Reliability assessment of man-machine systems subject to mutually dependent machine degradation and human errors

Reliability Engineering and System Safety 190 (2019) 106504

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Reliability assessment of man-machine systems subject to mutually dependent machine degradation and human errors Che Haiyanga,b, Zeng Shengkuia,b, Guo Jianbina,b, a b

T

⁎

School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China Science and Technology on Reliability and Environmental Engineering Laboratory, Beijing 100191, China

ARTICLE INFO

ABSTRACT

Keywords: Man-machine systems Machine degradation Human error Mutual dependence Piecewise-deterministic Markov process Reliability modeling

Many man-machine systems experience machine degradation and human errors, which may be mutually dependent and have detrimental effects on system reliability. On the one hand, machine degradation will increase fatigue inducing conditions and result in more human errors. On the other hand, human errors usually cause shock loads on a machine and accelerate its degradation. Therefore, machine degradation and human errors aggravate each other. To model the mutual dependence, we develop a Piecewise-deterministic Markov process modeling framework, which can incorporate machine degradation and human errors to evaluate the system reliability. In the framework, the machine degradation is described by a multi-state model with a Semi-Markov process, where the times of transitions due to the mutual dependence are time-varying random variables; a mathematical model is developed to evaluate the human error rate under the effect of fatigue-recovery, where human errors occur according to a nonhomogeneous Poisson process. A Monte Carlo simulation algorithm is implemented to compute the reliability. The turret of a lathe operated by a worker is presented to illustrate the effectiveness of the reliability model.

1. Introduction System reliability refers to the dependability of the performance of machine subsystem and human subsystem [1]. Man-machine systems (MMSs) may be influenced by machine degradation and human errors, possibly dependent. Therefore, human factors cannot be ignored when estimating the system reliability [2], nor can the dependence. Human factors play an important role in MMSs and human errors have serious consequences on system reliability [2]. The failure or degradation of a system is mainly caused by human errors, which can occur as a result of performing tasks non-sequentially, applying wrong force or torque, misalignments, and etc. [3,4]. Yang et al. [5] consider human errors as random shocks that can result in machine degradation or complete failure. Lathe is a key equipment in manufacturing systems, and its reliability is widely studied in literatures [6–8]. The failures of the turret in a lathe are mainly caused by the human errors [6]. The analysis above suggests that human errors contribute to machine degradation, while machine degradation may accelerate the occurrence of human errors. Degraded machines will lead to a higher level of workload and pressure, or a more demand of the worker's physical capacity, which can directly or indirectly affect worker fatigue [9,10]. Fatigue is considered as the loss of a worker's efficiency [11] ⁎

and is described as feelings of tiredness [9]. Fatigue is known as a contributor to human error generation [12], and Myszewski [13] indicates that a higher degree of fatigue will cause a higher rate of human error. Therefore, machine degradation and human errors are mutually dependent, since human errors contribute to machine degradation and a degraded machine increases the accumulation rate of worker fatigue and then leads to more human errors. The mutual dependence results from complex man-machine interactions and it poses new challenging issues to evaluate the reliability of MMSs. To consider the dependence that the degraded machine leads to more human errors, it is necessary to study the mechanism of human error. One way to address human error and human reliability in systems is the human reliability analysis (HRA) method [14]. The HRA models consider the effects of machines, working environment and worker's physical and mental capabilities on human error probability (HEP) through performance shaping factors (PSFs) [15], Error Producing Conditions (EPCs) [16,17] or Performance Influencing Factors (PIFs), which are utilized to modify the reference values to obtain task- and context-related estimates [18]. Human performance may be time dependent since PSFs change with time. The quantification of PSFs is important to evaluate the HEP. However, in general, these factors require an expert's judgment to be determined, and a precise determination is difficult [19].

Corresponding author. E-mail address: [email protected] (J. Guo).

https://doi.org/10.1016/j.ress.2019.106504 Received 24 November 2018; Received in revised form 17 April 2019; Accepted 17 May 2019 Available online 22 May 2019 0951-8320/ © 2019 Elsevier Ltd. All rights reserved.

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To solve the problem, many studies have investigated HEP through a Bayesian network (BN) which is attractive in the data shortage fields concerning the consequent reliance on subjective judgments [20]. Musharraf et al. [21–23], Norafneeza et al. [24,25], and Zhou et al. [26] use a BN approach to determine the PSFs to study the time-dependent human performance. In their studies, the BN modeling of PSFs enable dynamic updating through emerging information from virtual experiments or expert judgments. In the models above, the human errors will be influenced by many PSFs and the human reliability has been estimated accurately. In addition, the HEP is related to the conditions of equipment, the task characteristics, and the human properties. To reduce the subjective factors that arise from a lack of information, the failure mode, effects and critically analysis (FMECA) [27,28], hazard and operability analysis (HAZOP) [28], task analysis [18,29,30], and cognitive models [18,31] are introduced to analyze the potential human errors. These methods have been successfully applied in estimating the HEP. Many studies claim that many PSFs can be represented by fatigue or can affect fatigue directly, and they investigate the HEP based on the accumulation of fatigue [12,32]. Fatigue can probably be regarded as one of the most fundamental factors that affects worker performance particularly in manufacturing systems [33]. Many studies have provided a more in-depth analysis of the time-dependent human performance under the effect of worker fatigue. Givi et al. [12] and Jaber et al. [32] propose a novel analytical model to measure the error rate of a worker taking into account the effects of physical fatigue and its opposite, recovery. Myszewski [13] develops the mathematical model of the occurrence of human error in manufacturing processes, where a quantitative description of the effect of fatigue on human error is provided. In these models, machine degradation influences the HEP through fatigue. In literature, the most widely utilized approaches to evaluate the reliability of a MMS under the detrimental effects of human errors, are Fault Tree Analysis (FTA), Event Tree Analysis (ETA), Statistical reliability methods, Reliability Block Diagrams (RBD), and Markov Models [34]. Terano et al. [35] evaluate the reliability of a MMS based on FTA and ETA through combining the probabilistic treatment of machines and the fuzzy treatment of human errors. Dhillon et al. [36] and Liu et al. [37] build a Markov model to analyze the reliability of a MMS with human errors. They assume that the machine is binary (i.e., failed or perfect), while, in general, the performance of a machine will degrade in time [38]. Yang et al. [5] consider a multi-state production system with unilateral dependence, where the machine will be degraded by human errors. In the mentioned models, the failures of the MMS are due to machine failures or a human errors. However, they do not consider the impact of machine degradation on human errors and the human errors occur according to a homogeneous a Poisson process. The above-mentioned methods only consider the unilateral dependence between machine degradation and human errors when evaluating the reliability of MMSs. However, in fact, the machine degradation and human errors are mutually dependent during the dynamic failure generation process of a MMS, and ignoring the mutual dependence will overestimate the system reliability. In this paper, we propose a general reliability model of a MMS with the mutual dependence. Such model is supported by a Piecewise-deterministic Markov process (PDMP), which is a family of Markov processes involving deterministic evolution punctuated by random jumps. In our model, we integrate a machine degradation model and a human error model in the PDMP modeling framework to describe the mutual dependence. The machine degradation is described as a multi-state model (MSM) with a semiMarkov process. Moreover, this paper is an attempt to identify some of the factors that affect human error, i.e. fatigue-recovery, and provides a novel analytical human error model according to a nonhomogeneous Poisson process. A Monte Carlo (MC) simulation algorithm is proposed to realize the degradation transition process, and compute the reliability.

