Bayesian Prediction of the Overhaul Effect on a Repairable System with Bounded Failure Intensity

Abstract

This paper deals with the Bayes prediction of the future failures of a deteriorating repairable mechanical system subject to minimal repairs and periodic overhauls. To model the effect of overhauls on the reliability of the system a proportional age reduction model is assumed and the 2-parameter Engelhardt-Bain process (2-EBP) is used to model the failure process between two successive overhauls. 2-EBP has an advantage over Power Law Process (PLP) models. It is found that the failure intensity of deteriorating repairable systems attains a finite bound when repeated minimal repair actions are combined with some overhauls. If such a data is analyzed through models with unbounded increasing failure intensity, such as the PLP, then pessimistic estimates of the system reliability will arise and incorrect preventive maintenance policy may be defined. On the basis of the observed data and of a number of suitable prior densities reflecting varied degrees of belief on the failure/repair process and effectiveness of overhauls, the prediction of the future failure times and the number of failures in a future time interval is found. Finally, a numerical application is used to illustrate the advantages from overhauls and sensitivity analysis of the improvement parameter carried out.

1. Introduction

A repairable system is a system that, after failing to perform one or more of its functions satisfactorily, can be restored to satisfactory performance.

Most repairable mechanical systems are subjected to degradation phenomena with operating time, so that the failures become increasingly frequent with time. Such systems often undergo a maintenance policy. Maintenance extends system's lifetime or at least the mean time to failure, and an effective maintenance policy can reduce the frequency of failures and the undesirable consequences of such failures. Maintenance can be categorized into two classes: corrective and preventive actions. Corrective maintenance, called repair, is all actions performed to restore the system to functioning condition when it fails. Preventive maintenance is all actions performed to prevent failures when the system is operating. Corrective and preventive maintenance actions are generally classified in terms of their effect on the operating conditions of the system. Pham and Wang [1] classified them as perfect maintenance, minimal maintenance, imperfect maintenance, and worse maintenance. At one extreme is the assumption of perfect maintenance, that a system is restored to good-as-new condition after maintenance. At the other extreme is the bad-as-old assumption that the failure rate of a system is not enhanced by maintenance. In real-world situations, maintenance generally enhances the condition of the equipment at a level between these two extremes, that is, imperfect maintenance.

We consider a system that deteriorates with age and receives two kinds of maintenance actions: minimal repair and overhaul. When a failure occurs, minimal repair is carried out. The minimal repair is a corrective maintenance action that brings the repaired equipment to the conditions it was just before the failure occurrence (bad-as-old). Hence, the reliability of the system decreases with operating time until it reaches unacceptable values. When it reaches unacceptable values or at prefixed epochs, preventive maintenance action (overhaul) is performed so as to improve the system condition and hence reduce the probability of failure occurrence in the following interval. However, overhaul cannot return the system to “good-as-new”, and thus it can be treated as imperfect repair. When the overhaul is effective, the reliability of the system improves significantly. An overhaul usually consists of a set of preventive maintenance actions such as oil change, cleaning, greasing, and replacing some worn components of the system.

Many imperfect repair models have already been proposed [1]. Malik [2] proposed a general approach to model the improvement effect of maintenance, where each maintenance reduces the age of the unit in the view of the rate of occurrences of failures. Each maintenance is assumed to reduce proportionally the operating time elapsed from the previous maintenance. Malik's proportional age reduction (PAR) model for imperfect maintenance is a generalization of good-as-new and bad-as-old. On the basis of this general model, Shin et al. [3] have proposed a PAR model which assumes that each major overhaul reduces proportionally the age of the equipment by a fraction of the epoch of the overhaul. Shin's paper proposes a method of data analysis to estimate the parameters of the failure process and the maintenance effect for a repairable unit. Two parametric families of intensity functions are considered, power law [4–11] and log linear [12]. To model different effects of rejuvenation of the preventive maintenance, two classes of models have been proposed by Jack [13].

