Study and analysis on fuzzy quality control for the high-end manufacturing process based on Taguchi quality loss function

Abstract

Most high-end manufacturing systems have implemented online inspection and control. However, fuzzy attribute indexes can’t be monitored online. The establishment of fuzzy quality control chart cannot be ignored on high-end manufacturing process. Because of this, based on Taguchi quality loss function, this paper is focusing on the question of monitoring and measuring process capability. Firstly, According to the characteristics of fuzzy quality attributes, the fundamental problem is to solve the distribution of fuzzy quality control chart. It is proposed that the beta distribution is the statistical basis of the fuzzy control chart. Secondly, fuzzy set distance indexes were constructed on fuzzy control chart, after analyzing the Beta distribution characteristics. Finally, the model of fuzzy control chart and the calculation of process capability are put forward based on Taguchi quality loss function. The results show that Taguchi quality loss function was effective in construction of fuzzy quality control chart to decide if process capability is under control or not. By calculation of production process datas, this paper provides policy implications for monitoring and control of fuzzy attribute on manufacturing process. Methodologically, this paper verified a way that transferring the form of product qualification ratio to that can be used in fuzzy control chart.

Keywords

Fuzzy high-end manufacturing fuzzy set distance quality loss

1. Introduction

The objective of quality control is to monitor the production process. The On-line Test Technology can be used on most of the high-end manufacturing process, but fuzzy attribute indexes of high-end manufacturing process cannot be monitored or controlled on on-line, such as the ratio of the Standard Products which is fuzzy quality attribute index. The qualified rate could barely meet customer requirements. According to the theory of Taguchi quality loss function, the ratio of the standard products cannot stand for quality level completely. Closer the center of tolerance is, less the quality loss is, also higher the quality level is. So the rank of qualified products is the most characteristic index of fuzzy quality attribute [1].

The index of distance must be used to measure the distance from the center value, if fuzzy quality index can be controlled in the process. Further the distance is, larger the similarity index is, worse the product quality is. In contrary, closer the distance is, smaller the similarity index is, then better the product quality is. The fuzzy quality characteristic value is close to the statistical steady state, which is a kind of control chart based on fuzzy quality indicators for monitoring object index control chart. The range of similarity index is in [0, 1]. According to the random variation characteristics of index value, the study of this paper begins with the Beta distribution in the application of control chart.

2. Literature review

2.1 The Taguchi quality loss function

Taguchi (1961), who is a Japanese quality management expert, think that the quality of products and the quality loss are closely related, because the quality does not meet the requirements, and all the losses are caused by the producers, users and society. Taguchi used a monetary unit to measure the quality of the product. Greater the quality loss, worse the quality of the product is, smaller the quality loss is, better the quality of the product is [2].

As an example, the color uniformity is a very sensitive quality index, we set $m$ as the basic specification in the design and the corresponding tolerance $\Delta u=\pm$ 5. According to the conformance quality standard, if the tolerance is in the range $\left[{m-5,m+5}\right]$ , it is qualified, otherwise, it is unqualified. In the Fig. 1a, there is the given conformity quality level. The value of $u$ presents “mutation” and “jump”, but in users’ feeling, it does not exist this artificial demarcation point. In the Fig. 1b, there is the applicability quality level. The value of $u$ presents a smooth curve in the range $\left[{m-5,m+5}\right]$ . Figure 2a and b are these two color uniformity characteristic value of the probability density curve.

Figure 1.

(a) Conformity quality level; (b) Applicability quality level.

Figure 2.

(a) Conformity quality characteristics probability density; (b) Applicability quality characteristics probability density curve.

Comparing traditional conformance quality standard with these two probability density curve in Fig. 2a and b, we can see that the color uniformity characteristic values are uniform distributed in the range $[m-5,m+5]$ , and it does not classify qualified products. In Fig. 2b, most of the color uniformity characteristic values focus on around of $u=m$ , which point reflects a higher applicability. Therefore, in the view of applicable quality, the color uniformity presented by Fig. 2b, is significantly better than that presented in Fig. 2a. By further quantitative calculation, we also can quantitatively compare the difference between them as well [3].

The change form a sole “static point” to a continuous “qualified rate curve” reflects a more sophisticated method of quantity analysis for the quality standard [4].

2.2 Theory of fuzzy set distance

The fuzzy similarity is an important index which describes the similarity degree of two fuzzy sets [5]. Suppose fuzzy set $A$ referring to the standard products, and all samples form a fuzzy set $B$ . Assuming the subordinate functions of sets $A$ and $B$ were defined, we can calculate the fuzzy similarity of $A$ and $B$ , $N(A,B)$ . $N(A,B)$ is an actual number. Thus, we may transfer fuzzy mathematical information to actual values. The mathematical definition of $N(A,B)$ is as follows:

Assuming $F(U)$ is the algebra of domain $U$ , $A B$ and $C\in F(U)$ , if $N:F(U)\times F(U)\to[0,1]$ , satisfying the conditions [6]:

$\displaystyle\text{\Pisymbol{pzd}{172}}\,N(A,B)=N(B,A);$ $\displaystyle\text{\Pisymbol{pzd}{173}}\,N(A,A)=1,N(U,\phi)=0;$ (1) $\displaystyle\text{\Pisymbol{pzd}{174}}\,A\subseteq B\subseteq C\Rightarrow N(% A,C)\leqslant N(A,B)\wedge N(B,C)$

then, $N(A,B)$ is Similarity of fuzzy set $A$ and $B$ , $N$ is fuzzy Similarity function of $F(U)$ . The value of fuzzy similarity $N(A,B)$ is between [0, 1]. $N(A,B)$ is closer to 0, the fuzzy set $A$ more similar to set $B$ . Otherwise, $N(A,B)$ is closer to 1, the fuzzy set $A$ more different from set $B$ [7]. Depending on the actual status, we can flexibly construct a new similarity function in order to meet various requirements. When fuzzy similarity method is applied on quality control, quality fluctuation of the production process is reflected by the changes of fuzzy similarity indexes in a way [8].

2.3 Characteristics of Beta distribution

The characteristic of Beta distribution is that the range of random variables is (0, 1), such as Unqualified rate, market share [9]. The Beta distribution describes the quality characteristics, which not only reflects the form of the distribution of the quality characteristics in a unified form, but also describe the law of mass distribution accurately, which is better than the quality characteristics of the normal distribution. So Beta distribution is better than normal distribution for quality characteristics. The Beta distribution can approximate to a variety of symmetric and biase distributions including the normal distribution by adjusting the parameters [10].

The density function expression of the Beta distribution is as follows:

$\displaystyle f(x:a,b)=\left\{{\begin{array}[]{l}\frac{\Gamma(a+b)}{\Gamma(a)% \Gamma(b)}x^{a-1}(1-x)^{b-1}0<x<1\\ 0\\ \end{array}}\right\}$ (2)

In Eq. (2), $\Gamma(r)=\int_{0}^{+\infty}x^{r-1}e^{-x}dx$ . When the distribution parameters and values are different, the density function curve of the Beta distribution has different characteristics, and the density function curve of the Beta distribution is shown in Fig. 3 [11].

Figure 3.

The change of density function curve when parameters are taken from different values.

When the distribution of Beta is a single peak, it is more suitable to fit the random changes of the index of fuzzy set distance in this paper, and which is shown in Figs 4 and 5.

Figure 4.

Distribution density function of Beta, ( $a=$ 2, $b=$ 40, 20, 10, 5, 2).

Figure 5.

Distribution density function of Beta, ( $a=$ 2, 5, 10, 20, 40, $b=$ 5).

With the decrease of $b$ , the peak of the density function curve is moved to the right, and the function is reduced from the left deviation. With the decrease of $a$ , the peak is moved to the left, the function decreases from the right deviation and the kurtosis is smaller. When the parameters $a$ and $b$ are relatively small, the kurtosis is smaller and the variance is larger. When $a$ and $b$ are large, the kurtosis is larger and the variance is smaller. The changes reflect that the Beta distribution density function is symmetric on the parameters $a$ and $b$ [12]. Mathematical expectation of random variables in Beta distribution are shown below:

$\displaystyle EX=a_{1}=\frac{a}{a+b}$ (3)

When $b/a\to\infty$ , the mean of the overall distribution tends to be 0. When $b/a\to 0$ , the mean tends to be 1. When $b=a$ , the mean is 1/2.

Variance of random variables in Beta distribution are shown below:

$\displaystyle\textit{VarX}=\mu_{2}=\frac{ab}{(a+b)^{2}(a+b+1)}$ (4)

It can be shown that the influence of the parameters $a$ and $b$ is equivalent, and the greater the sum of the variance, the smaller the variance of the overall distribution [13].

Skewness coefficient are shown below:

$\displaystyle\beta_{1}=\frac{\mu_{3}}{\mu_{2}^{3/2}}=\frac{2(b-a)\sqrt{a+b+1}}% {(a+b+2)\sqrt{ab}}$ (5)

When $a=b$ , the skewness coefficient of the overall distribution is 0. When $b>a$ , $\beta_{1}>0$ , distribution is right-hand deviation. When $b<a$ , $\beta_{1}<0$ , which is left-hand deviation. The coefficient of skewness varies with $a$ and $b$ . Kurtosis coefficient are shown below:

$\displaystyle\beta_{2}=\frac{\mu_{4}}{\mu_{2}^{2}}=\frac{3(2a^{2}+2b^{2}-2ab+a% ^{2}b+ab^{2})(a+b+1)}{ab(a+b+2)(a+b+3)}$ (6)

It can be shown that the parameters $a$ and $b$ the influence on the kurtosis coefficient of the overall distribution are equivalent [14].

3. Research method

3.1 Simulation analysis of maximum likelihood estimation under Beta distribution

The foundation for control chart based on index of fuzzy set distance is to estimate the estimators of parameters $a$ and $b$ under Beta distribution. The samples $u_{1},u_{2},\cdots,u_{n}$ , is taken from $\textit{Beta}(a,b)$ , the maximum likelihood function of the Beta distribution is be shown below [15]:

$\displaystyle L(a,b)=\left(\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\right)^{n}% \prod\limits_{i=1}^{n}{u_{i}^{a-1}}\prod\limits_{i=1}^{n}{(1-u_{i})^{b-1}}$ (7) $\displaystyle\ln L=n\ln\left(\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\right)+(a-% 1)\sum\limits_{i=1}^{n}{u_{i}}+(b-1)\sum\limits_{i=1}^{n}{(1-u_{i})}$ (8)

Using Matlab, random number can be generated under Beta distribution. When parameter of Beta distribution, $a=$ 1.5 and $b=$ 11, the sample statistics of parameters $a$ and $b$ are simulated, which are shown as Tables 1 and 2.

