Construct Six Sigma DMAIC Improvement Model for Manufacturing Process Quality of Multi-Characteristic Products

Abstract

After a product has undergone a manufacturing process, it usually has several important quality characteristics. When the process quality of all quality characteristics meets the requirements of the quality level, the process quality of the product can be guaranteed to satisfy customers’ needs. A large number of studies have pointed out that good process quality can raise product yield and product value; at the same time, it can reduce the ratio of rework and scrap, achieve the effect of energy saving and waste reduction, and contribute to the sustainable operation of enterprises as well the environment. Since the six sigma method combines the statistical analysis method of manufacturing cost and production data, it is a useful tool for process improvement and process quality enhancement. Therefore, this paper adopted the six sigma-define, measure, analyze, improve and control (DMAIC) improvement process to lift the manufacturing process quality of multi-characteristic products. Besides, the Taguchi process capability index is one of the commonly used tools for quality assessment in the industry. Not only can it reflect the process loss, but it also can ensure the process yield when the index value is large enough. Consequently, this paper discussed the relationship between the Taguchi process capability index and the six sigma quality level. Meanwhile, the entire six sigma DMAIC improvement process was built on the basis of the process capability index and developed by the method of statistical quality control. Hence, the proposed method is very convenient for process engineers to apply, as well as is helpful for enterprises to move toward the goal of smart manufacturing and sustainability.

1. Introduction

Numerous studies have indicated that a product usually has multiple important quality characteristics after undergoing a manufacturing process. When the process quality of all quality characteristics reaches the requirements of the quality level, the process quality of the product can be guaranteed to meet customers’ needs [1,2]. The so-called products here include important components or some semi-finished products. In addition, many studies have also pointed out that good process quality not only can level up product yield and product value, but can also reduce the ratio of rework and scrap. Moreover, it can also enhance the availability rate and prolong the maintenance time, thereby achieving the effect of energy saving and waste reduction, which will contribute to the sustainable management of enterprises and the environment [3,4,5,6,7,8]. Accordingly, how to promote the processing quality of products with multiple quality characteristics is apparently an important issue.

The six sigma method is a useful tool used by the industry to improve the process as well as the process quality. It was originally proposed by Motorola in 1986 [9,10]. In terms of the manufacturing process, the six sigma method combines the statistical analysis method of manufacturing cost and production data, enabling process engineers to define, measure, analyze, improve, and control, abbreviated as the DMAIC improvement process to enhance process and process quality [11,12,13]. After Motorola, American General Electric Company (GE)also began to promote this method vigorously in 1995, not only achieving remarkable results, but also doubling its operating performance. Subsequently, Ford Motor Company of the United States and ASUS of Taiwan also followed up one after another and began to adopt this management model, so that they greatly enhanced their performance of operation and management.

Under the assumption of normality, let random variable $X$ represent the process distribution of a quality characteristic, and then obey the normal distribution with mean $\mu$ and standard deviation $\sigma$, denoted as $X~N\left(\mu ,\sigma \right)$. According to many studies, when the standard deviation ($\sigma$) of the process is one-sixth of the tolerance ($\sigma =d/6$), and the process average ($\mu$) deviated from the target value ($T$) is less than $1.5 \sigma$ ($\left|\mu -T\right|\le 1.5 \sigma$), the process quality of the product can be said to meet the quality level of six sigma ($6 \sigma$). $d$ is a half of the length of the specification interval, i.e.,$d=\left(USL-LSL\right)/2$; process target value $T$ is the midpoint of the specification limits, i.e., $T=\left(USL+LSL\right)/2$, where LSL is the lower specification limit and USL is the upper specification limit. At this time, the process defect rate is 3.4 parts per million [14,15].

In addition to the six sigma method, the process capability index (PCI) is also a process quality evaluation tool commonly used in the industry [16,17,18,19,20,21,22,23,24,25,26]. It is an evaluation and analysis tool of process quality for internal engineers, as well as a convenient and effective communication tool for business personnel and suppliers [27]. According to the Taguchi loss function, Chan et al. [28] proposed the Taguchi process capability index $C_{pm}$, which can fully reflect the process loss. It is one of the most widely used process quality assessment tools in the manufacturing industry [28,29,30]. Its definition is illustrated as follows:

$$C_{pm}=\frac{d}{3\sqrt{{\left(\mu -T\right)}^2+{\sigma }^2}}$$

(1)

where ${\left(\mu -T\right)}^2+{\sigma }^2=E{(X-T)}^2$, the expected value of Taguchi loss.

