Optimal maintenance decisions under imperfect inspection

The process industry is increasingly making use of Risk Based Inspection (RBI) techniques to develop cost and/or safety optimal inspection plans. This paper proposes an adaptive Bayesian decision model to determine these optimal inspection plans under uncertain deterioration. It uses the gamma stochastic process to model the corrosion damage mechanism and Bayes’ theorem to update prior knowledge over the corrosion rate with imperfect wall thickness measurements. This is very important in the process industry as current non-destructive inspection techniques are not capable of measuring the exact material thickness, nor can these inspections cover the total surface area of the component. The decision model finds a periodic inspection and replacement policy, which minimizes the expected average costs per year. The failure condition is assumed to be random and depends on uncertain operation conditions and material properties. The combined deterioration and decision model is illustrated by an example using actual plant data of a pressurized steel vessel.     

Two probabilistic life-cycle maintenance models for deteriorating civil infrastructures

The purpose of this paper is to describe and compare two maintenance models for deteriorating civil infrastructures that can be used to insure an adequate level of reliability at minimal life-cycle cost. These models, referred to Rijkswaterstaat's model and Frangopol's model have been independently developed by the authors of this paper and their associates. The former model has been applied by the Netherlands Ministry of Transport, Public Works and Water Management (Rijkswaterstaat). It can be used for justification and optimisation of maintenance measures. The latter model contributed to the further development of the bridge management methodology that has been set up by the UK Highways Agency.     

A joint spare part and maintenance inspection optimisation model using the Delay-Time concept

Spare parts and maintenance are closely related logistics activities where maintenance generates the need for spare parts. When preventive maintenance is present, it may need more spare parts at one time because of the planned preventive maintenance activities. This paper considers the joint optimisation of three decision variables, e.g., the ordering quantity, ordering interval and inspection interval. The model is constructed using the well-known Delay-Time concept where the failure process is divided into a two-stage process. The objective function is the long run expected cost per unit time in terms of the three decision variables to be optimised. Here we use a block-based inspection policy where all components are inspected at the same time regardless of the ages of the components. This creates a situation that the time to failure since the immediate previous inspection is random and has to be modelled by a distribution. This time is called the forward time and a limiting but closed form of such distribution is obtained. We develop an algorithm for the optimal solution of the decision process using a combination of analytical and enumeration approaches. The model is demonstrated by a numerical example.     

Explicit formulas for the variance of discounted life-cycle cost

In life-cycle costing analyses, optimal design is usually achieved by minimising the expected value of the discounted costs. As well as the expected value, the corresponding variance may be useful for estimating, for example, the uncertainty bounds of the calculated discounted costs. However, general explicit formulas for calculating the variance of the discounted costs over an unbounded time horizon are not yet available. In this paper, explicit formulas for this variance are presented. They can be easily implemented in software to optimise structural design and maintenance management. The use of the mathematical results is illustrated with some examples.     

A survey of the application of gamma processes in maintenance

This article surveys the application of gamma processes in maintenance. Since the introduction of the gamma process in the area of reliability in 1975, it has been increasingly used to model stochastic deterioration for optimising maintenance. Because gamma processes are well suited for modelling the temporal variability of deterioration, they have proven to be useful in determining optimal inspection and maintenance decisions. An overview is given of the rich theoretical aspects as well as the successful maintenance applications of gamma processes. The statistical properties of the gamma process as a probabilistic stress-strength model are given and put in a historic perspective. Furthermore, methods for estimation, approximation, and simulation of gamma processes are reviewed. Finally, an extensive catalogue of inspection and maintenance models under gamma-process deterioration is presented with the emphasis on engineering applications.     

A methodology for the prioritization of operating experience in nuclear power plants

The processing of operating experience at nuclear power plants is critically important to safe and reliable operations because it represents the process by which important external information is incorporated into the organization. Nuclear power plants typically receive between 800 and 1000 pieces of operating experience every year, of which 70–80 meet the criteria for a more extensive review. This paper deals with the prioritization of these items. Specifically, a prioritization methodology utilizing multiattribute utility theory has been developed. What sets this methodology apart from other techniques employing multiattribute methods is its emphasis on deliberations to achieve consensus among objectives and preferences among those objectives. Along with an explanation of this methodology, the results of its application to the prioritization of operating experience at a nuclear power plant are presented. Lastly, the results of a workshop that was held at MIT are presented. The workshop demonstrated the feasibility of the prioritization methodology and the validity of the case study results.     

A Gamma–normal series truncation approximation for computing the Weibull renewal function

This paper presents a series truncation approximation for computing the Weibull renewal function. In the proposed model, the n-fold convolution of the Weibull Cdf is approximated by a mixture of the n-fold convolutions of Gamma and normal Cdfs. The mixture weight can be optimally determined and fitted into a very accurate linear function of Weibull shape parameter β. Major advantages of the proposed model include:

(a) The proposed model and its parameters can be directly written out. Using the proposed model, the renewal density and variance functions can be easily evaluated.

(b) The proposed model includes Gamma and normal series truncation models as its special cases. It is easy to be implemented in Excel. The series converges fairly fast.

(c) Over the range of β(0.87,8.0), the maximum absolute error is smaller than 0.01; and over , the maximum absolute error is smaller than 0.0037.

(d) The model can be easily extended to non-Weibull case with some additional work.

Cost–benefit optimization for maintained structures by a renewal model

The paper develops objective functions for optimizing design and maintenance strategies of aging structural components. It introduces a repair model with or without preceding inspections. Objective functions are derived for maintenance by inspection and repairs (renewals). Earlier formulations for independent repair and failure events are extended to dependent no-repair/repair and failure events. This can make formulations more realistic because no-repair/repair and failure events depend on the same deterioration processes, at the expense of numerically more elaborate computations. In addition, finite renewal (repair) times independent of failure times are considered.     

Optimal design of sensor networks for vehicle detection, classification, and monitoring

The tracking and identification of vehicles for the purpose of surveillance is a widespread application. Observations from a network of sensors can be used to make decisions regarding the identity of vehicles, as well as their trajectories. Generally, the information provided by a sensor network is limited, so vehicles may be misclassified, go undetected, and/or their trajectories may not be determined uniquely. Often, assumptions are made regarding, for example, traffic composition and possible vehicle trajectories. Because the performance of a sensor network can be sensitive to these assumptions, the conclusions made by the network about the identity and trajectory of vehicles can be highly inaccurate. In this paper, these assumptions are treated as possible models of reality that are subsequently evaluated in a decision framework. Mathematical models for vehicle movement and sensor behavior are developed. Candidate designs for the sensor network are considered, where each design is defined by the number, location, and range of the sensors. Methods from decision theory are used to determine the optimal design for the sensor network.     

A comparison of models for measurable deterioration: An application to coatings on steel structures

Steel structures like bridges, tanks and pylons are exposed to outdoor weathering conditions. In order to prevent them from corrosion they are protected by organic coating systems. This paper focuses on modelling the deterioration of the organic coating layer that protects steel structures from corrosion. Only if there is sufficient knowledge of the condition of the coating on these structures, maintenance actions can be done in the most efficient way. Therefore the course of the deterioration of the coating system and its lifetime, which is also of importance for doing maintenance, have to be assessed accurately. In this paper, three different stochastic processes, viz. Brownian motion with non-linear drift, the non-stationary gamma process and a two-stage hit-and-grow physical process, are fitted to two real data sets. In this way we are the first who compare the three stochastic processes empirically on criteria such as goodness-of-fit, computational convenience and ease of implementation. The first data set is based on expert judgement; the second consists of inspection results. In the first case the model parameters are obtained by a least-squares approach, in the second case by the method of maximum likelihood. A meta-analysis is performed on the two-stage hit-and-grow model by means of fitting Brownian motion and gamma process to the outcomes of this model.     

Gamma function tables for the estimation of the mean and standard deviation of the weibull distribution

This paper provides tabulated numerical values for the gamma function necessary for the analytical estimation of the mean and standard deviation of the Weibull distribution. The tables produced also include estimates of the Weibull coefficient of variation, a quantity very important for maintenance optimization calculations when Glasser's charts are employed.     