Table 1

The sample statistics of parameter $a$

Maximum likelihood	Mean	Median	Standard	Skewness	Kurtosis	Minimum	Maximum	Percentile	Percentile
estimation $n$	value		deviation	coefficien	coefficient			0.315%	99.865%
$n=$ 20	1.7287	1.6090	0.5708	1.5539	7.7914	0.6782	7.6738	0.7676	5.1938
$n=$ 50	1.5811	1.5447	0.3028	0.8455	4.3924	0.8503	3.4798	0.9481	2.8317
$n=$ 100	1.5446	1.5267	0.2046	0.5875	3.7455	0.9887	2.6232	1.0571	2.3767
$n=$ 1000	1.5038	1.5021	0.0621	0.1909	3.1990	1.2734	1.7110	1.3252	1.6952
$n=$ 5000	1.5006	1.5004	0.0275	0.0564	2.9941	1.4080	1.6062	1.4187	1.5874

Table 2

The sample statistics of parameter $b$

Maximum likelihood	Mean	Median	Standard	Skewness	Kurtosis	Minimum	Maximum	Percentile	Percentile
estimation $n$	value		deviation	coefficien	coefficient			0.315%	99.865%
$n=$ 20	13.0325	12.0722	4.8791	1.6519	8.4877	4.1770	57.4055	4.8654	42.4793
$n=$ 50	11.7322	11.4254	2.5736	0.9015	4.7306	5.5471	28.8579	6.4039	22.9363
$n=$ 100	11.3967	11.2301	1.7472	0.6111	3.7184	6.5815	21.6125	7.3421	18.3382
$n=$ 1000	11.0379	11.0258	0.5207	0.1857	3.1102	9.3383	113.3059	9.6009	12.7573
$n=$ 5000	11.0042	11.0033	0.2321	0.0839	3.0497	10.1359	11.9659	10.3329	11.7538

When the sample size is large ( $n=$ 5000), based on order statistics of the maximum likelihood estimation method, the gap between Mean and median of sample estimates, and the parameter $a$ and $b$ is very small. It is proved that the maximum likelihood estimation based on the order statistics is better. It means that the random changes of fuzzy set distance index can be described by the Beta distribution.

3.2 Analysis of quality control chart under Beta distribution

The statistics $(\bar{\mu},\bar{\sigma})$ of fuzzy quality chart can be derived from of Beta distribution. If $\bar{\mu}$ and $\bar{\sigma}$ is independent, Joint distribution function is shown as below:

$\displaystyle P(\bar{\mu}\leqslant c,\bar{\sigma}\leqslant d)=P(\bar{\mu}% \leqslant c)P(\bar{\sigma}\leqslant d)$ (9)

In for Eq. (9), $c$ and $d$ is a certain constant, then it is be verified that the Simulation of $\bar{\mu}$ and $\bar{\sigma}$ is shown in Table 3. The overall distribution is Beta (1.5, 11), and the amount of the simulated subsample is 50, modeling 10000 times.

Table 3

The value of $P(\bar{\mu}\leqslant c,\bar{\sigma}\leqslant d)$ , $P(\bar{\mu}\leqslant c)$ and $P(\bar{\sigma}\leqslant d)$

$P(\bar{\mu}\leqslant c,\bar{\sigma}\leqslant d)$	C $=$ 0.09	C $=$ 0.10	C $=$ 0.11	C $=$ 0.12	C $=$ 0.13	C $=$ 0.14	C $=$ 0.15	$P(\bar{\sigma}\leqslant d)$
$d=$ 0.05	0.0012	0.0038	0.0044	0.0049	0.0049	0.0049	0.0049	0.0049
$d=$ 0.06	0.0037	0.0195	0.0442	0.0561	0.0586	0.0588	0.0588	0.0588
$d=$ 0.07	0.0047	0.0394	0.1366	0.2263	0.2616	0.2696	0.2707	0.2708
$d=$ 0.08	0.0052	0.0458	0.1986	0.4073	0.5463	0.5881	0.5949	0.5958
$d=$ 0.09	0.0053	0.0474	0.2166	0.4942	0.7327	0.8296	0.8527	0.8554
$d=$ 0.10	0.0053	0.0474	0.2188	0.5125	0.7854	0.9213	0.9616	0.9679
$d=$ 0.11	0.0053	0.0474	0.2190	0.5146	0.7931	0.9378	0.9860	0.9949
$P(\bar{\mu}\leqslant c)$	0.0053	0.0474	0.2190	0.5148	0.7937	0.9396	0.9898

In Table 3, it is be shown that $P(\bar{\mu}\leqslant c,\bar{\sigma}\leqslant d)\neq P(\bar{\mu}\leqslant c)P(% \bar{\sigma}\leqslant d)$ . According to the principle of Shewhart control chart, the control chart of $\bar{\mu}$ and $\bar{\sigma}$ are determined in steady state at the same time, the production process is controlled in statistic stability. There is an exception both the control charts, and it is abnormal. Another control chart is no longer available even if there is no exception. The process must be readjusted until the two chart are in steady state at the same time.

