Reliability Analysis of Repairable Systems with Dependent Component Failures under Partially Perfect Repair Qingyu Yang & Nailong Zhang Department of Industrial and System Engineering Wayne State University Detroit, MI 48202, USA Yili Hong Department of Statistics Virginia Tech Blacksburg, VA 2406, USA Abstract: Existing reliability models for repairable systems with a single component can be well applied for a range of repair actions from perfect repair to minimal repair. Establishing reliability models for multi-component repairable systems, however, is still a challenge problem when considering the dependency of component failures. This paper focuses on a special repair assumption, called partially perfect repair, for repairable systems with dependent component failures, where only the failed component is repaired to as good as new condition. A parametric reliability model is proposed to capture the statistical dependency among different component failures, in which the oint distribution of the latent component failure time is established using copula functions. The model parameters are estimated by using the maximum likelihood method, and the maximum likelihood function is calculated based on the conditional probability. Based on the proposed reliability model, statistical hypothesis testing procedures are developed to determine the dependency of component failures. The developed methods are illustrated with an application in a cylinder head assembling cell that consists of multiple stations.
Keywords: Competing risks, recurrent events, Archimedean copula, Gaussian copula, lognormal, Weibull. ACRONYMS ML Maximum likelihood cdf Cumulative distribution function pdf AIC NLV Probability density function Akaike information criterion Negative log-likelihood value NOTATIONS N( t ) The number of system failures of all components occurring in the time interval (0, t ] r ( t ) The most recently observed failure time of component occurring before time t K τ The number of failure types The predetermined ending time for failure observation (i.e., last follow up time) 2
T i Random variables to denote the failure time of the th i failure t i Observed failure time of the th i failure ( ) a t, the age of component at time t since its last repair D The random vector of the latent ages to failure of all components D = [,,, ] T, where D k is the latent age to failure of the i th D D2 D K component µσ, µ = [ µ,, µ ] T K, and σ = [ σ,, σ ] T K, denoting the location, and scale parameters in the Weibull marginals, respectively M A lower triangular matrix with positive diagonal elements ω A matrix of angles with dimension K ( K ) C u u K (,, ) Copula function with K marginal distributions λ Lθ ( ) ˆθ Σ The association parameter in the Gumbal-Hougaard copula Likelihood function for unknown parameter vector θ ML estimator of θ Covariance matrix of multivariate lognormal distribution and Gaussian copula Λ Correlation matrix of multivariate lognormal distribution and Gaussian copula 3
. Introduction A repairable system is defined as a system that can be restored to fully satisfactory operation by parts replacements or adustments when failing to perform one or more of its functions satisfactorily []. Traditional study on repairable systems mainly focuses on reliability models for single-component repairable systems under different repair actions. A comprehensive review on statistical methods of single-component repairable systems is provided by Lindqvist [2]. In recent years, more attention has been put on the reliability modeling for multi-component systems under competing risks. The competing risks model assumes that each component s failure will cause the whole system s failure, thus a repair action should be taken whenever a failure occurs regardless of the failure type. A real example of a repairable competing risks system can be found in [3]. For reliability models under competing risks, most of the existing research assumes statistical independency of component failure [3]-[2]. Thus, the reliability analysis of the entire system subect to competing risks can be simplified by analyzing each component individually. When considering component failure dependency, most existing reliability models [3]-[20] assumes that, when a failure of one component occurs, it will result in a possible shock to the other components with a certain probability. In this paper, we establish reliability modeling of repairable complex systems subect to statistically dependent competing 4
