An improved attribute control chart for monitoring non-conforming proportion in high quality processes

Processes with very low rate of nonconformities are frequently observed in practice. These processes are known as “high quality processes”. Traditionally, the study of the rate of nonconformities was carried out using the conventional 3-sigma p control chart (Shewhart), constructed by the normal approximation. But this p chart suffers a serious inaccuracy in the modeling process and in control limits specification when the true rate of nonconforming items is small. This paper shows that, with simple adjustments to the control limits of the p-chart, achieving equal or even better improvement while still working on the original data scale, is feasible. In particular, an improved p chart which can provide a large improvement over the usual p chart for attributes is presented. This new chart, based on the Cornish–Fisher quantile correction, is also better than a previous simpler correction proposed in the literature. The improved p chart is compared with both, normal-based chart and modified p chart with one correction term and the benefits of including a new term of correction for monitoring high-quality processes is illustrated with real data.     

A demerit-fuzzy rating system,monitoring scheme and classification for manufacturing processes

For monitoring online manufacturing processes, the proportion of weights imposed on each type of product’s defects (nonconformities or demerits) has a profoundly effective impact on control charts’ performance. Apparently, the demerit-chart approach is superior than the widely-used c-chart scheme, because it allows us to place relative precise weights (real numbers) on defects according to their distinctly inferior degrees affecting the product quality so that the abnormal variations of processes can be literally exposed. However, in many applications, the seriousness of defects is evaluated partially or entirely by the inspectors’ perceptive judgement or knowledge, so with the precise-weight assignment, the demerit rating mechanism is considered to be somewhat constrained and subjective which inevitably leads to the targeted manufacturing process with limited and possibly biased information for online surveillance. To cope with the drawback, a demerit-fuzzy rating system and monitoring scheme is proposed in this paper. We first incorporate fuzzy weights (fuzzy numbers) to properly reflect the severity measures of defects which are categorized linguistically. Then, based on properties of fuzzy set theory and proposed approaches for fuzzy-number ranking, we develop the demerit-fuzzy charting scheme which is capable of discriminating process conditions into multi-intermittent statuses between in-control and out-of-control. This approach improves the traditional process control techniques with the binary-classification restraint for the process conditions. Finally, the proposed demerit-fuzzy rating system, monitoring scheme, and classification is elucidated by an application in garment industry to monitor textile-stitching nonconformities conditions.     

Circle formation of weak robots and Lyndon words

In this paper a discrete-time dynamic random graph process is studied that interleaves the birth of nodes and edges with the death of nodes. In this model, at each time step either a new node is added or an existing node is deleted. A node is added with probability p together with an edge incident on it. The node at the other end of this new edge is chosen based on a linear preferential attachment rule. A node (and all the edges incident on it) is deleted with probability q=1−p. The node to be deleted is chosen based on a probability distribution that favors small-degree nodes, in view of recent empirical findings. We analyze the degree distribution of this model and find that the expected fraction of nodes with degree k in the graph generated by this process decreases asymptotically as k−1−(2p/2p−1).     

Development of fuzzy process accuracy index for decision making problems

Several statistical decision making tools and methods are available to organize evidence, evaluate risks, and aid in decision making. Process capability indices are the summary statistics to point out the process performance. In this paper, these indices are analyzed to obtain a new decision making tool. Process accuracy index (C_a) measures the degree of process centering and gives alerts when the process mean departures from the target value. It focuses on the location of process mean and the distance between mean and target value. We modify the traditional process accuracy index to obtain a new tool under fuzziness. With the proposed tool, specification limits and process mean can be defined as triangular or trapezoidal fuzzy numbers. The proposed tool is illustrated to solve a supplier selection problem.     

