# Reliability Parameter Interval Estimation of NC Machine Tools considering Working Conditions

Authors: Li Hongzhou; Sun Lixia
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Aiming at the problem that the parameter interval estimation of NC machine tool’s reliability model considering working conditions established by Hongzhou is difficult to implement, given that it has several independent variables, an improved interval estimation method based on Bootstrap is proposed. Firstly, the two-step estimation method was used to calculate the point estimation of NC machine tool’s reliability parameter in test field, based on whichB resamplings are generated based on the point estimation. The reliability parameter’s point estimation of the resamplings was obtained by maximum likelihood estimation. Permutation of B point estimations was made in ascending order and the interval estimations were obtained by the α quantile of the permutation. Case study indicated that the location and length of the interval estimation of NC machine tools’ reliability parameter, under different levels of working condition covariates, vary obviously.


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## Body

## 1. Introduction

The reliability model of NC machine tools considering working conditions was established in [1], and the point estimations of the shape parameter, scale parameter, and the coefficients of working condition covariates were obtained by the two-step estimation method. To make NC machine tool’s reliability evaluation more accurate, the interval estimations of the shape parameter, scale parameter, and the coefficients of working condition covariates should be made. The commonly used interval estimation methods include Fisher information matrix method [2], likelihood ratio interval estimation [3, 4], pivot method [5], and maximum likelihood interval estimation [6]. Each of the above interval estimation methods usually establishes, respectively, the interval estimation formula with only one independent variable. However, there are several independent variables of the reliability model in [1], including Time between Failure (TBF) and working condition covariates, which make it difficult for the above methods to calculate the model parameters’ interval estimation.The Bootstrap method [7] only depends on a lot of resamples to calculate interval estimation overcoming the shortcomings of the other interval estimation methods which need to construct the complex formula.Aiming at the above problem, a reliability model parameter’s interval estimation method of NC machine tools considering working conditions based on the Bootstrap method is proposed in this paper. Firstly, parameter’s point estimation based on the NC machine tools’ test sample is obtained by two-step estimation method; the resampling is obtained by Bootstrap based on the parameter point estimations of the NC machine tools’ test sample. The parameter point estimations of each resampling are calculated by the maximum likelihood estimation. The parameter point estimations under every working condition level are obtained by conversion equation. Finally, the feasibility of the proposed method is validated in the case study.


## 2. Reliability Model of NC Machine Tools considering Working Conditions

For convenience, the failure rate function of NC machine tools considering working conditions in [1] is expressed as follows:(1)λtX=mηtηm-1·exp⁡βX-X1,where t is the Time between Failures (a random variable) of NC machine tools; X=(X1,X2,…,Xi,…,Xn), X is the vector of working condition covariates, which affects the failure rate of NC machine tools, and Xi is the ith covariate, such as cutting force, environment temperature, and number of tool changes or vibration; m is the shape parameter under X1, and m>0; η is the scale parameter under X1, η>0; β=(β1,β2,…,βi,…,βn) is the vector of X’s coefficients, which reflect the covariates’ influences on the failure rate function, and βi is the coefficient of Xi.So, the reliability function of NC machine tools considering working conditions is expressed as follows:(2)Rt,X=exp-tηmexpβX-X1.Suppose that the failure rate function of two-parameter Weibull distribution of NC machine tools under covariateXi is(3)λtXi=mηitηim-1,where m is the shape parameter; ηi is the scale parameter under covariate Xi.Substituting (3) in (1) gets the scale parameter ηi:(4)ηi=ηexpβ/mXi-X1.


## 3. Reliability Parameter Interval Estimations of NC Machine Tools considering Working Conditions

### 3.1. Bootstrap Method

Bootstrap method generates new samples by drawing samples from the original samples, obtaining the so-called resamplings, which can be used for parameter’s interval estimation. The basic idea of the Bootstrap resampling is as follows [8, 9].SupposeX=x1,x2,…,xk is a sample from population Fx∣θ with parameter θ which is equivalent to the maximum likelihood estimation θ^.Based onFx∣θ, B Bootstrap resamplings {X1,X2,…,Xb,…,XB} are drawn, where the size of each resampling is l, and Xb=x1b,x2b,…,xlb is the bth Bootstrap resampling. Based on each Bootstrap resampling, the parameters’ estimations are calculated to be {θ^1,θ^2,…,θ^B}.Arranging{θ^1,θ^2,…,θ^B} in ascending order obtains {θ^∗1,θ^∗2,…,θ^∗B}, and the interval estimation of parameter at the confidence level α is as follows:(5)θ^lo,θ^up=θ^∗α/2×B,θ^∗1-α/2×B,where θ^lo is the lower limit of the interval estimation; θ^up is the upper limit of the interval estimation.


