# Optimization of Aeroengine Shop Visit Decisions Based on Remaining Useful Life and Stochastic Repair Time

Authors: Jing Cai; Xin Li; Xi Chen
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

Considering the wide application of condition-based maintenance in aeroengine maintenance practice, it becomes possible for aeroengines to carry out their preventive maintenance in just-in-time (JIT) manner by reasonably planning their shop visits (SVs). In this study, an approach is proposed to make aeroengine SV decisions following the concept of JIT. Firstly, a state space model (SSM) for aeroengine based on exhaust gas temperature margin is developed to predict the remaining useful life (RUL) of aeroengine. Secondly, the effect of SV decisions on risk and service level (SL) is analyzed, and an optimization of the aeroengine SV decisions based on RUL and stochastic repair time is performed to carry out JIT manner with the requirement of safety and SL. Finally, a case study considering two CFM-56 aeroengines is presented to demonstrate the proposed approach. The results show that predictive accuracy of RUL with SSM is higher than with linear regression, and the process of SV decisions is simple and feasible for airlines to improve the inventory management level of their aeroengines.


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## Body

## 1. Introduction

Both purchase cost and maintenance cost of civil aeroengines are very expensive. For example, CFM56-5 is about 4.5 million dollars, and its shop visit (SV) cost is up to one million dollars per visit [1, 2]. According to the definition given by the world airlines technical operations glossary (WATOG), an aeroengine removal is classified as an SV whenever the subsequent engine maintenance performed prior to reinstallation entails separation of pairs of major mating flanges or removal of a disk, hub, or spool. SV is due to two main reasons: (a) ultralimit of exhaust gas temperature margin (EGTM) or (b) repair/replacement of life limited part (LLP). EGTM is the most important measure of aeroengine health, and once EGTM has dropped to the threshold value (usually 0), which means that there is a potential for failure, an SV is required to restore the aeroengine. A spare aeroengine is needed when an SV occurs, so the airlines try to use as few spare aeroengines as possible by decreasing utilization of installed aeroengines in order to avoid that several SVs occur simultaneously [3]. In fact, one spare aeroengine is sufficient if there is no overlap of SVs, which is similar to the concept of just-in-time (JIT) if a repaired aeroengine back from shop is regarded as a spare aeroengine. With the development of condition-based maintenance (CBM) in aeroengine maintenance practice, it is now possible to implement the concept of JIT through reasonably planning the aeroengine SVs by considering an essential condition that the previous aeroengine will return from the shop before the next aeroengine needs a spare aeroengine. Therefore, the repair time (i.e., SV duration) and remaining useful life (RUL) are key factors for an SV decision. Because repair time is always stochastic due to differences in repair range, repair level, transport conditions, and so on, the aeroengine SV decision will be optimized based on both RUL and stochastic repair time in this study.There has been a lot of research on RUL prediction [4–11], but very few studies considering RUL prediction of aeroengines have appeared in the literatures. A hybrid PSO-SVM-based model for the RUL prediction of aeroengines was described [12]. An approach is presented based on the so-called shapelet extraction to estimate the RUL of turbofan engines [13]. EGTM is taken as a measure of aeroengines health to predict the RUL of aeroengines [9, 14–16]. There are many papers dealing with JIT inventory management, but JIT for an aeroengine SV has not been considered. In order to reasonably plan aeroengine SVs and smooth the SV rate, the performance ranking of aeroengines has been researched [9, 17–20]. On the basis of the performance ranking, an aeroengine maintenance cost was developed for a scheme of aeroengine fleet with different quantity of spare aeroengines in a finite period of time [21]. Further, SV prediction of aeroengines was focused on and the take-off exhaust gas temperature was considered for SVs decisions [22]. An approach was proposed for aeroengine preventive repair based on an LLP life time distribution [23]. Recently, a graphical technique was presented [24], which considers a rule for decision-making based on both condition-based reliability function and a stochastic/fixed lead time. However, no papers have been published considering the aeroengine SV decisions based on the concept of JIT. Therefore, in this study, according to the concept of JIT, the RUL of aeroengine would not only be predicted but also be applied to make SV decisions to avoid the overlap of SVs and reduce the number of spare aeroengines; furthermore, stochastic repair time of aeroengine is also considered as a major factor during making SV decisions.Obviously, SV ahead of schedule leads to life utilization loss of an aeroengine, but, on the other hand, postponing SV may result in failure and expensive downtime. Therefore, the effect of SV decision on risk and service level (SL) should be analyzed, and then aeroengine SV decisions should be optimized based on RUL and stochastic repair time, in order to minimize the number of spare aeroengines and to maintain required safety and SL. The rest of this paper is organized as follows: a state space model (SSM) for RUL prediction is developed firstly; then the optimization of the aeroengine SV decisions is presented. Finally, a case study using real data is developed, and the conclusions from the work presented in this study and suggestions for future research are given.


## 2. RUL Prediction of Aeroengine

EGTM is a kind of time series data which measures performance deterioration of aeroengine, and the SSM approach offers a very general and powerful framework to provide best estimates and forecast performance trends by modeling such data. An SSM based on EGTM data will be developed in this study and used for the RUL prediction.


