A general p -value-based approach for testing quality by considering fuzzy hypotheses

Abstract

In testing the capability of industrial processes, the researcher considers and tests the vague hypothesis “the capability index is low” against the vague hypothesis “the capability index is high”. But, two fuzzy concepts low and high are usually formulated by two crisp hypotheses in traditional quality tests. In this paper, we formulate these two fuzzy concepts by considering two complement fuzzy sets. Afterwords, a new p-value-based approach is considered for testing the mentioned fuzzy hypotheses which is constructed on the basis of two capability indices C_p and C_pm. This new approach has several advantages over the common p-value methods for testing fuzzy hypotheses. The main one is depending the result of this new approach on both null and alternative fuzzy hypotheses, while the common p-value-based methods are according to only the null fuzzy hypothesis. To clarify the potential of the proposed approach in the process of capability analyses, two applied examples are given based on two real-world data set.

Keywords

Boundary of fuzzy hypothesis weighted density Taguchi capability index maximum likelihood estimator

1 Introduction and background

After the inception of the fuzzy set concept by Zadeh [40], many authors have applied fuzzy set theory to different areas of quality control and especially process capability analyses. Yongting [38] introduced the first notion of fuzzy quality and an index for measuring fuzzy quality in process analyses. Then, Sadeghpour [30] compared Yongting’s index with the traditional capability indices C_p and C_pk in case of measurement error occurrence. Lee [16] calculated the membership function of fuzzy capability index C_pk when observations are triangular fuzzy numbers. In [4], Chen et al. investigated the fuzzy capability index C_pm for selecting the best supplier. By considering two triangular fuzzy numbers as fuzzy specification limits, Parchami et al. [19] introduced the membership functions of several fuzzy capability indices. This idea followed by Parchami et al. [22], Parchami and Mashinchi [23] and Ramezani et al. [27] to determine the fuzzy confidence interval and test on the introduced fuzzy capability indices. Tsai and Chen [35] tested the capability hypotheses based on index C_p by considering fuzzy observations. Parchami and Mashinchi [20] estimated the classical process capability indices using Buckley’s fuzzy estimation approach. Chen and Chen [3] analysed multi-process capability plot by using fuzzy inference. Amirzadeh et al. [1] constructed p-charts on the basis of Yongting’s fuzzy quality for precise data. Kahraman and Kaya [9] introduced the fuzzy process capability indices for quality control of irrigation water. Kaya and Kahraman [14] investigated on the effects of robustness in process capability analyses and also they developed the fuzzy capability indices for decision making problems in [13]. Parchami and Mashinchi [21] introduced a new generation of process capability indices on the basis of fuzzy specification limits. Kaya and Kahraman [11] investigated on process capability analysing based on fuzzy measurements and fuzzy control charts. Kaya and Kahraman [12] worked on the process capability analyses by considering fuzzy parameters. Sadeghpour and Moradi [31] monitored the process capability using the process capability plots based on fuzzy measurements. Moradi and Sadeghpour [18] proposed a general multi-variate capability index based on fuzzy tolerance region for precise data. In this paper, another efficient and simple p-value-based method for testing quality is developed in this paper by considering fuzzy capability hypothesis and fuzzy incapability hypothesis.

This paper is organized as follows. After presenting the preliminary concepts of the fuzzy hypothesis, the boundary of fuzzy hypothesis and the probability measure under fuzzy hypothesis, a new p-value-based approach is briefly reviewed in Section 2. The motivation of using fuzzy capability hypothesis for testing quality is discussed in Section 3. In Sections 4 and 5, the new p-value-based method is developed for testing conventional capability indices C_p and C_pm, where the considered hypotheses are fuzzy rather than crisp. Two real-world data are given to illustrate the proposed capability tests. Finally, the conclusions are provided and some ideas for future works are proposed.

2 Preliminaries

2.1 Fuzzy hypothesis and its boundary

First, we briefly review some basic concepts about fuzzy hypotheses from Taheri and Behboodian [32] and Parchami et al. [24], which are used in Sections 4 and 5 for testing quality by process capability indices.

Definition 1.Any hypothesis of the form “ $\tilde{H} : θ$ is H (θ)” is called a fuzzy hypothesis, where “θ is H (θ)” implies that θ is in a fuzzy set of Θ, the parameter space, with membership function H (θ).

Definition 2. (a) Fuzzy hypothesis “ $\tilde{H} : θ$ is H (θ)” is called a right one-sided fuzzy hypothesis, if there exists θ₁ ∈ Θ so that H (θ) =1 for θ ≥ θ₁, and H is a non-decreasing function of θ for θ < θ₁.

(b) Fuzzy hypothesis “ $\tilde{H} : θ$ is H (θ)” is called a left one-sided fuzzy hypothesis, if there exists θ₁ ∈ Θ so that H (θ) =1 for θ ≤ θ₁, and H is a non-increasing function of θ for θ > θ₁.

