{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,6]],"date-time":"2026-06-06T19:41:40Z","timestamp":1780774900843,"version":"3.54.1"},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,2,28]],"date-time":"2023-02-28T00:00:00Z","timestamp":1677542400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,2,28]],"date-time":"2023-02-28T00:00:00Z","timestamp":1677542400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"CSIR-HRDG","award":["09\/895(0014)\/2019-EMR-I"],"award-info":[{"award-number":["09\/895(0014)\/2019-EMR-I"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Int J Comput Intell Syst"],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Zadeh introduced fuzzy sets to study imprecision in real life after which many generalizations have been developed in literature. Fuzzy numbers is the major research area of study because of its needfulness for modeling qualitative and imprecise continuous transitions. Most of the time, data involved in multi-criteria decision making (MCDM) will be in the form of fuzzy numbers due to qualitative and continuous deforming criteria. Different methods of defining total ordering on the class of fuzzy numbers have important role in MCDM to find the preference order of alternatives. Many total ordering techniques for various types of piecewise linear fuzzy numbers such as triangular (3-sided), trapezoidal (4-sided), pentagonal (5-sided), hexagonal (6-sided) and so on are available in the literature. In this paper, a generalized \u2018n\u2019gonal linear fuzzy number (n-sided) as a generalization of triangular (3-sided), trapezoidal (4-sided), pentagonal (5-sided), hexagonal (6-sided) and so on is defined and a method of defining total ordering on the class of generalized \u2018<jats:italic>n<\/jats:italic>\u2019gonal linear fuzzy numbers (<jats:italic>n<\/jats:italic>-sided) which generalizes total ordering methods defined for triangular (3-sided), trapezoidal (4-sided), pentagonal (5-sided), hexagonal (6-sided) and so on in the literature has been proposed and analyzed. Further, a similarity measure on \u2018<jats:italic>n<\/jats:italic>\u2019 gonal linear fuzzy numbers using the proposed midpoint score function is also defined and the applicability of the proposed operations, total ordering method and similarity measure on \u2018<jats:italic>n<\/jats:italic>\u2019 gonal linear fuzzy numbers in MCDM is shown by comparing with some other methods in the literature.<\/jats:p>","DOI":"10.1007\/s44196-022-00180-8","type":"journal-article","created":{"date-parts":[[2023,3,2]],"date-time":"2023-03-02T11:29:21Z","timestamp":1677756561000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Total Ordering on Generalized \u2018n\u2019 Gonal Linear Fuzzy Numbers"],"prefix":"10.1007","volume":"16","author":[{"given":"Lakshmana Gomathi Nayagam","family":"Velu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5783-7915","authenticated-orcid":false,"given":"Bharanidharan","family":"Ramalingam","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,2,28]]},"reference":[{"issue":"3","key":"180_CR1","doi-asserted-by":"publisher","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","volume":"8","author":"LA Zadeh","year":"1965","unstructured":"Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338\u2013353 (1965). https:\/\/doi.org\/10.1016\/S0019-9958(65)90241-X","journal-title":"Inf. Control"},{"issue":"1","key":"180_CR2","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1016\/0165-0114(78)90029-5","volume":"1","author":"LA Zadeh","year":"1978","unstructured":"Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1(1), 3\u201328 (1978). https:\/\/doi.org\/10.1016\/0165-0114(78)90029-5","journal-title":"Fuzzy Sets Syst."