Fuzzy testing decision-making model for intelligent manufacturing process with Taguchi capability index

Abstract

As the Internet-of-Things matures, technologies for the measurement and collection of production data are also improving. What is needed next is an effective information analysis model to aid in the timely adjustment of manufacturing parameters to optimize production. Production data analysis models must be continuously and properly utilized to monitor and maintain process quality. Improving product quality helps to lengthen service life, decrease scrap and rework, and reduce the social losses caused by malfunctions and maintenance. The Taguchi capability index C_pm can fully reflect the losses and yield of processes and is a convenient and effective tool to evaluate and analyze process data in the industry. As it contains unknown parameters, we derived the upper confidence limit (UCL) of C_pm based on collected production data. Due to the fuzzy uncertainties that are common in measurement data, we then used the UCL of the index to construct a fuzzy membership function and propose a fuzzy testing decision-making model to determine whether processes are in need of improvement. Before the proposed fuzzy test methods became full-fledged, we used the concept of sample size and the rules of statistical testing to explain the motivation underlying those methods. In fact, the sample size influences the risk of misjudgment in UCL, and in practice, sample sizes are rarely large due to cost and time considerations, thereby they produce larger UCL with a corresponding decrease in accuracy and increase in risk of misjudgment. The fuzzy test methods proposed in this study are based on statistical inference, and judgement is aided by expertise, thus are capable of solving the problems associated with larger UCL. Therefore, the theoretical foundation of this fuzzy testing decision-making model is the UCL, it can lower the chance of misjudgment caused by sampling errors and increase evaluation accuracy.

Keywords

Fuzzy testing decision-making model intelligent manufacturing Taguchi capability index membership function of fuzzy number

1 Introduction

In this section, we introduce the research background of this paper, using literature to indicate why we adopt Taguchi capability index as a decision making methodology. We also explain why we need to develop the fuzzy testing model. Further, we examine the research motivation and aims to show the advantages and main contributions; subsequently, a research framework is set up.

As noted by Lin et al. [1], Internet-of-Things (IoT) environments are becoming more prevalent and mature in major cities and manufacturing bases around the world. Technologies for the measurement and collection of production data are also improving as they can greatly enhance manufacturing, monitoring and management technologies. This not only accelerates the development of intelligent manufacturing but also greatly increases the quality of products under intelligent manufacturing conditions. Chen et al. [2] maintained that increasing process quality lowers scrap and rework rates, lengthens product life span, and reduces product maintenance, which can in turn reduce environmental pollution. Along with the above mentioned, process capability indices (PCIs) are a convenient and effective tool to evaluate and analyze production quality data in the industry [3 –6].

With regard to process quality assessment, Chan et al. [7] proposed a Taguchi capability index that can fully reflect process losses based on the Taguchi loss functions. The denominator of Taguchi capability index is the expected value of Taguchi loss functions [8, 9]. Huang et al. [10] pointed out greater product quality can reduce the costs incurred by product malfunctions and maintenance and social losses such as environmental pollution based on the Taguchi loss functions. According to Chen and Huang [11], when process is functional then the relationship between Taguchi capability index and process yield is Yield % ≥2Φ (3C_pm) -1. Moreover, this index can fully reflect the losses and yield of processes and is a convenient and effective tool to evaluate and analyze process quality data in the industry. At the same time, it assists internal engineers in analyzing processes and provides direction for the improvement of processes with poor quality [12 –19]. Therefore, we need Taguchi capability index as a decision making methodology.

The Taguchi capability index contains unknown parameters, which means that it must be estimated using sample data. Chen [19] stated that the sample size n is rarely large for the sake of practical and cost considerations and that statistical inference results will vary with sample size n due to sampling errors. Moreover, uncertainties are likely to exist in measurement data. For these reasons, we derived the upper confidence limit of the Taguchi capability index to construct a fuzzy membership function to increase testing accuracy and overcome uncertainty in measurement data. Based on the fuzzy membership function, we then proposed a fuzzy testing decision-making model to determine whether processes are in need of improvement. As the theoretical foundation of this fuzzy testing decision-making model is the upper confidence limit, it can lower the chance of misjudgment caused by sampling errors and increase the accuracy of decisions on whether improvements are needed, thereby forming more accurate intelligent manufacturing. Besides, it can increase product value and company profits [8, 20].

Before the proposed fuzzy test methods based on the concept of Chen [21] became full-fledged, we used the concept of sample size and the rules of statistical testing to explain the motivation underlying those methods. As noted by Chen [22], the sample size influences the risk of misjudgment in upper confidence limit, and in practice, sample sizes are rarely large due to cost and time considerations, thereby they produce larger upper confidence limit with a corresponding decrease in accuracy and increase in risk of misjudgment. The fuzzy test methods proposed in this study are based on statistical inference, and judgement is aided by expertise, thus they are capable of solving the problems coupled with larger upper confidence limit. There are three advantages in fuzzy test methods developed in this study, which can be shown as following:

The Taguchi capability index is a convenient and effective tool that can fully reflect process losses by evaluating and analyzing process quality in the industry; at the same time, it assists internal engineers in analyzing processes and provides direction for the improvement of processes with poor quality.

The fuzzy test methods proposed in this study are based on statistical inference, and judgement is aided by expertise, thus they are capable of solving the problems coupled with larger upper confidence limit.

