The economic design of cumulative sum charts used to maintain current control of non-normal process means

This paper proposes a method for the economic design of cusum charts to maintain the current control of the non-normal process means. The economic design involves the determination of the design parameters that minimize a relevant cost function. An expression for the expected loss-cost function for the process is defined. An algorithm for near-optimal determination of the design parameters is presented. A computer program is designed for the application of the algorithm, and deriving the values of the design parameters and loss-cost function of an economic design of cusum chart to control non-normal process means. Also a numerical example is provided. Finally, a simplified scheme, which would be appliable at the workshop level is presented.     

Economic model of x̄-CHART under non-normality and measurement errors

This paper investigates the effect of non-normality and measurement errors on the economic design of $\text{x}\text{?}\text{-}\text{control}$ charts. The measurable quality characteristic of the product is assumed to be non-normally distributed random variable. The process is subject to a single assignable cause with exponentially distributed occurrence time. This assignable cause shifts the process from in-control state to out-of-control state. Each observation involves some deviation from true value due to measurement error. This deviation, characterized by bias and imprecision, is considered to be a normally distributed random variate. The production cycle for the process model consists of: (1 ) the in-control period, (2) the out-of-control period due to occurrence of the assignable cause, (3) the search period due to false alarm, and (4) the search and correction time due to true alarm. An expected-cost model is formulated which comprises the fixed and variable cost of sampling, the cost of searching for the assignable cause when it does and does not exist, and the adjustment and correction costs. The economic design of $\text{x}\text{-}\text{?}\text{rmchart}$ involves optimal determination of the design parameters; the sample size, the sampling interval and the control limits coefficients so as to minimize the expected total cost. The optimal value of the design parameters are obtained using a computerized search technique. Consequently the effect of non-normality parameters and measurement errors on the design parameters and on the loss-cost function is explained through numerical examples.     

A teaching–learning based optimization approach for economic design of X-bar control chart

Being simple to use X-bar control chart has been most widely used in industry for monitoring and controlling manufacturing processes. Measurements of a quality characteristic in terms of samples are taken from the production process at regular interval and the sample means are plotted on this chart. Design of a control chart involves the selection of three parameters, namely the sample size (n), the sampling interval (h) and the width of control limits (k). In case of economic design, these three control chart parameters are selected in such a manner that the total cost of controlling the process is the least. The effectiveness of this design depends on the accuracy of determination of these three parameters. In this paper, a new efficient and effective optimization technique named as teaching–learning based optimization (TLBO) has been used for the global minimization of a loss cost function expressed as a function of three variables n, h and k in an economic model of X-bar chart based on unified approach. In this work, the TLBO algorithm has been modified to simplify the tuning of teaching factor. A MATLAB computer program has been developed for this purpose. A numerical example has been solved and the results are found to be better than the earlier published results. Further, the sensitivity analysis using fractional factorial design and analysis of variance have been carried out to identify the critical process and cost parameters affecting the economic design.     

Economic statistical design of non-uniform sampling scheme X bar control charts under non-normality and Gamma shock using genetic algorithm

This paper presented an approach which simultaneously considered the properties of cost and quality based on the Burr distribution and the non-unifampling scheme. The objective was to determine three parameters, namely, sample size, sampling interval between successive samples, and control limits, when an X bar chart monitors a manufacturing process with Gamma (λ, 2) failure characteristic and non-normal data. The design parameters of the X bar control charts can be obtained through the genetic algorithm (GA) method. An example was also adopted to indicate the solution procedure and sensitivity analyses. The results show that an increase of skewness coefficient (α₃) results in a slight decrease for sample size (n) while an increase of kurtosis coefficient (α₄) leads to a wider control limit width.     

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.     

The Updatable \overline{X} & S Control Charts

The tools for Statistical Process Control (SPC) should be continuously improved in order to continuously improve the product quality. This article proposes a scenario for continuously improving the & S control charts during the use of the charts. It makes use of the information collected from the out-of-control cases in a manufacturing process to update the charting parameters (i.e. the sample size, sampling interval and control limits) step-by-step. Consequently, the resultant control charts (called the updatable charts) become more and more effective to detect the mean shift δ_μ and standard deviation shift δ_σ for the particular process. The updatable charts are able to considerably reduce the average value of the loss function due to the occurrences of the out-of-control cases. Noteworthily, unlike the designs of the economic control charts, the designs of the updatable charts only require limited number of specifications that can be easily decided.     

