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• Two-kind combined reliability systems are introduced in terms of two reliability systems: (n , f , k) and < n , f , k >. • Some important concepts such as irreducible structure and structure equivalence are introduced. • We present an important criterion: four-class combined reliability systems are needed to be considered in essential. • A coordinate system is introduced for representing the working and failure regions for the combined reliability systems. • A procedure for calculations of the combined reliability systems is given. In the paper, two-element (f , k) and < f , k > combined systems are developed, and two-kind combined reliability systems and some related new concepts such as irreducible structure and structure equivalence are introduced, which can be used not only in real practical situations, but also enrich the reliability contents. We proved that four-class combined reliability systems need to be considered actually, which facilitates the handling of these combined reliability systems. Meanwhile, a coordinate system with two axes, the number of total failed components and the number of maximal consecutive failed components, is introduced in the paper for representing the working and failed areas for the combined reliability systems, which brings out a useful tool for understanding the working and failure mechanisms, and system state transition ways, for the combined reliability systems. To summarize the reliability calculation, a procedure for calculations of the combined reliability systems is presented, which is an efficient guidance for these reliability computations. Finally, two numerical examples are given to illustrate the computation procedure in which the detailed steps, special on using Finite Markov Chain Imbedding Approach (FMCIA), are shown. We believe that the present paper can shed light on researches on combined reliability systems. [ABSTRACT FROM AUTHOR]

• A new optimal maintenance model is proposed for a three-unit parallel production system. • The goal of maintenance policy is to maximize the system's yields. • We obtained the expectation yields of the system and found the optimal maintenance policy by comparing the expectation yields. • A procedure for calculations of the optimal maintenance policy is given. Many production facilities consist of multiple parallel units to facilitate the completion of production plans, yet these devices are inevitably prone to malfunctions with increased operating time. Therefore, developing the optimal maintenance policy to maximize the production capacity of production facilities is a crucial issue. This paper focuses on determining the optimal maintenance policy for a three-unit production system, aiming to maximize its yield. The optimal solution is provided based on the given maintenance policy, and a general optimal maintenance policy for the system is derived. The properties related to the optimal maintenance policy are analyzed. The proposed model offers two notable contributions when compared to existing research. Firstly, a maintenance policy based on the number of visits and yields is developed. Secondly, the parallel assumption of the system and the assumption that the subsystems satisfy any distribution are in line with reality, enriching the research on maintenance strategies for production systems. To illustrate the results of the model, a numerical example is presented. [ABSTRACT FROM AUTHOR]

A warranty product often has an interval structure, resulting in different warranty periods for each component due to their distinct degradation paths. From the manufacturer's perspective, it is reasonable to establish appropriate warranty periods for each part to achieve cost savings. This paper introduces a dependent series configuration consisting of a main component and an auxiliary component. The failure of the main component leads to system failure, while the failure of the auxiliary component only causes a random damage to the main component. These two components follow different Wiener degradation paths, and then their first passage times (FPTs) follow disparate inverse Gaussian distributions. Assuming a pre-deterministic warranty period, W , for the main component, and η W , 0 < η < 1 , for the auxiliary component, the warranty period can be divided into two stages: [ 0 , η W ] and [ η W , W ]. In this context, the paper proposes warranty claim policies for replacement, objective-oriented imperfect repair (OOIR), and refund. The corresponding warranty costs sustained by manufacturer and consumer within these two stages are considered. Finally, a numerical example is presented to illustrate the warranty costs sustained by manufacturer and consumer under different scenarios, as well as the total warranty costs. • The degradation paths, warranty periods and warranty costs of the series system is analyzed. • The warranty costs sustained by manufacturer and consumer under different scenarios is analyzed. • The interdependence between various components is considered. [ABSTRACT FROM AUTHOR]

• In a shock environment, the system's self-healing mechanism is considered. • The damage evolution effect of shock is proposed. • In a dynamic self-healing environment, two competing failure modes are presented. • The change in degradation rates is considered. We consider a system subject to two distinct failure modes, i.e., a soft failure subject to gradual degradation, and a hard failure caused by cumulative shocks, whichever comes first. This type of systems has wide applications and has been extensively studied in the literature. However, little attention has been paid to the dynamic self-healing mechanism in such systems. With the emergence of intelligent systems and complex architectures, materials with intrinsic self-healing mechanism have the ability to recover damage under external random shocks. In order to illustrate practical significance of the proposed model, reinforced concrete beams (RC-beam) with self-healing capsules in dependent competing failure processes due to cumulative crack evolution effect and rebar performance degradation are taken as the application background. In this paper, the change of degradation rates and damage evolution effect are considered based on the design of critical failure threshold and self-healing mechanism. This study proposes a novel reliability model for a system experiencing dependent competing degradation and cumulative shock failure in a dynamic self-healing environment. A procedure is developed to evaluate the reliability based on this model. We demonstrate the fitness of proposed model through numerical simulation and sensitivity analysis. [ABSTRACT FROM AUTHOR]

