國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
第五章 結論與未來研究方向
根據 EWMA 管制圖之管制界限應隨時間變動，藉由縮減前期管制界限的範圍，
改善傳統 EWMA 管制圖對於製程初期異常的敏感度，本研究所採用的變動管制界限 形式，為各期管制統計量分配的 α/2 和 1 − α/2 分位數，由於各期管制統計量分配 型態未知，因此利用傅立葉級數近似各期管制統計量之累積分佈函數，進而求得分位 數作為變動管制界限。研究結果顯示，此種變動管制界限對於提升管制圖在製程初期 偵測能力之效果，主要取決於加權常數 λ，一般而言，加權常數越小對於改善製程初 期之偵測能力效果越好，且縮減的抽樣次數固定，例如：當 λ = 0.05 時，在各種參 數設定和不同偏移程度之下，大約皆能縮減 5 次抽樣即能偵測出製程為失控狀態，因 此能有效提升對於製程微小偏移的偵測能力；而偏移幅度較大時，能縮減抽樣次數至 1 ∼ 2 次，若以縮減效果而言，隨樣本數或 α 越大效果越好，且當製程為厚尾分配 (如：雙指數分配) 時，對於改善製程初期偵測能力的效果更為明顯。
針對後續研究方向以及待改進方法，在此提出以下建議：
1. 以往無母數 EWMA 管制圖相關文獻中，大多未提及變動管制界限之效果，例 如：Hackl and Ledolter (1991) 所提出對於單一觀測值，給定一組參考樣本 Y1, Y2, · · · , Yg−1 計算其標準化排序
Rt = 2
g (R∗t − g + 1 2 ) Rt∗ = 1 +
g−1
X
i=1
I(Xt> Yi)
其中，Rt 為 iid 離散型均勻分配，依據 Rt 建構 EWMA 管制程序監控製程中 心值，其乃利用模擬方法求得適當固定管制界限，因此可將本文所提出之方法運 用於此，建構變動管制界限並探討其效果。
2. Steiner (1999) 認為當 EWMA 管制圖之管制界限隨時間變動僅相似於快速起
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
始反應的性質，欲使 EWMA 管制圖具備快速起始反應之性質，進而提出藉由 調整項 FIRadj = 1 − (1 − f∗)1+a(k−1) 縮減前期管制界限以提升偵測能力，則新 的管制界限如下：
U CLk = µ0+ L σX 1 − (1 − f∗)1+a(k−1) r λ
2 − λ(1 − (1 − λ)2k) LCLk = µ0− L σX 1 − (1 − f∗)1+a(k−1)
r λ
2 − λ(1 − (1 − λ)2k) 其中 f∗ 及 a 為可調整常數，Steiner 認為大約 20 個觀察值之後快速起始反應 就沒有什麼效果，使得 a = (−2/log(1 − f∗) − 1)/19，而 f∗ 其目的即為調整 起始值介於目標值及管制界限之間，大多使用 0.5。本研究僅提出無母數管制 圖建構變動管制界限之方法及觀察結果，因此可進一步研究管制參數 α 變動方 式，以達快速起始反應之功能。
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
參 考 文 獻
Abate, J. and Whitt, W. (1992), “The Fourier–Series Method for Inverting Trans-forms of Probability Distributions”, Queueing systems, 10(1), 5–87.
Alloway, J. and Raghavachari, M. (1991), “Control Chart Based on the Hodges–Lehmann Estimator”, Journal of Quality Technology, 23(4), 336–347.
Amin, R. W., Reynolds, M. R. Jr., and Bakir, S. T. (1995), “Nonparametric Qual-ity Control Charts Based on the Sign Statistic”, Communications in Statistics
— Theory and Methods, 24(6), 1597–1623.
Amin, R. W. and Searcy, A. J. (1991), “A Nonparametric Exponentially Weighted Moving Average Control Scheme”, Communications in Statistics — Simulation and Computation, 20(4), 1049–1072.
Arnold, H. J. (1965), “Small Sample Power of the one Sample Wilcoxon Test for Non–Normal Shift Alternatives”, The Annals of Mathematical Statistics, 36(6), 1767–1778.
Bakir, S. T. (2004), “A Distribution–Free Shewhart Quality Control Chart Based on Signed–Ranks”, Quality Engineering, 16(4), 613–623.
(2006), “Distribution–Free Quality Control Charts Based on Signed–Rank–Like Statistics”, Communications in Statistics — Theory and Methods, 35(4), 743–757.
Bakir, S. T., Chakraborti, S., and Van der Laan, P. (2001), “Nonparametric Con-trol Charts: An Overview and Some Results”, Journal of Quality Technology, 33(3), 304–315.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
Bakir, S. T. and Reynolds, M. R. Jr. (1979), “A Nonparametric Procedure for Process Control Based on Within–Group Ranking”, Technometrics, 21(2), 175–
183.
