Dynamic artificial neural network-based reliability considering operational context of assets

Dynamic Artificial Neural Network-based reliability considering operational context of assets. J. Izquierdoa,b,∗∗, A. Crespo Márquezb,∗, J. Uribetxebarriaa aIK4-Ikerlan Technology Research Centre, Operations and Maintenance Technologies Area, 20500 Gipuzkoa, Spain bIndustrial Organization and Business Management I, School of Engineering, University of Seville, Camino de los5 Descubrimientos s/n, 41092 Seville, Spain Abstract Assets reliability is a key issue to consider in the maintenance management policy and given its importance several estimation methods and models have been proposed within the reliability engineering discipline. However, these models involve certain assumptions which are the source of different uncertainties inherent to the estimations. An important source of uncertainty is the operational context in which the assets operate and how it affects the different failures. Therefore, this paper contributes to the reduction of the uncertainty coming from the operational context with the proposal of a novel method and its validation through a case study. The proposed model specifically addresses changes in the operational context by implementing dynamic capabilities in a new conception of the Proportional Hazards Model. It also allows to model interactions among working environment variables as well as hidden phenomena thanks to the integration within the model of artificial neural network methods. Keywords: Dynamic Reliability, Proportional Hazards Model, Artificial Neural Networks, Operational Context, Maintenance Management, Epistemic Uncertainty 1. Introduction10 With the emerging of engineered systems in the 19th century so did arise the need for a scientific discipline dealing with their reliability, this discipline began to be known as reliability engineering in the 1950s [1]. The assumptions under which traditional techniques of reliability were developed, simplify many of the real-life boundary conditions; the motivation lying behind these simplifications, was the pressure of upholding the rigid and fast technological advances in the industries which initially showed15 more interest in reliability engineering (maritime, military, aircraft and Oil & Gas industries) [2]. As the independent scientific discipline that reliability engineering has become, it measures the reliability by quantitative metrics and controls throughout the product lifecycle [3]. According to Zio [1], the purpose of reliability engineering is to provide a collection of formal methods aiming at exploring the relation among system operation and failure, and it is in this discipline where this paper intends to20 contribute with the novel model later on proposed. 1.1. Motivation and research Reliability is a key issue to address by comprehensively analyzing the asset performance under the effects of different uncertainties during its life cycle [4]. The uncertainties affecting the asset performance have generally been classified into two types (see [3–7]):25 •Aleatory uncertainty. Considered to be the result of a fundamental randomness in the natural phenomena of the world, it results in uncertainty referring to the inherent physical behavior of the system. •Epistemic uncertainty. Associated with the lack of knowledge, it refers to the uncertainty related to the completeness and accuracy in the understanding of the failure process.30 ∗Principal corresponding author, Tel.: +34 954 487215. ∗∗ Corresponding author, Tel.: +34 943 712400. Email addresses: [email protected] (J. Izquierdo), [email protected] (A. Crespo Márquez), [email protected] (J. Uribetxebarria) Preprint submitted to Reliability Engineering & System Safety March 27, 2019 As stated before, traditional reliability models involve assumptions and simplifications[8]; these are the sources of significant uncertainties causing the inability of the reliability models to properly describe the true behavior of the system [5]. Some important assumptions that introduce epistemic uncertainty in the models are the independence among components, the renewal assumption o the constant operational condition and external factors35 [9]. Such premises and suppositions are the cause of epistemic uncertainty, reducible by integrating new information on the contrary of aleatory uncertainty, which comes from the stochastic character of the problem [10]. Therefore, if the models-based methods aim at accurately quantify the reliability, it is a requisite for them to integrate the effect of epistemic uncertainties [3]. It is within this motivation where the scope of the paper lies; more precisely, the authors have developed a novel model for reliability40 estimation that reduces epistemic uncertainty referring to operational conditions of the assets. A more realistic estimation of reliability, through a model integrating operating environment information, will enable more effective and better-customized maintenance policies [11]. The benefits of such models gain special interest in the early stages of the asset life-cycle, where it is relevant to reflect the context into which they will perform [12]. This aspect is a key pillar to take into