分数阶离散灰色模型及其在备件需求预测中的应用

针对某型新概念武器装备缺乏可比对的现有装备，备件需求历史数据少，对装备本身保障特性缺乏了解等问题，提出应用分数阶GM(r,1)模型进行备件需求预测的方法。应用矩阵扰动理论证明了GM(r,1)模型的扰动界小于GM(1,1)模型的扰动界。利用1阶累加矩阵及其矩阵乘法运算推导出p阶累加矩阵。应用分数阶差分方程理论，将p阶累加矩阵推广到r分数阶累加矩阵，建立分数阶累加灰色模型GM(r,1)。通过矩阵求逆运算，得到r分数阶累减矩阵，简化了r分数阶累减计算方法。应用遗传算法确定GM(r,1)模型最优阶数，利用GM(r,1)模型预测维修备件需求，并通过实际数据实验，表明GM(r,1)模型比GM(1,1)模型具有更好的预测性能。

Fractional Order Discrete Grey Model and Its Application in Spare Parts Demand Forecasting

A method of applying fractional GM (r,1) to forecast the demand of the spare parts is proposed for a new concept weapon because of the lack of comparable existing equipment,less historical data on spare parts demand,and the lack of understanding the supportability of equipment.The perturbation bound of GM (r,1) is proven to be smaller than the perturbation bound of GM (1,1) by using the matrix perturbation theory.p-order cumulative matrix is obtained by the first-order cumulative matrix and its matrix multiplication.Based on fractional order differential equation theory,the p-order accumulative matrix is extended to the r fractional order accumulative matrix,and a fractional accumulative gray model GM (r,1) is established.r fractional order difference matrix is obtained by matrix inversion,and the calculation method of r fractional difference is simplified.The optimal value of r in GM (r,1) is determined through genetic algorithm (GA).The GM (r,1) model is applied to forecast the demand of spare parts.The experimental results show that GM (r,1) model has better prediction performance than GM (1,1)model.