{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,28]],"date-time":"2026-06-28T04:59:19Z","timestamp":1782622759529,"version":"3.54.5"},"reference-count":49,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2018,7,4]],"date-time":"2018-07-04T00:00:00Z","timestamp":1530662400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2019,6]]},"DOI":"10.1007\/s12190-018-1202-z","type":"journal-article","created":{"date-parts":[[2018,7,4]],"date-time":"2018-07-04T04:17:45Z","timestamp":1530677865000},"page":"27-50","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["An intuitionistic fuzzy information measure of order-\n                \n                  \n                \n                $$(\\alpha , \\beta )$$\n                \n                  \n                    \n                      (\n                      \u03b1\n                      ,\n                      \u03b2\n                      )\n                    \n                  \n                \n               with a new approach in supplier selection problems using an extended VIKOR method"],"prefix":"10.1007","volume":"60","author":[{"given":"Rajesh","family":"Joshi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Satish","family":"Kumar","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2018,7,4]]},"reference":[{"key":"1202_CR1","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1016\/S0165-0114(86)80034-3","volume":"20","author":"KT Atanassov","year":"1986","unstructured":"Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87\u201396 (1986)","journal-title":"Fuzzy Sets Syst."},{"key":"1202_CR2","volume-title":"On Measures of Information and Their Characterization","author":"J Aczel","year":"1975","unstructured":"Aczel, J., Daroczy, Z.: On Measures of Information and Their Characterization. Academic Press, New York (1975)"},{"key":"1202_CR3","first-page":"305","volume":"118","author":"P Burillo","year":"2001","unstructured":"Burillo, P., Bustince, H.: Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets. Fuzzy Sets Syst. 118, 305\u2013316 (2001)","journal-title":"Fuzzy Sets Syst."},{"key":"1202_CR4","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1016\/0020-0255(93)90073-U","volume":"67","author":"D Bhandari","year":"1993","unstructured":"Bhandari, D., Pal, N.R.: Some new information measures for fuzzy sets. Inf. Sci. 67, 209\u2013228 (1993)","journal-title":"Inf. Sci."},{"key":"1202_CR5","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1002\/hfm.20252","volume":"21","author":"FE Boran","year":"2011","unstructured":"Boran, F.E., Genc, S., Akay, D.: Personnel selection based on intuitionistic fuzzy sets. Hum. Factors Ergon. Manuf. Serv. Ind. 21, 493\u2013503 (2011)","journal-title":"Hum. Factors Ergon. Manuf. Serv. Ind."},{"key":"1202_CR6","doi-asserted-by":"publisher","first-page":"4207","DOI":"10.1016\/j.ins.2010.07.009","volume":"180","author":"T Chen","year":"2010","unstructured":"Chen, T., Li, C.: Determining objective weights with intuitionistic fuzzy entropy measures: a comparative analysis. Inf. Sci. 180, 4207\u20134222 (2010)","journal-title":"Inf. Sci."},{"key":"1202_CR7","doi-asserted-by":"publisher","first-page":"531","DOI":"10.1007\/BF00933438","volume":"27","author":"ATW Chu","year":"1979","unstructured":"Chu, A.T.W., Kalaba, R.E., Spingarn, K.: A comparison of two methods for determining the weights of belonging to fuzzy sets. J. Optim. Theor. Appl. 27, 531\u2013538 (1979)","journal-title":"J. Optim. Theor. Appl."},{"key":"1202_CR8","doi-asserted-by":"publisher","first-page":"983","DOI":"10.1057\/jors.1985.177","volume":"36","author":"EU Choo","year":"1985","unstructured":"Choo, E.U., Wedley, W.C.: Optimal criterian weights in repetitive multicriteria decision making. J. Oper. Res. Soc. 36, 983\u2013992 (1985)","journal-title":"J. Oper. Res. Soc."