In contrast to the classical approach used by them, Bayes approach has been used by several authors as it helps in incorporating prior information and/or technical knowledge on the failure mechanism and on the overhaul effectiveness into the inferential procedure. Pulcini [14] deals with the statistical analysis, from a Bayes viewpoint, of the failure data of repairable mechanical units subjected to minimal repairs and periodic overhauls. The effect of overhauls on the reliability of the system is modeled by PAR model and the power law process (PLP) is used to model the failure process between two successive overhauls. Pulcini provided Bayes point and interval estimation of model parameters (and functions thereof), as well as testing procedures on the effectiveness of the performed overhauls. Further, Pulcini [15] deals with the prediction, from a Bayes viewpoint, of future failures of repairable mechanical units subjected to minimal repairs and periodic overhauls. In [15], also PLP is used to model the failure process between two successive overhaul epochs, and the effect of overhauls on the reliability of the system is modeled by proportional age reduction model.

In PLP models, the increasing failure intensity tends to infinity as the system age increases. However, it is noted that the failure intensity of deteriorating repairable systems attains a finite bound when beginning from a given system age, repeated minimal repair actions are combined with some overhauls performed in order to oppose the growth of failure intensity with the operating time. The average behavior of the intensity function due to the consecutive steps with increasing intensity between two subsequent overhauls results in globally constant asymptotic intensity. If such data is analyzed through models with unbounded increasing failure intensity, such as the PLP, then pessimistic estimates of the system reliability will arise, and incorrect preventive maintenance policy may be defined.

Engelhardt and Bain [16] were the first to suggest NHPP with an increasing bounded intensity, called Engelhardt-Bain process (EBP). But the mathematical simplicity of their 1-parameter model makes it inadequate to analyze failure data in many cases. Most recently, Pulcini [17] proposed a 2-parameter NHPP called the bounded intensity process (BIP) to fit the failure data of deteriorating repairable systems showing bounded intensity function. Attardi and Pulcini [18] proposed a new bounded intensity process called 2-parameter Engelhardt-Bain process (2-EBP) whose failure intensity function is given by 𝜆(𝑡)𝑡=𝜂𝑡+𝜃,𝜂,𝜃>0,𝑡≥0 (1) The failure intensity in (1) tends to its asymptote more slowly than the BIP intensity and is in a way a compromise between the PLP and BIP models. When 𝜂≡1/𝜃, the intensity in (1) degenerates into the EBP intensity.

This paper deals with the Bayes prediction of the future failures of a deteriorating repairable mechanical system subjected to minimal repairs and periodic overhauls. To model the effect of overhauls on the reliability of the system, a proportional age reduction model is assumed, and the 2-parameter Engelhardt-Bain process (2-EBP) is used to model the failure process between two successive overhauls. On the basis of the observed data and of a number of suitable prior densities reflecting varied degrees of belief on the failure/repair process and effectiveness of overhauls, the prediction of the future failure times and the number of failures in a future time interval is found. Finally, a numerical application is used to illustrate the advantages of overhauls.

2. Basic Assumptions

(i)Failure rate of the system is an increasing function of time that attains a finite bound as 𝑡 tends to infinity. (ii)System is subjected to two kinds of maintenance actions: minimal repair and overhaul.(iii)The times to perform maintenance actions are ignored. (iv)Minimal repair will restore the failure rate only to bad-as-old condition. But overhauls will improve the system to a condition between bad-as-old and good-as-new. (v)The failure density function is not changed by overhauls.(vi)The quality of an overhaul is dependent on improvement factor 𝜌(0≤𝜌≤1).(vii)The improvement parameter 𝜌 has a uniform value over all the overhaul actions. (viii)The 𝑗th overhaul reduces the age 𝑡 of the system by a fraction 𝜌 of the epoch 𝑥_𝑗 of the overhaul.(ix)The effect of overhauls on the reliability of the system is modeled by proportional age reduction model, and the 2-parameter Engelhardt-Bain process (2-EBP) is used to model the failure process between two successive overhauls, say (𝑥_𝑗,𝑥_𝑗+1).(x)𝑥₁,𝑥₂,…,𝑥_𝑘 are the 𝑘 overhaul epochs, which may coincide with failure times.

3. Model Formulation

Let 𝑡₁<𝑡₂<⋯<𝑡_𝑛 denote the 𝑛 failure times of the repairable system observed till 𝑇. If 𝑇≡𝑡_𝑛, that is, the process is observed till 𝑛th failure, then it is failure-truncated sampling, and 𝑇 is a random variable. If 𝑇>𝑡_𝑛, then it is time-truncated sampling, where 𝑇 is a prefixed quantity, and 𝑛 is a random variable.

