Economic Design of Variable Sampling Interval X-Bar Control Charts for Monitoring Correlated Non-Normal Samples Seyed Taghi Akhavan Niaki, Ph.D.1 Professor of Industrial Engineering, Sharif University of Technology P.O. Box 11155-9414 Azadi Ave., Tehran 1458889694 Iran Phone: (+9821) 66165740, Fax: (+9821) 66022702, E-Mail:
[email protected] Fazlollah Masoumi Gazaneh, M.Sc. Department of Industrial Engineering, Islamic Azad University (South-Tehran Branch), Tehran, Iran, E-Mail:
[email protected] Moslem Toosheghanian, M.Sc. Department of Industrial Engineering, Iran University of Science and Technology
[email protected]
Abstract Recent studies have shown the X-bar control chart with variable sampling interval detects shifts in the process mean faster than the traditional X-bar chart. These studies are usually based on the assumption that the process data are independently and normally distributed. However, many situations in practice violate these assumptions. In this study, a methodology is developed to economically design a variable sampling interval X-bar control chart that takes into consideration correlated non-normal sample data. An example is provided to illustrate the solution procedure. A sensitivity analysis on the input parameters (i.e., the cost and the process parameters) is performed taking into account the non-normality and the correlation on the optimal design of the chart.
Keywords: X-bar control chart; Economic design; Variable sampling interval; Non-normality; Correlated process data; Genetic algorithms 1
Corresponding author
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1. Introduction and literature survey Control charts are the most important tools in statistical quality control environments for continuous improvement of the quality of the items being produced and the X chart is the first that was originally introduced by Shewhart in 1924 (Shewhart 1931). The design of the Shewhart X control charts requires the determination of three design parameters: the sample size ( n ), the
sampling interval ( h ), and the control limits width ( k ). Shewhart considered these parameters fixed, however recent studies have shown that varying the sampling interval (VSI), the sample size (VSS), the sampling interval jointly with the sample size (VSSI), or all design parameters at the same time, including k , (VP) reduce the detection time of process changes (see for example, Reynolds et al., 1988; Baxley, 1996; Prabhu et al., 1993,1994; Costa, 1994, 1997, 1999). Further, a survey of studies on VSR charts is presented in Tagaras (1998). Duncan (1956) proposed the first economic model to determine the optimal values of n ,
h , and k of an X chart. Since then, the economical design of control charts has received increasing attention in the literature (e.g., Montgomery, 1980; Vance, 1983; Woodall, 1986; Pignatiello and Tsai, 1988; Niaki et al. 2010). The usual approach in an economical design is to develop a cost model for a special type of industrial process, and then derive the optimal parameters by minimizing the expected cost per unit of time. While Duncan’s (1956) and Chiu’s (1975) models for the economic design of control charts have been widely studied in practice, in the former the process remains in operation during the period of search for the assignable cause and in the latter the process is ceased. In the design process of the Shewhart X control chart, one assumes the measurements (observations) within the subgroups are independently and normally distributed. However, these assumptions may not hold in many practices. For example, when a production process consists of 2
multiple but similar units on a single part, such as several cavities on a single casting, multiple pins on an integrated circuit chip, or multiple contact pads on a single machine mount, the collected measurements within a subgroup are usually correlated (Grant and Leavenworth 1988). Neuhardt (1987) considered the effect of correlation within a subgroup on the performances of control charts. Yang and Hancock (1990) employed Neuhardt’s idea to conduct simulation studies to specify the effect of correlated data on the performances of X , R, S and S2 charts. Liu et al. (2002) used the correlation model of Yang and Hancock (1990) to develop a minimum-loss design of fixed sample size and sampling interval (FSI) charts for correlated data. Chen and Chiou (2005) combined Yang and Hancock’s correlation model (1990) with the cost model of Bai and Lee (1998) to make the economic design of the VSI charts for correlated data. Recently, Chen et al. (2007) incorporated the Yang Hancock’s correlation