Mathematical Problems in Engineering

Volume 2018, Article ID 9734189, 12 pages

https://doi.org/10.1155/2018/9734189

## Condition-Based Predictive Order Model for a Mechanical Component following Inverse Gaussian Degradation Process

^{1}School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China^{2}Key Laboratory of Modern Measurement and Control Technology, Ministry of Education, Beijing Information Science and Technology University, Beijing 100192, China

Correspondence should be addressed to Jianxin Xu; nc.ude.upwn@xjux

Received 3 April 2018; Revised 13 August 2018; Accepted 26 August 2018; Published 10 September 2018

Academic Editor: Paolo Crippa

Copyright © 2018 Cheng Wang 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

An efficient condition-based predictive spare ordering approach is the key to guarantee safe operation, improve service quality, and reduce maintenance costs under a predefined lower availability threshold. In this paper, we propose a condition-based predictive order model (CBPO) for a mechanical component, whose degradation path is modeled as inverse Gaussian (IG) process with covariate effect. The CBPO is dependent on the remaining useful life (RUL), random lead-time, speed-up lead-time degree, and availability threshold. RUL estimation is obtained through the IG degradation process at each inspection time. Both regular lead-time and expedited lead-time considered in RUL-based spare ordering policy can be cost-effective and reduce losses caused by unexpected failure. Speed-up lead-time can meet the urgent needs for spare parts on site. The decision variable of CBPO is the spare ordering time. Based on the CBPO under the lower availability threshold constraint, the objective of this study is to determine the optimal spare ordering time such that the expected cost rate is minimized. Finally, a case study of the mechanical spindle is presented to illustrate the proposed model and sensitivity analysis on critical parameters is performed.

#### 1. Introduction

With the development of science and technology, the safety and effective perception of some products, especially for some high-end products, become more and more important. Product degradation condition prediction and efficient spare ordering are the key to guarantee safe operation, improve service quality, and reduce maintenance costs. It is of great importance to avoid the failure of a highly reliable product, especially for aircraft engine, high-end machine tools, high-speed train, wind turbines, and so on. In order to prevent or reduce product unexpected failure, various preventive maintenance (PM) policies have been proposed over the past several decades [1]. In PM policies, there are mainly two categories: time-based maintenance (TBM) policies and condition-based maintenance (CBM) policies. For TBM, much attention has been paid to lifetime distribution models. For CBM, decision is made according to degradation level or health condition. More recently, rapid development of computer-based condition monitoring (CBCM) technologies (e.g., advanced sensors) and Internet of Things (IoT) make it possible to track the product health in a real-time manner and have further facilitated efficient potential failure detection and CBM practices. Comparing with TBM, CBM is a more promising maintenance policy since it emphasizes combining data-driven reliability models with condition monitoring data. Therefore, CBM has received considerable attention in both academia and industry [2–9].

Most CBM policies studied in the literatures assume that at any time there is an unlimited supply of available spares for replacement. However, this assumption is generally unrealistic and unpractical when available spares are limited and/or delivery lead times are much longer. When spares are expensive, scarce, and with higher and random lead times, it is important to consider shortage cost and holding cost. Therefore, when to order spare is a key factor to make tradeoff between shortage cost and holding cost. In other words, reasonable spare ordering time can minimize the entire maintenance cost in engineering practice. Motivated by addressing this type of problem, some order policies have been extensively researched. Wang* et al*. [10] proposed a joint optimization of CBM and spare ordering management for a single-unit system. Chien and Chen [11] presented an age-based spare ordering policy with random lead-time. Chien [12] presented an optimal spares ordering policy based on the optimal number of minimal repairs with random lead-time. Louit* et al*. [13] presented an order policy based on remaining useful life (RUL) estimation through the assessment of the component age and condition indicators. Godoy* et al*. [14] presented a graphic technique that considered a rule for decision based on both condition-based reliability function and a random/fixed lead-time. Panagiotidou [15] proposed a joint optimization of spares ordering and maintenance policies for multiple identical times. Wang* et al*. [16] proposed a prognostics-based spare ordering and system replacement policy with random lead-time for a deteriorating system. Chen* et al*. [17] proposed a joint optimization of replacement and spare ordering for critical rotary component based on collected condition monitoring signals. Cai* et al*. [18] proposed an appointment policy of spares based on (s,S) policy. Among them, prognostics-based order policy is few.