Numerous studies have pointed out that there is an inequality relationship [31,32] between index $C_{pm}$ and process yield when process capability is sufficient. This inequality relationship is described as follows:

$$Yield\%\ge 2 \Phi (3 C_{pm})-1$$

(2)

where $\Phi (\cdot )$ is the standard normal cumulative distribution function (CDF). As mentioned above, after a product is developed, designed and manufactured, it usually has multiple important quality characteristics. When the process quality of all quality characteristics meets the requirements of the quality level, the process quality of the product can be guaranteed to satisfy the customers’ needs. As the value of index $C_{pm}$ is larger, not only can it reflect the process loss, but it also can ensure the process yield when the index value is large enough. Obviously, the Taguchi process capability index is a process capability index that is very suitable for practical application. Next, this paper will develop a process capability analysis chart using $C_{pm}$, which can evaluate all the quality characteristics of a product at the same time, so that it is convenient for the industry to grasp and monitor the complete product quality. Besides, through the relationship between $C_{pm}$ and the six sigma quality level, the industry can master the product process, as well as the six sigma quality level. Therefore, based on this index, this paper develops the six sigma DMAIC improvement process, so that it can be offered to the industry for reference while improving process quality.

Additionally, when the process quality is enhanced, the ratio of rework and scrap can be reduced, which not only can decrease production costs and increase product value, but can also achieve the effect of energy saving and waste reduction. As a result, the industry can improve its competitiveness and achieve the goal of sustainable operation.

The remainder of this paper is organized as follows. Section 2 explores the relationship between the Taguchi process capability index and the six sigma quality level. Meanwhile, the tolerances of all quality characteristics are converted into $0\pm 1$. In Section 3 and Section 4, the six sigma DMAIC improvement process is developed based on the Taguchi process capability index. Section 3 introduces step 1 (Define) and develops the method in step 2 (Measure). Next, Section 4 develops the model of Analyze, Improve, and Control. Then, this paper uses an application example to demonstrate the six sigma DMAIC improvement process in Section 5. Finally, Section 6 is the conclusion.

2. Taguchi Capability Index and Six Sigma Quality Level

Without loss of generality, this paper assumed that the product has q important quality characteristics. Let random variable $X_h$ represent the process distribution of the h^th quality characteristic. Under the assumption of normal process, random variable $X_h$ obeys the normal process distribution with mean ${\mu }_h$ and standard deviation ${\sigma }_h$, denoted as $X_h$~$N\left({\mu }_h,{\sigma }_h\right)$, h $=$ 1, 2, …, q. In order to develop a process capability evaluation analysis chart for multi-characteristic products for the six sigma DMAIC improvement process, this paper applied the variable transformation of standardized concepts and let random variable be expressed as follows:

$$Y_h=\frac{X_h-T_h}{d_h},$$

(3)

where $T_h$ is the target value and $d_h$ is the tolerance. Then, $Y_h$ obeys the normal process distribution with mean ${\delta }_h$ and standard deviation ${\gamma }_h$, denoted as $Y_h$~$N\left({\delta }_h,{\gamma }_h\right)$, where ${\gamma }_h$ is called the precision index, and ${\delta }_h$ is called the accuracy index, representing the ratio of the process deviation from the target value, displayed as follows:

$$\mathrm{precision} \mathrm{index}: {\delta }_h=\frac{{\mu }_h-T_h}{d_h};$$

(4)

$$\mathrm{accuracy} \mathrm{index}: {\gamma }_h=\frac{{\sigma }_h}{d_h}.$$

(5)

Obviously, the process tolerances of all quality characteristics are converted into $0\pm 1$, and the Taguchi capability index $C_{pmh}$, evaluating the h^th quality characteristic, can be expressed as follows:

$$C_{pmh}=\frac{1}{3\sqrt{{\delta }_h^2+{\gamma }_h^2}}.$$

(6)

Based on Chen et al. [33], the product Taguchi capability index can be defined below:

$$C_{pm}=\frac{1}{3} {\Phi }^{-1}\left\{1-{\displaystyle \sum _{h=1}^q\left(1-\Phi \left(3 C_{pmh}\right)\right)}\right\}.$$

(7)

When we require the value of the Taguchi process capability index $C_{pm}$ of the product to be at least $C_T$, then the value of the Taguchi process capability index for each quality characteristic can be required to be at least $C$. According to Equation (7), value $C$ can be expressed as follows:

$$C=\frac{1}{3} {\Phi }^{-1}\left\{1-\frac{1-\Phi \left(3 C_T\right)}{q}\right\}.$$

(8)

In addition, when the process quality level reaches $k$-sigma, then $\left|\delta \right|\le 1.5/k$ and $\gamma =1/k$. The relationship between the indices and the $k$-sigma quality level is illustrated as follows:

$$C_{pm}\ge C_{pm}\left(k\right)=\frac{1}{3\sqrt{{\left(1.5/k\right)}^2+{\left(1/k\right)}^2}}=\frac{k}{3\sqrt{3.25}}.$$

(9)

When

$$C_{pm}\left(k\right)=\frac{k}{3\sqrt{3.25}}=C_T.$$

(10)

then according to Equation (7), value $C$ can be re-expressed as follows:

$$C=\frac{1}{3} {\Phi }^{-1}\left\{1-\frac{1-\Phi \left(k/\sqrt{3.25}\right)}{q}\right\}.$$

(11)

According to the above-mentioned, when the process quality level of the product is required to reach $k$-sigma, the value of Taguchi process capability index $C_{pm}$ of the product is required to be at least $C_T$. In order to reach this requirement, the value of process capability index $C_{pmh}$ for each quality characteristic can be required to be at least $C$ as displayed in Equation (8). Values $C_T$ corresponding to $C$ with $k$-sigma quality level ($k=4,5,6$) under q quality characteristics are shown in

Table 1. Values $C_T$ corresponding to $C$ with $k$-sigma quality level ($k=4,5,6$) under q quality characteristics.

q	4 Sigma	5 Sigma	6 Sigma
1	0.74	0.92	1.11
2	0.83	0.99	1.17
3	0.87	1.03	1.21
4	0.91	1.06	1.23
5	0.93	1.08	1.25
6	0.95	1.10	1.27
7	0.97	1.11	1.28
8	0.98	1.13	1.29
9	0.99	1.14	1.30
10	1.00	1.15	1.31

below.

3. Define and Measure

As mentioned earlier, this paper introduces step 1 (Define) and develops the method in step 2 (Measure) in this section. First, step 1 (Define) is introduced as follows:

3.1. Define

Obviously, the center of this paper is based on the Taguchi process capability index, aiming at the manufacturing process of multi-characteristic products and developing a six sigma DMAIC process quality improvement model. Therefore, the improvement theme of step 1 is defined as follows:

To improve the process capability of multi-characteristic products, the specific operation in practice is to:

(1)

raise the value of Taguchi process capability index $C_{pm}$ of a product to at least $C_T$; and

(2)

increase the value of Taguchi process capability index $C_{pmh}$ of a quality characteristic to at least $C$.

According to the above-mentioned two definitions for practical operation, the method of step 2 (Measure) is developed as follows:

3.2. Measure

As mentioned above, Taguchi process capability index $C_{pm}$ can reflect both process loss and process yield, and it is one of the most widely used process quality assessment tools in the manufacturing industry. According to the above-mentioned, if the value of Taguchi process capability index $C_{pmh}$ for each quality characteristic is required to be at least $C$, then it can be ensured that the value of Taguchi process capability index $C_{pm}$ of the product is at least $C_T$. Therefore, this paper regarded the accuracy index as the abscissa (x-axis) and the precision index as the ordinate (y-axis) to establish a multi-characteristic process capability analysis chart (MPCAC/$C_{pmh}$), used as a measure tool for the six sigma DMAIC process quality improvement model. According to Equation (6), first let $x={\delta }_h$ and $y={\gamma }_h$, then

$$C_{pmh}\left(x,y\right)=\frac{1}{3\sqrt{x^2+y^2}}=C.$$

(12)

The contours of $C_{pmh}$ form a process capability analysis chart can be shown in Figure 1:

Clearly, the contour of $C_{pmh}=C$ and the x-axis ($y=0$) form an acceptable process capability block named $Z_A$ as follows:

$$Z_A=\left\{(x,y)|x^2+y^2\le 1/{(3 C)}^2,-\frac{1}{3 C}\le x\le \frac{1}{3 C},y\ge 0\right\}.$$

(13)

Since the index has unknown parameters, it must be estimated from a sample. Let $Y_{h,1},\dots ,Y_{h,n}$ be a random sample from $N\left({\delta }_h,{\gamma }_h\right)$, and then the estimators of ${\delta }_h$ and ${\gamma }_h$ can be shown as follows:

$${\delta }_h^{\ast }=\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{Y}_i} \mathrm{and} {\gamma }_h^{2\ast }=\frac{1}{n-1}{\displaystyle {\sum }_{i=1}^n{\left(Y_{h,i}-{\overline{Y}}_h\right)}^2}.$$

Under the assumption of normality, we let

$$T_n=\frac{\sqrt{n}\left({\delta }_h^{\ast }-{\delta }_h\right)}{{\gamma }_h^{\ast }} \mathrm{and} K_n=\frac{\left(n-1\right){\gamma }_h^{2\ast }}{{\gamma }_h^2},$$