risks due to degradation. Pan and Balakrishnan [2] studied a reliability model where the bivariate degradation of two statistically dependent performance characteristics are modeled by a gamma process. Other research [22]-[24] discussed the dependency between failures due to random shocks and degradation processes. However, these methods cannot deal with the dependency of failures due to different degradation processes. To deal with the dependency of competing risks in complex systems, copula functions have been applied in the literature [25]-[28]. These works, however, are only capable of dealing with competing systems that are either non-repairable or perfectly repaired. In this paper, we study repairable statistically dependent competing risks systems under a special repair action called partially perfect repair, i.e., only the failed component, rather than the whole system, is replaced whenever a failure happens [5], [7]. The partially perfect repair action is general for many complex multi-component engineering systems in real applications. For a simple example, if the bumper of a vehicle has a failure, only the failed bumper will be repaired, while the other parts of the vehicle will still be used. On the other hand, under most situations, it is more economical to replace the failed component (e.g., bumper), rather than repair it, due to high labor cost. For repairable competing risks systems under a perfect repair assumption, after each repair action, the system will have the same failure mechanism as the original one. In 5
contrast, under partially perfect repair actions, the replacement of a failed component affects the failure mechanism of all the other components that will be continually used in the system. Thus, the methods developed in the literature cannot be directly applied. In this research, a parametric statistical model is established to capture the statistical dependency of repairable multiple component failures under partially perfect repair assumptions. In the proposed model, the oint distribution of components latent ages to failure is established using copula functions. In particular, two types of copulas, i.e., Archimedean copula [29], and Gaussian copula [30], are applied to study the overall statistical dependency and pairwise statistical dependencies among different components, respectively. The framework of the paper is as follows. After the introduction, Section 2 proposes a general statistical model for multiple statistically dependent competing risks. When different marginal distributions of latent ages to failure are selected, different parametric forms of the proposed model are investigated in Sections 3. In Section 4, the model parameters are estimated by using the maximum likelihood (ML) method, and the ML function is calculated based on the conditional probability. Hypothesis testing procedures are developed in Section 5 to determine the failure dependency. Section 6 applies the proposed methods to an application of an automobile cylinder head assembly cell. Finally, Section 7 gives the summary. 6
2. Statistical multiple dependent competing risks model We consider a single repairable multi-component system, i.e., the failure data are collected from only one realization of the repairable system. The research results can be easily extended to multiple systems. Suppose the system consists of K different components or subsystems corresponding to K different types of failures. The system is under competing risk, and the repair action is assumed to be a partially perfect repair. In addition, it is assumed that two or more failures do not occur simultaneously. The observed failure events include failure time t, t 2,..., and associated failure types, which occur before a pre-determined time τ. The counting process, denoted by N( t ), records the number of failures regardless of failure types occurring in the time interval (0, t ]. Consider a system that consists of K new components starting to work at time 0. Because the components are under competing risks, the system fails if any component fails, while the failure time of all the other components cannot be observed. The latent age to failure of each component is denoted in this paper as a random variable D ; =, 2,..., K, with the oint survival function S(d,, d K ) = Pr(D > d,, D K > d K ). Let r ( t ) be the most recent failure time of component before time t. The running time of component at time t since its last replacement, which is defined as age and is denoted as a ( t ), can be calculated as a ( t) = t r ( t). Note that a ( t ) is a left-continuous function. The next failure time of the system on or after time t is 7
determined as the minimum value of { r ( t) + D, =,2,..., K } under the condition that D > a ( t), =,2,..., K. Fig. illustrates the first two failures in a competing risks T system. Let vector d,,..., d, K be the first realization of the latent ages to failures D. The first failure is determined by the minimal value of d ;, i i =,..., K. Suppose the first failure is due to component i, and occurs at time t (Fig. left). Under partially perfect repair assumptions, only component i is replaced, and the most recent failure times are updated as ri ( t) = t, and r ( t ) = 0, i, =,..., K. The second failure can be calculated as the minimum value of { d2, i + t, d2, ; i, =,..., K} (Fig. right), where T d2,,..., d 2, K is the second realization of D. d, d 2, r ( t ) = 0 r ( t 2) = 0 t 2 d,i d 2,i r ( t ) = 0 i t r ( t ) = t i 2 d,k d 2,K r ( ) 0 K t = r3 ( t 2) = 0 Fig.. Illustrations of the first (left), and the second (right) failures in a competing risks system. 3. Parametric forms for the proposed model The oint distribution of the random vector D describes the statistical failure mechanism of multiple components, and thus captures their statistical failure dependency. In this section, parametric models are proposed to characterize the oint distribution of D. 8