Note that the ordinary hypothesis “H : θ ≤ θ₀” is a left one-sided fuzzy hypothesis with the membership function H (θ) =1 at θ ≤ θ₀, and zero otherwise, i.e., the indicator function of the crisp set {θ : θ ≤ θ₀}.

Definition 3. The boundary of the fuzzy hypothesis $\tilde{H}$ is a fuzzy subset of Θ with membership function H_b defined as follows,

(a) $H_{b} (θ) = {\begin{matrix} H (θ) & f o r & θ \leq θ_{1}, \\ 0 & f o r & θ > θ_{1} \end{matrix}}$ , if $\tilde{H}$ is right one-sided fuzzy hypothesis,

(b) $H_{b} (θ) = {\begin{matrix} H (θ) & f o r & θ \geq θ_{1}, \\ 0 & f o r & θ < θ_{1} \end{matrix}}$ , if $\tilde{H}$ is left one-sided fuzzy hypothesis.

Example 1. Let X be an exponential random variable with unknown mean λ; i.e. $f (x; λ) = \frac{1}{λ} e^{- \frac{x}{λ}}, x > 0, λ > 0 .$

Suppose that $\begin{matrix} H (λ) = {\begin{matrix} e^{- (λ - 3)^{2}} & if & λ \leq 3, \\ 1 & if & λ > 3 . \end{matrix} \end{matrix}$

Then, the hypothesis “ $\tilde{H} : λ$ is H (λ)” is a right one-sided fuzzy hypothesis. So, by Definition 3, $H_{b} (λ) = {\begin{matrix} e^{- (λ - 3)^{2}} & if & λ \leq 3, \\ 0 & if & λ > 3, \end{matrix}$ is the boundary of the fuzzy hypothesis $\tilde{H}$ (see Fig. 1).

2.2 The probability measure under fuzzy hypothesis

Definition 4. [34] Let X be a real-valued random variable with the p.d.f. or p.m.f. f (x ; θ). The weighted probability density function of X, under the fuzzy hypothesis “ $\tilde{H} : θ$ is H (θ)”, is defined by $f (x; \tilde{H}) = \int_{θ} H^{*} (θ) f (x; θ) d θ,$ in which $H^{*} (θ) = \frac{H (θ)}{\int_{θ} H (θ) d θ}$ and ∫_θH (θ) dθ< ∞. Replace integration by summation when the support of H is discrete.

Remark 1. If H is the crisp hypothesis “H : θ = θ₀”, then $f (x; \tilde{H}) = f (x; θ_{0})$ .

In one-parameter exponential family, the introduced weighted p.d.f. $f (x; \tilde{H})$ in Definition 4 is a p.d.f., since $f (x; \tilde{H})$ is non-negative and $\int_{x} f (x; \tilde{H}) dx = 1$ . One of the advantages of Definition 4 is that, the weighted p.d.f. can integrate all possible probability density functions with different weights. The value of H^* (θ) can be understood as the weight of f (x ; θ), and the weighted p.d.f. can let different possible f (x ; θ)’s play different roles in this integration.

2.3 The new p-value-based approach for testing fuzzy hypotheses

After considering Definitions 1 and 3, one can assert that “ ${\tilde{H}}_{b} : θ is H_{b} (θ)$ ” is a fuzzy hypothesis, and therefore we can generalize the classical p-value for testing one-sided fuzzy hypotheses based on the weighted p.d.f. of the test statistic, under the fuzzy hypothesis of boundaries. In the following, it is assumed that we have a random sample X = (X₁,. . . , X_n) with the observed value x = (x₁,. . . , x_n) from the p.d.f. (p.m.f.) f (x ; θ) and θ is the parameter of interest. Moreover, T = T (X₁,. . . , X_n) is the related test statistic.

Definition 5. Consider the left one-sided fuzzy hypothesis “ ${\tilde{H}}_{0} : θ$ is H₀ (θ)” against the right one-sided fuzzy hypothesis “ ${\tilde{H}}_{1} : θ$ is H₁ (θ)” which imply that “θ is low” and “θ is high”, respectively (see Definition 2).

(a) In testing the left one-sided fuzzy hypothesis ${\tilde{H}}_{0}$ against the right one-sided fuzzy hypothesis ${\tilde{H}}_{1}$ , the p-value is

$\begin{matrix} a) p - value & = & P_{{\tilde{H}}_{0 b}} (T \geq t) \\ = & \int_{θ} H_{0 b}^{*} (θ) P_{θ} (T \geq t) d θ, \end{matrix}$ (1)

(b) In testing the right one-sided fuzzy hypothesis ${\tilde{H}}_{1}$ against the left one-sided fuzzy hypothesis ${\tilde{H}}_{0}$ , the p-value is

$\begin{matrix} b) p - value & = & P_{{\tilde{H}}_{1 b}} (T \leq t) \\ = & \int_{θ} H_{1 b}^{*} (θ) P_{θ} (T \leq t) d θ, \end{matrix}$ (2) where t is the observed value of test statistic T and $H_{j b}^{*} (θ) = \frac{H_{j b} (θ)}{\int_{θ} H_{j b} (θ) d θ}$ , j = 0, 1. A replacement of integration by summation is needed in discretecase.

