雙次抽樣平均數和變異數管制圖設計之研究 - 政大學術集成
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(2) 謝辭 在兩年的研究所生涯裡，學習到了許多知識也成長了許多，最終完成了論文， 這一切都要感謝學習生涯裡遇到的許多人對我的提攜與幫助。 首先要感謝我的論文指導老師 楊素芬教授，老師花費了許多心思和時間指 導和修正我的論文，在老師的訓練與培養下，讓我對統計與品管的專業知識與應 用更加的精進，也讓我更加的熟練統計軟體的操作與程式撰寫，同時也讓我更積 極與負責任的去處理所面對到的任何事務。 再來要感謝我的口試委員 黃榮臣教授、曾勝滄教授與葉小蓁教授，三位老 師在口試時給我的建議與指點，使得論文更加的完善。. 治 政 也感謝我的同學與朋友，陪我一起上課與休閒娛樂，一起渡過了兩年的碩士 大 立 生活。 ‧ 國. 學. 最後，要感謝我的家人，感謝你們默默的支持我進修碩士，讓我能夠專心的. ‧. 完成學業。. y. sit. io. n. al. er. 致謝。. Nat. 本研究承蒙科技部補助，計畫編號 MOST 104-2118-M-004 -006 -MY2，謹此. Ch. engchi. i Un. v. 吳信宏. 謹致. 中華民國一百零五年七月 i.
(3) Abstract Control charts are effective tools for detecting manufacturing processes and service processes. Nowadays, much of the data in manufacturing or service industries comes from processes having non-normal or unknown distributions. The commonly used Shewhart control charts, which depend heavily on the normality assumption, are not appropriately used for this situation. In this paper, we propose a standardized dynamic double sampling asymmetric EWMA mean control chart (SDDS EWMA-AM chart), a standardized dynamic double sampling asymmetric EWMA. 政 治 大 SDDS EWMA-AM and SDDS EWMA-AV charts) to monitor process mean, variance 立. variance control chart (SDDS EWMA-AV chart), and their combined charts (joint. ‧ 國. 學. and both shifts, respectively. The charts based on the double sampling procedure and two simple distribution-free transformed statistics are used for non-normal. ‧. distribution of a quality variable. The performance of the proposed charts and that of. Nat. sit. y. some existing distribution-free mean and variance charts are compared. Further, a. n. al. er. io. non-normal service times example from the service system of a bank branch is used to. i Un. v. illustrate the applications of the proposed charts and to compare detection. Ch. engchi. performance with the existing distribution-free mean and variance control charts. The charts we proposed show superior detection performance compared to the existing distribution-free mean and variance charts. Thus they are recommended.. Keywords: Double sampling; Average run length; Binomial distribution ii.
(4) Contents 1. Introduction .............................................................................................................. 1 2. The SDDS EWMA-AM Chart ................................................................................ 3 2.1. Construction of the SDDS EWMA-AM Chart .......................................... 3 2.2. Detection Performance of the SDDS EWMA-AM Chart .............................. 7 2.3. Performance Comparison with Existing Control Charts ............................ 14 2.4. Example ........................................................................................................... 32 3. The SDDS EWMA-AV Chart................................................................................ 39. 政 治 大 3.2. Detection Performance 立 of the SDDS EWMA-AV Chart ............................. 43 3.1. Construction of the SDDS EWMA-AV Chart .............................................. 39. ‧ 國. 學. 3.3. Performance Comparison with Existing Control Charts ............................ 49 3.4. Example ........................................................................................................... 63. ‧. 4. The Joint SDDS EWMA-AM and SDDS EWMA-AV Charts ........................... 72. Nat. sit. y. 4.1. Construction of the Joint SDDS EWMA-AM and SDDS EWMA-AV. n. al. er. io. Charts ................................................................................................................. 72. i Un. v. 4.2. Detection Performance of the Joint SDDS EWMA-AM and SDDS. Ch. engchi. EWMA-AV Charts ............................................................................................ 76 4.3. Detection Performance of the Joint EWMA-AM and EWMA-AV Charts ................................................................................................................. 94 4.4. Performance Comparison with Existing Control Charts .......................... 104 4.5. Example ......................................................................................................... 129 5. Conclusions ........................................................................................................... 140 References ................................................................................................................. 141. iii.
(5) List of Table Table 1. Parameters of the SDDS EWMA-AM chart with ,. ,. and. for various. . ………………………..……….…. 9. given. Table 2. Parameters of the SDDS EWMA-AM chart with various distributions,. ,. and. and 500 for . ……...…..……... 10. given. Table 3. The out-of-control ARL of SDDS EWMA-AM chart with. and. 0.1. ……………..………………………………. 11. under. Table 4. The out-of-control ARL of SDDS EWMA-AM chart with. and. 政 治 大 Table 5. The out-of-control ARL of SDDS EWMA-AM chart with 立. 0.2. ………..………………………………….… 12. under. and. 0.3. ……..………………...…………………….. 12. ‧ 國. 學. under. Table 6. The out-of-control ARL of SDDS EWMA-AM chart with. and. ‧. 0.4. ……………………………..…………...….. 13. under. Nat. and. sit. y. Table 7. The out-of-control ARL of SDDS EWMA-AM chart with. 0.5. ………..……………………………………. 13. n. al. er. io. under. i Un. v. Table 8. Performance comparison of the SDDS EWMA-AM chart and the. Ch. EWMA-AM chart with. e n g c, h i. and. under. 0.1. …………………………………………………………….….. 15 Table 9. Performance comparison of the SDDS EWMA-AM chart and the EWMA-AM chart with. ,. and. under. 0.3. ………………………………………………………………... 15 Table 10. Performance comparison of SDDS EWMA-AM chart and EWMA-AM chart with Table 11. The values of. ,. and. and corresponding. iv. under. 0.5. .... 16. under various distributions. ... 18.
(6) Table 12. Performance comparison of the SDDS EWMA-AM chart, the EWMA-AM chart, the NLE chart and the CEW chart with. ,. and. under N( , 1). …………………………………………...……. 19 Table 13. Performance comparison of the SDDS EWMA-AM chart, the SL charts and the SC charts with. ,. and. under. N( , 1). ………………………………………………………………….. 19 Table 14. Performance comparison of the SDDS EWMA-AM chart and the Shewhart, DSVSI, VP, VSSI, VSS, and the VSI. charts with. ,. under N( , 1). …………………………………. 20. and. 治 政 Table 15. Performance comparison of the SDDS EWMA-AM 大 chart and the Shewhart, 立charts with DS, and the VP , and ‧ 國. 學. under N( , 1). …………………………………………………………... 21. under. sit. y. and. ). …………………………………………………………... 24. io. al. er. DE( ,. ,. Nat. chart and the SL charts with. ‧. Table 16. Performance comparison of the SDDS EWMA-AM chart, the EWMA-AM. n. Table 17. Performance comparison of the SDDS EWMA-AM chart, the SL chart and the SC chart with DE(. Ch. e n, g c h i. i Uandn. v. under. , 1). ………………………………………………………….…. 25. Table 18. Performance comparison of the SDDS EWMA-AM chart, the NLE chart, the CEW chart and the EWM chart with under. ,. and. . ……………………………………………….. 25. Table 19. Performance comparison of the SDDS EWMA-AM chart, the Shewhart, the DS, and the VP under t(4). charts with. ,. and. . …………………………………………………….…. 26. v.
(7) Table 20. Performance comparison of the SDDS EWMA-AM chart, the Shewhart, the DS, and the VP. charts with. ,. and. . ……………………………………………………..…. 27. under t(10). Table 21. Performance comparison of the SDDS EWMA-AM chart, the Shewhart, the DS, and the VP. charts with. ,. and. under t(30). . ……………………………………………………... 28. Table 22. Performance comparison of the SDDS EWMA-AM chart and the EWMA-AM chart with Unif. ,. and. under. 政 治. …………………………………. 30 大. and Exp(1). 立. 學. ‧ 國. Table 23. Performance comparison of the SDDS EWMA-AM chart, the NLE chart, the CEW chart and the EWM chart with. ,. and. . …………………………………….. 31. under. ‧. Table 24. The in-control service times from 10 counters in a bank branch. ……….. 33. y. Nat. n. al. er. , 15. ………………………………………………………... 34. io. for t=1, 2,. sit. Table 25. The plotting statistics of the SDDS EWMA-AM chart with. Ch. i Un. v. Table 26. The new service times on the SDDS EWMA-AM chart with. engchi. . …………………………………………………………….. 37 Table 27. Parameters of the SDDS EWMA-AV chart with for various. ,. ,. and. . ……………………………. 43. Table 28. Parameters of the SDDS EWMA-AV chart with distributions,. ,. and. given. ,. ,. vi. for various. . ………………………... 45. Table 29. Parameters of the SDDS EWMA-AV chart with distributions given. and. and. for various . ………….. 46.
