Research Article Condition-Based Predictive Order

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Sep 10, 2018 - 1 School of Mechanical Engineering, Northwestern Polytechnical University, .... expedited lead-time, and be the standard deviation of ... hazards model (PHM) [39, 40]. .... inspection times by choosing randomly or by the prior ..... 1.8. 2. Speed-up lead-time degree. 57. 58. 59. 60. 61. 62. 63. 64.
Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 9734189, 12 pages https://doi.org/10.1155/2018/9734189

Research Article Condition-Based Predictive Order Model for a Mechanical Component following Inverse Gaussian Degradation Process Cheng Wang 1 2

,1 Jianxin Xu ,1 Hongjun Wang,2 and Zhenming Zhang1

School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China 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; [email protected] 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. 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 leadtime 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, highspeed 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

2 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 agebased 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

Mathematical Problems in Engineering 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 𝑙, 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., 𝜌𝑠 > 𝜌ℎ . (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 leadtime distribution, let 𝑤(𝑡) be the probability density function (PDF) of regular lead-time, 𝑊(𝑡) be the cumulative distribution function (CDF) of regular lead-time, 𝑊(𝑡) be the survival function, 𝜇𝑟 be the mean of regular lead-time, and 𝜎𝑟 be the standard deviation of regular lead-time. In expedited leadtime distribution, let ℎ(𝑡) be the PDF of expedited lead-time, 𝐻(𝑡) be the CDF of expedited lead-time, 𝐻(𝑡) be the survival function, 𝜇𝑒 be the mean of expedited lead-time, and 𝜎𝑒 be the standard deviation of expedited lead-time. And in general, we assume that 𝜇𝑟 is bigger than 𝜇𝑒 , i.e., 𝜇𝑟 > 𝜇𝑒 .

Mathematical Problems in Engineering

3

(vii) Replacements are made perfectly and do not affect the component’s characteristics. (viii) The spare in stock does not degenerate or fail.

𝐹𝑙 (𝑡) = Pr {𝑇 < 𝑡} = Pr {𝑋 (𝑡) ≥ 𝑙}

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 𝑋(𝑡) be degradation of the mechanical component at time 𝑡, and then degradation process {𝑋(𝑡), 𝑡 > 0} is called an IG process if it satisfies the following properties: (a) 𝑋(0) = 0 with probability one; (b) 𝑋(𝑡) has independent increments on nonoverlapping intervals, i.e., 𝑋(𝑡) − 𝑋(𝑠) and 𝑋(𝑢) − 𝑋(V) are independent for 𝑡 > 𝑠 > 𝑢 > V ≥ 0; (c) 𝑋(𝑡)−𝑋(𝑠) is subject to the IG distribution 𝐼𝐺(𝜇[Λ(𝑡)− Λ(𝑠)], 𝜂[Λ(𝑡)−Λ(𝑠)]2) for 𝑡 > 𝑠 ≥ 0, where Λ(𝑡) is a monotone increasing function with Λ(0) = 0, 𝜇 > 0 and 𝜂 > 0 are constants, and 𝐼𝐺(𝑎, 𝑏), 𝑎 > 0, 𝑏 > 0 denotes the IG distribution with PDF 2

𝑓 (𝑦; 𝑎, 𝑏) = (

𝑏 (𝑦 − 𝑎) 𝑏 1/2 ), ) exp (− 3 2𝜋𝑦 2𝑎2 𝑦

(1)

where 𝑦 > 0 and CDF 𝐹 (𝑦; 𝑎, 𝑏) = Φ [√

We define the first passage time to the failure threshold 𝑙 as 𝑇 = inf {𝑡 ≥ 0 | 𝑋(𝑡) ≥ 𝑙}. Based on (2), then 𝑇 follows the IG distribution with CDF given by

𝑏 𝑦 ( − 1)] 𝑦 𝑎 (2)

2𝑏 𝑏 𝑦 + exp ( ) Φ [√ ( + 1)] , 𝑎 𝑦 𝑎 where 𝑦 > 0. 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 Λ(𝑡) as follows: Λ (𝑡, 𝑍𝑡 ) = 𝜓 (𝑍𝑡 ) (Λ (𝑡))ℎ(𝑍𝑡 ) ,

(3)

where 𝜓(𝑍𝑡 ) is nonnegative function such that 𝜓(𝑍0 ) = ℎ(𝑍0 ) = 1. For the sake of simplification, we assume ℎ(𝑍𝑡 ) = 1 and 𝜓(⋅) 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 𝑡, 𝑍𝑡 = 𝑧𝑡 , the shape function of the IG process is defined by Λ (𝑡, 𝑧𝑡 ) = Λ (𝑡) exp (∑𝛾𝑖 𝑍𝑖 (𝑡)) ,

