Fuzzy Evaluation Model for Operational Performance of Air Cleaning Equipment

Abstract

Global warming has led to the continuous deterioration of the living environment, in which air quality directly affects human health. In addition, the severity of the COVID-19 pandemic has further increased the attention to indoor air quality. Indoor clean air quality is not only related to human health but also related to the quality of the manufacturing environment of clean rooms for numerous high-tech processes, such as semiconductors and packaging. This paper proposes a comprehensive model for evaluating, analyzing, and improving the operational performance of air cleaning equipment. Firstly, three operational performance evaluation indexes, such as the number of dust particles, the number of colonies, and microorganisms, were established. Secondly, the 100(1$-$ $\alpha$)% upper confidence limits of these three operational performance evaluation indexes were deduced to construct a fuzzy testing model. Meanwhile, the accumulated value of $\varphi$ was used to derive the evaluation decision-making value. The proposed model can help companies identify the key quality characteristics that need to be improved. Furthermore, the competitiveness of cooperative enterprises towards smart manufacturing can be strengthened, so that enterprises can not only fulfill their social responsibilities while developing the economy but also take into account the sustainable development of enterprises and the environment.

1. Introduction

The gradual popularization and maturation of the Internet of Things (IOT) have led to continuous advancements in big data analysis techniques [1,2,3,4]. Manufacturing companies can increase product quality and value through data management and even reduce the adverse effects of the production process on humans and environments [5,6,7].

The manufacturing environments of many high-tech processes, such as semiconductors, packaging, and testing, require cleanrooms to increase process quality and yield. Among cleanroom equipment, air-cleaning equipment plays a crucial role; its operational performance influences the safety, hygiene, and health of the workers in the cleanroom, as well as the process quality and yield of the products. The objective of this study is to establish a comprehensive fuzzy evaluation, analysis, and improvement model for the operational performance of air-cleaning equipment. Relevant studies have used the number of dust particles of various sizes per cubic foot (${\mathrm{ft}}^3$) of air as standards [8,9]. For instance, ISO Class 4 standards set the following limits for different sizes of dust particles [10]:

• No more than $U_1=$ 10,000 ${\mathrm{particles}/\mathrm{ft}}^3$ of dust particles $\ge 0.1 \mathsf{\mu }\mathrm{m}$
• No more than $U_2=$ 2370 ${\mathrm{particles}/\mathrm{ft}}^3$ of dust particles $\ge 0.2 \mathsf{\mu }\mathrm{m}$
• No more than $U_3=$ 1020 ${\mathrm{particles}/\mathrm{ft}}^3$ of dust particles $\ge 0.3 \mathsf{\mu }\mathrm{m}$
• No more than $U_4=$ 352 ${\mathrm{particles}/\mathrm{ft}}^3$ of dust particles $\ge 0.4 \mathsf{\mu }\mathrm{m}$
• No more than $U_5=$ 83 ${\mathrm{particles}/\mathrm{ft}}^3$ of dust particles $\ge 0.5 \mathsf{\mu }\mathrm{m}$

There are also similar limits for bacteria and microorganisms. For instance, there may be no more than 1 colony forming unit (cfu) of bacteria per 10 ${\mathrm{ft}}^3$ and no more than 25 cfu of microorganisms per 10 ${\mathrm{ft}}^3$. The numbers of dust particles, bacterial colonies, and microbial colonies per unit of air thus present clear measures of the operational performance of the cleaning equipment. In this study, we let h $=$ 1 represent operational performance factor 1 (the number of dust particles), h $=$ 2 represent operational performance factor 2 (the number of bacterial colonies), and h $=$ 3 represent operational performance factor 3 (microorganism content). Based on the above, we let j represent various categories of operational performance factors. For example, h $=$ 1 and $j=$ 1 indicate the number of dust particles $\ge 0.1 \mathsf{\mu }\mathrm{m}$ per cubic foot (${\mathrm{ft}}^3$) of air. Next, we let $U_{hj}$ represent the upper limit (standard) of the content of category j of operational performance factor h, where h $=$ 1, 2, 3 and $j=$ 1, 2,…, $m_h$. We then let the random variable $X_{hj}$ denote the content of category j of operational performance factor h in the air. Drawing on the work of Li et al. [11], we define $Y_{hj}=X_{hj}/U_{hj}$, meaning that the upper limits of the content of each category j of each operational performance factor h are all set to 1.0. If we let ${\delta }_h$ represent the total mean of operational performance factor h and let ${\gamma }_h^2$ denote the variance in the operational performance factor h, then according to Chang et al. [12], the operational performance index of factor h is defined as follows:

