{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T05:34:23Z","timestamp":1772343263871,"version":"3.50.1"},"reference-count":56,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,5,29]],"date-time":"2021-05-29T00:00:00Z","timestamp":1622246400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Scientific Research at King Khalid University, Saudi Arabia","award":["R. G. P. 2\/71\/41"],"award-info":[{"award-number":["R. G. P. 2\/71\/41"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we investigate the multi-criteria decision-making complications under intuitionistic fuzzy hypersoft set (IFHSS) information. The IFHSS is a proper extension of the intuitionistic fuzzy soft set (IFSS) which discusses the parametrization of multi-sub attributes of considered parameters, and accommodates more hesitation comparative to IFSS utilizing the multi sub-attributes of the considered parameters. The main objective of this research is to introduce operational laws for intuitionistic fuzzy hypersoft numbers (IFHSNs). Additionally, based on developed operational laws two aggregation operators (AOs), i.e., intuitionistic fuzzy hypersoft weighted average (IFHSWA) and intuitionistic fuzzy hypersoft weighted geometric (IFHSWG), operators have been presented with their fundamental properties. Furthermore, a decision-making approach has been established utilizing our developed aggregation operators (AOs). Through the established approach, a technique for solving decision-making (DM) complications is proposed to select sustainable suppliers in sustainable supply chain management (SSCM). Moreover, a numerical description is presented to ensure the validity and usability of the proposed technique in the DM process. The practicality, effectivity, and flexibility of the current approach are demonstrated through comparative analysis with the assistance of some prevailing studies.<\/jats:p>","DOI":"10.3390\/e23060688","type":"journal-article","created":{"date-parts":[[2021,5,31]],"date-time":"2021-05-31T00:22:15Z","timestamp":1622420535000},"page":"688","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":54,"title":["Robust Aggregation Operators for Intuitionistic Fuzzy Hypersoft Set with Their Application to Solve MCDM Problem"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2656-8679","authenticated-orcid":false,"given":"Rana Muhammad","family":"Zulqarnain","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Management and Technology, Sialkot Campus, Lahore 51310, Pakistan"}]},{"given":"Imran","family":"Siddique","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan"}]},{"given":"Rifaqat","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Arts, Muhayil, King Khalid University, Abha 61413, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8522-1942","authenticated-orcid":false,"given":"Dragan","family":"Pamucar","sequence":"additional","affiliation":[{"name":"Department of Logistics, Military academy, University of Defence in Belgarde, Pavla Jurisica Sturma 33, 11000 Belgrade, Serbia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3583-9434","authenticated-orcid":false,"given":"Dragan","family":"Marinkovic","sequence":"additional","affiliation":[{"name":"Faculty of Mechanical and Transport Systems, Technische Universit\u00e4t Berlin, 10623 Berlin, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9657-0889","authenticated-orcid":false,"given":"Darko","family":"Bozanic","sequence":"additional","affiliation":[{"name":"Military Academy, University of Defence in Belgarde, Pavla Jurisica Sturma 33, 11000 Belgrade, Serbia"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,29]]},"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 Sets Syst."},{"key":"ref_3","first-page":"507","article-title":"Cubic intuitionistic fuzzy sets and its fundamental properties","volume":"33","author":"Garg","year":"2019","journal-title":"J. Mult. Valued Logic Soft Comput."},{"key":"ref_4","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 Sets Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2056","DOI":"10.4028\/www.scientific.net\/AMR.1070-1072.2056","article-title":"Entropy and Distance Measure of Intuitionistic Fuzzy Soft Sets","volume":"1070\u20131072","author":"Li","year":"2014","journal-title":"Adv. Mater. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/S0898-1221(99)00056-5","article-title":"Soft Set Theory First Results","volume":"37","author":"Molodtsov","year":"1999","journal-title":"Comput. Math. Appl."