From the above results, by using the $3\sigma$ principle of the Shewhart control chart [16], the center line and the control limit of $\bar{\mu}$ control chart are shown as:

$\displaystyle\textit{UCL}_{\bar{\mu}}=\bar{\mu}_{U}$ $\displaystyle\textit{CL}_{\bar{\mu}}=\bar{\mu}_{M}$ (10) $\displaystyle\textit{LCL}_{\bar{\mu}}=\bar{\mu}_{L}$

Similarly, the center line and the control limit of $\bar{\sigma}$ control chart are shown as:

$\displaystyle\textit{UCL}_{\bar{\mu}}=\bar{\sigma}_{U}$ $\displaystyle\textit{CL}_{\bar{\mu}}=\bar{\sigma}_{M}$ (11) $\displaystyle\textit{LCL}_{\bar{\mu}}=\bar{\sigma}_{L}$

Under Beta (1.5, 11), the amount of the simulated subsample is 50, modeling 10000 times. The change of the Statistic $\bar{\mu}$ and $\bar{\sigma}$ are shown in Figs 4 and 5.

Figure 6.

Frequency histogram of $\bar{\mu}$ .

Figure 7.

Frequency histogram of $\bar{\sigma}$ .

In the Figs 6 and 7, the median of mean estimators is 0.1196 and the median of variance estimators is 0.0871. In Beta (1.5, 11), $\mu=$ 0.12 and $\sigma=$ 0.0884. It is slightly smaller than the true value of the parameter. The 99.865% and 0.135% quantiles of mean estimators are respectively 0.1576 and 0.0862. The 99.865% and 0.135% quantiles of variance estimators are respectively 0.1256 and 0.0564. When the process in steady state, the index of fuzzy set distance obeying Beta (1.5, 11). Generating 10000 sets random number, each subgroup is $n$ . Then 10000 estimators of $\bar{\mu}$ and $\bar{\sigma}$ can be calculated separately, and a run length of control chart $\bar{\mu}$ and $\tilde{\sigma}$ can also be calculated. Only when the process is abnormal, the ARL of control chart is obviously less than ARL (0) in steady state.

3.3 Optimized analysis on process capability based on lose of product quality

According Boyles (2002) and Hubele (2006), it is restrictive to analyze process capability by single process capability index, more considerations need to get involved [17]. According to the content of research, different values of $\mu$ and $\sigma$ have significant reflection on process capability. Thus, the method of analyzing process capability from Hubele (2004) and the method of optimized analysis on process capability from Flaig (2010) can be used for reference [18].

Under the Asymmetric quality loss function given by Li and Chou (2001)

$\displaystyle L(x)=\left\{\begin{array}[]{ll}k_{1}(x-L)^{2}&x\leqslant L\\ k_{2}(x-L)^{2}&x\geqslant L\\ \end{array}\right.$ (12)

In this Equation, $k_{1}=A_{1}/d_{L}^{2}$ , $k_{2}=A_{2}/d_{U}^{2}$ . $A_{1}$ is the quality loss at the lower limit of common difference. $A_{2}$ is the quality loss at the upper limit of common difference. Obviously, in the common situation of asymmetric common difference, $k_{1}\neq k_{2}$ . For the situation that fuzzy set distance follows $\textit{Beta}(a,b)$ distribution, the average quality loss is [19]:

$\displaystyle E[L(x)]=\int_{0}^{T_{U}}k_{2}x^{2}f(x;a,b)dx$ (13)

Obviously, the lower the average quality loss is, the better the process capability is. In Eq. (13), $k_{2}$ is constant, the value of $E[L(x)]$ is determined by $a$ and $b$ from $\textit{Beta}(a,b)$ . In other words, it is determined by mean value $\mu$ and standard deviation $\sigma$ . In Eq. (12), the value of $\mu$ should approach $L$ which is the ideal value of fuzzy set distance. Ideally, $\mu$ and $L$ should equal. Then, when $\sigma$ decreases, $E[L(x)]$ decreases. The relationship among $\mu$ , $\sigma$ , and $E[L(x)]$ is shown in Eq. (14) [20].

$\displaystyle E[L(x)]=\int_{0}^{T_{U}}k_{2}x^{2}f(x;a,b)dx$ (14)

When $T_{U}=1$ , $E[L(x)]=k_{2}(\mu^{2}+\sigma^{2})$ . Then as $\mu$ and $\sigma$ decrease, $E[L(x)]$ also decreases. When $T_{U}<1$ , get the Equation below.

$\displaystyle E[L(x)]=\frac{a(a+1)}{(a+b)(a+b+1)}C(a+2,b;T_{U})k_{2}$ (15)

In Eq. (15), $C(a,b;T_{U})=\int_{0}^{T_{U}}f(x;a,b)dx$ . Under the consideration of Eq. (14), $E[L(x)]$ can be transformed into the function of $\mu$ and $\sigma$ . Due to the complexity of function calculation, the relationship is shown in Fig. 8.

