Product infant failure risk modeling based on quality variation propagation and functional failure dependency

Abstract

Product infant failure is the most crucial part of product quality risk that critically affects customer satisfaction. However, most of manufacturers lack sufficient understanding of the formation mechanism of product infant failure and quantitative infant failure risk modeling technology. These issues cause the inefficiency of warranty policy in preventing and controlling infant failure risk. Therefore, first, on the basis of the reverse mapping process of axiomatic design, an infant failure risk formation chain, that is, “process quality variation—physical defect—functional vulnerability—infant failure,” is proposed in this study by considering quality variation propagation and functional failure dependency to determine the inherent formation and dependent enlarging process of the risk. Second, on the basis of the risk formation chain, the infant failure risk is modeled from inherent risk and dependent risk. Specifically, the inherent infant failure risk is computed based on the stream of quality variation in production, whereas the dependent infant failure risk is computed by a Bayesian network by considering the functional failure dependency. Finally, a case study of an illustrative electromechanical system is introduced to verify the applicability of the proposed method. Result shows that the proposed method has a better performance in infant failure risk modeling.

Keywords

Infant failure risk analysis risk formation chain stream of variation functional failure dependency

Introduction

Quality risk is specifically emphasized for product quality improvement according to the newly released ISO 9001:2015 standard.¹ Infant failures can sharply decrease the customer satisfaction and increase warranty costs,² which are the most critical object of product quality risk management. However, most of manufacturers lack sufficient knowledge of product infant failure mechanism and quantitative infant failure risk modeling technology.³ These issues result in the weakness of warranty policy in preventing and controlling infant failure risk. For example, if a proactive infant failure risk analysis for Samsung Galaxy Note 7 exists,⁴ then the painful damage due to battery explosion may be remarkably reduced or avoided.

Specifically, infant failures are concentrated embodiments of accumulation of product manufacturing quality variation,⁵ and these failures are the important sources of product quality risk.⁶ Regarding the quality variation propagation and its influence on infant failure, Huang and Shi⁷ proposed the stream of variation (SoV) method to analyze the transmission, connection, and accumulation of quality variations in multistage manufacturing process. This method provides theoretical foundation for the analysis of quality variation propagation. Jiang and Murthy⁶ investigated the effects of quality variations in manufacturing on product failure and proposed quality data-based reliability modeling in both parametric and non-parametric cases. Thornton et al.⁸ proposed the variation risk management to reduce the effect of quality variation on the final quality of a product. He et al.⁹ presented a multilayered model structured as “part, component, and system levels” to determine the infant failure rate by quantifying the holistic quality variations based on the manufacturing process. Abellan-Nebot¹⁰ proposed a new approach modeling the dimensions of ceramic tiles based on an improved SoV model that helps manufacturers improve process and quality control. Liu et al.¹¹ proposed an improved failure mode and effect analysis (FMEA) based on cluster analysis and foreground theory. The weight of the whose risk factor is objectively obtained based on the risk assessment information. These studies are helpful to identify the key quality variations related to infant failure and analyze the probability of the infant failure risk by clarifying the manufacturing variation propagation process. However, production variation is inevitable,^8,12 and infant failure cannot be eliminated completely. Moreover, once infant failure derived from inherent manufacturing uncertainties exists, infant failure will spread due to the effect of functional failure dependency.¹³ Therefore, accurately analyzing the functional failure dependency for inherent infant failure is another crucial work to expound the mechanism of infant failure risk.

Furthermore, some studies consider functional failure dependency with each other. Krus and Lough¹⁴ presented a function-based failure propagation (FBFP) method to analyze dependencies of failures through the functions present in a system. Tumer and Smidts¹⁵ introduced the functional failure identification and propagation (FFIP) analysis framework. This framework models the behavior of components and failure conditions of functions, such that functional failures and their dependencies can be assessed using a system model through behavioral simulation. Anastasiadis et al.¹⁶ constructed an aging and statistical dependence model among the system components by associating individual shock processes with potentially overlapping subsystems composed of groups of components to handle dependent failures. Barua et al.¹⁷ proposed a risk modeling methodology for dynamic systems based on a Bayesian network (BN). This methodology is used in representing the dependencies among variables graphically and capturing the changes of variables over time to address time-dependent failures in risk estimation of oil/gas and petrochemical plants. Xing et al.¹⁸ proposed a new combinatorial method of dynamic failure tree (DFT) and Markov-based method for the reliability analysis of the systems subject to functional dependencies. He et al.¹⁹ proposed relational tree to analyze the physical decomposition and functional dependencies of infant failure. Wang et al.²⁰ modeled and analyzed the reliability of phased-mission system subject to multiple functional dependence groups using a Markov chain–based methodology and considered propagated failures with global and selective effects. Chiacchio et al.²¹ believed that using the Priority-AND gate in the dynamic fault tree could accurately interpret the stochastic dependence of components in the system, while it is necessary to check the consequences of its failure logic and system consistency behavior. These studies are beneficial for accurately analyzing the enlarging effect of infant failure risk.

Generally, process quality variations and functional failure dependency should be considered simultaneously for infant failure risk assessment. Therefore, a novel risk modeling approach for product infant failure by considering quality variation propagation and functional failure dependency is presented in this work. First, an infant failure risk formation chain, that is, “process quality variation—physical defect—functional vulnerability—infant failure,” is proposed based on the reverse mapping process of axiomatic design (AD) to identify the formation mechanism of infant failure risk. Second, the risk is categorized into two aspects, namely, inherent and dependent risks, based on the risk formation chain. Finally, the inherent infant failure risk is analyzed based on the stream of quality variation in production, whereas the dependent infant failure risk is analyzed using a BN that considers functional failure dependency.

