適應性(平均值)X管制圖變動管制參數之評估

An Evaluative Study on Adaptive (average)X Control Charts Under Various Combinations of Variable Parameters

林裕章(Yu-Chang Lin) ； 周昭宇(Chao-Yu Chou)

《管理學報》 21卷3期 (2004/06) Pp. 375-389

https://doi.org/10.6504/JOM.2004.21.03.06

摘要

適應性管制圖的管制參數(抽樣間隔h、樣本數n和管制界限係數k)是根據目前製程的狀況來調整，若製程相當穩定，則採取較寬鬆的管制程序；反之，則採取較嚴格的管制程序。本文旨在探討適應性管制圖在變動不同管制參數的組合下，其偵測製程平均數偏移之績效。研究結果顯示：變動樣本數n對於偵測較小幅度平均數偏移較為靈敏，變動抽樣間隔h對於偵測較大幅度平均數偏移效果較佳，而管制界限係數k與樣本數n同時變動時，其偵測效果將會有顯著地改善。另外，VP管制圖(變動h,n,k) 對於製程平均數小幅度偏移的偵測效果較佳；而VSI管制圖(變動h)與標準Shewhart管制圖(固定管制參數)對於製程平均數發生中至大幅度偏移的偵測效果較佳。

關鍵字

統計製程管制 ； 管制圖 ； 適應性管制圖 ； 變動管制參數

並列摘要

The averages control charts (or (average)X charts) are widely applied in statistical process control. The conventional (average)X charts are static; i.e., the three chart parameters (including the sample size n, the sampling interval between successive samples h, and the control limit coefficient k) are fixed in the duration of operation. Recently developed adaptive (average)X charts have been shown to give substantially faster detection of most process shifts than the conventional (average)X charts. These adaptive (average)X charts include the variable sampling intervals (VSI) (average)X chart, the variable sample size (VSS) (average)X chart, variable sample size and sampling intervals (VSSI) (average)X chart, and the variable parameters (VP) (average)X chart. In the operation of adaptive (average)X charts, if a sample point falls in the central region (i.e., the point is close to the target), then it is reasonable to relax the control by waiting more time to take the next sample (i.e., using the long sampling interval h1), decreasing the size of the next sample (i.e., using small sample size n1), and/or plotting the next sample point on the chart with wide control limits (i.e., using wide action limit coefficient k1). If the sample point falls in the warning region (i.e., the point is far away from the target but still within the action limits), then it is reasonable to tighten the control by waiting less time to take the next sample (i.e., using the short sampling interval h2), increasing the size of the next sample (i.e., using large sample size n2), and/or plotting the next sample point on the chart with narrow control limits (i.e., using narrow action limit coefficient k2). If the sample point falls outside the action (or control) limits, then the process may be out of control caused by the assignable cause(s).
In this paper, a comparative study is conducted to evaluate the performance of seven (average)X control charts (see Table 1): the standard Shewhart (SS) (average)X chart and the adaptive (average)X charts (i.e., VP(h), VP(n), VP(h,n), VP(h,k), VP(n,k) and VP(h,n,k) (average)X charts). Traditionally, average number of samples to signal (ANSS), average number of observations to signal (ANOS), average time to signal (ATS) and adjusted average time to signal (AATS), have been generally employed as performance indicators to evaluate the effectiveness of various adaptive control schemes. When the process is in control, ATS may be used to develop the measures of the false alarm rate for a chart. When the process is out of control, ANSS, AATS and ANOS may be used to measure the performance of a chart. The equations for calculating ANSS, ANOS, ATS and AATS of the adaptive (average)X charts are derived using the Markov chain approach and are shown in Appendix. We compare the adaptive (average)X charts with the SS (average)X chart in the detection ability of process mean shifts, and the results are shown in Table 2-5. We also compare the AATS and ANOS for the adaptive (average)X charts with various n2, k1 and h2 when the process mean occur the small (δ=0.25) and large (δ=2.5) shifts, the results are shown in Figure 4-6.
This paper is concluded that the influence of three chart parameters, varying sample size n can increase the sensitiveness of small shifts in the process. Varying sampling interval h can increase the effectiveness of large shifts in the process. The improved effect is more significantly that combine control limit coefficient k with n the parameters varies simultaneously. In addition, From Table 2-5 and Figure 4-6, we have the following observations:
1. If the process mean has the small shifts, then the VP(h,n,k) chart has the better ability to detect the shifts and also incurs the lower sampling costs. The performance of the VP(n,k) and VP(h,n) (average)X charts are secondly followed, and the other charts are the worse ones.
2. If the process mean has the large shifts, then the VP(h) and the Shewhart (fixed parameters) charts have similar effectiveness in detecting the shifts and sampling costs. The performances of the VP(h) and SS (average)X charts are obviously better than the other adaptive charts.

並列關鍵字

statistical process control ； SPC ； control chart ； adaptive control chart ； variable control parameters

參考文獻

Amin, R. W.,Miller, R. W.(1993).A Robustness Study of □ Charts with Variable Sampling Intervals.Journal of Quality Technology.25,36-44.

Bai, D. S.,Lee, K. T.(2002).Variable Sampling Interval □ Control Charts with an Improved Switching rule.International Journal of Production Economics.76,189-199.

Costa, A. F. B.(1994).□ Charts with Variable Sample Size.Journal of Quality Technology.26,155-163.

Costa, A. F. B.(1997).□ Chart with Variable Sample Size and Sampling Intervals.Journal of Quality Technology.29,197-204.

Costa, A. F. B.(1998).Joint □ and R Charts with Variable parameters.IIE Transactions.30,505-514.