$$O_{Ph}={\displaystyle \frac{1-{\delta }_h}{{\gamma }_h}},h=1,2,3.$$

(1)

The above equation indicates that the smaller the total mean ${\delta }_h$ and variance ${\gamma }_h^2$ of operational performance factor h, the greater the value of the operational performance index $O_{Ph}$. In other words, a lower mean content of dust particles, bacterial colonies, and microorganisms per unit of air, along with a smaller variance, indicates that the operational performance of the air cleaning equipment is reliably good. In contrast, a smaller value of $O_{Ph}$ represents poorer operational performance of the air cleaning equipment. The operational performance index $O_{Ph}$ has a one-to-one mathematical relationship with the qualification ratio of operational performance.

Based on the three mentioned research factors, we develop an evaluation, analysis, and improvement model for the operational performance of air cleaning equipment to aid manufacturers in identifying critical quality characteristics in need of improvement. This model also enables process engineers to identify items with poor performance based on evaluation data, find the cause of poor performance, and devise improvement strategies. By following this procedure, experience can be accumulated to create a smart improvement database. This database can serve as a reference for designing new generations of products and establishing a supplier selection model for critical materials or components. The proposed model thus supports progress towards smart manufacturing, facilitating sustainable development while pursuing economic growth. For example, during the design stage, it is essential to consider not only cost but also the recycling rate of the entire product. In the manufacturing stage, investing additional resources in technology research and development can help optimize the process, minimize costs, and ease carbon emissions resulting from rework and scrap. Additionally, such investments can lower costs and carbon emission losses associated with maintenance after the product is sold [1,5].

As the proposed indices contain unknown parameters, we derive the 100(1$-$$\alpha$)% upper confidence limits of the indices to mitigate the risk of misjudgment due to sampling errors [13,14,15]. We use the upper confidence limits to establish statistical testing rules based on the significance level. Researchers have pointed out that in the era of Industry 4.0, companies strive for swift responses, which generally mean small sample sizes [15,16,17]. In this case, accumulated data and experience can be used to formulate fuzzy test rules based on the confidence intervals [18,19,20]. The incorporation of confidence intervals offers both scientific rigor and convenience.

The remainder of this paper is organized as follows. In Section 2, we derive the upper confidence limit of the proposed index based on sample data. In Section 3, we use the upper confidence limits of three indices to establish a one-tailed fuzzy testing model. Section 4 introduces a case to illustrate the application of the fuzzy testing model proposed in this paper. Section 5 presents our conclusions.

2. Upper Confidence Limit of Index ${\mathit{O}}_{\mathit{P}\mathit{h}}$

The numbers of dust particles, bacterial colonies, and microorganisms per unit of air represent the operational performance of air cleaning equipment. These three characteristics are all categorized as the smaller-the-better type, so we proposed an operational performance evaluation index of operational performances for factor h in accordance with the concepts presented by Chang et al. [12], as shown in Equation (1). Assuming a normal process, the qualification ratio of operational performance p is defined as follows:

$$p=p\left(Y_h\le 1\right)=p\left(Z\le {\displaystyle \frac{1-{\delta }_h}{{\gamma }_h}}\right)=\Phi \left(O_{Ph}\right),$$

(2)

where $Z=\left(Y_h-{\delta }_h\right)/{\gamma }_h$ follows a standard normal distribution, and $\Phi (•)$ is the cumulative distribution function of the standard normal distribution.

