Extended composite importance measures for multi-state systems with epistemic uncertainty of state assignment

Mechanical Systems and Signal Processing 109 (2018) 305–329

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Extended composite importance measures for multi-state systems with epistemic uncertainty of state assignment Tangfan Xiahou, Yu Liu ⇑, Tao Jiang School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, PR China Center for System Reliability and Safety, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, PR China

a r t i c l e

i n f o

Article history: Received 21 September 2017 Received in revised form 23 December 2017 Accepted 10 February 2018

Keywords: Multi-state system (MSS) Evidential networks Extended composite importance measures Epistemic uncertainty Component state assignment

a b s t r a c t Importance measures of multi-state systems have been intensively investigated from different perspectives in the past few years as the results are able to provide a valuable guidance for effective reliability improvement and enhancement. The state assignment is oftentimes conducted to identify the state of a multi-state system when features and/or knowledge related to the health condition of the particular system are collected. However, due to the scarcity of sensor data, limited accuracy of sensing techniques, and vague/conflicting judgments from experts, conducting the state assignment is imprecise and inevitably produces epistemic uncertainty. In this paper, some composite importance measures of multi-state systems are extended by considering the epistemic uncertainty associated with component state assignment. To take account of such epistemic uncertainty, the proposed method contains three basic steps: (1) propagate the epistemic uncertainty associated with component state assignment to the reliability function of a multi-state system by dynamic evidential network models, (2) evaluate the intervals of the conditional reliability by inputting hard evidences and/or vacuous evidence into the tailored dynamic evidential network models, and (3) compute the extended composite importance measures by constructing a pair of optimization problems and properly handling the dependency among input intervals. A numerical example of a multi-state bridge system together with an engineering example of a feeding control system of CNC lathes is exemplified to demonstrate the impact of the epistemic uncertainty on the importance measures of components and their rankings. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Importance measures are effective tools to identify the weak components of an engineered system from the reliability and/or structure perspectives [1]. The results gained from importance measures can provide valuable insights to reliability

Abbreviations: MSS, Multi-State System; ET, Evidence Theory; TBM, Transferable Belief Model; EMC, Evidential Markov Chain; EN, Evidential Network; DEN, Dynamic Evidential Network; CIM, Composite Importance Measure; CBIM, Composite Birnbaum Importance Measure; MRAW, Multi-state Reliability Achievement Worth; MRRW, Multi-state Reliability Reduction Worth; MFV, Multi-state Fussel-Vesely; E-CIM, Extended Composite Importance Measure; E-CBIM, Extended Composite Birnbaum Importance Measure; E-MRAW, Extended Multi-state Reliability Achievement Worth; E-MRRW, Extended Multi-state Reliability Reduction Worth; E-MFV, Extended Multi-state Fussel-Vesely. ⇑ Corresponding author. Postal Address: No. 2006, Xiyuan Avenue, West Hi-Tech Zone, Chengdu, Sichuan 611731, PR China. E-mail address: [email protected] (Y. Liu). https://doi.org/10.1016/j.ymssp.2018.02.021 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.

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Nomenclature CðtÞ the state combination of all the components at time instant t /ðÞ system structure function ACj l component C l in state j C ðtÞ Aj l component C l in state j at time instant t state space of component C l SC l mðACj l jACi l Þ transition mass of component C l transiting from state i to state j within a basic time interval mC l ðtÞ mass distribution of component C l at time instant t C ðtÞ ConðAj l Þ conditional reliability at time instant t on condition that component C l in state j RðtÞ system reliability at time instant t C l ðtÞ PRðAj Þ partial reliability of a system at time instant t PDfa > bg the degree of preference of the interval a over the interval b

improvement and maintenance planning of a system [1,2]. Many importance measures have been developed from various angles [1], and they have been successfully implemented to complex engineered systems, such as nuclear power plant [3], mechanism models [4], reconfigurable systems [5], and so forth. Among the existing reliability importance measures, the Birnbaum importance measure was the most popular one used for system reliability design [6]. The Birnbaum importance measure quantifies the most critical component to the system reliability. Other reliability importance measures, such as the Risk Reduction Worth (RRW), Risk Achievement Worth (RAW), and the Fussell-Vesely importance measure (FV) [7], were also studied from different implications of reliability improvement or decrease. More recently, Natvig et al. [8] presented a new importance for repairable and non-repairable systems. Zio et al. [9] studied a joint importance measure based on the partial derivative on the reliability of a group of components. Peng et al. [10] introduced a criticality measure for degrading components. Alieea et al. [11] proposed a new Birnbaum importance to non-coherent systems. Kuo and Zhu [1,12] summarized the importance measures into c-types and p-types, and they applied importance measures to the component assignment problem (CAP). Nevertheless, the traditional importance measures were based on the premise that a system and its components can only be in one of only two possible states, either fully functioning or completely failed. As engineered systems become more sophisticated, the traditional reliability methods for binary-state systems, however, fail to characterize the complicated deteriorating process of systems with multi-state nature. By introducing more than two states, from completely functioning down to completely failed, multi-state system (MSS) models are able to more accurately characterize the complicated behaviors of a system [13–15]. In the context of MSSs, many novel approaches, such as the simulation-based method [3,16,17], the multi-valued decision diagram method [18,19], the stochastic processes [15], the universal generating function [20], and the recursive algorithm [21], have been developed to facilitate the reliability assessment of MSSs. The importance measures of MSSs have also received considerable concerns, and a set of new importance measures of MSSs have been defined from various perspectives. Griffith [22] firstly introduced the concept of multi-state system performance, and studied an importance measure to quantify the MSS performance improvement due to the component performance improvement. Levitin and Lisnianski [23] proposed a partial derivative method to examine how a component performance may influence the availability of an MSS. Levitin et al. [24] introduced a new MSS importance measure based on the performance restriction. Zio and Podofillini [3] presented Birnbaum, Fussell-Vesely, RAW, RRW unavailability measures for MSSs. In their study, the Monte-Carlo simulation (MC) was used to emulate the stochastic behaviors of multistate components. Ramirez-Marquez and Coit [2] proposed composite importance measures (CIMs) to analyze how a specific multi-state component may affect the reliability of an MSS. Lisnianski et al. [25] studied a new sensitivity measure for aging components based on the definition of importance measures. Si et al. [26] put forth an integrated importance measure of multi-state systems to study how the transition intensities of components affect the loss of system performance. Dui et al. [27] proposed a cost-based integrated importance measure for the preventive maintenance of MSSs. Among all the existing importance measures of MSSs, CIMs proposed by Coit [2] have received the most widespread applications, such as the network allocation [4], network resilience [28], power industry [29], and so forth, because they are capable of evaluating the influence of all the states of a particular component, not a single state, on the reliability of an MSS. Our focus on this work is placed on the CIMs due to their popularity in a sizable amount of engineering applications. Nevertheless, epistemic uncertainty, caused by lack of sufficient data and vague/conflicting knowledge, is inevitable in engineering practices, and manipulating epistemic uncertainty is a challenging task to reliability assessment of complex systems [30,31]. Some non-probabilistic methods were used to represent epistemic uncertainty, such as the evidence theory (ET) [32,33], the fuzzy set theory [34], the interval theory [35], and the imprecise probability theory [36]. Generally, the ET, as one of the representations of epistemic uncertainty, have received considerable attentions in the field of information fusion [37], fault diagnosis [38,39], decision making [40] and so forth. Some attempts have been made to study the influence of epistemic uncertainty on the reliability analysis of MSSs in the context of the ET. Sallak [41] generalized the universal

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generating function in the framework of the ET to study the reliability assessment of MSSs under imprecise knowledge of components’ state distributions. Simon and Webber [42] introduced the evidential network (EN) model, which can be regarded as a generalized model of Bayesian networks, to infer the state distribution of MSSs under both aleatory and epistemic uncertainties. In their work, the conditional dependencies among components and the entire system were quantified by using the conditional belief mass functions, and then, the system reliability was evaluated by the imprecise utility function. Most recently, Simon and Bicking [43] developed a hybrid EN model to evaluate the system reliability by using the pbox theory to represent the uncertainties of components’ reliabilities. To explore the impact of epistemic uncertainty on the important measures of binary-state systems, Baraldi et al. [44] studied a new importance ranking measure under the ET when epistemic uncertainty affects the probability density function. Based on the framework of the ET, Sallak et al. [45] extended the Birnbaum importance, RRW, RAW, and FV by considering the total lack of knowledge of components’ reliabilities. The extended importance measures were eventually converted into the affined arithmetic problem to avoid the error explosion problem. As shown in their studies, the epistemic uncertainty associated with components’ reliability could influence the importance ranking of components. Nonetheless, to the best of our knowledge, the importance measures of MSSs under epistemic uncertainty have not been reported yet. On the other hand, even though advanced sensor technologies have been pervasively adopted in engineering practices to track the condition of complex degrading systems in recent years [37,46], it is still not one hundred percentage guaranteed to identify the true health state of a system due to the limited accuracy and monitoring capability of sensors and/or the insufficient knowledge from experts [38,39]. It, therefore, introduces epistemic uncertainty to the state assignment of an MSS. As reported in [44,45], for binary-state systems, the epistemic uncertainty could eventually have a significant impact on the component importance measures. Thereby, our motivation is to explore the importance measures for MSSs when the epistemic uncertainty associated with component state assignment cannot be completely avoided. However, unlike binary-state systems reported in [44,45], extending the importance measures under epistemic uncertainty associated with component state assignment in the context of MSSs is not straightforward, and it raises three challenges to be addressed: (1) the lack of an effective tool to quantify the epistemic uncertainty associated with state assignment, (2) the high computational complexity in evaluating the reliability of an MSS, as well as the conditional reliability which is defined as the reliability of the system on the condition that a specific component is assumed to be at a particular state, and (3) the treatment of the dependency among the inputs of importance measure evaluation. It is noteworthy that even though the EN was developed by Simon and Webber in [42] to deal with both aleatory and epistemic uncertainties associated with component states. Nevertheless, it is a static model that can only evaluate the system reliability at a particular time instant, rather than a period of time. Hence, the epistemic uncertainty propagation with the evolution of components’ states cannot be well characterized to facilitate the evaluation of importance measures. To overcome the above obstacles, in this paper, a dynamic evidential network (DEN) combining the evidential Markov chain (EMC) and the EN is tailored to quantify the epistemic uncertainty associated with component state assignment, to propagate the epistemic uncertainty with the evolution of components’ states, and to overcome the high computational complexity in the system reliability evaluation. Furthermore, the total belief and plausibility theories are utilized to deal with the dependency among the output intervals of the DEN. Some well-known CIMs are extended by introducing a pair of optimization formulations to gain the upper and lower bounds of these importance measures. The rest of this paper is rolled out as follows. Section 2 introduces the basic background of problems to be studied in this work. Section 3 briefly reviews the fundamentals of the ET, and the dynamic evidential network model is tailored to model our specific problem. The system reliability and conditional reliability which are the inputs of importance measures are evaluated by the DEN in Section 4. The new formulations for four typical CIMs in the context of epistemic uncertainty are proposed in Section 5. Two illustrative examples, together with a set of comparative studies, are presented to demonstrate the validity and applicability of the extended CIMs in Section 6. Our conclusions and a summary of future works are given in Section 7.