### 3.2. Reliability Parameter Interval Estimation’s Step of NC Machine Tools considering Working Conditions

Step 1.
Point estimations, includingm^, η^, and β^, of the shape parameter m, the scale parameter η, and coefficients of working condition covariates β are calculated by two-step interval estimation in [1], according to the fault information and the working conditions corresponding to fault information obtained from the test field.Step 2.
B Bootstrap resamplings with the fault information and the working conditions are obtained by sampling based on the point estimations m^, η^, and β^ in Step 1.Step 3.
The maximum likelihood estimation method is adopted to estimate parametersβ, m, and η estimation in (1) for each resampling, then point estimations m^b, η^b, and β^b are obtained, and the subscript b is the bth resampling, b=1,2,…,B.The likelihood function is given as(6)Lη,m=∏i=1rfti,Xi∏i=r+1nRti,Xi=∏i=1rexp⁡-tiηmexp⁡βXi-X1·mηtiηm-1·exp⁡βXi-X1·∏i=r+1nexp-tiηmexp⁡βXi-X1.Take the logarithm of both sides in (6); then(7)LnLη,m=rln⁡m-rmln⁡η+m-1ln⁡t1t2…tr-t1ηm+t2ηm+⋯+trηm+βX1+X2+⋯+Xr-rX1+∑i=r+1n-tiηmexp⁡βXi-X1.Take the partial derivatives of the parametersβ, η, and m in (7), respectively, and then(8)∂LnLη,m∂β1=1-rX1+∑i=r+1n-tiηm1-rX1·exp⁡βXi-X1∂LnLη,m∂βi=βXi+∑i=r+1n-tiηmXiexpβXi-X1i=2,3,…,n∂LnLη,m∂η=-rmη+mt1mη-m-1+mt2mη-m-1+⋯+mtrmη-m-1+∑i=r+1nmtimη-m-1exp⁡βXi-X1∂LnLη,m∂m=rm-rln⁡η+ln⁡t1t2⋯tr-t1ηmln⁡t1η+t2ηmln⁡t2η+⋯+trηmln⁡trη+∑i=r+1n-tiηmlntiηexp⁡βXi-X1.Since (8) have no analytical solutions, Newton-Raphson [10] numerical algorithm is used to estimate parameters β, η, and m.Step 4.
The scale parameterηbi of two-parameter Weibull distribution of NC machine tools under covariate Xi in the bth resampling is obtained based on (4). (9)ηbi=ηbexp⁡βb/mbXi-X1b=1,2,…,B,i=1,2,…,u,where u represents number of covariates’ levels.Step 5.
According to the scale parameterηbi obtained by Step 4 and shape parameter m obtained by Step 3, the MTBF of NC machine tools under covariate Xi in the bth resampling is(10)MTBFbi=η^bi∗Γ1+1m^bb=1,2,…,B,i=1,2,…,u.Step 6.
The shape parametersm^1,…,m^B obtained by Step 3 are arranged in ascending order and then get sequence(11)m^∗1,…,m^∗B.Step 7.
The scale parametersη^bi,i=1,2,…,u obtained by Steps 3 and 4 are arranged, respectively, in ascending order and then get sequence(12)η^∗11,…,η^∗1b,…,η^∗1Bη^∗21,…,η^∗2b,…,η^∗2B⋮η^∗u1,…,η^∗ub,…,η^∗uB.Step 8.
The coefficientsβ^1,…,β^k of working condition covariates obtained by Step 3 are arranged in ascending order and then get sequence(13)β^∗i1,…,β∗iB,i=1,2,…,k.Step 9.
M
T
B
F
b
i
,
i
=
1
,
2
,
…
,
u obtained by Step 5 are arranged in ascending order and then get sequence(14)MTBF∗i1,…,MTBF∗iBi=1,2,…,u.Step 10.
According to Step6, set up confidence level 1-α and solve and round B·α/2 and B·1-α/2, respectively; then the interval estimations of the shape parameter are obtained as follows:(15)m^∗B·α/2,m^∗B·1-α/2,where m^∗B·α/2 is the lower limit of interval estimation of the shape parameter at the confidence level 1-α. m^∗B·1-α/2 is the upper limit of interval estimation of the shape parameter at the confidence level 1-α.Step 11.
According to Step7, set up confidence level 1-α and solve and round B·α/2 and B·1-α/2, respectively; then the interval estimations of the scale parameter under the ith covariate level are obtained: (16)η^∗i,B·α/2,η^∗i,B·1-α/2i=1,2,…,u,where η^∗i,B·α/2 is the lower limit of interval estimation of the scale parameter under the ith covariate level at the confidence level 1-α. η^∗i,B·1-α/2 is the upper limit of interval estimation of the shape parameter under the ith covariate level at the confidence level 1-α.Step 12.
According to Step8, set up confidence level 1-α and solve and round B·α/2 and B·1-α/2, respectively; then the interval estimations of the coefficients of the ith working condition covariate are obtained: (17)β^∗i,B·α/2,β^∗i,B·1-α/2i=1,2,…,k,where β^∗i,B·α/2 is the lower limit of interval estimation of the coefficients of the ith working condition covariate at the confidence level 1-α. β^∗i,B·1-α/2 is the upper limit of interval estimation of the coefficients of the ith working condition covariate at the confidence level 1-α.Step 13.
According to Step9, set up confidence level 1-α and solve and round B·α/2 and B·1-α/2, respectively; then the interval estimations of the MTBF of NC machine tools under the ith covariate level are got: (18)MTBF∗i,B·α/2,MTBF∗i,B·1-α/2i=1,2,…,u,where MTBF∗i,B·α/2 is the lower limit of interval estimation of the MTBF of NC machine tools under the ith covariate level at the confidence level 1-α. MTBF∗i,B·1-α/2 is the upper limit of interval estimation of the MTBF of NC machine tools under the ith covariate level at the confidence level 1-α.


