{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T23:28:59Z","timestamp":1780442939161,"version":"3.54.1"},"reference-count":57,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,3]],"date-time":"2025-01-03T00:00:00Z","timestamp":1735862400000},"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>The symmetrical linear Diophantine fuzzy Hamacher aggregation operators play a fundamental role in many decision-making applications. The selection of a cyber security system is of paramount importance for maintaining digital assets. It necessitates a comprehensive review of threat landscapes, vulnerability assessments, and the specific needs of the organization in order to ensure the implementation of effective security measures. Smart grid (SG) technology uses modern communication and monitoring technologies to enhance the management and regulation of electricity production and transmission. However, greater dependence on technology and connection creates new vulnerabilities, exposing SG communication networks to large-scale attacks. Unlike previous surveys, which often give broad overviews of SG design, our research goes a step further, giving a full architectural layout that includes major SG components and communication linkages. This in-depth review improves comprehension of possible cyber threats and allows SGs to analyze cyber risks more systematically. To determine the best cybersecurity strategies, this study introduces a multi-criteria group decision-making (MCGDM) approach using the linear Diophantine fuzzy Hamacher prioritized aggregation operator (LDFHPAO). In real-world applications, aggregation operators (AOs) are essential for information fusion. This research presents innovative prioritized AOs designed to address MCGDM problems in uncertain environments. We developed the LDF Hamacher prioritized weighted average (LDFHPWA) and LDF Hamacher prioritized weighted geometric (LDFHPWG) operators, which address the shortcomings of traditional operators and provide a more robust modeling approach for MCGDM challenges. This study also outlines key characteristics of these new prioritized AOs. An MCGDM approach incorporating these operators is proposed and demonstrated to be effective through an example that compares and selects the optimal cybersecurity.<\/jats:p>","DOI":"10.3390\/sym17010070","type":"journal-article","created":{"date-parts":[[2025,1,3]],"date-time":"2025-01-03T11:50:10Z","timestamp":1735905010000},"page":"70","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Prioritized Decision Support System for Cybersecurity Selection Based on Extended Symmetrical Linear Diophantine Fuzzy Hamacher Aggregation Operators"],"prefix":"10.3390","volume":"17","author":[{"given":"Muhammad Zeeshan","family":"Hanif","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Riphah International University, I-14, Islamabad 44000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5362-9916","authenticated-orcid":false,"given":"Naveed","family":"Yaqoob","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Riphah International University, I-14, Islamabad 44000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"662","DOI":"10.3390\/jcp3040031","article-title":"Cyberattacks in smart grids: Challenges and solving the multi-criteria decision-making for cybersecurity options, including ones that incorporate artificial intelligence, using an analytical hierarchy process","volume":"3","author":"Bouramdane","year":"2023","journal-title":"J. Cybersecur. Priv."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"495","DOI":"10.5370\/JEET.2017.12.2.495","article-title":"Methodology of cyber security assessment in the smart grid","volume":"12","author":"Woo","year":"2017","journal-title":"J. Electr. Eng. Technol."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Ansari, M., and Asrari, A. (2020, January 17\u201320). Reaction to detected cyberattacks in smart distribution systems. Proceedings of the 2020 IEEE Power & Energy Society Innovative Smart Grid Technologies Conference (ISGT), Washington, DC, USA.","DOI":"10.1109\/ISGT45199.2020.9087771"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Alkuwari, A.N., Al-Kuwari, S., and Qaraqe, M. (2022, January 20\u201322). Anomaly detection in smart grids: A survey from cybersecurity perspective. Proceedings of the 2022 3rd International Conference on Smart Grid and Renewable Energy (SGRE), Doha, Qatar.","DOI":"10.1109\/SGRE53517.2022.9774221"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"57542","DOI":"10.1109\/ACCESS.2021.3071263","article-title":"A review of rule learning-based intrusion detection systems and their prospects in smart grids","volume":"9","author":"Liu","year":"2021","journal-title":"IEEE Access"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/s11761-022-00349-1","article-title":"Detection of cyber attacks in smart grids using SVM-boosted machine learning models","volume":"16","author":"Alwageed","year":"2022","journal-title":"Serv. Oriented Comput. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Zhai, F., Yang, T., Zhao, B., and Chen, H. (2022). Privacy-preserving outsourcing algorithms for multidimensional data encryption in smart grids. Sensors, 22.","DOI":"10.3390\/s22124365"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Miller, M.Z., Griendling, K., and Mavris, D.N. (2012, January 16\u201319). Exploring human factors effects in the smart grid system of systems demand response. Proceedings of the 2012 7th International Conference on System of Systems Engineering (SoSE), Genova, Italy.","DOI":"10.1109\/SYSoSE.2012.6384176"},{"key":"ref_9","unstructured":"Fredman, D. (2018). A Human Side of the Smart Grid: Behavior-Based Energy Efficiency from Renters Using Real-Time Feedback and Competitive Performance-Based Incentives, The University of Vermont and State Agricultural College."