{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,18]],"date-time":"2026-07-18T05:06:13Z","timestamp":1784351173775,"version":"3.55.0"},"reference-count":37,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,8,14]],"date-time":"2020-08-14T00:00:00Z","timestamp":1597363200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the National Social Science Foundation of China","award":["17CGL058"],"award-info":[{"award-number":["17CGL058"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The shortest path problem is a topic of increasing interest in various scientific fields. The damage to roads and bridges caused by disasters makes traffic routes that can be accurately expressed become indeterminate. A neutrosophic set is a collection of the truth membership, indeterminacy membership, and falsity membership of the constituent elements. It has a symmetric form and indeterminacy membership is their axis of symmetry. In uncertain environments, the neutrosophic number can more effectively express the edge distance. The objectives in this study are to solve the shortest path problem of the neutrosophic graph with an edge distance expressed using trapezoidal fuzzy neutrosophic numbers (TrFNN) and resolve the edge distance according to the score and exact functions based on the TrFNN. Accordingly, the use of a circle-breaking algorithm is proposed to solve the shortest path problem and estimate the shortest distance. The feasibility of this method is verified based on two examples, and the rationality and effectiveness of the approach are evaluated by comparing it with the Dijkstra and Bellman algorithms.<\/jats:p>","DOI":"10.3390\/sym12081360","type":"journal-article","created":{"date-parts":[[2020,8,14]],"date-time":"2020-08-14T08:28:35Z","timestamp":1597393715000},"page":"1360","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Shortest Path Solution of Trapezoidal Fuzzy Neutrosophic Graph Based on Circle-Breaking Algorithm"],"prefix":"10.3390","volume":"12","author":[{"given":"Lehua","family":"Yang","sequence":"first","affiliation":[{"name":"College of Electronics and Information Science, Fujian Jiangxia University, Fuzhou 350108, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dongmei","family":"Li","sequence":"additional","affiliation":[{"name":"College of Foreign Languages, Fujian Jiangxia University, Fuzhou 350108, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ruipu","family":"Tan","sequence":"additional","affiliation":[{"name":"College of Electronics and Information Science, Fujian Jiangxia University, Fuzhou 350108, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1145\/77600.77615","article-title":"Faster algorithms for the shortest path problem","volume":"37","author":"Ahuja","year":"1990","journal-title":"J. ACM"},{"key":"ref_2","unstructured":"Dubois, D., Prade, H., and Yager, R.R. (1997). Fuzzy Information Engineering: A Guided Tour of Applications, John Wiley and Sons."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Biswas, S.S., Alam, B., and Doja, M.N. (2013, January 1\u20133). Intuitionistic Fuzzy Real Time Multigraphs for Communication Networks: A Theoretical Model. Proceedings of the 2013 AASRI Conference on Parallel and Distributed Computing and Systems, Singapore.","DOI":"10.1016\/j.aasri.2013.10.066"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/s10479-011-1004-2","article-title":"New models for the robust shortest path problem: Complexity, resolution and generalization","volume":"207","author":"Gabrel","year":"2013","journal-title":"Ann. Oper. Res."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/j.jda.2014.11.008","article-title":"The shortest path problem in the Kn\u00f6del graph","volume":"31","author":"Grigoryan","year":"2015","journal-title":"J. Discret. Algorithms"},{"key":"ref_6","first-page":"74","article-title":"Isolated Single Valued Neutrosophic Graphs","volume":"11","author":"Broumi","year":"2016","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_7","first-page":"35","article-title":"Interval-Valued Possibility Quadripartitioned Single Valued Neutrosophic Soft Sets and Some Uncertainty Based Measures on Them","volume":"14","author":"Rajashi","year":"2016","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/978-3-540-76290-4_18","article-title":"Fuzzy shortest path problem. Monte Carlo methods in fuzzy optimization","volume":"222","author":"Buckley","year":"2007","journal-title":"Stud. Fuzziness Soft Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1231","DOI":"10.1016\/j.asoc.2011.11.011","article-title":"Fuzzy Dijkstra algorithm for shortest path problem under uncertain environment","volume":"12","author":"Deng","year":"2012","journal-title":"Appl. Soft Comput."},{"key":"ref_10","first-page":"23","article-title":"Multiple-attribute decision-making method under a single-valued neutrosophic hesitant fuzzy environment","volume":"24","author":"Ye","year":"2014","journal-title":"J. Intell. Syst."},{"key":"ref_11","first-page":"3","article-title":"Multi-valued Neutrosophic Sets and its Application in Multi-criteria Decision-making Problems","volume":"10","author":"Peng","year":"2015","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1157","DOI":"10.1007\/s00521-014-1787-6","article-title":"Trapezoidal neutrosophic set and its application to multiple attribute decision-making","volume":"26","author":"Ye","year":"2015","journal-title":"Neural Comput. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1615\/Int.J.UncertaintyQuantification.2016018441","article-title":"An improved score function for ranking neutrosophic sets and its application to decision making process","volume":"6","author":"Nancy","year":"2016","journal-title":"Int. J. Uncertain. Quantfication"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"300","DOI":"10.7763\/IJMO.2017.V7.602","article-title":"Computing Minimum Spanning Tree in Interval Valued Bipolar Neutrosophic Environment","volume":"7","author":"Broumi","year":"2017","journal-title":"Int. J. Model. Optim."