## 3.1. Bootstrap Method

Bootstrap method generates new samples by drawing samples from the original samples, obtaining the so-called resamplings, which can be used for parameter’s interval estimation. The basic idea of the Bootstrap resampling is as follows [8, 9].SupposeX=x1,x2,…,xk is a sample from population Fx∣θ with parameter θ which is equivalent to the maximum likelihood estimation θ^.Based onFx∣θ, B Bootstrap resamplings {X1,X2,…,Xb,…,XB} are drawn, where the size of each resampling is l, and Xb=x1b,x2b,…,xlb is the bth Bootstrap resampling. Based on each Bootstrap resampling, the parameters’ estimations are calculated to be {θ^1,θ^2,…,θ^B}.Arranging{θ^1,θ^2,…,θ^B} in ascending order obtains {θ^∗1,θ^∗2,…,θ^∗B}, and the interval estimation of parameter at the confidence level α is as follows:(5)θ^lo,θ^up=θ^∗α/2×B,θ^∗1-α/2×B,where θ^lo is the lower limit of the interval estimation; θ^up is the upper limit of the interval estimation.


## 3.2. Reliability Parameter Interval Estimation’s Step of NC Machine Tools considering Working Conditions

Step 1.
Point estimations, includingm^, η^, and β^, of the shape parameter m, the scale parameter η, and coefficients of working condition covariates β are calculated by two-step interval estimation in [1], according to the fault information and the working conditions corresponding to fault information obtained from the test field.Step 2.
B Bootstrap resamplings with the fault information and the working conditions are obtained by sampling based on the point estimations m^, η^, and β^ in Step 1.Step 3.
The maximum likelihood estimation method is adopted to estimate parametersβ, m, and η estimation in (1) for each resampling, then point estimations m^b, η^b, and β^b are obtained, and the subscript b is the bth resampling, b=1,2,…,B.The likelihood function is given as(6)Lη,m=∏i=1rfti,Xi∏i=r+1nRti,Xi=∏i=1rexp⁡-tiηmexp⁡βXi-X1·mηtiηm-1·exp⁡βXi-X1·∏i=r+1nexp-tiηmexp⁡βXi-X1.Take the logarithm of both sides in (6); then(7)LnLη,m=rln⁡m-rmln⁡η+m-1ln⁡t1t2…tr-t1ηm+t2ηm+⋯+trηm+βX1+X2+⋯+Xr-rX1+∑i=r+1n-tiηmexp⁡βXi-X1.Take the partial derivatives of the parametersβ, η, and m in (7), respectively, and then(8)∂LnLη,m∂β1=1-rX1+∑i=r+1n-tiηm1-rX1·exp⁡βXi-X1∂LnLη,m∂βi=βXi+∑i=r+1n-tiηmXiexpβXi-X1i=2,3,…,n∂LnLη,m∂η=-rmη+mt1mη-m-1+mt2mη-m-1+⋯+mtrmη-m-1+∑i=r+1nmtimη-m-1exp⁡βXi-X1∂LnLη,m∂m=rm-rln⁡η+ln⁡t1t2⋯tr-t1ηmln⁡t1η+t2ηmln⁡t2η+⋯+trηmln⁡trη+∑i=r+1n-tiηmlntiηexp⁡βXi-X1.Since (8) have no analytical solutions, Newton-Raphson [10] numerical algorithm is used to estimate parameters β, η, and m.Step 4.
The scale parameterηbi of two-parameter Weibull distribution of NC machine tools under covariate Xi in the bth resampling is obtained based on (4). (9)ηbi=ηbexp⁡βb/mbXi-X1b=1,2,…,B,i=1,2,…,u,where u represents number of covariates’ levels.Step 5.
