Reliability loss–oriented product assembling quality risk modeling approach based on reliability–quality–reliability chain

Abstract

Assembling variations caused by assembling system reliability degradation are the root causes of poor-assembled product reliability, and the reliability loss is the barometer of assembling quality risks. However, few studies have integrated reliability loss with assembling quality risk analysis. Therefore, a modeling approach of assembling quality risk regarding product reliability loss is expounded in this work. First, the assembling quality risk is divided into systematic, exterior, and interior risks, and a formation mechanism of assembling quality risk is presented. Second, the fusion framework of big operational data in assembling is described, an assembling reliability–quality–reliability chain is established, and the parameters of the chain constitute assembling system reliability (R), assembling process quality (Q), and assembled product reliability (R). Third, on the basis of the reliability–quality–reliability chain, the risk priority number is adopted to quantify the assembling quality risk, which is extended by quantifying the undetectable rate of the assembling system, the occurrence possibility of process variations, and the product reliability loss. Finally, an assembling quality risk of a circuit maintenance cover of a car is conducted to validate the effectiveness and advancement of the presented approach. The result shows that the proposed method can systematically quantify the assembling quality risk.

Introduction

Reliability is the significant dimension of quality, and the decline in manufacturing quality is an important factor leading to product reliability degradation. ¹ Therefore, product reliability modeling and assurance must be integrated into the manufacturing process. ² Assembling quality is directly related to the product reliability; thus, the high quality of assembling is a prerequisite to output qualified products with high reliability. ³ Therefore, to ensure that reliable products are manufactured in batch productions with low risks, manufacturing engineers have to continuously improve the assembling quality. ⁴

The newly released ISO9001:2015 standard emphasizes the application of risk-based thinking on the quality assurance of products, thereby making integrated risk management an unavoidable trend. ⁵ Therefore, to reduce potential assembling risk with regard to product reliability, assembling quality should be controlled with the aid of risk thinking. During the assembling phase, the transmission and accumulation of process variations are the main factors leading to the product reliability degradation,^6,7 but the focus should limit not only the assembling operation process but also the uncertainty of the upstream assembling system and the downstream assembled products. Currently, an analysis model that combines risk analysis and assembling quality has not received sufficient attention, and few studies and applicable methods are available regarding the integration of risk analysis and assembling quality control. ⁸

Currently, quality risk has two main definitions: cause-oriented and result-oriented thinking. Cause-oriented thinking considers quality risk as the cause of quality decline; ⁹ however, result-oriented thinking considers quality risk as the potential result of quality variations. ¹⁰ Correspondingly, assembling quality risk modeling has two main ideas. For cause-oriented modeling, Lee et al. ¹¹ proposed a method combining the Petri net framework and simulation technology to conduct risk modeling and assessment for distributed manufacturing systems. Liu et al. ¹² proposed a method based on comprehensive failure mode and affect analysis to evaluate the risk under uncertain conditions to improve the reliability and quality. But the capability to identify and analyze adverse factors of these methods is bound to decline with the gradual improvement of process technology until it is out of line with the production of modern enterprises. ¹³ For the result-oriented method, Thornton and colleagues^14,15 proposed a risk-oriented management approach called variation risk management for quantifying the assembling quality problem. Zhang et al. ¹⁶ used the positive and negative aspects of the quality variation analysis method to innovate quality risk analysis technology and to suggest a new mass loss function. He et al. ⁴ proposed a quantitative risk method in consideration of product reliability by analyzing the low quality in assembling processes. Based on the cost of quality risk, Mo et al. ¹⁷ sort the process errors that lead to quality risks and to determine the priority of risk control to reduce the quality risk of the composite fiber–reinforced plastic assembly process.

For the assembling process, process variations are the main sources of assembling quality risks. The studies related to the impact of process variations on assembling quality include the following. Jin and Shi ¹⁸ developed an effective variation transfer model for multistage assembling processes. Huang et al. ¹⁹ proposed a stream of variation (SOV) model and a dimensional displacement model, wherein the variation control and prediction method based on SOV theory improved the automotive assembling reliability. Wang et al. ²⁰ proposed a positioning variation model of aircraft panel assembly based on elastic deformation theory to improve the aerodynamic performance of the aircraft. Mei et al. ²¹ proposed an assembly variation modeling and analysis method based on elasticity mechanics and interval method to solve the problem of insufficient information available in the variation source. However, most of the current literature considers the adverse effect of assembling variations on assembling process quality only; the impact of assembling process quality on the reliability of the final product is still ignored. ²² For the error compensation of thin-walled workpieces, Si and Wang ²³ proposed a research method of error compensation to improve the reliability of thin-walled products. Jiang et al. ²⁴ considered the influence of workpiece variation accumulation on the current process. Based on the analysis result of machining variation, a workpiece positioning adjustment method under multi-tolerance conditions is proposed to reduce the adverse effect of variation on process quality. Research attention is given to assembling process variations only; assembling system failures and product manufacturing defects are not being considered. This condition is not conducive to prevent assembling quality risk from the source. In summary, a precondition required for comprehensively solving the problem of product reliability loss caused by poor-assembling quality is the establishment of a quantitative relationship between assembling system failure, assembling process variation, product manufacturing defects, and assembling quality risk.