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1615\/Int.J.UncertaintyQuantification.2017020416","article-title":"Algorithms for interval neutrosophic multiple attribute decision-making based on MABAC, similarity measure, and EDAS","volume":"7","author":"Peng","year":"2017","journal-title":"Int. J. Uncertain. Quantfication"},{"key":"ref_16","unstructured":"Smarandache, F. (1998). A Unifying Field in Logics: Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probability and Statistics, American Research Press. [4th ed.]."},{"key":"ref_17","first-page":"10","article-title":"Single valued neutrosophic sets","volume":"20","author":"Wang","year":"2012","journal-title":"Tech. Sci. Appl. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1309","DOI":"10.1007\/s13042-016-0505-3","article-title":"A ranking method of single valued neutrosophic numbers and its applications to multi-attribute decision making problems","volume":"8","author":"Deli","year":"2017","journal-title":"Int. J. Mach. Learn. Cyb."},{"key":"ref_19","first-page":"403","article-title":"Neutrosophic sets: On overview","volume":"2","author":"Broumi","year":"2018","journal-title":"New Trends Neutrosophic Theor. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"59","DOI":"10.4028\/www.scientific.net\/AMM.859.59","article-title":"Shortest Path Problem under Bipolar Neutrosphic Setting","volume":"859","author":"Broumi","year":"2017","journal-title":"Appl. Mech. Mater."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4941","DOI":"10.1007\/s00500-018-3140-y","article-title":"A novel interval-valued neutrosophic AHP with cosine similarity measure","volume":"22","author":"Bolturk","year":"2018","journal-title":"Soft Comput."},{"key":"ref_22","first-page":"40","article-title":"Distance measure based MADM strategy with interval trapezoidal neutrosophic numbers","volume":"19","author":"Biswas","year":"2018","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_23","first-page":"478","article-title":"Expansions and reductions on neutrosophic classical soft set","volume":"22","author":"Deli","year":"2018","journal-title":"S\u00fcleyman Demirel U. J. Nat. Appl. Sci."},{"key":"ref_24","first-page":"131","article-title":"Operators on single valued trapezoidal neutrosophic numbers and SVTN-group decision making","volume":"22","author":"Deli","year":"2018","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"291","DOI":"10.3233\/JIFS-151677","article-title":"Some weighted geometric operators with SVTrN-numbers and their application to multi-criteria decision making problems","volume":"32","author":"Deli","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1016\/j.cogsys.2018.10.023","article-title":"A hybrid neutrosophic multiple criteria group decision making approach for project selection","volume":"57","author":"Basset","year":"2019","journal-title":"Cogn. Syst. Res."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1427","DOI":"10.1007\/s12652-017-0548-7","article-title":"Neutrosophic AHPDelphi Group decision making model based on trapezoidal neutrosophic numbers","volume":"9","author":"Basset","year":"2017","journal-title":"J. Amb. Intel. Hum. Comp."},{"key":"ref_28","first-page":"5","article-title":"Neutrosophic shortest path problem","volume":"23","author":"Kumar","year":"2018","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_29","first-page":"216","article-title":"Shortest path problem under interval valued neutrosophic setting","volume":"8","author":"Broumi","year":"2019","journal-title":"Int. J. Adv. Trends Comput. Sci. Eng."},{"key":"ref_30","first-page":"851","article-title":"Solving methods for the shortest path problem based on trapezoidal fuzzy neutrosophic numbers","volume":"34","author":"Tan","year":"2019","journal-title":"Control Decis."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1007\/s40747-019-0101-8","article-title":"Shortest path problem using Bellman algorithm under neutrosophic environment","volume":"5","author":"Broumi","year":"2019","journal-title":"Complex Intell. Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"21","DOI":"10.54216\/IJNS.030104","article-title":"Application of Pentagonal Neutrosophic Number in Shortest Path Problem","volume":"3","author":"Chakraborty","year":"2020","journal-title":"Int. J. Neutrosophic Sci."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"18","DOI":"10.54216\/IJNS.020104","article-title":"Uncertainty: Two probabilities for the three states of neutrosophy","volume":"2","author":"Schweizer","year":"2020","journal-title":"Int. J. Neutrosophic Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"19","DOI":"10.54216\/IJNS.010104","article-title":"A Direct Model for Triangular Neutrosophic Linear Programming","volume":"1","author":"Edalatpanah","year":"2020","journal-title":"Int. J. Neutrosophic Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"88717","DOI":"10.1109\/ACCESS.2020.2990912","article-title":"Research on the Shortest Path Solution Method of Interval Valued Neutrosophic Graphs Based on the Ant Colony Algorithm","volume":"8","author":"Yang","year":"2020","journal-title":"IEEE Access"},{"key":"ref_36","first-page":"40","article-title":"Algorithm of breaking circle for minimum tree","volume":"4","author":"Guan","year":"1975","journal-title":"J. Math. Pract. Theory"},{"key":"ref_37","first-page":"15","article-title":"Islanding Algorithm of Distribution System with Distributed Generations based on Circle-breaking Algorithm","volume":"35","author":"Zeng","year":"2019","journal-title":"Power Sys. Eng."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/8\/1360\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:01:00Z","timestamp":1760176860000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/8\/1360"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,14]]},"references-count":37,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2020,8]]}},"alternative-id":["sym12081360"],"URL":"https:\/\/doi.org\/10.3390\/sym12081360","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,14]]}}}