Imperfect Repair Model Research Articles

The ENBIS‐21 quality and reliability engineering international special issue

The ENBIS-21 Quality and Reliability Engineering International Special Issue is related to the contributions to the 21st Annual Conference of the European Network for Business and Industrial Statistics (ENBIS) held online from 13 to 15 September, 2021. The topics covered by the 12 papers included in this special issue fall within the scope of ENBIS, whose ultimate goal is to bridge the gap between methodology and application in the field of business and industrial statistics and foster networking between statisticians in academic and commercial settings. It becomes evident from the papers on this issue that classical methods of industrial statistics are still having a wide range of applications and expansions towards complex and big data structures as well as they are increasingly cross-fertilized by artificial intelligence and machine learning perspectives and approaches. The issue is opened by the contribution of Yang et al.1 in the Statistical Process Monitoring (SPM) of data in the form of tensors. The authors develop a hierarchical in-situ process monitoring approach for anomaly detection, which is applied to thermal video streams acquired during a metal additive manufacturing process for highly complex parts. A more traditional SPM approach based on generalized likelihood ratio (GLR) control charts is elaborated by Rizzo and Di Bucchianico2 to set up tailor-made monitoring for high-purity (high-quality) processes and deals with issues arising in GLR control charts for discrete processes. Along the line of developing statistical methods for complex data structures, Gril et al.3 set up a tensor-on-tensor regression model able to dynamically predict repetitive human motions helpful in avoiding unwanted collisions at the interface with robots in industrial assembly tasks. Chuquin et al.4 present a straightforward extension of the K-means clustering to spatially correlated functional data, with an application in the analysis of the normalized difference vegetation index in a large region of Ecuador. From a more general perspective, Vanacore et al.5 address the problem of assessing prediction performance in the multi-class classification of imbalanced data sets. In the optimal Design of Experiments (DoE) Pesce et al.6 offer new results that are oriented to bias reduction in large datasets and protection against confounders. Whereas, Kirchhoff et al.7 use DoE as the initial solution for the optimization of shift planning in high-bay warehouse operations based on Gaussian process models with low-rank correlation kernels. Risk assessment and management is the topic of the following three papers. Testik and Unlu8 compare traditional and fuzzy failure modes and effect analysis to rank risks in different areas of testing and calibration laboratories. Meynaoui et al.9 deal with an accidental scenario in a sodium-cooled fast nuclear reactor by means of a global sensitivity analysis of the input parameter distribution used in numerical simulator outputs to model physical phenomena. Lai and Zwetsloot10 offer a data-driven ensemble ranking system of product quality, which is validated in the identification of high-risk manufacturers in a solar industry case study. The final two papers are devoted to the Reliability and Maintenance of repairable systems. El-Aroui and Gaudoin11 propose a general goodness-of-fit test for imperfect repair models that has the favorable property of being asymptotically distribution-free for renewal and power law processes. Fouladirad et al.12 develop a new bounded transformed gamma process model to describe and predict degradation phenomena by relaxing the typical assumption that the degradation level can grow indefinitely, which is often unrealistic for technological units. The proposed model is also applied to a set of wear measures of cylinder liners that equip a diesel engine for marine propulsion. We are deeply grateful to the journal staff for their professional assistance in the editorial process, as well as to the anonymous referees for the time and effort spent to provide careful and constructive comments, and to all the authors, who made revisions on a very tight timescale.

Imperfect repair models: Sequential goodness‐of‐fit testing based on predictive performances