2. Problem descriptions 2.1. Multi-state systems Many engineered systems can be viewed as MSSs by assigning their degradation processes into several discrete damage severities or health states [46]. For instance, the degradation of a gear cracking process can be roughly classified into several discrete states from perfectly functioning to completely failed, say ‘‘normal”, ‘‘slight crack”, ‘‘medium crack”, ‘‘serious crack”, and ‘‘completely worn out”. Similar treatments can be found in various engineering systems, such as power generating systems [47] and networked systems [48]. Without loss of generality, an MSS is composed of M components which belong to N subsystems. The states of component C l (l ¼ 1; 2; . . . ; M), subsystem Sn (n ¼ 1; 2; . . . ; N), and the entire system S at time instant t are denoted as C l ðtÞ 2 NC l , Sn ðtÞ 2 NSn , and SðtÞ 2 NS , respectively, where NC l ¼ f0; 1; 2; . . . ; nC l g, NSn ¼ f0; 1; . . . ; nSn g, and NS ¼ f0; 1; . . . ; nS g. Among these states, State 0 is the worst state, whereas nC l , nSn , and nS are the best states of component C l , subsystem Sn , and the system, respectively. The state combinations of all the components at time instant t can be denoted as CðtÞ ¼ fC 1 ðtÞ; C 2 ðtÞ; . . . ; C M ðtÞg.

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The state of the entire system can be determined by the components’ state combinations and the system structure function, denoted as /ðCðtÞÞ. In addition to the above definitions of MSSs, some basic assumptions regarding the studied MSSs are summarized as follows: (1) To quantitatively characterize the degradation behavior of an MSS, the stochastic evolutions of its components need to be first studied. With a memoryless assumption that the probability of a future state of component C l is s-independent of its previous state, the evolution process of a component can be characterized by a Markov process. In this study, we will focus on the case where the deterioration of a component can be governed by a homogenous discrete-time Markov chain. The state transition probability of multi-state component C l from its state i to j (0 6 j 6 i 6 nC l ) within a C

basic time interval Dt is denoted as pi;jl ¼ PrfC l ðt þ DtÞ ¼ jjC l ðtÞ ¼ ig. The corresponding one-step state transition matrix of component C l can be written as:

2

C

l p0;0 6 C 6 p l 6 1;0 PC l ¼ 6 6 .. 6 . 4 C pnCl ;0 l

where

PnCl

Cl j¼i pi;j

0

0

0

C

0

0

.. .

..

.. .

l p1;1

C

pnCl ;1 l

.

3 7 7 7 7; 7 7 5

C

   pnCl ;nC l

ð1Þ

l

¼ 1:0 (i ¼ 0; 1; . . . ; nC l ). The state probability distribution of component C l at any time instant t is C

C

C

denoted as a vector of pC l ðtÞ ¼ ðp0 l ðtÞ; p1l ðtÞ; . . . ; pnCl ðtÞÞ. Thus, the state probability distribution of component C l after l

k

a duration of kDt (k ¼ 1; 2 . . .) can be computed by pC l ðt þ kDtÞ ¼ pC l ðtÞ  ðPC l Þ . (2) The structure function /ðCðtÞÞ of the entire MSS with respect to the states of all the components is deterministic and known. Any combination of components’ states corresponds to one of all the possible system states. Moreover, the structure function /ðCðtÞÞ is coherent, that is, any improvement to component states cannot lead to a deterioration of system state. (3) The reliability of an MSS is defined as the probability that the system state is no less than a threshold d at any time instant [20,24], which can be mathematically written as: RðtÞ ¼ Prf/ðCðtÞÞ P dg. (4) Maintenance activities are not considered in the present study, and thus a transition from a lower state to a better state does not exist. 2.2. Epistemic uncertainty associated with state assignment In real-world situations, condition monitoring techniques can be used to track the health state of multi-state components/systems [37,46]. However, due to the inaccuracy of sensing techniques, lack of sufficient knowledge, and vague/conflicting judgments from engineers, the state of deteriorated components/systems cannot be assigned accurately [37–39,49]. Such phenomenon has been commonly encountered in condition monitoring and fault diagnosis of rotating machinery even if many advanced sensors and algorithms have been used [38,39,50]. On the other hand, there is no strict criterion to determine the number of the states of degrading components/systems as well as the boundaries of states in engineering practices [51,52]. The state assignment of components/systems is oftentimes conducted by experts based on the sensitive features extracted from raw data [53]. The results, of course, may vary from person to person, introducing the epistemic uncertainty to component/system state assignment. Fig. 1 illustrates the epistemic uncertainty produced in the process of state assignment. In Fig. 1(a), as different experts or condition monitoring techniques may identify the state of components/systems based on their own criteria or mechanism, the epistemic uncertainty arise when the judgements/results from multiple experts or algorithms are conflicting. Furtherly, in the field of sensor data-based condition monitoring, it is extremely difficult to categorize the sensor data which are close to boundaries of states to a particular state among all the possible states. It, thus, introduces the epistemic uncertainty associated with state assignment due to the inaccuracy of sensing techniques as shown in Fig. 1(b). The evidence theory (ET) can tackle the above imprecise knowledge or information by assigning a non-probabilistic measure for the lack of knowledge. As compared to other non-probabilistic theories, such as the interval theory, the possibility theory, and the fuzzy set theory, which are also widely used to represent epistemic uncertainty, the ET offers a uniform framework to lump together various sources of evidences by handling conflicting evidences and updating the nonprobabilistic measure [38]. For instance, the ET was implemented to fault diagnosis [38,39] when the epistemic uncertainty arises from feature extraction, and it was also utilized to fuse multiple experts’ judgements for decision-making [37,40]. In view of its strong capability in terms of handling the imprecise knowledge or information, the ET is used in this work to quantify the epistemic uncertainty associated with state assignment. As depicted in Fig. 1, by introducing a non-singleton subset of states, say, State ½0; 1; . . . ; n, the ET can represent the uncertainty among the singletons without any inclination to select a particular state. It is worth noting that State ½0; 1; . . . ; n is not a new state of components/systems. Rather, it is

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Fig. 1. The epistemic uncertainty associated with state assignment. (a) The epistemic uncertainty from the conflicting judgements of multiple experts. (b) The epistemic uncertainty from the inaccuracy of condition monitoring techniques.

an unassigned state, representing the epistemic uncertainty due to the limited ability in terms of assigning the singletons based on the readings from condition monitoring and experts’ vague/conflicting judgements. Due to the epistemic uncertainty resulting from the state assignment, the evidential network models will be implemented to infer the state distribution of an MSS. The detail procedures of constructing the evidential networks for our studied problem will be introduced in the ensuing section. 3. Dynamic evidential network models for MSS In this section, by combining the EMC with the EN, a tailored DEN is implemented to assess the system reliability and conditional reliability. The EMC is first introduced to quantify the epistemic uncertainty associated with component state assignment, and then, the EN is used to graphically represent the relations between components’ states and system states. Before starting with our proposed model, the basic knowledge of the ET is briefly reviewed. 3.1. The fundamentals of the evidence theory The evidence theory (ET), as a generalized extension to probability theory, was firstly introduced by Dempster [54], and was further improved by Shafer [55]. Due to its capability of representing epistemic uncertainty, the ET has widespread applications in the fields of information fusion [37], fault diagnosis [38,39] and reliability analysis [56]. In this work, the ET is used as a tool to quantify the epistemic uncertainty associated with component state assignment as mentioned in Section 2. In the framework of the ET, the state space of component C l is called the frame of discernment, denoted as

XCl ¼ f0; 1; . . . ; nC l g. Let ACj l represents component C l being in state j, and it takes value from the power set C

SC l ¼ f0; 1; . . . ; nC l ; sC l g, where sC l represents the unassigned state, i.e., state ½0; 1; . . . ; nC l . A mass function mðAj l Þ can be then P C C defined as a mapping function from SC l to ½0; 1, where ACl # S mðAj l Þ ¼ 1 and mð;Þ ¼ 0. The mass function mðAj l Þ represents j

Cl

C

C

the belief assigned to the hypothesis that the truth lies in set Aj l without further dividing this belief to a subset of Aj l . Some properties of the mass function are as follows: C

C

(1) If mðAj l Þ > 0, then Aj l is a focal set to the frame of discernment XC l . Particularly, if all the focal sets are unique singletons, the mass function in the framework of the ET degenerates to Bayesian probability. (2) mð;Þ ¼ 0 satisfies if one accepts the closed world assumption. However, it can be relaxed to an open world assumption since the frame of discernment may be incomplete. In this study, we comply with the assumption of mð;Þ ¼ 0. C

C

(3) In the ET, a vacuous evidence for component C l corresponds to mðsC l Þ ¼ 1 and mðAj l Þ ¼ 0 (Aj l –sC l ), meaning the total ignorance (total uncertainty) for the states of component C l . Likewise, a hard evidence is represented by assigning the C

entire mass to a unique singleton, that is, mðAj l Þ ¼ 1 (j 2 XC l ). C

The belief function and the plausibility function are two useful tools to quantify the confidence of a particular event Aj l in the framework of ET, and they are, respectively, defined as: C

BelðAj l Þ ¼

X Cl

C

C

C

mðAi l Þ 8Ai l ; Aj l # SCl ; Cl

;–Ai # Aj

ð2Þ

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T. Xiahou et al. / Mechanical Systems and Signal Processing 109 (2018) 305–329

and

X

C

PlðAj l Þ ¼ Cl

C

C

C

mðAi l Þ 8Ai l ; Aj l # SCl :

ð3Þ

Cl

Ai \Aj –; C

The belief function quantifies the amount of mass which supports Aj l . It can be interpreted as the degree of belief that the C

C

truth lies in Aj l . The plausibility function quantifies the maximum amount of the potential support which can be given to Aj l . It can be interpreted as the mass that fails to doubt

