{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T13:18:06Z","timestamp":1775481486608,"version":"3.50.1"},"reference-count":73,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T00:00:00Z","timestamp":1768694400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Bialystok University of Technology and funded by the Ministry of Science and Higher Education","award":["grant WZ\/WI-IIT\/2\/25"],"award-info":[{"award-number":["grant WZ\/WI-IIT\/2\/25"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Normalization is a critical step in Multiple-Criteria Decision Analysis (MCDA) because it transforms heterogeneous criterion values into comparable information. This study examines normalization techniques through the lens of entropy, highlighting how criterion data structure shapes normalization behavior and ranking stability within TOPSIS (Technique for Order Preference by Similarity to Ideal Solution). Seven widely used normalization procedures are analyzed regarding mathematical properties, sensitivity to extreme values, treatment of benefit and cost criteria, and rank reversal. Normalization is treated as a source of uncertainty in MCDA outcomes, as different schemes can produce divergent rankings under identical decision settings. Shannon entropy is employed as a descriptive measure of information dispersion and structural uncertainty, capturing the heterogeneity and discriminatory potential of criteria rather than serving as a weighting mechanism. An illustrative experiment with ten alternatives and four criteria (two high-entropy, two low-entropy) demonstrates how entropy mediates normalization effects. Seven normalization schemes are examined, including vector, max, linear Sum, and max\u2013min procedures. For vector, max, and linear sum, cost-type criteria are treated using either linear inversion or reciprocal transformation, whereas max\u2013min is implemented as a single method. This design separates the choice of normalization form from the choice of cost-criteria transformation, allowing a cleaner identification of their respective contributions to ranking variability. The analysis shows that normalization choice alone can cause substantial differences in preference values and rankings. High-entropy criteria tend to yield stable rankings, whereas low-entropy criteria amplify sensitivity, especially with extreme or cost-type data. These findings position entropy as a key mediator linking data structure with normalization-induced ranking variability and highlight the need to consider entropy explicitly when selecting normalization procedures. Finally, a practical entropy-based method for choosing normalization techniques is introduced to enhance methodological transparency and ranking robustness in MCDA.<\/jats:p>","DOI":"10.3390\/e28010114","type":"journal-article","created":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T08:23:56Z","timestamp":1768811036000},"page":"114","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Entropy and Normalization in MCDA: A Data-Driven Perspective on Ranking Stability"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2249-7217","authenticated-orcid":false,"given":"Ewa","family":"Roszkowska","sequence":"first","affiliation":[{"name":"Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15-351 Bialystok, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Greco, S., Ehrgott, M., and Figueira, J.R. (2016). Multiple Criteria Decision Analysis: State of the Art Surveys, Springer. [2nd ed.]. International Series in Operations Research & Management Science.","DOI":"10.1007\/978-1-4939-3094-4"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Matsatsinis, N., and Grigoroudis, E. (2018). Disaggregation Approaches for Multicriteria Classification: An Overview. Preference Disaggregation in Multiple Criteria Decision Analysis: Essays in Honor of Yannis Siskos, Springer International Publishing. Multiple Criteria Decision Making.","DOI":"10.1007\/978-3-319-90599-0"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"397","DOI":"10.3846\/20294913.2011.593291","article-title":"Multiple Criteria Decision Making (MCDM) Methods in Economics: An Overview","volume":"17","author":"Zavadskas","year":"2011","journal-title":"Technol. Econ. Dev. Econ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1016\/j.ejor.2015.05.032","article-title":"Multiple Criteria Decision Aiding for Finance: An Updated Bibliographic Survey","volume":"247","author":"Zopounidis","year":"2015","journal-title":"Eur. J. Oper. Res."