Control charts are powerful tools to monitor process. Design of control chart is usually performed through two ways, statistical and economical approaches Sometimes, a relationship, commonly called profile, between a response variable and one or more explanatory variables is defined for process monitoring. Various control charts proposed up to now to monitor Profiles have not been designed economically. The importance of an economic design of Profiles is significant. This is due to difference in some concepts and numerous of parameters. In this paper, for one of the monitoring approaches, which use three exponentially weighted moving average control charts simultaneously, cost function is presented and this approach is redesigned economically. In addition, the average run length calculating model is developed using the Markov chain method. This economic design is applied for a numerical example and solved by Nelder-Mead downhill Simplex method and the optimal values of parameters are calculated.

In some applications a single variable, either a process variable or product variable, characterizes the state of the process. In the other applications, multiple variables characterize the state of the process. However, in some practical situations, the quality of a process or product is characterized by a relationship between two or more variables instead of by the distribution of a single quality characteristic. This relationship, which can be Linear, nonLinear or even a complicated model, is referred to as profile by researchers. Up to now, several methods have been proposed for monitoring Simple Linear Profiles in both Phases I and II. In this paper, for improving phase II monitoring of Linear Profiles, a method has been proposed which applies Cumulative Sum control charts. Average run length criterion and simulation studies are used in order to evaluate the performance of the proposed method. The results show the suitable performance of the proposed method. Finally, the effect of reference value on the performance of the proposed method is evaluated.

In some applications of statistical process monitoring, a quality characteristic can be characterized by Linear regression relationships between several response variables and one explanatory variable, which is referred to as a “ multivariate Simple Linear profile. ” It is usually assumed that the process parameters are known in Phase II. However, in most applications, this assumption is violated; the parameters are unknown and should be estimated based on historical data sets in Phase I. This study aims to compare the effect of parameter estimation on the performance of three Phase II approaches for monitoring multivariate Simple Linear Profiles, designated as MEWMA, MEWMA_3 and MEWMA∕  2. Three metrics are used to accomplish this objective: AARL, SDARL and CVARL. The superior method may be different in terms of the AARL and SDARL metrics. Using the CVARL metric helps practitioners make reliable decisions. The comparisons are carried out under both in-control and out-of-control conditions for all competing approaches. The corrected limits are also obtained by a Monte Carlo simulation in order to decrease the required number of Phase I samples for parameter estimation. The results reveal that parameter estimation strongly affects the in-control and out-of-control performance of monitoring approaches, and a large number of Phase I samples are needed to achieve a parameter estimation that is close to the known parameters. The simulation results show that the MEWMA and MEWMA∕  2 methods perform better than the MEWMA_3 method in terms of the CVARL metric. However, the superior approach is different in terms of AARL and SDARL.

If the quality of a process is described using a Linear functional relationship betweenthe response variable and independent variables, such a relationship is called the profile. Today, with the development of manufacturing technologies, multistage processes have found a special position in manufacturing companies and industriesIn the present paper, we consider a multistage process with AR(1) auto-correlatedSimple Linear profile in each stage and address the effect of both auto-correlationand cascade property on the efficiency of common monitoring procedures. Toeliminate the effect of auto-correlation, we used a transformation method as a remedial measure at first. Then, an approach based on the U statistic is applied toeliminate the cascade effect. Next, a modified T 2 control chart is proposed to monitor the process in the second stage. The performance of the proposed controchart is evaluated in terms of the average run length criterion. The simulationstudies show that the proposed control chart perform satisfactorily.

Assuming a first-order auto-regressive model for the auto-correlation structure between observations, in this paper, a transformation method is first employed to eliminate the effect of auto-correlation. Then, a maximum likelihood estimator (MLE) of a step change in the parameters of the transformed model is derived and three separate EWMA control charts are used to monitor the parameters of the profile. The performance of the proposed change-point estimator is next compared to the one of the built-in change-point estimator of EWMA control chart through some simulation experiments. The results show that the proposed MLE of the change point accurately estimates the true change point and outperforms the built-in estimator of EWMA chart for almost all shift values and auto-correlation coefficients, while the built-in estimator of EWMA chart, in general, underestimates the true change point.