C Aj l

[57]. These two quantities, i.e.,

upper bounds which the probability of the occurrence of event

C Aj l

C BelðAj l Þ

and

could lie in, and the relation

C PlðAj l Þ, offer the lower and C C C BelðAj l Þ 6 PrfAj l g 6 PlðAj l Þ

always holds. Suppose that XC l and XC lþ1 are two frames of discernments of component C l and component C lþ1 , respectively, the total plausibility law (TPL) can be used to construct the casual link between component C l and component C lþ1 , and it is formulated as:

X

C

PlðAj lþ1 Þ ¼

XC lþ1 jXCl

Pl

C

C

C

ðAj lþ1 jAj l Þ  mðAj l Þ:

ð4Þ

Cl

Aj # SC

l

Likewise, for the belief function, one has: C

BelðAj lþ1 Þ ¼

X

XC lþ1 jXCl

Bel

C

C

C

ðAj lþ1 jAj l Þ  mðAj l Þ:

ð5Þ

Cl

Aj # SC

l

It is noted that Eqs. (4) and (5) are the basis of the evidential networks to be used in the ensuing sections [58]. 3.2. Evidential Markov chain By the framework of the ET, a non-singleton subset of states, i.e., state ½0; 1; . . . ; nC l , is introduced to represent the epistemic uncertainty associated with the state assignment of component C l . The mass function of the non-singleton subset quantifies the amount of epistemic uncertainty. Furthermore, the transition between a singleton states j (j 2 NC l ) and a non-singleton state sC l is used to model the stochastic behaviors of components C l over time. Thus, the matrix of one-step C

transition probability in Markov model can be generalized under the ET by replacing the transition probability pi;jl with the transition mass

C C mðAj l jAi l Þ,

and it is so-called the evidential Markov chain (EMC) [50,59]. The corresponding matrix of

transition mass can be written as:

2

C

C

mðA0l jA0 l Þ



0

3

0

7 6 6 mðAC0l jAC1l Þ mðAC1l jAC1l Þ    mðACs l jAC1l Þ 7 Cl 7 6 7; M ¼6 7 6 .. .. .. .. 7 6 . . . . 5 4 Cl Cl Cl Cl Cl Cl mðA0 jAsC Þ mðA1 jAsC Þ    mðAsC jAsC Þ Cl

l

where

P C

Aj l

l

C C mðAj l jAi l Þ # SC

¼ 1 for any

l

vector of mC l ðtÞ ¼

l

C Ai l

ð6Þ

l

# SC l . The state mass distribution of component C l at time instant t is denoted as a

C ðtÞ C ðtÞ ðmðA0 l Þ; mðA1 l Þ; . . . ; mðAsCCl ðtÞ ÞÞ. l

In the same fashion as Eq. (1), the state mass distribution of component C l

after a duration of kDt (k ¼ 1; 2; . . .) can be computed by [59]: k

mCl ðt þ kDtÞ ¼ mC l ðtÞ  ðMC l Þ :

ð7Þ Cl

It should be noted that the transition mass matrix M of component C l can be estimated by collecting sensor data of a particular component C l during a time horizon of interest T. Given the mass distributions of component C l , i.e., mC l ðtÞ (t ¼ 0; Dt; 2Dt; . . . ; T), which can be obtained by sensor data, the transition mass matrix can be calculated by the total mass theory as follows [59]:

PT1 C C mðAj l jAi l Þ

¼P j2SC

C ðtþDtÞ

C ðtÞ

mðAj l Þ  mðAi l Þ ; PT1 C l ðtþDtÞ C ðtÞ Þ  mðAi l Þ t¼0 mðAj

t¼0 l

ð8Þ C

where the denominator of Eq. (8) is the total mass of the transitions starting with state Ai l during the time horizon of interest C

C

T, whereas the numerator of Eq. (8) represents the total mass of the transition from state Ai l to state Aj l within one basic time interval Dt.

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3.3. Evidential networks To evaluate the state mass distribution of the entire system, the evidential network (EN) can be used as it is able to graphically represent the relations among system’s states and components’ states. In general, an EN is composed of two elements, i.e., the topological structure and the conditional belief mass tables (CBMTs) [42]. The topological structure of a specific system is a directed acyclic graph (DAG) consisting of n nodes, denoted as fX 1 ; X 2 ; . . . ; X n g, and z directed edges. Each node X i (i ¼ 1; 2; . . . ; n) represents a variable that can manifest all the singleton and non-singletons under the ET. A directed edge from X j to X i can be used to represent the casual relation between the two nodes. Hence, X j can be viewed as a parent node of X i , whereas X i is a child node of X j , denoted as paðX i Þ ¼ X j . In an EN, the topological structure of the EN reflects the dependence among nodes, whereas the CBMTs provide a quantitative measure of the strengths of these dependences. When a node X i has parent nodes, it has a CBMT, denoted as mðX i jpaðX i ÞÞ, representing the mass distribution of X i on the condition of paðX i Þ. A node without parent nodes is a root node, and one has to specify the marginal mass distribution to this node. To characterize an MSS by an EN model, components, subsystems, and the entire system can be all represented by nodes in the EN. Suppose that a MSS consists of M components which belong to N subsystems, a set of nodes, denoted as H ¼ fC 1 ; C 2 ; . . . ; C M ; S1 ; S2 ; . . . ; SN ; Sg, can be used to construct an EN model. A series-parallel MSS with four components is given in Fig. 2 to demonstrate the construction of the EN model for a particular MSS. The EN contains seven nodes, i.e., fC 1 ; C 2 ; C 3 ; C 4 ; S1 ; S2 ; Sg, where the root nodes C 1 , C 2 , C 3 , and C 4 denote components 1, 2, 3, and 4, respectively, whereas the nodes S1 and S2 represent subsystem 1 and 2, respectively. The node S represents the entire system. The nodes C l (l ¼ 1; 2; . . . ; 4) can take any value in the set of SC l ¼ f0; 1; . . . ; nC l ; sC l g (l ¼ 1; 2; . . . ; 4). It should be noted that if the mass to state sC l (l ¼ 1; 2; . . . ; 4) of nodes C l (l ¼ 1; 2; . . . ; 4) is zero, the EN model degenerates to the Bayesian network model. A representative CBMT for a serial configuration, e.g., components C 1 and C 2 , is given in Table 1, whereas the CBMT for a parallel configuration, e.g., subsystem S1 and component C 3 , is presented in Table 2. As shown in Tables 1 and 2, if the states of the parent nodes are fully known, the state of the corresponding child node can be determined. For example, mðS1 ¼ 0jC 1 ¼ 0; C 2 ¼ 2Þ ¼ 1 in the 3rd row of Table 1 represents that subsystem S1 is in state 0 if components C 1 and C 2 are in state 0 and state 2, respectively. However, if the states of some parent nodes are uncertain, the state of their corresponding child nodes may be uncertain. For instance, mðS1 ¼ sS1 jC 1 ¼ 2; C 2 ¼ sC 2 Þ ¼ 1 in the 12th row of Table 1. Nevertheless, for parallel configuration, if subsystem S1 is in state 2 and component C 3 is in unassigned state sC 3 , subsystem S2 will be in state 2 as shown in the 12th row of Table 2. Because the state of subsystem S2 corresponds to the best state of subsystem S1 and component C 3 . Based on these two CBMTs for series and parallel configurations, the relations among components’ states, subsystems’ states, and system states can be completely determined. Nonetheless, the aforementioned EN is only a static model representing the states of components, subsystem, and the entire system at a particular time instant. With the purpose of characterizing the stochastic behaviors of an MSS, the concept of time slice can be introduced into the EN model. A time slice EN model can be used to represent an MSS at a particular time instant, and the MSS after degradation in a basic time interval can be represented by another adjacent time slice. Hence, a dynamic evidential network (DEN), which is composed of multiple chronological time slices of a repetitive EN model, can be constructed to characterize the stochastic behavior of an MSS over time. The temporal links from one time slice to the next can be quantified by the transition mass matrix of the corresponding EMC. Therefore, an DEN model for an MSS is composed of a set of nodes, denoted as HðtÞ ¼ fC 1 ðtÞ; C 2 ðtÞ; . . . ; C M ðtÞ; S1 ðtÞ; S2 ðtÞ; . . . ; SN ðtÞ; SðtÞg, where t ¼ 0; 1; . . . ; T, T is the time horizon of interest. C l ðtÞ ðl ¼ 1; 2; . . . ; MÞ, Sn ðtÞ ðn ¼ 1; 2; . . . ; NÞ, and SðtÞ represent the states of component Cl, subsystem Sn, and

C1

1

C2

S1

2 4

C3

C4

S2

3 S Fig. 2. A series-parallel MSS and its topological structure.

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Table 1 The CBMT for a serial configuration. C1

C2

mðS1 jC 1 ; C 2 Þ S1 ¼ 0

S1 ¼ 1

S1 ¼ 2

S1 ¼ sS1

0 0 0 0 1 1 1 1 2 2 2 2 sC 1 sC 1 sC 1 sC 1

0 1 2 sC 2 0 1 2 sC 2 0 1 2 sC 2 0 1 2 sC 2

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

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

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

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

Table 2 The CBMT for a parallel configuration. S1

C3

mðS2 jS1 ; C 3 Þ S2 ¼ 0

S2 ¼ 1

S2 ¼ 2

S2 ¼ sS2

0 0 0 0 1 1 1 1 2 2 2 2 sS1 sS1 sS1 sS1

0 1 2 sC 3 0 1 2 sC 3 0 1 2 sC 3 0 1 2 sC 3

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

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

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

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

the system at any time slice t, respectively. The DEN for the four-component MSS in Fig. 2 can be constructed by linking the time slices of the EN model in a chronological order as depicted in Fig. 3. 4. Reliability and conditional reliability assessment for MSS In this section, the reliability and conditional reliability of an MSS under epistemic uncertainty are evaluated because these values will serve as the inputs of importance measures developed in the next section. In an DEN, the mass distributions of the root nodes are computed by the EMC. Referring to the notations of MSSs in Section 3.2, the initial mass distribution of component Cl at t ¼ 0 is mC l ð0Þ, which can be viewed as the marginal mass distribution of node C l ð0Þ at the first time slice. The marginal mass distribution of node C l ðtÞ at the time slice t ¼ kDt can be, then, calculated by Eq. (7), which characterizes the degradation process of component C l . In order to characterize the degradation process of the entire system, the marginal mass distribution of all the child nodes X i ðtÞ (X i ðtÞ ¼ S1 ðtÞ; S2 ðtÞ; . . . ; SN ðtÞ; SðtÞ) at time slice t should be computed. The inference algorithm of the EN can be used here to gain the marginal mass distribution of child node X i ðtÞ (X i ðtÞ ¼ S1 ðtÞ; S2 ðtÞ; . . . ; SN ðtÞ; SðtÞ), and it is written as [60]:

mðX i ðtÞÞ ¼

X paðX i ðtÞÞ2HðtÞ

mðX i ðtÞjpaðX i ðtÞÞÞ 

Y

! mðpaðX i ðtÞÞÞ ;

ð9Þ

paðX i ðtÞÞ2HðtÞ

where paðX i ðtÞÞ is a set of the parent nodes of node X i ðtÞ; mðX i ðtÞjpaðX i ðtÞÞÞ is the CBMT of node X i ðtÞ at time slice t. Hence, the marginal mass distributions of all the child nodes computed by Eq. (9) can be further generalized as:

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T. Xiahou et al. / Mechanical Systems and Signal Processing 109 (2018) 305–329

…

t=0

t=1

…

t=2

t=T

Time Slice t

Fig. 3. An DEN model for the series-parallel MSS.

