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Approximately three decades ago, the fuzzy quality was proposed by Yongting as a flexible and powerful tool to quality modeling. Furthermore, in order to more compliance of products with customer requirements, the probability of fuzzy quality was introduced as a process capability index to evaluate manufacturing processes based on the fuzzy quality. The diversity of fuzzy quality membership functions as well as the complexity of the distribution of Yongting index estimator hindered the scientific progress in the field of statistical inference. To cover a part of this deprivation, testing the capability of productive process on the basis of the flexible fuzzy quality is proposed in this paper by Monte Carlo simulation. Hence, the objective of this research is estimate the critical value by Monte Carlo simulation approach for normal data on the basis of fuzzy quality. A case study is provided in pipe manufacturing industries to show the performance of approach.

This article presents a general approach for quality test to make a decision based on the extended process capability indices. Usual methods in measuring the quality of a manufactured product have widely focused on the precise specification limits, but in this study the lower and upper specification limits are considered as non-precise (fuzzy). Extended process capability indices are common tools to quantify the performance of a normal process where lower and upper specification limits are fuzzy. The main purpose of this study is to propose a new statistical approach to estimate a critical value for analyzing the capability of a manufacturing process using the extended process capability index to determine whether the process meets customer requirement. To show the real application of the proposed method, a case study at the glass bottle manufacturing industry is provided and the obtained theoretical results are tabulated for making reliable decisions in different situations.

Usually, the statistical estimators (or mathematical functions) are the base of scientific decision making. In applied situations, at least one of the parameters or variables of the decision function may be fuzzy valued, instead of real valued. In such vague situations, one way to perform the calculations is using extension principle approach which has a complex form. Recently, two software packages have been freely available to perform some facilities in calculation and computation based on fuzzy numbers. In other words, these two software packages have the role of the first calculator on fuzzy numbers. This paper discussed and compared two software packages “FuzzyNumbers” and “Calculator.LR.FNs” which were recently published on CRAN by Gagolewski and Parchami, respectively. These packages have the ability of installation on R software, and in fact they propose some useful instruments and functions to the users for drawing and easily using arithmetic operators on the set of fuzzy numbers. For the convenience of the readers, the proposed methods and functions have been presented with several numerical examples to help in better understanding.

Analysis of variance is an important method in exploratory and confirmatory data analysis when explanatory variables are discrete and response variables are continues and independent from each other. The simplest type of analysis of variance is one-way analysis of variance for comparison among means of several populations. In this paper, we extend one-way analysis of variance to a case where observed data are non-symmetric triangular fuzzy observations rather than real numbers. Meanwhile, a case study on the car battery length-life is provided on the basis on the proposed method.

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.

The extension principle is one of the most fundamental principles in the fuzzy set theory. It provides a powerful technique in order to extend the domain of a classical function from real numbers to fuzzy sets. In the present paper, we propose a new program in R software to graphically show the behavior of the simplest type of the extension principle, where the function is one-to-one. Numerical examples are given to show the results of the proposed program by import-export figures and animations.

In hypotheses testing, such as other statistical problems, we may confront imprecise concepts. One case is a situation in which the hypotheses of interest are imprecise. In this paper, we recall and redefine some concepts about testing fuzzy hypotheses and then we provide a minimax approach to the problem of testing fuzzy hypotheses by using crisp (non-fuzzy) data. We give some illustrative/numerical examples, by which we study the effect of fuzziness by using the power functions of minimax tests.

Testing one-way analysis of variance (ANOVA) is used for experimental data analysis in which there is a continuous response variable and a single independent classification variable. In this paper, we extend one-way ANOVA to a case where observed data are imprecise numbers rather than real numbers. Several fast computable formulas are calculated for symmetric triangular and normal fuzzy data. Similar to the classical testing ANOVA, the total observed variation in the response variable is explained as the sum of observed variation due to the effects of the classification variable and the observed variation due to random error. A real case is given to clarify the proposed method.

Three organic fertilizers of manure sheep, sewage sludge and municipal waste were mixed with soil at the rates of 0%, 1.25% and 2.5% (w/w), respectively. In a greenhouse experiment, corn seed were grown on pots of 3 kg treated contaminated soils and irrigated. Sixty days after sowing, the aerial plant parts were harvested and analyzed for Cadmium (Cd) and Lead (Pb) contents. Since all of data in this research are fuzzy, we need an extended version of analysis of variance (ANOVA) to investigate on these fuzzy observations. In this paper, as a method to compare several populations, the fuzzy analysis of variance (FANOVA) has been used where the collected data considered fuzzy rather than crisp numbers and therefore all calculations are based on FANOVA method. Although, ANOVA based on vague data can lead to a fuzzy decision, but measuring the vagueness of this fuzzy decision is one of advantages of proposed method from the applied point of view.