(8) Table 30. The out-of-control ARL of SDDS EWMA-AV chart with. and. 0.1. …………………………………………….. 47. under. Table 31. The out-of-control ARL of SDDS EWMA-AV chart with. and. 0.2. …………………………………………….. 47. under. Table 32. The out-of-control ARL of SDDS EWMA-AV chart with. and. 0.3. …………………………………………….. 48. under. Table 33. The out-of-control ARL of SDDS EWMA-AV chart with. and. 0.4. …………………………………………….. 48. under. Table 34. Performance comparison of the SDDS EWMA-AV chart, the EWMA-AV. 治 政 chart and the New EWMA chart with under 大 and 立 0.1. ……………………………………………………………….. 50 ‧ 國. 學. Table 35. Performance comparison of the SDDS EWMA-AV chart, the EWMA-AV and. under. ‧. chart and the New EWMA chart with. sit. y. Nat. 0.4. ……………………………………………………………….. 50. io. al. and. n. EWMA chart with Table 37. The values of. er. Table 36. Performance comparison of the SDDS EWMA-AV chart and the New. Ch. 0.3. ….… 51. under. iv. U n various distributions. ... 53 e n g c h i under. and corresponding. Table 38. Performance comparison of the SDDS EWMA-AV chart, the Shewhart R, the Shewhart S, the VSI S, the DS S and the DSVSI S charts with ,. and. under N(0,. ). ………………… 54. Table 39. Performance comparison of the SDDS EWMA-AV chart, the NLE, the CEW, the EWMA-AV, IRC and ISC charts with and. under N(0,. ,. ). ………………………………………….. 54. vii.
(9) Table 40. Performance comparison of the SDDS EWMA-AV chart, the NP-M charts and the Khoo and Lim’s IRC chart with under N(0,. ,. and. ). ……………………………………………….. 55. Table 41. Performance comparison of the SDDS EWMA-AV chart, the SL charts and the SC charts with N(0,. ,. and. under. ). ………………………………………………………………… 55. Table 42. Performance comparison of the SDDS EWMA-AV chart, the EWMA-AV, the Khoo and Lim’s IRC and the NP-M charts with. ,. ). ……………………………………. 57. 治 政 Table 43. Performance comparison of the SDDS EWMA-AV 大 chart and the SL charts 立 and. under DE(0,. with. ,. and. ). …….. 58. under DE(0,. ‧ 國. 學. Table 44. Performance comparison of the SDDS EWMA-AV chart, the SL charts and and. under. ). ……………………………………………………………….. 58. sit. y. Nat. DE(0,. ,. ‧. the SC charts with. io. al. n. CEW chart and the EWM chart with under. Ch. er. Table 45. Performance comparison of the SDDS EWMA-AV chart, the NLE chart, the , v i. and. n 60 U i e.n…………………………………………… h gc. Table 46. Performance comparison of the SDDS EWMA-AV chart, the EWMA-AV, the Khoo and Lim’s IRC and the NP-M charts with and. under. ,. . ………………………………. 61. Unif. Table 47. Performance comparison of the SDDS EWMA-AV chart, the NLE chart, the CEW chart and the EWM chart with under. ,. and. . ………………………………………………… 62. viii.
(10) Table 48. Performance comparison of the SDDS EWMA-AV chart, the EWMA-AV and the Khoo and Lim’s IRC charts with. ,. and. under Exp( ). ………………………………………………… 63 Table 49. The values of statistics. ,. , j=1, 2,. , 5,. and. for the. in-control service times. ……………………………………………………... 65. Table 50. The plotting statistics of the SDDS EWMA-AV chart with t=1, 2,. for. , 15. ………………………………………………………….… 66. Table 51. The plotting statistics of the SDDS EWMA-AV chart with. for. the new service times. …………………………………………………… 69. (4, 6, 5) under various (. 學. ‧ 國. 治 政 Table 52. Coefficients of the control limits of the joint 大 SDDS EWMA-AM and SDDS 立 EWMA-AV charts with , and ( , , )= ). …………………………………….… 77. ,. ,. and (. ,. ,. )=. io. al. ). …………………………………...… 78. ,. er. (6, 12, 8) under various (. sit. y. Nat. EWMA-AV charts with. ‧. Table 53. Coefficients of the control limits of the joint SDDS EWMA-AM and SDDS. n. Table 54. Coefficients of the control limits of the joint SDDS EWMA-AM and SDDS. Ch EWMA-AV charts with (8, 16, 10) under various (. e n g c ,h i. i Un. v. and (. ,. ,. )=. ). ……………………………………. 79. ,. Table 55. Coefficients of the control limits of the joint charts using Method 1 with and. under various distributions and (. ,. ,. ). ……………………………………………………………………….. 80 Table 56. Coefficients of the control limits of joint charts using Method 2 with ,. and (. ,. ,. )=(8, 16, 10) under various. distributions. ……………………………………………………………... 81. ix.
(11) Table 57. Coefficients of the control limits of joint charts using Method 1 with various ,. and (. ,. ,. ) (4, 6, 5) under various. distributions. ……………………………………………………………... 82 Table 58. The out-of-control ARL, 25th, 50th, 75th percentiles of run length and E(n) of the joint charts using Method 1 with under (. ,. ,. and. ) (8, 16, 10) and (. ,. ) (0.1, 0.1). ……………. 84. Table 59. The out-of-control ARL, 25th, 50th, 75th percentiles of run length and E(n) of the joint charts using Method 1 with. and. , ) (0.1, 0.4). ……………. 85 治 政 Table 60. The out-of-control ARL, 25th, 50th, 75th percentiles 大 of run length and E(n) 立 of the joint charts using Method 1 with and ,. under (. ,. ,. ) (8, 16, 10) and (. ,. ) (8, 16, 10) and (. ,. 學. ‧ 國. under (. ) (0.3, 0.1). ……………. 86. ‧. Table 61. The out-of-control ARL, 25th, 50th, 75th percentiles of run length and E(n). ,. y ) (8, 16, 10) and (. ,. io. sit. ,. and. ) (0.3, 0.4). ……………. 87. er. under (. Nat. of the joint charts using Method 1 with. al. n. Table 62. The out-of-control ARL, 25th, 50th, 75th percentiles of run length and E(n). i n CMethod of the joint charts using 1 with U hengchi under (. ,. ,. ) (8, 16, 10) and (. ,. v. and. ) (0.5, 0.1). ……………. 88. Table 63. The out-of-control ARL, 25th, 50th, 75th percentiles of run length and E(n) of the joint charts using Method 1 with under (. ,. ,. and. ) (8, 16, 10) and (. ,. ) (0.5, 0.4). ………….… 89. Table 64. The out-of-control ARL, 25th, 50th, 75th percentiles of run length and E(n) of the joint charts using Method 1 with under (. ,. ,. ) (4, 6, 5) and (. x. and ,. ) (0.4, 0.3). ………………. 90.
(12) Table 65. The out-of-control ARL, 25th, 50th, 75th percentiles of run length and E(n) of the joint charts using Method 2 with under (. ,. ,. ) (8, 16, 10) and (. and ,. ) (0.1, 0.1). ……………. 91. Table 66. The out-of-control ARL, 25th, 50th, 75th percentiles of run length and E(n) of the joint charts using Method 2 with under (. ,. ,. ) (8, 16, 10) and (. and ,. ) (0.5, 0.1). ……………. 92. Table 67. The out-of-control ARL, 25th, 50th, 75th percentiles of run length and E(n) of the joint charts using Method 2 with. and. , ) (0.4, 0.3). ………………. 93 治 政 Table 68. Coefficients of the control limits of the joint 大 EWMA-AM and EWMA-AV 立 and charts with under various ,. ,. ,. ) (4, 6, 5) and (. 學. (. ,. ‧ 國. under (. ). ………………………………………………………….. 95. sit. y. for various distributions. . ……………………………………………………………. 95. io. er. and. and. Nat. charts with. ‧. Table 69. Coefficients of the control limits of the joint EWMA-AM and EWMA-AV. al. n. Table 70. The out-of-control ARL, 25th, 50th, 75th percentiles of run length of the. i n joint EWMA-AM andC EWMA-AV charts with U hengchi under. ,. v. ,. and. . ………………………... 97. Table 71. The out-of-control ARL, 25th, 50th, 75th percentiles of run length of the joint EWMA-AM and EWMA-AV charts with under. ,. ,. and. . ………………………... 98. Table 72. The out-of-control ARL, 25th, 50th, 75th percentiles of run length of the joint EWMA-AM and EWMA-AV charts with under. ,. xi. ,. and. . ………………………... 99.