Then $T_n$ is distributed as t-distribution with $n-1$ degree of freedom, denoted by $t_{n-1}$, and $K_n$ is distributed as chi-square distribution with $n-1$ degree of freedom, denoted by ${\chi }_{n-1}^2$. To derive the $\left(1-\alpha \right)\times 100\%$ upper confidence limits on index $C_{pmh}$, this paper defines $E_{L{\delta }_h}=\left\{T_n\ge -t_{\alpha /2;n-1}\right\}$, $E_{U{\delta }_h}=\left\{T_n\le t_{\alpha /2;n-1}\right\}$, $E_{L{\gamma }_h}=\left\{K_n\le {\chi }_{1-\alpha /2;n-1}^2\right\}$, and $e_h={\gamma }_h^{\ast }{t}_{\alpha /2;n-1}/\sqrt{n}$ where $t_{\alpha /2;n-1}$ is the upper $\alpha /2$ quintile of $t_{n-1}$, and ${\chi }_{1-\alpha /2;n-1}^2$ is the lower $1-\alpha /2$ quintile of ${\chi }_{n-1}^2$. Next, this paper solves the $100\left(1-\alpha \right)\%$ upper confidence limit $U{C}_{pmh}$ of index $C_{pmh}$ and its corresponding coordinate point $(x_h,y_h)$ based on Case 1: ${\delta }_h^{\ast }-e_h>0$, Case 2: ${\delta }_h^{\ast }-e_h\le 0\le {\delta }_h^{\ast }+e_h$, and Case 3: ${\delta }_h^{\ast }+e_h<0$. Subsequently, according to the location of the coordinate point $(x_h,y_h)$, this paper measures the process quality of quality characteristic h. The measure rules are as follows:

When $(x_h,y_h)$$\in$$Z_{\mathrm{A}}$, then $C_{pmh}\ge C$ means that the process quality level of quality characteristic h reaches the required level.

When $(x_h,y_h)$$\notin$$Z_{\mathrm{A}}$, then $C_{pmh}<C$ means the process quality level of quality characteristic h has not reached the required level and must be improved. This paper refers to the quality characteristic h which needs to be improved as Critical to Quality (CTQ).

Case 1: ${\delta }_h^{\ast }-e_h>0$

Obviously, when ${\delta }_h^{\ast }-e_h>0$, then we can conclude that ${\delta }_h>0$ and $C_{pmh}=1/\left(3\sqrt{{\delta }_h^2+{\gamma }_h^2}\right)$. Thus, the estimator $C_{pmh}$ of is $C_{pmh}^{\ast }=1/\left(3\sqrt{{\delta }_h^{\ast 2}+{\gamma }_h^{\ast 2}}\right)$. In fact, $P\left(E_{U{\delta }_h}\right)=P\left(E_{L{\gamma }_h}\right)=1-\alpha /2$ and $P\left(E_{U{\delta }_h}^C\right)=P\left(E_{L{\gamma }_h}^C\right)=\alpha /2$. According to Boole’s inequality and DeMorgan’s theorem, we can learn the relationship of $P\left(E_{U\delta h}\cap E_{L\gamma h}\right)\ge 1-P\left(E_{U\delta h}^C\right)-P\left(E_{L\gamma h}^C\right)$, and then

$$P\left\{{\delta }_h\ge {\delta }_h^{*}-t_{\alpha /2;n-1}\times \left(\frac{{\gamma }_h^{\ast }}{\sqrt{n}}\right),{\gamma }_h\ge \sqrt{\frac{n-1}{{\chi }_{1-\alpha /2;n-1}^2}}{\gamma }_h^{\ast }\right\}=1-\alpha .$$

(14)

Therefore, the confidence region can be shown as follows:

$$C{R}_1=\left\{\left({\delta }_h,{\gamma }_h\right)|{\delta }_h\ge {\delta }_h^{\ast }-t_{\alpha /2;n-1}\times \left(\frac{{\gamma }_h^{\ast }}{\sqrt{n}}\right),{\gamma }_h\ge \sqrt{\frac{n-1}{{\chi }_{1-\alpha /2;n-1}^2}}{\gamma }_h^{\ast }\right\}.$$

(15)

Obviously, $C_{pmh}$ is the function of $\left({\delta }_h,{\gamma }_h\right)$. This paper employed mathematical programming (MP) to solve the $100\left(1-\alpha \right)\%$ upper confidence limit $U{C}_{pmh}$ and its corresponding coordinate point, taking $C_{pmh}$ as the objective function and $C{R}_1$ as the feasible solution area, as shown below:

$$\{\begin{array}{l}U{C}_{pmh}=Max 1/\left(3\sqrt{{\delta }_h^2+{\gamma }_h^2}\right)\\ s.t. \left({\delta }_h,{\gamma }_h\right)\in C{R}_1\end{array}.$$

(16)