### 2.1. State Space Model

EGTM is defined as(1)EGTM=EGTRED-EGTE,where EGTRED is the red line temperature of aeroengine provided by manufacturers and EGTE is the exhaust gas temperature of aeroengine with full power take-off in standard conditions (sea-level pressure and turnover temperature).Given the observations up to timeT flight cycles (FC), the observed and actual EGTM value at time t (t=1,2,…,T) are, respectively, represented by yt and xt. Because there is the observation noise, yt can be expressed by the following equation:(2)yt=xt+εt.It is supposed that observation noise has Gaussian distribution, that is,εt~N(0,σε2), where variance σε2 needs to be estimated. Throughout the whole life cycle of an aeroengine, EGTM has characteristic of piecewise linear degradation, that is, at the beginning of putting into operation, the degradation of EGTM is fast because of initial wear, and, after a period of time, the degradation of EGTM gradually slows down. So the degradation rate of EGTM changes over time, and a piecewise linear degradation model is suitable for describing the degradation of EGTM [25]. Therefore, a linear growth model is adopted as state equations to describe the aeroengine degradation path [26]:(3)xt=xt-1+βt-1+υxυx~N0,σx2,βt=βt-1+υβυβ~N0,σβ2,where βt is the rate of change for EGTM degradation; υx and υβ are the process noises with Gaussian distributions, where variances σx2 and σβ2 need to be estimated. Using (2)-(3), a Gaussian linear SSM for describing aeroengine performance degradation can be obtained as follows:(4)yt=FtHt+εtεt~N0,σε2Ht=GtHt-1+wtwt~N0,Σ,where Ht=xtβt, Σ=σx200σβ2, Gt=1101, Ft=10.


### 2.2. Bayesian State Estimation and Prediction

It is assumed that initial informationH0∣D0 is subject to the distribution N(m0,C0), where D0 is the initial prior information, m0∈R2 and C0∈R2×2. At time t, the available information set is Dt=yt,Dt-1. Given that the posterior probability distribution of Ht-1∣Dt-1 is N(mt-1,Ct-1) at time t-1, so that the prior probability distribution of Ht∣Dt-1 can be expressed by [26](5)Ht∣Dt-1~Nat,Rt,where at=Gtmt-1, Rt=GtCt-1Gt′+Σ.A one-step predictive observed EGTMYt is given by(6)Yt∣Dt-1~Nft,Qt,where ft=Ft′at, Qt=FtRt-1Ft′+σε2.Hence, the posterior probability distribution ofHt∣Dt at time t is given by(7)Ht∣Dt~Nmt,Ct,where(8)mt=at+Atet,Ct=Rt-AtQtAt′,At=RtFtQt-1,et=yt-ft.Based on the above method, up to operation timeT, HT∣DT~N(mT,CT) can be obtained. It is assumed that mT=μxμβ, CT=∑xx∑βx∑βx∑ββ, and the k-step predictive HT+k is given using the following equations:(9)HT+k∣DT=xT+kβT+kDT~NaTk,RTk.a T ( k ) and RT(k) can be recursively calculated by(10)aTk=GT+kaTk-1RTk=GT+kRTk-1GT+k′+Σ.Based on (5) and (9), k-step predictive EGTM value xT+k∣DT~N(μT(k),σT2(k)) can be obtained, where μT(k)=μx+k·μβ, σT2(k)=∑xx+k2∑ββ+2k∑βx+kσx+(k-1k2k-1/6)σβ.Hence the probability density function (pdf) ofxT+k can be expressed as(11)πxT+k=12πσTkexp⁡-xT+k-μTk22σT2k.


### 2.3. RUL Distribution Function

When EGTM drops to the threshold valuexF, there is a potential for failure, and preventive maintenance needs to be performed for the aeroengine. So at time T, the cumulative distribution function (CDF) and pdf of the k-step predictive RUL can be expressed, respectively, by(12)FTk=∫-∞xFπxT+kdxT+k=ϕxF-μTkσTk=ϕxF-μx-k·μβ∑xx+k2∑ββ+2k∑βx+kσx+k-1k2k-1/6σβ(13)fTk=∂FTk∂k=18πexp⁡-xF-μx-kμβ22σT2kσT2k′μx+kμβ-xF-2μβσT2kσT3k,where σT2k′=2k∑ββ+2∑βx+σx+(6k2-6k+1/6)σβ.


## 2.1. State Space Model

EGTM is defined as(1)EGTM=EGTRED-EGTE,where EGTRED is the red line temperature of aeroengine provided by manufacturers and EGTE is the exhaust gas temperature of aeroengine with full power take-off in standard conditions (sea-level pressure and turnover temperature).Given the observations up to timeT flight cycles (FC), the observed and actual EGTM value at time t (t=1,2,…,T) are, respectively, represented by yt and xt. Because there is the observation noise, yt can be expressed by the following equation:(2)yt=xt+εt.It is supposed that observation noise has Gaussian distribution, that is,εt~N(0,σε2), where variance σε2 needs to be estimated. Throughout the whole life cycle of an aeroengine, EGTM has characteristic of piecewise linear degradation, that is, at the beginning of putting into operation, the degradation of EGTM is fast because of initial wear, and, after a period of time, the degradation of EGTM gradually slows down. So the degradation rate of EGTM changes over time, and a piecewise linear degradation model is suitable for describing the degradation of EGTM [25]. Therefore, a linear growth model is adopted as state equations to describe the aeroengine degradation path [26]:(3)xt=xt-1+βt-1+υxυx~N0,σx2,βt=βt-1+υβυβ~N0,σβ2,where βt is the rate of change for EGTM degradation; υx and υβ are the process noises with Gaussian distributions, where variances σx2 and σβ2 need to be estimated. Using (2)-(3), a Gaussian linear SSM for describing aeroengine performance degradation can be obtained as follows:(4)yt=FtHt+εtεt~N0,σε2Ht=GtHt-1+wtwt~N0,Σ,where Ht=xtβt, Σ=σx200σβ2, Gt=1101, Ft=10.