(4)

𝑖

where 𝛾𝑖 is regression parameters defining the effects of the covariate of the IG process and 𝑍(𝑡) = (𝑍1 (𝑡), 𝑍2 (𝑡), ...) is a vector of time-dependent covariates. Let the constant 𝑙 (𝑙 > 0) be the failure threshold of the process {𝑋(𝑡), 𝑡 > 0}. We assume that the degradation follows an IG process model with function Λ(𝑡) and parameters 𝜇 and 𝜂. Since 𝑋(0) = 0, 𝑋(𝑡) yields 𝐼𝐺(𝜇Λ(𝑡), 𝜂Λ(𝑡)2 ) with mean 𝜇Λ(𝑡) and variance 𝜇3 Λ(𝑡)/𝜂.

= 1 − 𝐹 (𝑙; 𝜇Λ (𝑡) , 𝜂Λ (𝑡)2 ) 𝜂 𝑙 = Φ (√ (Λ (𝑡, 𝑧𝑡 ) − )) 𝑙 𝜇 − exp (

(5)

2𝜂Λ (𝑡, 𝑧𝑡 ) 𝜂 𝑙 ) Φ (−√ (Λ (𝑡, 𝑧𝑡 ) + )) , 𝜇 𝑙 𝜇

where parameters 𝜂 and 𝜇 are positive and Φ(⋅) 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 Λ(𝑡) is a linear function, i.e., Λ(𝑡) = 𝑡. (b) The external force can affect the degradation process of the mechanical component. Based on (4), Λ(𝑡) under the external force can be expressed by Λ (𝑡, 𝑧𝑡 ) = Λ (𝑡) exp (𝛾𝑍 (𝑡)) .

(6)

Let 𝑍(𝑡) denote the state of operating environment at time 𝑡, and {𝑍(𝑡), 𝑡 ≥ 0} take values in 𝐸 = {0, 1, 2, ..., 𝑟}. (c) Under the external force, the degradation path 𝑋(𝑡) 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., meantime-to-failure (MTTF), can be easily formulated by ∞



0

0

𝑇𝑙 = 𝐸 (𝑇) = ∫ (1 − 𝐹𝑙 (𝑡)) d𝑡 = ∫ 𝐹𝑙 (𝑡) d𝑡.

(7)

In addition, according to the nature of IG process, we denote 𝜃 = {𝜇, 𝜂, 𝛾} for the IG model parameters, and we ̂ 𝜂̂, 𝛾̂} by maximum obtain parameters estimation 𝜃̂ = {𝜇, likelihood estimation (MLE) at each time 𝑡𝑘 based on the CBCM data 𝑋0:𝑘. 3.2. RUL Estimation. Let 𝑋0:𝑘 = {𝑥0 , 𝑥1 , . . . , 𝑥𝑘 } denote the degradation observation at 0 = 𝑡0 < 𝑡1 < ⋅ ⋅ ⋅ < 𝑡𝑘 , which can be irregularly spaced, and 𝑥𝑖 = 𝑋(𝑡𝑖 ) represents the degradation measurement of the mechanical component at time 𝑡𝑖 . Therefore, using the first passage time of the degradation process, we define the RUL, abbreviated as 𝑅𝑘 that corresponds to 𝑅𝑘 = inf {𝑟𝑘 > 0 : 𝑋(𝑡𝑘 + 𝑟𝑘 ) ≥ 𝑙 | 𝑋0:𝑘}. Based on the definition of the RUL and condition on the degradation data 𝑋0:𝑘 , the distribution function of the RUL can be obtained by the following: 𝐹𝑅𝑘 (𝑟𝑘 ; 𝑋0:𝑘 ) = Pr {𝑋 (𝑡𝑘 + 𝑟𝑘 ) ≥ 𝑙 | 𝑋0:𝑘 } = Pr {𝑋 (𝑡𝑘 + 𝑟𝑘 ) − 𝑥𝑘 ≥ 𝑙 − 𝑥𝑘 }

4

Mathematical Problems in Engineering

Observed degradation data X 0:k at time t k

Parameters estimation k = {̂k , k , k }

RUL estimation

via MLE

Optimal spare ordering time

Availability threshold

∗ tko

Decision-making ∗ according to tko

Next inspection time

no

Order model (CBPO)

Random lead-time

Speed-up lead-time degree



tko = tk+1 ? yes End

Figure 1: Flow chart of the implementation process of order model at each time 𝑡𝑘 .