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Siksnelyte, I., Zavadskas, E.K., Streimikiene, D., and Sharma, D. (2018). An Overview of Multi-Criteria Decision-Making Methods in Dealing with Sustainable Energy Development Issues. Energies, 11.","DOI":"10.3390\/en11102754"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Stoj\u010di\u0107, M., Zavadskas, E.K., Pamu\u010dar, D., Stevi\u0107, \u017d., and Mardani, A. (2019). Application of MCDM Methods in Sustainability Engineering: A Literature Review 2008\u20132018. Symmetry, 11.","DOI":"10.3390\/sym11030350"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"25","DOI":"10.31181\/dma1120237","article-title":"A Comprehensive Review of Multiple Criteria Decision-Making (MCDM) Methods: Advancements, Applications, and Future Directions","volume":"1","author":"Sahoo","year":"2023","journal-title":"Decis. Mak. Adv."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Hwang, C.-L., and Yoon, K. (1981). Methods for Multiple Attribute Decision Making, Springer. Lecture Notes in Economics and Mathematical Systems.","DOI":"10.1007\/978-3-642-48318-9"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.31181\/dmame210402001a","article-title":"Comparative Analysis of the Normalization Techniques in the Context of MCDM Problems","volume":"4","author":"Aytekin","year":"2021","journal-title":"Decis. Mak. Appl. Manag. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Chakraborty, S., Chatterjee, P., and Das, P.P. (2024). Normalization Techniques. Multi-Criteria Decision-Making Methods in Manufacturing Environments, Apple Academic Press.","DOI":"10.1201\/9781003377030"},{"key":"ref_11","first-page":"251","article-title":"Normalization Affects the Results of MADM Methods","volume":"11","year":"2001","journal-title":"Yugosl. J. Oper. Res."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1007\/s00158-004-0473-1","article-title":"The Effect of Normalization Norms in Multiple Attribute Decision Making Models: A Case Study in Gear Material Selection","volume":"29","author":"Milani","year":"2005","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.eswa.2019.06.035","article-title":"Effects of Normalization on the Entropy-Based TOPSIS Method","volume":"136","author":"Chen","year":"2019","journal-title":"Expert Syst. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1007\/s10479-018-2945-5","article-title":"Normalization in TOPSIS-Based Approaches with Data of Different Nature: Application to the Ranking of Mathematical Videos","volume":"296","author":"Liern","year":"2021","journal-title":"Ann. Oper. Res."},{"key":"ref_15","unstructured":"Camarinha-Matos, L.M., Adu-Kankam, K.O., and Julashokri, M. (2018). Selection of Normalization Technique for Weighted Average Multi-Criteria Decision Making. Proceedings of the Technological Innovation for Resilient Systems, Springer International Publishing."},{"key":"ref_16","first-page":"857","article-title":"MACONT: Mixed Aggregation by Comprehensive Normalization Technique for Multi-Criteria Analysis","volume":"31","author":"Wen","year":"2020","journal-title":"Informatica"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1007\/978-981-16-7414-3_6","article-title":"Why Does the Choice of Normalization Technique Matter in Decision-Making","volume":"Volume 407","author":"Kulkarni","year":"2022","journal-title":"Multiple Criteria Decision Making"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Krishnan, A.R. (2022). Past Efforts in Determining Suitable Normalization Methods for Multi-Criteria Decision-Making: A Short Survey. Front. Big Data, 5.","DOI":"10.3389\/fdata.2022.990699"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Mukhametzyanov, I.Z. (2023). Normalization of Multidimensional Data for Multi-Criteria Decision Making Problems: Inversion, Displacement, Asymmetry, Springer International Publishing. International Series in Operations Research & Management Science.","DOI":"10.1007\/978-3-031-33837-3"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Shih, H.-S., and Olson, D.L. (2022). Rank Reversal in TOPSIS. TOPSIS and Its Extensions: A Distance-Based MCDM Approach, Springer International Publishing.","DOI":"10.1007\/978-3-031-09577-1"},{"key":"ref_21","first-page":"105","article-title":"Sensitivity Study of TOPSIS and COPRAS Methods with Respect to Normalization Techniques","volume":"10","year":"2022","journal-title":"Balt. J. Mod. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Malefaki, S., Markatos, D., Filippatos, A., and Pantelakis, S. (2025). A Comparative Analysis of Multi-Criteria Decision-Making Methods and Normalization Techniques in Holistic Sustainability Assessment for Engineering Applications. Aerospace, 12.","DOI":"10.3390\/aerospace12020100"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Chakraborty, S., and Yeh, C.