},{"key":"180_CR3","unstructured":"Klir, B.G.J.: Yuan: Fuzzy sets and fuzzy logic theory and applications. Prentice Hall Inc., Upper Saddle River (1995)"},{"issue":"3","key":"180_CR4","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1016\/0165-0114(89)90205-4","volume":"31","author":"K Atanassov","year":"1989","unstructured":"Atanassov, K., Gargov, G.: Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31(3), 343\u2013349 (1989)","journal-title":"Fuzzy Sets Syst."},{"key":"180_CR5","doi-asserted-by":"crossref","unstructured":"Yager, R.R.: Pythagorean fuzzy subsets. In: 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA\/NAFIPS), pp. 57\u201361 (2013)","DOI":"10.1109\/IFSA-NAFIPS.2013.6608375"},{"key":"180_CR6","doi-asserted-by":"publisher","unstructured":"Cuong, B.: Picture fuzzy sets. Journal of Computer Science and Cybernetics 30 (2015). https:\/\/doi.org\/10.15625\/1813-9663\/30\/4\/5032","DOI":"10.15625\/1813-9663\/30\/4\/5032"},{"key":"180_CR7","doi-asserted-by":"publisher","unstructured":"Zhang, W.-R.: Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. In: NAFIPS\/IFIS\/NASA \u201994. Proceedings of the First International Joint Conference of The North American Fuzzy Information Processing Society Biannual Conference. The Industrial Fuzzy Control and Intellige, pp. 305\u2013309 (1994). https:\/\/doi.org\/10.1109\/IJCF.1994.375115","DOI":"10.1109\/IJCF.1994.375115"},{"key":"180_CR8","first-page":"287","volume":"24","author":"F Smarandache","year":"2004","unstructured":"Smarandache, F.: Neutrosophic set - a generalization of the intuitionistic fuzzy set. Int. J. Pure Appl. Math. 24, 287\u2013298 (2004)","journal-title":"Int. J. Pure Appl. Math"},{"key":"180_CR9","first-page":"185","volume":"3","author":"F Augustin","year":"2015","unstructured":"Augustin, F.: A nonagonal fuzzy number and its arithmetic operation. Int. J. Math. Appl. 3, 185\u2013195 (2015)","journal-title":"International Journal of Mathematics And its Applications"},{"issue":"4","key":"180_CR10","first-page":"49","volume":"6","author":"M A","year":"2009","unstructured":"A, M.: On some structures of fuzzy numbers. Iranian J. Fuzzy Syst. 6(4), 49\u201359 (2009)","journal-title":"Iranian Journal of Fuzzy System"},{"key":"180_CR11","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1007\/s44196-022-00066-9","volume":"15","author":"L Jin","year":"2022","unstructured":"Jin, L., Xu, Y.-Q., Chen, Z.-S., Mesiar, R., Yager, R.R.: Relative basic uncertain information in preference and uncertain involved information fusion. Int. J. Comput. Intell. Syst. 15, 12 (2022)","journal-title":"Int. J. Comput. Intell. Syst"},{"key":"180_CR12","doi-asserted-by":"publisher","first-page":"154","DOI":"10.1007\/s44196-021-00002-3","volume":"14","author":"H Wang","year":"2021","unstructured":"Wang, H., Liu,Y., Liu,  F., Lin, J.: Multiple attribute decision-making method based upon intuitionistic fuzzy partitioned dual maclaurin symmetric mean operators. Int. J. Comput. Intell. Syst. 14, 154 (2021)","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"180_CR13","unstructured":"M. Subha, R. Shanmugapriya, M.L. Suresh, S. Karthik.: Generalized nonagonal fuzzy number and its application in assignment problem. Math. Statistician Eng. Appl. 71 (2022)"},{"key":"180_CR14","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1007\/s44196-021-00059-0","volume":"15","author":"J Bi","year":"2022","unstructured":"Bi, J., Zhou, G., Zhou, Y., Luo, Q., Deng, W.: Artificial electric field algorithm with greedy state transition