(wileyonlinelibrary.com) DOI: 10.1002/qre.1692 Published online in Wiley Online Library
Shewhart Control Charts for MonitoringReliability with Weibull LifetimesAlireza Faraz,a*,† Erwin M. Sanigab and Cedric Heuchennea,c‡
In this paper, we present Shewhart-typeZand S2 control charts for monitoring individual or joint shifts in the scale and shapeparameters of a Weibull distributed process. The advantage of this method is its ease of use and flexibility for the case wherethe process distribution is Weibull, although the method can be applied to any distribution. We illustrate the performance ofour method through simulation and the application through the use of an actual data set. Our results indicate that Z and S2
Keywords: control charts; reliability; Weibull distribution
1. Introduction
The Weibull distribution enjoys importance in survival analysis and modeling failure times. It has also been used to describeproduct properties such as strength, elongation, and resistance. The distribution is characterized by two parameters, shapeand scale. In reliability and survival studies, different aspects of lifetime variables can be described by these two parameters.
Initially, Weibull1 showed that material breaking strength can be modeled with the Weibull distribution. Since then, the Weibulldistribution has been widely used in modeling products lifetimes with increasing or decreasing failure rates. A report by Weibull2 listsover 1000 references to the application of the Weibull model in solving a variety of problems from many different disciplines. Forexample, the strength distribution of brittle and composite materials has been investigated by Fok et al.,3 Basu et al.,4 and Durhamand Padgett.5 Padgett and Spurrier6 and Padgett et al.7 studied the failures of carbon fiber composites. Keshvan et al.8 modeledthe fracture strength distribution of glass and the distribution of the size of Antarctic icebergs with the Weibull distribution,respectively. More examples on the application of the Weibull distribution in other subjects can be found in Murthy et al.9
For monitoring and improving processes, it is likely that Shewhart X and R (or S2) control charts are the most common among a widerange of alternatives. Nevertheless, when the quality variable follows an asymmetric distribution such as the Weibull distribution, thesecontrol charts can be misleading. Padgett and Spurrier’s6 simulation studies showed that when the process distribution is Weibull, thetype I error rate of X and R charts is significantly higher than what it is designed for. Moreover, they concluded that monitoringpercentiles is even more important than monitoring the process mean and variance. Hence, they proposed using Shewhart control chartsfor monitoring the percentiles of the Weibull distribution, assuming that the scale and shape parameters were unknown. Bai and Choi10
presented a heuristic method for designing X and R control charts for skewed distributions including the Weibull. Ramalhoto andMorais11 designed Shewhart control charts with fixed and variable sampling interval schemes for detecting shifts in the scale parameter.They assumed that the shape and threshold parameters are known and fixed. Nichols and Padgett12 presented a bootstrap approach toestablish lower and upper control limits for monitoring Weibull percentiles. Their results indicate that their chart detects out-of-control(OOC) situations quicker than Padgett and Spurrier’s.6 Recently, Pascual and Zhang13 proposed Shewhart control charts for monitoringchanges in the Weibull shape parameter based on the range of a random sample from the smallest extreme value distribution. Thesecharts can be used to check the validity of the fixed shape parameter assumption that is made in some control charts for monitoring
aHEC Management School, University of Liège, Liège, 4000, BelgiumbDepartment of Business Administration, University of Delaware, Newark, Delaware 19716, USAcInstitute of Statistics, Biostatistics and Actuarial Sciences, Université catholique de Louvain, Louvain-la-Neuve, 1348, Belgium*Correspondence to: Alireza Faraz, HEC Management School, University of Liège, Liège 4000, Belgium.E-mail: [email protected]†Alireza Faraz ([email protected]) acknowledges financial support from Fonds de la Recherche Scientifique—Fonds National de la Recherche Scientifique (F.R.S.-FNRS; postdoctoral researcher grant from ‘F.R.S.-FNRS’, Belgium) and from Interuniversity Attraction Poles (IAP) research network P7/06 of the Belgian Government(Belgian Science Policy).‡C. Heuchenne ([email protected]) acknowledges financial support from IAP research network P7/06 of the Belgian Government (Belgian Science Policy) andfrom the contract ‘Projet d’Actions de Recherche Concertees’ 11/16-039 of the ‘Communaute francaise de Belgique’, granted by the ‘Academie universitaire Louvain’.
the scale parameter. Furthermore, they designed one-sided and two-sided control charts. They concluded that the one-sided charts havebetter performance in detecting specific shifts in the shape parameters.
