{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,5]],"date-time":"2026-01-05T15:06:24Z","timestamp":1767625584858,"version":"build-2065373602"},"reference-count":63,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,11]],"date-time":"2020-03-11T00:00:00Z","timestamp":1583884800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The Pythagorean probabilistic hesitant fuzzy set (PyPHFS) is an effective, generalized and powerful tool for expressing fuzzy information. It can cover more complex and more hesitant fuzzy evaluation information. Therefore, based on the advantages of PyPHFSs, this paper presents a new extended fuzzy TOPSIS method for dealing with uncertainty in the form of PyPHFS in real life problems. The paper is divided into three main parts. Firstly, the novel Pythagorean probabilistic hesitant fuzzy entropy measure is established using generalized distance measure under PyPHFS information to find out the unknown weights information of the attributes. The second part consists of the algorithm sets of the TOPSIS technique under PyPHFS environment, where the weights of criteria are completely unknown. Finally, in order to verify the efficiency and superiority of the proposed method, this paper applies some practical examples of the selection of the most critical fog-haze influence factor and makes a detailed comparison with other existing methods.<\/jats:p>","DOI":"10.3390\/e22030318","type":"journal-article","created":{"date-parts":[[2020,3,12]],"date-time":"2020-03-12T04:13:57Z","timestamp":1583986437000},"page":"318","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":59,"title":["Entropy Based Pythagorean Probabilistic Hesitant Fuzzy Decision Making Technique and Its Application for Fog-Haze Factor Assessment Problem"],"prefix":"10.3390","volume":"22","author":[{"given":"Bushra","family":"Batool","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}]},{"given":"Mumtaz","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7474-5115","authenticated-orcid":false,"given":"Saleem","family":"Abdullah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8616-8829","authenticated-orcid":false,"given":"Shahzaib","family":"Ashraf","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6113-3689","authenticated-orcid":false,"given":"Ronnason","family":"Chinram","sequence":"additional","affiliation":[{"name":"Algebra and Applications Research Unit, Department of Mathematics and Statistics, Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla 90110, Thailand"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1421","DOI":"10.1016\/j.scitotenv.2018.01.195","article-title":"Research and application of a novel hybrid air quality early-warning system: A case study in China","volume":"626","author":"Li","year":"2018","journal-title":"Sci. Total."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"740","DOI":"10.1016\/j.scitotenv.2018.11.368","article-title":"The association between high ambient air pollution exposure and respiratory health of young children: A cross sectional study in Jinan, China","volume":"656","author":"Chen","year":"2019","journal-title":"Sci. Total. Environ."},{"key":"ref_3","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_4","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_5","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.inffus.2015.03.002","article-title":"Fuzzy multiattribute group decision making based on intuitionistic fuzzy sets and evidential reasoning methodology","volume":"27","author":"Chen","year":"2016","journal-title":"Inf. Fusion"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1002\/int.21584","article-title":"Pythagorean membership grades, complex numbers, and decision making","volume":"28","author":"Yager","year":"2013","journal-title":"Int. J. Intell. Syst."},{"key":"ref_7","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_8","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":"2013","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_9","first-page":"529","article-title":"Hesitant fuzzy sets","volume":"25","author":"Torra","year":"2010","journal-title":"Int. J. Intell. Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3971","DOI":"10.3233\/JIFS-17811","article-title":"Pythagorean hesitant fuzzy sets and their application to group decision making with incomplete weight information","volume":"33","author":"Khan","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1007\/s10700-016-9257-5","article-title":"Consensus building with a group of decision makers under the hesitant probabilistic fuzzy environment","volume":"16","author":"Xu","year":"2017","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/S0019-9958(72)90199-4","article-title":"A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory","volume":"20","author":"Termini","year":"1972","journal-title":"Inf. Control"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/0165-0114(96)84611-2","article-title":"Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets","volume":"78","author":"Burillo","year":"1996","journal-title":"Fuzzy Sets Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1016\/S0165-0114(98)00402-3","article-title":"Entropy for intuitionistic fuzzy sets","volume":"118","author":"Szmidt","year":"2001","journal-title":"Fuzzy Sets Syst."},{"key":"ref_15","first-page":"15","article-title":"Entropy for intuitionistic fuzzy set theory and mass assignment theory","volume":"10","author":"Szmidt","year":"2004","journal-title":"Notes Ifss"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1002\/int.20131","article-title":"Fuzzy entropy on intuitionistic fuzzy sets","volume":"21","author":"Hung","year":"2006","journal-title":"Int. J. Intell. Syst."},{"key":"ref_17","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_18","doi-asserted-by":"crossref","first-page":"729","DOI":"10.30948\/afmi.2017.13.6.729","article-title":"Decision making with distance and cosine similarity measures for intuitionistic hesitant fuzzy sets","volume":"13","author":"Wang","year":"2017","journal-title":"Ann. Fuzzy Math. Inform."