Research On Bayesian Reliability Assessment For NC Machine Tools Based On Grid Approximation

In recent years, there is a trend in domestic NC(Numerical Control) machine tools which is characterized by small-lot production, accelerating pace of products’ update and reliability levels’ rising year by year. And this trend is more obvious in domestic high-end NC machine tools. Thus for a field reliability test of a new model of NC machine tools, there cannot be many available products of the same model and a long period of test time like traditional reliability tests, and people cannot observe failures that frequently like before. The above facts indicate that: the decrease in number of failures observed in NC machine tools’ reliability tests and the corresponding decrease in size of data sample is an inevitable trend.The following problem has arisen already in engineering practice, that is, for some models of NC machine tools developed under the state key science and technology project, the sizes of data sample generated in tests are so small, that the classic statistical methods, which rely on a large data sample, cannot be utilized due to severe bias. This problem is named briefly as the small-sample problem of NC machine tools.The small-sample problem of NC machine tools arose comparatively later compared with aeronautics and astronautics industries, and the corresponding solutions are not fully developed.Recently, scholars in NC machine tools industry started to borrow experiences from aeronautics, astronautics, nuclear power and military equipment industries, and adopted Bayesian reliability modeling and assessment method to solve small-sample problem of NC machine tools. Since a NC machine tool is a complex and repairable system, its data form and reliability model is different from those of success/failure system, such as rockets and missiles.Thus, in process of exploring Bayesian reliability modeling and assessment techniques independently by experts in machine tools industry:(a) some solutions to the existing problems need to be improved;(b) for some new phenomenon which has happened already, the corresponding new problems need to be presented, the corresponding mathematical models should be established, and the corresponding solutions need to be proposed. The above discussions are objectives of this dissertation, which are introduced concretely as follows:(1) The first stage of a Bayesian reliability modeling and assessment method is to build the prior distributions of the reliability models’ parameters. However, most literatures skip this stage by few words which briefly state that the prior distributions are given by experts according to their experience and information of similar products. Although giving prior distribution in this way is beyond questioning in assessing reliability of astronautic products, parameters of NC machine tools’ reliability model do not have obvious physical meanings compared with success rate of launching rockets. Therefore, to present prior distributions of parameters directly by experts cannot avoid a large subjective bias. Actually, it is a special research area composed of techniques of eliciting expert judgment, and many scholars have proposed systematic and structured expert-judgment process.A method of indirectly building Weibull parameters’ prior distributions is proposed aimed at the above problems, which consists of two stages:(a) multi-source prior information of NC machine tools and other definitions are presented, and process of eliciting expert judgment is designed, which is suitable for NC machine tools and obtains the quantified result of expert judgment;(b)mathematical method of transforming the expert-judgment result into the Weibull parameters’ prior distributions is proposed. In combination with the real case, the proposed method is applied, which achieves the fusion of multi-source prior information and the expert experience, and decreases the subjective bias as a result of experts directly giving prior distributions.(2) The second stage of a Bayesian reliability modeling and assessment method is to calculate reliability model parameters’ posterior distributions. For two-parameter Weibull distribution, this stage encounters the problem of high-dimension integration which has no closed form and makes the calculation complicated. At present, many literatures adopt the Markov chain Monte Carlo(MCMC) algorithm to solve this problem. However, most of the literatures referring to the MCMC algorithm do not present the concrete algorithm, since MCMC algorithm is a general name for many concrete algorithms, and not all of the algorithms can solve the problem of NC machine tools.Aimed at the above problem, a bivariate Metropolis algorithm, a member of MCMC algorithm family, is developed independently to calculate posterior distributions of Weibull parameters. Parameters of the algorithm are presented such as the proposal distribution and acceptance probability; iterative procedure of the algorithm is presented; and Matlab code of the algorithm is presented. In combination with the real case, parameter estimators and mean time between failures(MTBF) are calculated.(3) Some literatures adopt software WinBUGS to solve the complex calculation of reliability model parameters’ posterior distributions, but few of them introduce the operations of WinBUGS in details. Actually, if a user needs to use WinBUGS to solve problems of NC machine tools, he should have a background in Bayes statistics, learn some knowledge of programming language BUGS, and grasp skills to describe non-standard distributions, which is specific for WinBUGS.Aimed at the above problem, each step of WinBUGS is introduced in details; the Bayesian reliability model is described in BUGS language; the “zeros trick” for describing non-standard distributions is introduced and proved. In combination with real case, BUGS code is given and process of operations is described. Finally, parameter estimators and MTBF of NC machine tools are obtained. In addition, the fact is pointed out that the WinBUGS used slice sampling, one of MCMC algorithms, to calculate posterior distributions.(4) Any MCMC algorithm, programmed by user or WinBUGS, has common problems:(a) although an MCMC algorithm is highly accurate, the non-standard distributions will cause unstable and uncertain factors in random-sampling process, which may cause algorithm’s crash;(b) principles of MCMC algorithms are complex, which complicates the process of solving problems by independently programming or software. Thus, MCMC algorithms are not fit for widely application of Bayesian methods in NC machine tools’ engineering field.Aimed at the above problems, the grid approximation method is adopted, and the probability mass functions of parameters are defined which discretize the continuous prior distributions. Discrete-formed formula for calculating parameters’ posterior distributions is derived, which solves the calculation difficulty of high-dimension integration. In combination with real case, estimators of parameters and MTBF of NC machine tools are obtained. Comparisons among grid approximation, Metropolis algorithm and WinBUGS are made, and results indicate that errors of MTBF estimators of these three methods are smaller than 0.03 hours, which demonstrates that the developed grid approximation method is as good as MCMC algorithms in calculation accuracy, simple in principle and easy in programming. The grid approximation method is helpful for widely application of Bayesian methods in NC machine tools’ engineering field.(5) It is a new phenomenon for NC machine tools which generate zero failures in a reliability test, and it is a new problem to implement reliability modeling and assessment under zero failures. Up to now, no literatures have been found to describe and solve this problem. Zero-failure problem is a classic problem in other industries and nearly all of the corresponding solutions are Bayesian methods. These methods provide references for solving zero-failure problem of NC machine tools.Aimed at the above problems, data form of NC machine tools’ zero-failure problem is proposed, and corresponding Bayes statistical model is established. A second method of eliciting process of expert judgment and building Weibull parameters’ prior distributions is proposed. In combination with real case, the software WinBUGS and the developed grid approximation method are applied respectively to calculating parameter estimators and MTBF. The results indicate that: error of MTBF estimators between the two methods is smaller than 1 hour, which again demonstrates that the developed grid approximation method is practical, highly-accurate and suitable for engineering application.(6) To answer some questions on “subjectivity” of Bayesian methods, the testing strategy for Bayesian methods is proposed. In combination with real case, comparison between Bayesian method and classic method is implemented, of which the results indicate that when sample size n≤10, the Bayesian method is more objective, accurate and closer to the realities compared with classic method.(7) The first independently developed method of “eliciting expert judgment and building Weibull parameters’ prior distributions” and the independently developed grid approximation method is integrated and made into a B/S(Browser/ Server) structured software, of which the name is: Bayesian reliability modeling and assessment system for NC machine tools.