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1

Remaining useful life prediction of equipment considering dynamic thresholds under the influence of maintenance

Li Kangning, Ren Li'na, Li Xueliang, Wang Ziqian

Eksploatacja i Niezawodność

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2024

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Vol. 26, no. 1

art. no. 174903

EN

A novel approach for predicting remaining useful life (RUL) is proposed for situations where maintenance threshold and failure threshold exhibit dynamic behavior due to uncertainties in degradation and the influence of detection strategies during maintenance processes. The approach introduces maintenance threshold error to establish a multi-stage maintenance-impact degradation model with dynamic maintenance threshold based on the Wiener process. This model considers the impact of maintenance on degradation rate, amount, and path. Moreover, by using the first hitting time (FHT) and introducing failure threshold error to reflect the dynamic behavior of the failure threshold, the formula for predicting equipment RUL is derived. The model parameters are estimated using both the maximum likelihood estimation (MLE) approach and Bayesian formula. The proposed approach was validated with simulation data and gyroscope degradation data, and the results demonstrate its ability to effectively enhance the precision of equipment RUL prediction.

2

Quasi-periodic Inspection and Preventive Maintenance Policy Optimisation for a system with Wiener Process degradation

Gao Wenke, Wang Yu, Zhang Xiwen, Wang Zhuangzhuang

Eksploatacja i Niezawodność

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2023

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Vol. 25, no. 2

art. no. 162433

EN

Periodic inspection policy is performed for some degradation systems to check their degradation states, whereas it is usually difficult to implement on time due to impact of some random factors. Inspections and some maintenance actions are implemented in an inspection window with random, and thus how to optimize the inspection windows and the degradation threshold of the system to perform preventive maintenance (PM) are beneficial in practice. To this end, an optimisation of quasi-periodic inspection and PM policy with inspection window is proposed for a degradation system whose degradation followed Wiener process with a linear drift. Assume that PM can change the degradation rate and inspections are randomly performed in each inspection window. After optimisation, the optimal interval of the inspection window, the degradation threshold of PM and PM policy are determined by minimising the long-term running cost rate of the system. Finally, modeling and optimisation are illustrated using the degradation process of an axial piston pump, and the sensitivity analysis of some key parameters is conducted.

3

Reliability assessment for micro inertial measurement unit based on accelerated degradation data and copula theory

Chi Bingjia, Wang Yashun, Hu Jingwen, Zhang Shufeng, Chen Xun

Eksploatacja i Niezawodność

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2022

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Vol. 24, no. 3

554--563

EN

With its extensive use in industry, assessing the reliability of the micro inertial measurment unit (MIMU) has become a pressing need. Unfortunately, the MIMU is made up of several components, and the degradation processes of each are intertwined, making it difficult to assess the MIMU’s reliability and remaining useful life. In this research, we offer a reliability assessment approach for the MIMU, which has long-lifetime and multiple performance characteristics (PCs), based on accelerated degradation data and copula theory.Each PC model of MIMU is constructed utilizing drift Brownian motion to depict accelerated degradation process. The copula function is used to model the multivariate dependent accelerated degradation test data and to describe the dependency between multiple MIMU performance parameters. The particle swarm optimization algorithm is used to estimate the unknown parameters in the multi-dependent ADT model. Finally, the storage test and simulation example on MIMU’s accelerated degradation data verify the feasibility and effectiveness of the proposed method.

4

An approach in determining the critical level of degradation based on results of accelerated test

Hoang Anh D., Vintr Zdenek, Valis David, Mazurkiewicz Dariusz

Eksploatacja i Niezawodność

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2022

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Vol. 24, no. 2

330--337

EN

Nowadays, systems are more complex and require high reliability for their components, especially critical system components. Therefore, to avoid serious damage, system are often replaced before the actual failure. The replaced parts are considered to have “soft failure”, and the limit in which the parts are replaced is known as the critical level of the degradation process. Determining the appropriate value of the critical level for a product is an important problem in their exploitation, as well as for predicting the Mean Time to Failure (MTTF) or Remaining Useful Lifetime (RUL) of this product based on the degradation data by the mathematical models. In this article, an approach in determining the critical levels based on failure data from an accelerated test is introduced. This approach is applied with the degradation process of Light-Emitting Diodes (LED) in an accelerated test and a type of Wiener process-based model is used to predict the MTTF or RUL of LED based on their degradation data and the found critical level.

