A multi-component and multi-failure mode inspection model based on the delay time concept

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Abstract

The delay time concept and the techniques developed for modelling and optimising plant inspection practices have been reported in many papers and case studies. For a system comprised of many components and subject to many different failure modes, one of the most convenient ways to model the inspection and failure processes is to use a stochastic point process for defect arrivals and a common delay time distribution for the duration between defect the arrival and failure of all defects. This is an approximation, but has been proven to be valid when the number of components is large. However, for a system with just a few key components and subject to few major failure modes, the approximation may be poor. In this paper, a model is developed to address this situation, where each component and failure mode is modelled individually and then pooled together to form the system inspection model. Since inspections are usually scheduled for the whole system rather than individual components, we then formulate the inspection model when the time to the next inspection from the point of a component failure renewal is random. This imposes some complication to the model, and an asymptotic solution was found. Simulation algorithms have also been proposed as a comparison to the analytical results. A numerical example is presented to demonstrate the model.

Introduction

Inspections are important activities in any preventive maintenance program. When plant items are being inspected, potential defects may be identified and removed to prevent future failures from occurring. For any planned maintenance shutdown, it is estimated that up to 80% of activities result from defects identified or previously reported, [1]. Thus, the determination of inspection intervals is one of the key decisions of a maintenance manager. Traditionally, for time-based maintenance, the interval between planned maintenance interventions is determined by the managers’ experience or by the original equipment manufacturer's recommendation [2]. Evidence has been provided that such practice is sub-optimal and conservative [3]. On the other hand, many researchers have developed models to optimise inspection intervals under various modelling scenarios (see the reviews of [4], [5]), but few have actually been used. The delay time concept and associated models originally proposed by Christer [6] are exceptional, in that case studies have been conducted and reported, e.g. [7], [8], [9], [10], [11], [12], [13]. For a recent review of the delay time inspection models, see [14].

The delay time concept considers the failure process as a two stage process from new to an initial point that a defect can be identified by an inspection, and then from that point to failure if the defect is unattended. The time from the initial point of an identifiable defect to failure is called the delay time of the defect. If an inspection is carried out during the delay time of the defect, the defect should/could be removed, depending on the quality of the inspection and defect removal.

The delay time models can be divided into two categories: a complex system model and a component tracking model, where the former refers to a system with many components and failure modes and the latter refers to a single component subject to a single failure mode. A majority of the delay time models previously reported were complex system-based (see [3], [7], [8], [10], [11] and [15], [16], [17] for some examples), but few have dealt with single component models [18], [19], [20]. For complex systems, an approximation was made so that defect arrivals from all components are grouped and modelled by a stochastic point process, such as an HPP or an NHPP. It was also assumed that the delay times of all defects follow one identical distribution. If the numbers of components and failure modes are large, this approximation is good and can be justified mathematically [21]. If, however, a system has only a few major components with a few major failure modes, then the above approximation will be invalid.

We propose in this paper a multi-component-based delay time model, which is different from the previous complex system and single component models, in that (1) several components are considered individually but at the same time to form the subsystem or system model, (2) there can be more than one failure mode of a component, (3) inspections are scheduled for the subsystem or system rather than for each individual component, and (4) there could be possible opportunity inspections at the component failure times to check other components within the same subsystem. None of these situations have been considered before, and thus this model offers maintenance managers a useful tool for determining the optimal plant inspection intervals for cases similar to those considered here. In the following, we start with some necessary modelling assumptions and notation. Then, we present an analytical model of the intended system assuming all components and failure modes are independent. If there are opportunistic inspections at the time of a component's failure to check other components, then the model becomes very complicated, and therefore, we resort to a simulation-based algorithm. Finally, we illustrate the model in a comprehensive numerical example.

Section snippets

Assumptions

  • 1.

    Consider a multi-component system with several subsystems. Each subsystem may have several components.

  • 2.

    A component is the lowest level of the product tree and may have several failure modes.

  • 3.

    Each failure mode has its own initial and delay time probability density functions (pdfs).

  • 4.

    If a failure occurs according to a failure mode, then the complete component is replaced or repaired to an ‘as new’ condition. This corrective maintenance implies that the component is renewed. This is termed a failure

Probabilities of failures and defect identifications

In general, either a component fails between inspections before any defect is identified, or a defect is identified at an inspection before failure by any failure mode. By our assumptions, the component is completely renewed in both cases. For a component with K failure modes, and following the assumption that failure modes are independent, for the time to the first renewal L, we have, for tn−1⩽t<tnRL(t)=P(L>t)=P(Umin>tn1,Tmin>t)=k=1KP(Uk>tn1,Tk>t)=k=1KP(Uk>tn1,Hk>tUk)=k=1K[P(Uk>tn1)P(U

Exact formulation of failure distribution with start of the renewals at failures

In Eqs. (15), (16), we have used an assumption that all renewal cycles start at an inspection; i.e., as if the start time is zero and inspections are performed at tin=i, n=1, 2, 3 …. The assumption actually only applies to a case, where the previous renewal is an inspection renewal due to a defect removal. But for a failure renewal, the inspections start again at ti1=τix, where x (o⩽x<τi) is the time of the previous failure measured from the inspection just before the failure (let us call it

Simulation analysis

In this section, we use simulations to validate our previous modelling developments. Because of the flexibility and the powerful nature of simulation, we further assume that there are opportunistic checks at the time of a component failure on the other components within the subsystem. Then the random time of failure becomes an opportunistic inspection and an analytical solution is even more difficult, though possible.

Below, we describe a simulation procedure for a subsystem with opportunistic

Numerical example

The example below represents subsystem 2 shown in Fig. 1. Initially, we assume each component has only one failure mode for the purpose of a comparison between the approximated analytical and exact simulation results. We then add one more failure mode to component 22 to observe the effect of competing failure modes. To enable a quick comparison, we assume that both the initial and the delay time distributions for all failure modes of components 21 and 22 are exponential i.e.,g211(u)=α211eα211u,

Conclusions

In this paper, we develop a delay-time-based model to determine the optimal inspection interval for a system with a number of components and failure modes. The model fills the gap in theory that is not covered by previous delay-time-based studies. This model, if programmed into a software package, can be readily used to advise plant maintenance managers of the best inspection service interval. Theoretically speaking, there is no limitation on the number of components and failure modes given

Acknowledgements

The research reported here is partially supported by EPSRC under Grant nos. EP/C54658X/1 and EP/G023042/1 and by the Ontario Centre of Excellence and Natural Sciences and Engineering Research Council of Canada. The authors also want to thank Dr. Matthew Carr for his valuable comments on an earlier version of this paper.

References (23)

  • A.H. Christer et al.

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