Product reliability–oriented optimization design of time-between-events control chart system for high-quality manufacturing processes

Abstract

In high-quality manufacturing processes, the yield is no more the only issue, and the quality control approach that also considers the product’s actual performance gradually becomes the focus. Reliability is a critical dimension of quality, and its degradation in usage is always determined by the manufacturing quality. To mitigate the degradation of reliability, this article presents a product reliability–oriented optimization design of the time-between-events control chart system, where the quantitative impact of process quality on the product reliability is analyzed. First, critical-to-reliability process parameters are identified, and a product reliability degradation model is proposed considering the effect of process variations and defects. Second, the observed event and the used statistics are determined to prearrange the time-between-events chart system. Third, all individual time-between-events charts are systematically optimized to minimize the expected batch product reliability degradation caused by potential process shifts, where the cost and statistical performance are considered as constraints. The result of the case study shows that the proposed design can reduce 27.02% of batch product reliability degradation due to manufacturing process quality, and this model can also save the operating cost on the basis of attaining minimum reliability degradation in certain situations.

Introduction

To maintain competitive edge, manufacturers must strive for high product quality and reliability regarding increasing customer requirements.^1,2 Nowadays, a high-quality manufacturing process with low defective percentage and process shift rate can be achieved more easily than decades ago.^3,4 In this background, the focus of quality control cannot be restricted only to the yield anymore. The actual reliability is severely dependent on the manufacturing quality and the effectiveness of process control.^5,6 Thus, enhancing the product reliability is a worthy improvement direction in process control. In addition, the newest ISO 9001:2015 standard emphatically advocates the integration of the quality management system with the management of other important product characteristics, ⁷ which implies that reliability assurance should be coordinated with traditional quality control activities.^6,8

Many studies have shed light on the correlativity between the manufacturing quality and the product reliability. Jiang and Murthy ⁹ discussed the impacts of component non-conformance and assembly error on the product reliability based on Weibull distribution. Bebbington et al. ¹⁰ used polynomial functions to establish various product hazard rate models. Jin et al. ¹¹ applied the Six Sigma methodology and presented a closed-loop control framework for estimating infant failures of products. In the field of semiconductor manufacturing, Roesch ¹² discussed the connotations of yield, quality, and reliability and then proposed a new bathtub curve that measures fallout to predict batch reliability. Kim ¹³ focused on the effect of manufacturing defect accumulation on the product infant failure and presented a defect-based reliability model. He et al. ¹⁴ integrated the rough set and the fuzzy TOPSIS (Technique for Order Preference by Similarities to Ideal Solution) to mine the correlation between quality data and reliability data from product lifecycle. Halabi et al. ¹⁵ introduced typical reliability and risk assessments into product development and proposed a comprehensive methodology to improve product reliability. He et al. ¹⁶ used the weighted association rule mining to search for the root causes of product infant failure in the manufacturing process. Although these studies contribute to the batch reliability analysis based on process data, quantitative reliability–oriented quality monitoring approaches remain underdeveloped.

In the field of statistical process control (SPC), various control charts are widely used to keep manufacturing in control by detecting process shifts. Therefore, designing a control chart plan considering product reliability is a practical way to strengthen reliability assurance in manufacturing. However, studies on reliability-oriented control chart are rare. Panagiotidou and Nenes ¹⁷ noticed the influence of process shift on product failure and used an optimal adaptive variable-parameter Shewhart chart to mitigate this effect. He et al. ¹⁸ proposed a convergent conditional expected value (CEV)-based chart to control the censored process variables for reliability improvement. In addition, another alternative methodology is to apply proper control charts after determining the process parameters that influence reliability. Thus, many advanced charts can be used for various manufacturing processes, including single-station process, ¹⁹ integrated process with correlative parameters, ²⁰ and multi-station parallel process. ²¹ Furthermore, the quality cost incurred by non-conformance can be used to qualitatively reflect the batch reliability; therefore, reliability is usually involved in the economical design of control chart and other process control plan, and such considerations are usually computed by the processing cost^22,23 or Taguchi loss.^24,25 However, the introduction of the quantitative quality–reliability model is still ignored, which limits the pertinence of reliability-oriented control.

Notably, reliability degradation under the in-control state cannot be improved only by adjusting the monitoring method, so the detection of the out-of-control state must be emphasized. In addition, the control chart using the time-between-events (TBE) data is more applicable than traditional Shewhart charts in the high-quality processes. ²⁶ Thus, this article proposes a practical reliability–oriented design of the TBE control chart system to enhance the effectiveness of using process quality monitoring to achieve the reliability improvement, and the main contributions are as follows:

1. The critical-to-reliability (CTR) characteristics are defined and a quantitative model between process quality and reliability of the produced product is developed, which makes it possible to carry out reliability-oriented quality control for high-quality manufacturing processes.

2. The TBE chart system is prearranged to monitor the CTR process parameters, and all the individual TBE charts are optimized systematically to minimize the negative effect of process shift on batch reliability, which provides a feasible technical means to realize reliability-oriented quality control for high-quality manufacturing processes.

The remainder of this article is organized as follows. In section “TBE control chart system predesign for enhancing product reliability,” a batch product reliability estimation model is presented based on the process data and the preliminary design of the TBE chart system and necessary assumptions are clarified. In section “Joint optimization model of the TBE chart system,” the involved mathematical optimization model is explained and the proposed approach is described. In section “Case study,” a case study is conducted to evaluate the performance of the chart system. Finally, section “Conclusion” presents the conclusions with a summary and directions for future research.

