Fuzzy Process Capability Indices Research Articles

Fuzzy control charts for individual observations to analyze variability in health monitoring processes

In healthcare monitoring, the quality of medical services and patient outcomes are significantly influenced by the unnatural variations emphasizing the importance of effective control and monitoring strategies. In such scenarios the quality is compromised when a shift in the medical process is not detected timely. In this study, we commenced a comprehensive analysis of the variability present in the patient's hematocrit levels. We introduced a novel approach using fuzzy control charts for individual measurements to detect shift within hematocrit levels effectively. For the initial forecasting of hematocrit level variability, we employed the exponential smoothing method by using R software. Following this, we proposed a set of fuzzy control charts intended for individual measurements, including the fuzzy moving average control chart, fuzzy weighted moving control chart, and fuzzy moving range control chart. These control charts are defined with fuzzy control rules, allowing them to analyze and interpret the small shifts in the process precisely. Also, to measure the process's capability for providing a deeper understanding of the processes, fuzzy process capability indices are introduced. A Monte Carlo simulation approach is employed to obtain the performance metrics. Finally, a case study sourced from Kaggle was conducted to evaluate the performance of the proposed fuzzy control charts for assessing hematocrit levels.

Design and analysis of Cpm and Cpmk indices for uncertainty environment by using two dimensional fuzzy sets

Process capability analysis (PCA) is an important stage to check variability of process by using process capability indices (PCIs) that are very effective statistics to summarize process’ performance. Traditional PCIs can produce some incorrect results and declare misinterpretation about process’ quality if the process includes uncertainties. Additionally, definitions of process’ parameters with exact values is not possible when there are uncertainty caused by measurement errors, sensitivities of measuring instruments or quality engineers’ hesitancies. Although the fuzzy set theory (FST) has been successfully used in PCA, it is the first time to use of Pythagorean fuzzy sets (PFSs) to model uncertainties of process more than traditional fuzzy sets in PCA. Since the PFSs has two-dimensional configurations by defining membership and non-membership values, they also have a huge ability to model uncertainty that arises from the human’s thinking and hesitancies, and has brought flexibility, sensitivity and reality for PCA. In this paper, specification limits (SLs), mean (μp), standard deviation (σ) and target value (T) main parameters of PCIs have been analyzed by using PFSs and Pythagorean fuzzy process capability indices (PFPCIs) for two well-known PCIs such as ( C ˜ pm ) and ( C ˜ pmk ) have been derived. The Pythagorean ( C ˜ pm ) and ( C ˜ pmk ) indices have also been applied and tested on some numerical examples based on real case applications from manufacturing industry. The obtained results show that PFPCIs provide wider knowledge about capability of process and to obtain more realistic results. As a result of considering all possibilities about the process, it has been concluded that the process is incapable. In light of this information, the results obtained using different fuzzy set extensions for (Cpm) and (Cpmk) indices can be compared.

A Novel Approach of Fuzzy Control Chart with Fuzzy Process Capability Indices Using Alpha Cut Triangular Fuzzy Number

Statistical Quality Control is a valuable strategy that applies to the statistical technique for monitoring a manufacturing system under particular situations. On the other hand, the fuzzy set theory is an ideal instrument to cope with an unclear situation. The existing studies are restricted, and there is still mystery behind the unclear data. This paper deals with technique: namely, the fuzzy control chart based on fuzzy process capability indices (FCPI) using triangular fuzzy numbers (TFNs). Alpha cut theory is applied in statistical quality control for fuzzy process control industrial application. This is a five-phase study that deals with the control chart using capability indices. The numerical example is also performed using the proposed technique. This paper would help to better assess/understand the manufacturing system data and would explore the application of the fuzzy control techniques.

A Non-Invasive Method to Evaluate Fuzzy Process Capability Indices via Coupled Applications of Artificial Neural Networks and the Placket–Burman DOE

The capability analysis of a process against requirements is often an instrument of change. The traditional and fuzzy process capability approaches are the most useful statistical techniques for determining the intrinsic spread of a controlled process for establishing realistic specifications and use for comparative processes. In the industry, the traditional approach is the most commonly used instrument to assess the impact of continuous improvement projects. However, these methods used to evaluate process capability indices could give misleading results because the dataset employed corresponds to the final product/service measures. This paper reviews an alternative procedure to assess the fuzzy process capability indices based on the statistical methodology involved in the modeling and design of experiments. Firstly, a model with reasonable accuracy is developed using a neural network approach. This model is embedded in a graphic user interface (GUI). Using the GUI, an experimental design is carried out, first to know the membership function of the process variability and then include this variability in the model. Again, an experimental design identifies the improved operating conditions for the significative independent variables. A new dataset is generated with these operating conditions, including the minimum error reached for each independent variable. Finally, the GUI is used to get a new prediction for the response variable. The fuzzy process capability indices are determined using the triangular membership function and the predicted response values. The feasibility of the proposed method was validated using a random data set corresponding to the basis weight of a papermaking process. The results indicate that the proposed method provides a better overview of the process performance, showing its true potential. The proposed method can be considered non-invasive.