## 2.2. Bayesian State Estimation and Prediction

It is assumed that initial informationH0∣D0 is subject to the distribution N(m0,C0), where D0 is the initial prior information, m0∈R2 and C0∈R2×2. At time t, the available information set is Dt=yt,Dt-1. Given that the posterior probability distribution of Ht-1∣Dt-1 is N(mt-1,Ct-1) at time t-1, so that the prior probability distribution of Ht∣Dt-1 can be expressed by [26](5)Ht∣Dt-1~Nat,Rt,where at=Gtmt-1, Rt=GtCt-1Gt′+Σ.A one-step predictive observed EGTMYt is given by(6)Yt∣Dt-1~Nft,Qt,where ft=Ft′at, Qt=FtRt-1Ft′+σε2.Hence, the posterior probability distribution ofHt∣Dt at time t is given by(7)Ht∣Dt~Nmt,Ct,where(8)mt=at+Atet,Ct=Rt-AtQtAt′,At=RtFtQt-1,et=yt-ft.Based on the above method, up to operation timeT, HT∣DT~N(mT,CT) can be obtained. It is assumed that mT=μxμβ, CT=∑xx∑βx∑βx∑ββ, and the k-step predictive HT+k is given using the following equations:(9)HT+k∣DT=xT+kβT+kDT~NaTk,RTk.a T ( k ) and RT(k) can be recursively calculated by(10)aTk=GT+kaTk-1RTk=GT+kRTk-1GT+k′+Σ.Based on (5) and (9), k-step predictive EGTM value xT+k∣DT~N(μT(k),σT2(k)) can be obtained, where μT(k)=μx+k·μβ, σT2(k)=∑xx+k2∑ββ+2k∑βx+kσx+(k-1k2k-1/6)σβ.Hence the probability density function (pdf) ofxT+k can be expressed as(11)πxT+k=12πσTkexp⁡-xT+k-μTk22σT2k.


## 2.3. RUL Distribution Function

When EGTM drops to the threshold valuexF, there is a potential for failure, and preventive maintenance needs to be performed for the aeroengine. So at time T, the cumulative distribution function (CDF) and pdf of the k-step predictive RUL can be expressed, respectively, by(12)FTk=∫-∞xFπxT+kdxT+k=ϕxF-μTkσTk=ϕxF-μx-k·μβ∑xx+k2∑ββ+2k∑βx+kσx+k-1k2k-1/6σβ(13)fTk=∂FTk∂k=18πexp⁡-xF-μx-kμβ22σT2kσT2k′μx+kμβ-xF-2μβσT2kσT3k,where σT2k′=2k∑ββ+2∑βx+σx+(6k2-6k+1/6)σβ.


## 3. Optimization of the Aeroengine SV Decisions

### 3.1. SV Decision Policy

As mentioned above, SV is due to two main reasons. According to (12), the time of ultralimit of EGTM obeys a certain distribution; however, the time of repair/replacement of LLP is a fixed value, but it can be described by a normal distribution with small variance. Thus, RUL is a probability distribution function no matter whether the aeroengine removal is due to ultralimit of EGTM or repair/replacement of LLP.JIT inventory management requires that the aeroengines should undergo SV in turn, so the different aeroengines have the different times of putting into operation. It is assumed that the #1 and #2 aeroengines have operated forT1 and T2 FC, respectively, and the corresponding predictive RUL of #1 and #2 aeroengine is P and Z, respectively. Based on the historical EGTM data, using (5)–(13), the pdf of P and Z, respectively, represented by f1,T1(p) and f2,T2(z), can be obtained and shown in the Figure 1(a), where subscripts “1” and “2” indicate #1 and #2 aeroengines, respectively (similarly later in the text).Figure 1
(a) The pdf of the predictive RUL. (b) SV decision policy.
(a) (b)In order to reduce the number of spare aeroengines with the requirement of safety and SL, an SV decision policy is proposed and illustrated with the curves of the two obtained predictive pdfs in Figure1(b). It is assumed that the SV time of #1 aeroengine will be after k FC, for example, k=215 FC, so the predictive probability of the risk of #1 aeroengine is equal to the blue shaded area in Figure 1(b). Let U be the repair time random variable and s(u) be its pdf. If u=400 FC, the return time of #1 aeroengine from shop will be after 615 FC, so the corresponding predictive probability of the shortage of #2 aeroengine is equal to the green-shaded area in Figure 1(b). If the shortage occurs, the airline must spend much money on renting an aeroengine urgently.Based on the analysis above, it can be found that variablek has effect on the life utilization and risk of #1 aeroengine as well as the SL of #2 aeroengine. For example, taking k as a smaller value would lead to a higher life utilization loss of #1 aeroengine; on the other hand, a bigger value of k would result in the higher risk of #1 aeroengine and lower SL of #2 aeroengine.


