Extra Quadratic Loss Research Articles

Design of a t‐chart using generalized multiple dependent state sampling

AbstractIn this article, a new t‐chart based on generalized multiple dependent state (GMDS) sampling is proposed for efficient monitoring of a process by assuming that the time between events follows the exponential distribution. The proposed t‐chart has two pair of control limits and utilizes the past sample information with the current sample information. The control chart coefficients are estimated by considering different values of the in‐control average run lengths. The proposed t‐chart is compared with the existing chart by using the out‐of‐control average run length and extra quadratic loss function. The comparison reveals that the proposed charting strategy has better shift detection ability in process mean. An illustrative example is given for the practical implementation of the proposed chart.

Location charts based on ranked set sampling for normal and non‐normal processes

AbstractControl charts, based on ranked set sampling schemes, had been proposed recently for efficient monitoring of process location. All the proposals in the literature are based on the ideal assumption of normally distributed quality characteristics. No study as of yet investigated the performance of location charts based on ranked set sampling for non‐normal processes. In this study, we investigated the location chart based on simple random sampling (SRS) and three well‐known rank‐based schemes ie, ranked set sampling (RSS), median ranked set sampling (MRSS), and extreme ranked set sampling (ERSS), considering normal and a variety of non‐normal parent distributions. Both heavy‐tailed symmetric and skewed cases have been considered in this study. The performance of the charts is evaluated using average run length (ARL), extra quadratic loss (EQL), and relative ARL (RARL) measures. A real life example is also presented that details the monitoring of pH levels in water for an experiment conducted to study the reproduction of Mysids. The study will help quality practitioners to choose the chart based on an efficient sampling scheme for normal and non‐normal processes.

An adaptive approach to EWMA dispersion chart using Huber and Tukey functions

AbstractRandom causes are vital part of every process in manufacturing and nonmanufacturing environments, and these do not affect the product features. Special causes, on the other hand, come because of some burden(s) in a process and requires special attention; otherwise, it ruins the products excellence. Special causes are categorized into small, moderate, and large shifts and are handled by statistical quality control charts. The Shewhart control chart is well known for large shifts, while the cumulative sum and exponentially weighted moving average are more effective in detecting small to moderate shifts. However, in practice, many processes require the simultaneous monitoring of both the small to the large shifts. In this study, we have designed an adaptive EWMA for dispersion parameter in connection with Huber and Tukey's bisquare functions. The performance measures used in this study include average run length, extra quadratic loss, relative average run length, and performance‐comparison index. We have observed that the study proposals are good competitors to the other counter parts for an efficient monitoring of shifts of varying amounts. An illustrative example using real data is given to demonstrate the implementation of the study proposal.

Modified side-sensitive synthetic double sampling monitoring scheme for simultaneously monitoring the process mean and variability

This paper develops a new synthetic double sampling (SDS) monitoring scheme based on the modified side-sensitive (MSS) design for jointly monitoring the process mean and variability. We first give the operation of the proposed scheme and secondly, the closed-form expression of the probability of declaring a specific sampling stage as “conforming” or “nonconforming”. Thirdly, we evaluate the performance of the newly proposed scheme in terms of the zero- and steady-state out-of-control average run-length, standard deviation of the run-length, average number of observations to signal, average extra quadratic loss, average ratio of the average run-length and performance comparison index. Finally, we compare the performance of the new monitoring scheme with some existing monitoring schemes. It is observed that the proposed scheme has attractive run-length properties and outperforms the existing joint X¯ and S SDS monitoring scheme as well as all other competing schemes in many situations.