strategy for spherical multiple traveling salesmen problem. Int. J. Comput. Intell. Syst. 15, 9 (2022)","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"180_CR15","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1007\/s41066-020-00212-8","volume":"6","author":"A Chakraborty","year":"2021","unstructured":"Chakraborty, A., Maity, S., Jain, S., Mondal, S., Alam, S.: Hexagonal fuzzy number and its distinctive representation, ranking, defuzzification technique and application in production inventory management problem. Granul. Comput. 6, 507\u2013521 (2021). https:\/\/doi.org\/10.1007\/s41066-020-00212-8","journal-title":"Granul. Comput."},{"issue":"3","key":"180_CR16","doi-asserted-by":"publisher","first-page":"1464","DOI":"10.1016\/j.eswa.2010.07.055","volume":"38","author":"V Lakshmana Gomathi Nayagam","year":"2011","unstructured":"Lakshmana Gomathi Nayagam, V., Muralikrishnan, S., Sivaraman, G.: Multi-criteria decision-making method based on interval-valued intuitionistic fuzzy sets. Expert Syst. Appl. 38(3), 1464\u20131467 (2011). https:\/\/doi.org\/10.1016\/j.eswa.2010.07.055","journal-title":"Expert Syst. Appl."},{"key":"180_CR17","first-page":"1","volume":"32","author":"VLG Nayagam","year":"2019","unstructured":"Nayagam, V.L.G., Ponnialagan, D., Jeevaraj, S.: Similarity measure on incomplete imprecise interval information and its applications. Neural Comput. Appl. 32, 1\u201313 (2019)","journal-title":"Neural Comput. Appl"},{"key":"180_CR18","doi-asserted-by":"publisher","DOI":"10.1016\/j.engappai.2022.105401","volume":"116","author":"Y Rong","year":"2022","unstructured":"Rong, Y., Yu, L., Niu, W., Liu, Y., Senapati, T., Mishra, A.R.: Marcos approach based upon cubic fermatean fuzzy set and its application in evaluation and selecting cold chain logistics distribution center. Eng. Appl. Artif. Intell. 116, 105401 (2022). https:\/\/doi.org\/10.1016\/j.engappai.2022.105401","journal-title":"Eng. Appl. Artif. Intell."},{"issue":"S2","key":"180_CR19","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1002\/cplx.21783","volume":"21","author":"V Lakshmana Gomathi Nayagam","year":"2016","unstructured":"Lakshmana Gomathi Nayagam, V., Jeevaraj, S., Geetha, S.: Total ordering for intuitionistic fuzzy numbers. Complexity 21(S2), 54\u201366 (2016)","journal-title":"Complexity"},{"key":"180_CR20","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1016\/j.eswa.2016.05.003","volume":"60","author":"S Jeevaraj","year":"2016","unstructured":"Jeevaraj, S., Dhanasekaran, P., et al.: A linear ordering on the class of trapezoidal intuitionistic fuzzy numbers. Expert Syst. Appl. 60, 269\u2013279 (2016)","journal-title":"Expert Syst. Appl."},{"issue":"6","key":"180_CR21","doi-asserted-by":"publisher","first-page":"3209","DOI":"10.3233\/IFS-152064","volume":"30","author":"V Nayagam","year":"2016","unstructured":"Nayagam, V., Dhanasekaran, P., Jeevaraj, S.: A complete ranking of incomplete trapezoidal information. J. Intell. Fuzzy Syst. 30(6), 3209\u20133225 (2016)","journal-title":"Journal of Intelligent and Fuzzy Systems"},{"key":"180_CR22","doi-asserted-by":"publisher","first-page":"4283","DOI":"10.1007\/s00500-019-04193-y","volume":"24","author":"G Sivaraman","year":"2019","unstructured":"Sivaraman, G., Vishnukumar, P., Raj, M.E.A.: Mcdm based on new membership and non-membership accuracy functions on trapezoidal-valued intuitionistic fuzzy numbers. Soft Comput. 24, 4283\u20134293 (2019)","journal-title":"Soft. Comput."