Partially observed systems are common in practice, where the exact operational state of each component is inaccessible or obscured, yet aggregate functioning component numbers can be obtained by inspection or monitoring. Existing methodologies that incorporate observation data for reliability evaluation often lack the versatility for diverse system architectures and component interdependencies. To bridge these gaps, this study establishes a reliability updating framework applicable to general system configurations and dependency structures. This approach fully leverages partially observed order statistics data – specifically aggregate survival counts. By introducing the concepts of conditional ordered lifetime distributions and a general survival signature, this framework facilitates the derivation of closed-form reliability expressions for partially observed systems. An illustrative example is presented to elucidate the methodology and to demonstrate the generality, interpretability, and practical applicability of the proposed approach.

Previous research on event log data analysis has primarily focused on identifying critical and frequent events, as well as qualitatively assessing correlations between event occurrences. However, the probabilistic behavior of frequently occurring events over time remains poorly understood. Through an in-depth exploratory analysis, we reveal that the (log) inter-arrival times of events follow a bimodal mixture distribution, suggesting the presence of transitions between latent states. To better understand the data-generating mechanism underlying these frequent events, we employ Markov Modulated Renewal Processes (MMRPs), a type of hidden Markov model, to capture the patterns exhibited in the inter-arrival times between successive events. Due to limitations in record precision, some inter-arrival times are recorded as zero. To address this issue, we propose a simple data imputation algorithm to generate non-zero inter-arrival times, facilitating inference on the inter-arrival time distributions and the underlying MMRPs. The effectiveness of the algorithm is validated using synthetic data. Finally, we evaluate the proposed model on real manufacturing system data, uncovering key insights into system states.

This article pioneers an order statistics framework for analyzing systems reliability with dependent components, uniquely integrating survival signature computation and order statistics-derived probabilistic expressions. A Copula-based dependence modeling approach is utilized to characterize interdependencies within the system. Comparative analyses are conducted between the proposed framework and three established reliability assessment methodologies: structure function-based method; signature-based method; and survival signature-based method.

Structural reliability integrates the design variables over the safety region characterized by a positive limit state function when the reliability of the entire system is of concern. Calculating the structure function can be challenging for high-dimensional systems or intricate system architectures. In order to enhance the efficiency of time-dependent system reliability assessment, we emulate the integral formulation in structural reliability. To elaborate further, we treat each individual unit's lifetime variable as a design variable and subsequently perform calculations involving multiple integrals. Given the ordered nature of unit failure times, we leverage the order statistics distribution to simplify the multiple integrals into a double integral, and then multiply this result by the survival signature to obtain reliability. A two-terminal nine-unit network system configuration is illustrated to assess the performance and effectiveness of the proposed method.

The reliability-related design has been a crucial chain in complex and reliability-critical engineering systems. It serves as a preventive countermeasure to catastrophic failures caused by uncertainties. To keep up with this rapidly developing field, this paper presents a novel metaheuristic, based on the variational Bayesian inference (VBI) to efficiently solve the optimal reliability design. Specifically, the reliability-redundancy allocation problem (RRAP). The proposed metaheuristic starts from a primary population, then leverages VBI to fully excavate the information of feasible individuals from the previous generation, in order to produce the next-generation population. This process is iterated until the solution converges to the optimal decision scheme. In addition, we set up an automatic stratification strategy, so that the new individuals can approach the optimal solution faster. Furthermore, we divide RRAP into a reliability optimization problem (ROP) and a redundancy allocation problem (RAP). This not only reduces the dimension of decision variables, but also speeds up the convergence. ROP and RAP are solved sequentially and iteratively until the preset stopping condition is satisfied. The case studies showcase that the proposed approach can obtain the optimal or near-optimal solution within a reasonable period.

Prime numbers are roots of integers (1). Over millennia, nobody has been able to predict where prime numbers sprout (2) or how they spread. It has been believed that primes grow like weeds among natural numbers. This study nevertheless establishes the Periodic Table of Primes (PTP) using four prime numbers of 2, 3, 5, and 7. We identify 48 integers out of a period of 2×3×5×7=210 to be the roots of all primes and composites without factors of 2, 3, 5, and 7, each of which is an offspring of the 48 integers uniquely allocated on the PTP. The PTP provides a platform to make the study of primes clearer and easier. In other words, the 48 integers serve as the genes of primes, and among them 15 pairs serve as the genes of twin primes.

Estimating rare events in structural engineering demands intensive computational resources, and the complexity of the performance function poses challenges to accurate probability estimation. To efficiently conduct the structural reliability analysis on rare events, we propose a novel variational Bayesian Monte Carlo (VBMC)-based subset simulation (SS) method. The proposed approach can be divided into two aspects: the Monte Carlo simulation (MCS) is first applied to estimate the first-layer unconditional failure probability, then VBMC is employed to estimate subsequent conditional failure probabilities. Finally, the original small failure probability is obtained by the product of the first-layer unconditional failure probability and a series of conditional failure probabilities. The proposed approach inherits the merits of SS as well as VBMC, which converts the tricky rare event estimation into a series of high-frequency event estimations and leverages VBMC instead of the canonical Markov chain to produce the independent and informative next-generation failure-conditional samples. Four case studies, including high-dimensional and discrete state space scenarios, are performed to illustrate the feasibility and generality of the proposed method.