account when designing45 the maintenance strategy, and it has been addressed throughout the later on reviewed literature (see Subsection 1.2). It is this main pillar which gives birth to the research process conducted in this paper and summarized in figure 1. The conducted research, aiming at integrating the operational context effect on reliability assessment, have followed two courses; in the last phase, they converge into a comprehensive model that50 gathers the benefits from the researches regarding both lines. These research lines face two problems derived from the main motivation. The research represented in the bottom line aims at modeling the effects of changing operational environment, it deals with the reality of the same asset performing in very different environments and stress levels during its life-cycle. However, the research course represented by the top line of the process deals with the problem of modeling unknown interactions among the different55 parameters of the operational environment of the asset. Finally, the convergence of both lines, hence the merge of two innovative models, leads to the proposal of a novel model which not only integrates the dynamism of operating environments but the possible unknown interactions among its variables as well. Figure 1: Research Process 1.2. Related works. The study of the affection that the operational environment has on asset management and reliability60 engineering aspects has received considerable attention lately. More specifically, interesting contributions regarding assets reliability can be found, for example in Okaro and Tao [13] the reliability of a subsea compression system is analyzed under operational covariate stresses, in Peng et al. [14] an approach taking into account dynamic conditions is proposed, for cable failure the operational context is explicitly modeled in Tang et al. [15] and also for traction transformers in Lin et al. [16]. However, reliability assessment65 is not the only aspect of asset management addressed in the literature with the affection of operational context to failures as a common underlying cause. For instance, spare parts estimation depending on the operational environment is studied in [17–19], availability assessment considering different weather conditions in [20] and maintainability analysis with context variables in [21, 22]. In the related literature the authors have gone through, the Proportional Hazards Model (PHM)70 is a recurrent model utilized for analyzing the effects of different variables of the operational context, therefore this model plays a key role as a starting point in the research. It was first introduced by Cox in 1972 [23] (it is also known as Cox model) and it was proposed for reliability studies by the author 2 himself and by other researchers [24, 25]. An early literature review of the first attempts of using the model, along with a case study exemplifying its proper use, can be seen in [26]; through the paper, the75 work of Bendell facilitated the posterior applications of the model in reliability engineering. Afterward, several applications of the model can be found in many different industries and systems. The model have been proposed for the rail industry to predict rolling stock reliability, for instance in the door system [27] or the Heating Ventilating and Air Conditioning (HVAC) system [11]. It is also proposed for the wind energy sector, to model failure rate in wind turbines [28] and to assess the turbines stress condition80 [29]. The research in the cutting tool industry also provides interesting applications of the model [30, 31]. And among others, the model have also been proposed for bearings reliability prediction [32], for the machinery in the processing industry [33], for electronic components [34, 35] and for piping infrastructure [36]. Another recurrent approach found in the literature for integrating epistemic information regarding the85 operational context of the assets is the application of Artificial Neural Networks (ANN). As stated by Xu [37], the attractiveness of neural networks is mainly due to the fact that no ’a priori’ assumption regarding the models is needed, and they have the advantage of representing complex nonlinear relationships as Marugán et al. [38] write. To date, several studies have begun to investigate the application of ANN for reliability problems, e.g.: in Al-Garni et al. [39] they are compared with the Weibull regression model,90 in Fink et al. [40] they are proposed to approach reliability prediction problems as time series, Beg et al. [41] present a comparison of several ANN-based models at estimating the probability of failure and Santosh et al. [42] propose an interesting approach for time to failure estimation combining Weibull regression models with ANNs. In particular, the mentioned work of Santosh et al. [42] is a demonstration of the benefits of combining95 statistical models and ANN methods. Consistent with this approach it is important to emphasize the work of Faraggi and Simon [43] whose proposed model combines both of the previous approaches, ANNs architecture and the PHM. This ANN extension of the Cox model was conceived for studying rightcensored survival data [43] in the field