The rest of this paper is organized as follows. Section 2 provides the models of machine degradation, worker fatigue, and human errors, and analyzes their dependence. Section 3 presents the PDMP model of MMS with machine degradation, human errors, and their dependencies. The proposed MC simulation for reliability estimation is presented in Section 4. Section 5 presents an illustrative study taken from a typical manufacturing system, which is a turret operated by a worker. Finally, Section 6 concludes the work and makes some suggestions for further work. 2. System description The manufacturing system which we focus on is a typical MMS with multi-state machines. The machines may experience mutually dependent machine degradation and human errors. At the meantime, the workers may be affected by a fatiguing type of mechanism, and the fatigue will accelerate the occurrence of human errors. The models of machine degradation, worker fatigue, and human errors are described and the dependence is analyzed in the following sections. 2.1. Notation list The notation used in formulating the models is now listed. Acronyms MMS MSM PDMP MC CET CENT FMECA HAZOP Notations S ti λi,j(ti) fa(t) Re(τa) α αi β fa(i, t) Y(t) Y′(t) N(t) pij S′ ti,m λ(i,m),(j, n)(ti,m) m i, j (ti, m )

μ(t) μi(t) μ0 γ Z(t) Tk Zk

f(i,m),(j, η

n)(ti,m|t)

Man-machine system Multi-state model Piecewise-deterministic Markov process Monte Carlo Cumulative error resulting in a transition Cumulative error not resulting in a transition failure mode, effects and critically analysis hazard and operability analysis The set of machine degradation states The residence time of the machine being in the current state i The transition rate from state i to state j The fatigue accumulated by time t The residual fatigue after a rest break of the duration τa The fatigue parameter The fatigue parameter with the machine in state i The recovery parameter The fatigue accumulated in state i by time t Machine degradation state Machine degradation state considering human errors The number of human errors by time t The probability of the change from state i to state j due to human errors The state space of the machine degradation considering human errors The residence time of the machine being in the current state (i, m) The transition rate from state (i, m) to state (j, n) The transition rate after m CENT The intensity of human errors at time t The intensity of human errors with the machine in state i The initial intensity of human errors The dependence factor The degradation process of the whole MMS The time of the kth jump The state of Z(t) after kth jump of Y′(t) The failure states set of the MMS The transition probability density function The relative increment of transition rates due to an occurred CENT

2.2. Model descriptions 2.2.1. Machine degradation models To study the basic characteristics and evolvement rules of the failures of MMSs, we focus more on a long-term machine degradation of the manufacturing systems. In this paper, as the same assumptions in [39–41], the maintenance is not considered and we assume that the 2

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machine will continue to operate until it completely fails. Based on the assumption, we study the mutual dependence between machine degradation and human errors, where machine degradation and human errors aggravate each other. The considered machines in manufacturing systems consist of multiple states which represent different functioning conditions [42]. Therefore, they belong to the multi-state machines. The multi-state machines have been widely studied to improve the efficiency and productivity [43]. Markov processes are widely utilized to model machine degradation processes, and we follow the assumptions of the Markov property used in [44,45]. The following assumptions on MSMs are adopted from [45]. Fig. 1. The behavior of the fatigue-recovery process over repeated work-rest cycle [32].

1 The machine degradation process takes values from a finite number of states denoted by S = {0, 1, ...,M } , where state 0 is a complete failure state and state M is a perfect performing state. The machine will perform partially in the intermediate degradation states i (0 < i < M). 2 The transition rate from state i to state j, λi, j(ti), i > j, characterizes the probability of the degradation transition, where ti is the residence time of the machine in the current state i since its last transition. 3 The initial state of the machine is M.