},{"key":"ref_7","first-page":"589","article-title":"Fuzzy soft sets","volume":"9","author":"Maji","year":"2001","journal-title":"J. Fuzzy Math."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Ali, G., Akram, M., Koam, A.N.A., and Alcantud, J.C.R. (2019). Parameter Reductions of Bipolar Fuzzy Soft Sets with Their Decision-Making Algorithms. Symmetry, 11.","DOI":"10.3390\/sym11080949"},{"key":"ref_9","first-page":"677","article-title":"Intuitionistic fuzzy soft sets","volume":"9","author":"Maji","year":"2001","journal-title":"J. Fuzzy Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2944","DOI":"10.3934\/math.2020190","article-title":"TOPSIS method based on correlation coefficient for solving decision-making problems with intuitionistic fuzzy soft set information","volume":"5","author":"Garg","year":"2020","journal-title":"AIMS Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"563745","DOI":"10.1155\/2015\/563745","article-title":"Entropy Measures for Interval-Valued Intuitionistic Fuzzy Sets and Their Application in Group Decision-Making","volume":"2015","author":"Wei","year":"2015","journal-title":"Math. Probl. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"6656858","DOI":"10.1155\/2021\/6656858","article-title":"TOPSIS Method Based on the Correlation Coefficient of Interval-Valued Intuitionistic Fuzzy Soft Sets and Aggregation Operators with Their Application in Decision-Making","volume":"2021","author":"Zulqarnain","year":"2021","journal-title":"J. Math."},{"key":"ref_13","first-page":"931","article-title":"A robust aggregation operators for multi-criteria decision-making with intuitionistic fuzzy soft set environment","volume":"25","author":"Arora","year":"2017","journal-title":"Sci. Iran."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"973","DOI":"10.1007\/s00500-020-05193-z","article-title":"Pythagorean fuzzy interactive Hamacher power aggregation operators for assessment of express service quality with entropy weight","volume":"25","author":"Wang","year":"2021","journal-title":"Soft Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"6245","DOI":"10.3233\/JIFS-182570","article-title":"Continuous interval-valued Pythagorean fuzzy aggregation operators for multiple attribute group decision making","volume":"36","author":"Wang","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"131962","DOI":"10.1109\/ACCESS.2019.2938706","article-title":"Interval-valued q-rung orthopair 2-tuple linguistic aggregation operators and their applications to decision making process","volume":"7","author":"Wang","year":"2019","journal-title":"IEEE Access"},{"key":"ref_17","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_18","doi-asserted-by":"crossref","first-page":"415","DOI":"10.3233\/FI-2016-1415","article-title":"Fundamental Properties of Pythagorean Fuzzy Aggregation Operators","volume":"147","author":"Peng","year":"2016","journal-title":"Fundam. Inform."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"886","DOI":"10.1002\/int.21809","article-title":"A New Generalized Pythagorean Fuzzy Information Aggregation Using Einstein Operations and Its Application to Decision Making","volume":"31","author":"Garg","year":"2016","journal-title":"Int. J. Intell. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.1002\/int.20498","article-title":"Intuitionistic fuzzy geometric aggregation operators based on einstein operations","volume":"26","author":"Wang","year":"2011","journal-title":"Int. J. Intell. Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1080\/03081070600574353","article-title":"Some geometric aggregation operators based on intuitionistic fuzzy sets","volume":"35","author":"Xu","year":"2006","journal-title":"Int. J. Gen. Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1109\/TFUZZ.2012.2189405","article-title":"Intuitionistic Fuzzy Information Aggregation Using Einstein Operations","volume":"20","author":"Wang","year":"2012","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1007\/s40314-019-0764-1","article-title":"Group decision-making method based on prioritized linguistic intuitionistic fuzzy aggregation operators and its fundamental properties","volume":"38","author":"Arora","year":"2019","journal-title":"Comput. Appl. Math."