Design Of Control Charts Research Articles

The np Chart with Guaranteed In‐control Average Run Lengths

We evaluate the in‐control performance of the np‐control chart with estimated parameter conditional on the Phase I sample. We then apply the bootstrap method to adjust the control chart limits to guarantee the desired in‐control average run length (ARL0) value in the monitoring stage. The adjusted limits ensure that the ARL0 would take a value greater than the desired value (say, B) with a certain specified probability, that is, Pr(ARL0 > B) = 1 − ρ. The results indicate that adjusting control limits is not always necessary. We present a method to design control charts such that in control and out of control run lengths are guaranteed with pre specified probabilities. This method is an improvement of the classical statistical design approach employing constraints on in control and out of control ARL because, with this approach, there is a substantial probability that the actual run length in control may be too small. In addition, using the ARL approach may result in an actual out of control run length that is too large. Some numerical examples illustrate the efficacy of this design method. Copyright © 2016 John Wiley & Sons, Ltd.

Economic and statistical design of Vp control charts for finite-horizon processes

ABSTRACTIn manufacturing environments where the production horizon for a specific product can be limited to a few hours or shifts, statistical process monitoring based on control charts is strategic to cut scrap, rework costs, and meet due dates. In this article, a Markov chain model is proposed to design a fully adaptive Shewhart control chart in a process with finite production horizon. The proposed Markov chain model allows the exact computation of several statistical performance metrics, as well as the expected cost of the monitoring and operation process for any adaptive Shewhart control chart with an unknown but finite number of inspections. Illustrative examples show the implementation of the Vp chart in short runs producing a finite batch of products.

Economic and economic-statistical designs of the T2 control charts with SVSSI sampling scheme

ABSTRACTRecent studies have shown that the adaptive T2 chart with two different sampling interval and three sample sizes (SVSSI) shows a good performance in detecting small to large shifts in the process mean. This paper investigates the economic and economic statistical designs of the SVSSI T2 charts. We use the Markov chain approach to developing the cost model proposed by Costa and Rahim (Journal of applied statistics 2001; 28: 875–885). A genetic algorithm approach is used to find the optimal solutions. Using numerical examples, we illustrate the performance of the proposed model and compare the statistical, economic, and economic statistical designs of the SVSSI T2 chart with respect to the economic and statistical criteria. Furthermore, we compare the performance of the SVSSI T2 chart with the other T2 control schemes.

Adaptive c-chart with estimated parameter

ABSTRACTIn the design of control charts, it is usually assumed that process parameters are known. However, in many practical applications the values of these parameters are unknown and should be estimated using historical in-control process observations. In this study, the performance of adaptive c-chart with estimated parameter is evaluated. It is demonstrated that by increasing the size and the number of samples in estimating the process parameter, the performance of the chart converges to that of the known parameter case. Finally the best phase I sampling scenarios are presented to make the chart with the estimated parameter perform as well as the chart with the known parameter.

Monitoring process variability: a hybrid Taguchi loss and multiobjective genetic algorithm approach

The common consideration on economic model is that there is knowledge about the risk of occurrence of an assignable cause and the various cost parameters that does not always adequately describe what happens in practice. Hence, there is a need for more realistic assumptions to be incorporated. In order to reduce cost penalties for not knowing the true values of some parameters, this paper aims to develop a bi-objective model of the economic-statistical design of the S control chart to minimize the mean hourly loss cost while minimizing out-of-control average run length and maintaining reasonable in-control average run length considering Taguchi loss function. The purpose of Taguchi loss function is to reflect the economic loss associated with variation in, and deviations from, the process target or the target value of a product characteristic. In contrast to the existing modeling approaches, the proposed model and given Pareto-optimal solution sets enables the chart designer to obtain solutions that is effective even for control chart design problems in uncertain environments. A comparison study with a traditional economic design model reveals that the proposed chart presents a better approach for quality system costs and the power of control chart in detecting the assignable cause.

Economic and economic–statistical designs of VSI Bayesian control chart using Monte Carlo method and ABC algorithm

ABSTRACTThe usual practice in using a Bayesian control chart to monitor a process is done by taking samples from the process with fixed sampling intervals. Recent studies on traditional control charts have shown that variable sampling interval (VSI) scheme compared to classical scheme (fixed ratio sampling, FRS) helps practitioners to detect process shifts more quickly. In this paper, the effectiveness of VSI scheme on performance of Bayesian control chart has been studied, based on economic (ED) and economic–statistical designs (ESD). Monte Carlo method and artificial bee colony algorithm have been utilized to obtain optimal design parameters of Bayesian control chart (sample size, sampling intervals, warning limit and control limit) since the statistic of this approach does not have any specified distribution. Finally, VSI Bayesian control chart has been compared to FRS Bayesian and VSI X-bar approaches based on ED and ESD, separately. According to the results, it has been found that the performance of VSI Bayesian scheme is better than FRS Bayesian and VSI X-bar approaches.