-H. (2009). A Simulation Comparison of Normalization Procedures for TOPSIS. Proceedings of the 2009 International Conference on Computers and Industrial Engineering (CIE39), Troyes, France, 6\u20139 July 2009., IEEE Institute of Electrical and Electronics Engineers.","DOI":"10.1109\/ICCIE.2009.5223811"},{"key":"ref_24","first-page":"871","article-title":"Investigation of Different Normalization Methods for TOPSIS","volume":"32","author":"Liao","year":"2012","journal-title":"Beijing Ligong Daxue Xuebao"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1007\/978-3-030-78288-7_13","article-title":"Assessing Normalization Techniques for TOPSIS Method","volume":"Volume 626","author":"Ferreira","year":"2021","journal-title":"Technological Innovation for Applied AI Systems"},{"key":"ref_26","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. Tech. J."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1016\/j.asoc.2017.07.033","article-title":"Entropy-Based Weights on Decision Makers in Group Decision-Making Setting with Hybrid Preference Representations","volume":"60","author":"Yue","year":"2017","journal-title":"Appl. Soft Comput."},{"key":"ref_28","first-page":"3564835","article-title":"Effectiveness of Entropy Weight Method in Decision-Making","volume":"2020","author":"Zhu","year":"2020","journal-title":"Math. Probl. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"264","DOI":"10.3390\/encyclopedia2010018","article-title":"Entropy","volume":"2","author":"Tsallis","year":"2022","journal-title":"Encyclopedia"},{"key":"ref_30","first-page":"547","article-title":"On Measures of Entropy and Information","volume":"Volume 4","year":"1961","journal-title":"Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, Berkeley, CA, USA, 20 June\u201330 July 1960"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1142\/S0219622016300019","article-title":"Development of TOPSIS Method to Solve Complicated Decision-Making Problems\u2014An Overview on Developments from 2000 to 2015","volume":"15","author":"Zavadskas","year":"2016","journal-title":"Int. J. Inf. Technol. Decis. Mak."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1016\/j.eswa.2017.02.016","article-title":"A Bibliometric-Based Survey on AHP and TOPSIS Techniques","volume":"78","author":"Zyoud","year":"2017","journal-title":"Expert Syst. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1016\/j.mcm.2006.03.023","article-title":"An Extension of TOPSIS for Group Decision Making","volume":"45","author":"Shih","year":"2007","journal-title":"Math. Comput. Model."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Chen, B., Wang, J., Zhao, H., and Principe, J.C. (2016). Insights into Entropy as a Measure of Multivariate Variability. Entropy, 18.","DOI":"10.3390\/e18050196"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"601","DOI":"10.15388\/Informatica.2006.158","article-title":"Evaluation of Ranking Accuracy in Multi-Criteria Decisions","volume":"17","author":"Zavadskas","year":"2006","journal-title":"Informatica"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"76","DOI":"10.31181\/dmame210402076i","article-title":"Specific Character of Objective Methods for Determining Weights of Criteria in MCDM Problems: Entropy, CRITIC and SD","volume":"4","author":"Mukhametzyanov","year":"2021","journal-title":"Decis. Mak. Appl. Manag. Eng."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"4207","DOI":"10.1016\/j.ins.2010.07.009","article-title":"Determining Objective Weights with Intuitionistic Fuzzy Entropy Measures: A Comparative Analysis","volume":"180","author":"Chen","year":"2010","journal-title":"Inf. Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2185","DOI":"10.1007\/s12665-015-4208-y","article-title":"On the Sensitivity of Entropy Weight to Sample Statistics in Assessing Water Quality: Statistical Analysis Based on Large Stochastic Samples","volume":"74","author":"Wu","year":"2015","journal-title":"Environ. Earth Sci."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"He, D., Xu, J., and Chen, X. (2016). Information-Theoretic-Entropy Based Weight Aggregation Method in Multiple-Attribute Group Decision-Making. Entropy, 18.","DOI":"10.3390\/e18060171"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1471","DOI":"10.1016\/j.jmrt.2020.12.114","article-title":"Revealing the Benefits of Entropy Weights Method for Multi-Objective Optimization in Machining Operations: A Critical Review","volume":"10","author":"Kumar","year":"2021","journal-title":"J. Mater. Res. Technol."