Therefore, a reliability loss–oriented assembling quality risk analysis model based on a reverse assembling reliability–quality–reliability (RQR) chain is developed, and the main contributions are as follows:

1. Based on the relationship between assembling system, assembling process, and assembled product, the connotation and formation mechanism of assembly quality risk are expounded from the perspective of risk origin, transmission, and influence.

2. The quantitative metric and fusion method of the big operational data of assembling for risk quantification is explained, and the assembly RQR chain is proposed.

3. The risk priority number (RPN) is extended by analyzing big operational data of assembling to synthesize risk.

The remainder of this article is organized as follows. The connotation and formation mechanism of assembling quality risk are expounded in section “Analysis basics of assembling quality risk.” The data fusion framework and assembling RQR chain is proposed in section “Assembling RQR chain based on big operational data of assembling.” The RPN extended to quantify the assembling quality risk is given in section “Assembling quality risk modeling based on extended RPN,” and a case study is provided in section “Case study.” Finally, conclusions and directions for the future research are given in section “Conclusion.”

Analysis basics of assembling quality risk

Connotation of assembling quality risk

According to the systems’ engineering principle, the assembling stage comprises three main management objects: assembling system, assembling process, and assembled product. The three objects can affect each other, and any variation or error in the management objects will lead to an increase in product assembling quality risk. The connotation of assembling quality risk is shown in Figure 1.

Figure 1.

Connotation of product assembling quality risk.

As shown in Figure 1, the top level reflects the interrelationships between the three management objects in assembling phase, and the three latter levels expound the causes and control ideas of assembling quality risk from top to bottom. As shown in the first level, the assembling system, assembling process, and assembled product can affect each other. Meanwhile, the second and third levels show that when the assembling system fails, system reliability is degraded, and a systemic risk is formed. When the equipment begins to operate, the systemic risk affects the assembling process, resulting in the process quality decline and the occurrence of key process variable (KPV) assembling variations, and the exterior risk is formed. This risk is the main manifestation and the most important component of the assembling quality risk in the assembling stage. It can be viewed as a narrow version of assembling quality risk. With the operating multi-station assembling process, variations are gradually transmitted and accumulated, and the exterior risk gradually enlarges and acts on the assembled products, resulting in production defects and the formation of interior risks. The three risks work together to constitute assembling quality risk, which ultimately leads to product reliability loss. Therefore, assembling quality risk originates from the assembling system, appears in the assembling process, and acts on the assembled product.

Formation mechanism of assembling quality risk

Product assembling quality risk is an integrated risk. The formation mechanism is shown in Figure 2.

Figure 2.

Formation mechanism of product assembling process quality risk.

As shown in the left part of Figure 2, in the actual engineering operation, the influence of external stresses often leads to assembling equipment defects, which cause assembling system reliability degradation. The existence of such defects will affect the assembling process quality and lead to process variations. This risk is caused by assembling system defects and is called systemic risk, from which assembling quality risk originates. However, identifying systemic risk in an actual assembling stage is difficult. As shown in the middle part of the figure, systemic risk leads to assembling variations, and with the proceeding multi-station assembling process, the previous station process variations will continue to be transmitted downward. Meanwhile, the new variations in the next station produce and cause accumulated variations. According to axiomatic domain mapping, the key quality characteristics (KQCs) can be mapped to the KPVs. Once the KPV variations occur, they will gradually accumulate, resulting in KQC variations and creating exterior risk. The process variation can be discovered and the exterior risk can be controlled by reducing the process variation. As shown in the right part of the figure, after the assembling process, the assembled product is formed. The exterior risk will affect the final product reliability; specifically, the actual reliability of the product is declined compared with the design process. Then, product internal defects occur, and interior risk forms. However, the interior risk can be discovered only after the product has been assembled, which causes substantial economic losses to the company. Among the risks, the systemic risk is the hardest to control because of its fuzziness. Exterior risk is the kernel of the assembling quality risk and easy to control. Meanwhile, any attempt to control interior risk is futile because by then, product reliability can no longer be improved, and large economic losses will be incurred. Thus, exterior risk is most valuable in the control of assembling quality risk.

Assembling RQR chain based on big operational data of assembling

The fusion framework of big operational data in assembling

With the continuous development of intelligent manufacturing, sensors for process data acquisition and processing are becoming increasingly intelligent, which could help manufacturers collect more data for analysis easily.

The big operational data of the assembling collected by smart sensors reflect the origin, formulation, and propagation of assembling quality risk. Therefore, the big operational data of assembling process can be integrated and analyzed to quantify the risk, and the risk analysis results can be used to find the weak links in the assembling stage, which provides a possible means to control the assembling quality risk and makes the proactive control feasible. The process of using the big operational data of assembling is shown in Figure 3.

