Assessing the reliability of components with micro- and nano-structures when they are part a multi-scale system

Reliability Engineering and System Safety 138 (2015) 13–20

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Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress

Assessing the reliability of components with micro- and nano-structures when they are part a multi-scale system Nader Ebrahimi n, Mahmoud Shehadeh Division of Statistics, NIU, Dekalb, IL 60115, United States

art ic l e i nf o

a b s t r a c t

Article history: Received 19 June 2013 Received in revised form 25 October 2014 Accepted 15 January 2015 Available online 29 January 2015

Products with macro- and meso-scales components have grown into multi-scale systems where some of their components have shrunk to a smaller scale such as micro- and nano-structures. In practice, engineers may be unable to accurately assess their designs to improve the reliability of components in multi-scale systems, because component parts with micro- and nano-structures cannot be realized directly in system design phase. Thus, all kinds of design processes such as engineering verifications for physics or functions become increasingly complex in a muti-scale system. For this shortcoming, we present a parametric Bayesian method that enables engineers to assess indirectly the reliability and the quality of a component with either a micro-structure or a nano-structure using experimental data on failure time of the multi-scale system. Our proposed Bayesian approach is flexible in allowing a general model for distributions of failure times of a multi-scale system and its components. Our method is applied to a simulated data set. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Competing risks Hazard rate function Sub-hazard rate function Multi-scale system Exponential distribution Weibull distribution

1. Introduction Multi-scale systems consist of components from two or more length scales (nano-, micro-, meso- or macro-scales). Examples of such systems are precision instruments [1], fiber optics [2], nanorobots [3] and many more. Generally speaking, any system is characterized by components functions, components interfaces and components arrangements. However, in multi-scale systems design principles and fabrication processes are very complex. Because components should be designed in such a way that they are conceptually and model-wise compatible with other scale components they interface with. A challenging problem here is that while engineers may be able to design such components, often they have difficulty accurately assessing their designs, for example sounding rockets on NASA mission [4], to improve the reliability of components and consequently the reliability of multi-scale systems. The difficulty arises from a situation that components with either micro- and nano-structures cannot be realized directly in system design phase. In fact, with existing knowledge, we often cannot identify the correct component with either micro- or nano-structure that caused failure of a multi-scale system. As an example, nanorobotics refers to the nanotechnology engineering discipline of designing and building nanorobots. Nanorobots are often

n

Corresponding author. E-mail address: [email protected] (N. Ebrahimi).

http://dx.doi.org/10.1016/j.ress.2015.01.015 0951-8320/& 2015 Elsevier Ltd. All rights reserved.

constructed using nano-scale and micro-scale components. To design a reliable nanorobot one needs reliable components and frequently it is impossible to identify the correct nano-scale component that causes failure of the nanorobot. Thus, engineers need statistical tools to assess reliability of components without having access to such information. We should mention that assessing the reliability of a multi-scale system, without considering the structure of the system, was considered by some authors, see for example [5,6]. In this paper we develop a parametric Bayesian approach in the framework of cause specific hazard rate function to assess the reliability and the quality of the design of components with microand nano-structures, when experimental data on failure time of the multi-scale system is available. The notion of cause specific hazard rate function arises in statistical analysis of masked survival data under a competing risks framework. In survival data analysis it is often the case that systems under study can experience any one of several possible causes of failures referred to as competing risks and failure for each system being due to only one failure cause. It is also possible that some of the systems that fail during the experiment have a cause of failure that is only belonging to a certain subset of all possible causes of failure. In this case we say that the actual cause is grouped masked. Masked survival data are quite common in most industrial and medical applications. Crowder [7] provides a recent review of competing risks problem and Sen et al. [8] and Flehinger et al. [9] provide review of masking problem. For more references see also [10–16] and many references cited there.

14

N. Ebrahimi, M. Shehadeh / Reliability Engineering and System Safety 138 (2015) 13–20

Also, for some interesting industrial applications of competing risks see [17,18]. The paper is organized as follows. In Section 2 we describe the model and obtain several properties of it. In Section 3 we present in detail our Bayesian method to assess the reliability of components from failure data on the multi-scale system. For our purpose we consider Exponential and Weibull models for each component. Because of scarcity and secrecy of the available data in this area, in Section 4 we illustrate our methodology using a simulated data. Concluding remarks are given in Section 5.