! 8 Y X > > > mðSn ðtÞjpaðSn ðtÞÞÞ  mðpaðSn ðtÞÞÞ for all the subsystems > > < paðS Þ2HðtÞ paðSn Þ2HðtÞ n ! mðX i ðtÞÞ ¼ ; > Y X > > > mðSðtÞjpaðSðtÞÞÞ  mðpaðSðtÞÞÞ for the entire system > : paðSÞ2HðtÞ

ð10Þ

paðSÞ2HðtÞ

where paðSn ðtÞÞ and paðSðtÞÞ are the parent nodes of Sn ðtÞ and SðtÞ, respectively; mðpaðSi ðtÞÞÞ is the mass distribution of all the parent nodes to node Sn ðtÞ (n ¼ 1; 2; . . . ; N); mðpaðSðtÞÞÞ is the mass distribution of all the parent nodes of node SðtÞ; mðSn ðtÞjpaðSn ðtÞÞÞ and mðSðtÞjpaðSðtÞÞÞ are the CBMTs of nodes Sn ðtÞ and SðtÞ, respectively. Referring to the definition of the reliability of MSSs (Assumption 3) in Section 2.1, the interval bounds of the reliability of an MSS at this particular time slice can be calculated by putting the belief and plausibility functions into the mass distribution mðSðtÞÞ of node SðtÞ at time slice t (t ¼ 0; 1; 2; . . . ; T). Put another way, the lower bound of the reliability of the entire system, denoted as Belð/ðCðtÞÞ P dÞ, can be evaluated by:

X

Belð/ðCðtÞÞ P dÞ ¼

SðtÞ

mðAj Þ;

ð11Þ

SðtÞ Aj Pfdg

whereas the upper bound, denoted as Plð/ðCðtÞÞ P dÞ, is formulated as:

X

Plð/ðCðtÞÞ P dÞ ¼

SðtÞ

mðAj Þ:

ð12Þ

SðtÞ SðtÞ ðAj PfdgÞ[ðAj \fdg–;Þ

Hence, the reliability of the system at any time instant t falls into the interval of ½Belð/ðCðtÞÞ P dÞ; Plð/ðCðtÞÞ P dÞ. For SðtÞ

example, suppose that the reliability of the system in Fig. 3 is defined as the probability that the system state Aj SðtÞ

ðAj

# f0; 1; 2; ½0; 1; 2gÞ is no less than the state 1, i.e., d ¼ 1, the lower bound of the system reliability is Belð/ðCðtÞÞ P SðtÞ

1Þ ¼ mðAj f2gÞ þ

SðtÞ

¼ f1gÞ þ mðAj

SðtÞ mðAj

¼ f2gÞ,

whereas

the

upper

bound

is

SðtÞ

Plð/ðCðtÞÞ P 1Þ ¼ mðAj

SðtÞ

¼ f1gÞ þ mðAj

¼

¼ f½0; 1; 2gÞ. C ðtÞ

In our study, the conditional reliability of an MSS, denoted as Prf/ðCðtÞ P djAj l Þg, is the reliability of the system at time instant t on the condition that component C l at a particular state j. Furthermore, considering the epistemic uncertainty associated with component state assignment, the conditional reliability becomes interval-valued, denoted as C ðtÞ

C ðtÞ

½Belð/ðCðtÞÞ P djAj l Þ; Plð/ðCðtÞÞ P djAj l Þ, and it can also be evaluated by the DEN model effectively. In this study, we compute the interval of the conditional reliability by inputting hard evidences or vacuous evidences into the DEN model. By specifying the state of a particular component, the belief and plausibility functions of the conditional reliability can be, respectively, formulated as: C ðtÞ

C ðtÞ

Belð/ðCðtÞÞ P djAj l Þ ¼ Belð/ðCðtÞÞ P djmðAj l Þ ¼ 1Þ;

ð13Þ

and C ðtÞ

C ðtÞ

Plð/ðCðtÞÞ P djAj l Þ ¼ Plð/ðCðtÞÞ P djmðAj l Þ ¼ 1Þ:

ð14Þ

314

T. Xiahou et al. / Mechanical Systems and Signal Processing 109 (2018) 305–329 C ðtÞ

In general, a hard evidence to component C l at time instant t, denoted as mðAj l Þ ¼ 1 (j 2 NC l ), represents that the state of C ðtÞ

C ðtÞ

component C l is ascertained. On the contrary, a vacuous evidence of component C l , denoted as mðAj l Þ ¼ 1 (Aj l

¼ fsC l g), is

the total ignorance to the state of component C l . Thus, by putting a hard evidence or a vacuous evidence of component C l together with the mass distributions of the remaining components into a DEN model, the interval of the conditional reliaC ðtÞ

C ðtÞ

bility ½Belð/ðCðtÞÞ P djAj l Þ; Plð/ðCðtÞÞ P djAj l Þ, for all C l 2 fC 1 ; C 2 ; . . . ; C M g can be assessed. 5. The extended composite importance measures for MSS The CIMs defined by Coit [2] focus on investigating how a specific multi-state component affect the reliability of an MSS. A variety of CIMs have been introduced to measure positive and/or negative impacts from a component on the system reliability. The positive impact can be interpreted as the reliability increment of a system generated by the state improvement of a particular component. The importance measures, such as MRAW, belong to this category. On the contrary, the importance measures, such as MRRW and MFV, evaluate the negative impact, that is, the system reliability decrement produced by the degradation of a particular component. The CBIM is a particular importance measure which can examine the impact from both positive and negative angles. The CIMs have gained much popularity in the field of networked systems [28], and power systems [29] due to its strong ability to identify the weakness of a system. In this study, four commonly used CIMs, i.e., CBIM, MRAW, MRRW, and MFV, are extended by taking account of the aforementioned epistemic uncertainty associated with component state assignment. 5.1. Extended CBIM In the context of MSSs, the composite Birnbaum importance measure (CBIM) quantifies the criticality of a component to the system reliability. The CBIM of component C l at time instant t can be mathematically expressed as [2]: nC l X C ðtÞ jPrf/ðCðtÞÞ P djAj l g  Prf/ðCðtÞÞ P dgj

IBC l ðtÞ ¼

j¼0

;

jNCl j  1

ð15Þ

where jNC l j is the cardinality of NC l , and the denominator restricts the value of IBC l ðtÞ to be in the range of ½0; 1. Due to the epistemic uncertainty associated with component state assignment, the system reliability Prf/ðCðtÞÞ P dg and the condiC ðtÞ

tional reliability Prf/ðCðtÞÞ P djAj l g in Eq. (15) become interval-valued, denoted as ½Belð/ðCðtÞÞ P dÞ; Plð/ðCðtÞÞ P dÞ and ½Belð/ðCðtÞÞ P

C ðtÞ djAj l Þ; Plð/ðCðtÞÞ

C ðtÞ

P djAj l Þ, respectively, as introduced in Section 4. Such uncertainty will propagate

to the CBIM, and the extended CBIM becomes an interval value too. Thanks to the inferencing capability of the DEN, the reliability interval value ½Belð/ðCðtÞÞ P dÞ; Plð/ðCðtÞÞ P dÞ can be assessed by putting the components mass distributions into the DEN. Likewise, as introduced in Section 4, putting the hard/vacuous evidences of a particular component C l into the C ðtÞ

DEN model, the conditional reliability given that component C l being in state Aj l CBIM (E-CBIM) of component C l , denoted as ~IB ðtÞ, can be formulated as:

can also be evaluated. Hence, the extended

Cl

~IB ðtÞ ¼ Cl

sC

l X 1 a~ ðAjCl ðtÞ Þ; jSC l j  1 j¼0

ð16Þ

where

a~ ðAjCl ðtÞ Þ ¼ j½Belð/ðCðtÞÞ P djAjCl ðtÞ Þ; Plð/ðCðtÞÞ P djAjCl ðtÞ Þ  ½Belð/ðCðtÞÞ P dÞ; Plð/ðCðtÞÞ P dÞj:

ð17Þ

The denominator jSC l j  1 of Eq. (16) is the normalized factor to restrict the value of ~IBC l ðtÞ to be in the range of [0,1]. jSC l j is the cardinality of SC . Apparently, evaluating ~IB ðtÞ is equivalent to seek the minimum and maximum values of the quantity l

Cl

a~ ðAjCl ðtÞ Þ, for all ACj l ðtÞ # SC l . The interval arithmetic reported in [45] can be used to assess the a~ ðACj l ðtÞ Þ. However, the range of a~ ðAjCl ðtÞ Þ may be overestimated due to the repeated variables in a~ ðACj l ðtÞ Þ. This is so-called dependency problem in interval ~ ðAjC l ðtÞ Þ can be addressed by the TPL. The detail procedures of conarithmetic [61]. In this work, the dependency problem in a trolling the dependency problem of the interval of system reliability ½Belð/ðCðtÞÞ P dÞ; Plð/ðCðtÞÞ P dÞ and the interval of C ðtÞ C ðtÞ ~ ðAjC l ðtÞ Þ are provided in the Appendix A. Thus, a ~ ðAjC l ðtÞ Þ conditional reliability ½Belð/ðCðtÞÞ P djAj l Þ; Plð/ðCðtÞÞ P djAj l Þ in a can be equivalently written as:

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T. Xiahou et al. / Mechanical Systems and Signal Processing 109 (2018) 305–329

   C l ðtÞ ~ aðAj Þ ¼ ½Belð/ðCðtÞÞ P djAjCl ðtÞ Þ; Plð/ðCðtÞÞ P djAjCl ðtÞ Þ  ð1  mðAjCl ðtÞ ÞÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}  part 1 hX i X C ðtÞ C ðtÞ C ðtÞ C ðtÞ  Belð/ðCðtÞÞ P djAi l Þ  mðAi l Þ; Plð/ðCðtÞÞ P djAi l Þ  mðAi l Þ  i–j i–j |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð18Þ

part 2

where the part 1 (annotated with brackets underneath) in Eq. (18) represents the interval of the conditional reliability, whereas the part 2, namely the interval of the partial reliability, represents the interval of the system reliability excluding C ðtÞ

