Mixed redundancy, which combines active and cold-standby redundancy, can improve reliability design flexibility but substantially increases computational complexity. Consequently, it is rarely used in multi-objective reliability-redundancy allocation problems (MRRAPs), as balancing conflicting objectives proves challenging. To address it, this paper formulates a bi-objective RRAP (BRRAP) with mixed redundancy, aiming to maximize system reliability while minimizing cost. The reliability of cold-standby and mixed redundant subsystems is precisely evaluated using continuous time Markov chain models. Although swarm intelligence algorithms are widely used for MRRAPs because of their global search capability and implementation simplicity, their stochastic updating mechanism often leads to weak local exploitation and premature convergence. To overcome this limitation, an importance measure (IM)-based two-stage BRRAP optimization framework is developed, which iteratively combines swarm intelligence-based global search with IM-guided local refinement. By adjusting Pareto solutions from both reliability and redundancy perspectives, the IM-based local optimization effectively pushes the Pareto front toward higher reliability and lower cost. Experiments based on four benchmarks demonstrate that the proposed framework improves solution quality, convergence, and diversity of Pareto fronts.

Remaining useful life (RUL) prediction is fundamental to prognostics and health management (PHM) in industrial systems. In practical deployment, rotating machinery operates over long service lifecycles under continuously evolving loads and rotational speeds. Such variability, combined with long-term acquisition of monitoring data, gives rise to non-stationary degradation patterns and distributional shifts. These characteristics challenge conventional deep learning-based RUL models, which are typically trained on static datasets and lack adaptability to sequentially arriving operating conditions. Moreover, most existing approaches lack explicit mechanisms for predictive uncertainty quantification in dynamic task environments, restricting their reliability in risk-aware maintenance decision-making. To address these challenges, this paper proposes a hybrid Bayesian learning framework for uncertainty-aware continual RUL prediction in non-stationary industrial settings. The framework employs Bayesian neural networks (BNNs) to jointly model aleatoric and epistemic uncertainties and integrates prior-guided Bayesian knowledge transfer with likelihood-guided generative replay for continual learning. A conditional diffusion-based replay mechanism is introduced to synthesize representative pseudo-samples, together with a dual-uncertainty-driven sample selection strategy that retains informative historical knowledge without storing raw data. Extensive experiments on multiple run-to-failure bearing and gear datasets under diverse operating conditions demonstrate that the proposed method achieves superior RUL prediction accuracy, enhanced robustness to distributional shifts, and more reliable uncertainty quantification in continual learning scenarios, underscoring its suitability for long-term industrial monitoring and predictive maintenance.

Knowledge distillation has recently been proven to be effective for model compression and acceleration of point cloud-based 3D object detection. However, the complementary network pruning is often overlooked during knowledge distillation. In this paper, we propose a pre-pruned distillation framework that combines network pruning and knowledge distillation to better transfer knowledge from the teacher to the student. To maintain the feature consistency between the student and the teacher, we train a teacher model and then generate a compact student model by structural channel pruning. Then, we employ multi-source knowledge distillation to transfer both mid-level and high-level information to the student model. Additionally, to improve the object detection performance of the student model, we propose a soft pivotal position selection mask to emphasize the features of the foreground regions during distillation. We conduct experiments on both pillarand voxel-based 3D object detectors on the Waymo datasets, demonstrating the effectiveness of our approach in compressing point cloud-based 3D detectors.

