{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T05:24:41Z","timestamp":1771478681003,"version":"3.50.1"},"reference-count":46,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,7,1]],"date-time":"2021-07-01T00:00:00Z","timestamp":1625097600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Qinglin Project of Jiangsu Province","award":["Qinglin Project of Jiangsu Province"],"award-info":[{"award-number":["Qinglin Project of Jiangsu Province"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Interval-valued T-spherical fuzzy set (IVTSFS) handles uncertain and vague information by discussing their membership degree (MD), abstinence degree (AD), non-membership degree (NMD), and refusal degree (RD). MD, AD, NMD, and RD are defined in terms of closed subintervals of [0,\u00a01] that reduce information loss compared to the T-spherical fuzzy set (TSFS), which takes crisp values from [0,\u00a01] intervals; hence, some information may be lost. The purpose of this manuscript is to develop some Hamacher aggregation operators (HAOs) in the environment of IVTSFSs. To do so, some Hamacher operational laws based on Hamacher t-norms (HTNs) and Hamacher t-conorms (HTCNs) are introduced. Using Hamacher operational laws, we develop some aggregation operators (AOs), including an interval-valued T-spherical fuzzy Hamacher (IVTSFH) weighted averaging (IVTSFHWA) operator, an IVTSFH-ordered weighted averaging (IVTSFHOWA) operator, an IVTSFH hybrid averaging (IVTSFHHA) operator, an IVTSFH-weighted geometric (IVTSFHWG) operator, an IVTSFH-ordered weighted geometric (IVTSFHOWG) operator, and an IVTSFH hybrid geometric (IVTSFHHG) operator. The validation of the newly developed HAOs is investigated, and their basic properties are examined. In view of some restrictions, the generalization and proposed HAOs are shown, and a multi-attribute decision-making (MADM) procedure is explored based on the HAOs, which are further exemplified. Finally, a comparative analysis of the proposed work is also discussed with previous literature to show the superiority of our work.<\/jats:p>","DOI":"10.3390\/axioms10030145","type":"journal-article","created":{"date-parts":[[2021,7,1]],"date-time":"2021-07-01T12:03:27Z","timestamp":1625141007000},"page":"145","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Approach to Multi-Attribute Decision-Making Methods for Performance Evaluation Process Using Interval-Valued T-Spherical Fuzzy Hamacher Aggregation Information"],"prefix":"10.3390","volume":"10","author":[{"given":"Yun","family":"Jin","sequence":"first","affiliation":[{"name":"Tourism Department, Wuxi Vocational College of Science and Technology, Wuxi 214028, China"}]},{"given":"Zareena","family":"Kousar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Riphah Institute of Computing & Applied Sciences (RICAS), Riphah International University (Lahore Campus), Lahore 54000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1438-6413","authenticated-orcid":false,"given":"Kifayat","family":"Ullah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Riphah Institute of Computing & Applied Sciences (RICAS), Riphah International University (Lahore Campus), Lahore 54000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3871-3845","authenticated-orcid":false,"given":"Tahir","family":"Mahmood","sequence":"additional","affiliation":[{"name":"Department of Mathematics & Statistics, International Islamic University Islamabad, Islamabad 44000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7094-8097","authenticated-orcid":false,"given":"Nimet","family":"Yapici Pehlivan","sequence":"additional","affiliation":[{"name":"Department of Statistics, Science Faculty, Sel\u00e7uk University, Konya 42031, Turkey"}]},{"given":"Zeeshan","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics & Statistics, International Islamic University Islamabad, Islamabad 44000, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0165-0114(86)80034-3","article-title":"Intuitionistic fuzzy sets","volume":"20","author":"Atanassov","year":"1986","journal-title":"Fuzzy Set Syst."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Yager, R.R. (2013, January 24\u201328). Pythagorean fuzzy subsets. Proceedings of the 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA\/NAFIPS), Edmonton, AB, Canada.","DOI":"10.1109\/IFSA-NAFIPS.2013.6608375"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1222","DOI":"10.1109\/TFUZZ.2016.2604005","article-title":"Generalized Orthopair fuzzy sets","volume":"25","author":"Yager","year":"2016","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/0165-0114(89)90205-4","article-title":"Interval valued intuitionistic fuzzy sets","volume":"31","author":"Atanassov","year":"1989","journal-title":"Fuzzy Set Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/0020-0255(75)90036-5","article-title":"The concept of a linguistic variable and its application to approximate reasoning\u2014I","volume":"8","author":"Zadeh","year":"1975","journal-title":"Inf. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"444","DOI":"10.1002\/int.21790","article-title":"Fundamental properties of interval-valued Pythagorean fuzzy aggregation operators","volume":"31","author":"Peng","year":"2016","journal-title":"Int. J. Intell. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"5225","DOI":"10.3233\/JIFS-169806","article-title":"Interval valued q-rung orthopair fuzzy sets and their properties","volume":"35","author":"Joshi","year":"2018","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"61","DOI":"10.3233\/FI-2019-1755","article-title":"Complex interval-valued intuitionistic fuzzy sets and their aggregation operators","volume":"164","author":"Garg","year":"2019","journal-title":"Fundam. Inform."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3595","DOI":"10.1007\/s10462-019-09772-x","article-title":"Novel aggregation operators and ranking method for complex intuitionistic fuzzy sets and their applications to decision-making process","volume":"53","author":"Garg","year":"2020","journal-title":"Artif. Intell. Rev."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2302","DOI":"10.1109\/TFUZZ.2019.2897961","article-title":"Linguistic interval-valued Atanassov intuitionistic fuzzy sets and their applications to group decision making problems","volume":"27","author":"Garg","year":"2019","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.31181\/dmame2003001s","article-title":"A rough set approach for forecasting models","volume":"3","author":"Sharma","year":"2020","journal-title":"Decis. Mak. Appl. Manag. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1142\/S0218488518500046","article-title":"A linear programming method based on an improved score function for interval-valued Pythagorean fuzzy numbers and its application to decision-making","volume":"26","author":"Garg","year":"2018","journal-title":"Int. J. Uncertain. Fuzziness Knowl. Based Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"70","DOI":"10.31181\/dmame2003070r","article-title":"Certain properties of soft multi-set topology with applications in multi-criteria decision making","volume":"3","author":"Riaz","year":"2020","journal-title":"Decis. Mak. Appl. Manag. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/s40747-019-0103-6","article-title":"On some distance measures of complex Pythagorean fuzzy sets and their applications in pattern recognition","volume":"6","author":"Ullah","year":"2020","journal-title":"Complex Intell. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1426","DOI":"10.1002\/int.21985","article-title":"Some q-rung orthopair fuzzy Heronian mean operators in multiple attribute decision making","volume":"33","author":"Wei","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1572","DOI":"10.1002\/int.22108","article-title":"Some cosine similarity measures and distance measures between q-rung orthopair fuzzy sets","volume":"34","author":"Liu","year":"2019","journal-title":"Int. J. Intell. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"162","DOI":"10.31181\/rme200101162k","article-title":"Risk analysis of cutting system under intuitionistic fuzzy environment","volume":"1","author":"Kushwaha","year":"2020","journal-title":"Rep. Mech. Eng."},{"key":"ref_19","first-page":"409","article-title":"Picture fuzzy sets","volume":"30","author":"Cuong","year":"2014","journal-title":"J. Comput. Sci. Cybern."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Liu, P., Munir, M., Mahmood, T., and Ullah, K. (2019). Some similarity measures for interval-valued picture fuzzy sets and their applications in decision making. Information, 10.","DOI":"10.3390\/info10120369"},{"key":"ref_21","first-page":"51","article-title":"Some Generalized Distance and Similarity Measures for Picture Hesitant Fuzzy Sets and Their Applications in Building Material Recognition and Multi-Attribute Decision Making","volume":"51","author":"Jan","year":"2019","journal-title":"Punjab Univ. J. Math."},{"key":"ref_22","first-page":"84","article-title":"Multi-attribute decision making based on averaging aggregation operators for picture hesitant fuzzy sets","volume":"23","author":"Ullah","year":"2018","journal-title":"Tech. J."},{"key":"ref_23","first-page":"55","article-title":"The Fuzzy Cross-Entropy for Picture Hesitant Fuzzy Sets and Their Application in Multi Criteria Decision Making","volume":"52","author":"Mahmood","year":"2020","journal-title":"Punjab Univ. J. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"7041","DOI":"10.1007\/s00521-018-3521-2","article-title":"An approach toward decision-making and medical diagnosis problems using the concept of spherical fuzzy sets","volume":"31","author":"Mahmood","year":"2019","journal-title":"Neural. Comput. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Ullah, K., Hassan, N., Mahmood, T., Jan, N., and Hassan, M. (2019). Evaluation of investment policy based on multi-attribute decision-making using interval valued T-spherical fuzzy aggregation operators. Symmetry, 11.","DOI":"10.3390\/sym11030357"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Ullah, K., Mahmood, T., and Jan, N. (2018). Similarity measures for T-spherical fuzzy sets with applications in pattern recognition. Symmetry, 10.","DOI":"10.3390\/sym10060193"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Garg, H., Munir, M., Ullah, K., Mahmood, T., and Jan, N. (2018). Algorithm for T-spherical fuzzy multi-attribute decision making based on improved interactive aggregation operators. Symmetry, 10.","DOI":"10.3390\/sym10120670"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1647","DOI":"10.1007\/s00500-019-03993-6","article-title":"Correlation coefficients for T-spherical fuzzy sets and their applications in clustering and multi-attribute decision making","volume":"24","author":"Ullah","year":"2020","journal-title":"Soft Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"10208","DOI":"10.1109\/ACCESS.2019.2963260","article-title":"Divergence measure of T-Spherical Fuzzy Sets and its applications in Pattern Recognition","volume":"8","author":"Wu","year":"2019","journal-title":"IEEE Access"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"44","DOI":"10.31181\/rme200101044p","article-title":"Normalized weighted Geometric Dombi Bonferoni Mean Operator with interval grey numbers: Application in multicriteria decision making","volume":"1","author":"Pamucar","year":"2020","journal-title":"Rep. Mech. Eng."