{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T20:34:07Z","timestamp":1770064447811,"version":"3.49.0"},"reference-count":58,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T00:00:00Z","timestamp":1679443200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002383","name":"King Saud University","doi-asserted-by":"publisher","award":["RSP2023R18"],"award-info":[{"award-number":["RSP2023R18"]}],"id":[{"id":"10.13039\/501100002383","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The intuitionistic hesitant fuzzy set (IHFS) is an enriched version of hesitant fuzzy sets (HFSs) that can cover both fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs). By assigning membership and non-membership grades as subsets of [0, 1], the IHFS can model and handle situations more proficiently. Another related theory is the theory of set pair analysis (SPA), which considers both certainties and uncertainties as a cohesive system and represents them from three aspects: identity, discrepancy, and contrary. In this article, we explore the suitability of combining the IHFS and SPA theories in multi-attribute decision making (MADM) and present the hybrid model named intuitionistic hesitant fuzzy connection number set (IHCS). To facilitate the design of a novel MADM algorithm, we first develop several averaging and geometric aggregation operators on IHCS. Finally, we highlight the benefits of our proposed work, including a comparative examination of the recommended models with a few current models to demonstrate the practicality of an ideal decision in practice. Additionally, we provide a graphical interpretation of the devised attempt to exhibit the consistency and efficiency of our approach.<\/jats:p>","DOI":"10.3390\/sym15030778","type":"journal-article","created":{"date-parts":[[2023,3,23]],"date-time":"2023-03-23T03:07:57Z","timestamp":1679540877000},"page":"778","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["Multiple-Attribute Decision Making Based on Intuitionistic Hesitant Fuzzy Connection Set Environment"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4926-722X","authenticated-orcid":false,"given":"Wajid","family":"Ali","sequence":"first","affiliation":[{"name":"Department of Mathematics, Air University, E-9, Islamabad 44000, Pakistan"}]},{"given":"Tanzeela","family":"Shaheen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Air University, E-9, Islamabad 44000, Pakistan"}]},{"given":"Iftikhar Ul","family":"Haq","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Air University, E-9, Islamabad 44000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1406-6188","authenticated-orcid":false,"given":"Hamza Ghazanfar","family":"Toor","sequence":"additional","affiliation":[{"name":"Biomedical Engineering Department, Riphah International University, Islamabad 46000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1483-1801","authenticated-orcid":false,"given":"Faraz","family":"Akram","sequence":"additional","affiliation":[{"name":"Biomedical Engineering Department, Riphah International University, Islamabad 46000, Pakistan"}]},{"given":"Saeid","family":"Jafari","sequence":"additional","affiliation":[{"name":"Mathematical and Physical Science Foundation, 4200 Slagelse, Denmark"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5215-1834","authenticated-orcid":false,"given":"Md. Zia","family":"Uddin","sequence":"additional","affiliation":[{"name":"Software and Service Innovation, SINTEF Digital, 0373 Oslo, Norway"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3479-3606","authenticated-orcid":false,"given":"Mohammad Mehedi","family":"Hassan","sequence":"additional","affiliation":[{"name":"Information Systems Department, College of Computer and Information Sciences, King Saud University, Riyadh 11543, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Zhang, F., Ma, W., and Ma, H. (2023). Dynamic Chaotic Multi-Attribute Group Decision Making under Weighted T-Spherical Fuzzy Soft Rough Sets. Symmetry, 15.","DOI":"10.3390\/sym15020307"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40314-021-01742-w","article-title":"Approaches to multi-attribute decision-making based on picture fuzzy Aczel\u2013Alsina average aggregation operators","volume":"41","author":"Senapati","year":"2022","journal-title":"Comput. Appl. Math."},{"key":"ref_3","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_4","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"2018","journal-title":"Inf. Control."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Sarfraz, M., Ullah, K., Akram, M., Pamucar, D., and Bo\u017eani\u0107, D. (2022). Prioritized Aggregation Operators for Intuitionistic Fuzzy Information Based on Aczel\u2013Alsina T-Norm and T-Conorm and Their Applications in Group Decision-Making. Symmetry, 14.","DOI":"10.3390\/sym14122655"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Jin, H., Hussain, A., Ullah, K., and Javed, A. (2022). Novel Complex Pythagorean Fuzzy Sets under Aczel\u2013Alsina Operators and Their Application in Multi-Attribute Decision Making. Symmetry, 15.","DOI":"10.3390\/sym15010068"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5493","DOI":"10.1002\/int.22520","article-title":"Why do we need q-rung orthopair fuzzy sets? Some evidence established via mass assignment","volume":"36","author":"Shaheen","year":"2021","journal-title":"Int. J. Intell. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/0165-0114(94)90229-1","article-title":"New operations defined over the intuitionistic fuzzy sets","volume":"61","author":"Atanassov","year":"1994","journal-title":"Fuzzy Sets Syst."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Atanassov, K.T. (2012). On Intuitionistic Fuzzy Sets Theory, Springer.","DOI":"10.1007\/978-3-642-29127-2"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2031","DOI":"10.1007\/s10489-017-1035-8","article-title":"A nonlinear-programming methodology for multi-attribute decision-making problem with interval-valued intuitionistic fuzzy soft sets information","volume":"48","author":"Garg","year":"2018","journal-title":"Appl. Intell."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"177","DOI":"10.3846\/20294913.2014.984253","article-title":"Intuitionistic fuzzy generalized probabilistic ordered weighted averaging operator and its application to group decision making","volume":"22","author":"Zeng","year":"2016","journal-title":"Technol. Econ. Dev. Econ."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"224","DOI":"10.5391\/IJFIS.2013.13.3.224","article-title":"Intuitionistic Fuzzy Theta-Compact Spaces","volume":"13","author":"Eom","year":"2013","journal-title":"Int. J. Fuzzy Log. Intell. Syst."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Liu, P., Mahmood, T., and Khan, Q. (2017). Multi-attribute decision-making based on prioritized aggregation operator under hesitant intuitionistic fuzzy linguistic environment. Symmetry, 9.","DOI":"10.3390\/sym9110270"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.ins.2019.12.008","article-title":"Aggregation of infinite chains of intuitionistic fuzzy sets and their application to choices with temporal intuitionistic fuzzy information","volume":"514","author":"Alcantud","year":"2020","journal-title":"Inf. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/0165-0114(94)00343-6","article-title":"Correlation of interval-valued intuitionistic fuzzy sets","volume":"74","author":"Bustince","year":"1995","journal-title":"Fuzzy Sets Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1279","DOI":"10.1016\/j.mcm.2009.06.010","article-title":"Correlation coefficient of interval-valued intuitionistic fuzzy sets and its application to multiple attribute group decision making problems","volume":"50","author":"Park","year":"2009","journal-title":"Math. Comput. Model."},{"key":"ref_17","first-page":"94","article-title":"New Topological Operator Over Intuitionistic Fuzzy Sets","volume":"1","author":"Atanassov","year":"2022","journal-title":"J. Comput. Cogn. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"6899","DOI":"10.1016\/j.eswa.2008.08.042","article-title":"Multicriteria fuzzy decision-making method based on a novel accuracy function under interval-valued intuitionistic fuzzy environment","volume":"36","author":"Ye","year":"2009","journal-title":"Expert Syst. Appl."},{"key":"ref_19","first-page":"113","article-title":"A new vector valued similarity measure for intuitionistic fuzzy sets based on OWA operators","volume":"16","author":"Fei","year":"2019","journal-title":"Iran. J. Fuzzy Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"11671","DOI":"10.1016\/j.eswa.2011.03.048","article-title":"Gray relational analysis method for intuitionistic fuzzy multiple attribute decision making","volume":"38","author":"Wei","year":"2011","journal-title":"Expert Syst. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1016\/j.eswa.2018.08.046","article-title":"A new similarity\/distance measure between intuitionistic fuzzy sets based on the transformed isosceles triangles and its applications to pattern recognition","volume":"116","author":"Jiang","year":"2019","journal-title":"Expert Syst. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/int.20386","article-title":"Generalized aggregation operators for intuitionistic fuzzy sets","volume":"25","author":"Zhao","year":"2010","journal-title":"Int. J. Intell. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"118832","DOI":"10.1016\/j.eswa.2022.118832","article-title":"Intuitionistic fuzzy geometric aggregation operators in the framework of Aczel-Alsina triangular norms and their application to multiple attribute decision making","volume":"212","author":"Senapati","year":"2023","journal-title":"Expert Syst. Appl."},{"key":"ref_24","first-page":"529","article-title":"Hesitant fuzzy sets","volume":"25","author":"Torra","year":"2010","journal-title":"Int. J. Intell. Syst."