New Shewhart-type synthetic bar{X} control schemes for non-normal data

In this paper, Burr-type XII bar{X} synthetic schemes are proposed as an alternative to the classical bar{X} synthetic schemes when the assumption of normality fails to hold. First, the basic design of the Burr-type XII bar{X} synthetic scheme is developed and its performance investigated using exact formulae. Secondly, the non-side-sensitive and side-sensitive Burr-type XII bar{X} synthetic schemes are introduced and their zero-state and steady-state performances, in terms of the average run-length and expected extra quadratic loss values, are investigated using a Markov chain approach. Thirdly, the proposed schemes are compared to the existing classical runs-rules and synthetic bar{X} schemes. It is observed that the proposed schemes have very interesting properties and outperform the competing schemes in many cases under symmetric and skewed underlying process distributions. Finally, an illustrative real-life example is given to demonstrate the design and implementation of the proposed Burr-type XII bar{X} synthetic schemes.

Double exponentially weighted moving average control chart with supplementary runs-rules

ABSTRACTEnhancing the sensitivity of control charts for detecting small process shift is desirable and may be done in different ways. In this paper, we propose the double exponentially weighted moving average (DEWMA) control chart using the 2-of-2 and 1-of-1 or 2-of-2 runs-rules schemes for monitoring process mean shifts. The performances of the resulting control schemes are investigated in terms of the average run-length (), standard deviation of the run-length () and selective percentiles of the run-length (). In addition, the overall performance of the proposed runs-rules charts are examined and compared with those of the existing charts in terms of the , average extra quadratic loss (), average ratio of the average run-length () and performance comparison index () values. It is observed that the proposed control schemes are more efficient in many situations and improve the ability of the existing DEWMA schemes in detecting small, moderate and large mean shifts.

An Assorted Design for Joint Monitoring of Process Parameters: An Efficient Approach for Fuel Consumption

Due to high fuel consumption, we face the problem of not only the increased cost, but it also affects greenhouse gas emission. This paper presents an assorted approach for monitoring fuel consumption in trucks with the objective to minimize fuel consumption. We propose a control charting structure for joint monitoring of mean and dispersion parameters based on the well-known max approach. The proposed joint assorted chart is evaluated through various performance measures such as average run length, extra quadratic loss, performance comparison index, and relative average run length. The comparison of the proposed chart is carried out with existing control charts, including a combination of X and S, the maximum exponentially weighted moving average (Max-EWMA), combined mixed exponentially weighted moving average-cumulative sum (CMEC), maximum double exponentially weighted average (MDEWMA), and combined mixed double EWMA-CUSUM (CMDEC) charts. The implementation of the proposed chart is presented using real data regarding the monitoring of fuel consumption in trucks. The outcomes revealed that the joint assorted chart is very efficient to detect different kinds of shifts in process behaviors and has superior performance than its competitor charts.

Side-sensitive synthetic double sampling <i><span style="text-decoration: overline">X</span><i> control charts

This paper develops two new synthetic double sampling (SDS) charts based on the revised side-sensitive (RSS) and modified side-sensitive (MSS) schemes for monitoring the location process parameter. In this paper, we first give the operation of the proposed charts and secondly, the exact form expression of the probability of declaring a sampling stage as 'conforming'. Thirdly, we investigate the zero-state and steady-state performances of the proposed charts in terms of the out-of-control average run-length, average sample size, average number of observations to signal and average extra quadratic loss. Finally, we compare the performance of the new charts with some well-known charts. It is observed that the proposed charts have attractive zero-state and steady-state properties and outperform the existing SDS chart and all other competing charts in many situations. An example of a real-life application is given to facilitate the design and implementation of the proposed charts. [Received: 28 February 2017; Revised: 14 March 2018; Revised: 28 August 2018; Accepted: 4 October 2018]

Optimal CUSUM and adaptive CUSUM charts with auxiliary information for process mean

ABSTRACTThe CUSUM chart is good enough to detect small-to-moderate shifts in the process parameter(s) as it can be optimally designed to detect a particular shift size. The adaptive CUSUM (ACUSUM) chart provides good detection over a range of shift sizes because of its ability to update the reference parameter using the estimated process shift. In this paper, we propose auxiliary-information-based (AIB) optimal CUSUM (OCUSUM) and ACUSUM charts, named AIB-OCUSUM and AIB-ACUSUM charts, using a difference estimator of the process mean. The performance comparisons between existing and proposed charts are made in terms of the average run length (ARL), extra quadratic loss and integral relative ARL measures. It is found that the AIB-OCUSUM and AIB-ACUSUM charts are more sensitive than the AIB-CUSUM and ACUSUM charts, respectively. Moreover, the AIB-ACUSUM chart surpasses the AIB-OCUSUM chart when detecting a range of mean shift sizes. Illustrative examples are given to support the theory.

