本文提出一种基于贝叶斯后验分布异常数据处理的复杂装备成本估算灰色GM(0, N)模型（多变量无导数灰色预测模型。GM表示Grey Model，灰色模型，0表示无导数项，N表示有N个变量），旨在提高复杂装备成本估算的准确性和可靠性。以液体火箭为研究案例，首先通过“3σ准则”剔除了可能存在的异常数据样本，提升了数据集的质量和模型的准确性。随后，利用贝叶斯估计方法计算了每单位研制成本的关键参数（如起飞质量）的均值和方差，为后续的异常值检测提供了科学依据。在此基础上，构建了分数阶累加的灰色GM(0, N)模型，并通过最小二乘法确定了各项未知参数，实现了对目标火箭研制成本的精确估算。实验结果表明，与多元线性回归模型和普通灰色GM(0, N)模型相比，本文提出的方法具有更高的估算精度和更好的鲁棒性。具体而言，本文方法的预测误差仅为2.6375%，而多元线性回归模型和普通灰色GM(0, N)模型的误差分别为20.7169%和14.2128%。此外，该方法不仅适用于液体火箭研制成本估算，还可以推广到其他复杂工程系统的成本估算问题中，为相关领域的科学研究和技术发展提供了有益参考。
A Bayesian posterior distribution-based abnormal data processing GM(0, N) model (Multivariate non-derivative grey prediction model. GM represents Grey Model, grey model, 0 indicates no derivative terms, N indicates there are N variables) is proposed for complex equipment cost estimation, aiming to enhance the accuracy and reliability of cost prediction for sophisticated systems. Taking rocket systems as a research case, potential abnormal data samples were first eliminated using the three-sigma criterion to improve dataset quality and model precision. Subsequently, Bayesian estimation methods were employed to calculate the mean and variance of critical development cost-per-unit parameters (e.g., takeoff mass), providing a scientific foundation for subsequent outlier detection. A fractional-order accumulation GM(0, N) model was subsequently constructed, where the unknown parameters were determined through the least squares method, enabling precise development cost estimation for target rocket configurations. Experimental results demonstrate that compared with multivariate linear regression models and conventional GM(0, N) models, the proposed method achieves superior estimation accuracy and enhanced robustness. Specifically, the prediction error of this methodology was reduced to 2.6375%, whereas the errors of multivariate linear regression and conventional GM(0, N) models reached 20.7169% and 14.2128% respectively. Furthermore, this methodology not only applies to liquid rocket development cost estimation but can also be extended to cost prediction problems in other complex engineering systems, providing valuable references for scientific research and technological development in related fields.
[1] 吴诗辉,刘晓东,贺波,等.装备体系费用评估问题研究[C]//国防科技大学系统工程学院.第五届体系工程学术会议论文集——数智时代的体系工程.空军工程大学装备管理与无人机工程学院;,2023:347-353.DOI:10.26914/c.cnkihy.2023.056292.
[2] 吴利丰,刘思峰,方世力,等.相似信息优先的复杂装备费用测算模型[J].系统工程与电子技术,2014,36(10):2024-2028.
[3] 刘思峰,灰色系统理论及其应用(第五版).江苏省,南京航空航天大学,2010-10-27.
[4] 吴利丰,于亮,文朝霞.预测复杂装备研制费用的GM(0,N)模型[J].中国管理科学, 2019,27(07):203-207. DOI:10.16381/j.cnki.issn1003-207x.2019.07.020.
[5] 陈伟.基于灰色关联分析的某变电站绿色施工评价研究[J].工程机械与维修,2024,(12):97-99.
[6] 魏东涛,刘晓东,丁刚,等.基于熵理论的复杂装备费用测算模型[J].北京航空航天大学学报,2022,48(05):816-823.DOI:10.13700/j.bh.1001-5965.2020.0678.
[7] 方世力,吴利丰,刘思峰,等.复杂装备费用驱动因子识别的效应比重极大熵模型[J].系统工程,2014,32(10):149-153.
[8] 罗党,王小雷,孙德才,等.含时间周期项的离散灰色DGM(1,1,T)模型及其应用[J].系统工程理论与实践,2020,40(10):2737-2746.
[9] 王鹏.国产大飞机在东南亚市场的挑战与机遇[J].大飞机,2024,(04):48-52.
[10] 王江三,高伟,席林军.基于寿命周期的航空装备定费用设计与控制[J].航空财会,2024,6(05):45-49.DOI:10.19840/j.cnki.FA.2024.05.010.
[11] Ma X ,Xiang M ,Jiang X , et al.Bayesian survival analysis of interactions between truck platoons and surrounding vehicles through a two-dimensional surrogate safety measure[J].Accident Analysis and Prevention,2025,213107945-107945.
[12] 贾琼.基于贝叶斯网络的飞机通信导航系统故障检测方法[J].长江信息通信,2024,37(12):136-138.DOI:10.20153/j.issn.2096-9759.2024.12.040.
[13] 岳盼想,李孝鹏,周文明,等.可重复使用运载火箭可靠性经济性分析研究初探[J].质量与可靠性,2024,(04):36-38+43.
[14] 朱雄峰,刘阳,刘鹰,等.猎鹰-9运载火箭发射成本研究[J].国际太空,2023,(11):35-40.
[15] Cost of launching heavy-lift Angara rockets estimated at $50 mln to $100 mln - designers[J].Interfax : Russia & CIS Military Newswire,2020,
[16] Space and Rocket Building; Cost of launching heavy-lift Angara rockets estimated at $50 mln to $100 mln - designers[J].Interfax : Russia & CIS Defense Industry Weekly,2020.
[17] 郑山锁,丁明松,郑捷,等.基于多元线性回归方法的砌体结构加固费用估算模型研究[J].地震工程学报,2018,40(01):6-13.
[18] Mingchi L ,Dubo H ,Shengxiang S .Multivariable Case Adaptation Method of Case-Based Reasoning Based on Multi-Case Clusters and Multi-Output Support Vector Machine for Equipment Maintenance Cost Prediction[J].IEEE ACCESS,2021,9151960-151971.
[19] Liu S ,Chunwu Y ,Dazhi C .Weapon equipment management cost prediction based on forgetting factor recursive GM (1,1) model[J].Grey Systems: Theory and Application,2020,10(1):38-45.
[20] 赵玉婷.基于工程估算法的某型座舱显示器寿命周期费用估算研究[J].工程建设与设计,2020,(22):156-157.DOI:10.13616/j.cnki.gcjsysj.2020.11.272.
[21] Xiao Z L ,Yang P D ,Sun X D , et al.The Optimiza-tion Algorithm of Aviation Equipment Maintenance Cost Forecast and its Applied Research[J].Advanced Materials Research,2013,2542(760-762):1851-1855.
[22] Cui Z ,Zheng M ,Wang J , et al.Reliability Analysis of a Three-Engine Simultaneous Pouring Control System Based on Bayesian Networks Combined with FMEA and FTA[J].Applied Sciences,2023,13(20).
[23] 吴燕生.中国航天运输系统智能飞行技术发展展望[J].宇航学报,2023,44(03):313-321.
[24] Ding C ,Wang J ,Chen X , et al.Robust parameter design for rocket boosters using deep Gaussian process[J].Aerospace Science and Technolo-gy,2025,161110143-110143.
[25] 赵剑,黄悦琛,李海阳,等.垂直起降运载火箭返回轨迹不确定性优化[J].航空学报,2021,42(11):262-276.
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