In this paper a maximum likelihood estimator for the extended capability index is obtained, where fuzzy quality is considered under an exponential process. Moreover, the consistency, biassednes and the mean square of error for the proposed estimator are discussed. Also, an equal tails (1−a)100% confidence interval is constructed for the proposed estimator. Finally, a numerical example is given to clarify the point and interval estimations of the proposed capability index based on fuzzy quality.

Process capability indices (PCIs) provide numerical measures on whether a process conforms to the defined manufacturing capability prerequisite. These have been successfully applied by companies to compete with and to lead high-profit markets by evaluating the quality and productivity performance. The PCI Cp compares the output of a process to the specification limits (SLs) by forming the ratio of the width between the process SLs with the width of the natural tolerance limits which is measured by six process standard deviation units. As another common PCI, Cpm incorporates two variation components which are variation to the process mean and deviation of the process mean from the target. A meaningful generalized version of above PCIs is introduced in this paper which is able to handle in a fuzzy environment. These generalized PCIs are able to measure the capability of a fuzzy-valued process in producing products on the basis of a fuzzy quality. Fast computing formulas for the generalized PCIs are computed...

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.

In quality control, such as other statistical problems, we may confront imprecise concepts. One case is a situation in which specification limits are two fuzzy sets. In such a fuzzy environment, the product is not qualified with a two valued Boolean view, but to some degree depending on the quality level of the product and the strictness of the decision maker. This matter can be cause to a justified judgment in decision making on manufacturing processes. The generalized process capability indices $C_{\widetilde{p}}$, $C_{\widetilde{pk}}$ and $C_{\widetilde{pm}}$ can be helpful and necessary for measuring the fuzzy quality in an in-control process. In this paper, the generalized Taguchi index $C_{\widetilde{pm}}$ is proposed to provide an assessment of the ability of the fuzzy process to be clustered around the target value. Considering fuzzy specification limits we present four approximate 1001-γ% confidence intervals for the generalized process capability index $C_{\widetilde{pm}}$ in this paper. Then, we obtain several fast computable 1001-γ% confidence intervals for $C_{\widetilde{pm}}$, where the fuzzy specification limits are linear, normal or elliptic. An industrial example is given to show the performance of the proposed 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.

Process capability indices (PCIs) have been proposed in the manufacturing industry to provide numerical measures on process capability, which are effective tools for quality assurance. Usual practices in measuring production quality have focused on the precise specification limits (SLs). If vagueness is involved 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 article, similar to the traditional PCIs, we develop a fuzzy analogue by operations defined on fuzzy SLs and we extend the traditional PCIs, using an imprecise (fuzzy) quality. Some interesting relations among these extended capability indices are proved. A real industrial application and numerical examples are given to clarify the method.

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.

Fuzzy process capability indices are used to determine whether a production process is capable of producing items within fuzzy specification tolerance. In this paper we test a fuzzy process capability index Cp, where instead of precise quality we have two membership functions for specification limits. Numerical examples are given to show the performance of the method.

In testing statistical hypotheses, as in other statistical problems, we may be confronted with fuzzy concepts. This paper deals with the problem of testing hypotheses, when the hypotheses are fuzzy and the data are crisp. We first introduce the notion of fuzzy p-value, by applying the extension principle and then present an approach for testing fuzzy hypotheses by comparing a fuzzy p-value and a fuzzy significance level, based on a comparison of two fuzzy sets. Numerical examples are also provided to illustrate the approach.

In this paper, on the basis of Zadeh’s probability measure of fuzzy events, the p-value concept is generalized for testing fuzzy hypotheses. We prove that the introduced p-value has uniform distribution over (0, 1) when the null fuzzy hypothesis is true. Then, based on such a p-value, a procedure is illustrated to test various types of fuzzy hypotheses. Several applied examples are given to show the performance of the method.

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.

Fuzzy set theory models situations in which the uncertainty is due to the non-precise (fuzzy) environment. One such case is testing the hypotheses problem where hypotheses are fuzzy rather than crisp and the data are crisp. Pais and Benton (1997) [1] present a suitable amount of cadmium absorption which is more appropriate for modeling by a fuzzy subset. In this paper, on real-world agricultural data generated by Ivani (2007) [2] , the suitableness of the mean absorption cadmium in the plant from the polluted soil is tested at the given significance level. Due to uncertainty in the suitable amount of cadmium absorption, in order to test this amount, we use fuzzy hypotheses testing instead of classical hypotheses testing. The fuzzy p -value approach is used for this test to conclude whether or not the mean absorption of cadmium coincides with the proposed amounts by Pais and Benton [1] . As expected, in fuzzy hypotheses testing, the degree of acceptance or rejection of the null fuzzy hypothesis is computed for each treatment of pollution. The data are from two plants: radish and cress, which are experimented on in soil polluted with CdNO3 salt. The results showed that using classical hypotheses testing may lead to contradictory decisions, and the proposed fuzzy hypotheses testing is a rational substitute for classical hypotheses testing when the environment is in a vague status.

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.