Lognormal and Weibull distributions are from the log-location-scale family, and are commonly-used as failure-time distributions. In this paper, they are chosen as the marginal distribution of random vector D. It is well known that, when the oint distribution of a random vector is multivariate lognormal, the marginal distributions are also lognormal. As for a multivariate Weibull distribution, we use a copula function to link the oint distributions to the marginal distributions. 3. Parametric forms for multivariate lognormal distribution When the oint distribution of random vector D is multivariate lognormal, the oint probability density function (pdf) is calculated as T p( d,, d K ; µσ, ) = exp /2 [ log( ) ] [ log( ) ] K /2 d µ Σ d µ () (2 π ) Σ 2 The model parameters θ include µ and Σ. K µ R, and K K Σ R are the mean, and covariance matrix of the multivariate lognormal, respectively. 3.2 Parametric forms for a multivariate Weibull distribution via Archimedean copula The Archimedean family of copulas are frequently used for the construction of multivariate distributions due to their simple forms [29] C( u,, uk ) = ψ ψ ( u) + + ψ ( uk ) (2) 9
where ψ is the generator of the Archimedean copula. Different generators will generate different Archimedean copulas. For example, ψ ( t) / λ = ( + t), and ψ = / λ ( t) exp( t ) are generators for Clayton, and Gumbel-Hougaard copulas, respectively. In this paper, the Gumbel-Hougaard copula is selected as an example to illustrate the application of the Archimedean copula family in the proposed reliability model. The K-dimensional Gumbel-Hougaard copula has the form { } λ λ C( u,, uk ) exp [ log( u )] [ log( uk )] λ = + + (3) where λ is called the association parameter. To construct the multivariate Weibull distribution, we choose the log-location-scale form [3] of the Weibull distribution as marginal distributions of D, i.e., log( d ) µ F ( d; µ, σ ) = Φ SEV σ (4) where Φ ( x SEV ) = exp[ exp( x )] is the cumulative distribution function (cdf) of the standard smallest extreme value distribution. Let F( d; µ, σ ) in (4) be u ; =,..., K, and substitute (4) into (3); then the oint cdf of random vector D following a multivariate Weibull distribution is obtained as λ λ K log( d ) µ F( d,, dk ; µ,σ, λ) = exp log Φ SEV (5) = σ 0
In this multivariate Weibull distribution constructed via the Gumbel-Hougaard copula, the model parameters θ = ( µ,σ, λ). 3.3 Parametric forms for multivariate Weibull distribution via the Gaussian copula The Gaussian copula is a special copula taking advantage of the pdf of the multivariate normal distribution. Specifically, a Gaussian copula has the form Gauss C ( u,,, u K ) = Φ Σ[ Φ ( u),, Φ ( u K )] (6) where Φ is the inverse of the cdf of the standard normal distribution, and Φ Σ is the cdf of a multivariate normal distribution with zero mean vector, and its covariance matrix equals its correlation matrix. The Gaussian copula density function is given as [32] T Φ ( u) Φ ( u ) Gauss c ( u,, uk ) = exp ( ) /2 2 Σ I (7) Σ Φ ( uk ) Φ ( uk ) where Σ is the correlation matrix, and I is the identity matrix. When applying a Gaussian copula to construct the oint Weibull distribution, the survival function of random vector D is obtained as = Gauss K µσσ K K d dk S( d,, d ;,, ) f ( x,, x ) dx dx (8) Gauss where f ( x,, x K ) denotes the pdf of the oint Weibull distribution obtained by the chain rule, i.e.,
K Gauss Gauss C du du f ( x,, xk ) = u u dx dx K Gauss = c ( u, u ) f ( x ) f ( x ) K K K K, (9) and f ( ) denotes the pdf of the marginal. In the multivariate Weibull distribution constructed via the Gaussian copula, the model parameters θ = ( µ,σσ, ). 4. Parameters estimation based on ML approach An ML method is developed in this paper to estimate model parameters. To implement the ML approach, we first calculate the likelihood function. Suppose the th i failure is due to component k, and occurs at time t i. The latent age to failure of the failed component k is equal to a ( t ), while the latent ages of all other components k i should be larger than a ( t ); k, =,..., K. Thus, the unconditional probability to i observe failure i is calculated as Pr( D = a ( t ), D > a ( t ); k, =,..., K) k k i i S[ a ( t ),, D,, a ( t )] (0) = i k K i D k Dk = ak ( ti ) Note that (0) gives the probability at a given time t for a continuous random variable. Although technically Pr( X = t) = 0 for a continuous random variable X with pdf f ( t ), Pr( X = t) can be interpreted as Pr( t X t + dt) = f ( t) dt, which is proportional to f ( t ). We ignore dt only for notational convenience in the calculation of the likelihood function for all continuous random variables in the rest of the paper. 2