According to the scale parameterηbi obtained by Step 4 and shape parameter m obtained by Step 3, the MTBF of NC machine tools under covariate Xi in the bth resampling is(10)MTBFbi=η^bi∗Γ1+1m^bb=1,2,…,B,i=1,2,…,u.Step 6.
The shape parametersm^1,…,m^B obtained by Step 3 are arranged in ascending order and then get sequence(11)m^∗1,…,m^∗B.Step 7.
The scale parametersη^bi,i=1,2,…,u obtained by Steps 3 and 4 are arranged, respectively, in ascending order and then get sequence(12)η^∗11,…,η^∗1b,…,η^∗1Bη^∗21,…,η^∗2b,…,η^∗2B⋮η^∗u1,…,η^∗ub,…,η^∗uB.Step 8.
The coefficientsβ^1,…,β^k of working condition covariates obtained by Step 3 are arranged in ascending order and then get sequence(13)β^∗i1,…,β∗iB,i=1,2,…,k.Step 9.
M
T
B
F
b
i
,
i
=
1
,
2
,
…
,
u obtained by Step 5 are arranged in ascending order and then get sequence(14)MTBF∗i1,…,MTBF∗iBi=1,2,…,u.Step 10.
According to Step6, set up confidence level 1-α and solve and round B·α/2 and B·1-α/2, respectively; then the interval estimations of the shape parameter are obtained as follows:(15)m^∗B·α/2,m^∗B·1-α/2,where m^∗B·α/2 is the lower limit of interval estimation of the shape parameter at the confidence level 1-α. m^∗B·1-α/2 is the upper limit of interval estimation of the shape parameter at the confidence level 1-α.Step 11.
According to Step7, set up confidence level 1-α and solve and round B·α/2 and B·1-α/2, respectively; then the interval estimations of the scale parameter under the ith covariate level are obtained: (16)η^∗i,B·α/2,η^∗i,B·1-α/2i=1,2,…,u,where η^∗i,B·α/2 is the lower limit of interval estimation of the scale parameter under the ith covariate level at the confidence level 1-α. η^∗i,B·1-α/2 is the upper limit of interval estimation of the shape parameter under the ith covariate level at the confidence level 1-α.Step 12.
According to Step8, set up confidence level 1-α and solve and round B·α/2 and B·1-α/2, respectively; then the interval estimations of the coefficients of the ith working condition covariate are obtained: (17)β^∗i,B·α/2,β^∗i,B·1-α/2i=1,2,…,k,where β^∗i,B·α/2 is the lower limit of interval estimation of the coefficients of the ith working condition covariate at the confidence level 1-α. β^∗i,B·1-α/2 is the upper limit of interval estimation of the coefficients of the ith working condition covariate at the confidence level 1-α.Step 13.
According to Step9, set up confidence level 1-α and solve and round B·α/2 and B·1-α/2, respectively; then the interval estimations of the MTBF of NC machine tools under the ith covariate level are got: (18)MTBF∗i,B·α/2,MTBF∗i,B·1-α/2i=1,2,…,u,where MTBF∗i,B·α/2 is the lower limit of interval estimation of the MTBF of NC machine tools under the ith covariate level at the confidence level 1-α. MTBF∗i,B·1-α/2 is the upper limit of interval estimation of the MTBF of NC machine tools under the ith covariate level at the confidence level 1-α.


