Economic design of variable sampling intervals T2 control charts using genetic algorithms

Control charting is a graphical expression and operation of statistical hypothesis testing. In this paper, we develop the economic design of the variable sampling intervals (VSI) T² control chart to determine the values of the five test parameters of the chart (i.e. the sample size, the long sampling interval, the short sampling interval, the warning limit, and the control limit) such that the expected total cost, associated with the test procedure, is minimized. The genetic algorithm (GA) is employed to search for the optimal values of the five test parameters of the VSI T² chart, and an example is provided to illustrate the solution procedure. Sensitivity analysis is then carried out to investigate the effects of model parameters on the solution of the economic design.     

Integrated inventory control and inspection policies with deterministic demand

In this paper, we develop integrated inventory inspection models with and without replacement of nonconforming items. Inspection policies include no inspection, sampling inspection, and 100% inspection. We consider a buyer who places an order from a supplier. When a lot is received, the buyer uses some type of inspection policy. The fraction nonconforming is assumed to be a random variable following a beta distribution. Both the order quantity and the inspection policy are decision variables. In the inspection policy involving determining sampling plan parameters, constraints on the buyer and manufacturer risks are set in order to obtain a fair plan for both parties. A solution procedure for determining the operating policies for inventory and inspection consisting of order quantity, sample size, and acceptance number is proposed. Numerical examples are presented to conduct a sensitivity analysis for important model parameters and to illustrate important issues about the developed models.     

可变抽样区间二元双抽样广义方差控制图

将可变抽样区间特性融入二元双抽样广义方差控制图,构建可变抽样区间二元双抽样广义方差控制图及其监控方法.采用马尔科夫链方法完成控制图调整的平均报警时间和平均报警时间性能指标的计算,进而建立用控制图参数设计的优化模型,并通过遗传算法完成模型求解.通过与可变抽样区间的二元广义方差合成控制图、二元双抽样广义方差控制图以及基于Cornish-Fisher修正的二元双抽样广义方差控制图的判异性能对比,验证该控制图的优势,并通过加工实例对性能优势进行进一步说明.     

A combined synthetic and np scheme for detecting increases in fraction nonconforming

The applications of attribute control charts cover a wide variety of manufacturing processes in which quality characteristics cannot be measured on a continuous numerical scale or even a quantitative scale. The np control chart is an attribute chart used to monitor the fraction nonconforming p of a process. This chart is effective for detecting large process shifts in p. The attribute synthetic chart is also proposed to detect p shifts. It utilizes the information about the time interval or the Conforming Run Length (CRL) between two nonconforming samples. During the implementation of a synthetic chart, a sample is classified as nonconforming if the number d of nonconforming units falls beyond a warning limit. Unlike the np chart, the synthetic chart is more powerful to detect small and moderate p shifts. This article proposes a new scheme, the Syn-np chart, which comprises a synthetic chart and an np chart. Since the Syn-np chart has both the strength of the synthetic chart for quickly detecting small p shifts and the advantage of the np chart of being sensitive to large p shifts, it has a better and more uniform overall performance. Specifically, it is more effective than the np chart and synthetic chart by 73% and 31%, respectively, in terms of Weighted Average of Average Time to Signal (WAATS) over a wide range of p shifts under different conditions.     

Comparison of control charts for low defective rate

It is well known that the conventional p control chart constructed by the normal approximation for the binomial distribution suffers a serious inaccuracy in the monitor process when the true rate of nonconforming items is small. A similar problem also arises in the binomial confidence interval estimation. Adjusted confidence intervals are established in the literature to improve the coverage probability when the binomial proportion is small. In this paper, a new p control chart based on an adjusted confidence interval is established, which can substantially improve the existing control charts when the nonconforming rate is small.     