On designing CUSUM charts using ratio-type estimators for monitoring the location of normal processes

Control chart is one of the important tools in statistical process control (SPC) that plays a significant role in monitoring and identifying disturbances of any production process. The Shewhart, cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) are commonly used control charts for detecting process shifts. CUSUM and EWMA chart are more sensitive in detecting smaller shifts whereas the typical Shewhart chart is known to be only sensitive to large process shifts. The present study incorporates ratio-type estimators based on auxiliary information in CUSUM structure as a substitute of simple mean estimator for monitoring process location. These estimators are more efficient than the simple mean estimator in the presence of high correlation between the study and the auxiliary variables. Average run length (ARL), standard deviation of run length (SDRL) and extra quadratic loss (EQL) are used to measure the performances of the proposed charts. The performance comparison of the proposed chart with the existing CUSUM, CUSUM-FIR and many other control charts are made by using out-of-control ARL. The comparison revealed the superiority of the suggested charts over the other existing charts. An illustrative example is also provided for the performance evaluation of the proposed charts.

On auxiliary information-based control charts for autocorrelated processes with application in manufacturing industry

Multivariate autoregressive (MAR) models are an attractive choice for applications in the processes related to finance, medical, and industry. For the monitoring of such processes, control chart is the most important and widely used tool of statistical process control tool kit. Moreover, the presence of auxiliary information helps in better estimation of different process parameters. The literature on use of auxiliary variables in control charts assumes independence of observations. In practice, we may come across processes dealing with autocorrelated outcomes. In such situations, a control chart usually produces high false alarms and exhibits slow detection of shifts when the process is out-of-control. This study intends to suggest some auxiliary information-based Shewhart charts for autocorrelated univariate and bivariate AR(1) processes. The proposed structures take into account the autocorrelation structure and offer more effective designs of control charts for efficient process monitoring. The performance measures used in this study are based on run length measures such as average run length, extra quadratic loss, relative average run length and performance comparison index. A detailed performance analysis is carried out to sort out the best performing charts. In addition, we have considered an application from a manufacturing process to demonstrate the implementation of the proposed charting structures in real scenario.

Simultaneous monitoring of linear profile parameters under progressive setup

Abstract In many manufacturing or service processes, we come across different quality characteristics that govern the process behavior. These characteristics are categorized as the main quality characteristics (study variables) and the supporting or explanatory characteristics. There is always a possibility that some of the explanatory variables offer a relationship with the study variable which is known as profiles. The monitoring of study variable which is linearly associated with an explanatory variable is termed as simple linear profiles. In this study, we intend to design an efficient memory type structure based on progressive mean for the simultaneous monitoring of linear profile parameters. The performance of proposed scheme (PM_3) and its counterparts (i.e. EWMA_3 chart, Hotelling T 2 chart, EWMA/R chart and Shewhart_3 chart) are evaluated using some useful performance measures such as average run length (ARL), relative average run length (RARL), sequential relative average run length (SRARL), extra quadratic loss (EQL) and sequential extra quadratic loss (SEQL). In the presence of shifts in linear profile parameters, the findings depict that PM_3 chart has better detection ability as compared to counterpart charts. A case study related to Queen size problem is also discussed to highlight the importance of the newly proposed control chart.