Figure 8.

The curve surface of $E[L(x)]$ based on Eq. (15).

In Fig. 8, as $\mu$ and $\sigma$ decreases, $E[L(x)]$ decreases. In addition, the marginal impact of average quality loss decline is similar.

3.4 Optimized analysis on process capability based on product value

Based on Taguchi quality loss function, product value can be seen as the level of process capability. In other words, the higher the product value is, the higher the level of process capability is. The function of product value is shown below [21].

$\displaystyle E[W(X)]=\int_{0}^{T_{U}}\left(W-\frac{W}{T_{U}^{2}}x^{2}\right)f% (x;a,b)dx$ (16)

In Eq. (16), $W_{L}$ and $T_{U}^{2}$ is constant. Thus, $\mu$ and $\sigma$ of $\textit{Beta}(a,b)$ determine the value of $E[W(X)]$ . In other words, $\mu$ and $\sigma$ determine the value of $E[W(X)]$ . So, in order to get higher product value, some effects should be taken on $\mu$ and $\sigma$ . The get the following Equation [22].

$\displaystyle E[W(X)]=\textit{WC}(a,b;T_{U})-\frac{W}{T_{U}^{2}}\frac{a(a+1)}{% (a+b)(a+b+1)}C(a+2,b;T_{U})$ (17)

In Eq. (17), then $C(a,b;T_{U})=\int_{0}^{T_{U}}f(x;a,b)dx$ . Under the consideration of Eq. (17), $E[W(X)]$ can be transformed into the function of $\mu$ and $\sigma$ . This function is shown in Fig. 9.

Figure 9.

Curved surface chart of Eq. (17).

In Fig. 9, as $\sigma$ decreases, $E[W(X)]$ decreases. When $\mu$ decreases, $E[W(X)]$ increases. The margin impact of product value by decline of $\mu$ is larger than the margin impact of product value by decline of $\sigma$ . Although, their order of magnitude are different. Under the effect of $\mu$ and $\sigma$ , in asymmetric deviation the level of quality improvement under the change of $\mu$ is higher than the level of quality improvement under the change of $\sigma$ .

Table 4

The sample data of high-end manufacturing process

Serials	Number of	Number of	Number of	Number of	Number of
	standard class	secondary class	third class	inferior class	samples
1	121	46	14	5	186
2	130	51	16	7	204
3	145	48	18	8	129
4	139	37	23	4	213
5	109	61	29	3	202
6	98	29	40	11	188
7	85	44	8	10	157
8	105	78	11	9	203
9	118	58	10	8	194
10	140	79	28	7	254
11	110	64	25	8	197
12	136	49	12	6	203
13	130	45	19	5	199
14	115	35	15	12	177
15	120	25	27	10	182
16	139	68	8	2	157
17	106	75	31	3	215
18	140	51	35	6	232
19	125	23	18	12	178
20	118	39	26	15	198
21	127	18	19	5	169
22	167	55	17	9	248
23	150	70	19	7	246
24	181	33	9	6	229
25	117	28	31	11	187
26	130	35	23	31	219
27	167	39	21	2	229
28	85	41	26	17	179
29	129	58	10	15	212
30	110	61	20	7	198

4. Results and discussion

4.1 Calculation case

The data are from real high-end manufacturing process. This company’s products are classified by the quality experts at four levels: Standard class (no defect or only one invisible small defect), Second class (only one visible small defect and do not affect the use), Third class (only one visible big defect and do not affect the use), Inferior class (existing defect and affect the use).

In the production process, the data in per pried of half hour will be gathered, and the samples number of inspection is variable. An example date list as follow.

In the case, there are four product classes. The closest to the center value which represents the optimal product quality is Standard classes. The index of distance must be used to measure the distance from the center value as fuzzy quality index. The fuzzy quality characteristic value is close to the statistical steady state, which is a kind of control chart based on fuzzy quality indicators for monitoring object index control chart. The normal distribution is not applicable, if the control chart is used to monitor the process. The Beta distribution is applicable in the application of control chart.

To transfer these classified dates to indexes of the fuzzy similarity, we firstly set four quality classes (standard class, secondary class, third class, inferior class) respectively corresponding to four sections of fuzzy similarity index: [0, 0.25], [0.25, 0. 5], [0.5, 0.75], [0.75, 1]. Moreover, we also set the fuzzy similarity indexes conforming to the uniform distribution. Based on above conditions, we can apply these dates to estimate the parameters of the Beta distribution, establish the fuzzy similarity control chart for $\bar{\mu}$ and $\bar{\sigma}$ .