The rest of this article is organized as follows. Section “Infant failure risk formation chain” formulates an infant failure risk formation chain. Section “Infant failure risk modeling based on the risk formation chain” proposes a quantitative risk modeling approach for the infant failure. Section “Case study” presents the case study of an illustrative electromechanical system (EMS). Finally, section “Conclusion” presents the conclusions.

Infant failure risk formation chain

Notations

The notations used in this article are defined as follows:

$R$	The infant failure risk of a product
$R_{i}$	The infant failure risk of the ith component
$R_{I, i}$	The inherent infant failure risk of the ith component
$R_{D, i}$	The dependent infant failure risk of the ith component
$n$	The number of components of the product
$λ_{I, i}$	The inherent infant failure rate of the ith component
$T_{I, i}$	The terminal time of infant failure of component i
$IF L_{I, i}$	The inherent infant failure losses of the ith component
$λ_{D, i}$	The dependent infant failure rate of the system when only the ith component is functioning
$T_{D, i}$	The terminal time of the infant failure of component i
$IF L_{D, i}$	The dependent infant failure losses of the ith system when only the ith component is functioning
$IF P_{I, i}$	The inherent infant failure probability of the ith component
$R_{ijKQC}$	The reliability of the jth KQC variation that does not cause infant failure to the ith component
$R_{ij} (E)$	The reliability of the exterior defects relevant to the jth KQC that does not cause failure to the ith component
$R_{ij} (I)$	The reliability of the interior defects relevant to the jth KQC that does not cause failure to the ith component
$T_{ij}$	The one-sided tolerance of the jth KQC
$μ_{f}$ , $σ_{f}$	The mean and standard deviation of the jth KQC, respectively
$N_{ij}$	The number of interior defects associated with the jth KQC
$P (η_{ij})$	The Poisson distribution with mean $η_{ij}$
$φ_{ij}$	The detection probability of the defects associated with the jth KQC in the detection phase
$a_{ij}$ , $b_{ij}$	The scale and shape parameters of Weibull distribution
$C_{ri}$	The replacement cost of component i
$C_{w}$	The maintenance labor cost per time unit
$Δ t_{i}$	The maintenance time for the ith component
$IF P_{D, i}$	The dependent infant failure probability of the system when only the ith component fails
$X$	The state variable of input events
$Y$	The state variable of the output event
$IF P_{Sub j, D i}$	The infant failure probability of the jth subsystem that is dependent on the ith component
$F_{j p a_{k}}$	The kth parent of the jth subsystem
$F_{Sub j}$	The jth subsystem
$F_{C i}$	The ith component
$IF P_{C i}$	The infant failure probability of the ith component
$F_{p a_{l}}$	The lth parent of the system function
$F_{Sub j, D i}$	The jth subsystem dependent on the ith component
$C_{1 i}$	The penalty cost of the system shutdown caused by the infant failure of component i
$C_{2 i}$	The cost of safety effect of the system shutdown caused by the infant failure of component i
$C_{3 i}$	The cost of environmental effect of the system shutdown caused by the infant failure of component i

Connotation of infant failure risk

Product quality is the output of design and development process, which is embodied in usage by reliability. The key aim of product quality risk management is to ensure a stable manufacturing process and a highly reliable usage period. In addition, most of the infant failures are determined by the production quality variations and manufacturing defects, which seriously affect customer satisfaction in early usage. Therefore, infant failure risk is an important and specific part of product quality risk. On this basis, the connotations of infant failure risk must be further clarified as follows:

Risk sources are inevitable manufacturing quality variations. Therefore, after a product has been manufactured, a potential occurrence probability of infant failure may exist.

When the product is tested in its early usage, the potential probability gradually results in the occurrence of infant failure due to the stimulation of environmental stress, and this failure may trigger other failures, such as failure dependency in different extents, thereby expanding the severity of the infant failure.

Therefore, in this study, the infant failure risk is defined as the comprehensive measurement of quality variation and functional failure dependency. That is, more quality variations indicate a stronger failure dependency and higher infant failure risk. Therefore, quantitatively expounding the formation mechanism of infant failure risk is important.

Infant failure risk formation chain based on AD

From the perspective of system engineering, Professor Suh²² proposed the AD theory. This theory is a bidirectional domain mapping relationship model composed of customer, functional, physical, and process domains. The forward mapping of AD is the product design and development process, and the output is the manufactured product. Meanwhile, the reverse mapping of AD is the risk source identification process. In the reverse mapping process, the process quality should be monitored to ensure a reasonable physical architecture of a product. By achieving the physical requirements, the functional requirements will also be satisfied. Then, in its early usage stage, the product function may behave normally. On the basis of the reverse mapping process of AD, a risk formation chain, that is, “Process Quality Variation—Physical Defect—Functional Vulnerability —Infant Failure,” is proposed to identify the formation mechanism of infant failure risk, as shown in Figure 1.

Figure 1.

Infant failure risk formation chain.

