J.M. van Noortwijk

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.

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.

Gamma processes and peaks-over-threshold distributions for time-dependent reliability

In the evaluation of structural reliability, a failure is defined as the event in which stress exceeds a resistance that is liable to deterioration. This paper presents a method to combine the two stochastic processes of deteriorating resistance and fluctuating load for computing the time-dependent reliability of a structural component. The deterioration process is modelled as a gamma process, which is a stochastic process with independent non-negative increments having a gamma distribution with identical scale parameter. The stochastic process of loads is generated by a Poisson process. The variability of the random loads is modelled by a peaks-over-threshold distribution (such as the generalised Pareto distribution). These stochastic processes of deterioration and load are combined to evaluate the time-dependent reliability.

The influence of temporal uncertainty of deterioration on life-cycle management of structures

In the life-cycle management of infrastructure systems, the decisions regarding the time and frequency of inspection, maintenance and replacement are confounded by sampling and temporal uncertainties associated with deterioration of the structural resistance. To account for these uncertainties, probabilistic models of deterioration have been developed under two broad categories, namely the random variable model and the stochastic process model. This paper presents a conceptual exposition of these two models and highlights their profound implications on age-based and condition-based preventive maintenance policies. The stochastic gamma process model of deterioration proposed here is more versatile than the random rate model commonly used in structural reliability literature.

Expert judgment in maintenance optimization

A comprehensive method for the use of expert opinion for obtaining lifetime distributions required for maintenance optimization is proposed. The method includes procedures for the elicitation of discretized lifetime distributions from several experts, the combination of the elicited expert opinion into a consensus distribution, and the updating of the consensus distribution with failure and maintenance data. The development of the method was prompted by the lack of statistical training of the experts and the high demands on their time. The use of a discretized life distribution provides more flexibility, is more comprehendible by the experts in the elicitation stage, and greatly reduces the computation in the combination and updating stages. The methodology is Bayes, using the Dirichlet distribution as the prior distribution for the elicited discrete lifetime distribution. Methods are described for incorporating information concerning the expertise of the experts into the analysis. >

Optimal inspection decisions for the block mats of the Eastern-Scheldt barrier

Abstract To prevent the southwest of The Netherlands from flooding, the Eastern-Scheldt storm-surge barrier was constructed, has to be inspected and, when necessary, repaired. Therefore, one is interested in obtaining optimal rates of inspection for which the expected maintenance cost is minimal and the barrier is safe. For optimisation purposes, a maintenance model was developed for part of the sea-bed protection of the Eastern-Scheldt barrier, namely the block mats. This model enables optimal inspection decisions to be determined on the basis of the uncertainties in the process of occurrence of scour holes and, given that a scour hole has occurred, of the process of current-induced scour erosion. The stochastic processes of scour-hole initiation and scour-hole development was regarded as a Poisson process and a gamma process, respectively. Engineering knowledge was used to estimate their parameters.

Discounted cost model for condition-based maintenance optimization

This paper presents methods to evaluate the reliability and optimize the maintenance of engineering systems that are damaged by shocks or transients arriving randomly in time and overall degradation is modeled as a cumulative stochastic point process. The paper presents a conceptually clear and comprehensive derivation of formulas for computing the discounted cost associated with a maintenance policy combining both condition-based and age-based criteria for preventive maintenance. The proposed discounted cost model provides a more realistic basis for optimizing the maintenance policies than those based on the asymptotic, non-discounted cost rate criterion.

Renewal theory with exponential and hyperbolic discounting

To determine optimal investment and maintenance decisions, the total costs should be minimized over the whole life of a system or structure. In minimizing life-cycle costs, it is important to account for the time value of money by discounting and to consider the uncertainties involved. This article presents new results in renewal theory with costs that can be discounted according to any discount function that is nonincreasing and monotonic over time (such as exponential, hyperbolic, generalized hyperbolic, and no discounting). The main results include expressions for the first and second moment of the discounted costs over a bounded and unbounded time horizon as well as asymptotic expansions for nondiscounted costs.

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).

Bayesian updating of a prediction model for sewer degradation

Sewer degradation is mainly a stochastic process. The future condition of sewers can be predicted using models based on condition states. In The Netherlands, the SPIRIT model is being developed combining expert opinion and visual inspections. In this model the likelihood function of condition states is updated with inspections. A Dirichlet distribution is used to describe ‘subjective’ prior knowledge, i.e. expert knowledge. The results show that the model can be solved analytically reducing calculation time. In addition, the weight of experts and inspections is determined on the basis of prior information and data instead of estimated by subjective expert knowledge.