新型碱性阻挡层抛光液在300mm铜布线平坦化中的应用

研发了一种新型碱性阻挡层抛光液,其不含通用的腐蚀抑制剂和不稳定的氧化剂（通常为H2O2）。Cu/Ta/TEOS去除速率和选择性实验结果表明,此阻挡层抛光液对Cu具有较低的去除速率,能够保护凹陷处的Cu不被去除,而对阻挡层（Ta）和保护层（TEOS）具有较高的去除速率,利于碟形坑和蚀坑的修正。在300mm铜布线CMP中应用表明,此碱性阻挡层抛光液为高选择性抛光液,能够实现Ta/TEOS/Cu的选择性抛光,对碟形坑（dishing）和蚀坑（erosion）具有较强的修正作用,有效地消除了表面的不平整性,与去除速率及选择性实验结果一致,能够用于多层铜布线阻挡层的抛光。     

Integro-differential Chapman-Kolmogorov equation for continuous-jump Markov processes and its use in problems of multi-component renewal impulse process excitations

In the present paper the applications of the integro-differential Chapman-Kolmogorov equation to the problems of pure-jump stochastic processes and continuous-jump response processes are discussed. The pure-jump processes considered herein are the counting Poisson process, a two-state jump process, and a multi-state jump process. The differential equations governing the Markov state probabilities are obtained from the degenerate, pure differential form, of the general, integro-differential Chapman-Kolmogorov equation, with the aid of the jump probability intensity functions. The continuous-jump response process is the response of a dynamic system to a multi-component renewal impulse process excitation. The excitation consists of a number of n statistically independent random trains of impulses, each of which is driven by an Erlang renewal process with parameters ν_j,k_j. Each of the impulse processes is characterized by an auxiliary zero-one jump stochastic process, which consists of k_j negative exponential distributed phases. The Markov states for the whole problem are determined by the coincidences of the phases of the individual jump processes. Thus the response probability distribution may be characterized by a joint probability density-discrete distribution of the state variables of the dynamic system and of the states of the pertinent Markov chain. The explicit integro-differential equations governing the joint probability density-discrete distribution of the response are obtained from the general forward integro-differential Chapman-Kolmogorov equation, after the determination of the jump probability intensity functions for the continuous-jump and pure-jump processes.     

Model selection for a class of stochastic processes or random fields with bounded range

Methods are developed for finding an optimal model for a non-Gaussian stationary stochastic process or homogeneous random field under limited information. The available information consists of: (i) one or more finite length samples of the process or field; and (ii) knowledge that the process or field takes values in a bounded interval of the real line whose ends may or may not be known. The methods are developed and applied to the special case of non-Gaussian processes or fields belonging to the class of beta translation processes. Beta translation processes provide a flexible model for representing physical phenomena taking values in a bounded range, and are therefore useful for many applications. Numerical examples are presented to illustrate the utility of beta translation processes and the proposed methods for model selection.     

Decolorization of reactive textile dyes using water falling film dielectric barrier discharge

Decolorization of reactive textile dyes Reactive Black 5, Reactive Blue 52, Reactive Yellow 125 and Reactive Green 15 was studied using advanced oxidation processes (AOPs) in a non-thermal plasma reactor, based on coaxial water falling film dielectric barrier discharge (DBD). Used initial dye concentrations in the solution were 40.0 and 80.0 mg/L. The effects of different initial pH of dye solutions, and addition of homogeneous catalysts (H₂O₂, Fe²⁺ and Cu²⁺) on the decolorization during subsequent recirculation of dye solution through the DBD reactor, i.e. applied energy density (45-315 kJ/L) were studied. Influence of residence time was investigated over a period of 24 h. Change of pH values and effect of pH adjustments of dye solution after each recirculation on the decolorization was also tested. It was found that the initial pH of dye solutions and pH adjustments of dye solution after each recirculation did not influence the decolorization. The most effective decolorization of 97% was obtained with addition of 10 mM H₂O₂ in a system of 80.0 mg/L Reactive Black 5 with applied energy density of 45 kJ/L, after residence time of 24 h from plasma treatment. Toxicity was evaluated using the brine shrimp Artemia salina as a test organism.     