As a nonlinear measure, Lempel–Ziv complexity (LZC) can be considered as a suitable parameter for characterizing bearing health status by measuring the complexity of vibration signals. However, in continuous monitoring scenario under noisy condition, all components of a multicomponent bearing signal are not equally sensitive toward a change of LZC value. As a result, a direct application of LZC for bearing health monitoring not only suffers from its inefficient early fault warning but also fails to infer the fault progression. In this article, instead of direct utilization of a whole vibration signal, its fundamental component (FC) sensitive to LZC calculation is separated with the help of continuously adjustable parameterized tunable$Q$factor wavelet transform (TQWT). In this context, a study based on sparsity indices has been done for$Q$factor selection of TQWT. Since TQWT uses an oscillation-based bearing FC separation scheme for LZC calculation, the proposed measure is termed as oscillatory Lempel–Ziv complexity (OLZC). Two experimental cases are used for validation. Performance of OLZC is compared with original LZC, representative sparsity indices and recently proposed multiscale symbolic Lempel–Ziv complexity. Results demonstrate that the proposed OLZC can not only overcome the limitations of the original LZC but also performs better than other indices in comparison to continuous monitoring of bearing health.

The input of low-quality feedstocks triggers the interdependence between workpiece quality and machine reliability, which will further adversely impact the performance of manufacturing systems. Considering the interconnected manufacturing system structures, our primary goal is to provide an effective method to compute the performance of networked manufacturing systems suffering from machine and low-quality feedstock interdependence. The strength of our work first lies in the model for the compound degradation process of machines and dissemination of low-quality feedstocks, which enables us to construct a response chain to model the interdependence between machines and feedstocks in the manufacturing network. Then, the second strength is the effective algorithm for the computation of route connectivity and quality loss of a manufacturing network based on the interdependence model. A computational experiment shows our models and algorithm can work well for evaluating the operational performance of manufacturing networks.

In this article, a fault diagnosis scheme based on multiscale diversity entropy (MDE) and extreme learning machine (ELM) is presented. First, a novel entropy method called diversity entropy (DE) is proposed to quantify the dynamical complexity. DE utilizes the distribution of cosine similarity between adjacent orbits to track the inside pattern change, resulting in better performance in complexity estimation. Then, the proposed DE is extended to multiscale analysis called MDE for a comprehensive feature description by combining with the coarse gaining process. Third, the obtained features using MDE are fed into the ELM classifier for pattern identification of rotating machinery. The effectiveness of the proposed MDE method is verified using simulated signals and two experimental signals collected from the bearing test and the dual-rotator of the aeroengine test. The analysis results show that our proposed method has the highest classification accuracy compared with three existing approaches: sample entropy, fuzzy entropy, and permutation entropy.

Bayesian importance measures (BIMs) are useful tools for quantifying the contribution of an edge to the up or down state of the network. This article investigates BIMs for the K-terminal networks under the assumption that the failures of edges occur according to a branching process in which the total number of the failed edges follows a saturated Lagrangian Poisson distribution (SLPD). First, we derive two types of BIM equations when the total number of the failed edges follows a certain probability distribution. Both BIMs are represented in terms of network spectra that depend only on the network structure. It is also found that both BIMs are equivalent as they lead to the identical ranking order of network edges. Next, when the total number of the failed edges has an SLPD, several unique properties of BIMs rankings are explicitly derived. We further prove that under certain conditions, the rankings based on the BIMs belong to the structural ranking, namely, the spectra-based rankings solely depend on the network structure. Finally, the numerical analyses of a transportation network show that the BIMs can effectively measure the edge importance for the medium and relatively large networks.

Fierce global competition drives automated manufacturing systems (AMSs) to be increasingly complex, which poses significant challenges on performance analysis and production control. The multistage production via serial stations will lead to the propagation of failures in AMSs, which will affect system performance by triggering complex competitions among multiple failure modes. Although machine performance and product quality have been considered, very little has been done to investigate the effect of imperfect quality inspection on competing failures. Focusing on a time balance serial AMS, this article presents a new competing failure model to investigate the complex interactions among machine failures, product quality, and inspection process, which enables the characterizations of time-delayed propagation of failure, accumulation of degradation, and dynamics of states in serial AMSs. In order to further analyze the impact of competing behaviors on system performance, we have also developed decision diagram models and algorithms, which are evaluated and validated on serial AMSs with imperfect inspection, revealing the characteristic of multistate interactions. Experimental results show that the proposed methods have strong potentials for performance modeling and analysis of serial AMSs and also demonstrate general applicability for manufacturing decision making.

