The Optimization Method Research On The Non-homogeneous GM(1,1) Model And Grey Verhulst Model

Grey system theory founded by Professor Deng in the early 1980 s,research on the“part of known information, part of unknown information ”and “small sample,poor information” uncertain system, generation and develop the “part of known information”,extract the valuable information to correctly describe and control the actual system’s opreation rules and its’ evolution of operation rules. There are many examples of poor information in the actual production process, among them, the GM model is the most widely used prediction model, including GM(1,1) basic model, non-homogeneous GM(1,1) model, DGM(1,1) model, grey Verhulst model, etc..Although GM prediction model has made some achievements, but its essence still exist mainly in research and application which still lack of innovation theory and method. Therefore, many scholars have carried on different aspects to each kind of GM forecast model. In this paper, the optimazation method based on the theory of the grey differential equation and the white differential equation is studied from two aspects,which are the grey Verhulst model of first-order accumulated generating sequence in Logistic curve form and the non-homogeneous GM(1,1) model of the original sequence in non-homogeneous index form. The main work of this paper is as follows:(1) For the non-homogeneous GM(1,1) model, start with the fact of the accuracy of the model, this paper integral the both sides of whitenization differential equation at the same time that based on the 1-AGO form of approximate non-homogeneous exponential sequence. After that the grey differential equation is obtained so that the background value has been constructed more reasonable and parameter a can be solved. Then let we can find the parsing expressions of parameterα and β through the target function as the minimum of the sum of the square of the relative error. Combine with the above two steps, a new optimization model is offered.(2) According to the error analysis of grey Verhulst model, the concept of optimizing the grey derivative and the parameters at the same time is proposed to develop a new prediction model. Firstly, let k be a continuous value directly and take the derivative of the first order accumulated function. After that the original sequence of grey differential equation must be replaced by the derivative function obtained in the previous step, thus the optimal grey differential equation which matches with whitenization differential equation and the parameter a can be got. Then we can find the parsing expressions of parameter α and β through the least square method used intarget function to archive the dual optimization of the parameters and the translational constant. Combine with the above two steps, a new grey Verhulst model can be obtained. The result shows that the model is simulated and predicted with complete coincidence to meet Logistic function curve. Finally, experiments show the validity and rationality of this method, they also illustrate the correctness of the conclusion.