{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T10:15:54Z","timestamp":1783937754825,"version":"3.55.0"},"reference-count":45,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,5,7]],"date-time":"2024-05-07T00:00:00Z","timestamp":1715040000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Initially, fuzzy sets and intuitionistic fuzzy sets were used to address real-world problems with imprecise data. Eventually, the notion of the hesitant fuzzy set was formulated to handle decision makers\u2019 reluctance to accept asymmetric information. However, in certain scenarios, asymmetric information is gathered in terms of a possible range of acceptance and nonacceptance by players rather than specific values. Furthermore, decision makers exhibit some hesitancy regarding this information. In such a situation, all the aforementioned expansions of fuzzy sets are unable to accurately represent the scenario. The purpose of this article is to present asymmetric information situations in which the range of choices takes into account the hesitancy of players in accepting or not accepting information. To illustrate these problems, we develop matrix games that consider the payoffs of interval-valued intuitionistic hesitant fuzzy elements (IIHFEs). Dealing with these types of fuzzy programming problems requires a significant amount of effort. To solve these matrix games, we formulate two interval-valued intuitionistic hesitant fuzzy programming problems. Preserving the hesitant nature of the payoffs to determine the optimal strategies, these two problems are transformed into two nonlinear programming problems. This transformation involves using mathematical operations for IIHFEs. Here, we construct a novel aggregation operator of IIHFEs, viz., min-max operators of IIHFEs. This operator is suitable for applying the developed methodology. The cogency and applicability of the proposed methodology are verified through a numerical example based on the situation of conflict between hackers and defenders to prevent damage to cybersecurity. To validate the superiority of the proposed model along with the computed results, we provide comparisons with the existing models.<\/jats:p>","DOI":"10.3390\/sym16050573","type":"journal-article","created":{"date-parts":[[2024,5,7]],"date-time":"2024-05-07T09:03:38Z","timestamp":1715072618000},"page":"573","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["A Nonlinear Programming Approach to Solving Interval-Valued Intuitionistic Hesitant Noncooperative Fuzzy Matrix Games"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9085-6184","authenticated-orcid":false,"given":"Shuvasree","family":"Karmakar","sequence":"first","affiliation":[{"name":"Department of Mathematics, Kazi Nazrul University, Asansol 713340, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4746-5369","authenticated-orcid":false,"given":"Mijanur Rahaman","family":"Seikh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Kazi Nazrul University, Asansol 713340, India"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,7]]},"reference":[{"key":"ref_1","unstructured":"Neumann, J.V., and Morgenstern, O. (1947). Theory of Games and Economic Behaviour, Princeton University Press."},{"key":"ref_2","unstructured":"Dubois, D., and Prade, H. (1980). Fuzzy Sets and Systems: Theory and Applications, Academic Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1016\/j.ins.2021.05.019","article-title":"The two-person and zero-sum matrix game with probabilistic linguistic information","volume":"570","author":"Mi","year":"2021","journal-title":"Inf. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1016\/j.ejor.2015.05.011","article-title":"On solving matrix games with pay-offs of triangular fuzzy numbers: Certain observations and generalizations","volume":"246","author":"Chandra","year":"2015","journal-title":"Eur. J. Oper. Res."},{"key":"ref_5","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_6","doi-asserted-by":"crossref","first-page":"1112","DOI":"10.1109\/TFUZZ.2010.2065812","article-title":"Mathematical-programming approach to matrix games with payoffs represented by Atanassov\u2019s interval-valued intuitionistic fuzzy sets","volume":"18","author":"Li","year":"2010","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Li, D.F. (2014). Decision and Game Theory in Management with Intuitionistic Fuzzy Sets, Springer.","DOI":"10.1007\/978-3-642-40712-3"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1080\/03081079.2017.1413100","article-title":"Interval-valued intuitionistic fuzzy matrix games based on archimedean t-conorm and t-norm","volume":"47","author":"Xia","year":"2018","journal-title":"Int. J. Gen. Syst."