},{"key":"180_CR23","first-page":"555","volume":"7","author":"YLP Thorani","year":"2012","unstructured":"Thorani, Y.L.P., Rao, P., Nowpada, R.S.: Ordering generalized trapezoidal fuzzy numbers. Int. J. Contemporary Math. Sci. 7, 555\u2013573 (2012)","journal-title":"International journal of contemporary mathematical sciences"},{"issue":"3","key":"180_CR24","doi-asserted-by":"publisher","first-page":"413","DOI":"10.1016\/j.camwa.2008.10.090","volume":"57","author":"S Abbasbandy","year":"2009","unstructured":"Abbasbandy, S., Hajjari, T.: A new approach for ranking of trapezoidal fuzzy numbers. Comput. Math. Appl. 57(3), 413\u2013419 (2009). https:\/\/doi.org\/10.1016\/j.camwa.2008.10.090","journal-title":"Comput. Math. Appl."},{"key":"180_CR25","first-page":"783","volume":"12","author":"H Ganesh","year":"2017","unstructured":"Ganesh, H., Suresh, M.: Ordering of generalised trapezoidal fuzzy numbers based on area method using euler line of centroids. Adv. Fuzzy Math. 12, 783\u2013791 (2017)","journal-title":"Adv. Fuzzy Math."},{"key":"180_CR26","doi-asserted-by":"publisher","unstructured":"Kamble, A.: Some notes on pentagonal fuzzy numbers, 2320\u20133250 (2017). https:\/\/doi.org\/10.22457\/ijfma.v13n2a2","DOI":"10.22457\/ijfma.v13n2a2"},{"key":"180_CR27","first-page":"5","volume":"7","author":"ATG Selvaraj","year":"2022","unstructured":"Selvaraj A, T.G., Saravanakumar, G.: Ranking of linear and non-linear hexagonal fuzzy number through haar wavelet. Int. J. Mech. Eng. 7, 5 (2022)","journal-title":"Int. J. Mech. Eng"},{"key":"180_CR28","doi-asserted-by":"publisher","first-page":"15","DOI":"10.9790\/5728-10321522","volume":"10","author":"P Rajarajeswari","year":"2014","unstructured":"Rajarajeswari, P., Sudha, A.S.: Ordering generalized hexagonal fuzzy numbers using rank, mode, divergence and spread. IOSR J. Math. 10, 15\u201322 (2014)","journal-title":"IOSR J. Math."},{"key":"180_CR29","doi-asserted-by":"publisher","unstructured":"Lakshmana Gomathi Nayagam, V., J.M., K, S.: Hexagonal fuzzy number inadvertences and its applications to mcdm and hffls based on complete ranking by score functions. Comput. Appl. Math. 39 (2020). https:\/\/doi.org\/10.1007\/s40314-020-01292-7","DOI":"10.1007\/s40314-020-01292-7"},{"key":"180_CR30","doi-asserted-by":"publisher","unstructured":"Gunasekar T., T.J., .A, S.: Ranking of linear and non-linear octagonal fuzzy number through haarwavelet. J. Jilin Univ. (Engineering and Technology Edition) 41 (2022). https:\/\/doi.org\/10.17605\/OSF.IO\/HNQ6F","DOI":"10.17605\/OSF.IO\/HNQ6F"},{"key":"180_CR31","first-page":"1144","volume":"9","author":"P Rajarajeswari","year":"2019","unstructured":"Rajarajeswari, P., Menaka, G.: Ordering octoganal fuzzy numbers using rank, mode, divergence and spread. J. Inform. Comput. Sci. 9, 1144\u20131152 (2019)","journal-title":"J. Inform. Comput. Sci."},{"key":"180_CR32","first-page":"6","volume":"118","author":"MAJ Venkatachalapathy","year":"2018","unstructured":"Venkatachalapathy, M.A.J., Samuel, A.E.: A study on solving octagonal fuzzy numbers using modified vogel\u2019s approximation method. Int. J. Pure Appl. Math. 118, 6 (2018)","journal-title":"Int. J. Pure Appl. Math."},{"key":"180_CR33","first-page":"12702","volume":"13","author":"P Kirtiwant","year":"2018","unstructured":"Kirtiwant P. Ghadle, S.M.I.: A new ranking on generalized octagonal fuzzy numbers. Int. J. Appl. Eng. Res. 13, 12702\u201312709 (2018)","journal-title":"Int. J. Appl. Eng. Res."