{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:25:33Z","timestamp":1760149533387,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,21]],"date-time":"2023-08-21T00:00:00Z","timestamp":1692576000000},"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>A significant addition to fuzzy set theory for expressing uncertain data is an n,m-th power root fuzzy set. Compared to the nth power root, Fermatean, Pythagorean, and intuitionistic fuzzy sets, n,m-th power root fuzzy sets can cover more uncertain situations due to their greater range of displayed membership grades. When discussing the symmetry between two or more objects, the innovative concept of an n,m-th power root fuzzy set over dual universes is more flexible than the current notion of an intuitionistic fuzzy set, a Pythagorean fuzzy set, and a nth power root fuzzy set. In this study, we demonstrate a number of additional operations on n,m-th power root fuzzy sets along with a number of their special aspects. Additionally, to deal with choice information, we create a novel weighted aggregated operator called the n,m-th power root fuzzy weighted power average (FWPAmn) across n,m-th power root fuzzy sets and demonstrate some of its fundamental features. To rank n,m-th power root fuzzy sets, we also define the score and accuracy functions. Moreover, we use this operator to identify the countries with the best standards of living and show how we can select the best option by contrasting aggregate results using score values. Finally, we contrast the results of the FWPAmn\u00a0operator with the square-root fuzzy weighted power average (SR-FWPA), the nth power root fuzzy weighted power average (nPR-FWPA), the Fermatean fuzzy weighted power average (FFWPA), and the n,m-rung orthopair fuzzy weighted power average (n,m-ROFWPA) operators.<\/jats:p>","DOI":"10.3390\/sym15081617","type":"journal-article","created":{"date-parts":[[2023,8,22]],"date-time":"2023-08-22T00:46:22Z","timestamp":1692665182000},"page":"1617","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Novel Approach to Multi-Criteria Decision-Making Based on the n,mPR-Fuzzy Weighted Power Average Operator"],"prefix":"10.3390","volume":"15","author":[{"given":"Tareq","family":"Hamadneh","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al Zaytoonah University of Jordan, Amman 11733, Jordan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9417-2695","authenticated-orcid":false,"given":"Hariwan Z.","family":"Ibrahim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Education, University of Zakho, Zakho 42002, Iraq"}]},{"given":"Mayada","family":"Abualhomos","sequence":"additional","affiliation":[{"name":"Department Basic Science, Applied Science Private University Applied Science Research Center (ASRC), Amman 11733, Jordan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6416-8097","authenticated-orcid":false,"given":"Maha Mohammed","family":"Saeed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}]},{"given":"Gharib","family":"Gharib","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan"}]},{"given":"Maha","family":"Al Soudi","sequence":"additional","affiliation":[{"name":"Department of Basic Scientific Sciences, Applied Science Private University, Amman 11931, Jordan"}]},{"given":"Abdallah","family":"Al-Husban","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid 21110, Jordan"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,21]]},"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":"701","DOI":"10.37394\/23206.2022.21.82","article-title":"Multi-fuzzy rings","volume":"21","author":"Saadeh","year":"2022","journal-title":"Wseas Trans. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.knosys.2016.11.019","article-title":"Fuzzy time series forecasting based on optimal partitions of intervals and optimal weighting vectors","volume":"118","author":"Chen","year":"2017","journal-title":"Knowl.-Based Syst."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"v","DOI":"10.1142\/S0218488523020014","article-title":"Fuzzy optimization and algorithms in autonomous systems","volume":"31","author":"Er","year":"2023","journal-title":"Int. J. Uncertainty Fuzziness -Knowl.-Based Syst."},{"key":"ref_5","unstructured":"Vovan, T. (2023). Granular Computing, Springer."},{"key":"ref_6","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_7","doi-asserted-by":"crossref","unstructured":"Yager, R.R. (2013, January 24\u201328). Pythagorean fuzzy subsets. Proceedings of the 2013 Joint IFSAWorld Congress and NAFIPS Annual Meeting (IFSA\/NAFIPS), Edmonton, AB, Canada.","DOI":"10.1109\/IFSA-NAFIPS.2013.6608375"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1061","DOI":"10.1002\/int.21676","article-title":"Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets","volume":"29","author":"Zhang","year":"2014","journal-title":"Int. J. Intell. Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3495","DOI":"10.1016\/j.apm.2013.11.045","article-title":"An extended compromise ratio model with an application to reservoir flood control operation under an interval-valued intuitionistic fuzzy environment","volume":"38","author":"Hashemi","year":"2014","journal-title":"Appl. Math. Model."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3346","DOI":"10.1007\/s10489-018-1152-z","article-title":"Distance measures for connection number sets based on set pair analysis and its applications to decisionmaking process","volume":"48","author":"Garg","year":"2018","journal-title":"Appl. Intell."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.knosys.2014.04.014","article-title":"A complete pattern recognition approach under Atanassov\u2019s intuitionistic fuzzy sets","volume":"66","author":"Chu","year":"2014","journal-title":"Knowl.-Based Syst."},{"key":"ref_12","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_13","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1007\/s12652-019-01377-0","article-title":"Fermatean fuzzy sets","volume":"11","author":"Senapati","year":"2020","journal-title":"J. Ambient. Intell. Humaniz. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1222","DOI":"10.1109\/TFUZZ.2016.2604005","article-title":"Generalized orthopair fuzzy sets","volume":"25","author":"Yager","year":"2017","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_15","first-page":"3653225","article-title":"SR-fuzzy sets and their weighted aggregated operators in applications to decision-making","volume":"2022","author":"Ibrahim","year":"2022","journal-title":"J. Funct. Spaces"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"99562","DOI":"10.1109\/ACCESS.2022.3207184","article-title":"n,m-Rung orthopair fuzzy sets with applications to multicriteria decision making","volume":"10","author":"Ibrahim","year":"2022","journal-title":"IEEE Access"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"350","DOI":"10.5391\/IJFIS.2022.22.4.350","article-title":"nth power root fuzzy sets and its topology","volume":"22","author":"Ibrahim","year":"2022","journal-title":"Int. J. Fuzzy Log. Intell. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3045","DOI":"10.3233\/JIFS-210167","article-title":"A topological reduction for predicting of a lung cancer disease based on generalized rough sets","volume":"41","year":"2021","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1007\/s00500-021-06456-z","article-title":"Topological approach to generalized soft rough sets via near concepts","volume":"26","author":"Ali","year":"2022","journal-title":"Soft Comput."},{"key":"ref_20","first-page":"4723233","article-title":"Economic decision-making using rough topological structures","volume":"2023","author":"Taher","year":"2023","journal-title":"J. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3073","DOI":"10.32604\/iasc.2023.034234","article-title":"Approximations by ideal minimal structure with chemical application","volume":"36","author":"Hosny","year":"2023","journal-title":"Intell. Autom. Soft Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"12459","DOI":"10.3233\/JIFS-201706","article-title":"A new type of generalized picture fuzzy soft set and its application in decision making","volume":"40","author":"Lu","year":"2021","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"9082","DOI":"10.3934\/math.2023455","article-title":"Medical diagnosis for the problem of Chikungunya disease using soft rough sets","volume":"8","year":"2023","journal-title":"AIMS Math."},{"key":"ref_24","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_25","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_26","doi-asserted-by":"crossref","first-page":"1224","DOI":"10.1016\/j.knosys.2011.05.013","article-title":"Intuitionistic fuzzy ordered weighted distance operator","volume":"24","author":"Zeng","year":"2011","journal-title":"Knowl.-Based Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"958","DOI":"10.1109\/TFUZZ.2013.2278989","article-title":"Pythagorean membership grades in multicriteria decision making","volume":"22","author":"Yager","year":"2014","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_28","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. Informaticae"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1234","DOI":"10.1002\/int.21827","article-title":"A novel correlation coefficients between Pythagorean fuzzy sets and its applications to decisionmaking processes","volume":"31","author":"Garg","year":"2016","journal-title":"Int. J. Intell. Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1234","DOI":"10.1002\/int.21979","article-title":"Linguistic Pythagorean fuzzy sets and its applications in multi attribute decision making process","volume":"33","author":"Garg","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1080\/0952813X.2018.1467497","article-title":"Generalised pythagorean fuzzy geometric interactive aggregation operators using Einstein operations and their application to decision making","volume":"30","author":"Garg","year":"2018","journal-title":"J. Exp. Theor. Artif. Intell."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Shahzadi, G., Akram, M., and Al-Kenani, A.N. (2020). Decision making approach under pythagorean fuzzy Yager weighted operators. Mathematics, 8.","DOI":"10.3390\/math8010070"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Waseem, N., Akram, M., and Alcantud, J.C.R. (2019). Multiattribute decision-making based on m-polar fuzzy Hamacher aggregation operators. Symmetry, 11.","DOI":"10.3390\/sym11121498"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1016\/j.engappai.2019.05.012","article-title":"Fermatean fuzzy weighted averaging\/geometric operators and its application in multi-criteria decision-making methods","volume":"85","author":"Senapati","year":"2019","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1487724","DOI":"10.1155\/2023\/1487724","article-title":"Applications of nth power root fuzzy sets in multicriteria decision making","volume":"2023","author":"Ibrahim","year":"2023","journal-title":"J. Math."},{"key":"ref_36","first-page":"4189527","article-title":"Evaluating sustainable liveable city via multi-MCDM and hopfield neural network","volume":"2020","author":"Chen","year":"2020","journal-title":"Math. Probl. Eng."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Yadav, N., Yadav, A., Bansal, J., Deep, K., and Kim, J. (2019). Harmony Search and Nature Inspired Optimization Algorithms. Advances in Intelligent Systems and Computing, Springer.","DOI":"10.1007\/978-981-13-0761-4"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1504\/IJBSR.2016.073688","article-title":"A fuzzy MCDM approach for choosing a tourism destination in Portugal","volume":"10","author":"Filipe","year":"2016","journal-title":"Int. J. Bus. Syst. Res."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Wu, C.K., Wang, C.-N., and Le, T.K.T. (2022). Fuzzy multi criteria decision making model for agritourism location selection: A case study in vietnam. Axioms, 11.","DOI":"10.3390\/axioms11040176"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"97677","DOI":"10.1109\/ACCESS.2022.3206391","article-title":"n,mth power root fuzzy set and its applications to topology and decision-making","volume":"10","author":"Saeed","year":"2022","journal-title":"IEEE Access"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/8\/1617\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:38:55Z","timestamp":1760128735000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/8\/1617"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,21]]},"references-count":40,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["sym15081617"],"URL":"https:\/\/doi.org\/10.3390\/sym15081617","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,8,21]]}}}