TBE control chart system predesign for enhancing product reliability

Product reliability degradation modeling based on process quality data

Because of the inevitable process variations, the reliability of a product is consistently lower than its design value, and an appropriate measurement is needed to describe the degree of such degradation. In most modern manufacturing enterprises, if a product failure occurring in the warranty period is caused by its own quality problems (instead of other external reasons), then after-sales activities will lead to a significant increase in the total cost. Considering this phenomenon, the product reliability degradation due to manufacturing, denoted by $\mathrm{\Delta }R$ , is expressed as follows

$$\mathrm{\Delta }R=R_0\left(t_w\right)-R\left(t_w\right)$$

(1)

where $R(·)$ and $R_0(·)$ express the degraded reliability and design reliability at time t, respectively, and $t_w$ is the length of the warranty.

Then, the manufacturing variables related to reliability must be identified. The general quality can be expressed by the conformance of key quality characteristics (KQCs). As an indispensable indicator of quality, reliability is also determined by a group of CTR characteristics. Thus, identifying controllable CTR parameters in manufacturing is crucial.

The axiomatic design theory presented by Suh ²⁷ can be applied to seek the CTR process parameters, as shown in Figure 1. By analyzing customer requirements, product-level CTR characteristics are determined. Then, CTR function modules are determined and decomposed into the function requirement tree. Furthermore, the proposed tree is successively mapped to identify structural components and final CTR process parameters. On this basis, the quality data of the identified CTR parameters can be used to evaluate the reliability degradation.

Figure 1.

Identification of CTR process parameters using domain mapping and decomposition.

Dimensional variation and internal defect are the two main indicators of process quality. ³ The accumulation of dimensional variations results in an accelerated product crash by the original mechanism that has been considered in design stage, whereas internal defect reduces product reliability by introducing a new failure mode. Both process variations and defects may accumulate and remain in delivered products, so their effects should be considered in the process of establishing the reliability degradation model. Specific steps are as follows:

• Step 1. Only CTR dimensional variations are considered. As dimensional variations cannot change the failure mechanism, the degraded reliability can be expressed by a new function $R_V(t)$ , which has the similar distribution with $R_0(t)$ . Considering m types of CTR dimensional variations and Q types of independent CTR characteristics, the correlation between CTR characteristic deviations and dimensional variations is obtained

$$Y_l={\psi }_l+{\mathbf{a}}_l^T\mathbf{V}+{\mathbf{b}}_l^T\mathbf{z}+{\mathbf{z}}^T{\rho }_l\mathbf{V},l=1,2,\dots ,Q$$

(2)

where $Y_l$ denotes the deviation of CTR characteristics l; ${\psi }_l$ is the constant; $\mathbf{V}=(V_1,V_2,\dots ,V_m{)}^T$ and $\mathbf{z}=(z_1,z_2,\dots ,z_m{)}^T$ are the vectors of the CTR dimensional variation and noise, respectively; ${\mathbf{a}}_l\mathrm{and}{\mathbf{b}}_l$ are the column vectors defining the linear effects of $\mathbf{V}\mathrm{and}\mathbf{z}$ , respectively; and ${\rho }_l$ is the matrix defining the effects of interactions between $\mathbf{V}\mathrm{and}\mathbf{z}$ .

The deviation of product KQCs often results in economic losses for handling customer complaints. A reasonable assumption can be posed that the loss caused by deviations of CTR characteristics mainly reflects in the increased cost for handling unwanted product failures in warranty. Therefore, the relationship between the CTR characteristic deviations and reliability degradation can be obtained

$$\sum _{l=1}^Q{c}_l{Y}_l^2=[1-R_V\left(t_w\right)]c_w-[1-R_0\left(t_w\right)]c_w$$

(3)

where $c_w$ is the average warranty cost.

Thus, the degraded reliability can be obtained

$$R_V\left(t_w\right)=R_0\left(t_w\right)-\frac{1}{c_w}\sum _{l=1}^Q{c}_l{[({\mathbf{a}}_l^T+{\mathbf{z}}^T{\rho }_l)\mathbf{V}+{\mathbf{b}}_l^T\mathbf{z}+{\psi }_l]}^2$$

(4)

• Step 2. The effects of CTR internal defects are analyzed. Let $D_i$ denote the number of the ith type of defects and assume that all defects of the same type follow the same failure distribution $G_i(t)$ . Therefore, the reliability of all CTR internal defects $R_D(t)$ can be evaluated as follows

$$R_D\left(t\right)=\underset{i=1}{\overset{n}{\Pi }}{[1-G_i\left(t\right)]}^{D_i}$$

(5)

where n is the number of types of internal defects.

Based on the competing risk principle, the final degraded reliability can be computed by $R(t)=R_V(t)R_D(t)$ . Thus, the reliability degradation of a single product is expressed as

$$\begin{array}{cc}\multicolumn{1}{c}{\mathrm{\Delta }R}& =R_0\left(t_w\right)-R_V\left(t_w\right)R_D\left(t_w\right)\hfill \\ \multicolumn{1}{c}{}& =R_0\left(t_w\right)-\{R_0\left(t_w\right)-\frac{1}{c_w}\sum _{l=1}^Q{c}_l{[({\mathbf{a}}_l^T+{\mathbf{z}}^T{\rho }_l)\mathbf{V}+{\mathbf{b}}_l^T\mathbf{z}+{\psi }_l]}^2\}\underset{i=1}{\overset{n}{\Pi }}{[1-G_i\left(t_w\right)]}^{D_i}\hfill \end{array}$$

(6)

Batch reliability degradation is the expected value of reliability degradation of a single product, as denoted by $E(\mathrm{\Delta }R)$ . In view of the complexity of equation (6), the Monte Carlo simulation is applied to estimate $E(\mathrm{\Delta }R)$ on the basis of determining the probability density functions of CTR parameters. Computer programs in C or other languages for executing the Monte Carlo simulation have been highly popular. Thus, this article does not cover them here.

Predesigned reliability-oriented TBE chart system

The following assumptions are involved:

• Assumption 1. The analyzed system only consists of all identified CTR stations (processes). Without loss of generality, only one machine is considered in a single station.