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Brikaa, M.G., Zheng, Z., and Ammar, E.S. (2020). Resolving indeterminacy approach to solve multi-criteria zero-sum matrix games with intuitionistic fuzzy goals. Mathematics, 8.","DOI":"10.3390\/math8030305"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1109\/TFUZZ.2017.2651103","article-title":"Ambika methods for solving matrix games with Atanassov\u2019s intuitionistic fuzzy pay-offs","volume":"26","author":"Verma","year":"2018","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"5127","DOI":"10.1007\/s10462-021-10014-2","article-title":"Matrix games with linguistic intuitionistic fuzzy payoffs: Basic results and solution methods","volume":"54","author":"Verma","year":"2021","journal-title":"Artif. Intell. Rev."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"123398","DOI":"10.1016\/j.eswa.2024.123398","article-title":"Type-2 interval-valued intuitionistic fuzzy matrix game and application to energy vehicle industry development","volume":"249","author":"Dong","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_13","first-page":"155","article-title":"A novel defuzzification approach of type-2 fuzzy variable to solving matrix games: An application to plastic ban problem","volume":"18","author":"Seikh","year":"2021","journal-title":"Iran. J. Fuzzy Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3077","DOI":"10.1007\/s40747-021-00475-x","article-title":"A q rung orthopair fuzzy non cooperative game method for competitive strategy group decision making problems based on a hybrid dynamic experts\u2019 weight deter-mining model","volume":"7","author":"Yang","year":"2021","journal-title":"Complex Intell. Syst."},{"key":"ref_15","first-page":"5","article-title":"Analytical Hierarchy Process in modeling and solving matrix games in neutrosophic environment and its application in military problems","volume":"19","author":"Bigdeli","year":"2023","journal-title":"Mil. Sci. Tactics"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"921","DOI":"10.1007\/s00500-021-06559-7","article-title":"Solution of matrix games with payoffs of single-valued trape-zoidal neutrosophic numbers","volume":"26","author":"Seikh","year":"2022","journal-title":"Soft Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1007\/s41066-022-00327-0","article-title":"Interval neutrosophic matrix game-based approach to counter cybersecurity issue","volume":"8","author":"Seikh","year":"2023","journal-title":"Granul. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Khalil, A.M., Cao, D., Azzam, A., Smarandache, F., and Alharbi, W.R. (2020). Combination of the single-valued neutrosophic fuzzy set and the soft set with applications in decision-making. Symmetry, 12.","DOI":"10.3390\/sym12081361"},{"key":"ref_19","first-page":"529","article-title":"Hesitant fuzzy sets","volume":"25","author":"Torra","year":"2010","journal-title":"Int. J. Intell. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/j.ijar.2010.09.002","article-title":"Hesitant fuzzy information aggregation in decision making","volume":"52","author":"Xia","year":"2011","journal-title":"Int. J. Approx. Reason."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2128","DOI":"10.1016\/j.ins.2011.01.028","article-title":"Distance and similarity measures for hesitant fuzzy sets","volume":"181","author":"Xu","year":"2011","journal-title":"Inform. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1007\/s40815-018-00599-2","article-title":"Weighted interval-valued hesitant fuzzy sets and its application in group decision making","volume":"21","author":"Zeng","year":"2019","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10489-022-03442-2","article-title":"Linguistic Pythagorean hesitant fuzzy matrix game and its application in multi-criteria decision making","volume":"53","author":"Jana","year":"2023","journal-title":"Appl. Intell."},{"key":"ref_24","first-page":"25","article-title":"Solving matrix games with hesitant fuzzy pay-offs","volume":"17","author":"Seikh","year":"2020","journal-title":"Iran. J. Fuzzy Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3854","DOI":"10.15837\/ijccc.2020.3.3854","article-title":"Matrix game with payoffs represented by triangular dual hesitant fuzzy numbers","volume":"15","author":"Yang","year":"2020","journal-title":"Int. J. Comput. Commun. Control"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"5085","DOI":"10.3233\/JIFS-222466","article-title":"Matrix games involving interval-valued hesitant fuzzy linguistic sets and its application to electric vehicles","volume":"44","author":"Naqvi","year":"2023","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_27","first-page":"670285","article-title":"Interval-valued intuitionistic hesitant fuzzy aggregation operators and their application in group decision-making","volume":"2013","author":"Zhang","year":"2013","journal-title":"J. Appl. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"62","DOI":"10.13189\/ms.2014.020202","article-title":"New operations over interval-valued intuitionistic hesi-tant fuzzy set","volume":"2","author":"Broumi","year":"2014","journal-title":"Math. Stat."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.ejor.2015.06.047","article-title":"Interval-valued intuitionistic hesitant fuzzy choquet integral based TOPSIS method for multi-criteria group decision making","volume":"248","author":"Joshi","year":"2016","journal-title":"Eur. J. Oper. Res."