Equation (0) only accounts for the probability of an observed failure at time t i regardless of previous failure data. The conditional probability of observing failure i given all the previous i- failures is solely determined by the ages of all component after the repair action of the (i-) th failure. Specifically, the likelihood of failure i, i =,2,, N( τ), conditioned on all the previous i- failures, can be calculated as L i [ ( ),,,, ( )] S a ti Dk ak ti D = S[ a ( t ),, a ( t )] k Dk = ak ( ti ) + + i K i () where, S( ) is the oint survival function of the latent ages to failures. For example, consider the first two failures illustrated in Fig.. After the repair action of the first failure at time t, the age of the failed component i is updated to a ( t + + ) = 0, while the ages of all the other components are given as a ( t ) = t, i i, =,..., K. As the second failure is due to component one, and occurs at time t 2, the likelihood of the second failure conditioned on the first failure can be calculated as Pr( D = a ( t ), D > a ( t );, =,..., K D > 0, D > t ; k i, k =,..., K ), where 2 2 i k ai ( t2) = t2 t, and ak ( t2 ) = t2; k i, k =,..., K. As there is no failure observed from t to the predetermined ending time τ, the N ( τ ) likelihood LN ( τ ) + can be calculated as L Pr{ D > [ τ r ( τ )],..., D > [ τ rk ( τ )]} = (2) [ ( ),..., ( )] K N ( τ ) + + + S a tn ( τ ) ak tn ( τ ) 3
S[ a ( t ),..., a ( t )] =. S[ a ( t ),..., a ( t )] τ K τ + + N ( τ ) K N ( τ ) Combining the results in () and (2), the following Proposition can be used to calculate the likelihood function based on the observed failure data. Proposition. Given the observed failure data, the likelihood function can be calculated by N ( τ ) + L( θ ) = Li (3) i= where L can be calculated based on (), and (2) for i =, 2,, N( τ ), and i i = N( τ ) +, respectively. The estimated model parameters ˆθ are obtained by maximizing Lθ ( ). Based on the ML theory [33], [34], the estimated parameters ˆθ are asymptotically normally distributed with a large sample assumption. In the following sections, we will discuss how to apply the developed ML methods for different parametric models. 4. ML method applied for multivariate lognormal distribution When the oint distribution of D is selected as multivariate lognormal, the oint survival function S( ) can be calculated by integrating (), i.e., = d dk S ( dµ ;, Σ) p ( xµ ;, Σ) dx dx. (4) K 4
The likelihood function Lθ ( ) can be obtained by substituting (4) into () and (2). To calculate L i, the first order partial derivative of the survival function in (4) needs to be evaluated. We extend the first order partial derivative of the survival function as S[ a ( ti ),, Dk,, ak ( ti )] D k D = a ( t ) k k i = f ( a ( t )) Pr( D > a ( t ); k D = a ( t )) k k i i k k i (5) In (5), fk ( ak ( t i )) denotes the marginal lognormal probability density of the latent age to failure of component k. The detailed calculation of the conditional probability Pr( D > a ( t ); k D = a ( t )) is given in the Appendix. i k k i A challenge to implement the parameter estimation of the multivariate lognormal comes from the positive definite constraint of the covariance matrix Σ. We use Cholesky decomposition to achieve this constraint. Specifically, if M is a lower triangular matrix with positive diagonal elements, Σ = M T M is positive definite. Therefore, we directly estimate the lower triangular matrix ˆM, from which the covariance estimator ˆΣ can be calculated. 4.2 ML method applied for the multivariate Weibull distribution via the Archimedean copula When the oint Weibull function is constructed via a Gumbel-Hougaard copula, we need to substitute the survival function of random vector D into () and (2) to implement the ML approach. The cdf of random vector D is given in (5). Based on (5), 5
the survival function can be calculated using (25), and the details to obtain (25) are given in the Appendix. Thus, the likelihood function Lθ ( ) is calculated by substituting the survival function (25) into () and (2), where θ = ( µσ,, λ). A scale-invariant variable, called Kendall s tau, and calculated as τ = 4 E( C( U, V )), is generally used to measure the dependency in the bivariate C copula. For the Gumbal-Hougaard copula, Kendall s tau can be calculated based on association parameter λ [29], i.e., τ = λ. (6) Kendall s tau can only capture the overall statistical dependency of component failures. To capture the pairwise dependencies of component failures, the Gaussian copula function is applied to derive the oint distribution of random vector D, which will be discussed in the next section. 4.3 ML method applied to a multivariate Weibull distribution via the Gaussian copula Similar to the parameters estimation for the multivariate lognormal distribution, the positive definite constraint is also required for the multivariate normal distribution. The Cholesky decomposition, however, fails to construct the positive definite correlation matrix due to the additional constraint that requires a correlation matrix equal to the covariance matrix. To tackle this problem, we use a hypersphere decomposition proposed 6
