Variable 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.

Economic-Statistical Design of Integrated Model of VSI Control Chart and Maintenance Incorporating Multiple Dependent State Sampling

The mutualistic relationship between quality control and maintenance of production systems has contributed to the development of the integration model of statistical process control (SPC) and maintenance. Such models are not only beneficial to the improvement of production quality, but also to the reduction of maintenance cost. This paper presents an integrated model of SPC and maintenance that applies a VSI & MDSS $\bar X$ control chart designing based on the variable sampling interval (VSI) scheme and multiple dependent/deferred state sampling (MDSS) policy. A Markov chain approach is employed to model the integrated process, and the average total cost per unit time is derivated and solved by a developed genetic algorithm subject to statistical quality constraints. A shrimp canning production process example is given to illustrate the application of integrated model. Finally, a comparative study using the design of experiments approach is conducted to demonstrate the effectiveness of the suggested model.

Detecting a shift in variance using economically designed VSI control chart with combined attribute-variable inspection

Recently, a MIX S 2 control chart using both attribute and variable inspections is proposed by Ho and Quinino (2016) to monitor the process variability. This paper considers the variable sampling interval (VSI) MIX S 2 control chart in which the sampling interval between two successive sampling points is varied based on the value of the preceding complementary sample variance to improve the performance of the original MIX S 2 control chart. Additionally, a unified economical design model for the process is established with the cost function derived. The performance of the VSI chart is compared to that of the original static chart. The numerical results show that comparing to original MIX S 2 control chart, the VSI chart yields better performance in terms of less expected cost per unit of time.

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.

An adaptive exponentially weighted moving average chart for the mean with variable sampling intervals

AbstractThe AEWMA control chart is an adaptive EWMA (exponentially weighted moving average) type chart that combines the Shewhart and the classical EWMA schemes in a smooth way. To improve the detection performance of the FSI (fixed sampling interval) AEWMA control chart in terms of the ATS(average time to signal), this paper proposes a new VSI (variable sampling interval) AEWMA control chart. A Markov chain approach is used to calculate the ATS values of the new VSI AEWMA control chart, and comparative results show that the proposed control chart performs better than the standard FSI AEWMA control chart and than other VSI control charts over a wide range of shifts.

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.

대용변수를 이용한 가변형 부분군 채취 간격 X 관리도의 경제적 설계

In many cases, an control chart which is based on the performance variable is used in industrial fields. However, if the performance variable is too costly or impossible to measure and a less expensive surrogate variable is available, the process may be more efficiently controlled using surrogate variables. In this paper, we propose a model for the economic design of a VSI (Variable Sampling Interval) control chart using a surrogate variable that is linearly correlated with the performance variable. The total average profit model is constructed, which involves the profit per cycle time, the cost of sampling and testing, the cost of detecting and eliminating an assignable cause, and the cost associated with production during out-of-control state. The VSI control charts using surrogate variables are expected to be superior to the Shewhart FSI (Fixed Sampling Interval) control charts using surrogate variables with respect to the expected profit per unit cycle time from economic viewpoint.

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.

Non-normality and combined double sampling and variable sampling interval control charts

A combination of double sampling the chart and the variable sampling interval of the chart (DSVSI chart) increases the sensitivity in the small shift detection. A usual assumption of control charts is that the process observations are normally distributed. However, this assumption may not be true for some processes in the real industry. This paper presented the performance of DSVSI chart under non-normality and compared it with the Shewhart chart and the variable parameters chart. The compared results show that the DSVSI chart has the best performance in detecting small process mean shifts.

Using VSI Loss Control Charts to Monitor a Process with Incorrect Adjustment

The article considers the optimum variables control scheme for a process with incorrect adjustment. Incorrect adjustment of a process may result in shifts in process mean and/or variance, ultimately affecting the quality of products and creating a loss. For the process with incorrect adjustment, we construct the variable sampling interval (VSI) and Z S 2 control charts to monitor the shifts in the process mean and variance and with minimum loss. The minimum loss variable sampling interval control charts is measured by the average loss per unit time derived by a Markov chain approach. An example is given to show the application and the performance of the proposed control charts. Furthermore, the comparison of the performance of the VSI loss control charts and the fixed sampling interval (FSI) loss control charts suggested that the VSI charts outperformed that of FSI charts.

