{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T18:15:35Z","timestamp":1779300935251,"version":"3.51.4"},"reference-count":54,"publisher":"MDPI AG","issue":"24","license":[{"start":{"date-parts":[[2021,12,13]],"date-time":"2021-12-13T00:00:00Z","timestamp":1639353600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"China National Natural Science Foundation Grant","award":["U1933202"],"award-info":[{"award-number":["U1933202"]}]},{"name":"China Equipment Pre-research Field Foundation Grant","award":["MJ-2020-Y-011"],"award-info":[{"award-number":["MJ-2020-Y-011"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Spare parts are one of the important components of the equipment comprehensive support system. Spare parts management plays a decisive role in achieving the desired availability with the minimum cost. With the equipment complexity increasing, the price of spare parts has risen sharply. The traditional spare parts management makes the contradiction between fund shortage and spare parts shortage increasingly prominent. Based on the analysis of the multi-echelon and multi-indenture spare parts support model VARI-METRIC (vary multi-echelon technology for recoverable item control, VARI-METRIC), which is widely used by troops and enterprises in various countries, the model is mainly used in high system availability scenarios. However, in the case of low equipment system availability, the accuracy and cost of model inventory prediction are not ideal. This paper proposed the multi-level spare parts optimization model, which is based on the demand-supply steady-state process. It is an analytical model, which is used to solve the low accuracy problem of the VARI-METRIC model in the low equipment system availability. The analytical model is based on the multi-level spare parts support process. The article deduces methods for solving demand rate, demand\u2013supply rate, equipment system availability, and support system availability. The marginal analysis method is used in the model to analyze the spare parts inventory allocation strategy\u2019s current based cost and availability optimal value. Finally, a simulation model is established to evaluate and verify the model. Then, the simulation results show that, when the low availability of equipment systems are 0.4, 0.6, the relative errors of the analytical model are 3.54%, 3.86%, and its costs are 0.52, 1.795 million \u00a5 RMB. The experiment proves that the inventory prediction accuracy of the analytical model is significantly higher than that of the VARI-METRIC model in low equipment system availability. Finally, the conclusion and future research directions are discussed.<\/jats:p>","DOI":"10.3390\/s21248324","type":"journal-article","created":{"date-parts":[[2021,12,14]],"date-time":"2021-12-14T01:22:05Z","timestamp":1639444925000},"page":"8324","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["The Demand Supply Steady-State Process-Based Multi-Level Spare Parts Optimization"],"prefix":"10.3390","volume":"21","author":[{"given":"Jiaju","family":"Wu","sequence":"first","affiliation":[{"name":"College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China"},{"name":"Institute of Computer Application, China Academy of Engineering Physics, Mianyang 621900, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huijun","family":"Liu","sequence":"additional","affiliation":[{"name":"Institute of Computer Application, China Academy of Engineering Physics, Mianyang 621900, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongfu","family":"Zuo","sequence":"additional","affiliation":[{"name":"College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zheng","family":"Cheng","sequence":"additional","affiliation":[{"name":"Institute of Computer Application, China Academy of Engineering Physics, Mianyang 621900, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yonghui","family":"Yang","sequence":"additional","affiliation":[{"name":"Institute of Computer Application, China Academy of Engineering Physics, Mianyang 621900, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yongqi","family":"Ma","sequence":"additional","affiliation":[{"name":"Institute of Computer Application, China Academy of Engineering Physics, Mianyang 621900, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2477-118X","authenticated-orcid":false,"given":"Linggang","family":"Kong","sequence":"additional","affiliation":[{"name":"Institute of Computer Application, China Academy of Engineering Physics, Mianyang 621900, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,13]]},"reference":[{"key":"ref_1","first-page":"5","article-title":"Some problems of equipment integrated support foundation and technology","volume":"24","author":"Yongli","year":"2010","journal-title":"Armored Forces Eng. Coll. J."