The 2-EBP model is an NHPP whose failure intensity is of the form 𝜆(𝑡)𝑡=𝜂𝑡+𝜃,𝜂,𝜃>0,𝑡≥0, (2) which is an increasing bounded intensity function with the operating time 𝑡, equal to 0 at 𝑡=0 and approaching an asymptote of 𝜂 as 𝑡 increases. So, the parameter 𝜂 is the asymptotic value of the intensity function: 𝜂≡𝜆_∞, and the parameter 𝜃 is a measure of the initial increasing rate of 𝜆(𝑡): the smaller 𝜃 is, the faster the failure intensity increases until it approaches 𝜂.

Thus, the initial failure intensity, that is, the intensity function till the first overhaul epoch 𝑥₁, is𝜆₁(𝑡)𝑡=𝜂𝑡+𝜃,𝑡≤𝑥₁. (3) The conditional intensity function at a generic time 𝑡in the interval (𝑥_𝑗,𝑥_𝑗+1), given 𝑥_𝑗 is, 𝜆_𝑗+1𝑡∣𝑥_𝑗=𝜆₁𝑡−𝜌𝑥_𝑗=𝜂𝑡−𝜌𝑥_𝑗𝑡−𝜌𝑥_𝑗+𝜃;𝑥_𝑗<𝑡≤𝑥_𝑗+1. (4) The expected number of failures between two successive overhaul epochs is 𝐸𝑁𝑥_𝑗,𝑥_𝑗+1=^𝑥𝑗+1_𝑥𝑗𝜆_𝑗+1𝑡∣𝑥_𝑗𝑥𝑑𝑡=𝜂_𝑗+1−𝑥_𝑗𝑥−𝜃log^𝑗+1−𝜌𝑥_𝑗+𝜃𝑥_𝑗−𝜌𝑥_𝑗.+𝜃 (5) The cumulative value of expected number of failures from 0 up to a generic time 𝜏 in the interval (𝑥_𝑖,𝑥_𝑖+1) is𝑁_(𝜏)=^𝑖−1_𝑗=0𝐸𝑁𝑥_𝑗,𝑥_𝑗+1+𝐸𝑁𝑥_𝑖,𝑥_𝜏,𝑥_𝑖≤𝜏≤𝑥_𝑖+1,𝑥₀=0=𝜂^𝑖−1_𝑗=0𝑥_𝑗+1−𝑥_𝑗𝑥−𝜃log^𝑗+1−𝜌𝑥_𝑗+𝜃𝑥_𝑗−𝜌𝑥_𝑗+𝜃+𝜂𝜏−𝑥_𝑖−𝜃log𝜏−𝜌𝑥_𝑖+𝜃𝑥_𝑖−𝜌𝑥_𝑖.+𝜃 (6)

4. Likelihood Function

  The likelihood function based on observed data: 𝑡₁,𝑡₂,…,𝑡_𝑛,𝑇, is

The conditional probability that the unit of age 𝑡_𝑖−1 will fail before 𝑡_𝑖 is𝐹𝑡_𝑖∣𝑡_𝑖−1=𝑃𝑇≤𝑡_𝑖∣𝑇>𝑡_𝑖−1=𝐹𝑡_𝑖−𝐹𝑡_𝑖−1𝐹𝑡_𝑖−1=−1−𝐹𝑡_𝑖+1−𝐹𝑡_𝑖−1𝐹𝑡_𝑖−1=𝐹𝑡_𝑖−1−𝐹𝑡_𝑖𝐹𝑡_𝑖−1=1−𝐹𝑡_𝑖𝐹𝑡_𝑖−1. (7) The conditional reliability just before the 𝑖th failure will be𝐹𝑡_𝑖∣𝑡_𝑖−1=1−𝐹𝑡_𝑖∣𝑡_𝑖−1=𝐹𝑡_𝑖𝐹𝑡_𝑖−1=𝑒^{−∫𝑡𝑖𝑡𝑖−1𝑟(𝑢)𝑑𝑢}, (8) and the conditional pdf is 𝑓(𝑡_𝑖∣𝑡_𝑖−1).

