between Prostate Cancer Mortality and Dietary Fat Assumption Reprinted from: Symmetry 2019 , 11 , 330, doi:10.3390/sym11030330 . . . . . . . . . . . . . . . . . 489 Yingcang Ma, Xiaohong Zhang, Xiaofei Yang, Juanjuan Zhang and Hu Zhao Generalized Neutrosophic Extended Triplet Group Reprinted from: Symmetry 2019 , 11 , 327, doi:10.3390/sym11030327 . . . . . . . . . . . . . . . . . 496 Muhammad Gulistan and Nasruddin Hassan A Generalized Approach towards Soft Expert Sets via Neutrosophic Cubic Sets with Applications in Games Reprinted from: Symmetry 2019 , 11 , 289, doi:10.3390/sym11020289 . . . . . . . . . . . . . . . . . 512 Lilian Shi and Yue Yuan Hybrid Weighted Arithmetic and Geometric Aggregation Operator of Neutrosophic Cubic Sets for MADM Reprinted from: Symmetry 2019 , 11 , 278, doi:10.3390/sym11020278 . . . . . . . . . . . . . . . . . 538 Chengdong Cao, Shouzhen Zeng and Dandan Luo A Single-Valued Neutrosophic Linguistic Combined Weighted Distance Measure and Its Application in Multiple-Attribute Group Decision-Making Reprinted from: Symmetry 2019 , 11 , 275, doi:10.3390/sym11020275 . . . . . . . . . . . . . . . . . 548 Wen Jiang, Zihan Zhang and Xinyang Deng Multi-Attribute Decision Making Method Based on Aggregated Neutrosophic Set Reprinted from: Symmetry 2019 , 11 , 267, doi:10.3390/sym11020267 . . . . . . . . . . . . . . . . . 559 Majid Khan, Muhammad Gulistan, Naveed Yaqoob, Madad Khan and Florentin Smarandache Neutrosophic Cubic Einstein Geometric Aggregation Operators with Application to Multi-Criteria Decision Making Method Reprinted from: Symmetry 2019 , 11 , 247, doi:10.3390/sym11020247 . . . . . . . . . . . . . . . . . 572 Muhammad Aslam and Mansour Sattam Aldosari Inspection Strategy under Indeterminacy Based on Neutrosophic Coefficient of Variation Reprinted from: Symmetry 2019 , 11 , 193, doi:10.3390/sym11020193 . . . . . . . . . . . . . . . . . 596 Aliya Fahmi, Fazli Amin, Madad Khan and Florentin Smarandache Group Decision Making Based on Triangular Neutrosophic Cubic Fuzzy Einstein Hybrid Weighted Averaging Operators Reprinted from: Symmetry 2019 , 11 , 180, doi:10.3390/sym11020180 . . . . . . . . . . . . . . . . . 604 vii Jun Ye and Wenhua Cui Neutrosophic Compound Orthogonal Neural Network and Its Applications in Neutrosophic Function Approximation Reprinted from: Symmetry 2019 , 11 , 147, doi:10.3390/sym11020147 . . . . . . . . . . . . . . . . . 633 Majdoleen Abu Qamar and Nasruddin Hassan An Approach toward a Q-Neutrosophic Soft Set and Its Application in Decision Making Reprinted from: Symmetry 2019 , 11 , 139, doi:10.3390/sym11020139 . . . . . . . . . . . . . . . . . 642 Muhammad Aslam A Variable Acceptance Sampling Plan under Neutrosophic Statistical Interval Method Reprinted from: Symmetry 2019 , 11 , 114, doi:10.3390/sym11010114 . . . . . . . . . . . . . . . . . 660 Chiranjibe Jana and Madhumangal Pal A Robust Single-Valued Neutrosophic Soft Aggregation Operators in Multi-Criteria Decision Making Reprinted from: Symmetry 2019 , 11 , 110, doi:10.3390/sym11010110 . . . . . . . . . . . . . . . . . 667 viii About the Special Issue Editor Florentin Smarandache is Professor of Mathematics at the University of New Mexico, United States. He was awarded his MSc in Mathematics and Computer Science from the University of Craiova, Romania, PhD in Mathematics from the State University of Kishinev, and was a Postdoctor in Applied Mathematics at the Okayama University of Sciences, Japan. He is the founder of neutrosophy (generalization of dialectics), neutrosophic set, logic, probability and statistics, and since 1995, has published hundreds of papers on neutrosophic physics, superluminal and instantaneous physics, unmatter, absolute theory of relativity, redshift and blueshift due to the medium gradient and refraction index besides the Doppler effect, paradoxism, outerart, neutrosophy as a new branch of philosophy, law of included