Based on the weights of the range data, and of the every sample can be evaluated, and the formulas are as follows:

$\displaystyle\bar{X}=\int_{0}^{0.25}{\frac{\alpha_{1}x}{0.25}}d_{x}+\int_{0.25% }^{0.5}{\frac{\alpha_{2}x}{0.25}d_{x}}+\int_{0.5}^{0.75}{\frac{\alpha_{3}x}{0.% 25}}d_{x}+\int_{0.75}^{1}{\frac{\alpha_{4}x}{0.25}}d_{x}$ $\displaystyle S=\int_{0}^{0.25}{\frac{\alpha_{1}(x-\bar{X})}{0.25}^{2}d_{x}+% \int_{0.25}^{0.5}{\frac{\alpha_{2}(x-\bar{X})^{2}}{0.25}}}d_{x}+\int_{0.5}^{0.% 75}{\frac{\alpha_{3}(x-\bar{X})^{2}}{0.25}}d_{x}{}+\int_{0.75}^{1}{\frac{% \alpha_{4}(x-\bar{X})^{2}}{0.25}d_{x}}$ (18)

In these, $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$ are frequencies of various categories products in every sample, then the formulas can be simplified:

$\displaystyle\bar{X}=0.125\alpha_{1}+0.375\alpha_{2}+0.625\alpha_{3}+0.875% \alpha_{4}$ $\displaystyle S=\frac{1}{0.75}[\alpha_{1}\bar{X}^{3}+(\alpha_{1}-\alpha_{2})(0% .25-\bar{X})^{3}+(\alpha_{2}-\alpha_{3})(0.5-\bar{X})^{3}+(\alpha_{3}-\alpha_{% 4})(0.75-\bar{X})^{3}{}+\alpha_{4}(1-\bar{X})^{3}]$ (19)

According to above function, we can calculate and. The result is listed as following.

Table 5

$\bar{X}$ and $S$ of closeness index on Porcelain product

Samples	$\bar{X}$	$S$	Samples	$\bar{X}$	$S$
1	0.235623	0.193874	16	0.255863	0.183472
2	0.297324	0.203840	17	0.292247	0.204648
3	0.224609	0.218474	18	0.237485	0.183589
4	0.250001	0.194748	19	0.203788	0.214648
5	0.252376	0.224243	20	0.217831	0.204758
6	0.204723	0.203764	21	0.262238	0.214855
7	0.273271	0.235758	22	0.257134	0.234658
8	0.351093	0.189664	23	0.294427	0.194748
9	0.251200	0.214758	24	0.241359	0.204794
10	0.263780	0.264264	25	0.230967	0.217985
11	0.273633	0.194764	26	0.211738	0.224637
12	0.214529	0.225754	27	0.27837	0.189581
13	0.292389	0.194648	28	0.237846	0.204648
14	0.243261	0.204783	29	0.212734	0.225759
15	0.283680	0.215324	30	0.264327	0.194758

$\bar{\mu}$ control chart and $\bar{\sigma}$ control chart. Based on $\bar{X}$ and $S$ listed above, we use the data to calculate centerline, upper control limits and lower control limits of $\bar{\mu}$ chart and $\bar{\sigma}$ chart. The calculate centerline, upper control limits and lower control limits for $\bar{\mu}$ control chart:

$\displaystyle\textit{UCL}_{\bar{\sigma}}=0.3660$ $\displaystyle\textit{CL}_{\bar{\sigma}}=0.2537$ $\displaystyle\textit{LCL}_{\bar{\sigma}}=0.1413$

The calculate centerline, upper control limits and lower control limits for $\bar{\sigma}$ control chart:

$\displaystyle\textit{UCL}_{\bar{\sigma}}=0.2753$ $\displaystyle\textit{CL}_{\bar{\sigma}}=0.2092$ $\displaystyle\textit{LCL}_{\bar{\sigma}}=0.1432$

Use the data of the Table 5 as sample dates to trace points in the $\tilde{\mu}$ control chart and $\bar{\sigma}$ control chart as following Fig. 10.

Figure 10.

$\bar{X}$ and $S$ control chart of fuzzy set distanc.

In the Fig. 10, there are shown that the points all are between the upper control limits and lower control limits. The process is stable on the whole. We can understand if the process is controlled from the $\bar{\mu}$ control chart and $\bar{\sigma}$ control chart.

4.2 Process capability index under Beta distribution

According to abnormal distribution, analyzing process capability indexes. Set $a=$ 0.135%. Then get and correspond to 0.135% and 99.865% quintiles of fuzzy set distance, respectively. Then process capability index under Beta distribution ${C}^{\prime}_{P}$ is [23]:

$\displaystyle{C}_{p}^{\prime}=\frac{T}{\xi_{1-\alpha}-\xi_{\alpha}}$ (20)

In Eq. (20), $T$ is the selection range of production. Based on control chart of fuzzy set distance, technical standard is usually $0_{-0}^{+T_{U}}(T_{U}\leqslant 1)$ , and $T=T_{U}$ . $T_{U}$ is the upper limit of common difference. When the middle of common difference and the middle of quality feature distribution are different, define skew process capability index ${C}^{\prime}_{\textit{PK}}$ as below [24]:

$\displaystyle{C}^{\prime}_{\textit{PK}}=\frac{T-\left|{\xi_{0.5}-M}\right|}{% \xi_{1-\alpha}-\xi_{\alpha}}$ (21)

In Eq. (21), $\xi_{0.5}$ is the median of Beta distribution. $M$ is the tolerance center. According to these formulas, when quality characteristic value follows normal distribution, ${C}^{\prime}_{P}$ and ${C}^{\prime}_{\textit{PK}}$ equal to $C_{P}$ and $C_{\textit{PK}}$ , respectively.