AbstractImperfect repair (IR) modelling has attracted a lot of attention during the last decades. To assess and predict repairable systems' reliability, analysts have to choose among a plethora of models: renewal processes, non‐homogeneous Poisson processes (NHPPs), Brown–Proschan, Kijima, ARA, ARI, Quasi‐Renewal (QR), Trend‐Renewal and so forth. Choosing an appropriate model for a given failures, dataset is an important practical issue. The fit of an IR model can be assessed using goodness‐of‐fit (GoF) tests but very few have been proposed and a good fit to past data does not guarantee good reliability predictions. This work proposes general GoF tests for IR models based on sequential (or on‐line) assessment of times‐to‐failures forecasts. The suggested predictive‐sequential (or prequential) GoF tests have a low computational complexity and a common test statistic for several IR models. These tests have been proved to be asymptotically distribution‐free (ADF) for renewal and power‐law processes (PLPs). Our numerical simulations and preliminary theoretical results highly suggest that this ADF property still holds for several other IR models. The prequential tests are much easier to use than the already known bootstrap tests, and a simulation study shows that they are also slightly more powerful. The simulations also show that the prequential tests are powerful to identify classes of similar appropriate models but are much less powerful to distinguish models belonging to the same class. A comparison of MTTF estimates show that models from the same class give close reliability predictions and that the tests are able to reject models which would yield to dramatic errors in reliability predictions.

An Imperfect Repair Model with Delayed Repair under Replacement and Repair Thresholds

Based on the extended geometric process, a repair replacement model of a degradation system is studied, in which the delayed repair time depends on the working time after the last repair. Replacement and repair thresholds describe when the system will be replaced and when the system can be repaired, respectively. Two kinds of replacement policies are studied. One policy is jointly determined by the moment of the Nth failure and the first hitting time of the working time after the last repair for the replacement threshold, and the system is replaced, whichever occurs first; the other is the special case of the first policy, and the system is replaced when the working time after the last repair first hits the replacement threshold. The exact expressions of the long-run average cost rate are obtained. The optimal policies exist and can be ascertained by numerical methods. Finally, numerical examples are presented to demonstrate the application of the results obtained in the paper.

On the delayed worse-than-minimal repair model and its application to preventive replacement

Abstract Minimal repair and other imperfect repair models have been intensively studied in the literature. Much less attention has been payed to the ‘worse than minimal’ repair problem, although it often occurs in practice due to the adverse effects of previous repairs, environmental and internal shocks, etc. To model this type of repair, we define a new point process that behaves as the non-homogeneous Poisson process up to a certain event or time (minimal repairs) and then it becomes the generalized Polya process of repairs (worse than minimal repairs). The corresponding replacement policy is defined and the optimal solutions that minimize the long run expected cost rate are analyzed. The replacement can be executed univariately either after the given time T or the given number of repairs (on the k-th failure). Moreover, the system can be also replaced by implementing the bivariate strategy, that is, after the time T or on the k-th failure, whichever comes first. The detailed numerical examples illustrate our findings. It is shown that the k-strategy outperforms the T -strategy (lower cost rates), whereas the bivariate strategy is not worse than the best univariate strategy.

Unobserved heterogeneity in stable imperfect repair models

This study investigates the effect of heterogeneity on the failures of repairable systems that undergo imperfect repairs, which are extensively used in reliability engineering. When considering a group of similar systems, the assumption that the repair processes are independent and identically distributed becomes questionable owing to the unobserved heterogeneity in these systems. The basic model we consider is the Kijima type II virtual age process with constant repair efficiency and a Weibull baseline distribution. We use the frailty models to study the heterogeneity between the systems and, in particular, the gamma-distributed frailty is investigated. We thus derive the asymptotic properties of the mixed repair process and corresponding likelihood estimates, and then evaluate the effects on the model parameter estimation process when heterogeneity is erroneously ignored. Furthermore, when the model is established correctly by accounting for the gamma distribution, we find that the maximum likelihood estimator is inconsistent and propose an alternative approach. Three case studies are presented to illustrate the benefits of taking account of unobserved heterogeneity in the estimation of the aging speed and reliability of assets and in scheduling preventive maintenance activities.

Steady-state imperfect repair models

Imperfect maintenance models are widely used in reliability engineering. This paper discusses relevant asymptotic properties for the steady-state virtual age processes. It is shown that the limiting distributions of age, the residual lifetime and the spread that describe an ordinary renewal process can be generalized to the stable virtual age process, although the cycles of the latter are not independent. Asymptotic distributions of the virtual age at time t, as well as of the virtual ages at the start and the end of a cycle containing t (as t tends to infinity) are explicitly derived for two popular in practice imperfect maintenance models, namely, the Arithmetic Reduction of Age (ARA) and the Brown–Proschan (BP) models. Some applications of the obtained results to maintenance optimization are discussed.