The theoretical foundation of this fuzzy testing decision-making model is the upper confidence limit, it can lower the chance of misjudgment caused by sampling errors, and thereby increase evaluation accuracy.

Tunn et al. [23] and Tseng et al. [24] reported that the electronics industry in Taiwan maintains a complete and highly clustered ecological chain in the global supply of information-and-communications technology and thus holds a solid and crucial position in the global electronics industry. An important aspect of this industry is integrated-circuit (IC) packaging, in which processed wafers are cut into die and then coated in materials such as resin, ceramic, and metal. Their small size enables electronic components to be closer together, which accelerates circuit operations, and IC packaging protects chips from contamination. ICs are therefore essential components of electronic devices. Colloidal warpage is an important quality characteristic of the IC-packaging molding process. Excessive colloidal warpage can affect the subsequent processes of IC products or the functions of the final product. We thus took colloidal warpage in IC packaging as an example to demonstrate the fuzzy testing decision-making model proposed in this study.

The remainder of this paper is organized as follows. Section 2 derives the upper confidence limit of the Taguchi capability index using mathematical planning, and Section 3 constructs the fuzzy membership function based on this upper confidence limit. Next, a fuzzy testing decision-making model is proposed based on the fuzzy membership function to determine whether processes are in need of improvement. Section 4 uses the colloidal warpage in the molding process of IC packaging as an example to demonstrate the application of the proposed method. Finally, Section 5 provides Conclusions and future research.

2 Upper confidence limit of taguchi capability index

Ruczinski [25] and Chen and Huang [11] reported that when process capability is sufficient, Taguchi capability index C_pm has an unequal relationship with process yield, such that Yield % ≥2Φ (3C_pm) - 1, where Φ (·) denotes the cumulative distribution function of standard normal distribution [26]. Many paper assumes that the relevant quality characteristic of a manufacturing process X is distributed normally, i.e. X ∼ N (μ, σ²)[7 , 27]. Then $C_{pm} = \frac{d}{3 \sqrt{(μ - T)^{2} + σ^{2}}}$ (1) where (μ - T) ² + σ² = E (X - T) ² is the expected value of the Taguchi loss function, μ is process mean, σ is process standard deviation, T is the target value, d = (USL - LSL)/ - 2 is half of the length of the specification interval, and USL and LSL represent the upper and lower specification limits [9, 19]. We let Y = (X - T)/ - d, which is distributed normally with mean δ = (μ - T)/ - d and variance γ² = σ²/ - d, i.e. Y ∼ N (δ, γ²). Thus, the Taguchi capability index C_pm can be rewritten as follows: $C_{pm} = \frac{1}{3 \sqrt{δ^{2} + γ^{2}}}$ (2)

If we let (Y₁, ⋯ , Y_j, ⋯ , Y_n) be a random sample, then the sample mean and sample variance can be obtained as follows: $\bar{Y} = \frac{1}{n} \sum_{j = 1}^{n} Y_{j} and S_{n}^{2} = \frac{1}{n} \sum_{j = 1}^{n} {(Y_{j} - \bar{Y})}^{2}$

Since $\bar{Y}$ and $S_{n}^{2}$ are the maximum likelihood estimates (MLEs) of δ and γ², respectively, $C_{pm}^{*}$ is the MLE of C_pm. Therefore, the estimator of Taguchi capability index C_pm proposed by Boyles [12] can be expressed as follows: $C_{pm}^{*} = \frac{1}{3 \sqrt{{\bar{Y}}^{2} + S_{n}^{2}}}$ (3)

Under the assumption of normality, we let $Z = \frac{\bar{Y} - δ}{γ / \sqrt{n}} and K = \frac{{nS}_{n}^{2}}{γ^{2}}$ then random variables Z and K are distributed as N (0, 1) and $χ_{n - 1}^{2}$ , respectively. Therefore, the probability of events of Z and K can be shown respectively as following:

$P {- Z_{0.5 - \sqrt{1 - α} / - 2} \leq Z \leq Z_{0.5 - \sqrt{1 - α} / - 2}} = \sqrt{1 - α}$ and

$P {χ_{0 . 5 - \sqrt{1 - α} / - 2; n - 1}^{2} \leq K \leq χ_{0 . 5 + \sqrt{1 - α} / - 2; n - 1}^{2}} = \sqrt{1 - α}$ where $Z_{0.5 - \sqrt{1 - α} / - 2}$ is the upper $0.5 - \sqrt{1 - α} / - 2$ quantile of N (0, 1), $χ_{a; n - 1}^{2}$ is the upper a quantile of $χ_{n - 1}^{2}$ , and $a = 0.5 - \sqrt{1 - α} / - 2$ or a =0.5+ $\sqrt{1 - α} / - 2$ . Since $\bar{Y}$ and $S_{n}^{2}$ are mutually independent, then Z and K are also mutually independent. From these relationships, we can further obtain the following:

$P {\begin{matrix} - Z_{0 . 5 - \sqrt{1 - α} / - 2} \leq Z \leq Z_{0 . 5 - \sqrt{1 - α} / - 2}, \\ χ_{0 . 5 - \sqrt{1 - α} / - 2; n - 1}^{2} \leq K \leq χ_{0 . 5 + \sqrt{1 - α} / - 2; n - 1}^{2} \end{matrix}} = 1 - α$

