An economic design of double sampling $\overline{X}$ charts for correlated data using genetic algorithms

Abstract

Double sampling $\overline{X}$ control chart (DS) which is a Shewhart-type chart can reduce sample size and detect small process shift fast. In process monitoring of real industry, observations may be interdependent and correlated, and an original DS design will occur high cost for the wrong determination of the process state. In this study, an economic design model of DS is developed based on Yang and Hancock’s assumption of correlation and Lorenzen and Vance’s cost model to determine sample size, sampling interval, and coefficients of control limits and warring limits. The genetic algorithms (GAs) are applied to solve this economic design model of DS for the determination of the optimal parameters. A real example of IC packaging process is given to illustrate the model application. Sensitivity analysis shows the influence of different model parameters on the DS chart designs.

Introduction

Statistical methods are often used in process monitoring, where the technology of control chart is one of the major tools. When an assignable cause occurs, the process will have variation. The control chart can detect the process variation and signal to operators immediately for finding out the cause of variation.

In 1942, Dr. Shewhart developed $\overline{X}$ control chart, in which $\mu \pm 3\sigma$ is used to set control limits for monitoring process means. Shewhart’s $\overline{X}$ control chart is as easy to set up as it is to use. Therefore, the chart is widely applied in industries. However, the capability of this control chart is limited in smaller mean shift, and there have been many studies to improve this disadvantage, which in turn encouraged the modification of Shewhart control chart, such as variable sample size $\overline{X}$ charts (VSS) that improve the sample size in small shift processes (Costa, 1994), as well as variable sampling intervals $\overline{X}$ charts (VSI) and variable parameters $\overline{X}$ charts (VP) that improve the sensitivity in detection of small mean shift (Costa, 1999, Reynolds et al., 1988). In addition, CUSUM and EWMA have shown good performance in small shift process control, but their control procedures are complicated and they are not as easy to use as Shewhart’s control chart.

Depending on the production process, the needs for process control are different. The proper selection of control chart for process control increases production efficiency effectively. Daudin (1992) applied the concept of double sampling plans to Shewhart’s $\overline{X}$ control chart and adopted the two-stage Shewhart’s $\overline{X}$ control chart to monitor process mean so that it was called as double sampling $\overline{X}$ control chart (DS). DS has successfully improved the sensitivity in the detection of small mean shift processes and reduced the sample size (Daudin, 1992, He et al., 2002). Costa (1994) discussed that the sample size of DS is more economic than that of VSS in detecting small mean shift processes. For process monitoring with higher inspection costs or destructive tests, DS is a good choice.

When control chart is used for process control, the observations must be independent and must follow the assumptions of normal distribution. If these assumptions are violated, control charts will be easy to wrong determine the process state, leading to production efficiency reduction. Neuhardt (1987) pointed out that for process of the manufacturing industry, the observations may violate the assumption of independence and may have correlation. For example, data collection from multiple but similar measurements on a single product such as several cavities on a single casting, multiple pins on an integrated circuit chip or multiple contact pads on a single machine amount, and measurements of the collected data may be correlated (Grant & Leavenworth, 1996). By further developing the ideas of Neuhardt, 1987, Yang and Hancock, 1990 explored the process control performance of Shewhart’s control chart in process control with correlated data. At the end, it is found that there is a higher probability of occurrence of false alarms in control chart when the observations are highly correlated. A higher probability of occurrence of false alarms easily increases the process control cost and then redesign of the control chart is required for the cost reduction. From this, if DS is used to control a process with correlated samples, it has to be redesigned.

Duncan (1956) started by constructing his model and solving for three key parameters, the control limit coefficient of Shewhart’s $\overline{X}$ control chart, sample size and sampling interval. Duncan’s approach (1956) is called economic design of control chart. Lorenzen and Vance (1986) provided one unified approach to the economic design of control charts. They constructed a general cost model that applied to all control charts, regardless of the statistic used. Later, several studies (e.g., De Magalhães et al., 2001, Montgomery et al., 1995, Prabhu et al., 1997, Torng et al., 1995 etc.) tried to modify Lorenzen and Vance’s (1986) model to determine the design of other control charts and to lower the costs of process control. In the past, DS only dealt with statistical design methods and did not take the concept of costs into account, resulting in failure to reduce DS’s process control costs. Besides, the optimal design of correlated data has not been investigated.

The design of DS $\overline{X}$ control chart has to determine the control limit coefficients and sample size of two-stage charts, warring limit coefficient and sampling interval. This study intends to explore the design method of DS for correlated data from economic point of view and to modify the cost model of Lorenzen and Vance (1986) based on DS’s control procedure. The correlation model of Yang and Hancock (1990) and the assumption of normal distribution are included in the cost model. In the past, there were many successful examples of solving economic designs of control chart using genetic algorithms, including Celano and Fichera, 1999, Chou et al., 2006, Vommi and Seetala, 2007. Hence, genetic algorithms are used to determine DS parameters of minimized costs. Finally, a real case of IC packaging is used to illustrate the application of the model and the solving procedure. Through sensitivity analysis, the effects of variation in cost model parameters on DS are studied.