基于数据间内在关联性的自适应模糊聚类模型

提出了一种新的模糊聚类模型(Fuzzy C-means clustering model, FCM), 称为自适应模糊聚类(Adaptive FCM, AFCM). 和现有的大多数模糊聚类方法不同的是, AFCM考虑了数据集中全体数据的内在关联性, 模型中引入了自适应度向量W和自适应指数p. 其中, W在迭代过程中是自适应的, p是一个给定参数. W和p共同作用调控聚类过程. AFCM同时输出三组参数: 模糊隶属度集U, 自适应度向量W, 以及聚类原型集V. 本文给出了两组数据实验验证AFCM的性能. 第1组实验验证AFCM的聚类性能, 以FCM为比较对象. 实验表明 AFCM可以得到更好的聚类质量, 而且通过合理选择自适应指数p, AFCM和FCM在时间复杂性上保持同一水平. 第2组实验检验了AFCM的离群点挖掘性能, 以目前常用的基于密度的LOF为比较对象. 实验表明AFCM算法具有极大的计算效率优势, 且AFCM得到的离群点是全局的, 反映的是离群点和整个数据集的关系, 离群点涵盖的信息也更丰富. 文章指出, AFCM在挖掘大数据集和实时数据中的离群点应用方面, 以及获得高质量的聚类结果的应用方面, 特别在聚类的同时需要挖掘离群点的应用方面具有独特的优势.     

Ranking L-R fuzzy number based on deviation degree

This paper proposed a novel approach to ranking fuzzy numbers based on the left and right deviation degree (L-R deviation degree). In the approach, the maximal and minimal reference sets are defined to measure L-R deviation degree of fuzzy number, and then the transfer coefficient is defined to measure the relative variation of L-R deviation degree of fuzzy number. Furthermore, the ranking index value is obtained based on the L-R deviation degree and relative variation of fuzzy numbers. Additionally, to compare the proposed approach with the existing approaches, five numerical examples are used. The comparative results illustrate that the approach proposed in this paper is simpler and better.     

Process capability analyses based on fuzzy measurements and fuzzy control charts

Process performance can be analyzed by using process capability indices (PCIs), which are summary statistics to depict the process location and dispersion successfully. Although they are very usable statistics, they have some limitations which prevent a deep and flexible analysis because of the crisp measurements and specification limits (SLs). If the specification limits or measurements are expressed by linguistic variables, traditional PCIs cause some misleading results. In this paper, the fuzzy set theory is used to add more information and flexibility to process capability analyses (PCA). For this aim, linguistic definition of the quality characteristic measurements are converted to fuzzy numbers and fuzzy PCIs are produced based on these measurements and fuzzy specification limits (SLs). Also fuzzy control charts are derived for fuzzy measurements of the related quality characteristic. They are used to increase the accuracy of PCA by determining whether or not the process is in statistical control. The fuzzy formulation of the indices C_p and C_pk, which are the most used two traditional PCIs, are produced when SLs and measurements are both triangular (TFN) and trapezoidal fuzzy numbers (TrFN). The proposed methodologies are applied in a piston manufacturer in Konya’s Industrial Area, Turkey.     

Some remarks on the lattice of fuzzy intervals

In this paper we study the connections between three related concepts which have appeared in the fuzzy literature: fuzzy intervals, fuzzy numbers and fuzzy interval numbers (FIN’s). We show that these three concepts are very closely related. We propose a new definition which encompasses the three previous ones and proceeds to study the properties ensuing from this definition. Given a reference lattice (X, ?), we define fuzzy intervals to be the fuzzy sets such that their p-cuts are closed intervals of (X, ?). We show that, given a complete lattice (X, ?), the collection of its fuzzy intervals is a complete lattice. Furthermore we show that, if (X, ?) is completely distributive, then the lattice of its fuzzy intervals is distributive. Finally we introduce a new inclusion measure, which can be used to quantify the degree in which a fuzzy interval is contained in another, an approach which is particularly valuable in engineering applications.     

Modelling time series when mean and variability both change

An extended least-squares method is described which allows us to model the location and scale of a process parametrically without specifying any parametric form for the distribution of the errors. The degree of the associated polynomials is chosen using a step-down method on a table of p-values. A pseudo-likelihood ratio test is given. The concepts of upper and lower return levels are extended to non-stationary series. The method is applied to annual means and extremes of Auckland temperatures. Evidence is found that the mean is changing non-linearly and the variance is also changing for all three series.     

Application of non-normal p-norm trapezoidal fuzzy number in reliability evaluation of electrical substations

This paper presented a non-normal p-norm trapezoidal fuzzy number–based fault tree technique to obtain the reliability analysis for substations system. Due to uncertainty in the collected data, all the failure probabilities are represented by non-normal p-norm trapezoidal fuzzy number. In this paper, the fault tree incorporated with the non-normal p-norm trapezoidal fuzzy number and minimal cut sets approach are used for reliability assessment of substations. An example of 66/11 kV substation is given to demonstrate the method. Further, fuzzy risk analysis problems are described to find out the probability of failure of each components of the system using linguistic variables, which could be used for managerial decision making and future system maintenance strategy.     