Optimization of the Number and Frequency of Repairs

When managing the production process at an enterprise, it is important to correctly develop a policy for overhauls of machinery and equipment. Usually for this purpose, enterprises use the standards for the repairs frequency recommended in the literature. In publications on the reliability theory, repair policy is considered as an optimization problem. In publications on the theory of reliability, this problem is considered as optimization. In this case, usually the criterion of optimization is some kind of relative indicator, such as availability rate or average operating costs per unit of time. However, such criteria do not fully reflect the economic interests of enterprises – owners of equipment. In a market economy, these enterprises are focused on maximizing their market value. In this paper, we propose to form a repair policy in which the market value of the enterprise, and therefore all its assets, will be most. The corresponding mathematical model is the best use underling the valuation theory. It also takes into account the physical deterioration of machines during their operation, as well as the possibility of random marginal failures, leading to the need to stop using the machine. In this case, the improvement of the technical condition of the machine after its overhaul is described using the Kijima imperfect repair model. The calculations carried with our model allow establishing how the optimal number and timing of repairs depend on such parameters as repair cost and its restoration factor, the rate of physical wear of machines, and the intensity of their resource failures. We show how to use the constructed model to take into account the effect of inflation rates and income and property tax rates on the repair policy.

Stochastic comparisons of imperfect maintenance models for a gamma deteriorating system

This paper compares two imperfect repair models for a degrading system, with deterioration level modeled by a non homogeneous gamma process. Both models consider instantaneous and periodic repairs. The first model assumes that a repair reduces the degradation of the system accumulated from the last maintenance action. The second model considers a virtual age model and assumes that a repair reduces the age accumulated by the system since the last maintenance action. Stochastic comparison results between the two resulting processes are obtained. Furthermore, a specific case is analyzed, where the two repair models provide identical expected deterioration levels at maintenance times. Finally, two optimal maintenance strategies are explored, considering the two models of repair.

A probabilistic model for maintaining and optimising the life-cycle performance of deteriorating structures

Benefiting from a devised imperfect repair model and given a cost structure, this paper addresses the problem of determining an optimal inspection and threshold-type repair policy for systems whose performance is described by a Wiener process. The system is monitored at periodic times and preventive maintenance actions are carried out in response to the observed system state. The approach can deal with failures defined by performance or regulations. Precisely speaking, failure is defined by a critical set such that the first entry of the performance process to the critical set implies system failure. This approach is typically appropriate for lifecycle models as a specific performance requirement is, or is close to being violated. Since there is a random amount of maintenance, and on the other hand each maintenance incurs a cost, using the renewal-reward theorem, this paper aims at joint determination of an optimal inspection and repair policy providing a right balance between the amount of maintenance and the increasing cost. Benefiting from an imperfect repair model, the presented probabilistic model provides a framework for further developments.

A model of system limiting availability under imperfect maintenance

PurposeThe purpose of this paper is to explore the impact of the Kijima Type II imperfect repair model on the availability of repairable systems (RS). Since many individuals are interested in measuring the extent to which the system will be available after it has been run for a long time, the specific interest in this study is in the steady-state (limiting) availability behavior of such systems. Furthermore, the authors study the impact of age-based preventive maintenance (PM) on the RS performance.Design/methodology/approachBecause of the complexity of the underlying assumptions of the Kijima Type II model, the authors use simulation modeling to estimate the system availability. Based on preliminary simulation results, the availability function achieves a steady-state value greater than zero. The system steady-state availability is then estimated from the simulation output by computing the average of the availability estimates beyond the initial transient period. Next, the authors develop a meta-model to convert the system reliability and maintainability parameters into the coefficients of the limiting availability estimate without the simulation effort. Using a circumscribed central composite experimental design, the authors confirm the accuracy of the meta-model.FindingsThe results show that the meta-model is robust, and provides good estimates of the system limiting availability. Also, the authors find that when using a Kijima Type II model for a system repair process, age-based PM can improve the steady-state availability value. Therefore, an optimal age-based PM policy that maximizes the system’s steady-state availability can be identified.Originality/valueIn practice, it is important to study the system steady-state availability because many individuals, i.e. engineers, are more interested in measuring the extent to which the system will be available after it has been run for a long time. Therefore, this study represents a significant addition to the body of knowledge related to virtual age modeling, in that it incorporates a Kijima type II model and considers system steady-state availability.