(13) Table 73. The out-of-control ARL, 25th, 50th, 75th percentiles of run length of the joint EWMA-AM and EWMA-AV charts with under. ,. and. . ………………………. 100. ,. Table 74. The out-of-control ARL, 25th, 50th, 75th percentiles of run length of the joint EWMA-AM and EWMA-AV charts with under. ,. and. . ………………………. 101. ,. Table 75. The out-of-control ARL, 25th, 50th, 75th percentiles of run length of the joint EWMA-AM and EWMA-AV charts with. ,. and. , . ………………………. 102 治 政 Table 76. The out-of-control ARL, 25th, 50th, 75th percentiles 大 of run length of the 立 joint EWMA-AM and EWMA-AV charts with , and under. 學. ‧ 國. under. . ………………………. 103. ,. ‧. Table 77. Detection performance (ARL) of the joint SDDS charts using Method 1 and ,. under. ,. . ………. 105. io. er. and. sit. y. Nat. 2, and the joint EWMA-AM and EWMA-AV charts with. al. Table 78. Detection performance (ARL) of the joint SDDS charts using Method 1 and. n. iv n C 2, and the joint EWMA-AM EWMA-AV charts with h eand ngchi U and. under. ,. , . ………. 106. Table 79. Detection performance (ARL) of the joint SDDS charts using Method 1 and the joint EWMA-AM and EWMA-AV charts with and. under. ,. , . …………………. 107. Table 80. Detection performance (ARL) of the joint SDDS charts using Method 1 and the joint EWMA-AM and EWMA-AV charts with and. under. ,. xii. , . …………………. 108.
(14) Table 81. Detection performance (ARL) of the joint SDDS charts using Method 1 and the joint EWMA-AM and EWMA-AV charts with and. under. , . …………………. 109. ,. Table 82. Detection performance (ARL) of the joint SDDS charts using Method 1 and the joint EWMA-AM and EWMA-AV charts with and. under. , . ………………….. 110. ,. Table 83. Detection performance (ARL1) of the joint SDDS charts using Method 1 and 2, joint EWMA-AM and EWMA-AV charts, and Shewhart under N( ). …………... 113 治 政 Table 84. Detection performance (ARL ) of the joint SDDS 大 charts using Method 1 立 and 2, the Max chart, the GLR chart and the Fisher chart with charts with. and. under N(. 學. and. ‧ 國. 1. ). ………………………………………. 115. ‧. Table 85. Detection performance (ARL1) of the joint SDDS charts using Method 1,. under N(. ). …………………………. 117. io. er. and. sit. y. Nat. the MEW, the NCS, the WLC, the ELR and the CEW charts with. al. Table 86. Detection performance (ARL1) of the joint SDDS charts using Method 1,. n. iv n C the SC and the SL charts h with e n g c h i Uand N(. under. ). ……………………………………………………………... 119. Table 87. Detection performance (ARL1) of the joint SDDS charts using Method 1, the joint DS. and S charts, the standard, the TSS and the VSS joint. and R charts with. and. under N(. ). ……... 120. Table 88. Detection performance (ARL1) of the joint SDDS charts using Method 1 and 2, and the joint EWMA-AM and EWMA-AV charts with and. under DE( ,. xiii. ,. ). ………………. 123.
(15) Table 89. Detection performance (ARL1) of the joint SDDS charts using Method 1, the SC and the SL charts with DE(. and. under. ). ………………………………………………………….. 124. ,. Table 90. Detection performance (ARL1) of the joint SDDS charts using Method 1 and 2, and the joint EWMA-AM and EWMA-AV charts with and. under. , . …….. 126. Unif. Table 91. Detection performance (ARL1) of the joint SDDS charts using Method 1 and 2, and the joint EWMA-AM and EWMA-AV charts with. ,. . ………………….. 127 治 政 Table 92. Detection performance (ATS) of the joint SDDS 大 charts using Method 1 and 立 the Lower-sided Gamma chart with and under the and. under Exp( ). ‧ 國. 學. Gamma distribution. ……………………………………………………. 129 using. , 15. ……………………………………………. 131. sit. y. Nat. Method 1 for t=1, 2,. ‧. Table 93. The plotting statistics of the joint SDDS charts with. io. al. Method 2 for t=1, 2,. using. er. Table 94. The plotting statistics of the joint SDDS charts with. , 15. ……………………………………………. 132 using. Table 96. The plotting statistics of the joint SDDS charts with. using. n. iv n C Table 95. The plotting statistics of the SDDS charts with h ejoint ngchi U. Method 1 for the new service times. …………………………………… 134. Method 2 for the new service times. …………………………………… 136. xiv.
(16) List of Figures Figure 1. The SDDS EWMA-AM chart for t=1, 2,. , 15. ……………………...… 35. Figure 2. The SDDS EWMA-AM chart for t=1, 2,. , 25. …..……….…................ 38. Figure 3. The EWMA-AM chart. ..…………………………………………………. 38 Figure 4. The transformed Shewhart. chart. ……………………………...…...… 38. Figure 5. The SDDS EWMA-AV chart for t=1, 2,. , 15. ……………………….... 67. Figure 6. The SDDS EWMA-AV chart for t=1, 2,. , 25. ………………………… 70. Figure 7. The EWMA-AV chart. …………………………………………………… 70. 政 治 大 Figure 9. The transformed EWMA-S chart. ……………………………………...… 71 立. Figure 8. The transformed Shewhart S chart. ………………………………………. 71. , 15. ………….… 131. Figure 11. The joint SDDS charts using Method 2 for t=1, 2,. , 15. ……………. 132. Figure 12. The joint SDDS charts using Method 1 for t=1, 2,. ‧. ‧ 國. 學. Figure 10. The joint SDDS charts using Method 1 for t=1, 2,. , 25. ……………. 135. Nat. , 25. ……………. 137. sit. y. Figure 13. The joint SDDS charts using Method 2 for t=1, 2,. n. al. Figure 15. The transformed Shewhart. Ch. er. io. Figure 14. The joint EWMA-AM and EWMA-AV charts. ……………………….. 139. i Un. v. and S charts. …………………………... 139. engchi. xv.
(17) 1. Introduction Control charts are commonly used tools to detect signal in process and improve the quality of service processes and manufacturing processes. In the past years, more and more statistical process control techniques have been developed under normality assumption. For example, The double sampling. charts (Daudin (1992)) for. monitoring process mean; the combined mean and variance charts based on generalized likelihood ratio test and Fisher method of combining tests (Deng (2009)) for monitoring process mean and variance; the ELR chart based on generalized likelihood ratio test (Zhang et al. (2010)) for monitoring process mean and variance;. 治 政 improved R chart and improved S chart (Zhang (2014)) 大 for monitoring the process 立 variance. ‧ 國. 學. The double sampling. charts (Daudin (1992)) offer better statistical efficiency chart, and the double sampling. ‧. to detect process mean shift than the Shewhart. sit. y. Nat. scheme has high flexibility in designing the control chart. There have been a few. io. al. chart (Carot et al. (2002)); double sampling. er. studies in this area, like combined double sampling and variable sampling interval and S chart (He and Grigoryan (2006));. n. iv n C combined double sampling and variable interval S chart (Lee et al. (2012)). h e nsampling gchi U. However, much of service process data come from non-normal or unknown distributions and previous studies based on traditional Shewhart control charts and normality assumption are not suitable for this situation. Some research about non-normal distributions with known form has been done, for example, a gamma chart was proposed for monitoring time between events (Zhang et al. (2007)); double sampling. charts under non normality (Torng and Lee (2009)) was proposed for. monitoring process mean; a distribution-free EWMA control chart based on an nonparametric goodness-of-fit test (Zou and Tsung (2010)) was developed for 1.
(18) monitoring process mean and variance; the SL chart based on Lepage statistic (Mukherjee and Chakraborti (2012)) was applied for monitoring process mean and variance; the SC chart based on Cucconi statistic (Chowdhury et al. (2014)) was developed for monitoring process mean and variance; nonparametric control charts based on Sukhatme test and Mood test (Ghute (2014)) were developed for monitoring process variance. Nonparametric approaches like the NLE chart (Zou and Tsung (2010)), the SL chart, the SC chart, the NP-S and the NP-M charts (Ghute (2014)) are not easy for practitioners to apply, because they don’t quite understand how to implement the. 治 政 schemes with a proper way. Control chart designed for 大specific distributions like the 立 gamma chart can’t be adopted for unknown distribution. Yang and Arnold (2014) ‧ 國. 學. proposed distribution-free EWMA-AM and EWMA-AV charts based on simple. ‧. statistics to monitor process mean and variance, and Yang and Arnold (2015). sit. y. Nat. proposed a new EWMA chart based on a simple statistic of arcsin transformation to. io. al. approaches, and have better detection ability.. er. monitor process variance. These approaches are easier than above nonparametric. n. iv n C In this paper, for the improvement ability of the EWMA-AM and h e nofgdetection chi U. EWMA-AV charts, we propose a standardized dynamic double sampling asymmetric EWMA mean (SDDS EWMA-AM) control chart to monitor process mean, a standardized dynamic double sampling asymmetric EWMA variance (SDDS EWMA-AV) control chart to monitor process variance, and the joint SDDS EWMA-AM and SDDS EWMA-AV charts to monitor shifts in the process mean and variance simultaneously. The proposed charts based on double sampling scheme are expected to have better detection ability than the EWMA-AM chart and EWMA-AV chart proposed by Yang and Arnold (2014) and other existing mean and variance 2.