The closer to the origin is, the larger the value of $C_{pmh}$ is. Consequently, the maximum value is generated at the coordinate point in the lower left corner. The upper confidence limit $U{C}_{pmh}$ and its corresponding coordinate point $(x_h,y_h)$ are displayed as follows:

$$U{C}_{pmh}=1/3\sqrt{{\left({\delta }_h^{*}-t_{\alpha /2;n-1}\times \left(\frac{{\gamma }_h^{\ast }}{\sqrt{n}}\right)\right)}^2+{\left(\sqrt{\frac{n-1}{{\chi }_{1-\alpha /2;n-1}^2}}{\gamma }_h^{\ast }\right)}^2};$$

(17)

$$\left(x_h,y_h\right)=\left({\delta }_h^{\ast }-t_{\alpha /2;n-1}\times \left(\frac{{\gamma }_h^{\ast }}{\sqrt{n}}\right),\sqrt{\frac{n-1}{{\chi }_{1-\alpha /2;n-1}^2}}{\gamma }_h^{\ast }\right).$$

(18)

Case 2: ${\delta }_h^{\ast }-e_h\le 0\le {\delta }_h^{\ast }+e_h$

For the same reason, when ${\delta }_h^{\ast }-e_h\le 0\le {\delta }_h^{\ast }+e_h$, then we can conclude that ${\delta }_h=0$ and $C_{pmh}=1/\left(3{\gamma }_h\right)$. Thus, the estimator of $C_{pmh}$ is $C_{pmh}^{\ast }=1/\left(3{\gamma }_h^{\ast }\right)$. Therefore, the confidence region can be shown as follows:

$$C{R}_2=\left\{\left({\delta }_h,{\gamma }_h\right)|0,{\gamma }_h\ge \sqrt{\frac{n-1}{{\chi }_{1-\alpha /2;n-1}^2}}{\gamma }_h^{\ast }\right\}.$$

(19)

In the same way, $C_{pmh}$ is the function of $\left({\delta }_h,{\gamma }_h\right)$. This paper applied mathematical programming (MP) to solve the $100\left(1-\alpha \right)\%$ upper confidence limit $U{C}_{pmh}$ and its corresponding coordinate point, taking $C_{pmh}$ as the objective function and $C{R}_2$ as the feasible solution area, as illustrated below:

$$\{\begin{array}{l}U{C}_{pmh}=Max \frac{1}{3{\gamma }_h}\\ s.t. \left({\delta }_h,{\gamma }_h\right)\in C{R}_2\end{array}.$$

(20)

Given that it is closer to the original point, the value of $C_{pmh}$ gets larger. As a result, the maximum value is generated at the coordinate point in the lower left corner. The upper confidence limit $U{C}_{pmh}$ and its corresponding coordinate point are expressed as follows:

$$U{C}_{pmh}=1/\left(3{\gamma }_h^{\ast }\sqrt{\frac{n-1}{{\chi }_{1-\alpha /2;n-1}^2}}\right);$$

(21)

$$\left(x_h,y_h\right)=\left(0,\sqrt{\frac{n-1}{{\chi }_{1-\alpha /2;n-1}^2}}{\gamma }_h^{\ast }\right).$$

(22)

Case 3: ${\delta }_h^{\ast }+e_h<0$

Obviously, when ${\delta }_h^{\ast }+e_h<0$, then we can conclude that ${\delta }_h<0$ and $C_{pmh}=1/\left(3\sqrt{{\delta }_h^2+{\gamma }_h^2}\right)$. Thus, the estimator of $C_{pmh}$ is $C_{pmh}^{\ast }=1/\left(3\sqrt{{\delta }_h^{\ast 2}+{\gamma }_h^{\ast 2}}\right)$. In fact, $P\left(E_{L{\delta }_h}\right)=P\left(E_{L{\gamma }_h}\right)=1-\alpha /2$ and $P\left(E_{L{\delta }_h}^C\right)=P\left(E_{L{\gamma }_h}^C\right)=\alpha /2$. According to Boole’s inequality and DeMorgan’s theorem, when the relationship of $P\left(E_{L{\delta }_h}\cap E_{L{\gamma }_h}\right)\ge 1-P\left(E_{L{\delta }_h}^C\right)-P\left(E_{L{\gamma }_h}^C\right)$ is learned, then

$$P\left\{{\delta }_h\le {\delta }_h^{*}+t_{\alpha /2;n-1}\times \left(\frac{{\gamma }_h^{\ast }}{\sqrt{n}}\right),{\gamma }_h\ge \sqrt{\frac{n-1}{{\chi }_{1-\alpha /2;n-1}^2}}{\gamma }_h^{\ast }\right\}=1-\alpha .$$

(23)

Therefore, the confidence region can be shown as follows:

$$C{R}_3=\left\{\left({\delta }_h,{\gamma }_h\right)|{\delta }_h\le {\delta }_h^{\ast }+t_{\alpha /2;n-1}\times \left(\frac{{\gamma }_h^{\ast }}{\sqrt{n}}\right),{\gamma }_h\ge \sqrt{\frac{n-1}{{\chi }_{1-\alpha /2;n-1}^2}}{\gamma }_h^{\ast }\right\}.$$