• Assumption 2. Only one process is out of control at any moment.

• Assumption 3. Only one assignable cause exists in a particular CTR process, and its occurrence rate is independent and known.

• Assumption 4. Only one CTR process parameter is analyzed and monitored in a single CTR process.

• Assumption 5. The value of CTR dimensional variations and the number of internal defects follow independent Gaussian and Poisson distributions with known parameters, respectively.

• Assumption 6. The system continues operating during the search for possible assignable causes, and the time needed is a known constant.

The TBE chart is presented to monitor the occurrence of random events in high-quality processes. Comparing with traditional charts, the sampling policy in a TBE chart can be set at any value. The definition of “event” also has different interpretations even for a fixed sampling policy. The occurrence of the event can be modeled by a homogeneous Poisson process. If the occurrence rate of a defined event is $\upsilon$ according to historical data, the TBE data must follow the exponential distribution with the parameter $\upsilon$ . The TBE data are the concerned statistics.

On this basis, the TBE chart system is prearranged to control each CTR process by independent TBE chart. The time interval between two successive events is monitored. For a CTR process where the dimensional variation $V$ is the observation, assume that $V$ follows normal distribution $N({\mu }_0+\mathrm{\Delta }\mu ,({\sigma }_0+\mathrm{\Delta }\sigma {)}^2)$ , and event ${\Omega }_V$ can be denoted as the condition that $V$ fails out of the interval $({\mu }_0-\zeta {\sigma }_0,{\mu }_0+\zeta {\sigma }_0)$ , where $\zeta$ is the control limit width. Similarly, for a CTR process where the internal defect is focused, assume that the defect number D follows a Poisson distribution $P({\varphi }_0+\mathrm{\Delta }\varphi )$ , where ${\varphi }_0$ is usually extremely small. Thus, denote the event ${\Omega }_D$ as the condition that $D>0$ to enhance the feasibility of monitoring procedure.

Joint optimization model of the TBE chart system

Input parameters

In addition to the necessary parameters for establishing the quality–reliability model, the following parameters must be specified.

Process parameters
${\mu }_{0,k},{\mu }_{\delta ,k},{\sigma }_{0,k},{\sigma }_{\delta ,k}$	Mean and standard deviation under in-control and out-of-control states of the kth CTR dimensional variation
${\varphi }_{0,i},{\varphi }_{\delta ,i}$	Mean under in-control and out-of-control states of the ith CTR internal defect
${\upsilon }_{V,k}^0$ , ${\upsilon }_{V,k}^1$ , ${\upsilon }_{D,i}^0$ , ${\upsilon }_{D,i}^1$	In-control and out-of-control event occurrence rates
${\lambda }_{V,k}$ , ${\lambda }_{D,i}$	Occurrence rate of assignable cause
$h_{V,k},h_{D,i}$	Probability of out-of-control occurrence
$g_{V,k},g_{D,i}$	Time constant for searching and eliminating the assignable cause
$\tau$	Minimum allowable $\mathrm{AT}{S}_0$ of the TBE chart system
Chart cost parameters
$A_{V,k},A_{D,i}$	Average cost associated with one false alarm
$B_{V,k},B_{D,i}$	Average cost of searching and eliminating the assignable cause
$W_{V,k},W_{D,i}$	Average cost for observing and plotting an event
$\omega$	Maximum allowable average operating cost $C_{\mathrm{TBE}}$ of the TBE chart system

Optimization model

Because assignable causes always increase the occurrence rate of the event, TBE data, $S_{V,k}$ and $S_{D,i}$ , are larger-the-better characteristics. Therefore, only the lower control limit (LCL) $\mathrm{LC}{L}_{V,k},\mathrm{LC}{L}_{D,i}$ is considered. Upon the event occurrence, the observed values of $S_{V,k}$ and $S_{D,i}$ are plotted on the chart. The point falling below the LCL indicates a possible increase in the event occurrence rate. Thus, a process shift may have occurred, and action should be taken for identifying and removing the assignable cause. On the contrary, the process is considered in the in-control state, and no action is needed.

A vital truth is that the reliability degradation in the in-control state cannot be improved except by enhancing machine capability. Thus, detection optimization of the out-of-control state must be the emphasized to attain the reliability-oriented monitoring. Here, this article chooses to optimize the TBE chart system, characterized as the combination of $\mathrm{LC}{L}_{V,k}\mathrm{and}\mathrm{LC}{L}_{D,i}$ , to minimize the batch reliability degradation increment index $E(\Theta )$ . The average cost and statistical performance are constraints. This problem can be conducted by the following model

$$\mathrm{Objective}\mathrm{function}:E(\Theta )=\mathrm{minimum}$$

(7)

$$\mathrm{Constraint}\mathrm{function}:\mathrm{AT}{S}_{0,\mathrm{TBE}}\ge \tau$$

(8)

$$C_{\mathrm{TBE}}\le \omega$$

(9)

The design variables are $\mathrm{LC}{L}_{V,k}(k=1,2,\dots ,m)$ and $\mathrm{LC}{L}_{D,i}(i=1,2,\dots ,n)$ .

The calculation of the involved parameters will be described in the following subsection.

System in-control $\mathrm{AT}{S}_{0,\mathrm{TBE}}$

The $\mathrm{AT}{S}_{0,\mathrm{TBE}}$ of the TBE chart system is the average time to signal when all processes are in control, and it can be estimated by the false alarm rate of each process.