A variable parameter Shewhart control scheme for joint monitoring of process mean and variance

This paper presents a new Statistical Process Control model for the economic optimization of a variable-parameter control chart monitoring a process operation where two assignable causes may occur, one affecting the mean and the other the variance of the process. Therefore, it is possible for the process to operate in statistical control, when none of the two assignable causes has occurred, or under the effect of one or both the assignable causes. By making the assumption that the occurrence rate of each assignable cause is exponential, a Markov chain approach is utilized to determine the probabilities that the process operates at any of the above possible states. The model uses an economic (or an economic/statistical) optimization criterion for the time to the next sampling instance, the size of the next sample, as well as the control limits of the inspection. That is, all design parameters of the control scheme are selected so as to minimize the total expected quality-related costs. The superiority of the proposed model is estimated by comparing its expected quality control cost vs. the outcome of the Fp (Fixed-parameter) Shewhart control chart, the Variable Sample Size (VSS) control chart, the Variable Sampling Interval (VSI) and the Variable Sample Size and Sampling Interval (VSSI) control chart, for a benchmark of examples. The numerical investigation indicates that the economic improvement of the proposed model may be significant.     

A variable parameter Shewhart control scheme for joint monitoring of process mean and variance

This paper presents a new Statistical Process Control model for the economic optimization of a variable-parameter control chart monitoring a process operation where two assignable causes may occur, one affecting the mean and the other the variance of the process. Therefore, it is possible for the process to operate in statistical control, when none of the two assignable causes has occurred, or under the effect of one or both the assignable causes. By making the assumption that the occurrence rate of each assignable cause is exponential, a Markov chain approach is utilized to determine the probabilities that the process operates at any of the above possible states. The model uses an economic (or an economic/statistical) optimization criterion for the time to the next sampling instance, the size of the next sample, as well as the control limits of the inspection. That is, all design parameters of the control scheme are selected so as to minimize the total expected quality-related costs. The superiority of the proposed model is estimated by comparing its expected quality control cost vs. the outcome of the Fp (Fixed-parameter) Shewhart control chart, the Variable Sample Size (VSS) control chart, the Variable Sampling Interval (VSI) and the Variable Sample Size and Sampling Interval (VSSI) control chart, for a benchmark of examples. The numerical investigation indicates that the economic improvement of the proposed model may be significant.     

Adaptive CUSUM control chart with variable sampling intervals

The standard cumulative sum chart (CUSUM) is widely used for detecting small and moderate process mean shifts, and its optimal detection ability for any pre-specified mean shift has been demonstrated by its equivalence to continuous sequential tests. In real practice, the assumption of knowing the true mean shift in prior cannot be always met. So it is desirable to design a procedure that is efficient for detecting a range of future expected but unknown mean shifts. Adaptive CUSUM control chart, which can continuously adjust itself by a one-step forecasting operator, has been proposed to detect efficiently and robustly for a range of mean shifts in the early literature. Moreover, in terms of sampling time to signal, control chart with the VSI (variable sampling intervals) feature can detect the process changes more quickly than the traditional FSI (fixed sample intervals) chart. In this paper, a new CUSUM control chart which is based on both adaptive and VSI features is discussed. Also, a two-dimensional Markov chain model is developed to evaluate its run-time performance.     

Synthetic double sampling np control chart for attributes

The quality of a product, based on the number of non-conforming items can be controlled using the np chart. This paper proposes a synthetic double sampling (DS) np chart which comprises two sub-charts, i.e. the DS np and conforming run length (CRL) sub-charts. For the zero-state case, the synthetic DS np chart surpasses its standard counterpart, i.e. the synthetic np and the basic DS np chart, and other np type charts like the standard np, combined synthetic and np (Syn-np), variable sample size (VSS) np, exponentially weighted moving average (EWMA) np and cumulative sum (CUSUM) np charts, for detecting increases in the fraction of non-conforming items p, for most shift sizes. The synthetic DS np chart also performs reasonably well in the steady-state case in comparison with other charts mentioned above. Thus, among the competing charts, the synthetic DS np chart stands out as one of the best charts.     

A flexible process-capability-qualified resubmission-allowed acceptance sampling scheme

This paper introduces a new variables-acceptance-sampling scheme for resubmitted lots, based on process-capability-index (C_pmk) sampling information. The scheme competently evaluates both the process yield and the potential process loss of the submitted lots. Vital criteria and decision rules, by which inspected lots are approved in the resubmitted sampling strategy, include required sample size for inspection, critical acceptance levels stipulated for quality standards, and risks to producers and consumers. To obtain these vital criteria, the operating function of the proposed sampling scheme is derived based on the exact sampling distribution of the C_pmk estimator. In terms of the given rules and criteria, the resubmitted sampling plans provide greater insights than traditional single sampling plans. Finally, our proposed process-capability-qualified resubmission-allowed sampling strategy is evaluated on an industrial example.     