},{"key":"1202_CR9","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1016\/S0019-9958(72)90199-4","volume":"20","author":"A Luca De","year":"1972","unstructured":"De Luca, A., Termini, S.: A definition of non-probabilistic entropy in the setting of fuzzy set theory. Inf. Control 20, 301\u2013312 (1972)","journal-title":"Inf. Control"},{"key":"1202_CR10","first-page":"255","volume":"36","author":"ZQ Feng","year":"2012","unstructured":"Feng, Z.Q., Liu, C.G.: On similarity based approximate reasoning in interval-valued fuzzy environments. Informatica 36, 255\u2013262 (2012)","journal-title":"Informatica"},{"key":"1202_CR11","unstructured":"Fan, Z.P.: Complicated multiple attribute decision making: theory and applications. Ph.D. Dissertation, Northeastern university, Shenyang (1996)"},{"issue":"4","key":"1202_CR12","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1002\/int.20131","volume":"21","author":"WL Hung","year":"2006","unstructured":"Hung, W.L., Yang, M.S.: Fuzzy entropy on intuitionistic fuzzy sets. Int. J. Intell. Syst. 21(4), 443\u2013451 (2006)","journal-title":"Int. J. Intell. Syst."},{"key":"1202_CR13","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61580-1","volume-title":"Group Decision Making Under Multiple Criteria: Methods and Applications","author":"CL Hwang","year":"1987","unstructured":"Hwang, C.L., Lin, M.J.: Group Decision Making Under Multiple Criteria: Methods and Applications. Springer, Berlin (1987)"},{"issue":"5","key":"1202_CR14","doi-asserted-by":"publisher","first-page":"627","DOI":"10.1142\/S0218488503002375","volume":"11","author":"WL Hung","year":"2003","unstructured":"Hung, W.L.: A note on entropy of intuitionistic fuzzy sets. Int. J. Uncertain. Fuzziness Knowl. Syst. 11(5), 627\u2013633 (2003)","journal-title":"Int. J. Uncertain. Fuzziness Knowl. Syst."},{"key":"1202_CR15","doi-asserted-by":"publisher","first-page":"184","DOI":"10.1016\/j.engappai.2016.10.018","volume":"57","author":"A Hafezalkotob","year":"2017","unstructured":"Hafezalkotob, A., Hafezalkotob, A.: Interval target-based VIKOR method supported on interval distance and preference degree for machine selection. Eng. Appl. Artif. Intell. 57, 184\u2013196 (2017)","journal-title":"Eng. Appl. Artif. Intell."},{"issue":"C","key":"1202_CR16","doi-asserted-by":"publisher","first-page":"302","DOI":"10.1016\/j.ress.2017.06.014","volume":"167","author":"J Huang","year":"2017","unstructured":"Huang, J., Li, Z.S., Liu, H.-C.: New approach for failure mode and effect analysis using linguistic distribution assessments and TODIM method. Reliab. Eng. Syst. Saf. 167(C), 302\u2013309 (2017)","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"1202_CR17","first-page":"147","volume":"9","author":"A Haldar","year":"2014","unstructured":"Haldar, A., Ray, A., Benerjee, D., Ghosh, S.: Resilient supplier selection under a fuzzy environment. Int. J. Manag. Sci. Eng. Manag. 9, 147\u2013156 (2014)","journal-title":"Int. J. Manag. Sci. Eng. Manag."},{"issue":"3","key":"1202_CR18","first-page":"253","volume":"27","author":"R Joshi","year":"2016","unstructured":"Joshi, R., Kumar, S.: A new approach in multiple attribute decision making using \n                    \n                      \n                    \n                    $$R$$\n                    \n                      \n                        R\n                      \n                    \n                  -norm entropy and Hamming distance measure. Int. J. Inf. Manag. Sci. 27(3), 253\u2013268 (2016)","journal-title":"Int. J. Inf. Manag. Sci."},{"key":"1202_CR19","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1016\/j.fiae.2017.06.004","volume":"9","author":"R Joshi","year":"2017","unstructured":"Joshi, R., Kumar, S.: Parametric \n                    \n                      \n                    \n                    $$(R, S)$$\n                    \n                      \n                        \n                          (\n                          R\n                          ,\n                          S\n                          )\n                        \n                      \n                    \n                  -norm entropy on intuitionistic fuzzy sets with a new approach in multiple attribute decision making. Fuzzy Inf. Eng. 9, 181\u2013203 (2017)","journal-title":"Fuzzy Inf. Eng."