For the cause specific hazards, we can choose the popular proportional hazards (PH) model where the sub-hazard rate functions in the group i are proportional to each other with proportionality factors hij, that is, hij ðtÞ ¼ hij hi0 ðtÞ;

hij ðtÞ ¼ lim

Pðt r T r t þ h; C ij j T Z tÞ ; h

h-0

In (2.1), Cij means that the multi-scale system failure is due to the component j from the group i; j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4. It should be noted that by the law of total probability, the overall hazard rate function is Gi 4 X X f ðtÞ hij ðtÞ: ¼ F ðtÞ i ¼ 1 j ¼ 1

ð2:2Þ

P i Now, if we define hi þ ðtÞ ¼ Gj ¼ 1 hij ðtÞ; i ¼ 1; 2; 3; 4. Then, hi þ ðtÞ can be interpreted as follows. By combining all the components in the group i into a single super-component, which fails if any of components in this group fails. This super-component has hazard rate function hi þ ðtÞ; i ¼ 1; 2; 3; 4. From Eqs. (2.1) and (2.2) the conditional probability that the failure of a multi-scale system is due to the component j in the group i given T ¼t is given by PðC ij j T ¼ tÞ ¼

hij ðtÞ ; hðtÞ

ð2:3Þ

j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4. Eq. (2.3) gives vital information to engineers about how good the reliability design is for the component j in the group i. For example, if an engineer has two different designs for this component, then the design which gives smaller probability, using Eq. (2.3), is a better reliability design. To clarify ð1Þ

ð2Þ

this, suppose under two designs, hij ðtÞ ¼ hij ðtÞ for j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4, except the component m in the lth group. For this jT ð1Þ ¼ tÞ r PðC ð2Þ jT ð2Þ ¼ tÞ. Then, one can component, suppose PðC ð1Þ lm lm ð1Þ

ð2Þ

hðtÞ ¼

Gi 4 X X

hij hi0 ðtÞ ¼

i¼1j¼1

4 X

hi þ hi0 ðtÞ;

ð2:5Þ

i¼1

and PðC ij jT ¼ tÞ ¼

hij hi0 ðtÞ ; 4 P hi þ hi0 ðtÞ

ð2:6Þ

i¼1

where hi þ ¼

PGi

j¼1

PðC ij j T ¼ tÞ hij ¼ ; PðC ij0 j T ¼ tÞ hij0

hij , i ¼ 1; 2; 3; 4, respectively. Also, ja j0 ; j; j0 ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4:

ð2:7Þ

That is, odds of failing from the component j in the group i in comparison from the component j0 in the same group is free from the time of failure of the multi-scale system and depends only on hij and hij0 . 3. Bayesian method

j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4: ð2:1Þ

hðtÞ ¼

ð2:4Þ

Note that in (2.4), hi0 ðtÞ is the baseline sub-hazard rate function specific to the group i, and is allowed to vary across different groups. The PH model may be specialized if a reason exists to assume that hi0 ðtÞ follows a particular form, say the exponential hazard rate function. From (2.4), Eqs. (2.2) and (2.3) reduce to

2. Model formulation Consider a multi-scale system that consists of M components. We divide these components into four groups. Without loss of generality assume that the first G1 components have macrostructures and they are in group 1. The second G2 components have meso-structures and are in group 2 and the third G3 components have micro-structures and are in group 3, and the remaining G4 ¼ M  G1  G2 G3 components have nano-structures and are in group 4. In order to understand the basic idea, we first need to lay down some notations. Let T be the failure time of a multi-scale system having the hazard rate function h(t) and the survival function or Rt the reliability function F ðtÞ ¼ PðT 4 tÞ ¼ expð  0 hðuÞ duÞ, then one key ingredient in our formulation is the cause specific hazard rate function (sometimes referred to as the sub-hazard rate function), representing the instantaneous risk of failing from the component j in group i and defined as

j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4:

show that hℓm ðtÞ r hℓm ðtÞ. That is, the instantaneous risk of failing from the component m in the group ℓ under the first design is smaller than instantaneous risk of failing from the same component under the second design. Here the superscript ðiÞ stands for the design i; i ¼ 1; 2.

