{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:51:31Z","timestamp":1760241091543,"version":"build-2065373602"},"reference-count":49,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2019,12,4]],"date-time":"2019-12-04T00:00:00Z","timestamp":1575417600000},"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>Condition-based maintenance (CBM) is a promising technique for a wide variety of deteriorating systems. Condition-based maintenance\u2019s effectiveness largely depends on the quality of condition monitoring. The majority of CBM mathematical models consider perfect inspections, in which the system condition is assumed to be determined error-free. This article presents a mathematical model of CBM with imperfect condition monitoring conducted at discrete times. Mathematical expressions were derived for evaluating the probabilities of correct and incorrect decisions when monitoring the system condition at a scheduled time. Further, these probabilities were incorporated into the equation of the Shannon entropy. The problem of determining the optimal preventive maintenance threshold at each inspection time by the criterion of the minimum of Shannon entropy was formulated. For the first time, the article showed that Shannon\u2019s entropy is a convex function of the preventive maintenance threshold for each moment of condition monitoring. It was also shown that the probabilities of correct and incorrect decisions depend on the time and parameters of the degradation model. Numerical calculations show that the proposed approach to determining the optimal preventive maintenance threshold can significantly reduce uncertainty when deciding on the condition of the monitoring object.<\/jats:p>","DOI":"10.3390\/e21121193","type":"journal-article","created":{"date-parts":[[2019,12,5]],"date-time":"2019-12-05T03:16:36Z","timestamp":1575515796000},"page":"1193","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Optimization of Condition Monitoring Decision Making by the Criterion of Minimum Entropy"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7555-2473","authenticated-orcid":false,"given":"Ahmed","family":"Raza","sequence":"first","affiliation":[{"name":"Projects and Maintenance Section, The Private Department of the President of the United Arab Emirates, Abu Dhabi 000372, UAE"}]},{"given":"Vladimir","family":"Ulansky","sequence":"additional","affiliation":[{"name":"Research and Development Department, Mathematical Modelling &amp; Research Holding Limited, London W1W 7LT, UK"},{"name":"Department of Electronics, Robotics, Monitoring Technology, and IoT, National Aviation University, 03058 Kyiv, Ukraine"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,4]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","first-page":"110205","DOI":"10.1109\/ACCESS.2019.2931136","article-title":"Machine learning based file entropy analysis for ransomware detection in backup systems","volume":"7","author":"Lee","year":"2019","journal-title":"IEEE Access"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1097","DOI":"10.1109\/JBHI.2017.2711487","article-title":"Maximum-entropy-rate selection of features for classifying changes in knee and ankle dynamics during running","volume":"22","author":"Einicke","year":"2018","journal-title":"IEEE J. Biomed. Health Inform."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1577","DOI":"10.1109\/ACCESS.2017.2779475","article-title":"Belief reliability distribution based on maximum entropy principle","volume":"6","author":"Zu","year":"2017","journal-title":"IEEE Access"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1109\/TSMC.2013.2296276","article-title":"Intelligent prognostics for battery health monitoring using the mean entropy and relevance vector machine","volume":"44","author":"Li","year":"2014","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1016\/j.ymssp.2016.05.036","article-title":"Multipoint optimal minimum entropy deconvolution and convolution fix: Application to vibration fault detection","volume":"82","author":"McDonald","year":"2017","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2069","DOI":"10.1007\/s00170-017-0252-y","article-title":"A hybrid health condition monitoring method in milling operations","volume":"92","author":"Liu","year":"2017","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1536","DOI":"10.3182\/20130619-3-RU-3018.00051","article-title":"Evaluation of minimal data size by using entropy, in a HMM maintenance manufacturing use","volume":"46","author":"Robles","year":"2013","journal-title":"IFAC Proc. Vol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1109\/TIFS.2019.2916406","article-title":"Fingerprint entropy and identification capacity estimation based on pixel-level generative modelling","volume":"15","author":"Yankov","year":"2019","journal-title":"IEEE Trans. Inf. Forensics Secur."