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Monta\u00f1ez, R., Golob, E., and Xu, S. (2020). Human cognition through the lens of social engineering cyberattacks. Front. Psychol., 11.","DOI":"10.3389\/fpsyg.2020.01755"},{"key":"ref_11","unstructured":"Ray, J.R. (2022, December 02). Training Programs to Increase Cybersecurity Awareness and Compliance in Non-Profits. Available online: https:\/\/scholarsbank.uoregon.edu\/items\/8a6c47a2-8e7d-4bc6-92df-719fb3ede12f."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Loi, M., and Christen, M. (2020). Ethical frameworks for cybersecurity. The Ethics of Cybersecurity, Springer.","DOI":"10.1007\/978-3-030-29053-5"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1109\/TDSC.2015.2446492","article-title":"Secure and private data aggregation for energy consumption scheduling in smart grids","volume":"14","author":"Rahman","year":"2015","journal-title":"IEEE Trans. Dependable Secur. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Albasrawi, M.N., Jarus, N., Joshi, K.A., and Sarvestani, S.S. (2014, January 21\u201325). Analysis of reliability and resilience for smart grids. Proceedings of the 2014 IEEE 38th Annual Computer Software and Applications Conference, Vasteras, Sweden.","DOI":"10.1109\/COMPSAC.2014.75"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Kanca, A.M., and Sa\u011firo\u011flu, \u015e. (2021, January 2\u20133). Sharing cyber threat intelligence and collaboration. Proceedings of the 2021 International Conference on Information Security and Cryptology (ISCTURKEY), Ankara, Turkey.","DOI":"10.1109\/ISCTURKEY53027.2021.9654328"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Majumder, M., and Majumder, M. (2015). Multi criteria decision making. Impact of Urbanization on Water Shortage in Face of Climatic Aberrations, Springer.","DOI":"10.1007\/978-981-4560-73-3"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Triantaphyllou, E., and Triantaphyllou, E. (2000). Multi-Criteria Decision Making Methods, Springer.","DOI":"10.1007\/978-1-4757-3157-6"},{"key":"ref_18","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_19","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_20","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_21","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_22","doi-asserted-by":"crossref","first-page":"1700","DOI":"10.2991\/ijcis.d.210531.002","article-title":"Improved Knowledge Measures for q-Rung Orthopair Fuzzy Sets","volume":"14","author":"Khan","year":"2021","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"628","DOI":"10.1002\/int.22313","article-title":"Knowledge measure for the q-rung orthopair fuzzy sets","volume":"36","author":"Khan","year":"2021","journal-title":"Int. J. Intell. Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"93497","DOI":"10.1109\/ACCESS.2021.3091179","article-title":"q-Rung orthopair fuzzy modified dissimilarity measure based robust VIKOR method and its applications in mass vaccination campaigns in the context of COVID-19","volume":"9","author":"Khan","year":"2021","journal-title":"IEEE Access"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"5417","DOI":"10.3233\/JIFS-190550","article-title":"Linear Diophantine fuzzy set and its applications towards multi-attribute decision-making problems","volume":"37","author":"Riaz","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_26","first-page":"5548033","article-title":"Linear Diophantine fuzzy Einstein aggregation operators for multi-criteria decision-making problems","volume":"1","author":"Iampan","year":"2021","journal-title":"J. Math."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Ayub, S., Shabir, M., Riaz, M., Mahmood, W., Bozanic, D., and Marinkovic, D. (2022). Linear diophantine fuzzy rough sets: A new rough set approach with decision making. Symmetry, 14.","DOI":"10.3390\/sym14030525"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3097","DOI":"10.3233\/JIFS-223844","article-title":"Linear diophantine multi-fuzzy aggregation operators and its application in digital transformation","volume":"45","author":"Jeevitha","year":"2023","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"29859","DOI":"10.1109\/ACCESS.2022.3151684","article-title":"Some linear Diophantine fuzzy similarity measures and their application in decision making problem","volume":"10","author":"Mohammad","year":"2022","journal-title":"IEEE Access"},{"key":"ref_30","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_31","first-page":"276","article-title":"In Progress in Cybernatics and Systems Research","volume":"3","author":"Hamacher","year":"1978","journal-title":"Uber Logische Verknunpfungenn Unssharfer Aussagen Und Deren Zugenhorige Bewertungsfunktione"},{"key":"ref_32","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_33","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1016\/j.asoc.2014.10.007","article-title":"Hesitant fuzzy Hamacher aggregation operators for multicriteria decision making","volume":"26","author":"Tan","year":"2015","journal-title":"Appl. Soft Comput."},{"key":"ref_34","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_35","first-page":"189","article-title":"Pythagorean fuzzy Hamacher aggregation operators and their application to multiple attribute decision making","volume":"21","author":"Wu","year":"2017","journal-title":"Int. J. Knowl. Based Intell. Eng. Syst."},{"key":"ref_36","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_37","doi-asserted-by":"crossref","first-page":"32111","DOI":"10.1109\/ACCESS.2024.3367117","article-title":"Fuzzy Decision Support Systems for Selection of NEA Detection Technologies Under Non-Linear Diophantine Fuzzy Hamacher Aggregation Information","volume":"12","author":"Shams","year":"2024","journal-title":"IEEE Access"},{"key":"ref_38","first-page":"315","article-title":"An approach to intuitionistic fuzzy multi-attribute group decision making based on hybrid Einstein aggregation operators","volume":"9","author":"Zhang","year":"2016","journal-title":"Int. J. Intell. Inf. Database Syst."