A new approach to design efficient univariate control charts to monitor the process mean

AbstractDesigning of efficient control charts for monitoring manufacturing processes and identifying assignable cause of variations is of constant interest in statistical process control. The present study incorporates neoteric ranked set sampling to design more efficient Shewhart, cumulative sum, and exponentially weighted moving average control charts for monitoring the mean of normal process. Using Monte Carlo simulations, the performance of the proposed charts is evaluated and compared with competing control charts on the basis of average run length and standard deviation of the run length for detection of shift of a specific magnitude. The extra quadratic loss is used to assess the overall performance of the control charts to detect shifts of various magnitudes. To further support the findings, a simulation study is carried out which revealed the supremacy of the proposed control charts over the existing charts. A real data set from a combined cycle power plant is used to demonstrate the application of the proposed control charts.

A progressive approach to joint monitoring of process parameters

Abstract Process monitoring is a continuous phenomenon and it needs careful attention for an improved quality of output. Location and dispersion parameters play a vital role in regulating every process and it requires a timely detection of any change in their stable behaviors. Nowadays, practitioners prefer a single charting setup that offers better ability to detect joint shifts in the process parameters. In this study, we propose a new parametric memory-type charting structure based on progressive mean under max statistic, namely Max-P chart, for the joint monitoring of location and dispersion parameters. Assuming normality of the quality characteristics of interest this study provides an extensive comparison between the proposed chart and some existing schemes for joint monitoring of process location and dispersion parameters. We use run length properties for the performance analysis of different schemes under investigation in this study. These properties include individual and overall measures (average run length, standard deviation run length, extra quadratic loss, relative average run length, and performance comparison index) for comparative analysis. The study findings reveal that the newly proposed Max-P monitoring scheme offers relatively better performance to detect shifts in the process parameter(s). A real-life application of is also included from electrical engineering where the monitoring of the voltage of photovoltaic system is desired. The proposed scheme also offers better detection ability to identify special causes in the parameters of electrical process.

EWMA-type scheme for monitoring location parameter using auxiliary information

Abstract Control charts are statistical methods used to detect shifts in the location parameter of a process that is monitored over time. Here, we propose an improvement to the performance of the classical exponentially weighted moving average (EWMA) control charts, by making use of auxiliary information that is correlated with the process variable. We present an EWMA P and its modifications: EWMA-type control charts based on a product estimator where the location parameter of the process is monitored using an auxiliary variable. The charts are developed using different sampling schemes: simple random sampling, ranked set sampling (RSS) and median ranked set sampling (MRSS), and we evaluate their performance using average run length, and other performance measures such as extra quadratic loss and relative average run length. It is observed that the proposed control charts are performing better than the classical EWMA control chart in monitoring shifts in the location parameter of a process, especially when a strong negative correlation exists between the process and the auxiliary variables. They are particularly efficient in detecting small to moderate shifts in the process.

Monitoring type‐2 censored reliability data in multistage processes

AbstractIn this paper, we propose control charts to monitor the Weibull scale parameter of type‐2 censored reliability data in multistage processes. A cumulative sum control chart and 2 exponentially weighted moving average control charts based on conditional expected values are devised to detect decreases in the mean level of reliability‐related quality characteristic. The proposed control schemes are based on standard smallest extreme value distributions derived from Weibull processes to effectively account for the cascade property, which is the main characteristic of multistage processes. Subsequently, simulation study is conducted to evaluate the performance of the control charts using average run length criterion. Extra quadratic loss, performance comparison index, and relative average run length are also used to compare the detect ability of our proposed monitoring procedures. Moreover, sensitivity analysis is done to study the impact of failure number in the sample size and to investigate the robustness of the proposed monitoring procedures against the shift in the previous stage. Finally, a real case study in a glass bottle–making company is investigated to illustrate the performance of the competing control charts. The results reveal the superiority of the cumulative sum control chart.

