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In statistical quality control, as in other statistical problems, we may be confronted with fuzzy concepts. This paper deals with the problem of process capability estimation, when the observation and the specification limits are fuzzy rather than crisp. In other words, this paper illustrate how a researcher can use “FuzzyNumbers” Package in R software to exactly plot the membership function of capability indices based on fuzzy data and crisp specification limits.

Process capability indices provide numerical measures to compare the output of a process to client's expectations. However, most of the existing researches have used traditional distribution frequency method by using a single sample due to assess process capability. An alternative to this approach is to use the Bayesian method. In this paper, we utilize a Bayesian approach based on subsamples to check process capability via capability index Cpk. As a new suggestion, we used the informative normal prior distribution and the characteristics of sufficient statistic of the parameter to drive the posterior distribution. The capability test is done, and the posterior probability p, for which the process under investigation is capable, is derived both based on the most popular index Cpk. Finally, a numerical example is given to clarify the method. Copyright © 2013 John Wiley & Sons, Ltd.

In some cases, the specification limits of a quality characteristic should be defined under uncertain information. In the literature, process capability analyses have been handled by using type-1 fuzzy sets under fuzziness up to now. In this paper, we develop the concept of type-2 fuzzy quality and use it in the calculation of process capability. Lower and upper specification limits are considered as type-2 fuzzy sets. We define the addition and subtraction operations for introducing interval type-2 fuzzy process capability indices. We can calculate the nonconforming percentage of the production using the developed equations. We present some numerical examples to show the applicability of the proposed equations.

Most of the traditional methods for assessing the capability of manufacturing processes are dealing with crisp quality. In this paper we discuss the fuzzy quality and introduce fuzzy process capability indices, where instead of precise quality we have two membership functions for specification limits. These indices are necessary when the specification limits are fuzzy and they are helpful for comparing manufacturing processes with fuzzy specification limits. Some interesting relations among the introduced indices are obtained. Numerical examples are given to clarify the method.

Most of the traditional methods for assessing the capability of manufacturing processes are dealing with crisp quality. In quality control, such as other statistical problems, we may confront imprecise concepts. One case is a situation in which specification limits (SLs) are imprecise. In this situation, the fuzzy process capability indices (PCIs) [image omitted], [image omitted] and [image omitted] are necessary for measuring the fuzzy quality in an in-control process. These fuzzy capability indices are also helpful for comparing manufacturing processes with SLs. The fuzzy capability index [image omitted] is used to provide an assessment of the ability of the fuzzy process to be clustered around the target value. The emphasis on the use of [image omitted] over the other two fuzzy indices, [image omitted] and [image omitted], is due to its definition that provides indications of both the process variability and deviation of process mean from a specified target. In this article, by using triangular fuzzy SLs, we present four approximate [image omitted] fuzzy confidence regions for the fuzzy PCI [image omitted]. A numerical example is given to show the performance of the method.

This paper deals with the problem of testing statistical hypotheses when both the hypotheses and data are fuzzy. To this end, we first introduce the concept of fuzzy p-value and then develop an approach for testing fuzzy hypotheses by comparing a fuzzy p-value and a fuzzy significance level. Numerical examples are provided to illustrate the approach for different cases.

In quality control, we may confront imprecise concepts. One case is a situation in which upper and lower specification limits (SLs) are imprecise. If we introduce vagueness into SLs, we face quite new, reasonable and interesting processes, and the ordinary capability indices are not appropriate for measuring the capability of these processes. In this paper, similar to the traditional process capability indices (PCIs), we develop a fuzzy analogue by a distance defined on a fuzzy limit space and introduce PCIs, where instead of precise SLs we have two membership functions for upper and lower SLs. These indices are necessary when SLs are fuzzy, and they are helpful for comparing manufacturing process with fuzzy SLs. Some interesting relations among these introduced indices are proved. Numerical examples are given to clarify the method.

After the fuzzy set theory was introduced and developed, many studies have been realized to combine quality control methods and fuzzy set theory. This chapter is including the categorization of most essential works on fuzzy process capability indices in the following four main categories: (1) Lee et al.’s method and its extensions: This class deals with the method of modeling and estimating the membership function of process capability indices where all data and specifications are fuzzy numbers; (2) Parchami et al.’s method and its extensions: This class deals with the problem of obtaining fuzzy process capability indices based on fuzzy specification limits and crisp data by extension principle approach; (3) Kaya and Kahraman’s method and its extensions: This class deals with the problem of estimating the classical process capability indices by a triangular shaped fuzzy number when both specifications and data are crisp; (4) Yongting’s method and its extensions: This class deals with introducing process capability indices based on fuzzy quality where the data and parameters are crisp. After presenting the basic idea of the main works, all related studies briefly reviewed in each class. Some numerical examples are presented to show the applicability of the proposed methods.

The impreciseness happens in the quality control same as other statistical problems. In quality control where specification limits are better expressed by fuzzy sets, the generalized process capability indices C p , C pk and C pm can be helpful and necessary for measuring the capability. We propose a generalized form of Taguchi index C pm to assess the ability of the fuzzy process to be clustered around the target value. An approximate 100(1−γ)% confidence interval for the generalized process capability index C pm by using fuzzy specification limits is also presented.

In this paper, a new method is proposed for testing fuzzy hypotheses based on the following two generalized p-values: (1) the generalized p-value of null fuzzy hypothesis against alternative fuzzy hypothesis and (2) the generalized p-value of alternative fuzzy hypothesis against null fuzzy hypothesis. In the proposed method, each generalized p-value is formulated on the basis of Zadeh’s probability measure of fuzzy events. The introduced p-value method has several advantages over the common p-value methods for testing fuzzy hypotheses. A few illustrative examples and also an agricultural example, based on a real-world data set, are given to clarify the proposed method.

It is more appropriate that many industrial products be evaluated and qualified by an imprecise (fuzzy) quality. By this idea the products could be evaluated using two membership functions for specification limits rather than two real numbers used in classical quality control. This idea leads the researchers to be able to deal with the vague process capability indices modeled as triangular fuzzy numbers. In this paper, we discuss on such fuzzy qualities and review some fuzzy process capability indices. Then we will bring them up to analyze several educational systems, such as comparing capability indices of two or more teachers, schools, and so on. The idea of this paper could be applied in other similar evaluation schemes as well.