As shown in Figure 1, when the reasonable design scheme is fixed, the product has transferred into the manufacturing stage. The components are machined and assembled based on the process requirements. In view of the variations of people, machines, materials, methods, environment, and measurements involved in the process, the key quality characteristics (KQCs) will have quality variations in machining and assembly process presented as non-conformities. These quality variations accumulate and propagate between stations and cause physical defects of the components, namely, exterior defects (e.g. excess of tolerance and deformation) and interior defects (e.g. cracks and residual stress). Moreover, these defective components are the infant failure risk carriers. Besides, preliminary identification of KQC can be finished based on the engineering experiences or powerful statistical analysis tool, such as the design of experiment. After the product has been transferred into its early usage, components with physical defects may develop vulnerability of their functionalities and inherent infant failure probabilities. Under the condition of actual stress, the vulnerable components gradually begin to degrade, which causes critical failures of these degraded components. Meanwhile, the infant failures of these components may further cause dependent function failure of system with dependency effects (e.g. series, parallel, hot spare (HSP), cold spare (CSP), and other functional dependencies). Furthermore, the failure may result in relatively severe consequences, such as penalty cost of system shutdown. This issue is the formation process of infant failure risk. Hence, quantitative risk modeling should be urgently proposed based on the risk formation chain to prevent and mitigate infant failure risk effectively.

Infant failure risk modeling based on the risk formation chain

Framework of infant failure risk modeling

Workflow of the modeling

The propagation process from quality variation to infant failure and the deterioration of infant failure consequence can be explained using the proposed infant failure risk formation chain. In this section, the infant failure risk modeling framework is presented based on the forward process of the risk formation chain, as shown in Figure 2.

Figure 2.

Workflow of infant failure risk modeling.

First, infant failure risk-related data, including product design, production historical data (i.e. manufacturing quality variation and product quality inspecting data), and reports of accident and incidents in the early usage (i.e. warranty data) should be collected based on the big industrial production data using sensors or searched using cloud network (Figure 2). The infant failure risk database should be established by sorting these data.

Second, DFT should be established based on the risk formation chain for infant failure risk analysis. By analyzing the warranty data in the infant failure risk database, the top events and related components of the DFT can be determined. Then, the failure correlation of each component can be determined by analyzing the product design data in the database. Finally, the dynamic infant failure tree (DIFT) can be established based on the correlations among the components.

Third, after the qualitative relationships are determined among components through the DIFT, the conceptual quantitative model should be built subsequently to characterize the infant failure risk generally. The specific analysis is discussed in section “Conceptual quantitative model.”

Finally, on the basis of the conceptual model, the infant failure risk can be categorized into two different aspects, namely, inherent and dependent. The inherent infant failure risk is mainly caused by quality variation, whereas the dependent infant failure risk is mainly caused by the functional failure dependency. Their detailed modeling approaches are explained in sections “Analysis of inherent infant failure risk considering quality variation ” and “Analysis of dependent infant failure risk considering functional failure dependency,” respectively.

Conceptual quantitative model

On the basis of the analysis of the front part of infant failure risk chain in Figure 1, inherent infant failure probabilities are present in defective components after the product is produced. This phenomenon is due to the KQC variations in manufacturing process, which constitutes bottom events of the DIFT. When the product is tested in its early usage, the defective components should fail under various environmental stresses; thus, bottom events occur and losses, such as maintenance cost, may arise. Furthermore, as explained by the rear part of the infant failure risk chain, the inherent failures of the component may propagate to the entire system, which may result in dependent infant failures of the system caused by failure dependencies. After the occurrence of dependent infant failure of the system, problems, such as insurance costs, customer dissatisfaction, and environment pollution, may arise. Thus, the two aspects of the infant failure risk of a product are expressed as follows

\begin{matrix} R = \sum_{i = 1}^{n} R_{i} \\ = \sum_{i = 1}^{n} (R_{I, i} + R_{D, i}) \end{matrix}

(1)

where R_i is the infant failure risk of the ith component and $R_{I, i}$ is the inherent infant failure risk of the ith component. Inherent infant failure risk is due to the cumulative delivery of manufacturing quality deviations that cause the product to have manufacturing defects. Furthermore, defective products have inherent infant failure risk. $R_{D, i}$ is the inherent dependent failure risk of the ith component. Dependent infant failure risk can be considered as the possibility of infant failure of a subsystem or system due to functional failure correlation after an infant failure of component, which often have complicated formation. And n is the number of components of the product.

The objective of infant failure risk modeling is to reduce the losses due to infant failure in advance. Thus, the cost-based risk modeling is adopted for each component to analyze the inherent and dependent infant failure risks. Specifically, the inherent infant failure risk can be represented as

R_{I, i} = (\int_{0}^{t} λ_{I, i} (t) dt) \times IF L_{I, i}

(2)

where $λ_{I, i} (t)$ is the inherent infant failure rate of the ith component, and IFL_I,i is the inherent infant failure losses of the ith component. In addition, the time interval of the infant failure risk is $t \in (0, T_{I, i})$ , where T_I,i is the terminal time of infant failure of component i and can be presented as T_I,I = t(curvature of λ _I,i (t) = 0). The detailed analysis of the inherent infant failure risk is shown in section “Analysis of inherent infant failure risk considering quality variation.”

When a product is tested in its early usage, the failure of one component due to environmental stress may result in other failures in the system because of the functional failure dependency. After the dependent infant failure of the system occurs, problems, such as insurance costs, customer loss, and mass of released harmful contaminants into the environment, may arise. Thus, the dependent infant failure risk of the ith component can be expressed as follows

R_{D, i} = (\int_{0}^{t} λ_{D, i} (t) dt) \times IF L_{D, i}

(3)

where λ_D,i(t) is the dependent infant failure rate of the system when only the ith component is functioning, and IFL_D,i is the dependent infant failure losses of the ith system when only the ith component is functioning. The time interval of the infant failure risk is $t \in (0, T_{D, i})$ , where T_D,i is the terminal time of the infant failure of component i and can be expressed as $T_{D, i} = t (curvature of λ_{D, i} (t) = 0)$ .

