A fuzzy approach to determine process quality for one-sided specification with imprecise data

Abstract

Process quality is a crucial determinant of client satisfaction and affects a product’s value; therefore, maintaining quality control is a vital aspect of corporate sustainability and development. Many researchers have developed evaluation models for process quality. The majority of such research assumes that collected measurement data are precise, but fuzziness and stochastic uncertainty are unavoidable features of any collected data. When the measurements of a quality characteristic are insufficiently precise, a crisp-based approach is not suitable for the assessment of process quality. This study endeavored to use one-sided Six Sigma quality indices as measurement tools to accurately reflect process yield and quality levels. Taking Buckley’s approach into consideration, we extend the crisp estimators from the indices into fuzzy estimators. We then develop a fuzzy hypothesis testing method for one-sided Six Sigma quality indices, with the intent of increasing reliability of evaluation for process quality levels. Finally, we present a real-world case to illustrate implementation of the proposed approach, demonstrating its effectiveness and practical applicability.

Introduction

As the global economy continues to develop, companies find themselves in an increasingly competitive environment, challenged by major shifts in their respective industries. Thus, quality control is an increasingly important component of competitiveness, especially for the manufacturing industry.^1,2 Process quality not only affects a product’s value but is also a crucial determinant of client satisfaction. When Motorola was threatened by competition on the “process quality” front, it rolled out a Six Sigma campaign with the goal of improving its process quality, effectively lowering its product defect rate, and raising profitability.^3,4 It is clear that as long as competition exists within an industry, quality control will remain a cornerstone of corporate sustainability and development.

Motorola’s Six Sigma ( $6\sigma$ ) reaches 3.4 defects per million opportunities (DPMO) when the process mean is allowed to shift by $1.5\sigma$ off process target $T$ .^5–8 Wu et al. ⁹ and Chen et al. ¹⁰ indicate that a lower shift means lower expected loss. Given this, it is clear that reaching process quality $k\sigma$ level allows for a $1.5\sigma$ shift in the process mean. However, this is specifically for situations where the process quality characteristic is a nominal-the-best (NTB) type for two-sided specification tolerance and is not suitable for one-sided specification tolerance, that is, larger-the-better (LTB) or smaller-the-better (STB) characteristics. More specifically, when the quality characteristic is LTB, the target value T is theoretically infinitely large ( $\infty$ ); $T-\mu \le 1.5\sigma$ is thus virtually impossible. In the same vein, when the quality characteristic is STB, the target value T is theoretically zero. However, as process mean $\mu$ approaches zero, more advanced technologies and process expenses become necessary. Thus, $\mu -T\le 1.5\sigma$ is also virtually impossible. In light of this, with regard to STB or LTB quality characteristics for one-sided specification tolerance, the rules for $k\sigma$ process quality are defined as follows: $\mu -(k-1.5)\sigma \ge \mathrm{LSL}$ if $\mathrm{LTB}$ and $\mu +(k-1.5)\sigma \le \mathrm{USL}$ if $\mathrm{STB}$ , where $\mathrm{USL}$ and $\mathrm{LSL}$ are, respectively, the upper and lower specification limits.

Since the Six Sigma model is capable of effectively raising process quality, many statisticians and quality engineers have researched the correlation between process capability indices (PCIs) and Six Sigma quality levels and developed models for process quality evaluation.^11–17 Based on the results of their investigations, Chen et al. ¹⁰ came up with Six Sigma quality indices (SSQIs) $Q_{\mathrm{pu}}$ and $Q_{\mathrm{pl}}$ which are suitable for the evaluation of process quality with one-sided specification tolerance.

The recent rise in environmental awareness has led to proliferation of the concept of a circular economy, with increased corporate advocacy for waste reduction and the efficient leveraging of resources.^18–20 A circular economy is not merely a system where resources are reused and recycled; rather, it is a system that can help companies realize environmental protection goals by fulfilling all four Rs of the 4R principle: reduce, reuse, recycle, and recover. Chen et al. ²¹ stated that a rise in quality can reduce the proportion of process write-offs and reworking, and concurrently increase product life cycles, which in turn reduce the proportion of maintenance and resource inputs. This minimizes pollution and damage to the environment. The manufacturing industry aims to meet environmental demands by effectively raising process quality through sustainable production practices and circular economic strategies.

In summary, increasing process quality is not only a matter of corporate sustainability and competitiveness but also a gauge of whether a company is able to effectively reduce its waste and protect the environment. Thus, assessing process quality to improve product quality is a topic of immediate relevance for manufacturers. This study employs the one-sided SSQIs $Q_{\mathrm{pu}}$ and $Q_{\mathrm{pl}}$ as tools for evaluating process quality. In practice, population parameters $\mu$ and $\sigma$ are usually unknown and need to be estimated based on randomly collected samples. Unfortunately, fuzziness and stochastic uncertainty are unavoidable features of collected data. Therefore, in order to lower measurement errors in the sample as well as errors in the estimated process parameters, this study adopts the approach proposed by Buckley ²² and presents a set of confidence intervals producing a triangular-shaped fuzzy number for the estimation of one-sided SSQIs. This study further develops a method for fuzzy hypothesis testing for one-sided SSQIs.

Measuring process quality using confidence intervals for one-sided SSQIs reduces the risk of misjudgment caused by sampling error. In general, misjudgment risk in confidence intervals is influenced by sample size. The length of the confidence interval widens with decreases in sample size n, which increases the risk of misjudgment. Cost and time considerations mean that large sample sizes are rare. To overcome this limitation, the fuzzy hypothesis testing proposed in this study is based on statistical inference to judge process quality by expertise. In view of this, the proposed approach not only reduces the risk of misjudgment but also overcomes the issue of imprecise data.

