Sampling Interval Control Charts Research Articles

A combined adaptive double sampling and variable sampling interval control chart for monitoring three-level products

Three-level control charts have been developed for monitoring processes where the quality of products is characterized by classifying the product characteristic into three discrete levels. In this article, a combined double sampling and variable sampling interval (DSVSI) attribute control chart is developed for monitoring three-level products. This research is motivated by recent studies done on combining the DSVSI scheme with traditional attribute charts and their results recommend the application of the DSVSI attribute charts. The proposed DSVSI three-level chart is compared with the existing standard, adaptive, and EWMA three-level charts. The results approve the superiority of the proposed scheme and recommend it as an efficient three-level control chart in application. An example is also provided to illustrate the application of the proposed chart.

Time to signal distribution of multivariate bayesian control chart with dual sampling scheme

The ability of a control chart to detect a shift in the process can be determined by the average run length (ARL), a widely used performance indicator for single sampling interval (SSI) control charts. When it comes to variable sampling interval (VSI) schemes, average time to signal (ATS) is utilized instead of ARL. Although analysis of ARL has been extensively studied for SSI schemes, calculation of the ATS for VSI control charts is still in its infancy due to several theoretical challenges. In this context, the paper derives closed-form expressions for time to signal distribution and the ATS for a spatial design of VSI multivariate Bayesian control chart referred to as the dual sampling scheme (DSS). In a DSS strategy, the process is being monitored initially with a longer sampling interval, which is changed to the shorter sampling interval at a designed switching point. Unlike previous approximate developments where the switching point is ignored, the paper derives exact expressions for computation of time to signal distribution and ATS within such a Bayesian framework. Derivations of the ATS are based on an intuitively pleasing definition of an artificial absorbing state and coded values of the posterior probability using Markov chain theory.

Integration of quadratic loss function approach in the economic design of VSI control chart with combined attribute‐variable inspection

AbstractThis paper proposes a variable sampling interval control chart for monitoring the process mean using combined attribute‐variable inspection. An economic design of the proposed chart using Taguchi's quadratic loss function is presented. The proposed model minimizes the total expected cost per unit time while minimizing out‐of‐control average time‐to‐signal and maintaining reasonable in‐control average time‐to‐signal. The numerical results indicate that, compared with the corresponding fixed sampling interval control chart scheme, the proposed control chart yields better performance in terms of both shorter time to signal a process shift and less total expected cost per unit time. Sensitivity analyses of the cost model are also carried out by using a fractional factorial experimental design.

Optimal Variable Sample Size and Sampling Interval Control Chart for the Process Mean based on Expected Average Time to Signal

Optimal Variable Sample Size and Sampling Interval Control Chart for the Process Mean based on Expected Average Time to Signal

A combined variable sampling interval and double sampling control chart with auxiliary information for the process mean

In recent years, the suitable use of auxiliary information technique in control charts has shown an improved run length performance compared to control charts that do not have this feature. This article proposes a combined variable sampling interval (VSI) and double sampling (DS) chart using the auxiliary information (AI) technique (called VSIDS-AI chart, hereafter). The plotting-statistic of the VSIDS-AI chart requires information from both the study and auxiliary variables to efficiently detect process mean shifts. The charting statistics, optimal design and performance assessment of the VSIDS-AI chart are discussed. The steady-state average time to signal (ssATS) and steady-state expected average time to signal (ssEATS) are considered as the performance measures. The ssATS and ssEATS results of the VSIDS-AI chart are compared with those of the DS AI, variable sample size and sampling interval AI, exponentially weighted moving average AI (EWMA-AI) and run sum AI (RS-AI) charts. The results of comparison show that the VSIDS-AI chart outperforms the charts under comparison for all shift sizes, except the EWMA-AI and RS-AI charts for small shift sizes. An illustrative example is provided to demonstrate the implementation of the VSIDS-AI chart.

Optimal design of the variable sampling interval np control chart for monitoring service process

As customer satisfaction is a key success factor for a service system, the main purpose of this paper is to design a control chart for monitoring the service process. The variable sampling interval (VSI) control charts are useful and have important applications in service industries, which sample at a higher rate when there is evidence of a change in the process, and are thus able to detect process changes faster than the traditional fixed sampling interval (FSI) control charts. This paper evaluates and compares the performance of the VSI np control chart with the average time to signal (ATS) properties, when the process parameters are known and estimated, the results demonstrate that these performances of the VSI np chart are quite different when the number of samples used during Phase I is small. Then we suggest new dedicated chart parameters for the VSI np chart with estimated parameter, to have approximately the same in-control ATS values as in the known parameter case.