The false alarm rate of a single process can be approximately calculated by the product of the type I error probability and the event occurrence rate. The type I error probabilities ${\alpha }_{V,k}$ and ${\alpha }_{D,i}$ can be expressed as

$$\begin{array}{cc}\multicolumn{1}{c}{{\alpha }_{V,k}}& =P(S_{V,k}\le \mathrm{LC}{L}_{V,k}|\mathrm{in}-\mathrm{control})\hfill \\ \multicolumn{1}{c}{}& =1-e^{-{\upsilon }_{V,k}^0\mathrm{LC}{L}_{V,k}}\hfill \end{array}$$

(10)

$${\alpha }_{D,i}=1-e^{-{\upsilon }_{D,i}^0\mathrm{LC}{L}_{D,i}}$$

(11)

Therefore, the probability that a chart in a process produces a false alarm per time unit can be approximately evaluated by ${\upsilon }_{V,k}(1-e^{-{\upsilon }_{V,k}\mathrm{LC}{L}_{V,k}})$ or ${\upsilon }_{D,i}(1-e^{-{\upsilon }_{D,i}\mathrm{LC}{L}_{D,i}})$ . Therefore, the false alarm probability that the entire system per unit time $P_{0,\mathrm{TBE}}$ can be expressed as

$$\begin{array}{cc}\multicolumn{1}{c}{P_{0,\mathrm{TBE}}}& =1-\underset{k=1}{\overset{m}{\Pi }}(1-{\upsilon }_{V,k}^0{\alpha }_{V,k})\underset{i=1}{\overset{n}{\Pi }}(1-{\upsilon }_{D,i}^0{\alpha }_{D,i})\hfill \\ \multicolumn{1}{c}{}& =1-\underset{k=1}{\overset{m}{\Pi }}[1-{\upsilon }_{V,k}^0(1-e^{-{\upsilon }_{V,k}^0\mathrm{LC}{L}_{V,k}})]\hfill \\ \multicolumn{1}{c}{}& \underset{i=1}{\overset{n}{\Pi }}[1-{\upsilon }_{D,i}^0(1-e^{-{\upsilon }_{D,i}^0\mathrm{LC}{L}_{D,i}})]\hfill \end{array}$$

(12)

Finally

$$\mathrm{AT}{S}_{0,\mathrm{TBE}}=1/P_{0,\mathrm{TBE}}$$

(13)

Expected length of an operational cycle of a single process T

A single process consists of successive operational cycles, and each cycle comprises the in-control period $t_1$ , out-of-control period $t_2$ , and period for searching and eliminating the assignable cause $t_3$ . For simplicity, the calculation of T in the process of introducing the kth CTR dimensional variation is demonstrated as the example, and the results of processes of CTR internal defects are directly provided.

Because it obeys exponential distribution, and the process continues operating during the search for the possible assignable cause, therefore

$$t_{1,V,k}=1/{\lambda }_{V,k}$$

(14)

$$t_{1,D,i}=1/{\lambda }_{D,i}$$

(15)

The expected length of the out-of-control period can be calculated by the inverse of probability that the chart monitoring the out-of-control process gives a signal per unit time

$$t_{2,V,k}=\frac{1}{{\upsilon }_{V,k}^1(1-e^{-{\upsilon }_{V,k}^1\mathrm{LC}{L}_{V,k}})}$$

(16)

$$t_{2,D,i}=\frac{1}{{\upsilon }_{D,i}^1(1-e^{-{\upsilon }_{D,i}^1\mathrm{LC}{L}_{D,i}})}$$

(17)

The expected time for searching and eliminating the assignable cause is a known constant. Thus

$$t_{3,V,k}=g_{V,k}$$

(18)

$$t_{3,D,i}=g_{D,i}$$

(19)

The expected length of an operational cycle of the considered CTR process is

$$T_{V,k}=t_{1,V,k}+t_{2,V,k}+t_{3,V,k}$$

(20)

$$T_{D,i}=t_{1,D,i}+t_{2,D,i}+t_{3,D,i}$$

(21)

Expected average cost for operating TBE system per unit time $C_{\mathrm{TBE}}$

The average cost for operating TBE system can be obtained by summing the cost of individual charts. Taking the process that introduces the kth CTR dimensional variation as the example, the expected number of false alarms $M_{V,k}$ can be estimated by

$$M_{V,k}={\upsilon }_{V,k}^0{\alpha }_{V,k}{t}_{1,V,k}=\frac{{\upsilon }_{V,k}^0(1-e^{-{\upsilon }_{V,k}^0\mathrm{LC}{L}_{V,k}})}{{\lambda }_{V,k}}$$

(22)

$$M_{D,i}={\upsilon }_{D,i}^0{\alpha }_{D,i}{t}_{1,D,i}=\frac{{\upsilon }_{D,i}^0(1-e^{-{\upsilon }_{D,i}^0\mathrm{LC}{L}_{D,i}})}{{\lambda }_{D,i}}$$

(23)

The expected event number $N_{D,i}$ observed from the occurrence of the assignable cause to the elimination of the out-of-control condition can be calculated as

$$\begin{array}{cc}\multicolumn{1}{c}{N_{V,k}}& =(t_{2,V,k}+t_{3,V,k}){\upsilon }_{V,k}^1(1-e^{-{\upsilon }_{V,k}^1\mathrm{LC}{L}_{V,k}})\hfill \\ \multicolumn{1}{c}{}& =1+g_{V,k}{\upsilon }_{V,k}^1(1-e^{-{\upsilon }_{V,k}^1\mathrm{LC}{L}_{V,k}})\hfill \end{array}$$

(24)

$$N_{D,i}=1+g_{D,i}{\upsilon }_{D,i}^1(1-e^{-{\upsilon }_{D,i}^1\mathrm{LC}{L}_{D,i}})$$

(25)

Thus, the expected average cost for operating TBE system per unit time $C_{\mathrm{TBE}}$ is

$$\begin{array}{cc}\multicolumn{1}{c}{C_{\mathrm{TBE}}=}& \sum _{k=1}^m\left[\frac{M_{V,k}{A}_{V,k}+B_{V,k}+(M_{V,k}+N_{V,k})W_{V,k}}{T_{V,k}}\right]\hfill \\ \multicolumn{1}{c}{}& +\sum _{i=1}^n\left[\frac{M_{D,i}{A}_{D,i}+B_{D,i}+(M_{D,i}+N_{D,i})W_{D,i}}{T_{D,i}}\right]\hfill \end{array}$$

(26)

Batch reliability degradation increment index $E(\Theta )$

The increment index of batch reliability degradation caused by a single assignable cause, denoted by $\Theta$ , is computed by the product of the reliability degradation increment in out-of-control period I and the time length from the occurrence of the assignable cause to the elimination of the out-of-control condition $t_2+t_3$ . The expected value of $\Theta$ is denoted by $E(\Theta )$ .