of statistical medicine where most of its applications reside (see for instance [44, 45]).100 However this is not the only model in the literature that brings together both techniques; the work of Biganzoli et al. [46] focus on partial logistic models with ANN and feed forward ANN for modeling the hazard function, in Ripley et al. [47] seven survival models based on neural networks (both continuous and discrete time models) are described and Mazini et al. [48] proposes the use of neural networks to create a health condition model for wind turbines. As previously stated, the applications of these models105 remain almost exclusively in the discipline of medical statistics; nonetheless, there are already in the literature some attempts to extrapolate them to reliability engineering [49, 50]. When reliability engineering is considered within the organizations, it is related with maintenance management and therefore, with maintenance activities [51]. According to Crespo [52], the maintenance activities can either be classified as corrective maintenance, performed after the failure has happened;110 or preventive maintenance, dispatched at predefined intervals or according to certain criteria previously established. Each type of actions have an associated cost, corrective cost (CC) and preventive cost (PC), being on a general basis the CC greater [53]. Generally both types of costs include direct costs associated to manpower, replacement parts and materials, however the corrective costs not only is unplanned but also include indirect costs associated to penalization in terms of operation losses, impact on quality,115 environment or security among others [52]. In order to optimize the costs associated to maintenance activities, reliability modeling has been proposed as a useful tool [54]. In the literature can be found recent works that propose launching preventive maintenance activities according to a defined reliability threshold [55–57] being this the scenario intended for the later proposed model. 1.3. Scientific contribution120 Based on the motivation stated in Subsection 1.1 and the related works presented in Subsection 1.2, the research conducted by the authors has led to the development of a novel model for reliability estimation. As it is represented in figure 1, the main contribution of this paper is the proposal of a Dynamic Artificial Neural Network-based Reliability model which is a result of combining the concepts proposed for the ANN-based models and the Dynamic PHM, being the later a novel proposal in this125 paper as well. The value proposition of the model is the improvement of reliability estimations, these better estimations rendered by the proposed model are mainly due to directly addressing three simplifications that other models have not dealt with altogether: 3 A. The Dynamic ANN-based model not only takes into account the reliability driver but the operational130 context information as well. By integrating this information the model is built-up upon more complete knowledge of failure mechanisms, therefore, the epistemic uncertainty associated with the model is reduced. B. The dynamic capabilities of the model directly address the reality of assets performing in changing operating contexts. The novel model here proposed has the capability of taking into account changes135 in the operational context and the length in the time of those changes, thus the model enhances reliability estimations in production peaks, seasonality, and other similar working environments with high variability in its operational conditions. C. The proposed model brings in the advantages of integrating ANN. The ANN methods allow modeling unknown interactions between the different variables of the operational environment, without140 having to previously define them, as well as hidden phenomena which may be triggering or influencing the failure modes. One important contribution of the research process is the proposal of the Dynamic PHM, which deals with items A and B. However, the apex of the scientific contribution of the research here presented is in the Dynamic ANN-based reliability model, which has the potential of addressing the three aforementioned145 key points into a single model. 1.4. Overview The remainder of the paper consists of three sections, the following section presents the development of the Dynamic ANN-based model, then the model is tested on a case study in section 3 and finally the results and conclusions are discussed in the last section.150 The development of the model in section 2 has followed the same research process previously presented in figure 1, and therefore the section is structured in a similar way. First, in order to set the foundations, the main introduction to PHM is presented, then in the following subsections 2.2 and 2.3 individual solutions for the top line and the bottom line of the applied research progress in figure 1 are respectively developed. Special emphasis is made on subsection 2.3 due to the novelty of the proposal stated in the155 previous subsection 1.3. And finally subsection 2.4 presents the main contribution of the paper, the Dynamic Artificail Neural Network-based reliability model which is conceived, and thus presented, as