In this paper, we propose a new approach for monitoring a Weibull process that is simple, is adaptable to any asymmetric distribution,and has good statistical properties. We assume that the scale and the shape parameters are unknown as they generally are in practiceand can be estimated by initial in-control samples. In our approach, we transform the in-control Weibull distribution to a standardnormal distribution and subsequently monitor the process by Shewhart-type control charts. The paper is organized as follows.
In the next section, we review the Weibull distribution and its properties. A method to transform the Weibull to a normaldistribution is described. Section 3 constructs the control charts for the purpose of process monitoring. An illustrative example isgiven in section 4, and concluding remarks are provided in the last section.
2. The Weibull distribution
Let us assume that the lifetime variable (T) follows a Weibull distribution with the scale parameter η and the shape parameter β, thatis, T~W (η> 0, β> 0).
2.1. A brief review
The probability density function (pdf), cumulative distribution function (cdf), and survival function of the random variable T are givenby (Murthy et al 9)
f t; η; βð Þ ¼ β�t β�1ð Þ
ηβexp � t
η
� �β" #
; t > 0 (1)
F tð Þ ¼ Pr T≤tð Þ ¼ 1� exp � t
η
� �β( )
; t > 0 (2)
F tð Þ ¼ Pr T > tð Þ ¼ 1� F tð Þ ¼ exp � t
η
� �β( )
; t > 0 (3)
It is clear that the shape of the density function depends only on the shape parameter, and the scale parameter has no effect. Ifβ =1, the Weibull distribution is identical to the exponential distribution with parameter 1/η. If β =2, it is identical to a Rayleighdistribution. The hazard and cumulative hazard functions are given as follows9:
h tð Þ ¼ f tð Þ1� F tð Þ ¼
βtβ�1
ηβ(4)
H tð Þ ¼ ∫t0h xð Þdx ¼ t
η
� �β
(5)
Like the pdf, the shape of the hazard function depends only on the shape parameter β. The hazard function h(t) is constant if β =1,decreasing if β< 1, and increasing if β> 1. It is easy to show that the kth moment about the origin is given by9
E Tk� � ¼ ηk�Γ 1þ k
β
� �(6)
Hence, the mean and variance of T are given by
E Tð Þ ¼ η�Γ 1þ 1
β
� �(7)
Var Tð Þ ¼ η2 Γ 1þ 2
β
� �� Γ2 1þ 1
β
� �� �(8)
Furthermore, the p-percentile of T is given as follows:
tp ¼ η � ln 1� pð Þ½ ��1=β (9)
2.2. Weibull to normal transformation
DefinitionLet T be a continuous random variable. The cdf F(t) = Pr(T ≤ t) is uniformly distributed on the interval (0,1). Let U be uniformlydistributed on (0,1) and F(T) =U. Then, the random variable T= F�1(U) has the cdf F(T). That is, Pr(T≤ t) = Pr(U ≤ F(t)) = F(t).