(19) charts. In practice, the double sampling scheme can be applied in a process with slow production rate and high-cost sampling, such as the service industry with service times monitoring, in healthcare with disease infection control and health surveillance. We can make a software with the double sampling procedure, and install it in an automatic manufacturing system. The paper is organized as follows. In Section 2, we propose SDDS EWMA-AM chart to detect the process mean, its performance is calculated by simulation and compared with the existing mean charts, and a non-normal service times example in. 治 政 Yang and Arnold (2014) is applied. In Section 3, we propose 大 SDDS EWMA-AV chart 立 to detect the process variance, its performance is calculated by simulation and ‧ 國. 學. compared with the existing variance charts, and the same service times example in. ‧. Yang and Arnold (2014) is applied. In Section 4, we propose the joint SDDS. sit. y. Nat. EWMA-AM and SDDS EWMA-AV charts to detect the process mean and variance. io. er. simultaneously, calculate their performance using simulation and compare detection. al. performance with the existing charts, and the example of same service times is. n. iv n C applied. Section 5, we summarize the and provide recommendation. h efindings ngchi U 2. The SDDS EWMA-AM Chart 2.1.. Construction of the SDDS EWMA-AM Chart We use the double sampling procedure and follow the approach of EWMA-AM. chart in Yang and Arnold (2014) to construct the SDDS EWMA-AM chart and to monitor the process mean. Denote a critical quality variable observation at time t.. , j=1, 2,. has an in-control mean 3. and t=1, 2, and variance. , be the jth . To construct.
(20) the SDDS EWMA-AM chart, at the stage 1, we take a sample of size. ,. , from an IC process and define ,. P(. ), for j = 1, 2,. = The statistic. ,. , t=1, 2,. ), t=1, 2,. .. (1). .. (2). follows a binomial distribution with parameters ( depends on the distribution of. ), and. . Thus, the statistic. 政 治 大 ,. is defined as follows:. 立. . Hence the mean and variance of E(. Nat. Var(. ; that is,. are. )=. (3). ‧. ‧ 國. , be the mean of. .. 學. Let the starting value,. , t=1, 2,. , and. y. based on the statistic. ,. ) =. .. io. sit. the value of. ~B(. ,. (4). n. al. er. Thus, in stage 1 the dynamic double sampling (DDS) EWMA-AM chart. Ch. i Un. v. constructed based on the mean and variance of the statistic varying upper control limit (. engchi. (. ) and lower control limit (. ), upper warning limit (. ), lower warning limit (. with time ), central line ) are , ,. ,. (5) , , 4.
(21) where. and. ,. appropriately chosen coefficients for the. ,. ,. ,. ,. are. and. ,. respectively. Let the SDDS EWMA-AM Chart be used by the standardized form. Hence the plotting statistic at stage 1 is defined as follows: .. (6). The stage 1 standardized control limits, warning limits and central line are given by. 立. , 政 治 大 ,. ‧ 國. ,. al. n. ,. ,. ,[. engchi U. ], then the. v ni. the warning region. Ch. y. io. process is in control. If. ,. the central region. , then the process. sit. ,(. ,. er. the action region. is out of control. If. ‧. .. Nat. If. (7). 學. ,. ,. , then taking a second sample of size. , ,. , and the procedure is entering stage 2. Let. the number of taking second sample in the double sampling procedure,. that is, setting. =1 when the first time we take the second sample in the procedure,. and adding 1 to procedure.. when the next time we need to take the second sample in the. will be corresponding to some time t, let f be the corresponding. function, and the time where we take the second sample can be expressed as t = f( ) for. =1, 2,. .. In the stage 2, similarly,. and 5. ( j =. ,. ,. ,.
(22) and. =1, 2,. ) are calculated from equation (1), and define =. ~B(. ,. ),. =1, 2,. ,. ~B( The statistic. ,. ),. =1, 2,. . (8). follows a binomial distribution with parameters (. and statistic. follows a binomial distribution with parameters (. Thus, the statistic. based on the statistic. ,. ). ,. ).. is defined as follows: ,. 立. Let the starting value,. =1, 2, . 政 ,治 大 , be the mean. (9) of. ; that is, are. E(. ,. .. (10). sit. y. )=. Nat. Var(. )=. ‧. ‧ 國. 學. . Hence the mean and variance of. n. al. er. io. Thus, in the stage 2 the dynamic double sampling (DDS) EWMA-AM chart. i Un. v. constructed based on the mean and variance of the statistic varying upper control limit ( (. Ch. engchi. ), central line (. with time. ) and lower control limit. ) are defined as follows: ：. ,. ：. ,. (11). ： where. ,. ,. , and. chosen coefficients for the bounds for. and. ,. and. , are appropriately , and. , respectively. 6. and. are upper.
(23) Let the SDDS EWMA-AM chart be used by the standardized form. Hence the plotting statistic at the stage 2 is defined as follows: .. (12). In the stage 2, the standardized control limits and central line are given by , ,. (13) .. If. , ( , 政 治 大 the central region. the action region. 立. process is out-of-control. If. , then the. ,[. ,. ],. ‧ 國. 學. then the process is in-control.. ‧. 2.2.. ,. Detection Performance of the SDDS EWMA-AM Chart. y. Nat. io. sit. To measure the detection performance of the SDDS EWMA-AM chart, we. n. al. er. calculated the average run length (ARL) using the computer simulation of the process. Ch. with 10,000 times. The in-control ARL, depends on the values of. ,. ,. engchi ,. i Un. v. , of the SDDS EWMA-AM chart. ,. ,. ,. ,. ,. ,. . The. average sample size of the double sampling procedure is , where of. (14). is the probability of taking the second sample in whole process. The value also can be calculated by the computer simulation and the in-control E(n). should be small than the sample size of the single sampling chart,. , for comparing. the detection performance with a single sampling control chart. However, there is no restriction for the out-of-control E(n). The choice of the design parameters has a high 7.
(24) flexibility for an acceptably fixed. , and we use the direct search approach to. calculate the unique coefficients (. ,. ,. ,. The direct search approach with. ,. ,. given. ,. ) by simulation. ,. ,. and. is. described as follows: Step 1: Only considering the stage 1 chart, denote stage 1, we search. such that. be the. 1480 and. . Let. at be the. solution. Step 2: Only considering the stage 1 chart, we search 740. Let. Step 3: We search. and. , ,. .. y. io. n. a and l C h. er. be the solution.. Step 4: We search. (15). ‧. ,. Nat. and. .. 政 such治 that 大. 學. ‧ 國. 立. E(n). Let. be the solution and. sit. such that. given. such that , en gchi. i Un. v. , , , where and. and. are upper limits for. and. , respectively. Let. be the solution. The combination of (. 2 charts with. ,. ,. ,. ,. ,. ) in the stage 1 and stage. is just the corresponding solution and the solution is. unique. 8.
(25) Table 1. Parameters of the SDDS EWMA-AM chart with ,. ,. and. given. for various. .. No.. E(n) 4. 6. 5. 3.10. 2.44. 1.86. 1.46. 2.71. 2.14. 4.76. 371.89. 2. 0.1. 6. 12. 8. 3.00. 2.54. 1.68. 1.42. 2.63. 2.23. 7.74. 371.24. 3. 0.1. 8. 16. 10. 3.00. 2.54. 1.83. 1.55. 2.52. 2.13. 9.83. 370.65. 4. 0.2. 4. 6. 5. 2.95. 2.57. 1.77. 1.54. 2.60. 2.26. 4.73. 368.31. 5. 0.2. 6. 12. 8. 2.92. 2.61. 1.64. 1.46. 2.62. 2.34. 7.76. 367.30. 6. 0.2. 8. 16. 10. 2.89. 2.67. 1.76. 1.63. 2.42. 2.24. 9.76. 374.84. 7. 0.3. 4. 6. 5. 2.88. 2.70. 1.73. 1.62. 2.47. 2.32. 4.71. 373.44. 8. 0.3. 6. 12. 8. 2.86. 2.71. 1.60. 1.52. 2.53. 2.40. 7.83. 368.89. 9. 0.3. 8. 16. 10. 2.84. 2.71. 1.73. 1.65. 2.39. 2.28. 9.95. 370.10. 10. 0.4. 4. 6. 5. 2.80. 2.72. 1.68. 2.42. 4.78. 369.50. 11. 0.4. 6. 12. 8. 2.81. 2.44. 7.76. 368.81. 12. 0.4. 8. 16. 10. 2.30. 9.87. 371.41. 13. 0.5. 3. 6. 4. 2.81 立. 1.63 2.49 治 政 1.57 1.53大2.50 2.74 2.75 1.71 1.68 2.35. 2.74. 2.74. 1.64. 1.64. 2.38. 2.38. 3.81. 374.04. 14. 0.5. 4. 15. 0.5. 6. 16. 0.5. 8. ‧ 國. 0.1. 學. 1. 5. 2.76. 2.76. 1.66. 1.66. 2.51. 2.51. 4.79. 373.77. 12. 8. 2.76. 2.76. 1.55. 1.55. 2.48. 2.48. 7.83. 371.62. 16. 10. 2.76. 2.76. 1.68. 1.68. 2.35. 2.35. 9.81. 371.36. Nat. n. al. er. io. sit. y. ‧. 6. Ch. engchi. 9. i Un. v.