Analysis of inherent infant failure risk considering quality variation

As shown in equation (1), the inherent infant failure risk of a component can be analyzed based on the inherent infant failure rate $λ_{I, i} (t)$ and inherent infant failure losses $IF L_{I, i}$ . Moreover, the quality variation is the most important factor that should be considered when estimating the inherent infant failure rate.

Inherent infant failure rate computation based on SoV

In multistage manufacturing process, quality variation of KQCs exists and propagates in each station and may cause physical defects (i.e. exterior and interior defects) of the component and result in infant failure. The inherent infant failure rate of the ith component can be expressed as follows

λ_{I, i} (t) = \frac{\frac{d (IF P_{I, i} (t))}{dt}}{1 - IF P_{I, i} (t)}

(4)

IF P_{I, i} = 1 - Π_{j = 1}^{m} R_{ijKQC}

(5)

where $IF P_{I, i}$ is the inherent infant failure probability of the ith component; $R_{ijKQC}$ is the reliability of the jth KQC variation that does not cause infant failure to the ith component, which are assumed to be independent with each other and can be regarded as a series relationship;²³ and m is the number of KQCs relevant to the ith component.

The reliability of the jth KQC variation that does not cause infant failure to the ith component, which is characterized by the exterior and interior defects during time t, can be defined as follows

\begin{array}{l} R_{i j K Q C} = \Pr (\begin{matrix} e x t e r i o r d e f e c t s c a u s e n o f a i l u r e, \\ i n t e r i o r d e f e c t s c a u s e n o f a i l u r e \end{matrix} | d u r i n g t i m e t) \\ = R_{i j} (E) \times R_{i j} (I) \end{array}

(6)

where $R_{ij} (E)$ and $R_{ij} (I)$ represent the reliability of the exterior and interior defects, respectively, relevant to the jth KQC that does not cause failure to the ith component before time t.

The deviation of $Δ X_{ij} (t)$ and the tolerance of the KQC can be obtained according to the process model $X_{ij} (t)$ of the jth KQC after n stations, as presented by Chen and Jin²³ as follows

f (x) = {\begin{matrix} (Δ X_{ij} (t) - T_{ij}) if Δ X_{ij} (t) > 0 \\ (T_{ij} - Δ X_{ij} (t)) if Δ X_{ij} (t) < 0 \end{matrix}

(7)

where $Δ X_{ij} (t) = X_{ij} (t) - X_{ij 0}$ , $X_{ij 0}$ is the standard value of the jth KQC according to the design requirements of the process domain, and $T_{ij}$ represents the one-sided tolerance of the jth KQC. According to the Law of Large Numbers, when the amount of data is large enough, the data follow the normal distribution. Therefore, by determining the mean $μ_{f}$ and standard deviation $σ_{f}$ of n samples of the jth KQC, $R_{ij} (E)$ can be defined as

\begin{matrix} R_{ij} (E) & = p {f (x) < 0) \\ = p {\frac{f (x) - μ_{f}}{σ_{f}} < - \frac{μ_{f}}{σ_{f}}} \\ = Φ (- β) \end{matrix}

(8)

$R_{ij} (I)$ in equation (5) can be identified using the following calculations. The number of interior defects associated with the jth KQC is assumed to be $N_{ij}$ , and $R_{ij} (I)$ is then expressed as follows

\begin{array}{l} R_{i j} (I) = \sum_{m = 0}^{\infty} \Pr (N_{i j} = m, n o n e o f m o c c u r b y t i m e t) \\ = \sum_{p = 0}^{\infty} \sum_{m = 0}^{p} (\Pr (N_{i j} = m, n o n e o f m o c c u r b y t i m e t | N_{i j} = p) \\ \Pr (N_{i j} = p) \\ = E {(1 - θ_{i j} (t) (1 - φ_{i j}))}^{N_{i j}} \end{array}

(9)

where $N_{ij}$ is the number of interior defects associated with the jth KQC generated after n stations and is commonly characterized and quantified by a Poisson distribution $P (η_{ij})$ , where $η_{ij}$ is the mean of the Poisson distribution, namely the defect density associated with the jth KQC; $φ_{ij}$ is the detection probability of the defects associated with the jth KQC in the detection phase; and $θ_{ij} (t)$ is the probability for the interior defects relevant to the jth KQC with failure or being detected at time t during usage and commonly follows a Weibull distribution,²³ with probability density function (PDF) $θ_{ij} (t) = 1 - \exp (- (t / a_{ij}))^{b_{ij}}$ , where $a_{ij}$ and $b_{ij}$ are the scale and shape parameters. And $τ$ indicates whether the defect can pass

τ = {\begin{matrix} 1 If the product passes the test \\ 0 If the product fails the test \end{matrix}

(10)

Therefore

\begin{array}{l} R_{i j} (I) = \frac{E {{[1 - θ_{k} (t) (1 - φ_{i j})]}^{N_{i j}}}}{\Pr (τ = 1)} \\ = \frac{\sum_{m = 0}^{\infty} {[1 - θ_{k} (t) (1 - φ_{i j})]}^{m} \Pr (N_{i j} = m)}{\sum_{n = 0}^{\infty} {(1 - φ_{i j})}^{n} \Pr (N_{i j} = n)} \\ = \exp {- η_{i j} (1 - φ_{i j}) (1 - \exp {(- (t / a_{i j}))}^{b_{i j}})} \end{array}

(11)

From this model, the inherent infant failure probability of a component can be calculated.