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Faizi, S., Sa\u0142abun, W., and Nawaz, S. (2021). Best-Worst method and Hamacher aggregation operations for intuitionistic 2-tuple linguistic sets. Expert Syst. Appl., 115088.","DOI":"10.1016\/j.eswa.2021.115088"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1179","DOI":"10.1109\/TFUZZ.2006.890678","article-title":"Intuitionistic fuzzy aggregation operators","volume":"15","author":"Xu","year":"2007","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_26","first-page":"92","article-title":"An evaluation of a third-party logistics provider: The application of the rough Dombi-Hamy mean operator","volume":"3","author":"Sinani","year":"2020","journal-title":"Decis. Mak. Appl. Manag. Eng."},{"key":"ref_27","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_28","first-page":"224","article-title":"Pythagoren fuzzy soft set and its application","volume":"41","author":"Peng","year":"2015","journal-title":"Comput. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1050","DOI":"10.3934\/math.2020073","article-title":"A novel entropy measure of pythagorean fuzzy soft sets","volume":"5","author":"Athira","year":"2020","journal-title":"AIMS Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"5545","DOI":"10.3233\/JIFS-202781","article-title":"Aggregation operators of Pythagorean fuzzy soft sets with their application for green supplier chain management","volume":"40","author":"Zulqarnain","year":"2021","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Riaz, M., Sa\u0142abun, W., Farid, H.M.A., Ali, N., and W\u0105tr\u00f3bski, J. (2020). A Robust q-Rung Orthopair Fuzzy Information Aggregation Using Einstein Operations with Application to Sustainable Energy Planning Decision Management. Energies, 13.","DOI":"10.3390\/en13092155"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Faizi, S., Sa\u0142abun, W., Ullah, S., Rashid, T., and Wi\u0119ckowski, J. (2020). A New Method to Support Decision-Making in an Uncertain Environment Based on Normalized Interval-Valued Triangular Fuzzy Numbers and COMET Technique. Symmetry, 12.","DOI":"10.3390\/sym12040516"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Bashir, Z., Rashid, T., W\u0105tr\u00f3bski, J., Sa\u0142abun, W., and Malik, A. (2018). Hesitant probabilistic multiplicative preference relations in group decision making. Appl. Sci., 8.","DOI":"10.3390\/app8030398"},{"key":"ref_34","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_35","doi-asserted-by":"crossref","unstructured":"Sa\u0142abun, W., W\u0105tr\u00f3bski, J., and Shekhovtsov, A. (2020). Are MCDA Methods Benchmarkable? A Comparative Study of TOPSIS, VIKOR, COPRAS, and Promethee II Methods. Symmetry, 12.","DOI":"10.3390\/sym12091549"},{"key":"ref_36","first-page":"168","article-title":"Extension of Soft Set to Hypersoft Set, and then to Plithogenic Hypersoft Set","volume":"22","author":"Smarandache","year":"2018","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_37","first-page":"34","article-title":"Plithogenic Fuzzy Whole Hypersoft Set: Construction of Operators and their Application in Frequency Matrix Multi Attribute Decision Making Technique","volume":"28","author":"Rana","year":"2019","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_38","first-page":"271","article-title":"Generalized Aggregate Operators on Neutrosophic Hypersoft Set","volume":"36","author":"Zulqarnain","year":"2020","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_39","unstructured":"Zulqarnain, R.M., Xin, X.L., and Saeed, M. (2021). A Development of Pythagorean fuzzy hypersoft set with basic operations and decision-making approach based on the correlation coefficient. Theory and Application of Hypersoft Set, Pons Publishing House Brussels. Chapter 5."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"6634991","DOI":"10.1155\/2021\/6634991","article-title":"Development of TOPSIS Technique under Pythagorean Fuzzy Hypersoft Environment Based on Correlation Coefficient and Its Application towards the Selection of Antivirus Mask in COVID-19 Pandemic","volume":"2021","author":"Zulqarnain","year":"2021","journal-title":"Complexity"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2021\/5531830","article-title":"Selection of an Effective Hand Sanitizer to Reduce COVID-19 Effects and Extension of TOPSIS Technique Based on Correlation Coefficient Under Neutrosophic Hypersoft Set","volume":"2021","author":"Samad","year":"2021","journal-title":"Complexity"},{"key":"ref_42","first-page":"134","article-title":"Some Fundamental Operations on Interval Valued Neutrosophic Hypersoft Set with Their Properties","volume":"40","author":"Zulqarnain","year":"2021","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_43","first-page":"12","article-title":"Decision-Making Approach Based on Correlation Coefficient with its Properties Under Interval-Valued Neutrosophic hypersoft set environment","volume":"40","author":"Zulqarnain","year":"2021","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2732","DOI":"10.3934\/math.2021167","article-title":"Extension of TOPSIS method under intuitionistic fuzzy hypersoft environment based on correlation coefficient and aggregation operators to solve decision making problem","volume":"6","author":"Zulqarnain","year":"2021","journal-title":"AIMS Math."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"72","DOI":"10.31181\/oresta1901060c","article-title":"A two-phase fuzzy AHP-fuzzy TOPSIS model for supplier evaluation in manufacturing environment","volume":"2","author":"Chatterjee","year":"2019","journal-title":"Oper. Res. Eng. Sci. Theory Appl."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"79","DOI":"10.5937\/vojtehg68-24805","article-title":"A role, significance and approaches for studying logistics","volume":"68","author":"Milenkov","year":"2020","journal-title":"Vojn. Glas."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"39","DOI":"10.31181\/oresta2003034z","article-title":"Modelling Procedure for the Selection of Steel Pipes Supplier by Applying Fuzzy AHP Method","volume":"3","author":"Zavadskas","year":"2020","journal-title":"Oper. Res. Eng. Sci. Theory Appl."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"W\u0105tr\u00f3bski, J., Sa\u0142abun, W., and Ladorucki, G. (2017, January 3\u20135). The Temporal Supplier Evaluation Model Based on Multicriteria Decision Analysis Methods. Proceedings of the Asian Conference on Intelligent Information and Database Systems, Kanazawa, Japan.","DOI":"10.1007\/978-3-319-54472-4_41"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"82","DOI":"10.31181\/dmame180182v","article-title":"A Rough Multicriteria Approach for Evaluation of the Supplier Criteria in Automotive Industry","volume":"1","author":"Vasiljevic","year":"2018","journal-title":"Decis. Mak. Appl. Manag. Eng."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"W\u0105tr\u00f3bski, J., and Sa\u0142abun, W. (2016, January 4\u20136). Green Supplier Selection Framework Based on Multi-Criteria Decision-Analysis Approach. Proceedings of the International Conference on Sustainable Design and Manufacturing, Chania, Greece.","DOI":"10.1007\/978-3-319-32098-4_31"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"61","DOI":"10.31181\/rme200101061t","article-title":"Force analysis and kinematic optimizationof a fluid valve driven by shape memory alloys","volume":"1","author":"Todorov","year":"2020","journal-title":"Rep. Mech. Eng."},{"key":"ref_52","first-page":"107","article-title":"The Vehicle Routing Problem with Stochastic Demands in an Urban Area\u2014A Case Study","volume":"18","year":"2020","journal-title":"Facta Univ. Ser. Mech. Eng."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"355","DOI":"10.5937\/vojtehg69-29629","article-title":"Model for selecting a route for the transport of hazardous materials using a fuzzy logic system","volume":"69","author":"Chatterjee","year":"2021","journal-title":"Vojn. Glas."},{"key":"ref_54","first-page":"341","article-title":"A Vikor and Topsis Focused Reanalysis of the Madm Methods Based on Logarithmic Normalization","volume":"18","author":"Zolfani","year":"2020","journal-title":"Facta Univ. Ser. Mech. Eng."},{"key":"ref_55","first-page":"882","article-title":"On clustering approach to intuitionistic fuzzy sets","volume":"22","author":"Zhang","year":"2007","journal-title":"Contr. Decis."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"3775","DOI":"10.1016\/j.ins.2008.06.008","article-title":"Clustering algorithm for intuitionistic fuzzy sets","volume":"178","author":"Xu","year":"2008","journal-title":"Inf. Sci."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/6\/688\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:10:29Z","timestamp":1760163029000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/6\/688"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,29]]},"references-count":56,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["e23060688"],"URL":"https:\/\/doi.org\/10.3390\/e23060688","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,29]]}}}