### 3.2. Optimization Formulation

Based on (9), T1 FC after putting into operation, k-step predictive EGTM value of #1 aeroengine can be estimated as x1,T1+k~Nμ1,T1k,σ1,T12k. It is assumed that FREQ is the accepted probability of risk of aeroengine; therefore, based on (12), the following inequality needs to be satisfied:(14)ϕxF-μ1,T1kσ1,T1k≤FREQ.Similarly,T2 FC after putting into operation, z-step predictive EGTM value of #2 aeroengine can be estimated as x2,T2+z~Nμ2,T2z,σ2,T22z, and, according to (13), the corresponding pdf of #2 aeroengine, represented by f2,T2(z), can be obtained.Therefore, if #2 aeroengine has operated forT2 FC and #1 aeroengine will be taken to the shop after k FC, the predictive probability of the SL of #2 aeroengine, represented by SLT2(k), is [24](15)SLT2k=Pk+U≤Z=1-∫-∞∞∫-∞z-ksuduf2,T2zdz.The following inequality needs to be satisfied:(16)SLT2k≥PSUP,where PSUP is the accepted probability of SL.Therefore, the SV timek can be optimized by the following inequalities:(17)ϕxF-μ1,T1kσ1,T1k≤FREQ1-∫-∞∞∫-∞z-ksuduf2,T2zdz≥PSUP.


## 3.1. SV Decision Policy

As mentioned above, SV is due to two main reasons. According to (12), the time of ultralimit of EGTM obeys a certain distribution; however, the time of repair/replacement of LLP is a fixed value, but it can be described by a normal distribution with small variance. Thus, RUL is a probability distribution function no matter whether the aeroengine removal is due to ultralimit of EGTM or repair/replacement of LLP.JIT inventory management requires that the aeroengines should undergo SV in turn, so the different aeroengines have the different times of putting into operation. It is assumed that the #1 and #2 aeroengines have operated forT1 and T2 FC, respectively, and the corresponding predictive RUL of #1 and #2 aeroengine is P and Z, respectively. Based on the historical EGTM data, using (5)–(13), the pdf of P and Z, respectively, represented by f1,T1(p) and f2,T2(z), can be obtained and shown in the Figure 1(a), where subscripts “1” and “2” indicate #1 and #2 aeroengines, respectively (similarly later in the text).Figure 1
(a) The pdf of the predictive RUL. (b) SV decision policy.
(a) (b)In order to reduce the number of spare aeroengines with the requirement of safety and SL, an SV decision policy is proposed and illustrated with the curves of the two obtained predictive pdfs in Figure1(b). It is assumed that the SV time of #1 aeroengine will be after k FC, for example, k=215 FC, so the predictive probability of the risk of #1 aeroengine is equal to the blue shaded area in Figure 1(b). Let U be the repair time random variable and s(u) be its pdf. If u=400 FC, the return time of #1 aeroengine from shop will be after 615 FC, so the corresponding predictive probability of the shortage of #2 aeroengine is equal to the green-shaded area in Figure 1(b). If the shortage occurs, the airline must spend much money on renting an aeroengine urgently.Based on the analysis above, it can be found that variablek has effect on the life utilization and risk of #1 aeroengine as well as the SL of #2 aeroengine. For example, taking k as a smaller value would lead to a higher life utilization loss of #1 aeroengine; on the other hand, a bigger value of k would result in the higher risk of #1 aeroengine and lower SL of #2 aeroengine.


## 3.2. Optimization Formulation

Based on (9), T1 FC after putting into operation, k-step predictive EGTM value of #1 aeroengine can be estimated as x1,T1+k~Nμ1,T1k,σ1,T12k. It is assumed that FREQ is the accepted probability of risk of aeroengine; therefore, based on (12), the following inequality needs to be satisfied:(14)ϕxF-μ1,T1kσ1,T1k≤FREQ.Similarly,T2 FC after putting into operation, z-step predictive EGTM value of #2 aeroengine can be estimated as x2,T2+z~Nμ2,T2z,σ2,T22z, and, according to (13), the corresponding pdf of #2 aeroengine, represented by f2,T2(z), can be obtained.Therefore, if #2 aeroengine has operated forT2 FC and #1 aeroengine will be taken to the shop after k FC, the predictive probability of the SL of #2 aeroengine, represented by SLT2(k), is [24](15)SLT2k=Pk+U≤Z=1-∫-∞∞∫-∞z-ksuduf2,T2zdz.The following inequality needs to be satisfied:(16)SLT2k≥PSUP,where PSUP is the accepted probability of SL.Therefore, the SV timek can be optimized by the following inequalities:(17)ϕxF-μ1,T1kσ1,T1k≤FREQ1-∫-∞∞∫-∞z-ksuduf2,T2zdz≥PSUP.


## 4. Case Study

The initial EGTM data of two CFM56-5B aeroengines in one of the Chinese airlines are shown in Figures2 and 3. The threshold value xF is 0 and the two aeroengines were removed, respectively, after 2080 and 2253 FC. #1 aeroengine was put into operation 200 FC earlier than #2 aeroengine; that is, T1-T2 = 200 FC.Figure 2
EGTM data of #1 CFM56-5B aeroengine.Figure 3
EGTM data of #2 CFM56-5B aeroengine.