Definition 1 allows us to generate different continuous distributions from the uniform distribution. For example, it can be easilyshown that if U is uniformly distributed on (0,1), then the random variable T= η(�lnU)1/β, η> 0 , β> 0, follows a Weibull distributionwith the desired scale and shape parameters η and β, respectively. Furthermore, Definition 1 allows us to convert any uniformdistribution to any desired continuous distribution and vice versa. For example, Villanueva et al.14 transformed a normal distributionto a Weibull distribution. However, in this paper, we are interested in converting random Weibull observations to observations havingthe standard normal distribution, N(0,1). The standard normal cdf is defined as follows:
Φ zð Þ ¼ 1
2þ 1ffiffiffiffiffiffi
2πp ∫
z0e
�u2
2 du ; z ≥ 0 (10)
In statistics, it is easier to rewrite Eqn (10) in closed form based on the error function erf as follows15:
Φ zð Þ ¼ 1
21þ erf
zffiffiffi2
p� �
(11)
where erf zð Þ ¼ 1ffiffiπ
p ∫z�ze
�u2
du. The derivative of the error function and its inverse are given as follows15:
d
dzerf zð Þ ¼ 2ffiffiffi
πp exp �z2
� �(12)
d
dzerf�1 zð Þ ¼
ffiffiffiπ
p2
exp � erf�1 zð Þ� �2n o(13)
Now, let F(t) and Φ(z) be the cdf of the Weibull and standard normal distributions, respectively. According to Definition 1, we definetwo new uniform variables ut= F(t) and uz Φ(z). In order to transform the Weibull to a standard normal distribution, we match Eqns (2)
and (11) and by solving this equation with respect to z, we have
z ¼ WN t; η; βð Þ ¼ffiffiffi2
perf�1 1� 2exp � t
η
� �β( ) !
(14)
Note that this method can be applied to transform any nonnormal and continuous distribution to the standard normal distribution.Figure 1 shows the pdf of the three different Weibull distributions and their transformation to the standard normal distribution. For
example, suppose that a process follows the in-control Weibull distributionW (η0 =1, β0 =1.5).W (η1 = 2, β0 = 1.5) indicates a shift in thescale parameter, and W (η1 = 2, β1 = 1) indicates a shift in both Weibull parameters. Figure 2 shows the normal probability plot for theOOC Weibull distributions with respect to the normal transformation WN(t, η0, β0) for a sample of 50 observations. It is apparent thatwhen the Weibull parameters change, the transformed normal distribution would be a long-tailed distribution.
2.3. The relationship between the Weibull and normal parameters
In this section, we derive the relationship between the Weibull parameters and the mean and variance of the standard normaldistribution. First, we need to calculate the inverse of the WN(t, η, β) function and its derivative as follows:14
t ¼ W�1N z; η; βð Þ ¼ η �ln
1� erf zffiffi2
p �
2
0@
1A
24
351=β
(15)
dt
dz¼
ffiffiffi2
π
rηβ
exp � z2
2
n o1� erf zffiffi
2p � �ln
1� erf zffiffi2
p �
2
0@
1A
24
351= β�1ð Þ
(16)
Figure 1. Probability density functions for different Weibull and standard normal distributions
Figure 2. Normal probability plot for (a) W(2,1) and (b) W(2,1.5)
A. FARAZ, E. M. SANIGA AND C. HEUCHENNE
By using Eqns (1), (15), and (16), we obtain the pdf of the standard normal distribution and its mean and variance as a function ofthe Weibull parameters as follows:
φ z; η; βð Þ ¼ f W�1N z; η; βð Þ� � dt
dz(17)
μ ¼ E Z; η; βð Þ ¼ ∫zφ z; η; βð Þdz (18)
σ2 ¼ Var Z; η; βð Þ ¼ ∫ z � μð Þ2φ z; η; βð Þdz (19)
Equations (18) and (19) allow us to measure the actual shift size in the mean and variance of the standard normal distribution as aresult of any shift in the Weibull parameters. Note that when the Weibull process is in control, we have μ0 = 0 and σ2
0 = 1. If theWeibull process shifts toW(η1, β1), then the corresponding shift in the mean and variance of the normal distribution can be calculatedby μ1 =E (Z, η1, β1) and σ2
1 = Var (Z, η1, β1), respectively. It is obvious that a shift in the scale or shape parameter results in a shift in themean and/or standard deviation of the normal process. Hence, in the next section, we propose a joint monitoring scheme forcontrolling the normal process parameters.
