Optimal inspection intervals for safety systems with partial inspections

The introduction of International Standard IEC 61508 and its industry-specific derivatives sets demanding requirements for the definition and implementation of life-cycle strategies for safety systems. Compliance with the Standard is important for human safety and environmental perspectives as well as for potential adverse economic effects (eg, damage to critical downstream equipment or a clause for an insurance or warranty contract). This situation encourages the use of reliability models to attain the recommended safety integrity levels using credible assumptions. During the operation phase of the safety system life cycle, a key decision is the definition of an inspection programme, namely its frequency and the maintenance activities to be performed. These may vary from minimal checks to complete renewals. This work presents a model (which we called ρβ model) to find optimal inspection intervals for a safety system, considering that it degrades in time, even when it is inspected at regular intervals. Such situation occurs because most inspections are partial, that is, not all potential failure modes are observable through inspections. Possible reasons for this are the nature and the extent of the inspection, or potential risks generated by the inspection itself. The optimization criterion considered here is the mean overall availability A _o , but also taking into account the requirements for the safety availability A _s . We consider several conditions that ensure coherent modelling for these systems: sub-systems decomposition, k-out-of-n architectures, diagnostics coverage (observable/total amount of failure modes), dependent and independent failures, and non-negligible inspection times. The model requires an estimation for the coverage and dependent-failure ratios for each component, global failure rates, and inspection times. We illustrate its use through case studies and compare results with those obtained by applying previously published methodologies.     

A Simple Dynamic Method for Obtaining Inspection Schedules

A method for obtaining inspection schedules is proposed for situations where it is difficult to quantify the costs associated with inspections and undetected failure, or when these costs vary in time. After each inspection, the next inspection is scheduled so that there is only a small, specified probability that a failure goes undetected for longer than a specified time. Some properties of the inspection times are derived, and numerical illustrations are given for some particular cases.     

A delay-time-based maintenance model of a multi-component system

There is a well established literature on delay-time modelsof regular inspection policies where inspections may or maynotbe perfect, and where the initial point u of a defect arisesas a homogeneous Poisson process. This paper extends the modellingin two ways. The first is to include the observed practice wherethe multi-component system is inspected not only on a plannedbasis, but also when a component fails. The second extensionis to use a nonhomogeneous Poisson process to describe defectarrivals in the system. An inspection–replacement modelbased upon these two extensions is then developed for a multi-componentsystem. The total expected cost per unit time is minimized withrespect to theinspection intervals and the system replacementtime. The likelihood function of the time of failures and thenumber of defects found at inspections is established, in orderto estimate model parameters based upon routinely collectedmaintenance data. As a special case of the general model, aninspection model—based upon a homogeneous Poisson processof defects arising—is also proposed, which has a relativelysimple structure. Both simulated and real-life data of failuresand defects identified at inspections are used to test the modelsand parameter-estimating procedure.     

Optimal scheduling of non-perfect inspections

^** Corresponding author. Email: romulo.zequeira{at}utt.fr^*** Email: christophe.berenguer{at}utt.fr In this paper, we study the determination of optimal inspectionpolicies when three types of inspections are available: partial,perfect and imperfect. Perfect inspections diagnose withouterror the system state. The system can fail because of threecompeting failure types: I, II and III. Partial inspectionsdetect without error type I failures. Failures of type II canbe detected by imperfect inspections which have non-zero probabilityof false positives. Partial and imperfect inspections are madeat the same time. Type III failures are detectable only by perfectinspections. If the system is found failed in an inspection,a repair is made which renders the system in a good-as-new condition.The system is preventively maintained following an age-basedpolicy. Preventive maintenance actions return the system toa good-as-new condition. We consider cost contributions of inspections,repairs, preventive maintenance and periods of unavailability.The model presented permits to determine the optimal (constant)inter-inspection period for partial, imperfect and perfect inspectionsand the optimal times of preventive maintenance actions.     

An Optimal Vehicle-fleet Inspection Schedule

The purpose of this paper is to demonstrate how the relation between the mean arrival rate of break-downs conditional upon the inspection frequency of equipment can be estimated in practice. The work was carried out for a large, urban transit authority operating a fleet of approximately 2000 buses undertaking about 80 million kilometres per year. A model relating total downtime of buses incurred due to inspections and repairs per unit time to inspection frequency is developed, and the optimal inspection frequency which maximizes bus availability is determined.     