A variable sample size and sampling interval control chart for monitoring the process mean using auxiliary information

A variable sample size and sampling interval (VSSI) control chart using the auxiliary information (VSSI AI) is proposed for efficiently monitoring the process mean. The control chart’s statistics, optimal design and implementation are discussed. The average time to signal (ATS) and expected ATS (EATS) criteria are adopted to evaluate the performance of the VSSI AI chart. The ATS and EATS performances of the VSSI AI chart, exponentially weighted moving average (EWMA) chart, EWMA chart with AI (EWMA AI) and the synthetic chart with AI (Syn AI) are compared and explored for common patterns. The proposed VSSI AI chart shows a better performance than the EWMA, EWMA AI and Syn AI charts, in terms of the ATS and EATS criteria. An example is also presented to illustrate the implementation of the VSSI AI chart.

A Markov chain approach for double-objective economic statistical design of the variable sampling interval control chart

ABSTRACTAn chart with variable sampling interval (VSI) has been shown superior to the traditional chart with fixed ratio sampling (FRS). A constraint approach is not an efficient method for Economic Statistical Design (ESD) of VSI control charts because statistical properties are of the same importance as economic properties and should be optimized simultaneously. Then, a Multi-Objective Genetic Algorithm for ESD is proposed for identifying the Pareto optimal solutions of control chart design. The proposed method allows the practitioner to be provided with a set of optimal designs rather than a single solution, and they can select the locally optimal solution according to the process information. Through an illustrative example, the advantages of the proposed approach are shown by providing a list of viable optimal solutions and graphical representations, which indicate the advantage of flexibility and adaptability of our approach. Performance measures of these adaptive control charts are obtained through a Markov chain approach.

The Effect of Measurement Errors on the Performance of Variable Sample Size and Sampling Interval Control Chart

The effect of measurement errors on adaptive and non-adaptive control charts has been occasionally considered by researchers throughout the years. However, that effect on the variable sample size and sampling interval (VSSI) control charts has not so far been investigated. In this paper, we evaluate the effect of measurement errors on the VSSI control charts. After a model development, the effect of measurement errors and multiple measurements on the performance of VSSI scheme are evaluated in terms of the out-of-control average time to signal (ATS) criterion, which is obtained using a Markov Chain approach. At last, a real case is presented to show the application of the proposed scheme.

Economic design based on a multivariate Bayesian VSI chart with a dual control limit

ABSTRACTIn this paper, a multivariate Bayesian variable sampling interval (VSI) control chart for the economic design and optimization of statistical parameters is designed. Based on the VSI sampling strategy of a multivariate Bayesian control chart with dual control limits, the optimal expected cost function is constructed. The proposed model allows the determination of the scheme parameters that minimize the expected cost per time of the process. The effectiveness of the Bayesian VSI chart is estimated through economic comparisons with the Bayesian fixed sampling interval and the Hotelling's T2 chart. This study is an in-depth study on a Bayesian multivariate control chart with variable parameter. Furthermore, it is shown that significant cost improvement may be realized through the new model.

Monitoring the coefficient of variation using a variable sample size and sampling interval control chart

ABSTRACTThis article proposes a CV chart by using the variable sample size and sampling interval (VSSI) feature to improve the performance of the basic CV chart, for detecting small and moderate shifts in the CV. The proposed VSSI CV chart is designed by allowing the sample size and the sampling interval to vary. The VSSI CV chart's statistical performance is measured by using the average time to signal (ATS) and expected average time to signal (EATS) criteria and is compared with that of existing CV charts. The Markov chain approach is employed in the design of the chart.

Economic design of variable sampling interval sign control chart

In this article, we have proposed an economic design of variable sampling interval (VSI) control chart based on sign statistic. An expression for expected loss per unit time is obtained and minimized with respect to design parameters. The performance of this economic design is compared with economic design of fixed sampling interval (FSI) sign chart to control location of process characteristic. Though the power of VSI sign chart is slightly lower than that of FSI sign chart design, it appears to be dominant over FSI sign chart design for moderate shifts. The effect of change in input parameters on loss cost is observed for different out of control probabilities of quality characteristic under VSI sign chart scheme. Change in shift parameter, penalty cost parameter, and time to detect the shift have considerable effect on loss cost from the design. The design is fairly robust to the changes in other parameters.