Re ( a ) = fa (t ) e

where Re(τa) is the residual fatigue after a rest break of duration τa. Re equal to 0 or 1 represents complete recovery (no residual fatigue) or no recovery (maximum fatigue), respectively. α is a fatigue parameter, which represents the speed of fatigue accumulation. β is a recovery parameter, which controls the speed of recovery alleviation. A low value of α/β means a slow fatigue accumulation/ recovery alleviation, while a high value means a fast one. We assume that the initial value of fatigue is zero at the beginning of work in each weekday. The behavior of the fatigue-recovery process during each weekday is illustrated in Fig. 1. In Eq. (2), fatigue always accumulates from an initial value of zero, which means full recovery attained in the previous rest break. In practice, rest breaks will separate work cycles and are usually short during each weekday. Thus, a full recovery cannot occur. As made in [32], the residual fatigue, Re(τa), needs to carry forward into cycle a + 1 and (1) can be rewritten as

2.2.2. Worker fatigue and recovery models Fatigue is multidimensional and can be described as lack of energy, physical discomfort, physical effort, lack of motivation, and sleepiness [46]. The effect of fatigue on performance has been demonstrated in many industrial sectors as an increase of error rate [47], loss of throughput [48], performance degradation [49], and injuries and accidents [50]. Worker fatigue can be classified into two major categories: physical fatigue and mental fatigue [19]. Physical fatigue is defined as the reduction of performance in the muscular system while mental fatigue is defined as a feeling of weariness, reduction of alertness, and reduced mental performance [46]. Stranks et al. [51] and Arellano et al. [46] suggest that worker fatigue and human error rate are related to mental conditions. Many factors affect the accumulation of worker fatigue. Ji et al. [10] conclude that the work characteristics such as the work environment can be represented by worker fatigue. Givi et al. [12] summarize the PSFs that can directly or indirectly affect worker fatigue, i.e. task repetitiveness and complexity, workload, skill level, stress, and etc. In addition, task time is an important factor that influences worker fatigue [13,19,52]. Therefore, the accumulation process of worker fatigue can be modeled by a general model as

fa (t ) = Fa (t ; )

faa + 1 (t ) = Re ( a ) + (1

e

t,

Re ( a )) (1

e

(ta tra)

),

(4)

where ta is the production time of the cycle a, and tra is determined by projecting the value of Re(τa) on the fatigue curve as

tra =

ln(1

Re ( a ))/

(5)

2.2.3. Human error model Dhillon [56] summarizes the common human errors in engineering processes as: maintenance error, operator error, design error, handling error, and so on. In this paper, human error is considered as operator error defined as a lapse or mistake in performing a machine that results in machine degradation. They can accelerate machine degradation discretely and have been described as random shocks [5,57–59]. In addition, human errors are influenced mostly by the work characteristics such as workload and work environment, which can be represented by worker fatigue. We use the assumptions to build the human error model. The first two assumptions are taken from [60], while the third assumption is first proposed in this paper.

(1)

where fa(t) is the fatigue measure and Fa( · ) is the accumulation path of fatigue. Vector θ consists of a set of random parameters that represent the effects of PSFs on fatigue. It should be noted that Eq. (1) is a general fatigue model, while, for a specific case, the form of Fa( · ) can be specified to describe the case-specific fatigue mechanism. The mental fatigue is not directly measurable and must be inferred [19], and a quantitative model of mental fatigue is difficult to build. However, as for the models to quantify fatigue, Ahrache et al. [53] concluded that existing studies focus mainly on the estimation of physical fatigue. Some researchers asserted that physical fatigue accumulates exponentially with time [54,55], and Konz et al. [55] suggested that the form of the recovery function in ergonomics is also exponential. From the discussion above, we can find that fatigue is a wide ranging term and it is impossible to capture all aspects of fatigue in a systematic study [19]. In this paper, we focus on the accumulation process of physical fatigue, and use the assumptions made in [32] on the following fatigue and recovery functions:

fa (t ) = 1

(3)

a,

1 Human errors can cause shock damage to the machine. According to the magnitude of shock damage, human errors can be divided into two types: extreme error and cumulative error, and these two types of errors are mutually exclusive. 2 Extreme errors lead to machine failure immediately, while cumulative errors degrade the machine gradually. 3 Human errors occur based on a nonhomogeneous Poisson process with the intensity function μ(t). The effect of fatigue on human error is considered as µ (t ) = µ 0 + fa (t ) , where μ0 is the initial intensity and γ is the dependence factor. μ0 is referred to the nominal human error probability (NHEP) in many HRA models, and it can be determined through the HRA methods.

(2) 3

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fatigue parameter will increase when the machine goes to a more degraded state. Similarly, Eq. (4) can be rewritten as

faa + 1 (i, t ) = Re ( a ) + (1

Re ( a )) (1

e

i (ta tra)

).

(7)

Finally, the intensity function of human errors with the machine in state i, μi(t), can be expressed as

µi (t ) = µ0 + fa (i , t ).

2.3.2. Human errors’ impacts on machine degradation Human errors may cause the extreme damage which leads to the system failure suddenly, or the cumulative damage which makes the transition to a more degraded state except for the failure state or aggravates the degradation condition without a change of state. Therefore, human errors can be divided into three categories: extreme error, cumulative error resulting in a transition (CET), and cumulative error not resulting in a transition (CENT). A CET can result in the state variable Y(t) changing from state Y (t ) = i to a more degraded state Y (t ) = j with a probability pij, i > j, j ≠ 0. pii and pi0 denote the probability that the human error is a CENT and an extreme error respectively. In addition, the aging degradation due to the production process can also change the state to a more degraded one. The degradation process considering human errors, Y′(t), is a continuous-time Markov process and the times between transitions are random variables. Therefore, Y′(t) is a semi-Markov process based on its definition in [63], and it is described in Fig. 3. The layers indicate the degradation states Y(t), and the numbers in each layer indicate the number of CENT occurred during the residence time in the current state, N(t). The system state Y′(t) can be then represented by the pair (Y(t), N(t)). The state space of the degradation process Y′(t) can be represented by S = {(c, d), c S , c 0, d } {(c, d ), c = 0, d = 0, 1} . The transitions described by black solid lines are caused by the aging degradation process, where the transition to a more degraded state occurs and N(t) is set to 0. The transitions described by black dotted lines are caused by CET, where the degradation condition will go to a more degraded state with N(t) setting to 1. The transitions described by blue thick solid lines are due to CENT, where the degradation layer do not change and N(t) will be increased by one. The transitions described by red dashed lines are due to extreme errors, which cause Y′(t) to be (0, 1). In this model, the transition rate denoted by λ(i, m), (j, n)(ti, m) is residence time-dependent, where ti, m is the residence time of a machine in the current state (i, m). For a non-failure state (i, m), the transition rates can be divided into the following four types:

Fig. 2. The mutual dependence between human errors and machine degradation.