= Φ (√

𝜂 𝑙 − 𝑥𝑘 )) (Λ (r𝑘 , 𝑧𝑟𝑘 ) − 𝑙 − 𝑥𝑘 𝜇

− exp (

2𝜂Λ (r𝑘 , 𝑧𝑟𝑘 )

⋅ Φ (−√

𝜇

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

)

Step 2. Parameters estimation: estimate the degradation ̂ 𝜂̂, 𝛾̂} by MLE according to model parameters 𝜃̂ = {𝜇, the nature of IG process. The available estimation samples are obtained from the observed degradation data of the mechanical component till time 𝑡𝑘 , i.e., 𝑋0:𝑘 .

𝜂 𝑙 − 𝑥𝑘 )) . (Λ (r𝑘 , 𝑧𝑟𝑘 ) + 𝑙 − 𝑥𝑘 𝜇 (8)

For (8), the mean of 𝑅𝑘 can be easily obtained as follows: ∞

𝐸 (𝑅𝑘 ) = ∫ (1 − 𝐹𝑅𝑘 (𝑟𝑘 ; 𝑋0:𝑘)) d𝑡 0



(9)

= ∫ 𝐹𝑅𝑘 (𝑟𝑘 ; 𝑋0:𝑘 ) d𝑡, 0

where 𝐹𝑅𝑘 (𝑟𝑘 ; 𝑋0:𝑘 ) 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 3. RUL estimation: based on the real-time estimated ̂ 𝜂̂, 𝛾̂} in Step 2, we can derive the parameters 𝜃̂ = {𝜇, RUL distribution of the mechanical component at time 𝑡𝑘 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 𝑡0𝑘 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 𝑡0𝑘 is not equal to the next inspection time 𝑡𝑘+1 , 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

Mathematical Problems in Engineering

5 rk Lr

State 1

0

t Replacement

H T

tko

tk

rk Lr

State 2 0

tko

tk

T S

Speed-up

t Replacement

Le

rk

S State 3 0

tk

T tko

t Replacement

Figure 2: Possible order-replacement states of one cycle.

𝑃𝑠2 = Pr {𝑡𝑜𝑘 + 𝐿 𝑟 > 𝑡𝑘 + 𝑟𝑘 and 0 < 𝑡𝑜𝑘 − 𝑡𝑘 < 𝑟𝑘 }

the flow chart of the implementation process of order model at each time 𝑡𝑘 . 4.2. Order Policy. The order policy is made up of regular order policy and expedited order policy. When the scheduled spare ordering time occurs before the component failure time, regular order policy is triggered. When the scheduled spare ordering time occurs after the component failure time, expedited order policy is triggered. The detailed order policy for the component is depicted in Figure 2, where 𝑟𝑘 is a remaining time, 𝑇 is the component failure time, 𝐿 𝑟 is regular lead-time, 𝐿 𝑒 is expedited lead-time, 𝑡𝑘 is the current inspection time, 𝑡𝑜𝑘 is the scheduled spare ordering time, 𝐻 is the holding time, and 𝑆 is the shortage time. State 1. If the failed component occurs after 𝑡𝑜𝑘 and the spare is delivered before the component failure, the failed component will not be replaced until the residual lifetime is equal to zero. Let 𝑃𝑠1 denote the probability of State 1 and corresponding CDF can be shown as

=∫

𝑡𝑜𝑘 −𝑡𝑘

𝑡𝑘 +𝑢−𝑡𝑜𝑘



0

d𝑊 (𝑡) d𝐹𝑅𝑘 (𝑢) ,



𝑡𝑜𝑘 −𝑡𝑘





𝑡𝑘 +𝑢−𝑡𝑜𝑘

d𝑊 (𝑡) d𝐹𝑅𝑘 (𝑢) ,

(11)

where 𝑡, 𝑢 denote integral variable, respectively. State 3. The mechanical component failure occurs before 𝑡𝑜𝑘 ; an expedited order is made and the failed component is replaced by the spare as soon as the spare is delivered. Let 𝑃𝑠3 denote the probability of State 3 and corresponding CDF can be shown as 𝑃𝑠3 =

Pr {𝑡𝑜𝑘



𝑡𝑜𝑘 −𝑡𝑘

0

0

> 𝑡𝑘 + 𝑟𝑘 } = ∫ ∫

d𝐹𝑅𝑘 (𝑡) d𝐻 (𝑢) ,

(12)

where 𝑡, 𝑢 denote integral variable, respectively. 4.3. Model of Objective Function. The main goal of this section is to build the expected cost rate model under lower availability threshold constraint. 4.3.1. Cost Rate Model

𝑃𝑠1 = Pr {𝑡𝑜𝑘 + 𝐿 𝑟 < 𝑡𝑘 + 𝑟𝑘 and 0 < 𝑡𝑜𝑘 − 𝑡𝑘 < 𝑟𝑘 } ∞

=∫

(10)

(1) The Expected Replacement Cycle Length. As shown in Figure 2 and formulation of the order-replacement states, a replacement cycle can be formulated as

where 𝑡, 𝑢 denote integral variable, respectively.