Analytical properties of an imperfect repair model and application in preventive maintenance scheduling

The paper considers repairable systems under imperfect repair. The failure rate of a new system is assumed to follow a Weibull distribution and the repair efficiency is characterized by a Kijima type II virtual age model named Arithmetic Reduction of Age with infinite memory. An analytical approach to obtain the distribution of the inter-failure times is presented. The existence of a stationary regime is highlighted and the limiting distributions are explicitly derived. In this context, an optimal age-based preventive maintenance policy can be implemented. Three approaches are proposed, considering a static, a dynamic or a failure limit policy. Numerical simulations are presented to illustrate the policies.

Warranty claims forecasting based on a general imperfect repair model considering usage rate

Because manufacturers of products sold with any type of warranty incur additional costs from servicing customer warranty claims, accurately forecasting the number of warranty claims has become increasingly important to manufacturers. Under the assumption of Weibull distributed time to failure, we propose two models based on a generalised renewal process that consider product usage rate to forecast the number of failures of products sold with a two-dimensional warranty policy. The accelerated failure time model is used to investigate the effects of the product usage rate on system reliability. The maximum likelihood estimation combined with a nonlinear constrained programming method is used to estimate the parameters of the proposed models. We conduct data experiments based on both simulation and real data collected from an excavator manufacturer in China to test the performance of the proposed models. The results indicate that the models incorporating a variable usage rate more accurately forecast the number of failures than those models based on a nominal usage rate.

Dynamics of an optimal maintenance policy for imperfect repair models

A preventive maintenance policy that considers information provided by observing the failure history of a repairable system is proposed. For a system that is to be operated for a long time, it is shown that the proposed policy will have a lower expected cost than a periodical one which does not take into account the failure history. Statistical inference using both maximum likelihood point estimates and bootstrap confidence intervals is discussed. The proposed policy is applied to a real situation involving maintenance of off-road engines owned by a Brazilian mining company. A simulation study compares the performance between the maintenance policy proposed and the periodical one.

ARA and ARI imperfect repair models: Estimation, goodness-of-fit and reliability prediction

An appropriate maintenance policy is essential to reduce expenses and risks related to equipment failures. A fundamental aspect to be considered when specifying such policies is to be able to predict the reliability of the systems under study, based on a well fitted model. In this paper, the classes of models Arithmetic Reduction of Age and Arithmetic Reduction of Intensity are explored. Likelihood functions for such models are derived, and a graphical method is proposed for model selection. A real data set involving failures in trucks used by a Brazilian mining is analyzed considering models with different memories. Parameters, namely, shape and scale for Power Law Process, and the efficiency of repair were estimated for the best fitted model. Estimation of model parameters allowed us to derive reliability estimators to predict the behavior of the failure process. These results are a valuable information for the mining company and can be used to support decision making regarding preventive maintenance policy.

On the optimal degree of imperfect repair

A simple cost-wise comparison between the minimal and perfect repair of a system is discussed first using a relevant example. The main focus of this note, however, is on imperfect (general) repair. The best repair for our system in this case is defined as the one that corresponds to the optimal level (extent) of repair actions that minimize the long-run expected cost per unit of time. This complex optimization problem is considered for a specific imperfect repair model (Kijima II), using the developed earlier asymptotic approach to the corresponding virtual age modelling. It is shown that the optimal solution exists when the failure rate of a system tends to infinity as t tends to infinity and the corresponding cost function decreases sufficiently fast. An example illustrating the optimization procedure is considered.