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"988","DOI":"10.1016\/j.asoc.2015.10.040","article-title":"A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems","volume":"38","author":"Garg","year":"2016","journal-title":"Appl. Soft Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1093","DOI":"10.1007\/s13042-015-0433-7","article-title":"Interval valued hesitant fuzzy uncertain linguistic aggregation operators in multiple attribute decision making","volume":"7","author":"Wei","year":"2016","journal-title":"Int. J. Mach. Learn. Cybern."},{"key":"ref_21","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_22","doi-asserted-by":"crossref","unstructured":"Kumar, K., and Garg, H. (2018). Prioritized linguistic interval-valued aggregation operators and their applications in group decision-making problems. Mathematics, 6.","DOI":"10.3390\/math6100209"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Wang, P., Wei, G., Wang, J., Lin, R., and Wei, Y. (2019). Dual hesitant q-Rung orthopair fuzzy hamacher aggregation operators and their applications in scheme selection of construction project. Symmetry, 11.","DOI":"10.3390\/sym11060771"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1667","DOI":"10.1080\/1331677X.2019.1638808","article-title":"Pythagorean fuzzy Muirhead mean operators in multiple attribute decision making for evaluating of emerging technology commercialization","volume":"32","author":"Tang","year":"2019","journal-title":"Econ.-Res.-Ekon. istra \u017eivanja"},{"key":"ref_25","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_26","first-page":"445","article-title":"Correlation coefficients of hesitant fuzzy sets and their application based on fuzzy measures","volume":"7","author":"Meng","year":"2015","journal-title":"Cogn."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1002\/int.22067","article-title":"Hesitant Pythagorean fuzzy Maclaurin symmetric mean operators and its applications to multiattribute decision-making process","volume":"34","author":"Garg","year":"2019","journal-title":"Int. J. Intell. Syst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1119","DOI":"10.1007\/s12559-016-9419-3","article-title":"Group decision making with dual hesitant fuzzy preference relations","volume":"8","author":"Zhao","year":"2016","journal-title":"Cogn. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/s12559-016-9436-2","article-title":"Distance and aggregation-based methodologies for hesitant fuzzy decision making","volume":"9","author":"Farhadinia","year":"2017","journal-title":"Cogn. Comput."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.engappai.2018.03.019","article-title":"A robust correlation coefficient measure of dual hesitant fuzzy soft sets and their application in decision making","volume":"72","author":"Arora","year":"2018","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Garg, H., and Arora, R. (2017). Distance and similarity measures for dual hesitant fuzzy soft sets and their applications in multicriteria decision making problem. Int. J. Uncertain. Quantif., 7.","DOI":"10.1615\/Int.J.UncertaintyQuantification.2017019801"},{"key":"ref_32","first-page":"1","article-title":"Hesitant fuzzy linguistic arithmetic aggregation operators in multiple attribute decision making","volume":"13","author":"Wei","year":"2016","journal-title":"Iran. J. Fuzzy Syst."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Garg, H. (2018). Hesitant Pythagorean fuzzy sets and their aggregation operators in multiple attribute decision-making. Int. Uncertain. Quantif., 8.","DOI":"10.1615\/Int.J.UncertaintyQuantification.2018020979"},{"key":"ref_34","first-page":"1","article-title":"Group decision making approach based on possibility degree measure under linguistic interval-valued intuitionistic fuzzy set environment","volume":"13","author":"Garg","year":"2018","journal-title":"J. Ind. Manag."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1016\/j.apm.2012.02.008","article-title":"An avoiding information loss approach to group decision making","volume":"37","author":"Yue","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1002\/int.21796","article-title":"A novel approach based on similarity measure for Pythagorean fuzzy multiple criteria group decision making","volume":"31","author":"Zhang","year":"2016","journal-title":"Int. Intell. Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1016\/j.asoc.2015.12.020","article-title":"Pythagorean fuzzy TODIM approach to multi-criteria decision making","volume":"42","author":"Ren","year":"2016","journal-title":"Appl. Soft Comput."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"989","DOI":"10.1002\/int.21814","article-title":"Pythagorean fuzzy Choquet integral based MABAC method for multiple attribute group decision making","volume":"31","author":"Peng","year":"2016","journal-title":"Int. J. Intell. Syst."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1016\/j.ins.2015.10.012","article-title":"Multicriteria Pythagorean fuzzy decision analysis: A hierarchical QUALIFLEX approach with the closeness index-based ranking methods","volume":"330","author":"Zhang","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"991","DOI":"10.1002\/int.21880","article-title":"Pythagorean fuzzy information measures and their applications","volume":"32","author":"Peng","year":"2017","journal-title":"Int. J. Intell. Syst."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"5937376","DOI":"10.1155\/2017\/5937376","article-title":"Some generalized Pythagorean fuzzy Bonferroni mean aggregation operators with their application to multiattribute group decision-making","volume":"2017","author":"Zhang","year":"2017","journal-title":"Complexity"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.asoc.2017.06.034","article-title":"The new extension of TOPSIS method for multiple criteria decision making with hesitant Pythagorean fuzzy sets","volume":"60","author":"Liang","year":"2017","journal-title":"Appl. Soft Comput."