Figure 3.

The flow chart of big operational data fusion in assembling.

As shown in this figure, the use process of big data can be divided into four stages, namely, data acquisition, data transmission, data filtering, and data fusion and integration: First, collection of data includes multiple data types such as equipment, process, and product data. Second, the data are transmitted through industrial Ethernet. Third, based on the type of data needed by the assembling RQR chain, the big data are filtered. For this article, the required data mainly include the degradation data of the assembling system (such as the clamping force data of the fixture and the positioning pin positioning data) and the assembling process quality data (such as the assembling process variation data of work in process (WIP)). Then, data fusion and integration are needed to use complex data for risk analysis and quantification. Finally, based on the analysis result, the risk control strategy is formulated and implemented from the perspectives of equipment maintenance and process control to achieve continuous reduction of assembling quality risk.

Assembling RQR chain

The big operational data of the assembling could provide solid data support for risk assessment algorithms, and thus, a bidirectional chain structure assembling RQR chain is proposed with the aid of the collected data, and its conceptual model is shown in Figure 4.

Figure 4.

Conceptual model of assembling RQR chain.

As shown in the upper portion of Figure 4, the assembling system, the assembling process, and the assembled product can affect each other during the multi-station assembling. To quantify the assembling quality risk by using the big operational data of assembling, three quantifiable objects are proposed: assembling system reliability (R), assembly process quality (Q), and assembly product reliability (R). The three objects constitute the assembling RQR chain.

As indicated in Figure 4, the left-to-right positive assembling RQR chain is a result-oriented risk transfer chain that describes the process of assembling quality risk creation, transmission, and presentation. The effect of external factors leads to assembling quality risk, and then, it is transmitted along the forward assembling RQR chain; the reverse RQR chain (right to left) is a cause-oriented risk-tracking chain that provides solutions for risk quantification and control. When product reliability loss occurs, it can be traced according to the reverse assembling RQR chain for risk traceability.

Assembling quality risk modeling based on extended RPN

Modeling framework of assembling quality risk

The analysis framework that is based on an extension of RPN is proposed for assembling quality risk. This framework is shown in Figure 5 and includes risk identification, quantification, and ranking.

Figure 5.

Framework of product assembling quality risk quantification based on RPN.

• Step 1. Risk identification

Collect real-time assembling data with smart sensors to build data foundation. Then, establish assembling RQR chain and identify the risk source.

• Step 2. Risk quantification

Considering the impact of actual data on the objectivity of risk quantification, the RPN method is extended to quantify product assembling quality risk in this study. The parameters are quantified by the big operational data of assembling. (The detail is discussed in section “Computation of RPN.”).

• Step 3. Risk ranking

According to the value of $S,O,\mathrm{and}D$ , the assembling quality risk value $R_A$ is synthesized. Then, the risk rating table is developed by engineering experience and literature, and the assembling quality risk is ranked. The risk level can be obtained.

Computation of RPN

Combined with the big operational data and the mechanism of assembling quality risk, the extended RPN calculation framework is proposed and shown in Figure 6.

Figure 6.

Extended RPN calculation framework.

As shown in Figure 6, starting from the three management objects of the RQR chain, the traditional RPN value is extended to determine the value of the parameters $S,O,\mathrm{and}D$ . The undetectable rate D of the risk is linked to assembling system reliability, the occurrence O of the risk is linked to assembling process quality, and the consequence severity S of the risk is linked to assembled product reliability. Then, product assembling quality risk can be quantified in equation (1)

$$R_A=S\times O\times D$$

(1)

where $R_A$ is the assembling quality risk and $S,O,\mathrm{and}D$ indicates the consequence severity, occurrence probability, and undetectable rate of assembling quality risk, respectively.

The novel calculation methods of $S,O,\mathrm{and}D$ are as follows.

Severity S computation based on reliability loss

The difference between the actual product reliability and the design reliability is called as reliability loss and results from product assembling quality risk. The most significant component of product reliability loss is product assembling defect. Therefore, the consequence severity S of the assembling quality risk is determined by the assembled product reliability $R_P$ . Specifically, the direct cause of the product reliability loss is the assembling process variation, and the amount of the reliability loss is determined by the KPV variation amount in each station and the station weight. Therefore, S can be quantified by a comprehensive measurement of the KPV variation amount in each station and station weight

$$S=\mathrm{RL}=\sum _{i=1}^n{K}_i{Y}_i$$

(2)

where $\mathrm{RL}$ is the product reliability loss, $K_i$ is the weight coefficient, and $Y_i$ is the KPV variation on the ith measurement point on each station. $Y_i$ consists of two parts: the KPV variation on the ith station $V_i$ and the measuring KPV variation on the ith station $Z_i$ . Therefore

$$Y_i=D_i·V_i+Z_i$$

(3)