Optimal production run time for a deteriorating production system under an extended inspection policy

A deteriorating production system is subjected to random deterioration from an in-control state to an out-of-control state with a general shift distribution. In order to reduce the defective items, part inspection policy, under which production inspections are performed only at the end of the production run, and full inspection policy are both considered in the literature. Moreover, the former dominates the latter. Since the product produced towards the end of a production cycle are more likely to be defective, it can further economize the inspection costs that they are directly reworked without inspection. In this paper, we propose an extended product inspection policy for a deteriorating production system. Product inspections are performed in the middle of a production cycle, and after the inspection, all products produced until the end of the production run are fully reworked. Based on the model, we show that there exists a production run time and a corresponding unique inspection policy such that the expected total cost per item per cycle is minimized. Finally, numerical examples are provided to illustrate our extended inspection policy, and indicate that such product inspection model will reduce the quality-related cost than part inspection does.     

Utilizing the information theory of entropy to solve an off-line inspection problem

This study presents an off-line inspection problem for a batch produced from a process subject to random failures and exhibiting manufacturing variations. The objective of this paper is to develop an inspection policy in which units should be inspected in a particular order to find the transition unit in the batch under a required confidence level. This study develops an algorithm to compute the expected number of inspections. This approach uses the information theory of entropy to select an un-inspected unit to be inspected, and effectively minimizes the uncertainty of the transition unit in the production batch. A numerical example illustrates the proposed off-line inspection policy, and the effects of model parameters on the expected inspection number are investigated. The numerical example in this study indicates that full inspection is required when the required confidence level is one or the process has larger manufacturing variations.     

A Markov decision algorithm for optimal inspections and revisions in a maintenance system with partial information

This paper deals with a gradually deteriorating equipment whose actual degree of deterioration can be revealed by inspections only. An inspection can be succeeded by a revision depending on the system's degree of deterioration. In the absence of inspections and revisions, the working condition of the system evolves according to a Markov chain whose changes of state are not observable with the possible exception of a breakdown. Examples of this model include production machines subject to stochastic breakdowns, and maintenance of communication systems. The cost structure of the model consists of inspection, revision and operating costs. It is intuitively reasonable that in many applications a simple control-limit rule will be optimal. Such a rule prescribes a revision only when inspection reveals that the degree of deterioration has exceeded some critical level. A special-purpose Markov decision algorithm operating on the class of control-limit rules is developed for the computation of an average cost optimal schedule of inspections and revisions.     

Utilizing an approximative solution to obtain the real optimal solution for a production and inspection model

Previous studies have proposed an integrated production and inspection model for a deteriorating production system whose process is characterized by an exponential distribution. Since the simultaneous determination of the optimal scheduled inspection times and the optimal production run length is difficult, an approximative production and inspection solution is obtained under the condition that the optimal inspection policy is equally-spaced. That is, obtaining an approximative production run length and number of inspections. This study further investigates this approximative solution and demonstrates that how to utilize it to obtain the real optimal solution more efficiently.     

修理延迟的设备修如旧模型的可用度

本文通过将使用时间在可能发生故障的时刻划分成时间区间,对修如旧模型下的多部件系统,在修理延迟条件下,对各类故障在同一时间区间里相容的条件下导出了任意给定时刻设备的可用度函数,并对文[3]中的算法做了改进,得到了更精确更合理的结果     

An Optimal Complete Inspection Plan for Critical Multicharacteristic Components

In this paper, a new inspection plan for critical multicharacteristic components is presented. A mathematical model that depicts and represents the plan has been developed. An algorithm is proposed to determine the optimal number of repeat inspections and sequence characteristics for inspection that minimizes the expected total cost. The expected total cost consists of the cost of false acceptance (cost of Type II error), cost of false rejection (cost of Type I error), and the cost of inspection. Empirical comparisons with the Raouf et al. model on randomly generated problems have been conducted. The results have shown that the proposed plan performs better in terms of the expected total cost on 82% of the generated problems for the assumed specific parameters. The reduction in the expected total cost ranges from 0.1-10%.     

A delay-time model for repairable machinery: maximum likelihood estimation of optimum inspection intervals

A delay-time model is presented for a complex repairable systemin which defects arise according to a nonhomogeneous Poissonprocess. The focus of this paper is not on the resulting probabilisticmodel itself, but on the estimation of the model parametersand their errors from records of failure times and number ofdefects found at inspections of a machine which has been operatedfor some time under some (usually suboptimal) inspection regime.The likelihood function of the observed failure and inspectiondata is derived, and a general approach for estimating the optimuminspection interval is discussed. An example relating to themaintenance of medical equipment is used to illustrate the techniquesdescribed in the paper.     

具有随机效应的Wiener加速退化模型的可靠性研究

研究了有随机效应的Wiener退化模型基于加速退化数据的统计推断问题.利用广义枢轴量方法得到了模型参数和感兴趣可靠性指标的广义置信区间.说明了不含随机效应的Wiener退化模型的统计推断问题是有随机效应的Wiener退化模型的特殊情况.蒙特卡罗模拟结果显示文中提出的区间估计有较好的覆盖比例.最后利用LED加速退化数据说...     

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.     

