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.     

Optimization for condition-based maintenance with semi-Markov decision process

The semi-Markov decision model is a powerful tool in analyzing sequential decision processes with random decision epochs. In this paper, we have built the semi-Markov decision process (SMDP) for the maintenance policy optimization of condition-based preventive maintenance problems, and have presented the approach for joint optimization of inspection rate and maintenance policy. Through numerical examples, the improvement of this method is compared with the scheme, which optimizes only over the inspection rate. We also find that under a special case when the deterioration rate at each failure stage is the same, the optimal policy obtained by SMDP algorithm is a dynamic threshold-type scheme with threshold value depending on the inspection rate.     

How heterogeneity influences condition-based maintenance for gamma degradation process

In many applications, units from the same population exhibit heterogeneity that they degrade with different rates due to random factors. This article studies how this heterogeneity in degradation influences condition-based maintenance (CBM) policy. Many CBM polices are developed based on gamma process because it is popularly used to characterise monotone degradation processes. In this study, we also model the unit’s degradation by gamma process. To account for the heterogeneity among units’ degradation, we incorporate a random effect parameter in the gamma process. Then the optimal policy for CBM is obtained through Markov decision process. We show that when heterogeneity exists, the transition probability of degradation state depends on both unit’s age and observed degradation level. And consequently, the optimal maintenance policy is a monotone control limit policy. We conduct extensive numerical experiments to validate and demonstrate our findings in depth.     

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

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.     

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.     

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.     

Maintenance scheduling of a manufacturing system subject to deterioration

This paper presents an integrated model for the joint determination of both optimal inspection strategy and optimal repair policy for a manufacturing system whose resulting output is subject to system state. An appropriate maintenance strategy is essential to optimize revenue from a manufacturing system which is in continuous operation and subject to deterioration. The optimum policy balances the amount of maintenance required to increase availability against the loss of revenue arising from the down time: insufficient maintenance leads to an increase in the number of defective items, low profit and low maintenance cost; excessive maintenance results in a reduction in the proportion of defective items, high profit and high maintenance cost. In this paper, an intensity control model adapted to partial information provides an optimal inspection intensity and repair degree of the system as an optimal control process to yield maximum revenue. The solution is obtained through formulating an equivalent deterministic Hamilton-Jacobi equation. A numerical example is provided to illustrate the behavior of the optimal control process. The optimal control process determines a solution to both optimum inspection frequency and optimal replacement policy which results in an optimal production run length of the system.     

Optimum condition-based maintenance policies for deteriorating systems with partial information

Systems are considered which deteriorate as time goes on and whose conditions can be observed. The optimum maintenance policy with respect to cost is determined, based on a continuous deterioration process, by formulating the decision process as a discrete Markov decision problem. Examples from the field of civil engineering are given. The relation between the type of deterioration and condition-based optimum inspection intervals, optimum repair level, minimum average maintenance costs and mean time to repair is shown, using some numerical results.     

Multinomial-exponential reliability function: a software reliability model

The multinomial-exponential reliability function (MERF) was developed during a detailed study of the software failure/correction processes. Later on MERF was approximated by a much simpler exponential reliability function (EARF), which keeps most of MERF mathematical properties, so the two functions together makes up a single reliability model. The reliability model MERF/EARF considers the software failure process as a non-homogeneous Poisson process (NHPP), and the repair (correction) process, a multinomial distribution. The model supposes that both processes are statistically independent.The paper discusses the model's theoretical basis, its mathematical properties and its application to software reliability. Nevertheless it is foreseen model applications to inspection and maintenance of physical systems. The paper includes a complete numerical example of the model application to a software reliability analysis.     

Online maintenance policy for a deteriorating system with random change of mode

Most of maintenance policies proposed in the literature for gradually deteriorating systems, consider a stationary deterioration process. This paper is an attempt to take into account stochastically deteriorating systems which are subject to a sudden change in their degradation process. A technical device subject to gradual degradation is considered. It is assumed that the level of degradation can be resumed by a single scalar variable. An online maintenance decision rule is proposed, which makes it possible to take into account in real time the online information available on the operating mode of the system as well as its actual deterioration level. We show the efficiency of considering online decision rules for maintenance with respect to traditional maintenance policies based on a static alarm threshold. Numerical simulations are given, to assess and optimize the performance of the maintained system from its asymptotic unavailability point of view. It is compared to the results obtained with classical control-limit maintenance policies.     

A condition-based maintenance policy with non-periodic inspections for a two-unit series system

This paper considers a condition-based maintenance policy for a two-unit deteriorating system. Each unit is subject to gradual deterioration and is monitored by sequential non-periodic inspections. It can be maintained by good as new preventive or corrective replacements. Every inspection or replacement entails a set-up cost and a component-specific unit cost but if actions on the two components are combined, the set-up cost is charged only once. A parametric maintenance decision framework is proposed to coordinate inspection/replacement of the two components and minimize the long-run maintenance cost of the system. A stochastic model is developed on the basis of the semi-regenerative properties of the maintained system state and the associated cost model is used to assess and optimize the performance of the maintenance model. Numerical experiments emphasize the interest of a control of the operation groupings.     