### 4.1. RUL Prediction

Set observation time data asT and observed EGMT data as Y, and the correlation coefficient of T and Y is given by [27] (18)ρTY′=ETY′-μTμYET2-μT2EY2-μY2,where μT and μY are mean of T and Y, respectively, and E(·) is the expectation function.Substituting the initial EGTM data into (18), the correlation coefficients can be obtained as(19)ρ1,TY′=-0.9365ρ2,TY′=-0.9476.Since the correlation coefficients are very close to −1, the performance degradation process can approximately be regarded as linear degradation, as shown in Figures2 and 3. Therefore, a linear growth model can be adopted as state equations to describe the aeroengine degradation path as (3).The software package dlm is anR package for Bayesian analysis of dynamic linear models [26, 28], which is applied in this study to estimate the aeroengine degradation state and the unknown model parameters. For example, if T1 is equal to 1200 FC, that is, T2=1000 FC, with function dlmMLE, variances σε2, σx2, and σβ2 of #1 aeroengine can be estimated as follows:(20)σ1,ε2=15.19,σ1,x2=0.0119,σ1,β2=1.019×10-15.And variancesσε2, σx2, and σβ2 of #2 aeroengine can be estimated as follows:(21)σ2,ε2=20.88,σ2,x2=0.0002,σ2,β2=2.22×10-9.Then, with function dlmFilter, the following can be obtained:(22)x1,1200∣D1200~N16.37,0.8112,β1,1200∣D1200~N-0.022,0.0032x2,1000∣D1000~N31.46,0.5662,β2,1000∣D1000~N-0.0173,0.0022.Finally, with the function dlmForecast, the predictive distribution of EGTM value can be obtained. For example, the pdfs of #1 aeroengine predictive EGTM atp=300 and 600 FC are described as blue-dashed and blue-dotted curve, respectively, in Figure 4, and their corresponding probabilities of predictive EGTM below 0 at p=300 and 600 FC are equal to the green-shaded area and the red-shaded area, respectively, in Figure 4. The corresponding predictive parameters and expected RUL are shown in Tables 1 and 2, respectively. Meanwhile, in order to verify the correctness of expected RUL, the linear regression as a reference method is also applied and the corresponding results are shown in Figure 5 and Table 2.Table 1
Predictive parameters of #1 aeroengine.
Predicted parameters k-step predictionwhen T1=1200 (FC) k-step predictionwhen T1=1500 (FC) p = 300 p = 600 p = 200 p = 500 μ 1 , T 1 p 9.87 3.36 6.30 2.43 σ 1 , T 1 p 2.93 4.62 2.07 2.85 F 1 , T 1 p 3.78 × 10−4 0.2335 0.0012 0.1969Table 2
Expected RUL of #1 and #2 aeroengine.
Aeroengine T 1 / T 2 (FC) Actual RUL (FC) Excepted RUL with SSM (FC) SSM prediction error Excepted RUL with linear regression (FC) Linear regression prediction error #1 1200 880 754 14.31% 699 20.57% #1 1500 580 625 7.76% 528 8.97% #2 1000 1253 1734 38.4% 1868 49.1% #2 1300 953 1045 9.65% 728 23.61%Figure 4
EGTM prediction (T1=1200 FC) with SSM for #1.Figure 5
EGTM prediction (T1=1200 FC) with linear regression for #1.The pdfs of #2 aeroengine predictive EGTM atz=800 and 1500 FC are described as blue-dashed and blue-dotted curve, respectively, in Figures 6 and 7, and the corresponding predicted parameters and expected RUL are shown in Tables 3 and 2, respectively.Table 3
Predictive parameters of #2 aeroengine.
Predicted parameters z-step predictionwhen T2=1000 (FC) z-step predictionwhen T2=1300 (FC) z = 800 z = 1500 z = 500 z = 900 μ 2 , T 2 z 16.85 4.22 12.27 3.27 σ 2 , T 2 z 3.16 6.32 3.51 8.45 F 2 , T 2 z 4.85 × 10−8 0.2522 2.36 × 10−4 0.35Figure 6
EGTM prediction (T2=1000 FC) with SSM for #2.Figure 7
EGTM prediction (T2=1000 FC) with linear regression for #2.Figures8 and 9 show the pdf of #1 aeroengine predictive EGTM at time p=200 and 500 FC, and Figures 10 and 11 show the pdf of #2 aeroengine predictive EGTM at time z=500 and 900 FC. The corresponding predicted parameters and expected RUL are shown in Tables 1–3.Figure 8
EGTM prediction (T1=1500 FC) with SSM for #1.Figure 9
EGTM prediction (T1=1500 FC) with linear regression for #1.Figure 10
EGTM prediction (T2=1300 FC) with SSM for #2.Figure 11
EGTM prediction (T2=1300 FC) with linear regression for #2.From Table2, it can be found that SSM can predict RUL more accurately than linear regression. Moreover, it is clear that the predictive accuracy of #1 aeroengine RUL at time T1=1500 FC is 6.55% higher than at time T1=1200 FC, and the predictive accuracy of #2 aeroengine RUL at time T2=1300 FC is 28.75% higher than at time T2=1000 FC. So with increasing historical EGTM data, the predictive accuracy of the RUL is getting higher, which is illustrated with #2 aeroengine in Figure 12.Figure 12
Predictive life of #2 aeroengine with different timeT2.