by Rebonato [35], which provides an approach to construct the valid correlation matrix satisfying all the constraints. Specifically, based on an arbitrary matrix of angles ω of dimension K ( K ), matrix B of dimension K K is constructed by the following rules. Bi, = cosωi, sin ωi, k (if =,, K ) k= Bi, = sin ωi, k (if = K) k= (7) where the subscripts i and are used to represent the element s position in matrix B. Then T Σ = B B is a valid correlation matrix satisfying both constraints. Therefore, the likelihood function is a function of θ = ( µσω,, ). In addition, the correlation matrix is an immediate measure of failure dependency. 5. Hypothesis testing for dependency Based on the proposed reliability model for the competing risks systems, statistical hypothesis testing procedures are developed in this section to determine the component failure dependencies. In Section 5., a dependency test based on the multivariate lognormal distribution is proposed. Then in Section 5.2, a dependency test for the multivariate Weibull distribution derived by the Archimedean copula and the Gaussian copula are discussed, respectively. 7
5. A dependency test for the multivariate lognormal distribution In the multivariate lognormal distribution, the dependency information is captured by the correlation matrix Λ, which can be calculated based on covariance matrix Σ : Λ i, Σ i, = i = Σ Σ i, i, ;,,, K (8) where Λ, and i, Σ are the elements of the i, th i row, and the th column in Λ, and Σ, respectively. In addition, T Σ = M M, and M is the lower triangular matrix. Based on the correlation matrix Λ, a hypothesis testing procedure is developed to determine the statistical dependency among latent ages to failures of components. H 0 : component failures i, are statistically independent. H : component failures i, are statistically dependent. (9) When the asymptotic result is applied, we use a normal approximation to construct the test statistics. The test statistics W are used to test the dependency between i, component failures i and, which equals the estimate of correlation Λ ˆ i, divided by its estimated standard error var( Λ ˆ i, ), i.e., W =Λˆ / var( Λ ˆ ) (20) i, i, i, where the asymptotic variance var( Λ ˆ i, ) is calculated using the delta method, and the detailed calculation is given in the Appendix. In hypothesis testing (9), H 0 is reected if Wi, Z α / 2 >, or Wi, <, where α is the test significance level, and Z / 2 α /2 is the Z α 8
upper quantile of the standard normal distribution. Based on hypothesis testing (9), pairwise statistical dependencies between different component failures can be tested. 5.2 A dependency test for multivariate Weibull distributions In this section, a statistical dependency test for oint distribution D constructed via the Archimedean copula is first discussed. The Archimedean copula can capture the overall statistical dependency, which is determined by the global association parameter λ. We developed the following hypothesis testing procedure to test the overall failure dependency, i.e., to see whether Kendall s tau is equal to 0. H 0 : all failure types are statistically independent. H : not all failure types are statistically independent. (2) Here, the asymptotic test statistic W overall is constructed using the estimate of Kendall s tau ˆ τ divided by its estimated standard error var( ˆ τ ). H 0 is reected if Woverall > or Woverall < /2. As Kendall s tau is a function of the association Z α /2 Z α parameter λ, the variance of the estimate of Kendall s tau ˆ τ can be calculated by using the delta method. dτ T var( ˆ) ( ) var( ˆ dτ τ = λ)( ) dλ dλ var( ˆ λ) = ˆ4 λ λ = ˆ λ (22) 9
where var( ˆ λ ) denotes the asymptotic variance of ˆλ, which is calculated in the same way as the asymptotic covariance matrix of ˆM in the Appendix. Based on (22), the asymptotic test statistic of the overall dependency is given as W overall ˆ τ ˆ2 λ =. (23) var( ˆ λ) For the multivariate Weibull distribution constructed via the Gaussian copula, the correlation matrix determines the pairwise dependencies of latent ages to component failures. Thus, the test procedure is the same as that discussed for the multivariate lognormal distribution in Section 5.. The only difference here is that the correlation matrix Λ is a function of the angle matrix ω for hypersphere decomposition, rather than a function of lower triangular matrix M in Cholesky decomposition. Specifically, the asymptotic variance of the correlation coefficient Λ ˆ i, in (25) is calculated as ˆ Λ Λ var( Λ ) = ( ) Σˆ ( ) ω ω i, T i, i, ωˆ ω= ωˆ. (24) The test statistics for Multivariate Weibull via the Gaussian copula has the same form as (20). Thus, hypothesis testing (9) can be used here to test the pairwise statistical dependencies among different failure types. 6. An application in a cylinder head production process We consider an application from an automobile power-train plant in the United States. The plant has a manufacturing cell that consists of four stations, which are denoted by 20