A new intelligent SOFM-based sampling plan for advanced process control

Sample measurement inspecting for a process parameter is a necessity in semiconductor manufacturing because of the prohibitive amount of time involved in 100% inspection while maintaining sensitivity to all types of defects and abnormality. In current industrial practice, sample measurement locations are chosen approximately evenly across the wafer, in order to have all regions of the wafer equally well represented, but they are not adequate if process-related defective chips are distributed with spatial pattern within the wafer.In this paper, we propose the methodology for generating effective measurement sampling plan for process parameter by applying the Self-Organizing Feature Map (SOFM) network, unsupervised learning neural network, to wafer bin map data within a certain time period. The sampling plan specifies which chips within the wafer need to be inspected, and how many chips within the wafer need to be inspected for a good sensitivity of 100% wafer coverage and defect detection. We finally illustrate the effectiveness of our proposed sampling plan using actual semiconductor fab data.     

Optimal selection of sampling locations and c-chart parameters

This paper presents a method for designing a statistical process contral program over the attributes of a multi-stage discrete part manufacturing system. The objective of the program is to minimize the sum of the costs of bad quality, inspection, searching for special causes of variation, process adjustment, and reworking of rejected items. The costs are affected by two interacting levels of decision, macro and micro. The macro level selects the points in the process at which quality control is exercised, referred to here as quality control windows. The micro level determines the control chart parameters, namely frequency of sampling, the sample size, and the chart range.     

A generalized Conforming Run Length control chart for monitoring the mean of a variable

The attribute Conforming Run Length (CRL) control chart has attracted increasing research interests in Statistical Process Control (SPC). It decides the process status based on the interval or distance between two nonconforming units. This article proposes a Generalized CRL chart (namely GCRL chart) for monitoring the mean of a measurable quality characteristic x under 100% inspection. To run a GCRL chart, each unit will be classified as a passing or nonpassing unit depending on whether the sample value of x falls within or beyond a pair of lower and upper inspection limits LIL and UIL. When a nonpassing unit is detected, the GCRL chart checks the distance between the current and last nonpassing units in order to determine the process status (in control or out of control). The inspection limits LIL and UIL are determined by an optimization design. The GCRL chart not only solves a dead-corner problem suffered by the conventional CRL chart, but also considerably outperforms the latter for detecting mean shifts. The most interesting finding is that the attribute GCRL chart excels the variable X chart to a significant degree in SPC for variables. It suggests that the simple attribute chart may replace the variable chart in some SPC applications. The design of the GCRL chart has to be carried out by a computer program, but the design can be completed almost in no time in a personal computer.     

Extending fuzzy and probabilistic clustering to very large data sets

Approximating clusters in very large (VL=unloadable) data sets has been considered from many angles. The proposed approach has three basic steps: (i) progressive sampling of the VL data, terminated when a sample passes a statistical goodness of fit test; (ii) clustering the sample with a literal (or exact) algorithm; and (iii) non-iterative extension of the literal clusters to the remainder of the data set. Extension accelerates clustering on all (loadable) data sets. More importantly, extension provides feasibility—a way to find (approximate) clusters—for data sets that are too large to be loaded into the primary memory of a single computer. A good generalized sampling and extension scheme should be effective for acceleration and feasibility using any extensible clustering algorithm. A general method for progressive sampling in VL sets of feature vectors is developed, and examples are given that show how to extend the literal fuzzy (c-means) and probabilistic (expectation-maximization) clustering algorithms onto VL data. The fuzzy extension is called the generalized extensible fast fuzzy c-means (geFFCM) algorithm and is illustrated using several experiments with mixtures of five-dimensional normal distributions.     