multiple-middle, multispace and multistructure, quantum paradoxes, degree of dependence and independence between neutrosophic components, refined neutrosophic set, neutrosophic over/under/offset, plithogenic set, neutrosophic triplet and duplet structures, quadruple neutrosophic structures, and Dezert–Smarandache theory (DSmT) to many peer-reviewed international journals and many books, and has presented papers and plenary lectures at many international conferences around the world. [http://fs.unm.edu/FlorentinSmarandache.htm] ix xi Preface to ”New types of Neutrosophic Set/Logic/Probability, Neutrosophic Over-/Under-/ Off-Set, Neutrosophic Refined Set, and 5 heir Extension to Plithogenic Set/Logic/Probability, with Applications” Department ȱ of ȱ Mathematics, ȱ University ȱ of ȱ New ȱ Mexico, ȱ 705 ȱ Gurley ȱ Ave., ȱ Gallup, ȱ NM ȱ 87301, ȱ USA; ȱ smarand@unm.edu ȱ Received: ȱ 07 ȱ November ȱ 2019; ȱ Published: ȱ 18 ȱ November ȱ 2019 ȱ DZ ȱ In ȱ this ȱ paper ȱ we ȱ review ȱ all ȱ thirty Ȭ seven ȱ neutrosophic ȱ and ȱ plithogenic ȱ papers ȱ published ȱ by ȱ Symmetry ȱ journal ȱ within ȱ the ȱ special ȱ issue ȱ “New ȱ types ȱ of ȱ Neutrosophic ȱ Set/Logic/Probability, ȱ Neutrosophic ȱ Over Ȭ /Under Ȭ /Off Ȭ Set, ȱ Neutrosophic ȱ Refined ȱ Set, ȱ and ȱ their ȱ Extension ȱ to ȱ Plithogenic ȱ Set/Logic/Probability, ȱ with ȱ Applications” ȱ (2019). ȱ ¢ DZ ȱ neutrosophy, ȱ neutrosophic ȱ set, ȱ neutrosophic ȱ logic, ȱ neutrosophic ȱ probability, ȱ neutrosophic ȱ statistics, ȱ neutrosophic ȱ overset, ȱ neutrosophic ȱ underset, ȱ neutrosophic ȱ offset, ȱ degree ȱ of ȱ dependence ȱ and ȱ independence ȱ between ȱ components, ȱ refined ȱ neutrosophic ȱ set, ȱ law ȱ of ȱ included ȱ multiple Ȭ middle, ȱ neutrosophic ȱ bipolar ȱ and ȱ tripolar ȱ and ȱ multipolar ȱ sets, ȱ neutrosophic ȱ algebraic ȱ structures, ȱ neutrosophic ȱ triplet ȱ algebraic ȱ structures, ȱ neutrosophic ȱ extended ȱ triplet ȱ algebraic ȱ structures, ȱ quadruple ȱ neutrosophic ȱ algebraic ȱ structures, ȱ plithogeny, ȱ plithogenic ȱ set, ȱ plithogenic ȱ logic ȱȱ The ȱ fields ȱ of ȱ neutrosophic ȱ and ȱ plithogenic ȱ sets, ȱ logic, ȱ measure, ȱ probability, ȱ and ȱ statistics ȱ have ȱ been ȱ developed ȱ and ȱ explored ȱ extensively ȱ in ȱ the ȱ last ȱ few ȱ years ȱ because ȱ of ȱ their ȱ multiple ȱ practical ȱ applications. ȱ The ȱ neutrosophic ȱ components ȱ of ȱ truth ȱ (T), ȱ indeterminacy ȱ (I), ȱ and ȱ falsehood ȱ (F) ȱ are ȱ symmetric ȱ in ȱ form, ȱ since ȱ T ȱ is ȱ symmetric ȱ to ȱ its ȱ opposite ȱ F ȱ with ȱ respect ȱ to ȱ I, ȱ which ȱ acts ȱ as ȱ an ȱ axis ȱ of ȱ symmetry ȱ between ȱ T–I–F. ȱ This ȱ Special ȱ Issue ȱ invited ȱ state Ȭ of Ȭ the Ȭ art ȱ papers ȱ on ȱ new ȱ topics ȱ related ȱ to ȱ neutrosophic ȱ theories ȱ and ȱ applications, ȱ including: ȱ —Studies ȱ of ȱ corner ȱ cases ȱ of ȱ neutrosophic ȱ sets/probabilities/statistics/logics, ȱ such ȱ as: ȱ x neutrosophic ȱ intuitionistic ȱ sets ȱ (which ȱ are ȱ different ȱ from ȱ intuitionistic ȱ fuzzy ȱ sets), ȱ neutrosophic paraconsistent ȱ sets, ȱ neutrosophic ȱ faillibilist ȱ sets, ȱ neutrosophic ȱ paradoxist ȱ sets, ȱ neutrosophic, pseudo Ȭ paradoxist ȱ sets, ȱ neutrosophic ȱ tautological ȱ sets, ȱ neutrosophic ȱ nihilist ȱ sets, ȱ neutrosophic dialetheist ȱ sets, ȱ and ȱ neutrosophic ȱ trivialist ȱ sets; x neutrosophic ȱ intuitionistic ȱ probability ȱ and ȱ statistics, ȱ neutrosophic ȱ paraconsistent ȱ probability and ȱ statistics, ȱ neutrosophic ȱ faillibilist ȱ probability ȱ and ȱ statistics, ȱ neutrosophic ȱ paradoxist probability ȱ and ȱ statistics, ȱ neutrosophic ȱ pseudo Ȭ paradoxist ȱ probability ȱ and ȱ statistics, neutrosophic ȱ tautological ȱ probability ȱ and ȱ statistics, ȱ neutrosophic ȱ nihilist ȱ probability ȱ and statistics, ȱ neutrosophic ȱ dialetheist ȱ probability ȱ and ȱ statistics, ȱ and ȱ neutrosophic ȱ trivialist probability ȱ and ȱ statistics; ȱ ȱ xii x neutrosophic ȱ paradoxist ȱ logic ȱ (or ȱ paradoxism), ȱ neutrosophic ȱ pseudo