If the data is time truncated, the joint pdf of the failure times (𝑡₁,𝑡₂,…,𝑡_𝑛) is𝑓𝑡₁,𝑡₂,…,𝑡_𝑛=𝑓𝑡₁𝑓𝑡₂∣𝑡₁⋯𝑓𝑡_𝑛∣𝑡₁,𝑡₂,…,𝑡_𝑛−1×𝐹𝑇∣𝑡_𝑛=𝜆𝑡₁⋅𝑒^{−∫𝑡10𝜆(𝑢)𝑑𝑢}⋅𝜆𝑡₂⋅𝑒^{−[∫𝑡20𝜆(𝑢)𝑑𝑢−∫𝑡10𝜆(𝑢)𝑑𝑢]}⋯𝜆𝑡_𝑛⋅𝑒^{−[∫𝑡𝑛0𝜆(𝑢)𝑑𝑢−∫𝑡0𝑛−1𝜆(𝑢)𝑑𝑢]}⋅𝑒^{−∫𝑇𝑡𝑛𝜆(𝑢)𝑑𝑢}=^𝑛_𝑖=1𝜆𝑡_𝑖⋅𝑒^{−∫𝑇0𝜆(𝑢)𝑑𝑢}, (9a) where 𝑇 is the truncation time.

As 𝑥₁,𝑥₂,…,𝑥_𝑘 are the 𝑘 overhaul epochs, they provide disjoint fixed intervals (𝑥₀,𝑥₁),(𝑥₁,𝑥₂),…,(𝑥_𝑘−1,𝑥_𝑘). Since the overhaul epochs are prescheduled, they are not random. Let the time interval (𝑥_𝑗,𝑥_𝑗+1) be the (𝑗+1)th period.

From (4) and (9a), for 𝑡∈(𝑥_𝑗,𝑥_𝑗+1), the joint pdf of failure times in (𝑥_𝑗,𝑥_𝑗+1) is =_{𝑖∶𝑡𝑖∈𝑥𝑗,𝑥𝑗+1}𝜆_𝑗+1𝑡_𝑖∣𝑥_𝑗⋅𝑒^{−∫𝑥𝑥𝑗𝑗+1𝜆𝑗+1(𝑡∣𝑥𝑗)𝑑𝑡}. (9b)Since the overhaul epochs are prescheduled, the failure times in disjoint epoch periods are independent, and therefore, the joint pdf of the failure times (𝑡₁,𝑡₂,…,𝑡_𝑛) is simply the product form of (9b), that is,𝑓𝑡₁,𝑡₂,…,𝑡_𝑛==𝐿data∣𝜂,𝜃,𝜌_{𝑘𝑗=0𝑡𝑖∈𝑥𝑗,𝑥𝑗+1}𝜆_𝑗+1𝑡_𝑖∣𝑥_𝑗⋅𝑒^{−∑𝑘𝑗=0∫𝑥𝑥𝑗𝑗+1𝜆𝑗+1(𝑡∣𝑥𝑗)𝑑𝑡}=𝜂^𝑛𝑘_{𝑗=0𝑡𝑖∈𝑥𝑗,𝑥𝑗+1}𝑡_𝑖−𝜌𝑥_𝑗𝑡_𝑖−𝜌𝑥_𝑗+𝜃⋅𝑒^{−𝜂∑𝑘𝑗=0[(𝑥𝑗+1−𝑥𝑗)−𝜃log((𝑥𝑗+1−𝜌𝑥𝑗+𝜃)/(𝑥𝑗−𝜌𝑥𝑗+𝜃))]}, (10) where 𝑥_𝑘+1=𝑇.