In reality, prognostic largely focuses on estimating the RUL of the component by using life distribution or the available degradation data. Therefore, RUL estimation [19–26] can offer adequate lead-time for the maintainer to implement the essential maintenance actions ahead of failure. The impact of the dynamic environment (e.g., temperature, stress, humidity, or dust) on the RUL is indicated by covariates [3, 13, 27, 28]. In CBM policies, prognostic is based on product’s degradation process through available degradation data. Stochastic models with continuous-state are widely used to characterize the degradation process. Notable among them, the Wiener process [26, 29], the Gamma process [30], and the inverse Gaussian (IG) process [28, 31–33], including their variants, attract significant attention because of their nice mathematical properties and clear physical interpretations. CBM policies based on the Wiener process [26, 34, 35], the Gamma process [36, 37], and the IG process [38] have been extensively investigated. The IG process is a very effective method to characterize the random effects and covariates and has been frequently used to model the monotone degradation process. For most mechanical components, the path of degradation increases monotonically. Based on the merit of IG process, the mechanical component’s degradation path is modeled as IG process with covariates.

This paper aims to develop a condition-based predictive order model (CBPO) for a mechanical component subject to IG degradation process with covariate effect. Covariates provide an indication of the external state and affect the degradation process of the mechanical component. When the degradation level of the component attains a predetermined threshold, the component fails. The proposed model depends on the RUL, random lead-time, speed-up lead-time degree, and availability threshold. The decision variable of the model is the spare ordering time. RUL estimation is obtained through the IG degradation process at each inspection time. Random lead-time includes both regular lead-time and expedited lead-time to be cost-effective and reduce losses caused by unexpected failure. Speed-up lead-time is triggered when the component failure occurs before the ordered spare is delivered. Based on the proposed model under the lower availability threshold constraint, what we aim is to find the optimal spare ordering time minimizing the expected cost rate. This paper differs from the existing works in following aspects. (a) Expedited order is considered in RUL-based spare ordering policy for the mechanical component subject to IG degradation process with covariate effect. (b) Speed-up lead-time is triggered when the component failure occurs before the ordered spare is delivered. (c) On the basis of cost rate objective function, we add the availability threshold constraint.

The remainder of this paper is organized as follows. In Section 2, we describe some assumptions. Section 3 introduces IG process and presents prediction method of RUL based on IG process. Based on RUL, a novel spare ordering model for the mechanical component is developed in Section 4. In Section 5, a case study of the mechanical spindle is given to illustrate the proposed order model and sensitivity analysis on critical parameters is performed. Section 6 concludes this paper and offers possible research in future.

#### 2. Assumptions

The spare ordering problem is considered under the following assumptions:(i)When degradation level of the component exceeds the predetermined failure threshold , the component fails.(ii)The mechanical component's health condition can be detected perfectly by CBCM.(iii)The mechanical component begins operating at time 0.(iv)When the mechanical component is failed, the expedited order is triggered immediately. That is to say, it is not to consider the downtime and the corresponding shortage cost between the failed time and spare ordering time.(v)Shortage cost per unit time is bigger than holding cost per unit time due to the component shutdown affecting the production progress and custom service negatively, i.e., .(vi)The lead-time is a generally distributed random variable. There are two types of lead-time, i.e., regular lead-time and expedited lead-time. In regular lead-time distribution, let be the probability density function (PDF) of regular lead-time, be the cumulative distribution function (CDF) of regular lead-time, be the survival function, be the mean of regular lead-time, and be the standard deviation of regular lead-time. In expedited lead-time distribution, let be the PDF of expedited lead-time, be the CDF of expedited lead-time, be the survival function, be the mean of expedited lead-time, and be the standard deviation of expedited lead-time. And in general, we assume that is bigger than , i.e., .(vii)Replacements are made perfectly and do not affect the component's characteristics.(viii)The spare in stock does not degenerate or fail.