Consider a system consisting of multistate components that perform the same function, and each component occupies a location in the system. The deterioration processes of components differ due to different workloads, usage rates, or en-vironmental stresses that are associated with the locations. This article proposes a condition-based maintenance policy, in which the component reallocation (CR) with distinct assignments of com-ponents to locations and the preventive replacement of system are dynamically implemented based on the system state. A Markov decision process (MDP) is formulated to optimize the proposed condition-based multi-CR maintenance policy by determining the actions for each system state that minimize the expected long-run system maintenance cost. In the current studies on the MDP for maintenance optimization, the actions mainly include component replacement, imperfect repair, and system replacement. In this article, including CRs of distinct assignments as actions and con-sidering multiple times of CRs increase the action space of the MDP significantly. An enumeration-based value iteration algorithm and a genetic-algorithm-based value iteration algorithm are proposed. Numerical experiments on k-out-of-n:G systems and Monte Carlo simulation tests show the effectiveness of CRs on reducing the system maintenance cost and extending system lifetime and provide structural insights on the optimal maintenance policy.

Time‐dependent reliability‐based design optimization (RBDO) is a computationally tough problem that needs to be addressed urgently. The difficulty of solving the time‐dependent RBDO mainly comes from the time‐dependent reliability analysis involved in probabilistic constraints, which itself is one of the thorny problems in the reliability community and makes the computational cost become much more onerous. In this paper, a deep‐learning‐assisted approach is proposed to solve the time‐dependent RBDO. The proposed approach leverages the classification capability of the deep learning, and constructs the alternative model for the actual probabilistic constraint function in the so‐called augmented reliability space, so as to make the trained alternative model accurate wherever it will be invoked. Moreover, a sequential sampling technique utilizing the classification probability provided by the deep learning is proposed to further reduce the computational cost. Then, the time‐dependent reliability analysis involved in the time‐dependent RBDO is conducted by the cheaper alternative model instead of the original computing‐intensive probabilistic constraint function, which evidently reduces the computational burden. The presented examples showcase the performance of the proposed approach. Especially, for the complicated engineering application, the proposed approach saves about 10% of the computational cost compared with the existing methods.

A real system may go through several states ranging from full performance to complete failure, carrying out infinite partial performances during service time. This article portrays such a nonbinary state of systems via the fuzzy membership function. Then, the relation between the system reliability under the binary state and that under the fuzzy state is deduced, on which a multitype component allocation problem (MCAP) is investigated to search for the optimal permutation of different types of components to maximize the fuzzy system reliability. After that, we inherit the exploration ability of genetic algorithm (GA) and the exploitation ability of Birnbaum importance (BI) to propose a fuzzy-BI-based two-stage approach combined with GA in order to deal with the MCAP under fuzzy state assumption efficiently and accurately. The k-out-of-n systems in both low and high dimensions are presented to illustrate the effectiveness of the proposed algorithm and demonstrate the similarity and difference between the MCAP under binary state and that under fuzzy state. The experimental results showcase that the proposed approach outperforms the existing state-of-the-art approaches for MCAP.

Reliability-based design optimization (RBDO) is one of the most crucial techniques in complex and reliability-critical engineering systems. This has been a research hotspot over the past few decades. RBDO allows us to take into consideration the uncertainties from various sources, during the early design stage. It provides a design that satisfies various constraints as well as the reliability of the designed system performing the expected functions. Such capabilities can serve as countermeasures against the foreseeable uncertainties in the manufacturing or application process. Following a short preliminary overview of RBDO as well as the canonical strategies, this article sets out to review how surrogate models have been explored to streamline RBDO. This is done through a systematic study, outlining their respective advantages as well as disadvantages, and discussing the problems that need to be solved.

RGB and CYMK are two major coloring schemes currently available for light colors and pigment colors, respectively. Both systems use letter-based color codes that require a large range of values to represent different colors. The problem is that these two systems are hard to use for manipulating any operations involving combinations of colors, and they lack the capacity for inter-changeability or unification. Based on prime number theory and Goldbach's conjecture, this study presents a universal color system (C235) using a number-based structure to encode, compute and unify all colors on a color wheel. The proposed C235 system offers a unified representation for the efficient encoding and effective manipulation of color. It can be applied to designing a high-rate LCD system and colorizing objects with multiple attributes and DNA codons, opening the door to manipulating colors and lights for even broader applications.

his paper investigates the robust reliability design of a complex system with imprecise model parameters. The considered system is supposed to have multiple types of components. The random failure times of different types of components are independently distributed, while the i.i.d. assumption holds for failure times of components of the same type. The challenge lies in the involvement of imprecise parameters in the decision-making, which makes the redundancy allocation problem (RAP) increasingly complex as the injection of uncertainty. To address this gap, an efficient system reliability analysis method based on the survival signature is employed, in order to take its advantage of separating the probabilistic information from the embedded structural information. To facilitate the decision-making, a robust counterpart of RAP is constructed by using the min-max regret framework. After that, the corresponding min-max optimization problem is first decomposed into several subproblems, and then solved in an iterative way. The superiority of the proposed method is demonstrated and validated by several case studies.

