{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T20:53:49Z","timestamp":1771102429160,"version":"3.50.1"},"reference-count":54,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,6,26]],"date-time":"2020-06-26T00:00:00Z","timestamp":1593129600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,6,26]],"date-time":"2020-06-26T00:00:00Z","timestamp":1593129600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2020,9]]},"DOI":"10.1007\/s40314-020-01228-1","type":"journal-article","created":{"date-parts":[[2020,6,26]],"date-time":"2020-06-26T09:36:09Z","timestamp":1593164169000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":58,"title":["A new picture fuzzy information measure based on shannon entropy with applications in opinion polls using extended VIKOR\u2013TODIM approach"],"prefix":"10.1007","volume":"39","author":[{"given":"Vikas","family":"Arya","sequence":"first","affiliation":[]},{"given":"Satish","family":"Kumar","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,6,26]]},"reference":[{"issue":"2","key":"1228_CR1","first-page":"54","volume":"2","author":"SI Amalendu","year":"2019","unstructured":"Amalendu SI, Das S, Kar S (2019) An approach to rank picture fuzzy numbers for decision making problems. Decis Mak Appl Manag 2(2):54\u201364","journal-title":"Decis Mak Appl Manag"},{"key":"1228_CR2","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0165-0114(86)80034-3","volume":"20","author":"KT Atanassov","year":"1986","unstructured":"Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87\u201396","journal-title":"Fuzzy Sets Syst"},{"key":"1228_CR3","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-7908-1870-3","volume-title":"Intuitionistic fuzzy sets","author":"KT Atanassov","year":"1999","unstructured":"Atanassov KT (1999) Intuitionistic fuzzy sets. Springer Verlag, New York, NY, USA"},{"key":"1228_CR4","unstructured":"Benayoun R, Roy B, Sussman B (1966) ELECTRE: Une methode pour guider le choix en presence de points de vue multiples, Note de travail 49. SEMAMETRA International, Direction Scienti que"},{"key":"1228_CR5","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1016\/j.fss.2015.06.017","volume":"281","author":"I Bloch","year":"2015","unstructured":"Bloch I (2015) Fuzzy sets for image processing and understanding. Fuzzy Sets Syst 281:280\u2013291","journal-title":"Fuzzy Sets Syst"},{"issue":"11","key":"1228_CR6","doi-asserted-by":"crossref","first-page":"268","DOI":"10.3390\/sym9110268","volume":"9","author":"C Bo","year":"2017","unstructured":"Bo C, Zhang X (2017) New operations of picture fuzzy relations and fuzzy comprehensive evaluation. Symmetry 9(11):268","journal-title":"Symmetry"},{"key":"1228_CR7","first-page":"477","volume-title":"Operational research 84","author":"JP Brans","year":"1984","unstructured":"Brans JP, Mareschel V (1984) PROMETHEE : a new family of outranking methods in multicriteria analysis. In: Brans JP (ed) Operational research 84. North-Holland, New York, pp 477\u2013490"},{"key":"1228_CR8","doi-asserted-by":"crossref","first-page":"4207","DOI":"10.1016\/j.ins.2010.07.009","volume":"180","author":"T Chen","year":"2010","unstructured":"Chen T, Li C (2010) Determinig objective weights with intutionistic fuzzy entropy measures: a comparative analysis. Inf Sci 180:4207\u20134222","journal-title":"Inf Sci"},{"key":"1228_CR9","doi-asserted-by":"crossref","unstructured":"Cuong BC, Kreinovich V (2013) Picture Fuzzy Sets-a new concept for computational intelligence problems, Third World Congress on Information and Communication Technologies (WICT 2013) IEEE, 1\u20136","DOI":"10.1109\/WICT.2013.7113099"},{"issue":"4","key":"1228_CR10","first-page":"409","volume":"30","author":"BC Cuong","year":"2014","unstructured":"Cuong BC (2014) Picture fuzzy sets. J Comput Sci Cybern 30(4):409\u2013420","journal-title":"J Comput Sci Cybern"},{"key":"1228_CR11","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.eswa.2014.07.026","volume":"42","author":"BC