Economic statistical design of a T2 control chart with double warning lines

Recent studies have shown that enhancing the common T² control chart by using variable sample sizes (VSS) and variable sample intervals (VSI) sampling policies with a double warning line scheme (DWL) yields improvements in shift detection times over either pure VSI or VSS schemes in detecting almost all shifts in the process mean. In this paper, we look at this problem from an economical perspective, certainly at least as an important criterion as shift detection time if one considers what occurs in the industry today. Our method is to first construct a cost model to find the economic statistical design (ESD) of the DWL T² control chart using the general model of Lorenzen and Vance (Technometrics 1986; 28 :3–11). Subsequently, we find the values of the chart parameters which minimize the cost model using a genetic algorithm optimization method. Cost comparisons of Fixed ratio sampling, VSI, VSS, VSIVSS with DWL, and multivariate exponentially weighted moving average (MEWMA) charts are made, which indicate the economic efficacy of using either VSIVSS with DWL or MEWMA charts in practice if cost minimization is of interest to the control chart user. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimal T2 Control Chart with a Double Sampling Scheme – An Alternative to the MEWMA Chart

Recent studies have shown that a double sampling (DS) scheme yields improvements in detection times of process shifts over variable ratio sampling (VRS) methods that have been extensively studied in the literature. Additionally, a DS scheme is more practical than some of the VRS methods since the sampling interval is fixed. In this paper, we investigate the effect of double sampling on cost, a criterion as important as detection rate. We study economic statistical design of the DS T² chart (ESD DS T²) so that designs are found that are economically optimal but yet meet desired statistical properties such as having low probabilities of false searches and high probabilities of rapid detection of process shifts. Through an illustrative example, we show that relatively large benefits can be achieved in a comparison with the classical T² chart and the statistical DS T² charts with our ESD DS T² approach. Furthermore, the economic performance of the ESD DS T² charts is favorably compared to the MEWMA and other VRS T² control charts in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

Design Strategies for the Multivariate Exponentially Weighted Moving Average Control Chart

The multivariate exponentially weighted moving average (MEWMA) control chart has received significant attention from researchers and practitioners because of its desirable properties. There are several different approaches to the design of MEWMA control charts: statistical design; economic–statistical design; and robust design. In this paper a review and comparison of these design strategies is provided.Copyright © 2004 John Wiley & Sons, Ltd.     

The economic and economic-statistical designs of the Hotelling's T2 chart based on the expected average run length

Existing economic and economic-statistical designs require practitioners to specify the Mahalanobis Distance Shift Size (MDSS) as an exact value. However, practitioners may find it difficult to specify this distance. This article proposes the economic and economic-statistical designs of the Hotelling's T² chart, where practitioners do not have to specify the MDSS. Adopting optimal design parameters based on the wrong MDSS results in a significant increase in cost. In comparison, adopting the optimal design parameters based on the proposed methodology results in a slight increase in cost. This article also studies the effects of different input parameters and statistical constraints.     

Optimal designs of the multivariate synthetic chart for monitoring the process mean vector based on median run length

The average run length (ARL) is usually used as a sole measure of performance of a multivariate control chart. The Hotelling's T², multivariate exponentially weighted moving average (MEWMA) and multivariate cumulative sum (MCUSUM) charts are commonly optimally designed based on the ARL. Similar to the case of univariate quality control, in multivariate quality control, the shape of the run length distribution changes in accordance to the magnitude of the shift in the mean vector, from highly skewed when the process is in‐control to nearly symmetric for large shifts. Because the shape of the run length distribution changes with the magnitude of the shift in the mean vector, the median run length (MRL) provides additional and more meaningful information about the in‐control and out‐of‐control performances of multivariate charts, not given by the ARL. This paper provides a procedure for optimal designs of the multivariate synthetic T² chart for the process mean, based on MRL, for both the zero and steady‐state modes. Two Mathematica programs, each for the zero state and steady‐state modes are given for a quick computation of the optimal parameters of the synthetic T² chart, designed based on MRL. These optimal parameters are provided in the paper, for the bivariate case with sample sizes, nin{4, 7, 10}. The MRL performances of the synthetic T², MEWMA and Hotelling's T² charts are also compared. Copyright © 2011 John Wiley & Sons, Ltd.     