Inherent infant failure losses

On the basis of the theory of failure cost analysis, the inherent infant failure losses of the ith component can be presented as

IF L_{I, i} = C_{ri} + C_{W} Δ t_{i}

(12)

where C_ri is the replacement cost of component i, C_w is the maintenance labor cost per time unit, and $Δ t_{i}$ is the maintenance time for the ith component.

From this model, the inherent infant failure risk of component can be obtained.

Analysis of dependent infant failure risk considering functional failure dependency

As shown in equation (2), similar to the calculation of inherent infant failure risk, the dependent risk of a component can be analyzed based on the dependent infant failure rate $λ_{D, i} (t)$ and dependent infant failure losses $IF L_{D, i}$ .

Dependent infant failure rate computation based on a BN

The dependent infant failure rate of the system when only the ith component fails can be expressed as follows

λ_{D, i} (t) = \frac{\frac{d (IF P_{D, i} (t))}{dt}}{1 - IF P_{D, i} (t)}

(13)

where $IF P_{D, i}$ is the dependent infant failure probability of the system when only the ith component fails and can be analyzed using a BN. The main strength of a BN lies in its capability to express and examine dependence relationships and uncertainties inherent in a complex problem.²⁵ In this section, the infant failure probability is computed by examining the complex failure dependency with the aid of a BN. The computation steps are shown as follows.

Step 1: modeling structure of the BN

The BN topology can be modeled on the basis of the nodes of their perspective functions (i.e. system, subsystems, and components) and dependence relationships. Each node is a random variable that can determine whether a failure occurs or not. If the value of a node is “S,” then no failure occurs. If a failure occurs, then the value of that node becomes “F.”

The infant failure probabilities of root nodes (components) are simply the inherent infant failure probabilities analyzed in equation (3). The conditional probability tables (CPTs) represent failure dependency relationships of these nodes.

Step 2: modeling the CPT of the BN

After the structure of the BN is determined, the dependence relationships among nodes are quantified by constructing the CPTs attached to the nodes using the DFT.

For the BN, the graphic structure represents the qualitative relationship between an individual node and its parent and child nodes. Meanwhile, the CPT with each non-root node represents the quantitative functional failure dependency relationships (i.e. series, parallel, HSP, CSP, common failure causes, and cascading failure) between nodes. The CPT can be acquired using the DFT model to characterize the functional failure relationships between the nodes of the established BN. Subsequently, the conditional probability distribution of various gates should be explained briefly, which can be transferred to the CPT with its corresponding BN models.

AND

Let $X = (X_{1}, X_{2}, \dots, X_{m})$ , where $X_{i}$ (i = 1, 2,…, m) denotes the state variable of input events, and m is the number of input events to the gate. Let Y be the state variable of the output event. The state space of each variable is {1, 2,…, n + 1}. The $Y = Max (X_{1}, X_{2}, \dots, X_{m})$ under a certain state combination of input events.

The notation and meanings of all variables and their corresponding state spaces are similar to the aforementioned AND gate. Let $r = Min (X_{1}, X_{2}, . . ., X_{m})$ The conditional probability distribution of the output of OR Gate is expressed as follows:

P (Y = j | X) = {\begin{matrix} 1, j = r \\ 0, j \neq r \end{matrix}

(14)

CSP

According to the failure mechanism of CSP, the failure distribution of the spare is related to its failure rate and the failure time of the primary component. Let x and y be the states of primary input A and spare B, respectively, and λ is the failure rate of A and B. Obviously, only wheny > x, P(B = y|A = x) makes sense. It is assumed that the failure rate of spare part and component is subject to an exponential distribution;²⁶ then $P (A = x) = \int_{(x - 1) Δ}^{x Δ} λ e^{- λ τ} d τ$ , and $P (B = y, A = x) = \int_{(x - 1) Δ}^{x Δ} \int_{(y - 1) Δ}^{y Δ} λ e^{- λ (t - τ)} λ e^{- λ τ} d τ d t$ ; the conditional probability distribution of B is

HSP

Despite the difference in failure mechanisms between the AND gate and HSP, the conditional probability distribution of the output nodes for HSP is identical; thus, they are not repeated.

Functional dependency

For the common cause of failure, the functional dependent gate (FDG) can be used. Meanwhile, in the case that the FDG has only one trigger event A, the CPT of the output B dependent of A under the condition that A has occurred will be an identity matrix. Its CPT is expressed as follows

P (B = j | A = i) = {\begin{matrix} 1, j = i \\ 0, j \neq i \end{matrix} i, j = 1, 2, \dots, n + 1

(15)

Meanwhile, the FDG has two or more trigger events, and an intermediate node is inserted between the output node and all the trigger nodes. The CPT of the intermediate node depends on the relationship between the intermediate node and all the trigger nodes, whereas that of the output node will also be an identity matrix and is only dependent on the intermediate node.

Step 3: probabilities of system-dependent infant failure

After the probability calculation of the inherent infant failure of the ith component and determination of CPT, the probability of the system-dependent infant failure can be determined from the bottom to the top using the BN inference, supposing that only the ith component fails. First, the infant failure probability of the jth subsystem (as a child node) that is dependent on the ith component (as a parent node) can be identified using $IF P_{Sub j, D i}$ based on the marginal probability in the BN, as shown in the following

\begin{array}{l} I F P_{S u b j, D i} = \sum_{k = 1}^{r} P (F_{S u b j} = F | F_{j p a_{k}} = S, F_{C i} = F) \\ P (F_{j p a_{k}} = S) I F P_{C i} \end{array}

(16)

where $F_{j p a_{k}}$ is the kth parent of the jth subsystem, except for the ith component; r is the parent number; $F_{Sub j}$ is the jth subsystem; $F_{C i}$ is the ith component; and $IF P_{C i}$ is the infant failure probability of the ith component.