C ðtÞ

C ðtÞ

C ðtÞ

the term of ½Belð/ðCðtÞÞ P djAj l Þ  mðAj l Þ; Plð/ðCðtÞÞ P djAj l Þ  mðAj l Þ. It should be noted that the interval of conditional reliability and its corresponding interval of partial reliability are independent of one another as the reliability is con~ ðACj l ðtÞ Þ, a constrained ditioned on the independent states. In order to compute the lower and upper bounds of the a ~ ðAjC l ðtÞ Þ with a narrower bound. The detail procedures of formulating optimization problem is formulated in this work to get a the optimization problem are as follows: C ðtÞ

C ðtÞ

C ðtÞ

Step 1: Let the variable ConðAj l Þ represent the conditional reliability, i.e., Prf/ðCðtÞÞ P djAj l g, and the variable PRðAj l Þ P C l ðtÞ C ðtÞ represent the partial reliability of an MSS, i.e., g  PrfAi l g excluding the term of i–j Prf/ðCðtÞÞ P djAi C ðtÞ

C ðtÞ

C ðtÞ

Prf/ðCðtÞÞ P djAj l g  PrfAj l g. The variable ConðAj l Þ takes its value from the part 1 of Eq. (18), whereas the variable C ðtÞ PRðAj l Þ

takes its value from the part 2 of Eq. (18).

~ ðAjC l ðtÞ Þ (AjC l ðtÞ # SC l ) is equivalent to searching the minimum and maximum Step 2: Seeking the lower and upper bounds of a a

C ðtÞ

C ðtÞ

C ðtÞ

C ðtÞ

values of f j ðConðAj l Þ; mðAj l Þ; PRðAj l ÞÞ (Aj l a

C ðtÞ

C ðtÞ

# SC l ), and it results a pair of the following optimization problems:

C ðtÞ

C ðtÞ

C ðtÞ

C ðtÞ

min ðmaxÞ f j ðConðAj l Þ; mðAj l Þ; PRðAj l ÞÞ ¼ jConðAj l Þ  ð1  mðAj l ÞÞ  PRðAj l Þj C ðtÞ

C ðtÞ

C ðtÞ

Belð/ðCðtÞÞ P djAj l Þ 6 ConðAj l Þ 6 Plð/ðCðtÞÞ P djAj l Þ X X C ðtÞ C ðtÞ C ðtÞ Belð/ðCðtÞÞ P djAi l Þ  mðAi l Þ 6 PRðAj l Þ 6

s: t:

C l ðtÞ

Ai

# SC ;i–j

X

C ðtÞ

Aj l

C l ðtÞ

Ai

l

C ðtÞ mðAj l Þ

# SC

C ðtÞ

C ðtÞ

Plð/ðCðtÞÞ P djAi l Þ  mðAi l Þ

# SC ;i–j l

¼1

l

ð19Þ Consequently, the E-CBIM of each component can be expressed as:

~IB ðtÞ ¼ Cl

1 jSCl j  1

1 ¼ jSCl j  1

X C l ðtÞ

Aj

~ ðAjCl ðtÞ ÞÞ; maxða ~ ðAjC l ðtÞ ÞÞ ½minða

# SC

X C ðtÞ

Aj l

l

a

C ðtÞ

C ðtÞ

C ðtÞ

a

C ðtÞ

C ðtÞ

C ðtÞ

ð20Þ

½minðf j ðConðAj l Þ; mðAj l Þ; PRðAj l ÞÞÞ; maxðf j ðConðAj l Þ; mðAj l Þ; PRðAj l ÞÞÞ

# SC

l

It is noteworthy that the resulting optimization problem in Eq. (19) can be resolved by the existing optimization algorithms, such as the sequential quadratic programming, the interior-point algorithm, trust-region-reflective algorithm. In our illustrative cases, the trust-region-reflective algorithm in the MATLAB optimization toolkit was chose to identify the ~ ðAjC l ðtÞ Þ. If the resulting optimization problem is highly dimensional, the advanced optimization lower and upper bounds of a algorithms, such as the genetic algorithm and the particle swarm optimization, can be used in lieu of the trust-regionreflective algorithm. 5.2. Extended MRAW The MRAW quantifies the maximum increment of the system reliability by upgrading a particular component. It is mathematically defined as [2]:

IMRAW ðtÞ ¼ 1 þ Cl

X 1 C ðtÞ maxð0; bðAj l ÞÞ; jNC l j  1 j2N

ð21Þ

Cl

where C ðtÞ

C ðtÞ

bðAj l Þ ¼

Prf/ðCðtÞÞ P djAj l g  Prf/ðCðtÞÞ P dg Prf/ðCðtÞÞ P dg

;

ð22Þ

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T. Xiahou et al. / Mechanical Systems and Signal Processing 109 (2018) 305–329 C ðtÞ

for any Aj l C ðtÞ bðAj l Þ

# NC l . By taking account of the epistemic uncertainty produced by component state assignments, the quantity

is no longer a crisp value. Rather, it can be formulated as an interval value by the belief and plausibility functions as

follows:

~ C l ðtÞ Þ ¼ bðA j

C ðtÞ

C ðtÞ

½Belð/ðCðtÞÞ P djAj l Þ; Plð/ðCðtÞÞ P djAj l Þ  ½Belð/ðCðtÞÞ P dÞ; Plð/ðCðtÞÞ P dÞ ½Belð/ðCðtÞÞ P dÞ; Plð/ðCðtÞÞ P dÞ

:

ð23Þ

Hence, the extended MRAW (E-MRAW) can be defined as:

~IMRAW ðtÞ ¼ 1 þ Cl

where the value of

1 jSCl j  1

X C l ðtÞ

Aj

~ C l ðtÞ ÞÞ Hð0; bðA j

# SC

~ l ÞÞ; Hð0; bðA j C ðtÞ

ð24Þ

l

~ C l ðtÞ Þ, and one has: depends on the range of the interval quantity bðA j

8 > < ½a; b if a P 0 Hð0; ½a; bÞ ¼ ½0; b if a < 0 and b > 0 : > : 0 if b 6 0

ð25Þ

5.3. Extended MFV The MFV quantifies the contribution of a particular component to the system failure. The MFV of component C l can be mathematically defined as [2]:

IMFV C l ðtÞ ¼

X 1 C ðtÞ maxð0; bðAj l ÞÞ; jNC l j  1 j2N

ð26Þ

Cl

~ C l ðtÞ Þ is interval-valued as shown in Eq. (23), the extended MFV (E-MFV) can be written as: Similarly, as bðA j

~IMFV ðtÞ ¼ Cl

1 jSCl j  1

X C l ðtÞ

Aj

# SC

~ l ÞÞ: Hð0; bðA j C ðtÞ

ð27Þ

l

5.4. Extended MRRW The MRRW measures the potential decrease of system reliability caused by the decrement of a particular component’s states. The relation between the MRRW and MFV is as follows [2]:

IMRRW ðtÞ ¼ Cl

1 1  IMFV C l ðtÞ

:

ð28Þ

Hence, the extended MRRW (E-MRRW) can be expressed as:

~IMRRW ðtÞ ¼ Cl

1 : ~ 1  IMFV C l ðtÞ

ð29Þ

~ C l ðtÞ Þ is the core of the aforementioned three It should be noted that the evaluating the lower and upper bounds of bðA j extended importance measures. In the same manner, a pair of constrained optimization problems can be formulated to seek ~ C l ðtÞ Þ should be handled ~ C l ðtÞ Þ, for any AC l ðtÞ # SC . However, the dependency problem of the bðA the lower and upper bounds of bðA j

j

j

l

~ l Þ is addressed by the TPL (see the Appendix before constructing the optimization problems. Likewise, the dependency of bðA j ~ C l ðtÞ Þ, the lower and upper bounds of bðA ~ C l ðtÞ Þ (AC l ðtÞ # SC ) can be found by: A). After controlling the dependency problem of bðA C ðtÞ

j

b

C ðtÞ

C ðtÞ

C ðtÞ

ConðAj

C ðtÞ

min ðmaxÞ f j ðConðAj l Þ; mðAj l Þ; PRðAj l ÞÞ ¼ s: t:

j

C l ðtÞ

C ðtÞ

C ðtÞ

ConðAj l

C ðtÞ

ÞmðAj l

Þ C ðtÞ

ÞþPRðAj l

Ai

C ðtÞ

X

C ðtÞ

Aj l

C l ðtÞ

# SC ;i–j

# SC

Ai

l

C ðtÞ

mðAj l Þ ¼ 1 l

l

1

Þ

Belð/ðCðtÞÞ P djAj l Þ 6 ConðAj l Þ 6 Plð/ðCðtÞÞ P djAj l Þ X X C ðtÞ C ðtÞ C ðtÞ Belð/ðCðtÞÞ P djAi l Þ  mðAi l Þ 6 PRðAj l Þ 6 C l ðtÞ

j

C ðtÞ

C ðtÞ

Plð/ðCðtÞÞ P djAi l Þ  mðAi l Þ

# SC ;i–j l

ð30Þ

T. Xiahou et al. / Mechanical Systems and Signal Processing 109 (2018) 305–329

317

~ C l ðtÞ Þ, denoted as ½minðbðA ~ C l ðtÞ ÞÞ; maxðbðA ~ C l ðtÞ ÞÞ, are equal to Consequently, the lower and upper bounds of bðA j j j b

C ðtÞ

C ðtÞ

C ðtÞ

b

C ðtÞ

C ðtÞ

C ðtÞ

½ðminðf j ðConðAj l Þ; mðAj l Þ; PRðAj l ÞÞ; maxðf j ðConðAj l Þ; mðAj l Þ; PRðAj l ÞÞ. To sum up, a flowchart of computing the extended CIMs under epistemic uncertainty is given in Fig. 4. It contains four major steps. The DEN model for the studied MSS is constructed in Step 1 by considering the epistemic uncertainty associated with component state assignment (see the details in Section 3). With the DEN model, the intervals of the system reliability and conditional reliability can be assessed in Step 2 (see the details in Section 4). By controlling the dependency between the intervals of the system reliability and conditional reliability in Step 3, the extended CIMs can be consequentially converted to the corresponding optimization problems in Step 4 to identify the lower and upper bounds (see the details in Section 5). In order to identify the weak components of an MSS, components can be ranked by the importance measures. In this work, as the CIMs become interval-valued, the existing interval ranking methods can be implemented to rank the components based on the E-CIMs. A quantity of interval ranking methods have been reported in literature, such as the pessimistic ranking rule [62], the optimistic ranking rule [62], the compromised ranking rule [45], the preference ranking rule [63], the possibility degree ranking rule [64], and etc. In this work, components are ranked by three rules, i.e., the pessimistic and optimistic rules where only the lower and upper bounds of the E-CIMs is used for ranking, respectively, and the preference ranking rule which takes account of both the lower and upper bounds of the E-CIMs. The preference ranking rule is summarized as follows: Let a ¼ ½a1 ; a2  and b ¼ ½b1 ; b2  denote two interval values, the degree of the interval a being greater than b is referred as the degree of preference of a over b, and can be mathematically written as [63]:

PDfa > bg ¼

maxð0; a2  b1 Þ  maxð0; a1  b2 Þ : ða2  a1 Þ þ ðb2  b1 Þ

ð31Þ

In the same manner, the degree of preference of b over a can be defined as [63]:

PDfb > ag ¼

maxð0; b2  a1 Þ  maxð0; b1  a2 Þ : ða2  a1 Þ þ ðb2  b1 Þ

ð32Þ

It is obvious that PDfa > bg þ PDfb > ag ¼ 1. If PDfa > bg > PDfb > ag, i.e., PDfa > bg > 0:5, then a is said to be superior to b, and can be denoted as a  b. If PDfa > bg ¼ PDfb > ag ¼ 0:5, then a is said to be indifferent to b, denoted as a  b. If PDfa > bg < 0:5, then a is said to be inferior to b, denoted by a b. In this study, the preference ranking rule is applied to rank components based on the E-CIMs. If ~IC  ~IC holds, component C l is said to be more important than component l

lþ1

C lþ1 . Similarly, ~IC l  ~IC lþ1 and ~IC l ~IC lþ1 can be interpreted as component C l being equally important with and less important than component C lþ1 , respectively.

Estimate the transition mass matrix of the EMM by Eq. (8)

Step 1

Build the static EN of the studied MSS Construct the tailored DEN by combining the EMM and the static EN in a chronological order

Step 2

Compute the interval of the MSS reliability by Eqs. (11) and (12) Compute the interval of the conditional reliability by inputting the hard or vacuous evidences

Step 3

Handle the dependency between the reliability and conditional reliability by Eqs. (A.1) and (A.2)

Step 4

Convert to the optimization problem

Compute the extended CIMs Fig. 4. The flowchart of computing the extended CIMs.

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6. Illustrative examples Two illustrative examples are given in this section to examine the validity and effectiveness of the proposed method. A multi-state bridge system is firstly designed to elaborate the detail procedures of calculating the E-CIMs, and the proposed method is further implemented to an engineering example of a feeding control system of CNC lathes. In these examples, components will be ranked in a descending order, i.e., a component with a lower ranking order is more important. 6.1. A multi-state bridge system The configuration of a multi-state bridge system is shown in Fig. 5. The system consists of five components each of which has three possible states NC l ¼ f0; 1; 2g (l ¼ 1; 2; . . . ; 5). An amount of masses is assigned to the state sC l ¼ ½0; 1; 2 (l ¼ 1; 2; . . . ; 5) to represent the epistemic uncertainties associated with component state assignment. The reliability of the multi-state bridge sysSðtÞ

tem is defined as the probability that the system state, i.e., Aj , is no less than state 1, i.e., d ¼ 1. Suppose that components 1, 2, and 5 are exactly the same, whereas components 3 and 4 are identical. The transition mass matrices of all the components within a basic time interval Dt ¼ 1 month are set to be MC 1 ¼ MC 2 ¼ MC 5 ¼ M1 and MC 3 ¼ MC 4 ¼ M2 , where

3 3 2 1 0 0 0 1 0 0 0 6 0:08 0:91 0 0:01 7 6 0:08 0:9 0 0:02 7 7 7 6 6 2 M1 ¼ 6 7 and M ¼ 6 7: 4 0:01 0:08 0:9 0:01 5 4 0:01 0:11 0:87 0:01 5 2

0:09 0:18

0

0:73

0:23

0

0

0:77

All the components are in the brand new condition before the system is put into use, and thus, the mass distributions of all the components at time instant t ¼ 0 (unit: month) are mC l ð0Þ ¼ ð0; 0; 1; 0Þ (l ¼ 1; 2; . . . ; 5). The mass distributions of all the components at any time t ¼ kDt can be calculated by Eq. (7) and the results are shown in Fig. 6. The mass distributions of all the components can be viewed as the inputs of the EN model of the bridge system. The static EN model of the bridge system can be constructed based on the minimum path sets as introduced in [42]. In this bridge system, there are four minimum path sets, i.e., P1 ¼ fC 1 ; C 3 g, P 2 ¼ fC 1 ; C 4 ; C 5 g, P3 ¼ fC 2 ; C 4 g, and P1 ¼ fC 2 ; C 3 ; C 5 g. The bridge system can be, then, decomposed into two simplified models as shown in Fig. 7, where Model 1 considers component C 1 as one of the components in the minimum path sets while component C 2 is as one of the components in the minimum path sets in Model 2. On this basis, the EN models for each of the two models can be built up as elaborated in Section 3.3. The two EN models can be linked by a parallel connection, and thus the EN model for the entire bridge system can be constructed as shown in Fig. 8. Precisely speaking, nodes S1 , S2 , S3 and nodes S4 , S5 , S6 represent the subsystems of Model 1 and Model 2, respectively. Node S denotes the entire system. The CBMTs of nodes S1 , S3 , S4 , and S6 are tabulated in Table 2, whereas the CBMTs of nodes S2 , S5 and S are given in Table 1. Consequently, the DEN model of the multi-state bridge system can be obtained by linking all the time slices of the EN model in a chronological order. In this work, the basic time interval of the DEN is one month and the total time horizon of interest, i.e., T, is set to be 40 months. By putting the mass distributions of all the components into the DEN, the upper and lower bounds of the system reliability can be calculated by Eqs. (11) and (12), and the results are shown in Fig. 9. Furthermore, when the hard or vacuous evidences are put into the DEN of the bridge system, the conditional reliability with respect to each component state can be inferred by Eqs. (13) and (14), and they are depicted in Fig. 10. As depicted in Figs. 9 and 10, the system reliability and conditional reliability of all the components are interval-valued when the epistemic uncertainty associated with component state assignment cannot be voided. By putting the intervals of the system reliability and the conditional reliabilities into the proposed formulas of importance measures, i.e., Eqs. (16), (24), (27), and (29), and resolving the resulting optimization problems, one can get the proposed four E-CIMs as shown in Fig. 11. As observed from Fig. 11, all the E-CIMs are interval-valued too. These interval values provide an answer to the influence of epistemic uncertainty on the CIMs. Furthermore, for the E-CBIM, the importance measures of all the components start from zero at time t ¼ 0 and end up with zero as time goes to infinite. Because the Birnbaum importance measure can examine the importance of a component from both the positive and negative angles. At the very beginning when the system is put into use, the decrement from only a single component does not lead to the failure of the bridge system, thus the E-CBIMs of all the components start with zero. In the same manner, one component recovering from a worse state to a better state can’t make system work again if the system is very aged, thus the values of all the E-CBIMs approach to zero at this moment. For

1

3 5

2

4

Fig. 5. A multi-state bridge system.

319

1

1

0.8

0.8

State State State State

0.6 0.4

Mass Distribution

Mass Distribution

T. Xiahou et al. / Mechanical Systems and Signal Processing 109 (2018) 305–329

0 1 2 [0,1,2]

0.2 0

State 0 State 1 State 2 State [0,1,2]

0.6 0.4 0.2

0

10

20 t (months)

30

0

40

0

10

20

30

40

t (months)

(a)

(b)

Fig. 6. The mass distributions of all the components in the multi-state bridge system, (a) Components C 1 , C 2 , and C 5 , (b) Components C 3 and C 4 .

5

5

4

3 2

1

4

3

(a)

(b)

Fig. 7. Decomposing the bridge system into two models: (a) Model 1, (b) Model 2.

C5

C4

C3

C5

S1

S4 S2

C1

S5

S3

S6

C2

S Fig. 8. The EN model of the multi-state bridge system [42].

the E-MFV and E-MRRW, they both quantify the importance of a component from a negative perspective, and thus, the decrement of the system reliability caused by a component declines with the decrease of the system reliability. The EMFV and E-MRRW go to zero when the system reliability is close to zero. The E-MRAW quantifies the importance of a component from a positive side, thus the system reliability increment is the largest when the system becomes very aged. The importance ranking can provide the priority of degraded components to be maintained and/or improved [27]. As mentioned earlier in Section 5, three ranking rules, i.e., the pessimistic rule, the optimistic rule, and the preference rule, can be used as criteria to rank all the components based on the results of the four types of E-CIMs. As shown in Figs. 12, 13, and 14, the importance rankings of components may not be always exactly the same for all the three rules, and therefore, the epistemic uncertainty associated with component state assignment indeed influence the importance ranking. On the other hand, the importance rankings of components may not be consistent for all the four importance measures as they have

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1 Upper Bound Lower Bound

Reliability

0.8 0.6 0.4 0.2 0

0

10

20

30

40

t (months) Fig. 9. Reliability bounds of the multi-state bridge system.

distinct underlying physical meanings. Furthermore, the importance rankings of components C 1 and C 3 are varying with respect to time for the pessimistic and optimistic rules. Because the epistemic uncertainty propagated from component state assignment is varying over time. However, all the three rules of the E-CIMs consistently reveal that component C 5 is the least important among all the components in the system. In this work, the dependency among the reliability and the conditional reliability of an MSS is addressed. Overlooking such dependency may generate a wider interval bound for the proposed importance measures. The four types of E-CIMs on the condition that the dependency is controlled or ignored are compared to illustrate our arguments. If the dependency is overlooked, Eqs. (19) and (30) will, respectively, change to: a

C ðtÞ

C ðtÞ

min ðmaxÞ f j ðConðAj l Þ; RðtÞÞ ¼ jConðAj l Þ  RðtÞj C ðtÞ

C ðtÞ

C ðtÞ

ð33Þ

C ðtÞ

ð34Þ

Belð/ðCðtÞÞ P djAj l Þ 6 ConðAj l Þ 6 Plð/ðCðtÞÞ P djAj l Þ

s: t:

Belð/ðCðtÞÞ P dÞ 6 RðtÞ 6 Plð/ðCðtÞÞ P dÞ and b

C ðtÞ

C ðtÞ

min ðmaxÞ f j ðConðAj l Þ; RðtÞÞ ¼

ConðAj l

ÞRðtÞ

RðtÞ

C ðtÞ

C ðtÞ

Belð/ðCðtÞÞ P djAj l Þ 6 ConðAj l Þ 6 Plð/ðCðtÞÞ P djAj l Þ

s: t:

Belð/ðCðtÞÞ P dÞ 6 RðtÞ 6 Plð/ðCðtÞÞ P dÞ ~ ðACj l ðtÞ Þ or where the variable RðtÞ represent the reliability of an MSS at time instant t. After the lower and upper bounds of a ~ C l ðtÞ Þ are identified by the resulting optimization problems, the E-CIMs which the dependency is ignored can be computed bðA j

by Eqs. (18) and (24). The results of the E-CIMs when the dependency problem is ignored or controlled are illustrated in

1

1

Bound of State 2 Bound of State [0,1,2]

0.6

Conditional Reliability

Conditional Reliability

0.8

0.4

0.2

0

0

10

20 t (months)

(a)

30

40

1 Bound of State Bound of State Bound of State Bound of State

0.8 0.6 0.4 0.2 0

0

10

20

30

0 1 2 [0,1,2]

Conditi onal Reliabilit y

Bound of State 0 Bound of State 1

40

Bound of State 0 Bound of State 1

0.8

Bound of State 2 Bound of State [0,1,2]

0.6 0.4 0.2 0

0

10

20

t (months)

t (months)

(b)

(c)

30

40

Fig. 10. The conditional reliabilities of all the components in multi-state bridge system: (a) Components C 1 and C 2 ; (b) Components C 3 and C 4 , (c) Component C 5 .

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Extended MRAW

Extended CBIM

0.3 0.2 0.1 0

81 61

10

20 t (months)

30

2

1.5

1

1

4

7

t (months)

41 21 1

0

Extended MRAW

100

0.4

40

1

10

20

30

40

t (months)

0.7

3 Extended MRRW

Extended MFV

0.6

0.4

0.2

0

1

10

20

30

2.5 2 1.5 1

40

1

10

t (months)

20

30

40

t (months)

Interval values of C1 & C2

Interval values of C3 & C4

Interval values of C5

Pessimistic Ranking

1 2 3

Pessimistic Ranking

0

8

20

3

0

4

20 t (months)

(a) E-CBIM

(b) E-MRAW

2 3

4

2

40

1

0

1

t (months)

20

40

Pessimistic Ranking

Pessimistic Ranking

Fig. 11. The four types of E-CIMs for the multi-state bridge system.

1 2 3

0 3

t (months)

20 t (months)

(d) E-MRRW

(c) E-MFV C1 & C 2

40

C3 & C4

C5

Fig. 12. The pessimistic ranking for all the components in the bridge system.

40

T. Xiahou et al. / Mechanical Systems and Signal Processing 109 (2018) 305–329

1

Optimistic Ranking

Optimistic Ranking

322

2 3

0

9

20

40

1 2 3

0

8

t (months)

2 3

8

40

(b) E-MRAW Optimistic Ranking

Optimistic Ranking

(a) E-CBIM 1

0

20

t (months)

20

1 2 3

40

0

5

20

t (months)

t (months)

(c) E-MFV

(d) E-MRRW C 1 & C2

C3 & C4

40

C5

1

Preference Ranking

Preference Ranking

Fig. 13. The optimistic ranking for all the components in the bridge system.

2 3

20

30

3

40

0

20 t (months)

(a) E-CBIM

(b) E-MRAW

1 2 3

0

2

t (months)

Preference Ranking

Preference Ranking

0

1

20

40

40

1 2 3

0

20

40

t (months)

t (months)

(d) E-MRRW

(c) E-MFV C 1 & C2

C 3 & C4

C5

Fig. 14. The preference ranking for all the components in the bridge system.

Fig. 15. As shown in Fig. 15, the proposed method can yield narrower bounds as compare to the case where the dependency is not properly addressed.

6.2. The feeding control system of CNC lathes The DL series horizontal lathes have four axes as shown in Fig. 16, that is, X axis of tool head lateral movement, Z axis of tool head longitudinal movement, U1 axis of left gang tool movement, and U2 axis of right gang tool movement. The Z, U1, and U2 axes feeding have the same modules as X axis. Thus, only X axis feeding control system is analyzed in our study as the

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0.5

0.5

0.4

0.4

0.4

0.3 0.2

0.3 0.2 0.1

0.1 0

Extended CBIM

0.5

Extended CBIM

Extended CBIM

same procedure is applicable to the remaining axes. For X axis feeding control system, a signal generated by 611D-type servo driven module (MO) is transmitted through electric wire (EW) to control the motor (MT) in the X axis feeding control system, and a speed feedback device (SF) is used to return the speed of X axis to servo driven module (MO), while the grating scales (GR) report the straightness of X axis to MO so as to adjust the feeding speed and direction. The components EW and GR are viewed as binary components as they are merely consisting of electronic devices. The MO, MT, and SF are treated as three-state components for they all contain electric motors, which can work at an intermediate state rather than two extreme states, i.e., either perfectly working or completely failed. The components MT and SF are

0

10

20

30

0

40

0

10

20

(a)

0

10

20

30

40

30

40

30

40

30

40

t (months)

(c) 70

151 101 51 10

20 t (months)

30

1

10

20

30

21

1

40

1

Extended MRRW

2 1.5 30

2.5 2 1.5 1

40

3 2.5 2 1.5 1

1

10

20 t (months)

30

40

1

(i)

0.6

0.6

0.6

0.4 0.2

0 10

20 t (months)

30

40

Extended MFV

0.75

Extended MFV

0.75

0.2

0.4 0.2

0

1

10

20

30

40

(k) Independent

1

10

20 t (months)

t (months)

(j)

20 t (months)

0.75

1

10

( h)

0.4

20

(f)

3.5

3

(g)

0

10

t (months)

(e)

3.5

2.5

20 t (months)

101

41

t (months)

3

10

201

1

40

(d)

1

301

Extended MRRW

1

Extended MRAW

61

201

Extended MRAW

Extended MRAW Extended MRRW

0

40

(b)

400

3.5

Extended MFV

30

t (months)

250

1

0.2 0.1

t (months)

1

0.3

(l) Dependent

Fig. 15. The comparison of the E-CIMs when dependency among input intervals is addressed or ignored. (a), (d), (g) and (j) for Components C 1 and C 2 ; (b), (e), (h), and (k) for component C 3 and C 4 , (c), (f), (i), and (l) for Component C 5 .

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T. Xiahou et al. / Mechanical Systems and Signal Processing 109 (2018) 305–329

assembled together but due to the complex working environment and inaccuracy sensing techniques, an uncertain state [0,1,2] has to be added to the state spaces of components MT and SF to represent the epistemic uncertainty associated with component state assignment. For the X feeding control system, apart from the state [0,1,2], another uncertain state [0,1] is introduced because as long as one of the three-state components works in state 1, the system cannot work in state 2. Hence, once one of components falls into state 1, say the state combination CðtÞ ¼ ðC MO ðtÞ; C SF ðtÞ; C MT ðtÞ; C EW ðtÞ; C GR ðtÞÞ ¼ ð1; ½0; 1; 2; 2; 2; 2Þ, the system must be in state [0,1] rather than in state [0,1,2]. The reliability of the X feeding control system is defined as the probability that the system state being greater than state 0, i.e., d ¼ 1, the time horizon of interest is set to be 15 months (T ¼ 15). The epistemic uncertainties associated with the state assignment of components MT and SF are quantified by their respective EMCs, the transition mass matrices MMT and MSF can be estimated by Eq. (7) with sensor data. The stochastic behaviors of the remaining components without epistemic uncertainty are modeled by the discrete-time Markov chain, and the corresponding transition probability matrices for these components can be also estimated by sensor data [65]. The transition mass matrices and transition probability matrices are as follows:

2 M

MT

1

0

0

0

3

6 0:03 0:77 0 0:20 7 6 7 ¼6 7; 4 0:10 0:08 0:70 0:12 5

2 SF

M

1

0

0

3

0

6 0:07 0:83 0 0:01 7 6 7 ¼6 7; 4 0:02 0:13 0:75 0:10 5

2

MO

P

0:10 0:08 0:01 0:81 0:12 0:11 0:01 0:76    1 0 1 0 ¼ ; and PGR ¼ : 0:05 0:95 0:1 0:9

3 1 0 0 6 7 ¼ 4 0:08 0:92 0 5; 0:01 0:03 0:96



PEW

The corresponding EN model of the X feeding control system is given in Fig. 17. Based on the functional relations of all components to the X feeding control system, the states of the X feeding control system with respect to each combination of components’ states can be tabulated in Table 3. Table 3 consists of 2  2  3  4  4 ¼ 192 rows, corresponding to all the possible components’ states combinations. For each components’ states combination, if the five components’ states are all known, the system state is then completely determined. However, if component SF or MT is in the uncertain state ½0; 1; 2 and the other components are in their best states, the system state is uncertain. Based on the CBMT of the X feeding control system, the relation among components’ states and system states can be determined. By linking the repetitive time slice EN models in a chronological order, the DEN model of the X feeding control system can be constructed. The interval of the reliability of the X feeding control system can be computed by the DEN model together SðtÞ

SðtÞ

with Eqs. (11) and (12). In this case, the lower bound of the system reliability is Belð/ðCðtÞÞ P dÞ ¼ mðA1 Þ þ mðA2 Þ, whereas the upper bound of system reliability is Plð/ðCðtÞÞ P dÞ ¼

SðtÞ mðA1 Þ

þ

SðtÞ mðA2 Þ

þ

SðtÞ mðA½0;1 Þ

þ

SðtÞ mðA½0;1;2 Þ.

Consequently,

the conditional reliability can be evaluated by the DEN model together with Eqs. (13) and (14). Putting the intervals of the system reliability and conditional reliability into Eqs. (16), (24), (27), and (29), and resolving the optimization problems ~ C l ðtÞ Þ, the four E-CIMs can be computed as shown in Fig. 18. ~ ðAC l ðtÞ Þ or bðA of a j

j

I/O

Control Console

EW

GR

MO

MT

SF X Feeding

Z Feeding

Fig. 16. The feeding control system of DL series CNC lathes.

U1 & U2 Feeding

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T. Xiahou et al. / Mechanical Systems and Signal Processing 109 (2018) 305–329

CEW

CGR

CSF

CMO

CMT

S Fig. 17. The EN model of the X feeding control system.