5

Energy of Taut Strings Accompanying Random Walk

Lifshits Mikhail A., Siuniaev Anatoly A.

Probability and Mathematical Statistics

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2021

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Vol. 41, Fasc. 1

9--23

EN

We consider the kinetic energy of the taut strings accompanying trajectories of a Wiener process and a random walk. Under certain assumptions on the band width, it is shown that the energy of a taut string accompanying a random walk within a band satisfies the same strong law of large numbers as proved earlier for a Wiener process and a fixed band width. New results for Wiener processes are also obtained.

6

On background driving distribution functions (BDDF) for some selfdecomposable variables

Jurek Zbigniew J.

Mathematica Applicanda

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2021

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Vol. 49, no. 2

85--109

EN

Many classical variables (statistics) are selfdecomposable. They admit the random integral representations via Levy processes. In this note are given formulas for their background driving distribution functions (BDDF). This may be used for a simulation of those variables. Among the examples discussed are: gamma variables, hyperbolic characteristic functions, Student t-distributions, stochastic area under planar Brownian motions, inverse Gaussian variable, logistic distributions, non-central chi-square, Bessel densities and Fisher z-distributions. Found representations might be of use in statistical applications.

PL

Wiele klasycznych modeli probabilistycznych opiera sie o zmienne losowe samorozkładalne. Maja one losowe reprezentacje całkowe oparte o procesy Lévy’ego. W tej notatce podano wzory dla ich kierujących (generujących) dystrybuant. Takich reprezentacji można używać do symulacji tych zmiennych. Wśród omawianych przykładów są: rozkłady t-Studenta, pole stochastyczne pod planarnymi ruchami Browna, odwrotny rozkład Gaussa, rozkłady logistyczne, niecentralny rozkład chi-kwadrat, rozkład Bessela i rozkłady statystyk Z-Fishera. Podane reprezentację mogą być przydatne w statystyce.

7

Accelerated degradation analysis based on a random-effect Wiener process with one-order autoregressive errors

Li Junxing, Wang Zhihua, Liu Chengrui, Qiu Ming

Eksploatacja i Niezawodność

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2019

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Vol. 21, no. 2

246--255

EN

For highly reliable and long-life products, accelerated degradation test (ADT) is often an effective and attractive way to assess the reliability. To analyze the accelerated degradation data, it has been well recognized that it is necessary to incorporate three sources of variability including the temporal variability, the unit-to-unit variability and measurement errors into the ADT model. The temporal variability can be properly described by the Wiener process. However, the randomness of the initial degradation level, which is an important part of the unit-to-unit variability, has been often neglected. In addition, regarding the measurement errors, current ADT models often assumed them to follow a mutually independent normal distribution and ignored the autocorrelation that may probably exist in them. These problems lead to a poor accuracy for reliability evaluation in some situation. Thus, a random-effect Wiener process-based ADT model considering one-order autoregressive (AR(1)) errors is proposed. Then closed-form expressions for the failure time distribution (FTD) is derived based on the concept of first hitting time (FHT). A statistical inference method is adopted to estimate unknown parameters. Finally, a comprehensive simulation study and a practical application are given to demonstrate the rationality and effectiveness of the proposed model.