In the process of producing, fuzzy set distance is variable. According to this feature. Based on Beta distribution, the relations among the mean value of fuzzy set distance, the standard deviation of fuzzy set distance, and distribution index $a$ and $b$ are shown below:

$\displaystyle a=\frac{\mu\left({\mu-\mu^{2}-\sigma^{2}}\right)}{\sigma^{2}},b=% \frac{\left({1-\mu}\right)\left({\mu-\mu^{2}-\sigma^{2}}\right)}{\sigma^{2}}$ (22)

When $\mu$ and $\sigma$ change, process capability index changes, then analyze the feature of process capability index.Assume common difference $T=$ 0.5, in other words, technical standard is $0_{0}^{+0.5}$ . Then get the curve surface chart of ${C}^{\prime}_{P}$ , when $\mu$ and $\sigma$ changes. When $\mu$ and $\sigma$ changes, is shown in Fig. 11.

When the lower deviation of technical standard $d_{L}=$ 0, and the upper deviation of technical standard $d_{U}=$ 0.5, $\mu$ and $\sigma$ should follow some restrictions. In Fig. 11, the lower curve surface reveals the effects of these restrictions. When $\sigma$ decreases, ${C}^{\prime}_{P}$ increases. It means the change of ${C}^{\prime}_{P}$ reflects process capability. Meanwhile, when $\mu$ decreases to the ideal value of fuzzy set distance, ${C}^{\prime}_{P}$ increases at the first, then decreases a little bit. And lower the value of $\mu$ is, better the process capability is. According to the analysis above, the change of ${C}^{\prime}_{P}$ can reveal the change of process capability in some kind, but not properly. Therefore, ${C}^{\prime}_{P}$ is restrictive when used as the index of revealing process capability.

By using the same method, get frequency distribution of ${\hat{C}}^{\prime}_{P}$ and ${\hat{C}}^{\prime}_{\textit{PK}}$ relatively. Furthermore, after simulation, get the sample statistic value of ${\hat{C}}^{\prime}_{P}$ and ${\hat{C}}^{\prime}_{\textit{PK}}$ , shown in Table 6.

Table 6

Estimate value of ${\hat{C}}^{\prime}_{P}$ and ${\hat{C}}^{\prime}_{\textit{PK}}$

Statistic	C $=$ 0.09	C $=$ 0.10	C $=$ 0.11	C $=$ 0.12	C $=$ 0.13	C $=$ 0.14	C $=$ 0.15
${\hat{C}}^{\prime}_{P}$ ( $n=$ 20)	1.06189	1.02831	0.21676	0.974853	4.57097	0.57851	2.25435
${\hat{C}}^{\prime}_{P}$ ( $n=$ 50)	1.02568	1.01278	0.12713	0.653427	3.80332	0.67745	1.75105
${\hat{C}}^{\prime}_{P}$ ( $n=$ 100)	1.01439	1.00853	0.08798	0.457785	3.39333	0.77271	1.40713
${\hat{C}}^{\prime}_{P}$ ( $n=$ 1000)	1.00134	1.00080	0.027296	0.174460	3.10742	0.90594	1.12863
${\hat{C}}^{\prime}_{\textit{PK}}$ ( $n=$ 20)	0.74066	0.72389	0.14259	0.827304	4.31304	0.38870	1.57933
${\hat{C}}^{\prime}_{\textit{PK}}$ ( $n=$ 50)	0.71707	0.71024	0.08494	0.503901	3.52968	0.44853	1.15525
${\hat{C}}^{\prime}_{\textit{PK}}$ ( $n=$ 100)	0.70913	0.70616	0.05928	0.377703	3.33335	0.53376	0.99427
${\hat{C}}^{\prime}_{\textit{PK}}$ ( $n=$ 1000)	0.70058	0.70016	0.01847	0.127817	3.05023	0.64028	0.78421

Figure 11.

When $\mu$ and $\sigma$ get different value, the curve surface of ${C}^{\prime}_{P}$ .

This paper analyzes process capability by the features of random variable in fuzzy set distance. Based on reference [25], extend the definition of process capability index ${\hat{C}}^{\prime}_{P}$ and $C_{\textit{PK}}$ . Then, get two process capability indexes ${\hat{C}}^{\prime}_{P}{\hat{C}}^{\prime}_{\textit{PK}}$ , which follow Beta distribution. However, these indexes are restrictive for leveling process capability. Then, with optimized analysis on process capability of rate of rejection, product quality loss, and product value, get the way to optimize overall parameters of Beta distribution. At last, introduce ${\hat{C}}^{\prime}_{P}$ and ${\hat{C}}^{\prime}_{\textit{PK}}$ as the estimate value of and relatively. Due to complexity of these estimate values, Monte Carlo method is introduced to get distribution simulation, then take deep analysis on these distribution.

5. Conclusion

By the above comprehensive analysis and evaluation, the conclusion can be summed up, as following:

For the control of fuzzy quality attributes, normal distribution is not applicable. It is necessary to design unconventional fuzzy quality control charts.

It is feasible that Taguchi’s quality loss function is as the theoretical support of fuzzy quality control chart.