### 4.2. Stochastic Repair Time Determination

As mentioned Section1, the repair time scatters since it is affected by many factors. Based on the historical SV data, with the general distribution fitting method and Kolmogorov-Smirnov test, the results show that the normal distribution has a good fit with P value of 0.91198 for the repair time, as shown in Figure 13. And the pdf of the repair time is(23)su=12πσue-1/2u-μu/σu2,μu=329,σu=69.14.Figure 13
Goodness-of-fit test for aeroengine repair time.
(a) (b) (c) (d)


### 4.3. SV Decision

Carrying out SV is a complex work and preparations should be made in advance; in order not to affect the airlines operation, an aeroengine SV decision is always made approximately 2 weeks (50 FC) ahead of its real SV time. So the shortest interval between two consecutive optimizations of SV time should be less than 50 FC. However, because the performance deterioration of an aeroengine is a gradual process, overmuch frequent optimizations of SV time is unnecessary, especially at the early stage of operation. So the following rules would be adopted in this study:(1)
If the optimal SV timek is more than 150 FC, #1 aeroengine will continue to be used and the next time of optimizing SV time k is after 100 FC.(2)
If the optimal SV timek is between 100 FC and 150 FC, #1 aeroengine will continue to be used and the next time of optimizing SV time k is after 50 FC.(3)
If the optimal SV timek is between 50 FC and 100 FC, the SV decision should be made in (k-50) FC and #1 aeroengine needs to be taken to the shop in k FC.(4)
If the optimal SV timek is less than 50 FC, the SV decision should be made immediately and #1 aeroengine needs to be taken to the shop ink FC.According to inequalities (17), the effect of SV time k on the risk of #1 aeroengine and the SL of #2 aeroengine needs to be analyzed simultaneously in order to obtain the optimal SV time. It is assumed that FREQ=1×10-6 and PSUP = 99.9%. Based on (12), the failure probability distribution of #1 aeroengine at different operation time can be obtained, as shown in Figure 14(a). For example, the red-solid curve in Figure 14(a) represents failure probability distribution of #1 aeroengine at operation time T1=1200 FC. And the failure probability distribution changes over time because the predictive parameters as shown in Table 1 are constantly being updated. Because FT1(k)≤FREQ should be satisfied, the optimal SV time k on the view of risk can be obtained at different operation time T1, as shown in Figure 14(b). From Figure 14(b), it can be known that the more FREQ, the less optimal SV time.Figure 14
F T 1 ( k ) at different operation time T1.
(a) (b)In the same way, based on inequality (15), the service level of #2 aeroengine at different operation time can be obtained, as shown in Figure 15(a). Because SLT2(k)≥PSUP should be satisfied, the optimal SV time k on the view of service level can be obtained at different operation time T2, as shown in Figure 15(b). From Figure 15(b), it can be known that the more PSUP, the less optimal SV time.Figure 15
S L T 2 ( k ) at different operation time T2.
(a) (b)Based on Figure14, it can be found that when T1=1200 FC, the optimal SV time is 310 FC because F1200(310) is equal to 1×10-6. However, at the same time, that is, when T2=T1-200 FC = 1000 FC, it can be found from Figure 15 that the optimal SV time is 923 FC because SL1000(923) is equal to 99.9%. Therefore, according to inequalities (17), the optimal k is 310 FC. Because the optimal k is more than 150 FC, the next time of optimizing SV time is after 100 FC.However, over time, at operation timeT1=1800 FC (T2=1600 FC), F1800(111) is equal to 1×10-6 and SL1600(159) is equal to 99.9%, so the optimalk is 111 FC (between 100 FC and 150 FC) and the next time of optimizing SV time is after 50 FC. And then when T1=1850 FC (T2=1650 FC), F1850(57) is equal to 1×10-6 and SL1650(108) is equal to 99.9%, so the optimalk is 57 FC (between 50 FC and 100 FC) and the SV decision should be made in 7 FC. The SV decision process can be shown in Table 4.Table 4
SV decision process.
T 1 (FC) 1200 1300 1400 1500 1600 1700 1800 1850 T 2 (FC) 1000 1100 1200 1300 1400 1500 1600 1650 k whenFT1(k)=1×10-6 310 289 277 242 210 157 111 57 k whenSLT2(k) = 99.9% 923 639 514 342 303 257 159 108 Decision Continue, optimize SV time every 100 FC Continue, optimize SV time after 50 FC SV decision should be made in 7 FCThe study case indicates that (1) the predictive accuracy of the RUL is getting higher with increasing historical EGTM data, (2) the SV decision process is dynamic and the optimal SV time changes over time, and (3) both PSUP and FREQ have an effect on SV decisions as well as RUL and repair time, and the more FREQ or PSUP, the less optimal SV time; that is, the requirements of lower risk or higher service level means that aeroengine visits shop earlier.