3. Monitoring lifetime variables
In this section, we present the control procedure for monitoring the Weibull process. Suppose that Ti1, Ti2,…, Tin are n random samplesfrom the Weibull distributionW (η, β) at sampling time i (i= 1, 2,…). In practice, the Weibull parameters are unknown, and they usuallyshould be estimated from past data using the usual approach of takingm rational samples of size n from the in-control process. Thereare different methods to estimate the Weibull parameters, and readers are referred to Murthy et al.9 Let η0 and β0 be the estimatedscale and shape parameters, respectively. Then, Zi1 =WN(Ti1, η0, β0),…, Zin =WN(Tin, η0, β0) define random samples from the standardnormal distribution N(0,1). The ith subgroup mean and variance are calculated and then are charted on the Shewhart control chartsZ and S2 for the monitoring purpose. A search for assignable causes begins when at least one of the charts signals. The upper, center,and lower control limits of the Z and S2 charts are given by
where zp and χ2p are the p-percentile of the standard normal and the chi-square distributions, respectively. We assume without loss ofgenerality that both charts have the same type I error rate γ.
The joint probability of type I (α) and type II (β) error probabilities is calculated as follows:16
α ¼ 2γ� γ2 (22)
β ¼ βz�βs (23)
where
βz ¼ 1� Φzγ=2 � μ1
ffiffiffin
p
σ1
� �� Φ
zγ=2 þ μ1
ffiffiffin
p
σ1
� �(24)
βs ¼ F χ21�γ=2 n–1ð Þ=σ21; n–1
�� F χ2γ=2 n–1ð Þ=σ2
1; n–1 �
(25)
and F(x,v) is the cdf of the central chi-square distribution at point x and v degrees of freedom. We then calculate the in-control andOOC average run length (ARL) of the joint control charts as follows:
ARL0 ¼ 1=α (26)
ARL1 ¼ 1=β (27)
Equations (20) to (25) can be adjusted to one-sided Z and S2 charts to detect shifts in a specific direction more quickly. Fordetecting small shifts, one can employ cumulative sum or exponentially weighted moving average charts.
4. Illustrative examples
Suppose that the breaking strength of a fibrous composite follows the Weibull distribution with parameters η0 =1 andβ0 =1.5. UsingEqns (18) and (19), Tables I and II give the corresponding shifts in the mean and standard deviation of the normal variable for thegiven shifts η1, β1 = 0.25(0.25)3, 4, and 5, respectively. The results indicate that when the Weibull process is in control (η0 =1,β0 =1.5), the transformed observations (the normal process) follow a standard normal distribution, that is, μ0 = 0 and σ0 = 1, and a shiftin the scale and/or shape parameter can cause a shift in the mean and/or variance of the normal variable.
If only the Weibull shape parameter changes (η0, β1), the imposed shift sizes μ1 and σ1 depend on the direction and severity of β1.Positive shifts in the shape parameter (β1> 1.5) cause very small to small changes in the normal process mean varying from 3 up to19%, while the normal standard deviation decreases 13 to 66%. For negative shifts in the shape parameter (β1< 1.5; especially whenthe shape of the Weibull hazard function changes from increasing to decreasing (β1< 1)), the normal distribution dispersion increasesby at least 124% with a maximum change equal to 851% for β1 = 0.25. Again, except for the case where β1 = 0.25, negative changes inthe shape parameter do not have severe effects on the normal process mean.
If we fix the shape parameter and vary the scale parameter, that is, (η1,β0), negative shifts in the scale again cause greater shifts inthe normal parameters than the corresponding positive shifts in the scale parameter. Moreover, the larger the shift is, the greater arethe changes in the normal parameters. For example, when η1 = 5, then the normal mean shifts to 3.99, and the normal standarddeviation increases by 324%.
If both Weibull parameters change, drifts in β do not affect the normal process mean as long as there are small shifts in the scaleparameter. Furthermore, there are some cases that even large shifts in the parameters result in very small shifts (less than 10%) in thenormal standard deviation but moderate to large shifts in the normal process mean.
Table III gives the ARL of the joint Z and S2 control charts for Tables I and II. We fix the type I error rate γ= 0.00135 to ensure that thejoint error rate α= 0.0027 or ARL0 = 370, which is the usual “three-sigma” Shewhart equivalent. The results indicate that joint Z and S2
charts have good performance in detecting different shifts in the Weibull parameters. We do caution that the performance fordetecting very small shifts in the shape parameters is poor when the scale parameter is in control. Detection of tiny shifts is a notuncommon shortcoming of Shewhart-type charts but not a shortcoming that has prevented them from enjoying an overwhelmingpopularity in practice because of their ease of use and the benefits that they provide.