stations, 2, 3, and 4, to protect proprietary sensitivity information. A cylinder head is assembled by going through all stations sequentially, and all stations move synchronized with each other. The statistical failure dependency among these 4 stations is not well understood, thus we will investigate the dependency based on the proposed methodology. Fig. shows the cumulative number of station failures collected from the cylinder head manufacturing process for about one and a half years. 70 Cumulative number of failures 60 50 40 30 20 Station Station 2 Station 3 Station 4 0 0 0 000 2000 3000 4000 5000 6000 7000 8000 Time/Hours Fig. 2. Failure data from stations, 2, 3, and 4. Given the failure time data of the manufacturing cell, the statistical reliability model is established when the marginal distributions are selected as lognormal and Weibull, 2
respectively. The ML method is used to estimate the model paremeters. All the compuation is done in the R environment. For the numerical optimization methods, we applied both Nelder-Mead and simulated annealing methods. The results obtained from these two methods are very close. Based on the developed reliability model, the hypothesies testing methods can be used to investigate the dependency among stations, 2, 3, and 4. 6. A reliability model with a multivariate lognormal distribution When the oint distribution of D is selected as a multivariate lognormal distribution, the statistical reliability model is established, and the model parameters are estimated by maximizing the likelihood function in (3). The parameter estimates, θ ˆ = ( µ ˆ, M ˆ ), are [ ] µ ˆ = 2.84 2.76 2.0 3.08 2.39 0 0 0 ˆ -0.7355 2.7503 0 0 M = 0.335 0.647 2.8293 0 0.8835 0.6660 0.0409 2.048 The estimate of the covariance matrix, calculated as Σˆ = Mˆ M ˆ T, is given as 5.3449 -.7004 0.7665 2.0425 ˆ -.7004 8.054.520.820 Σ = 0.7665.520 8.5267 0.8360 2.0425.820 0.8360 5.2852 To test the statistical dependency of component failures, the ML estimate of covariance coefficient Λ is calculated using (8), and the asymptotic variance of i, Λ, T ˆ i 22
is obtained based on (26) in the Appendix. The test statistics W obtained by (20) are i, used to calculate the p-values. Table I lists calculated p-values and test results of pairwise statistical dependencies with a 95% confidence level, where indicates statistical dependency, and 0 indicates statistical independency. From Table I, it can be seen that stations (, 2) and stations (, 4) fail statistically dependently. Table I p-values, and statistical dependency tests for the multivariate lognormal distribution Station 2 3 4-0.000 0.437 0 0.0096 2 - - 0.208 0 0.3488 0 3 - - - 0.5468 0 4 - - - - 6.2 A dependency test for the multivariate Weibull distriubtions When the Gumbel-Hougaard copula is applied to derive the oint Weibull distribution, the ML estimates of the parameters in the marginal Weibull distribution [ µσ ˆ, ˆ ] T is calculated as T [ ] T [ ] µ ˆ = 4.40 3.97 4.27 5.20 σˆ =.54.80.87.33 The estimate of association parameter ˆ λ =.0076. Based on (23), Woverall is calculated as.4285. The p-value is 0.532, which indicates that the null hypothesis is not reected. Thus, the dependency test result shows that all failure types are statistically independent. 23
When applying the Gaussian copula to construct the oint distribution, the model parameters θ = ( µσω,, ), including the parameters from both Weibull marginals, and the multivariate normal. The ML estimates are obtained as [ ] [ ] µ ˆ = 3.85 4.22 4.40 4.76 σˆ =.86.92 2.00.52 -.0769 -.4669-2.207-2.0370-2.025-0.8623 ωˆ = -2.897-2.0342-3.0887-0.9554-2.869 -.7982 Based on (7) of the hypersphere decomposition, the estimate of ˆB is calculated based on ˆω as 0.474-0.094-0.5229-0.7025 ˆ -0.4495 0.388 0.5255-0.632 B = -0.5802 0.364-0.7276-0.0385 0.5773 0.7864-0.0495-0.24 Then the correlation matrix estimate ˆΛ in the Gaussian copula is calculated by Λˆ = Bˆ B ˆ T as.0000-0.0920 0.0992 0.3782 ˆ -0.0920.0000 0.04 0.460 Λ = 0.0992 0.04.0000-0.0043 0.3782 0.460-0.0043.0000 To test the statistical dependency of component failures, Table II lists the p-values and test results of pairwise statistical dependency for the Gaussian copula based on (24), and (20). Table II T T 24