𝑈

State 2. The failed component occurs after 𝑡𝑜𝑘 and the spare is not delivered when the component failure occurs. To reduce the wait time after the component failure, we speed up the lead-time. The failed component is replaced by the spare as soon as the spare is delivered. Let 𝑃𝑠2 denote the probability of State 2 and corresponding CDF can be shown as

{𝑡𝑘 + 𝑟𝑘 , { { { = {𝑡𝑜𝑘 + 𝐿 𝑟 , { { {𝑜 {𝑡𝑘 + 𝐿 𝑒 ,

if 𝑡𝑜𝑘 + 𝐿 𝑟 < 𝑡𝑘 + 𝑟𝑘 and 0 < 𝑡𝑜𝑘 − 𝑡𝑘 < 𝑟𝑘 , if 𝑡𝑜𝑘 + 𝐿 𝑟 > 𝑡𝑘 + 𝑟𝑘 and 0 < 𝑡𝑜𝑘 − 𝑡𝑘 < 𝑟𝑘 , if 𝑡𝑜𝑘 > 𝑡𝑘 + 𝑟𝑘 ,

where 𝑈 is a replacement cycle length.

(13)

6

Mathematical Problems in Engineering

Since holding time only occurs in State 1, the expected holding time 𝐸𝐻 during the replacement cycle is 𝐸𝐻 = 𝐸 [𝑡𝑘 + 𝑟𝑘 − 𝑡𝑜𝑘 − 𝐿 𝑟 ] =∫



𝑡𝑜𝑘 −𝑡𝑘



𝑡𝑘 +𝑢−𝑡𝑜𝑘

0

(𝑡𝑘 + 𝑢 − 𝑡𝑜𝑘 − 𝑡) d𝑊 (𝑡) d𝐹𝑅𝑘 (𝑢) ,

(14)

where 𝑡, 𝑢 denote integral variable, respectively. Furthermore, since the shortage time may occur in State 2 and State 3, the expected shortage time 𝐸𝑆 during the replacement cycle is 𝐸𝑆 =

𝐸 [𝑡𝑜𝑘 + 𝐿 𝑟 − 𝑡𝑘 − 𝑟𝑘 ] + 𝐸 [𝐿 𝑒 ] 1+𝜅 ∞

=



∫𝑡𝑜 −𝑡 ∫𝑡 +𝑢−𝑡𝑜 (𝑡𝑜𝑘 + 𝑡 − 𝑡𝑘 − 𝑢) d𝑊 (𝑡) d𝐹𝑅𝑘 (𝑢) 𝑘

𝑘

𝑘

𝑘

1+𝜅

(15)

4.3.2. Availability. Availability is an important performance index of repairable components and hereafter denoted by 𝐴. Based on key renewal theorem in Ross [41], 𝐴 can be defined to be such that 𝐴 (𝑡𝑜𝑘 ) =

𝑇𝑙 𝐸 (run time) = 𝐸 (run time) + 𝐸 (down time) 𝑇𝑙 + 𝐸𝑆

1 = 1 + 𝐸𝑆/𝑇𝑙

where 𝐸(run time) and 𝐸(down time) correspond to the expected run time and down time. 𝐸(run time) is equal to 𝑇𝑙 and 𝐸(down time) is equal to 𝐸𝑆. Based on (19), we can find that 𝐴 decreases as 𝐸𝑆 increases. 4.3.3. Cost Rate Model Subject to Availability. Consider the availability as a constraint attached to the cost rate model. Modifying (18), the cost rate model subject to availability is as follows:



+ ∫ 𝑢𝐹𝑅𝑘 (𝑡𝑜𝑘 − 𝑡𝑘 ) ℎ (𝑢) d𝑢, 0 ≤ 𝜅 < ∞

𝐶𝑅 (𝑡𝑜𝑘 ) =

0

where 𝑡, 𝑢 denote integral variable, respectively, and 𝜅 denotes speed-up lead-time degree. Based on the above analyses and using expression (13), the expected replacement cycle length can be formulated as ∞

