{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,22]],"date-time":"2026-05-22T05:07:54Z","timestamp":1779426474292,"version":"3.53.1"},"reference-count":29,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2026,5,15]],"date-time":"2026-05-15T00:00:00Z","timestamp":1778803200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>To address the challenges of difficult value quantification, lack of market benchmarks, and scarcity of historical data for embedded software amidst the intelligent transformation of equipment systems, this study develops a scientific price estimation method based on functional capability contribution. A nonlinear pricing model is constructed to accurately characterize the two-stage evolution of software price: diminishing marginal utility during the mature technology accumulation stage and exponential growth during the technical bottleneck breakthrough stage. To ensure the consistency of pricing logic between hardware and software, a penalty function is innovatively designed to modify the standard likelihood function, effectively transforming practical business logic into a model regularization term. Parameter estimation is achieved by employing a Bayesian inference framework integrated with operational constraints, utilizing Markov Chain Monte Carlo (MCMC) sampling to realize robust posterior inference under small-sample constraints. Empirical analysis demonstrates that the proposed method achieves superior cross-domain data transfer performance compared to traditional baseline models, with a Leave-One-Out Cross-Validation (LOOCV) Mean Absolute Percentage Error (MAPE) of 21.2%. This research provides a practical value-oriented price estimation method for embedded equipment software pricing.<\/jats:p>","DOI":"10.3390\/a19050396","type":"journal-article","created":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T12:36:40Z","timestamp":1779107800000},"page":"396","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Bayesian Inference Algorithm for Equipment Software Price Estimation Based on Nonlinear Contribution Models"],"prefix":"10.3390","volume":"19","author":[{"given":"Tian","family":"Meng","sequence":"first","affiliation":[{"name":"Naval University of Engineering, Wuhan 430000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Guoping","family":"Jiang","sequence":"additional","affiliation":[{"name":"Naval University of Engineering, Wuhan 430000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,5,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.jmsy.2023.07.002","article-title":"Production quality prediction of multistage manufacturing systems using multi-task joint deep learning","volume":"70","author":"Wang","year":"2023","journal-title":"J. Manuf. Syst."},{"key":"ref_2","first-page":"18","article-title":"A cyber-physical systems architecture for industry 4.0-based manufacturing systems","volume":"3","author":"Lee","year":"2015","journal-title":"Manuf. Lett."},{"key":"ref_3","first-page":"42","article-title":"Challenges and Countermeasures of Equipment Pricing Work Under the Background of Digital Construction","volume":"7","author":"Zhang","year":"2025","journal-title":"Aviat. Financ. Account."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Harrell, F.E. (2015). Regression Modeling Strategies: With Applications to Linear Models, Logistic Regression, and Survival Analysis, Springer. [2nd ed.].","DOI":"10.1007\/978-3-319-19425-7"},{"key":"ref_5","first-page":"34","article-title":"Research on Quality Management of Embedded Software for Avionics Equipment","volume":"1","author":"Tang","year":"2022","journal-title":"Aeronaut. Stand. Qual."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Nandini, K.P., and Seshikala, G. (2021). Role of Embedded Computing Systems in Biomedical Applications\u2013Opportunities and Challenges. Proceedings of the 2021 IEEE International Conference on Distributed Computing, VLSI, Electrical Circuits and Robotics (DISCOVER), IEEE.","DOI":"10.1109\/DISCOVER52564.2021.9663646"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1109\/JPROC.2009.2039631","article-title":"Challenges and Solutions for Embedded and Networked Aerospace Software Systems","volume":"98","author":"Sharp","year":"2010","journal-title":"Proc. IEEE"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"452","DOI":"10.1007\/s12599-009-0075-y","article-title":"Pricing Strategies of Software Vendors","volume":"1","author":"Lehmann","year":"2009","journal-title":"Bus. Inf. Syst. Eng."},{"key":"ref_9","first-page":"21","article-title":"Construction of Data Asset Value Evaluation Model\u2014Based on Multi-period Excess Earnings Method","volume":"23","author":"Chen","year":"2021","journal-title":"Financ. Account. Mon."},{"key":"ref_10","first-page":"501","article-title":"Software Cost Estimation Method Based on Weighted Analogy","volume":"45","author":"Zhao","year":"2018","journal-title":"Comput. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"511","DOI":"10.33545\/27068919.2020.v2.i4h.447","article-title":"The Review of Software Cost Estimation Model: SLIM","volume":"2","author":"Ghafory","year":"2020","journal-title":"Int. J. Adv. Acad. Stud."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Santos, M.V., and Soares, M.S. (2025). Cost Estimation for Agile Software Development in Small and Medium-Sized Enterprises. Proceedings of the Simp\u00f3sio Brasileiro de Qualidade de Software (SBQS), SBC.","DOI":"10.5753\/sbqs_estendido.2025.13396"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Tsunoda, M., Monden, A., Matsumoto, K., Ohiwa, S., and Oshino, T. (2012). Analysis of Attributes Relating to Custom Software Price. Proceedings of the 2012 Fourth International Workshop on Empirical Software Engineering in Practice, IEEE.","DOI":"10.1109\/IWESEP.2012.19"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Yerastova, V.V., and Oleshchenko, L.M. (2020). Forecasting Software Market Price Using Back Propagation Neural Network, \u0412\u0447\u0435\u043d\u0456 \u0417\u0430\u043f\u0438\u0441\u043a\u0438.","DOI":"10.32838\/2663-5941\/2020.5\/11"},{"key":"ref_15","first-page":"173","article-title":"A Case Study Research on Software Cost Estimation Using Experts\u2019 Estimates, Wideband Delphi, and Planning Poker Technique","volume":"8","author":"Gandomani","year":"2014","journal-title":"Int. J. Softw. Eng. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Acito, F. (2023). Ordinary Least Squares Regression. Predictive Analytics with KNIME: Analytics for Citizen Data Scientists, Springer Nature.","DOI":"10.1007\/978-3-031-45630-5"},{"key":"ref_17","unstructured":"Auer, M. (2004). Analogy-Based Software Cost Estimation: Improving Explicability, Effectiveness and Efficiency, Technische Universit\u00e4t Wien."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1007\/BF03160297","article-title":"The Optimal Price Policy of Congener Software Product","volume":"6","author":"Ren","year":"2001","journal-title":"Wuhan Univ. J. Nat. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Nizovkina, N.G. (2005). Quality and Price Evaluation on the Software Market. Proceedings of the 9th Russian-Korean International Symposium on Science and Technology (KORUS 2005), IEEE.","DOI":"10.1109\/KORUS.2005.1507937"},{"key":"ref_20","first-page":"103645","article-title":"A systematic approach to evaluate the functional contribution of embedded software in complex industrial systems","volume":"138","author":"Martinez","year":"2022","journal-title":"Comput. Ind."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5057","DOI":"10.1109\/TNNLS.2017.2785792","article-title":"Efficient Unsupervised Parameter Estimation for One-Class Support Vector Machines","volume":"29","author":"Ghafoori","year":"2018","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"e2300990120","DOI":"10.1073\/pnas.2300990120","article-title":"Machine Learning for Parameter Estimation","volume":"120","author":"Kutz","year":"2023","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1038\/s41534-021-00497-w","article-title":"A Machine Learning Approach to Bayesian Parameter Estimation","volume":"7","author":"Nolan","year":"2021","journal-title":"npj Quantum Inf."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1111\/j.1745-3984.1986.tb00241.x","article-title":"Maximum Likelihood and Bayesian Parameter Estimation in Item Response Theory","volume":"23","author":"Lord","year":"1986","journal-title":"J. Educ. Meas."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1016\/S0022-2496(02)00028-7","article-title":"Tutorial on Maximum Likelihood Estimation","volume":"47","author":"Myung","year":"2023","journal-title":"J. Math. Psychol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"789","DOI":"10.1016\/0043-1354(92)90010-2","article-title":"Model-Parameter Estimation Using Least Squares","volume":"26","author":"Rittmann","year":"1992","journal-title":"Water Res."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1348","DOI":"10.1002\/bjs.10895","article-title":"LASSO Regression","volume":"105","author":"Ranstam","year":"2018","journal-title":"J. Br. Surg."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Saleh, A.K.M.E., Arashi, M., and Kibria, B.M.G. (2019). Theory of Ridge Regression Estimation with Applications, John Wiley & Sons.","DOI":"10.1002\/9781118644478"},{"key":"ref_29","unstructured":"Wang, Y.L., Shi, X.M., Zhao, Q., Liu, H.B., and Liang, J.P. (2025). Bayesian Inference of Ammunition Consumption for Armored Vehicle Targets Based on Gamma Distribution. Syst. Eng. Electron., 615\u2013626."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/19\/5\/396\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,22]],"date-time":"2026-05-22T04:30:45Z","timestamp":1779424245000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/19\/5\/396"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,15]]},"references-count":29,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2026,5]]}},"alternative-id":["a19050396"],"URL":"https:\/\/doi.org\/10.3390\/a19050396","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,15]]}}}