Equivalently,

$\begin{matrix} P {\bar{Y} - Z_{0.5 - \sqrt{1 - α} / - 2} \times (\frac{γ}{\sqrt{n}}) \leq δ, \\ \leq \bar{Y} + Z_{0.5 - \sqrt{1 - α} / - 2} \times (\frac{γ}{\sqrt{n}}) \\ \sqrt{\frac{n}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}} \times S_{n} \leq γ \leq \sqrt{\frac{n}{χ_{0.5 - \sqrt{1 - α} / - 2; n - 1}^{2}}} \times S_{n}} \\ = 1 - α . \end{matrix}$

If we let (y₁, ⋯ , y_j, ⋯ , y_n) be the observed value of (Y₁, ⋯ , Y_j, ⋯ , Y_n), then the observed values of $\bar{Y}$ and S_n respectively are $\bar{y} = \frac{1}{n} \sum_{i = 1}^{n} y_{i} and s_{n} = \sqrt{\frac{1}{n} \sum_{j = 1}^{n} {(y_{j} - \bar{y})}^{2}}$

Therefore, the observed value of the confidence region can be shown as follows:

$CR = {(δ, γ) | \begin{matrix} \bar{y} - \frac{Z_{0.5 - \sqrt{1 - α} / - 2}}{\sqrt{n}} \times γ \leq δ \leq \bar{y} + \frac{Z_{0.5 - \sqrt{1 - α} / - 2}}{\sqrt{n}} \times γ, \\ \sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}} \leq γ \leq \sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 - \sqrt{1 - α} / - 2; n - 1}^{2}}} \end{matrix}}$

In fact, index C_pm is a function of (μ, σ). Chen et al. [12] reported that the object function is C_pm (δ, γ) and CR is the feasible solution area. Thus, the mathematical programming (MP) model can be written as follows: ${\begin{matrix} {UC}_{pm} = Max {[3 \sqrt{δ^{2} + γ^{2}}]}^{- 1} \\ subject to (δ, γ) \in CR \end{matrix}$ (5)

Obviously, for any (δ, γ) ∈ CR and γ≠ $\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}}$ , then

$\begin{matrix} C_{pm} (δ, γ) \leq C_{pm} (δ, \sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}}) \\ = {[3 \sqrt{δ^{2} + {(\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}})}^{2}}]}^{- 1} \end{matrix}$ (6)

Based on Equation (6) and as noted by Chen et al. [19], the object function UC_pm and the feasible solution area can be rewritten as following: ${\begin{matrix} {UC}_{pm} = Max {[3 \sqrt{δ^{2} + {(\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}})}^{2}}]}^{- 1} \\ subject to \bar{y} - e_{z} \leq δ \leq \bar{y} + e_{z} \end{matrix}$ (7) where $e_{z} = \frac{z_{0.5 - \sqrt{1 - α} / 2} \times s_{n}}{\sqrt{χ_{0.5 + \sqrt{1 - α} / 2; n - 1}^{2}}}$ (8)

Situation 1: $\bar{y} - e_{z} > 0$

Obviously, for any $\bar{y} - e_{z} \leq δ \leq \bar{y} + e_{z}$ and $δ \neq \bar{y} - e_{z}$ ,

$\begin{matrix} C_{pm} (δ, \sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}}) \\ \leq C_{pm} (\bar{y} - e_{z}, \sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}}) \\ = {[3 \sqrt{{(\bar{y} - e_{z})}^{2} + {(\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}})}^{2}}]}^{- 1} \end{matrix}$ (9)

In this situation, the upper confidence limit of C_pm is

${UC}_{pm} = {[3 \sqrt{{(\bar{y} - e_{z})}^{2} + {(\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}})}^{2}}]}^{- 1}$ (10)

Situation 2: $\bar{y} - e_{z} \leq 0 \leq \bar{y} + e_{z}$

Obviously, for any $\bar{y} - e_{z} \leq δ \leq \bar{y} + e_{z}$ and δ ≠ 0,

$\begin{matrix} C_{pm} (δ, \sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}}) \\ \leq C_{pm} (0, \sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}}) \\ = {[3 (\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}})]}^{- 1} \end{matrix}$ (11)

In this situation, the upper confidence limit of C_pm is ${UC}_{pm} = {[3 (\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}})]}^{- 1}$ (12)

Situation 3: $\bar{y} + e_{z} < 0$

Obviously, for any $\bar{y} - e_{z} \leq δ \leq \bar{y} + e_{z}$ and $δ \neq \bar{y} + e_{z}$ ,

$\begin{matrix} C_{pm} (δ, \sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}}) \\ \leq C_{pm} (\bar{Y} + e_{z}, \sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}}) \\ = {[3 \sqrt{{(\bar{y} + e_{z})}^{2} + {(\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}})}^{2}}]}^{- 1} \end{matrix}$ (13)

In this situation, the upper confidence limit of C_pm is ${UC}_{pm} = {[3 \sqrt{\begin{matrix} {(\bar{y} + \frac{z_{0.5 - \sqrt{1 - α} / 2} \times s_{n}}{\sqrt{χ_{0.5 + \sqrt{1 - α} / 2; n - 1}^{2}}})}^{2} + \\ {(\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}})}^{2} \end{matrix}}]}^{- 1}$ (14)