The assumption that human errors obey a nonhomogeneous Poisson distribution is based on human error model in [13], where the occurrence of human error is a discrete, Poisson-type, distributed random variable. In addition, according to the descriptions of the dependence in [61], we assume that the intensity function is a linear function of the fatigue level, µ (t ) = µ 0 + fa (t ) . The dependence factor γ describes the impact of the current fatigue level on the intensity of human errors. 2.3. Dependence analysis For a MMS, there are many man-machine interactions where human errors and machine degradation will occur. Moreover, as shown in Fig. 2, the human errors and machine degradation process are mutually dependent. On the one hand, human errors are discretely distributed contributions to machine degradation (line a). On the other hand, workers are influenced by the fatiguing type of mechanism, a degraded machine will increase the accumulation rate of worker fatigue (line b), and the high-level magnitude of fatigue will leads to more human errors (line c). Due to the mutual dependence, the human errors can cause the degradation of a machine, and then accelerate the fatigue accumulation process, and finally facilitate the occurrence of human errors recursively. 2.3.1. Machine degradation's impacts on human errors In this paper, when operating a perfect machine, the workload and work environment are assumed to be at standard level. However, the machine degradation will increase the workload and the work environment will become worse, which may increase the fatigue inducing conditions. The rate of fatigue increases with the increasing fatigue inducing conditions [62]. For a manufacturing machine, such as the turret in a lathe, its degradation will lead to a lower machining precision and a noisier work environment (such as the noise of turret vibration). Therefore, worker must pay more attention when operating a degraded machine to ensure the same machining accuracy as a perfect machine. The increased attention will cause a heavier workload and greater stress, which will increases the accumulation rate of worker fatigue. In addition, fatigue can probably be regarded as one of the most fundamental factors that strongly affect worker performance and the occurrence of human errors [33]. Therefore, the machine degradation leads to a higher workload and a worse work environment, and then increases the fatigue rate, and finally increases the human error probability. To consider the impacts of machine degradation on worker fatigue, we assume that the fatigue parameter α in Eq. (2) is dependent on the degradation state of the machine. Eq. (2) can then be rewritten as

fa (i , t ) = Fa (i , t ) = 1

e

it,

(8)

(i, m),(0,1)

(ti, m ) = µi (ti, m) pi,0 (ti, m),

(9)

The occurrence rate of an extreme shock which will lead to state (0, 1), i.e., the machine will fail immediately; (i, m),(i, m + 1)

(ti, m) = µi (ti, m) pi, i (ti, m),

(10)

The occurrence rate of a CENT which will not change the degradation layer and cause the machine to go to state (i, m + 1) ; (i, m),(j,1)

(ti, m) = µi (ti, m) pi, j (ti, m ), i > j,

(11)

The occurrence rate of a CET which will change the degradation layer to the jth layer and cause the machine to go to state (j, 1); and (i, m),(j,0)

(ti , m ) =

m i, j (ti, m ),

(12)

The rate of transition due to the aging degradation process which will cause the machine to make the transition to state (j, 0). The impact of CENT on the degradation process is considered in Eq. (12) by utilizing the superscript m, which is the number of CENT occurring during the residence time in the current state. Eq. (12) is a general formulation, i.e., the transition rate depends on the number of CENT. The first three types Eqs. (9)–(11) depend on the probability of a human error being an extreme error, CENT, and CET, respectively; the

(6)

where αi is the fatigue parameter with the machine in state i, and the 4

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Fig. 3. Degradation process and human errors.

last type of transition rate Eq. (12) depends on the cumulative damage of CENT. In this model, we assume that the damage caused by CENT can influence the degradation transition departing from the current state, which is taken from [45].The impacts of the CENT are reflected in the change of transition rates. A machine will fail if the degradation state reaches layer 0. Therefore, the space of the machine failure states can be denoted by M = {(0, 0), (0, 1)} .

N (i = (xi , yi ), (dx , yj ), dt ) = P (X k + 1 k

= i, t

(

ie

it,

×S,

(13)

Q ( (i = (x i , yi ), t ), (dx , yj )) = P (X k + 1 |Tk + 1 Tk [t , t + dt ], Zk = i ) = P (Y k + 1 = yj |X k + 1 [x , x + dx ], Tk + 1 × P (X k + 1

.

[x , x + dx ]|Tk + 1

Tk

[x , x + dx ], Y k + 1 = yj Tk

[t , t + dt ], Zk = i )

.

[t , t + dt ], Zk = i )

Since the fatigue function is right continuous and the left limits exist, the value of X k + 1 must be in the interval [x , x + dx ] given Tk + 1 Tk [t , t + dt ]. Then we can obtain

Therefore, Z(t) is a PDMP according to the definition in [64,65], the reason for which can be described through the following two aspects: (a) based on Eq. (14), we can get Z (t ) = (Zk , t Tk ) , for t [Tk, Tk + 1], where φ satisfies (y, t + s ) = ( (y , t ), s ) , ∀t, s ≥ 0, y ∈ E, and φ(y, t) is a right continuous function with left limits; (b) {Zn, Tn}n ≥ 0 is a Markov renewal process on the space E × +. The probability of the transition from state Zk to state j in the time interval [Tk, Tk + t ] given {Zi, Ti}i ≤ k can be expressed as

Zk

(17)

(18)

(0, 0) ) for Y (t )

E, j

[x , x + dx ], Y k + 1 = yj

where Q(φ(i, t), (dx, yj)) is the conditional probability that Zk + 1 = j given Tk + 1 Tk = t and Zk = i , and dFi(t) is the conditional probability distribution of Tk + 1 Tk = t given Zk = i . In other words, dFi(t) is the conditional probability that the transition will be out of state i after holding time t given Zk = i . Based on the dependence between machine degradation and human errors, Q(φ(i, t), (dx, yj)) can be derived as

(14)

,j

0,

, |Tk + 1 Tk [t , t + dt ], Zk = i ) × P (Tk + 1 Tk [t , t + dt ]|Zk = i ) = Q ( (i , t ), (dx , yj )) dFi (t )

[Tk , Tk + 1].