Remark 2. When the hypothesis is crisp rather than fuzzy, the membership function of the fuzzy boundary is reduced to the indicator function of the single-point boundary θ₀. Afterwords, the introduced p-values for testing one-sided fuzzy hypotheses in Definition 5 are respectively reduced to the classical p-values P_{θ
₀} (T ≥ t) and P_{θ
₀} (T ≤ t), where θ₀ is the boundary of the null hypothesis; see [7].

The following decision rule is an extension of the decision rule provided in [6] for testing fuzzy hypotheses.

Decision rule: In testing fuzzy hypotheses ${\tilde{H}}_{0}$ against ${\tilde{H}}_{1}$ , accept ${\tilde{H}}_{1}$ with confidence factor $\frac{p_{10}}{p_{01} + p_{10}}$ if p₀₁ < p₁₀; otherwise accept ${\tilde{H}}_{0}$ with confidence factor $\frac{p_{01}}{p_{01} + p_{10}}$ , where p₀₁ is the p-value of testing ${\tilde{H}}_{0}$ against ${\tilde{H}}_{1}$ , and p₁₀ is the p-value of testing ${\tilde{H}}_{1}$ versus ${\tilde{H}}_{0}$ .

Several advantages of the proposed method over the common p-value methods have been listed in [25]. The main advantage is that, the proposed method is based on both null and alternative fuzzy hypotheses, while the common p-value-based approaches for testing fuzzy hypotheses are based on only fuzzy null hypothesis; see e.g. [7] and [24].

3 Motivation of testing fuzzy hypotheses in quality control

Process capability indices have been proposed as numerical measures on process capability in manufacturing industries. In the study of process capability testing, to judge if the process is capable or not, one can test the capability by considering the null hypothesis “H₀ : C_p ≤ c₀ (the process is not capable)”, against “H₁ : C_p > c₀ (the process is capable)”, where c₀ is a predetermined capability requirement [15]. Table 8.3 on Page 354 of [17] includes some recommended guidelines for minimum values of the capability indices which are commonly used as a standard minimal criteria in testing capability. For instance in testing the above hypotheses, he recommends to consider c₀ = 1.33 as the standard minimal criteria for an existing process and consider c₀ = 1.50 for a new process. But considering unremitting changes in manufacturing processes, we don’t have a useful criteria for actually measuring the amount of the refurbishment of a process. Therefore, one may be actually confused in determining a suitable value for the standard minimal criteria c₀ in common testing “H₀ : C_p ≤ c₀” against “H₁ : C_p > c₀”, which has an important rule in the result of test. One way to avoid the above confusion is considering fuzzy hypotheses ${\begin{matrix} {\tilde{H}}_{0} : C_{p} is low (process is not capable), \\ {\tilde{H}}_{1} : C_{p} is high (process is capable), \end{matrix}$ (3) instead of the common classical hypotheses for testing capability in which the linguistic and non-precise concepts low and high are respectively defined by membership functions $H_{0} (C_{p}) = {\begin{matrix} 1 & if & 0 < C_{p} \leq 1.33 \\ \frac{1.50 - C_{p}}{1.50 - 1.33} & if & 1.33 < C_{p} \leq 1.50 \\ 0 & if & 1.50 < C_{p}, \end{matrix}$ and H₁ (C_p) =1 - H₀ (C_p) (see Fig. 2). Another advantage of considering the fuzzy hypotheses (3) instead of common precise hypotheses, is using from a gradually (fuzzy) boundary for defining the sets of “capable” and “incapable” parameter spaces in a capability test. In other words, there is not an eruption in the boundaries of fuzzy hypotheses (3) as it behaves in classical hypotheses, which can lead the user to a more justified judgement in decision making on manufacturing processes (compare Figs. 2 with 3).

We are going to test fuzzy hypotheses (3) in Sections 4 and 5 by the proposed p-value-based method in Subsection 2.3 according to two capability indices C_p and C_pm.