Based on the above, the 100 (1 - α)% upper confidence limit of C_pm is a function of α and can be shown as follows: ${UC}_{pm} (α) = {\begin{matrix} {[3 \sqrt{{(\bar{y} - \frac{z_{0.5 - \sqrt{1 - α} / 2} \times s_{n}}{\sqrt{χ_{0.5 + \sqrt{1 - α} / 2; n - 1}^{2}}})}^{2} + {(\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}})}^{2}}]}^{- 1}, \bar{y} - e_{z} > 0 \\ {[3 (\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}})]}^{- 1}, \bar{y} - e_{z} \leq 0 \leq \bar{y} + e_{z} \\ {[3 \sqrt{{(\bar{y} + \frac{z_{0.5 - \sqrt{1 - α} / 2} \times s_{n}}{\sqrt{χ_{0.5 + \sqrt{1 - α} / 2; n - 1}^{2}}})}^{2} + {(\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}})}^{2}}]}^{- 1}, \bar{y} + e_{z} < 0 \end{matrix}$ (15)

3 Fuzzy testing decision-making model

Before developing the fuzzy hypothesis testing method, we refer to Chen et al. [2] and examine the process of statistical hypothesis testing. In applying upper confidence limit to statistical hypothesis testing, and the rules are as following:

If UC_pm≥C, then do not reject H₀ and conclude that process is capable (C_pm≥C).

If UC_pm<C, then reject H₀ and conclude that process is incapable (C_pm<C).

Based on Pearn and Chen [28], the value of C can be 1.00 or 1.33. When C = 1.00, then a process is called “inadequate.” if C_pm<1.00. A process is called “capable” if C_pm≥1.00. When C = 1.33, then a process is called “satisfactory” if C_pm≥1.33. In fact, the 100 (1 - α)% upper confidence limit UC_pm is the decreasing function of sample size n, and it can be rewritten as UC_pm (n) for fix α. Let n′ < n″ be UC_pm (n″)<C<UC_pm (n′), then

C<UC_pm (n′) with n = n′, then H₀ is rejected, and C_pm≥ C is concluded.

UC_pm (n″)< C with n = n″, then reject H₀ and conclude that C_pm< C.

It is clear that sample size n influences the statistical inference results. Thus, this paper will develop the fuzzy testing decision-making model based on the above rules and the method introduced by Buckley [29] and expanded by Chen [21]. As described by Chen et al. [2], the α - cuts of the half-triangular fuzzy number ${\tilde{C}}_{pmo}^{*}$ is ${\tilde{C}}_{pmo}^{*}$ [α]=[UC_pm (1) , UC_pm (α)] for 0.01 ≤ α ≤ 1, where $C_{pmo}^{*}$ = $1 / - 3 \sqrt{{\bar{y}}^{2} + s_{n}^{2}}$ and

$\begin{matrix} {UC}_{pm} (1) \\ = {\begin{matrix} {[3 \sqrt{{(\bar{y})}^{2} + {(\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5; n - 1}^{2}}})}^{2}}]}^{- 1}, \bar{y} - e_{z} > 0 \\ {[3 (\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5; n - 1}^{2}}})]}^{- 1}, \bar{y} - e_{z} \leq 0 \leq \bar{y} + e_{z} \\ {[3 \sqrt{{(\bar{y})}^{2} + {(\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5; n - 1}^{2}}})}^{2}}]}^{- 1}, \bar{y} + e_{z} < 0 \end{matrix} \end{matrix}$ (16)

Recall that all of the α - cuts of half-triangular fuzzy number ${\tilde{C}}_{pmo}^{*}$ [α] for 0 ≤ α < 0.01 are equal to ${\tilde{C}}_{pmo}^{*}$ [0.01]. Thus, the half-triangular fuzzy number of UC_pm is ${\tilde{C}}_{pmo}^{*}$ =, Δ (C_M, C_R), where C_M=UC_pm(1) and C_R=UC_pm(0.01) can be shown as follows: $C_{R} = {\begin{matrix} {[3 \sqrt{{(\bar{y} - \frac{z_{0.00125} \times s_{n}}{\sqrt{χ_{0.99875; n - 1}^{2}}})}^{2} + {(\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.99875; n - 1}^{2}}})}^{2}}]}^{- 1}, \\ \bar{y} - e_{z} > 0 \\ {[3 (\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.99875; n - 1}^{2}}})]}^{- 1}, \bar{y} - e_{z} \leq 0 \leq \bar{y} + e_{z} \\ {[3 \sqrt{{(\bar{y} + \frac{z_{0.00125} \times s_{n}}{\sqrt{χ_{0.99875; n - 1}^{2}}})}^{2} + {(\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.99875; n - 1}^{2}}})}^{2}}]}^{- 1}, \\ \bar{y} + e_{z} < 0 \end{matrix}$ (17)

It follows that the membership function of the fuzzy number ${\tilde{C}}_{pmo}^{*}$ is $η (x) = {\begin{matrix} 0 if x < C_{M} \\ 1 if x = C_{M} \\ α if C_{M} < x \leq C_{R} \\ 0 if C_{R} < x \end{matrix}$ (18) where C_pm is determined by UC_pm(α)=x; i.e.,