PL

W przypadku wysoce niezawodnych produktów o długim cyklu życia, przyspieszone badanie degradacji (ADT) często stanowi skuteczny i atrakcyjny sposób oceny niezawodności. Jak wiadomo, analiza danych z przyspieszonej degradacji wymaga włączenia do modelu ADT trzech źródeł zmienności, w tym zmienności czasowej, zmienności między jednostkami i błędów pomiarowych. Zmienność czasową można odpowiednio opisać za pomocą procesu Wienera. Jednak losowość początkowego poziomu degradacji, który stanowi ważną część zmienności między jednostkami, jest często w badaniach pomijana. Ponadto, w odniesieniu do błędów pomiaru, obecne modele ADT często zakładają, że mają one wzajemnie niezależne rozkłady normalne, ignorując możliwą autokorelację. Problemy te prowadzą w niektórych sytuacjach do niskiej trafności oceny niezawodności. W związku z powyższym, zaproponowano model ADT oparty na procesie Wienera z efektem losowym, w którym uwzględniono błędy autoregresyjne pierwszego rzędu (AR (1)). Następnie, w oparciu o pojęcie pierwszego czasu przejścia, wyprowadzono wyrażenia w postaci zamkniętej dla rozkładu czasu uszkodzenia (FTD). Do oszacowania nieznanych parametrów przyjęto metodę wnioskowania statystycznego. Na koniec przedstawiono kompleksowe studium symulacyjne i wskazano praktyczne zastosowanie modelu w celu wykazania jego racjonalności i skuteczności.

8

Residual life prediction for highly reliable products with prior accelerated degradation data

Wang H.-W., Teng K.-N.

Eksploatacja i Niezawodność

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2016

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Vol. 18, no. 3

379--389

EN

To precisely predict the residual life for functioning products is a key of carrying out condition based maintenance. For highly reliable products, it is difficult to obtain abundant degradation data to precisely predict the residual life under normal stress levels. Thus, how to make use of historical degradation data to improve the accuracy of the residual life prediction is an interesting issue. Accelerated degradation testing, which has been widely used to evaluate the reliability of highly reliable products, can provide abundant accelerated degradation data. In this paper, a residual life prediction method based on Bayesian inference that takes accelerated degradation data as prior information was studied. A Wiener process with a time function was used to model degradation data. In order to apply the random effects of all the parameters of a Wiener process, the non-conjugate prior distributions were considered. Acceleration factors were introduced to convert the parameter estimates from accelerated stress levels to normal stress levels, so that the proper prior distribution types of the random parameters can be selected by the Anderson-Darling statistic. A Markov Chain Monte Carlo method with Gibbs sampling was used to evaluate the posterior means of the random parameters. An illustrative example of self-regulating heating cable was utilized to validate the proposed method.

PL

Precyzyjne przewidywanie trwałości resztkowej użytkowanego produktu stanowi klucz do prawidłowego utrzymania ruchu w oparciu o bieżący stan techniczny (condition-based maintenance).W przypadku produktów o wysokiej niezawodności, trudno jest uzyskać ilość danych degradacyjnych, która umożliwiałaby precyzyjne prognozowanie trwałości resztkowej przy normalnym poziomie obciążeń. Dlatego też bardzo ważnym zagadnieniem jest wykorzystanie historycznych danych degradacyjnych umożliwiających zwiększenie trafności prognozowania trwałości resztkowej. Przyspieszone badania degradacyjne, które powszechnie wykorzystuje się do oceny niezawodności wysoce niezawodnych produktów, mogą dostarczać bogatych danych o przyspieszonej degradacji. W przedstawionej pracy badano metodę prognozowania trwałości resztkowej opartą na wnioskowaniu bayesowskim, w którym jako uprzednie informacje wykorzystano dane z przyspieszonych badań degradacji. Dane degradacyjne modelowano za pomocą procesu Wienera z funkcją czasu. Aby móc zastosować efekty losowe wszystkich parametrów procesu Wienera, rozważano niesprzężone rozkłady a priori. Wprowadzono współczynniki przyspieszenia , które pozwoliły na przekształcenie szacowanych wartości parametrów z poziomu obciążeństosowanych w próbie przyspieszonej do poziomu obciążeń normalnych, co umożliwiło wybór odpowiednich typów parametrów losowych rozkładu a priori zwykorzystaniem statystyki testowej Andersona-Darlinga. Metodę Monte Carlo opartą na łańcuchach Markowa z próbnikiem Gibbsa wykorzystano do oceny średnich a posteriori parametrów losowych. Proponowaną metodę zweryfikowano na postawie przykładu samoregulującego przewodu grzejnego.