},{"key":"ref_2","first-page":"6","article-title":"Analysis of spare parts management system","volume":"5","author":"Fang","year":"2005","journal-title":"Equip. Manag. Maint."},{"key":"ref_3","first-page":"135","article-title":"Equipment Spare Parts Management and Cost Control","volume":"12","author":"Li","year":"2013","journal-title":"Enterp. Sci. Technol. Dev."},{"key":"ref_4","first-page":"227","article-title":"Spare parts management: A review of forecasting research and extensions","volume":"21","author":"Boylan","year":"2010","journal-title":"IMA J. Manag. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/S0377-2217(97)00039-8","article-title":"Multicriteria inventory classification using a genetic algorithm","volume":"105","author":"Guvenir","year":"1998","journal-title":"Eur. J. Oper. Res."},{"key":"ref_6","first-page":"14","article-title":"Spare parts decision analysis\u2014The missing link in CMMSs (Part II)","volume":"17","author":"Exton","year":"2002","journal-title":"J. Maint. Asset Manag."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"6035","DOI":"10.1016\/j.aej.2021.03.039","article-title":"A supply chain model with service level constraints and strategies under uncertainty","volume":"60","author":"Bhuniya","year":"2021","journal-title":"Alexandria Eng. J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1986","DOI":"10.1007\/s40815-021-01073-2","article-title":"The Selection of the Sustainable Suppliers by the Development of a Decision Support Framework Based on Analytical Hierarchical Process and Fuzzy Inference System","volume":"23","author":"Omair","year":"2021","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Sardar, S.K., Sarkar, B., and Kim, B. (2021). Integrating Machine Learning, Radio Frequency Identification, and Consignment Policy for Reducing Unreliability in Smart Supply Chain Management. Processes, 9.","DOI":"10.3390\/pr9020247"},{"key":"ref_10","first-page":"26","article-title":"A risk estimation method for the full life of weapons and equipment","volume":"1","author":"Guozhu","year":"2000","journal-title":"Sci. Res. Manag."},{"key":"ref_11","first-page":"54","article-title":"Research on performance of closed-loop support system for equipment system repairable components","volume":"50","author":"Dan","year":"2014","journal-title":"Comput. Eng. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1016\/j.cja.2014.04.021","article-title":"Configuration model of partial repairable spares under batching ordering policy on inventory state","volume":"27","author":"Ruan","year":"2014","journal-title":"Chin. J. Aeronaut."},{"key":"ref_13","first-page":"111","article-title":"Modeling and optimization of multi-echelon inventory for repairable spares based on birth and death process","volume":"39","author":"Liu","year":"2017","journal-title":"Natl. Def. Sci. Technol. Univ. J."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1287\/opre.16.1.122","article-title":"METRIC: A multi-echelon technique for recoverable item control","volume":"16","author":"Sherbrooke","year":"1968","journal-title":"Oper. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1247","DOI":"10.1287\/mnsc.31.10.1247","article-title":"A multi-echelon inventory model for a repairable item with one-for-one replenishment","volume":"31","author":"Graves","year":"1985","journal-title":"Manag. Sci."},{"key":"ref_16","first-page":"1749","article-title":"Multi-echelon inventory optimization of spare parts under imperfect maintenance conditions","volume":"27","author":"Kou","year":"2021","journal-title":"Comput. Integr. Manuf. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"628","DOI":"10.1287\/opre.29.4.628","article-title":"Operations Management & Operations Research in Inventory Management: A Review and Critique","volume":"29","author":"Silver","year":"1981","journal-title":"Oper. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/S0925-5273(01)00174-8","article-title":"An overview of recent literature on spare parts inventories","volume":"76","author":"Kennedy","year":"2002","journal-title":"Int. J. Prod. Econ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1057\/jors.2008.173","article-title":"Forecasting for inventory planning: A 50-year review","volume":"60","author":"Syntetos","year":"2009","journal-title":"J. Oper. Res. Soc."