Beta distribution as statistical basis of fuzzy control model is applicable. When the range of similarity index is in [0, 1], it is a kind of control chart based on fuzzy quality indicators for monitoring object index.

The sufficient mathematical calculus and statistical simulation is used to prove the correctness and rationality of the proposed hypothesis.

This paper conducts the fuzzy quality control under Beta distribution, using Taguchi quality loss function, and fuzzy set distance method. And it is discussed that process is controlled or not based on case study. It can be drawn a conclusion that the fuzzy control charts of fuzzy set distance are applicable based on Taguchi quality loss function.

Footnotes

Acknowledgments

Project supported by Social Science Foundation of Beijing, China (14JGC105); the project of “real training” for the 2017–2018 academic year of the Beijing Municipal Education Committee; the project of Beijing Knowledge Management Base (71F1810916).

References

Cheng

, Fuzzy process control: construction of control charts with fuzzy numbers [J], Fuzzy Sets and Systems 154 (2003), 287–303.

Taleb

and Limam

, Fuzzy and probabilistic control charts [J], International Journal of Production Research 40 (2012), 2849–2863.

Franceschini

and Romano

, Control chart for linguistic variables: a method based on the use of linguistic quantifiers [J], Journal of Production 37 (1999), 3791–3801.

Thomas Foster

, Design for an SPRT control scheme based on linguistic data [J], International Journal of Production Research 41(6) (2003), 1299–1309.

Wang

L.R.

and Rowlands

, An approach of fuzzy logic evaluation and control in SPC [J], Quality and Reliability Engineering International 16 (2000), 91–98.

Cheng

, Fuzzy process control: construction of control charts with fuzzy numbers [J], Fuzzy Sets and Systems 154 (2015), 287–303.

Imm

B.-Y.

Lee

J.H.

and Lee

S.H.

, Sensory quality index (SQI) for commercial food products, Food Quality and Preference 22 (2011), 748–752.

Gülbay

Kahraman

and Ruan

, Probability theory and fuzzy logic are complementary rather than competitive [J], Technometrics 37(3) (1995), 271–276.

Cheng

, Fuzzy process control: construction of control charts with fuzzy numbers [J], Fuzzy Sets and Systems 154 (2015), 287–303.

10.

Franceschini

and Romano

, Control chart for linguistic variables: a method based on the use of linguistic quantifiers [J], Journal of Production 37(16) (1999), 3791–3801.

11.

Discriminant models based on sensory evaluations: Single assessors versus panel average, P.M. Granittoa, b, F. Biasioli, Food Quality and Preference, 2015.

12.

Zadeh

L.A.

, Probability theory and fuzzy logic are complementary rather than competitive [J], Technometrics 37(3) (1995), 271–276.

13.

Zhou

and Sun

, Nussbaum-based fuzzy adaptive nonlinear fault-tolerant control for hypersonic vehicles with diverse actuator faults, Aerospace Science and Technolgy 71 (2017), 432–440.

14.

Woodall

W.H.

Tsui

K.L.

and Tucker

G.R.

, A review of statistical and fuzzy control charts based on categorical data [J], Frontiers in Statistical Quality Control 5 (2007), 83–89.

15.

and Nie

, Robust fuzzy quality function deployment based on the mean-end-chain concept: Service station evaluation problem for rail catering services, European Journal of Operational Research 263 (2017), 974–995.

16.

Sakthivel

Saravanakumar

Y.K.

and Anthoni

S.M.

, Finite-time resilient reliable sampled-data control for fuzzy systems with randomly occurring uncertainties, Fuzzy Sets and Systems 329 (2017), 1–18.

17.

Thomsona

D.M.H.

Crocker

and Marketo

C.G.

, Linking sensory characteristics to emotions: An example using dark chocolate [J], Food Quality and Preference 12 (2010), 1117–1125.

18.

Senturk

and Erginel

, Development of fuzzy and control charts using a-Cuts [J], Information Sciences 179 (2009), 1542–1551.

19.

Jaeger

S.R.

and Axten

L.G.

, PLS discriminant analysis applied to conventional sensory profiling data [J], Food Quality and Preference 235 (2016), 234–241.

20.

F.-J.

Low

C.-Y.

and Cheng

S.-S.

, Design for an SPRT control scheme based on inguistic data [J], International Journal of Production Research 41 (2013), 1299–1309.

21.

Gülbay

and Kahramana

, An alternative approach to fuzzy control charts: Direct fuzzy approach [J], Information Sciences 177 (2007), 1463–1480.

22.

Senturk

and Erginel

, Development of fuzzy and control charts using a-Cuts [J], Information Sciences 179 (2016), 1542–1551.

23.

Gülbay

Kahraman

and Ruan

, a-Cut fuzzy control charts for linguistic data [J], International Journal of Intelligent Systems 19 (2014), 1173–1195.

24.

Castagliola

Celano

Costa

and Fichera

, Study on fuzzy and probabilistic control charts [J], International Journal of Production Research 40(12) (2002), 2849–2863.

25.

Miao

Wei

C.Q.

Z.F.

and Jiang

Z.B.

, A robust MEWMA control chart based on FAST-MCD algorithm [J], Advanced Materials Research 13 (2013), 562–564, 2013.