On Model Selection for Autocorrelated Processes in Statistical Process Control

Using traditional control charts to monitor autocorrelated processes is not beneficial, because it will lead us to misleading detections in the processes. One of the methods used to deal with the control charts for autocorrelated process is the model‐based approach. It uses an adequate time series model that fits the process and uses the residuals as monitoring statistics. For the said purpose, it is important to pick a suitable model that can adequately be used for different designs of control charts under specific time series model. This study intends to do the same for three popular types of charts namely Shewhart, exponentially weighted moving average, and cumulative sum. The models covered in this study include AR(1), MA(1), and ARMA(1,1) as the potential models to fit the process of interest. We have focused on two performance aspects namely efficiency and robustness. Average run length is used as a performance measure for different in‐control and out‐of‐control states of the autocorrelated processes under varying levels of autocorrelation. An application example based on a real data set is also included in the study to highlight the importance of the study proposals. Copyright © 2016 John Wiley & Sons, Ltd.

A generalized version of Banerjee and Rahim model in economic and economic statistical designs of multivariate control charts under generalized exponential shock model

ABSTRACTIn this article, a method for economic and economic-statistical design of multivariate (T-squared) control charts under a two-parameter generalized exponential shock model and uniform sampling scheme which is based on primary work of Banerjee and Rahim (1988) and Yang and Rahim (2006) was developed. A sensitivity analysis was conducted to study the effects of process, and cost parameters on design parameters and designs were compared. Comparison showed that the difference between sampling interval and minimum cost in these two designs is negligible and economic-statistical design provides smaller average run length (ARL) under control.

Statistically constrained economic design of maximum double EWMA control charts based on loss functions

A maximum double exponentially weighted moving average (EWMA) chart, called the MaxDEWMA chart, is used for fast detection of small and moderate shifts in the process mean and/or variance. A loss model that combines Lorenzen and Vance’s cost model with linear, quadratic and exponential loss functions is applied in this work to develop the statistical constraints of the MaxDEWMA chart. The optimal decision variables, namely sample size n, sampling interval time h, control limit width L and smoothing constant , are obtained by minimizing the expected cost function. Via simulations, the MaxDEWMA charts have smaller expected cost than the MaxEWMA charts based on different loss functions. In addition, adopting different loss functions impacts the optimal expected cost and decision variables. Sensitivity analyses reveal that a larger magnitude of the mean and/or variance shifts leads to a smaller sample size and a more frequent sampling interval.

An extension of Banerjee and Rahim model in economic and economic statistical designs for multivariate quality characteristics under Burr XII distribution

ABSTRACTThe design parameters of the economic and economic statistical designs of control charts depend on the distribution of process failure mechanism or shock model. So far, only a small number of failure distributions, such as exponential, gamma, and Weibull with fixed or increasing hazard rates, have been used as a shock model in the economic and economic statistical designs of the Hotelling T2 control charts. Due to both theoretical and practical aspects, the lifetime of the process under study may not follow a distribution with fixed or increasing hazard rate. A proper alternative for this situation may be the Burr distribution, in which the hazard rate can be fixed, increasing, decreasing, single mode, or even U-shaped. In this research article, economic and economic statistical designs of the Hotelling T2 control charts under the Burr XII shock models under two uniform and non uniform sampling schemes were proposed, constructed, and compared. The obtained design models were implemented by a numerical example, and a sensitivity analysis was conducted to evaluate the effect of changing parameters of shock model distribution on the optimum values of the proposed design models. The results showed that first the proposed designs under non uniform sampling scheme perform better and second the optimum values of the designs are not significantly sensitive to changing of the Burr XII distribution parameters. We showed that the obtained design models are also true for the beta Burr XII shock model.