## 4. Case Study

The parameter interval estimations are made according to the test data in [1] which is shown in Table 1 for convenience to discuss.Table 1
Failure data of NC machine tools under different working conditions covariates.
Workpiece Name
Cutting force/KN
Number of tool changes/(N/h)
Cutting Fluid
Temperature/°C
TBF/h
Data type
Flywheel
0.35
2
1
20
437
1
Flywheel
0.35
2
1
20
1896
1
Flywheel
0.35
2
1
20
340
1
Flywheel
0.35
2
1
20
244
1
Flywheel
0.35
2
1
20
249
1
Flywheel
0.35
2
1
20
898
1
Flywheel
0.35
2
1
20
1148
0
Cylinder Block
0.43
17
0
21
158
1
Cylinder Block
0.43
17
0
21
67
1
Cylinder Block
0.43
17
0
21
242
1
Cylinder Block
0.43
17
0
21
107
1
Cylinder Block
0.43
17
0
21
155
1
Cylinder Block
0.43
17
0
21
1717
1
Cylinder Block
0.43
17
0
21
812
1
Cylinder Block
0.43
17
0
21
724
1
Cylinder Head
0.54
4
0
22
316
1
Cylinder Head
0.54
4
0
22
99
1
Cylinder Head
0.54
4
0
22
1419
1
Cylinder Head
0.54
4
0
22
1430
1
Cylinder Head
0.54
4
0
22
225
1
Cylinder Head
0.54
4
0
22
773
1
Cylinder Head
0.54
4
0
22
843
0
Mould
0.78
10
1
19
398
1
Mould
0.78
10
1
19
29
1
Mould
0.78
10
1
19
401
1
Mould
0.78
10
1
19
1148
1
Mould
0.78
10
1
19
1012
1
Mould
0.78
10
1
19
733
1
Mould
0.78
10
1
19
1717
1
Mould
0.78
10
1
19
773
1
Mould
0.78
10
1
19
445
0
Flywheel Housing
0.81
4
1
20
348
1
Flywheel Housing
0.81
4
1
20
167
1
Flywheel Housing
0.81
4
1
20
1232
1
Flywheel Housing
0.81
4
1
20
1118
1
Flywheel Housing
0.81
4
1
20
633
1
Flywheel Housing
0.81
4
1
20
382
1
Flywheel Housing
0.81
4
1
20
321
1
Flywheel Housing
0.81
4
1
20
576
0
Cylinder
0.84
6
0
19
58
1
Cylinder
0.84
6
0
19
37
1
Cylinder
0.84
6
0
19
58
1
Cylinder
0.84
6
0
19
592
1
Cylinder
0.84
6
0
19
1008
1
Cylinder
0.84
6
0
19
365
1
Cylinder
0.84
6
0
20
1144
1
Cylinder
0.84
6
0
20
1430
1
Cylinder
0.84
6
0
20
373
1
Cylinder
0.84
6
0
20
659
1
Connect-ing Plate
1.03
14
1
22
234
1
Connect-ing Plate
1.03
14
1
22
175
1
Connect-ing Plate
1.03
14
1
22
190
1
Connect-ing Plate
1.03
14
1
22
151
1
Connect-ing Plate
1.03
14
1
22
18
1
Connect-ing Plate
1.03
14
1
22
530
1
Connect-ing Plate
1.03
14
1
22
349
1
Connect-ing Plate
1.03
14
1
22
526
1
Connect-ing Plate
1.03
14
1
22
368
1
Connect-ing Plate
1.03
14
1
22
174
1According to Step1, the working condition covariates’ coefficients of the batch of NC machine tools in Table 1 are β^1=1.142,β^2=0.049, and the parameters of the baseline failure rate function are m=1.2121,η=1156 when cutting force Fc=0.35KN and number of tool changes Nh=2.The interval estimations of reliability parameters of the batch of NC machine tools are calculated by Steps2–13. The result is shown in Table 2, where 1-α = 97.5% and B=1000.Table 2
Reliability parameters interval estimations of NC machine tools considering working condition covariate.
Parameter
Interval estimation
m
[
1.03,1.57
]
η
[
758.72,1797.42
]
β
1
[
-
0.11,2.45
]
β
2
[
-
0.00,0.11
]The interval estimation of the scale parameterη and MTBF under each working condition covariant level is calculated by Steps 2–13. The result is shown in Table 3.Table 3
Reliability parameter interval estimations of NC machine tools under each working condition covariate level.
Covariate level
F
c/KN
N
h n/h
η interval estimation
MTBF interval estimation
1
0.35
2
[
758.72,1797.42
]
[
713.97,1649.99
]
2
0.43
17
[
342.42,1035.87