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"3464","DOI":"10.1002\/int.22423","article-title":"A novel approach to MADM problems using Fermatean fuzzy Hamacher aggregation operators","volume":"36","author":"Hadi","year":"2021","journal-title":"Int. J. Intell. Syst."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1007\/s41066-019-00209-y","article-title":"Intuitionistic fuzzy Dombi aggregation operators and their application to multiple attribute decision-making","volume":"6","author":"Seikh","year":"2021","journal-title":"Granul. Comput."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"3000","DOI":"10.1002\/int.22183","article-title":"Pythagorean Dombi fuzzy aggregation operators with application in multicriteria decision-making","volume":"34","author":"Akram","year":"2019","journal-title":"Int. J. Intell. Syst."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"851","DOI":"10.3233\/JIFS-191763","article-title":"Fermatean fuzzy TOPSIS method with Dombi aggregation operators and its application in multi-criteria decision making","volume":"39","author":"Aydemir","year":"2020","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/j.ijar.2007.08.009","article-title":"Prioritized aggregation operators","volume":"48","author":"Yager","year":"2008","journal-title":"Int. J. Approx. Reason."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3846\/20294913.2012.762951","article-title":"Intuitionistic fuzzy prioritized operators and their application in multi-criteria group decision making","volume":"19","author":"Yu","year":"2013","journal-title":"Technol. Econ. Dev. Econ."},{"key":"ref_45","first-page":"3095","article-title":"Intuitionistic fuzzy Einstein prioritized weighted average operators and their application to multiple attribute group decision making","volume":"9","author":"Verma","year":"2015","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_46","first-page":"466","article-title":"Prioritized averaging\/geometric aggregation operators under the intuitionistic fuzzy soft set environment","volume":"25","author":"Arora","year":"2018","journal-title":"Sci. Iran"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/s41066-018-0093-6","article-title":"Pythagorean fuzzy prioritized aggregation operators and their application to multi-attribute group decision making","volume":"4","author":"Khan","year":"2019","journal-title":"Granul. Comput."},{"key":"ref_48","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_49","doi-asserted-by":"crossref","first-page":"13897","DOI":"10.1007\/s00500-021-06308-w","article-title":"A novel approach to MADM problems using Fermatean fuzzy Hamacher prioritized aggregation operators","volume":"25","author":"Jan","year":"2021","journal-title":"Soft Comput."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"e12712","DOI":"10.1111\/exsy.12712","article-title":"A hybrid decision-making framework under complex spherical fuzzy prioritized weighted aggregation operators","volume":"38","author":"Akram","year":"2021","journal-title":"Expert Syst."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Riaz, M., Farid, H.M.A., Aslam, M., Pamucar, D., and Bozani\u0107, D. (2021). Novel approach for third-party reverse logistic provider selection process under linear Diophantine fuzzy prioritized aggregation operators. Symmetry, 13.","DOI":"10.3390\/sym13071152"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"11201","DOI":"10.3934\/math.2022627","article-title":"Sustainable thermal power equipment supplier selection by Einstein prioritized linear Diophantine fuzzy aggregation operators","volume":"7","author":"Farid","year":"2022","journal-title":"Aims Math."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"e13104","DOI":"10.1111\/exsy.13104","article-title":"Novel distance and entropy definitions for linear Diophantine fuzzy sets and an extension of TOPSIS (LDF-TOPSIS)","volume":"40","year":"2023","journal-title":"Expert Syst."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"9726","DOI":"10.1002\/int.23003","article-title":"Extension of GRA method for multiattribute group decision making problem under linguistic Pythagorean fuzzy setting with incomplete weight information","volume":"37","author":"Khan","year":"2022","journal-title":"Int. J. Intell. Syst."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"5839410","DOI":"10.1155\/2023\/5839410","article-title":"Decision-Making Based on Spherical Linear Diophantine Fuzzy Rough Aggregation Operators and EDAS Method","volume":"2023","author":"Qiyas","year":"2023","journal-title":"J. Math."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Karuppiah, K., Sankaranarayanan, B., Ali, S.M., and Marimuthu, U. (2024). Integrated approach for offshore wind turbine site selection: Implications for sustainability in power supply chain. Energies, 17.","DOI":"10.3390\/en17143419"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"109085","DOI":"10.1016\/j.engappai.2024.109085","article-title":"Intuitionistic fuzzy Sugeno-Weber decision framework for sustainable digital security assessment","volume":"137","author":"Hussain","year":"2024","journal-title":"Eng. Appl. Artif. Intell."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/70\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:22:40Z","timestamp":1759918960000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/70"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,3]]},"references-count":57,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["sym17010070"],"URL":"https:\/\/doi.org\/10.3390\/sym17010070","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,3]]}}}