Fault diagnosis of bearings using classification techniques plays an important role in industrial applications, and, hence, has received increasing attention. Recently, significant efforts have been made to develop various methods for bearing fault classification and the application of Case Western Reserve University (CWRU) data for validation has become a standard reference to test the fault classification algorithms. However, a systematic research for evaluating bearing fault classification performance using the CWRU data is still lacking. This paper aims to provide a comprehensive benchmark analysis of the CWRU data using various entropy and classification methods. The main contribution of this paper is applying entropy-based fault classification methods to establish a benchmark analysis of entire CWRU datasets, aiming to provide a proper assessment of any new classification methods. Recommendations are provided for the selection of the CWRU data to aid in testing new fault classification algorithms, which will enable the researches to develop and evaluate various diagnostic algorithms. In the end, the comparison results and discussion are reported as a useful baseline for future research.

Traditional importance measures seldom consider how the number of failed components influences the network reliability. This paper proposes two importance measures under the circumstance that the failure sequence of the components follows a counting process. The first importance measure aims to assess the contribution of the individual component (edge) to the network failure. The second evaluates the contribution of the individual component to the network functionality. Both importance measures are time-dependent functions, and their values are jointly determined by the network structure and the distribution of the number of failed components at a particular time. We prove that the proposed importance measures are able to generate consistent rankings based on edge's impact on the network reliability behavior. When networks possess special structure or the number of failed edges follows the special distribution, the rankings are coincident with the results generated from some traditional importance measures. When component's failure sequence follows a saturated nonhomogeneous Poisson process, the proposed importance measures are equivalent to the structural importance measure as time approaches zero or infinite. Finally, numerical examples are provided to demonstrate the application and performance of the proposed measures.

Importance measure can be used to identify the most vulnerable components with respect to system functionality or failure. Traditional importance measures may not be effective to evaluate the contribution of an individual component if its reliability value does not fall in the full range between 0 and 1. Based on the cost-reliability relation, this paper proposes a generalized Birnbaum importance measure (GBIM) to quantify the contribution of individual components to system reliability improvement by considering reliability range, manufacturing complexity, and technology feasibility. Since GBIM possesses several unique features in terms of guiding system reliability optimization, in this paper, we further develop a GBIM-based genetic algorithm to solve a type of optimal reliability allocation problem. The numerical studies show that both the computational efficiency and the near global optimality based on GBIM outperforms the methods using the traditional importance measures.

The continuous state system is a special kind of a system in which the states of the system and its components have continuous values, ranging from perfect functioning to complete failure. This paper introduces the performance improvement for a continuous state system, which can be used to measure the improvement of systems performance comparing pre- and postmaintenance time. The probabilistic characteristics of performance improvement are discussed in detail. Then, the performance improvement for multicomponent maintenance and corresponding calculation method are also put forward to establish the objective function for maintenance optimization. Third, a maintenance optimization model for such a system is studied, and corresponding performance improvement based genetic algorithm is provided to search a near global optimal solution. Finally, two numerical examples and an oil transportation system application case study are implemented to verify the effectiveness of the proposed method.

Redundant systems, which usually consist of a number of same/similar components (or modules), have been used in various critical infrastructures to ensure the system's normal function. Usually, system reliability can be improved with the adoption of additional components or redundancies. Typically, two types are mainly included, i.e., majority voters and standby redundancies (referred to as SPARE gates for simplicity). Nevertheless, with the increment of redundancies, the consumed cost or space requirement also grows. This study considers a tradeoff between cost and reliability in order to pursue the cost-effective optimal design. The relationships between the total cost and corresponding parameters are discussed thoroughly. Besides, in order to determine the cost-effective design at the expense of a unit of cost, a revised evaluation standard is proposed (referred to as R per Cost). In this paper, we perform a cost-effective analysis of majority voters with different implementations; and we also perform the analyses of SPARE gates (here, a warm spare gate and a cold spare gate are mainly focused). For deriving corresponding total cost, algorithms are presented to predict the necessary failure time of components. In this line, cost-effective analyses of several case studies are performed.