Borror, C. M., Champ, C. W., and Rigdon, S. E. (1998), “Poisson EWMA Control Charts”, Journal of Quality Technology, 30(4), 352–361.
Chakraborti, S. and Eryilmaz, S. (2007), “A Nonparametric Shewhart–Type Signed–Rank Control Chart Based on Runs”, Communications in Statistics:
Simulation and Computation, 36(2), 335–356.
Chakraborti, S. and Van de Wiel, M. A. (2008), “A Nonparametric Control Chart Based on The Mann–Whitney Statistic”, Institute of Mathematical Statistics, 1, 156–172.
Chandrasekaran, S., English, J. R., and Disney, R. L. (1995), “Modeling and Analysis of EWMA Control Schemes with Variance–Adjusted Control Limits”, IIE Transactions, 27(3), 282–290.
Crowder, S. V. (1987), “A Simple Method for Studying Run–Length Distributions of Exponentially Weighted Moving Average Charts”, Technometrics, 29(4), 401–
407.
Hackl, P. and Ledolter, J. (1991), “A Control Chart Based on Ranks”, Journal of Quality Technology, 23(2), 117–124.
(1992), “A New Nonparametric Quality Control Technique”, Communica-tions in Statistics — Simulation and Computation, 21(2), 423–443.
Knoth, S. (2003), “EWMA Schemes with Nonhomogeneous Transition Kernels”, Sequential Analysis, 22(3), 241–255.
(2005), “Fast Initial Response Features for EWMA Control Charts”, Sta-tistical Papers, 46(1), 47–64.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
Lucas, J. M. and Crosier, R. B. (1982), “Fast Initial Response for CUSUM Qual-ity–Control Schemes: Give Your CUSUM a Head Start”, Technometrics, 24(3), 199–205.
Lucas, J. M. and Saccucci, M. S. (1990), “Exponentially Weighted Moving Average Control Schemes: Properties and Enhancements”, Technometrics, 32(1), 1–12.
MacGregor, J. F. and Harris, T. J. (1993), “The Exponentially Weighted Moving Variance”, Journal of Quality Technology, 25, 106–106.
McDonald, D. (1990), “A CUSUM Procedure Based on Sequential Ranks”, Naval Research Logistics, 37(5), 627–646.
Montgomery, D. C. (1991), Introduction to Statistical Quality Control, New York : Wiley, 2 edition.
Reynolds, M. R. Jr. (1975), “A Sequential Signed–Rank Test for Symmetry”, The Annals of Statistics, 3(2), 382–400.
Rhoads, T. R., Montgomery, D. C., and Mastrangelo, C. M. (1996), “A Fast Ini-tial Response Scheme for the ExponenIni-tially Weighted Moving Average Control Chart”, Quality Engineering, 9(2), 317–327.
Roberts, S. W. (1959), “Control Chart Tests Based on Geometric Moving Aver-ages”, Technometrics, 3, 239–250.
Robinson, P. B. and Ho, T. Y. (1978), “Average Run Lengths of Geometric Moving Average Charts by Numerical Methods”, Technometrics, 20(1), 85–93.
Steiner, S. H. (1999), “Exponentially Weighted Moving Average Control Charts with Time–Varying Control Limits and Fast Initial Response”, Journal of Qual-ity Technology, 31, 75–86.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
Tarter, M. E. and Kronmal, R. A. (1976), “An Introduction to The Implementa-tion and Theory of Nonparametric Density EstimaImplementa-tion”, American Statistician, 30(3), 105–112.
Woodall, W. H. (2000), “Controversies and Contradictions in Statistical Process Control”, Journal of Quality Technology, 32(4), 341–378.
Woodall, W. H. and Montgomery, D. C. (1999), “Research Issues and Ideas in Statistical Process Control”, Journal of Quality Technology, 31, 376–386.
Yourstone, S. A. and Zimmer, W. J. (2007), “Non–Normality and the Design of Control Charts for Averages”, Decision Sciences, 23(5), 1099–1113.
‧
‧
偶數。則可根據 composite Simpson’s rule 誤差公式求得| I | =
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
第 II 部分，為估計機率誤差
| II | ≤ hn 3
Aˆzn−1− Azn−1 max
ξ∈(0,x)f0(ξ)
(1 + 4 + 2 + · · · + 4 + 1)
= x
Aˆzn−1 − Azn−1
max
ξ∈(0,x)f0(ξ)
‧
‧
cos(nx) cos(mx) dx
+ bn Z π
−π
sin(nx) cos(mx) dx
其中 sin(mx)
cos(nx) cos(mx) dx =
Theorem 1 (Fourier convergence). 已知 f 為一分段平滑的週期函數，則
(SNf )(x) = f (x+) + f (x−)