{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,2]],"date-time":"2026-09-02T05:20:36Z","timestamp":1788326436062,"version":"build-2803163510"},"reference-count":49,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2026,8,31]],"date-time":"2026-08-31T00:00:00Z","timestamp":1788134400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["62503232"],"award-info":[{"award-number":["62503232"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"award":["62503232"],"award-info":[{"award-number":["62503232"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]},{"name":"College Students\u2019 Innovation Training Projects of Nanjing Forestry University","award":["202610298066Z"],"award-info":[{"award-number":["202610298066Z"]}]},{"award":["202610298066Z"],"award-info":[{"award-number":["202610298066Z"]}],"id":[{"id":"https:\/\/ror.org\/03m96p165","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Distance measures in random permutation set (RPS) theory are crucial for characterizing inconsistency among permutation-based information distributions. However, existing RPS discrepancy measures do not explicitly distinguish ordering conflicts according to their positional importance under propensity semantics. To address this issue, this paper proposes a new RPS distance, termed MEOWA-KTC, by combining maximum-entropy-based ordered weighted averaging (MEOWA) weights with Kendall\u2019s tau coefficient (KTC). Specifically, MEOWA-KTC constructs a top-weighted similarity between permutation events by using KTC to evaluate the ordinal consistency of corresponding sub-permutations and MEOWA weights controlled by an adjustable orness parameter to emphasize discrepancies at leading positions. Additionally, a spectral correction is applied to ensure that the proposed distance satisfies the metric axioms. Numerical examples and ablation results demonstrate the positional sensitivity of the proposed distance and the respective contributions of MEOWA weighting and KTC. Based on this distance, a fusion model is further developed to derive source support degrees and fusion weights from pairwise RPS distances. In the threat-assessment application, the proposed method produces stable decisions and generally larger decision margins than the benchmark methods. Monte Carlo experiments further demonstrate its robustness to mass-distribution and permutation-order noise.<\/jats:p>","DOI":"10.3390\/e28090970","type":"journal-article","created":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T12:41:18Z","timestamp":1788266478000},"page":"970","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["MEOWA-KTC: A New Distance Measure for Random Permutation Sets Based on MEOWA Weights and Kendall\u2019s Tau Coefficient"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-7825-067X","authenticated-orcid":false,"given":"Chengyi","family":"Jin","sequence":"first","affiliation":[{"name":"College of Information Science and Technology & Artificial Intelligence, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"methodology","vocabulary":"credit"},{"role":"software","vocabulary":"credit"},{"role":"validation","vocabulary":"credit"},{"role":"writing-original-draft","vocabulary":"credit"},{"role":"visualization","vocabulary":"credit"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-8691-8785","authenticated-orcid":false,"given":"Luyuan","family":"Chen","sequence":"additional","affiliation":[{"name":"College of Information Science and Technology & Artificial Intelligence, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"methodology","vocabulary":"credit"},{"role":"supervision","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"},{"role":"validation","vocabulary":"credit"}]},{"given":"Hao","family":"Li","sequence":"additional","affiliation":[{"name":"College of Information Science and Technology & Artificial Intelligence, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"software","vocabulary":"credit"},{"role":"methodology","vocabulary":"credit"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,31]]},"reference":[{"key":"ref_1","unstructured":"Feller, W. (1991). An Introduction to Probability Theory and Its Applications, John Wiley & Sons."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Solaiman, B., Gu\u00e9riot, D., Almouahed, S., Alsahwa, B., and Boss\u00e9, \u00c9. (2021). A new hybrid possibilistic-probabilistic decision-making scheme for classification. Entropy, 23.","DOI":"10.3390\/e23010067"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"102736","DOI":"10.1016\/j.inffus.2024.102736","article-title":"A survey of evidential clustering: Definitions, methods, and applications","volume":"115","author":"Zhang","year":"2025","journal-title":"Inf. Fusion"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"202204","DOI":"10.1007\/s11432-024-4563-x","article-title":"Measure-based uncertainty with Dempster-Shafer structure","volume":"68","author":"Zhou","year":"2025","journal-title":"Sci. China Inf. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2010","DOI":"10.1109\/TCYB.2025.3545195","article-title":"Z-Number Generation Model and Its Application in a Rule-Based Classification System","volume":"55","author":"Li","year":"2025","journal-title":"IEEE Trans. Cybern."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"121310","DOI":"10.1016\/j.ins.2024.121310","article-title":"New distance measures of complex Fermatean fuzzy sets with applications in decision making and clustering problems","volume":"686","author":"Liu","year":"2025","journal-title":"Inf. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"100764","DOI":"10.1016\/j.spasta.2023.100764","article-title":"Fuzzy clustering of spatial interval-valued data","volume":"57","author":"Federico","year":"2023","journal-title":"Spat. Stat."