(24)

Similarly, $C_{pmh}$ is the function of $\left({\delta }_h,{\gamma }_h\right)$. This paper employed mathematical programming (MP) to solve the $100\left(1-\alpha \right)\%$ upper confidence limit $U{C}_{pmh}$ and its corresponding coordinate point, taking $C_{pmh}$ as the objective function and $C{R}_3$ as the feasible solution area, as shown below:

$$\{\begin{array}{l}U{C}_{pmh}=Max 1/\left(3\sqrt{{\delta }_h^2+{\gamma }_h^2}\right)\\ s.t. \left({\delta }_h,{\gamma }_h\right)\in C{R}_3\end{array}.$$

(25)

The closer to the origin is, the larger the value of $C_{pmh}$ is. Accordingly, the maximum value is generated at the coordinate point in the lower left corner. The upper confidence limit $U{C}_{pmh}$ and its corresponding coordinate point $(x_h,y_h)$ are expressed as follows:

$$U{C}_{pmh}=1/3\sqrt{{\left({\delta }_h^{*}+t_{\alpha /2;n-1}\times \left(\frac{{\gamma }_h^{\ast }}{\sqrt{n}}\right)\right)}^2+{\left(\sqrt{\frac{n-1}{{\chi }_{1-\alpha /2;n-1}^2}}{\gamma }_h^{\ast }\right)}^2};$$

(26)

$$\left(x_h,y_h\right)=\left({\delta }_h^{\ast }+t_{\alpha /2;n-1}\times \left(\frac{{\gamma }_h^{\ast }}{\sqrt{n}}\right),\sqrt{\frac{n-1}{{\chi }_{1-\alpha /2;n-1}^2}}{\gamma }_h^{\ast }\right).$$

(27)

To conclude, this paper defined the coordinate point $(x_h,y_h)$ corresponding to the upper confidence limit $U{C}_{pmh}$ as the measure coordinate point of quality characteristic h as follows:

$$\left(x_h,y_h\right)=\{\begin{array}{l}\left({\delta }_h^{\ast }-t_{\alpha /2;n-1}\times \left(\frac{{\gamma }_h^{\ast }}{\sqrt{n}}\right),\sqrt{\frac{n-1}{{\chi }_{1-\alpha /2;n-1}^2}}{\gamma }_h^{\ast }\right),{\delta }_h^{\ast }-e_h>0\\ \left(0,\sqrt{\frac{n-1}{{\chi }_{1-\alpha /2;n-1}^2}}{\gamma }_h^{\ast }\right),{\delta }_h^{\ast }-e_h\le 0\le {\delta }_h^{\ast }+e_h\\ \left({\delta }_h^{\ast }+t_{\alpha /2;n-1}\times \left(\frac{{\gamma }_h^{\ast }}{\sqrt{n}}\right),\sqrt{\frac{n-1}{{\chi }_{1-\alpha /2;n-1}^2}}{\gamma }_h^{\ast }\right),{\delta }_h^{\ast }+e_h<0\end{array}.$$

(28)

Then, quality engineers can measure whether the location of the coordinate point $(x_h,y_h)$ falls in the acceptable process capability block $Z_{\mathrm{A}}$ and quickly find out the characteristic of Critical to Quality (CTQ) that needs to be improved according to the measure rules.

4. Analyze, Improve and Control

Section 3 defines and develops the measure method. Then, in this Section, the model of Analyze, Improve and Control are described as follows:

4.1. Analyze

Obviously, the characteristic of Critical to Quality (CTQ) that needs to be improved can be easily found through the above-mentioned measure of MPCAC/$C_{pmh}$. Then, this paper defined three process capability improvement blocks $Z_{B^{-}}$, $Z_{B^{+}}$, and $Z_C$ in MPCAC/$C_{pmh}$ as follows:

$$Z_{B^{-}}=\left\{(x_h,y_h)|x_h^2+y_h^2>1/{(3 C)}^2,x_h<-\frac{1}{3 C},y_h\ge 0\right\};$$

(29)

$$Z_{B^{-}}=\left\{(x_h,y_h)|x_h^2+y_h^2>1/{(3 C)}^2,x_h>\frac{1}{3 C},y_h\ge 0\right\};$$

(30)

$$Z_C=\left\{(x_h,y_h)|y_h\ge \frac{1}{3 C}\right\}.$$

(31)

Figure 2 summarizes the above three process capability improvement blocks $Z_{B^{-}}$, $Z_{B^{+}}$, and $Z_C$ on MPCAC/$C_{pmh}$. Then, the measure coordinate point $(x_h,y_h)$ of CTQ characteristic h which needs to be improved falls into the location of the above-mentioned process capability improvement blocks. According to the characteristics of the aforementioned accuracy index ${\delta }_h$ and precision index ${\gamma }_h$, a preliminary analysis and improvement direction is proposed as follows:

(1)

$\left(x_h,y_h\right)\in$$Z_{B^{-}}$ indicates that the process accuracy of quality characteristic h is insufficient (skewed to the left), and it is necessary to adjust the machine parameters to move the process to the middle, so as to enhance process yield and reduce process loss.