where the measurement matrix $D_i$ is determined by the measuring point placement information, that is, the number and location of the measuring points at station i. $V$ is the vector sum of the vectors $X,U,\mathrm{and}W$ multiplied by the corresponding coefficient matrix, where $X(j-1)$ is the state vector produced by the $(j-1)$ th station, $U(j)$ is the input vector produced by the jth station, and $W(j)$ is a random vector produced by the j th station, respectively. $V$ can be expressed in equation (4)

$$V=A(j-1)X(j-1)+B(j)U(j)+W(j)$$

(4)

where $A(j)$ is defined to represent the coordinate system transformation from the $(j-1)$ th station to the j station, and $B(j)$ is the input matrix that shows the effect of fixture variation on positioning variation. They can be expressed as follows

$$A_i(j)=\left[\begin{array}{cccc}\cos \mathrm{\Delta }{\delta }_i(j)\hfill & 0\hfill & \sin \mathrm{\Delta }{\delta }_i(j)\hfill & \mathrm{\Delta }{x}_i(j)\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-\sin \mathrm{\Delta }{\delta }_i(j)}& 0\hfill & \cos \mathrm{\Delta }{\delta }_i(j)\hfill & \mathrm{\Delta }{z}_i(j)\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]=\left[\begin{array}{cc}{R}_i(j)\hfill & T_i(j)\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right]$$

(5)

where $R_{\mathrm{ij}}$ is a $3\times 3$ rotation matrix that represents the coordinate system transformation from the $(j-1)$ th station to the jth station; $T_{\mathrm{ij}}$ is a $3\times 1$ translational vector that represents the coordinate system transformation from the $(j-1)$ th station to the jth station

$$B(j)=\left[\begin{array}{cccc}{M}_{A_1,P_1}{Q}_{P_1,P_2}(j)\hfill & 0^{3\times 4}\hfill & \cdots \hfill & 0^{3\times 4}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{M_{A_1,P_1}{Q}_{P_1,P_2}(j)}& 0^{3\times 4}\hfill & \cdots \hfill & 0^{3\times 4}\hfill \\ \multicolumn{1}{c}{0^{3\times 4}}& S_{t+1}(j)\hfill & \cdots \hfill & 0^{3\times 4}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{0^{3\times 4}}& 0^{3\times 4}\hfill & \cdots \hfill & S_{t+s}(j)\hfill \end{array}\right]$$

(6)

where $P_1\mathrm{and}{P}_2$ are the four-way and the two-way positioning pins of each part among each component, where $i=t+1,t+2,...t+s$ . The expressions of the matrices $Q_{P_1,P_2},S_{t+i},\mathrm{and}{M}_{A,P}$ are as follows

$$Q_{P_1,P_2}(j)=\left[\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\frac{\sin \beta }{L\cos (\alpha -\beta )}}& \frac{-\cos \beta }{L\cos (\alpha -\beta )}\hfill & \frac{-\sin \beta }{L\cos (\alpha -\beta )}\hfill & \frac{\cos \beta }{L\cos (\alpha -\beta )}\hfill \end{array}\right]$$

(7)

$$S_{t+i}(j)=M_{A_{t+i},P_1^{t+i}}{Q_{P_1^{t+i},}}_{P_2^{t+i}}(j)$$

(8)

$$M_{A,P}=\left[\begin{array}{ccc}1\hfill & 0\hfill & l_z(A,P)\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & -l_x(A,P)\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right]$$

(9)

where $L$ is the distance between the two positioning pins $P_1\mathrm{and}{P}_2$ on the fixture, $\alpha$ is the angle between the two positioning pins and the x-axis, and $\beta$ is the angle between the long axis of the two-way positioning slot and the x-axis.

After the aforementioned process, $Y$ is obtained and equation (3) can be rewritten as equation (10) to clearly describe the relationship between the input variation of each station and the final variation after the assembling process is completed

$$Y=\mathrm{TU}+\Psi$$

(10)

where

$$\Psi =\left(\begin{array}{cccc}{D}_1\hfill & 0\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{D_2·{\mathbf{\Phi }}_{2,1}}& D_2\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{D_n·{\mathbf{\Phi }}_{n,1}}& D_n·{\mathbf{\Phi }}_{n,2}\hfill & \cdots \hfill & D_n\hfill \end{array}\right)·\left(\begin{array}{c}{W}_1\hfill \\ \multicolumn{1}{c}{W_2}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{W_n}\end{array}\right)+\left(\begin{array}{c}{V}_1\hfill \\ \multicolumn{1}{c}{V_2}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{V_n}\end{array}\right)$$

(11)

$$T=\left(\begin{array}{cccc}{D}_1·B_1\hfill & 0\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{D_2·{\mathbf{\Phi }}_{2,1}·B_1}& D_2·B_2\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{D_n·{\mathbf{\Phi }}_{n,1}·B_1}& D_n·{\mathbf{\Phi }}_{n,2}·B_2\hfill & \cdots \hfill & D_n·B_n\hfill \end{array}\right)$$