},{"key":"1202_CR20","doi-asserted-by":"publisher","DOI":"10.1007\/s40314-017-0491-4","author":"R Joshi","year":"2017","unstructured":"Joshi, R., Kumar, S.: An \n                    \n                      \n                    \n                    $$(R, S)$$\n                    \n                      \n                        \n                          (\n                          R\n                          ,\n                          S\n                          )\n                        \n                      \n                    \n                  -norm fuzzy information measure with its applications in multiple attribute decision making. Comput. Appl. Math. (2017). \n                    https:\/\/doi.org\/10.1007\/s40314-017-0491-4","journal-title":"Comput. Appl. Math."},{"key":"1202_CR21","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1007\/s40304-017-0109-6","volume":"5","author":"R Joshi","year":"2017","unstructured":"Joshi, R., Kumar, S.: A new exponential fuzzy entropy of order \n                    \n                      \n                    \n                    $$(\\alpha, \\beta )$$\n                    \n                      \n                        \n                          (\n                          \u03b1\n                          ,\n                          \u03b2\n                          )\n                        \n                      \n                    \n                   and its application in multiple attribute decision making. Commun. Math. Stat. 5, 213\u2013229 (2017)","journal-title":"Commun. Math. Stat."},{"key":"1202_CR22","doi-asserted-by":"publisher","first-page":"579","DOI":"10.1016\/j.asoc.2014.11.036","volume":"28","author":"H-C Li","year":"2015","unstructured":"Li, H.-C., You, J.-X., You, X.-Y., Shan, M.-M.: A novel approach for failure mode and effects analysis using combination weighting and fuzzy VIKOR method. Appl. Soft Comput. 28, 579\u2013588 (2015)","journal-title":"Appl. Soft Comput."},{"key":"1202_CR23","unstructured":"Opricovic, S.: Multi-criteria optimization of civil engineering systems. Ph.D. Thesis, University of Belgrade, Belgrade (1998)"},{"key":"1202_CR24","unstructured":"Renyi, A.: On measures of entropy and information. In: Proceedigs of 4th Barkley Symposium on Mathematical Statistics and Probability, vol. 1, p. 547. University of California Press (1961)"},{"key":"1202_CR25","first-page":"1037","volume":"31","author":"E Szmidt","year":"2002","unstructured":"Szmidt, E., Kacprzyk, J.: Using intuitionistic fuzzy sets in group decision-making. Control Cybern. 31, 1037\u20131054 (2002)","journal-title":"Control Cybern."},{"issue":"2","key":"1202_CR26","doi-asserted-by":"publisher","first-page":"335","DOI":"10.15388\/Informatica.2015.51","volume":"26","author":"D Stanujkic","year":"2015","unstructured":"Stanujkic, D.: Extension of ARAS method for decision making problems with interval-valued triangular fuzzy numbers. Informatica 26(2), 335\u2013355 (2015)","journal-title":"Informatica"},{"key":"1202_CR27","volume-title":"The Analytical Hierarchy Process","author":"TL Saaty","year":"1980","unstructured":"Saaty, T.L.: The Analytical Hierarchy Process. McGraw-Hill, New York (1980)"},{"key":"1202_CR28","doi-asserted-by":"publisher","first-page":"651","DOI":"10.1108\/BIJ-11-2014-0109","volume":"23","author":"AK Sahu","year":"2016","unstructured":"Sahu, A.K., Datta, S., Mahapatra, S.S.: Evaluation and selection of resilient suppliers in fuzzy environment: exploration of fuzzy-VIKOR. Benchmarking 23, 651\u2013673 (2016)","journal-title":"Benchmarking"},{"key":"1202_CR29","unstructured":"Sambuc, R.: Function \n                    \n                      \n                    \n                    $$\\phi $$\n                    \n                      \n                        \u03d5\n                      \n                    \n                  -Flous, Application al\u2019aide au Diagnostic en Pathologie Thyroidienne. These de Doctoraten Medicine, University of Marseille (1975)"},{"key":"1202_CR30","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1016\/j.patrec.2006.07.004","volume":"28","author":"IK Vlachos","year":"2007","unstructured":"Vlachos, I.K., Sergiadis, G.D.: Intuitionistic fuzzy information\u2014applications to pattern recognition. Pattern Recognit. Lett. 28, 197\u2013206 (2007)","journal-title":"Pattern Recognit. Lett."