a combination of the previous models. After laying down the theoretical dimensions of the model, the main topic covered in section 3 is the performance of the model in a case study as an illustrative example of its use and potential. The section160 consist of four subsections: in the first one the database and the case study are presented, then the optimization of the proposed model for the case study is explained, then the performance of the model is tested in terms of reliability estimations in the third and lastly cost-performance metrics are analyzed in the fourth subsection. Finally, section 4, ties together up the various conclusions withdrawn during the research process,165 the model development and its application to the case study. Also some research lines worth of further investigation are presented here. 2. Proposed model As previously stated, the proposed model takes as starting point the PHM and the concept of partial likelihood that Cox proposed along with it [23]. It is the concept of partial likelihood the one that will170 enable to integrate ANN methods with the PHM and therefore the section development follows this logic; starting by introducing the PHM with the partial likelihood, continuing with the ANN-based PHM, then the Dynamic extension of the Weibull PHM and finally the Dynamic ANN-based reliability model. 2.1. Weibull Proportional Hazards Model The most common description of the PHM is the definition of the hazard rate provided by equation 1.175 In equation 1, the failure rate of a system is expressed as the product of a baseline hazard rate h0(t), as a function of the operating time of the asset t; and a term incorporating the operational context variables in an exponential form in which each one of the covariates X= (X1, ..., Xk)is multiplied by a parameter β= (β1, ..., βk)describing its effect. h(t, X) = h0(t)exp   k X j=1 βjXj (1) 4 Table 1: Nomenclature definition Nomenclature XVector of covariates of the operational context XiVector of covariates of the i-th operational context Xij j-th variable of the i-th operational context βVector of parameters of Cox model βjj-th parameter associated to j-th covariate knumber of covariates, i.e. variables of the operational context XVector of mean values of the covariates h(t, X)Hazard function depending on time and context variables h0(t)Baseline hazard function for the null covariates vector hX(t)Baseline hazard function for the mean values covariates vector h(t, Xi)hazard function for the i-th operational context depending on time HR Hazard ratio between two identical systems operating in different contexts TVector of failure times t(i)i-th failure time sorted in increasing order pnumber of failure times Riset of systems at risk at the i-th failure time LiConditional probability of i-th time LPartial likelihood γShape parameter of a Weibull distribution αScale parameter of a Weibull distribution WWeights vector for the ANN whyz Weight of the connection between node yin hidden layer hwith node zin the next hidden layer BBiases vector of the ANN bhz Bias of the node zin the hidden layer h g(x)Hyperbolic tangent function θhz Output of the node zin the hidden layer h HNumber of hidden layer, included the output layer ZhNumber of nodes in hidden layer h YhNumber of nodes in hidden layer h G(X,W,B)Output of the ANN depending on the covariates, the weights and the biases of the network R(t, Xi)Reliability function in the i-th operational context depending on time CIntegration constant that enable the dynamic capabilities of the model tiTime of the i-th change of operational context CiIntegration constant with the information i-th change of operational context The baseline hazard function, h0(t), represents the hazard of a system operating in a context described180 by the null vectors of variables, X=0∈Rk, and it can be parametric following certain distribution or of an unspecified form; and the operational context variables can be either a naturally variable or an indicator variable. Given two identical systems operating in two different operational contexts, that would be X= (X1, ..., Xk)and X0= (X0 1, ..., X0 k), being X0the one with a higher risk; it is defined the Hazard Ratio185 (HR) between the two of them as: HR =h(t, X0) h(t, X)= h0(t)exp Pk j=1 βjX0 j h0(t)exp Pk j=1 βjXj=exp   k X j=1 βj(X0 j−Xj) (2) It can be observed from equation 2 that HR is independent of the system operating time. Thus, considering a special case of the HR in which the comparison is between a system operating in an environment where the covariates vector take mean values X= (X1, ..., Xk)and the same system operating in a context Xi= (Xi1, ..., Xik), the hazard function of the system operating in Xiwould be expressed by190 equation 3, where the hazard function is decomposed into a baseline hazard function in the mean values of the covariates (hX(t)), and into the exponential part where the deviations from the mean value of each 5 covariate are taken into account, instead of the covariates values themselves. h(t, Xi) = hX(t)exp   k X j=1 βjXij −Xj (3) One of the main advantages of the Cox model is that the estimation of the parameters in the exponential part, β= (β1, ..., βk), is performed independently from the