Development of an optimization method for the GM(1, N) model

The multi-variable grey model represented by GM(1,N) is an important causal relationship forecasting model. However, since the structure of GM(1,N) is more complicated than that of the single-variable grey forecasting model GM(1,1), it is difficult to build a useful GM(1,N) model. The simulation and prediction errors of GM(1,N) models are usually greater than those of GM(1,1) models. The modeling process of the traditional GM(1,N) model is studied and three serious defects are observed in terms of "modeling mechanism", "parameter estimation" and "model structure", which are the major causes of low model precision. In this paper, a novel GM(1,N) model is proposed to improve the performance of the traditional GM(1,N) model by introducing a linear correction term and a grey action quantity term to the traditional GM(1,N) model. The new model has more reasonable modeling process and more stable structure, which solves the three defects of the traditional GM(1,N) model. In addition, the new model is entirely compatible with the single variable discrete grey forecasting model DGM(1,1)and the multi-variable grey forecasting model GM(0,N). To verify the effectiveness of the development, the new model is used to simulate the tensile strength of a material. The mean relative simulation and prediction percentage errors of the new model are 0.0707% and 5.7369%, in comparison with those of the traditional precision GM(1,N) model and the classical GM(1,1) model, which are 6.0011%, 18.4280% and 1.1020%, 12.5190% respectively. The findings show that the new model has the best performance, which on one hand testifies the correctness of the defect analysis, and on the other hand validates the effectiveness of the structure reform of the traditional GM(1,N) model.