As mentioned in Section 1, masking always occurs in this situation because the exact cause of failure of the multi-scale system that failed will not be known if the actual cause was the components from either group 3 or group 4. Here we may only observe a set of possible causes that we refer to as “masking set”. The straightforward approach to handling this situation is to introduce masking probabilities qðsjC ij ; tÞ ¼ P(masking set S ¼ sj multi-scale system failed due to the component j in the i-th group and T ¼ tÞ into the model. It is usually assumed that these probabilities do not depend on t, i.e. qðsjC ij ; tÞ ¼ qðsjC ij Þ which is a reasonable assumption. Because, component parts having micro- and nano-structures cannot be identified by engineers no matter what the lifetime of a multi-scale system is. [19,20] also cited industrial examples where it is reasonable to make such assumption. In addition to the above assumption, we also assume the following conditions hold. (a) A multi-scale system that failed from the component that belongs to either group 1 or group 2 were observed to fail from that component. This means qðS ¼ C ij j C ij Þ ¼ 1;

j ¼ 1; …; Gi ; i ¼ 1; 2:

Note that engineers usually are able to identify whether any component belonging to either group 1 or group 2 failed or not. Therefore the condition (a) is a valid condition. (b) Define the masking sets s1 ¼ {all the components in group 3}, s2 ¼{all the components in group 4}, and s3 ¼ {all the components in groups 3 and 4}. If the actual cause of a multi-scale system is a component from group 3, then engineers usually have a hard time identifying this component and simply claim that it is one of the components from group 3, i.e., here the actual cause is masked by s1. However, if the actual cause of failure is a component from group 4, then the actual cause is masked by either s2 or s3 and

N. Ebrahimi, M. Shehadeh / Reliability Engineering and System Safety 138 (2015) 13–20

observed data related to the cause of failure is either the set s2 or the set s3. Mathematically,

and

qðS ¼ s1 jC 3j Þ ¼ 1;

F ðtÞ ¼ exp@  @

j ¼ 1; …; G3 ;

ð3:1Þ

0

0

Gi 4 X X

15

11 hij At A;

i¼1j¼1

qðS ¼ s2 jC 4j Þ ¼ 1  qðS ¼ s3 jC 4j Þ ¼ p; qðS ¼ s1 jC lj Þ ¼ 0;

j ¼ 1; …; G4 ;

ð3:2Þ

j ¼ 1; …; Gℓ ; ℓ ¼ 1; 2; 4;

ð3:3Þ

j ¼ 1; …; Gℓ ; ℓ ¼ 1; 2; 3; i ¼ 2; 3:

ð3:4Þ

and qðS ¼ si jC lj Þ ¼ 0;

respectively. To extract information about each component from groups 3 and 4, P i we write θi ¼ Gj ¼ 1 hij ; i ¼ 1; 2; 3; 4 and hij ¼ αij θ i ; j ¼ 1; …; Gi ; i ¼ 3; 4, PGi where α ¼ 1 and αij Z 0; j ¼ 1; …; Gi ; i ¼ 3; 4: Using (3.6), ij j¼1 the likelihood function reduces to L ¼ ðexpð  ðθ1 þ θ2 þ θ3 þ θ4 ÞVÞÞ

(c) Assume the “symmetry condition”, qðS ¼ s1 jC 3j Þ ¼ qðS ¼ s1 jC 3j0 Þ; for any j; j0 such that j aj0 and j; j0 ¼ 1; …; G3 ; qðS ¼ si jC 4j Þ ¼ qðS ¼ si jC 4j0 Þ;

for any j; j0 such that j aj0 and j; j0 ¼ 1; …; G4 ; i ¼ 2; 3. This is a reasonable assumption, because engineers usually cannot distinguish between components in group 3 or group 4. Suppose that we have n multi-scale systems under study. For the multi-scale system i; i ¼ 1; …; n, the ðyi ; δi ; wi Þ is observed, where yi is the actual failure time or the censoring time depends on whether δi ¼ 1 or zero respectively. Also, wi is the cause (or set of causes) of failure identified by engineers. For our setup if the actual cause of failure is the component from either group 1 or group 2 then the exact cause is observed by engineers. However, if the exact cause is in group 3, then the observed cause is S ¼ s1 . If the exact cause is in group 4, then the observed cause is either S ¼ s2 or S ¼ s3 . Under conditions (a)–(c), the joint likelihood function can be written as L ¼ ðΠ ni¼ 1 ðF ðyi ÞÞÞ Π ni¼ 1 ½hwi ðyi ÞIðwi ϵA1 Þ þ h3 þ ðyi ÞIðwi ¼ s1 Þ þph4 þ ðyi Þ Iðwi ¼ s2 Þ þð1  pÞh4 þ ðyi Þ Iðwi ¼ s3 Þδi ;

ð3:5Þ

where A1 ¼ fall the components in groups 1 and 2g. Note that wi in the first term is singleton, and can be any component j from the group ℓ; j ¼ 1; …; Gℓ and ℓ ¼ 1; 2. Generally speaking, from Eq. (3.5) it is obvious that nothing can be learned directly from the data about the components hazard rate functions in groups 3 and 4.What can be estimated are h3 þ ðtÞ and h4 þ ðtÞ. However, knowledge such as say h3 þ ðtÞ is noticeably greater than h4 þ ðtÞ can be valuable. We will provide more details in Sections 3.1 and 4. Under the proportional hazards assumption (2.4), Eq. (3.5) reduces to L ¼ ðΠ ni¼ 1 F ðyi ÞÞΠ ni¼ 1 ½hwi h10 ðyi ÞIðwi ϵAn1 Þ þ hwi h20 ðyi ÞI ðwi ϵAn2 Þ þ h3 þ h30 ðyi ÞIðwi ¼ s1 Þ þ ph4 þ h40 ðyi ÞIðwi ¼ s2 Þ þ ð1  pÞh4 þ h40 ðyi ÞIðwi ¼ s3 Þδi ;

ð3:6Þ

n

where Aℓ ¼ fall the components in the group ℓg; ℓ ¼ 1; 2. 3.1. Constant sub-hazard functions In Eq. (2.4) assume that hi0 ðtÞ ¼ 1; i ¼ 1; 2; 3; 4. That is, hij ðtÞ ¼ hij ;

j ¼ 1; …; Gi ;

i ¼ 1; 2; 3; 4:

From Eq. (2.5), the overall hazard rate and survival functions are hðtÞ ¼

Gi 4 X X i¼1j¼1

hij ;

r 1j G2 r 2j r 3 r4 r5 r4 þ r5 1 ; ð3:7Þ Π Gj ¼ 1 ðh1j Þ Π j ¼ 1 ðh2j Þ ðθ 3 Þ ½p ð1  pÞ ðθ 4 Þ Pn where V ¼ i ¼ 1 yi , r ij is the number of multi-scale systems that failed because of the component j from the group i, j ¼ 1; …; Gi ; i ¼ 1; 2, r i is the number of multi-scale systems that failed because of a component from the group i; i ¼ 1; 2; 3, r 4 is the number of multiscale systems that failed because of a component from group 4 and their causes masked by s2, and r 5 is the number of multi-scale systems that failed because of a component from group 4 and their causes masked by s3 : In order to carry out the Bayesian analysis, we assume the prior specification for unknown parameters,

p  Betaðη1 ; η2 Þ hij  Gammaðaij ; bij Þ; j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4: Then, from (3.7) it is clear that the posterior distributions are p  Betaðη1 þ r 4 ; η2 þ r 5 Þ hij  Gammaðaij þ r ij ; bij þ VÞ;

j ¼ 1; …; Gi ; i ¼ 1; 2:

The posterior distribution of θ3 given ðy1 ; δ1 Þ; …; ðyn ; δn Þ and α31 ; …; α3ðG3  1Þ is Gamma with parameters r 3 þa3 þ  ðG3  1Þ and P 3 PG3 ðV þ Gj ¼ 1 b3j α3j Þ, where a3 þ ¼ j ¼ 1 a3j and the joint posterior distribution of α31 ; …; α3ðG3  1Þ is Dirichlet distribution with parameters a31 ; …; a3G3 . Since all conditionals are in standard forms, the joint posterior distribution of θ3 and α31 ; …; α3ðG3  1Þ can be easily generated using Markov Chain Monte Carlo (MCMC) algorithms, see [21] for more details about MCMC. Finally, the posterior distribution of θ4 given α41 ; …; α4ðG4  1Þ and ðy1 ; δ1 Þ; …; ðyn ; δn Þ is Gamma with parameters ðr þ r 5 þ a4 þ  ðG4  1ÞÞ and P 4 PG4 4 ðV þ Gj ¼ 1 b4j α4j Þ, where a4 þ ¼ j ¼ 1 a4j , the joint posterior distribution of α41 ; …; α4ðG4  1Þ is Dirichlet distribution with parameters a41 ; …; a4G4 and the joint can be easily generated using MCMC. To clarify further the point regarding non-informativeness of the data, we note that here information about αij ; j ¼ 1; …; Gi ; i ¼ 3; 4 only comes from our prior information about hij ; j ¼ 1; …; Gi ; i ¼ 3; 4 and for these parameters the data is noninformative. As we mentioned in Section 2, Eq. (2.3) gives vital information to engineers about the reliability of each component of a multiscale system. To get such information about each component of a multi-scale system from the failure data set, one can easily generate a sample from PðC ij jT ¼ tÞ; j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4 by generating samples from hij ; j ¼ 1; 2; …; Gi ; i ¼ 1; 2; θ3 ; θ4 and αij ; j ¼ 1; …; ðGi  1Þ; i ¼ 3; 4. See Section 4 for more clarification. Note that from Eq. (2.6), PðC ij j T ¼ tÞ ¼ hij =ðθ1 þ θ2 þ θ3 þθ4 Þ. 3.2. Weibull model for sub-hazards functions Suppose sub-hazards rate functions are hij ðtÞ ¼ hij βij t βij  1 ;

j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4:

ð3:8Þ

From Eq. (2.2), the overall hazard rate and survival functions are hðtÞ ¼

Gi 4 X X i¼1j¼1

hij βij t βij  1

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N. Ebrahimi, M. Shehadeh / Reliability Engineering and System Safety 138 (2015) 13–20

Table 1 Failure data on multi-scale system with 10 components (Cij is the component j from the group i). Run no.

yi

δi

wi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

0.705581317 0.28198097 0.634789122 0.149674742 0.062876631 0.524544663 1.004285292 0.303119943 0.459878896 0.915752175 0.351598247 1.18268863 0.815949007 0.345899528 0.250682006 0.111666313 0.593673616 0.108262617 0.35497659 0.969094721 0.459951616 0.802897806 0.133936712 0.713445145 0.859236286 0.541797238 0.355695614 0.460387409 0.429567667 0.763692185 0.511479664 0.042770508 0.109859023 0.837828609 0.240554381 0.585157942 0.152239072 0.038151755 0.590683967 0.235450475 0.307028112 0.076223107 0.626679207 0.742255722 0.006996808 0.879928729 0.238089866 1.071353489 0.646552453 0.296829063 0.096648863 1.232415081 1.133698619 0.75927259 0.365155673 1.036037701 0.269859894 0.8837854 0.720211235 0.130068069 1.083473027 0.738043898 0.492783926 0.590885326 0.398970086 1.220641385 1.001736495 0.179119319 0.376671227 0.233482238 0.540095373 0.413201365 1.059821158 0.316733262 0.760389591

0 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 1 1 1 0 1 1

NA C12 Group Group Group C11 NA Group Group Group C11 NA C22 C11 C21 Group Group C22 Group NA Group C12 Group Group C11 C12 C12 Group Group Group C11 Group Group NA Group Group Group C12 Group C12 Group Group C12 NA Group C21 C12 NA Group C21 Group NA NA NA Group NA Group NA C12 C11 NA NA C22 C22 Group NA NA Group Group C11 Group C12 NA Group Group

3 4 3 or group 4

3 3 3

3 3 3 or group 4 3 3 or group 4 3 or group 4

3 or group 4 3 or group 4 3 3 or group 4 3 3 3 3 3 or group 4 4 3

3 or group 4

3 or group 4 3

3 3 or group 4

3

3 3 3

3 3 or group 4

N. Ebrahimi, M. Shehadeh / Reliability Engineering and System Safety 138 (2015) 13–20

17

Table 1 (continued )

and

Run no.

yi

δi

wi

76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.893938566 0.129900817 0.825300529 0.046951303 0.375217856 0.824745891 0.709124117 0.47492974 0.971931737 0.559080846 0.446610213 0.020562046 1.069722519 0.361893782 0.125362612 0.625982275 0.257928297 0.297881101 0.517893308 0.383731253 0.588814308 0.285127701 0.323814531 0.266688413 0.50695353

0 1 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1

NA C22 NA Group C12 Group NA Group C11 Group Group Group NA Group Group C22 C22 Group C11 C21 C11 C12 Group Group C11

