^{1}

^{1}

^{1}

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.

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 [

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 [

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 [

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 [

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.

Failure rate of the system is an increasing function of time that attains a finite bound as

System is subjected to two kinds of maintenance actions: minimal repair and overhaul.

The times to perform maintenance actions are ignored.

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.

The failure density function is not changed by overhauls.

The quality of an overhaul is dependent on improvement factor

The improvement parameter

The

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

Let

The 2-EBP model is an NHPP whose failure intensity is of the form

Thus, the initial failure intensity, that is, the intensity function till the first overhaul epoch

The likelihood function based on observed data:

The conditional probability that the unit of age

If the data is time truncated, the joint pdf of the failure times

As

From (

Suppose that the analyst is able to anticipate the following:

the asymptotic value

the time

Assuming the prior independence of the parameters

By combining the likelihood (

Let

Given the observed data, we are interested in predicting the

Sensitivity analysis is carried out with respect to the prior information on

Consider the following hypothetical data for illustrative purpose.

The failure times

Failure times and overhaul epochs.

202 | 265 | 300* | 363 | 508 | 571 | 600* | 755 | 770 | 818 | 868 |
---|---|---|---|---|---|---|---|---|---|---|

900* | 999 | 1054 | 1068 | 1108 | 1200* | 1230 | 1268 | 1330 | 1376 | 1447 |

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 (

Suppose that analyst is able to anticipate a prior mean

In addition, from previous experiences, the analyst possesses a vague belief that the failure intensity, at the time _{r} through the exponential density having mean

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

Now, suppose that the above failure process is time truncated at

major overhaul is performed at

no overhaul is performed at

All further repair actions are assumed to be minimal repairs. The remaining observed values, that is,

In Table

Comparison of the occurred failure times

Actual | Overhaul at | No overhaul at | |||

1 | 1230 | 1204.3 | 1421.0 | 1202.4 | 1342.0 |

2 | 1268 | 1226.1 | 1514.0 | 1216.4 | 1420.0 |

3 | 1330 | 1255.2 | 1588.4 | 1236.6 | 1486.0 |

4 | 1376 | 1286.5 | 1653.6 | 1259.5 | 1546.8 |

5 | 1447 | 1318.4 | 1713.5 | 1283.5 | 1603.7 |

In Table

Comparison of occurred failure times with bayes prediction.

Actual | 90% Lower Limit | Median Value | 90% Upper Limit | ||
---|---|---|---|---|---|

13 | 1230 | 1204.3 | 1258.1 | 1421.0 | |

14 | 1268 | 1233.4 | 1275.7 | 1417.0 | |

15 | 1330 | 1271.0 | 1308.0 | 1435.2 | |

16 | 1376 | 1332.7 | 1366.1 | 1482.0 | |

17 | 1447 | 1378.5 | 1409.4 | 1518.1 |

Table

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

Posterior Mean | 0.95 Upper credibility limit | |
---|---|---|

Overhaul at | 5.327 | 11 |

No Overhaul at | 7.612 | 14 |

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

Tables

Sensitivity analysis with respect to the prior mean of

Percentage Deviation | |||
---|---|---|---|

−1% | 0.594 | 7.599 | 5.359 |

+1% | 0.606 | 7.618 | 5.312 |

−2% | 0.588 | 7.588 | 5.386 |

+2% | 0.612 | 7.628 | 5.287 |

−3% | 0.582 | 7.578 | 5.410 |

+3% | 0.618 | 7.639 | 5.261 |

Sensitivity analysis with respect to the standard deviation of

Percentage deviation | |||
---|---|---|---|

−1% | 0.2574 | 7.610 | 5.352 |

+1% | 0.2626 | 7.614 | 5.322 |

−2% | 0.2548 | 7.596 | 5.367 |

+2% | 0.2652 | 7.619 | 5.308 |

−3% | 0.2522 | 7.591 | 5.381 |

+3% | 0.2678 | 7.625 | 5.292 |

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.

Total no. of failures

Total no. of overhauls

Parameters of 2-EBP

Improvement factor

Time at which the failure intensity is

Initial failure intensity

Conditional intensity function at a generic time

Expected number of failures between two successive overhaul epochs

Prior pdf on

Prior pdf on

Prior pdf on

Prior gamma parameters of

Prior gamma parameters of

Prior beta parameters of

Prior mean and variance of

Prior mean and variance of

Prior mean and variance of

Joint prior pdf of

Joint posterior pdf of

Expected number of failures in the future time interval

Nonhomogeneous poisson process

2-parameter Engelhardt-Bain process

Power law process.

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