Shewhart-type control charts for variation in phase I data analysis

Control charts for variation play a key role in the overall statistical process control (SPC) regime. We study the popular Shewhart-type S², S and R control charts when the mean and the variance of a normally distributed process are both unknown and are estimated from m independent samples (subgroups) each of size n. This is the Phase I setting. Current uses of these charts do not recognize that in this setting the signalling events are statistically dependent and that m comparisons are made with the same control limits simultaneously. These are important issues because they affect the design and the performance of the control charts. The proposed methodology addresses these issues (which leads to working with the joint distribution of a set of dependent random variables) by calculating the correct control limits, so that the false alarm probability (FAP), defined as the probability of at least one false alarm, is at most equal to some given nominal value FAP₀. To aid practical implementation, tables are provided for the charting constants for each Phase I chart, for an FAP₀ of 0.01 and 0.05, respectively. An illustrative example is given.     

Design of attribute control charts for multistage production systems-based on cost criteria

This paper develops a method to design attribute control charts, p-charts and c-charts, for multistage production systems based on quality cost criteria. The production systems are structured and formulated as a dynamic programming model. Each productions station and its inspection station is viewed as a combined station and is treated as the stage in the dynamic programming model. The model has quality distribution as the state variable and the stage transformation of this model is also developed. the stage return of this dynamic programming model is the quality cost of implementing an attribute control chart to that stage and the decision variables are the control chart parameters, n-sample size and control limits. Methods to combine quality distributions for finding type I and type II errors are then developed. As the dynamic programming model has quality distribution as state variable, conventional methods of solving dynamic programming problems are not applicable. A method, for solving this model, is thus developed based on branch and bound approach. The method itself may be applied to other dynamic programming problems where conventional methods are inefficient or not applicable. An example of the foundry process of cylinder liners is included to illustrate the developed method.     

On efficient use of auxiliary information for control charting in SPC

In Statistical Process Control (SPC), monitoring of the process dispersion has a major impact on the performance of processes like manufacturing, management and services. Control charts act as the most important SPC tool, used to differentiate between common and special cause variations in the process. The use of auxiliary information can enhance the detection ability of control charts and hence an efficient monitoring of process parameter(s) can be done. This study deals with the Shewhart type variability control charts based on auxiliary characteristics for the non-cascading processes, assuming stability of auxiliary parameters. The control chart structures of these variability charts are provided and their performance evaluations are carried out in terms of average run length (ARL), relative average run length (RARL) and extra quadratic loss (EQL) under the normal and t distributed process environments. The comparisons have been made among different variability charts and superiorities are established based on their detection abilities for different amounts of shifts in process dispersion. An illustrative example is also provided in support of the theory, and finally the study ends with concluding remarks and suggestions for future research.     

The economic design of control charts with repair cost depending on detection delay

This study extends Duncan's [1] model to two different manufacturing process models in which the processes continue and discontinue in operations during the search for the assignable cause. A more realistic assumption considered in this paper is that the cost of repair and the net hourly out-of-control income are functions of detection delay. In the continuous model, detection delay is defined as the elapsed time from the time when the shift of the process occurs until it is identified by control charts and the assignable cause is eliminated. The discontinuous model defines detection delay as the time interval from the occurrence of the process shift to the completion of testing a set of samples and interpreting the results. An efficient procedure is developed to determine the optimal designs without using any approximation approach. Thus, the proposed procedure can obtain the truly optimal designs rather than those approximate designs determined by Duncan [1] and other subsequent researchers. This paper illustrates several numerical examples and makes some relevant comparisons. The results indicate that this optimal solution procedure is more accurate than that of Panagos et al. [2]. Also, detection delay is sensitive to the economic design of control charts.     

A new monitoring design for uni-variate statistical quality control charts

In this research, an iterative approach is employed to analyze and classify the states of uni-variate quality control systems. To do this, a measure (called the belief that process is in-control) is first defined and then an equation is developed to update the belief recursively by taking new observations on the quality characteristic under consideration. Finally, the upper and the lower control limits on the belief are derived such that when the updated belief falls outside the control limits an out-of-control alarm is received. In order to understand the proposed methodology and to evaluate its performance, some numerical examples are provided by means of simulation. In these examples, the in and out-of-control average run lengths (ARL) of the proposed method are compared to the corresponding ARL’s of the optimal EWMA, Shewhart EWMA, GEWMA, GLR, and CUSUM[11] methods within different scenarios of the process mean shifts. The simulation results show that the proposed methodology performs better than other charts for all of the examined shift scenarios. In addition, for an autocorrelated AR(1) process, the performance of the proposed control chart compared to the other existing residual-based control charts turns out to be promising.     