P (Zk + 1 = j , Tk + 1 [Tk , Tk + t ]|{Zi, Ti }i k) = P (Zk + 1 = j , Tk + 1 [Tk , Tk + t ]|Zk ) k

0, dt

N (i = (xi , yi ), (dx , yj ), dt ) = P (X k + 1

where E is a space which combines and S = S × . Let Tk, k denote the time of the kth jump, which may be caused by the transition due to the original degradation or a human error, and then Zk = Z (Tk ) = (X (Tk ), Y (Tk )) = (X k , Y k) . The evolution of Z(t) between two consecutive jumps of Y′(t) is determinate and will not change, because during this period there is no occurrence of human errors and no changes of the degradation state. The evolution can be obtained as

Z (t ) = (X (t ), Y (t )) = (f a (i, t ), (0, 0)) =

, dx

Eq. (16) can be expressed as Eq. (17)

Let Z(t) denote the degradation process of the whole MMS:

E=

Tk

(16)

3. The mutual dependence model based on PDMPs

Z (t ) = (X (t ) = fa (Y (t ), t ), Y (t ) = (Y (t ), N (t )))

[x , x + dx ], Y k + 1 = yj , Tk + 1

.

[t , t + dt ]| Zk = i ) , yi , yj E , x i , dx

P (X k + 1

[x , x + dx ]|Tk + 1

Tk

(19)

[t , t + dt ], Zk = i ) = 1,

and Eq. (17) can be expressed as

Q ( (i = (x i , yi ), t ), (dx , yj )) = P (X k + 1

[x , x + dx ], Y k + 1 = yj

|Tk + 1 Tk [t , t + dt ], Zk = i ) = P (Y k + 1 = yj |Tk + 1 Tk [t , t + dt ], Zk = i )

. (20)

4. Reliability estimation

(15)

{Zn, Tn}n ≥ 0 can be described by a semi-Markov kernel as

In this paper, a PDMP is utilized to model the mutual dependence 5

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between the machine degradation and human error. Due to the complex PDMP, an analytical solution is difficult to obtain, while we can obtain the reliability by Monte Carlo (MC) simulation [64].

(c) Calculate the rate of the transition that make the machine depart from the current state using Eq. (A.3) based on Eqs. (9)–(12). (d) Sample a residence time t′ from (23) based on Eq. (A.3). (e) Set t = t + t , Calculate fa(i, t) by using Eq. (7), Calculate μi(t) through Eq. (8), and Calculate Eqs. (9)–(12), and then calculate λ(i, m)(t′) by using Eq. (A.3). (f) Sample a machine state (j, n) by using Eq. (24). (g) Update the system state Z = (fa (i, t ), (j, n)) . (h) Update the fatigue parameter = j .

4.1. Basics of Monte Carlo simulation Based on the semi-Markov kernel of {Zn, Tn}n ≥ 0 (Eq. (17)), the system reliability at time t can be calculated by the MC simulation method. To replicate the life process of the MMS, we need to sample its holding time by dFi(t) and arrival state by Q(φ(i, t), (dx, yj)) repeatedly. The expressions of dFi(t) and Q(φ(i, t), (dx, yj)) can be derive as Eqs. (21) and (22) respectively, and Appendix A provides a more detailed proof of the expressions.

dFi (t ) =

(i, m )

ti, m

(ti, m)exp

(i, m)

0

Q ( (i = (xi , yi ), t ), (dx , yj )) =

(s ) ds .

(i, m),(j, n) (i, m )

(a) When t < tmax, if Z , set n = n + 1, and go to Step 4; otherwise go back to the Step 2. (b) If t ≥ tmax, go to Step 4.

(21)

(ti, m )

(ti, m)

Step 3: Complete a cycle of simulation

.

Step 4: Complete Nmax cycles of simulation Set k = k + 1. If k < Nmax, reset t = 0 , t = 0 , and Z = (0, (M , 0)) , and go back to the Step 2; otherwise go to Step 5. Step 5: Calculate the system reliability

(22)

where λ(i, m)(ti, m) is the rate of transition that causes the machine to depart from (i, m) in the infinitesimal time interval (t + ti, m, t + ti, m + dti, m ) , and λ(i, m), (j, n)(ti, m) is the rate of transition that causes the machine to depart from (i, m) to (j, n)in the infinitesimal time interval (t + ti, m, t + ti, m + dti, m ) . In the MC procedure, for a machine in any non-failure state (i, m) at any time t, the process will sample the residence time at state (i, m), ti, m, based on Eq. (21) first, and then determine the arrival state (j, n) from state (i, m) corresponding to Eq. (22). The procedure will be executed repeatedly until the accumulated residence time reaches the predefined time, or the MMS reaches the failure states = { , M} .