Multi-state system models are more practical and valuable when describing complex engineering structures than binary-state system models. Modelling of linear consecutive- (k , l) -out-of- n : F systems with shared components under non-homogeneous Markov dependence is discussed in this work. The proposed model is an extension of both "consecutive- k -out-of- n : F system" and "consecutive- k -out-of- n : F system with shared components between adjacent subsystems". All the components in the system are assumed to have three states and are non-homogeneous Markov-dependent. Using the method of sum of disjoint products (SDP), the system reliability formula is derived by summing the reliability values for all disjoint cases. The well-known finite Markov chain imbedding approach (FMCIA) is used for deriving the reliability formula for each disjoint case. For the special cases of systems with homogeneous Markov-dependent or independent components, the corresponding results are deduced. Finally, some numerical examples are presented to illustrate all the results developed here. • Consecutive-(k,l)-out-of-n: F systems with shared components are modelled. • Using the method of SDP and FMCIA, the system reliability formula is derived. • The method is also fit for homogeneous Markov-dependent or independent cases. • Numerical examples demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Due to the complex structures and the multi-functionality of modern products, there are usually two or more performance characteristics which can reflect a product's degradation states. The degradation processes corresponding to these performance characteristics are dependent in general, which brings challenges to the degradation data analysis. In this paper, a gamma process based degradation model is developed for the bivariate dependent degradation data, where the dependency between the two degradation processes is captured by a common random effect naturally. The expectation maximization algorithm is employed to estimate the model parameters. Then, a real-time prediction method for a product's remaining useful life is proposed using the Bayesian method. Finally, both the simulation study and the case study are provided for illustration, whose results demonstrate that the proposed model as well as the corresponding inference methods does work well. • A gamma process based bivariate degradation model is proposed. • The dependency between the two PCs is captured by a common random effect naturally. • The MLEs of the model parameters are obtained via the EM algorithm. • Analytical formulas are presented for the RUL prediction. • The effectiveness is validated via simulation and case studies. [ABSTRACT FROM AUTHOR]

Reliability analysis for k -out-of- n systems has received much attention in the field of reliability due to its practical importance. In this work, models for linear and circular k -out-of- n : F systems with shared components, consisting of m subsystems, are discussed. The finite Markov chain imbedding approach cannot be used directly to derive reliability of the considered systems because of the common area existing between adjacent subsystems. For this reason, we first derive the probability that the number of failed components is a given value in the m common areas. Then, all disjoint cases, each of which is regarded as a series system consisting of m subsystems with different initial state probability distributions, are obtained, with which the n i th step transition rate matrix is calculated by using the Markov chain technique, for i = 1 , 2 , ... , m. Finally, the reliability of the system is obtained by summing the reliabilities of all cases along with the finite Markov chain imbedding approach. Some numerical examples are presented to illustrate the efficiency of the proposed reliability evaluation method. • Linear and circular k -out-of- n : F systems with shared components are modelled. • A method combining the finite Markov chain imbedding approach with some theoretical analysis is proposed. • Reliability calculation rules for both two considered systems are described based on our method. • Numerical examples are presented to illustrate the efficiency of the proposed reliability evaluation method. [ABSTRACT FROM AUTHOR]

l First/second-order discrete-time aggregated semi-Markov processes are studied. l Multi-point/interval bounded-covering availabilities are introduced for discrete-time systems. l Formulas are derived of the new indices for these aggregated semi-Markov processes. Research on reliability indices has received considerable attention recently in the field of reliability. In this work, two new reliability indices — multi-point bounded-covering availability and multi-interval bounded-covering availability — are introduced for discrete-time systems. Their explicit formulas are derived for discrete-time first- and second-order aggregated semi-Markov systems, whose state spaces are divided into three subsets corresponding to perfect functioning states, imperfect functioning states and failure states, respectively. Finally, some numerical examples are presented to illustrate the developed results. The results established here will also be useful for many practical reliability systems. [ABSTRACT FROM AUTHOR]

• We develop dynamic maintenance policies for redundant systems. • We establish the CBM optimization model and propose improved RL methods. • We validate the robustness of the proposed CBM strategy via comparisons. • We illustrate the applicability of the proposed framework via arbitrary structures. The condition-based maintenance (CBM) decision-making for redundant systems has attracted increasing attention. Most existing studies are dedicated to k -out-of- n redundant systems and the search of the optimal maintenance policy is efficient for low-dimensional CBM. In practical applications, complex system structures and failure criteria are commonly observed, posing challenges for searching the optimal CBM policy. This paper studies the optimal CBM strategy for redundant systems with arbitrary system structures using improved reinforcement learning, considering failure and economic dependences. The decisions of imperfect repair and replacement of failed components are considered dynamically, and an efficient solution method of dynamic maintenance strategy is investigated via improved reinforcement learning incorporating re-learning and pre-learning processes. Numerical studies are conducted and the results indicate that the proposed method is effective in reducing the maintenance cost and efficient in searching the optimal CBM strategy for redundant systems. [ABSTRACT FROM AUTHOR]