{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,6]],"date-time":"2026-08-06T03:06:21Z","timestamp":1785985581076,"version":"3.56.0"},"reference-count":58,"publisher":"MDPI AG","issue":"13","license":[{"start":{"date-parts":[[2022,6,28]],"date-time":"2022-06-28T00:00:00Z","timestamp":1656374400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004281","name":"National Science Centre","doi-asserted-by":"publisher","award":["2021\/41\/B\/HS4\/01296"],"award-info":[{"award-number":["2021\/41\/B\/HS4\/01296"]}],"id":[{"id":"10.13039\/501100004281","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The Pythagorean fuzzy sets conveniently capture unreliable, ambiguous, and uncertain information, especially in problems involving multiple and opposing criteria. Pythagorean fuzzy sets are one of the popular generalizations of the intuitionistic fuzzy sets. They are instrumental in expressing and managing hesitant under uncertain environments, so they have been involved extensively in a diversity of scientific fields. This paper proposes a new Pythagorean entropy for Multi-Criteria Decision-Analysis (MCDA) problems. The entropy measures the fuzziness of two fuzzy sets and has an influential position in fuzzy functions. The more comprehensive the entropy, the more inadequate the ambiguity, so the decision-making established on entropy is beneficial. The COmplex PRoportional ASsessment (COPRAS) method is used to tackle uncertainty issues in MCDA and considers the singularity of one alternative over the rest of them. This can be enforced to maximize and minimize relevant criteria in an assessment where multiple opposing criteria are considered. Using the Pythagorean sets, we represent a decisional problem solution by using the COPRAS approach and the new Entropy measure.<\/jats:p>","DOI":"10.3390\/s22134879","type":"journal-article","created":{"date-parts":[[2022,6,29]],"date-time":"2022-06-29T01:48:38Z","timestamp":1656467318000},"page":"4879","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["The Group Decision-Making Using Pythagorean Fuzzy Entropy and the Complex Proportional Assessment"],"prefix":"10.3390","volume":"22","author":[{"given":"Parul","family":"Thakur","sequence":"first","affiliation":[{"name":"Yogananda School of AI, Computer and Data Sciences, Faculty of Engineering and Technology, Shoolini University, Solan 173229, Himachal Pradesh, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5736-4014","authenticated-orcid":false,"given":"Bart\u0142omiej","family":"Kizielewicz","sequence":"additional","affiliation":[{"name":"Research Team on Intelligent Decision Support Systems, Department of Artificial Intelligence and Applied Mathematics, Faculty of Computer Science and Information Technology, West Pomeranian University of Technology in Szczecin, ul. \u017bo\u0142nierska 49, 71-210 Szczecin, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6534-3264","authenticated-orcid":false,"given":"Neeraj","family":"Gandotra","sequence":"additional","affiliation":[{"name":"Yogananda School of AI, Computer and Data Sciences, Faculty of Engineering and Technology, Shoolini University, Solan 173229, Himachal Pradesh, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0834-2019","authenticated-orcid":false,"given":"Andrii","family":"Shekhovtsov","sequence":"additional","affiliation":[{"name":"Research Team on Intelligent Decision Support Systems, Department of Artificial Intelligence and Applied Mathematics, Faculty of Computer Science and Information Technology, West Pomeranian University of Technology in Szczecin, ul. \u017bo\u0142nierska 49, 71-210 Szczecin, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Namita","family":"Saini","sequence":"additional","affiliation":[{"name":"Yogananda School of AI, Computer and Data Sciences, Faculty of Engineering and Technology, Shoolini University, Solan 173229, Himachal Pradesh, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7076-2519","authenticated-orcid":false,"given":"Wojciech","family":"Sa\u0142abun","sequence":"additional","affiliation":[{"name":"Research Team on Intelligent Decision Support Systems, Department of Artificial Intelligence and Applied Mathematics, Faculty of Computer Science and Information Technology, West Pomeranian University of Technology in Szczecin, ul. \u017bo\u0142nierska 49, 71-210 Szczecin, Poland"},{"name":"National Institute of Telecommunications, Szachowa 1, 04-894 Warsaw, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Gandotra, N., Bajaj, R.K., and Mathew, J. (2015, January 3\u20135). On Ranking in Triangular Intuitionistic Fuzzy Multi-criteria Decision Making under (a, \u00df)-cut with `Useful\u2019 Parametric Entropy. Proceedings of the 2015 Fifth International Conference on Advances in Computing and Communications (ICACC), Gwalior, India.","DOI":"10.1109\/ICACC.2015.33"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.procs.2017.12.007","article-title":"Multi-criteria decision making with triangular intuitionistic fuzzy number based on distance measure & parametric entropy approach","volume":"125","author":"Saini","year":"2018","journal-title":"Procedia Comput. Sci."