Monitoring cascade processes using VSI EWMA control charts

AbstractThe paper considers the variables process control scheme for cascade process. We construct variable sampling interval (VSI) EWMA and EWMA control charts to effectively monitor the input variable and the output variable produced by a cascade process. Numerical analysis results demonstrate that the performance of the VSI control charts is much better than the fixed sampling interval (FSI) control charts in detecting small and median shifts. Copyright © 2009 John Wiley & Sons, Ltd.

Process control using VSI cause selecting control charts

The article considers the variable process control scheme for two dependent process steps with incorrect adjustment. Incorrect adjustment of a process may result in shifts in process mean, process variance, or both, ultimately affecting the quality of products. We construct the variable sampling interval (VSI) $${Z_{\overline{X}}-Z_{S_X^2}}$$ and $${Z_{\bar{{e}}}-Z_{S_e^2}}$$ control charts to effectively monitor the quality variable produced by the first process step with incorrect adjustment and the quality variable produced by the second process step with incorrect adjustment, respectively. The performance of the proposed VSI control charts is measured by the adjusted average time to signal derived using a Markov chain approach. An example of the cotton yarn producing system shows the application and performance of the proposed joint VSI $${Z_{\overline{X}} -Z_{S_X^2 }}$$ and $${Z_{\bar{{e}}} -Z_{S_e^2 }}$$ control charts in detecting shifts in mean and variance for the two dependent process steps with incorrect adjustment. Furthermore, the performance of the VSI $${Z_{\overline{X}}-Z_{S_X^2 }}$$ and $${Z_{\bar{{e}}} -Z_{S_e^2 }}$$ control charts and the fixed sampling interval $${Z_{\overline{X}} -Z_{S_X^2 }}$$ and $${Z_{\bar{{e}}} -Z_{S_e^2 }}$$ control charts are compared by numerical analysis results. These demonstrate that the former is much faster in detecting small and median shifts in mean and variance. When quality engineers cannot specify the values of variable sampling intervals, the optimum VSI $${Z_{\overline{X}}-Z_{S_X^2 }}$$ and $${Z_{\bar{{e}}} -Z_{S_e^2 }}$$ control charts are also proposed by using the Quasi-Newton optimization technique.

Controlling over-adjusted process means and variances using VSI cause selecting control charts

Process adjusted unnecessarily is a common problem in statistical process control. Incorrect adjustment of a process may result in shifts in process mean, process variance, or both, ultimately affecting the quality of products. The article considers the variable process control scheme for two dependent process steps with incorrect adjustment. We construct the variable sampling interval (VSI) Z X ¯ - Z S 2 and Z e ¯ - Z S e 2 control charts in order to effectively monitor the quality variable produced by the first process step with incorrect adjustment and the quality variable produced by the second process step with incorrect adjustment, respectively. The performance of the proposed VSI control charts is measured by the adjusted average time to signal (AATS) derived using a Markov chain approach. An example of process control for automobile braking system shows the application and performance of the proposed joint VSI Z X ¯ - Z S 2 and Z e ¯ - Z S e 2 control charts in detecting small and median shifts in mean and variance for the two dependent process steps with incorrect adjustment. The performance of the VSI Z X ¯ - Z S 2 and Z e ¯ - Z S e 2 control charts and the fixed sampling interval (FSI) Z X ¯ - Z S 2 and Z e ¯ - Z S e 2 control charts are compared via the numerical analysis results. These demonstrate that the former is much faster in detecting shifts in mean and variance. Whenever quality engineers cannot specify the values of variable sampling intervals, the optimal VSI Z X ¯ - Z S 2 and Z e ¯ - Z S e 2 control charts are suggested. Furthermore, the impacts of misusing Z Y ¯ - Z S y 2 charts to monitoring the process mean and variance in the second step are also investigated.