Design of control charts with m‐of‐m runs rules

This paper investigates economic–statistical properties of the X? charts supplemented with m‐of‐m runs rules. An out‐of‐control condition for the chart is either a point beyond a control limit or a run of m‐of‐m successive points beyond a warning limit. The sampling process is modeled by a Markov chain with 2m states. The steady‐state probability for each state and the average run length (ARL) from each state of the Markov chain are derived in explicit formulas. Then the stationary average run length (SALR) is derived so as to develop an economic–statistical model. Using this model, the design parameters are optimized by minimizing the cost function with constraints on the average time to signal (ATS). The X? chart supplemented with m‐of‐m runs rules is compared with the Shewhart X? chart in terms of the SARL and the cost function. Sensitivity of the design parameters with respect to the cost function is also analyzed. General guidelines for implementing the X? chart with m‐of‐m runs rules are presented from those observations. It should be emphasized that supplementing run rules may provide feasible and efficient solutions even if the sample size is limited, while the Shewhart X? chart may not. Copyright © 2009 John Wiley & Sons, Ltd.     

Economic and economic-statistical designs of the side sensitive group runs chart with auxiliary information

This article studies the economic and economic-statistical designs of the auxiliary information (AI) based side sensitive group runs (SSGR-AI) chart. The regression estimator that consists of information not only from the primary variable but also from the auxiliary variable is integrated into the control charting statistic. Optimal designs of the SSGR-AI chart, for the minimization of the expected cost function with and without statistical constraints, are developed based on (i) average run length (ARL) and (ii) expected average run length (EARL). Furthermore, sensitivity analyses are conducted, that is, the impact of various input parameters on the optimal parameters and costs for different values of correlation coefficients (ρ) between the primary and auxiliary variables are investigated. In addition, the effects of incorrect specification of the size of the shift on the optimal cost of the SSGR-AI chart are studied. The comparative study reveals that the SSGR-AI chart is superior to the exponentially weighted moving average-AI (EWMA-AI) and synthetic-AI (Syn-AI) charts, for both designs, by giving the smallest costs.     

An Economic Approach to the Management of High‐Quality Processes

Modern manufacturing developments have forced researchers to investigate alternative quality control techniques for high‐quality processes. The cumulative count of conforming (CCC) control chart is a powerful alternative approach for monitoring high‐quality processes for which traditional control charts are inadequate. This study develops a mathematical model for the economic design of the CCC control chart and presents an application of the proposed model. On the basis of the results of the application, the economic and classical CCC control chart designs of the CCC control chart are compared. The optimal design parameters for different defective fractions are tabulated, and a sensitivity analysis of the model is presented for the CCC control chart user to determine the optimal economic design parameters and minimum hourly costs for one production run according to different defective fractions, cost, time, and process parameters. Copyright © 2012 John Wiley & Sons, Ltd.     

Toward identifying the source of mean shifts in multivariate SPC: a neural network approach

To identify the source(s) of process shifts under a multivariate setting is a challenging problem. Though some statistical techniques have been proposed, they are limited or restricted in their level of success and ease of use. In this paper, we propose a neural-network based identifier (NNI) to detect process mean shifts as well as indicate the variable(s) responsible for the shifts in a process where variables are correlated. Various network configurations and training strategies were investigated to develop an effective network. This research demonstrates how the NNI with a simple network structure, i.e. without any hidden layers, can perform superiorly to the Hotelling T ² chart and comparably to the MEWMA chart in detecting small to moderate shifts for bivariate processes. The run length analysis also indicates that the NNI performs much more stably than the Hotelling T ² chart and the MEWMA chart. One of the great advantages of this approach is that the proposed identifier, aided with the NNI output chart, can indicate the source(s) of the shift(s), i.e. the variable(s) responsible for the shift(s). The NNI output chart allows this monitoring scheme to easily interpret the underlying structures of the process variables.     