Ȭ paradoxist ȱ logic ȱ (or neutrosophic ȱ pseudo Ȭ paradoxism), ȱ and ȱ neutrosophic ȱ tautological ȱ logic ȱ (or ȱ neutrosophic tautologism) ȱ [1]; ȯ Refined ȱ neutrosophic ȱ set ȱ components ȱ (T, ȱ I, ȱ and ȱ F), ȱ which ȱ are ȱ refined/split ȱ into ȱ many ȱ neutrosophic ȱ subcomponents: ȱ (T 1 , ȱ T 2 , ȱ ...; ȱ I 1 , ȱ I 2 , ȱ ...; ȱ F 1 , ȱ F 2 , ȱ ...) ȱ [2]; ȱ ȯ Degrees ȱ of ȱ dependence ȱ and ȱ independence ȱ between ȱ neutrosophic ȱ components: ȱ T, ȱ I, ȱ and ȱ F ȱ as ȱ independent ȱ components, ȱ leave ȱ room ȱ for ȱ incomplete ȱ information ȱ (when ȱ their ȱ superior ȱ sum ȱ < ȱ 1), ȱ paraconsistent ȱ and ȱ contradictory ȱ information ȱ (when ȱ the ȱ superior ȱ sum ȱ > ȱ 1), ȱ or ȱ complete ȱ information ȱ (sum ȱ of ȱ components ȱ = ȱ 1). ȱ For ȱ technical ȱ and ȱ engineering ȱ proposals, ȱ the ȱ classical ȱ unit ȱ interval ȱ [0,1] ȱ is ȱ used. ȱ For ȱ single Ȭ valued ȱ neutrosophic ȱ logic, ȱ the ȱ sum ȱ of ȱ the ȱ components ȱ is: ȱ 0 ȱǂȱ t ȱ + ȱ i ȱ + ȱ f ȱǂȱ 3 ȱ when ȱ all ȱ three ȱ components ȱ are ȱ independent; ȱ 0 ȱǂȱ t ȱ + ȱ i ȱ + ȱ f ȱǂȱ 2 ȱ when ȱ two ȱ components ȱ are ȱ dependent, ȱ while ȱ the ȱ third ȱ one ȱ is ȱ independent ȱ from ȱ them; ȱ 0 ȱǂȱ t ȱ + ȱ i ȱ + ȱ f ȱǂȱ 1 ȱ when ȱ all ȱ three ȱ components ȱ are ȱ dependent. ȱ When ȱ three ȱ or ȱ two ȱ of ȱ the ȱ components ȱ T, ȱ I, ȱ and ȱ F ȱ are ȱ independent, ȱ one ȱ leaves ȱ room ȱ for ȱ incomplete ȱ information ȱ (sum ȱ < ȱ 1), ȱ paraconsistent ȱ and ȱ contradictory ȱ information ȱ (sum ȱ > ȱ 1), ȱ or ȱ complete ȱ information ȱ (sum ȱ = ȱ 1). ȱ If ȱ all ȱ three ȱ components ȱ T, ȱ I, ȱ and ȱ F ȱ are ȱ dependent ȱ then, ȱ similarly, ȱ one ȱ leaves ȱ room ȱ for ȱ incomplete ȱ information ȱ (sum ȱ < ȱ 1) ȱ or ȱ complete ȱ information ȱ (sum ȱ = ȱ 1). ȱ In ȱ general, ȱ the ȱ sum ȱ of ȱ two ȱ components ȱ x ȱ and ȱ y ȱ that ȱ vary ȱ in ȱ the ȱ unitary ȱ interval ȱ [0,1] ȱ is ȱ 0 ȱǂȱ x ȱ + ȱ y ȱ ǂȱ 2 ȱ – ȱ d°(x, ȱ y), ȱ in ȱ which ȱ d°(x, ȱ y) ȱ is ȱ the ȱ degree ȱ of ȱ dependence ȱ between ȱ x ȱ and ȱ y ȱ [3]; ȱ Neutrosophic ȱ overset ȱ (when ȱ some ȱ neutrosophic ȱ component ȱ is ȱ > ȱ 1) ȱ is ȱ observed, ȱ for ȱ example, ȱ when ȱ an ȱ employee ȱ works ȱ overtime ȱ and ȱ deserves ȱ a ȱ degree ȱ of ȱ membership ȱ > ȱ 1, ȱ with ȱ respect ȱ to ȱ an ȱ employee ȱ that ȱ only ȱ works ȱ regular ȱ full Ȭ time ȱ and ȱ whose ȱ degree ȱ of ȱ membership ȱ = ȱ 1. ȱ Neutrosophic ȱ underset ȱ (when ȱ some ȱ neutrosophic ȱ component ȱ is ȱ < ȱ 0) ȱ is ȱ observed, ȱ for ȱ example, ȱ when ȱ an ȱ employee ȱ causes ȱ more ȱ damage ȱ than ȱ benefit ȱ to ȱ his ȱ company ȱ and ȱ deserves ȱ a ȱ degree ȱ of ȱ membership ȱ < ȱ 0, ȱ with ȱ respect ȱ to ȱ an ȱ employee ȱ that ȱ produces ȱ benefit ȱ to ȱ the ȱ company ȱ and ȱ has ȱ a ȱ degree ȱ of ȱ membership ȱ > ȱ 0. ȱ Neutrosophic ȱ offset ȱ occurs ȱ when ȱ some ȱ neutrosophic ȱ components ȱ are ȱ off ȱ the ȱ interval ȱ [0,1] ȱ (i.e., ȱ some ȱ neutrosophic ȱ component ȱ > ȱ 1 ȱ and ȱ some ȱ neutrosophic ȱ component ȱ < ȱ 0). ȱ —Then, ȱ similarly, ȱ neutrosophic ȱ logic/measure/probability/statistics, ȱ and ȱ so ȱ on, ȱ were ȱ extended ȱ to, ȱ respectively, ȱ neutrosophic ȱ over Ȭ /under Ȭ /off Ȭ logic, ȱ measure, ȱ probability, ȱ statistics, ȱ and ȱ so ȱ on ȱ [4,5]; ȱ ȯ Neutrosophic ȱ tripolar ȱ set ȱ and ȱ neutrosophic ȱ multipolar ȱ set ȱ and, ȱ consequently, ȱ the ȱ neutrosophic ȱ tripolar ȱ graph ȱ and ȱ neutrosophic ȱ multipolar ȱ graph ȱ [6]; ȱ —N Ȭ norm ȱ and ȱ N Ȭ conorm ȱ [7]; ȱ —Neutrosophic ȱ measure ȱ and ȱ neutrosophic ȱ probability ȱ (chance ȱ that ȱ an ȱ event ȱ occurs, ȱ indeterminate ȱ chance ȱ of ȱ occurrence, ȱ chance ȱ that ȱ the ȱ event ȱ does ȱ not ȱ occur) ȱ [8]; ȱ ȯ Law ȱ of ȱ included ȱ multiple Ȭ middle ȱ (as ȱ middle ȱ part ȱ of ȱ refined ȱ neutrosophy): ȱ (<A>; ȱ <neutA 1 >, ȱ <neutA 2 >, ȱ ...; ȱ <antiA>) ȱ [9]; ȱ —Neutrosophic ȱ statistics ȱ (indeterminacy ȱ is ȱ introduced ȱ into ȱ classical ȱ statistics ȱ with ȱ respect ȱ to ȱ the ȱ sample/population ȱ characteristics, ȱ or ȱ with ȱ respect ȱ to ȱ the ȱ individuals ȱ that ȱ only ȱ partially ȱ belong ȱ to ȱ a ȱ sample/population, ȱ or ȱ with ȱ respect ȱ to ȱ the ȱ neutrosophic ȱ probability ȱ distributions) ȱ [10]; ȱ —Neutrosophic ȱ precalculus ȱ and ȱ neutrosophic ȱ calculus ȱ [11]; ȱ —Refined ȱ neutrosophic ȱ numbers ȱ (a ȱ + ȱ b 1 I 1 ȱ + ȱ b 2 I 2 ȱ + ȱ ... ȱ + ȱ b n I n ), ȱ in ȱ which ȱ I 1 , ȱ I 2 , ȱ ..., ȱ I n ȱ are ȱ sub Ȭ indeterminacies ȱ of ȱ indeterminacy ȱ I; ȱ (t,i,f) Ȭ neutrosophic ȱ graphs ; ȱ thesis–antithesis–neutrothesis, ȱ and ȱ neutrosynthesis; ȱ neutrosophic ȱ axiomatic ȱ system; ȱ neutrosophic ȱ dynamic ȱ systems; ȱ symbolic ȱ neutrosophic ȱ logic; ȱ (t, ȱ i, ȱ f) Ȭ neutrosophic ȱ structures; ȱ i Ȭ neutrosophic ȱ structures; ȱ refined ȱ literal ȱ xiii indeterminacy; ȱ quadruple ȱ neutrosophic ȱ algebraic ȱ structures; ȱ and ȱ multiplication ȱ law ȱ of ȱ sub Ȭ indeterminacies ȱ [12]; ȱ —Theory ȱ of ȱ neutrosophic ȱ evolution: ȱ degrees ȱ of ȱ evolution, ȱ indeterminacy ȱ or ȱ neutrality, ȱ and ȱ involution ȱ [13]; ȱ —Plithogeny ȱ as ȱ generalization ȱ of ȱ dialectics ȱ and ȱ neutrosophy; ȱ plithogenic ȱ set/logic/probability/statistics ȱ (as ȱ generalization ȱ of ȱ fuzzy, ȱ intuitionistic ȱ fuzzy, ȱ neutrosophic ȱ set/logic/probability/statistics) ȱ [14]; ȱ —Neutrosophic ȱ psychology ȱ (neutropsyche; ȱ refined ȱ neutrosophic ȱ memory: ȱ conscious, ȱ aconscious, ȱ unconscious; ȱ neutropsychic ȱ personality; ȱ Eros/Aoristos/Thanatos; ȱ and ȱ neutropsychic ȱ crisp ȱ personality) ȱ [15]; ȱ ȯ Neutrosophic ȱ applications ȱ in ȱ artificial ȱ intelligence, ȱ information ȱ systems, ȱ computer ȱ science, ȱ cybernetics, ȱ theory ȱ methods, ȱ mathematical ȱ algebraic ȱ structures, ȱ applied ȱ mathematics, ȱ automation, ȱ control ȱ systems, ȱ big ȱ data, ȱ engineering, ȱ electrical, ȱ electronic, ȱ philosophy, ȱ social ȱ science, ȱ psychology, ȱ biology, ȱ engineering, ȱ operational ȱ research, ȱ management ȱ science, ȱ imaging ȱ science, ȱ photographic ȱ technology, ȱ instruments, ȱ instrumentation, ȱ physics, ȱ optics, ȱ economics, ȱ mechanics, ȱ neurosciences, ȱ radiology ȱ nuclear, ȱ interdisciplinary ȱ applications, ȱ multidisciplinary ȱ sciences, ȱ and ȱ so ȱ on ȱ [16]. ȱ The ȱ Special ȱ Issue ȱ “New ȱ types ȱ of ȱ Neutrosophic ȱ Set/Logic/Probability, ȱ Neutrosophic ȱ Over Ȭ /Under Ȭ /Off Ȭ Set, ȱ Neutrosophic ȱ Refined ȱ Set, ȱ and ȱ their ȱ Extension ȱ to ȱ Plithogenic ȱ Set/Logic/Probability, ȱ with ȱ Applications” ȱ comprises ȱ 37 ȱ papers ȱ focusing ȱ on ȱ topics ȱ such ȱ as: ȱ neutrosophic ȱ set; ȱ neutrosophic ȱ rings; ȱ neutrosophic ȱ quadruple ȱ rings; ȱ idempotents; ȱ neutrosophic ȱ extended ȱ triplet ȱ group; ȱ hypergroup; ȱ semihypergroup; ȱ neutrosophic ȱ extended ȱ triplet ȱ group; ȱ neutrosophic ȱ extended ȱ triplet ȱ semihypergroup ȱ (NET Ȭ semihypergroup); ȱ NET Ȭ hypergroup; ȱ neutrosophic ȱ offset; ȱ uninorm; ȱ neutrosophic ȱ offuninorm; ȱ neutrosophic ȱ offnorm; ȱ neutrosophic ȱ offconorm; ȱ implicator; ȱ prospector; ȱ n Ȭ person ȱ cooperative ȱ game; ȱ ordinary ȱ single ȱ valued ȱ neutrosophic ȱ (co)topology; ȱ ordinary ȱ single ȱ valued ȱ neutrosophic ȱ subspace; ȱ΅Ȭ level; ȱ ordinary ȱ single ȱ valued ȱ neutrosophic ȱ neighborhood ȱ system; ȱ ordinary ȱ single ȱ valued ȱ neutrosophic ȱ base; ȱ ordinary ȱ single ȱ valued ȱ neutrosophic ȱ subbase; ȱ fuzzy ȱ numbers; ȱ neutrosophic ȱ numbers; ȱ neutrosophic ȱ symmetric ȱ scenarios; ȱ performance ȱ indicators; ȱ financial ȱ assets; ȱ neutrosophic ȱ extended ȱ triplet ȱ group; ȱ neutrosophic ȱ quadruple ȱ numbers; ȱ refined ȱ neutrosophic ȱ numbers; ȱ refined ȱ neutrosophic ȱ quadruple ȱ numbers; ȱ multigranulation ȱ neutrosophic ȱ rough ȱ set; ȱ nondual; ȱ two ȱ universes; ȱ multiattribute ȱ group ȱ decision ȱ making; ȱ nonstandard ȱ analysis; ȱ extended ȱ nonstandard ȱ analysis; ȱ monad; ȱ binad; ȱ left ȱ monad ȱ closed ȱ to ȱ the ȱ right; ȱ right ȱ monad ȱ closed ȱ to ȱ the ȱ left; ȱ pierced ȱ binad; ȱ unpierced ȱ binad; ȱ nonstandard ȱ neutrosophic ȱ mobinad ȱ set; ȱ neutrosophic ȱ topology; ȱ nonstandard ȱ neutrosophic ȱ topology; ȱ visual ȱ tracking; ȱ neutrosophic ȱ weight; ȱ objectness; ȱ weighted ȱ multiple ȱ instance ȱ learning; ȱ neutrosophic ȱ triangular ȱ norms; ȱ residuated ȱ lattices; ȱ representable ȱ neutrosophic ȱ t Ȭ norms; ȱ De ȱ Morgan ȱ neutrosophic ȱ triples; ȱ neutrosophic ȱ residual ȱ implications; ȱ infinitely ȱ ǗȬ distributive; ȱ probabilistic ȱ neutrosophic ȱ hesitant ȱ fuzzy ȱ set ȱ (PNHFS); ȱ decision Ȭ making; ȱ Choquet ȱ integral; ȱ e Ȭ marketing; ȱ Internet ȱ of ȱ Things; ȱ neutrosophic ȱ set; ȱ multicriteria ȱ decision Ȭ making ȱ techniques; ȱ intuitionistic ȱ fuzzy ȱ parameters; ȱ uncertainty ȱ modeling; ȱ neutrosophic ȱ goal ȱ programming ȱ approach; ȱ shale ȱ gas ȱ water ȱ management ȱ system, ȱ and ȱ many ȱ more. ȱ Molodtsov ȱ originated ȱ soft ȱ set ȱ theory ȱ that ȱ provided ȱ a ȱ general ȱ mathematical ȱ framework ȱ for ȱ handling ȱ uncertainties, ȱ in ȱ which ȱ one ȱ meets ȱ the ȱ data ȱ by ȱ an ȱ affixed ȱ parameterized ȱ factor ȱ during ȱ information ȱ analysis ȱ as ȱ differentiated ȱ to ȱ fuzzy ȱ as ȱ well ȱ as ȱ neutrosophic ȱ set ȱ theory. ȱ The ȱ main ȱ objective ȱ of ȱ the ȱ first ȱ paper ȱ [17] ȱ is ȱ to ȱ lay ȱ a ȱ foundation ȱ for ȱ providing ȱ a ȱ new ȱ approach ȱ of ȱ a ȱ single Ȭ valued ȱ neutrosophic ȱ soft ȱ tool ȱ that ȱ considers ȱ many ȱ problems ȱ that ȱ contain ȱ uncertainties. ȱ In ȱ the ȱ present ȱ study, ȱ new ȱ aggregation ȱ operators ȱ of ȱ single Ȭ valued ȱ neutrosophic ȱ soft ȱ numbers ȱ have, ȱ so ȱ far, ȱ not ȱ yet ȱ been ȱ applied ȱ for ȱ ranking ȱ the ȱ alternatives ȱ in ȱ decision Ȭ making ȱ problems. ȱ To ȱ this ȱ proposed ȱ work, ȱ a ȱ single Ȭ valued ȱ neutrosophic ȱ soft Ȭ weighted ȱ arithmetic ȱ averaging ȱ (SVNSWA) ȱ operator ȱ and ȱ single Ȭ valued ȱ neutrosophic ȱ soft Ȭ weighted ȱ geometric ȱ averaging ȱ (SVNSWGA) ȱ operator ȱ have ȱ been ȱ used ȱ to ȱ compare ȱ two ȱ single Ȭ valued ȱ neutrosophic ȱ soft ȱ numbers ȱ (SVNSNs) ȱ for ȱ aggregating ȱ different ȱ single Ȭ valued ȱ xiv neutrosophic ȱ soft ȱ input ȱ arguments ȱ in ȱ neutrosophic ȱ soft ȱ environments. ȱ Then, ȱ its ȱ related ȱ properties ȱ have ȱ been ȱ investigated. ȱ Finally, ȱ a ȱ practical ȱ example ȱ for ȱ medical ȱ diagnosis ȱ problems ȱ is ȱ provided ȱ to ȱ test ȱ the ȱ feasibility ȱ and ȱ applicability ȱ of ȱ the ȱ proposed ȱ work. ȱ The ȱ acceptance ȱ sampling ȱ plan ȱ plays ȱ an ȱ important ȱ role ȱ in ȱ maintaining ȱ the ȱ high ȱ quality ȱ of ȱ a ȱ product. ȱ The ȱ variable ȱ control ȱ chart, ȱ using ȱ classical ȱ statistics, ȱ helps ȱ in ȱ making ȱ acceptance ȱ or ȱ rejection ȱ decisions ȱ about ȱ the ȱ submitted ȱ lot ȱ of ȱ the ȱ product. ȱ Furthermore, ȱ the ȱ sampling ȱ plan, ȱ using ȱ classical ȱ statistics, ȱ assumes ȱ that ȱ complete ȱ or ȱ determinate ȱ information ȱ are ȱ available ȱ for ȱ a ȱ lot ȱ of ȱ the ȱ products. ȱ However, ȱ in ȱ some ȱ situations, ȱ data ȱ may ȱ be ȱ ambiguous, ȱ vague, ȱ imprecise, ȱ and ȱ incomplete ȱ or ȱ indeterminate. ȱ In ȱ this ȱ case, ȱ the ȱ use ȱ of ȱ neutrosophic ȱ statistics ȱ can ȱ be ȱ applied ȱ to ȱ guide ȱ the ȱ experimenters. ȱ In ȱ the ȱ second ȱ paper ȱ [18], ȱ the ȱ authors ȱ proposed ȱ a ȱ new ȱ variable ȱ sampling ȱ plan ȱ using ȱ the ȱ neutrosophic ȱ interval ȱ statistical ȱ method. ȱ The ȱ neutrosophic ȱ operating ȱ characteristic ȱ (NOC) ȱ is ȱ derived ȱ using ȱ a ȱ neutrosophic ȱ normal ȱ distribution. ȱ An ȱ optimization ȱ solution ȱ is ȱ also ȱ presented ȱ for ȱ the ȱ proposed ȱ plan ȱ under ȱ the ȱ neutrosophic ȱ interval ȱ method. ȱ The ȱ effectiveness ȱ of ȱ the ȱ proposed ȱ plan ȱ is ȱ compared ȱ with ȱ the ȱ plan ȱ under ȱ classical ȱ statistics. ȱ Tables ȱ are ȱ presented ȱ for ȱ practical ȱ use, ȱ and ȱ a ȱ real ȱ example ȱ is ȱ given ȱ to ȱ explain ȱ the ȱ neutrosophic ȱ fuzzy ȱ variable