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1909","DOI":"10.3233\/JIFS-16092","article-title":"A group decision making approach in inter-val-valued intuitionistic fuzzy environment with confidence levels","volume":"31","author":"Yuan","year":"2016","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1007\/s41066-018-0077-6","article-title":"Entropy of interval-valued intuitionistic hesitant fuzzy set and its application to group decision making problems","volume":"3","author":"Joshi","year":"2018","journal-title":"Granul. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.eswa.2018.12.015","article-title":"Interval-valued intuitionistic hesitant fuzzy entropy based VIKOR method for industrial robots selection","volume":"121","author":"Narayanamoorthy","year":"2019","journal-title":"Expert Syst. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"5229","DOI":"10.3233\/JIFS-190205","article-title":"Multiple attribute decision making based on probabilistic inter-val-valued intuitionistic hesitant fuzzy set and extended TOPSIS method","volume":"37","author":"De","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"110213","DOI":"10.1016\/j.asoc.2023.110213","article-title":"A data-driven group emergency decision-making method based on interval-valued intuitionistic hesitant fuzzy sets and its application in COVID-19 pandemic","volume":"139","author":"Du","year":"2023","journal-title":"Appl. Soft Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1007\/s10100-019-00638-9","article-title":"Hesitant interval-valued intuitionistic fuzzy-linguistic term set approach in prisoners\u2019 dilemma game theory using TOPSIS: A case study on hu-man-trafficking","volume":"28","author":"Bhaumik","year":"2020","journal-title":"Cent. Eur. J. Oper. Res."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1651","DOI":"10.1109\/TFUZZ.2017.2740201","article-title":"Measures of probabilistic interval-valued intuitionistic hesitant fuzzy sets and the application in reducing excessive medical examination","volume":"26","author":"Zhai","year":"2018","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"157","DOI":"10.15388\/Informatica.2018.162","article-title":"An approach to decision making with interval-valued in-tuitionistic hesitant fuzzy information based on the 2-additive shapley function","volume":"29","author":"Zhang","year":"2018","journal-title":"Informatica"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"25706","DOI":"10.1109\/ACCESS.2020.2971102","article-title":"Strategic mapping of youth unemployment with interval valued intuitionistic hesitant fuzzy DEMATEL based on 2-tuple linguistic values","volume":"8","author":"Zhang","year":"2020","journal-title":"IEEE Access"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1007\/s41066-019-00191-5","article-title":"Intuitionistic interval-valued hesitant fuzzy matrix games with a new aggregation operator for solving management problem","volume":"6","author":"Bhaumik","year":"2021","journal-title":"Granul. Comput."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/s41066-016-0023-4","article-title":"An overview of interval-valued intuitionistic fuzzy information aggrega-tions and applications","volume":"2","author":"Xu","year":"2017","journal-title":"Granul. Comput."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"143","DOI":"10.3233\/IFS-130985","article-title":"Properties of interval-valued hesitant fuzzy sets","volume":"27","author":"Chen","year":"2014","journal-title":"J. Telligent Fuzzy Syst."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1179","DOI":"10.1109\/TFUZZ.2006.890678","article-title":"Intuitionistic fuzzy aggregation operator","volume":"15","author":"Xu","year":"2007","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/0377-2217(90)90375-L","article-title":"Multiobjective programming in optimization of the interval objective function","volume":"48","author":"Ishibuchi","year":"1990","journal-title":"Eur. J. Oper. Res."},{"key":"ref_44","unstructured":"(2023, June 27). National Crime Records Bureau, Available online: https:\/\/ncrb.gov.in\/en\/Crime-in-India-2020."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1227","DOI":"10.1007\/s10489-020-01759-4","article-title":"Solving matrix games based on Ambika method with hesitant fuzzy information and its application in the counter-terrorism issue","volume":"51","author":"Xue","year":"2021","journal-title":"Appl. Intell."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/5\/573\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:40:52Z","timestamp":1760107252000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/5\/573"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,7]]},"references-count":45,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["sym16050573"],"URL":"https:\/\/doi.org\/10.3390\/sym16050573","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,7]]}}}