4 Testing quality based on index C_p

Process capability indices, which establish the relationships between the actual process performance and the manufacturing specifications, have been a focus of research in quality assurance and capability analysis for the last 20 years [15]. In [10], Kane define the first process capability index as

$C_{p} = \frac{USL - LSL}{6 σ},$ (4) where USL is the upper specification limit, LSL is the lower specification limit and σ is the process standard deviation. To statistical inference on C_p based on a taken random sample of assembling line, we assume that X = (X₁,. . . , X_n) be a random sample with the observed value x = (x₁,. . . , x_n), where X_i’s have independently normal distribution with unknown mean μ and unknown variance σ², i.e. $X_{i} \overset{iid}{\sim} N (μ, σ^{2})$ . In testing fuzzy hypotheses ${\begin{matrix} {\tilde{H}}_{0} : C_{p} is H_{0} (C_{p}) (C_{p} is low), \\ {\tilde{H}}_{1} : C_{p} is H_{1} (C_{p}) (C_{p} is high), \end{matrix}$ suppose that the membership functions of fuzzy hypotheses are defined by $H_{0} (C_{p}) = {\begin{matrix} 1 & if & 0 < C_{p} \leq 1.33 \\ \frac{1.50 - C_{p}}{1.50 - 1.33} & if & 1.33 < C_{p} \leq 1.50 \\ 0 & if & 1.50 < C_{p}, \end{matrix}$ and H₁ (C_p) =1 - H₀ (C_p), respectively (see Fig. 2). In [5], Chou and Owen obtain the p.d.f. of $\hat{C_{p}} = \frac{USL - LSL}{6 \sqrt{\frac{1}{n - 1} \sum_{i = 1}^{n} (X_{i} - \bar{X})^{2}}} = \frac{USL - LSL}{6 S}$ under normality assumption by $f (x; C_{p}) = \frac{2 {(\sqrt{\frac{n - 1}{2}} C_{p})}^{n - 1}}{Γ (\sqrt{\frac{n - 1}{2}})} x^{- n} e^{\frac{- (n - 1) C_{p}^{2}}{2 x^{2}}}, x > 0,$ which is indicated the value of p.d.f. of random variable $\hat{C_{p}}$ in point x and C_p is the parameter of distribution. Therefore, by considering Definition 4, the weighted probability density function of $\hat{C_{p}}$ , under the fuzzy hypothesis ${\tilde{H}}_{0}$ , is $\begin{matrix} f (x; {\tilde{H}}_{0}) = \int_{C_{p}} H_{0}^{*} (C_{p}) f (x; C_{p}) {dC}_{p} \\ = \frac{\int_{0}^{1.33} f (x; C_{p}) {dC}_{p} + \int_{1.33}^{1.50} \frac{1.50 - C_{p}}{1.50 - 1.33} f (x; C_{p}) {dC}_{p}}{1.33 + \frac{0.17}{2}} \\ = \frac{2 \times 10^{2}}{283} \int_{0}^{1.33} \frac{2 {(\sqrt{\frac{n - 1}{2}} C_{p})}^{n - 1}}{Γ (\sqrt{\frac{n - 1}{2}})} x^{- n} e^{\frac{- (n - 1) C_{p}^{2}}{2 x^{2}}} {dC}_{p} \\ + \frac{2 \times 10^{4}}{17 \times 283} \int_{1.33}^{1.50} (1.50 - C_{p}) \\ \frac{2 {(\sqrt{\frac{n - 1}{2}} C_{p})}^{n - 1}}{Γ (\sqrt{\frac{n - 1}{2}})} x^{- n} e^{\frac{- (n - 1) C_{p}^{2}}{2 x^{2}}} {dC}_{p} \\ = \frac{2^{2} \times 10^{2} {(\frac{n - 1}{2})}^{\frac{n - 1}{2}}}{283 x^{n} Γ (\sqrt{\frac{n - 1}{2}})} {\int_{0}^{1.33} C_{p}^{n - 1} e^{\frac{- (n - 1) C_{p}^{2}}{2 x^{2}}} {dC}_{p} \\ + \frac{10^{2}}{17} \int_{1.33}^{1.50} (1.50 - C_{p}) C_{p}^{n - 1} e^{\frac{- (n - 1) C_{p}^{2}}{2 x^{2}}} {dC}_{p}}, \end{matrix}$

which is shown by bold line in Fig. 4.

One can calculate the p-value of testing ${\tilde{H}}_{0}$ against ${\tilde{H}}_{1}$ as follows

$\begin{matrix} p_{01} & = & P_{{\tilde{H}}_{0 b}} (\hat{C_{p}} \geq c_{p}) \\ = & \int_{C_{p}} H_{0 b}^{*} (C_{p}) P_{C_{p}} (\hat{C_{p}} \geq c_{p}) d C_{p} \\ = & \frac{\int_{1.33}^{1.50} \frac{1.50 - C_{p}}{1.50 - 1.33} P_{C_{p}} (\frac{σ}{S} C_{p} \geq c_{p}) d C_{p}}{\int_{1.33}^{1.50} H_{0 b} (C_{p}) d C_{p}} \\ = & \frac{2 \times 10^{4}}{17^{2}} \int_{1.33}^{1.50} (1.50 - C_{p}) \\ P_{C_{p}} (χ_{n - 1}^{2} \leq (n - 1) {(\frac{C_{p}}{c_{p}})}^{2}) d C_{p}, \end{matrix}$ (5) in which $χ_{n - 1}^{2} = \frac{(n - 1) S^{2}}{σ^{2}}$ has chi-square distribution with n - 1 degrees of freedom, $c_{p} = \frac{USL - LSL}{6 s}$ is the observed value of test statistic $\hat{C_{p}}$ and s is the observed value of the standard deviation S. Similarly, the p-value of testing ${\tilde{H}}_{1}$ against ${\tilde{H}}_{0}$ isequal to