ȱ sampling ȱ plan ȱ in ȱ the ȱ industry. ȱ A ȱ neutrosophic ȱ set ȱ was ȱ proposed ȱ as ȱ an ȱ approach ȱ to ȱ study ȱ neutral, ȱ uncertain ȱ information. ȱ It ȱ is ȱ characterized ȱ through ȱ three ȱ memberships, ȱ T, ȱ I ȱ and ȱ F, ȱ such ȱ that ȱ these ȱ independent ȱ functions ȱ stand ȱ for ȱ the ȱ truth, ȱ indeterminate, ȱ and ȱ false Ȭ membership ȱ degrees ȱ of ȱ an ȱ object. ȱ The ȱ neutrosophic ȱ set ȱ presents ȱ a ȱ symmetric ȱ form, ȱ since ȱ truth ȱ enrolment ȱ T ȱ is ȱ symmetric ȱ to ȱ its ȱ opposite ȱ false ȱ enrolment ȱ F ȱ with ȱ respect ȱ to ȱ indeterminacy ȱ enrollment ȱ I ȱ that ȱ acts ȱ as ȱ an ȱ axis ȱ of ȱ symmetry. ȱ The ȱ neutrosophic ȱ set ȱ was ȱ further ȱ extended ȱ to ȱ a ȱ Q Ȭ neutrosophic ȱ soft ȱ set, ȱ which ȱ is ȱ a ȱ hybrid ȱ model ȱ that ȱ keeps ȱ the ȱ features ȱ of ȱ the ȱ neutrosophic ȱ soft ȱ set ȱ in ȱ dealing ȱ with ȱ uncertainty ȱ and ȱ the ȱ features ȱ of ȱ a ȱ Q Ȭ fuzzy ȱ soft ȱ set ȱ that ȱ handles ȱ two Ȭ dimensional ȱ information. ȱ In ȱ the ȱ next ȱ paper ȱ [19], ȱ the ȱ authors ȱ discuss ȱ some ȱ operations ȱ of ȱ Q Ȭ neutrosophic ȱ soft ȱ sets, ȱ such ȱ as ȱ subset, ȱ equality, ȱ complement, ȱ intersection, ȱ union, ȱ AND ȱ operation, ȱ and ȱ OR ȱ operation. ȱ The ȱ authors ȱ also ȱ define ȱ the ȱ necessity ȱ and ȱ possibility ȱ operations ȱ of ȱ a ȱ Q Ȭ neutrosophic ȱ soft ȱ set. ȱ Several ȱ properties ȱ and ȱ illustrative ȱ examples ȱ are ȱ discussed. ȱ Then, ȱ the ȱ authors ȱ define ȱ the ȱ Q Ȭ neutrosophic ȱ set ȱ aggregation ȱ operator ȱ and ȱ use ȱ it ȱ to ȱ develop ȱ an ȱ algorithm ȱ for ȱ using ȱ a ȱ Q Ȭ neutrosophic ȱ soft ȱ set ȱ in ȱ decision Ȭ making ȱ issues ȱ that ȱ have ȱ indeterminate ȱ and ȱ uncertain ȱ data, ȱ followed ȱ by ȱ an ȱ illustrative ȱ real Ȭ life ȱ example. ȱ Neural ȱ networks ȱ are ȱ powerful ȱ universal ȱ approximation ȱ tools. ȱ They ȱ have ȱ been ȱ utilized ȱ for ȱ functions/data ȱ approximation, ȱ classification, ȱ pattern ȱ recognition, ȱ as ȱ well ȱ as ȱ their ȱ various ȱ applications. ȱ Uncertain ȱ or ȱ interval ȱ values ȱ result ȱ from ȱ the ȱ incompleteness ȱ of ȱ measurements, ȱ human ȱ observations, ȱ and ȱ estimations ȱ in ȱ the ȱ real ȱ world. ȱ Thus, ȱ a ȱ neutrosophic ȱ number ȱ (NsN) ȱ can ȱ represent ȱ both ȱ certain ȱ and ȱ uncertain ȱ information ȱ in ȱ an ȱ indeterminate ȱ setting ȱ and ȱ imply ȱ a ȱ changeable ȱ interval ȱ depending ȱ on ȱ its ȱ indeterminate ȱ ranges. ȱ In ȱ NsN ȱ settings, ȱ however, ȱ existing ȱ interval ȱ neural ȱ networks ȱ cannot ȱ deal ȱ with ȱ uncertain ȱ problems ȱ with ȱ NsNs. ȱ Therefore, ȱ the ȱ next ȱ study ȱ [20] ȱ proposes ȱ a ȱ neutrosophic ȱ compound ȱ orthogonal ȱ neural ȱ network ȱ (NCONN) ȱ for ȱ the ȱ first ȱ time, ȱ containing ȱ the ȱ NsN ȱ weight ȱ values, ȱ NsN ȱ input ȱ and ȱ output, ȱ and ȱ hidden ȱ layer ȱ neutrosophic ȱ neuron ȱ functions, ȱ to ȱ approximate ȱ neutrosophic ȱ functions/NsN ȱ data. ȱ In ȱ the ȱ proposed ȱ NCONN ȱ model, ȱ single ȱ input ȱ and ȱ single ȱ output ȱ neurons ȱ are ȱ the ȱ transmission ȱ notes ȱ of ȱ NsN ȱ data, ȱ and ȱ hidden ȱ layer ȱ neutrosophic ȱ neurons ȱ are ȱ constructed ȱ by ȱ the ȱ compound ȱ functions ȱ of ȱ both ȱ the ȱ Chebyshev ȱ neutrosophic ȱ orthogonal ȱ polynomial ȱ and ȱ the ȱ neutrosophic ȱ sigmoid ȱ function. ȱ In ȱ addition, ȱ illustrative ȱ and ȱ actual ȱ examples ȱ are ȱ provided ȱ to ȱ verify ȱ the ȱ effectiveness ȱ and ȱ learning ȱ performance ȱ of ȱ the ȱ proposed ȱ NCONN ȱ model ȱ for ȱ approximating ȱ neutrosophic ȱ nonlinear ȱ functions ȱ and ȱ NsN ȱ data. ȱ The ȱ contribution ȱ of ȱ this ȱ study ȱ is ȱ that ȱ the ȱ proposed ȱ NCONN ȱ can ȱ handle ȱ the ȱ approximation ȱ problems ȱ of ȱ neutrosophic ȱ nonlinear ȱ functions ȱ and ȱ NsN ȱ data. ȱ However, ȱ the ȱ main ȱ advantage ȱ is ȱ that ȱ the ȱ proposed ȱ NCONN ȱ implies ȱ a ȱ simple ȱ learning ȱ algorithm, ȱ higher ȱ speed ȱ learning ȱ convergence, ȱ and ȱ higher ȱ learning ȱ accuracy ȱ in ȱ indeterminate/NsN ȱ environments. ȱ In ȱ the ȱ following ȱ paper ȱ [21], ȱ a ȱ new ȱ concept ȱ of ȱ the ȱ triangular ȱ neutrosophic ȱ cubic ȱ fuzzy ȱ numbers ȱ (TNCFNs), ȱ their ȱ scores, ȱ and ȱ accuracy ȱ functions ȱ are ȱ introduced. ȱ Based ȱ on ȱ TNCFNs, ȱ some ȱ new ȱ xv Einstein ȱ aggregation ȱ operators, ȱ such ȱ as ȱ triangular ȱ neutrosophic ȱ cubic ȱ fuzzy ȱ Einstein ȱ weighted ȱ averaging ȱ (TNCFEWA), ȱ triangular ȱ neutrosophic ȱ cubic ȱ fuzzy ȱ Einstein ȱ ordered ȱ weighted ȱ averaging ȱ (TNCFEOWA), ȱ and ȱ triangular ȱ neutrosophic ȱ cubic ȱ fuzzy ȱ Einstein ȱ hybrid ȱ weighted ȱ averaging ȱ (TNCFEHWA) ȱ operators ȱ are ȱ developed. ȱ Furthermore, ȱ their ȱ application ȱ to ȱ multiple Ȭ attribute ȱ decision ȱ making ȱ with ȱ triangular ȱ neutrosophic ȱ cubic ȱ fuzzy ȱ (TNCF) ȱ information ȱ is ȱ discussed. ȱ Finally, ȱ a ȱ practical ȱ example ȱ is ȱ given ȱ to ȱ verify ȱ the ȱ developed ȱ approach ȱ and ȱ to ȱ demonstrate ȱ its ȱ practicality ȱ and ȱ effectiveness. ȱ The ȱ existing ȱ sampling ȱ plans ȱ that ȱ use ȱ the ȱ coefficient ȱ of ȱ variation ȱ (CV) ȱ are ȱ designed ȱ under ȱ classical ȱ statistics. ȱ These ȱ available ȱ sampling ȱ plans ȱ cannot ȱ be ȱ used ȱ for ȱ sentencing ȱ if ȱ the ȱ sample ȱ or ȱ the ȱ population ȱ has ȱ indeterminate, ȱ imprecise, ȱ unknown, ȱ incomplete, ȱ or ȱ uncertain ȱ data. ȱ In ȱ the ȱ next ȱ paper ȱ [22], ȱ the ȱ authors ȱ introduce ȱ the ȱ neutrosophic ȱ coefficient ȱ of ȱ variation ȱ (NCV) ȱ first. ȱ The ȱ authors ȱ design ȱ a ȱ sampling ȱ plan ȱ based ȱ on ȱ the ȱ NCV. ȱ The ȱ neutrosophic ȱ operating ȱ characteristic ȱ (NOC) ȱ function ȱ is ȱ then ȱ given ȱ and ȱ used ȱ to ȱ determine ȱ the ȱ neutrosophic ȱ plan ȱ parameters ȱ under ȱ some ȱ constraints. ȱ The ȱ neutrosophic ȱ plan ȱ parameters ȱ such ȱ as ȱ neutrosophic ȱ sample ȱ size ȱ and ȱ neutrosophic ȱ acceptance ȱ number ȱ are ȱ determined ȱ through ȱ the ȱ neutrosophic ȱ optimization ȱ solution. ȱ The ȱ efficiency ȱ of ȱ the ȱ proposed ȱ plan ȱ under ȱ the ȱ neutrosophic ȱ statistical ȱ interval ȱ method ȱ with ȱ the ȱ sampling ȱ plan ȱ under ȱ classical ȱ statistics ȱ is ȱ compared. ȱ A ȱ real ȱ example, ȱ which ȱ has ȱ indeterminate ȱ data, ȱ is ȱ given ȱ to ȱ illustrate ȱ the ȱ proposed ȱ plan. ȱ Neutrosophic ȱ cubic ȱ sets ȱ (NCs) ȱ are ȱ a ȱ more Ȭ generalized ȱ version ȱ of ȱ neutrosophic ȱ sets ȱ (Ns) ȱ and ȱ interval ȱ neutrosophic ȱ sets ȱ (INs). ȱ Neutrosophic ȱ cubic ȱ sets ȱ are ȱ better ȱ placed ȱ to ȱ express ȱ consistent, ȱ indeterminate, ȱ and ȱ inconsistent ȱ information, ȱ which ȱ provides ȱ a ȱ better ȱ platform ȱ to ȱ deal ȱ with ȱ incomplete, ȱ inconsistent, ȱ and ȱ vague ȱ data. ȱ Aggregation ȱ operators ȱ play ȱ a ȱ key ȱ role ȱ in ȱ daily ȱ life ȱ and ȱ in ȱ relation ȱ to ȱ science ȱ and ȱ engineering ȱ problems. ȱ In ȱ the ȱ following ȱ paper ȱ [23], ȱ the ȱ authors ȱ define ȱ the ȱ algebraic ȱ and ȱ Einstein ȱ sum, ȱ multiplication ȱ and ȱ scalar ȱ multiplication, ȱ and ȱ score ȱ and ȱ accuracy ȱ functions. ȱ Using ȱ these ȱ operations, ȱ the ȱ authors ȱ defined ȱ geometric ȱ aggregation ȱ operators ȱ and ȱ Einstein ȱ geometric ȱ aggregation ȱ operators. ȱ First, ȱ they ȱ define ȱ the ȱ algebraic ȱ and ȱ Einstein ȱ operators ȱ of ȱ addition, ȱ multiplication, ȱ and ȱ scalar ȱ multiplication, ȱ then ȱ the ȱ score ȱ and ȱ accuracy ȱ function ȱ to ȱ compare ȱ neutrosophic ȱ cubic ȱ values, ȱ and ȱ afterwards ȱ the ȱ neutrosophic ȱ cubic ȱ weighted ȱ geometric ȱ operator ȱ (NCWG), ȱ neutrosophic ȱ cubic ȱ ordered ȱ weighted ȱ geometric ȱ operator ȱ (NCOWG), ȱ neutrosophic ȱ cubic ȱ Einstein ȱ weighted ȱ geometric ȱ operator ȱ (NCEWG), ȱ and ȱ neutrosophic ȱ cubic ȱ Einstein ȱ ordered ȱ weighted ȱ geometric ȱ operator ȱ (NCEOWG) ȱ over ȱ neutrosophic ȱ cubic ȱ sets. ȱ A ȱ multicriteria ȱ decision Ȭ making ȱ method ȱ is ȱ developed ȱ as ȱ an ȱ application ȱ for ȱ these ȱ operators. ȱ This ȱ method ȱ is ȱ then ȱ applied ȱ to ȱ a ȱ daily ȱ life ȱ problem. ȱ Multiattribute ȱ decision ȱ making ȱ refers ȱ to ȱ the ȱ decision Ȭ making ȱ problem ȱ of ȱ selecting ȱ the ȱ optimal ȱ alternative ȱ or ȱ sorting ȱ the ȱ scheme ȱ when ȱ considering ȱ multiple ȱ attributes, ȱ which ȱ is ȱ widely ȱ used ȱ in ȱ engineering ȱ design, ȱ economy, ȱ management, ȱ military, ȱ and ȱ so ȱ on. ȱ But ȱ in ȱ real ȱ applications, ȱ the ȱ attribute ȱ information ȱ of ȱ many ȱ objects ȱ is ȱ often ȱ inaccurate ȱ or ȱ uncertain, ȱ so ȱ it ȱ is ȱ very ȱ important ȱ for ȱ us ȱ to ȱ find ȱ a ȱ useful ȱ and ȱ efficient ȱ method ȱ to ȱ solve ȱ the ȱ problem. ȱ A ȱ neutrosophic ȱ set ȱ is ȱ proposed ȱ from ȱ philosophical ȱ point ȱ of ȱ view ȱ to ȱ handle ȱ inaccurate ȱ information ȱ efficiently, ȱ and ȱ a ȱ single Ȭ valued ȱ neutrosophic ȱ set ȱ (SVNS) ȱ is ȱ a ȱ special ȱ case ȱ of ȱ neutrosophic ȱ set, ȱ which ȱ is ȱ widely ȱ used ȱ in ȱ actual ȱ field ȱ applications. ȱ In ȱ the ȱ next ȱ paper ȱ [24], ȱ a ȱ new ȱ method ȱ based ȱ on ȱ aggregating ȱ a ȱ single Ȭ valued ȱ neutrosophic ȱ set ȱ is ȱ proposed ȱ to ȱ solve ȱ a ȱ multiattribute ȱ decision Ȭ making ȱ problem. ȱ Firstly, ȱ a ȱ neutrosophic ȱ decision ȱ matrix ȱ is ȱ obtained ȱ by ȱ expert ȱ assessment, ȱ then ȱ a ȱ score ȱ function ȱ of ȱ single Ȭ valued ȱ neutrosophic ȱ sets ȱ (SVNSs) ȱ is ȱ defined ȱ to ȱ obtain ȱ the ȱ positive ȱ ideal ȱ solution ȱ (PIS) ȱ and ȱ the ȱ negative ȱ ideal ȱ solution ȱ (NIS). ȱ Then, ȱ all ȱ alternatives ȱ are ȱ aggregated ȱ based ȱ on ȱ the ȱ TOPSIS ȱ method ȱ to ȱ make ȱ a ȱ decision. ȱ Finally, ȱ numerical ȱ examples ȱ are ȱ given ȱ to ȱ verify ȱ the ȱ feasibility ȱ and ȱ rationality ȱ of ȱ the ȱ method. ȱ The ȱ aim ȱ of ȱ the ȱ next ȱ paper ȱ [25] ȱ is ȱ to ȱ present ȱ a ȱ multiple Ȭ attribute ȱ group ȱ decision Ȭ making ȱ (MAGDM) ȱ framework ȱ based ȱ on ȱ a ȱ new ȱ single Ȭ valued ȱ neutrosophic ȱ linguistic ȱ (SVNL) ȱ distance ȱ measure. ȱ By ȱ unifying ȱ the ȱ idea ȱ of ȱ the ȱ weighted ȱ average ȱ and ȱ ordered ȱ weighted ȱ average ȱ into ȱ a ȱ single Ȭ valued ȱ neutrosophic ȱ linguistic ȱ distance, ȱ the ȱ authors ȱ first ȱ developed ȱ a ȱ new ȱ SVNL ȱ weighted ȱ distance ȱ xvi measure, ȱ namely ȱ a ȱ SVNL ȱ combined ȱ and ȱ weighted ȱ distance ȱ (SVNLCWD) ȱ measure. ȱ The ȱ focal ȱ characteristics ȱ of ȱ the ȱ devised ȱ SVNLCWD ȱ are ȱ its ȱ ability ȱ to ȱ combine ȱ both ȱ the ȱ decision Ȭ makers’ ȱ attitudes ȱ toward ȱ the ȱ importance ȱ as ȱ well ȱ as ȱ the ȱ weights ȱ of ȱ the ȱ arguments. ȱ Various ȱ desirable ȱ properties ȱ and ȱ families ȱ of ȱ the ȱ developed ȱ SVNLCWD ȱ were ȱ contemplated. ȱ Moreover, ȱ a ȱ MAGDM ȱ approach ȱ based ȱ on ȱ the ȱ SVNLCWD ȱ was ȱ formulated. ȱ Lastly, ȱ a ȱ real ȱ numerical ȱ example ȱ concerning ȱ a ȱ low Ȭ carbon ȱ supplier ȱ selection ȱ problem ȱ was ȱ used ȱ to ȱ describe ȱ the ȱ superiority ȱ and ȱ feasibility ȱ of ȱ the ȱ developed ȱ approach. ȱ Neutrosophic ȱ cubic ȱ sets ȱ (NCSs) ȱ can ȱ express ȱ complex ȱ multiattribute ȱ decision Ȭ making ȱ (MADM) ȱ problems ȱ with ȱ its ȱ interval ȱ and ȱ single Ȭ valued ȱ neutrosophic ȱ numbers ȱ simultaneously. ȱ The ȱ weighted ȱ arithmetic ȱ average ȱ (WAA) ȱ and ȱ geometric ȱ average ȱ (WGA) ȱ operators ȱ are ȱ common ȱ aggregation ȱ operators ȱ for ȱ handling ȱ MADM ȱ problems. ȱ However, ȱ the ȱ neutrosophic ȱ cubic ȱ weighted ȱ arithmetic ȱ average ȱ (NCWAA) ȱ and ȱ neutrosophic ȱ cubic ȱ geometric ȱ weighted ȱ average ȱ (NCWGA) ȱ operators ȱ may ȱ result ȱ in ȱ some ȱ unreasonable ȱ aggregated ȱ values ȱ in ȱ some ȱ cases. ȱ In ȱ order ȱ to ȱ overcome ȱ the ȱ drawbacks ȱ of ȱ the ȱ NCWAA ȱ and ȱ NCWGA, ȱ a ȱ new ȱ neutrosophic ȱ cubic ȱ hybrid ȱ weighted ȱ arithmetic ȱ and ȱ geometric ȱ aggregation ȱ (NCHWAGA) ȱ operator ȱ is ȱ developed, ȱ and ȱ its ȱ suitability ȱ and ȱ effectiveness ȱ are ȱ investigated ȱ in ȱ the ȱ next ȱ paper ȱ [26]. ȱ Then, ȱ the ȱ authors ȱ established ȱ a ȱ MADM ȱ method ȱ based ȱ on ȱ the ȱ NCHWAGA ȱ operator. ȱ Finally,