Development of a Fuzzy Exponentially Weighted Moving Average Control Chart with an α-level Cut for Monitoring a Production Process

Statistical quality control is a useful approach that applies to statistical techniques for monitoring a production system. These charts are effective to monitor the process under certain conditions. On the other hand, the fuzzy set theory is an appropriate tool to deal with an uncertain situation. This paper is fourfold. First of all, triangular fuzzy numbers with an α-level cut technique are used for each sample. The α-level cut technique is sensitive to satisfy the process requirement. Second, a fuzzy exponentially weighted moving average (FEWMA) control chart is proposed with the α-level cut technique. The proposed FEWMA detects small shifts under uncertain situations while using a unity technique for samples. Third, the fuzzy target-focused process capability index (FCpm) index is proposed to measure the fuzzy process performance. Then, a case study is presented to monitor a pumice block plant using the FEWMA control chart with the α-level cut and measure the process performance with the FCpm index. Comparative studies are also presented. By using the proposed FEWMA control chart with the α-level cut, the accuracy and the flexibility of control specification limits are reported for the case study.

Fuzzy process capability indices for simple linear profile

Process capability indices are numerical tools that quantify how well a process can meet customer requirements, specifications or engineering tolerances. Fuzzy logic is incorporated to deal imprecise, incomplete data along with uncertainty. This paper develops two fuzzy methods for measuring the process capability in simple linear profiles for the circumstances in which lower and upper specification limits are imprecise. To guide practitioners, numerical example is provided.

Fuzzy process capability indices based on imprecise observations induced from non-normal distributions

This study has developed an approach to statistical process control indices for imprecise observations induced from an arbitrary population. For this purpose, most popular non-normal process capability index was extended based on a proposed triangular fuzzy distance between fuzzy numbers. The specific limits and targets are also considered as fuzzy numbers in this method. The order between the proposed fuzzy process capability indices was also investigated. A degree of belonging was proposed to verify the degree of process conditions of the proposed fuzzy process capability indices. The feasibility and effectiveness of the proposed method were also examined using a simulation study.

A New Fuzzy Process Capability Index for Asymmetric Tolerance Interval

Process capability indices are used to evaluate the performance of the manufacturing process. When the specification limits and the target value are not precise, the authors cannot use the traditional methods to assess the capability of the process. For the processes with asymmetric tolerance intervals, some fuzzy process capability indices have been introduced such as and . In some cases, these indices may fail to account the process performance. To overcome the problem with them, the authors propose two new fuzzy indices in the case that the specification limits and the target value are fuzzy while the data are crisp. Also, the authors present an application example to demonstrate effectiveness and performance of the proposed indices.

Process Capability Analysis Using Interval Type-2 Fuzzy Sets

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.

Introducing Triangular Fuzzy Process Capability Indices to Evaluate Performance of Continuous Production Process

Introducing Triangular Fuzzy Process Capability Indices to Evaluate Performance of Continuous Production Process

Process Capability Indices as Fuzzy Numbers

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.

Measuring Loss-Based Process Capability IndexLeand Its GenerationLe′′with Fuzzy Numbers

Loss-based process capability indices are appropriate and realistic tools in order to measure the process capability. Among them, indexLeand its generationLe′′are well-known loss-based process capability indices, whose concepts are based on the worth (the opposite concept of loss). Sometimes, in order to calculateLeandLe′′there are some uncertainties in observations, so fuzzy logic can be employed to manage the uncertainties. This paper investigates fuzzification of process capability indexL~eand its generationL~e′′. In order to find the membership function of process capability indicesL~eandL~e′′, theα-cuts of fuzzy observation were employed. Then with an example of fuzzy process capability index,L~eandL~e′′were calculated and compared. Results showed that fuzzyL~e′′was more sensitive compared withL~eand was increased while the target departs (asymmetric tolerance). This example also showed that, with departure from the target, variation of fuzzyL~e′′and consequently its fuzziness were increased.

Type-2 fuzzy process capability indices for non-normal processes

Type-2 fuzzy process capability indices for non-normal processes

A fuzzy Taguchi loss-based process capability index

Process capability indices are appropriate tools to measure the inherent capability of a process, but most of them do not take into account the losses of a process. By integrating the Taguchi loss function and a process capability index (Cpk), a new process capability index called fuzzy Taguchi loss-based process capability index (FTPCI) was developed. Due to the definitions of a capable process or value of standard loss, the fuzzy approach to the result of the index Cpk and Taguchi losses have been proposed. In order to construct this new fuzzy process capability index, Mamdani fuzzy system consisting of the Taguchi loss and the index Cpk was employed. Finally, an example on the usage of this process capability index illustrated the results.