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"103106","DOI":"10.1016\/j.inffus.2025.103106","article-title":"An empirical survey-type analysis of uncertainty measures for the fusion of crisp and fuzzy bodies of evidence","volume":"121","author":"Barhoumi","year":"2025","journal-title":"Inf. Fusion"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Zhou, X., Chen, L., Mao, E., and He, Y. (2026). ODE-BPA: A Novel Basic Probability Assignment Generation Method Based on OTSU and Deng Entropy. Entropy, 28.","DOI":"10.3390\/e28060634"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Abell\u00e1n, J., L\u00f3pez-Gay, A., Ben\u00edtez, M.I.A., and Castellano, F.J.G. (2026). A comparison of algorithms to achieve the maximum entropy in the theory of evidence. Entropy, 28.","DOI":"10.3390\/e28020247"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"102538","DOI":"10.1016\/j.inffus.2024.102538","article-title":"Evidence representation of uncertain information on a frame of discernment with semantic association","volume":"111","author":"Deng","year":"2024","journal-title":"Inf. Fusion"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"11861","DOI":"10.1007\/s10489-022-03960-z","article-title":"A framework for the fusion of non-exclusive and incomplete information on the basis of D number theory","volume":"53","author":"Deng","year":"2023","journal-title":"Appl. Intell."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1427","DOI":"10.1109\/TKDE.2026.3652983","article-title":"A CDGFN-based quantum multisource information fusion with its application in time series classification","volume":"38","author":"Yu","year":"2026","journal-title":"IEEE Trans. Knowl. Data Eng."},{"key":"ref_14","first-page":"17848","article-title":"An adaptive quantum circuit of Dempster\u2019s rule of combination for uncertain pattern classification","volume":"38","author":"Xiao","year":"2025","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"129992","DOI":"10.1016\/j.eswa.2025.129992","article-title":"A dependence assessment method based on quantum model of mass function in human reliability analysis","volume":"299","author":"Su","year":"2026","journal-title":"Expert Syst. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"10246","DOI":"10.1109\/TPAMI.2024.3438349","article-title":"Random permutation set reasoning","volume":"46","author":"Deng","year":"2024","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"109034","DOI":"10.1016\/j.fss.2024.109034","article-title":"A new orthogonal sum in random permutation set","volume":"490","author":"Wang","year":"2024","journal-title":"Fuzzy Sets Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"837","DOI":"10.1109\/TKDE.2024.3484009","article-title":"Order-2 probabilistic information fusion on random permutation set","volume":"37","author":"Zhou","year":"2024","journal-title":"IEEE Trans. Knowl. Data Eng."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"102083","DOI":"10.1016\/j.inffus.2023.102083","article-title":"Conjunctive and disjunctive combination rules in random permutation set theory: A layer-2 belief structure perspective","volume":"102","author":"Zhou","year":"2024","journal-title":"Inf. Fusion"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"13125","DOI":"10.1007\/s11071-023-08506-7","article-title":"Marginalization in random permutation set theory: From the cooperative game perspective","volume":"111","author":"Zhou","year":"2023","journal-title":"Nonlinear Dyn."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1242","DOI":"10.1109\/TFUZZ.2026.3654276","article-title":"DDHRPS: A Data-Driven Hierarchical Method for Constructing Random Permutation Set From the Perspective of Layer-2 Belief Structure","volume":"34","author":"Chen","year":"2026","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"122329","DOI":"10.1016\/j.ins.2025.122329","article-title":"OCCM-RPS: Ordered credal C-means clustering based on random permutation set","volume":"717","author":"Chen","year":"2025","journal-title":"Inf. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"114112","DOI":"10.1016\/j.engappai.2026.114112","article-title":"A new distance approach to measure the evidence conflict in orderable set and its applications","volume":"169","author":"Su","year":"2026","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1016\/j.ins.2023.01.105","article-title":"The distance of random permutation set","volume":"628","author":"Chen","year":"2023","journal-title":"Inf. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/S1566-2535(01)00026-4","article-title":"A new distance between two bodies of evidence","volume":"2","author":"Jousselme","year":"2001","journal-title":"Inf. Fusion"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"105701","DOI":"10.1016\/j.engappai.2022.105701","article-title":"Permutation Jensen\u2013Shannon divergence for Random Permutation Set","volume":"119","author":"Chen","year":"2023","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"121784","DOI":"10.1016\/j.eswa.2023.121784","article-title":"Symmetric Renyi-Permutation divergence and conflict management for random permutation set","volume":"238","author":"Chen","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3037","DOI":"10.1109\/TII.2024.3523536","article-title":"Multiview Uncertainty-Aware Fusion for Human Activity Recognition via Dempster\u2013Shafer Theory","volume":"21","author":"Dong","year":"2025","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1199","DOI":"10.1109\/TNNLS.2023.3329712","article-title":"Representation of imprecision in deep neural networks for