Cuong","year":"2015","unstructured":"Cuong BC, Son HL (2015) Some selected problems of modern soft computing. Exp Syst Appl 42:51\u201366","journal-title":"Exp Syst Appl"},{"key":"1228_CR12","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1007\/s41066-016-0024-3","volume":"2","author":"S Das","year":"2017","unstructured":"Das S, Kar S, Pal T (2017) Robust decision making using intuitionistic fuzzy numbers. Granul Comput 2:41\u201354","journal-title":"Granul Comput"},{"key":"1228_CR13","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/S0019-9958(72)90199-4","volume":"20","author":"A De Luca","year":"1972","unstructured":"De Luca A, Termini S (1972) A definition of a non-probabilitic entropy in the setting of fuzzy sets theory. Inf Control 20:301\u2013312","journal-title":"Inf Control"},{"issue":"9","key":"1228_CR14","first-page":"2421","volume":"9","author":"Y Deng","year":"2012","unstructured":"Deng Y (2012) D numbers: theory and applications. J Inf Comput Sci 9(9):2421\u20132428","journal-title":"J Inf Comput Sci"},{"key":"1228_CR15","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/j.knosys.2012.12.014","volume":"42","author":"ZP Fan","year":"2013","unstructured":"Fan ZP, Zhang X, Chen FD, Liu Y (2013) Extended TODIM method for hybrid multiple attribute decision making problems. Knowl Based Syst 42:40\u201348","journal-title":"Knowl Based Syst"},{"key":"1228_CR16","doi-asserted-by":"crossref","first-page":"5275","DOI":"10.1007\/s13369-017-2625-9","volume":"42","author":"H Garg","year":"2017","unstructured":"Garg H (2017) Some picture fuzzy aggregation operators and their applications to multicriteria decision-making. Arab J Sci Eng 42:5275\u20135290","journal-title":"Arab J Sci Eng"},{"key":"1228_CR17","first-page":"113","volume":"16","author":"L Gomes","year":"1991","unstructured":"Gomes L, Lima M (1991) TODIM: Basics and application to multicriteria ranking of projects with environmental impacts. Found Comput Decis Sci 16:113\u2013127","journal-title":"Found Comput Decis Sci"},{"issue":"5","key":"1228_CR18","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1016\/j.patcog.2008.10.002","volume":"42","author":"Y Guo","year":"2009","unstructured":"Guo Y, Cheng HD (2009) New neutrosophic approach to image segmentation. Pattern Recognit 42(5):587\u2013595","journal-title":"Pattern Recognit"},{"issue":"4","key":"1228_CR19","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1002\/int.20131","volume":"21","author":"WL Hung","year":"2006","unstructured":"Hung WL, Yang MS (2006) Fuzzy entropy on intuitionistic fuzzy sets. Int J Intell Syst 21(4):443\u2013451","journal-title":"Int J Intell Syst"},{"key":"1228_CR20","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-48318-9","volume-title":"Multiple attribute decision making: methods and applications","author":"CL Hwang","year":"1981","unstructured":"Hwang CL, Yoon K (1981) Multiple attribute decision making: methods and applications. Springer, Berlin"},{"key":"1228_CR21","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1007\/s40819-018-0486-x","volume":"4","author":"R Joshi","year":"2018","unstructured":"Joshi R, Kumar S (2018a) A new parametric intuitionistic fuzzy entropy and its applications in multiple attribute decision making. Int J Appl Comput Math 4:52\u201374","journal-title":"Int J Appl Comput Math"},{"issue":"5","key":"1228_CR22","doi-asserted-by":"crossref","first-page":"5624","DOI":"10.1007\/s40314-018-0656-9","volume":"37","author":"R Joshi","year":"2018","unstructured":"Joshi R, Kumar S (2018b) An intuitionistic fuzzy $$(\\delta,\\gamma )$$-norm entropy with its application in supplier selection problem. Comput Appl Math 37(5):5624\u20135649","journal-title":"Comput Appl Math"},{"issue":"1\u20132","key":"1228_CR23","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s12190-018-1202-z","volume":"60","author":"R Joshi","year":"2019","unstructured":"Joshi R, Kumar S (2019) An intuitionistic