{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T00:34:19Z","timestamp":1770683659882,"version":"3.49.0"},"reference-count":32,"publisher":"SAGE Publications","issue":"3","license":[{"start":{"date-parts":[[2016,1,20]],"date-time":"2016-01-20T00:00:00Z","timestamp":1453248000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"published-print":{"date-parts":[[2016,3]]},"abstract":"<jats:p>Neutrosophic numbers (NNs), which contain determinate part and indeterminate part, are considered as a useful tool to express the indeterminate evaluation information which is common in real situation. The TODIM which is the short of Interactive and Multicriteria Decision Making in Portuguese can take the bounded rationality of decision makers into consideration based on the prospect theory. However, the classical TODIM method can only process the multiple attribute decision making (MADM) problems in which the attribute values are crisp numbers. The purpose of the paper is to give a new method to handle multiple attribute group decision making problems with neutrosophic numbers, which means we combine the TODIM with neutrosophic numbers to deal with MADM problems. In this paper, we firstly introduce the definition, the properties and the operational laws of the neutrosophic numbers, and then the possibility degree function is introduced concisely. Secondly, we represent the idea and steps of TODIM, and then we extend the TODIM method to the multiple attribute group decision making (MAGDM) with neutrosophic numbers. Finally, we use an illustrative example to demonstrate the practicality and effectiveness of the proposed method.<\/jats:p>","DOI":"10.3233\/ifs-151889","type":"journal-article","created":{"date-parts":[[2016,3,1]],"date-time":"2016-03-01T10:49:25Z","timestamp":1456829365000},"page":"1773-1781","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":26,"title":["An extended multiple attribute group decision-making TODIM method based on the neutrosophic numbers"],"prefix":"10.1177","volume":"30","author":[{"given":"Maocong","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Development Planning and Discipline Construction, Shandong Normal University, Jinan Shandong, China"}]},{"given":"Peide","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan Shandong, China"}]},{"given":"Lanlan","family":"Shi","sequence":"additional","affiliation":[{"name":"School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan Shandong, China"}]}],"member":"179","published-online":{"date-parts":[[2016,1,20]]},"reference":[{"key":"e_1_3_2_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0019-9958(65)90241-X"},{"key":"e_1_3_2_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-0114(86)80034-3"},{"key":"e_1_3_2_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(89)90205-4"},{"key":"e_1_3_2_5_2","doi-asserted-by":"publisher","DOI":"10.3846\/tede.2010.18"},{"key":"e_1_3_2_6_2","first-page":"601","article-title":"Overview on fuzzy multi-criteria decision-making approach","volume":"23","author":"Wang J.Q.","year":"2008","unstructured":"WangJ.Q., Overview on fuzzy multi-criteria decision-making approach, Control and Decision 23 (2008), 601\u2013606.","journal-title":"Control and Decision"},{"key":"e_1_3_2_7_2","first-page":"321","article-title":"Aggregation operators on intuitionistic trapezoidal fuzzy number and its application to multi-criteria decision making problems","volume":"20","author":"Wang J.Q.","year":"2009","unstructured":"WangJ.Q. and ZhangZ.H., Aggregation operators on intuitionistic trapezoidal fuzzy number and its application to multi-criteria decision making problems, Journal of Systems Engineering and Electronics 20 (2009), 321\u2013326.","journal-title":"Journal of Systems Engineering and Electronics"},{"key":"e_1_3_2_8_2","first-page":"773","article-title":"Method of intuitionistic trapezoidal fuzzy number for multi-attribute group decision","volume":"25","author":"Wan S.P.","year":"2010","unstructured":"WanS.P. and DongJ.Y., Method of intuitionistic trapezoidal fuzzy number for multi-attribute group decision, Control and Decision 25 (2010), 773\u2013776.","journal-title":"Control and Decision"},{"key":"e_1_3_2_9_2","unstructured":"SmarandacheF. Neutrosophy: Neutrosophic probability set and logic American Research Press Rehoboth 1999."