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Roszkowska, E., and Wachowicz, T. (2024). Impact of Normalization on Entropy-Based Weights in Hellble Development in Thewig\u2019s Method: A Case Study on Evaluating Sustaina Education Area. Entropy, 26.","DOI":"10.3390\/e26050365"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"864","DOI":"10.1108\/03684921011046627","article-title":"Study on Risk Assessment of Water Security of Drought Periods Based on Entropy Weight Methods","volume":"39","author":"Dong","year":"2010","journal-title":"Kybernetes"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1108\/MEDAR-11-2016-0100","article-title":"Corporate Sustainability Measurement Based on Entropy Weight and TOPSIS: A Turkish Banking Case Study","volume":"25","author":"Aras","year":"2017","journal-title":"Meditari Account. Res."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1108\/IJOES-06-2019-0101","article-title":"Multi-Criteria Decision-Making in the Evaluation of Environmental Quality of OECD Countries: The Entropy Weight and VIKOR Methods","volume":"36","author":"Dang","year":"2019","journal-title":"Int. J. Ethics Syst."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Wang, Z.-X., Li, D.-D., and Zheng, H.-H. (2018). The External Performance Appraisal of China Energy Regulation: An Empirical Study Using a TOPSIS Method Based on Entropy Weight and Mahalanobis Distance. Int. J. Environ. Res. Public Health, 15.","DOI":"10.3390\/ijerph15020236"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"2263","DOI":"10.1016\/j.rser.2009.06.021","article-title":"Review on Multi-Criteria Decision Analysis Aid in Sustainable Energy Decision-Making","volume":"13","author":"Wang","year":"2009","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Zardari, N.H., Ahmed, K., Shirazi, S.M., and Yusop, Z.B. (2015). Weighting Methods and Their Effects on Multi-Criteria Decision Making Model Outcomes in Water Resources Management, Springer.","DOI":"10.1007\/978-3-319-12586-2"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1991","DOI":"10.1007\/s10115-021-01588-y","article-title":"Location Selection by Multi-Criteria Decision-Making Methods Based on Objective and Subjective Weightings","volume":"63","year":"2021","journal-title":"Knowl. Inf. Syst."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1007\/s11869-020-00930-7","article-title":"Comprehensive Evaluation of Urban Air Quality Using the Relative Entropy Theory and Improved TOPSIS Method","volume":"14","author":"Lin","year":"2021","journal-title":"Air Qual. Atmos. Health"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"115","DOI":"10.21859\/ijtmgh-040406","article-title":"An Entropy (Shannon) Based Approach for Determining Importance Weights of Influencing Factors in Selecting Medical Tourism Destinations","volume":"4","author":"Nakhaei","year":"2016","journal-title":"Int. J. Travel Med. Glob. Health"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1016\/S0377-2217(97)00246-4","article-title":"A Weight-Assessing Method with Habitual Domains","volume":"110","author":"Tzeng","year":"1998","journal-title":"Eur. J. Oper. Res."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/0309-1708(82)90040-9","article-title":"Model Selection in Multiobjective Decision Making for River Basin Planning","volume":"5","author":"Duckstein","year":"1982","journal-title":"Adv. Water Resour."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/S0377-2217(98)00141-6","article-title":"A Subjective and Objective Integrated Approach to Determine Attribute Weights","volume":"112","author":"Ma","year":"1999","journal-title":"Eur. J. Oper. Res."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1080\/09638180.2016.1234402","article-title":"Differential Weighting of Objective Versus Subjective Measures in Performance Evaluation: Experimental Evidence","volume":"27","author":"Dai","year":"2018","journal-title":"Eur. Account. Rev."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Qu, W., Li, J., Song, W., Li, X., Zhao, Y., Dong, H., Wang, Y., Zhao, Q., and Qi, Y. (2022). Entropy-Weight-Method-Based Integrated Models for Short-Term Intersection Traffic Flow Prediction. Entropy, 24.","DOI":"10.3390\/e24070849"},{"key":"ref_56","first-page":"102","article-title":"A Simulation Based Comparative Study of Normalization Procedures in Multiattribute Decision Making","volume":"Volume 6","author":"Chakraborty","year":"2007","journal-title":"Proceedings of the 6th Conference on 6th WSEAS Int. Conf. on Artificial Intelligence, Knowledge Engineering and Data Bases"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1016\/j.matdes.2014.09.022","article-title":"A State-of-the-Art Survey on the Influence of Normalization Techniques in Ranking: Improving the Materials Selection Process in Engineering Design","volume":"65","author":"Jahan","year":"2014","journal-title":"Mater. Des."