Table IV presents the data given in Padgett and Spurrier.6 Here, 20 samples of size n=5 were taken to monitor the breakingstrength of carbon fibers. The data are measured in gigapascal. The first 10 samples are taken while the Weibull process was assumedin control. We used the wblfit function in MATLAB to get maximum likelihood estimates that are η0 = 3.2 and β0 = 4.80. Thecorresponding transformed normal variables with the subgroup averages and standard deviations are also given in Table IV. Figure 3
shows the joint Z and S2 control charts for the carbon fiber data. The S2 chart gives an OOC signal on sample 13, just three samplesafter in-control samples, indicating a significant increase in σ2. The Z chart expresses no alarm.
Tables I and II can also be used as a guideline to interpret the OOC signals. Having a stable process mean and significant increase inthe variance suggests a dramatic negative change in the Weibull shape parameter, β1< 4.8. Samples 14–20 indicate a negative shift inμ and a periodic increase in σ2. Therefore, one can conclude that a negative shift in both Weibull parameters has occurred. Byanalyzing the last 10 samples, the maximum likelihood estimates of the Weibull parameters are η1 = 2.60 and β1 = 2.00.
Figure 3. The joint control charts for carbon fiber data
A. FARAZ, E. M. SANIGA AND C. HEUCHENNE
5. Concluding remarks
It has been shown that there are problems with applying Shewhart Z and S2 control charts to monitor Weibull processes and otherasymmetrically distributed processes. In this paper, we have provided a new approach in monitoring the Weibull and other
asymmetric processes with the use of Shewhart Z and S2 control charts used jointly. We have shown how these charts can beconstructed for monitoring the Weibull scale and shape parameters. We illustrated the performance of our method through simulateddata and its use using a real data set given in Padgett and Spurrier.6 We showed that a change in Weibull parameters results indifferent changes in the mean and variance of the transformed normal variables and that the charts have good performance indetecting shifts in the Weibull parameters. Finally, we showed how one can interpret OOC signals given by these charts.
This method can be considered as an asymmetric counterpart to Shewhart’s control charts that enjoy great popularity inapplications because of their ease of design, use, and strength in process monitoring.
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Authors' biographies
Alireza Faraz is an F.R.S-F.N.R.S researcher at HEC management School-University of Liège, Belgium. His research interests areStatistical Process Control, Fuzzy Sets and Systems, Industrial Statistics, Six Sigma, and Total Quality Management. He holds a PhDin Industrial Engineering from Tarbiat Modares University (Iran) and a M.Sc. degree in Statistics from Isfahan University of Technology(Iran). He has published around 40 papers in ISI journals like European Journal of Operation Research, Fuzzy Sets and Systems, Qualityand Reliability Engineering International, Quality and Quantity, Statistical Papers, Journal of Statistical Computation and Simulation,Applied Statistics, and in proceedings of international conferences.
Erwin Saniga is a Dana Johnson Professor of Information Technology, Professor of Operations Management Department at theUniversity of Delaware. He received his PhD from the Department of Management Science at the Pennsylvania State University. Hehas published more than 40 papers in the fields of management science and statistics in research journals including Institute forIndustrial Engineering Transactions, Management Science, Journal of the American Statistical Association, Technometrics, and Journalof Quality Technology. He is currently an associate editor of the Journal Economic Quality Control and on the editorial board of theInternational Journal of Quality and Productivity Management.
Cédric Heuchenne is a professor of Statistics at HEC Management School - University of Liège and also teaches at the CatholicUniversity of Louvain. His research interests focus on nonparametric statistical inference for comlex data structures and especiallyincomplete data. Many examples of these data are provided not only by health sciences, industry, and astronomy but also Economicsand Finance. His work led to various communications, chapters of books, and internationally peer-reviewed articles. He holds aMaster’s degree in Applied Sciences and a PhD in Statistics from the Catholic University of Louvain.