(a) Calculate R(t) by the equation, R (t max ) = 1 (b) The sample variance [67] can be R (t ) (1 R (tmax )) varR (tmax ) = max(N . 1)

(i, m )

(s ) ds = ln

1 , a1

5.1. Case study In this paper, we consider the turret of a conventional lathe operated by a worker, which is a typical MMS. Based on the FMECA of the lathe in [68], a conventional lathe includes three major components i.e. feed mechanism, turret and head stock assembly, and the head stock assembly is a more critical component. However, for a MMS, the critical components judged only by FMECA cannot provide sufficient information due to ignoring potentially hazardous errors. As discussed in [6], the turret failures are mainly caused by human errors. The HAZOP [28], or cognitive task analysis [18,29,30] may be helpful and their combination with FMECA [27,28] can provide more adequate information. Workers machine parts by controlling the speed and direction of the turret, and there are frequent man-machine interactions in the operation of turret. The worker fatigue will increase and there may be many human errors. The turret will fail more easily when considering potentially hazardous errors. For the MMS, the aging degradation and human errors contribute to the degradation process of the turret [5]. A turret consists of many mechanical components which will degrade with the progressing of machining, such as damage of the pin, screw, and box of clamping and the loose of the screw [6]. In addition, the failures or degradation of the turret may be caused by human errors [6]. Workers machine parts by controlling the turret, and the human errors may include applying a higher spindle speed, applying a wrong turret feed speed or direction. Such human errors will result in a sudden shock load to the turret, and then cause a looser screw or the damage of pin, screw and box of clamping if the shock load is huge enough. Therefore, the human errors may cause a more degraded state (such as a screw becoming looser) and even a failed state (such as the damage of pin or screw). A relatively small shock load will increase the transition rate to a more degraded state while it cannot change the degradation state of the turret. Therefore, human errors contribute to turret degradation and sometimes cause the turret failure. On the other hand, the degradation of the mechanical components will increase the intensity of the turret vibrations and decrease the machining accuracy and cause a nosier work environment. Therefore, producing a qualified part becomes more difficult and a worker must

(23)

and (j, n) will be determined as a*, which satisfies a* 1 k=0

(i, m), k (ti, m )

< a2

(i, m )

( ti , m )

a* k=0

(i, m), k (ti, m ),

through

5. Case study and results analysis

Two random numbers a1 and a2, which are uniformly distributed in the interval [0, 1], are sampled to determine the residence time ti, m at state (i, m) and the arrival state (j, n) from state (i, m) respectively. As the sampling method used in [66], ti, m can be determined through ti, m

obtained

max

4.2. The simulation procedure

0

1 n. Nmax

(24)

where a* is one state of all possible arrival states from state (i, m). The state a* is determined by going through all possible outgoing states from state (i, m) until Eq. (24) is satisfied. The framework of the reliability assessment based on the MC procedure is describes as follows: Step 1: Initialize the parameters of the MC simulation and the reliability model. (a) Set predefined time horizon tmax, maximum number of replications Nmax, initial cycle number k = 0 , the number of failures n = 0 , initial system time t = 0 , initial system state Z = (0, (M , 0)) . (b) Input the human error parameters (i.e. μ0 and γ), the fatigue parameters (i.e. α0, α1, ..., αM, β), the work-break schedule of workers, the transition rates due to human errors P, and the transition rates due to degradations λ. Step 2: Sample and update system state (a) Calculate the accumulation of worker fatigue based on the workbreak schedule using Eq. (7). (b) Calculate the human error rate using Eq. (8), calculate the transition rates due to human errors using Eqs. (9)–(11), and calculate the transition rates due to aging degradation process using Eq. (12). 6

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pay more attention to a better control of turret to machine the part. Then the accumulation rate of worker fatigue will increase, which leads to more human errors. Consequently, the turret experiences degradation due to production process and human errors, the degraded turret increases the accumulation rate of worker fatigue, the high level of worker fatigue results in more human errors, and human errors accelerate the turret degradation. The turret degradation and human errors are mutually dependent, which influences the system reliability significantly.

To analyze worker fatigue, we use the work-break schedule in Fig. 4, which is taken from Givi [12] for a Toyota production line. The total working time is 7.58 h (465 min) except for two short breaks of lengths 10 min and one long break of length 45 min. Therefore, the turret is operated for 7.58 h each day. The turret degrades on a longterm scale, while worker fatigue accumulates on a short-term scale. As shown in Fig. 5, we combine the turret degradation and worker fatigue in the PDMP framework. The worker operates the turret every day until the turret goes to state 0, and the worker experiences the fatigue-recovery process each day. In the beginning of each workday, worker fatigue is assumed to be 0. In addition, the accumulation rate of fatigue is higher when operating a more degraded turret, and then the rate of human errors will also increase consequently. The fatigue rate increases at t1 and t2, which are the time when the turret degrades from state 3 to state 2 and from state 2 to state 1 respectively. The corresponding parameters of the turret degradation and human errors are shown in Table 1. Then, the simulation procedure for this model can be proposed in Appendix B.

5.2. Model development The degradation process of the turret is modeled by a four-state, continuous-time Markov chain. The set of degradation states can be denoted as S = {0, 1, 2, 3} , where state 0 is the complete failure state and state 3 is the perfect state. The turret will be functioning until Y (t ) = 0 . The degraded turret will vibrate, and the intensity of the vibrations of turret at states 2 and 1 is evaluated by experts as “smooth” and “rough” respectively. λ32, λ21, and λ1, 0 are the transition rates. Human errors can degrade the turret from its current state i to a lower state j and even to state 0. The transition rate pij can be written as Eq. (25), which is taken from [5].

pij =

9 × (0.1)(i j + 1) 1 (0.1) (i + 1)

5.3. Results and analysis A Monte Carlo simulation for 105 replications is utilized to evaluate the reliability of the manufacturing system. As shown in Fig. 6, we evaluate the system reliability under three different types of conditions: (a) the unilateral dependence that human errors can accelerate the turret degradation and the human error follows a homogeneous Poisson process (i.e. = 0 ); (b) the unilateral dependence where human errors can accelerate the turret degradation, and the human errors follow a = 0.002 , nonhomogeneous Poisson process (i.e., and 3); (c) the mutual dependence between the turret 1 = 2 = 3 = 6.4e = 0.002, and degradation and the human errors (i.e., 3, 3 = 6.4e 3 ). In general, the reliability of the 1 = 0.19, 2 = 9.4e MMS decreases as a stronger dependence is considered. As expected, the sequence of the system reliability from high to low in order is: considering dependence (a), then dependence (b), and finally

(25)

As the assumption in [45], the corresponding degradation transition rates after m CENT are also assume to be m i, j ( t i, m )

= (1 + )m

i, j (ti, m ),

(26)

where η is the relative increment of transition rates due to an occurred CENT. Eq. (26) is utilized to characterize the accumulated impacts of such human errors and to describe the increase of the transition rates after one CENT happens. In the case study, for the sake of simplicity, the values of η for each CENT are assumed to be the same, whereas the model can handle different η for different states of the turret vibrations.