(12)

$${\mathbf{\Phi }}_{i,j}=\{\begin{array}{c}{\mathbf{A}}_{i-1}·{\mathbf{A}}_{i-2}·\dots ·{\mathbf{A}}_j,i>j\hfill \\ \multicolumn{1}{c}{\mathbf{I},i=j}\end{array};\mathbf{Y}=\left(\begin{array}{c}{Y}_1\hfill \\ \multicolumn{1}{c}{Y_2}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{Y_n}\end{array}\right);\mathbf{U}=\left(\begin{array}{c}{U}_1\hfill \\ \multicolumn{1}{c}{U_2}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{U_n}\end{array}\right)$$

(13)

Then, the analytic hierarchy process is used to determine the weight of each station k.

On the basis of the axiomatic domain, the $B_1$ user layer criteria, the $B_2$ functional layer criteria, and the $B_3$ structure layer criteria are determined and a judgment matrix $P$ is established. Then, a judgment matrix is established for the standard layer. The weight coefficient matrix is calculated for each station. Finally, the coefficient matrix $K$ is constructed to facilitate calculation.

Occurrence O calculation based on frequency of process variation

The reverse assembled RQR chain is traced back to the product assembling process, the low quality of which is the main reason for the reliability loss of downstream products. Specifically, when the assembling process quality $Q$ decreases, different degrees of process variable variations occur. Relatively large degrees of KPV variations will occur with certain probability laws, and these types of variations will result in product reliability loss. Therefore, the relatively large assembling KPV variation is the direct cause of product reliability loss, and assembling quality risk will also be generated with the same probability law. Thereby, the occurrence probability $O$ of risk is related to the assembling process quality $Q$ , and the amount of the occurrence $O$ is determined by the occurrence probability of the relatively large assembling variation of the KPVs. To satisfy the relationship, the following steps are conducted.

First, monitor KPV changes at each state in real time. Then, the minimum KPV variations (modulus of KPV variation threshold vector, hereafter called threshold) on each station that will lead to product reliability loss are determined via a consultation with experts’ experiences and referral to historical data of quality risk accidents.

Second, the probability of the variations is quantified. Given that the occurrence of KPV assembling process variations on each station is random and each KPV is independent, the occurrence probability of variation is subject to the Poisson distribution. Then, according to the KPV variation threshold condition of each station, the Poisson distribution is used in following equation (14)

$$o_i(\Vert Y_i\Vert \ge t_i)=\sum _{k=1}^{n}P(X=k)=\sum _{k=1}^n\frac{{\lambda }^k}{k!}{e}^{-\lambda }$$

(14)

where $o_i(\Vert Y_i\Vert \ge t_i)$ is the occurrence probability of the modulus of KPV variation vector not less than the threshold (recorded as event $\mathbf{\Omega }$ ) on the ith station, $\Vert Y_i\Vert$ is the modulus of KPV variation vector on the ith measurement point on each station, t_i is the KPV variation threshold on the ith measurement point on each station, $X$ is the number of occurrences of $\mathbf{\Omega }$ events, and $\lambda$ is the expectation and variance of the Poisson distribution.

Finally, the modulus of KPV variation vector not less than the threshold generated during the entire assembling process is recorded as $\mathbf{\Sigma }$ event as long as it is in one station. Since, as long as it is one station, $\mathbf{\Omega }$ event occurs when $\mathbf{\Sigma }$ event occurs. That is, if $\mathbf{\Omega }$ event does not occur, then $\mathbf{\Sigma }$ event does not occur. The relationship is satisfied in the following equation (15)

$$O=o=1-\underset{i-0}{\overset{n}{\Pi }}\overline{o_i}(\Vert Y_i\Vert \ge t_i)$$

(15)

where $O$ is the occurrence probability of assembling quality risk, $o$ is the occurrence probability of the $\mathbf{\Sigma }$ event, and $\overline{o_i}(\Vert Y_i\Vert \ge t_i)$ is the occurrence probability of $\mathbf{\Omega }$ event that does not occur.

Undetectable rate D assessment based on average running length

Relying on the RQR chain reverse operation mechanism, the risk is traced further. The source of assembling quality risk is assembling system reliability degradation. Specifically, when the assembling system reliability $R_S$ decreases, the operating state of the device becomes unstable, which leads to assembling process quality decline. Specifically, the detectable rate of the assembling system reliability degradation can represent the detectable rate of the assembling quality risk. When the detectable rate of $R_S$ is at a low level, the KPV cannot be monitored in time and the assembling system cannot operate normally, thereby leading to risks. In other words, the longer the uptime of the assembling system, the bigger the amount of the detectable rate of $R_S$ and the higher the detectable rate of the product assembling quality risk.