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Munir, M., Kalsoom, H., Ullah, K., Mahmood, T., and Chu, Y.M. (2020). T-spherical fuzzy Einstein hybrid aggregation operators and their applications in smulti-attribute decision making problems. Symmetry, 12.","DOI":"10.3390\/sym12030365"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/S0020-0255(03)00072-0","article-title":"On the use of Hamacher\u2019s t-norms family for information aggregation","volume":"153","author":"Oussalah","year":"2003","journal-title":"Inform. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"505","DOI":"10.3233\/IFS-131019","article-title":"Intuitionistic fuzzy Hamacher aggregation operators and their application to multiple attribute decision making","volume":"27","author":"Huang","year":"2014","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1007\/s40998-018-0167-0","article-title":"Intuitionistic fuzzy hamacher aggregation operators with entropy weight and their applications to multi-criteria decision-making problems","volume":"43","author":"Garg","year":"2019","journal-title":"Iran. J. Sci. Tech. Trans. Electr. Eng."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1109\/TFUZZ.2013.2248736","article-title":"Some Hamacher aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their application to group decision making","volume":"22","author":"Liu","year":"2013","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2229","DOI":"10.3233\/JIFS-172262","article-title":"Pythagorean fuzzy Hamacher prioritized aggregation operators in multiple attribute decision making","volume":"35","author":"Gao","year":"2018","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"57","DOI":"10.3233\/FI-2019-1794","article-title":"Pythagorean fuzzy Hamacher power aggregation operators in multiple attribute decision making","volume":"166","author":"Wei","year":"2019","journal-title":"Fundam. Inform."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"103259","DOI":"10.1016\/j.engappai.2019.103259","article-title":"Some q-rung Orthopair fuzzy Hamacher aggregation operators and their application to multiple attribute group decision making with modified EDAS method","volume":"87","author":"Darko","year":"2020","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Donyatalab, Y., Farrokhizadeh, E., Shishavan, S.A.S., and Seifi, S.H. (2020). Hamacher Aggregation Operators Based on Interval-Valued q-Rung Orthopair Fuzzy Sets and Their Applications to Decision Making Problems. International Conference on Intelligent and Fuzzy Systems, Springer.","DOI":"10.1007\/978-3-030-51156-2_54"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Jana, C., and Pal, M. (2019). Assessment of enterprise performance based on picture fuzzy Hamacher aggregation operators. Symmetry, 11.","DOI":"10.3390\/sym11010075"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"570","DOI":"10.1007\/s40815-020-00803-2","article-title":"Evaluation of the Performance of Search and Rescue Robots Using T-spherical Fuzzy Hamacher Aggregation Operators","volume":"22","author":"Ullah","year":"2020","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_42","first-page":"1","article-title":"Pythagorean fuzzy interactive Hamacher power aggregation operators for assessment of express service quality with entropy weight","volume":"25","author":"Wang","year":"2020","journal-title":"Soft Comput."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Zhu, W.B., Shuai, B., and Zhang, S.H. (2020). The Linguistic Interval-Valued Intuitionistic Fuzzy Aggregation Operators Based on Extended Hamacher T-Norm and S-Norm and Their Application. Symmetry, 1.","DOI":"10.3390\/sym12040668"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"5755","DOI":"10.3233\/JIFS-18266","article-title":"Dual hesitant bipolar fuzzy Hamacher aggregation operators and their applications to multiple attribute decision making","volume":"37","author":"Gao","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1431","DOI":"10.1002\/sd.2096","article-title":"A multi-criteria sustainable supplier selection framework based on neutrosophic fuzzy data and entropy weighting","volume":"28","author":"Zeng","year":"2020","journal-title":"Sustain. Dev."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/s41066-018-0089-2","article-title":"Linguistic intuitionistic fuzzy Hamacher aggregation operators and their application to group decision making","volume":"4","author":"Tang","year":"2019","journal-title":"Granul. Comput."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/3\/145\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:24:45Z","timestamp":1760163885000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/3\/145"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,1]]},"references-count":46,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,9]]}},"alternative-id":["axioms10030145"],"URL":"https:\/\/doi.org\/10.3390\/axioms10030145","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,1]]}}}