Economic design of VSI control charts

Duncan considered the design parameters of Shewhart control charts in 1956 based on an economic view-point, and ensured that an economic design control chart actually lowers the cost compared with a Shewhart control chart. His research method has been widely used in subsequent studies on the subject and focused on a fixed-sampling interval (FSI). However, it has been discovered that variable-sampling-interval (VSI) control charts are substantially quicker than FSI control charts in detecting shifts in the process. In an earlier study, an economic design of a VSI control chart with a single assignable cause for a continuous flow has been proposed. This research extends a single assignable cause to multiple-assignable causes to construct an economic model of VSI control charts. A numerical example is provided to demonstrate the effectiveness of the proposed model. The result indicates that VSI is more effective than FSI control charts.

Using VSI EWMA charts to monitor dependent process steps with incorrect adjustment

The article considers the variable process control scheme for two dependent process steps with incorrect adjustment. Incorrect adjustment of a process may result in shifts in process mean, ultimately affecting the quality of products. We construct variable sampling interval (VSI) EWMAZX and EWMAZe control charts to effectively monitor the quality variable produced by the first process step with incorrect adjustment and the quality variable produced by the second process step with incorrect adjustment. The performance of the proposed VSI control charts is measured by the adjusted average time to signal (AATS) derived using a Markov chain approach. An example of the automobile braking system with incorrect adjustment shows the application and performance of the proposed VSI EWMAZX and EWMAZe control charts in detecting shifts in process mean. Furthermore, the performance of the proposed VSI EWMAZX and EWMAZe control charts and the fixed sampling interval (FSI) EWMAZX and EWMAZe control charts are compared by numerical analysis results. These demonstrate that the former is much faster in detecting small and median shifts in mean. The optimum VSI EWMAZX and EWMAZe control charts are also proposed using optimization technique to minimize AATS when quality engineers cannot specify the values of variable sampling intervals. It has been found that the optimum VSI EWMAZX and EWMAZe control charts always work better than the VSI EWMAZX and EWMAZe control charts.

Optimization of design parameters for control charts with multiple assignable causes

Duncan's economic model of Shewhart's original x¯ chart has established its optimal and economic application for processes with the Markovian failure characteristic. As the sample statistics show some indications of process variations, the variable-sampling-interval (VSI) control charts perform more effectively than the fixed sampling interval (FSI) ones due to a higher frequency in the sampling rate. Regarding the economic design of control charts, most studies have been dedicated to the FSI scheme. In 1998, Bai & Lee considered the production process with a single assignable cause and proposed an economic VSI design for a general x¯ control chart. However, in real cases, there are multiple assignable causes in the production process. Therefore, concerning the operation characteristics of the real industry, this research develops an economic model for the VSI control chart with multiple assignable causes based on stochastic and statistics theory and determines the optimal design parameters of the chart. A numerical example is also provided to demonstrate the effectiveness of the proposed model and the result indicates that VSI performs more effectively than a FSI control chart.

Optimal Design of VSI ―X Control Charts for Monitoring Correlated Samples

This paper develops an economic design of variable sampling interval (VSI)―X control charts in which the next sample is taken sooner than usual if there is an indication that the process is off-target. When designing VSI―X control charts, the underlying assumption is that the measurements within a sample are independent. However, there are many practical situations that violate this hypothesis. Accordingly, a cost model combining the multivariate normal distribution model given by Yang and Hancock with Bai and Lee's cost model is proposed to develop the design of VSI charts for correlated data. An evolutionary search method to find the optimal design parameters for this model is presented. Also, we compare VSI and traditional ―X charts with respect to expected cost per unit time, utilizing hypothetical cost and process parameters as well as various correlation coefficients. The results indicate that VSI control charts outperform the traditional control charts for larger mean shift when correlation is present. In addition, there is a difference between the design parameters of VSI charts when correlation is present or absent. Copyright © 2005 John Wiley & Sons, Ltd.