Table 3 The CBMT for X feeding control system. C EW

C GR

C MO

C SF

C MT

0 0 0 0 0 .. . 2 2 2 2 2

0 0 0 0 0 .. . 2 2 2 2 2

0 0 0 0 0 .. . 2 2 2 2 2

0 0 0 0 1 .. . 2 sSF sSF sSF sSF

0 1 2 sMT 0 .. . sMT 0 1 2 sMT

mðSjC EW ; C GR ; C MO ; C SF ; C MT Þ S¼0

S¼1

S¼2

S ¼ ½0; 1

S ¼ ½0; 1; 2

1 1 1 1 1 .. . 0 1 0 0 0

0 0 0 0 0 .. . 0 0 0 0 0

0 0 0 0 0 .. . 0 0 0 0 0

0 0 0 0 0 .. . 0 0 1 0 0

0 0 0 0 0 .. . 1 0 0 1 1

25 Bound of Bound of Bound of Bound of Bound of

0.8 0.6 0.4

Extended MRAW

Extended CBIM

1 MO SF MT EW GR

0.2 0

3

6 9 t (months)

12

15

11 6 1

3

6

9

12

15

12

15

t (months)

3 Extended MRRW

Extended MFV

16

1 0

0.6

0.4

0.2

0

21

1

3

6

9 t (months)

12

15

2.5 2 1.5 1

1

3

6

9

t (months)

Interval values of MT

Interval values of SF

Interval values of EW

Interval values of GR

Interval values of MO

Fig. 18. The E-CIMs for the five components of the X feeding control system.

T. Xiahou et al. / Mechanical Systems and Signal Processing 109 (2018) 305–329

1 2 3 4 5

1 2

Optimistic Ranking

Optimistic Ranking

326

0

5 6

10

3 4 5 1

15

5 6

1 2 3 4 5 5 6

15

(b) E-MRAW

1 2

Optimistic Ranking

Optimistic Ranking

(a) E-CBIM

1

10 t (months)

t (months)

10

3 4 5 1

15

5 6

10

15

t (months)

t (months)

(c) E-MFV

(d) E-MRRW MO MT

MT MO MT

SF

EW

GR

1 2 3 4 5

Preference Ranking

Preference Ranking

Fig. 19. The optimistic rankings for the five components of X feeding control system.

0

5 6

10

1 2 3 4 5 1

15

5

5

7

10

15

(b) E-MRAW Preference Ranking

Preference Ranking

(a) E-CBIM

1 2 3 4 5 1

7

t (months)

t (months)

10

15

1 2 3 4 5 1

5

t (months)

(c) E-MFV

MT MO

7

10

15

t (months)

SF

MO MT

EW

(d) E-MRRW GR

Fig. 20. The preference rankings for the five components of X feeding control system.

As shown in Fig. 18, although only two components, i.e., components MT and SF, possess the epistemic uncertainty associated with state assignment, the E-CIMs of all the components are interval-valued. Because the epistemic uncertainties of some components have propagated to the system reliability measures and further to the importance measures of the remaining components. The three importance ranking rules introduced earlier in Section 5 are utilized to rank the five components. The results from the optimistic ranking rule and the preference ranking rule are given in Figs. 19 and 20. Nevertheless, the pessimistic

327

T. Xiahou et al. / Mechanical Systems and Signal Processing 109 (2018) 305–329

ranking rule cannot provide any useful insight to the importance rankings of all the components, because the minimum values of some importance measures are the same, such as E-MRAW, E-MRRW, and E-MFV (see Fig. 18). The phenomenon is due to the vulnerability of the X feeding control system as the failure of any one of the five components will cause the failure of the entire system. Recovering only one component from a worse state to a better state is ineffective to the system reliability improvement. As shown in Figs. 19 and 20, the E-MRAW, E-MRRW, and E-MFV provide the different rankings if different ranking rules are used, while the E-CBIM give the same rankings for the two rules. These results show that importance measures defined from different perspectives may result distinct importance rankings of all the components, meanwhile the importance rankings of components are also influenced by the ranking rule to be used. Furthermore, for the importance ranking of each individual component of the X feeding control system changes over time as observed in Fig. 18. Hence, from the perspective of maintenance planning, the priority of conducting maintenance for derated components may also be time-dependent. 7. Conclusions and future works In this paper, four CIMs, namely CBIM, MRAW, MRRW, and MFV, were extended by taking account of the epistemic uncertainty associated with component state assignment. The DEN combined with the EMC and EN was proposed to calculate the system reliability and conditional reliability. After handling the dependency between the intervals of the inputs of the proposed importance measures, a pair of optimization problems is formulated to compute the lower and upper bounds of the ECIMs. As demonstrated in two illustrative examples, due to the epistemic uncertainty associated with component state assignment, the proposed importance measures become interval values, rather than crisp values from the traditional CIMs, and they are able to quantify the potential variations caused by the epistemic uncertainty. Additionally, by manipulating the dependency of interval inputs, the proposed approach can yield narrower interval bounds for the E-CIM than the case where the dependency is ignored. It is worth mentioning that there are still some challenges to be addressed in our future works. Firstly, in the present work, the DEN model is a discrete-time DEN model, our future work will devote to exploring the continuous-time DEN model. Secondly, maintenance activities which can recover a system and its components to better states are not considered in our work, and the present work can be further extended to repairable multi-state systems. Lastly, the proposed method will be applied to maintenance decision-making and other fields in industry as long as epistemic uncertainty cannot be avoided in importance analysis. Acknowledgements The authors greatly acknowledge grant support from the Science Challenge Project under contract number TZ2018007 and the National Natural Science Foundation of China under contract number 71371042. The comments and suggestions from all the reviewers and the editor are very much appreciated. Appendix A The detail procedures of treating the dependency of the system reliability and conditional reliability are as follows: STEP 1: Compute the lower bound of the system reliability by the total belief theory:

X

Belð/ðCðtÞÞ P dÞ ¼ A

X

SðtÞ

mð/ðCðtÞÞ ¼ Ak Þ ¼

SðtÞ Pfdg k

A

X

SðtÞ C ðtÞ PfdgA l k i

# SC

0

SðtÞ

C ðtÞ

C ðtÞ

mð/ðCðtÞÞ ¼ Ak jAi l Þ  mðAi l Þ l

1

X B X C ðtÞ SðtÞ C ðtÞ C mð/ðCðtÞÞ ¼ Ak jAi l ÞA  mðAi l Þ @

¼

C ðtÞ

Ai l

# SC

SðtÞ

l

X

¼

C l ðtÞ

Ai

# SC

Ak Pfdg

ðA:1Þ C ðtÞ

C ðtÞ

Belð/ðCðtÞÞ P djAi l Þ  mðAi l Þ l

C ðtÞ

C ðtÞ

¼ Belð/ðCðtÞÞ P djAj l Þ  mðAj l Þ þ

X C ðtÞ C ðtÞ Belð/ðCðtÞÞ P djAi l Þ  mðAi l Þ i–j

328

T. Xiahou et al. / Mechanical Systems and Signal Processing 109 (2018) 305–329

STEP 2: Compute the upper bound of the system reliability by the total plausibility theory: X SðtÞ

Plð/ðCðtÞÞ P dÞ ¼

mð/ðCðtÞÞ ¼ Ak Þ

SðtÞ

SðtÞ

ðAk PfdgÞ[ðAk \fdg–;Þ

X

X

¼

SðtÞ SðtÞ C ðtÞ ðAk PfdgÞ[ðAk \fdg–;ÞA l i

0

X B @

¼

C l ðtÞ

Ai

¼

# SC

X

C l ðtÞ

Ai

SðtÞ

C ðtÞ

C ðtÞ

mð/ðCðtÞÞ ¼ Ak jAi l Þ  mðAi l Þ

# SC

1

l

X

SðtÞ C ðtÞ C C ðtÞ mð/ðCðtÞÞ ¼ Ak jAi l ÞA  mðAi l Þ

ðA:2Þ

SðtÞ SðtÞ ðA PfdgÞ[ðA \fdg–;Þ k k

l

C ðtÞ

C ðtÞ

Plð/ðCðtÞÞ P djAi l Þ  mðAi l Þ

# SC

l

C ðtÞ

C ðtÞ

¼ Plð/ðCðtÞÞ P djAj l Þ  mðAj l Þ þ

X C ðtÞ C ðtÞ Plð/ðCðtÞÞ P djAi l Þ  mðAi l Þ i–j

STEP 3: The interval of the system reliability, i.e., ½Belð/ðCðtÞÞ P dÞ; Plð/ðCðtÞÞ P dÞ, can be equivalent written as: C ðtÞ

½Belð/ðCðtÞÞ P dÞ; Plð/ðCðtÞÞ P dÞ ¼ ½Belð/ðCðtÞÞ P djAj l Þ; Plð/ðCðtÞÞ " # X X C l ðtÞ C l ðtÞ C l ðtÞ C l ðtÞ C l ðtÞ C l ðtÞ Belð/ðCðtÞÞ P djAi Þ  mðAi Þ; Plð/ðCðtÞÞ P djAj Þ  mðAi Þ P djAj Þ  mðAj Þ þ i–j

ðA:3Þ

i–j

~ C l ðtÞ Þ with Eq. (A.3), ~ ðAjC l ðtÞ Þ and bðA STEP 4: Consequently, by replacing the interval ½Belð/ðCðtÞÞ P dÞ; Plð/ðCðtÞÞ P dÞ in a j ~ C l ðtÞ Þ can be, respectively, written as: a~ ðACl ðtÞ Þ and bðA j  j  C l ðtÞ ~ aðAj Þ ¼ ½Belð/ðCðtÞÞ P djAjCl ðtÞ Þ; Plð/ðCðtÞÞ 

C ðtÞ

C ðtÞ

P djAj l Þ  ð1  mðAj l ÞÞ " #  X X  C ðtÞ C ðtÞ C ðtÞ C ðtÞ  Belð/ðCðtÞÞ P djAi l Þ  mðAi l Þ; Plð/ðCðtÞÞ P djAi l Þ  mðAi l Þ ;  i–j i–j

ðA:4Þ

and

~ C l ðtÞ Þ ¼ bðA j

hP C ðtÞ mðAj l Þ

þ

i–j

1 C l ðtÞ

Belð/ðCðtÞÞPdjAi

C ðtÞ

ÞmðAi l

C ðtÞ

½Belð/ðCðtÞÞPdjAj l

Þ;

P i–j

C ðtÞ

Plð/ðCðtÞÞPdjAi l C ðtÞ

Þ;Plð/ðCðtÞÞPdjAj l

C ðtÞ

ÞmðAi l

i  1:

ðA:5Þ

Þ

Þ

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