Denote by $E(\mathrm{\Delta }R{)}_0$ the batch reliability degradation when all processes are in control. If an assignable cause occurs in the process related to CTR parameter X, its probability density function will change from $f_0(x)\mathrm{to}{f}_{\delta }(x)$ , and the related batch reliability degradation $E(\mathrm{\Delta }R{)}_X$ can be expressed by the Monte Carlo method mentioned above. Thus

$$I_{V,k}=E{\left(\mathrm{\Delta }R\right)}_{V,k}-E{\left(\mathrm{\Delta }R\right)}_0$$

(27)

$$I_{D,i}=E{\left(\mathrm{\Delta }R\right)}_{D,i}-E{\left(\mathrm{\Delta }R\right)}_0$$

(28)

Based on Assumption 2, $E(\Theta )$ can be obtained by

$$\begin{array}{cc}\multicolumn{1}{c}{E\left(\Theta \right)=}& \sum _k^m{h}_{V,k}{I}_{V,k}(t_{2,V,k}+t_{3,V,k})\hfill \\ \multicolumn{1}{c}{}& +\sum _i^n{h}_{D,i}{I}_{D,i}(t_{2,D,i}+t_{3,D,i})\hfill \end{array}$$

(29)

Optimization algorithm

This article uses a dynamic search algorithm to determine the optimal combination of design variables $\mathrm{LC}{L}_{V,k}(k=1,2,\dots ,m)$ and $\mathrm{LC}{L}_{D,i}(i=1,2,\dots ,n)$ . Type I error probabilities ${\alpha }_{V,k}$ and ${\alpha }_{D,i}$ , which are limited to a finite interval of $[0,1]$ , are selected to replace $\mathrm{LC}{L}_{V,k}$ and $\mathrm{LC}{L}_{D,i}$ as the actual optimized object, respectively, to improve the efficiency of optimization search

$$\mathrm{LC}{L}_{V,k}=\frac{-\ln (1-{\alpha }_{V,k})}{{\upsilon }_{V,k}^0}$$

(30)

$$\mathrm{LC}{L}_{D,i}=\frac{-\ln (1-{\alpha }_{D,i})}{{\upsilon }_{D,i}^0}$$

(31)

Thus, this nonlinear programming problem aims to identify the optimal values for $({\alpha }_{V,1},\dots ,{\alpha }_{V,m},{\alpha }_{D,i},\dots ,{\alpha }_{D,n})$ that will jointly minimize the expected reliability degradation $E(\mathrm{\Delta }{R}_M)$ and simultaneously ensure $\mathrm{AT}{S}_{0,\mathrm{TBE}}\ge \tau \mathrm{and}{C}_{\mathrm{TBE}}\le \omega$ . The entire optimization search can be executed based on the MATLAB optimization toolbox according to the following steps:

• Step 1. Specify all the involved parameters.

• Step 2. Initialize $({\alpha }_{V,1},\dots ,{\alpha }_{V,m},{\alpha }_{D,1},\dots ,{\alpha }_{D,n})$ (e.g. $(0.01,\dots ,0.01)$ ) and set their lower and upper bounds as $[0,1]$ .

• Step 3. Set an appropriate maximum iteration number (e.g. 10,000).

• Step 4. Input the object function and constraint functions and output the optimal value $E(\Theta )$ and optimal design point $({\alpha }_{V,1},\dots ,{\alpha }_{V,m},{\alpha }_{D,1},\dots ,{\alpha }_{D,n})$ .

• Step 5. Calculate the lower control limit of each individual control chart by equations (30) and (31).

Case study

Background

The cylinder head is one of the most important parts of a modern engine, and the machining precision is a crucial factor which significantly decides the useful life and reliability of the engine. A poor-quality cylinder head will cause unwanted failures of the engine. In a Chinese enterprise, the manager hopes to enhance the stability of engine reliability by optimizing the quality control in cylinder head manufacturing. Therefore, the proposed approach is applied.

First, four CTR stations are identified by the axiomatic method. The schematic is shown in Figure 2, and the process requirements are listed in Table 1. Thereinto, two types of CTR dimensional variations, $V_1$ and $V_2$ , are related with the gas leakage rate $Y_1$ , which is the only product CTR characteristic. Two types of CTR internal defect, $D_1$ and $D_2$ , are involved.

Figure 2.

Schematic of the involved CTR stations.

Table 1.

KQCs of cylinder head and design specifications.

Station no.	Machining process	Specification (mm)	CTR parameter
D1	Casting the blank		$D_1$
V1	Milling face M and A	Distance between M and A = 28.3 ± 0.05	$V_1$
D2	Drilling hole B	$\mathrm{Distance}\mathrm{of}\mathrm{B}=5.1_0^{+0.15}$	$D_2$
V2	Milling groove S	Radius of S = 9.8 ± 0.08	$V_2$

CTR: critical-to-reliability.