5. Bayesian Procedure

5.1. Informative Prior

Suppose that the analyst is able to anticipate the following: (a)the asymptotic value 𝜆_∞≡𝜂 of the intensity function, (b)the time 𝑡_𝑟 at which the failure intensity is 𝑟 times the asymptotic value (𝑟<1), that is,𝜆𝑡_𝑟𝜆_∞=𝑟;𝑥_𝑖<𝑡_𝑟<𝑥_𝑖+1⟹𝑡_𝑟=(1−𝑟)𝜌𝑥_𝑖+𝑟𝜃.1−𝑟 (11) Now, we assume a Gamma density for 𝜂 and 𝑡_𝑟, both for the mathematical tractability and flexibility. Therefore, 𝑔𝜂=𝑏^𝑎Γ(𝑎)𝜂^𝑎−1𝑔𝑡exp−𝑏𝜂,𝑎,𝑏>0,_𝑟=𝑑^𝑐Γ(𝑐)𝑡_𝑟^𝑐−1exp−𝑑𝑡_𝑟,𝑐,𝑑>0, (12) where the gamma parameters are related to the prior mean and standard deviation by𝜇𝑎=^𝜂𝜎_𝜂²𝜇,𝑏=^𝜂𝜎²_𝜂,𝜇𝑐=^𝑡𝜎_𝑡²𝜇,𝑑=^𝑡𝜎²_𝑡. (13) The analyst can use the fact that a 𝜌 value closer to 0 (to 1) indicates less (more) effectiveness of the overhaul actions to formulate an informative prior on 𝜌. We assume that the analyst is able to anticipate both a prior mean and a standard deviation on 𝜌, say 𝜇_𝜌and𝜎_𝜌, then we formalize this information through a beta density 𝑔𝜌=𝜌^𝑝−11−𝜌^𝑞−1𝛽𝑝,𝑞,0≤𝜌≤1, (14) where the values of the parameters related to the prior mean and standard deviation are𝑝=1−𝜇_𝜌𝜇²_𝜌𝜎²_𝜌−𝜇_𝜌,𝑞=𝑝1−𝜇_𝜌𝜇_𝜌. (15)

5.2. Joint Prior Density

Assuming the prior independence of the parameters  (𝜂,𝜃,𝜌), and using (11) to find density of 𝜃, the joint prior density 𝑔(𝜂,𝜃,𝜌) is given by 𝑔𝜂,𝜃,𝜌∝𝜂^𝑎−1𝑒^−𝑏𝜂(1−𝑟)𝜌𝑥_𝑖+𝑟𝜃1−𝑟^𝑐−1×𝑒^{−𝑑(((1−𝑟)𝜌𝑥𝑖+𝑟𝜃)/(1−𝑟))}𝜌^𝑝−11−𝜌^𝑞−1. (16)

5.3. Joint Posterior Density on Model Parameters

By combining the likelihood (10) with the joint prior density (16), the joint posterior density on the model parameters is given by𝜋=1𝜂,𝜃,𝜌∣data𝐷𝜂^𝑛⋅^𝑘_{𝑗=0𝑡𝑖∈𝑥𝑗,𝑥𝑗+1}𝑡_𝑖−𝜌𝑥_𝑗𝑡_𝑖−𝜌𝑥_𝑗+𝜃⋅𝑒^{−𝜂∑𝑘𝑗=0[(𝑥𝑗+1−𝑥𝑗)−𝜃log((𝑥𝑗+1−𝜌𝑥𝑗+𝜃)/(𝑥𝑗−𝜌𝑥𝑗+𝜃))]}⋅𝜂^𝑎−1⋅𝑒^−𝑏𝜂⋅(1−𝑟)𝜌𝑥_𝑖+𝑟𝜃1−𝑟^𝑐−1⋅𝑒^{−𝑑(((1−𝑟)𝜌𝑥𝑖+𝑟𝜃)/(1−𝑟)))}⋅𝜌^𝑝−1⋅1−𝜌^𝑞−1, (17) where𝐷=Γ(𝑛+𝑎)^∞₀¹₀𝑓𝜃,𝜌∣data𝑏+∑^𝑘_𝑗=0𝑥_𝑗+1−𝑥_𝑗−𝜃log𝑥_𝑗+1−𝜌𝑥_𝑗+𝜃/𝑥_𝑗−𝜌𝑥_𝑗+𝜃^𝑛+𝑎𝑓=𝑑𝜌𝑑𝜃,𝜃,𝜌∣data^𝑘_{𝑗=0𝑡𝑖∈𝑥𝑗,𝑥𝑗+1}𝑡_𝑖−𝜌𝑥_𝑗𝑡_𝑖−𝜌𝑥_𝑗⋅+𝜃(1−𝑟)𝜌𝑥_𝑖+𝑟𝜃1−𝑟^𝑐−1⋅𝑒^{−𝑑(((1−𝑟)𝜌𝑥𝑖+𝑟𝜃)/(1−𝑟))}⋅𝜌^𝑝−1⋅1−𝜌^𝑞−1. (18)