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Torra, V., and Narukawa, Y. (August, January 20\u201324). On hesitant fuzzy sets and decision. Proceedings of the 2009 IEEE International Conference on Fuzzy Systems, Jeju Island, Republic of Korea.","DOI":"10.1109\/FUZZY.2009.5276884"},{"key":"ref_26","unstructured":"Mahmood, T., Ullah, K., Jan, N., Deli, I., and Khan, Q. (2017). Some aggregation operators for bipolar-valued hesitant fuzzy information based on Einstein operational laws. J. Eng. Appl. Sci., 36."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Mo\u010dko\u0159, J., and H\u00fdnar, D. (2021). On unification of methods in theories of fuzzy sets, hesitant fuzzy set, fuzzy soft sets and intuitionistic fuzzy sets. Mathematics, 9.","DOI":"10.3390\/math9040447"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"118276","DOI":"10.1016\/j.eswa.2022.118276","article-title":"Ranked hesitant fuzzy sets for multi-criteria multi-agent decisions","volume":"209","author":"Alcantud","year":"2022","journal-title":"Expert Syst. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"3057","DOI":"10.1007\/s40815-022-01250-x","article-title":"Research on Green Supplier Selection Based on Hesitant Fuzzy Set and Extended LINMAP Method","volume":"24","author":"Zhang","year":"2022","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/s10726-011-9261-7","article-title":"Some hesitant fuzzy aggregation operators with their application in group decision making","volume":"22","author":"Xia","year":"2013","journal-title":"Group Decis. Negot."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"14005","DOI":"10.1007\/s00500-020-04776-0","article-title":"Generalized hesitant fuzzy rough sets (GHFRS) and their application in risk analysis","volume":"24","author":"Shaheen","year":"2020","journal-title":"Soft Comput."},{"key":"ref_32","first-page":"1","article-title":"Hesitant fuzzy multi-attribute decision making based on TOPSIS with entropy-weighted method","volume":"9","author":"Sun","year":"2015","journal-title":"Manag. Sci. Eng."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1007\/s10700-021-09366-9","article-title":"Multiple attribute decision-making method based on projection model for dual hesitant fuzzy set","volume":"21","author":"Ni","year":"2022","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_34","first-page":"3051","article-title":"Generalized power average operator of hesitant fuzzy numbers and its application in multiple attribute decision making","volume":"9","author":"Liu","year":"2013","journal-title":"J. Comput. Inf. Syst."},{"key":"ref_35","first-page":"1165","article-title":"Power Aggregation Operators and Similarity Measures Based on Improved Intuitionistic Hesitant Fuzzy Sets and their Applications to Multiple Attribute Decision Making","volume":"126","author":"Mahmood","year":"2021","journal-title":"Comput. Model. Eng. Sci."},{"key":"ref_36","unstructured":"Zhao, K.Q. (1989). Set pair and set pair analysis-a new concept and systematic analysis method, National Conference on System Theory and Regional Planning."},{"key":"ref_37","unstructured":"Jiang, Y.L., Xu, C.F., Yao, Y., and Zhao, K.Q. (2004, January 26\u201329). Systems information in set pair analysis and its applications. Proceedings of the 2004 International Conference on Machine Learning and Cybernetics (IEEE Cat. No. 04EX826), Shanghai, China."},{"key":"ref_38","first-page":"1009","article-title":"Application of the set pair analysis theory in multiple attribute decision-making","volume":"6","author":"ChangJian","year":"2007","journal-title":"J. Mech. Strength"},{"key":"ref_39","first-page":"2222","article-title":"Set pair analysis method of containing target constraint mixed interval multi-attribute decision-making","volume":"Volume 226","author":"Zhang","year":"2012","journal-title":"Applied Mechanics and Materials"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Xie, Z., Zhang, F., Cheng, J., and Li, L. (2013, January 28\u201329). Fuzzy multi-attribute decision making methods based on improved set pair analysis. Proceedings of the 2013 Sixth International Symposium on Computational Intelligence and Design, Hangzhou, China.","DOI":"10.1109\/ISCID.2013.209"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1319","DOI":"10.1007\/s40314-016-0402-0","article-title":"TOPSIS method based on the connection number of set pair analysis under interval-valued intuitionistic fuzzy set environment","volume":"37","author":"Kumar","year":"2018","journal-title":"Comput. Appl. Math."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"2112","DOI":"10.1007\/s10489-017-1067-0","article-title":"Connection number of set pair analysis based TOPSIS method on intuitionistic fuzzy sets and their application to decision making","volume":"48","author":"Kumar","year":"2018","journal-title":"Appl. Intell."