A modified variable sample size and sampling interval controlchart

In this article, a modified version of variable sample size and sampling interval chart using three sampling intervals and two sample sizes, called the VSSIt control chart, is proposed to overcome the weakness of the standard Shewhart (SS) control chart in detecting small and moderate shifts. In order to evaluate the performance of the proposed scheme, average time to signal (ATS), average number of observations to signal (ANOS), and average number of samples to signal (ANSS) are used. Using this control chart can reduce the ATS value around 68, 27, 7, and 1.83 time unit (sampling interval of the SS chart) in detecting shift of size 0.25, 0.5, 0.75, and 1 standard deviation, respectively. The performance of the constructed adaptive chart is compared with various adaptive charts such as variable sample size (VSS), variable sampling interval (VSI), variable sample size and sampling interval (VSSI), and special variable sample size and sampling interval (SVSSI) \( \overline{x} \) control chart. It is shown that the proposed chart always outperforms SS, VSI, VSS, and VSSI schemes. It also can detect shift of size 0.25 standard deviation quicker than SVSSI scheme. Finally, the optimal parameters of the proposed chart are recommended.

The variable sampling interval control chart for finite-horizon processes

The requirement to be globally competitive requires companies to have a high level of flexibility to allow for the production of a large variety of products. To limit work-in-process, decision makers periodically schedule according to a make-to-order management strategy i.e. the production of finite batches of the same product code. Scheduling calls for frequent set up activities, which require the reconfiguration of a manufacturing process, and allows manufacturers to switch between different codes. This can limit the production horizon of one product code to a few hours or shifts. In this context, efficient online quality control monitoring using control charts is strategic to eliminate scrap or rework and to meet the demand at the time specified by the production plan. The design of control charts for a process with a limited production horizon is a challenge for statistical process control practitioners. Under this framework, this article investigates the issues related to the implementation of the Variable Sampling Interval (VSI) Shewhart control chart in a process with finite production horizon. When the production horizon is finite, the statistical properties of a control chart are known to be a function of the number of scheduled inspections. In the case of a VSI control chart, the quality practitioner cannot fix the number of inspections a priori due to the stochastic nature of the sampling interval selection. Therefore, the aim of this article is to propose a new Markov chain approach for the exact computation of the statistical performance of the VSI control chart in processes with an unknown and finite number of inspections. The proposed approach is general and does not depend on the monitored sample statistic. With reference to the process mean monitoring, an extensive numerical analysis compares the performance of the VSI chart to the Variable Sample Size and Fixed Sampling Rate z charts. Numerical results show that the VSI chart outperforms other charts for moderate to large shift sizes. An illustrative example shows the implementation of the VSI chart in a short run producing a finite batch of mechanical parts.

An economic design of the VSSI control charts for the means of positively Skewed distributions

Variable sample size and sampling interval (VSSI) control charts have been shown superior to standard Shewhart (SS) control charts for detection of a small or moderate shift in process mean. However, they might utilize more resources by a more frequent sampling rate and a large sample size to improve its performance. Recently, some economic models were used to express the long-run cost per hour of operating the VSSI control charts and gain insight into the way to design the charts. The usual assumption for the models is the normality of the underlying data or measurements. However, this assumption may not be true in practice. In this paper, an economic design of the VSSI control charts for skewed non-normal data is conducted by using the Markov chain approach and genetic algorithms. Two types of VSSI charts are considered and compared with the SS charts over several numerical examples: the symmetric control limits and asymmetric control limits. Moreover, effects of non-normality on the performance of the VSSI charts with respect to the costs of operating the charts are studied. It is shown that the reduction in cost can be achieved by using the VSSI charts instead of the SS charts. However, an increase on the skewness coefficient results in a decrease on the cost savings. In addition, the asymmetric control limits is a better choice with respect to the costs and the false alarm rate. Key words: Skewed non-normal, control chart, genetic algorithms, economic design.