},{"key":"180_CR34","doi-asserted-by":"publisher","unstructured":"Nayagam, V.L.G., Murugan, J.: Hexagonal fuzzy approximation of fuzzy numbers and its applications in mcdm 7 (2021). https:\/\/doi.org\/10.1007\/s40747-020-00242-4","DOI":"10.1007\/s40747-020-00242-4"},{"issue":"4","key":"180_CR35","doi-asserted-by":"publisher","first-page":"3551","DOI":"10.3233\/JIFS-211635","volume":"42","author":"L Jin","year":"2022","unstructured":"Jin, L., Mesiar, R., Yager, R., Kaya, S.K.: Interval basic uncertain information and related aggregations in decision making. J. Intell. Fuzzy Syst. 42(4), 3551\u20133558 (2022). https:\/\/doi.org\/10.3233\/JIFS-211635","journal-title":"J. Intell. Fuzzy Syst."},{"key":"180_CR36","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1007\/s44196-022-00116-2","volume":"15","author":"G Li","year":"2022","unstructured":"Li, G., Yager, R.R., Zhang, X., Mesiar, R., Bustince, H., Jin, L.: Comprehensive rules-based and preferences induced weights allocation in group decision-making with bui. Int. J. Comput. Intell. Syst. 15, 4 (2022)","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"180_CR37","doi-asserted-by":"publisher","unstructured":"Jin, L., Mesiar, R., Yager, R., Kalina, M., \u0160pirkov\u00e1, J., Borkotokey, S.: Deriving efficacy from basic uncertain information and uncertain choquet integral. Int. J. General Syst. (2022). https:\/\/doi.org\/10.1080\/03081079.2022.2104268","DOI":"10.1080\/03081079.2022.2104268"},{"issue":"3","key":"180_CR38","doi-asserted-by":"publisher","first-page":"1673","DOI":"10.3233\/JIFS-141907","volume":"32","author":"J Li","year":"2017","unstructured":"Li, J., Z, W.: Fuzzy risk analysis based on the similarity measure of generalized trapezoidal fuzzy numbers. J. Intell. Fuzzy Syst. 32(3), 1673\u20131683 (2017)","journal-title":"Journal of Inteligent Fuzzy Systems"},{"key":"180_CR39","doi-asserted-by":"publisher","unstructured":"Chakraborty, A., Pal, S., Mondal, S.P., Alam, S.: Nonlinear pentagonal intuitionistic fuzzy number and its application in epq model under learning and forgetting. Complex Intell. Syst. 8 (2022). https:\/\/doi.org\/10.1007\/s40747-021-00574-9","DOI":"10.1007\/s40747-021-00574-9"}],"container-title":["International Journal of Computational Intelligence Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s44196-022-00180-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s44196-022-00180-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s44196-022-00180-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,2]],"date-time":"2023-03-02T11:29:44Z","timestamp":1677756584000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s44196-022-00180-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,28]]},"references-count":39,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,12]]}},"alternative-id":["180"],"URL":"https:\/\/doi.org\/10.1007\/s44196-022-00180-8","relation":{},"ISSN":["1875-6883"],"issn-type":[{"value":"1875-6883","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,28]]},"assertion":[{"value":"10 November 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 December 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 February 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that there is no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable. This article does not contain any studies with human participants or animals performed by any of the authors. The authors assure that they have written entirely original works based on their own research and if the authors have used the work and\/or words of others this has been appropriately cited or quoted to the best of their knowledge.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical Approval and Consent to Participate"}},{"value":"Not applicable. This manuscript does not contain data from any individual person.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for Publication"}}],"article-number":"23"}}