The variable sampling interval run sum X‾ control chart

Traditional control charts for process monitoring are based on taking samples from the process at fixed length sampling intervals. More recently, research works focused on the use of variable sampling intervals (VSIs), where the lengths of the sampling intervals are varied according to the process quality. A short sampling interval is considered when the process quality indicates a possible out-of-control situation while a long sampling interval is considered, otherwise. In this paper, the VSI run sum (RS) $\bar{X}$ chart is proposed with its optimal scores and parameters determined using an optimization technique to minimize the out-of-control average time to signal (ATS) or the adjusted average time to signal (AATS). A Markov-chain method is used to evaluate both the ATS and AATS of the proposed chart, for the zero and steady state cases, respectively. Results show that the VSI RS $\bar{X}$ chart is considerably more efficient than the basic RS $\bar{X}$ chart. The VSI RS $\bar{X}$ chart performs generally well compared with other competing charts, such as the standard $\bar{X}$, synthetic $\bar{X}$, exponentially weighted moving average (EWMA) $\bar{X}$, VSI $\bar{X}$ and VSI EWMA $\bar{X}$ charts. The sensitivity of the VSI RS $\bar{X}$ chart can be enhanced further by adding more scoring regions or a head-start feature. An illustrative example is presented to explain the implementation of the proposed VSI RS $\bar{X}$ chart.     

A hybrid fuzzy adaptive sampling - Run rules for Shewhart control charts

In crisp run control rules, usually it is stated that a process moves very sharply from in-control condition to out-of-control act. This causes an increase in both false-alarm rate and control chart sensitivity. Moreover, the classical run control rules are not implemented on an intelligent sampling strategy that changes control charts’ parameters to reduce error probability when the process appears to have a shift in parameter values. This paper presents a new hybrid method based on a combination of fuzzified sensitivity criteria and fuzzy adaptive sampling rules, which make the control charts more sensitive and proactive while keeping false alarms rate acceptably low. The procedure is based on a simple strategy that includes varying control chart parameters (sample size and sample interval) based on current fuzzified state of the process and makes inference about the state of process based on fuzzified run rules. Furthermore, in this paper, the performance of the proposed method is examined and compared with both conventional run rules and adaptive sampling schemes.     

Integrating statistical and nonstatistical audit evidence using belief functions: A case of variable sampling

The main purpose of this article is to show how one can integrate statistical and nonstatistical items of evidence in the belief function framework. First, we use the properties of consonant belief functions to define the belief that the true mean of a variable lies in a given interval when a statistical test is performed for the variable. Second, we use the above definition to determine the sample size for a statistical test when a desired level of belief is needed from the sample. Third, we determine the level of belief that the true mean lies in a given interval when a statistical test yields certain values for the sample mean and the standard deviation of the mean for the variable. Finally, we use the auditing situation to illustrate the process of integrating statistical and nonstatistical items evidence. © 1994 John Wiley & Sons, Inc.     

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.     

An hybrid detection system of control chart patterns using cascaded SVM and neural network�Cbased detector

Early detection of unnatural control chart patterns (CCP) is desirable for any industrial process. Most of recent CCP recognition works are on statistical feature extraction and artificial neural network (ANN)-based recognizers. In this paper, a two-stage hybrid detection system has been proposed using support vector machine (SVM) with self-organized maps. Direct Cosine transform of the CCP data is taken as input. Simulation results show significant improvement over conventional recognizers, with reduced detection window length. An analogous recognition system consisting of statistical feature vector input to the SVM classifier is further developed for comparison.     

Using neural networks to detect the bivariate process variance shifts pattern

Most of the research in statistical process control has been focused on monitoring the process mean. Typically, it is also important to detect variance changes as well. This paper presents a neural network-based approach for detecting bivariate process variance shifts. Some important implementation issues of neural networks are investigated, including analysis window size, number of training examples, sample size, training algorithm, etc. The performance of the neural network, in terms of the ARL and run length distribution, is compared with that of traditional multivariate control charts. Through rigorous evaluation and comparison, our research results show that the proposed neural network performs substantially better than the traditional generalized variance chart and might perform better than the adaptive sizes control charts in the case that the out-of-control covariance matrix is not known in advance.