baseline hazard function. This is195 possible thanks to the concept of partial likelihood, it is defined as the product over every failure time of the conditional probability of failure of the system which actually failed at t(i). That is for a set of increasing ordered failure times T= (t(1), ..., t(i), ..., t(p)), let Ribe the set of systems at risk at the time t(i), i.e. subjects which have not failed nor been censored before t(i), then the conditional probability is described by equation 4 and therefore the partial likelihood by equation 5.200 Li=h(ti,Xi) Pl  Rih(ti,Xl)=exp(Pk j=1 βjXij −Xj) Pl  Riexp(Pk j=1 βjXlj −Xj)(4) L= p Y i=1 Li(5) The estimation of the parameters is performed by maximizing the partial likelihood in equation 5, traditionally by Newton-Raphson iterations; as it can be observed the effects of the variables of the operational context are estimated without making any assumptions regarding the baseline hazard function. However, it is interesting for the scope of the paper to introduce the full parametric version of the model in which the baseline hazard function in the mean values of the covariates is fitted to follow205 a two-parameter Weibull distribution. This alternative of the Cox model is represented in equation 6, where γis the shape parameter and αis the scale parameter, and it is known as Weibull PHM. h(t, Xi) = γ α.t αγ−1 .exp   k X j=1 βj(Xij −Xj) (6) 2.2. Neural-Network based reliability model In the previous section, it can be seen that the combination of the operational context variables is linear in the most simple form of the Weibull PHM, or “a priori” defined in other variations. Thus, this210 feature limits the suitability of the model for modeling unknown interactions or hidden phenomena and it is in this issue where the architecture of the ANNs provides value. The proposed model is based on feed-forward neural networks with only one output node, and the operational context variables, X= (X1, ..., Xk), will form the input nodes, a graphical representation can be seen in figure 2. The inputs and the output are connected through hidden layers with different nodes215 number. Every connection between nodes is associated with a weight parameter, all of them represented by the vector Wwhere each position is described by whyz, being hthe origin layer (starting in 0 for the input layer and ending in H for the output layer), yrepresent the node in the origin layer and zthe destination node in the next layer. Every node in the hidden layers is also associated with a bias term all of them represented by vector Bwhere each position is described by bhz being hthe hidden layer and220 zthe node in the corresponding hidden layer. In every neuron, also known as computational unit, an activation function is applied to the input received, in the proposed model the activation function g(x) is the hyperbolic tangent, which is described in equation 7 and have proofed to train faster than the sigmoid activation [44]. g(x) = exp(x)−exp(−x) exp(x) + exp(−x)(7) 6 Figure 2: ANN Representation Table 2: ANN equations Layer Output of the nodes in the layer Domain First hidden Layer (h=1) θ1z=g k X y=1 w0yzXy+b1z!∀z[1, Zh=1] Next hidden Layers θhz = g  Yh−1 X y=1 w(h−1)yzθ(h−1)y+bhz  ∀h[2,(H−1)] ∀z[1, Zh] Output layer (h=H) θH1= g  YH−1 X y=1 w(H−1)y1θ(H−1)y+bH1  - g(x) is the previously defined hyperbolic tangent function The equations that govern the ANN described in the paragraph above and represented in figure 2 can225 be seen in Table 2, the equations have been organized according to the different layers and the calculations that involve every one of them. The term θhz refers to the output of the neuron zof the hidden layer h. Considering the application of the equations on Table 2, it is possible to define the network as a non-linear complex function depending on the covariates vector X= (X1, ..., Xk), the weights of the connections between the nodes W, and the bias terms B. This function is denoted by G(X,W,B)and230 the Neural-Network based reliability model propose replacing the linear Pk j=1 βjXij in the Cox model by the output of the neural network function denoted by G(X,W,B). This proposal is described by equation 8, and with this model the partial likelihood function (L) to maximize in order to estimate the neural network parameters(weights and bias terms) would be equation 9. h(t, X) = h0(t)exp (G(X,W,B)) (8) L= p Y i=1 exp(G(Xi,W,B)) Pl  Riexp(G(Xl,W,B)) (9) 7 It is important to notice that in this model the parameters of the neural network are not obtained235 by training it but by maximizing the partial likelihood function, expressed in equation 9. Nonetheless, depending on the chosen architecture of the ANN the maximization of equation 9 may result in a very complex and computationally intensive problem; thus a genetic algorithm is proposed for maximizing the partial likelihood. 