Numerical example

First, the involved parameters are obtained from historical data and requirements of manufacturing, as follows:

• 1. Reliability modeling parameter

$$\begin{array}{c}{R}_0\left(t_w\right)=0.9,c_w=250,c_1=1600,\hfill \\ {\mathbf{a}}_1={(0.63,0.24)}^T,\mathbf{z}={(0,0)}^T,{\psi }_1=0.1,\hfill \\ G_1\left(t_w\right)=0.33,G_2\left(t_w\right)=0.47\hfill \end{array}$$

• 2. Process parameter

$$\begin{array}{c}{\zeta }_1={\zeta }_2=3,{\mu }_{0,1}=0,{\sigma }_{0,1}=0.0167,\hfill \\ {\mu }_{\delta ,1}=0.005,{\sigma }_{\delta ,1}=0.0271;{\mu }_{0,2}=0,\hfill \\ {\sigma }_{0,2}=0.0265,{\mu }_{\delta ,2}=0.009\hfill \\ {\sigma }_{\delta ,2}=0.0387,{\varphi }_{0,1}=0.004,{\varphi }_{\delta ,1}=0.08,\hfill \\ {\varphi }_{0,2}=0.003,{\varphi }_{\delta ,2}=0.045;{\upsilon }_{V,1}^0=0.005,\hfill \\ {\upsilon }_{V,1}^1=0.065,{\upsilon }_{V,2}^0=0.005\hfill \\ {\upsilon }_{V,2}^1=0.086,{\upsilon }_{D,1}^0=0.004,{\upsilon }_{D,1}^1=0.077,\hfill \\ {\upsilon }_{D,2}^0=0.003,{\upsilon }_{D,2}^1=0.074,{\lambda }_{V,1}=0.0001,\hfill \\ {\lambda }_{V,2}=0.0002\hfill \\ {\lambda }_{D,1}=0.00025,{\lambda }_{D,2}=0.00015,h_{V,1}=0.2,\hfill \\ h_{V,2}=0.3,h_{D,1}=0.25,h_{D,2}=0.25\hfill \\ g_{V,1}=0.6,g_{V,2}=0.8,g_{D,1}=0.7,g_{D,2}=0.5\hfill \end{array}$$

• 3. Chart cost parameter (unit: US$100)

$$\begin{array}{c}{A}_{V,1}=A_{V,2}=A_{D,1}=A_{D,2}=10;\hfill \\ B_{V,1}=15,B_{V,2}=20,B_{D,1}=25,B_{D,1}=20\hfill \\ W_{V,1}=W_{V,2}=W_{D,1}=W_{D,2}=0.5\hfill \end{array}$$

Reliability degradation estimation

The degradation model of a single product is

$$\begin{array}{cc}\multicolumn{1}{c}{\mathrm{\Delta }R}& =0.9-[0.9-6.4{(0.63V_1+0.24V_2+0.1)}^2]\hfill \\ \multicolumn{1}{c}{}& \times 0.{67}^{D_1}\times 0.{53}^{D_2}\hfill \end{array}$$

When the simulation number is set at 100,000, the in-control and out-of-control batch reliability degradation values are

$$\begin{array}{c}E{\left(\mathrm{\Delta }R\right)}_0=0.0672,E{\left(\mathrm{\Delta }R\right)}_{V,1}=0.0725,\hfill \\ E{\left(\mathrm{\Delta }R\right)}_{V,2}=0.0703,E{\left(\mathrm{\Delta }R\right)}_{D,1}=0.0879,\hfill \\ E{\left(\mathrm{\Delta }R\right)}_{D,2}=0.0678\hfill \end{array}$$

Thus, reliability degradation increments are

$$\begin{array}{c}{I}_{V,1}=0.0053,I_{V,2}=0.0031,\hfill \\ I_{D,1}=0.0207,I_{D,2}=0.0002\hfill \end{array}$$

Parameter calculation of the TBE chart system

The chart parameters are expressed by the type I error probability.

The $\mathrm{AT}{S}_{0,\mathrm{TBE}}$ is

$$\begin{array}{c}\mathrm{AT}{S}_{0,\mathrm{TBE}}=\frac{1}{P_{0,\mathrm{TBE}}}\hfill \\ =\frac{1}{1-(1-0.0027{\alpha }_{V,1})(1-0.0027{\alpha }_{V,2})(1-0.004{\alpha }_{D,1})(1-0.003{\alpha }_{D,2})}\hfill \end{array}$$

The expected cycle length T is

$$\begin{array}{c}{T}_{V,1}=10000+\frac{14.29}{1-{(1-{\alpha }_{V,1})}^{26.84}}+0.6\hfill \\ T_{V,2}=5000+\frac{22.03}{1-{(1-{\alpha }_{V,2})}^{17.19}}+0.8\hfill \\ T_{D,1}=4000+\frac{13}{1-{(1-{\alpha }_{D,1})}^{20}}+0.7\hfill \\ T_{D,2}=6667+\frac{22.73}{1-{(1-{\alpha }_{D,2})}^{15}}+0.5\hfill \end{array}$$

The expected average system cost $C_{\mathrm{TBE}}$ is

$$\begin{array}{cc}\multicolumn{1}{c}{C_{\mathrm{TBE}}=}& \frac{283.5{\alpha }_{V,1}-0.021{(1-{\alpha }_{V,1})}^{26.84}+15.521}{10000+(14.29/[1-{(1-{\alpha }_{V,1})}^{26.84}])+0.6}\hfill \\ \multicolumn{1}{c}{}& +\frac{141.75{\alpha }_{V,2}-0.018{(1-{\alpha }_{V,1})}^{17.19}+20.518}{5000+(22.03/[1-{(1-{\alpha }_{V,2})}^{17.19}])+0.8}\hfill \\ \multicolumn{1}{c}{}& +\frac{167.685{\alpha }_{D,1}-0.027{(1-{\alpha }_{D,1})}^{20}+25.527}{4000+(13/[1-{(1-{\alpha }_{D,1})}^{20}])+0.7}\hfill \\ \multicolumn{1}{c}{}& +\frac{209.685{\alpha }_{D,2}-0.022{(1-{\alpha }_{D,2})}^{15}+20.522}{6667+(22.73/[1-{(1-{\alpha }_{D,2})}^{15}])+0.5}\hfill \end{array}$$