$\begin{matrix} {[3 \sqrt{{(\bar{y} - \frac{z_{0.5 - \sqrt{1 - α} / 2} \times s_{n}}{\sqrt{χ_{0.5 + \sqrt{1 - α} / 2; n - 1}^{2}}})}^{2} + {(\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}})}^{2}}]}^{- 1} \\ = x, \bar{y} - e_{z} > 0, \end{matrix}$

$\begin{matrix} {[3 (\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}})]}^{- 1} \\ = x, \bar{y} - e_{z} \leq 0 \leq \bar{y} + e_{z}, \end{matrix}$

$\begin{matrix} {[3 \sqrt{{(\bar{y} + \frac{z_{0.5 - \sqrt{1 - α} / 2} \times s_{n}}{\sqrt{χ_{0.5 + \sqrt{1 - α} / 2; n - 1}^{2}}})}^{2} + {(\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}})}^{2}}]}^{- 1} \\ = x, \bar{y} + e_{z} < 0 \end{matrix}$

Figure 1 presents a diagram of membership function η (x) with vertical line x = C.

Fig. 1

Membership function η (x) and vertical line x = C.

Let set A_T be the area in the graph of η (x), such that $A_{T} = {(x, α) | C_{M} \leq x \leq {UC}_{pm} (α), 0 \leq α \leq 1}$ (19)

Thus, the area of set A_T can be calculated as follows: $a_{T} = \int_{C_{M}}^{C_{R}} η (x) d_{x}$ (20)

As noted by Buckley [29] and Chen et al. [2], calculating a_T directly through integration can be difficult. Therefore, based on Chen [21], we let j =⌊ 1000α ⌋, such that j = 0, 1, . . . , 1000 for 0 ≤ α ≤ 1, where ⌊1000α⌋ represents the largest integer less than or equal to 1000α. Let α = 0.001 × j and j = 0, 1, . . . , 1000, which indicate that 101 horizontal lines divide A_T into 1000 trapezoid-like blocks. Therefore, the jth block can be expressed as follows: $A_{Tj} = {\begin{matrix} (x, α) | C_{M} \leq x \leq {UC}_{pm} (0.001 \times j), \\ 0.001 \times (j - 1) \leq α \leq 0.001 \times j \end{matrix}}$ (21)

Therefore, the length of the jth horizontal line l_j can be described as follows: $l_{j} = {UC}_{pm} (0.001 \times j) - C_{M}, j = 10, . . ., 1000$ (22)

Obviously, l₀ = l₁ = . . . = l₉ = l₁₀, l₁₀₀₀ = 0 and A_Tj is a trapezoid-like block with lower base l_j-1, upper base l_j, and height 0.001. Thus, the approximate value of its area a_Tj can be written as follows: $a_{Tj} = (\frac{l_{j - 1} + l_{j}}{2}) \times (0.001), j = 1, . . ., 1000$ (23)

Therefore, $\begin{matrix} a_{T} = \sum_{j = 1}^{1000} a_{Tj} = \sum_{j = 1}^{1000} (0.001) \times (\frac{l_{j - 1} + l_{j}}{2}) \\ = 0.0095 \times l_{10} + 0.001 \times \sum_{j = 10}^{999} l_{j} \end{matrix}$ (24)

Let A_R be the area under the graph of η (x) to the right of the vertical line x=C. Let α = 0.001×h such that UC_pmo (0.001×h)=C. Then, $A_{R} = {\begin{matrix} (x, α) | C \leq x \leq {UC}_{pm} (α), \\ 0 \leq α \leq 0.001 \times h \end{matrix}}$ (25)

If we let the area of A_R be a_R, then $a_{R} = \int_{C}^{C_{R}} η (x) d_{x}$ (26)

Let α = 0.001 × j and j = 0, 1, . . . , h, which indicate that h + 1 horizontal lines divide A_R into h trapezoid-like blocks. Therefore, the jth block can be expressed as follows: $A_{Rj} = {\begin{matrix} (x, α) | C \leq x \leq {UC}_{pm} (0.001 \times j), \\ 0.001 \times (j - 1) \leq α \leq 0.001 \times j \end{matrix}}$ (27)

Therefore, the length of the jth horizontal line r_j, can be described as follows: $r_{j} = {UC}_{pm} (0.001 \times j) - C, j = 10, . . ., h$ (28)

Similarly, with r₀ = r₁ = . . . = r₁₀ and r_h = 0, the approximate value of area a_Rj can be written as follows: $a_{Rj} = (\frac{r_{j - 1} + r_{j}}{2}) \times (0.001), j = 1, . . ., h .$ (29)

Therefore, $\begin{matrix} a_{R} & = & \sum_{j = 1}^{h} a_{Rj} = \sum_{l = 1}^{h} (0.001) \times (\frac{r_{j - 1} + r_{j}}{2}) \\ = & 0.0095 \times r_{10} + 0.001 \times \sum_{j = 10}^{h - 1} r_{j} \end{matrix}$ (30)

As the area of A_T is equivalent to half of the area of the A_T proposed by Buckley [29], we get $\frac{a_{R}}{2 a_{T}} = \frac{0.0095 \times r_{10} + 0.001 \times \sum_{j = 10}^{h - 1} r_{j}}{0.0190 \times d_{10} + 0.002 \times \sum_{j = 10}^{999} d_{j}}$ (31)

Based on Chen et al. [3], we let 0<φ₁<φ₂< 0.5, and construct the fuzzy testing decision-making model rules as follows:

If $\frac{a_{R}}{2 a_{T}}$ ≤φ₁, then reject H₀ and conclude that C_pm< C.