𝐸𝑈 (𝑡𝑜𝑘 ) = 𝑡𝑘 + ∫ 𝐹𝑅𝑘 (𝑟𝑘 ; 𝑋0:𝑘 ) d𝑡 + 𝐸𝑆 = 𝑇𝑓 + 𝐸𝑆. (16) 0

In other words, the expected replacement cycle length can be denoted by the expected failure lifetime and the expected shortage time. (2) The Expected Replacement Cycle Cost. As shown in Figure 2, the expected cost per replacement cycle can be derived, which includes inspection cost 𝑘𝜌𝑖 , replacement cost 𝐶, regular ordering cost 𝐶𝑟 , regular delivery cost 𝐶𝑟𝑑 , expedited ordering cost 𝐶𝑒 , expedited delivery cost 𝐶𝑒𝑑 , speed-up lead-time cost 𝜅𝐶𝑒𝑑 , holding cost 𝜌ℎ 𝐸𝐻, shortage cost 𝜌𝑠 𝐸𝑆, and salvage value 𝐶𝑠V , where 𝐶𝑒 > 𝐶𝑟 , 𝐶𝑒𝑑 > 𝐶𝑟𝑑 . Therefore, by (10), (11), (12), (14), and (15), the expected cost per replacement cycle 𝐸𝑉(𝑡𝑜𝑘 ) can be expressed as 𝐸𝑉 (𝑡𝑜𝑘 ) = 𝑘 ⋅ 𝜌𝑖 + 𝐶 + 𝐶𝑟 ⋅ (𝑃𝑠1 + 𝑃𝑠2 ) + 𝐶𝑒 ⋅ 𝑃𝑠3 + 𝐶𝑟𝑑 ⋅ 𝑃𝑠1 + (𝐶𝑟𝑑 + 𝜅 ⋅ 𝐶𝑒𝑑 ) ⋅ 𝑃𝑠2 + 𝐶𝑒𝑑 ⋅ 𝑃𝑠3 + 𝜌ℎ (17) ⋅ 𝐸𝐻 + 𝜌𝑠 ⋅ 𝐸𝑆 − 𝐶𝑠V . (3) Cost Rate Model. Therefore, based on the renewal-reward theory in Ross [41], the expected cost rate is obtained as 𝐶𝑅 (𝑡𝑜𝑘 ) =

𝐸𝑉 (𝑡𝑜𝑘 ) , 𝐸𝑈 (𝑡𝑜𝑘 )

(18)

where 𝑡𝑘 < 𝑡𝑜𝑘 , 𝐸𝑈(𝑡𝑜𝑘 ) and 𝐸𝑉(𝑡𝑜𝑘 ) are, respectively, given by (16) and (17).

(19)

s.t.

𝐸𝑉 (𝑡𝑜𝑘 ) , 𝐸𝑈 (𝑡𝑜𝑘 )

(20)

𝐴 ≥ 𝐴∗

where 𝐴∗ is a predefined lower availability threshold. Based ∗ on (20), the optimal spare ordering time 𝑡𝑜𝑘 can be obtained at each inspection time 𝑡𝑘 simultaneously. What we aim is to ∗ seek 𝑡𝑜𝑘 by minimizing the expected cost rate under lower availability threshold.

5. Case Study 5.1. An Application on CJK1630 Spindle. The spindle is the core component of a computer numerical control (CNC) machine tool, and its running state directly affects the overall quality of the machine tool. The spindle rotation accuracy is one of the important parameters and directly affects the cylindricity of machining parts, axial dimension precision, face shape precision, and surface roughness. The degradation level of the spindle rotation accuracy increases monotonically with the run time. Considering the good properties of the IG process, we describe the degradation process of spindle rotation accuracy by the IG process. The external environment, such as external force, can accelerate the degradation of the spindle rotation accuracy. Therefore, the degradation process of the spindle rotation accuracy can be reasonably described by IG process with covariate effect. We can predict the RUL of the spindle through the available degradation data, which is collected by monitoring the health condition of the spindle in real-time. The mechanical spindle, in practice, is commonly expensive, complex, and highly reliable. Meanwhile, its lead-time is usually not predictable. In order to ensure the completion of the production tasks on time, the availability of the spindle does not fall below the predefined threshold. Above all else, we choose the CJK1630, a CNC lathe, as the subject for the experiment and apply the CBPO to a case study of the mechanical spindle. In order to test the health condition of the mechanical spindle, a CBCM

Mathematical Problems in Engineering

7 Table 1: CJK1630 spindle parameters.

Speed (r/min) 1400

Heat-balance temperature (∘ C) 55

Radial force (N) 140 Table 2: The degradation data with time variation. Mean radial error (𝜇𝑚) 11.8 11.9 12.1 12.3 12.4 12.7 12.9 13.0 13.2 13.6 13.8 13.9 14.1 14.2 14.6 14.8 15.0 15.4 15.5 15.6 16.0

CBCM time (𝑑) 21 22 23 24 25 26 27 28 29 30 40 45 50 55 60 65 70 75 80 85 90

Mean radial error (m)