6. Posterior Inference on the Expected Number of Failures in a Future Time Interval

Let 𝑀_𝜏 be the expected number of failures in the future time interval (𝑇,𝜏), 𝑀(𝜏){𝑁()=𝐸𝑇,𝜏)}=𝜂(𝜏−𝑇−𝜃log𝜏−𝜌𝑠+𝜃,𝑇−𝜌𝑠+𝜃 (19) where𝑥𝑠=𝑇ifoverhaulisperformedat𝑇,_𝑘,otherwise. (20) If no overhaul is performed after 𝑇, the conditional distribution of 𝑀_𝜏 is Poisson with mean 𝑀(𝜏) equal to the expected number of failures in the time interval (𝑇,𝜏), Pr𝑀_𝜏=[𝑀(𝜏=𝑚∣𝜂,𝜃,𝜌;data)]^𝑚exp[−𝑀(𝜏)],𝑀𝑚!Pr_𝜏=𝑚∣data=^∞₀^∞₀¹₀𝜋⋅[𝑀(𝜏𝜂,𝜃,𝜌)]^𝑚exp[−𝑀(𝜏)]=1𝑚!𝑑𝜌𝑑𝜃𝑑𝜂𝐷⋅Γ(𝑛+𝑎+𝑚)𝑚!^∞₀¹₀𝑓⋅𝑋𝜃,𝜌∣data_{𝜏(𝜃)𝑚}𝑏+∑^𝑘_𝑗=0𝑥_𝑗+1−𝑥_𝑗−𝜃log𝑥_𝑗+1−𝜌𝑥_𝑗+𝜃/𝑥_𝑗−𝜌𝑥_𝑗+𝜃+𝑋_𝜏^{(𝜃)𝑛+𝑎+𝑚}𝑑𝜌𝑑𝜃, (21)where𝑋_𝜏^(𝜃)=(𝜏−𝑇)−𝜃log𝜏−𝜌𝑠+𝜃.𝑇−𝜌𝑠+𝜃 (22) The posterior mean of 𝑀_𝜏 is given by the posterior expectation of 𝑀(𝜏), 𝐸𝑀_𝜏∣data=^∞_𝑚=0𝑚⋅Pr𝑀_𝜏=𝑚∣data. (23) After the transformation of variables, we have the following: 𝐸𝑀_𝜏=()!∣data𝑛+𝑎𝐷^∞₀¹₀𝑓⋅𝑋𝜃,𝜌∣data^𝜏(𝜃)𝑏+∑^𝑘_𝑗=0(𝑥_𝑗+1−𝑥_𝑗)−𝜃log𝑥_𝑗+1−𝜌𝑥_𝑗+𝜃/𝑥_𝑗−𝜌𝑥_𝑗+𝜃^𝑛+𝑎+1𝑑𝜌𝑑𝜃, (24)A conservative 𝛾 Bayes upper credibility limit for 𝑀_𝜏 is obtained by finding the smallest value of which 𝑚_𝑢 satisfiesPr𝑀_𝜏≤𝑚_𝑢∣data≥𝛾. (25)

7. Posterior Inference on the Future Failure Times

Given the observed data, we are interested in predicting the 𝑚th future failure time 𝑡_𝑛+𝑚, under the assumption that no overhaul is performed after 𝑇, 𝑀_𝑚=𝐸𝑁𝑇,𝑡_𝑛+𝑚=𝜂𝑡_𝑛+𝑚𝑡−𝑇−𝜃log^𝑛+𝑚−𝜌𝑠+𝜃,𝑇−𝜌𝑠+𝜃 (26) where𝑥𝑠=𝑇ifoverhaulisperformedat𝑇,_𝑘,otherwise. (27)𝑀_𝑚 is gamma distributed with shape parameter m and unit scale parameter. Thus, the conditional pdf of 𝑡_𝑛+𝑚, with 𝑡_𝑛+𝑚>𝑇, is given by𝑝𝑡_𝑛+𝑚=1∣𝜂,𝜃,𝜌;data𝜂Γ𝑚^𝑚𝑋_𝑚^{(𝜃)𝑚−1}exp−𝜂𝑋_𝑚^(𝜃)×𝑡_𝑛+𝑚−𝜌𝑠𝑡_𝑛+𝑚,−𝜌𝑠+𝜃 (28) where𝑋_𝑚^(𝜃)=𝑡_𝑛+𝑚𝑡−𝑇−𝜃log^𝑛+𝑚−𝜌𝑠+𝜃.𝑇−𝜌𝑠+𝜃 (29) The predictive density function of 𝑡_𝑛+𝑚 is given by ℎ𝑡_𝑛+𝑚∣data=^∞₀^∞₀¹₀𝜋𝜂,𝜃,𝜌⋅𝑝𝑡_𝑛+𝑚=1∣𝜂,𝜃,𝜌;data𝑑𝜌𝑑𝜃𝑑𝜂𝐷⋅Γ(𝑛+𝑎+𝑚)Γ(𝑚)×^∞₀¹₀𝑓⋅𝑋𝜃,𝜌∣data^{𝑚(𝜃)𝑚−1}.𝑡_𝑛+𝑚−𝜌𝑠/𝑡_𝑛+𝑚−𝜌𝑠+𝜃𝑏+∑^𝑘_𝑗=0𝑥_𝑗+1−𝑥_𝑗−𝜃log𝑥_𝑗+1−𝜌𝑥_𝑗+𝜃/𝑥_𝑗−𝜌𝑥_𝑗+𝜃+𝑋_𝑚^{𝑛+𝑎+𝑚}𝑑𝜌𝑑𝜃. (30)^(𝜃)A Bayes prediction interval of a given probability content 𝛾 is defined as the interval 𝐼 such that Pr{𝐼∣data}=_𝐼ℎ𝑡_𝑛+𝑚∣data𝑑𝑡_𝑛+𝑚+𝛾. (31)