Then, infant failure probabilities of the system dependent on the ith component are expressed as follows

\begin{array}{l} I F P_{D i} = \sum_{l = 1}^{s} P (F_{S} = F | F_{p a_{l}} = S, F_{S u b j, D i} = F) \\ P (F_{p a_{l}} = S) I F P_{S u b j, D i} \end{array}

(17)

where $F_{p a_{l}}$ is the lth parent of the system function, except for the jth subsystems; s is the parent number; $F_{Sub j, D i}$ is the jth subsystem dependent on the ith component; and $F_{S}$ denotes the system.

Dependent infant failure losses

On the basis of the system engineering view of the cost, the dependent infant failure losses of the system can be expressed as follows

IF L_{D, i} = C_{1 i} + C_{2 i} + C_{3 i}

(18)

where $C_{1 i}$ , $C_{2 i}$ , and $C_{3 i}$ are the penalty cost, cost of safety effect, and cost of environmental effect of the system shutdown caused by the infant failure of component i, respectively.

Case study

Background

Modern mechanical devices are characterized as multi-functional EMSs, which are composed of hundreds or thousands of components and are expected to be highly reliable during operation. However, EMSs often have high infant failure risk due to the increasing complexity of quality control in the production. This complexity is caused by excessive control parameters and stringent manufacture precision of EMSs. To aid the EMS manufacturers in China in assessing the risk of infant failure before shipment, this study uses the proposed method to model the infant failure risk of a typical EMS producer, its functional constitution, and key manufacturing processes, as shown in Figure 3.

Figure 3.

Functional relationship of a complex EMS and its key manufacturing processes: (a) functional relationship of the EMS and (b) manufacturing process of the control module 1.

As shown in Figure 3(a), the EMS is composed of control, power supply, powertrain, and hydraulic systems. The control system includes two control modules connected in parallel, which are used to perform the start–stop control of the main valve. It also sends signal to the hydraulic subsystem and controls the execution. The powertrain system is a key subsystem composed of a turbine, reducer, and pump. The power supply subsystem is composed of one main valve in the main work mode. Regarding its high infant failure rate, the manufacturing process of control module 1 is expounded, as shown in Figure 3(b). In addition, numerous unavoidable quality variations exist in the production process, including welding offset, nonconforming components, and assembly disorders. These variations are the main causes of the high infant failure of control module 1, and this failure may possibly cause cascading failures of other components due to the functional failure dependency effect.

The main purpose of this illustrative example is to model fully the infant failure risk of the EMS based on the manufacturing processes of components and identify its key components for infant failure risk mitigation. The data are collected from the production line of a Chinese EMS provider with the aid of their industrial engineers and field experts.

Numerical example

Data collection of infant failure risk

From the failure database, the components that suffer from infant failures are identified through warranty data analysis, and their design requirements and manufacturing parameters are acquired from product data management system and enterprise resource planning system. The infant failure risk data of control module 1 are shown in Table 1, and these collected data are crucial and provide the foundation to model the infant failure risk accurately.

Table 1.

Infant failure risk data of control module 1.

Component	KQCs	Manufacturingstation	Designspecifications	Infant failure losses
Component	KQCs	Manufacturingstation	Designspecifications	Inherent	Dependent
Controlmodule 1	Welding offset distanceof the connector	Welding	±0.03 mm	$50	$150
	Insulation strength ofthe electric wire	Electroplating	≥83 V
	Location tolerance ofthe power device	Assembling	1 ± 0.001 mm

KQC: key quality characteristic.

Analysis of infant failure risk

As shown in Figure 3, the second module of the control system is the hot backup for the first module. For the power supply system, the main valve will be opened by the hydraulic system after receiving the signal from the control system. Therefore, to ensure normal operation of the main valve, the hydraulic system must be at the normal working condition before the main valve starts to work. Consequently, the failure of the hydraulic system will result in the failure of the main valve. That is, a functional dependency exists between the hydraulic system and the main valve. The dynamic logic gates, including HSP and FDG, are employed to describe the sequential rules and dynamic behavior of the system. In this study, on the basis of the failure mechanism analysis of the system, the “complex EMS task failure” is selected as a top event for FT analysis. The DFT of this sample system is built to represent the infant failure risk, as shown in Figure 4.

Figure 4.

DFT model of the complex EMS.

In Figure 4, S represents the complex EMS task failure, Y1 represents the control system failure, Y2 is the powertrain system failure, Y3 represents the hydraulic system failure, and Y4 represents the power that is not transmitted to the sub-ordinate unit. The meanings of notations X1–X8 in Figure 4 are listed in Table 2.

Table 2.

Inherent infant failure probabilities of components based on KQC variations.