Fig. 4. The work-rest schedule during each weekday [12].

Fig. 5. The dependence between the turret degradation and the worker fatigue. Top figure: degradation process of the turret; bottom figure: fatigue-recovery process of the worker, where the blue thick solid lines denote the recovery process. 7

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(1) When t is small, the turret fails rarely and is usually in state 3. Hence, the accumulations of worker fatigue with dependence (b) and (c) are almost the same, as shown in Eq. (8). In addition, the worker fatigue is relative small and human errors occur rarely. Therefore, the reliabilities are almost the same before 800 h. (2) As t increases, the turret will change to more degraded states, and the accumulation of worker fatigue with dependence (c) will be accelerated significantly by the larger fatigue parameters α1 and α2. The high level of fatigue will lead to more human errors, and the human errors will aggravate the turret degradation. Therefore, the reliability decreases gradually after 800 h.

Table 1 The corresponding parameters for the reliability analysis of the MMS. Parameters

Value

Sources

λ32 λ21 λ10 α1 α2 α3 β μ0 η γ

3e-4/h 3e-4/h 3e-4/h 0.19 9.6e-3 6.4e-3 9.6e-3 5e-3 0.3 2e-3

Assumption Assumption Assumption [12] [12] [12] [12] Assumption Assumption Assumption

5.4. Sensitivity analysis dependence (c). At time t = 2000 h, the system reliabilities are 0.3558, 0.2881, and 0.2482, with sample variances equal to 2.292e-6, 2.051e-6, and 1.866e-6, respectively. The main mathematical and physical reasons for the phenomenon are:

The manufacturing system is subject to mutually dependent turret degradation and human errors. With the model developed in Section 5.2, the dependence factor γ in μ(t) and the relative increment factor η in im , j (ti, m ) are two important parameters. We perform a sensitive analysis of γ and η respectively, and analyze the sensitivity of the system reliability to these two parameters. The parameter γ represents the impact of worker fatigue on the intensity of the human errors. To investigate its impact on the system reliability, we evaluate the system reliability under four different levels of dependence (i.e. = 0, 0.001, 0.002, 0.005) and the results are plotted in Fig. 7. It can be seen that the declination of the curves of the system reliability become more rapid with γ increasing from 0 to 0.005. This is mainly due to the fact that worker fatigue have detrimental effects on human errors. When γ is relatively small, worker fatigue and the human errors are almost independent, and the occurrence rate of human errors is relatively small. Therefore, the reliability is high and decreases slowly. When γ is relatively large, the occurrence rate of human errors will increase as the worker fatigue accumulates, as shown in Eq. (8). Therefore, more human errors will occur, which accelerates the turret degradation, and then the system reliability decreases rapidly with time. The parameter η represents the impacts of the CENT on the transition rate of the current state. We take different values of η (i.e. = 02, 0.3, 0.4 ) to investigate the system reliability. As shown in Fig. 8, when η increases from 0.2 to 0.4, the system reliability decreases gradually. This can be explained by the fact that the transition rates to more degraded states will increase significantly with a higher value of η, as shown in Eq. (26). Hence, the transitions to the failure state will occur more. To investigate the sensitivity of the system reliability to the two

(1) When considering unilateral dependence (a), the human errors will cause the damage of turret, as shown in Eq. (25), and the rate of the transition to a more degraded state will be increased, as shown in Eq. (26). When considering unilateral dependence (b), the occurrence rate of human errors will increase with the accumulation of worker fatigue, and then the turret failure might occur more due to the high risks of human errors. Therefore, dependence (b) leads to a lower system reliability. (2) Due to the mutual dependence, the human errors may degrade the vibration state of turret, and workers are more likely to feel fatigue when using the degraded turret, as shown in Eq. (6), and then the intensity of human errors will increase, as shown in Eq. (8). Thus, the turret vibration is accelerated and the human errors are facilitated, on which condition, the extreme human errors may occur more often and the vibration state may reach state 0 earlier. Therefore, the system reliability considering the mutual dependence is the lowest. At 2000 h, the system reliability with dependence (c) is 13.85% lower than that with dependence (b), which indicates that the dependence of the accumulation of worker fatigue on the machine degradation has detrimental effects on system reliability. Fig. 6 shows that the two types of reliabilities are almost the same before 800 h and then decrease gradually. The main reasons for the phenomenon are as follows:

Fig. 7. The system reliabilities under different values of the dependence factor, γ.

Fig. 6. The reliabilities of the MMS under three different conditions. 8

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H. Che, et al.

Fig. 9. The reliabilities as a function of η and γ (at hour 800). Fig. 8. The system reliabilities under different values of the relative increment, η.