Most classical accelerated degradation test (ADT) planning models implicitly overlook the errors when measuring the degradation levels of the test units. However, the sensor measurement errors are inevitable and the magnitude of the errors may have a trend to increase over time due to sensor degradation. As a consequence improperly overlooking the sensor degradation in ADT planning could result in a test plan with unsatisfactory performance. This article addresses this issue by proposing a sequential ADT planning model that factors in sensor degradation. The system degradation level is periodically measured, based on which we dynamically adjust the stress level during ADT. We adopt a Bayesian framework that periodically updates the posterior distribution of model parameters considering the sensor degradation. An approximate Bayesian computation algorithm is developed to circumvent the difficulty of directly evaluating the complicated likelihood function in our problem. Numerical studies on a gas turbine reveal that our sequential model outperforms several traditional ADT designs that overlook the sensor degradation.

In a modular system, uncertainties will spread among coupled modules and cause system failure. To cope with this issue, the reliability-based design optimization (RBDO) of modular systems came into being. However, the solution of this design task is a nested triple-loop process, making the computational burden unaffordable for real-world systems. Thus, this paper endeavors to effectively mitigate this computational effort. The individual module feasible approach is first proposed to tackle the coupling effects of modules, whereby, the original optimization problem is converted into a conventional one. Then, the Bayesian-inference-based support vector machine is utilized to build the alternative model for the actual probabilistic constraint function, in the augmented reliability space. The alternative model is constructed using small number of model evaluations, which possesses enough precision everywhere in the augmented confidence region. Finally, the optimal decision scheme is obtained by solving the formulated conventional RBDO using the alternative model. The performance of the proposed method is investigated using several examples.

A population-based optimization algorithm combining the support vector machine (SVM) and importance sampling (IS) is proposed to achieve a global solution to optimal reliability design. The proposed approach is a greedy algorithm that starts with an initial population. At each iteration, the population is divided into feasible/infeasible individuals by the given constraints. After that, feasible individuals are classified as superior/inferior individuals in terms of their fitness. Then, SVM is utilized to construct the classifier dividing feasible/infeasible domains and that separating superior/inferior individuals, respectively. A quasi-optimal IS distribution is constructed by leveraging the established classifiers, on which a new population is generated to update the optimal solution. The iteration is repeatedly executed until the preset stopping condition is satisfied. The merits of the proposed approach are that the utilization of SVM avoids repeatedly invoking the reliability function (objective) and constraint functions. When the actual function is very complicated, this can significantly reduce the computational burden. In addition, IS fully excavates the feasible domain so that the produced offspring cover almost the entire feasible domain, and thus perfectly escapes local optima. The presented examples showcase the promise of the proposed algorithm.

With the popularization of big data, an increasing number of discrete event data have been collected and recorded during system operations. These events are usually stored in the form of event logs, which contain rich information of system operations and have potential applications in fault diagnosis and failure prediction. In manufacturing processes, various levels of correlations exist among the events, which can be used to predict the occurrence of failure events. However, two challenges remain to be solved for effective reliability analysis and failure prediction: (1) how to leverage various information from the event log to predict the occurrence of failure events and (2) how to model the effects of multiple correlations on the prediction. To address these issues, this paper proposes a novel reliability model, which integrates Cox proportional hazards (PHs) regression into survival analysis and association rule mining methodology. The model is used to evaluate the probability of failure event, which occurs within a certain period of time conditional on the occurrence history of correlated events. To estimate parameters and predict occurrence of failure events in the model, an effective algorithm is proposed based on piecewise‐constant time axis division, Cox PHs model, and maximum likelihood estimation. Unlike the existing literature, our model focuses on the interactions among events. The applicability of the proposed model is illustrated through a case study of a manufacturing company. Sensitivity analysis is conducted to illustrate the effectiveness of the proposed model.

Additive manufacturing is a revolutionary technology that offers a different pathway for material processing and design. However, innovations in either new materials or new processing technologies can seldom be successful without a synergistic combination. We demonstrate an in situ design approach to make alloys spatially modulated in concentration by using laser-powder bed fusion. We show that the partial homogenization of two dissimilar alloy melts—Ti-6Al-4V and a small amount of 316L stainless steel—allows us to produce micrometer-scale concentration modulations of the elements that are contained in 316L in the Ti-6Al-4V matrix. The corresponding phase stability modulation creates a fine scale–modulated β + α′ dual-phase microstructure that exhibits a progressive transformation-induced plasticity effect, which leads to a high tensile strength of ~1.3 gigapascals with a uniform elongation of ~9% and an excellent work-hardening capacity of >300 megapascals. This approach creates a pathway for concentration-modulated heterogeneous alloy design for structural and functional applications.