{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T16:15:23Z","timestamp":1771949723395,"version":"3.50.1"},"reference-count":57,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,24]],"date-time":"2020-04-24T00:00:00Z","timestamp":1587686400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key Research and Development Project of China","award":["2018YFF0214700"],"award-info":[{"award-number":["2018YFF0214700"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61672117"],"award-info":[{"award-number":["61672117"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Dempster\u2013Shafer evidence theory (DS theory) has some superiorities in uncertain information processing for a large variety of applications. However, the problem of how to quantify the uncertainty of basic probability assignment (BPA) in DS theory framework remain unresolved. The goal of this paper is to define a new belief entropy for measuring uncertainty of BPA with desirable properties. The new entropy can be helpful for uncertainty management in practical applications such as decision making. The proposed uncertainty measure has two components. The first component is an improved version of Dubois\u2013Prade entropy, which aims to capture the non-specificity portion of uncertainty with a consideration of the element number in frame of discernment (FOD). The second component is adopted from Nguyen entropy, which captures conflict in BPA. We prove that the proposed entropy satisfies some desired properties proposed in the literature. In addition, the proposed entropy can be reduced to Shannon entropy if the BPA is a probability distribution. Numerical examples are presented to show the efficiency and superiority of the proposed measure as well as an application in decision making.<\/jats:p>","DOI":"10.3390\/e22040487","type":"journal-article","created":{"date-parts":[[2020,4,24]],"date-time":"2020-04-24T11:42:14Z","timestamp":1587728534000},"page":"487","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["An Improved Total Uncertainty Measure in the Evidence Theory and Its Application in Decision Making"],"prefix":"10.3390","volume":"22","author":[{"given":"Miao","family":"Qin","sequence":"first","affiliation":[{"name":"School of Big Data and Software Engineering, Chongqing University, Chongqing 401331, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2568-9628","authenticated-orcid":false,"given":"Yongchuan","family":"Tang","sequence":"additional","affiliation":[{"name":"School of Big Data and Software Engineering, Chongqing University, Chongqing 401331, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Junhao","family":"Wen","sequence":"additional","affiliation":[{"name":"School of Big Data and Software Engineering, Chongqing University, Chongqing 401331, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.dss.2018.10.009","article-title":"Ensemble machine learning models for aviation incident risk prediction","volume":"116","author":"Zhang","year":"2019","journal-title":"Decis. Support Syst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"584","DOI":"10.1002\/int.22066","article-title":"A new divergence measure for basic probability assignment and its applications in extremely uncertain environments","volume":"34","author":"Fei","year":"2019","journal-title":"Int. J. Intell. Syst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1007\/s40815-017-0426-7","article-title":"Evidential supplier selection based on interval data fusion","volume":"20","author":"He","year":"2018","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_4","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_5","unstructured":"Feller, W. (1950). An Introduction to Probability Theory and Its Applications II, John Wiley & Sons."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1214\/aoms\/1177698950","article-title":"Upper and Lower Probabilities Induced by a Multi-Valued Mapping","volume":"38","author":"Dempster","year":"1967","journal-title":"Ann. Math. Stat."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Shafer, G. (1976). A Mathematical Theory of Evidence, Princeton University Press.","DOI":"10.1515\/9780691214696"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.firesaf.2016.10.003","article-title":"Reasoning on multi-sensor geographic smoke spread data for fire development and risk analysis","volume":"86","author":"Vandecasteele","year":"2016","journal-title":"Fire Saf. J."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"66813","DOI":"10.1109\/ACCESS.2020.2986807","article-title":"A Novel Failure Mode and Effects Analysis Model Using Triangular Distribution-Based Basic Probability Assignment in the Evidence Theory","volume":"8","author":"Zheng","year":"2020","journal-title":"IEEE Access"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/j.autcon.2016.09.003","article-title":"Perceiving safety risk of buildings adjacent to tunneling excavation: An information fusion approach","volume":"73","author":"Zhang","year":"2017","journal-title":"Autom. Constr."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.inffus.2018.04.003","article-title":"Multi-sensor data fusion based on the belief divergence measure of evidences and the belief entropy","volume":"46","author":"Xiao","year":"2019","journal-title":"Inf. Fusion"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/j.eswa.2015.09.006","article-title":"A hybrid fuzzy evaluation method for curtain grouting efficiency assessment based on an AHP method extended by D numbers","volume":"44","author":"Fan","year":"2016","journal-title":"Expert Syst. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"886","DOI":"10.1016\/j.ejor.2015.05.042","article-title":"A group evidential reasoning approach based on expert reliability","volume":"246","author":"Fu","year":"2015","journal-title":"Eur. J. Oper. Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1007\/s13735-017-0134-y","article-title":"Toward semantic content-based image retrieval using Dempster\u2013Shafer theory in multi-label classification framework","volume":"6","author":"Moghaddam","year":"2017","journal-title":"Int. J. Multimed. Inf. Retr."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2783","DOI":"10.1109\/TSMC.2016.2622247","article-title":"Hybrid Classification System for Uncertain Data","volume":"47","author":"Liu","year":"2017","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"618","DOI":"10.1109\/TFUZZ.2019.2911915","article-title":"Evidence combination based on credal belief redistribution for pattern classification","volume":"28","author":"Liu","year":"2019","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1109\/TFUZZ.2018.2869125","article-title":"BPEC: Belief-peaks evidential clustering","volume":"27","author":"Su","year":"2018","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1259","DOI":"10.1007\/s10489-019-01576-4","article-title":"Belief-peaks clustering based on fuzzy label propagation","volume":"50","author":"Meng","year":"2020","journal-title":"Appl. Intell."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1007\/s10479-014-1657-8","article-title":"Determining attribute weights to improve solution reliability and its application to selecting leading industries","volume":"245","author":"Fu","year":"2016","journal-title":"Ann. Oper. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"37926","DOI":"10.1109\/ACCESS.2020.2975989","article-title":"An Improved Method to Manage Conflict Data Using Elementary Belief Assignment Function in the Evidence Theory","volume":"8","author":"Li","year":"2020","journal-title":"IEEE Access"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/s10044-016-0575-6","article-title":"Evidence combination based on credibility and non-specificity","volume":"21","author":"Song","year":"2018","journal-title":"Pattern Anal. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"8207","DOI":"10.1007\/s00500-018-3455-8","article-title":"Combining conflicting evidence using the DEMATEL method","volume":"23","author":"Zhang","year":"2019","journal-title":"Soft Comput."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Martin, A. (2019). Conflict management in information fusion with belief functions. Information Quality in Information Fusion and Decision Making, Springer.","DOI":"10.1007\/978-3-030-03643-0_4"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/0165-0114(87)90088-1","article-title":"Properties of measures of information in evidence and possibility theories","volume":"24","author":"Dubois","year":"1987","journal-title":"Fuzzy Sets Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1016\/j.ijar.2020.01.004","article-title":"On properties of a new decomposable entropy of Dempster-Shafer belief functions","volume":"119","author":"Shenoy","year":"2020","journal-title":"Int. J. Approx. Reason."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"530","DOI":"10.1007\/s10489-015-0661-2","article-title":"Generalized evidence theory","volume":"43","author":"Deng","year":"2015","journal-title":"Appl. Intell."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"166410","DOI":"10.1109\/ACCESS.2019.2954132","article-title":"A Note on Incomplete Information Modeling in the Evidence Theory","volume":"7","author":"Zhou","year":"2019","journal-title":"IEEE Access"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1016\/j.chaos.2016.07.014","article-title":"Deng entropy","volume":"91","author":"Deng","year":"2016","journal-title":"Chaos Solitons Fractals"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Zheng, H., and Tang, Y. (2020). Deng Entropy Weighted Risk Priority Number Model for Failure Mode and Effects Analysis. Entropy, 22.","DOI":"10.3390\/e22030280"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"898","DOI":"10.1007\/s10489-016-0870-3","article-title":"An improved distance-based total uncertainty measure in belief function theory","volume":"46","author":"Deng","year":"2017","journal-title":"Appl. Intell."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"631","DOI":"10.15837\/ijccc.2017.5.2950","article-title":"An uncertainty measure for interval-valued evidences","volume":"12","author":"Jiang","year":"2017","journal-title":"Int. J. Comput. Commun. Control"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.knosys.2015.11.014","article-title":"A new distance-based total uncertainty measure in the theory of belief functions","volume":"94","author":"Yang","year":"2016","journal-title":"Knowl.-Based Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1672","DOI":"10.1007\/s10489-017-1024-y","article-title":"Uncertainty measure in evidence theory with its applications","volume":"48","author":"Wang","year":"2018","journal-title":"Appl. Intell."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/j.ijar.2012.08.004","article-title":"Generalized information theory for hints","volume":"54","author":"Pouly","year":"2013","journal-title":"Int. J. Approx. Reason."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"390","DOI":"10.1016\/j.knosys.2018.08.001","article-title":"Interval valued entropies for Dempster\u2013Shafer structures","volume":"161","author":"Yager","year":"2018","journal-title":"Knowl.