The remainder of this article is organized as follows. Section “Fuzzy estimator for $Q_{\mathrm{pu}}$ and $Q_{\mathrm{pl}}$ ” presents the fuzzy estimator for one-sided SSQIs and its membership function. In section “Fuzzy hypothesis testing for $Q_{\mathrm{pu}}$ and $Q_{\mathrm{pl}}$ ,” the fuzzy hypothesis testing for one-sided SSQIs is developed based on Buckley’s estimation approach. In section “Application example,” we present a case to illustrate the applicability of the proposed approach. Conclusions are summarized in section “Conclusion.”

Literature review

PCIs $C_{\mathrm{pu}}$ and $C_{\mathrm{pl}}$

A wide variety of methods exist for the evaluation of process quality. PCIs use numerical quantification to relate the mean and standard deviation to tolerance specifications. Thus, PCIs are widely applied to determine whether processes meet the required quality. For example, Chen et al. ²³ employed index $C_{\mathrm{pm}}$ and manufacturing time performance index to examine process performance and develop an outsourcing partner selection model, which helps companies select better contractors for long-term collaborations. Liao and Pearn ²⁴ modified the weighted standard deviation index $C_{\mathrm{pk}}(\mathrm{WSD})$ to propose an adequate measure of lognormal process capability. Otsuka and Nagata ²⁵ proposed a novel approach to process capability which controls part dimensions based on product performance. Zhou et al. ²⁶ proposed a sensitivity analysis of process capability evaluation for small-batch production runs. Chang et al. ²⁷ developed a process quality analysis chart based on an accuracy index and a precision index, which analyzes process quality determined by multiple quality characteristics. Ouyang et al. ²⁸ used index $C_{\mathrm{pm}}$ to develop a process capability analysis chart, which effectively determines deficiencies in process capability.

The most widely used PCIs for process performance and quality assessment are $C_{\mathrm{pu}}$ , $C_{\mathrm{pl}}$ , and $C_{\mathrm{pk}}$ , which were proposed by Kane. ²⁹ $C_{\mathrm{pu}}$ and $C_{\mathrm{pl}}$ are, respectively, used to assess process performance for STB and LTB quality characteristics, whereas $C_{\mathrm{pk}}$ is used to measure the process quality of NTB quality characteristics. As a result, $C_{\mathrm{pu}}$ and $C_{\mathrm{pl}}$ are also called unilateral specification indices, while $C_{\mathrm{pk}}$ is called the bilateral specification index. The three indices are defined as follows

$$\begin{array}{c}\multicolumn{1}{c}{C_{\mathrm{pu}}=\frac{\mathrm{USL}-\mu }{3\sigma },C_{\mathrm{pl}}=\frac{\mu -\mathrm{LSL}}{3\sigma },\mathrm{and}{C}_{\mathrm{pk}}}\\ \multicolumn{1}{c}{=\min (C_{\mathrm{pu}},C_{\mathrm{pl}})=\frac{d-|\mu -T|}{3\sigma }}\end{array}$$

where $\mu$ is the mean of a given process, $\sigma$ is the process standard deviation, $d$ is half the width of the specification interval (i.e. $d=(\mathrm{USL}-\mathrm{LSL})/2$ ), and process target $T$ is the midpoint of the specification limits (i.e. $T=(\mathrm{USL}+\mathrm{LSL})/2$ ).

Inconvenience of PCI in determining the process quality level

In the 1980s, the concept of Six Sigma quality control was pioneered by Motorola Inc. to serve as a tool for reducing production variance, eliminating defects, and improving quality. ³⁰ This process explicitly defines the quality goal of 3.4 DPMO, which is equivalent to a process yield of 99.99966%. Consequently, the Six Sigma model has been widely applied in various industries.^8,31–35 For this model, the 3.4-DPMO objective can only be achieved when the maximum shift of $\mu$ from $T$ is $1.5\sigma$ . Chen et al. ³⁶ further indicated that process quality can only reach the Six Sigma level ( $6\sigma$ ) when $|\mu -T|\le 1.5\sigma$ and $\sigma =d/6$ . For this reason, Chen et al. ¹¹ discussed the relationships among various quality levels and the indices $C_{\mathrm{pu}}$ , $C_{\mathrm{pl}}$ , and $C_{\mathrm{pk}}$ when the process mean shifts $1.5\sigma$ (see Table 1).

Table 1.

Relationships among quality level, and PCIs.

Quality level $k\sigma$	$C_{\mathrm{pu}}$	$C_{\mathrm{pl}}$	$C_{\mathrm{pk}}$
$6\sigma$	1.500	1.500	1.500
$5\sigma$	1.167	1.167	1.167
$4\sigma$	0.833	0.833	0.833
$3\sigma$	0.500	0.500	0.500

Many researchers have adopted the above relationships to determine whether process quality level meets requirements. However, since the value of $C_{\mathrm{pu}}$ (or $C_{\mathrm{pl}}$ , $C_{\mathrm{pk}}$ ) is not equal to the quality level reached, these researchers were only able to obtain a solution range and not the exact value of the quality level attained by the processes (see Table 1). This kind of result is inconvenient for decision-makers.