If φ₁≤ $\frac{a_{R}}{2 a_{T}}$ ≤φ₂, then make no decision regarding whether to reject/not reject.

If φ₂≤ $\frac{a_{R}}{2 a_{T}}$ ≤ 0.5, then do not reject H₀ and conclude that C_pm≥ C.

Based on the above fuzzy testing decision rules, we further provide a standardized procedure for practical purposes as following:

Specify the minimum capability required value C, and set significant level β.

Based on the sample data, we have calculation as $\bar{y}$ , s_n, $C_{pmo}^{*}$ , e_z, [ $\bar{y}$ -e_z, $\bar{y}$ +e_z]. Then, acoording to Equation (15) calculate UC_pm(α).

Based on Equation (16-18), calculate ${\tilde{C}}_{pmo}^{*}$ =Δ (C_M, C_R) and η (x).

Based on Equation (24) and (30), calculate a_T and a_R respectively, then calculate a_R/ - 2a_T based on Equation (31).

Make decision based on the fuzzy testing rules.

4 A practical application

At present, the electronics industry in Taiwan maintains a complete ecological chain in the global supply of information-and-communications technology, and the wafer foundry and IC packaging and testing industry in Taiwan have the highest output value in the world. As noted by Chen and Chang [8], the Taguchi capability index can fully reflect the losses and yield of processes and is a convenient and effective tool to evaluate and analyze process quality in the industry. Thus, the index C_pm is a commonly used indicator for the semiconductor industry in Taiwan. Based on the above standardized procedure, we illustrate the applicability of the fuzzy testing decision rules for the IC packaging.

Step 1: According to Pearn and Chen [28], C_pm≥1 indicates that the process is capable, while C_pm< 1 indicates that the process is incapable. Therefore, we adopt the null hypothesis H₀: C_pm≥ 1 (capable) versus the alternative hypothesis H₁:C_pm< 1 (incapable). As mentioned above, excessive colloidal warpage in the molding process of IC packaging can affect the subsequent processes of IC products or the functions of the final product. In order to demonstrate the practical application of our methodology, we selected 60 random samples (n = 60) for testing. The target value T =0, tolerance d=0.03 mm and set significant level β= 0.005.

Step 2: We calculate the relevant statistics based on these 60 samples: $\begin{matrix} \bar{y} = \frac{1}{n} \sum_{i = 1}^{n} y_{i} = 0.30, \\ s_{n} = \sqrt{\frac{1}{n} \sum_{j = 1}^{n} {(y_{j} - \bar{y})}^{2}} = 0.26, \\ C_{pmo}^{*} = \frac{1}{3 \sqrt{{\bar{y}}^{2} + s_{n}^{2}}} = \frac{1}{3 \sqrt{(0.30)^{2} + (0.26)^{2}}} \\ = 0.840, \\ e_{z} = \frac{z_{0.00125} \times s_{n}}{\sqrt{χ_{0.99875; 59}^{2}}} = \frac{2 . 806 \times 0 . 26}{\sqrt{94 . 075}} = 0.075, \\ [\bar{y} - e_{z}, \bar{y} + e_{z}] = [0.30 - 0.075, 0.30 + 0.075] \\ = [0.225, 0.375] . \end{matrix}$

Obviously, $\bar{y} - e_{z}$ , =0.225>0. Therefore, based on Equation (15), the upper confidence limit of C_pm is ${UC}_{pm} (α) = {[3 \sqrt{{(\bar{y} - \frac{z_{0.5 - \sqrt{1 - α} / 2} \times s_{n}}{\sqrt{χ_{0.5 + \sqrt{1 - α} / 2; n - 1}^{2}}})}^{2} + {(\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5 + \sqrt{1 - α} / - 2; n - 1}^{2}}})}^{2}}]}^{- 1}$

Step 3: The half-triangular fuzzy number of UC_pm is ${\tilde{C}}_{pmo}^{*}$ =, Δ (C_M, C_R), where

$\begin{matrix} C_{M} = {[3 \sqrt{{(\bar{y} - \frac{z_{0.5} \times s_{n}}{\sqrt{χ_{0.5; n - 1}^{2}}})}^{2} + {(\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.5; n - 1}^{2}}})}^{2}}]}^{- 1} \\ = {[3 \sqrt{{(0 . 3)}^{2} + {(\sqrt{\frac{60 \times (0 {. 26)}^{2}}{58 . 335}})}^{2}}]}^{- 1} \\ = 0.835, \\ C_{R} = {[3 \sqrt{{(\bar{y} - \frac{z_{0.00125} \times s_{n}}{\sqrt{χ_{0.99875; n - 1}^{2}}})}^{2} + {(\sqrt{\frac{n \times s_{n}^{2}}{χ_{0.99875; n - 1}^{2}}})}^{2}}]}^{- 1} \\ = {[3 \sqrt{{(0.30 - \frac{2.806 \times 0.26}{\sqrt{94.075}})}^{2} + {(\sqrt{\frac{60 \times (0.26)^{2}}{94.075}})}^{2}}]}^{- 1} \\ = 1.089 . \end{matrix}$