},{"key":"ref_3","unstructured":"Kumar, R., and Gandotra, N. A novel pythagorean fuzzy entropy measure using MCDM application in preference of the advertising company with TOPSIS approach. Mater. Today Proc., 2021. in press."},{"key":"ref_4","unstructured":"Saini, N., Gandotra, N., and Kumar, R. (2020, January 1\u20132). Multi Criteria Decision Making Under Fuzzy, Intuitionistic and Interval-Valued Intuitionistic Fuzzy Environment: A Review. Proceedings of the 3rd International Conference on Communications and Cyber Physical Engineering, ICCCE 2020, Delhi, India."},{"key":"ref_5","first-page":"338","article-title":"Information and control","volume":"8","author":"Zadeh","year":"1965","journal-title":"Fuzzy Sets"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Hernandez-Aguila, A., Garcia-Valdez, M., and Castillo, O. (2019). Graphical Representation of Intuitionistic Membership Functions for Its Efficient Use in Intuitionistic Fuzzy Systems. Recent Advances in Intuitionistic Fuzzy Logic Systems, Springer.","DOI":"10.1007\/978-3-030-02155-9_19"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1650037","DOI":"10.1142\/S1793524516500376","article-title":"An application of intuitionistic fuzzy sets in medicine","volume":"9","author":"Davvaz","year":"2016","journal-title":"Int. J. Biomath."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Atanassov, K.T. (1999). Intuitionistic Fuzzy Sets, Springer.","DOI":"10.1007\/978-3-7908-1870-3"},{"key":"ref_9","first-page":"26","article-title":"On some ways of determining membership and non-membership functions characterizing intuitionistic fuzzy sets","volume":"16","author":"Atanassov","year":"2010","journal-title":"Notes IFSs"},{"key":"ref_10","first-page":"1211","article-title":"Intuitionistic fuzzification functions","volume":"12","author":"Radhika","year":"2016","journal-title":"Glob. J. Pure Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/j.asoc.2019.01.041","article-title":"A group medical diagnosis model based on intuitionistic fuzzy soft sets","volume":"77","author":"Hu","year":"2019","journal-title":"Appl. Soft Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3041","DOI":"10.3233\/JIFS-190102","article-title":"A novel similarity degree of intuitionistic fuzzy sets induced by triangular norm and its application in pattern recognition","volume":"37","author":"Shi","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"665","DOI":"10.13189\/ms.2020.080606","article-title":"Measuring Given Partial Information about Intuitionistic Fuzzy Sets","volume":"8","author":"Arora","year":"2020","journal-title":"Math. Stat."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Greis, M., Schuff, H., Kleiner, M., Henze, N., and Schmidt, A. (2017, January 25\u201327). Input controls for entering uncertain data: Probability distribution sliders. Proceedings of the ACM on Human-Computer Interaction, Cancun, Mexico.","DOI":"10.1145\/3095805"},{"key":"ref_15","first-page":"1","article-title":"A new entropy and its properties based on the improved axiomatic definition of intuitionistic fuzzy entropy","volume":"12","author":"Jiang","year":"2018","journal-title":"Math. Probl. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1080\/18756891.2015.1129592","article-title":"Bootstrap Kuiper testing of the identity of 1D continuous distributions using fuzzy samples","volume":"8","author":"Nikolova","year":"2015","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"916","DOI":"10.1016\/j.eswa.2010.07.073","article-title":"Multicriteria decision making method based on intuitionistic fuzzy weighted entropy","volume":"38","author":"Wu","year":"2011","journal-title":"Expert Syst. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4148","DOI":"10.1016\/j.procs.2021.09.190","article-title":"A fuzzy assessment model for freestyle swimmers-a comparative analysis of the MCDA methods","volume":"192","author":"Dobryakova","year":"2021","journal-title":"Procedia Comput. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Kizielewicz, B., Shekhovtsov, A., and Sa\u0142abun, W. (2021, January 16\u201318). A new approach to eliminate rank reversal in the mcda problems. Proceedings of the International Conference on Computational Science, Krakow, Poland.","DOI":"10.1007\/978-3-030-77961-0_29"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"114186","DOI":"10.1016\/j.eswa.2020.114186","article-title":"Effects of the entropy weight on TOPSIS","volume":"168","author":"Chen","year":"2021","journal-title":"Expert Syst. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4578","DOI":"10.1016\/j.procs.2021.09.236","article-title":"Comparison of Fuzzy TOPSIS, Fuzzy VIKOR, Fuzzy WASPAS and Fuzzy MMOORA methods in the housing selection problem","volume":"192","author":"Kizielewicz","year":"2021","journal-title":"Procedia Comput. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Mardani, A., Zavadskas, E.K., Govindan, K., Amat Senin, A., and Jusoh, A. (2016). VIKOR technique: A systematic review of the state of the art literature on methodologies and applications. Sustainability, 8.","DOI":"10.3390\/su8010037"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1877","DOI":"10.1007\/s00521-021-06427-3","article-title":"The novel VIKOR methods for generalized Pythagorean fuzzy soft sets and its application to children of early childhood in COVID-19 quarantine","volume":"34","author":"Demir","year":"2022","journal-title":"Neural Comput. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1007\/s10100-021-00788-9","article-title":"Two-stage weighted PROMETHEE II with results\u2019 visualization","volume":"30","author":"Kuncova","year":"2022","journal-title":"Cent. Eur. J. Oper. Res."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Krouska, A., Kabassi, K., Troussas, C., and Sgouropoulou, C. (2022). Personalizing Environmental Awareness through Smartphones Using AHP and PROMETHEE II. Future Internet, 14.","DOI":"10.3390\/fi14020066"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Dezert, J., Tchamova, A., Han, D., and Tacnet, J.M. (2020, January 6\u20139). The SPOTIS rank reversal free method for multi-criteria decision-making support. Proceedings of the 2020 IEEE 23rd International Conference on Information Fusion (FUSION), Rustenburg, South Africa.","DOI":"10.23919\/FUSION45008.2020.9190347"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Shekhovtsov, A., Kizielewicz, B., and Sa\u0142abun, W. (2021, January 16\u201318). New rank-reversal free approach to handle interval data in mcda problems. Proceedings of the International Conference on Computational Science, Krakow, Poland.","DOI":"10.1007\/978-3-030-77980-1_35"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"35","DOI":"10.15388\/21-INFOR466","article-title":"A SWARA-CoCoSo-based approach for spray painting robot selection","volume":"33","author":"Kumar","year":"2022","journal-title":"Informatica"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1567","DOI":"10.1007\/s10462-021-09995-x","article-title":"Intuitionistic fuzzy soft decision making method based on CoCoSo and CRITIC for CCN cache placement strategy selection","volume":"55","author":"Peng","year":"2022","journal-title":"Artif. Intell. Rev."},{"key":"ref_30","first-page":"115","article-title":"Selection of fire position of mortar units using LBWA and Fuzzy MABAC model","volume":"4","year":"2021","journal-title":"Oper. Res. Eng. Sci. Theory Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"43","DOI":"10.12700\/APH.17.3.2020.3.3","article-title":"Ranking and assessment of the efficiency of social media using the fuzzy AHP-Z number model-fuzzy MABAC","volume":"17","author":"Bobar","year":"2020","journal-title":"Acta Polytech. Hung."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"947","DOI":"10.1007\/s41660-021-00192-5","article-title":"Solving material handling equipment selection problems in an industry with the help of entropy integrated COPRAS and ARAS MCDM techniques","volume":"5","author":"Goswami","year":"2021","journal-title":"Process. Integr. Optim. Sustain."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"115088","DOI":"10.1016\/j.eswa.2021.115088","article-title":"Best-Worst method and Hamacher aggregation operations for intuitionistic 2-tuple linguistic sets","volume":"181","author":"Faizi","year":"2021","journal-title":"Expert Syst. Appl."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Pamu\u010dar, D., Stevi\u0107, \u017d., and Sremac, S. (2018). A new model for determining weight coefficients of criteria in mcdm models: Full consistency method (fucom). Symmetry, 10.","DOI":"10.3390\/sym10090393"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2816","DOI":"10.1016\/j.egyr.2020.09.035","article-title":"Statistical evaluation of using the new generation of wind turbines in South Africa","volume":"6","author":"Mostafaeipour","year":"2020","journal-title":"Energy Rep."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"106916","DOI":"10.1016\/j.asoc.2020.106916","article-title":"A heuristic method to rank the alternatives in the AHP synthesis","volume":"100","author":"Lin","year":"2021","journal-title":"Appl. Soft Comput."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.acme.2017.04.011","article-title":"A new hybrid fuzzy MCDM approach for evaluation of construction equipment with sustainability considerations","volume":"18","author":"Ghorabaee","year":"2018","journal-title":"Arch. Civ. Mech. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"135310","DOI":"10.1016\/j.scitotenv.2019.135310","article-title":"Risk assessment of mega-city infrastructures related to land subsidence using improved trapezoidal FAHP","volume":"717","author":"Lyu","year":"2020","journal-title":"Sci. Total. Environ."