Inspection scheduling for imperfect production processes under free repair warranty contract

The paper considers scheduling of inspections for imperfect production processes where the process shift time from an ‘in-control’ state to an ‘out-of-control’ state is assumed to follow an arbitrary probability distribution with an increasing failure (hazard) rate and the products are sold with a free repair warranty (FRW) contract. During each production run, the process is monitored through inspections to assess its state. If at any inspection the process is found in ‘out-of-control’ state, then restoration is performed. The model is formulated under two different inspection policies: (i) no action is taken during a production run unless the system is discovered in an ‘out-of-control’ state by inspection and (ii) preventive repair action is undertaken once the ‘in-control’ state of the process is detected by inspection. The expected sum of pre-sale and post-sale costs per unit item is taken as a criterion of optimality. We propose a computational algorithm to determine the optimal inspection policy numerically, as it is quite hard to derive analytically. To ease the computational difficulties, we further employ an approximate method which determines a suboptimal inspection policy. A comparison between the optimal and suboptimal inspection policies is made and the impact of FRW on the optimal inspection policy is investigated in a numerical example.     

Inspection Policies When Duration of Checkings is Non-Negligible

Inspection policy models deal with stochastically failing systems, in which failure is detected by inspection only. Previous authors assume that the time required for checking is zero, while the current study deals with preparedness models when duration of checking is non-negligible. Costs are introduced for checking, unknown lost time and repairing (or replacing) the failed system. Furthermore, uncertainty probabilities associated with inspections are introduced. Optimal checking policies are calculated for three different objective functions: expected loss per cycle, per time unit and per unit of good time. The models are solved by differentiation and by dynamic programming. Numerical results indicate that the optimal policy for IFR distributions is characterized by decreasing checking time intervals, with a marked difference only between the first and second intervals.     

Modified profile likelihood approach for certain intraclass correlation coefficients

In this paper we consider the problem of constructing confidence intervals and confidence lower bounds for the intraclass correlation coefficient in an interrater reliability study where the raters are randomly selected from a population of raters. The likelihood function of the interrater reliability is derived and simplified, and the profile likelihood based approach is readily available for computing the confidence intervals of the interrater reliability. Unfortunately, the confidence intervals computed by using the profile likelihood function are in general too narrow to have the desired coverage probabilities. From the practical point of view, a conservative approach, if at least as precise as any existing method, is preferred since it gives the correct results with a probability higher than claimed. Under this rationale, we propose the so-called modified profile likelihood approach in this paper. Simulation study shows that, the proposed method in general has better performance than currently used methods.     

Inspection and maintenance optimization methods

Three methods of the optimal planning of the inspection and maintenance of offshore structures are described. The models are based on respectively: the maximization of the effect of inspections, measured by the total importance value of the errors detected, subject to a given total economical budget; the minimization of the total costs of obtaining respectively: a given importance value of errors detected or given numbers of inspections of various types. Special selections of the importance values of structural elements give problems of the maximization of the reliability of the structural system, or the minimization of the economical consequences of failures, or the minimization of the sum of the costs of inspections and failure-consequences, subject to a given total failure probability of the system.Different failure types of elements and time schedules of inspections can be included in the model.An extension of the incremental method of Fox is applied, and an evaluation measure is given for the calculation of bounds of the optimal objective value, or given numbers of inspections are planned by application of continuous linear programming with integral solutions.     

Realization of multi-input/multi-output switched linear systems from Markov parameters

This paper presents a four-stage algorithm for the realization of multi-input/multi-output (MIMO) switched linear systems (SLSs) from Markov parameters. In the first stage, a linear time-varying (LTV) realization that is topologically equivalent to the true SLS is derived from the Markov parameters assuming that the discrete states have a common MacMillan degree and a mild condition on their dwell times holds. In the second stage, stationary point set of a Hankel matrix with fixed dimensions built from the Markov parameters is examined. Splitting of this set into disjoint intervals and complements reveals linear time-invariant dynamics prevailing on these intervals. Clustering over a feature space permits recovery of the discrete states up to similarity transformations which is complete if a unimodality assumption holds and the discrete states satisfy a residence requirement. In the third stage, the switching sequence is estimated by three schemes. The first scheme is non-iterative in time. The second scheme is based on matching the estimated and the true Markov parameters of the SLS system over segments. The third scheme works also on the same principle, but it is a discrete optimization/hypothesis testing algorithm. The three schemes operate on different dwell time and model structure requirements, but the dwell time requirements are weaker than that needed to recover the discrete states. In the fourth stage, the discrete state estimates are brought to a common basis by a novel basis transformation which is necessary for predicting outputs to prescribed inputs. Robustness of the four-stage algorithm to amplitude bounded noise is studied and it is shown that small perturbations may only produce small deviations in the estimates vanishing as noise amplitude diminishes. Time complexities of the stages are also studied. A numerical example illustrates the derived results.