Optimal Estimation of the Condition of Production Equipment Based on the Analysis of Claims

The purpose of the study is to develop methods for optimal estimation of unknown non-random factors affecting the quality of production equipment by the criterion of maximum likelihood, as well as random factors by the criterion of minimum standard error based on the processing of information related to claims received using the mathematical theory of random point processes and the theory of statistical solutions.

The research method consists in applying the well-known hypothesis about the distribution of operating time for failure of technical systems in the form of an exponential distribution depending on the failure rate function. The fact is used that the corresponding distribution of the number of failures is distributed according to the Poisson law with the same function of failure rates. It is assumed that the intensity function depends not only on time, but also on a set of unknown non-random parameters, or on random parameters. It is emphasized that such factors may reflect the generalized state of the technical system, and information about this may be contained in the facts of product claims. The task of optimal estimation of the parameters on which the failure rate function depends is set. Since in this formulation of the problem, only the facts of filing claims, as well as the times of their presentation, are available for processing, the maximum likelihood function method is used for optimal estimation of nonrandom parameters, and the optimal Kalman filter is used for random parameters. The problem of optimal estimation of unknown parameters from a multiplicatively separable failure rate function, i.e. one that is representable as a product of a separate function of time and a vector function of unknown parameters, is considered. It is shown that for such a function, the optimal estimation problem is reduced to the problem of estimating a single scalar parameter that scales the time function. The well-known Kalman algorithm for continuous parameters is applied to the case of the observed process in the form of the number of claims’ events and the time of their occurrence. Examples of evaluation of both unknown and random factors are given for unified real data on tissue defects, and confirm the operability of the algorithms and their applicability for the simplest assessments of the condition of production equipment.

The new results include the formulation of the problem of studying a failure intensity function that depends on a set of unknown nonrandom parameters, the application of the maximum likelihood method and Kalman algorithm for optimal estimation of these parameters, and the proof that for a separable failure intensity function, the optimal estimation reduces to the estimation of a scalar quantity that scales the time-dependent intensity function.

The conclusion states that examples of assessment of factors affecting the function of the failure rate confirm the operability of the algorithm and its applicability for the simplest assessments of the condition of production equipment. A separate task is to develop analytical expressions for the failure rate function that depends on parameters, as well as methods for comparing estimates obtained by different methods. Solving these tasks will make it possible to develop methods for clarifying the condition of production equipment.

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