分数阶灰色预测模型及其在复杂装备费用测算中的应用研究

Due to the low quality and limited quantity of cost data on complex equipment cost estimation, establishing grey prediction model aimed to achieve to excavate the potential information of limited sample as much as possible, a new fractional grey modeling techniques is put forward by taking the "in between" idea from fractional order as the main line. Firstly, the defects and shortcomings of traditional grey prediction model are proved via the matrix perturbation theory, to address these problems, grey model with fractional accumulation and grey model with fractional derivative are developed by factorial formula and fractional Laplace transform respectively. Secondly, fractional order buffer operator is introduced via matrix algorithms, the mechanism of the order number impact on buffer intensity and the relationship between strengthening buffer operator and weakening buffer operator are revealed further. Thirdly, combining with new grey similar correlation, multi- variate buffer operator is constructed to solve the problem about cost data disru- pted by a variety of factors. Fourthly, the relationship between stability and sample size, priority of recent information and other mathematical properties of the new model are investigated further, and its advantages on prediction complex equipment costs are verified by theoretical proof and real examples. Finally, one complex equipment is selected as a case, the proposed model is used to estimate the life cycle cost of the equipment.

针对复杂装备费用估算中费用数据质量低、数量少的特点，以构建能够实现最少信息最大挖掘的灰色预测模型为目标，以分数阶的“in between”思想为研究主线，提炼出一套新的分数阶灰建模技术。首先通过矩阵扰动理论证明以往灰色预测模型的缺陷，为解决这些问题，分别运用阶乘公式和分数阶拉普拉斯变换提出分数阶累加灰色模型与分数阶导数灰色模型；接着运用矩阵运算法则，引入分数阶缓冲算子，并揭示阶数对缓冲强度的影响机理以及强、弱化缓冲算子之间的关系；再结合新型灰色相似关联度，构造多元缓冲算子，解决多种因素对费用数据的干扰问题；进一步探讨新模型稳定性与样本量的关系、模型的新信息优先性等一些数学性质，通过理论证明和实例验证揭示新模型在复杂装备费用测算方面的优势；最后，以我国某复杂装备作为实际案例进行研究，运用新模型测算该装备全寿命周期的费用。

以分数阶的“in between”思想为研究主线，分别提出了分数阶累加灰色模型、分数阶导数灰色模型、分数阶缓冲算子，实现了充分挖掘新信息和微调缓冲强度的目标，并探讨了新模型稳定性与样本量的关系等一些数学性质。构造的多元缓冲算子解决了多种因素对装备费用数据的干扰问题。再结合新型灰色相似关联度，提出相似信息优先的复杂装备费用预测模型。从理论上证明了基于相似关联度的GM(0,N)模型的建模优势。最后，通过实例验证了新模型在复杂装备费用预测中的实用性和有效性。本项目研究成果对完善灰色预测理论，拓展模型应用范围具有重要意义；运用本项目的模型预测新装备的费用，可以为装备发展建设提供决策支持。课题组在《Information Sciences》、《Applied Soft Computing》、《Energy》、《IEEE Transactions on Systems Man and Cybernetics: Systems》、《Applied Mathematics and Computation》、《IEEE/CAA Journal of Automatica Sinica》、《系统工程理论与实践》等国内外期刊发表论文28篇，其中SCI检索12篇(SCI一区2篇，二区3篇)，EI检索8篇。项目主持人以第一作者发表16篇，在科学出版社出版专著1部。研究成果获得省部级一等奖2次。