},{"key":"ref_20","first-page":"34","article-title":"System-oriented inventory models for spare parts","volume":"19","author":"Basten","year":"2014","journal-title":"Ser. Oper. Res. Manag. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/j.ejor.2010.05.048","article-title":"Inventory models with lateral transshipments: A review","volume":"210","author":"Paterson","year":"2011","journal-title":"Eur. J. Oper. Res."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"722","DOI":"10.1016\/j.omega.2011.06.008","article-title":"Spare parts classification and demand forecasting for stock control: Investigating the gap between research and practice","volume":"40","author":"Bacchetti","year":"2012","journal-title":"Omega"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/j.ejor.2012.03.004","article-title":"Review of inventory systems with a deterioration since 2001","volume":"221","author":"Bakker","year":"2012","journal-title":"Eur. J. Oper. Res."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ejor.2015.11.010","article-title":"Supply chain forecasting: Theory, practice, their gap and the future","volume":"252","author":"Syntetos","year":"2016","journal-title":"Eur. J. Oper. Res."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Liu, H.J., Wu, J.J., Liu, Z.J., Zhang, Y.F., Ma, Y.Q., and Meng, L.R. (2018). Research on Development of Spare Parts. Chin. Autom. Congr. (CAC), 2739\u20132743.","DOI":"10.1109\/CAC.2018.8623402"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1287\/opre.19.3.761","article-title":"Stationary Properties of a Two-Echelon Inventory Model for Low Demand Items","volume":"19","author":"Simon","year":"1971","journal-title":"Oper. Res."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1287\/mnsc.20.4.472","article-title":"A model for a multi-item, multi-echelon, multi-indenture inventory system","volume":"20","author":"Muckstadt","year":"1973","journal-title":"Manag. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1287\/mnsc.26.5.483","article-title":"Are multi-echelon inventory methods worth implementing in systems with low-demand-rate items","volume":"26","author":"Muckstadt","year":"1980","journal-title":"Manag. Sci."},{"key":"ref_29","unstructured":"Hillestad, R.J., and Carrillo, M.J. (1980). Models and Techniques for Recoverable Item Stockage When Demand and the Repair Process are Nonstationary\u2014Part I: Performance Measurement, RAND Corporation. N-1482-AF."},{"key":"ref_30","unstructured":"Hillestad, R.J. (1982). Dyna-METRIC: Dynamic Multi-Echelon Technique for Recoverable Item Control, RAND Corporation. R-2785-AF."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1287\/opre.34.2.311","article-title":"VARI-METRIC: Improved approximations for multi-indenture, multi-echelon availability models","volume":"34","author":"Sherbrooke","year":"1986","journal-title":"Oper. Res."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1002\/1520-6750(199202)39:1<29::AID-NAV3220390103>3.0.CO;2-L","article-title":"Multi echelon inventory systems with lateral supply","volume":"39","author":"Sherbrooke","year":"1992","journal-title":"Nav. Res. Logist."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1016\/S0377-2217(96)00279-2","article-title":"Models for multi-echelon repairable item inventory systems with limited repair capacity","volume":"96","author":"Diaz","year":"1997","journal-title":"Eur. J. Oper. Res."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/S0925-5273(02)00155-X","article-title":"Effects of finite repair capacity in multi-echelon, multi-indenture service part supply systems","volume":"79","author":"Sleptchenko","year":"2002","journal-title":"Int. J. Prod. Econ."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.ress.2013.05.006","article-title":"Multi-echelon, multi-indenture spare parts inventory control subject to system availability and budget constraints","volume":"119","author":"Costantino","year":"2013","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Van Houtum, G.J., and Kranenburg, B. (2015). Spare Parts Inventory Control under System Availability Constraints, Springer.","DOI":"10.1007\/978-1-4899-7609-3"},{"key":"ref_37","first-page":"227","article-title":"Performance evaluation model of two-level maintenance supply system under limited maintenance channel","volume":"35","author":"Xia","year":"2015","journal-title":"Theory Pract. Syst. Eng."