In some applications, performance of a process or quality of a product is characterized by a relationship between a response variable and one or more explanatory variables, referred to as pro, le in the literature. Certain methods have been developed to monitor various pro, les. On the other side, nowadays due to diversity of customer demand and short time for presenting products in market, manufacturing strategy is focused on short run processes characterized by high diversity and low volume. Therefore, statistical process control for such processes, due to inspection restrictions in a short period is a special practice. In such circumstances, control charts in Phase I cannot be performed and correct estimations are not available for estimating process mean and standard deviation. To overcome the situation, self-starting methods are developed to update the parameter estimations along with new observations and simultaneous checks of the out-of-control conditions. Hence, implementing traditional control charts for monitoring short run processes is not practical, and new methods and control charts should be developed to monitor such processes. In this paper with aggregating two above-mentioned subjects, quality characteristics which are pertained to short run processes and which are modeled by Simple Linear pro-, les, have been monitored. Suitable methods and new control charts are developed to monitor process e, ectively. In this paper, we focus on monitoring residuals and propose new control charts to monitor mean and dispersion of residuals simultaneously. In order to monitor residuals in short run processes whose quality characteristics are modeled by Simple Linear pro, les, we propose two control charts for monitoring mean and one control chart for monitoring standard deviation. Then, with combination of these control charts, we develop two distinct control charts named QMCC and TMCC to monitor mean and variance of residuals concurrently. Performance of the proposed control charts have been compared with competitor control chart using simulation studies and average run length (ARL) criterion. The results of simulation studies show that our proposed control charts in some parameters have better performance compared to competitive control chart under moderate and large shifts in terms of out-of-control ARLs.

In many processes in real practice at the start-up stages the process parameters are not known a priori and there are no initial samples or data for executing Phase I monitoring and estimating the process parameters. In addition, the practitioners are interested in using one control chart instead of two or more for monitoring location and variability of processes. In this paper, we consider a Simple Linear profile in which the relationship between a response variable and one explanatory characterizes the quality of a process. We proposed a self-starting Max-CUSUM control chart based on recursive residuals to monitor mean vector (including intercept and slope) and variability (variance of error term) of a Simple Linear profile simultaneously from the start-up stages of the process. We developed Max-CUSUM control chart to monitor Simple Linear profile in Phase II. Then, we compared our proposed control charts with the best one in the literature through simulation studies. The simulation results showed that our proposed control charts have better performance compared to competitive control charts under moderate and large shifts in terms of out-of-control (OC) ARLs. Finally, the application of the proposed self-starting control chart is illustrated through a real case in the leather industry.

In most of the researches in the area of profile monitoring, quality of a process is described by a relationship between a response variable and one explanatory variable, referred to as Simple Linear profile in the literature. Most of the papers in this field have assumed that observations within each profile are independent; however, the independency between the observations can be violated due to time collapse between two successive samplings in many real applications. On the other hand, usually real time of changes in process (change point) is different from the time control charts alarm the process is out-of-control. Finding the change point in the process saves time and money to find out root causes of the problem in the process. This paper specifically assumes that quality of process is modeled by using an AR (1) auto correlated Simple Linear profile. Then, the step change point of the process is estimated by using maximum likelihood and clustering methods after getting a signal from the T2 hotelling control chart in Phase II. Performance of the proposed methods is compared by using simulation studies. Finally, an application of the proposed methods is shown through a real case.

In many circumstances, the quality of a process or product is best characterized by a given mathematical function between a response variable and one or more explanatory variables that is typically referred to as profile. There are some investigations to monitor auto correlated Linear and nonLinear Profiles in recent years. In the present paper, we use the Linear mixed models to account autocorrelation within observations which is gathered on phase II of the monitoring process. We undertake that the structure of correlated Linear Profiles simultaneously has both random and fixed effects. The work enhanced a Hotelling’s T2 statistic, a multivariate exponential weighted moving average (MEWMA), and a multivariate cumulative sum (MCUSUM) control charts to monitor process. We also compared their performances, in terms of average run length criterion, and designated that the proposed control charts schemes could effectively act in detecting shifts in process parameters. Finally, the results are applied on a real case study in an agricultural field.