Optimal maintenance scheduling for a complex manufacturing system subject to deterioration

We address the problem of determining inspection strategy and replacement policy for a deteriorating complex multi-component manufacturing system whose state is partially observable. We develop inspection and replacement scheduling models and other simple maintenance scheduling models via employing an imperfect repair model coupled with a damage process induced by operational conditions. The system state in performance of the imperfectly repaired system is modelled using a proportional intensity model incorporating a damage process and a virtual age process caused by repair. The system is monitored at periodic times and maintenance actions are carried out in response to the observed system state. Decisions to perform imperfect repair and replacement are based on the system state and crossing of a replacement threshold. The model proposed here aims at joint determination of a cost-optimal inspection and replacement policy along with an optimal level of maintenance which result in low maintenance cost and high operational performance and reliability of the system. To demonstrate the use of the model in practical applications a numerical example is provided. Solutions to optimal system parameters are obtained and the response of the model to these parameters is examined. Finally some features of the model are demonstrated. The approach presented provides a framework so that different scenario can be explored.

A Joint Redundancy and Imperfect Maintenance Strategy Optimization for Multi-State Systems

The redundancy allocation problem has been extensively studied with the aim of determining optimal redundancy levels of components at various stages to achieve the required system reliability or availability. In most existing studies, failed elements are assumed to be as good as new after repair, from a failure perspective. Due to deterioration, the repaired element cannot always be restored to a virtually new condition unless replaced with a new element. In this paper, we present an approach of joint redundancy and imperfect maintenance strategy optimization for multi-state systems. Along with determining the optimal redundancy levels, the element replacement strategy under imperfect repair is also optimized simultaneously, so as to reach the desired availability with minimal average expenditure. A generalized imperfect repair model is proposed to characterize the stochastic behavior of multi-state elements (MSEs) after repair, and a replacement policy under which a MSE is replaced once it reaches the pre-determined number of failures is introduced. The cost-repair efficiency relation, which regards the imperfect repair efficiency as a function of assigned repair cost, is put forth to provide a flexibility of assigning repair efforts strategically among MSEs. The benefits of the proposed method compared to the existing ones are demonstrated and verified via an illustrative case study of a three-stage coal transportation system.

Bayesian Analysis of ARA Imperfect Repair Models

This article proposes a Bayesian analysis of a class of imperfect repair models, the ARA models. The choice of prior distributions and the computation of posterior distributions are discussed. The presentation is unified for all ARA models and many kinds of possible priors. A numerical study on the quality of the Bayesian estimators is presented, as well as a comparison with the maximum likelihood estimators. Finally, the approach is applied to a real data set.

An estimate of maintenance efficiency in Brown-Proschan imperfect repair model with bathtub failure intensity

Purpose: Estimate the maintenance efficiency in the Brown-Proschan model with the bathtub failure intensity. Design/methodology/approach: Empirical research through which we propose a framework to establish the characteristics of failure process and its influence on maintenance process. Findings: The main contribution of the present study is the reformulation of the Brown and Proschan model using the bathtub failure intensity Practical implications: Our model is defined by BP reformulation one using bathtub failure intensity. This form of intensity is presented like superposition of two NHPP and Homogeneous Poisson one. Originality/value: This is the follow on research on the study that employed the power-law-process type of failure intensity.

A General Imperfect Repair Model Considering Time-Dependent Repair Effectiveness

Kijima I and Kijima II models are two important imperfect repair models in literature. These two models use one constant parameter to represent the degree of repair, which is called Repair Effectiveness (RE) in this paper. We developed a more general imperfect repair model by extending the constant RE to a time-dependent function based on the virtual age process, where the Kijima models are special cases of the new model. A simulation method is developed to estimate the cumulative number of failures for the new model, and a Bayesian inference method is proposed to select the best imperfect repair model. Finally, to demonstrate the new model, a numerical example is provided. From this example, the new model shows a more accurate mean and a narrower confidence interval than that of the Non-Homogeneous Poisson Processes, and Kijima I and Kijima II models.