Process Capability indices (PCI) show that the Process conforms to the specification limits,when the product quality depends on more than one characteristic, Multivariate Process Capability indices (MCPI) are used. By modifying the Process capability indices, the Process incapability indices are created,these indices then provide information about the accuracy and precision of the Process separately. In the real world, in most cases, the parameters cannot be specified precisely,therefore, the use of fuzzy sets can solve this problem in statistical quality control. The purpose of this paper is to present, for the first time, a Multivariate Process incapability Index by considering the measurement error in a fuzzy environment. The presented index is shown for practical examples solved by considering Triangular Fuzzy Numbers,then the capability of the model is compared to the time when fuzzy logic is not used. The obtained results emphasize that ignoring the measurement error also leads to the incorrect calculation of Process capability, causing a lot of damage to manufacturing industries, especially high-tech ones.

Abstract:  In some practical applications, the quality of a product or Process is defined by a profile, which is a relationship between a response variable and one or more explanatory variables. Simple linear profiles (SLPs) are one of the various types of profiles in which the product or Process quality is related to a simple linear function between a response and an explanatory variable. In this article, a functional capability index for a simple linear profile with asymmetric tolerance is introduced. The performance of the proposed index and existing ones (  and ) are studied using numerical examples and simulation studies in terms of mean absolute error (MAE), mean square error (MSE) and absolute percentage error (APE) metrics. The results show that the new index performs better than the two existing indices. Furthermore, confidence intervals for the proposed index are constructed using three bootstrap methods, and their performance is evaluated using simulation studies. A real-world case study is presented to demonstrate the application of the proposed index.

How to profoundly change the beliefs of the polytheists during the age of revelation from the denial of the Prophet (pbuh) to the certainty of his Message and the acceptance of his full guardianship is a question worthy of attention. The order of verses about prophecy based on the order of revelation is the reason to lead a transformative Process and a special Quranic method in institutionalizing the belief of prophecy in the lives of Arabs. In the first stage, the Holy Quran in the first Makkī (Arabic: المکّیّ, suras revealed in Mecca) chapters has prepared the community from the motivational side to accept the truth of the Prophet’s (pbuh) prophecy and briefly stating the key issues surrounding it. The Holy Book of Quran at the time of the emergence of oppositions and doubts (the second stage) destroyed the arguments of the polytheists and explained the prophecy and proved its authenticity from a cognitive perspective in the second stage by presenting clear arguments. and the Holy Book of Quran in the third stage after Madanī (Arabic: المدنیّ, suras revealed in Medina) period has undertaken measures in order to fulfill the goals of the Massage by explaining the true position of the Prophet (pbuh) to deepen and operationalize the Prophetic belief through explaining practical duties for the Prophet (pbuh) and behavior based on the acceptance of Walaya (Arabic: ولایة, meaning “guardianship” or “governance”).

A profile is a relationship between a response variable and one or more independent variables being controlled during the time. Process Capability indices (PCI) are measured to evaluate the performance of Processes in producing conforming products. Despite frequent applications of profile and a variety of available methods to monitor its different types, little researches have been carried out on determining capability indices of profile Process. PCIs such as P C and PK C in profile state, are used to evaluate Process capability in producing conforming profiles. This paper presents a functional approach for nonlinear profiles which usually expressed as nonlinear regression. Thus, functions such as pertaining technical specification limits, mean and natural tolerance limits are determined as nonlinear profiles and also functional limit is applied to determine Functional Capability indices (FCI) P C and PK C of functional nonlinear profile. Easy calculation and the ability to calculate FC in each period and at each point are from the advantages of this method over the other methods.

Process capability indices show the ability of a Process to produce products according to the pre-specified requirements. Since final quality characteristics of a product are usually interrelated to its previous amounts in earlier workstations, one need to model and consider the relationship among them to assess the Process capability properly. Hence, conducting Process capability analysis in multivariate environment is inevitable; unfortunately, the analysis in multivariate environment is usually complex and requires extensive calculations. Sometimes it is preferable to simplify the analysis by assuming independency among quality characteristics and evaluating Process performance with respect to each individual quality characteristic using univariate Process capability indices such as CP, CPK, CPM, and CPMK. However, this simplification introduces some error in the analysis leading to under or overestimation of the Process capability index. This paper models the interrelationship among quality characteristics that are produced in different workstations to provide an overall Process capability index. Ridge residual regression is used as a vehicle to evaluate Process capability and helps quality engineers to provide a reasonable quality policy for controlling and reducing variation in quality characteristics.

The notion of fuzzy Process capability indices is studied by parchami et al. [1], were the specification limits are triangular fuzzy numbers. In this note, their results are revised for the general case, where the specification limits are L-R fuzzy intervals.