(26) Table 1 shows the coefficients of the control limits of the SDDS EWMA-AM chart for the combinations of the parameters with we consider. 0.1–0.5, ( ,. It can be seen that the when and. ,. and. . Here,. ) (3, 6, 4), (4, 6, 5), (6, 12, 8), (8, 16, 10).. increases,. ,. and. decrease and. ,. increase.. If the in-control quality variable is a symmetric distribution, then it is an asymmetric distribution, then the value of the value of. . If. will not be 0.5. We calculated. for the chi-square distribution with degree of freedom 3,. , and. the exponential distribution with mean 1, Exp(1). Table 2 shows the values of. for symmetric distributions and ( , and Exp(1).. Nat. given. n. Exp(1). and. 0.5. er. io. distribution symmetric. al. ,. sit. Table 2. Parameters of the SDDS EWMA-AM chart with various distributions,. ) (8, 16, 10) with. ‧. for. ,. y. with. 學. ‧ 國. 治 政 and the coefficients of the control limits of the SDDS 大EWMA-AM chart for their 立 corresponding distributions with . Here, we consider ( , , ) (4, 6, 5). i Un. and 500 for. .. v. E(n). 4. Ch. 1.72. 1.72. 2.60 2.60 4.66 507.85. 8. 16 10 2.79 2.76 1.70. 1.68. 2.34 2.32 9.77 373.64. 8. 16 10 2.79 2.75 1.70. 1.68. 2.34 2.31 9.81 368.09. 6. e n2.86 hi g c 2.86. 5. 10.
(27) For the out-of-control process, it is assumed that the mean proportion. has shifted and. . A small out-of-control ARL indicates superior. out-of-control detection performance of the control chart. Tables 3–7 show the out-of-control ARL and E(n) of the SDDS EWMA-AM chart with under. 0.1–0.9, three combinations of. and. and , and. 0.1,0.5(0.1), respectively. It can be seen that the out-of-control ARL decrease when the. is far away from. , the out-of-control E(n) increases for small or moderate shift and decrease. for large shift. If. and. increase (in-control E(n) increases), then the. out-of-control ARL decrease.. 立. 政 治 大. ‧ 國. 學. Table 3. The out-of-control ARL of SDDS EWMA-AM chart with under 0.1.. and. E(n). ARL. 371.89. 0.2. 5.81. 0.3. 6.39. 0.4. 6.66. a11.56 l3.99 Ch. 0.5. 6.59. 1.58. 9.08. 1.15. 11.93. 1.16. 0.6. 6.26. 1.25. 7.82. 1.05. 9.97. 1.05. 0.7. 5.65. 1.09. 6.73. 1.01. 8.72. 1.01. 0.8. 4.97. 1.03. 6.22. 1.00. 8.15. 1.00. 0.9. 4.30. 1.00. 6.02. 1.00. 8.00. 1.00. n. 10.73. 7.03. 9.83. 370.65. 13.51. 6.35. er. 4.76. io. 371.24. ARL. 0.1. 2.25. 7.74. E(n). sit. ARL. y. ‧. Nat. E(n). iv 11.29 2.38 n 14.52 U e10.43 n g c h i1.44 13.62. 11. 2.30 1.45.
(28) Table 4. The out-of-control ARL of SDDS EWMA-AM chart with under. and. 0.2.. E(n). ARL. E(n). ARL. E(n). ARL. 0.1. 6.36. 15.92. 11.49. 10.26. 14.02. 8.40. 0.2. 4.73. 368.31. 7.76. 367.30. 9.76. 374.84. 0.3. 5.69. 17.21. 10.70. 11.34. 12.84. 9.09. 0.4. 6.21. 5.61. 12.40. 3.61. 13.84. 2.97. 0.5. 6.52. 2.96. 13.53. 1.93. 13.60. 1.66. 0.6. 6.69. 1.97. 13.48. 1.32. 12.15. 1.22. 0.7. 6.75. 1.46. 12.29. 1.09. 10.26. 1.06. 0.8. 6.55. 8.65. 1.01. 0.9. 5.76. 8.08. 1.00. 治 1.00 1.05 政7.31 大 立 1.19. 9.88. 1.02. and. ‧. ‧ 國. 學. Table 5. The out-of-control ARL of SDDS EWMA-AM chart with under 0.3.. 6.90. 0.2. 6.00. 0.3. 4.71. 0.4. 5.78. a373.44 l C 20.55 h. 0.5. 6.54. 6.24. 0.6. 7.02. 0.7. io. 0.1. ARL. E(n). y. E(n). sit. ARL. ARL. 14.63. 3.24. 18.16. 2.62. 20.50. 11.38. 12.40. 14.24. 10.04. 7.83. 368.89. n. 6.03. er. Nat. E(n). i v9.95 n i U 13.57 e10.40 n g c h13.82. 370.10 10.92. 12.00. 4.29. 16.06. 3.29. 3.07. 13.46. 2.22. 17.63. 1.73. 7.16. 1.84. 14.48. 1.43. 17.49. 1.23. 0.8. 6.94. 1.29. 13.94. 1.12. 15.17. 1.06. 0.9. 5.88. 1.06. 11.55. 1.02. 10.88. 1.00. 12.
(29) Table 6. The out-of-control ARL of SDDS EWMA-AM chart with under. and. 0.4.. E(n). ARL. E(n). ARL. E(n). ARL. 0.1. 8.40. 2.67. 14.90. 1.76. 17.79. 1.71. 0.2. 7.10. 6.39. 12.63. 3.96. 15.65. 3.37. 0.3. 6.09. 21.98. 10.62. 14.36. 13.44. 11.80. 0.4. 4.78. 369.5. 7.76. 368.81. 9.87. 371.41. 0.5. 5.85. 22.81. 10.12. 15.23. 13.55. 12.33. 0.6. 6.57. 6.96. 11.34. 4.65. 16.07. 3.71. 0.7. 7.19. 3.42. 11.99. 2.34. 18.10. 1.85. 0.8. 7.93. 0.9. 8.85. 12.14 治 1.48 19.00 政 1.43 10.53 1.12大 16.51 立 2.09. 1.04. and. Nat. E(n). a l1.66 3.14 Ch. 17.11. 8.67. 0.2. 7.56. 0.3. 6.74. 14.93. 3.77. 0.4. 5.94. 23.02. 10.78. 15.07. 13.15. 12.41. 0.5. 4.79. 373.77. 7.83. 371.62. 9.81. 371.36. 0.6. 5.91. 22.97. 10.81. 15.13. 13.09. 12.56. 0.7. 6.78. 6.76. 12.84. 4.48. 14.92. 3.73. 0.8. 7.58. 3.10. 14.98. 2.06. 16.18. 1.84. 0.9. 8.67. 1.66. 17.09. 1.19. 15.27. 1.20. n U i e12.93 4.46 h ngc 14.93. 13. 2.07. 15.02. ARL 1.21. 6.88. 1.19. E(n). i v16.25. n. 0.1. ARL. er. io. ARL. sit. y. ‧. ‧ 國. 學. Table 7. The out-of-control ARL of SDDS EWMA-AM chart with under 0.5.. E(n). 1.25. 1.86.
(30) 2.3.. Performance Comparison with Existing Control Charts We compare the performance of the SDDS EWMA-AM chart with other existing. mean control charts under different distributions of quality variables. The average time to signal (ATS) and the adjusted ATS (AATS) are considered to measure the performance of a control chart if the variable sampling intervals are adopted. For fixed sampling intervals chart, the ATS is equal to the ARL when the time between two successive samples equals one time unit and the relationship of AATS and ATS is (see Daudin (1992)). For performance comparison, in-control AATS (. or. ) values of the competing charts are fixed at (or very close to) , and then. 學. ‧ 國. 治 政 an acceptable value, such as 370 or 500, with a fixed 大 sample size, 立 compare their out-of-control ARL or AATS.. If the in-control distribution of a quality variables is unknown, we consider , which depends on the distribution, for comparison. Tables. ‧. different values of. ,. io. ,. al. and. 0.1–0.9, under. er. EWMA-AM chart in Yang and Arnold (2014), for. sit. y. Nat. 8–10 show the detection performance of the SDDS EWMA-AM chart and the. 0.1, 0.3, 0.5, respectively. It can be seen that. n. iv n C the SDDS EWMA-AM chart has superior detection performance than h e n gout-of-control chi U the EWMA-AM chart for small to large shifts in mean, 0.1, 0.3, 0.5.. 14. 0.1–0.9, when.