(2)

$\left(x_h,y_h\right)\in$$Z_{B^{+}}$ shows that the process accuracy of quality characteristic h is insufficient (skewed to the right), and it is necessary to adjust the machine parameters to move the process to the middle, in order to improve process yield and diminish process loss.

(3)

$\left(x_h,y_h\right)\in$$Z_C$ means that the process precision of quality characteristic h is insufficient (excessive variation), and it is necessary to check whether the personnel changes are too large, the material supply is unstable, the operation is not in accordance with the standard operating procedures, or other factors, resulting in excessive process variation. Then, reasons are found and improvements are made, so that the process variation can be lowered to increase process yield and decrease process loss.

(4)

$\left(x_h,y_h\right)\in$$Z_{B^{-}}\cap Z_C$ demonstrates that the process accuracy of quality characteristic h is insufficient (skewed to the left), and the process precision is also insufficient (excessive variation). Therefore, it is necessary to refer to the improvement direction suggested by the preliminary analysis of (1) and (3) for enhancement.

(5)

$\left(x_h,y_h\right)\in$$Z_{B^{+}}\cap Z_C$, indicates that the process accuracy of quality characteristic h is insufficient (skewed to the right), and the process precision is also insufficient (excessive variation). Therefore, it is necessary to refer to the improvement directions suggested by the preliminary analysis of (2) and (3) for enhancement.

4.2. Improve

According to the improvement direction suggested by the preliminary analysis in the previous section, various methods of quality management are applied to the follow-up analysis depending on the situation, and then improvement strategies and methods, such as applying the Taguchi orthogonal table method to figure out the best combination of processing parameters, can be proposed. After the improvement is over, the improvement results can be verified with a statistical hypothesis test. The process of the statistical test is described as follows:

(1)

Null Hypothesis $H_0$: $C_{pmh}\ge C$ (good improvement effect)

(2)

Alternative Hypothesis $H_1$: $C_{pmh}<C$ (poor improvement effect)

(3)

Significance level $\alpha$ (generally speaking, $\alpha$ = 0.1, 0.05, 0.01)

(4)

The upper confidence limit of $C_{pmh}$ for the test statistic is expressed as follows:

$$U{C}_{pmh}=\{\begin{array}{l}1/3\sqrt{{\left({\delta }_h^{*}-t_{\alpha /2;n-1}\times \left(\frac{{\gamma }_h^{\ast }}{\sqrt{n}}\right)\right)}^2+{\left(\sqrt{\frac{n-1}{{\chi }_{1-\alpha /2;n-1}^2}}{\gamma }_h^{\ast }\right)}^2},{\delta }_h^{\ast }-e_h>0\\ 1/\left(3{\gamma }_h^{\ast }\sqrt{\frac{n-1}{{\chi }_{1-\alpha /2;n-1}^2}}\right),{\delta }_h^{\ast }-e_h\le 0\le {\delta }_h^{\ast }+e_h\\ 1/3\sqrt{{\left({\delta }_h^{*}+t_{\alpha /2;n-1}\times \left(\frac{{\gamma }_h^{\ast }}{\sqrt{n}}\right)\right)}^2+{\left(\sqrt{\frac{n-1}{{\chi }_{1-\alpha /2;n-1}^2}}{\gamma }_h^{\ast }\right)}^2},{\delta }_h^{\ast }+e_h<0\end{array}.$$

(5)

Test rules

(5-1)

When $U{C}_{pmh}\ge C$, then do not reject the null hypothesis $H_0$, which means that the effect of improvement is good.

(5-2)

When $U{C}_{pmh}<C$, then reject the null hypothesis $H_0$, which means that the effect of improvement is poor, so that it must be reviewed and continued to improve.

4.3. Control

According to the aforementioned test rules, when all the upper confidence limit values of the Taguchi process capability index for quality characteristic h are greater than C ($U{C}_{pmh}\ge C$), then it means that the Taguchi process capability index value of the product is greater than $C_T$. At this time, the process capability of the product has been improved to an acceptable quality level. Next, it is necessary to examine the best combination of all machine parameters in the improvement process, formulate standard operating procedures, and supplement various quality control methods, such as using checklists to ensure that various operations can implement standard operating procedures, in order to monitor quality as well as to ensure and maintain the process quality level of the product.