Integration of production quantity and control chart design in automotive manufacturing

This study presents a two-stage stochastic programming model for the determination of control limits in p-charts when a production process produces above a certain quantity. Consideration of production quantity needed along with control limit determination is important for the following competing two reasons: (1) Wider control limits make it difficult to detect the changes in the process, therefore producing excessive number of cars with paint defects. (2) Narrower control limits, on the other hand, increase the number of unnecessary interventions even if there is no deterioration in the process so that inspection costs increase. In both cases, quantity produced reduces due to defective products and unnecessary interventions. Therefore, it is important to design a control chart for proportion of defects that takes production quantity requirements into account. We consider the problem in an automotive manufacturing setting in which the cars are inspected for paint defects after paint operations.We formulate the problem as a two-stage stochastic programming model. In the first stage, control limit parameter k is decided for the p-chart and in the second stage, production quantity is determined that minimizes total quality-related and production costs. We solve the model by sample average approximation algorithm (SAA). In a numerical study, we investigate the effect of various factors on control limit parameter k and the total cost. Our numerical study shows that (i) an increase on the mean defect rate increases both the total cost and the total production quantity, (ii) effect of an increasing process variance to the control limit parameter k is significantly small, (iii) frequency of special cause occurrences affects the total cost significantly and (iv) all the experiments show that the commonly used 3σ control limits in practice are wider than required.

Integration of SPC and performance maintenance for supply chain system

In this paper, a supply chain system is viewed as a maintainable system, and the economic-statistical design of a likelihood ratio control chart with a maintenance application is considered for this system. The supply chain system is described by a three-state: normal state, warning state and failure state. A likelihood ratio control chart is used to monitor the system given that only categorical observations can be obtained. When the chart signals, a full inspection is performed to determine the actual system state (normal or warning), and preventive maintenance is immediately performed in the warning state. In addition, the supply chain system must be corrected upon failure (i.e. corrective maintenance), and should be maintained in a scheduled time (i.e. planned maintenance). A mathematical model is developed for the joint optimisation of the control chart parameters and planned maintenance time based on renewal theory. An example is presented to illustrate how to determine the optimal design parameters. We also investigate the effect of coefficients and statistical constraints on the decision variables and the expected cost.

CUSUM procedures with probability control limits for monitoring processes with variable sample sizes

ABSTRACTControl charts are usually designed with constant control limits. In this article, we consider the design of control charts with probability control limits aimed at controlling the conditional false alarm rate at the desired value at each time step. The resulting control limits are dynamic and thus are more general and capable of accommodating more complex situations in practice as compared with the use of a constant control limit. We consider the situation when the sample sizes are varying over time, with a primary focus on the CUmulative SUM (CUSUM)-type control charts. Unlike other methods, no assumptions about future sample sizes are required with our approach. An integral equation approach is developed to facilitate the design and analysis of the CUSUM control chart with probability control limits. The relationship between the CUSUM charts using probability control limits and the CUSUM charts with a fast initial response feature is investigated.

Optimal design of a distribution-free quality control scheme for cost-efficient monitoring of unknown location

Traditionally, a cost-efficient control chart for monitoring product quality characteristic is designed using prior knowledge regarding the process distribution. In practice, however, the functional form of the underlying process distribution is rarely known a priori. Therefore, the nonparametric (distribution-free) charts have gained more attention in the recent years. These nonparametric schemes are statistically designed either with a fixed in-control average run length or a fixed false alarm rate. Robust and cost-efficient designs of nonparametric control charts especially when the true process location parameter is unknown are not adequately addressed in literature. For this purpose, we develop an economically designed nonparametric control chart for monitoring unknown location parameter. This work is based on the Wilcoxon rank sum (hereafter WRS) statistic. Some exact and approximate procedures for evaluation of the optimal design parameters are extensively discussed. Simulation results show that overall performance of the exact procedure based on bootstrapping is highly encouraging and robust for various continuous distributions. An approximate and simplified procedure may be used in some situations. We offer some illustration and concluding remarks.

Economic design of multivariate Bayesian control chart with two sampling intervals

In this paper, the economic design of a multivariate Bayesian control chart with two sampling intervals for monitoring the process mean is developed. The economic design of traditional control charts such as T2, SPRT, EWMA, and CUSUM charts with variable sampling interval (VSI) has been developed in recent years to control the industrial processes by minimizing the average cost of the process, however, it has been proved that these traditional, non-Bayesian process control charts are not optimal. There has been some limited development of the economic design of Bayesian control charts with a single sampling interval (SSI), but no results regarding the economic design of the Bayesian control chart with two sampling intervals have been reported in the literature. The objective is to find the optimal control chart parameters, namely the sample size, sampling intervals and the control limits, minimizing the total long-run expected average cost per unit time. The proposed problem is formulated and solved in the semi-Markov decision process (SMDP) framework. A numerical example is provided to illustrate the whole procedure. The effectiveness of the newly proposed control chart is demonstrated through a comparison with the multivariate Bayesian control chart with a single sampling interval as well as with a recently developed MEWMA control chart. The results indicate that the proposed control chart has a considerably better performance.