},{"key":"ref_43","first-page":"2602376","article-title":"MOORA under Pythagorean fuzzy set for multiple criteria decision making","volume":"2018","author":"Xu","year":"2018","journal-title":"Complexity"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1002\/int.21941","article-title":"Pythagorean fuzzy LINMAP method based on the entropy theory for railway project investment decision making","volume":"33","author":"Xue","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_45","first-page":"3941847","article-title":"An approach to interval-valued hesitant fuzzy multiattribute group decision making based on the generalized Shapley-Choquet integral","volume":"2018","author":"Zhang","year":"2018","journal-title":"Complexity"},{"key":"ref_46","first-page":"8023013","article-title":"Pythagorean Fuzzy-Norm Information Measure for Multicriteria Decision-Making Problem","volume":"2018","author":"Guleria","year":"2018","journal-title":"Adv. Fuzzy Syst."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"5835","DOI":"10.1007\/s00500-018-3248-0","article-title":"Distance and similarity measures of hesitant fuzzy sets based on Hausdorff metric with applications to multi-criteria decision making and clustering","volume":"23","author":"Yang","year":"2019","journal-title":"Soft Comput."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Technol. J."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"2517","DOI":"10.1007\/s40815-018-0523-2","article-title":"Entropy for hesitant fuzzy sets based on Hausdorff metric with construction of hesitant fuzzy TOPSIS","volume":"20","author":"Hussain","year":"2018","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Hwang, C.L., and Yoon, K.S. (1981). Multiple Attribute Decision Methods and Applications, Springer.","DOI":"10.1007\/978-3-642-48318-9"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0165-0114(97)00377-1","article-title":"Extensions of the TOPSIS for group decision-making under fuzzy environment","volume":"114","author":"Chen","year":"2000","journal-title":"Fuzzy Sets Syst."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"11363","DOI":"10.1016\/j.eswa.2009.03.039","article-title":"A multi-criteria intuitionistic fuzzy group decision making for supplier selection with TOPSIS method","volume":"36","author":"Boran","year":"2009","journal-title":"Expert Syst. Appl."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1410","DOI":"10.1016\/j.fss.2007.11.004","article-title":"The interval-valued fuzzy TOPSIS method and experimental analysis","volume":"159","author":"Chen","year":"2008","journal-title":"Fuzzy Sets Syst."},{"key":"ref_54","first-page":"299","article-title":"TOPSIS-based nonlinear-programming methodology for multiattribute decision making with interval-valued intuitionistic fuzzy sets","volume":"18","author":"Li","year":"2010","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_55","first-page":"2544","article-title":"Extension of the TOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environment","volume":"35","author":"Park","year":"2011","journal-title":"Appl. Math."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"2119","DOI":"10.1016\/j.eswa.2011.07.119","article-title":"The LTOPSIS: An alternative to TOPSIS decision-making approach for linguistic variables","volume":"39","author":"Cables","year":"2012","journal-title":"Expert Syst. Appl."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.knosys.2013.05.011","article-title":"Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information","volume":"52","author":"Xu","year":"2013","journal-title":"Knowl.-Based Syst."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1162","DOI":"10.1002\/int.21623","article-title":"TOPSIS for hesitant fuzzy linguistic term sets","volume":"28","author":"Beg","year":"2013","journal-title":"Int. J. Intell. Syst."},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Khan, A.A., Ashraf, S., Abdullah, S., Qiyas, M., Luo, J., and Khan, S.U. (2019). Pythagorean fuzzy Dombi aggregation operators and their application in decision support system. Symmetry, 11.","DOI":"10.3390\/sym11030383"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1108","DOI":"10.1002\/int.22088","article-title":"Pythagorean fuzzy TOPSIS for multicriteria group decision-making with unknown weight information through entropy measure","volume":"34","author":"Biswas","year":"2019","journal-title":"Int. J. Intell. Syst."},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Barukab, O., Abdullah, S., Ashraf, S., Arif, M., and Khan, S.A. (2019). A New Approach to Fuzzy TOPSIS Method Based on Entropy Measure under Spherical Fuzzy Information. Entropy, 21.","DOI":"10.3390\/e21121231"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"983","DOI":"10.1002\/int.21908","article-title":"An Agent-Based Pythagorean Fuzzy Approach for Demand Analysis with Incomplete Information","volume":"33","author":"Baloglu","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.asoc.2016.04.010","article-title":"Deriving the priority weights from incomplete hesitant fuzzy preference relations based on multiplicative consistency","volume":"46","author":"Zhang","year":"2016","journal-title":"Appl. Soft Comput."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/3\/318\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:06:10Z","timestamp":1760173570000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/3\/318"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,11]]},"references-count":63,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["e22030318"],"URL":"https:\/\/doi.org\/10.3390\/e22030318","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2020,3,11]]}}}