Optimization results

In order to highlight the advantages of the proposed approach, this article compares the most important performance indexes, namely, the expected batch reliability degradation increment indexes $(E(\Theta ))$ , of the proposed reliability-oriented method and traditional Three Sigma design to directly reflect the innovation and effectiveness of the proposed method. The performance comparison strategy is widely used in control chart design studies.^28,29

The expected batch reliability degradation increment index $E(\Theta )$ is

$$\begin{array}{cc}\multicolumn{1}{c}{E\left(\Theta \right)}& =0.00106(t_{2,V,1}+t_{3,V,1})\hfill \\ \multicolumn{1}{c}{}& +0.00093(t_{2,V,2}+t_{3,V,2})\hfill \\ \multicolumn{1}{c}{}& +0.00518(t_{2,D,1}+t_{3,D,1})\hfill \\ \multicolumn{1}{c}{}& +0.00005(t_{2,D,2}+t_{3,D,2})\hfill \\ \multicolumn{1}{c}{}& =\frac{0.01515}{1-{(1-{\alpha }_{V,1})}^{26.84}}+\frac{0.02049}{1-{(1-{\alpha }_{V,2})}^{17.19}}\hfill \\ \multicolumn{1}{c}{}& +\frac{0.06734}{1-{(1-{\alpha }_{D,1})}^{20}}\hfill \\ \multicolumn{1}{c}{}& +\frac{0.00114}{1-{(1-{\alpha }_{D,2})}^{15}}+0.00504\hfill \end{array}$$

A traditional TBE chart system adopts the same type I error probability of 0.0027 for all individual charts, and the corresponding $\mathrm{AT}{S}_{0,\mathrm{TBE}}\mathrm{and}{C}_{\mathrm{TBE}}$ are 29,870 and 0.01442, respectively. Considering this, $\tau =27,000\mathrm{and}\omega =0.015$ are set as the constraints in the optimized system. Using MATLAB, the optimization results are obtained

$$\begin{array}{c}\mathrm{Station}\mathrm{V}1:\mathrm{LC}{L}_{V,1}=0.87400({\alpha }_{V,1}=0.0023602)\hfill \\ \mathrm{Station}\mathrm{V}2:\mathrm{LC}{L}_{V,2}=1.27076({\alpha }_{V,2}=0.0034298)\hfill \\ \mathrm{Station}\mathrm{V}3:\mathrm{LC}{L}_{D,1}=1.18654({\alpha }_{D,1}=0.0047349)\hfill \\ \mathrm{Station}4:\mathrm{LC}{L}_{D,2}=0.27401({\alpha }_{D,2}=0.0008217)\hfill \\ \mathrm{AT}{S}_{0,\mathrm{TBE}}=27000,C_{\mathrm{TBE}}=0.01424,E\left(\Theta \right)=1.44550\hfill \end{array}$$

Furthermore, the index $E(\Theta )$ of the traditional system is 1.98069. Thus, it is obvious that the optimized system reduces 27.02% of the reliability degradation and saves 1.25% cost in long-term operation. Next, a sensitivity analysis is conducted to investigate the system performance.

Sensitivity analysis

First, the effects of reliability parameter $G_i(t_w)$ and process parameters $({\varphi }_{0,i},{\varphi }_{\delta ,i},{\lambda }_{D,i},g_{D,i})$ are analyzed. Second, the cost parameter $A_{D,i}$ is focused, considering its impact on the operating cost when the minimum reliability degradation is achieved. In both studies, the traditional system, where the $\alpha$ value of each chart is 0.0027, is set for comparison.

Study I

This section analyzes the effects of parameter $G_i(t_w)$ and parameters $({\varphi }_{0,i},{\varphi }_{\delta ,i},{\lambda }_{D,i},g_{D,i})$ on the batch reliability improvement. For simplicity, a reconstructed process only including Stations D1 and D2 is analyzed in section “Numerical example.” Assume that $h_{D,1}=h_{D,2}=0.5$ . As shown below, each analyzed parameter has its low, nominal, and high values.

Parameter	Low	Nominal	High
$G_i(t_w)$	0.1	0.5	0.9
${\varphi }_{0,i}$	0.001	0.005	0.009
${\varphi }_{\delta ,i}$	0.01	0.05	0.09
${\lambda }_{D,i}$	0.0001	0.0005	0.0009
$g_{D,i}$	0.1	0.5	0.9

In each subcase, one selected parameter (active parameter) is set at its low value in Station D1 and high value in Station D2, and other parameters are set at nominal values in both stations. The cost parameters remain constant at $A_{D,1}=A_{D,2}=10,B_{D,1}=B_{D,2}=20,W_{D,1}=W_{D,2}=0.5$ . The constraint parameters are specified as $\tau =0.9\mathrm{AT}{S}_{0,\mathrm{TBE},\mathrm{traditional}}$ and $\omega =1.1C_{\mathrm{TBE},\mathrm{traditional}}$ .

To describe the effectiveness of the optimized system, the ratio r is calculated as follows

$$r=\frac{E{\left(\Theta \right)}_{\mathrm{optimized}}}{E{\left(\Theta \right)}_{\mathrm{traditional}}}$$

If the value of r is smaller than 1, then the optimized system will outperform the traditional system in terms of reliability degradation, and vice versa.

The results in Table 2 indicate that in all cases the proposed system shows an evident improvement in reliability-oriented control under different parameter combinations compared with the traditional system. In addition, based on Cases 1 and 3, the system performance is more sensitive to $G_i(t_w)$ and ${\varphi }_{\delta ,i}$ because the reliability degradation increment is determined by both the effects of out-of-control CTR variables and the length of the out-of-control period. Thus, these two parameters, which are important in product reliability modeling and shift degree measurement, are more influential.