(31) Table 8. Performance comparison of the SDDS EWMA-AM chart and the EWMA-AM chart with 0.1.. ,. SDDS EWMA-AM chart. E(n). and. under. EWMA-AM chart. ARL. ARL. 0.1. 9.83. 370.65. 370.30. 0.2. 13.51. 6.35. 10.40. 0.3. 14.52. 2.30. 4.90. 0.4. 13.62. 1.45. 3.30. 0.5. 11.93. 1.16. 2.50. 0.6. 9.97. 1.05. 2.10. 0.7. 8.72. 0.8. 8.15. 治 政 1.01 大 1.00. 0.9. 8.00. 1.00. 1.30. 立. 1.90 1.70. ‧ 國. 學. Table 9. Performance comparison of the SDDS EWMA-AM chart and the. y. sit. io. n. al. E(n). EWMA-AM chart. er. SDDS EWMA-AM chart. and. ‧. ,. Nat. EWMA-AM chart with 0.3.. Ch. ARL. e n g2.62 chi. v iARL n U. 0.1. 18.16. 0.2. 14.24. 10.04. 17.10. 0.3. 9.95. 370.10. 373.70. 0.4. 13.57. 10.92. 17.00. 0.5. 16.06. 3.29. 7.40. 0.6. 17.63. 1.73. 4.80. 0.7. 17.49. 1.23. 3.70. 0.8. 15.17. 1.06. 2.90. 0.9. 10.88. 1.00. 2.30. 15. 7.30. under.
(32) Table 10. Performance comparison of SDDS EWMA-AM chart and EWMA-AM chart with. ,. and. SDDS EWMA-AM chart. E(n). under. 0.5.. EWMA-AM chart. ARL. ARL. 0.1. 15.02. 1.21. 3.90. 0.2. 16.25. 1.86. 5.20. 0.3. 14.93. 3.77. 8.20. 0.4. 13.15. 12.41. 19.40. 0.5. 9.81. 371.36. 369.50. 0.6. 13.09. 12.56. 18.80. 0.7. 14.92. 3.73. 8.10. 0.8. 16.18. 0.9. 15.27. 立. 治 政 1.84 大 1.20. 5.20 3.90. ‧ 國. 學. If the in-control distribution of a quality variable is known, we compare their. ‧. out-of-control ARL for the shift scale. y. Nat. io. and corresponding. . Table 11 shows. under various in-control distributions.. n. al. sit. is corresponding to the value of. er. . The out-of-control the values of. . The process is in control when. i Un. v. For the IC symmetric normal distribution N(0, 1) of a quality variables with. Ch. engchi. in-control mean 0 and variance 1, whose corresponding. , Table 12 shows. the detection performance of the SDDS EWMA-AM chart, the EWMA-AM chart, the NLE chart and the CEW chart (Zou and Tsung (2010)) with and. ,. , Table 13 shows the detection performance of the SDDS EWMA-AM. chart, the SL chart (Mukherjee and Chakraborti (2012)) and SC chart (Chowdhury et al. (2014)) with. ,. and. , Table 14 shows the detection. performance of the SDDS EWMA-AM chart, Shewhart, DSVSI, VP, VSSI, VSS, VSI charts (Carot et al. (2002)) with scale. ,. ,. and the shift. . Table 15 shows the detection performance of the SDDS EWMA-AM 16.
(33) chart, Shewhart, DS, and VP and. charts (Torng and Lee (2009)) with. ,. .. It can be seen that under the standard normal distribution, (1) the SDDS EWMA-AM chart has superior out-of-control detection performance than the EWMA-AM, the NLE and the CEW charts with. for. (see. Table 12); (2) the SDDS EWMA-AM chart has superior out-of-control detection performance than the SL and the SC charts with. for. (see. Table 13); (3) the SDDS EWMA-AM chart has superior out-of-control detection performance than the Shewhart chart with. 立. ,. ,. ,. ,. ,. chart. ,. ,. ,. ,. ,. chart with. ,. ,. , for. y. al ,. chart with. for. n. chart with. , superior than the VP. Ch. ,. ,. ,. er. io. ,. ,. ,. , superior than the VSS ,. ,. , superior than the VSSI. Nat. ,. ,. ,. ‧. for. for. , superior. 學. ,. ‧ 國. , with. for 治 政, , 大. sit. than the DSVSI. chart with. , superior than the VSI. e ,n g c h i. i ,Un. v. ,. ,. for. (see Table 14); (4) the SDDS EWMA-AM chart has superior out-of-control detection performance than the Shewhart , superior than the DS , with. , ,. ,. chart with. for ,. chart with. , ,. ,. ,. ,. , ,. , ,. for. , superior than the DS. ,. superior than the VP ,. chart with. chart with. ,. for , 17. chart. for. ,. ,. ,. , superior than the VP ,. ,. ,.
(34) ,. for. or. Table 11. The values of. N( , 1). DE(. (see Table 15).. and corresponding. t(4). , 1). DE( ,. ). +. +. under various distributions.. t(10) +. t(30). Exp(1). Unif +. +. +. +. 0.00. 0.5000. 0.5000. 0.5000. 0.5000. 0.5000. 0.5000. 0.5000. 0.3916. 0.3679. 0.10. 0.5405. 0.5654. 0.5633. 0.5529. 0.5433. 0.5410. 0.5295. 0.4311. 0.4059. 0.20. 0.5790. 0.6233. 0.6241. 0.6043. 0.5862. 0.5810. 0.5589. 0.4733. 0.4489. 0.25. 0.5983. 0.6483. 0.6526. 0.6291. 0.6073. 0.6012. 0.5719. 0.4967. 0.4725. 0.30. 0.6175. 0.6737. 0.6802. 0.5865. 0.5193. 0.4971. 0.40. 0.6554. 0.7165. 0.5684. 0.5484. 0.50. 0.6913. 0.7531. 立 0.7742. 0.6154. 0.7411. 0.7052. 0.6957. 0.6441. 0.6201. 0.6070. 0.60. 0.7262. 0.7864. 0.8123. 0.7775. 0.7415. 0.7302. 0.6757. 0.6704. 0.75. 0.7736. 0.8268. 0.8574. 0.8259. 0.7898. 學. 0.6743. 0.7781. 0.7171. 0.7620. 0.7788. 1.00. 0.8409. 0.8782. 0.9093. 0.8849. 0.8548. 0.8452. 0.7891. 0.9074. 1.0000. 1.25. 0.8940. 0.9147. 0.9407. 0.9237. 0.9036. 0.8970. 0.8610. 1.0000. 1.0000. 1.50. 0.9335. 0.9403. 0.9598. 0.9491. 0.9378. 0.9351. 0.9327. 1.0000. 1.0000. 2.00. 0.9771. 0.9704. 0.9799. 0.9764. 0.9754. 0.9765. 1.0000. 1.0000. 1.0000. 2.50. 0.9940. 0.9856. 0.9885. 0.9880. 0.9906. 0.9926. 1.0000. 1.0000. 1.0000. 3.00. 0.9987. 0.9929. 0.9931. 0.9933. 0.9964. 1.0000. 1.0000. 1.0000. n. engchi U. 18. y. sit. io. Ch. er. Nat. al. ‧. ‧ 國. 0.6530 治 0.6279 0.6213 政 0.7311 0.6991 0.6680 大 0.6590. v ni. 0.9979.
(35) Table 12. Performance comparison of the SDDS EWMA-AM chart, the EWMA-AM chart, the NLE chart and the CEW chart with under N( , 1). SDDS EWMA-AM chart. E(n). ,. and. EWMA-AM chart. NLE chart. CEW chart. ARL. ARL. ARL. ARL. 0.00. 9.81. 371.36. 371.03. 369.00. 370.00. 0.25. 13.12. 12.89. 18.73. 98.00. 84.90. 0.50. 14.81. 4.03. 8.29. 36.10. 29.30. 0.75. 16.06. 2.14. 5.62. 20.10. 16.20. 1.00. 16.08. 1.51. 4.45. 14.10. 11.00. 1.50. 13.56. 1.10. 7.65. 6.28. 2.00. 10.44. 1.01. 4.57. 4.04. 3.00. 8.18. 2.08. 2.11. 立1.00. 政 治 3.44 3.07大 3.00. ‧ 國. 學. 4.66. 0.25. 5.91. 0.50. al. n. 0.00. SL chart m=100. ARL ARL i n C U h500.79 507.85 i e n g c h499.62. ARL. SC chart m=50. SC chart m=100. ARL. ARL. 513.00. 497.30. 509.40. y. SL chart m=50. er. io E(n). SL chart m=30. sit. Nat. SDDS EWMA-AM chart. ‧. Table 13. Performance comparison of the SDDS EWMA-AM chart, the SL charts and the SC charts with , and under N( , 1).. v ARL. 27.23. 369.08. 292.69. 257.60. 288.60. 253.60. 6.77. 8.37. 145.18. 94.69. 66.50. 92.20. 68.60. 1.00. 8.13. 2.88. 13.09. 9.09. 7.70. 8.50. 7.70. 1.50. 9.18. 1.69. 2.67. 2.32. 2.10. 2.20. 2.10. 19.