No	Component	KQCs	R_i(E)	$R_{i} (I) = \exp {- η_{i} (1 - φ_{i}) (1 - \exp (- t / a_{i})^{b_{i}})$
X1	Control module 1 failure	Location tolerance of the power device	0.98	$η_{1} = 2.1 φ_{1} = 0.8 a_{1} = 4500 b_{1} = 0.81$
		Welding offset distance of the connector	0.86	$η_{2} = 3.9 φ_{2} = 0.85 a_{2} = 5500 b_{2} = 0.59$
		Insulation strength of the electric wire	0.96	$η_{3} = 2.3 φ_{3} = 0.82 a_{3} = 4000 b_{3} = 0.85$
X2	Control module 2 failure	Location tolerance of the power device	0.97	$η_{1} = 1.9 φ_{1} = 0.85 a_{1} = 5000 b_{1} = 0.9$
		Welding precision of the connector	0.94	$η_{2} = 2.4 φ_{2} = 0.81 a_{2} = 4600 b_{2} = 0.83$
		Insulation strength of the electric wire	0.88	$η_{3} = 4.1 φ_{3} = 0.85 a_{3} = 5300 b_{3} = 0.61$
X3	Turbine failure	Axial stiffness of the sealing system	0.98	$η_{1} = 1.6 φ_{1} = 0.84 a_{1} = 4000 b_{1} = 0.92$
X3	Turbine failure	Intensity of the exhaust pipe	0.96	$η_{2} = 1.3 φ_{2} = 0.84 a_{2} = 4000 b_{2} = 0.92$
X4	Reducer failure	Surface hardness of the gears	0.93	$η_{1} = 1.8 φ_{1} = 0.80 a_{1} = 3800 b_{1} = 0.92$
X4	Reducer failure	Shaft diameter precision of the oil seal	0.95	$η_{2} = 2.1 φ_{2} = 0.85 a_{2} = 4200 b_{2} = 0.89$
X5	Pump failure	Fastener strength	0.90	$η_{1} = 4.1 φ_{1} = 0.83 a_{1} = 5200 b_{1} = 0.63$
X5	Pump failure	Bearing diametral clearance	0.92	$η_{2} = 2.4 φ_{2} = 0.87 a_{2} = 4500 b_{2} = 0.87$
X6	Main valve failure	Inside the nominal diameter	0.89	$η_{1} = 4.6 φ_{1} = 0.88 a_{1} = 5500 b_{1} = 0.59$
		Surface flatness of the flat spring	0.93	$η_{2} = 3.9 φ_{2} = 0.85 a_{2} = 5000 b_{2} = 0.78$
		Hardness of the valve diaphragm	0.95	$η_{3} = 3.5 φ_{3} = 0.84 a_{3} = 4800 b_{3} = 0.81$
X7	Oil pump failure	Camshaft strength	0.96	$η_{1} = 1.4 φ_{1} = 0.85 a_{1} = 3200 b_{1} = 0.93$
X7	Oil pump failure	Dreg content of injection nozzle	0.98	$η_{2} = 1.2 φ_{2} = 0.87 a_{2} = 3000 b_{2} = 0.95$
X8	Hydraulic motor failure	Strength of hydraulic support bar	0.93	$η_{1} = 2.0 φ_{1} = 0.89 a_{1} = 4000 b_{1} = 0.88$
		Out-of-roundness of the cylinder	0.90	$η_{2} = 2.8 φ_{1} = 0.85 a_{1} = 4700 b_{1} = 0.71$
		Surface roughness of bearing bush	0.95	$η_{3} = 1.9 φ_{3} = 0.82 a_{3} = 4200 b_{3} = 0.84$

KQC: key quality characteristic.

After the representation of infant failure risk, the inherent and dependent infant failure risks can be analyzed subsequently. First, for components X1–X8, the process variation data of their KQCs can be collected, as shown in Table 2, and equations (3)–(10) can be used in modeling these variation data to represent the inherent infant failure rate of each component.

The inherent infant failure rate of the components is used as the input information of the root nodes. The BN, which is transformed from DFT in section “Data collection of infant failure risk,” can be constructed using the “AgenaRisk. 6.2, Revision 2077” to model the dependent infant failure probabilities of the components using equations (15) and (16). The results are shown in Figure 5.

Figure 5.

Example of a manufacturing system.

From the collected failure cost data from the early usage of the EMS and its components, the inherent and dependent infant failure losses can be analyzed using equations (11) and (17).

Calculation of infant failure risk

After the inherent and dependent infant failure risk analysis, the infant failure risk can be calculated using equations (1)–(3). The results are shown in Figure 5. From the figure, the infant failure risks of the EMS and its components increase over time, and the infant failure risks of X8 “hydraulic motor failure,” X4 “reducer failure,” and X5 “pump failure” are the main contributors to the infant failure risk of the EMS. The high infant failure risks of X4 and X5 are possibly due to the quality variations of their KQCs during the manufacturing process. The high infant failure risk of X8 derives not only from the quality variations but also from the failure dependency with X6, which increases the infant failure risk.

Sensitivity analysis

A sensitivity analysis of the infant failure risk on the quality variation of KQCs and failure dependency for component X1 is performed to show the robustness of the proposed approach. Figure 6 shows the analysis results.

Figure 6.

Sensitivity analysis of infant failure risk for the component X1: (a) infant failure risk versus time with different reliabilities of exterior defects, (b) infant failure risk versus time with different interior densities η, and (c) infant failure risk versus time with different failure dependencies.