As the indices all contain two unknown parameters, they must be estimated using collected performance data. The estimations of these two unknown parameters are expressed as follows:

$${\delta }_h^{\ast }={\displaystyle \frac{1}{n{m}_h}}{\displaystyle \sum _{j=1}^{m_h}{\displaystyle \sum _{i=1}^n{Y}_{h,j,i}}}$$

(3)

and

$${\gamma }_h^{\ast }=\sqrt{{\displaystyle \frac{1}{n{m}_h}}{\displaystyle \sum _{j=1}^{m_h}{\displaystyle \sum _{i=1}^n{\left(Y_{h,j,i}-{\delta }_h^{\ast }\right)}^2}}}.$$

(4)

Based on these equations, the estimation of the associated evaluation indices is written as follows:

$$O_{Ph}^{\ast }={\displaystyle \frac{1-{\delta }_h^{\ast }}{{\gamma }_h^{\ast }}},h=1,2,3.$$

(5)

Under the assumption of a normal process, let the random variable $T_h$ be defined as follows:

$$T_h={\displaystyle \frac{\sqrt{n{m}_h}\left({\delta }_h^{\ast }-{\delta }_h\right)}{{\gamma }_h^{\ast }}}.$$

(6)

Thus, the probability distribution of the random variable $T_h$ is a t-distribution with $n{m}_h-1$ degrees of freedom, written as $T_h~t_{n{m}_h-1}$. Next, we set random variable $K_h$ as follows:

$$K_h={\displaystyle \frac{n{m}_h{\gamma }_h^{\ast 2}}{{\gamma }_h^2}}.$$

(7)

The probability distribution of the random variable $K_h$ is a chi-square distribution with $n{m}_h-1$ degrees of freedom, written as $K_h~{\chi }_{n{m}_h-1}^2$. To derive the upper confidence limit of the operational performance index $O_{Ph}$, we first define Event $E_1$ as follows:

$$E_1=\left\{{\delta }_h\ge {\delta }_h^{\ast }-{\displaystyle \frac{t_{\alpha /2;n{m}_h-1}}{\sqrt{n{m}_m}}}{\gamma }_h^{\ast }\right\}.$$

(8)

Clearly,

$$p\left(E_1\right)=p\left\{{\delta }_h\ge {\delta }_h^{\ast }-{\displaystyle \frac{t_{\alpha /2;n{m}_h-1}}{\sqrt{n{m}_m}}}{\gamma }_h^{\ast }\right\}=p\left\{T_h\ge -t_{\alpha /2;n{m}_h-1}\right\}=1-{\displaystyle \frac{\alpha }{2}}.$$

(9)

Next, we define Event $E_2$:

$$E_2=\left\{{\gamma }_h\ge \sqrt{{\displaystyle \frac{n{m}_h}{{\chi }_{1-\alpha /2;n{m}_h-1}^2}}}{\gamma }_h^{\ast }\right\}.$$

(10)

Similarly,

$$p\left(E_2\right)=p\left\{{\gamma }_h\ge \sqrt{{\displaystyle \frac{n{m}_h}{{\chi }_{1-\alpha /2;n{m}_h-1}^2}}}{\gamma }_h^{\ast }\right\}=p\left\{K_h\le {\chi }_{1-\alpha /2;n{m}_h-1}^2\right\}=1-{\displaystyle \frac{\alpha }{2}}.$$

(11)

We let the complement sets of Events $E_1$ and $E_2$ be $E_1^c$ and $E_2^c$, respectively. Thus,

$$p\left(E_1^c\right)=1-p\left(E_1\right)={\displaystyle \frac{\alpha }{2}} \mathrm{and} p\left(E_2^c\right)=1-p\left(E_2\right)={\displaystyle \frac{\alpha }{2}}.$$