Economic and Economic‐Statistical Designs of Phase II Profile Monitoring

In economic design of profiles, parameters of a profile are determined such that the total implementation cost is minimized. These parameters consist of the number of set points, n, the interval between two successive sampling, h, and the parameters of a control chart used for monitoring. In this paper, the Lorenzen–Vance cost function is extended to model the costs associated with implementing profiles. The in‐control and the out‐of‐control average run lengths, ARL₀ and ARL₁, respectively, are used as two statistical measures to evaluate the statistical performances of the proposed model. A genetic algorithm (GA) is developed for solving both the economic and the economic‐statistical models, where response surface methodology is employed to tune the GA parameters. Results indicate satisfactory statistical performance without much increase in the cost of implementation. Copyright © 2013 John Wiley & Sons, Ltd.     

Increasing the Sensitivity of Multivariate EWMA Control Chart

A multivariate exponentially weighted moving average (MEWMA) control chart is used for fast detection of small shifts in multivariate statistical quality control. However, for ease of computation, the MEWMA control chart statistics are computed based on the asymptotic form of their covariance matrix in most cases. Another reason that justifies the design of the MEWMA control chart using the asymptotic covariance matrix is that the chart will be insensitive at start-up since processes are more likely to be away from the target value when the control scheme is initiated due to start-up problems. However, if initial out-of-control conditions are deemed important for quick detection, then the MEWMA statistics should be computed based on the exact covariance matrix, as it leads to a natural fast initial response for the MEWMA chart. It will also be shown in this paper the importance of computing the MEWMA statistics based on the exact form of their covariance matrix to further enhance the MEWMA control chart's sensitivity for detecting small shifts. The MEWMA statistics based on the asymptotic and the exact form of their covariance matrix will be referred to as the asymptotic and the exact MEWMA statistics, respectively. Plots and factors that simplify the design of the exact MEWMA control chart are also given.     

Economic design of X charts for non-normally correlated data

When the X chart is applied to monitor a manufacturing process, three parameters should be determined: sample size, sampling interval between successive samples, and the control limits for the chart. In 1956, Duncan presented the first cost model to determine the three parameters for the X charts, which is called the economic design of X charts. This paper develops the economic design of X charts for non-normally correlated data. An example of juice production process is presented to illustrate the solution procedure. A sensitivity analysis is performed to show the effects of non-normality and correlation coefficient on the optimal design of the chart.     

The joint economic‐statistical design of X and R charts for nonnormal data

We consider the joint economic‐statistical design of X and R control charts under the assumption that the quality measurement and the in‐control time have Johnson and Weibull distributions. The Johnson distribution is general in that it can be made to fit all possible values of skewness and kurtosis. The four parameters—the sample size n, time h between successive samples, and the control factors k₁ and k₂ for the X and R charts—are determined so that the mean hourly loss‐cost is minimized under constraints on the Type I and II error probabilities. We have generalized the Costa model to accommodate the Johnson and Weibull distributions. Sensitivity to nonnormality, shift, and Weibull scale parameter is considered in our analysis. Our sensitivity analysis shows that the optimal design parameters are sensitive to nonnormality. Comparisons of the fully economic and economic‐statistical designs are given. Copyright © 2010 John Wiley & Sons, Ltd.     

Duncan's model for X̄‐control charts: sensitivity analysis to input parameters

Duncan's model is a well‐known procedure to build a control chart with specific reference to the production process it has to be applied to. Although many papers report true applications proving the procedure's noteworthy economic advantages over control charts set purely on the basis of standard statistical criteria, this method is often perceived only as an academic exercise. Perhaps the greater barrier preventing its practical application stems from the difficulty in making cost items explicit. In this paper a sensitivity analysis is proposed for misspecification in the cost parameters for optimal solutions of Duncan's model. While similar contributions published in the literature perform sensitivity analyses with a one‐factor‐at‐a‐time scheme, the original contribution of this paper is represented by the focus given on interactions among changes in values of different cost parameters. The results obtained here denote that all factors significantly affect optimal solutions through quite complicated interactions. This should not, in our opinion, discourage the implementation of Duncan's model, pointing conversely to its robust versions, already available in the current literature. Copyright © 2009 John Wiley & Sons, Ltd.     