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Nowak, W., and Guthke, A. (2016). Entropy-based experimental design for optimal model discrimination in the geosciences. Entropy, 18.","DOI":"10.3390\/e18110409"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Young, C., and Subbarayan, G. (2019). Maximum entropy models for fatigue damage in metals with application to low-cycle fatigue of Aluminum 2024-T351. Entropy, 21.","DOI":"10.3390\/e21100967"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2092","DOI":"10.1016\/j.microrel.2015.06.076","article-title":"Entropy-based sensor selection for condition monitoring and prognostics of aircraft engine","volume":"55","author":"Liu","year":"2015","journal-title":"Microelectron. Reliab."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Liu, L., Wang, S., Liu, D., and Peng, Y. (2016, January 19\u201321). Quantitative description of sensor data monotonic trend for system degradation condition monitoring. Proceedings of the Prognostics and System Health Management Conference (PHM-Chengdu), Chengdu, China.","DOI":"10.1109\/PHM.2016.7819924"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1322","DOI":"10.21595\/jve.2019.20255","article-title":"Bearing degradation assessment based on entropy with time parameter and fuzzy c-means clustering","volume":"21","author":"Liu","year":"2019","journal-title":"J. Vibroengineering"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3536","DOI":"10.1109\/TIM.2018.2881529","article-title":"Entropy-based local irregularity detection for high-speed railway catenaries with frequent inspections","volume":"68","author":"Wang","year":"2019","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_16","unstructured":"(2013, March 07). Aeronautical Design Standard Handbook. Condition-Based Maintenance System for US Army Aircraft: ADS-79D-HDBK. Available online: http:\/\/everyspec.com\/ARMY\/ADS-Aero-Design-Std\/ADS-79-HDBK_2013_49364\/."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1016\/j.ejor.2014.11.029","article-title":"Condition-based maintenance using the inverse Gaussian degradation model","volume":"243","author":"Chen","year":"2015","journal-title":"Eur. J. Oper. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"917","DOI":"10.2307\/1427108","article-title":"Inspection and maintenance policies of devices subject to deterioration","volume":"19","year":"1987","journal-title":"Adv. Appl. Probab."},{"key":"ref_19","first-page":"584","article-title":"Correction to: \u201cInspection and maintenance policies of devices subject to deterioration\u201d","volume":"27","year":"1995","journal-title":"Adv. Appl. Probab."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/S0951-8320(01)00148-X","article-title":"A condition-based maintenance policy for stochastically deteriorating systems","volume":"76","author":"Grall","year":"2002","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1016\/S0377-2217(02)00593-3","article-title":"Sequential condition-based maintenance scheduling for a deteriorating system","volume":"150","author":"Dieulle","year":"2003","journal-title":"Eur. J. Oper. Res."},{"key":"ref_22","unstructured":"Kallen, M.J., and Kuniewski, S.P. (2009). An adaptive condition-based maintenance policy with environmental factors. Risk and Decision Analysis in Maintenance Optimization and Flood Management, IOS Press."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1109\/TR.2002.1011518","article-title":"Continuous-time predictive-maintenance scheduling for a deteriorating system","volume":"51","author":"Grall","year":"2002","journal-title":"IEEE Trans. Reliab."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1016\/j.ress.2010.12.018","article-title":"A periodic inspection and replacement policy for systems subject to competing failure modes due to degradation and traumatic events","volume":"96","author":"Huynh","year":"2011","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1109\/TR.2015.2487578","article-title":"Condition-based maintenance with scheduling threshold and maintenance threshold","volume":"65","author":"Wang","year":"2016","journal-title":"IEEE Trans. Reliab."},{"key":"ref_26","unstructured":"Guo, C., Bai, Y., and Jia, Y. (2016, January 12\u201314). Maintenance optimization for systems with non-stationary degradation and random shocks. Proceedings of the 9th IMA International Conference on Modelling in Industrial Maintenance and Reliability, London, UK."},{"key":"ref_27","unstructured":"Liu, B., Xie, M., and Kuo, W. (2016, January 12\u201314). Condition-based maintenance for degrading systems with state-dependent operating cost. Proceedings of the 9th IMA International Conference on Modelling in Industrial Maintenance and Reliability, London, UK."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.ress.2012.01.006","article-title":"Safety constraints applied to an adaptive Bayesian condition-based maintenance optimization model","volume":"102","author":"Flage","year":"2012","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1080\/15732479.2011.563095","article-title":"Environmental information adaptive condition-based maintenance policies","volume":"8","author":"Deloux","year":"2012","journal-title":"Struct. Infrastruct. Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"983","DOI":"10.1109\/TR.2015.2417153","article-title":"Scheduling preventive maintenance as a function of an imperfect inspection interval","volume":"64","author":"He","year":"2015","journal-title":"IEEE Trans. Reliab."