},{"key":"e_1_3_2_10_2","doi-asserted-by":"publisher","DOI":"10.1515\/jisys-2014-0149"},{"key":"e_1_3_2_11_2","first-page":"43","article-title":"Misfire fault diagnosis method of gasoline engines using the cosine similarity measure of neutrosophic numbers","volume":"8","author":"Kong L.","year":"2015","unstructured":"KongL., WuY. and YeJ., Misfire fault diagnosis method of gasoline engines using the cosine similarity measure of neutrosophic numbers, Neutrosophic Sets and Systems 8 (2015), 43\u201346.","journal-title":"Neutrosophic Sets and Systems"},{"key":"e_1_3_2_12_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00521-014-1814-7"},{"key":"e_1_3_2_13_2","doi-asserted-by":"publisher","DOI":"10.3233\/IFS-2012-0572"},{"key":"e_1_3_2_14_2","first-page":"534","article-title":"Intuitionistic uncertain linguistic powered einstein aggregation operators and their application to multi-attribute group decision making","volume":"5","author":"Liu P.D.","year":"2015","unstructured":"LiuP.D. and ShiL.L., Intuitionistic uncertain linguistic powered einstein aggregation operators and their application to multi-attribute group decision making, Journal of Applied Analysis and Computation 5 (2015), 534\u2013561.","journal-title":"Journal of Applied Analysis and Computation"},{"key":"e_1_3_2_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.asoc.2013.12.010"},{"key":"e_1_3_2_16_2","doi-asserted-by":"publisher","DOI":"10.1080\/01969722.2014.929348"},{"key":"e_1_3_2_17_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2013.12.133"},{"key":"e_1_3_2_18_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2012.04.014"},{"key":"e_1_3_2_19_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcss.2012.07.001"},{"key":"e_1_3_2_20_2","doi-asserted-by":"publisher","DOI":"10.3233\/IFS-141142"},{"key":"e_1_3_2_21_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.knosys.2013.12.009"},{"key":"e_1_3_2_22_2","first-page":"113","article-title":"TODIM: Basics and application to multicriteria ranking of projects with environmental impacts","volume":"16","author":"Gomes L.F.A.M.","year":"1992","unstructured":"GomesL.F.A.M. and LimaM.M.P.P., TODIM: Basics and application to multicriteria ranking of projects with environmental impacts, Foundations of Computing and Decision Sciences 16 (1992), 113\u2013127.","journal-title":"Foundations of Computing and Decision Sciences"},{"key":"e_1_3_2_23_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejor.2007.10.046"},{"key":"e_1_3_2_24_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.eswa.2012.04.006"},{"key":"e_1_3_2_25_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.knosys.2012.12.014"},{"key":"e_1_3_2_26_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.knosys.2013.08.028"},{"key":"e_1_3_2_27_2","doi-asserted-by":"publisher","unstructured":"WangJ.Q. TODIM method with multi-valued neutrosophic set Control and Decision 2015 DOI: 10.13195\/j.kzyjc.2014.0467","DOI":"10.13195\/j.kzyjc.2014.0467"},{"key":"e_1_3_2_28_2","doi-asserted-by":"publisher","DOI":"10.1002\/cplx.21625"},{"key":"e_1_3_2_29_2","doi-asserted-by":"publisher","DOI":"10.3233\/IFS-141186"},{"key":"e_1_3_2_30_2","volume-title":"Neutrosophy: Neutrosophic probability, set, and logic","author":"Smarandache F.","year":"1998","unstructured":"SmarandacheF., Neutrosophy: Neutrosophic probability, set, and logic American Research Press, Rehoboth, USA, 1998."},{"key":"e_1_3_2_31_2","unstructured":"SmarandacheF. Introduction to neutrosophic measure neutrosophic integral and neutrosophic Probability Sitech & Education Publisher Craiova \u2013Columbus 2013."},{"key":"e_1_3_2_32_2","unstructured":"SmarandacheF. Introduction to neutrosophic statistics Sitech & Education Publishing 2014."},{"key":"e_1_3_2_33_2","doi-asserted-by":"publisher","DOI":"10.3233\/IFS-120724"}],"container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/IFS-151889","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.3233\/IFS-151889","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/IFS-151889","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,9]],"date-time":"2026-02-09T07:24:00Z","timestamp":1770621840000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/IFS-151889"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,1,20]]},"references-count":32,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2016,3]]}},"alternative-id":["10.3233\/IFS-151889"],"URL":"https:\/\/doi.org\/10.3233\/ifs-151889","relation":{},"ISSN":["1064-1246","1875-8967"],"issn-type":[{"value":"1064-1246","type":"print"},{"value":"1875-8967","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,1,20]]}}}