{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T05:04:03Z","timestamp":1764997443867,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,2,3]],"date-time":"2022-02-03T00:00:00Z","timestamp":1643846400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["879441522"],"award-info":[{"award-number":["879441522"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Multiple attribute group decision making (MAGDM) issues play important roles in our daily life. In order to solve the problem that decision makers (DMs) may feel hesitant to select the appropriate evaluation values from several possible values in the process of providing evaluations, fuzzy theory and its extensions are widely applied in MAGDM problems. In this study, we first proposed hesitant picture fuzzy sets (HPFSs), which is a combination of the hesitant fuzzy set and picture fuzzy set. Subsequently, we introduced a novel Schweizer\u2013Sklar t-norm and t-conorm operation rules of HPFSs and proposed a family of hesitant picture fuzzy Schweizer\u2013Sklar Maclaurin symmetric mean operators. To show the application procedure of the proposed method to practical MAGDM issues, a numerical example about enterprise informatization level evaluation was employed to elaborate the calculation process with the proposed method. Finally, through the parameter analysis, validity analysis, and comparative analysis with some existing methods, we found that our method is more superior in providing DMs a greater decision-making freedom and relaxing the constraints on expressing personal preferences. This study provides a general framework of the proposed method to MAGDM problems under hesitant picture fuzzy environment, which enriches the fuzzy theory and its applications.<\/jats:p>","DOI":"10.3390\/e24020238","type":"journal-article","created":{"date-parts":[[2022,2,4]],"date-time":"2022-02-04T11:35:17Z","timestamp":1643974517000},"page":"238","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Novel MAGDM Method Based on Hesitant Picture Fuzzy Schweizer\u2013Sklar Maclaurin Symmetric Mean Operators and Their Application"],"prefix":"10.3390","volume":"24","author":[{"given":"Tiedong","family":"Chen","sequence":"first","affiliation":[{"name":"School of Economics and Management, Beijing Jiaotong University, Beijing 100044, China"}]},{"given":"Long","family":"Ye","sequence":"additional","affiliation":[{"name":"School of Economics and Management, Beijing Jiaotong University, Beijing 100044, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1007\/s10699-020-09714-9","article-title":"The Santa fe institute and econophysics: A possible genealogy?","volume":"26","author":"Schinckus","year":"2021","journal-title":"Found. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3299","DOI":"10.1140\/epjst\/e2016-60099-y","article-title":"1996\u20132016: Two decades of econophysics: Between methodological diversification and conceptual coherence","volume":"225","author":"Schinckus","year":"2016","journal-title":"Eur. Phys. J.-Spec. Top."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"101378","DOI":"10.1016\/j.irfa.2019.101378","article-title":"When financial economics influences physics: The role of Econophysics","volume":"65","author":"Jovanovic","year":"2019","journal-title":"Int. Rev. Financ. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Gandotra, N., Kizielewicz, B., Anand, A., B\u0105czkiewicz, A., Shekhovtsov, A., W\u0105tr\u00f3bski, J., Rezaei, A., and Sa\u0142abun, W. (2021). New Pythagorean entropy measure with application in multi-criteria decision analysis. Entropy, 23.","DOI":"10.3390\/e23121600"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Kou, Y., Feng, X., and Wang, J. (2021). A novel q-rung dual hesitant fuzzy multi-attribute decision-making method based on entropy weights. Entropy, 23.","DOI":"10.3390\/e23101322"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"111174","DOI":"10.1016\/j.enpol.2019.111174","article-title":"Improving renewable energy policy planning and decision-making through a hybrid MCDM method","volume":"137","author":"Alizadeh","year":"2020","journal-title":"Energy Policy"},{"key":"ref_7","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_8","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 Set Syst."},{"key":"ref_9","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_10","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/j.asoc.2008.04.006","article-title":"Fractional programming methodology for multi-attribute group decision-making using IFS","volume":"9","author":"Li","year":"2009","journal-title":"Appl. Soft Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1109\/TFUZZ.2019.2908123","article-title":"Three-way group conflict analysis based on Pythagorean fuzzy set theory","volume":"28","author":"Lang","year":"2019","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_12","unstructured":"Cuong, B. (2013). Picture fuzzy sets-first results. Part 1. Neuro-Fuzzy Systems with Applications, Institute of Mathematics."},{"key":"ref_13","first-page":"2857","article-title":"Correlation coefficients for picture fuzzy sets","volume":"27","author":"Singh","year":"2014","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.eswa.2014.07.026","article-title":"DPFCM: A novel distributed picture fuzzy clustering method on picture fuzzy sets","volume":"2","author":"Son","year":"2015","journal-title":"Expert Syst. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1016\/j.asoc.2016.05.009","article-title":"Generalized picture distance measure and applications to picture fuzzy clustering","volume":"46","author":"Son","year":"2016","journal-title":"Appl. Soft Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"103956","DOI":"10.1016\/j.engappai.2020.103956","article-title":"A new similarity measure between picture fuzzy sets and its application","volume":"96","author":"Luo","year":"2020","journal-title":"Eng. Appl. Artif. Intel."