p-values, and statistical dependency test when using a Gaussian copula to construct the oint Weibull distribution. Station 2 3 4-0.279 0 0.377 0 0.00 2 - - 0.7674 0 0.2627 0 3 - - - 0.9646 0 4 - - - - Compared with the results from the multivariate lognormal distribution, the statistical dependency result of stations (, 2) is different, while the dependency results of the other 5 pairs are consistent. The difference is due to the different multivariate distributions applied in the statistical reliability model. After the hypothesis testing, if a group of components fails statistically independently from other components, the corresponding observed failure data can be used to estimate their statistical failure distributions individually. The statistical reliability model can be simplified, and the statistical reliability model can be fitted with better precision. Thus, we fit the reliability model for a multivariate lognormal distribution with the restirction that station 3 is statistically independent from all the others. Similarly, the reliability model for the multivariate Weibull distribution is also fitted with the restirction that both station 2 and station 3 are statistically independent from all the other stations. In the literature, the likelihood values, and the Akaike information criterion (AIC) values that take the number of parameters into account are generally used to compare different models for model fitting [26]. Table III lists the negative log-likelihood values (NLV), and AIC values for different reliability models when the oin distributions of D 25
are selected as a mulitvariate lognormal distribution, a mulitvariate Weibull distribution constructed via the Gumbel-Hougaard copula, and a mulitvariate Weibull distribution constructed via the Gaussian copula, respectively. The NLV for the corresponding models are 46.7,.609, and 0.546, respectively; and the AIC values are 232.234, 224.28, and 2239.092, respectively. As smaller NLV or AIC values indicate better model fitting, the performance of the multivariate Weibull distribution constructed via a Gaussian copula is better than the other two models. Thus, the results of the statistical dependency tests based on a reliabity model with Weiblull marginal distributions and a Gaussian copula is selected. We also compare the proposed statistical model that captures component statistical failure dependency to the model that assumes the statistical failure independency in Table III. Under the statistical failure independency assumption, the NLV are 47.963, and 3.42, and the AIC values are 23.926, and 2242.824, when lognormal, and Weibull are selected as the failure time distributions for all stations, respectively. Similar to models developed in this research, the reliability model with a Weibull distribution has a better performance. By comparing NLV and AIC values of the fitted models, it can be seen that the proposed statistical model that captures failure dependency fits the observed data best. Table III NLV, and AIC values for different reliability models multivariate lognormal multivariate Weibull distribution 26
distribution Our model (station 3 fails statistically independently from others) Gumbel-Hougaard copula Gaussian copula (stations 2&3 fail statistically independently from others) NLV: 46.7 AIC: 232.234 NLV:.609 AIC: 224.28 NLV: 0.546 AIC: 2239.092 Independent model NLV: 47.963 AIC: 23.926 NLV: 3.42 AIC: 2242.824 7. Summary In this paper, a general statistical reliability model is proposed for repairable multi-component systems considering statistical dependent competing risks under a partially perfect repair assumption. For the reliability analysis of repairable multi-component systems, most of the research in the literature assumes component failure statistical independence. The failure mechanism (marginal distribution) of each component can thus be estimated individually based on its failure data. This paper focuses on the reliability analysis of repairable systems with component failure dependency. In the developed model, copula functions are used to model the oint distribution of component failure times. Specifically, two types of copulas, i.e., the Archimedean copula, and the Gaussian copula, are applied to study the overall dependency, and pairwise dependencies among different components, respectively. 27