(12)

According to Boole’s inequality and DeMorgan’s laws [21],

$$p\left(E_1\cap E_2\right)\ge 1-p\left(E_1^c\right)-p\left(E_2^c\right)=1-\alpha .$$

(13)

Thus,

$$\begin{array}{ll}p\left(E_1\cap E_2\right)& =p\left\{{\delta }_h\ge {\delta }_h^{\ast }-{\displaystyle \frac{t_{\alpha /2;n{m}_h-1}}{\sqrt{n{m}_m}}}{\gamma }_h^{\ast },{\gamma }_h\ge \sqrt{{\displaystyle \frac{n{m}_h}{{\chi }_{1-\alpha /2;n{m}_h-1}^2}}}{\gamma }_h^{\ast }\right\}\\ & =p\left\{1-{\delta }_h\le 1-{\delta }_h^{\ast }+{\displaystyle \frac{t_{\alpha /2;n{m}_h-1}}{\sqrt{n{m}_m}}}{\gamma }_h^{\ast },{\gamma }_h\ge \sqrt{{\displaystyle \frac{n{m}_h}{{\chi }_{1-\alpha /2;n{m}_h-1}^2}}}{\gamma }_h^{\ast }\right\}\\ & =p\left\{{\displaystyle \frac{1-{\delta }_h}{{\gamma }_h}}\le \left({\displaystyle \frac{1-{\delta }_h^{\ast }}{{\gamma }_h^{\ast }}}+{\displaystyle \frac{t_{\alpha /2;n{m}_h-1}}{\sqrt{n{m}_h}}}\right)\sqrt{{\displaystyle \frac{{\chi }_{1-\alpha /2;n{m}_h-1}^2}{n{m}_h}}}\right\}\end{array}.$$

(14)

Based on Equations (1), (5), and (14),

$$p\left\{O_{Ph}\le \left(O_{Ph}^{\ast }+{\displaystyle \frac{t_{\alpha /2;n{m}_h-1}}{\sqrt{n{m}_h}}}\right)\sqrt{{\displaystyle \frac{{\chi }_{1-\alpha /2;n{m}_h-1}^2}{n{m}_h}}}\right\}\ge 1-\alpha .$$

(15)

Therefore, the 100(1$-$$\alpha$)% upper confidence limit of the operational performance index $O_{Ph}$ is defined as follows:

$$U{O}_{Ph}=\left(O_{Ph}^{\ast }+{\displaystyle \frac{t_{\alpha /2;n{m}_h-1}}{\sqrt{n{m}_h}}}\right)\sqrt{{\displaystyle \frac{{\chi }_{1-\alpha /2;n{m}_h-1}^2}{n{m}_h}}}.$$

(16)

3. Development of Fuzzy Testing Model

If the required value of the operational performance index $O_{Ph}$ is $R_h$, then null hypothesis $H_0$ and alternative hypothesis $H_1$ of the test are defined as follows:

$$\begin{array}{c}{H}_0:O_{Ph}\ge R_h;\\ H_1:O_{Ph}<R_h.\end{array}$$

We let the observed value of $Y_{h,j,i}$ be $y_{h,j,i}$, where h $=$ 1, 2, 3, $j=$ 1, 2,…, $m_h$, and $i=$1, 2, …, n. Thus, the observed values of the estimator $O_{ph}^{\ast }$ and the 100(1$-$$\alpha$)% upper confidence limit $U{O}_{Ph}$ are expressed as follows:

$$O_{ph0}^{\ast }={\displaystyle \frac{1-{\overline{y}}_h}{s_h}}$$

(17)

and

$$U{O}_{Ph0}=\left(O_{Ph0}^{\ast }+{\displaystyle \frac{t_{\alpha /2;n{m}_h-1}}{\sqrt{n{m}_h}}}\right)\sqrt{{\displaystyle \frac{{\chi }_{1-\alpha /2;n{m}_h-1}^2}{n{m}_h}}},$$