Thus, the membership function of the fuzzy number ${\tilde{C}}_{pmo}^{*}$ is $η (x) = {\begin{matrix} 0 if x < 0.835 \\ 1 if x = 0.835 \\ α if 0.835 < x \leq 1.089 \\ 0 if 1.089 < x \end{matrix}$ (32) where α is determined by $3 \sqrt{\begin{matrix} {(0.30 - \frac{z_{0.5 - \sqrt{1 - α} / 2}}{\sqrt{χ_{0.5 + \sqrt{1 - α} / 2; 59}^{2}}} \times 0.26)}^{2} \\ + {(\sqrt{\frac{60}{χ_{0.5 + \sqrt{1 - α} / - 2; 59}^{2}}} \times 0.26)}^{2} \end{matrix}} = x^{- 1}$

Therefore, membership function η (x) with vertical line x= 1 for this practical example is as shown in Fig. 2.

Fig. 2

Membership function η (x) with vertical line x = 1.

Step 4: Based on Equation (24), the a_T value can be calculated as follows: $a_{T} = 0.0095 \times l_{10} + 0.001 \times \sum_{j = 10}^{999} l_{j} = 97.226$ where l_j =, UC_pm (0.001 × j) -0.835 and j = 10, . . . , 1000. When h = 125, UC_pmo (0.001 × 125) =1. Based on Equation (30), the a_R value can be calculated as follows: $a_{R} = 0.0095 \times r_{10} + 0.001 \times \sum_{j = 10}^{124} r_{j} = 3.499$ where r_j = UC_pm (0.001 × j) -1, j = 10, . . . , 125.

Furthermore, based on Equation (31) we can compute $\frac{a_{R}}{2 a_{T}} = \frac{3.499}{2 \times 97.226} = 0.018 .$

Step 5: As emphasized by Chen [13], we take φ₁=0.2 and φ₂=0.5. Based on the fuzzy testing decision-making model rule (1), since a_R/ - 2a_T=0.018≤φ₁=0.2, we reject H₀ and conclude that the process is incapable (C_pm< 1). Based on statistical inference, we do not reject H₀ and conclude that process is capable. However, $C_{pmo}^{*} =$ 0.840 is much less than C_pm =1. This is because the sample size is not big enough, which results in a large sampling error. As emphasized by Chen et al. [3], timeliness is highly important in a competitive market. Sample size is rarely large, which is why we incorporated fuzzy testing. The proposed model indicates H₀ should be rejected and that the process is incapable. From a practical perspective, this result seems more reasonable.

5 Conclusions and future research

As noted by Lin et al. [1], Internet-of-Things (IoT) environments become more prevalent and mature, technologies for the measurement and collection of production data are also improving. Production data analysis models must be continuously and properly utilized to monitor and maintain process quality and technology upgraded to develop intelligent manufacturing. According to Chen et al. [2], improving product quality helps to lengthen service life, decrease scrap and rework, and reduce the social losses caused by malfunctions and maintenance. The Taguchi capability index can fully reflect the process loss and process yield and is a convenient and effective tool to evaluate and analyze process quality data in the industry.

Based on the concept of Chen [21], we used the concept of sample size and the rules of statistical testing to explain the motivation underlying those methods. As noted by Chen [22], the sample size influences the risk of misjudgment in upper confidence limit, and in practice, sample sizes are rarely large due to cost and time considerations, thereby they produce larger upper confidence limit with a corresponding decrease in accuracy and increase in risk of misjudgment. The fuzzy test methods proposed in this study are based on statistical inference, and judgement is aided by expertise, thus they are capable of solving the problems coupled with larger upper confidence limit.

Therefore, the theoretical foundation of this fuzzy testing decision-making model is the upper confidence limit, it can lower the chance of misjudgment caused by sampling errors and increase evaluation accuracy. Finally, we present a practical example to illustrate the applicability of the proposed method. This example demonstrates that from a practical perspective the proposed method derives more reasonable results than statistical inference methods.

As noted by Chen et al. [30] and Pearn and Cheng [31], a product generally is featured with a number of crucial quality characteristics, which may include unilateral and bilateral specifications. For a product to satisfy customer needs (i.e., be considered a good product), it must meet all requirements for process quality during production. However, there are some asymmetric tolerances we find in the process, which together with multi-process quality characteristics are the limitations of this study. In the future, the fuzzy testing decision-making model can be more functional and be effectively applied to the process analysis of asymmetric tolerances and multi-process quality characteristics.

References

Lin

K.P.

, Yu

C.M.

and Chen

K.S.

, Production data analysis system using novel process capability indices-based circular economy, Industrial Management & Data Systems (2019) (in press).

Chen

K.S.

, Wang

C.H.

and Tan

K.H.

, Developing a fuzzy green supplier selection model using six sigma quality indices, International Journal of Production Economics 212 (2019), 1–7.

Chen

K.S.

, Wang

K.J.

and Chang

T.C.

, A novel approach to deriving the lower confidence limit of indices Cpu, Cpl, and Cpk in assessing process capability, International Journal of Production Research 55(17) (2017), 4963–4981.