fuzzy information measure of order $$(\\alpha,\\beta )$$ with a new approach in supplier selection problems using an extended VIKOR method. J Appl Math Comput 60(1\u20132):27\u201350","journal-title":"J Appl Math Comput"},{"key":"1228_CR24","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/s40096-019-0291-7","volume":"13","author":"S Khan","year":"2019","unstructured":"Khan S, Abdullah S, Ashraf S (2019) Picture fuzzy aggregation information based on Einstein operations and their application in decision making. Math Sci 13:213\u2013229","journal-title":"Math Sci"},{"issue":"4","key":"1228_CR25","doi-asserted-by":"crossref","first-page":"2363","DOI":"10.3233\/JIFS-17500","volume":"33","author":"N Konwar","year":"2017","unstructured":"Konwar N, Debnath P (2017) Continuity and Banach contraction principle in intuitionistic fuzzy n-normed linear spaces. J Intell Fuzzy Syst 33(4):2363\u20132373","journal-title":"J Intell Fuzzy Syst"},{"key":"1228_CR26","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1016\/j.knosys.2013.08.028","volume":"53","author":"RA Krohling","year":"2013","unstructured":"Krohling RA, Pacheco AGC, Siviero ALT (2013) IF-TODIM: An intuitionistic fuzzy TODIM to multicriteria decision making. Knowl Based Syst 53:142\u2013146","journal-title":"Knowl Based Syst"},{"issue":"3","key":"1228_CR27","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1007\/s40314-019-0886-5","volume":"38","author":"RX Liang","year":"2019","unstructured":"Liang RX, He SS, Wang JQ, Chen K, Li L (2019) An extended MABAC method formulti-criteria group decision-making problems based on correlative inputs of intuitionistic fuzzy information. Comput Appl Math 38(3):112","journal-title":"Comput Appl Math"},{"issue":"2","key":"1228_CR28","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1057\/s41274-017-0182-y","volume":"69","author":"M Lin","year":"2017","unstructured":"Lin M, Xu Z, Zhai Y, Yao Z (2017) Multi-attribute group decision-making under probabilistic uncertain linguistic environment. J Oper Res Soc 69(2):157\u201370","journal-title":"J Oper Res Soc"},{"key":"1228_CR29","first-page":"1","volume":"11","author":"M Lin","year":"2019","unstructured":"Lin M, Wang H, Xu Z (2019) TODIM-based multi-criteria decision-making method with hesitant fuzzy linguistic term sets. Artif Intell Rev 11:1\u201325","journal-title":"Artif Intell Rev"},{"issue":"3","key":"1228_CR30","doi-asserted-by":"crossref","first-page":"2125","DOI":"10.1007\/s11071-019-04910-0","volume":"96","author":"M Lin","year":"2019","unstructured":"Lin M, Chen Z, Liao H, Xu Z (2019) ELECTRE II method to deal with probabilistic linguistic term sets and its application to edge computing. Nonlinear Dyn 96(3):2125\u20132143","journal-title":"Nonlinear Dyn"},{"key":"1228_CR31","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1016\/j.cie.2018.11.007","volume":"127","author":"HV Long","year":"2019","unstructured":"Long HV, Ali M, Son LH, Khan M, Tu DN (2019) A novel approach for fuzzy clustering based on neutrosophic association matrix. Comput Ind Eng 127:687\u2013697","journal-title":"Comput Ind Eng"},{"key":"1228_CR32","doi-asserted-by":"crossref","first-page":"6459","DOI":"10.1016\/j.eswa.2013.05.070","volume":"40","author":"R Lourenzutti","year":"2013","unstructured":"Lourenzutti R, Krohling RA (2013) A study of TODIM in a intuitionistic fuzzy and random environment. Exp Syst Appl 40:6459\u20136468","journal-title":"Exp Syst Appl"},{"key":"1228_CR33","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1016\/j.asoc.2017.06.055","volume":"60","author":"RX Nie","year":"2017","unstructured":"Nie RX, Wang JQ, Li L (2017) A shareholder voting method for proxy advisory firm selection based on 2-tuple linguistic picture preference relation. Appl Soft Comput 60:520\u2013539","journal-title":"Appl Soft Comput"},{"key":"1228_CR34","volume-title":"Multicriteria optimization of civil engineering systems","author":"S Opricovic","year":"1998","unstructured":"Opricovic