$\begin{matrix} p_{10} & = & P_{{\tilde{H}}_{1 b}} (\hat{C_{p}} \leq c_{p}) \\ = & \int_{c} H_{1 b}^{*} (C_{p}) P_{C_{p}} (\hat{C_{p}} \leq c_{p}) d C_{p} \\ = & \frac{\int_{1.33}^{1.50} (1 - \frac{1.50 - C_{p}}{1.50 - 1.33}) P_{C_{p}} (\frac{σ}{S} C_{p} \leq c_{p}) d C_{p}}{\int_{1.33}^{1.50} H_{1 b} (C_{p}) d C_{p}} \\ = & \frac{2 \times 10^{4}}{17^{2}} \int_{1.33}^{1.50} (C_{p} - 1.33) \\ P_{C_{p}} (χ_{n - 1}^{2} \geq (n - 1) {(\frac{C_{p}}{c_{p}})}^{2}) d C_{p} . \end{matrix}$ (6)

Decision rule in testing capability based on C_p: The process is capable with confidence factor $\frac{p_{10}}{p_{01} + p_{10}}$ if p₀₁ < p₁₀; otherwise it is incapable with confidence factor $\frac{p_{01}}{p_{01} + p_{10}}$ , where p₀₁ and p₁₀ are introduced by (5) and (6), respectively. Therefore in this method, the “confidence factor into the given decision” can be introduced by

$\begin{matrix} CF & = & \frac{p_{01}}{p_{01} + p_{10}} I (p_{01} \geq p_{10}) \\ + \frac{p_{10}}{p_{01} + p_{10}} I (p_{01} < p_{10}), \end{matrix}$ (7) where I (.) is the indicator function.

Example 2. (An application in manufacturing process of jet aircraft engine) A component part of a jet aircraft engine is manufactured by an investment casting process. The vane opening on this casting is an important quality characteristic and we will test its quality based on C_p by the proposed method in Subsection 4. The values given in Table 1 have been coded by using the last three digits of the dimension; e.g., 27.3 should be 0.50273 inch which are quoted from Page 611 of [17]. We are going to test fuzzy hypotheses ${\begin{matrix} {\tilde{H}}_{0} : C_{p} is H_{0} (C_{p}) (C_{p} is low), \\ {\tilde{H}}_{1} : C_{p} is H_{1} (C_{p}) (C_{p} is high), \end{matrix}$ where the membership functions of fuzzy hypotheses are defined by $\begin{matrix} H_{0} (C_{p}) = {\begin{matrix} 1 & if & 0 < C_{p} \leq 1.33 \\ \frac{1.50 - C_{p}}{1.50 - 1.33} & if & 1.33 < C_{p} \leq 1.50 \\ 0 & if & 1.50 < C_{p}, \end{matrix} \end{matrix}$ and H₁ (C_p) =1 - H₀ (C_p). Regarding to the collected sample data with size n = 100 which is quoted in Table 1, the sample mean and the sample standard deviation are $\bar{x} = 33.32$ and s = 3.299, respectively. It is reasonable to accept the normality assumption of the collected data from the proposed assemble line, whereas the Shapiro-Wilk test strongly confirms the normality assumption with p-value= 0.353. Considering specifications limits LSL = 16 and USL = 44, the estimated value of index C_p is equal to $c_{p} = \frac{USL - LSL}{6 s} = \frac{40 - 20}{6 \times 3.299} = 1.415 .$

So by Relation (5), one can compute the p-value of testing ${\tilde{H}}_{0}$ against ${\tilde{H}}_{1}$ based on the weighted p.d.f. of $\hat{C_{p}}$ under ${\tilde{H}}_{0 b}$ as follows

$\begin{matrix} p_{01} & = & P_{{\tilde{H}}_{0 b}} (\hat{C_{p}} \geq 1.415) \\ = & \frac{2 \times 10^{4}}{17^{2}} \int_{1.33}^{1.50} (1.50 - C_{p}) \\ P_{C_{p}} (χ_{99}^{2} \leq 99 {(\frac{C_{p}}{1.415})}^{2}) d C_{p} \\ = & 0.413, \end{matrix}$ (8) which is equivalent to the area of dark-grey surface in Fig. 5 where $χ_{99}^{2}$ has chi-square distribution with 99 degrees of freedom. To decide according to the proposed decision rule, we need to calculate the p-value of testing ${\tilde{H}}_{1}$ versus ${\tilde{H}}_{0}$ . Hence, one can similarly compute p₁₀ by Relation (6) as follows, which is equivalent to the area of light-grey surface in Fig. 5