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1007\/s40815-018-0538-8","article-title":"A novel fuzzy decision-making method using entropy weights-based correlation coefficients under intuitionistic fuzzy environment","volume":"21","author":"Joshi","year":"2019","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Yager, R.R. (2016). Properties and applications of Pythagorean fuzzy sets. Imprecision and Uncertainty in Information Representation and Processing, Springer.","DOI":"10.1007\/978-3-319-26302-1_9"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Yang, M.S., and Hussain, Z. (2018). Fuzzy entropy for pythagorean fuzzy sets with application to multicriterion decision making. Complexity.","DOI":"10.1155\/2018\/2832839"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Melliani, S., and Castillo, O. (2019). Recent Advances in Intuitionistic Fuzzy Logic Systems, Springer.","DOI":"10.1007\/978-3-030-02155-9"},{"key":"ref_43","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_44","doi-asserted-by":"crossref","first-page":"85","DOI":"10.3846\/1392-3730.2008.14.3","article-title":"Selection of the effective dwelling house walls by applying attributes values determined at intervals","volume":"14","author":"Zavadskas","year":"2008","journal-title":"J. Civ. Eng. Manag."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1016\/j.autcon.2009.12.006","article-title":"Model for a complex analysis of intelligent built environment","volume":"19","author":"Kaklauskas","year":"2010","journal-title":"Autom. Constr."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1486","DOI":"10.1109\/TEM.2019.2907673","article-title":"A novel approach integrating AHP and COPRAS under Pythagorean fuzzy sets for digital supply chain partner selection","volume":"68","author":"Buyukozkan","year":"2019","journal-title":"IEEE Trans. Eng. Manag."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"121208","DOI":"10.1016\/j.energy.2021.121208","article-title":"A new Pythagorean fuzzy-based decision-making method through entropy measure for fuel cell and hydrogen components supplier selection","volume":"234","author":"Alipour","year":"2021","journal-title":"Energy"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"81","DOI":"10.4995\/ijpme.2021.14734","article-title":"A comparison of TOPSIS, grey relational analysis and COPRAS methods for machine selection problem in the food industry of Turkey","volume":"9","year":"2021","journal-title":"Int. J. Prod. Manag. Eng."},{"key":"ref_49","first-page":"1641","article-title":"New hybrid multi-criteria decision-making DEMATELMAIRCA model: Sustainable selection of a location for the development of multimodal logistics centre","volume":"31","author":"Pamucar","year":"2018","journal-title":"Econ. Res.-Ekon. Istraz."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"4039","DOI":"10.29042\/2018-4039-4043","article-title":"Decision Making using COPRAS and ELECTRE Technique","volume":"8","author":"Borkar","year":"2018","journal-title":"HELIX"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1016\/j.omega.2015.05.013","article-title":"Comparative analysis of MCDM methods for the assessment of sustainable housing affordability","volume":"59","author":"Mulliner","year":"2016","journal-title":"Omega"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Chaurasiya, R., and Jain, D. (2022). Pythagorean fuzzy entropy measure-based complex proportional assessment technique for solving multi-criteria healthcare waste treatment problem. Granul. Comput., 1\u201314.","DOI":"10.1007\/s41066-021-00304-z"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"9045","DOI":"10.1007\/s00500-021-05842-x","article-title":"EDAS method for probabilistic linguistic multiple attribute group decision making and their application to green supplier selection","volume":"25","author":"Wei","year":"2021","journal-title":"Soft Comput."},{"key":"ref_54","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_55","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_56","doi-asserted-by":"crossref","first-page":"1050","DOI":"10.3934\/math.2020073","article-title":"A novel entropy measure of Pythagorean fuzzy soft sets","volume":"5","author":"Athira","year":"2020","journal-title":"AIMS Math."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"117936","DOI":"10.1016\/j.jclepro.2019.117936","article-title":"A novel VIKOR approach based on entropy and divergence measures of Pythagorean fuzzy sets to evaluate renewable energy technologies in India","volume":"238","author":"Rani","year":"2019","journal-title":"J. Clean. Prod."},{"key":"ref_58","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."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/13\/4879\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:39:51Z","timestamp":1760139591000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/13\/4879"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,28]]},"references-count":58,"journal-issue":{"issue":"13","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["s22134879"],"URL":"https:\/\/doi.org\/10.3390\/s22134879","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,28]]}}}