model (1990) with the cost model of Costa (2001) to develop an economic design of the VSSI charts for correlated data. If the size of subgroups (sample size) is large enough, the statistic X is approximately normal based on the central limit theorem. However, when a control chart is applied to monitor the process, unfortunately the sample size is usually not sufficiently large due to the sampling cost. As a result, due to non-normality, the traditional method of designing the control chart may decrease the ability of detecting assignable causes. Yourstone and Zimmer (1992) considered the Burr distribution to represent the non-normal distributions and statistically designed the X chart. Chen (2004) presented the economic design of VSI X control charts for non-normal data. Lin and Chou (2005) employed the Burr distribution to design both symmetric and asymmetric-limit VSSI X chart under non-normality. Recently, Lin and Chou (2007) studied the effect of nonnormality on the performances of adaptive X control charts (i.e., variable sample size, sampling intervals, and and/or action limit coefficients) and concluded the VP X control chart is more 3
effective than the other adaptive charts in detecting small process mean shifts under non-normal process data. In this research, the economic model of the VSI X chart for correlated non-normal process data is developed using Burr distribution and the correlation model presented in Yang and Hancock (1990). The FSI and VSI schemes are compared in terms of the long-run expected cost per hour and statistical performances for various levels of correlation. The remainder of the paper are as follows. In the next section, the concept of the VSI control charts is briefly discussed. The Burr distribution representing the various types of nonnormal distributions is introduced in section 3. The economic model based on the Burr distribution that works under correlated process data is developed in section 4. A solution method to find the optimal values of the design parameters of an asymmetric VSI X chart is presented in section 5. Sensitivity analysis on the process parameters and correlation coefficients are conducted in section 6. Finally, conclusions make up the last section.
2. The VSI X control chart Consider a process involving a single quality characteristic following a normal distribution with in-control mean 0 and in-control known variance 2 . To monitor the process mean using an X control chart, traditionally at a given sample point a sample of n independent n
observations (x i ; i 1, 2,..., n ) are taken and the sample mean x 1 n x i is calculated i 1
accordingly. If the plotted x falls within the interval 0 k
n , ( 0 is the centerline and k is
the control limit width) the process is considered in-control and the next sample is taken in a fixed interval of h1 time. Otherwise, a signal is sent to inform the operator to search for an
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assignable cause. This control chart is said to operate in a fixed sample-size ( n ) and fixed sampling interval (h1 ) (FSI) condition. When the VSI scheme is in use, the waiting time h until the next sampling point is a function of the current x value. In other words, if x I i ; i 1, 2 , then h hi where
I 1 0 k '1 , 0 k ' 2 , 0 k 1 0 k 2 n n n n
I 2 0 k ' 2 , 0 k 2 n n
(1)
with 0 k 2 k 1 , 0 k 2' k 1' , and h2 h1 0 . According to Chen (2004), the slight cost savings offered by the VSI schemes with more than two sampling interval sizes does not justify their use. As an example, Fig. 1 depicts a VSI X chart using two interval lengths of h1 and h2 .
0
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The sample means in Fig.1 are plotted against the time on the horizontal axis. The first sample mean falls within I 2 , so the next sampling interval is h2 . The second one falls within I 1 , which is close to the control limits. Hence, a shorter sampling interval is adopted to take the third sample and so on. The way the VSI X control charts work can improve the detection ability of FSI charts by shortening the length of time until a signal is occurred. Nonetheless, the complexity will increase when the number of divided areas becomes large. Further, traditional symmetric VSI charts can be easily obtained by setting k 1 k '1 .
3. The Burr distribution In this study, the Burr distribution is used to model non-normal process data. Burr (1942) proposed a simple probability distribution function that was capable to model various types of non-normal data. The cumulative distribution function of the two-parameter Burr distribution is
1 k 1 F y 1 y c 0
y0 y