$\begin{matrix} p_{10} & = & P_{{\tilde{H}}_{1 b}} (\hat{C_{p}} \leq 1.415) \\ = & \frac{2 \times 10^{4}}{17^{2}} \int_{1.33}^{1.50} (C_{p} - 1.33) \\ P_{C_{p}} (χ_{99}^{2} \geq 99 {(\frac{C_{p}}{1.415})}^{2}) d C_{p} \\ = & 0.378 . \end{matrix}$ (9)

Therefore p₀₁ > p₁₀, and so we accept ${\tilde{H}}_{0}$ against ${\tilde{H}}_{1}$ with confidence factor $CF = \frac{0.413}{0.413 + 0.378} = 0.522 .$

Note that based on the classical p-value method, one accepts ${\tilde{H}}_{0}$ against ${\tilde{H}}_{1}$ at any significance level α < p₀₁ = 0.413 (e.g., at significance level 0.01 or 0.05). Although in this example, the result of the proposed method coincides with the result of the classical significant test (e.g., at level 0.05), we assert that the proposed method is more reasonable than the classical significant test, since the proposed method is based on both null and alternative fuzzy hypotheses.

5 Testing quality based on index C_pm

In this section, the new p-value-based method is developed for testing capability index C_pm, where the considered hypotheses are fuzzy rather than crisp. One of the lacks of capability index C_p is that it dose not take account into the cost of failing to meet customers’ requirements. In [8], to handle this situation, Hsiang and Taguchi introduced capability index

$\begin{matrix} C_{pm} & = & \frac{USL - LSL}{6 \sqrt{σ^{2} + (μ - T)^{2}}} \\ = & \frac{C_{p}}{\sqrt{1 + {(\frac{μ - T}{σ})}^{2}}}, \end{matrix}$ (10) which involves the variation of production items with respect to the target value T and it is able to concentrate on measuring the ability of the process to cluster around target value. In [2] and [26], Boyles and Pearn et al. have respectively shown that the distributed of C_pm is $\hat{C_{pm}} \sim \frac{USL - LSL}{6 σ} \sqrt{\frac{n}{χ_{n, λ}^{2}}} = C_{p} \sqrt{\frac{n}{χ_{n, λ}^{2}}},$ by using (10), we can write $\hat{C_{pm}} \sim C_{pm} \sqrt{1 + \frac{λ}{n}} \sqrt{\frac{n}{χ_{n, λ}^{2}}} = C_{pm} \sqrt{\frac{n + λ}{χ_{n, λ}^{2}}},$ where $χ_{n, λ}^{2}$ denotes the non-central chi-square distribution with n degrees of freedom and the non-centrality parameter $λ = \frac{n (μ - T)^{2}}{σ^{2}}$ . In testing fuzzy hypotheses ${\begin{matrix} {\tilde{H}}_{0} : C_{pm} is H_{0} (C_{pm}) (C_{pm} is low), \\ {\tilde{H}}_{1} : C_{pm} is H_{1} (C_{pm}) (C_{pm} is high), \end{matrix}$ (11) suppose that fuzzy hypotheses are defined by membership functions $H_{0} (C_{pm}) = {\begin{matrix} 1 & if & 0 < C_{pm} \leq 1.33 \\ \frac{1.50 - C_{pm}}{1.50 - 1.33} & if & 1.33 < C_{pm} \leq 1.50 \\ 0 & if & 1.50 < C_{pm}, \end{matrix}$ and H₁ (C_pm) =1 - H₀ (C_pm). The p-value of testing ${\tilde{H}}_{0}$ against ${\tilde{H}}_{1}$ computed as follows

$\begin{matrix} p_{01} & = & P_{{\tilde{H}}_{0 b}} (\hat{C_{pm}} \geq c_{pm}) \\ = & \int_{C_{pm}} H_{0 b}^{*} (C_{pm}) P_{C_{pm}} (\hat{C_{pm}} \geq c_{pm}) d C_{pm} \\ = & \frac{\int_{1.33}^{1.50} \frac{1.50 - C_{pm}}{1.50 - 1.33} P_{C_{pm}} (C_{pm} \sqrt{\frac{n + λ}{χ_{n, λ}^{2}}} \geq c_{pm}) d C_{pm}}{\int_{1.33}^{1.50} H_{0 b} (C_{pm}) d C_{pm}} \\ = & \frac{2 \times 10^{4}}{17^{2}} \int_{1.33}^{1.50} (1.50 - C_{pm}) \\ P_{C_{pm}} (χ_{n, λ}^{2} \leq (n + λ) {(\frac{C_{pm}}{c_{pm}})}^{2}) d C_{pm}, \end{matrix}$ (12)

in which $c_{pm} = \frac{USL - LSL}{6 \sqrt{s^{2} + (\bar{x} - T)^{2}}}$ is the observed value of test statistic $\hat{C_{pm}}$ . Also, the p-value of testing ${\tilde{H}}_{1}$ against ${\tilde{H}}_{0}$ is