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.ress.2004.10.004","article-title":"Optimal maintenance decisions under imperfect inspection","volume":"90","author":"Kallen","year":"2005","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_32","unstructured":"Newby, M., and Dagg, R. (2002, January 19\u201321). Optimal inspection policies in the presence of covariates. Proceedings of the European Safety and Reliability Conference (ESREL\u201902), Lyon, France."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1002\/qre.1609","article-title":"A Bayesian approach to condition monitoring with imperfect inspections","volume":"31","author":"Ye","year":"2015","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"520","DOI":"10.3390\/en7020520","article-title":"Remaining useful life prediction of lithium-ion batteries based on the Wiener process with measurement error","volume":"7","author":"Tang","year":"2014","journal-title":"Energies"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.1239\/jap\/1067436098","article-title":"An inspection-repair-replacement model for a deteriorating system with unobservable state","volume":"40","author":"Lam","year":"2003","journal-title":"J. Appl. Probab."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/S0951-8320(02)00154-0","article-title":"Optimal inspection and preventive maintenance of units with revealed and unrevealed failures","volume":"78","author":"Berrade","year":"2002","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"716","DOI":"10.1016\/j.ejor.2011.12.003","article-title":"Maintenance scheduling of a protection system subject to imperfect inspection and replacement","volume":"218","author":"Berrade","year":"2012","journal-title":"Eur. J. Oper. Res."},{"key":"ref_38","first-page":"187","article-title":"Optimal scheduling of non-perfect inspections","volume":"17","author":"Zequeira","year":"2006","journal-title":"IMA J. Manag. Math."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.ress.2013.02.024","article-title":"Imperfect inspection and replacement of a system with a defective state. A cost and reliability analysis","volume":"120","author":"Berrade","year":"2013","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/j.ress.2016.08.009","article-title":"A review on condition-based maintenance optimization models for stochastically deteriorating system","volume":"157","author":"Alaswad","year":"2017","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_41","unstructured":"Walpole, R., Myers, R., Myers, S., and Ye, K. (2012). Probability and Statistics for Engineers and Scientists, Pearson Prentice Hall. [9th ed.]."},{"key":"ref_42","first-page":"58","article-title":"Analysis of equipment fault prediction based on metabolism combined model","volume":"2","author":"Ma","year":"2013","journal-title":"J. Mach. Manuf. Autom."},{"key":"ref_43","unstructured":"(1997, March 01). 80K-40 High Voltage Probe-Fluke Corporation. Available online: https:\/\/dam-assets.fluke.com\/s3fs-public\/80k40___iseng0900.pdf."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1109\/TSTE.2010.2049452","article-title":"An approach for condition-based maintenance optimization applied to wind turbine blades","volume":"1","author":"Besnard","year":"2010","journal-title":"IEEE Trans. Sustain. Energy"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/1099-1824(200001\/03)3:1<1::AID-WE28>3.0.CO;2-2","article-title":"A Summary of the fatigue properties of wind turbine materials","volume":"3","author":"Sutherland","year":"2000","journal-title":"Wind Energy"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Wang, Z., Xue, X., Yin, H., Jiang, Z., and Li, Y. (2018). Research progress on monitoring and separating suspension particles for lubricating oil. Complexity, 1\u20139.","DOI":"10.1155\/2018\/9356451"},{"key":"ref_47","unstructured":"Coronado, D., and Fisher, K. (2015, June 01). Condition Monitoring of Wind Turbines: State of the Art, User Experience, and Recommendations. Project Report. Available online: https:\/\/www.semanticscholar.org\/paper\/CONDITION-MONITORING-OF-WIND-TURBINES-%3A-STATE-OF-%2C-Coronado Fischer\/477fabdc00482a7f1265efc5fbc5ee15db66d353."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Sood, B., Severn, L., Osterman, M., Pecht, M., Bougaev, A., and McElfresh, D. (2012, January 11\u201315). Lithium-ion battery degradation mechanisms and failure analysis methodology. Proceedings of the 38th International Symposium for Testing and Failure Analysis, Phoenix, AZ, USA.","DOI":"10.31399\/asm.cp.istfa2012p0239"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"2911","DOI":"10.1109\/TII.2017.2684821","article-title":"RUL prediction of deteriorating products using an adaptive Wiener process model","volume":"13","author":"Zhai","year":"2017","journal-title":"IEEE Trans. Ind. Inform."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/12\/1193\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:40:12Z","timestamp":1760190012000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/12\/1193"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,4]]},"references-count":49,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2019,12]]}},"alternative-id":["e21121193"],"URL":"https:\/\/doi.org\/10.3390\/e21121193","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2019,12,4]]}}}