CBCM time (𝑑) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

32 30 28 26 24 22 20 18 16 14 12

Mean radial error (𝜇𝑚) 16.2 16.3 16.5 16.6 16.8 17.0 17.2 17.3 17.5 17.8 19.8 20.8 21.8 22.8 23.7 24.8 25.6 26.6 27.7 28.6 29.8

failure threshold failure time

degradation path 0

10

20

30

40

50

60

70

80

90

The condition monitoring time (d)

Figure 4: The degradation path of the mechanical spindle. Figure 3: The test platform of the mechanical spindle.

process was carried out by a set of dedicated test platforms each day. The test platform of the mechanical spindle is shown in Figure 3 and the corresponding test parameters are shown in Table 1. The inspection times are taken from zero to 90 with the measured frequency Δ𝑡 = 1, and the failure threshold is chosen to be 𝑙 = 29.8 with the corresponding lifetime 𝑇 = 90. The degradation data of the spindle rotating accuracy with time variation is shown in Table 2 and the corresponding degradation path of the spindle is shown in

Figure 4. In order to test the degradation of the spindle rotating accuracy under radial force, we consider that the covariate is 𝑍(𝑡𝑖 ) = {0, 1}; i.e., 𝑍(𝑡𝑖 ) = 0 indicates that at time 𝑡𝑖 the environment is “normal” and 𝑍(𝑡𝑖 ) = 1 indicates that at time 𝑡𝑖 the environment is “external force”, where the classification criterion is according to the experience of the experts. With the approach presented in Section 3.1 and observed degradation data, the estimated degradation model paramê 𝜂̂, 𝛾̂} can be obtained by MLE and partial results ters 𝜃̂ = {𝜇, are selected to show in Table 3.

Mathematical Problems in Engineering

The PDF of the lead-time

8 0.3 0.25

The mean of the expedited lead-time

0.2 The mean of the regular lead-time

0.15 0.1 0.05 0

0

5

10

15

20

25

30

35

40

The lead-time (d) The PDF of the expedited lead-time The PDF of the regular lead-time

Figure 5: The PDF of the lead-time. Condition monitoring tk =

1

Lower availability threshold (!∗=0.9700)

0.125

The optimal spare ordering time (Nok =61,CR=0.1203)

0.9

The spare ordering time under minimum cost rate (Nok =64, CR=0.1200)

0.12 45

50

55 60 65 70 The scheduled spare ordering time Nok

The expected availability

The expected cost rate

0.13

0.8 75

Figure 6: The decision-making result of the 45th condition monitoring time by the CBPO.

Table 3: Partial parameter estimation results for CJK1630 spindle. 𝑡𝑘 30 35 40 45 50 55

𝜇̂𝑘 0.0157 0.0157 0.0156 0.0156 0.0158 0.0157

𝜂̂𝑘 0.0275 0.0276 0.0278 0.0279 0.0274 0.0276

𝛾̂𝑘 1.9820 1.9764 1.9722 1.9690 1.9876 1.9786

For validating the effectiveness and applicability of the CBPO, the cost parameters are provided in Table 4. Based on historical order practice of the spindle, we suggest the lognormal distribution be used as the distribution of the leadtime in this case study. The PDF of the expedited lead-time can be written as ℎ(𝑡) = exp{−(ln 𝑡 − 𝜇ℎ )2 /2𝜎ℎ 2 }/𝑡𝜎ℎ √2𝜋, where 𝜇ℎ = 2.7, 𝜎ℎ = 0.1 and the mean of the expedited leadtime is 𝜇𝑒 = 15. Similarly, the PDF of the regular lead-time can be written as 𝑤(𝑡) = exp{−(ln 𝑡 − 𝜇𝑤 )2 /2𝜎𝑤 2 }/𝑡𝜎𝑤 √2𝜋, where 𝜇𝑤 = 3.4, 𝜎𝑤 = 0.1 and the mean of the regular lead-time is 𝜇𝑒 = 30. We plot ℎ(𝑡) and 𝑤(𝑡) in Figure 5 for illustration, respectively. Note that ℎ(𝑡) and 𝑤(𝑡) used here are obtained by consulting with supplier of the mechanical spindle. Next, we provide and discuss the decision-making results for the mechanical spindle in the case study. Specifically,