(18)

where ${\overline{y}}_h$ is the observed value of ${\delta }_h^{\ast }$, and $s_h^2$ is the observed value of ${\gamma }_h^{\ast 2}$, as shown below:

$${\overline{y}}_h={\displaystyle \frac{1}{n{m}_h}}{\displaystyle \sum _{j=1}^{m_h}{\displaystyle \sum _{i=1}^n{y}_{h,j,i}}}$$

(19)

and

$$s_h^2={\displaystyle \frac{1}{n{m}_h}}{\displaystyle \sum _{j=1}^{m_h}{\displaystyle \sum _{i=1}^n{\left(y_{h,j,i}-{\overline{y}}_h\right)}^2}}.$$

(20)

We present the following statistical testing rules:

(1)

When $U{O}_{Ph0}\ge R_h$, then do not reject $H_0$ and conclude that $O_{Ph0}\ge R_h$.

(2)

When $U{O}_{Ph0}<R_h$, then reject $H_0$ and conclude that $O_{Ph0}<R_h$.

We use Equation (18) and the above rules to develop our fuzzy testing and evaluation model. As suggested by Chen and Lin [18], the α-cuts of quasi-triangular fuzzy number ${\tilde{O}}_{Ph}$ is defined as follows:

$${\tilde{O}}_{Ph}\left[\alpha \right]=\{\begin{array}{l}\left[O_{Ph}\left(1\right),O_{Ph}\left(\alpha \right)\right], for 0.01\le \alpha \le 1\\ \left[O_{Ph}\left(1\right),O_{Ph}\left(0.01\right)\right], for 0\le \alpha \le 0.01\end{array},$$

(21)

where

$$O_{Ph}\left(1\right)=O_{Ph0}^{\ast }\sqrt{{\displaystyle \frac{{\chi }_{0.5;n{m}_h-1}^2}{n{m}_h}}};$$

(22)

$$U{O}_{Ph0}\left(\alpha \right)=\left(O_{Ph0}^{\ast }+{\displaystyle \frac{t_{\alpha /2;n{m}_h-1}}{\sqrt{n{m}_h}}}\right)\sqrt{{\displaystyle \frac{{\chi }_{1-\alpha /2;n{m}_h-1}^2}{n{m}_h}}}.$$

(23)

According to the abovementioned, the quasi-triangular fuzzy number is denoted as ${\tilde{O}}_{Ph}=\Delta \left(O_{PMh},O_{PRh}\right)$, where $O_{PMh}=O_{Ph}\left(1\right)$ and

$$O_{PRh}=\left(O_{Ph0}^{\ast }+{\displaystyle \frac{t_{0.005;n{m}_h-1}}{\sqrt{n{m}_h}}}\right)\sqrt{{\displaystyle \frac{{\chi }_{0.995;n{m}_h-1}^2}{n{m}_h}}}.$$

(24)

Thus, the fuzzy membership function of ${\tilde{O}}_{Ph}$ can be expressed as follows:

$${\eta }_h\left(x\right)=\{\begin{array}{ll}0& if x<O_{PMh}\\ 1& if x=O_{PMh}\\ \alpha & if O_{PMh}<x\le O_{PRh}\\ 0& if x>O_{PRh}\end{array},$$

(25)

where the value of $\alpha$ is determined using the following equation:

$$x=\left(O_{Ph0}^{\ast }+{\displaystyle \frac{t_{\alpha /2;n{m}_h-1}}{\sqrt{n{m}_h}}}\right)\sqrt{{\displaystyle \frac{{\chi }_{1-\alpha /2;n{m}_h-1}^2}{n{m}_h}}}.$$

(26)

Let Set $A_{Th}$ be the region bounded by the fuzzy membership function ${\eta }_h\left(x\right)$ and the x axis. Set $A_{Rh}$ is the area of $A_{Th}$ on the right side of the vertical line $x=R_h$, as shown in Figure 1:

Figure 1. Fuzzy membership function ${\eta }_h\left(x\right)$ and vertical line $x=R_h$.