## 4.1. RUL Prediction

Set observation time data asT and observed EGMT data as Y, and the correlation coefficient of T and Y is given by [27] (18)ρTY′=ETY′-μTμYET2-μT2EY2-μY2,where μT and μY are mean of T and Y, respectively, and E(·) is the expectation function.Substituting the initial EGTM data into (18), the correlation coefficients can be obtained as(19)ρ1,TY′=-0.9365ρ2,TY′=-0.9476.Since the correlation coefficients are very close to −1, the performance degradation process can approximately be regarded as linear degradation, as shown in Figures2 and 3. Therefore, a linear growth model can be adopted as state equations to describe the aeroengine degradation path as (3).The software package dlm is anR package for Bayesian analysis of dynamic linear models [26, 28], which is applied in this study to estimate the aeroengine degradation state and the unknown model parameters. For example, if T1 is equal to 1200 FC, that is, T2=1000 FC, with function dlmMLE, variances σε2, σx2, and σβ2 of #1 aeroengine can be estimated as follows:(20)σ1,ε2=15.19,σ1,x2=0.0119,σ1,β2=1.019×10-15.And variancesσε2, σx2, and σβ2 of #2 aeroengine can be estimated as follows:(21)σ2,ε2=20.88,σ2,x2=0.0002,σ2,β2=2.22×10-9.Then, with function dlmFilter, the following can be obtained:(22)x1,1200∣D1200~N16.37,0.8112,β1,1200∣D1200~N-0.022,0.0032x2,1000∣D1000~N31.46,0.5662,β2,1000∣D1000~N-0.0173,0.0022.Finally, with the function dlmForecast, the predictive distribution of EGTM value can be obtained. For example, the pdfs of #1 aeroengine predictive EGTM atp=300 and 600 FC are described as blue-dashed and blue-dotted curve, respectively, in Figure 4, and their corresponding probabilities of predictive EGTM below 0 at p=300 and 600 FC are equal to the green-shaded area and the red-shaded area, respectively, in Figure 4. The corresponding predictive parameters and expected RUL are shown in Tables 1 and 2, respectively. Meanwhile, in order to verify the correctness of expected RUL, the linear regression as a reference method is also applied and the corresponding results are shown in Figure 5 and Table 2.Table 1
Predictive parameters of #1 aeroengine.
Predicted parameters k-step predictionwhen T1=1200 (FC) k-step predictionwhen T1=1500 (FC) p = 300 p = 600 p = 200 p = 500 μ 1 , T 1 p 9.87 3.36 6.30 2.43 σ 1 , T 1 p 2.93 4.62 2.07 2.85 F 1 , T 1 p 3.78 × 10−4 0.2335 0.0012 0.1969Table 2
Expected RUL of #1 and #2 aeroengine.
Aeroengine T 1 / T 2 (FC) Actual RUL (FC) Excepted RUL with SSM (FC) SSM prediction error Excepted RUL with linear regression (FC) Linear regression prediction error #1 1200 880 754 14.31% 699 20.57% #1 1500 580 625 7.76% 528 8.97% #2 1000 1253 1734 38.4% 1868 49.1% #2 1300 953 1045 9.65% 728 23.61%Figure 4
EGTM prediction (T1=1200 FC) with SSM for #1.Figure 5
EGTM prediction (T1=1200 FC) with linear regression for #1.The pdfs of #2 aeroengine predictive EGTM atz=800 and 1500 FC are described as blue-dashed and blue-dotted curve, respectively, in Figures 6 and 7, and the corresponding predicted parameters and expected RUL are shown in Tables 3 and 2, respectively.Table 3
Predictive parameters of #2 aeroengine.
Predicted parameters z-step predictionwhen T2=1000 (FC) z-step predictionwhen T2=1300 (FC) z = 800 z = 1500 z = 500 z = 900 μ 2 , T 2 z 16.85 4.22 12.27 3.27 σ 2 , T 2 z 3.16 6.32 3.51 8.45 F 2 , T 2 z 4.85 × 10−8 0.2522 2.36 × 10−4 0.35Figure 6
EGTM prediction (T2=1000 FC) with SSM for #2.Figure 7
EGTM prediction (T2=1000 FC) with linear regression for #2.Figures8 and 9 show the pdf of #1 aeroengine predictive EGTM at time p=200 and 500 FC, and Figures 10 and 11 show the pdf of #2 aeroengine predictive EGTM at time z=500 and 900 FC. The corresponding predicted parameters and expected RUL are shown in Tables 1–3.Figure 8
EGTM prediction (T1=1500 FC) with SSM for #1.Figure 9
EGTM prediction (T1=1500 FC) with linear regression for #1.Figure 10
EGTM prediction (T2=1300 FC) with SSM for #2.Figure 11
EGTM prediction (T2=1300 FC) with linear regression for #2.From Table2, it can be found that SSM can predict RUL more accurately than linear regression. Moreover, it is clear that the predictive accuracy of #1 aeroengine RUL at time T1=1500 FC is 6.55% higher than at time T1=1200 FC, and the predictive accuracy of #2 aeroengine RUL at time T2=1300 FC is 28.75% higher than at time T2=1000 FC. So with increasing historical EGTM data, the predictive accuracy of the RUL is getting higher, which is illustrated with #2 aeroengine in Figure 12.Figure 12
Predictive life of #2 aeroengine with different timeT2.