Economic design of autoregressive moving average control chart using genetic algorithms

When designing control charts, it is usually assumed that the observations from the process at different time points are independent. However, this assumption may not be true for some production processes, e.g., the continuous chemical processes. The presence of autocorrelation in the process data can result in significant effect on the statistical performance of control charts. Jiang, Tsui, and Woodall (2000) developed a control chart, called the autoregressive moving average (ARMA) control chart, which has been shown suitable for monitoring a series of autocorrelated data. In the present paper, we develop the economic design of ARMA control chart to determine the optimal values of the test and chart parameters of the chart such that the expected total cost per hour is minimized. An illustrative example is provided and the genetic algorithm is applied to obtain the optimal solution of the economic design. A sensitivity analysis shows that the expected total cost associated with the control chart operation is positively affected by the occurrence frequency of the assignable cause, the time required to discover the assignable cause or to correct the process, and the quality cost per hour while producing in control or out of control, and is negatively influenced by the shift magnitude in process mean.     

Multi-criteria design of an X̄ control chart

Control chart designs are widely studied because control charts are not only costly used but also play an important role in improving firms' quality and productivity. Design of control charts refers to the selection of parameters, including sample size, control-limit width, and sampling frequency. In this paper, a possible combination of design parameters is considered as a decision-making unit; it is characterized by three attributes: hourly expected cost, average run length of process being controlled, and detection power of the chart designed with the selected parameters. Accordingly, optimal design of control charts can be formulated as a multiple criteria decision-making (MCDM) problem. To solve the MCDM problem, a solution procedure on the basis of data envelopment analysis is proposed. Finally, an industrial application is presented to illustrate the solution procedure. Also, adjustment to control chart design parameters is suggested when there are process improvements or process deteriorations.     

Multi-criteria design of an control chart

Control chart designs are widely studied because control charts are not only costly used but also play an important role in improving firms' quality and productivity. Design of control charts refers to the selection of parameters, including sample size, control-limit width, and sampling frequency. In this paper, a possible combination of design parameters is considered as a decision-making unit; it is characterized by three attributes: hourly expected cost, average run length of process being controlled, and detection power of the chart designed with the selected parameters. Accordingly, optimal design of control charts can be formulated as a multiple criteria decision-making (MCDM) problem. To solve the MCDM problem, a solution procedure on the basis of data envelopment analysis is proposed. Finally, an industrial application is presented to illustrate the solution procedure. Also, adjustment to control chart design parameters is suggested when there are process improvements or process deteriorations.     

Integrating economics into quality control charts

When control of a manufacturing process is needed, the common tool is Statistical Quality control (SQC). In the past, however, economic factors were the results after employing the SQC charts. Design of control charts refers to the specification of the sample size, the sampling frequency and the control limits for the chart. The authors have tested a model that uses economics as an integral part for the design of an X-bar control chart.

Douglas C. Montgomery developed a computer program for the optimal economic design of an X-bar control chart. The program is based on the cost model proposed by A.J. Duncan.

Montgomery's program was modified to select the optimal design parameters from a table of parameter values. Subroutines were developed to enable the user to enter the number of subgroups and the data points for each subgroup. The economically designed control chart is plotted using standard Graphical Kernel System (GKS) subroutines.     

Performance of t control charts in short runs with unknown shift sizes

Recently, control charts plotting a statistic having a Student’s t distribution have been proposed as an efficient solution to perform Statistical Process Control (SPC) in short production runs where the shift size of the in-control process mean from μ₀ to μ₁ is known a priori. The shift size is usually measured as a multiple δ of the in-control process standard deviation σ₀: but in practice, at the beginning of the production run, both the value of next shift δ and σ₀ are unknown. As a consequence, when the actual shift size differs from the value assumed at the chart design stage, the performance of the control chart can be seriously affected. To overcome this problem, this paper investigates the statistical performance of the Shewhart, EWMA and CUSUM t charts for short production runs when the shift size is unknown and modeled by means of a statistical distribution. An extensive numerical analysis allows the properties of the three charts to be compared and discussed when uniform and triangular distributions are used by quality practitioners to fit the unknown shift size. An illustrative example is utilized to demonstrate a practical implementation of the best performing among the three investigated charts.