SSQIs $Q_{\mathrm{pu}}$ and $Q_{\mathrm{pl}}$

In order to eliminate the inconveniences posed by PCIs, Chen et al. ¹⁰ proposed the SSQIs $Q_{\mathrm{pu}}$ and $Q_{\mathrm{pl}}$ to directly determine and analyze process quality level. Indices $Q_{\mathrm{pu}}$ and $Q_{\mathrm{pl}}$ are defined as follows

$$Q_{\mathrm{pl}}=\frac{\mu -\mathrm{LSL}}{\sigma }+1.5,\mathrm{for}\mathrm{LTB}$$

(1)

$$Q_{\mathrm{pu}}=\frac{\mathrm{USL}-\mu }{\sigma }+1.5,\mathrm{for}\mathrm{STB}$$

(2)

With regard to LTB characteristics, if $\mu -(k-1.5)\sigma \ge \mathrm{LSL}$ and process quality requirements are $k\sigma$ , then $Q_{\mathrm{pl}}=k$ and process yield is $\Phi (k-1.5)$ , where $\Phi (·)$ denotes the cumulative distribution function of the standard normal distribution. Similarly, for STB characteristics, if $\mu +(k-1.5)\sigma \le \mathrm{USL}$ and process quality requirements are $k\sigma$ , then $Q_{\mathrm{pu}}=k$ and process yield is $\Phi (k-1.5)$ . For this reason, Six Sigma indices $Q_{\mathrm{pu}}$ and $Q_{\mathrm{pl}}$ can precisely reflect process yield and quality level. In view of this, numerous researchers have used SSQIs to evaluate and analyze process quality.^9,37–39

Fuzzy estimator for ${\mathit{Q}}_{\mathrm{pu}}$ and ${\mathit{Q}}_{\mathrm{pl}}$

Collected data are often imprecise. Therefore, the values of related test statistics are also imprecise, which can lead to erroneous inferences. In order to assess process quality levels more accurately, this study adopts Buckley’s approach ²² to present a set of confidence intervals, producing a triangular-shaped fuzzy number for the estimation of one-sided SSQIs. First, we assumed that the process of STB or LTB quality characteristic $X$ is distributed normally, that is, $X~N({\mu }_X,{\sigma }_X^2)$ . Then, we let $X_1,\dots ,X_j,\dots ,X_n$ be a random sample, and ${\mu }^{*}$ denote the sample mean and ${\sigma }^{*}$ the sample standard deviation, as follows

$${\mu }^{*}=\frac{1}{n}\sum _{j=1}^n{X}_j\mathrm{and}{\sigma }^{*}=\sqrt{\frac{1}{n-1}\sum _{j=1}^n{(X_j-{\mu }^{*})}^2}$$

For one-sided Six Sigma quality index $Q_{\mathrm{pu}}$ or $Q_{\mathrm{pl}}$ , calculation of estimator $Q_{\mathrm{pi}}^{*}$ can be based on random samples, as follows

$$Q_{\mathrm{pi}}^{*}=\{\begin{array}{c}{Q}_{\mathrm{pl}}^{*}=\frac{{\mu }^{*}-\mathrm{LSL}}{{\sigma }^{*}}+1.5,i=1,\mathrm{for}\mathrm{LTB}\hfill \\ \multicolumn{1}{c}{Q_{\mathrm{pu}}^{*}=\frac{\mathrm{USL}-{\mu }^{*}}{{\sigma }^{*}}+1.5,i=2,\mathrm{for}\mathrm{STB}}\end{array}$$

(3)

Furthermore, let $Z_i=\sqrt{n}\times (-1{)}^i[(Q_{\mathrm{pi}}-1.5)-(Q_{\mathrm{pi}}^{*}-1.5)({\sigma }^{*}/\sigma )]$ and $K=[(n-1){\sigma }^{*2}/{\sigma }^2]$ . Under the assumption of normality, $Z_i$ follows a standard normal distribution (i.e. $N(0,1)$ ) and $K$ follows a chi-square distribution with $n-1$ degrees of freedom (i.e. ${\chi }_{n-1}^2$ ). Therefore, we can derive the following

$$P\{-z_{\alpha \prime /2}\le Z_i\le z_{\alpha \prime /2}\}=\sqrt{1-\alpha }$$

(4)

$$P\{{\chi }_{\alpha \prime /2,n-1}^2\le K\le {\chi }_{1-(\alpha \prime /2),n-1}^2\}=\sqrt{1-\alpha }$$

(5)

where $z_{\alpha \prime /2}$ is the upper $\alpha \prime /2$ quantile of $N(0,1)$ , ${\chi }_{\alpha \prime /2,n-1}^2$ is the lower $\alpha \prime /2$ quantile of the chi-square distribution with $n-1$ degrees of freedom, $\alpha \prime =1-\sqrt{1-\alpha }$ , and $\alpha$ represents the confidence level. Since ${\mu }^{*}$ and ${\sigma }^{*}$ are mutually independent under the normality assumption, $Z_i$ and $K$ are also mutually independent. From these relationships, we can further obtain the following

$$P\{-z_{\alpha \prime /2}\le Z_i\le z_{\alpha \prime /2},{\chi }_{\alpha \prime /2,n-1}^2\le K\le {\chi }_{1-(\alpha \prime /2),n-1}^2\}=1-\alpha$$

Equivalently,

$$P\left\{\left(Q_{pi}^{*}-1.5\right)\sqrt{\frac{{\chi }_{{\alpha }^{\prime }/2,n-1}^2}{n-1}}-\frac{z_{{\alpha }^{\prime }/2}}{\sqrt{n}}+1.5\le Q_{pi}\le \left(Q_{pi}^{*}-1.5\right)\sqrt{\frac{{\chi }_{1-({\alpha }^{\prime }/2),n-1}^2}{n-1}}+\frac{z_{{\alpha }^{\prime }/2}}{\sqrt{n}}+1.5\right\}\ge 1-\alpha$$

(6)

Adopting Buckley’s approach, ²² the $\alpha -\mathrm{cuts}$ of the triangular-shaped fuzzy number ${\tilde{Q}}_{\mathrm{pi}}^{*}$ can be obtained as follows

$${\tilde{Q}}_{\mathrm{pi}}^{*}\left[\alpha \right]=\{\begin{array}{c}[Q_{\mathrm{pi}1}^{*}\left(\alpha \right),Q_{\mathrm{pi}2}^{*}\left(\alpha \right)],0.01\le \alpha \le 1\hfill \\ \multicolumn{1}{c}{[Q_{\mathrm{pi}1}^{*}(0.01),Q_{\mathrm{pi}2}^{*}(0.01)],0\le \alpha \le 0.01}\end{array}$$