decision-making results at each time 𝑡𝑘 mainly include the ∗ optimal spare ordering time 𝑡𝑜𝑘 and corresponding expected ∗ cost rate 𝐶𝑅(𝑡𝑜𝑘 ). The decision-making results of the last 10 CBCM times by the CBPO are shown in Table 5 and the 45th data is selected to plot in Figure 6. From Table 5 and Figure 6, two observations can be made. First, the decision-making result can be real-time updated at each CBCM time. In other words, the CBPO is implemented dynamically according to the real-time health condition of the mechanical spindle. Second, the decision-making process is stopped at time 𝑡𝑜𝑘 = 61 since the optimal spare ordering time is equal to the next inspection time 𝑡𝑘 = 61. We will also find that the spare will ∗ be ordered at the optimal spare ordering time 𝑡𝑜𝑘 = 61. 5.2. Sensitivity Analysis on Critical Parameters. In this section, successive sensitivity analysis will be made for 𝜎𝑟 , 𝜎𝑒 , 𝜌𝑠 , 𝜌ℎ , 𝜅 to explore their influences on the decisionmaking results. We take the 45th CBCM time as an example, i.e., 𝑡𝑘 = 45. The results are summarized separately in Figures 7–9. Based on these experimental results, the following observation can be drawn. (1) In Figure 7, we can find that the optimal spare ∗ ordering time 𝑡𝑜𝑘 deceases a little as the standard deviation 𝜎𝑟 increases. The reason for this may lie in that the probabilities that the ordered spare can be delivered earlier or later than its mean lead-time will be along with the variation of the standard deviation and these probabilities are highly

Mathematical Problems in Engineering

9 Table 4: Cost parameters in the case study.

𝐶 10

𝐶𝑟 0.3

𝐶𝑒 1

𝐶𝑟𝑑 0.1

𝐶𝑒𝑑 0.2

𝜌𝑖 0.01

𝜌ℎ 0.05

𝜌𝑠 0.13

𝐶𝑠V 0.1

Table 5: Decision-making results at the last 10 CBCM times under availability threshold (𝐴∗ = 0.9700). 𝑡𝑘 ∗ 𝑡𝑜𝑘 ∗ 𝐶𝑅 (𝑡𝑜𝑘 )

45 61 0.1203

46 61 0.1204

47 61 0.1204

48 61 0.1206

49 61 0.1207

Condition monitoring tk =

66

66 65

64

64

51 60 0.1209

52 60 0.1210

53 61 0.1210

54 61 0.1212

Condition monitoring tk =



65

The optimal spare ordering time t ko



The optimal spare ordering time t ko

50 61 0.1208

63 62 61 60 59 58

62 61 60 59 58 57

57 56

63

3

3.5

4

4.5

5

5.5

6

Standard deviation of the regular lead-time L

56 1.5

2 2.5 3 3.5 4 Standard deviation of the expedited lead-time e

4.5

Figure 7: Sensitivity of optimal spare ordering time on 𝜎𝑟 and 𝜎𝑒 .

correlated with the expected shortage time and holding time among the mechanical spindle’s whole life. In the present ∗ case, the optimal spare ordering time 𝑡𝑜𝑘 decreases since the shortage cost is larger than holding cost. Meanwhile, the ∗ optimal spare ordering time 𝑡𝑜𝑘 is almost unchanged as the standard deviation 𝜎𝑒 increases. The reason is simply that when the inspection time is equal to 𝑡𝑘 = 45, the probability of State 3 is almost zero. (2) In Figure 8, we can find that the optimal spare ∗ ordering time 𝑡𝑜𝑘 decreases as the shortage cost per unit time 𝜌𝑠 increases. The reason is that a spare unit for replacement should be ordered more early to avoid machine shut down when the mechanical component has a high shortage cost. ∗ However, the optimal spare ordering time 𝑡𝑜𝑘 is almost unchanged as the holding cost per unit time 𝜌ℎ increases. This is also reasonable because 𝑡𝑜𝑘 increases under minimum cost rate as 𝜌ℎ increases while corresponding availability decreases. In order to make the availability not below the predefined threshold, one should place an order earlier. Therefore, the final result shows no change.

(3) In Figure 9, we can find that the optimal spare ∗ ordering time 𝑡𝑜𝑘 first increases and then decreases as speedup lead-time degree 𝜅 increases. Before the inflection point, as 𝜅 increases, the shortage time decreases and the availability increases. Since the speed-up cost is less than the cost of ∗ shortening the shortage time, 𝑡𝑜𝑘 needs to be moved backward. After the inflection point, with the increase of 𝜅, the shortage time further decreases and the availability further increases. Because the speed-up cost at this time has less and ∗ less impact on shortening the shortage time, 𝑡𝑜𝑘 needs to be moved forward. All the experimental results are intuitive and match our expectations.