},{"key":"1202_CR31","doi-asserted-by":"crossref","unstructured":"Verma, R.K., Sharma, B.D.: On intuitionistic fuzzy entropy of order-\n                    \n                      \n                    \n                    $$\\alpha $$\n                    \n                      \n                        \u03b1\n                      \n                    \n                  . Adv. Fuzzy Syst. Article ID 789890, 8 pages (2014)","DOI":"10.1155\/2014\/789890"},{"key":"1202_CR32","first-page":"1694","volume":"27","author":"J Wang","year":"2012","unstructured":"Wang, J., Wang, P.: Intuitionistic linguistic fuzzy multi-criteria decision-making method based on intuitionistic fuzzy entropy. Control Decis. 27, 1694\u20131698 (2012)","journal-title":"Control Decis."},{"key":"1202_CR33","doi-asserted-by":"publisher","first-page":"2063","DOI":"10.1016\/j.patrec.2005.03.018","volume":"26","author":"W Wang","year":"2005","unstructured":"Wang, W., Xin, X.: Distance measure between intuitionistic fuzzy sets. Pattern Recognit. Lett. 26, 2063\u20132069 (2005)","journal-title":"Pattern Recognit. Lett."},{"key":"1202_CR34","doi-asserted-by":"publisher","DOI":"10.3390\/su9050688","author":"K-Q Wang","year":"2017","unstructured":"Wang, K.-Q., Liu, H.-C., Liu, L., Huang, J.: Green supplier evaluation and selection using cloud model theory and the QUALIFLEX method. Sustainability (2017). \n                    https:\/\/doi.org\/10.3390\/su9050688","journal-title":"Sustainability"},{"key":"1202_CR35","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1080\/03081070600574353","volume":"35","author":"ZS Xu","year":"2006","unstructured":"Xu, Z.S., Yager, R.R.: Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gen. Syst. 35, 417\u2013433 (2006)","journal-title":"Int. J. Gen. Syst."},{"key":"1202_CR36","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1016\/j.inffus.2010.12.001","volume":"13","author":"M Xia","year":"2012","unstructured":"Xia, M., Xu, Z.S.: Entropy\/cross entropy-based group decision making under intuitionistic fuzzy environment. Inf. Fusion 13, 31\u201347 (2012)","journal-title":"Inf. Fusion"},{"key":"1202_CR37","doi-asserted-by":"publisher","first-page":"1179","DOI":"10.1109\/TFUZZ.2006.890678","volume":"15","author":"ZS Xu","year":"2007","unstructured":"Xu, Z.S.: Intuitionistic fuzzy aggregation operators. IEEE Trans. Fuzzy Syst. 15, 1179\u20131187 (2007)","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"1202_CR38","doi-asserted-by":"publisher","first-page":"254","DOI":"10.1016\/j.jclepro.2017.01.162","volume":"148","author":"F Xu","year":"2017","unstructured":"Xu, F., Liu, J., Lin, S., Yuan, J.: A VIKOR-based approach for assessing the service performance of electric vehicle sharing programs: a case study in Beijing. J. Clean. Prod. 148, 254\u2013267 (2017)","journal-title":"J. Clean. Prod."},{"key":"1202_CR39","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-29584-3","volume-title":"Intuitionistic Fuzzy Information Aggregation: Theory and Applications","author":"ZS Xu","year":"2012","unstructured":"Xu, Z.S., Cai, X.Q.: Intuitionistic Fuzzy Information Aggregation: Theory and Applications. Springer, Berlin (2012)"},{"key":"1202_CR40","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1142\/S0219622010003816","volume":"9","author":"ZS Xu","year":"2010","unstructured":"Xu, Z.S., Hu, H.: Projection models for intuitionistic fuzzy multiple attribute decision making. Int. J. Inf. Technol. Decis. Mak. 9, 267\u2013280 (2010)","journal-title":"Int. J. Inf. Technol. Decis. Mak."