According to Figure 1, Sets $A_{Th}$ and $A_{Rh}$ are defined as follows:

$$A_{Th}=\left\{\left(x,\alpha \right)|O_{PMh}\le x\le O_{Ph}(\alpha ),0\le \alpha \le 1\right\}$$

(27)

and

$$A_{Rh}=\left\{\left(x,\alpha \right)|R_h\le x\le O_{Ph}(\alpha ),0\le \alpha \le a\right\},$$

(28)

where $O_{Ph}(a)=R_h$. Let $d_{Th}$$=$$2\left(O_{PRh}-O_{PMh}\right)$ and $d_{Rh}$$=$$O_{PRh}-R_h$. Thus,

$${\displaystyle \frac{d_{Rh}}{d_{Th}}}={\displaystyle \frac{O_{PRh}-R_h}{2\left(O_{PRh}-O_{PMh}\right)}}={\displaystyle \frac{\left(O_{Ph0}^{\ast }+\frac{t_{0.005;n{m}_h-1}}{\sqrt{n{m}_h}}\right)\sqrt{\frac{{\chi }_{0.995;n{m}_h-1}^2}{n{m}_h}}-R_h}{2\left(\left(O_{Ph0}^{\ast }+\frac{t_{0.005;n{m}_h-1}}{\sqrt{n{m}_h}}\right)\sqrt{\frac{{\chi }_{0.995;n{m}_h-1}^2}{n{m}_h}}-O_{Ph0}^{\ast }\sqrt{\frac{{\chi }_{0.5;n{m}_h-1}^2}{n{m}_h}}\right)}}.$$

(29)

To summarize, the confidence interval-based fuzzy testing model proposed in this paper differs from traditional fuzz testing, which typically involves four main steps: fuzzification, rule-based approach, inference mechanism, and defuzzification. Instead, this method mainly determines the value of $\varphi$ through statistical inference, past accumulated production data, and expert experience to balance an increase in $\alpha$-risk [22,23]. For convenience, we replace R_h in Equation (29) with $FU{O}_{Ph}$, resulting in $d_{Rh}/d_{Th}=\varphi$, as shown in the following equation:

$${\displaystyle \frac{O_{PRh}-FU{O}_{Ph}}{2\left(O_{PRh}-O_{PMh}\right)}}=\varphi ,$$

(30)

which is equivalent to

$$FU{O}_{Ph}=O_{PRh}\left(1-2\varphi \right)+2\varphi O_{PMh},$$

(31)

(1)

where $FU{O}_{Ph}$ is the fuzzy upper confidence limit of the operational performance index $O_{Ph}$. When the fuzzy upper confidence limit $FU{O}_{Ph}$ is greater than or equal to the required value $R_h$ ($FU{O}_{Ph}\ge R_h$), then ${d_{Rh}/d}_{Th}$$\le$$\varphi$, indicating that $H_0$ is not rejected and the operational performance has reached the required level. Conversely, when $FU{O}_{Ph}$ is smaller than the required value $R_h$ ($FU{O}_{Ph}<R_h$), then ${d_{Rh}/d}_{Th}$$>$$\varphi$, meaning $H_0$ is rejected and the operational performance needs improvement. Based on these conditions, the fuzzy testing rules are made as follows: When the fuzzy upper confidence limit is smaller than the required value $R_h$ ($FU{O}_{Ph}<R_h$), reject $H_0$ and improve this characteristic.

(2)

When the fuzzy upper confidence limit is greater than or equal to the required value $R_h$ ($FU{O}_{Ph}\ge R_h$), do not reject $H_0$.

(3)

When all three operational performance factors meet the requirements (that is, $FU{O}_{Ph}$ is greater than or equal to the required value $R_h$, where h $=$ 1, 2, 3), the operational performance of the air cleaning equipment meets requirements.