8. Sensitivity Analysis

Sensitivity analysis is carried out with respect to the prior information on 𝜌, namely the prior mean 𝜇_𝜌 and the standard deviation 𝜎_𝜌, by evaluating 𝑀_𝜏: expected number of failures in the future time interval (𝑇,𝜏) when overhaul is performed at 𝑇 and not performed at 𝑇, over a reasonable range of values for 𝜇_𝜌 and 𝜎_𝜌. If the slight change in values of 𝜇_𝜌 and 𝜎_𝜌 does not have much effect on 𝑀_𝜏  then the proposed prior is robust.

9. Numerical Application

Consider the following hypothetical data for illustrative purpose.

The failure times (𝑛=18) and overhaul epochs are given in Table 1, major overhauls marked with*. We have assumed it to be a time-truncated sample with failures observed for 1500 units. Four major overhauls are assumed to be performed at times different from failure times.

Table 1 

Failure times and overhaul epochs.

The proportional age reduction 2-EBP model is adequate for this hypothetical data set in contrast to pre-existing proportional age reduction power law process model as its log-likelihood value obtained using (10) is −95.4606 which is greater than the log-likelihood value −96.1537 of power law process model.

Suppose that analyst is able to anticipate a prior mean 𝜇_𝜂=0.133 and 𝜎_𝜂=0.004(𝑎=4,𝑏=30).

In addition, from previous experiences, the analyst possesses a vague belief that the failure intensity, at the time 𝑡_𝑟=753 units, is nearly half its asymptotic value: 𝜆(753)/𝜆_∞=0.5.Then, he formalizes his prior knowledge on t_r through the exponential density having mean 𝜇_𝑡=753, so that 𝑐=1 and 𝑑=0.001. As 𝑥_𝑖<𝑡_𝑟<𝑥_𝑖+1, therefore 600<753<900. Hence, 𝑥_𝑖=600.

Again, the analyst possesses a vague belief that the overhaul actions are quite effective, and then he chooses the beta density for the improvement parameter with prior mean 𝜇_𝜌=0.6 and standard deviation 𝜎_𝜌=0.26(𝑝=1.5and𝑞=1).

Now, suppose that the above failure process is time truncated at 𝑇=1200 units. On the basis of the observed data, we predict the future failure times under two hypotheses:(a)major overhaul is performed at 𝑇, (b)no overhaul is performed at 𝑇.

All further repair actions are assumed to be minimal repairs. The remaining observed values, that is, {1230,1268,1330,and1447} have been used for comparison with the predicted values.

In Table 2, we compare the occurred failure times 𝑡_13+𝑚(𝑚=1,2,3,4,5) with the Bayes prediction of the 𝑚th future failure times under both the above hypotheses; in particular, the lower (𝑡_𝑙) and upper (𝑡_𝑢) limits of the 0.90 Bayes equal-tail credibility interval are shown. It clearly shows the advantages of performing overhauls as the future failure times are delayed considerably by overhaul.

Table 2 

Comparison of the occurred failure times 𝑡_13+𝑚(𝑚=1,2,3,4,5) with the Bayes prediction under the hypothesis that overhaul is or is not performed at 𝑇.