Optimal load rating-based inspection planning of corroded steel girders using Markov decision process

Steel girder bridges are vulnerable to corrosion. To maintain their safety above a predefined target level, the load rating can be computed from the inspection results and guide the following maintenance actions. Optimizing inspection and maintenance based on load ratings has substantial practical and economic relevance. Load rating-based strategies can be categorized based on whether the inspection interval and replacement criteria are fixed or flexible. Existing studies focus on fixed inspection intervals throughout the service life. In general, their results are not optimal for inspection planning. To reduce life-cycle cost, aged steel girders may be inspected and repaired in an adaptive manner. To this end, a method based on Markov decision process (MDP) is proposed to compare the life-cycle cost of four load rating-based policies (i.e. uniform or adaptive non-uniform inspection interval, and fixed or adaptive replacement threshold). Load rating-based inspection planning is formulated as MDP and the optimal plans are obtained using dynamic programming. The conventional approach to discretize states cannot accurately approximate the non-stationary deterioration process, while state augmentation is successful in doing this but will increase computational cost. A comparison of two approaches is made to investigate their effects on life-cycle cost. A bridge girder under corrosion attack is used as an illustrative example. The results show that the load rating-based plan with an adaptive non-uniform inspection interval and fixed replacement threshold obtained using the state augmentation technique can be near-optimal.     

Stochastic analysis of shock process and modeling of condition-based maintenance

The paper presents an analytical formulation for evaluating the maintenance cost of engineering systems that are damaged by shocks arriving randomly in time. The damage process is nonlinear in a sense that damage increments form an increasing sequence (i.e., accelerated damage) or a decreasing sequence (saturated damage) of random increments. Such processes are motivated from damage data collected from nuclear reactor components. To model the nonlinear nature of damage process, the paper proposes the use of non-homogeneous Poisson process for damage increments, which is in contrast with the common use of a renewal process for modeling the damage. The paper presents a conceptually clear and comprehensive derivation of formulas for computing the expected cost rate associated with a periodic inspection and preventive maintenance policy. Distinctions between the analysis of self-announced and latent failures are highlighted. The analytical model presented in this paper is quite generic and versatile, and it can be applied to optimize other types of maintenance policies.     

Modeling the failure data of a repairable equipment with bathtub type failure intensity

The paper deals with the reliability modeling of the failure process of large and complex repairable equipment whose failure intensity shows a bathtub type non-monotonic behavior. A non-homogeneous Poisson process arising from the superposition of two power law processes is proposed, and the characteristics and mathematical details of the proposed model are illustrated. A graphical approach is also presented, which allows to determine whether the proposed model can adequately describe a given failure data. A graphical method for obtaining crude but easy estimates of the model parameters is then illustrated, as well as more accurate estimates based on the maximum likelihood method are provided. Finally, two numerical applications are given to illustrate the proposed model and the estimation procedures.     

Closed-form analytical results for condition-based maintenance

Preventive maintenance is applied to improve the device availability or decrease the repair costs when the device failures are in deterioration (or aging) phase. Preventive maintenance can be made more efficient by periodic monitoring wherein the state of deterioration can be assessed. This leads to the notion of condition-based maintenance. In this paper, we study the condition-based maintenance, and derive closed-form expressions of system availability when the device undergoes both deterioration as well as Poisson type failures. These closed-form solutions enable us to find faster algorithms to determine the optimal inspection policy.     

Integrating quality and maintenance decisions in a production-inventory model for deteriorating items

In typical production-inventory models of deteriorating items, deterioration of the production process has not been considered. In this paper, a model is proposed in which both the produced items and the production equipment deteriorate. When the production system deteriorates, it shifts to an out-of-control state and begins to produce a proportion of defective items, necessitating corrective maintenance action. A model is formulated to integrate several realistic aspects, including item and process deterioration, varying demand and production rates, quality, inspection, and maintenance. A heuristic solution algorithm is developed to determine the production and inspection schedules, and a numerical example is solved.     

Modelling imperfect maintenance and the reliability of complex systems using superposed renewal processes

In problems of maintenance optimization, it is convenient to assume that repairs are equivalent to replacements and that systems or objects are, therefore, brought back into an as good as new state after each repair. Standard results in renewal theory may then be applied for determining optimal maintenance policies. In practice, there are many situations in which this assumption cannot be made. The quintessential problem with imperfect maintenance is how to model it. In many cases it is very difficult to assess by how much a partial repair will improve the condition of a system or object and it is equally difficult to assess how such a repair influences the rate of deterioration. In this paper, a superposition of renewal process is used to model the effect of imperfect maintenance. It constitutes a different modelling approach than the more common use of a virtual age process.     

Effects of maintenance policy on an imperfect production system under trade credit

The traditional production model development assumed that all products are perfect quality and did not consider maintenance, which is far from reality. In practice, the production process may shift randomly from an in-control state to an out-of-control state during a production run, i.e. process deterioration. This paper considers both preventive maintenance and corrective maintenance which are used to increase the system reliability. The objective of this paper is to determine the optimal production run time and maintenance frequency while minimising the total cost under process deterioration and trade credit. This paper develops a theorem and an algorithm to solve the problem described, provides numerical analysis to illustrate the proposed solution procedure, and discusses the impact of various system parameters. A real case of hi-tech manufacturer is used to verify the model. It predicts a 10.36% decrease in total cost if the preventive maintenance decision is considered.