S (1998) Multicriteria optimization of civil engineering systems. Faculty of Civil Engineering, Belgrade"},{"key":"1228_CR35","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","volume":"27","author":"CE Shannon","year":"1948","unstructured":"Shannon CE (1948) A mathematical theory of communications. Bell Syst Tech J 27:379\u2013423","journal-title":"Bell Syst Tech J"},{"key":"1228_CR36","first-page":"1","volume-title":"A unifying field in logics Neutrosophy: neutrosophic probability, set and logic","author":"F Smarandache","year":"1999","unstructured":"Smarandache F (1999) A unifying field in logics Neutrosophy: neutrosophic probability, set and logic. American Research Press, Rehoboth, pp 1\u2013141"},{"key":"1228_CR37","first-page":"1","volume-title":"A unifying field in logics: neutrosophic logic neutrosophy, neutrosophic set, neutrosophic probability and statistics","author":"F Smarandache","year":"2003","unstructured":"Smarandache F (2003) A unifying field in logics: neutrosophic logic neutrosophy, neutrosophic set, neutrosophic probability and statistics, 3rd edn. Xiquan, PhoenixPhoenix, pp 1\u2013156","edition":"3"},{"key":"1228_CR38","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1016\/S0165-0114(98)00244-9","volume":"114","author":"E Szmidt","year":"2000","unstructured":"Szmidt E, Kacprzyk J (2000) Distances between intuitionistic fuzzy sets. Fuzzy Set Syst 114:505\u2013518","journal-title":"Fuzzy Set Syst"},{"issue":"2","key":"1228_CR39","doi-asserted-by":"crossref","first-page":"331","DOI":"10.3846\/tede.2019.11412","volume":"26","author":"C Tian","year":"2019","unstructured":"Tian C, Peng J (2019) An integrated picture fuzzy ANP-TODIM multi-criteria decesion-making approach for tourism attraction recommendation. Technol Econ Dev Econ 26(2):331\u2013354","journal-title":"Technol Econ Dev Econ"},{"key":"1228_CR40","first-page":"1","volume":"29","author":"C Tian","year":"2019","unstructured":"Tian C, Peng J, Zhang W, Zhang S, Wang J (2019) Tourism environmental impact assessment based on improved AHP and picture fuzzy PROMETHEE II methods. Technol Econ Dev Econ 29:1\u201324","journal-title":"Technol Econ Dev Econ"},{"key":"1228_CR41","doi-asserted-by":"crossref","first-page":"106037","DOI":"10.1016\/j.cie.2019.106037","volume":"137","author":"C Tian","year":"2019","unstructured":"Tian C, Peng JJ, Zhang WY, Zhang S, Wang JQ (2019) Weighted picture fuzzy aggregation operators and their applications to multi-criteria decision-making problems. Comput Ind Eng 137:106037","journal-title":"Comput Ind Eng"},{"issue":"2","key":"1228_CR42","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/j.patrec.2006.07.004","volume":"28","author":"IK Vlachos","year":"2007","unstructured":"Vlachos IK, Sergiadis GD (2007) Intuitionistic fuzzy information- Applications to pattern recognition. Pattern Recognit Lett 28(2):197\u2013206","journal-title":"Pattern Recognit Lett"},{"key":"1228_CR43","first-page":"477","volume":"37","author":"C Wang","year":"2017","unstructured":"Wang C, Zhou X, Tu H, Tao S (2017) Some geometric aggregation operators based on picture fuzzy setsand their application in multiple attribute decision making. Ital J Pure Appl Math 37:477\u2013492","journal-title":"Ital J Pure Appl Math"},{"issue":"4","key":"1228_CR44","doi-asserted-by":"crossref","first-page":"491","DOI":"10.3846\/16111699.2016.1197147","volume":"17","author":"GW Wei","year":"2016","unstructured":"Wei GW (2016) Picture fuzzy cross-entropy for multiple attribute decision making problems. J Bus Econ Manag 17(4):491\u2013502","journal-title":"J Bus Econ Manag"},{"issue":"1","key":"1228_CR45","first-page":"77","volume":"15","author":"G Wei","year":"2018","unstructured":"Wei G (2018a) Some more similarity measures for picture fuzzy sets and their applications. Iran J Fuzzy Syst 15(1):77\u201389","journal-title":"Iran J Fuzzy