#### 3. Model Statements

##### 3.1. IG Process

In practice, the performance of numerous mechanical components degrades over time, which can be modeled by a stochastic process. In our research, we will employ the IG process to represent monotonic degradation process. Let be degradation of the mechanical component at time , and then degradation process is called an IG process if it satisfies the following properties:

(a) with probability one;

(b) has independent increments on nonoverlapping intervals, i.e., and are independent for ;

(c) is subject to the IG distribution , for , where is a monotone increasing function with , and are constants, and , , denotes the IG distribution with PDFwhere and CDFwhere .

To characterize the influence of the covariates on the degradation process, we use a model similar to a proportional hazards model (PHM) [39, 40]. The covariate modifies the baseline rates as follows:where is nonnegative function such that For the sake of simplification, we assume and to be an exponential function which regulates the impacts of the covariate on the degradation.

More precisely, for a given value of the covariate at time , , the shape function of the IG process is defined bywhere is regression parameters defining the effects of the covariate of the IG process and is a vector of time-dependent covariates.

Let the constant be the failure threshold of the process We assume that the degradation follows an IG process model with function and parameters and Since , yields with mean and variance .

We define the first passage time to the failure threshold as . Based on (2), then follows the IG distribution with CDF given bywhere parameters and are positive and is the standard normal distribution function.

To describe the IG process clearly, some assumptions are made as follows.

(a) We assume that the shape function is a linear function, i.e.,

(b) The external force can affect the degradation process of the mechanical component. Based on (4), under the external force can be expressed byLet denote the state of operating environment at time , and take values in .

(c) Under the external force, the degradation path can be determined by the current stress and the accumulated degradation. The external force not only has an effect on the degradation speed but also the degradation volatility.

According to (5), the mean of the lifetime, i.e., mean-time-to-failure (MTTF), can be easily formulated by

In addition, according to the nature of IG process, we denote for the IG model parameters, and we obtain parameters estimation by maximum likelihood estimation (MLE) at each time based on the CBCM data

##### 3.2. RUL Estimation

Let denote the degradation observation at , which can be irregularly spaced, and represents the degradation measurement of the mechanical component at time . Therefore, using the first passage time of the degradation process, we define the RUL, abbreviated as that corresponds to .

Based on the definition of the RUL and condition on the degradation data , the distribution function of the RUL can be obtained by the following:

For (8), the mean of can be easily obtained as follows:where denotes reliability function of the RUL.

#### 4. Order Model

##### 4.1. Order Process

To facilitate the implementation process of order model, the main procedures of the model at each time are summarized as follows.

*Step 1. *Parameters initialization: set the initial value of inspection times by choosing randomly or by the prior knowledge.

*Step 2. *Parameters estimation: estimate the degradation model parameters by MLE according to the nature of IG process. The available estimation samples are obtained from the observed degradation data of the mechanical component till time , i.e., .

*Step 3. *RUL estimation: based on the real-time estimated parameters in Step 2, we can derive the RUL distribution of the mechanical component at time immediately.

*Step 4. *Applying order model (i.e., CBPO): based on Step 3, random lead-time distribution, and speed-up lead-time degree, we can seek the optimal spare ordering time that minimizes the expected cost rate under lower availability threshold by (20).

*Step 5. *Decision-making: according to the result of Step 4, if the optimal spare ordering time is not equal to the next inspection time , then return to Step 1 and wait for the next inspection time for new decision. Otherwise, stop the procedure and place an order simultaneously. Figure 1 shows the flow chart of the implementation process of order model at each time