},{"key":"ref_38","first-page":"77","article-title":"Inventory optimization model of repairable parts of shipborne aircraft with limited maintenance capacity","volume":"41","author":"Zhang","year":"2015","journal-title":"J. Beijing Aeronaut. Astronaut. Univ."},{"key":"ref_39","first-page":"1065","article-title":"Time-varying availability evaluation modeling of incomplete repair parts under cannibalization","volume":"39","author":"Zhou","year":"2017","journal-title":"Syst. Eng. Electron. Technol."},{"key":"ref_40","first-page":"220319-1","article-title":"Availability evaluation modeling for K\/N (G) structural system with cannibalization","volume":"4","author":"Zhou","year":"2017","journal-title":"Chin. J. Aviat."},{"key":"ref_41","unstructured":"Liu, R.Y., Li, H., Li, Q.M., and Xu, L. (2016). Evaluation of availability for partial repairable exponential spares with cannibalization. Theory Pract. Syst. Eng., 1857\u20131862."},{"key":"ref_42","first-page":"57","article-title":"Influence of Cannibalization on Stock Investment of Air Materiel","volume":"12","author":"Ma","year":"2004","journal-title":"China J. Manag. Sci."},{"key":"ref_43","first-page":"17","article-title":"Inventory Optimization Method of Spare Parts Based on the EML Model","volume":"44","author":"Wang","year":"2019","journal-title":"China Fire Control Command. Control"},{"key":"ref_44","first-page":"1043","article-title":"Warship equipment\u2019s spare demand in consideration of maintenance","volume":"31","author":"Zhai","year":"2019","journal-title":"J. Nav. Univ. Eng."},{"key":"ref_45","first-page":"71","article-title":"Maintenance-oriented spare parts sharing service and allocation optimization","volume":"49","author":"Wang","year":"2021","journal-title":"Mach. Tool Hydraul."},{"key":"ref_46","first-page":"756","article-title":"Optimization of Multi-echelon Inventory Based on Component Commonality","volume":"47","author":"Dong","year":"2019","journal-title":"Comput. Digit. Eng."},{"key":"ref_47","first-page":"68","article-title":"Evaluation method of incomplete repair parts under lateral replenishment strategy","volume":"29","author":"Shao","year":"2017","journal-title":"Nav. Eng. Univ. J."},{"key":"ref_48","first-page":"141","article-title":"Application of OPUS 10 in the formulation of ship\u2019s initial spare parts list","volume":"12","author":"Yu","year":"2014","journal-title":"Equip. Manuf. Technol."},{"key":"ref_49","unstructured":"Sherbrooke, C.C. (2004). Optimal Inventory Modeling of Systems: Multi Echelon Techniques, Springer Science Business Media. [2nd ed.]."},{"key":"ref_50","first-page":"103","article-title":"Study on the optimal inventory of repairable spare parts based on marginal analysis method","volume":"14","author":"Guo","year":"2017","journal-title":"Value Eng."},{"key":"ref_51","first-page":"2363","article-title":"The dynamic configuration method of spare parts for system optimization","volume":"32","author":"Cai","year":"2010","journal-title":"Syst. Eng. Electron. Technol."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"245","DOI":"10.3901\/JME.2020.14.245","article-title":"Research on Multi-level Inventory Optimization Algorithm of Repairable Spare Parts Based on Two Improved Differential Evolution","volume":"56","author":"Gu","year":"2020","journal-title":"J. Mech. Eng."},{"key":"ref_53","first-page":"128","article-title":"Forecasting Method for Spare Parts Demand Based on the Analysis of Unstable Time Series","volume":"6","author":"Luo","year":"2016","journal-title":"Syst. Eng."},{"key":"ref_54","first-page":"36","article-title":"Decision the spare parts\u2019 reserves of weapon systems by using the method of genetic algorithms","volume":"17","author":"Zhang","year":"2005","journal-title":"Ord. Eng. Coll. J."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/24\/8324\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:46:31Z","timestamp":1760168791000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/24\/8324"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,13]]},"references-count":54,"journal-issue":{"issue":"24","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["s21248324"],"URL":"https:\/\/doi.org\/10.3390\/s21248324","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,13]]}}}