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"7801","DOI":"10.1007\/s00500-019-04398-1","article-title":"Multiattribute decision making based on the binary connection number in set pair analysis under an interval-valued intuitionistic fuzzy set environment","volume":"24","author":"Shen","year":"2020","journal-title":"Soft Comput."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"432","DOI":"10.1016\/j.sepro.2011.08.064","article-title":"Dynamic stochastic multi-criteria decision-making method based on cumulative prospect theory and set pair analysis","volume":"1","author":"Hu","year":"2011","journal-title":"Syst. Eng. Procedia"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"9172","DOI":"10.1109\/ACCESS.2019.2963484","article-title":"Distance Measures for Multiple-Attributes Decision-Making Based on Connection Numbers of Set Pair Analysis with Dual Hesitant Fuzzy Sets","volume":"8","author":"Liu","year":"2020","journal-title":"IEEE Access"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Wang, C.-N., Yang, F.-C., Vo, N.T.M., and Nguyen, V.T.T. (2022). Wireless communications for data security: Efficiency assessment of cybersecurity industry\u2014A promising application for UAVs. Drones, 6.","DOI":"10.3390\/drones6110363"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Dang, T.-T., Nguyen, N.-A., Nguyen, V.-T., and Dang, L.-T. (2022). A two-stage multi-criteria supplier selection model for sustainable automotive supply chain under uncertainty. Axioms, 11.","DOI":"10.3390\/axioms11050228"},{"key":"ref_48","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_49","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1016\/j.ins.2013.01.002","article-title":"Hesitant fuzzy power aggregation operators and their application to multiple attribute group decision making","volume":"234","author":"Zhang","year":"2013","journal-title":"Inf. Sci."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"724","DOI":"10.1109\/3468.983429","article-title":"The power average operator","volume":"31","author":"Yager","year":"2001","journal-title":"IEEE Trans. Syst. Man Cybern.-Part A Syst. Hum."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Ahmmad, J., Mahmood, T., Mehmood, N., Urawong, K., and Chinram, R. (2022). Intuitionistic Fuzzy Rough Aczel-Alsina Average Aggregation Operators and Their Applications in Medical Diagnoses. Symmetry, 14.","DOI":"10.3390\/sym14122537"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Mahmood, T., Ali, Z., Baupradist, S., and Chinram, R. (2022). TOPSIS Method Based on Hamacher Choquet-Integral Aggregation Operators for Atanassov-Intuitionistic Fuzzy Sets and Their Applications in Decision-Making. Axioms, 11.","DOI":"10.3390\/axioms11120715"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"456","DOI":"10.1007\/s13198-021-01468-2","article-title":"Efficient road traffic anti-collision warning system based on fuzzy nonlinear programming","volume":"13","author":"Peng","year":"2022","journal-title":"Int. J. Syst. Assur. Eng. Manag."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"5961","DOI":"10.32604\/cmc.2022.029484","article-title":"Optimum Design for the Magnification Mechanisms Employing Fuzzy Logic-ANFIS","volume":"73","author":"Huynh","year":"2022","journal-title":"CMC-COMPUTERS MATERIALS CONTINUA"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"3389","DOI":"10.1007\/s00542-020-05132-w","article-title":"Optimizing compliant gripper mechanism design by employing an effective bi-algorithm: Fuzzy logic and ANFIS","volume":"27","author":"Nguyen","year":"2021","journal-title":"Microsyst. Technol."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Patel, U.D., and Radenovi\u0107, S. (2022). An Application to Nonlinear Fractional Differential Equation via \u03b1-\u0393 F-Fuzzy Contractive Mappings in a Fuzzy Metric Space. Mathematics, 10.","DOI":"10.3390\/math10162831"},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Radenovic, S., Ali, W., Shaheen, T., ul Haq, I., Akram, F., and Toor, H. (2022). Multiple Attribute Decision-Making Based on Bonferroni Mean Operators under Square Root Fuzzy Set Environment. J. Comput. Cogn. Eng.","DOI":"10.47852\/bonviewJCCE2202366"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"5172679","DOI":"10.1155\/2022\/5172679","article-title":"A Novel SIR Approach to Closeness Coefficient-Based MAGDM Problems Using Pythagorean Fuzzy Aczel\u2013Alsina Aggregation Operators for Investment Policy","volume":"2022","author":"Shaheen","year":"2022","journal-title":"Discret. Dyn. Nat. Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/3\/778\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:00:59Z","timestamp":1760122859000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/3\/778"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,22]]},"references-count":58,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["sym15030778"],"URL":"https:\/\/doi.org\/10.3390\/sym15030778","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,22]]}}}