Mining fuzzy β-certain and β-possible rules from quantitative data based on the variable precision rough-set model

The rough-set theory proposed by Pawlak, has been widely used in dealing with data classification problems. The original rough-set model is, however, quite sensitive to noisy data. Ziarko thus proposed the variable precision rough-set model to deal with noisy data and uncertain information. This model allowed for some degree of uncertainty and misclassification in the mining process. Conventionally, the mining algorithms based on the rough-set theory identify the relationships among data using crisp attribute values; however, data with quantitative values are commonly seen in real-world applications. This paper thus deals with the problem of producing a set of fuzzy certain and fuzzy possible rules from quantitative data with a predefined tolerance degree of uncertainty and misclassification. A new method, which combines the variable precision rough-set model and the fuzzy set theory, is thus proposed to solve this problem. It first transforms each quantitative value into a fuzzy set of linguistic terms using membership functions and then calculates the fuzzy β-lower and the fuzzy β-upper approximations. The certain and possible rules are then generated based on these fuzzy approximations. These rules can then be used to classify unknown objects. The paper thus extends the existing rough-set mining approaches to process quantitative data with tolerance of noise and uncertainty.     

Soft computing and cooperative strategies for optimization

In the last years, metaheuristics have emerged as powerful algorithmic approaches which have been applied with great success to difficult combinatorial optimization problems. However, this does not mean that metaheuristics can be applied blindly to any new problem.In this contribution we showed how the most basic ingredients of Soft Computing, namely fuzzy sets and fuzzy rules, are used in the context of a simple metaheuristic and a cooperative strategy based on it, to obtain successful results for the p-median problem.     

考虑属性权重优化的犹豫模糊多属性决策方法

针对属性权重完全未知的犹豫模糊多属性决策问题, 提出一种属性权重多目标优化方法. 首先, 根据属性值的均值、方差以及属性间的关联度建立属性权重确定模型; 然后, 利用方案与犹豫模糊正理想点的相似度对方案进行排序; 最后, 通过算例分析表明了所提出方法的有效性和可行性.

     

基于蕴涵算子Lp的模糊推理的FMP反向三I算法

研究了基于蕴涵算子L_p模糊推理的FMP反向三I支持算法及α-反向三I支持算法,给出了FMP模型的反向三I算法及α-反向三I算法的计算公式。     

Fuzzy random renewal reward process and its applications

This paper studies a renewal reward process with fuzzy random interarrival times and rewards under the ?-independence associated with any continuous Archimedean t-norm ?. The interarrival times and rewards of the renewal reward process are assumed to be positive fuzzy random variables whose fuzzy realizations are ?-independent fuzzy variables. Under these conditions, some limit theorems in mean chance measure are derived for fuzzy random renewal rewards. In the sequel, a fuzzy random renewal reward theorem is proved for the long-run expected reward per unit time of the renewal reward process. The renewal reward theorem obtained in this paper can degenerate to that of stochastic renewal theory. Finally, some application examples are provided to illustrate the utility of the result.     

Parametric testing statistical hypotheses for fuzzy random variables

In this paper, a method is proposed for testing statistical hypotheses about the fuzzy parameter of the underlying parametric population. In this approach, using definition of fuzzy random variables, the concept of the power of test and p value is extended to the fuzzy power and fuzzy p value. To do this, the concepts of fuzzy p value have been defined using the \(\alpha \)-optimistic values of the fuzzy observations and fuzzy parameters. This paper also develop the concepts of fuzzy type-I, fuzzy type-II errors and fuzzy power for the proposed hypothesis tests. To make decision as a fuzzy test, a well-known index is employed to compare the observed fuzzy p value and a given significance value. The result provides a fuzzy test function which leads to some degrees to accept or to reject the null hypothesis. As an application of the proposed method, we focus on the normal fuzzy random variable to investigate hypotheses about the related fuzzy parameters. An applied example is provided throughout the paper clarifying the discussions made in this paper.     