},{"issue":"1","key":"1202_CR41","doi-asserted-by":"publisher","first-page":"60","DOI":"10.1109\/3477.826947","volume":"30","author":"R Yager","year":"2000","unstructured":"Yager, R.: Fuzzy medeling for intelligent decision making under uncertainty. IEEE Trans. Syst. Man Cybern. Part B Cybern. 30(1), 60\u201370 (2000)","journal-title":"IEEE Trans. Syst. Man Cybern. Part B Cybern."},{"key":"1202_CR42","doi-asserted-by":"publisher","first-page":"1906","DOI":"10.1016\/j.eswa.2014.10.004","volume":"42","author":"X-Y You","year":"2015","unstructured":"You, X.-Y., You, J.-X., Liu, H.-C., Zhen, L.: Group multi-criteria supplier selection using an extended VIKOR method with interval 2-tuple linguistic information. Experts Syst. Appl. 42, 1906\u20131916 (2015)","journal-title":"Experts Syst. Appl."},{"key":"1202_CR43","first-page":"338","volume":"8","author":"LA Zadeh","year":"1965","unstructured":"Zadeh, L.A.: Fuzzy sets. Inf. Comput. 8, 338\u2013353 (1965)","journal-title":"Inf. Comput."},{"key":"1202_CR44","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1016\/0022-247X(68)90078-4","volume":"23","author":"LA Zadeh","year":"1968","unstructured":"Zadeh, L.A.: Probability measures of fuzzy events. J. Math. Anal. Appl. 23, 421\u2013427 (1968)","journal-title":"J. Math. Anal. Appl."},{"issue":"6","key":"1202_CR45","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1016\/j.knosys.2009.06.007","volume":"22","author":"H Zhang","year":"2009","unstructured":"Zhang, H., Zhang, W., Mei, C.: Entropy of interval-valued fuzzy sets based on distance and its relationship with similarity measure. Knowl. Based Syst. 22(6), 449\u2013454 (2009)","journal-title":"Knowl. Based Syst."},{"issue":"13","key":"1202_CR46","doi-asserted-by":"publisher","first-page":"2751","DOI":"10.1016\/j.ins.2008.02.012","volume":"178","author":"L Zadeh","year":"2008","unstructured":"Zadeh, L.: Is there any need of fuzzy logic? Inf. Sci. 178(13), 2751\u20132779 (2008)","journal-title":"Inf. Sci."},{"key":"1202_CR47","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1016\/0020-0255(75)90036-5","volume":"8","author":"LA Zadeh","year":"1975","unstructured":"Zadeh, L.A.: The concept of linguistic variable and its application to approximate reasoning\u2014I. Inf. Sci. 8, 199\u2013249 (1975)","journal-title":"Inf. Sci."},{"issue":"18","key":"1202_CR48","doi-asserted-by":"publisher","first-page":"5355","DOI":"10.1007\/s00500-016-2118-x","volume":"21","author":"H Zhao","year":"2017","unstructured":"Zhao, H., You, J.-X., Liu, H.-C.: Failure mode and effect analysis using MULTIMOORA method with continuous weighted entropy under interval-valued intuitionistic fuzzy environment. Soft Comput. 21(18), 5355\u20135367 (2017)","journal-title":"Soft Comput."},{"key":"1202_CR49","doi-asserted-by":"publisher","DOI":"10.3390\/sym9090169","author":"J Zhao","year":"2017","unstructured":"Zhao, J., You, X.-Y., Liu, H.-C., Wu, S.-M.: An extended VIKOR method using intuitionistic fuzzy sets and combination weights for supplier selection. Symmetry (2017). \n                    https:\/\/doi.org\/10.3390\/sym9090169","journal-title":"Symmetry"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-018-1202-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s12190-018-1202-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-018-1202-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,7,3]],"date-time":"2019-07-03T19:07:36Z","timestamp":1562180856000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s12190-018-1202-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,7,4]]},"references-count":49,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2019,6]]}},"alternative-id":["1202"],"URL":"https:\/\/doi.org\/10.1007\/s12190-018-1202-z","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,7,4]]},"assertion":[{"value":"16 March 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 July 2018","order":2,"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 there is no conflict of interest regarding the publication of this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}