Using genetic algorithm and response surface methodology for statistically constrained optimization of VSI X-bar control charts under multiple assignable causes and non-normality

Variable sampling interval (VSI) control charts have been introduced with the aim of improving performance of traditional control charts. Usually, in the economic–statistical design of the VSI $$ \overline{X} $$ control charts, it is assumed that observations are normally distributed and process is subjected to only one assignable cause. However, in practice these assumptions could easily fail to hold, and results no longer could be realistic. This paper considers non-normal observations for the case of multiple assignable causes to develop a cost model for the economic design of VSI $$ \overline{X} $$ control chart. Being more applicable for all types of distributions, Burr distribution is employed for representing the distribution of non-normal process data. Since the proposed design consists of a complex nonlinear cost function that cannot be solved using a classical optimization method, genetic algorithm (GA) searching method as an efficient famous metaheuristic is employed to find the optimal values for the design parameters. Moreover, to improve the performances, response surface methodology is employed to calibrate GA parameters. The effectiveness of the proposed scheme is evaluated through a numerical example. Sensitivity analysis is also carried out to show the effects of cost and process parameters on the outputs of the model. Results show that in all cases, presented VSI model has better economical and statistical performances than its corresponding fixed sampling interval scheme.

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.

Switching properties of CUSUM charts for controlling mean vector

Abstract Some switching properties of multivariate control charts are investigated when theinterval between two consecutive sample selections is not xed but changes accordingto the result of the previous sample observation. Many articles showed that the perfor-mances of variable sampling interval control charts are more ecient than those of xedsampling interval control charts in terms of average run length (ARL) and average timeto signal (ATS). Unfortunately, the ARL and the ATS do not provide any informationon how frequent a switch is being made. We evaluate several switching properties oftwo sampling interval Shewhart and CUSUM procedures for controlling mean vectorof correlated quality variables.Keywords: Average number of switches to signal, average sampling interval, probabilityof switch. 1. Introduction The purpose of a control chart is to detect assignable causes of variation so that thesecauses can be found and eliminated. During the control process, one wishes to detect anydeparture from a satisfactory state as quickly as possible and identify which attributes areresponsible for the deviation. Thus, a good control chart detects shifts quickly in the processwhen the process is out-of-control state, and produces few false alarms when the process isin-control state.Many situations in industrial quality control involve a vector of measurements of twoor more related quality variables rather than a single quality variable. When the qualityvariables are correlated, one could obtain better sensitivity by using multivariate controlprocedures than separate control variables for each of the quality variables.Traditional sampling practice in using a control chart is to take samples from the processat xed sampling interval (FSI) and performances of control charts have been developedwhen the sampling interval between samples is xed. In recent years, application of variable

An economic design of combined double sampling and variable sampling interval control chart

A combination of double sampling X¯ chart and variable sampling intervals X¯ chart (DSVSI) can successfully reduce the detection time of small mean shift. Before a DSVSI X¯ chart is used, its design parameters should be determined. This study constructs an economic design model of DSVSI X¯ chart for the determination of the design parameters. Then, the Genetic Algorithms (GA) are used to solve the optimal designs of DSVSI X¯ chart. Finally, a numerical example is provided to illustrate the use of this model. A sensitivity analysis of the effects of model parameters on the optimal design of DSVSI X¯ chart is also performed.

Multivariate Control Charts for Monitoring the Mean Vector and Covariance Matrix with Variable Sampling Intervals

Traditional control charts for process monitoring are fixed sampling rate (FSR) control charts based on taking samples of fixed size with a fixed sampling interval between samples. Variable sampling rate (VSR) control charts vary the sampling rate as a function of the data from the process, and can detect most process changes significantly faster than traditional FSR control charts. The main focus of this article is on a type of VSR control chart called a variable sampling interval (VSI) control chart, but another type of VSR control chart based on sequential sampling is also considered. We consider the problem of detecting sustained or transient shifts in the mean or variability of a process. We investigate Shewhart control charts and multivariate exponentially weighted moving average (MEWMA) control charts based on sample means and on the sum of the squared deviations from target. We show that the combination of the standard MEWMA chart and an MEWMA-type chart based on squared deviations from target gives excellent overall performance. When the VSI feature is used with this combination, the performance is dramatically better than the performance of the standard FSR multivariate control charts that have traditionally been used.