4. An Illustrative Example

As mentioned previously, the numbers of dust particles, bacterial colonies, and microbial colonies per unit of air represent the operational performance of air cleaning equipment. When all three indices reach the required levels, it means that the air cleaning equipment has achieved the necessary operational performance. Suppose quality engineers stipulate that the values of these three indices must be at least 1.0 based on customer requirements. Therefore, the null hypothesis $H_0$ and the alternative hypothesis $H_1$ are defined as follows:

$$\begin{array}{c}{H}_0: O_{Ph}\ge 1.0,\\ H_1: O_{Ph}<1.0.\end{array}$$

Each of these indices contains two unknown parameters, so they must be estimated by collecting sample data [24]. To facilitate swift responses, the quality engineers take 20 samples. The relevant statistics based on the sample data are listed as follows:

1.

Number of dust particles (h $=$ 1, n $=$ 20, $m_1=$ 5, $\varphi =0.2$)

$$\begin{array}{c}{\overline{y}}_1={\displaystyle \frac{1}{100}}{\displaystyle \sum _{j=1}^5{\displaystyle \sum _{i=1}^{20}{y}_{1,j,i}}}=0.751,\\ s_1=\sqrt{{\displaystyle \frac{1}{100}}{\displaystyle \sum _{j=1}^5{\displaystyle \sum _{i=1}^{20}{\left(y_{1,j,i}-{\overline{y}}_1\right)}^2}}}=0.330,\\ O_{P10}^{\ast }={\displaystyle \frac{1-{\overline{y}}_1}{s_1}}=0.755,\\ O_{PR1}=\left(O_{P10}^{\ast }+{\displaystyle \frac{t_{0.005;99}}{\sqrt{100}}}\right)\sqrt{{\displaystyle \frac{{\chi }_{0.995;99}^2}{100}}}=1.199,\\ O_{PM1}=O_{P10}^{\ast }\times \sqrt{{\displaystyle \frac{{\chi }_{0.5;99}^2}{100}}}=0.748,\\ \mathrm{and}\\ FU{O}_{P1}=0.6\times O_{PR1}+0.4\times O_{PMh}=1.019.\end{array}$$

According to the fuzzy testing rules, the fuzzy upper confidence limit is greater than the required value of 1.0. Thus, we do not reject $H_0$, as its operational performance meets the requirements.

2.

Number of bacterial colonies (h $=$ 2, n $=$ 10, $m_2=$ 1, $\varphi =0.2$)

$$\begin{array}{c}{\overline{y}}_2={\displaystyle \frac{1}{20}}{\displaystyle \sum _{j=1}^1{\displaystyle \sum _{i=1}^{20}{y}_{2,j,i}}}=0.852,\\ s_2=\sqrt{{\displaystyle \frac{1}{20}}{\displaystyle \sum _{j=1}^1{\displaystyle \sum _{i=1}^{20}{\left(y_{2,j,i}-{\overline{y}}_2\right)}^2}}}=0.421,\\ O_{P20}^{\ast }={\displaystyle \frac{1-{\overline{y}}_2}{s_2}}=0.352,\\ O_{PR2}=\left(O_{P20}^{\ast }+{\displaystyle \frac{t_{0.005;19}}{\sqrt{20}}}\right)\sqrt{{\displaystyle \frac{{\chi }_{0.995;19}^2}{20}}}=1.377,\\ O_{PM2}=O_{P20}^{\ast }\times \sqrt{{\displaystyle \frac{{\chi }_{0.5;19}^2}{20}}}=0.337,\\ \mathrm{and}\\ FU{O}_{P2}=0.6\times O_{PR2}+0.4\times O_{PM2}=0.961.\end{array}$$

According to the fuzzy testing rules, the fuzzy upper confidence limit is smaller than the required value of 1.0. Thus, we reject $H_0$ and improve this characteristic.

3.

