A brief introduction to basic multivariate economic statistical process control

[摘要] ENGLISH ABSTRACT: Statistical process control (SPC) plays a very important role in monitoring and improvingindustrial processes to ensure that products produced or shipped to the customer meet therequired specifications. The main tool that is used in SPC is the statistical control chart. Thetraditional way of statistical control chart design assumed that a process is described by asingle quality characteristic. However, according to Montgomery and Klatt (1972) industrialprocesses and products can have more than one quality characteristic and their joint effectdescribes product quality. Process monitoring in which several related variables are ofinterest is referred to as multivariate statistical process control (MSPC). The most vital andcommonly used tool in MSPC is the statistical control chart as in the case of the SPC. Thedesign of a control chart requires the user to select three parameters which are: sample size,n , sampling interval, h and control limits, k.Several authors have developed control chartsbased on more than one quality characteristic, among them was Hotelling (1947) whopioneered the use of the multivariate process control techniques through the development of a2 T -control chart which is well known as Hotelling 2 T -control chart.Since the introduction of the control chart technique, the most common and widely usedmethod of control chart design was the statistical design. However, according to Montgomery(2005), the design of control has economic implications. There are costs that are incurredduring the design of a control chart and these are: costs of sampling and testing, costsassociated with investigating an out-of-control signal and possible correction of anyassignable cause found, costs associated with the production of nonconforming products, etc.The paper is about giving an overview of the different methods or techniques that have beenemployed to develop the different economic statistical models for MSPC.The first multivariate economic model presented in this paper is the economic design of theHotelling‟s 2 T -control chart to maintain current control of a process developed byMontgomery and Klatt (1972). This is followed by the work done by Kapur and Chao (1996)in which the concept of creating a specification region for the multiple quality characteristicstogether with the use of a multivariate quality loss function is implemented to minimize totalloss to both the producer and the customer. Another approach by Chou et al (2002) is alsopresented in which a procedure is developed that simultaneously monitor the process meanand covariance matrix through the use of a quality loss function. The procedure is based on the test statistic 2ln L and the cost model is based on Montgomery and Klatt (1972) as wellas Kapur and Chao‟s (1996) ideas. One example of the use of the variable sample sizetechnique on the economic and economic statistical design of the control chart will also bepresented. Specifically, an economic and economic statistical design of the 2 T -control chartwith two adaptive sample sizes (Farazet al, 2010) will be presented. Farazet al (2010)developed a cost model of a variable sampling size 2 T -control chart for the economic andeconomic statistical design using Lorenzen and Vance‟s (1986) model.There are several other approaches to the multivariate economic statistical process control(MESPC) problem, but in this project the focus is on the cases based on the phase II stadiumof the process where the mean vector, and the covariance matrix, have been fairly wellestablished and can be taken as known, but both are subject to assignable causes. This latteraspect is often ignored by researchers. Nevertheless, the article by Farazet al (2010) isincluded to give more insight into how more sophisticated approaches may fit in withMESPC, even if the mean vector, only may be subject to assignable cause.Keywords: control chart; statistical process control; multivariate statistical process control;multivariate economic statistical process control; multivariate control chart; loss function.

[发布机构] Stellenbosch University