Economic design of X̄ and R charts under Weibull shock models

This paper considers the problem of a continuous production process where both the mean and variance are simultaneously monitored by an X̄ and R chart respectively, and generalizes the model of Costa (IIE Transactions 1993; 25 (6):27–33). The product variable quality characteristic is assumed to be normally distributed and the process is subject to two independent assignable causes (such as tool wear‐out, overheating or vibration). One cause changes the process mean and the other changes the process variance. However, the occurrence of one kind of assignable cause does not preclude the occurrence of the other. It is also assumed that the occurrence times of the assignable causes are described by Weibull distributions with increasing failure rates. A cost model is developed and a non‐uniform sampling interval scheme is adopted. A two‐step search procedure is employed to determine the optimal design parameters. The relative contribution of the paper over the results obtained by Costa is addressed. A sensitivity analysis of the model is conducted and the cost savings associated with the use of non‐uniform sampling intervals instead of constant sampling intervals are evaluated. The economic design model is then extended to an economic–statistical design model for achieving desired levels of statistical performance while minimizing the expected cost. Performances of purely economic design and economic–statistical design are compared. Copyright © 2000 John Wiley & Sons, Ltd.     

Economic designs of constrained EWMA and combined EWMA-X¯ control schemes

This research presents a comparison between the cost performance of the Exponentially Weighted Moving Average (EWMA) and the combined EWMA-x¯ control chart schemes. In particular, we explore the impact of constraining the in-control average run length on the optimal cost performance of both schemes. Methodologically, we incorporate traditional expected cost models and study the robustness of the two approaches. In general, there appears to be minimal motivation to combine the use of both charts within the same application. The cost model for the combined chart is not a well-behaved function, and yields varying optimal parameters when the in-control average run length is constrained.     

Performance of the MEWMA-CoDa control chart in the presence of measurement errors

This paper can be considered as an extension of the work of Tran et al (for monitoring compositional data using a multivariate exponentially weighted moving average MEWMA-compositional data [CoDa] chart) by taking into account potential measurement errors that are known to highly affect production processes. A linearly covariate error model with a constant error variance is used to study the impact of measurement errors on the MEWMA-CoDa control chart. In particular, the influence of the device parameters (σ_M,b), the number of independent observations m, and the the number of variables p are investigated in terms of the MEWMA optimal couples (r,H) as well as in terms of their corresponding ARLs. A comparison between the Hotelling-CoDa T² and the proposed chart is made in order to show that the MEWMA-CoDa chart is more efficient in detecting shifts in the presence of measurement errors. A real-life example of muesli production, using multiple measurements for each composition, is used to estimate the parameters and also to demonstrate how the MEWMA-CoDa can handle measurement errors to detect shifts in the process.     

Economic design of the integrated multivariate EPC and multivariate SPC charts

The goal of engineering process control (EPC) is to minimize variability by adjusting some manipulative process variables. The goal of statistical process control (SPC) is to reduce variability by monitoring and eliminating assignable causes of variation. As suggested by Box and Kramer and others, it is possible to reduce both special cause and common cause variations by integrating EPC and SPC. In the integrated multivariate EPC (MEPC) and multivariate SPC (MSPC) charts, we propose some statistical and economic criteria, such as the average Euclidean distance from the target vector and the average quality cost (AQC) to evaluate the performance of the MEPC/MSPC charts. The traditional average run length (ARL), average Euclidean distance and AQC of three MSPC charts are investigated and compared. The results of the simulations show that the MEPC/MGWMA chart is more effective and more economical than both the MEPC/MEWMA chart and the MEPC/Hotelling multivariate chart in detecting small shifts of the mean vector. Copyright © 2006 John Wiley & Sons, Ltd.     

Univariate X¯ control charts for individual characteristics in a multinormal model

The early work on multivariate statistical process control was built upon Hotelling's T ² control chart which was developed to simultaneously monitor the means of correlated quality variables. This chart, however, has a drawback, namely, the problem of identifying the responsible variable(s) when an out-of-control signal occurs. One alternative is to use a separate X¯ control chart for each individual characteristic with equal risks, based on Bonferroni inequality. In this study, we show that, from an economic perspective, it may be desirable to have unequal type I risks for the individual charts, because of different inspection and restoration costs associated with each variable. We obtain their risk ratios, which are measures of relative importance of the variables monitored. Then, based on these risk ratios, we develop computer algorithms for finding the exact control limits for individual variables from a multinormal distribution, in the sense that the overall type I risk of the charts is equal to the desired value. Numerical studies show that the proposed methods give optimal or near-optimal results from an economic as well as statistical point of view.