Syst"},{"issue":"3","key":"1228_CR46","doi-asserted-by":"crossref","first-page":"555","DOI":"10.15388\/Informatica.2018.181","volume":"29","author":"G Wei","year":"2018","unstructured":"Wei G (2018b) TODIM method for picture fuzzy multiple attribute decision making. Informatica 29(3):555\u201366","journal-title":"Informatica"},{"key":"1228_CR47","doi-asserted-by":"crossref","first-page":"166138","DOI":"10.1109\/ACCESS.2019.2953316","volume":"7","author":"G Wei","year":"2019","unstructured":"Wei G, Zhang S, Lu J, Wu J, Wei C (2019) An extended bidirectional projection method for picture fuzzy MAGDM and its application to safety assessment of construction project. IEEE Access 7:166138\u201347","journal-title":"IEEE Access"},{"key":"1228_CR48","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1016\/j.ijar.2007.08.008","volume":"48","author":"ZS Xu","year":"2008","unstructured":"Xu ZS, Yager RR (2008) Dynamic intuitionistic fuzzy multi-attribute decision making. I J Approx Reason 48:246\u2013262","journal-title":"I J Approx Reason"},{"issue":"3","key":"1228_CR49","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1515\/jisys-2013-0075","volume":"23","author":"J Ye","year":"2014","unstructured":"Ye J (2014a) Single valued neutrosophic minimum spanning tree and its clustering method. J Intell Syst 23(3):311\u2013324","journal-title":"J Intell Syst"},{"issue":"4","key":"1228_CR50","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1515\/jisys-2013-0091","volume":"23","author":"J Ye","year":"2014","unstructured":"Ye J (2014b) Clustering methods using distance-based similarity measures of single-valued neutrosophic sets. J Intell Syst 23(4):379\u2013389","journal-title":"J Intell Syst"},{"issue":"8","key":"1228_CR51","doi-asserted-by":"crossref","first-page":"936","DOI":"10.1287\/mnsc.19.8.936","volume":"19","author":"PL Yu","year":"1973","unstructured":"Yu PL (1973) A class of solutions for group decision problems. Manag Sci 19(8):936\u2013946","journal-title":"Manag Sci"},{"key":"1228_CR52","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","volume":"8","author":"LA Zadeh","year":"1965","unstructured":"Zadeh LA (1965) Fuzzy sets. Inf Control 8:338\u2013353","journal-title":"Inf Control"},{"key":"1228_CR53","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1016\/0022-247X(68)90078-4","volume":"23","author":"LA Zadeh","year":"1968","unstructured":"Zadeh LA (1968) Probability measures of fuzzy events. J Math Anal Appl 23:421\u2013427","journal-title":"J Math Anal Appl"},{"issue":"2","key":"1228_CR54","doi-asserted-by":"crossref","first-page":"191","DOI":"10.3390\/math7020191","volume":"7","author":"S Zeng","year":"2019","unstructured":"Zeng S, Asharf S, Arif M, Abdullah S (2019) Application of exponential Jensen picture fuzzy divergence measure in multicriteria group decision making. Mathematics 7(2):191","journal-title":"Mathematics"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-020-01228-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-020-01228-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-020-01228-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,6,25]],"date-time":"2021-06-25T23:09:16Z","timestamp":1624662556000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-020-01228-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,26]]},"references-count":54,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,9]]}},"alternative-id":["1228"],"URL":"https:\/\/doi.org\/10.1007\/s40314-020-01228-1","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6,26]]},"assertion":[{"value":"11 July 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 May 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 June 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 June 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with ethical standards"}},{"value":"The authors declare that no conflict of interest exists regarding the publication of this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of interest"}}],"article-number":"197"}}