8 <

9 Gi 4 X = X βij F ðtÞ ¼ exp  hij t ; : ; i¼1j¼1

respectively. Now, assume that components in each group have a common shape parameter, βij ¼ βi ; j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4. By taking hi0 ðtÞ ¼ βi t βi  1 ; i ¼ 1; 2; 3; 4 we have a special case of the PGi general PH model given by (2.4). As before we take j ¼ 1 hij ¼ θi ; i ¼ 1; 2; 3; 4; hij ¼ αij θi ; j ¼ 1; …; Gi ; i ¼ 3; 4. From (3.6) the likelihood function is !!! 4 X r 1j G2 r2j β 1 L ¼ Π ni¼ 1 exp  θℓ yi ℓ Π Gj ¼ 1 h1j Π j ¼ 1 h2j ℓ¼1

β 1

Π ri 1¼ 1 ðyi Þβ1  1 Π ri 1¼þrr12þ 1 ðyi 2 β 1

3 3 3 4 4 3 3 or group 4

3

3 3

…; α3ðG3  1Þ given β4 and the data is Dirichlet distribution with parameters a41 ; …; a4G4 . Also, the posterior distribution of β1 given the data is proportional to Xn   ðr1j  1 þ a1j Þ β 1 β 1 g 1 ðβ1 ÞΠ ri 1¼ 1 y11 βr11 Π Gj ¼ y 1 þ b1j ; 1 ℓ¼1 ℓ the posterior distribution of β2 given the data is proportional to !  ðr2j  1 þ a2j Þ n   X β2  1 β2 r1 þ r2 r2 G2 β2 Π j ¼ 1 g 2 ðβ2 Þ Π i ¼ r1 þ 1 yi yℓ þ b2j ; ℓ¼1

the posterior distribution of β3 given the data is proportional to !  ðr3  ðG3  1Þ þ a3 þ Þ n   X β3  1 β3 r3 r1 þ r 2 þ r 3 g 3 ðβ3 Þβ3 Π i ¼ r1 þ r2 þ 1 yi yℓ þ b3 þ ℓ¼1

ðθ3 Þr3 pr4 ð1 pÞr5 ðθ4 Þr4 þ r5

Π ri ¼ r1 þ r2 þ r3 þ 1 yi 4

3

and finally the posterior distribution of β4 given the data is proportional to

β 1

ÞΠ ri 1¼þrr12þþrr23þ 1 yi 3

ðβr11 βr22 βr33 βr44 þ r5 Þ:

ð3:9Þ

In (3.9) without loss of generality assume that the first r1 failures are due to group 1, the second r2 failures are due to group 2, the third r3 failures are due to group 3, the remaining failures are due to group 4 and r ¼ r 1 þ r 2 þ r 3 þ r 4 þ r 5 . To carry out the Bayesian analysis, we need to prescribe prior models for various parameters. However, to assign prior models to hij and βi ; j ¼ 1; …; Gi and i ¼ 1; 2; 3; 4 difficulties arise because of introduction of shape parameters β1 ; β2 ; β3 and β4 which do not have any continuous conjugate priors. Thus, conditional on βi we assume that hij  Gammaðaij ; bij Þ; j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4 and βi  g i ; i ¼ 1; 2; 3; 4. It is easy to verify that the conditional posterior of P β hij given βi and the data is Gamma ðaij þ r ij ; bij þ nℓ ¼ 1 yℓi Þ; j ¼ 1; …; Gi ; i ¼ 1; 2 and the conditional posterior of θ3 given α31 ; …; α3ðG3  1Þ ; β3 and the data is Gamma ðr 3 þ a3 þ  ðG3  1Þ; Pn PG3 β3 ℓ ¼ 1 yℓ þ j ¼ 1 b3j α3j Þ. As in Section 3.1 the posterior distribution of p is Beta with parameters η1 þ r 4 and η2 þ r 5 and the conditional posterior of α31 ; …; α3ðG3  1Þ given β3 and the data is Dirichlet distribution with parameters a31 ; …; a3G3 . Similarly, the posterior distribution of θ4 given α41 ; …; α4ðG4  1Þ ; β4 and the data is P P 4 β Gamma ðr 4 þ r 5 þ a4 þ ðG4  1Þ; nℓ ¼ 1 yℓ4 þ Gj ¼ 1 b4j α4j Þ and α31 ;