$\begin{matrix} p_{10} & = & P_{{\tilde{H}}_{1 b}} (\hat{C_{pm}} \leq c_{pm}) \\ = & \int_{c} H_{1 b}^{*} (C_{pm}) P_{C_{pm}} (\hat{C_{pm}} \leq c_{pm}) d C_{pm} \\ = & \frac{2 \times 10^{4}}{17^{2}} \int_{1.33}^{1.50} (C_{pm} - 1.33) \\ P_{C_{pm}} (χ_{n, λ}^{2} \geq (n + λ) {(\frac{C_{pm}}{c_{pm}})}^{2}) d C_{pm} . \end{matrix}$ (13)

Note that p₀₁ and p₁₀ are two functions of unknown parameter λ in Relations (12) and (13). Therefore considering the invariance property of ML (maximum likelihood) estimators, one can easily present the ML estimates of p₀₁ and p₁₀ accordingly $\begin{matrix} \hat{p_{01}} & = & \frac{2 \times 10^{4}}{17^{2}} \int_{1.33}^{1.50} (1.50 - C_{pm}) P_{C_{pm}} (χ_{n, \frac{n (\bar{x} - T)^{2}}{s^{2}}}^{2} \\ \leq (n + n (\frac{\bar{x} - T}{s})^{2}) {(\frac{C_{pm}}{c_{pm}})}^{2}) d C_{pm} \end{matrix}$ (14) $\begin{matrix} \hat{p_{10}} & = & \frac{2 \times 10^{4}}{17^{2}} \int_{1.33}^{1.50} (C_{pm} - 1.33) P_{C_{pm}} (χ_{n, \frac{n (\bar{x} - T)^{2}}{s^{2}}}^{2} \\ \geq (n + \frac{n (\bar{x} - T)^{2}}{s^{2}}) {(\frac{C_{pm}}{c_{pm}})}^{2}) d C_{pm}, \end{matrix}$ (15) whereas $\bar{x}$ and s are respectively ML estimates of μ and σ, under the normality assumption.

Decision rule in testing capability based on C_pm: The process is capable with the estimated confidence factor $\frac{\hat{p_{10}}}{\hat{p_{01}} + \hat{p_{10}}}$ if $\hat{p_{01}} < \hat{p_{10}}$ ; otherwise it is incapable with the estimated confidence factor $\frac{\hat{p_{01}}}{\hat{p_{01}} + \hat{p_{10}}}$ , where $\hat{p_{01}}$ and $\hat{p_{10}}$ are introduced by (14) and (15), respectively. Therefore, the “confidence factor into the given decision” can be estimated by $\begin{matrix} \hat{CF} & = & \frac{\hat{p_{01}}}{\hat{p_{01}} + \hat{p_{10}}} I (\hat{p_{01}} \geq \hat{p_{10}}) \\ + & \frac{\hat{p_{10}}}{\hat{p_{01}} + \hat{p_{10}}} I (\hat{p_{01}} < \hat{p_{10}}) . \end{matrix}$