(7)

where

$$Q_{pi1}^{*}\left(\alpha \right)=\left(Q_{pi}^{*}-1.5\right)\sqrt{\frac{{\chi }_{{\alpha }^{\prime }/2,n-1}^2}{n-1}}-\frac{z_{{\alpha }^{\prime }/2}}{\sqrt{n}}+1.5$$

$$Q_{pi2}^{*}\left(\alpha \right)=\left(Q_{pi}^{*}-1.5\right)\sqrt{\frac{{\chi }_{1-({\alpha }^{\prime }/2),n-1}^2}{n-1}}+\frac{z_{{\alpha }^{\prime }/2}}{\sqrt{n}}+1.5$$

When $\alpha =1$ , then $Q_{\mathrm{pi}1}^{*}(1)=Q_{\mathrm{pi}2}^{*}(1)=(Q_{\mathrm{pi}}^{*}-1.5)\sqrt{\frac{{\chi }_{0.5,n-1}^2}{n-1}}+1.5$ . Thus, the triangular-shaped fuzzy number of $Q_{\mathrm{pi}}^{*}$ can further be expressed as follows

$$\mathrm{\Delta }{Q}_{\mathrm{pi}}^{*}=(Q_L,Q_M,Q_R)$$

(8)

where $Q_L=Q_{\mathrm{pi}1}^{*}(0.01)$ , $Q_R=Q_{\mathrm{pi}2}^{*}(0.01)$ , and $Q_M=(Q_{\mathrm{pi}}^{*}-1.5)\sqrt{\frac{{\chi }_{0.5,n-1}^2}{n-1}}+1.5$ . It is worth noting that $Q_M\ne Q_{\mathrm{pi}}^{*}$ , so any evaluation of process quality will present considerable inconvenience for decision-makers. In view of this, to make the approach more practical, we convert the value of $Q_M$ into $Q_{\mathrm{pi}}^{*}$ based on transformation of a random variable, and let $Q_{\mathrm{pi}}^{\prime *}=(Q_{\mathrm{pi}}^{*}-1.5)\sqrt{\frac{n-1}{{\chi }_{0.5;n-1}^2}}+1.5$ . Thus, the new transformation $\alpha -\mathrm{cuts}$ of the triangular-shaped fuzzy number ${\tilde{Q}}_{\mathrm{pi}}^{*}$ can be expressed as follows

$${\tilde{Q}}_{\mathrm{pi}}^{\prime *}\left[\alpha \right]=\{\begin{array}{c}[Q_{\mathrm{pi}1}^{\prime *}\left(\alpha \right),Q_{\mathrm{pi}2}^{\prime *}\left(\alpha \right)],0.01\le \alpha \le 1\hfill \\ \multicolumn{1}{c}{[Q_{\mathrm{pi}1}^{\prime *}(0.01),Q_{\mathrm{pi}2}^{\prime *}(0.01)],0\le \alpha \le 0.01}\end{array}$$

(9)

where

$$\begin{array}{c}\multicolumn{1}{c}{Q_{\mathrm{pi}1}^{\prime *}\left(\alpha \right)=(Q_{\mathrm{pi}}^{*}-1.5}\end{array})\sqrt{\frac{{\chi }_{\alpha \prime /2,n-1}^2}{{\chi }_{0.5,n-1}^2}}-\sqrt{\frac{n-1}{n}}\left(\frac{z_{\alpha \prime /2}}{\sqrt{{\chi }_{0.5,n-1}^2}}\right)+1.5,$$

$$\begin{array}{c}\multicolumn{1}{c}{Q_{\mathrm{pi}2}^{\prime *}\left(\alpha \right)=(Q_{\mathrm{pi}}^{*}-1.5)\sqrt{\frac{{\chi }_{1-(\alpha \prime /2),n-1}^2}{{\chi }_{0.5,n-1}^2}}}\\ \multicolumn{1}{c}{+\sqrt{\frac{n-1}{n}}\left(\frac{z_{\alpha \prime /2}}{\sqrt{{\chi }_{0.5,n-1}^2}}\right)+1.5.}\end{array}$$

According to equation (9), it is easy to see $Q_{\mathrm{pi}1}^{\prime *}\left(1\right)=Q_{\mathrm{pi}2}^{\prime *}\left(1\right)=Q_{\mathrm{pi}}^{*}$ when $\alpha =1$ . Thus, the new transformation triangular-shaped fuzzy number of $Q_{\mathrm{pi}}^{*}$ can further be expressed as follows

$$\mathrm{\Delta }{Q}_{\mathrm{pi}}^{\prime *}=({Q\prime }_L,{Q\prime }_M,{Q\prime }_R)$$

(10)

where $Q{\prime }_M=Q_{\mathrm{pi}}^{*}$ , ${Q^{\prime }}_L={Q^{\prime }}_{pi1}^{*}\left(0.01\right)=\left(Q_{pi}^{*}-1.5\right)\sqrt{\frac{{\chi }_{0.0025,n-1}^2}{{\chi }_{0.5,n-1}^2}}-\sqrt{\frac{n-1}{n}}\left(\frac{z_{0.0025}}{\sqrt{{\chi }_{0.5,n-1}^2}}\right)+1.5$ , and $Q{\prime }_R=Q_{\mathrm{pi}2}^{\prime *}(0.01)=(Q_{\mathrm{pi}}^{*}-1.5)\sqrt{\frac{{\chi }_{0.9975,n-1}^2}{{\chi }_{0.5,n-1}^2}}+\sqrt{\frac{n-1}{n}}\left(\frac{z_{0.9975}}{\sqrt{{\chi }_{0.5,n-1}^2}}\right)+1.5$ . Thus, the membership function of ${\tilde{Q}}_{\mathrm{pi}}^{\prime *}$ can be defined as follows

$${\eta }_{{\tilde{Q}}_{\mathrm{pi}}^{\prime *}}(x)=\{\begin{array}{cc}0,\hfill & \mathrm{if}x<{Q\prime }_L\hfill \\ \multicolumn{1}{c}{y\prime ,}& \mathrm{if}{Q\prime }_L\le x<Q_{\mathrm{pi}}^{*}\hfill \\ \multicolumn{1}{c}{1,}& \mathrm{if}x=Q_{\mathrm{pi}}^{*}\hfill \\ \multicolumn{1}{c}{y″,}& \mathrm{if}{Q}_{\mathrm{pi}}^{*}<x\le {Q\prime }_R\hfill \\ \multicolumn{1}{c}{0,}& \mathrm{if}{Q\prime }_R<x\hfill \end{array}$$