2.3. Dynamic Weibull-Proportional Hazards Model240 Taking as a starting point the hazard rate described by the Weibull PHM in equation 6, the reliability derived from that particular expression of the failure rate of a system can be calculated as follows: R(t, Xi) = exp −ˆh(t, Xi)dt(10) R(t, Xi) = exp  −ˆγ αt αγ−1 exp( k X j=1 β(Xij −Xj))dt (11) R(t, Xi) = exp  −t αγ exp( k X j=1 β(Xij −Xj)) + C (12) In the general reliability function of equation 12 the variables of the operational context do not depend on time, the shape and scale parameters are considered to be constants inherent to the technical characteristics of the system and the parameter Cis the integration constant.245 When the asset starts to operate in a context A, i.e. XA= (XA1, ..., XAk), the reliability of the asset is 100% and therefore R(t= 0 , X =XA)=1, by solving the equation for the integration constant CA it can be seen that its value equals 0for any value of XA= (XA1, ..., XAk). However after operating in context A during certain time t=t1the operational context changes to B which is described by XB= (XB1, ..., XBk), and the system reliability is going to be described now by equation 13.250 R(t, XB) = exp  −t αγ exp( k X j=1 β(XBj −Xj)) + CB (13) To obtain the unknown value of CBit is necessary to solve the equation R(t1, XA) = R(t1, XB), and the same reasoning will be followed to calculate system reliability if the asset changes in t=t2from operational context B to operational context C. Generalizing this logical process leads to the Equations 14 and 15. R(t, Xi) = exp  −t αγ exp( k X j=1 β(Xij −Xj)) + Ci (14) Ci=     0∀i= 0 ti αγexp(Pk j=1 β(Xij −Xj)) −exp(Pk j=1 β(X(i−1)j−Xj))+Ci−1∀i6= 0 (15) The Ciparameter is the term in which all the information regarding the operational context changes255 is stored, hence its recursiveness towards its value in the previous operational context. It acts as an indicator of the state of the system in terms of reliability, linking the new reliability curve to be followed with the previous evolution of the asset. In this parameter the sub-index iis for the operational context changes starting in the first context with i= 0. Taking a closer look at the different elements of the expression of the Ci∀i6= 0 it is possible to explain its meaning by dividing it into three terms:260 8 •ti αγ. The changes in the operational context do not affect the asset in the same way during the whole span of its operating time. The impact of the changes will depend on the time at which they happen and also on the characteristic of the asset. This term integrates the aforementioned details; the time at which the change happens is explicitly represented (ti) and asset characteristics are included in the characterization of the reliability curve, by the shape (β) and scale (α) parameters.265 •exp(Pk j=1 β(Xij −Xj)) −exp(Pk j=1 β(X(i−1)j−Xj)). This term integrates in the constant information regarding how different is the new operational context from the previous, the bigger the differences the more effect will have the constant on the new reliability curve which the system will follow. • Ci−1. The system state will also depend on previous changes it had gone through and therefore, it is270 this term the one that integrates it by gathering the information of the constant from the previous operational context change. As the changes in the operational context are not specified but defined according to time intervals, the model allows for a dynamic calculation of the reliability. Besides, the length of the time intervals are not defined, thus, they can be of any length and they do not have to be of the same duration; the longer the275 time interval the more aggregated the information and therefore the less accurate the estimations, but still better than not considering any change in the operational conditions. 2.4. Dynamic Artificial Neural Network-based reliability model By now the Neural Network based reliability has been introduced in subsection 2.2 and its main contribution is the possibility of modeling unknown interactions among the variables and hidden phenomena.280 Also the subsection 2.3 the Dynamic Weibul PHM that allows calculating system reliability taking into account changes in the operational context and their time span. Here in this subsection, the combination of both models is developed in order to achieve a model which comprehensively gathers the benefits from both approaches. Consider the hazard proposed by the Neural Network based reliability model, however, the baseline285 hazard is going to be adjusted to the mean values of the operational context, equation 16. From there, the calculation of the reliability follows the same logical process as in the Dynamic Weibull PHM. As it can be observed, the output of the Artificial Neural Net does not depend on time, thus, for integration purposes it is considered a constant. Then the following equations are derived through the same calculations as in the Dynamic Weibull Proportional Hazards section, the resulting reliability function is the one described290 in equation 18 and the value of the Citerm is described in equation 19. h(t, X) = hX(t)exp G((X−¯ X),W,B)(16) R(t, Xi) = exp −ˆγ αt αγ−1 exp(G((X−¯ X),W,B))dt!(17) R(t, Xi) = exp −t αγ exp(G((Xi−¯ X),W,B)) + Ci(18) Ci=   0∀i= 0 ti αγexp(G((Xi−¯ X),W,B)) −exp(G((Xi−1−¯ X),W,B))+Ci−1∀i6= 0 (19) With the proposal of this new model, it is possible not only to integrate operational context variables but to model the interactions among them that may trigger or condition failures. 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