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The EM algorithm is a powerful tool and generic useful device in a variety of problems for maximum likelihood estimation with incomplete data which usually appears in practice. Here, the term ``incomplete" means a general state and in different situations it can mean different meanings, such as lost data, open source data, censored observations, etc. This paper introduces an application of the EM algorithm in which the meaning of ``incomplete" data is non-precise or fuzzy observations. The proposed approach in this paper for estimating an unknown parameter in the parametric statistical model by maximizing the likelihood function based on fuzzy observations. Meanwhile, this article presents a case study in the electronics industry, which is an extension of a well-known example used in introductions to the EM algorithm and focuses on the applicability of the algorithm in a fuzzy environment. This paper can be useful for graduate students to understand the subject in fuzzy environment and moreover to use the EM algorithm in more complex examples.

‎Process capability indices are used widely throughout the world to give a quick indication of a process capability in a format that is easy to use and understand‎. ‎A process capability index $C_p$ that constructed for measuring the quality is an effective tool for assessing process capability‎, ‎since this index can reflect whether a centering process is capable of reproducing items meeting the specifications limits‎. ‎The minimax approach is proposed in this paper for testing capability on the basis of precision index Cp when the producer goal is avoiding the largest possible risk‎. ‎Motivations and benefits of proposing minimax approach are discussed for capability test‎. ‎Also‎, ‎the proposed method clarified by an industrial application.

Such as other statistical problems, we may confront with uncertain and fuzzy concepts in quality control. One particular case in process capability analysis is a situation in which specification limits are two fuzzy sets. In such a uncertain and vague environment, the produced product is not qualified with a two-valued Boolean view, but to some degree depending on the decision-maker strictness and the quality level of the produced product. This matter can be cause to a rational decision-making on the quality of the production line. First, a comprehensive approach is presented in this paper for modeling the fuzzy quality concept. Then, motivations and advantages of applying this flexible approach instead of using classical quality are mentioned.

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.