Markov model, and the human error model is integrated into a multistate machine degradation model to describe the mutual dependence. To the best of the authors’ knowledge, this is the first work on reliability modeling for a MMS with the mutual dependence. The PDMP framework is first introduced to reliability assessment of a MMS, and it provides an effective way to study the mutual dependence. In addition, through the framework, the dynamics of the failure generation process of a MMS is fully investigated, and hence this paper likely provide a more precise reliability assessment of a MMS. A turret of a lathe operated by a worker, which experiences turret vibrations and human errors, is considered as a realistic application to illustrate the effectiveness and modeling capabilities of the proposed model. The results and sensitive analysis show that the mutual dependence leads to a lower reliability performance. For future investigations, maintenance can be included in this model and optimal maintenance policies can be derived to enhance the system reliability. In this paper, we focused on the physical fatigue, and the mental fatigue should be considered in the future work. In addition, the effects of learn-forgetting can be considered to build the human error model.

parameters, we take values of γ within the range [0.001, 0.005], and η within the range [0.2, 0.4]. Fig. 9 shows the system reliabilities with different combinations of these two parameters. As shown in Fig. 8, the system reliability decreases no matter which parameter increases. In fact, a higher γ leads to a larger occurrence rate of human errors, which promotes the transitions to the failure state. Moreover, a higher relative increment factor η increase the transition rates to a more degraded state. 6. Conclusion Precise reliability assessment of a MMS necessitates consideration of the dependence between machine degradation and human errors due to the complexity of man-machine interactions. In this paper, we develop a general reliability model for a MMS with the mutual dependence by a PDMP modeling framework. In the framework, a human error model under the effects of worker fatigue-recovery is developed mathematically, a multi-state machine degradation model is proposed by a semiAppendix A

We will obtain the expressions of dFi(t) and Q(φ(i, t), (dx, yj)) to derive a MC procedure by using the transition probability density function f(i, m), f(i, m), (j, n)(ti, m|t)dti, m is the conditional probability that the transition from state (i, m) to state (j, n) occurs in the infinitesimal time interval (t + ti, m, t + ti, m + dti, m ) , given the system state (i, m) at time t. f(i, m), (j, n)(ti, m|t) can be derived as (A.1) by utilizing the transition rates Eqs. (9)–(12). (j, n)(ti, m|t).

f(i, m),(j, n) (ti, m |t ) = where P(i, satisfies

d P (i, m),(j, n) (ti, m |t ) = P (i, m) (ti, m |t ) dt

m)(ti, m|t)

d P (i, m) (ti, m |t ) = dt

(i, m ),(j, n)

(ti, m),

(A.1)

is the condition probability that no transitions occur in the time interval (t , t + ti, m) , given the system state (i, m) at time t. It

P (i, m) (ti, m |t )

(i, m )

(ti, m),

(A.2)

where λ(i, m)(ti, m) is the rate of transition that make the machine depart from (i, m) in the infinitesimal time interval (t + ti, m, t + ti, m + dti, m ) . λ(i, m)(ti, can be expressed as

m)

(i, m)

(ti , m ) =

For P(i,

(i , m )

m)(0|t)

P (i, m) (ti, m |t ) = exp

(i, m),(i , m ) (ti, m ).

(A.3)

equal to 1, we can obtain ti, m 0

(i, m )

(s ) ds .

(A.4)

Then dFi(t) can be expressed as

9

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H. Che, et al.

dFi (t ) =

dP (i, m) (ti, m |t ) = P (i, m) (ti, m |t )

(i, m )

( t i, m ) =

(i, m)

(ti, m)exp

ti, m (i, m )

0

(s ) ds ,

(A.5)

and Eq. (A.1) can be expressed as Eq. (A.6)

f(i, m),(j, n) (ti, m |t ) =

(i, m),(j, n)

ti, m

(ti, m)exp

(i, m)

0

(s ) ds .

(A.6)

To determine Q(φ(i, t), (dx, yj)), Eq. (A.6) can be expressed as Eq. (A.7)

f(i, m),(j, n) (ti, m |t ) = =

(i, m),(j, n)

(ti , m )

(i, m),(j, n) (ti, m)

(i, m)

(i, m) (ti, m)

(i, m )

(ti, m)exp[

ti, m 0

(ti, m)

(i, m )

(s ) ds]

, (A.7)

where π(i, m), (j, n)(ti, m) is the conditional probability that the transition will jump from state (i, m) after the residence time ti, m in state (i, m), and the arrival state will be (j, n). It satisfies (i, m ),(j , n) (ti, m )

ψ(i, (i, m )

=

(i, m )

m)(ti, m)

(ti, m) =

(i, m ),(j , n)

( t i, m )

(ti, m )

.

(A.8)

is the probability density function of the residence time ti,

(i, m)

(ti, m )exp

ti, m 0

(i, m )

m

in state (i, m), which is the same as dFi(t). It satisfies

(s ) ds .

(A.9)

Based on Eq. (20), Q ( (i = (xi , yi ), t ), (dx , yj )) can be derived as

Q ( (i = (x i , yi ), t ), (dx , yj )) = P (Y k + 1 = yj |Tk + 1 =

(i, m),(j, n)

(ti , m ) =

Tk

[t , t + dt ], Zk = i )

(i, m ),(j, n ) (ti, m )

. (A.20)

(i, m) (ti, m )

Appendix B The procedure of the MC simulation method is as follows: Set tmax , Nmax , n = 0 , and k = 0 . Input µ0 , , 1, 2, 3 , P (calculated by (25)), , , , and the work-break schedule.

While k < Nmax Initialize the time t = 0 , and the system state Z = (0, (3, 0)) . Set t = 0 (the residence time of a machine state). While t < tmax Sample a t from (23) based on (A.3), (9)–(12), and (26). Set t = t + t . Calculate fa (i, t ) by using (7), Calculate µi (t ) through Eq. (8), and Calculate (9)–(12), and (26), and then calculate Sample an arrival state (j, n) from (24). Set Z = (fa (i, t ), (j, n)) and = j .

(i, m) (t

) by (A.3).

,n=n+1 If Z Then break. End if. End while. Set k = k + 1. End while.

Calculate system reliability R(t) and the sample variance varR (tmax ) .

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