image classification","volume":"36","author":"Zhang","year":"2023","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"113821","DOI":"10.1016\/j.asoc.2025.113821","article-title":"Probabilistic transformation of basic probability assignment based on weighted visibility graph networks","volume":"184","author":"Tang","year":"2025","journal-title":"Appl. Soft Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"103286","DOI":"10.1016\/j.inffus.2025.103286","article-title":"An information fusion model of mutual influence between focal elements: A perspective on interference effects in Dempster\u2013Shafer evidence theory","volume":"124","author":"Gao","year":"2025","journal-title":"Inf. Fusion"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"104572","DOI":"10.1016\/j.inffus.2026.104572","article-title":"Basic Belief Assignment Determination Based on Contrastive Learning for Pattern Classification","volume":"136","author":"Li","year":"2026","journal-title":"Inf. Fusion"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"14709","DOI":"10.1109\/TPAMI.2023.3310594","article-title":"Higher order fractal belief R\u00e9nyi divergence with its applications in pattern classification","volume":"45","author":"Huang","year":"2023","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"102458","DOI":"10.1016\/j.inffus.2024.102458","article-title":"An effective multi-source data fusion approach based on \u03b1-divergence in belief functions theory with applications to air target recognition and fault diagnosis","volume":"110","author":"Liu","year":"2024","journal-title":"Inf. Fusion"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"414","DOI":"10.1109\/TVT.2023.3310938","article-title":"A data-driven maneuvering target tracking method aided with partial models","volume":"73","author":"Liu","year":"2023","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"102863","DOI":"10.1016\/j.techsoc.2025.102863","article-title":"Exploring obstacles to the use of unmanned aerial vehicles in emergency rescue: A BWM-DEMATEL approach","volume":"81","author":"Li","year":"2025","journal-title":"Technol. Soc."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"123786","DOI":"10.1016\/j.eswa.2024.123786","article-title":"Evidential software risk assessment model on ordered frame of discernment","volume":"250","author":"Chen","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Dempster, A.P. (2008). Upper and lower probabilities induced by a multivalued mapping. Classic Works of the Dempster-Shafer Theory of Belief Functions, Springer.","DOI":"10.1007\/978-3-540-44792-4_3"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Shafer, G. (2020). A Mathematical Theory of Evidence, Princeton University Press.","DOI":"10.2307\/j.ctv10vm1qb"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"109913","DOI":"10.1016\/j.fss.2026.109913","article-title":"Ternary coding of maximum Deng entropy","volume":"539","author":"Zhan","year":"2026","journal-title":"Fuzzy Sets Syst."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"109369","DOI":"10.1016\/j.cnsns.2025.109369","article-title":"Local entropy and nonextensivity of networks ensemble","volume":"152","author":"Li","year":"2026","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/j.chaos.2016.12.024","article-title":"Analyzing properties of Deng entropy in the theory of evidence","volume":"95","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"132004","DOI":"10.1016\/j.eswa.2026.132004","article-title":"A novel method for generating permutation mass functions using permutations to represent class bias","volume":"319","author":"Li","year":"2026","journal-title":"Expert Syst. Appl."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2650014","DOI":"10.1142\/S0218348X26500143","article-title":"A fractal-based permutation belief entropy for uncertainty measure in random permutation set","volume":"34","author":"Liu","year":"2026","journal-title":"Fractals"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"129513","DOI":"10.1016\/j.eswa.2025.129513","article-title":"BDOS: An orthogonal sum based on belief of permutation events and element distances in random permutation set","volume":"298","author":"Su","year":"2026","journal-title":"Expert Syst. Appl."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"4542","DOI":"10.15837\/ijccc.2022.1.4542","article-title":"Random permutation set","volume":"17","author":"Deng","year":"2022","journal-title":"Int. J. Comput. Commun. Control"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1109\/21.87068","article-title":"On ordered weighted averaging aggregation operators in multicriteria decisionmaking","volume":"18","author":"Yager","year":"1988","journal-title":"IEEE Trans. Syst. Man Cybern."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/0020-0255(94)00109-O","article-title":"Analytic properties of maximum entropy OWA operators","volume":"85","author":"Filev","year":"1995","journal-title":"Inf. Sci."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1093\/biomet\/30.1-2.81","article-title":"A new measure of rank correlation","volume":"30","author":"Kendall","year":"1938","journal-title":"Biometrika"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/9\/970\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,2]],"date-time":"2026-09-02T04:32:28Z","timestamp":1788323548000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/9\/970"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,31]]},"references-count":49,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2026,9]]}},"alternative-id":["e28090970"],"URL":"https:\/\/doi.org\/10.3390\/e28090970","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,31]]}}}