In Table 3, we compare the observed values of 𝑡_𝑛+1(𝑛=13,14,15,16,17) with the lower and upper limits of the Bayes 0.90 equal-tail credibility interval and with the posterior median obtained using (30) and (31). This allows the analyst to assess the adequacy of the selected model and the correctness of prior information. Table 3 shows that all occurred failures fall well within the Bayes prediction interval and are quite close to the respective posterior median, thus indicating the adequacy of the 2-EBP to model the given data set.

Table 3 

Comparison of occurred failure times with bayes prediction.

Table 4 clearly shows that the, 95% upper credibility limit for the number of failures in future time interval (1200, 1500) when major overhaul is performed at 𝑇=1200 is equal to 11 failures and the posterior mean = 5.327 failures, very close to the number of failures, namely 5, actually occurred in (1200, 1500).

Table 4 

Posterior mean and 0.95 upper credibility limit of the number of failures in the future time interval (1200, 1500).

On the contrary in the case of no overhaul 95% upper credibility limit for the number of failures in future time interval (1200, 1500) is equal to 14 failures which is more than that in major overhaul case, and the posterior mean is 7.612 which is again more than the value obtained in the major overhaul case. These results show the considerable advantages arising from performing a major overhaul at 𝑇.

Tables 5 and 6 show that the change in values of 𝜇_𝜌  and 𝜎_𝜌 does not have much effect on 𝑀_𝜏, so the proposed prior is robust.

Table 5 

Sensitivity analysis with respect to the prior mean of 𝜌.

Table 6 

Sensitivity analysis with respect to the standard deviation of 𝜌.

10. Conclusion

In this paper, the prediction of future failures of a deteriorating repairable mechanical system subject to minimal repairs and periodic overhauls has been done using Bayesian approach. The effect of overhauls on the reliability of the system has been modeled using a proportional age reduction model, and the failure process between two successive overhauls has been modeled using 2-parameter Engelhardt-Bain process (2-EBP). The prediction of the future failure times and the number of failures in a future time interval has been done on the basis of the observed data and of a number of suitable prior densities reflecting varied degrees of belief on the failure/repair process and effectiveness of overhauls. The advantages of overhauls have been highlighted using a numerical application and sensitivity analysis of the improvement parameter carried out.

Nomenclature

𝑛:	Total no. of failures
𝑘:	Total no. of overhauls
𝑥𝑖:	𝑖th overhaul epoch, 𝑖=1⋯𝑘
𝜂,𝜃:	Parameters of 2-EBP
𝜌:	Improvement factor
𝑡𝑟:	Time at which the failure intensity is 𝑟 times the asymptotic value
𝜆1(𝑡):	Initial failure intensity
𝜆𝑗+1(𝑡∣𝑥𝑗):	Conditional intensity function at a generic time 𝑡 in the interval (𝑥𝑗,𝑥𝑗+1)
𝐸{𝑁(𝑥𝑗,𝑥𝑗+1)}:	Expected number of failures between two successive overhaul epochs
𝑔(𝜂):	Prior pdf on 𝜂
𝑔(𝑡𝑟):	Prior pdf on 𝑡𝑟
𝑔(𝜌):	Prior pdf on 𝜌
𝑎,𝑏:	Prior gamma parameters of 𝜂
𝑐,𝑑:	Prior gamma parameters of 𝑡𝑟
𝑝,𝑞:	Prior beta parameters of 𝜌
𝜇𝜂,𝜎2𝜂:	Prior mean and variance of 𝜂
𝜇𝑡𝑟,𝜎2𝑡𝑟:	Prior mean and variance of 𝑡𝑟
𝜇𝜌,𝜎2𝜌:	Prior mean and variance of 𝜌
𝑔(𝜂,𝑡𝑟,𝜌):	Joint prior pdf of 𝜂,𝑡𝑟,𝜌
𝜋(𝜂,𝜃,𝜌∣data):	Joint posterior pdf of 𝜂,𝜃,𝜌
𝑀𝜏:	Expected number of failures in the future time interval (𝑇,𝜏)
𝑡𝑛+𝑚:	𝑚th future failure time
NHPP:	Nonhomogeneous poisson process
2-EBP:	2-parameter Engelhardt-Bain process
PLP:	Power law process.

Acknowledgment

The authors are grateful to the referees for their valuable comments and suggestions that improved this paper by making it more informative.