Mathematical Problems in Engineering

Volume 2015, Article ID 908742, 8 pages

http://dx.doi.org/10.1155/2015/908742

## Generalized Accelerated Failure Time Frailty Model for Systems Subject to Imperfect Preventive Maintenance

^{1}Department of Electrical & Information Engineering, CDHK, Tongji University, Shanghai 200092, China^{2}Department of Mathematics, Tongji University, Siping Road 1239, Shanghai 200092, China^{3}School of Electrical & Information Engineering, Tongji University, Shanghai 200092, China

Received 14 August 2014; Revised 13 October 2014; Accepted 13 October 2014

Academic Editor: Gang Li

Copyright © 2015 Huilin Yin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Imperfect preventive maintenance (PM) activities are very common in industrial systems. For condition-based maintenance (CBM), it is necessary to model the failure likelihood of systems subject to imperfect PM activities. In this paper, the models in the field of survival analysis are introduced into CBM. Namely, the generalized accelerated failure time (AFT) frailty model is investigated to model the failure likelihood of industrial systems. Further, on the basis of the traditional maximum likelihood (ML) estimation and expectation maximization (EM) algorithm, the hybrid ML-EM algorithm is investigated for the estimation of parameters. The hybrid iterative estimation procedure is analyzed in detail. In the evaluation experiment, the generated data of a typical degradation model are verified to be appropriate for the real industrial processes with imperfect PM activities. The estimates of the model parameters are calculated using the training data. Then, the performance of the model is analyzed through the prediction of remaining useful life (RUL) using the testing data. Finally, comparison between the results of the proposed model and the existing model verifies the effectiveness of the generalized AFT frailty model.

#### 1. Introduction

Condition-based maintenance (CBM) has continuously been an important issue in the area of maintenance strategy. As degradation processes before failure of many systems can be measured, CBM is more effective as corrective maintenance and time-based preventive maintenance in some aspects, such as catastrophic failure reduction and availability maximization [1, 2]. CBM program includes three steps: data acquisition, signal processing, and maintenance decision support [1]. Maintenance decision support is categorized into diagnostics and prognostics. Prognostics are often characterized by estimating the remaining useful life (RUL) of systems using available condition monitoring information. The RUL estimation ensures enough time to perform the necessary maintenance actions prior to failure [3, 4].

The proportional hazards model (PHM) [5, 6], which is a popular model in survival analysis, can be applied in reliability evaluation and maintenance optimization. It has shown its effectiveness for RUL prediction and CBM scheduling of industrial systems [7–9]. PHM is an effective CBM method due to its strength in handling the influence of variable operational conditions. The accelerated failure time (AFT) model [10, 11] is an important alternative to the PHM in survival analysis and has the advantage of being more intuitively interpretable than the PHM. However, the AFT method has been rarely applied in reliability-related fields. It is shown by reviewing and comparing the major mathematical models of survival analysis and reliability theory that both fields address the same mathematical problems [6]. Because of the unified mathematical models of both fields, the methods for survival analysis might be used for reliability analysis and related fields such as CBM. In this paper, we introduce the AFT model in reliability fields and investigate the AFT-based model for CBM.

Imperfect preventive maintenance (PM) restores a system to a better state but not “as good as new.” Imperfect PM activities are considerably common in industrial systems and there are many studies on the model of imperfect PM [12–17]. The extended PHM (EPHM) [16, 17] has been proved to be a superiorly effective method to predict RUL of systems subject to imperfect PM. Inspired by the EPHM, this paper investigates a new alternative model, the generalized AFT frailty model.

The rest of the paper is organized as follows. Section 2 introduces the generalized AFT frailty model and compares it with the existing model. Section 3 proposes the hybrid maximum likelihood- (ML-) expectation maximization (EM) algorithm for the calculation of the parameters in the model. Section 4 proves the effectiveness of the proposed model by the simulation experiment of RUL prediction for systems subject to imperfect PM. Finally, Section 5 concludes the paper.

#### 2. Generalized AFT Frailty Model

##### 2.1. Models

First, we introduce the Cox PHM, the AFT model, the generalized AFT model, and the PHM frailty model in sequence. Then, we propose the generalized AFT frailty model.

The Cox PHM [5] is a popular model in survival analysis, and it is an effective method for RUL prediction and CBM scheduling of industrial systems. Let be the hazard/failure function at time given the covariates ; the PHM is expressed aswhere is the baseline hazard/failure function, which considers the age of the system at the time of inspection. is the vector of the covariates and is the regression coefficient vector. In a CBM program, the systems’ degradation indicators are usually chosen to be the covariates.

The AFT model [10, 11, 18] is an important alternative to the PHM. Suppose is the failure time of system in cluster (or group) . There is a censoring random variable, which we denote as . We only observe and the linear regression (i.e., AFT) model iswhere is the vector of observed covariates, is the regression coefficient vector, and is the random error. This linear form of the AFT model deals with the regression relationship of the covariates and the failure times logarithms, which has the similar form to the generic linear regression model. In the PHM, the baseline failure function and the covariates in PHM are independent, which limits the modeling of some types of the survival data in medical research. Collett [19] has introduced the influence of the covariates in the baseline failure function and proposed the following generalized AFT model:In this model, the whole part of the covariates has been introduced into the baseline failure function. The effectiveness of this model has been verified by the modeling of the survival data in medical field [19].

PHM and AFT are suitable for the mutually independent failure time data. In reality, correlated or clustered failure time data are very common in the fields of survival analysis and reliability. For example, systems operate in the same environment with the same temperature and humidity. The shared environment of the subjects leads to the dependence among the observed failure time. Frailty is a good tool to represent the random effect shared by subjects in the same cluster (or group) and it induces dependence among the correlated or clustered failure time data. The PHM frailty model [20] iswhere is the frailty term and models the random effect that is shared by the systems in the th cluster (group). Frailty has been also introduced in the AFT model to represent the possible correlation among failure times [12]. Considering both the generalized AFT model (3) and the PHM frailty model (4), the following generalized AFT frailty model [21] is proposed:

##### 2.2. Comparison between the Generalized AFT Frailty Model and the Existing Model

The EPHM has been proposed by You et al. [16, 17] to model the failure likelihood for systems subject to imperfect PM activities. The EPHM has the following form:where is the hazard/failure function of the system after the th PM activity and before the th PM activity, is the th PM interval, and is the random local time between the th and the th PM activity. , where is the hazard rate increase factor (HRIF) due to the th PM activity, and is the age reduction factor (ARF) due to the th PM activity, . The HRIF modifies the increase rate of the hazard/failure rate of the system after an imperfect PM activity, and the ARF measures the extent to which the PM activity brings the system to a younger age.

Comparing (5) and (6), we observe that the HRIF and the frailty term act multiplicatively on the hazard/failure rate function and play the same role in the models. When the system starts operation after imperfect PM activities, the system has already aged somewhat. The ARF measures the extent to which the imperfect PM affects the effective age. The generalized AFT frailty model describes the hybrid influence of imperfect PM on both the hazard/failure rate and the effective age, which is shown in Figure 1.