9

Optimal and Suboptimal Control of a Standard Brownian Motion

Lefebvre M.

Archives of Control Sciences

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2016

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Vol. 26, no. 3

383--394

EN

The problem of optimally controlling a standard Brownian motion until a fixed final time is considered in the case when the final cost function is an even function. Two particular problems are solved explicitly. Moreover, the best constant control as well as the best linear control are also obtained in these two particular cases.

10

Solution of linear and nonlinear diffusion problems via stochastic differential equations

Bargieł M., Tory E. M.

Computer Science

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2015

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Vol. 16 (4)

415--428

EN

The equation for nonlinear diffusion can be rearranged to a form that immediately leads to its stochastic analog. The latter contains a drift term that is absent when the diffusion coefficient is constant. The dependence of this coefficient on concentration (or temperature) is handled by generating many paths in parallel and approximating the derivative of concentration with respect to distance by the central difference. This method works for one-dimensional diffusion problems with finite or infinite boundaries and for diffusion in cylindrical or spherical shells. By mimicking the movements of molecules, the stochastic approach provides a deeper insight into the physical process. The parallel version of our algorithm is very efficient. The 99% confidence limits for the stochastic solution enclose the analytical solution so tightly that they cannot be shown graphically. This indicates that there is no systematic difference in the results for the two methods. Finally, we present a direct derivation of the stochastic method for cylindrical and spherical shells.

11

Tractability of multi-parametric Euler and Wiener integrated processes

Lifshits M. A., Papageorgiou A., Woźniakowski H.

Probability and Mathematical Statistics

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2012

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Vol. 32, Fasc. 1

131--165

EN

We study average case approximation of Euler and Wiener integrated processes of d variables which are almost surely rk-times continuously differentiable with respect to the k-th variable and 0 ≤ rk ≤ rk+1. Let n(ε; d) denote the minimal number of continuous linear functionals which is needed to find an algorithm that uses n such functionals and whose average case error improves the average case error of the zero algorithm by a factor ε. Strong polynomial tractability means that there are nonnegative numbers C and p such that [formula]. We prove that the Wiener process is much more difficult to approximate than the Euler process. Namely, strong polynomial tractability holds for the Euler case iff ...[formula]. Other types of tractability are also studied.

12

O pewnych kryteriach wyboru inwestowania w akcje przy ustalonym poziomie ryzyka inwestora

Kozłowski E., Warowny T.

Badania Operacyjne i Decyzje

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2000

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Nr 2

5-13

PL

Przedmiotem artykułu są inwestycje w akcje. W modelu założono ciągłość zmian cen akcji w czasie, rozkład cen log-normalny, brak arbitrażu na rynku i liniowość funkcji kosztów transakcji. Przy tych założeniach inwestor, ustalając swoją skłonność do ryzyka, może tak dobierać rodzaje akcji i czas ich posiadania, że prawdopodobieństwo osiągnięcia ustalonego zysku będzie nie mniejszy niż ustalił.

EN

Each shareholder asks himself "which shares to choose in order to obtain a satisfying return?", "what risk is connected with the given investment?". We consider the probability of obtaining the return not smaller than the return from riskless bonds. We assumed lack of arbitrage and linearity of transactional cost function. It turned out that under these assumptions, the investor by defining his attitude to the risk can choose shares and time of realization of return in such a way that this constraint will be satisfied. We recall a definition and properties of the Wiener process which has huge meaning in stochastic modeling of capital markets. In order to describe the evolution of asset price we use geometric Brownian motion applied to investigation of the optimal strategy of choosing the moment of selling the shares.