As mentioned in section “Occurrence O calculation based on frequency of process variation,” the KPV variations in the list are monitored in real time. Once the variation threshold is exceeded, alarm can be issued. That is, as long as an $\mathbf{\Omega }$ event occurs, the assembling system deviates from the normal operating state. Therefore, the $\mathbf{\Omega }$ event is monitored using the control chart (called the $\mathbf{\Omega }$ -event control chart), where the average running length (ARL) indicates the controlled working time. Therefore, the longer the ARL of the $\mathbf{\Omega }$ -event control chart, the longer the normal operation of the process; that is, the longer the ARL of the $\mathbf{\Omega }$ -event control chart, the higher the detectable rate of the risk. Therefore, the ARL of the $\mathbf{\Omega }$ -event control chart is used to characterize the detectable rate D′. In the risk calculation process, the undetectable rate is needed. Thus, the definition of the undetectable rate D is

$$D=\frac{1}{D\prime }$$

(16)

$$D\prime =\mathrm{AR}{L}_{\mathbf{\Omega }}$$

(17)

where $\mathrm{AR}{L}_{\mathbf{\Omega }}$ is the $\mathrm{ARL}$ of the $\mathbf{\Omega }$ -event control chart.

This study considers the bilateral control limits, and the control chart is divided into five areas: the target area $T$ , the warning areas $W-$ and $W+$ , and the action areas $A-$ and $A+$ , respectively

$$\mathrm{Pr}(\overline{X}\in T)=p,\mathrm{Pr}(\overline{X}\in W+)=q_1,\mathrm{Pr}(\overline{X}\in W-)=q_2,\mathrm{Pr}(\overline{X}\in ((A+)\cup (A-)))=r$$

where $\overline{X}$ is the sample mean. Given that $\overline{X}~N(\mu ,{\sigma }^2/n)$

$$P=\Phi (B_2-\sqrt{n}\delta )+\Phi (B_2+\sqrt{n}\delta )-1$$

(18)

$$q_1=\Phi (B_1-\sqrt{n}\delta )-\Phi (B_2-\sqrt{n}\delta )$$

(19)

$$q_2=\Phi (B_1+\sqrt{n}\delta )-\Phi (B_2+\sqrt{n}\delta )$$

(20)

$$r=1-p-q_1-q_2$$

(21)

$$\delta =\frac{\mu -{\mu }_0}{\sigma }$$

(22)

The ARL calculation formula is as follows

$$\mathrm{AR}{L}_{\mathbf{\Omega }}=\frac{(1+q_1)(1+{\mathrm{q}}_2)}{1-q_1{q}_2-p(1+q_1)(1+{\mathrm{q}}_2)}$$

(23)

Case study

Background

In Chinese automobile manufacturing companies, product quality accidents frequently occur on the front cover of a car, which reduces the reliability of the car and results in substantial economic losses. Companies intend to adopt quality control methods in the assembling process to decrease the assembling quality risk and thus improve product reliability.

After a preliminary analysis of a company, technicians discover that quality accidents are related to the failure mechanism of the maintenance cover inside the front cover. In engineering practice, daily maintenance and vibration activities cause lid edge wear, and variations in its size during assembling accelerate wear, resulting in a gap between the lid and its corresponding counterpart clearance groove, causing dust accumulation in internal control circuit and resulting in malfunction.

The approach proposed in this study is used to identify and quantify the assembling quality risk of circuit maintenance cover. The specific assembling process design in this case study and the distribution of measurement points for each station are shown in Figure 7. Four components are assembled at stations I and II, and the final assembling is checked for surface finish, joint quality, and dimensional defects in station III.

Figure 7.

Assembling process and measurement point diagram.

Numerical examples

In this section, the proposed approach is verified with the circuit maintenance cover of a car.

Completing the risk source determination, the data results are shown in Table 1.

Table 1.

Initial coordinates of fixture pins and measurement points (unit: mm).

Category	Number	Coordinate	Number	Coordinate
Fixture pin	P 1	$(166.00,120.00)$	P 5	$(1066.00,400.90)$
	P 2	$(446.00,121.70)$	P 6	$(1300.00,401.00)$
	P 3	$(663.50,120.60)$	P 7	$(1066.00,150.00)$
	P 4	$(865.10,123.50)$	P 8	$(1300.50,150.50)$
Measurement point	M 1	$(10.00,400.00)$	M 4	$(1165.92,399.58)$
	M 2	$(612.50,600.71)$	M 5	$(9.87,401.21)$
	M 3	$(10.26,395.67)$	M 6	$(1164.39,124.55)$

Three stations comprise during the assembling process. Each station has two measurement points, and each point is represented by $M_i$ . The assembling process variation of each component is calculated by the SOV model.