Although the copula function method is also applied in the literature to study non-repairable systems, or systems under perfect repair action (replace the whole system when a failure happens), the problem studied in this paper is much more complex than those in the literature. When the whole system is replaced after a failure, the system will have the same failure mechanism as the original one. In contrast, when only the failed component is replaced, replacement of failed component affects the failure mechanism of the other components when considering failure dependency. Thus, the methods in the literature cannot be directly applied. Under competing risks assumptions, only the failed component is recorded as the latent ages to failures of other components cannot be observed. After a repair action under the partially perfect repair assumption, the failure mechanism of the new system and components will be changed. Thus, for a single repairable system in which the failure data can only be collected from a single realization, model parameters estimation is challenging. To tackle this problem, an ML method is developed in this research, and the ML function is calculated based on conditional probability. The partially perfect repair action is useful for many complex multi-component engineering systems when only the failed component is repaired, and the repair action is a replacement due to high labor cost. In addition, the partially perfect repair can be used to approximate the general imperfect repair actions. But the accuracy of approximation is 28
difficult to be evaluated under most situations. A future research topic is to consider repairable dependent competing risks under general repair actions. Hypothesis testing is developed in this paper to test the statistical dependency of component failures. The obtained statistical failure dependence provides more accurate information for reliability predictions, which can be used for system maintenance. It is also interesting to compare the maintenance performance with different repairable actions, e.g., perfect repair, partially perfect repair, and minimal repair. This work is another research topic that will be studied in the future. Appendix The conditional probability of the multivariate normal distribution Let X with mean µ X denote the latent age to failure of component k, and X 2 with mean µ X 2 denote the latent ages to failure of components, k, =,..., K. The variance and covariance matrix of X, X 2 are denoted by 2 σ, and Σ X 2, respectively. In addition, denote the covariance matrix between X and X 2 by Σ. X,X 2 The conditional distribution of X 2 conditioned on X = x is multivariate lognormal distributed with mean µ + Σ ( x µ ) Σ, and covariance matrix T X 2 X,X2 X X2 T 2 X X / X σ 2,X 2,X2 Σ Σ Σ. Thus, the conditional likelihood in (5) can be calculated as a cumulative probability of a multivariate normal distribution, which is well noted in the literature. 29
The calculation of the survival function of a multivariate Weibull constructed via the copula function It is well known that the cdf and survival function of a bivariate distribution has the following relationship. S( d, d ) = F(, ) [ F(, d ) + F( d, )] + F( d, d ) 2 2 2 Here, we extend this relationship from two dimensions to K dimension as S d µ,σ λ = F F + + F (25) ( ;, ) 0 ( ) K K where F k denotes the summation of cumulative probabilities of the vector d, in which there are exactly K k coordinates replaced by. For example, when K = 3, we can obtain F 2 F d d 2 F d 2 d 3 F d d 3 = (,, ) + (,, ) + (,, ). The calculation of the asymptotic covariance of the ML estimate by using the Delta method Based on ML theory, the ML estimate θ ˆ = ( µ ˆ, M ˆ ) is asymptotically normally distributed with a large-sample assumption [33]. Thus, the asymptotic covariance Σ ˆ θˆ for ˆθ is achievable from the local information matrix ( ˆ Ι θ ), i.e., Σˆ = Ι( θ ˆ). θˆ Ι( θ ˆ) is the negative of the Hessian matrix Hθ ( ) evaluated at θ = θ, ˆ which can be calculated as [33] 2 ɵ log( L( θ)) Iθ ( ) = T θ θ θ= θɵ 30
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[35] R. Rebonato and P. Jäckel, "The most general methodology for creating a valid correlation matrix for risk management and option pricing purposes," The Journal of Risk, vol. 2, pp. 7-28, 999. QingyuYang received the B.S., and M.S. degrees in Automatic Control and Intelligent System from the University of Science and Technology of China in 2000, and 2003 respectively; and the M.S. degree in Statistics, and the Ph.D. degree in Industrial Engineering, from the University of Iowa in 2007, and 2008 respectively. Currently, he is an Assistant Professor in the Department of Industrial and System Engineering at Wayne State University. His research interests include statistical data analysis, reliability and quality, and complex system modeling. He is a member of INFORMS and IIE. Nailong Zhang is a Ph.D. candidate in the Department of Industrial and Systems Engineering at Wayne State University. He received a B.S. degree in Mechanical Engineering from Harbin Institute of Technology, Harbin, China, in 2009. His research interests include reliability & maintenance engineering and optimization, especially reliability modeling and analysis of complex systems under competing risks. Yili Hong received a BS in statistics (2004) from University of Science and Technology of China; and MS, and PhD degrees in statistics (2005, 2009) from Iowa State University. He is currently an Assistant Professor in the Department of Statistics at Virginia Tech. His research interests include reliability data analysis, and engineering statistics. 35