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1098631","DOI":"10.1155\/2021\/1098631","article-title":"Picture fuzzy Maclaurin symmetric mean operators and their applications in solving multi-attribute decision-making problems","volume":"2021","author":"Ullah","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1007\/s12559-017-9523-z","article-title":"A novel picture fuzzy linguistic aggregation operator and its application to group decision-making","volume":"10","author":"Liu","year":"2018","journal-title":"Cogn. Comput."},{"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":"702","DOI":"10.1007\/s40815-017-0303-4","article-title":"Hesitant intuitionistic fuzzy aggregation operators based on the Archimedean t-norms and t-conorms","volume":"19","author":"Peng","year":"2017","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1007\/s40747-019-0104-5","article-title":"Hesitant Pythagorean fuzzy interaction aggregation operators and their application in multiple attribute decision-making","volume":"5","author":"Yang","year":"2019","journal-title":"Complex Intell. Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"11560","DOI":"10.1016\/j.eswa.2011.03.034","article-title":"A unified model between the weighted average and the induced OWA operator","volume":"38","author":"Merigo","year":"2011","journal-title":"Expert Syst. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.fss.2013.07.010","article-title":"A group decision making model based on a generalized ordered weighted geometric average operator with interval preference matrices","volume":"246","author":"Liu","year":"2014","journal-title":"Fuzzy Set. Syst."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Zhang, H., Zhang, R., Huang, H., and Wang, J. (2018). Some picture fuzzy Dombi Heronian mean operators with their application to multi-attribute decision-making. Symmetry, 10.","DOI":"10.3390\/sym10110593"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/s00521-015-2048-z","article-title":"Multiple attribute decision-making method based on some normal neutrosophic Bonferroni mean operators","volume":"28","author":"Liu","year":"2017","journal-title":"Neural Comput. Appl."},{"key":"ref_26","first-page":"59","article-title":"A second letter to Martin Folkes, Esq.; concerning the roots of equations, with demonstration of other rules of algebra","volume":"36","author":"Maclaurin","year":"1729","journal-title":"Philos Trans. Roy. Soc. Lond. Ser. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1043","DOI":"10.1002\/int.21911","article-title":"Pythagorean fuzzy Maclaurin symmetric mean operators in multiple attribute decision making","volume":"33","author":"Wei","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1016\/j.knosys.2012.02.004","article-title":"Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm","volume":"31","author":"Xia","year":"2012","journal-title":"Knowl.-Based Syst."},{"key":"ref_29","first-page":"2684","article-title":"Some generalized einstein aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their application to group decision making","volume":"22","author":"Liu","year":"2015","journal-title":"Sci. Iran"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1057\/s41274-017-0190-y","article-title":"Some intuitionistic fuzzy Dombi Bonferroni mean operators and their application to multi-attribute group decision making","volume":"69","author":"Liu","year":"2018","journal-title":"J. Oper. Res. Soc."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2235","DOI":"10.3233\/IFS-151699","article-title":"Multiple attribute group decision making methods based on intuitionistic fuzzy frank power aggregation operators","volume":"29","author":"Zhang","year":"2015","journal-title":"J. Intell. Fuzzy Syst."},{"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":"2014","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3801","DOI":"10.3233\/JIFS-18801","article-title":"Some Maclaurin symmetric mean aggregation operators based on Schweizer-Sklar operations for intuitionistic fuzzy numbers and their application to decision making","volume":"36","author":"Wand","year":"2019","journal-title":"J. Intell Fuzzy Syst."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Khan, Q., Gwak, J., Shahzad, M., and Alam, M.K. (2021). A novel approached based on T-spherical fuzzy Schweizer-Sklar power Heronian mean operator for evaluating water reuse applications under uncertainty. Sustainability, 13.","DOI":"10.3390\/su13137108"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"171","DOI":"10.3233\/IFS-151584","article-title":"Approaches to uncertain linguistic multiple attribute decision making based on dual Maclaurin symmetric mean","volume":"29","author":"Qin","year":"2015","journal-title":"J. Intell Fuzzy Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1016\/S0377-2217(99)00082-X","article-title":"On consistency of the weighted geometric mean complex judgement matrix in AHP","volume":"126","author":"Xu","year":"2000","journal-title":"Eur. J. Oper. Res."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"991","DOI":"10.1007\/s41066-020-00243-1","article-title":"Pythagorean fuzzy Schweizer and Sklar power aggregation operators for solving multi-attribute decision-making problems","volume":"6","author":"Biswas","year":"2021","journal-title":"Granul. Comput."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"713","DOI":"10.3233\/JIFS-161798","article-title":"Picture fuzzy aggregation operators and their application to multiple attribute decision making","volume":"33","author":"Wei","year":"2017","journal-title":"J. Intell Fuzzy Syst."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Wang, R., Wang, J., Gao, H., and Wei, G. (2019). Methods for MADM with picture fuzzy muirhead mean operators and their application for evaluating the financial investment risk. Symmetry, 11.","DOI":"10.3390\/sym11010006"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/2\/238\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:13:29Z","timestamp":1760134409000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/2\/238"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,3]]},"references-count":39,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["e24020238"],"URL":"https:\/\/doi.org\/10.3390\/e24020238","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,2,3]]}}}