(11)

where $y\prime$ and $y″$ are determined by the following equation

$$\{\begin{array}{c}(Q_{\mathrm{pi}}^{*}-1.5)\sqrt{\frac{{\chi }_{y\prime /2,n-1}^2}{{\chi }_{0.5,n-1}^2}}-\sqrt{\frac{n-1}{n}}\left(\frac{z_{y\prime /2}}{\sqrt{{\chi }_{0.5,n-1}^2}}\right)\hfill \\ \multicolumn{1}{c}{+1.5=x,\mathrm{for}{Q\prime }_L\le x<Q_{\mathrm{pi}}^{*}}\\ \multicolumn{1}{c}{(Q_{\mathrm{pi}}^{*}-1.5)\sqrt{\frac{{\chi }_{1-(y″/2),n-1}^2}{{\chi }_{0.5,n-1}^2}}+\sqrt{\frac{n-1}{n}}\left(\frac{z_{y″/2}}{\sqrt{{\chi }_{0.5,n-1}^2}}\right)}\\ \multicolumn{1}{c}{+1.5=x,\mathrm{for}{Q}_{\mathrm{pi}}^{*}<x\le {Q\prime }_R}\end{array}$$

Figure 1 plots the membership function of fuzzy estimator ${\tilde{Q}}_{\mathrm{pi}}^{\prime *}$ with $Q_{\mathrm{pi}}^{*}=4$ for $n=50$ , 70, and 100.

Figure 1.

Membership function for fuzzy estimator ${\tilde{Q}}_{\mathrm{pi}}^{\prime *}$ with $Q_{\mathrm{pi}}^{*}=4$ for $n=50$ , 70 and 100.

Fuzzy hypothesis testing for ${\mathit{Q}}_{\mathrm{pu}}$ and ${\mathit{Q}}_{\mathrm{pl}}$

When customers specify the process quality level must meet the $k\sigma$ level, the index $Q_{\mathrm{pi}}$ for quality characteristics must reach at least $k$ (i.e. $Q_{\mathrm{pi}}\ge k$ ). In order to ensure a rigorous assessment of process quality, the statistical approach to hypothesis testing is an effective means of determining whether the quality level of a given process is acceptable. Thus, we develop the hypothesis testing model for $Q_{\mathrm{pi}}$ , and statistical hypothesis testing can be expressed as follows

$$H_0:Q_{\mathrm{pi}}\ge k\mathrm{versus}{H}_1:Q_{\mathrm{pi}}<k$$

(12)

From the random sample, we could let $Q_{\mathrm{pi}}^{*}$ be the test statistic and determine the critical region $C=\{Q_{\mathrm{pi}}^{*}\le C_0\}$ , where $C_0$ is the critical value. Thus, the decision rule for this one-tail test is to reject $H_0$ if $Q_{\mathrm{pi}}^{*}\le C_0$ and fail to reject $H_0$ otherwise. In view of this, when the significance level of the test is set to $\beta$ , which is typically set at 0.01, 0.05, or 0.1, then $C_0$ is determined by

$$P\{Q_{\mathrm{pi}}^{*}\le C_0|Q_{\mathrm{pi}}^{*}=k\}=\beta$$

(13)

Since $\sqrt{n}(Q_{\mathrm{pi}}^{*}-1.5)$ follows a non-central t-distribution with a degree of freedom of $n-1$ , the non-centrality parameter ${\delta }_i$ of this distribution is $\sqrt{n}(Q_{\mathrm{pi}}-1.5)$ , denoted as $t{\prime }_{n-1}({\delta }_i)$ , $i=1,2$ . Thus, $C_0$ can be expressed as follows

$$C_0=\frac{{t\prime }_{\beta ,n-1}\left({\delta }_i\right)}{\sqrt{n}}+1.5$$

(14)

where ${\delta }_i=\sqrt{n}(k-1.5)$ , and $t{\prime }_{\beta ,n-1}({\delta }_i)$ is the lower $\beta$ quantile of the non-central t-distribution with $n-1$ degrees of freedom. Thus, we can find the critical value $C_0$ to make a decision for hypothesis testing of $Q_{\mathrm{pi}}$ . However, as mentioned in section “Fuzzy estimator for $Q_{\mathrm{pu}}$ and $Q_{\mathrm{pl}}$ ,” fuzziness and stochastic uncertainty are unavoidable features of collected data, and related test statistics will be imprecise, leading to erroneous inferences. Thus, we further developed fuzzy hypothesis testing for $Q_{\mathrm{pi}}$ .