-Based Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"909","DOI":"10.1080\/03610926.2018.1554123","article-title":"New definition of the cross entropy based on the Dempster-Shafer theory and its application in a decision-making process","volume":"49","author":"Khalaj","year":"2020","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"3302","DOI":"10.1002\/int.22195","article-title":"A total uncertainty measure for D numbers based on belief intervals","volume":"34","author":"Deng","year":"2019","journal-title":"Int. J. Intell. Syst."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/j.ijar.2018.09.001","article-title":"A correlation coefficient for belief functions","volume":"103","author":"Jiang","year":"2018","journal-title":"Int. J. Approx. Reason."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Gao, X., and Deng, Y. (2020). The Pseudo-Pascal Triangle of Maximum Deng Entropy. Int. J. Comput. Commun. Control, 15.","DOI":"10.15837\/ijccc.2020.1.3735"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.ijar.2017.10.010","article-title":"A new definition of entropy of belief functions in the Dempster\u2013Shafer theory","volume":"92","author":"Shenoy","year":"2018","journal-title":"Int. J. Approx. Reason."},{"key":"ref_41","first-page":"145","article-title":"On entropy of random sets and possibility distributions","volume":"1","author":"Nguyen","year":"1987","journal-title":"Anal. Fuzzy Inf."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1144","DOI":"10.1007\/s40815-019-00620-2","article-title":"A multiple-criteria decision-making method based on D numbers and belief entropy","volume":"21","author":"Xiao","year":"2019","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"3950","DOI":"10.1007\/s10489-018-1188-0","article-title":"Sensor dynamic reliability evaluation based on evidence theory and intuitionistic fuzzy sets","volume":"48","author":"Song","year":"2018","journal-title":"Appl. Intell."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/j.ijar.2017.09.008","article-title":"SELP: Semi\u2013supervised evidential label propagation algorithm for graph data clustering","volume":"92","author":"Zhou","year":"2018","journal-title":"Int. J. Approx. Reason."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1016\/j.knosys.2018.11.001","article-title":"A new pattern classification improvement method with local quality matrix based on K-NN","volume":"164","author":"Liu","year":"2019","journal-title":"Knowl.-Based Syst."},{"key":"ref_46","unstructured":"MacKay, D.J., and Mac Kay, D.J. (2003). Information Theory, Inference and Learning Algorithms, Cambridge University Press."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1080\/03081079308945088","article-title":"An uncertainty measure with monotonicity under the random set inclusion","volume":"21","author":"Maeda","year":"1993","journal-title":"Int. J. Gener. Syst."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1080\/03081078308960825","article-title":"Entropy and specificity in a mathematical theory of evidence","volume":"9","author":"Yager","year":"1983","journal-title":"Int. J. Gener. Syst."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Klir, G.J., and Parviz, B. (1992). A note on the measure of discord. Uncertainty in Artificial Intelligence, Elsevier.","DOI":"10.1016\/B978-1-4832-8287-9.50023-2"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/0888-613X(92)90009-O","article-title":"Uncertainty measures for evidential reasoning I: A review","volume":"7","author":"Pal","year":"1992","journal-title":"Int. J. Approx. Reason."},{"key":"ref_51","unstructured":"Klir, G.J., and Wierman, M.J. (2013). Uncertainty-Based Information: Elements of Generalized Information Theory, Physica."},{"key":"ref_52","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_53","doi-asserted-by":"crossref","first-page":"4359195","DOI":"10.1155\/2017\/4359195","article-title":"An improved belief entropy and its application in decision-making","volume":"2017","author":"Zhou","year":"2017","journal-title":"Complexity"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1016\/j.dss.2004.04.015","article-title":"Combining belief functions based on distance of evidence","volume":"38","author":"Yong","year":"2004","journal-title":"Decis. Support Syst."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/0020-0255(87)90007-7","article-title":"On the Dempster-Shafer framework and new combination rules","volume":"41","author":"Yager","year":"1987","journal-title":"Inf. Sci."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0167-9236(99)00084-6","article-title":"Combining belief functions when evidence conflicts","volume":"29","author":"Murphy","year":"2000","journal-title":"Decis. Support Syst."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"9562","DOI":"10.3390\/s140609562","article-title":"Novel algorithm for identifying and fusing conflicting data in wireless sensor networks","volume":"14","author":"Zhang","year":"2014","journal-title":"Sensors"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/4\/487\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:45:23Z","timestamp":1760363123000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/4\/487"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,24]]},"references-count":57,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["e22040487"],"URL":"https:\/\/doi.org\/10.3390\/e22040487","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,24]]}}}