After calculation

$$T=\left(\begin{array}{cccccccccccccccccc}0.956\hfill & 0.628\hfill & -0.628\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1.021\hfill & -1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1.05\hfill & 1.19\hfill & -1.1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0.96\hfill & 0.872\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0.519\hfill & -0.519\hfill & 0\hfill & 0\hfill & 1.268\hfill & 1\hfill & 1.268\hfill & -1.268\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0.039\hfill & -1\hfill & 0\hfill & 0\hfill & 0.47\hfill & 0\hfill & 0.47\hfill & -0.47\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 1.74\hfill & -1.74\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & -0.8\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0.519\hfill & -0.519\hfill & 0\hfill & 0\hfill & 1.268\hfill & 0\hfill & 0.853\hfill & -1.268\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0.759\hfill & 1\hfill & 0.759\hfill & -0.759\hfill \\ \multicolumn{1}{c}{0}& 0.039\hfill & -1\hfill & 0\hfill & 0\hfill & 0.47\hfill & 0\hfill & 0.753\hfill & -0.47\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0.52\hfill & 0\hfill & 1.05\hfill & -0.52\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & -1\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 1.76\hfill & -1.521\hfill & 1\hfill & 1.527\hfill & -1.527\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0.72\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & -1.26\hfill & 0.36\hfill & 0\hfill & -0.369\hfill & 1.369\hfill \end{array}\right)$$

$$\begin{array}{c}{Y}_1={\left[\begin{array}{cccccccccccc}1.215\hfill & 1.326\hfill & 0.652\hfill & 0.796\hfill & 0.783\hfill & 0.676\hfill & 0.213\hfill & 0.266\hfill & 0.325\hfill & 0.347\hfill & 0.268\hfill & 0.193\hfill \end{array}\right]}^T\hfill \\ Y_2={\left[\begin{array}{cccccccccccc}1.333\hfill & 1.375\hfill & 0.986\hfill & 0.856\hfill & 0.898\hfill & 0.774\hfill & 0.286\hfill & 0.267\hfill & 0.435\hfill & 0.371\hfill & 0.349\hfill & 0.217\hfill \end{array}\right]}^T\hfill \\ Y_3={\left[\begin{array}{cccccccccccc}1.756\hfill & 2.005\hfill & 1.032\hfill & 0.998\hfill & 0.963\hfill & 0.871\hfill & 0.302\hfill & 0.290\hfill & 0.501\hfill & 0.703\hfill & 0.397\hfill & 0.226\hfill \end{array}\right]}^T\hfill \end{array}$$

$$P=\left[\begin{array}{ccc}1\hfill & 3\hfill & 5\hfill \\ \multicolumn{1}{c}{\frac{1}{3}}& 1\hfill & 3\hfill \\ \multicolumn{1}{c}{\frac{1}{5}}& \frac{1}{3}\hfill & 1\hfill \end{array}\right]$$

$$k_1=0.75,k_2=1.23,\mathrm{and}{k}_3=0.98$$

The weight matrix is constructed from the variation vector

$$\begin{array}{c}{K}_1=\left[\begin{array}{cccccccccccc}0.75\hfill & 0.75\hfill & 0.75\hfill & 0.75\hfill & 0.75\hfill & 0.75\hfill & 0.75\hfill & 0.75\hfill & 0.75\hfill & 0.75\hfill & 0.75\hfill & 0.75\hfill \end{array}\right]\hfill \\ K_2=\left[\begin{array}{cccccccccccc}1.23\hfill & 1.23\hfill & 1.23\hfill & 1.23\hfill & 1.23\hfill & 1.23\hfill & 1.23\hfill & 1.23\hfill & 1.23\hfill & 1.23\hfill & 1.23\hfill & 1.23\hfill \end{array}\right]\hfill \\ K_3=\left[\begin{array}{cccccccccccc}0.98\hfill & 0.98\hfill & 0.98\hfill & 0.98\hfill & 0.98\hfill & 0.98\hfill & 0.98\hfill & 0.98\hfill & 0.98\hfill & 0.98\hfill & 0.98\hfill & 0.98\hfill \end{array}\right]\hfill \end{array}$$

The severity of the consequences of each station S is calculated

$$\begin{array}{c}{S}_1=R{L}_1=5.295,S_2=R{L}_2=10.021,S_3=R{L}_3=9.843\hfill \\ S=S_1+S_2+S_3=25.159\hfill \end{array}$$

According to the experts’ experiences and historical data, Table 2 is obtained.

Table 2.

Improved KPV variation threshold list.

Station number	Variation threshold
Station 1	$2.212$
Station 2	$2.509$
Station 3	$2.618$

According to the relevant information and data, $\lambda =0.56$ , then

$$\begin{array}{c}{o}_i(Y_i\ge t_i)=\sum _{k=1}^{n}P(X=k)=\sum _{k=1}^n\frac{{\lambda }^k}{k!}{e}^{-0.56}\hfill \\ o_1=o_2=o_3=\mathrm{EX}=\lambda =0.56\hfill \\ O=o=1-\underset{i=1}{\overset{3}{\Pi }}\overline{o_i}(\Vert Y_i\Vert \ge t_i)=0.824\hfill \end{array}$$

Finally, according to the historical data, the values of $p,q_1,\mathrm{and}{q}_2$ are obtained

$$p=0.996,q_1=0.001,\mathrm{and}{q}_2=0.001$$

Therefore

$$\begin{array}{c}D\prime =\mathrm{AR}{L}_{\mathbf{\Omega }}=499.5\hfill \\ D=\frac{1}{D\prime }=0.002\hfill \end{array}$$

Thus

$$R_A=S\times O\times D=0.0414$$

According to the engineering data and experts’ experiences, the risk ranking table is obtained, as shown in Table 3.