The performance of the system is often influenced by the performance of a particular component. Hence, it is important to identify the state changing of which component dominates the system performance changing in a maintenance process. Motivated by the Griffith importance measure (GIM) model, this paper proposed the generalized GIM, which extends the application of GIM to complex multi-state components, to evaluate the accurate contribution of the components in the changing of the system performance by considering the transition probabilities of the states of each component. Furthermore, the generalized GIM method is successfully applied in the continuous-state systems by extending the system structure function in GIM model. As a result, the expression of the performance of continuous system and the generalized GIM of the continuous-state components are proposed. A numerical example and an application to an oil transportation system are presented to illustrate how the proposed method works.

Importance measures in reliability engineering are used to identify weak components of a system and signify the roles of components in contributing to proper functioning of the system. Recently, an integrated importance measure (IIM) has been proposed to evaluate how the transition of component states affects the system performance based on the probability distributions and transition rates of component states. In the system operation phase, the bathtub curve presents the change of the transition rate of component states with time, which can be described by three different Weibull distributions. The behavior of a system under such distributions can be modeled by the semi-Markov process. So, based on the reported IIM equations of component states, this paper studies how the transition of component states affects system performance under the semi-Markov process. This measure can provide useful information for preventive actions (such as monitoring enhancement, construction improvement, etc.), and provide support to improve system performance. Finally, a simple numerical example is presented to illustrate the utilization of the proposed method.

Importance measures are widely used to characterize the roles of components in systems. The system lifetime can be divided into different life stages. Traditionally, importance measures do not consider the possible effect of the expected number of component failures over a system's lifetime and over different life stages, which, however, has a great effect on the system performance changes, and should therefore be taken into consideration. This paper extends the integrated importance measure (IIM) from unit time to system lifetime, and to different life stages. Based on the renewal functions of components, this measure can evaluate the changes of the system performance due to component failures. This generalization of the IIM describes which component is the most important to improve the performance of the system during the system lifetime and at different life stages. An example of the application of an oil transportation system is presented to illustrate the use of the generalized IIM.

This paper mainly focuses on the integrated importance measure (IIM) of component states for maintenance processes. To describe the impact of each component state in maintenance processes, a maintenance cost function of multi-state systems is defined at first. Second, considering the probability distributions, transition rates of the component states, and system maintenance costs, the IIM of component states is described. The corresponding characteristics of the IIM of the component states are discussed in both series systems and parallel systems. Then the relationships between IIM and Griffith importance, Wu importance, mean absolute deviation, and multi-state redundancy importance measures are also discussed. At last, a numerical example is given to demonstrate the IIM of component states. The results show that IIM can be used to identify the most important component state for the maintenance decision.

This paper mainly focuses on the integrated importance measure (IIM) of component states based on loss of system performance. To describe the impact of each component state, we first introduce the performance function of the multi-state system. Then, we present the definition of IIM of component states. We demonstrate its corresponding physical meaning, and then analyze the relationships between IIM and Griffith importance, Wu importance, and Natvig importance. Secondly, we present the evaluation method of IIM for multi-state systems. Thirdly, the characteristics of IIM of component states are discussed. Finally, we demonstrate a numerical example, and an application to an offshore oil and gas production system for IIM to verify the proposed method. The results show that 1) the IIM of component states concerns not only the probability distributions and transition intensities of the states of the object component, but also the change in the system performance under the change of the state distribution of the object component; and 2) IIM can be used to identify the key state of a component that affects the system performance most.