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/j.measurement.2017.12.019","article-title":"Comparative Study on the Evaluation and Benchmarking Information Hiding Approaches Based Multi-Measurement Analysis Using TOPSIS Method with Different Normalisation, Separation and Context Techniques","volume":"117","author":"Zaidan","year":"2018","journal-title":"Measurement"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"185","DOI":"10.15388\/Informatica.2014.10","article-title":"Comparative Analysis of Normalization Procedures in TOPSIS Method: With an Application to Turkish Deposit Banking Market","volume":"25","year":"2014","journal-title":"Informatica"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.mcm.2011.12.022","article-title":"On Rank Reversal and TOPSIS Method","volume":"56","author":"Lamata","year":"2012","journal-title":"Math. Comput. Model."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/j.cie.2019.04.023","article-title":"A New Approach to Avoid Rank Reversal Cases in the TOPSIS Method","volume":"132","author":"Aires","year":"2019","journal-title":"Comput. Ind. Eng."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1590\/0101-7438.2018.038.02.0331","article-title":"The Rank Reversal Problem in Multi-Criteria Decision Making: A Literature Review","volume":"38","author":"Aires","year":"2018","journal-title":"Pesqui. Oper."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"116821","DOI":"10.1016\/j.eswa.2022.116821","article-title":"Normalization Method for Quantitative and Qualitative Attributes in Multiple Attribute Decision-Making Problems","volume":"198","author":"Pena","year":"2022","journal-title":"Expert Syst. Appl."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1007\/s10726-025-09929-w","article-title":"Enhancing TOPSIS to Evaluate Negotiation Offers with Subjectively Defined Reference Points","volume":"34","author":"Wachowicz","year":"2025","journal-title":"Group Decis. Negot."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"2222","DOI":"10.1016\/j.procs.2020.09.259","article-title":"How the Normalization of the Decision Matrix Influences the Results in the VIKOR Method?","volume":"176","year":"2020","journal-title":"Procedia Comput. Sci."},{"key":"ref_66","unstructured":"Camarinha-Matos, L.M., Falc\u00e3o, A.J., Vafaei, N., and Najdi, S. (2016). Normalization Techniques for Multi-Criteria Decision Making: Analytical Hierarchy Process Case Study. Proceedings of the Technological Innovation for Cyber-Physical Systems, Springer International Publishing."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1590\/S0101-74382010000300010","article-title":"A Multicriteria Group Decision Model Aggregating the Preferences of Decision-Makers Based on Electre Methods","volume":"30","author":"Alencar","year":"2010","journal-title":"Pesqui. Oper."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"2051","DOI":"10.1016\/j.procs.2019.09.378","article-title":"Influence of Various Normalization Methods in PROMETHEE II: An Empirical Study on the Selection of the Airport Location","volume":"159","author":"Palczewski","year":"2019","journal-title":"Procedia Comput. Sci."},{"key":"ref_69","doi-asserted-by":"crossref","unstructured":"Roszkowska, E., Filipowicz-Chomko, M., \u0141yczkowska-Han\u0107kowiak, A., and Majewska, E. (2024). Extended Hellwig\u2019s Method Utilizing Entropy-Based Weights and Mahalanobis Distance: Applications in Evaluating Sustainable Development in the Education Area. Entropy, 26.","DOI":"10.3390\/e26030197"},{"key":"ref_70","doi-asserted-by":"crossref","unstructured":"Sa\u0142abun, W., W\u0105tr\u00f3bski, J., and Shekhovtsov, A. (2020). Are MCDA Methods Benchmarkable? A Comparative Study of TOPSIS, VIKOR, COPRAS, and PROMETHEE II Methods. Symmetry, 12.","DOI":"10.3390\/sym12091549"},{"key":"ref_71","first-page":"40","article-title":"The Mean Error Estimation of TOPSIS Method Using a Fuzzy Reference Models","volume":"7","year":"2013","journal-title":"J. Theor. Appl. Comput. Sci."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"36","DOI":"10.4028\/www.scientific.net\/AMR.204-210.36","article-title":"Rank Reversal and Rank Preservation in TOPSIS","volume":"204","author":"Kong","year":"2011","journal-title":"Adv. Mater. Res."},{"key":"ref_73","first-page":"219","article-title":"Propozycja Procedury Wspomagaj\u0105cej Wyb\u00f3r Metody Porz\u0105dkowania Liniowego","volume":"62","author":"Luty","year":"2015","journal-title":"Przegl\u0105d Stat."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/1\/114\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T05:31:59Z","timestamp":1769059919000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/1\/114"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,18]]},"references-count":73,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["e28010114"],"URL":"https:\/\/doi.org\/10.3390\/e28010114","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,18]]}}}