Table 3.

Assembling quality risk ranking table.

Risk level	Risk value
High	$\mathrm{More}\mathrm{than}0.05$
Middle	$0.005-0.05$
Low	$0-0.005$

$$0.005<R_A<0.05$$

Therefore, the risk is at middle risk level.

Comparative study

The traditional RPN method is not objective in terms of parameter value. Therefore, the extended RPN method is compared with the traditional RPN method.

In the traditional RPN method, $S,O,\mathrm{and}D$ of risk are divided into 10 different levels, represented by the Arabic numerals $1-10$ . Subjective estimation methods and experts’ experiences are used in determining the values of $S,O,\mathrm{and}D$ . The value-taking strategy has serious subjectivity. The value of the parameter cannot be measured by effective process data, and each parameter has a value range of only 10 intermittent numbers instead of a continuous range, resulting in the reduction of risk quantification result and the risk situation cannot be objectively reflected. However, in the extended RPN method, the aforementioned problems are effectively alleviated. In this work, the big operational data of assembling are fused, analyzed, and applied to the data results of the RPN method. The values of the three parameters are combined with the specific case. In the following section, the superiority of the method is indicated by a specific case.

Assume that there are two maintenance covers: a and b, and the two different RPN methods are used to determine the value of $S,O,\mathrm{and}D$ for comparison.

First, the traditional RPN method is used to determine the parameters of assembling quality risk of two covers. The values of $S_a,S_b,O_a,O_b,D_a,\mathrm{and}{D}_b$ are determined by expert scoring, and assembling quality risks of a and b are calculated. Their $S,O,\mathrm{and}D$ values have subtle differences, but the parameters of the two covers are at the same level, that is

$$S_a=S_b,O_a=O_b,D_a=D_b$$

The proposed RPN method is then used to determine the parameters of two covers by analyzing the big operational data of assembling. The $S_a,S_b,O_a,O_b,D_a,\mathrm{and}{D}_b$ are determined. The assembling quality risks of a and b are calculated.

Finally, the results obtained by the two methods are compared, as shown in Table 4.

Table 4.

Comparison of two methods’ risk calculation results table.

Case	Maintenance cover	S	O	D	RA	Comparative analysis	Risk level
1	$a$	$6$	$8$	$4$	$192$	$R_{\mathrm{Aa}}=R_{\mathrm{Ab}}$	Middle
	$b$	$6$	$8$	$4$	$192$		Middle
2	$a$	$21.06$	$0.822$	$0.0029$	$0.0502$	$R_{\mathrm{Aa}}<R_{\mathrm{Ab}}$	High
	$b$	$23.97$	$0.825$	$0.0023$	$0.0455$		Middle

When the traditional RPN quantifies the risk, the two covers’ risks are same and at middle level. Thus, the decision in actual engineering process is to implement middle-level risk prevention and control measures for both covers. When the risk is quantified by the extended RPN, the risk value of cover a is large and at the high risk level and the risk value of cover b is small and at the middle risk level. The decision in actual engineering process is to implement high risk prevention and control measures for a and middle risk prevention and control measures for b.

According to the aforementioned results, the proposed RPN is practical in engineering practice and has far-reaching significance for risk control.

Conclusion

This study considers product reliability loss, adopts risk thinking to deal with assembling quality, and presents the concept of product assembling quality risks. The use of the concepts of systematic, exterior, and interior risks from the assembling system, assembling process, and assembled product, respectively, is proposed for describing the connotation and formation mechanism of the assembling quality risk. Correspondingly, on the basis of the relationship between assembling system R, assembling process Q, and assembled product R, an assembling RQR chain is proposed. First, the assembling quality risk source is identified, and the traditional RPN is extended according to the assembling RQR chain; second, the values of the parameters $S,O,\mathrm{and}D$ are combined with the specific assembling process, and the risk results are ranked; finally, the proposed model is verified in the case of a Chinese automobile manufacturer. The analysis results show that, compared with the traditional RPN method, the proposed method provides more accurate and favorable analysis results for product assembling quality risk analysis and improvement and is accomplished to determine necessary quality control measures and quality improvement activities.

The following aspects are possible directions in the future work:

1. The contribution size of the three parameters in the extended RPN to the risk of assembling quality can be further explored.

2. According to the reverse operation mechanism of the assembling RQR chain, risk control measures can be researched.