Table 2.

Sensitivity study I.

Case	Active parameter		Optimal ${\alpha }_{D,i}$		$E(\Theta {)}_{\mathrm{optimized}}$	$E(\Theta {)}_{\mathrm{traditional}}$	Ratio r
			${\alpha }_{D,1}$	${\alpha }_{D,2}$
1	$G_1\left(t_w\right)=0.1$	$G_2\left(t_w\right)=0.9$	0.001507968	0.004492026	11.013	15.188	0.72511
2	${\varphi }_{0,1}=0.001$	${\varphi }_{0,2}=0.009$	0.003243968	0.002972880	12.856	14.273	0.90072
3	${\varphi }_{\delta ,1}=0.01$	${\varphi }_{\delta ,2}=0.09$	0.004090860	0.001909139	19.959	25.072	0.79607
4	${\lambda }_{D,1}=0.0001$	${\lambda }_{D,2}=0.0009$	0.003000018	0.002999969	13.862	15.380	0.90130
5	$g_{D,1}=0.1$	$g_{D,2}=0.9$	0.002999995	0.002999992	13.862	15.381	0.90124

Study II

In section “Numerical example,” the results indicate that the proposed method may save operating costs while minimizing reliability degradation. Thus, this study is conducted to analyze this phenomenon.

Similar to Study I, only Stations D1 and D2 are involved. The used parameters are assumed as follows

$$\begin{array}{c}{G}_1\left(t_w\right)=0.9,G_2\left(t_w\right)=0.1,\hfill \\ {\varphi }_{0,1}={\varphi }_{0,2}=0.005,{\varphi }_{\delta ,1}={\varphi }_{\delta ,2}=0.05,\hfill \\ {\lambda }_{D,1}={\lambda }_{D,2}=0.0005\hfill \\ h_{D,1}=h_{D,2}=0.5,g_{D,1}=0.5,g_{D,2}=0.5\hfill \end{array}$$

In actual conditions, most of the operating cost of a chart system derives from excluding false alarms, and this conclusion can be obtained based on equations (30) and (31). Thus, parameter $A_{D,i}$ is selected as the active parameter, and reference values of other cost parameters are $B_{D,1}=B_{D,2}=20,W_{D,1}=W_{D,2}=0.5$ To guarantee a sufficient value range of active parameters, the constraint parameter $\omega$ is relaxed to $\omega =1.5C_{\mathrm{TBE},\mathrm{traditional}}$ , whereas the constraint parameter $\tau =0.9\mathrm{AT}{S}_{0,\mathrm{TBE},\mathrm{traditional}}$ remains unchanged.

The ratio $u=C_{\mathrm{TBE},\mathrm{optimized}}/C_{\mathrm{TBE},\mathrm{traditional}}$ is applied to describe the effectiveness of the optimized system on the cost saving. The condition that u is smaller than 1 indicates that the optimized system can reduce costs on the basis of minimizing the reliability degradation, and vice versa.

Based on the results in Table 3, when $A_{D,1}$ becomes larger, the value of u increases and finally exceeds 1, whereas this condition does not occur even when $A_{D,2}$ becomes extremely small. This issue can be explained by the difference in the influence of Stations D1 and D2 on batch reliability. Given that the CTR defect in Station D1 can cause more reliability degradation, a large allowable false alarm can be allocated to the chart for Station D1 to detect the process shift quickly. Thus, it is possible for this optimized system to save costs.

Table 3.

Sensitivity study II.

Active parameter		$C_{\mathrm{TBE},\mathrm{optimized}}$	$C_{\mathrm{TBE},\mathrm{traditional}}$	Ratio u
$A_{D,1}=1$	$A_{D,2}=10$	0.014452	0.014915	0.9689925
$A_{D,1}=10$		0.014616	0.015002	0.9742615
$A_{D,1}=50$		0.015345	0.015392	0.9969527
$A_{D,1}=100$		0.016254	0.015877	1.0237275
$A_{D,1}=200$		0.018067	0.016846	1.0724820
$A_{D,1}=10$	$A_{D,2}=0.01$	0.014905	0.014572	0.9776426
	$A_{D,2}=0.1$	0.014906	0.014572	0.9776119
	$A_{D,2}=1$	0.014915	0.014576	0.9773056
	$A_{D,2}=100$	0.015877	0.015015	0.9456825
	$A_{D,2}=1000$	0.024632	0.019003	0.7715101

Conclusion

This article proposes an optimization design of the TBE chart system to reduce the product reliability degradation caused by manufacturing. By establishing the quantitative model between CTR process quality parameters (including dimensional variations and internal defects) and product reliability degradation, the control limits of each individual chart are jointly optimized to minimize batch reliability degradation in high-quality processes, and the proposed design is proved to observably improve batch reliability under different parameter combinations compared with the traditional design, which is derived from the complex but practical quality–reliability model proposed in this article. In addition, the proposed model can also save the system operating cost on the basis of achieving minimum reliability degradation in some cases, and this phenomenon is not rare.

On the contrary, this proposed optimization model also has two main limitations. The first one is that the application of this model may lead to slight increases in the system false rate and average operational cost, which are acceptable considering the model effect in reliability degradation mitigation. The second one is that the actual model of a specific system may be different from the model proposed in this study; however, the proposed optimizing procedure and ideology are generally applicable, which can adapt to different models.

Future works may focus on obtaining the optimal trade-off among chart operation cost, system false alarm rate, and batch reliability degradation. In our expected procedure, these three objects can be converted to the comprehensive control cost, which covers the manufacturing and usage stages of product lifecycle, and the related model is optimized to achieve the maximum possible manufacturing profit, where the product reliability is still a critical factor.