Applications to continuous-time processes of computational techniques for discrete-time renewal processes

For optimising maintenance, the total costs should be computed over a bounded or unbounded time horizon. In order to determine the expected costs of maintenance, renewal theory can be applied when we can identify renewals that bring a component back into the as-good-as-new condition. This publication presents useful computational techniques to determine the probabilistic characteristics of a renewal process. Because continuous-time renewal processes can be approximated with discrete-time renewal processes, it focusses on the latter processes. It includes methods to compute the probability distribution, expected value and variance of the number of renewals over a bounded time horizon, the asymptotic expansion for the expected value of the number of renewals over an unbounded time horizon, the approximation of a continuous renewal-time distribution with a discrete renewal-time distribution, and the extension of the discrete-time renewal model with the possibility of zero renewal times (in order to cope with an upper-bound approximation of a continuous-time renewal process).     

Optimal design of the inspection buffer for an unreliable process

Placing an inspection buffer immediately after an unreliable process subject to failure protects the process customers from receiving non-conforming products. Such protection is achieved by screening the buffer contents whenever a shift in the process is detected. Previous studies have suggested that, if properly designed, such buffers can reduce overall costs. In this paper we make two types of contributions to the study of inspection buffers. First, we refine the model proposed by Klastorin et al. (1993), by incorporating a more efficient way of sampling, a more realistic objective function and more accurate cost expressions. Second, we provide analytical results to assist in the calculation of the economically optimal buffer size. Specifically, these include conditions for determining whether an inspection buffer is at all justified and a theorem for bounding the search for the optimal buffer size. The performance of the search bound and the sensitivity of the model are examined computationally.     

Sampling inspection for the evaluation of time-dependent reliability of deteriorating systems under imperfect defect detection

The paper presents a sampling-inspection strategy for the evaluation of time-dependent reliability of deteriorating systems, where the deterioration is assumed to initiate at random times and at random locations. After initiation, defects are weakening the system's resistance. The system becomes unacceptable when at least one defect reaches a critical depth. The defects are assumed to initiate at random times modeled as event times of a non-homogeneous Poisson process (NHPP) and to develop according to a non-decreasing time-dependent gamma process. The intensity rate of the NHPP is assumed to be a combination of a known time-dependent shape function and an unknown proportionality constant. When sampling inspection (i.e. inspection of a selected subregion of the system) results in a number of defect initiations, Bayes’ theorem can be used to update prior beliefs about the proportionality constant of the NHPP intensity rate to the posterior distribution. On the basis of a time- and space-dependent Poisson process for the defect initiation, an adaptive Bayesian model for sampling inspection is developed to determine the predictive probability distribution of the time to failure. A potential application is, for instance, the inspection of a large vessel or pipeline suffering pitting/localized corrosion in the oil industry. The possibility of imperfect defect detection is also incorporated in the model.     

A multi-criteria decision model to determine inspection intervals of condition monitoring based on delay time analysis

In periodic monitoring, the main problem is determining the inspection interval of condition monitoring. For this problem, the decision variable is represented by the time of next inspection of condition monitoring. There are several studies that deal with prescribing inspection intervals. But only a few of these allow the decision maker to observe simultaneously more than one aspect. This does not accord with the natural tendency of the decision maker who desires to see the decision problem from a broader perspective, by having different viewpoints or dimensions of choices. Therefore, the main objective of this paper is to propose a decision model, which can simultaneously determine inspection intervals for condition monitoring regarding the failure behavior of equipment to be inspected, features of maintainability and decision maker preferences about cost and downtime.     

Measurements of the thermal gradient over EB-PVD thermal barrier coatings

Conventional two-layered structure thermal barrier coatings (TBCs), graded thermal barrier coatings (GTBCs) and graded thermal barrier coatings with micropores were prepared onto superalloy DZ22 tube by electron beam physical vapor deposition (EB-PVD). Thermal gradient of the TBCs was evaluated by embedding two thermal couples in the surfaces of the tube and the top coat at different surrounding temperatures with and without cooling gas flowing through the tube. The results showed that higher thermal gradient could be achieved for the GTBCs with micropores compared to the two-layered structure TBCs and GTBCs. However, after the samples were heated at 1050°C, the thermal gradient for the GTBCs with or without micropores decreased with the increase of heating time. On the other hand, the thermal gradient for the TBCs increased with the increase of heating time. Cross-section observations by scanning electron microscopy showed that the change in microstructure was the main reason for the change of the thermal gradient.