5. Application Example

With the prevalence of the Internet of Things and the gradual maturity of big data analysis technology, the smart manufacturing and management pursued by Industry 4.0 has gradually deepened in various industries. Particularly, the customization of home appliances is closely related to human life. In addition to remote monitoring, the remote control plays a critical role in smart home appliances, and frequency is an important quality characteristic of the remote control. Generally speaking, the remote control includes three quality characteristics—Frequency 1, Frequency 2, and Frequency 3. Take the production of an electronic company in central Taiwan, for example. The detection data of the three frequencies of the remote control produced by the company is $\left(X_{h,1},\dots ,X_{h,15}\right)$. Then, the detection data is converted into $\left(Y_{h,1},\dots ,Y_{h,15}\right)$ by Equation $Y_h=\left(X_h-T_h\right)/d_h$, transformed by variables. According to the data, ${\delta }_h^{\ast }$ and ${\gamma }_h^{\ast }$ are calculated respectively, and the coordinate point $(x_h,y_h)$ is displayed below in

Table 2. Tolerances and relevant evaluation data for three quality characteristics of the remote control.

Quality Characteristics	Tolerance (MHz)	${\mathit{\delta }}_{\mathit{h}}^{\ast }$	${\mathit{\gamma }}_{\mathit{h}}^{\ast }$	$\left({\mathit{x}}_{\mathit{h}},{\mathit{y}}_{\mathit{h}}\right)$
Frequency 1	903 $\pm$ 0.0005	−0.049	0.265	(−0.34, 0.19)
Frequency 2	914 $\pm$ 0.0005	−0.043	0.382	(0, 0.28)
Frequency 3	926 $\pm$ 0.0005	0.016	0.238	(0, 0.17)

.

Obviously, the sample size is n$=$15, and the number of quality characteristics is q $=$ 3. As mentioned earlier, according to Table 1, if the quality level is required to reach six sigma, then Define in step 1 requires that the product index $C_{pm}$ should be at least 1.11 and the values of $C_{pmh}$ in the three frequencies should be at least 1.21. Next, we drew the chart of the suggested direction for the preliminary analysis and improvement of MPCAC/for $C_{pmh}=1.21$, as well as plotted the three coordinate points of $(x_h,y_h)$ into the Figure 3 as follows:

Obviously, $(x_3,y_3)=\left(0,0.17\right)$$\in$$Z_{\mathrm{A}}$ indicates that Frequency 3 meets the requirement of the six sigma quality level. On the other hand, $(x_1,y_1)=\left(-0.34,0.19\right)$$\in$$Z_{B^{-}}$ means that the process of Frequency 1 is skewed to the left, that is to say, the average value is too small, and $\left(x_2,y_2\right)=\left(0,0.28\right)$ represents that the variation of Frequency 2 is too large, so both processes of Frequency 1 and Frequency 2 must be improved. From this case, it can be clearly seen that this paper mainly used the process capability analysis chart constructed by the process capability index to assist the implementation of the six sigma DMAIC improvement process, which can come up with specific Define and rapid and comprehensive measurement and immediately put forward improvement suggestions according to the coordinate points of the quality characteristics that do not reach the quality level. For example, this case is to improve the problem of excessive process variation and the average value is too small. Finally, the goal of maintaining process quality stability after improvement can be achieved by means of the establishment of SOP.

6. Conclusions

The Taguchi process capability index is one of the commonly used quality evaluation tools in the industry. Its denominator is the expected value of Taguchi loss function, so it can reflect process loss. In addition, when the index value meets the requirements of the quality level, there is an inequality relationship between the index and the process yield, that is to say, when the index value is large enough, the process yield can be guaranteed to be large enough as well. This paper established the relation between the Taguchi process capability index and the six sigma quality level. This relation can be provided to the industry for reference when they develop the definition of the six sigma DMAIC improvement model. Then, a multi-characteristic process capability analysis chart (MPCAC/$C_{pmh}$) was constructed as a tool for measuring, analyzing and finding the direction of improvement for the six sigma DMAIC process quality improvement model. It can be clearly seen from the case that this paper mainly employed the process capability analysis chart built on the process capability index to help carry out the six sigma DMAIC improvement process, so that specific Define as well as rapid and comprehensive measurement can be made. Furthermore, for quality characteristics which do not reach the quality level, suggestions for improvement are immediately proposed based on the position of their coordinate points. Apparently, the six sigma DMAIC process quality improvement model developed in this paper can help the industry promote the six sigma DMAIC process improvement process so as to enhance the product process quality. When the process quality is leveled up, the ratio of rework and scrap can be lowered, and the effect of energy saving and waste reduction can be achieved, which is helpful for enterprises to move towards the goal of smart manufacturing and sustainable operation.