Performance evaluation of continuous production process by trapezoidal fuzzy process capability indices

Process capability indices (PCIs) can be viewed as the effective and excellent means of measuring product quality and process performance. They are very useful statistical analysis tools to summarise process dispersion and location by using process capability analysis (PCA). However, they have some limitations which prevent a deep and flexible analysis because of the crisp measurements and specification limits (SLs). Fuzzy set theory can be used to add more flexibility to process capability analyses. In this study, the fuzzy and control charts are used to analyse the process of continuous production in terms of trapezoidal fuzzy numbers (TrFN). Also, fuzzy PCIs are produced when SLs and measurements are presented as trapezoidal fuzzy numbers. For this aim, a computer program is coded in MATLAB software. The fuzzy control charts are utilised in Yazd Fiber Production Plant. The results show that in continuous production processes, fuzzy measurements can yield a more accurate analysis. Furthermore, relying on the fuzzy capability indices, one can have a flexible analysis of the process performance.

Reply to “open problem of fuzzy confidence interval for fuzzy process capability index”

The recent paper by Parchami et al. [2] proposes an open problem: Let $ \hat{\tilde{C\!}}_p = T\left {\frac{{a_u-c_l }} {{6s}},\frac{{b_u-b_l }} {{6s}},\frac{{c_u-a_l }} {{6s}}} \right\$ be a point estimate of fuzzy process capability index $\\tilde{C}_p\$ as definition of Parchami et al. [2], where $ s = \sqrt {\frac{1}{{n-1}}\sum\limits_{i = 1}^n {\left {x_i-\bar x} \right^2}}\$. Is it true that: $ \mathop {\lim }\limits_{n \to \infty } \left[ {\hat{\tilde{C\!}}_{p \otimes } \sqrt {\frac{{\chi _{n-1,\alpha /2}^2 }} {{n-1}}},\hat{\tilde{C\!}}_{p \otimes } \sqrt {\frac{{\chi _{n-1,1-\alpha /2}^2 }} {{n-1}}}} \right] = \left\{ {\hat{\tilde{C\!}}_p } \right\}?\$ We modify their open problem and prove that “$ \mathop {\lim }\limits_{n \to \infty } \left[ {\hat{\tilde{C\!}}_{p \otimes } \sqrt {\frac{{\chi _{n-1,\alpha /2}^2 }} {{n-1}}},\hat{\tilde{C\!}}_{p \otimes } \sqrt {\frac{{\chi _{n-1,1-\alpha /2}^2 }} {{n-1}}} } \right] \cong \left\{ {\hat{\tilde{C\!}}_p } \right\}\$ {for large} n” is true.

The evaluation of educational systems: an application study

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.

Confidence Interval for Cpm Based on Dp,q Distance

Problem statement: A measurement control system ensures that measuring equipment and measurement processes are fit for their intended use and its importance in achieving product quality objectives. Approach: The manufacturing industries have been making an extensive effort to implement Statistical Process Control (SPC) in their plants and supply bases. Results: Capability indices derived from SPC have received increasing usage not only in capability assessments, but also in the evaluation of purchasing decisions. In most real life applications, real observations of continuous quantities are not precise numbers; in practice, they are more or less imprecise. Since observations of continuous random variables are imprecise the values of related test statistics become imprecise. In some cases Specification Limits (SLs) are not precise numbers and they are expressed in fuzzy terms, so that the classical capability indices could not be applied. Conclusion/Recommendations: In this study we obtain 100(1-α)% fuzzy confidence interval for Cpm fuzzy process capability index, where instead of precise quality we have two membership functions for specification limits.

Fuzzy confidence regions for the Taguchi capability index

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) , and 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 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 over the other two fuzzy indices, and , 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 fuzzy confidence regions for the fuzzy PCI . A numerical example is given to show the performance of the method.

A New Reliability Index Based On Fuzzy Process Capability Index for Travel Time in Multimodal Networks

The variability in travel times is a very important feature in determining the arrival of the passenger at the destination within a given travel time threshold and can be noted as one of the most important characteristic for travel time reliability that is an increasing concern of travelers, because it allows travelers to make better use of their own time. In this paper reliability in multimodal travel by consideration of focuses on the passenger travel times is investigated. For this aim, the variations passengers encounter in a multimodal network with respect to the capabilities of different services is analyzed. In order to identify the sources of variations and improve the service in an integrated manner a methodology based on process capability indices is presented. The fuzzy set theory is also integrated with process capability indices to increase the sensitiveness and flexibility of route and station reliability and to ease definition of specification limits. The proposed methodology is applied in Istanbul to investigate the reliability of travel times of multimodal travels of passengers using a specific subway line.