(36) Table 14. Performance comparison of the SDDS EWMA-AM chart and the Shewhart, DSVSI, VP, VSSI, VSS, and the VSI and under N( , 1). SDDS EWMA-AM. Shewhart. chart. chart. charts with. DSVSI chart. ,. VP. VSSI. chart. chart. VSS. VSI. chart. chart. .70. .00. AATS. AATS. 370.37. 370.37. 370.37. 370.37. 370.37. 280.60. 170.30. 202.60. 274.60. 277.20. 274.60. 28.34. 154.70. 52.63. 67.25. 130.20. 135.50. 141.50. 13.99. 80.72. 19.13. 23.38. 48.26. 53.12. 66.15. 8.68. 43.39. 8.45. 9.78. 16.58. 19.92. 30.72. 5.72. 24.46. 4.49. 5.07. 6.80. 8.60. 14.58. 4.15. 14.47. 2.79. 3.15. y. 82.03. 0.50. 4.89. 0.75. 5.37. 1.00. 5.78. 1.25. 6.21. 1.50. 6.59. 2.00. 7.31. 3.00. 8.33. AATS. 3.43. 4.69. 7.21. 2.45. 5.80. 1.51. sit. 4.29. AATS. 370.37. 1.70. 1.68. 2.35. 2.26. 1.50. 0.92. 0.85. 0.82. 1.24. 0.93. ‧ 國. 0.25. 立. AATS. ‧. 374.04. AATS. Nat. 1.17. n. al. Ch. engchi. 20. 學. 3.81. AATS. io. 0.00. 政 治 大. er. E(n). i Un. v.
(37) Table 15. Performance comparison of the SDDS EWMA-AM chart and the Shewhart, DS, and the VP under N( , 1).. charts with. SDDS EWMA-AM. Shewhart. chart. 0.75. 7.34. 1.00. 7.97. 1.50. 9.10. 2.00. 9.69. 3.00. 9.98. VP. chart. chart. chart. chart. AATS 政 治 大 370.40 370.40. AATS. 立 22.99. 373.77. AATS. AATS. AATS. 370.40. 370.40. 370.40. 23.33. 40.06. 80.75. 8.98. 5.87. 10.36. 3.29. 2.79. 2.48. 1.47. 1.85. 1.21. 0.64. 1.05. 0.76. 0.53. 0.70 0.65. 132.66. 37.64. 6.85. 32.90. 6.64. 3.27. 10.26. 2.62. 1.88. 4.00. 1.59. 0.89. 1.07. 0.84. 0.61. 0.58. 0.59. y. 6.67. VP. 0.77. 0.51. 0.50. 0.50. sit. 0.50. DS. 0.67. io. n. al. 0.50. er. 5.89. DS. ‧. 0.25. and. 學. 4.79. AATS. Nat. 0.00. chart. ‧ 國. E(n). ,. Ch. engchi. 21. i Un. v.
(38) For the IC symmetric double exponential distribution DE(0,. ) with mean 0. and variance 1 and IC symmetric DE(0, 1) with mean 0 and variance 2, and corresponding. , Table 16 shows the detection performance of the SDDS. EWMA-AM chart, the EWMA-AM chart and the SL charts (Mukherjee and Chakraborti (2012)) with. ,. and. under DE( ,. ),. Table 17 shows the detection performance of the SDDS EWMA-AM chart, the SL chart and the SC chart with. ,. It can be seen that under DE( ,. and. under DE(. , 1).. ), the SDDS EWMA-AM chart has superior. out-of-control detection performance than the EWMA-AM and the SL charts for. 治 政 (see Table 16); under DE( , 1), the大 SDDS EWMA-AM chart has 立 superior out-of-control detection performance than the SL and the SC charts for ‧ 國. 學. (see Table 17).. ‧. For the IC symmetric student's t distribution t(3), t(4), t(10), t(30) with degrees but have heavier. sit. y. Nat. of freedom 3, 4, 10, 30, respectively, and corresponding. io. al. er. tails than the standard normal distribution. Table 18 shows the detection performance. n. of the SDDS EWMA-AM chart, the NLE chart, the CEW chart and the EWM chart (Zou and Tsung (2010)). Ch with. e n g c h ,i. i Un. v. and. under. . Tables 19–21 show the detection performance of the SDDS EWMA-AM chart, the Shewhart, the DS, and the VP with. ,. t(30). and. charts (Torng and Lee (2009)). under t(4). , t(10). ,. , respectively. It can be seen that under. , the SDDS EWMA-AM chart has. superior out-of-control detection performance than the NLE, the CEW and the EWM charts for. (see Table 18); under t(4) 22. , the SDDS EWMA-AM.
(39) chart has superior out-of-control detection performance than the Shewhart chart with for. , superior than the DS. ,. ,. ,. for. ,. ,. ,. chart with. , superior than the VP ,. ,. than the VP ,. , ,. ,. for. ,. ,. or. ,. (see Table 19);. , the SDDS EWMA-AM chart has superior out-of-control detection. ,. , superior than the DS ,. io. for ,. ,. ,. or. a l,. for. ,. , superior than the VP ,. ,. Ch. or. v ni. ,. , chart with. , superior than the VP. n. ,. ,. sit. ,. chart with. for. Nat. ,. ,. ‧. ,. ,. er. for. ,. y. ,. ‧ 國. chart with. , superior than. 學. the DS. , , superior. ,. for. for ,. 政 治 大 performance than the Shewhart chart with for 立 under t(10). ,. , superior than the DS. chart with. ,. ,. ,. ,. ,. chart with. chart with. ,. , chart with. ,. ,. i U e n g c h(see Table 20); under t(30). ,. the SDDS EWMA-AM chart has superior out-of-control detection performance than the Shewhart chart with with. ,. for. ,. , superior than the DS ,. ,. , or. ,. chart with. for. ,. , superior than the DS , ,. ,. ,. , superior than the VP 23. for. ,. , superior than the VP. chart. ,. ,. chart with. , chart with. ,. , for. ,. ,.
(40) ,. ,. or. ,. ,. ,. for. (see Table 21). The heavier tails for the t distribution, the. superior out-of-control detection performance the SDDS EWMA-AM chart has.. Table 16. Performance comparison of the SDDS EWMA-AM chart, the EWMA-AM chart and the SL charts with , and under ). SDDS EWMA-AM chart. 394.53. 7.28. 5.07. 9.40. 207.50. 114.91. 8.00. 3.12. 7.03. 71.90. 28.81. 8.53. 2.33. 5.92. 25.56. 7.93. 8.95. 1.90. 5.30. 7.78. 3.19. 1.62. 4.93. y. 1.50. 18.34. 2.36. 1.80. 9.24. io. n. al. sit. 1.25. 495.25. 13.21. ‧ 國. 1.00. 6.39. ARL. ‧. 0.75. ARL. 學. 0.50. SL chart m=50. Nat. 0.25. SL chart m=30. 治 ARL 政 ARL 大487.67 4.66 立 507.85 494.85. E(n) 0.00. EWMA-AM chart. er. DE( ,. Ch. engchi. 24. i Un. v. 330.07.
(41) Table 17. Performance comparison of the SDDS EWMA-AM chart, the SL chart and the SC chart with SDDS EWMA-AM chart. E(n). ,. ARL. SC chart m=50. and SL chart m=50. under DE( , 1). SC SL chart chart m=100 m=100. ARL. ARL. ARL. ARL. 0.00. 4.66. 507.85. 492.7. 493.2. 509.6. 508.3. 0.25. 6.03. 22.13. 419.0. 403.1. 381.6. 366.9. 0.50. 6.78. 8.09. 240.4. 235.2. 191.0. 159.2. 1.00. 7.85. 3.37. 41.4. 36.1. 26.5. 19.9. 1.50. 8.62. 2.22. 7.2. 5.93. 4.8. 4.1. 2.00. 9.15. 1.69. 2.1 2.0 政 治 大. 1.8. 1.7. 立. ‧ 國. 學. Table 18. Performance comparison of the SDDS EWMA-AM chart, the NLE chart, the CEW chart and the EWM chart with , and under. .. ‧. NLE chart. CEW chart. sit. n. al. er. io E(n). ARL. EWM chart. y. Nat. SDDS EWMA-AM chart. 0.00. 9.81. 371.36. 0.25. 14.16. 6.04. 0.50. 15.94. 0.75. Ch. ARL. iv n U 127.00 ARL. e n372.00 gchi. ARL 365.00. 72.10. 73.50. 82.40. 2.17. 28.40. 28.60. 27.00. 15.97. 1.40. 17.90. 16.30. 15.20. 1.00. 14.70. 1.17. 13.30. 11.10. 10.60. 1.50. 11.98. 1.04. 8.42. 6.73. 6.62. 2.00. 10.13. 1.01. 6.04. 4.44. 4.90. 3.00. 8.82. 1.00. 3.74. 2.26. 3.31. 25.