Similar to ${\tilde{Q}}_{\mathrm{pi}}^{\prime *}[\alpha ]$ , the new transformation $\alpha -\mathrm{cuts}$ of the triangular-shaped fuzzy number for $C_0$ (i.e. $\tilde{C}{\prime }_0[\alpha ]$ ) can be re-expressed as follows

$$\tilde{C}{\prime }_0\left[\alpha \right]=\{\begin{array}{c}[{C\prime }_{01}\left(\alpha \right),{C\prime }_{02}\left(\alpha \right)],0.01\le \alpha \le 1\hfill \\ \multicolumn{1}{c}{[{C\prime }_{01}(0.01),{C\prime }_{02}(0.01)],0\le \alpha \le 0.01}\end{array}$$

(15)

where

$$C{\prime }_{01}\left(\alpha \right)=(C_0-1.5)\sqrt{\frac{{\chi }_{\alpha \prime /2,n-1}^2}{{\chi }_{0.5,n-1}^2}}-\sqrt{\frac{n-1}{n}}\left(\frac{z_{\alpha \prime /2}}{\sqrt{{\chi }_{0.5,n-1}^2}}\right)+1.5$$

$$C{\prime }_{02}\left(\alpha \right)=(C_0-1.5)\sqrt{\frac{{\chi }_{1-(\alpha \prime /2),n-1}^2}{{\chi }_{0.5,n-1}^2}}+\sqrt{\frac{n-1}{n}}\left(\frac{z_{\alpha \prime /2}}{\sqrt{{\chi }_{0.5,n-1}^2}}\right)+1.5$$

Clearly, $C{\prime }_{01}\left(1\right)=C{\prime }_{02}\left(1\right)=C_0$ . Thus, the new triangular-shaped fuzzy number of $C_0$ can be expressed as follows

$$\mathrm{\Delta }C{\prime }_0=({C\prime }_L,C_0,{C\prime }_R)$$

(16)

where $C{\prime }_L=C{\prime }_{01}(0.01)$ and $C{\prime }_R=C{\prime }_{02}(0.01)$ . Therefore, we have a fuzzy set ${\tilde{Q}}_{\mathrm{pi}}^{\prime *}$ for test statistic $Q_{\mathrm{pi}}^{*}$ and a fuzzy set $\tilde{C}{\prime }_0$ for the critical value $C_0$ . As mentioned above, the decision rule of hypothesis testing for $Q_{\mathrm{pi}}$ is to reject $H_0$ if $Q_{\mathrm{pi}}^{*}\le C_0$ . However, the test statistic $Q_{\mathrm{pi}}^{*}$ becomes a triangular-shaped fuzzy number $\mathrm{\Delta }{Q}_{\mathrm{pi}}^{\prime *}$ based on the consideration of imprecise output data. Thus, the final decision rule of fuzzy hypothesis testing for $Q_{\mathrm{pi}}$ will depend on the relationship between ${\tilde{Q}}_{\mathrm{pi}}^{\prime *}$ and $C_0$ . This is best understood through studying Figure 2. The vertex of ${\tilde{Q}}_{\mathrm{pi}}^{\prime *}$ is at $x=Q_{\mathrm{pi}}^{*}$ .

Figure 2.

Relationship of fuzzy estimator ${\tilde{Q}}_{\mathrm{pi}}^{\prime *}$ and $C_0$ in the final decision rule.

As seen in Figure 2, when $Q_{\mathrm{pi}}^{*}\ge C_0$ , then $Q_{\mathrm{pi}1}^{\prime *}(a)=C_0$ . Let $d_{\beta }=C_0-Q_{\mathrm{pi}}^{*}$ , thus $d_{\beta }$ can be expressed as follows

$$\begin{array}{c}\multicolumn{1}{c}{d_{\beta }=(Q_{\mathrm{pi}}^{*}-1.5}\end{array})(\sqrt{\frac{{\chi }_{(1-\sqrt{1-a})/2,n-1}^2}{{\chi }_{0.5,n-1}^2}}-1)-\sqrt{\frac{n-1}{n}}\left(\frac{z_{(1-\sqrt{1-a})/2}}{\sqrt{{\chi }_{0.5,n-1}^2}}\right)$$

(17)

Buckley ²² considered the ratio of two specific defined areas for testing with a triangular-shaped fuzzy number. In light of this, this study defines $A_T$ as the total area under the graph of ${\tilde{Q}}_{\mathrm{pi}}^{\prime *}$ and $A_R$ as the area under the graph of ${\tilde{Q}}_{\mathrm{pi}}^{\prime *}$ to the left of the vertical line through $C_0$ , where $0\le A_R/A_T\le 1$ . Furthermore, Buckley ²² chose two numbers $0<{\varphi }_1<{\varphi }_2<0.5$ , and the decision rules are as follows: (1) if $A_R/A_T\ge {\varphi }_2$ , then reject $H_0$ ; (2) if ${\varphi }_1<A_R/A_T<{\varphi }_2$ , then there is no decision on whether to reject or not reject $H_0$ ; and (3) if $A_R/A_T\le {\varphi }_1$ , then do not reject $H_0$ . Since the complexity of sampling distributions for ${\tilde{Q}}_{\mathrm{pi}}^{\prime *}$ greatly hinders computation of $A_T$ and $A_R$ , we propose an approximate approach to computing $A_T$ and $A_R$ and making a final decision. First, let $j=\lfloor 100\alpha \rfloor$ owing to $0.01\le \alpha \le 1$ , where $j$ is the greatest integer that is less than or equal to $100\alpha$ . Then, $\alpha \prime =1-\sqrt{1-0.01j}$ and $j=1,2,\dots ,100$ . In the circumstances given above, $d_j^{+}=Q_{\mathrm{pi}2}^{\prime *}(\alpha )-Q_{\mathrm{pi}}^{*}$ and $d_j^{-}=Q_{\mathrm{pi}}^{*}-Q_{\mathrm{pi}1}^{\prime *}(\alpha )$ can be, respectively, expressed as follows

$$\begin{array}{c}\multicolumn{1}{c}{d_j^{+}=(Q_{\mathrm{pi}}^{*}-1.5)(\sqrt{\frac{{\chi }_{0.5+(\sqrt{1-0.01j}/2),n-1}^2}{{\chi }_{0.5,n-1}^2}}-1)}\\ \multicolumn{1}{c}{+\sqrt{\frac{n-1}{n}}\left(\frac{z_{0.5-(\sqrt{1-0.01j}/2)}}{\sqrt{{\chi }_{0.5,n-1}^2}}\right)}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{d_j^{-}=(Q_{\mathrm{pi}}^{*}-1.5)(1-\sqrt{\frac{{\chi }_{0.5-(\sqrt{1-0.01j}/2),n-1}^2}{{\chi }_{0.5,n-1}^2}})}\\ \multicolumn{1}{c}{+\sqrt{\frac{n-1}{n}}\left(\frac{z_{0.5-(\sqrt{1-0.01j}/2)}}{\sqrt{{\chi }_{0.5,n-1}^2}}\right)}\end{array}$$