6. Conclusions and Future Research In this paper, CBPO is proposed for a mechanical component subject to IG degradation process with covariate effect. The CBPO, depending on RUL, random lead-time, speed-up lead-time degree, and availability threshold, can be realtime determined and make an accurate prediction for spare

10

Mathematical Problems in Engineering Condition monitoring tk =

61

Condition monitoring tk = 65 64

The optimal spare ordering time t ko

The optimal spare ordering time t ko





60

59

58

57

56

55

63 62 61 60 59 58

54 0.1

0.2 0.3 0.4 Shortage cost per unit time M

57 0.04

0.5

0.06

0.08 0.1 0.12 0.14 0.16 Holding cost per unit time B

0.18

0.2

The optimal spare ordering time t ko



Figure 8: Sensitivity of optimal spare ordering time on 𝜌𝑠 and 𝜌ℎ .

65 64 63 62 61 60 59 58 57

Condition monitoring tk =

0

0.2

0.4

0.6 0.8 1 1.2 1.4 Speed-up lead-time degree 

1.6

1.8

2

Figure 9: Sensitivity of optimal spare ordering time on 𝜅.

ordering time according to the actual health condition of the mechanical component in operation. The CBPO is illustrated as feasible and practical through a CNC lathe mechanical spindle case study. The sensitivity analysis results are intuitive and match our expectations. By using this model, the optimal spare ordering time with minimum expected cost rate under the lower availability threshold constraint can be calculated easily and effectively for enterprises. Inspection accuracy of the degradation characteristics for the mechanical components plays an important role in the CBPO. Engineers or managers should pay greater attention to improving the inspection accuracy. The higher inspection accuracy of the degradation characteristics for the mechanical components, the higher the prediction accuracy of spare ordering time. On the other hand, with reasonable speed-up lead-time degree, not only can the availability be improved, but maintenance costs can also be reduced. The approach proposed in this paper can provide guidance for

some systems beyond mechanical components, which are subject to IG degradation process or other degradation processes. The approach can prompt reasonable arrangements for production, improve product quality and service quality, reduce maintenance costs, and enhance market competitiveness. Although the CBPO can provide satisfactory results, there are still some research points that need to be investigated further in the future and future research may focus on the following topics. (a) Multi-spare ordering model based on RUL for systems following stochastic degradation process. (b) The CBPO under variable working condition that is challenging and worthwhile. (c) Spare ordering model based on the joint effect of aging and cumulative degradation, which is more practical.

Notations and Nomenclatures PM: 𝜇𝑒 : CBM: 𝜎𝑒 : TBM: 𝑤(𝑡): CBCM: 𝑊(𝑡): IoT: 𝜇𝑟 : RUL: 𝜎𝑟 : IG: 𝐶𝑟 : CBPO:

Preventive maintenance Finite mean of the expedited lead-time Condition-based maintenance Standard deviation of the expedited lead-time Time-based maintenance PDF of the regular lead-time Computer-based condition monitoring CDF of the regular lead-time Internet of Things Finite mean of the regular lead-time Remaining useful life Standard deviation of the regular lead-time Inverse Gaussian Regular spare ordering cost Condition-based predictive order model

Mathematical Problems in Engineering 𝐶𝑒 : PDF: 𝐶𝑟𝑑 : CDF: 𝐶𝑒𝑑 : PHM: 𝜌ℎ : MTTF: 𝜌𝑠 : MLE: 𝜌𝑖 : CNC: 𝐶: 𝐶𝑠V : 𝑘: 𝐿 𝑟: 𝑡𝑘 : 𝐿 𝑒: 𝑟𝑘 : 𝐸𝑈: 𝑡𝑜𝑘 : 𝐸𝑉: ∗ 𝑡𝑜𝑘 : 𝐶𝑅: 𝑇: 𝐴: 𝑇𝑙 : 𝐴∗ : ℎ(𝑡): 𝜅: 𝐻(𝑡): 𝑙:

Expedited spare ordering cost Probability density function Regular delivery cost Cumulative distribution function Expedited delivery cost Proportional hazards model Holding cost per unit time Mean-time-to-failure Shortage cost per unit time Maximum likelihood estimation Inspection cost for each time Computer numerical control Replacement cost Salvage value when 𝑟𝑘 = 0 Inspection times Regular lead-time Current inspection time Expedited lead-time Remaining time Expected replacement cycle length Scheduled spare ordering time Expected cost per replacement cycle Optimal spare ordering time Expected cost rate Failure time Availability Mean of lifetime Availability threshold PDF of the expedited lead-time Speed-up lead-time degree CDF of the expedited lead-time Predetermined failure threshold.

Data Availability The data used to support the findings of this study are included within the article.

Conflicts of Interest The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments This research is supported by the National Natural Science Foundation of China (Grant No. 51575055) and the National Science and Technology Major Project of China (Grant No. 2015ZX04001002).

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