Reliability evaluation and optimisation of imperfect inspections for a component with multi-defects

doi:10.1016/j.ress.2005.11.003

Reliability Engineering & System Safety

Volume 92, Issue 1, January 2007, Pages 65-73

https://doi.org/10.1016/j.ress.2005.11.003 Get rights and content

Abstract

In this paper, a model is developed to evaluate the reliability and optimise the inspection schedule for a multi-defect component. The model uses a non-homogeneous Poisson process (NHPP) method in conjunction with a delay time approach. The inspections are designed to detect any defects in the component, however it can be imperfect. The defect is a definable state before a functional failure happens to the component. Occurrences of defects are assumed to follow an NHPP and a defect will be minimally repaired if it is identified during an inspection. It is shown that the failures occurring in an interval of inspection will also follow an NHPP. The situation of imperfect inspections and non-constant inspection intervals are considered. An algorithm is presented to optimise the intervals of inspections in order to maximise the reliability of the component, and the properties of the algorithm are shown. A numerical example with parametric study is given to show the performance of the model and the algorithm.

Introduction

In many cases, before a functional failure occurs in a component, there exists a definable state, referred to as a defect or potential failure. If these defects are able to be detected during planned inspections and repair work is carried out, then its reliability can be significantly improved.

The time interval between the occurrence of a defect and its decay into a functional failure is usually called the P–F interval. The analysis of the P–F interval is a basic step in scheduling preventive maintenance (PM) for a system when employing reliability-centred maintenance (RCM) [1]. Usually, models of inspection are developed using a key parameter to define the states of the defect and failure, such as wear level, length of crack and erosion [2], [3], [4], [5]. In general, these models are application specific. Christer and Waller [6], however, considered the allowable time duration that could be spent in the state of defect and hence developed a so-called delay time model. A number of theoretical studies and applications have been conducted based on the concept of delay time [6], [7], [8]. Recently, the delay time model has been extended to optimise the scheduling of perfect inspections for multi-component systems [8]. In this model, however, a recursive formula was developed to calculate the expected number of failures over the inspection intervals and this led to difficulty in obtaining optimum schedules.

In practice, maintenance is imperfect. Considerable studies have been conducted on imperfect maintenance [9]. Badia et al. [10] have studied the optimal strategy for the maintenance of a single unit system with revealed and un-revealed failures assuming imperfect inspections. Within this work, revealed failures are defined as those failures which can be detected as soon as they occur, whereas, un-revealed failures are those that can only be detected during an inspection. Christer and Lee [11] later considered the downtime incurred during failures when imperfect maintenance is carried out and estimated the expected number of failures over a defined interval. Previously, Christer et al. [12] presented an approach to estimate the parameters for PM modelling. In this work the expected number of failures was predicted assuming a constant rate of defect occurrence.

Most of the previous research to obtain the optimal inspection interval has been based on economic assessments. Exceptionally, Christer [13] developed a model to evaluate the reliability through a recursive relationship. Based on Christer's model, Cerone [14] considered the problem of optimising inspection intervals through maximising reliability. However, in these papers perfect and periodic inspections were assumed.

In this paper, a model is developed to evaluate and optimise the reliability of a multi-defect component under inspection. The realistic scenario of imperfect inspections carried out at non-constant intervals is considered. It is assumed that the occurrence of defects follows a non-homogeneous Poisson process (NHPP). An algorithm is presented to optimise the intervals by maximising the reliability. A numerical example with parametric study is given to evaluate the performance of the model and the algorithm.

Section snippets

Problem statement

In this paper, it is considered that multi-defects may occur to a component within a period of time, and the arrival of defects is assumed to follow an NHPP. The time interval between the occurrence of a defect and its decay into a functional failure is referred to as delay time. It is also assumed that all defects are independent and have a common delay time distribution, and the delay time of a defect is independent of its time origin.

Inspections are assumed to be conducted at scheduled

Failures initiated during the inspection interval in which they occur

Suppose a component begins to work from t₀=0, and inspections are conducted at times, $t_{1}, t_{2}, \dots, t_{m}$ , measured from t₀. Consider the interval $[t_{j - 1}, t_{j}]$ . If N(t) denotes the number of defects within the interval $(t_{j - 1}, t]$ where $t ⩽ t_{j}$ , and X(t) denotes the failures in $(t_{j - 1}, t]$ which are caused by defects in $(t_{j - 1}, t]$ , then X(t) can be represented by $X (t) = \sum_{n = 1}^{N (t)} W (t, T_{n}, Y_{n}) .$ Here, $T_{n}$ is the arrival time of the nth defect in $(t_{j - 1}, t]$ and $Y_{n}$ is the delay time. These parameters are independent and identical

Reliability model

It has been shown in the previous section that the arrival of failures under inspection can still be considered as an NHPP. For an NHPP, the probability of no failure occurring during the interval $[0, t]$ is given by $R = \exp {- expectation of failure number in [0, t]} .$ It follows that under the condition of the component being functional at the time $t_{j - 1}$ , the reliability at instance $t_{j}$ can be represented by $R_{β} (t_{j} | component is reliable at t_{j - 1}) = \exp [- Λ_{β} (t_{j - 1}, t_{j})] (j ⩾ 1),$ where $t_{0} = 0$ .

It is noted that in the derivation

Reliability optimisation

In this paper, the problem of optimisation is, for a given period $[0, t]$ , to calculate the minimum number of cycles of inspections, $m^{*}$ , and the optimal instances of inspections, $T_{m}^{*}$ , in order to maximise the component reliability with the constraint of: $R_{β} (t, T_{m}^{*}) ⩾ p_{0},$ where $p_{0} (p_{0} < 1)$ is the required reliability.

To solve the problem, Eq. (24) is rewritten by taking the natural logarithm of both sides and using Eq. (20): $- \ln [R_{β} (t_{m + 1}, T_{m})] = E_{β} (t_{m + 1}, T_{m}) .$ Hence, the problem of maximising the component

Numerical example

An example is given to illustrate the performance of the model and algorithm. Firstly, let the delay time be exponentially distributed, i.e., $G (τ) = 1 - \exp (- 0.25 τ)$ as in previous examples [8], [13]. As Weibull's distribution can be used to represent many different types of failures, it is assumed that the occurrence of defects in a component is governed by Weibull's law with $α = 1.6$ and $L = 16$ . The detection rate, $β = 0.9$ , and the reliability at $t = 20$ time units is required to be at least $p_{0} = 0.7$ .

Using the

Concluding remarks

In this paper, a model is developed to evaluate and optimise the reliability of a component with imperfect and non-periodic inspections. An iterative algorithm is presented to optimise the intervals by maximising the reliability of the component. A numerical example and parametric study is given to show the performance of the model and the algorithm. It is relatively easy to program and quick to obtain the optimal policy of inspection and the maximum reliability of a component using the model

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