Thus, $d_j=d_j^{+}+d_j^{-}$ can be calculated as follows

$$d_j=\{\begin{array}{c}(Q_{\mathrm{pi}}^{*}-1.5\hfill \end{array})(\sqrt{\frac{{\chi }_{0.5+(\sqrt{1-0.01j}/2),n-1}^2}{{\chi }_{0.5,n-1}^2}}-\sqrt{\frac{{\chi }_{0.5-(\sqrt{1-0.01j}/2),n-1}^2}{{\chi }_{0.5,n-1}^2}})+2\sqrt{\frac{n-1}{n}}\left(\frac{z_{0.5-(\sqrt{1-0.01j}/2)}}{\sqrt{{\chi }_{0.5,n-1}^2}}\right),j=1,2,\dots ,100(Q_{\mathrm{pi}}^{*}-1.5)(\sqrt{\frac{{\chi }_{0.9975,n-1}^2}{{\chi }_{0.5,n-1}^2}}-\sqrt{\frac{{\chi }_{0.0025,n-1}^2}{{\chi }_{0.5,n-1}^2}})+2\sqrt{\frac{n-1}{n}}\left(\frac{z_{0.0025}}{\sqrt{{\chi }_{0.5,n-1}^2}}\right),j=0$$

(18)

However, if we let $r_j=C_0-Q_{\mathrm{pi}1}^{\prime *}(\alpha )$ , then $r_j=d_j^{-}+d_{\beta }$ . Obviously, $r_j$ can be expressed as follows

$$\begin{array}{c}\multicolumn{1}{c}{r_j=(Q_{\mathrm{pi}}^{*}-1.5)(1-\sqrt{\frac{{\chi }_{0.5-(\sqrt{1-0.01j}/2),n-1}^2}{{\chi }_{0.5,n-1}^2}})}\\ \multicolumn{1}{c}{+\sqrt{\frac{n-1}{n}}\left(\frac{z_{0.5-(\sqrt{1-0.01j}/2)}}{\sqrt{{\chi }_{0.5,n-1}^2}}\right)+d_{\beta }}\end{array}$$

(19)

Thus, when $\alpha =a$ , then $r_j=0$ . Similarly, let $h=\lfloor 100a\rfloor$ because of $0.01\le \alpha \le a$ , that is, $j=1,2,\dots ,h$ . Obviously, $r_j$ can be expressed as follows

$$r_j=\{\begin{array}{c}(Q_{\mathrm{pi}}^{*}-1.5)(1-\sqrt{\frac{{\chi }_{0.5-(\sqrt{1-0.01j}/2),n-1}^2}{{\chi }_{0.5,n-1}^2}})\hfill \\ \multicolumn{1}{c}{+\sqrt{\frac{n-1}{n}}\left(\frac{z_{0.5-(\sqrt{1-0.01j}/2)}}{\sqrt{{\chi }_{0.5,n-1}^2}}\right)+d_{\beta },j=1,2,\dots ,h}\\ \multicolumn{1}{c}{(Q_{\mathrm{pi}}^{*}-1.5)(1-\sqrt{\frac{{\chi }_{0.0025,n-1}^2}{{\chi }_{0.5,n-1}^2}})}\\ \multicolumn{1}{c}{+\sqrt{\frac{n-1}{n}}\left(\frac{z_{0.0025}}{\sqrt{{\chi }_{0.5,n-1}^2}}\right)+d_{\beta },j=0}\end{array}$$

(20)

As mentioned above, approximations of $A_T$ and $A_R$ can be obtained as follows

$$\begin{array}{c}\multicolumn{1}{c}{A_T=\sum _{j=1}^{100}{A}_{\mathrm{Tj}}=\sum _{j=1}^{100}\left(\frac{d_{j-1}+d_j}{2}\right)\times (0.01),}\\ \multicolumn{1}{c}{j=1,2,\dots ,100}\end{array}$$

(21)

$$\begin{array}{c}\multicolumn{1}{c}{A_R=\sum _{j=1}^h{A}_{\mathrm{Rj}}=\sum _{j=1}^h\left(\frac{r_{j-1}+r_j}{2}\right)\times (0.01),}\\ \multicolumn{1}{c}{j=1,2,\dots ,h}\end{array}$$

(22)

According to equations (21) and (22), we can calculate the ratio of $A_R$ to $A_T$ to make reliable decisions with our proposed fuzzy estimator ${\tilde{Q}}_{\mathrm{pi}}^{\prime *}$ . Thus, we choose two values ${\varphi }_1$ and ${\varphi }_2$ , such that $0<{\varphi }_1<{\varphi }_2<0.5$ , and the decision rules are as follows:

1. If $A_R/A_T\le {\varphi }_1$ , then do not reject $H_0$ . We can conclude that the process quality meets the $k\sigma$ level requirements.

2. If ${\varphi }_1<A_R/A_T<{\varphi }_2$ , then no decision on whether to reject or not reject $H_0$ . Obviously, it is not demonstrated that the process quality meets customer requirements.