β 1

g 4 ðβ4 Þβr44 þ r5 Π ri ¼ r1 þ r2 þ r3 þ 1 yi 4

n X ℓ¼1

!  ðr4 þ r5  ðG4  1Þ þ a4 þ Þ β

yℓ4 þ b4 þ

;

where g i ðβi Þ is the prior distribution for βi ; 1 ¼ 1; 2; 3; 4. Note that the posterior densities of β1 ; β2 ; β3 and β4 can be sampled numerically. For example if we use priors g i ðβi Þ which is proportional to ðβi Þ  γ , γ 40, then the posterior density will be log-concave and adaptive rejection sampling can be used.

4. Implementing the method using simulated data set Our simulated data set was generated from a hypothetical multiscale system having M¼10 components with G1 ¼ 2 components have macro-structures, G2 ¼ 2 components have meso-structures, G3 ¼ 3 have micro-structures and the remaining G4 ¼ 3 has nano-structures. We take h11 ¼ h12 ¼ 0:2; h21 ¼ h22 ¼ 0:1; h31 ¼ h32 ¼ h33 ¼ 0:2 and h41 ¼ h42 ¼ h43 ¼ 0:1. First we generate 100 observations from 10 independent exponentials with parameters 0.2, 0.2, 0.1, 0.1, 0.2, 0.2, 0.2, 0.1, 0.1 0.1, where each observation is the vector of 10 numbers generated from these 10 exponential distributions. Now we take the minimum of those 10 numbers that represents the failure time of a multi-scale system.

18

N. Ebrahimi, M. Shehadeh / Reliability Engineering and System Safety 138 (2015) 13–20

6

6

4

4

2

2

0

0 0.1

0.2

0.3

0.4

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h11

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h12

12

8 6

8

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4

2

0

0 0.00

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0.35

h22

h21

3.0

3.0

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1.0 0.0

0.0 0.0

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h32

h31

6

3.0 2.0

4

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2 0

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h33

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h41

5 4 3 2 1 0

5 4 3 2 1 0 0.0

0.1

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0.4

0.5

h42

0.0

0.1

0.2 h43

Fig. 1. The posterior distributions of hij ; j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4.

Table 2 The means, variances, and 95% credible intervals of hij ; j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4. hij

Mean

Variance

95% Credible interval

h11 h12 h21 h22 h31 h32 h33 h41 h42 h43

0.21013153 0.22881688 0.07946402 0.13018546 0.18569309 0.17773742 0.1803768 0.10939704 0.10493048 0.10534859

0.003749074 0.003703064 0.001253796 0.002110089 0.018970305 0.019421894 0.017932126 0.006858542 0.006646468 0.006354194

(0.114369504, 0.3453212) (0.125351556, 0.35823) (0.025158176, 0.1650131) (0.059550008, 0.2310402) (0.008514667, 0.4821761) (0.004957652, 0.4948249) (0.005581985, 0.4886899) (0.003330265, 0.3020893) (0.003779627, 0.288701) (0.002516821, 0.291148)

To simulate the censored observations we also need termina1 1 tion time which is generated from ð1:5 Þ þ ð1:5 Þ uniform (0, 1). Then we observe which ever comes first. Finally, we construct our wi

from our generated data simply by taking wi to be the component from either group 1 or group 2 depending on whether the δi ¼ 1 and i-th observation corresponds to the minimum which is the failure time of the component from group 1 or group 2. If δi ¼ 1 and the minimum is from group 3, then we take wi ¼ s1 and if it is from group 4, then we take wi ¼ s2 with probability 1/3 and wi ¼ s3 with probability 2/3. If δi ¼ 0, that is the observation is censo red, then we take wi ¼ NA. The resulting data is ðyi ; δi ; wi Þ; i ¼ 1; 2; …; 100. Table 1 gives our simulated data set, consists of 100 observations on yi with corresponding δi and wi. For example, for the run number 2 that represents the multi-scale system 2, δi ¼ 1 means that actual failure was observed for the multi-scale system 2 and its failure time is 0.28198097. The cause of failure is the second component in group 1. Now we apply the methodology described in Section 3.1 to our simulated data set. The steps followed for the posterior calculations are:

N. Ebrahimi, M. Shehadeh / Reliability Engineering and System Safety 138 (2015) 13–20

19

8 6 4 2 0

8 6 4 2 0 0.05

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p11

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12

15 10

8

5

4

0

0 0.00

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p22

5

4

4

3

3

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2

1

1

0

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5

8

4 3

6

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4

1

2

0

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−0.05

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p41

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8

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p42

0.0

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p43

Fig. 2. The posterior distributions of pnij ; j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4. Table 3 The means, variances, and 95% credible intervals of pnij ; j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4. pnij

Mean

Variance

95% Credible interval

pn11 pn12 pn21 pn22 pn31 pn32 pn33 pn41 pn42 pn43

0.14063468 0.15058787 0.05390686 0.08546857 0.12231032 0.1169814 0.11919425 0.07246028 0.06916504 0.06929072

0.001426943 0.001361545 0.000540617 0.00078745 0.007947234 0.008124108 0.007534258 0.002972227 0.002836261 0.002646659

(0.077067257, 0.2234476) (0.085461127, 0.2273385) (0.016935595, 0.1072563) (0.037853378, 0.1479884) (0.005636142, 0.3128789) (0.003487595, 0.3200499) (0.003430352, 0.3132885) (0.002373154, 0.198482) (0.002425095, 0.1919522) (0.00168571, 0.1862062)

(1) Following notations in Section 3.1, we assume prior parameters η1 ¼ 1; η2 ¼ 2; aij ¼ 1; i ¼ 1; 2; j ¼ 1; 2; aij ¼ 1; i ¼ 3; 4; j ¼ 1; 2; 3; b11 ¼ b12 ¼ 5; b21 ¼ b22 ¼ 10; b31 ¼ b32 ¼ b33 ¼ 5; b41 ¼ b42 ¼ b43 ¼ 10:(2)

For each parameter we carry out MCMC to obtain posterior dis tributions of h11 ; h12 ; h21 ; h22 ; h31 ; h32 ; h33 ; h41 ; h42 ; h43 which are given in Fig. 1a, b, c, d, e, f, g, h, i, and j, respectively. Table 2 gives the posterior summary statistics of hij ; j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4. As we said in Section 2, to get information about each component we need to compute pnij ¼ PðC ij jT ¼ tÞ. Fig. 2a–j gives the posterior distributions of pnij ; j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4; and Table 3 gives the posterior summary statistics for pnij ; j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4. From Table 3, it is evident that this probability is higher for components in group 1 and components in group 3. So one way to improve the reliability of the multi-scale system is to re-design components in groups 1 and 3. To determine the sensitivity of Bayesian analysis to choice of prior distributions, we used different values for aij and bij ; j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4: As anticipated, because components in groups 3 and 4 are generated from the same distributions (the data is very informative), our results were insensitive to prior distributions. In the next step, we generated different data sets

20

N. Ebrahimi, M. Shehadeh / Reliability Engineering and System Safety 138 (2015) 13–20

(different from the one reported in Table 1)with different values of hij ; j ¼ 1; …; Gi ; i ¼ 1; 2; 3; 4. As expected in all those cases, because some of the parameters related to groups 3 and 4 are identified only through the prior information and data is not informative for those parameters, at least to some extent our results were sensitive to prior distributions. In our opinion, this is not a problem at all. Because, one key advantage of Bayesian approach is it allows (coherent and optimal) incorporation of a prior information.

5. Concluding remarks In this paper, we have proposed a parametric Bayesian approach to analyzing failure data on a multi-scale system with masked caused of failure. The Bayesian approach provides a general framework for carrying out estimation of unknown parameters as well as assessing reliability of multi-scale system and its components. Our proposed method is very flexible when it comes to modeling failure time distributions of the multi-scale system and its components. If all the components with nano-structure are identical and all the components with micro-structure are identical as well, then our analysis is not sensitive to prior choice. However, if components with nano- or micro-structures are not identical, then prior distributions should be selected carefully. It must be mentioned that often a multi-scale system has nanostructure as well as micro-structure components that are identical. For example, in many sensors, nano-structure components are constructed using identical nanowires.

Acknowledgments We would like to thank both reviewers for very constructive comments that led to improvement of our paper. The work of both authors is partially supported by the National Science Foundation, DMS1208273. The work of the first author is also partially supported by the National Security Agency under Grant number H98230-11-1-0138.

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