Dharmasena

L.S.

and Zeephongsekul

, A new process capability index for multiple quality characteristics based on principal components, International Journal of Production Research 54(15) (2016), 4617–4633.

, Jia

, Liu

and You

, Yield-based capability index for evaluating the performance of multivariate manufacturing process, Quality and Reliability Engineering International 31(3) (2015), 419–430.

Kargar

, Mashinchi

and Parchami

, A bayesian approach to capability testing based on Cpk with multiple samples, Quality and Reliability Engineering International 30(5) (2014), 615–621.

Chan

L.K.

, Cheng

S.W.

and Spiring

F.A.

, A new measure of process capability: Cpm, Journal of Quality Technology 20(3) (1988), 162–175.

Chen

K.S.

and Chang

T.C.

, Construction and fuzzy hypothesis testing of Taguchi Six Sigma quality index, International Journal of Production Research (2019) (in press).

Chen

J.P.

and Chen

K.S.

, Comparing the capability of two process using Cpm, Journal of Quality Technology 36(3) (2004), 329–335.

10.

Huang

M.L.

, Chen

K.S.

and Li

R.K.

, Graphical analysis of capability of a process producing a product family, Quality & Quantity 39(5) (2005), 643–657.

11.

Chen

K.S.

and Huang

M.L.

, Process capability evaluation for the process of product families, Quality & Quantity 41(1) (2007), 151–162.

12.

Boyles

R.A.

, The Taguchi capability index, Journal of Quality Technology 23(1) (1991), 17–26.

13.

Chen

T.W.

, Lin

J.Y.

and Chen

K.S.

, Selecting a supplier by fuzzy evaluation of capability indices Cpm, International Journal of Advanced Manufacturing Technology 22(7-8) (2003), 534–540.

14.

Chen

J.P.

and Chen

K.S.

, Comparing the capability of two process using Cpm, Journal of Quality Technology 36(3) (2004), 329–335.

15.

Lin

G.H.

, Pearn

W.L.

and Yang

Y.S.

, A. Bayesian approach to obtain a lower bound for the Cpm index, Quality & Reliability Engineering International 21(6) (2005), 655–668.

16.

Chen

K.S.

, Huang

M.L.

and Hung

Y.H.

, Process capability analysis chart with the application of Cpm, International Journal of Production Research, 46(16) (2008), 4483–4499.

17.

Lin

G.H.

, A reliable procedure on performance evaluation - a large sample approach based on the estimated Taguchi capability index, International Journal of Engineering and Innovative Technology 3(5) (2013), 359–364.

18.

Otsuka

and Nagata

, Design method of Cpm-index based on product performance and manufacturing cost, Computers & Industrial Engineering 113 (2017), 921–927.

19.

Chen

K.S.

, Huang

C.F.

and Chang

T.C.

, A mathematical programming model for constructing the confidence interval of process capability index Cpm in evaluating process performance: An example of five-way pipe, Journal of the Chinese Institute of Engineers 40(2) (2017), 126–133.

20.

Yang

C.M.

, Lin

K.P.

and Chen

K.S.

, Confidence interval based fuzzy evaluation model for an integrated-circuit packaging molding process, Applied Science 9(13) (2019), 2623.

21.

Chen

K.S.

, Two-tailed Buckley fuzzy testing for operating performance index, Journal of Computational and Applied Mathematics 361 (2019), 55–63.

22.

Chen

K.S.

, Fuzzy testing of operating performance index based on confidence intervals, Annals of Operations Research (2019) (in press).

23.

Tunn

V.S.C.

, Bocken

N.M.P.

, van den Hende

E.A.

and Schoormans

J.P.L.

, Business models for sustainable consumption in the circular economy: An expert study, Journal of Cleaner Production 212 (2019), 324–333.

24.

Tseng

M.L.

, Chiang

J.H.

and Lan Lawrence

, Selection of optimal supplier in supply chain management strategy with analytic network process and choquet integral, Computer & Industrial Engineering 57(1) (2009), 330–340.

25.

Ruczinski

, The Relation Between. Cpm and the Degree of Includence. Ph. D. dissertation, University of Würzburg, Würzburg, 1996.

26.

Huang

C.F.

, Chen

K.S.

, Sheu

S.H.

and Sheu

T.S.

, Enhancement of axle bearing quality in sewing machines using six sigma, Proceedings of the Institution of Mechanical Engineers Part B - Journal of Engineering Manufacture 224(10) (2010), 1581–1590.

27.

K.T.

, Sheu

S.H.

and Chen

K.S.

, The evaluation of process capability for a machining center, International Journal of Advanced Manufacturing Technology 33(5-6) (2007), 505–510.

28.

Pearn

W.L.

and Chen

K.S.

, Multiprocess performance analysis: A case study, Quality Engineering 10(1) (1997), 1–8.

29.

Buckley

J.J.

, Fuzzy statistics: Hypothesis testing, Soft Computing 9(7) (2005), 512–518.

30.

Chen

K.S.

, Chen

S.C.

and Li

R.K.

, Process quality analysis of products, International Journal of Advanced Manufacturing Technology 19(8) (2002), 623–628.

31.

Pearn

W.L.

and Cheng

Y.C.

, Measuring production yield for processes with multiple characteristics, International Journal of Production Research 48(15) (2010), 4519–4536.