3. If $A_R/A_T\ge {\varphi }_2$ , then reject $H_0$ . Thus, we can conclude specifically that the process quality does not meet the specified $k\sigma$ level requirements.

Application example

To demonstrate practical application of the proposed fuzzy hypothesis testing for one-sided SSQIs, we present a case study of indium tin oxide (ITO) film manufacturing. ITO film is one of the main components in touchscreen devices, useful because of its conductive electrode functions, high transmittance, and low resistivity. Thus, the quality of the product is vital, as defects will ultimately result in poorly functioning touchscreens. In the following demonstration, we focus on an STB quality characteristic: distribution of resistivity. The specification limit for distribution of resistivity is set as follows: $\mathrm{USL}\le 5\mathrm{\%}$ . For a mass-produced ITO film product, a product engineer can collect data samples (100 measurements) at regular intervals from a stable process (i.e. in statistical control). Figure 3 presents the normal probability plot for the collected sample data. We further use the Anderson–Darling test to confirm the assumption of normality with $p\mathrm{-}\mathrm{value}>0.05$ . Therefore, it is reasonable to assume that the collected data are normally distributed.

Figure 3.

Normal probability plot for the distribution of resistivity sample data.

For these 100 measurements, we can calculate the sample mean ${\mu }^{*}=0.04261$ and sample standard deviation ${\sigma }^{*}=0.002334$ . Thus, $Q_{\mathrm{pi}}^{*}=Q_{\mathrm{pu}}^{*}=4.66714$ . The quality level of products must meet the $5\sigma$ level specified by the customer (i.e. $k=5$ ), and significance level $\beta$ was set to 0.05. Therefore, the critical value $C_0=4.60596$ can be obtained according to equation (14). Due to $Q_{p2}^{*}>C_0$ , we do not reject $H_0$ based on the crisp estimate $Q_{\mathrm{pi}}^{*}$ . However, fuzziness and stochastic uncertainty are unavoidable features of collected data. In view of this, we further undertake the fuzzy hypothesis testing for $Q_{\mathrm{pi}}$ to make a more reliable decision.

First, we can obtain $d_{\beta }=-0.06118$ based on $C_0$ and $Q_{\mathrm{pi}}^{*}$ . According to equation (17), we can further calculate $a=0.97778$ , that is, $h=\lfloor 100a\rfloor =97$ . In view of this, $A_T$ and $A_R$ can be obtained from equations (18), (20), (21), and (22), as follows:

$$\begin{array}{c}\multicolumn{1}{c}{A_T=\sum _{j=1}^{100}{A}_{\mathrm{Tj}}=(\frac{d_0+d_{100}}{2}+d_1+d_2+\cdots +d_{99})}\\ \multicolumn{1}{c}{\times 0.01=0.73475}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{A_R=\sum _{j=1}^{97}{A}_{\mathrm{Rj}}=(\frac{r_0+r_{97}}{2}+r_1+r_2+\cdots +r_{96})}\\ \multicolumn{1}{c}{\times 0.01=0.30171}\end{array}$$

Thus, we can further calculate $A_R/A_T=0.41063$ . When a decision-maker and engineer suggest the settings ${\varphi }_1=0.2$ and ${\varphi }_2=0.4$ , then $A_R/A_T\ge 0.4$ can be found based on the judgment criterion. Clearly, this results in rejection of the null hypothesis; in this situation, there is sufficient evidence that quality characteristics do not meet customer requirements. Thus, the manager should adopt measures to improve process quality.

Conclusion

Past research on the evaluation of process quality primarily used precise measurement data to analyze and evaluate process quality. However, measurement errors can easily prevent clear quantization of characteristic values. Moreover, products could have unclear product characteristics or other unclear elements that cause uncertainties in evaluation. Thus, this study endeavored to develop a more reliable evaluation of process quality through fuzzy inference to assess process quality in situations involving imprecise sample data. This was accomplished using one-sided SSQIs $Q_{\mathrm{pi}}$ . This study further presents a set of confidence intervals producing a triangular-shaped fuzzy number for $Q_{\mathrm{pi}}$ based on Buckley’s estimation approach. Thus, one-sided SSQIs $Q_{\mathrm{pi}}$ ’s crisp estimator $Q_{\mathrm{pi}}^{*}$ was extended to fuzzy estimator ${\tilde{Q}}_{\mathrm{pi}}^{*}$ , where ${\tilde{Q}}_{p1}^{*}={\tilde{Q}}_{\mathrm{pl}}^{*}$ and ${\tilde{Q}}_{p2}^{*}={\tilde{Q}}_{\mathrm{pu}}^{*}$ . Furthermore, when the test statistic is a fuzzy value, this study used the ratio of $A_R$ to $A_T$ (i.e. $A_R/A_T$ ) to perform fuzzy hypothesis testing for evaluation of process quality.

To illustrate the practical applicability of the proposed approach, we present an application example regarding distribution of resistivity for ITO film. In hypothesis testing with the crisp estimate $Q_{\mathrm{pi}}^{*}=4.66714$ , we do not reject null hypothesis $H_0:Q_{\mathrm{pi}}\ge 5$ because of $Q_{\mathrm{pi}}^{*}>C_0=4.60596$ . However, it is worth noting that $Q_{\mathrm{pi}}^{*}$ is only 0.06118 higher than $C_0$ . To this end, we further investigated the triangular-shaped fuzzy number of $Q_{\mathrm{pi}}^{*}$ under consideration of imprecise data and performed fuzzy hypothesis testing for $Q_{\mathrm{pi}}$ . Our results show that the process quality did not meet the required $5\sigma$ level because $A_R/A_T=0.41063\ge 0.4$ (i.e. null hypothesis rejected $H_0:Q_{\mathrm{pi}}\ge 5$ ). For this reason, the proposed quality assessment method for imprecise process data in this study could effectively assist manufacturers in making more reliable decisions.