The effects of the quality variation of KQCs on infant failure risk can be characterized and specified by the reliability of exterior defects and the density of interior defects. Therefore, in comparison with the original level of the reliability of the exterior defects $R (E) = [R_{1} (E), R_{2} (E), R_{3} (E)] = [0.98, 0.86, 0.96]$ , four levels of $R (E)$ with various shifts that are quantified by 10%, 5%, −5%, and −10% of the pre-mentioned $R (E) = [0.98, 0.86, 0.96]$ (when the shift result exceeds 1, it set as 1) are set to analyze the remarkable influence of the reliability of exterior defects $R (E)$ on infant failure risk (Figure 6(a)). Correspondingly, five infant failure risk curves are plotted under various reliabilities of exterior defects $R (E)$ to illustrate the correlated sensitivity and robustness. With the high reliability of exterior defects $R (E)$ (i.e. the negative shift by −10% and −5%), additional infant failures or malfunctions are expected around the early time scale, thereby indicating a high level of infant failure risk. Meanwhile, when the reliability of the exterior defects $R (E)$ of component X1 is high, as denoted by the red and green curves, a low level of infant failure risk is guaranteed. Similarly, in comparison with the original level of the density of interior defects $η = [η_{1}, η_{2}, η_{3}] = [2.1, 3.9, 2.3]$ , four levels of $η$ with various shifts that are quantified by 10%, 5%, −5%, and −10% of the pre-mentioned $η = [2.1, 3.9, 2.3]$ are set to analyze the remarkable influence of the density of interior defects on infant failure risk (Figure 6(b)). The correlated sensitivity and robustness are also illustrated.

In comparison with the normal failure dependency (with one hot backup X2 and considering the system-dependent failure), other three dependency conditions, namely, no failure dependency (with no hot backup and not considering the system-dependent failure), slight failure dependency (with two hot backups and considering the system-dependent failure), and strong failure dependency (with no hot backup and considering the system-dependent failure), are used to analyze the remarkable influence of the failure dependencies on infant failure risk (Figure 6(c)). Correspondingly, four infant failure risk curves are plotted under various failure dependencies to illustrate the correlated sensitivity and robustness. With a strong failure dependency, the infant failure of component X1 more likely to result in complex EMS failure and cause additional severe consequences, thereby indicating a high level of infant failure risk. By contrast, when the failure dependency of component X1 is slight and inexistence, as denoted by the blue and green curves, a low level of infant failure risk is guaranteed. These quantitative analysis conclusions will help decision makers of engine manufacturing to determine the effects of different manufacturing quality variation control measures and product design optimization approaches.

Comparative study

The infant failure risk of the EMS can simply be regarded as the expected infant failure costs, and the cost-based FMEA is a common method to represent these expected costs based on the risk perspective. Therefore, the infant failure risks of the EMS using the proposed method and the cost-based FMEA and the actual infant failure costs determined using warranty data are analyzed comparatively, as shown in Figure 7.

Figure 7.

Comparison of the infant failure risk based on the proposed method, cost-based FMEA, and actual condition.

From the figure, the proposed method has a better effect than the cost-based FMEA in predicting the infant failure risk (expected infant failure costs). The root mean square of the proposed method curve compared with the real cost curve is 36.4540, and the closeness degree of the cost-based FMEA curve compared with the real costs curve is 72.6348, considering that the real costs are high because several human or accidental conditions cause the infant failure. Moreover, the infant failure risk is underestimated using the cost-based FMEA because (1) infant failure is analyzed subjectively or using failure data directly without considering the effects of quality variations and (2) components are analyzed independently without considering failure dependency. Thus, the proposed method can model and estimate the infant failure risk accurately by simultaneously considering these two issues.

Another way to analyze the infant failure risk is to rank the criticality of the components based on the simulated expected cost values, where the components associated with the highest loss cost values are ranked as the most critical. Figure 8 illustrates the Pareto criticality ranking at the component level using the proposed method and the cost-based FMEA. From the figure, the top three critical components of infant failure analyzed using the proposed method are X8, X4, and X7, which account for 59.6% of the total expected cost of the EMS failure, whereas the top three critical components of infant failure analyzed using the cost-based FMEA method are X4, X3, and X5, which account for 55.3% of the total expected cost of the EMS failure. The result implies the need to implement relatively robust maintenance strategies, which are targeted at mitigating infant failures of these critical components. In this way, the total expected failure cost will be mitigated for the EMS. From the analysis of warranty data of infant failures of the EMS, 61% of the cost derives from infant failure of X8, X4, and X7, and 42% of the cost derives from X4, X3, and X5. This result implies that the proposed method has a better effect on ranking the criticality of components for infant failure risk mitigation compared with the cost-based FMEA method.

Figure 8.

Pareto analysis depicting the criticality ranking at the component level.

Conclusion

Product infant failure is the most crucial part of product quality risk that critically affects customer satisfaction. The failure is inherently caused by manufacturing quality variation and can be enlarged using failure dependency. Therefore, in this work, the formation mechanism of infant failure risk is primarily clarified, and on the basis of its mechanism, the infant failure risk is categorized into two aspects, namely, inherent and dependent risks. The inherent infant failure risk is analyzed based on the SoV by considering quality variation, whereas the dependent infant failure risk is analyzed based on a BN by considering functional failure dependency. Furthermore, a case study of an illustrative EMS is introduced to verify the applicability of the proposed method. The comparative result shows that the proposed method can model and predict the infant failure risk more accurately than the cost-based FMEA. Moreover, the Pareto criticality ranking implies that the proposed method has a better effect on ranking the criticality of components for infant failure risk mitigation.

However, this study still has some limitations that impede our future research plans. To improve the portability, two following conditions should be considered in future works:

Incorporate additional various factors, such as customer satisfaction, when considering the effect of infant failures.

Optimize the process control measures with the constraints of costs based on the results of infant failure risk modeling.

Footnotes

Handling Editor: Vesna Spasojević Brkić

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by grant 61473017 from the National Natural Science Foundation of China and a general project (no. 6140002050116HK01001) funded by the National Defense Pre-Research Foundation of China.

ORCID iD

Yihai He

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