## 4.2. Stochastic Repair Time Determination

As mentioned Section1, the repair time scatters since it is affected by many factors. Based on the historical SV data, with the general distribution fitting method and Kolmogorov-Smirnov test, the results show that the normal distribution has a good fit with P value of 0.91198 for the repair time, as shown in Figure 13. And the pdf of the repair time is(23)su=12πσue-1/2u-μu/σu2,μu=329,σu=69.14.Figure 13
Goodness-of-fit test for aeroengine repair time.
(a) (b) (c) (d)


## 4.3. SV Decision

Carrying out SV is a complex work and preparations should be made in advance; in order not to affect the airlines operation, an aeroengine SV decision is always made approximately 2 weeks (50 FC) ahead of its real SV time. So the shortest interval between two consecutive optimizations of SV time should be less than 50 FC. However, because the performance deterioration of an aeroengine is a gradual process, overmuch frequent optimizations of SV time is unnecessary, especially at the early stage of operation. So the following rules would be adopted in this study:(1)
If the optimal SV timek is more than 150 FC, #1 aeroengine will continue to be used and the next time of optimizing SV time k is after 100 FC.(2)
If the optimal SV timek is between 100 FC and 150 FC, #1 aeroengine will continue to be used and the next time of optimizing SV time k is after 50 FC.(3)
If the optimal SV timek is between 50 FC and 100 FC, the SV decision should be made in (k-50) FC and #1 aeroengine needs to be taken to the shop in k FC.(4)
If the optimal SV timek is less than 50 FC, the SV decision should be made immediately and #1 aeroengine needs to be taken to the shop ink FC.According to inequalities (17), the effect of SV time k on the risk of #1 aeroengine and the SL of #2 aeroengine needs to be analyzed simultaneously in order to obtain the optimal SV time. It is assumed that FREQ=1×10-6 and PSUP = 99.9%. Based on (12), the failure probability distribution of #1 aeroengine at different operation time can be obtained, as shown in Figure 14(a). For example, the red-solid curve in Figure 14(a) represents failure probability distribution of #1 aeroengine at operation time T1=1200 FC. And the failure probability distribution changes over time because the predictive parameters as shown in Table 1 are constantly being updated. Because FT1(k)≤FREQ should be satisfied, the optimal SV time k on the view of risk can be obtained at different operation time T1, as shown in Figure 14(b). From Figure 14(b), it can be known that the more FREQ, the less optimal SV time.Figure 14
F T 1 ( k ) at different operation time T1.
(a) (b)In the same way, based on inequality (15), the service level of #2 aeroengine at different operation time can be obtained, as shown in Figure 15(a). Because SLT2(k)≥PSUP should be satisfied, the optimal SV time k on the view of service level can be obtained at different operation time T2, as shown in Figure 15(b). From Figure 15(b), it can be known that the more PSUP, the less optimal SV time.Figure 15
S L T 2 ( k ) at different operation time T2.
(a) (b)Based on Figure14, it can be found that when T1=1200 FC, the optimal SV time is 310 FC because F1200(310) is equal to 1×10-6. However, at the same time, that is, when T2=T1-200 FC = 1000 FC, it can be found from Figure 15 that the optimal SV time is 923 FC because SL1000(923) is equal to 99.9%. Therefore, according to inequalities (17), the optimal k is 310 FC. Because the optimal k is more than 150 FC, the next time of optimizing SV time is after 100 FC.However, over time, at operation timeT1=1800 FC (T2=1600 FC), F1800(111) is equal to 1×10-6 and SL1600(159) is equal to 99.9%, so the optimalk is 111 FC (between 100 FC and 150 FC) and the next time of optimizing SV time is after 50 FC. And then when T1=1850 FC (T2=1650 FC), F1850(57) is equal to 1×10-6 and SL1650(108) is equal to 99.9%, so the optimalk is 57 FC (between 50 FC and 100 FC) and the SV decision should be made in 7 FC. The SV decision process can be shown in Table 4.Table 4
SV decision process.
T 1 (FC) 1200 1300 1400 1500 1600 1700 1800 1850 T 2 (FC) 1000 1100 1200 1300 1400 1500 1600 1650 k whenFT1(k)=1×10-6 310 289 277 242 210 157 111 57 k whenSLT2(k) = 99.9% 923 639 514 342 303 257 159 108 Decision Continue, optimize SV time every 100 FC Continue, optimize SV time after 50 FC SV decision should be made in 7 FCThe study case indicates that (1) the predictive accuracy of the RUL is getting higher with increasing historical EGTM data, (2) the SV decision process is dynamic and the optimal SV time changes over time, and (3) both PSUP and FREQ have an effect on SV decisions as well as RUL and repair time, and the more FREQ or PSUP, the less optimal SV time; that is, the requirements of lower risk or higher service level means that aeroengine visits shop earlier.


## 5. Conclusions

Reasonably planning aeroengine SVs can avoid the overlap of SVs and reduce the number of spare aeroengines; so according to the concept of JIT, an optimization of the aeroengine SV decision based on RUL and stochastic repair time is developed in this study, by analyzing the effect of SV decision on the risk and the SL. A case study of two CFM-56 aeroengines was given to demonstrate the proposed optimization, and the results show that (1) SSM can predict RUL more accurately than linear regression, and the expected RUL is closer to the actual RUL with increasing historical EGTM data; (2) the process of SV decisions is simple and feasible for airlines to implement; (3) the optimization of the aeroengine SV decisions not only takes full advantage of the CBM but also decreases the spare aeroengines with the requirement of safety and SL. In further study, an optimization of the aeroengine SV decisions for more than two aeroengines will be developed, and, in order to obtain more accurate optimization results, the economic analysis on aeroengine cost rate will be incorporated into the proposed optimization.



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*Source: 5648360-2016-12-14.xml*