An optimization design of the combined Shewhart-EWMA control chart

This article presents an optimization design of the combined Shewhart \( \overline{X} \) chart and exponentially weighted moving average (EWMA) chart (\( \overline{X} \)&EWMA chart in short) used in statistical process control (SPC). The design algorithm not only optimizes the charting parameters of the \( \overline{X} \) chart and EWMA chart, but also optimizes the allocation of detection power between the two charts’ elements, based on the loss function, so that the best overall performance can be achieved. The optimization design is carried out under the constraints on the false alarm rate and available resources. While the optimization design effectively improves the overall performance of the \( \overline{X} \)&EWMA chart over the entire process shift range, it does not increase the difficulty of understanding and implementing this combined chart. A 2k factorial experiment shows that the optimal \( \overline{X} \)&EWMA chart outperforms the main competitor, the basic \( \overline{X} \)&EWMA chart, by about 50 %, on average. Moreover, this article provides the SPC practitioners with a design table to facilitate the designs of the \( \overline{X} \)&EWMA charts. From this design table, the users can directly find the optimal values of the charting parameters according to the design specifications.

Optimising the design of tabular mean CUSUM control chart to enhance the detection of diminutive process shifts

Although the performance of the CUSUM chart in detecting small process shift is better than Shewhart chart, there exists cases in reality that require very quick response of detecting small process shift. This paper studies the influential factors that affect the CUSUM control chart design and its in-control and out-of-control average run length (ARL) values. The controllable factors studied are the slack value k; the decision interval h; and the Headstart or Fast Initial Response. These factors are varied at different values for the process expected shift (in standard deviation units) δ. Following this, a non-linear regression model is deduced to compute the ARL of the control chart at any combination of process and control chart variables. Optimisation model is utilised to achieve a specific value for the ARL through fixing the controllable factors. A computer program coded in Matlab workspace (Matlab and Statistics Toolbox Release, 2015) is used for the computation of the ARLs values.

Optimising the design of tabular mean CUSUM control chart to enhance the detection of diminutive process shifts

Although the performance of the CUSUM chart in detecting small process shift is better than Shewhart chart, there exists cases in reality that require very quick response of detecting small process shift. This paper studies the influential factors that affect the CUSUM control chart design and its in-control and out-of-control average run length (ARL) values. The controllable factors studied are the slack value k; the decision interval h; and the Headstart or Fast Initial Response. These factors are varied at different values for the process expected shift (in standard deviation units) δ. Following this, a non-linear regression model is deduced to compute the ARL of the control chart at any combination of process and control chart variables. Optimisation model is utilised to achieve a specific value for the ARL through fixing the controllable factors. A computer program coded in Matlab workspace (Matlab and Statistics Toolbox Release, 2015) is used for the computation of the ARLs values.

Cost comparisons of modifiedX̅chart for autocorrelated observations

This paper presents the economic design of modified X chart for autocorrelated data and comparison with the economic design of Shewhart’s X chart. An attempt has been made to counter autocorrelation by designing the modified X chart; as the cost of operating a process control system is an important element in the economic design of control charts. The economic designs of both modified and Shewhart X charts for autocorrelated observations are presented; using Lorenzen-Vance [1] cost model. The modified X chart is based upon sum of chi-squares and has simplicity like Shewhart X chart but more efficient than standard chart. Two sets of parameters suggested by Montgomery [2] are used to determine the optimal sampling intervals (h) and expected costs per hour. It shows that sampling interval (h) and expected cost/hour (C) depend upon various parameters of chart, used in this model. The expected costs/hour (C) for both the charts are computed for sample sizes of 2 and 4 at the various levels of correlation and process shifts in mean. It is concluded that at each level of correlation and shift in the process mean, the expected costs/hour of the modified X chart are lower than the Shewhart chart.

와이블분포하에서의 최소값 및 최대값 관리도의 설계

Department of Information and Statistics, Chungnam National UniversityStatistical process control techniques have been greatly implemented in industries for improving product quality and saving production costs. As a primary tool among these techniques, control charts are widely used to detect the occurrence of assignable causes. In most works on the control charts it considered the problem of monitoring the mean and variance, and the quality characteristic of interest is normally distributed. In some situations moni-toring of the minimum and maximum values is more important and the quality characteristic of interest is the Weibull distribution rather than a normal distribution. In this paper, we consider the statistical design of mini-mum and maximum control charts when the distribution of the quality characteristic of interest is Weibull. The proposed minimum and maximum control charts are applied to the wind data. The results of the application show that the proposed method is more effective than traditional methods.