]
[
320.05,958.97
]
3
0.54
4
[
685.85,1232.43
]
[
641.71,1151.08
]
4
0.78
10
[
462.05,733.07
]
[
432.67,683.51
]
5
0.81
4
[
532.58,1009.93
]
[
493.79,945.30
]
6
0.84
6
[
488.28,872.44
]
[
462.02,811.21
]
7
1.03
14
[
251.98,595.01
]
[
238.70,558.03
]For clearer, the interval estimations of the scale parameterη and MTBF under each working condition covariant level are shown in Figures 1 and 2.Figure 1
Scale parameter interval estimation under each working condition covariate level.Figure 2
MTBF interval estimation under each working condition covariate level.For comparison, model parameters and MTBF interval estimation of NC machine tools obtained by the traditional Bootstrap method, which does not consider the working conditions, are obtained. The detailed procedure of calculation by the tradition method is given in [11], and the corresponding result is shown in Table 4. The interval estimations mTra are shown in Figure 1. The interval estimation MTBFTra is shown in Figure 2.Table 4
Interval estimation obtained by the tradition Bootstrap method.
Parameter
Interval estimation
m
T
r
a
[
0.93,1.47
]
η
T
r
a
[
514.22,846.03
]
M
T
B
F
T
r
a
[
496.76,802.36
]It is seen from Tables1 and 2 and Figures 1 and 2 that the interval estimation obtained by the traditional method is only one interval estimation, and only under some particular working conditions are they similar to the interval estimation obtained by the new method (e.g., the cutting force Fc=0.84 KN and the number of tool changes Nh=6). There are obvious distinctions in the length and location of the interval estimation of the scale parameter and MTBF under different working condition covariant levels. When Fc=0.35 KN and Nh=2, the length of the interval estimation of the scale parameter and MTBF is longer and their locations are higher. When Fc=0.78 KN and Nh=10, the length of the interval estimation of the scale parameter and MTBF is shorter and their locations are lower. It can be concluded from the above facts that the new method can be used to calculate the reliability parameters’ interval estimation of NC machine tools considering working condition covariates.


## 5. Conclusions

Given that there may be two or more independent variables (e.g., TBF or working condition covariate) in the reliability model of NC machine tools considering working conditions covariates, this makes other interval estimation methods unfeasible to calculate the interval estimations of the reliability parameters. Considering this problem, the authors propose a new method for the interval estimation of NC machine tools considering working conditions covariates. The resamples are obtained based on the parameters of the test sample collected in the test field. Then the improved Bootstrap method is used to calculate parameter interval estimations of NC machine tools’ reliability model. The result of the case study indicated that the new method can be used to calculate reliability parameters’ interval estimation of the NC machine tools considering different working condition covariates which cannot be calculated using other methods and provides a more accurate basis for reliability evaluation.



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*Source: 4037903-2017-09-26.xml*