基于优选小波基和模糊SOM网络的模拟电路故障诊断

为了解决模拟电路故障诊断中的特征提取困难并实现对模拟电路故障模式准确的分类,提出一种优选小波基、模糊理论和自组织特征映射网络(SOM,self-organizing feature map)相结合的模拟电路故障诊断方法.该方法首先对模拟电路故障响应信号进行小波分解、提取能量值、均值和方差组成输入特征向量,同时采用余弦分离度评价小波变换在不同小波基函数下获取故障特征的有效性,据此选择余弦分离度最小的小波基分解的特征向量输入到自组织特征映射网络进行故障分类.仿真实验表明,利用余弦分离度选择的最优小波基能有效提高模拟电路故障特征提取,模糊神经网络能对故障模式进行精确分类.     

Feature extraction based on fuzzy class mean embedding (FCME) with its application to face and palm biometrics

In the local discriminant embedding (LDE) framework, the neighbor and class of data points were used to construct the graph embedding for classification problems. From a high-dimensional to a low-dimensional subspace, data points of the same class maintain their intrinsic neighbor relations, whereas neighboring data points of different classes no longer stick to one another. However, face images are always affected by variations in illumination conditions and different facial expressions in the real world. So, distant data points are not deemphasized efficiently by LDE and it may degrade the performance of classification. In order to solve above problems, in this paper, we investigate the fuzzy set theory and class mean of LDE, called fuzzy class mean embedding (FCME), using the fuzzy k-nearest neighbor (FKNN) and the class sample average to enhance its discriminant power in their mapping into a low dimensional space. In the proposed method, a membership degree matrix is firstly calculated using FKNN, then the membership degree and class mean are incorporated into the definition of the Laplacian scatter matrix. The optimal projections of FCME can be obtained by solving a generalized eigenfunction. Experimental results on the Wine dataset, ORL, Yale, AR, FERET face database and PolyU palmprint database show the effectiveness of the proposed method.     

Neuro-fuzzy model based on digital images for the monitoring of coffee bean color during roasting in a spouted bed

An adaptive-network-based fuzzy inference system based on color image analysis was used to estimate coffee bean moisture content during roasting in a spouted bed. The neuro-fuzzy model described the grain moisture changes as a function of brightness (L^*), browning index (BI) and the distance to a defined standard (ΔE). An image-capture device was designed to monitor color variations in the L^*a^*b^* space for high temperatures samples taken from the reactor. The proposed model was composed of three Gaussian-type fuzzy sets based on the scatter partition method. The neuro-fuzzy model was trained with the Back-propagation algorithm using experimental measurements at three air temperature levels (400, 450 and 500 °C). The performance of the neuro-fuzzy model resulted better compared to conventional methods obtaining a coefficient of determination > 0.98, a root mean square error < 0.002 and a modified Schwarz–Rissanen information criterion < 0. The simplicity of the model and its robustness against changes in the input variables make it suitable for monitoring on-line the roasting process.     

Simultaneous process mean and variance monitoring using artificial neural networks

Control chart patterns (CCPs) can be employed to determine the behavior of a process. Hence, CCP recognition is an important issue for an effective process-monitoring system. Artificial neural networks (ANNs) have been applied to CCP recognition tasks and promising results have been obtained. It is well known that mean and variance control charts are usually implemented together and that these two charts are not independent of each other, especially for the individual measurements and moving range (X–R_m) charts. CCPs on the mean and variance charts can be associated independently with different assignable causes when corresponding process knowledge is available. However, ANN-based CCP recognition models for process mean and variance have mostly been developed separately in the literature with the other parameter assumed to be under control. Little attention has been given to the use of ANNs for monitoring the process mean and variance simultaneously. This study presents a real-time ANN-based model for the simultaneous recognition of both mean and variance CCPs. Three most common CCP types, namely shift, trend, and cycle, for both mean and variance are addressed in this work. Both direct data and selected statistical features extracted from the process are employed as the inputs of ANNs. The numerical results obtained using extensive simulation indicate that the proposed model can effectively recognize not only single mean or variance CCPs but also mixed CCPs in which mean and variance CCPs exist concurrently. Empirical comparisons show that the proposed model performs better than existing approaches in detecting mean and variance shifts, while also providing the capability of CCP recognition that is very useful for bringing the process back to the in-control condition. A demonstrative example is provided.