Distribution‐free mixed cumulative sum‐exponentially weighted moving average control charts for detecting mean shifts

AbstractIn this paper, we propose distribution‐free mixed cumulative sum‐exponentially weighted moving average (CUSUM‐EWMA) and exponentially weighted moving average‐cumulative sum (EWMA‐CUSUM) control charts based on the Wilcoxon rank‐sum test for detecting process mean shifts without any distributional assumption of the underlying quality process. The performances of the proposed charts are measured through the average run‐length, relative mean index, average extra quadratic loss, and average ratio of the average run‐length and performance comparison index. It is found that the proposed charts perform better than its counterparts considered in this paper under non‐normal distributions and outperform the classical mixed CUSUM‐EWMA and EWMA‐CUSUM charts in many cases under the normal distribution. The effect of the phase I sample size is also investigated on the phase II performance of the proposed charts. A numerical illustration is given to demonstrate the implementation and simplicity of the proposed charts.

Monitoring of serially correlated processes using residual control charts

Control charts act as the most important tool for monitoring of process parameters. The assumption of independence that underpins the implementation of the charts is violated when process observations are correlated. The e ect of this issue can lead to the malfunctioning of the usual control charts by causing a large number of false alarms or slowing the detection ability of the chart in unstable situations. In this paper, weinvestigated the performance of the Mixed EWMA-CUSUM and Mixed CUSUM-EWMA charts for the ecient monitoring of autocorrelated data. The charts are applied to the residuals obtained from tting an autoregressive (AR) model to the autocorrelated observations. The performance of these charts is compared with the performances of the residual Shewhart, EWMA, CUSUM, combined Shewhart-CUSUM, and combinedShewhart-EWMA charts. Performance criteria such as Average Run Length (ARL) and Extra Quadratic Loss (EQL) are used for the evaluation and comparison of the charts. Illustrative examples are presented to demonstrate the application of the charts to serially correlated observations

Optimization design of the CUSUM and EWMA charts for autocorrelated processes

AbstractThe traditional control charts produce frequent false alarm signals in the presence of autocorrelation. The implementation of the modified chart scheme is a way of handling the problem of autocorrelation in control charts. In modified charts, the standard control limits of the traditional charts are adjusted to offset the influence because of the autocorrelation. The exponentially weighted moving average– and cumulative sum–modified charts are 2 widely used charts for monitoring autocorrelated data. These charts have design parameters in their formulation, and the choice of these parameters play significant roles in the detection of out‐of‐control situations. In reality, the magnitude of the mean shift is uncertain, and this leads to a difficulty in the choice of design parameters by the practitioner. The use of optimal parameters can enhance process performance in such situations. In this paper, we determine optimal design parameters for the charts using an exhaustive search procedure. In the optimization process, we determine the parameters that produce the smallest extra quadratic loss (EQL) value for each autocorrelation coefficient. This criterion measures the anticipated loss attributed to poor quality in the process. The loss in quality is lowered by minimizing the EQL and the combination of parameters in the chart that yields the smallest EQL has a better detection ability over the entire shift range. For the purpose of this work, we concentrate on autocorrelation that can be specifically modelled with autoregressive models. This article provides the practitioner with optimal parameters that can be used to enhance the overall effectiveness of the charts over an entire shift range.

Mixed EWMA-CUSUM and mixed CUSUM-EWMA modified control charts for monitoring first order autoregressive processes

In practice most processes are known to produce autocorrelated observations. Autocorrelation degrades the performance of control charts by producing frequent false alarms when the process is stable or makes the charts respond slowly to the detection of out-of-control states. The effects due to autocorrelation can be eliminated by using modified charts. In this procedure, the control limits of the traditional charts are adjusted to account for the autocorrelation. In this paper, we present the Mixed EWMA-CUSUM and Mixed CUSUM-EWMA modified charts for monitoring correlated data. The performance of these charts are compared to existing modified charts such as the Shewhart, CUSUM, EWMA, combined Shewhart-CUSUM and combined Shewhart-EWMA schemes using the average run length, extra quadratic loss and relative average run length measures. Examples are given to illustrate how the charts perform.