Example 3. (An application in electronic industry) In this example, we are going to apply the presented new p-value-based testing method in Section 5, on a real-world data set by considering fuzzy hypotheses on C_pm index. Data which are used in this example is quoted from [37]. An external AC/DC wall adapter is used to convert the AC line voltage into a regulated DC voltage or a current limited source in many electronic products. Line surges or faults in the adapter may result in over-voltage events which can damage sensitive electronic components within the product. In addition, portable products, such as digital cameras, cellular phones, portable computers, portable CD player applications and other consumer electronics, with removable battery packs pose special problems since the pack can be removed at any time. If the user removes a pack in the middle of charging, a large transient voltage spike can occur which can damage the product. Finally, the damage can result if the user plugs in the wrong adapter into the charging jack. The challenge of the product designer is improving the robustness of the design and avoid situations in which the product can be damaged due to unexpected, but unfortunately, likely events that will occur as the product is used. Therefore, the over-voltage protection IC (OVP-IC) is designed to protect sensitive electronic circuitry from over-voltage transients and power supply faults. This device senses an over-voltage condition and quickly disconnects the input voltage supply from the load before any damage can occur [37]. For a particular model of the OVP-IC, the tolerance of over-voltage threshold is 0.20 V, i.e. the lower specifications limit and the upper specification limit are LSL = 5.30 V and USL = 5.70 V, respectively. Also, suppose that the target value is equal to T = 5.50 V. We wish to test the following fuzzy hypotheses on the basis of the proposed method in Section 5 ${\begin{matrix} {\tilde{H}}_{0} : C_{pm} is H_{0} (C_{pm}) (C_{pm} is low), \\ {\tilde{H}}_{1} : C_{pm} is H_{1} (C_{pm}) (C_{pm} is high), \end{matrix}$ where the membership functions of fuzzy hypotheses are defined by $\begin{matrix} H_{0} (C_{pm}) = {\begin{matrix} 1 & if & 0 < C_{pm} \leq 1.33 \\ \frac{1.50 - C_{pm}}{1.50 - 1.33} & if & 1.33 < C_{pm} \leq 1.50 \\ 0 & if & 1.50 < C_{pm}, \end{matrix} \end{matrix}$ and H₁ (C_pm) =1 - H₀ (C_pm), respectively. Considering the collected sample data which is quoted in Table 2 from [37], the sample mean and the sample standard deviation are calculated as $\bar{x} = 5.482$ and s = 0.037, respectively. The Shapiro-Wilk test for normality is confirmed with p-value>0.1. Therefore, it is reasonable to accept the normality assumption of the collected data from assemble line. According to (10), the estimated value of capability index is equal to $\begin{matrix} c_{pm} & = & \frac{5.70 - 5.30}{6 \sqrt{s^{2} + (\bar{x} - T)^{2}}} \\ = & \frac{5.70 - 5.30}{6 \sqrt{0 . 037^{2} + (5.482 - 5.50)^{2}}} \\ = & 1.613 . \end{matrix}$

So, considering Relation (14) one can estimate the p-value of testing ${\tilde{H}}_{0}$ against ${\tilde{H}}_{1}$ as follows

$\begin{matrix} \hat{p_{01}} & = & \frac{2 \times 10^{4}}{17^{2}} \int_{1.33}^{1.50} (1.50 - C_{pm}) \\ \times P_{C_{pm}} (χ_{100, \frac{100 (5.482 - 5.50)^{2}}{0 . 037^{2}}}^{2} \leq \\ (100 + \frac{100 (5.482 - 5.50)^{2}}{0 . 037^{2}}) {(\frac{C_{pm}}{1.613})}^{2}) d C_{pm} \\ = & 0.030 . \end{matrix}$ (16)

To decide on the basis of the proposed decision rule, we need to estimate the p-value of testing ${\tilde{H}}_{1}$ versus ${\tilde{H}}_{0}$ , hence by Relation (15),

$\begin{matrix} \hat{p_{10}} & = & \frac{2 \times 10^{4}}{17^{2}} \int_{1.33}^{1.50} (C_{pm} - 1.33) \\ \times P_{C_{pm}} (χ_{100, \frac{100 (5.482 - 5.50)^{2}}{0 . 037^{2}}}^{2} \geq \\ (100 + \frac{100 (5.482 - 5.50)^{2}}{0 . 037^{2}}) {(\frac{C_{pm}}{1.613})}^{2}) d C_{pm} \\ = & 0.920 . \end{matrix}$ (17)

Therefore $\hat{p_{01}} < \hat{p_{10}}$ , and so we strongly accept ${\tilde{H}}_{1}$ against ${\tilde{H}}_{0}$ with the estimated confidence factor $\hat{CF} = \frac{0.920}{0.030 + 0.920} = 0.968 .$

Note that based on the classical p-value method, one accepts ${\tilde{H}}_{1}$ against ${\tilde{H}}_{0}$ at any significance level α > p₀₁ = 0.030. So, although the result of the classical p-value method for α > p₀₁ = 0.030 (e.g., at significance level α =0.05) coincides with the result of the proposed method in this paper, it is in conflict with the result of the proposed method for α < p₀₁ = 0.030 (e.g., at significance level α =0.01). This matter, i.e. dependence of the result to the given significance level α, is another weak-point of the classical p-value-based methods, while the proposed method in this paper dose not depend on the significance level whereas its decision rule is not a functionof α.

6 Conclusions and future works

In this paper, a new p-value-based approach was reviewed for testing statistical hypotheses when the hypotheses are fuzzy rather than crisp. In contrast with the common p-value-based methods for testing fuzzy hypotheses, the decision rule in the proposed method is based on two p-values: (1) the p-value of testing fuzzy null hypothesis versus fuzzy alternative hypothesis, and (2) the p-value of testing fuzzy alternative hypothesis versus fuzzy null hypothesis. Afterwords, two common capability tests on C_p and C_pm are developed by this method with assuming fuzzy capability/incapability hypotheses. Finally, two industrial examples were investigated to show how the proposed method can be implemented in real-world cases. Study on testing quality based on capability indices C_pk and C_pmk are two potential subjects for further research.

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