Numbers of microorganisms (h$=$3, n $=$ 10, $m_3=$ 1, $\varphi =0.2$)

$$\begin{array}{c}{\overline{y}}_3={\displaystyle \frac{1}{20}}{\displaystyle \sum _{j=1}^1{\displaystyle \sum _{i=1}^{20}{y}_{3,j,i}}}=0.752,\\ s_3=\sqrt{{\displaystyle \frac{1}{20}}{\displaystyle \sum _{j=1}^1{\displaystyle \sum _{i=1}^{20}{\left(y_{3,j,i}-{\overline{y}}_3\right)}^2}}}=0.388,\\ O_{P30}^{\ast }={\displaystyle \frac{1-{\overline{y}}_3}{s_3}}=0.639,\\ O_{PR3}=\left(O_{P30}^{\ast }+{\displaystyle \frac{t_{0.005;19}}{\sqrt{20}}}\right)\sqrt{{\displaystyle \frac{{\chi }_{0.995;19}^2}{20}}}=1.776,\\ O_{PM3}=O_{P30}^{\ast }\times \sqrt{{\displaystyle \frac{{\chi }_{0.5;19}^2}{20}}}=0.612,\\ \mathrm{and}\\ FU{O}_{P3}=0.6\times O_{PR3}+0.4\times O_{PM3}=1.311.\end{array}$$

According to the fuzzy testing rules, the fuzzy upper confidence limit is greater than the required value of 1.0. Thus, we do not reject, as the operational performance meets the requirements.

In this example, the operational performance of the air cleaning equipment does not meet the required level for the number of bacterial colonies, indicating that improvement is needed. Once this index is improved, the overall operational performance of the air cleaning equipment will attain the required level.

As noted above, if the statistical testing method is applied, the upper confidence limit for operational performance index 2 is 1.377. Since this value is greater than the required value of 1.0, $H_0$ is not rejected. Therefore, the decision is that no improvement is needed for operational performance index 2. However, the value of operational performance index 2 is 0.352, which is much smaller than the required value of 1.0 and is clearly unreasonable in practical terms.

If the method proposed in this paper is adopted, the fuzzy upper confidence limit for operational performance Index 2 is 0.961. Since this is less than the required value of 1.0, $H_0$ must be rejected. Therefore, the decision is that operational performance index 2 must be improved. Clearly, the method proposed in this paper is more reasonable in practical terms compared to the statistical testing method.

The result mentioned above is primarily due to the relatively small sample size n, which leads to a relatively long confidence interval. Nevertheless, in practice, businesses often need to make decisions even when the sample size n is very small in order to achieve quick response mechanisms [18,19,20]. The method proposed in this paper was developed to address this issue. When the sample size n is large, the confidence interval for the index is relatively short, and the error is relatively small, making the statistical testing method applicable [13,23].

5. Conclusions

Increasing awareness of environmental protection has highlighted the importance of air quality. Air cleaning equipment plays a crucial role in various industries. This study utilized fuzzy evaluation to help industries with high-tech processes, such as semiconductors, packaging, and testing, in enhancing the operational performance of their air cleaning equipment. It aims to reduce the adverse effects of the production process on humans and minimize the waste of production resources. This study adopted ISO Class 4 standards and standardized all upper limits of dust particle numbers to 1.0 to streamline data compilation and establish performance evaluation indices. Since the proposed evaluation indices all contain unknown parameters, we derived their 100(1$-$$\alpha$)% upper confidence limits to reduce the risk of misjudgment due to sampling errors and used these limits to establish fuzzy testing rules. When the sample size is large enough, the confidence interval is short and the error is small, allowing for the direct use of the statistical testing method. However, many studies have revealed that using upper confidence limit-based fuzzy testing allows for the incorporation of accumulated data and experience into decision rules to maintain a certain level of accuracy even with small sample sizes. This study supports manufacturers of air cleaning equipment in advancing towards smart manufacturing and promoting sustainable development while pursuing economic growth.