(42) Table 19. Performance comparison of the SDDS EWMA-AM chart, the Shewhart, the DS, and the VP. charts with. under t(4) . SDDS EWMA-AM chart. 0.75. 7.82. 1.00. 8.46. 1.50. 9.34. 2.00. 9.69. 3.00. 9.92. VP. chart. chart. chart. chart. 政 治AATS 大 370.10 369.60. AATS. 立 14.25. 373.27. AATS. AATS 370.27. 370.95. 99.83. 44.41. 105.07. 13.81. 5.53. 11.84. 3.34. 2.50. 2.29. 1.44. 1.59. 1.10. 0.60. 0.91. 0.75. 0.53. 0.71 0.70. 277.71. 45.51. 4.33. 131.91. 7.40. 2.11. 50.09. 2.57. 1.32. 17.66. 1.45. 0.77. 2.63. 0.74. 0.61. 0.79. 0.56. 0.73. 0.53. 0.50. 0.54. 0.68. io. n. al. Ch. 0.50. engchi. 26. i Un. v. AATS. 370.34. y. 7.05. VP. sit. 0.50. DS. er. 6.20. DS. ‧. 0.25. chart. and. 學. 4.79. Shewhart. Nat. 0.00. AATS. ‧ 國. E(n). ,.
(43) Table 20. Performance comparison of the SDDS EWMA-AM chart, the Shewhart, the DS, and the VP. 7.52. 1.00. 8.13. 1.50. 9.14. 2.00. 9.68. 3.00. 9.96. chart. chart. chart. chart. 政 治 大 AATS AATS 371.37. 19.84. 161.69. 41.34. 5.98. 44.35. 5.92. 2.83. 13.72. 2.60. 1.69. 5.08. 1.55. 0.86. 1.21. 0.82. 0.62. 0.60. 0.59. 0.50. 0.50. 0.51. n. al. Ch. engchi. 27. AATS 370.00. i Un. AATS. AATS. 370.69. 370.38. 68.15. 40.42. 85.61. 12.46. 5.82. 10.66. 3.01. 2.74. 2.45. 1.40. 1.80. 1.18. y. 0.75. VP. 0.64. 1.01. 0.75. sit. 6.79. VP. 371.77. ‧ 國. 0.50. DS. 0.53. 0.76. 0.68. 0.50. 0.66. 0.61. ‧. 5.97. 立 373.27. DS. 學. 0.25. chart. AATS. io. 4.79. Shewhart. Nat. 0.00. and. .. SDDS EWMA-AM chart. E(n). ,. er. under t(10). charts with. v.
(44) Table 21. Performance comparison of the SDDS EWMA-AM chart, the Shewhart, the DS, and the VP. charts with .. 0.75. 7.38. 1.00. 8.07. 1.50. 9.11. 2.00. 9.71. 3.00. 9.97. VP. chart. chart. chart. chart. 政 治 大 AATS AATS 371.10. 22.14. 140.31. 38.97. 6.70. 35.25. 5.81. 3.12. 10.97. 2.65. 1.79. 4.23. 1.57. 0.88. 1.10. 0.84. 0.61. 0.58. 0.59. 0.50. 0.50. 0.51. n. al. Ch. engchi. 28. AATS 369.80. i Un. AATS. AATS. 370.45. 370.16. 69.59. 39.87. 81.85. 11.31. 5.83. 10.41. 2.98. 2.77. 2.47. 1.26. 1.84. 1.20. y. 6.69. VP. 369.76. ‧ 國. 0.50. DS. 0.64. 1.04. 0.76. 0.53. 0.77. 0.70. 0.50. 0.62. 0.61. ‧. 5.93. DS. 學. 0.25. 立 373.27. AATS. io. 4.79. chart. Nat. 0.00. Shewhart. sit. SDDS EWMA-AM chart. E(n). and. er. under t(30). ,. v.
(45) For the IC symmetric uniform distribution Unif. with mean 0,. variance 1 and. , and the IC exponential distribution Exp(1) with mean 1,. variance 1 and. , Table 22 shows the detection performance of the. SDDS EWMA-AM chart and the EWMA-AM chart with and. under Unif. Unif. and Exp(1). and Exp(1). ,. . It can be seen that under. , the SDDS EWMA-AM chart has superior. out-of-control detection performance than the EWMA-AM chart for. (see. Table 22). , and corresponding 治 政 , Table 23 shows the detection performance 大 of the SDDS EWMA-AM 立 chart, the NLE chart, the CEW chart and the EWM chart (Zou and Tsung (2010)) with under. . It can be seen that. , the SDDS EWMA-AM chart has superior out-of-control. ‧. under. and. 學. ,. ‧ 國. For the chi-square distribution with degree of freedom 3,. y. sit. io. n. al. er. (see Table 23).. Nat. detection performance than the NLE, the CEW and the EWM charts for. Ch. engchi. 29. i Un. v.
(46) Table 22. Performance comparison of the SDDS EWMA-AM chart and the EWMA-AM chart with Unif distribution. ,. and. and Exp(1) Unif SDDS EWMA-AM chart. E(n). ARL. under. . Exp(1). EWMA-AM chart. ARL. SDDS EWMA-AM chart. E(n). ARL. EWMA-AM chart. ARL. 0.00. 9.81. 371.36. 371.03. 9.81. 368.09. 369.56. 0.10. 10.88. 87.31. 105.80. 11.16. 54.90. 69.63. 0.20. 12.01. 31.19. 39.96. 12.63. 16.61. 24.00. 0.25. 12.45. 11.20. 17.17. 0.30. 12.79. 7.96. 13.24. 0.40. 13.45. 28.97 13.14 治 政 15.96 22.64 13.49 大 15.22 14.44 立9.77. 4.43. 8.79. 0.50. 14.09. 6.64. 11.59. 14.96. 2.84. 6.43. 14.58. 4.80. 9.38. 15.13. 1.96. 4.98. 15.20. 3.25. 7.22. 13.99. 1.29. 3.63. 16.04. 1.97. 5.31. 8.00. 15.81. 1.39. 4.20. 8.00. 1.50. 13.59. 1.10. 3.43. 8.00. 2.00. 8.00. 1.00. 3.00. 8.00. 2.50. 8.00. 3.00. 8.00. 3.00. 8.00. a l1.00 1.00 Ch. y. n. 1.00. 2.00. 1.00. 2.00. 1.00. 2.00. 1.00. 2.00. 1.00 1.00. 2.00. sit. io. iv 3.00 8.00 n engchi U. 30. 2.00. er. Nat. 1.25. ‧. 1.00. ‧ 國. 0.75. 學. 0.60. 21.58.
(47) Table 23. Performance comparison of the SDDS EWMA-AM chart, the NLE chart, the CEW chart and the EWM chart with under SDDS EWMA-AM chart. E(n). ,. and. NLE chart. CEW chart. EWM chart. ARL. ARL. ARL. ARL. 0.00. 9.77. 373.64. 373.00. 120.00. 373.00. 0.25. 13.17. 11.23. 54.40. 65.00. 72.00. 0.50. 14.79. 2.97. 30.70. 29.40. 27.70. 0.75. 14.09. 1.39. 22.70. 17.10. 16.00. 1.00. 10.19. 1.03. 11.10. 1.50. 8.00. 1.00. 2.00. 8.00. 1.00. 17.90 11.80 政 12.30治 大 7.03. 3.00. 8.00. 1.00. 8.91. 4.65. 5.09. 5.11. 2.32. 3.42. 學 ‧. ‧ 國. 立. n. er. io. sit. y. Nat. al. Ch. 6.95. engchi. 31. i Un. v.
(48) 2.4.. Example. A non-normal service times data set from Yang and Arnold (2014) is used to illustrate the applications of the SDDS EWMA-AM chart. The in-control sampling service times. –. 15 days (t=1, 2, (. –. (unit: minutes) are measured from 10 counters every day for , 15) (see Table 24). For using double sampling scheme, we adopt. ),. –. (. ) and. To construct the SDDS EWMA-AM chart, , so. . is estimated by. can be calculated (see Table 24). Furthermore,. estimated by. is. 政 治 大. . Here, we assume the in-control data are close to. 立. 學. ‧ 國. population.. In reality, to reduce the error of estimation, the. and. are expected to be. estimated by large in-control data.. ‧. Thus, the SDDS EWMA-AM chart with. and. Nat. y. is. Stage 1 chart:. n. Ch. engchi. er. io. al. sit. constructed as follows:. i Un ,. v. , ,. (16). , Stage 2 chart: , .. 32. (17).
(49) Table 24. The in-control service times from 10 counters in a bank branch. t. Stage 1. Stage 2. 1. 0.88. 0.78. 5.06. 5.45. 0. 2.93. 6.11. 11.59. 1.20. 0.89. 3.21. 2. 2. 2. 3.82. 13.40. 5.16. 3.20. 1. 32.27. 3.68. 3.14. 1.58. 2.72. 7.71. 2. 3. 3. 1.40. 3.89. 10.88. 30.85. 2. 0.54. 8.40. 5.10. 2.63. 9.17. 3.94. 2. 4. 4. 16.80. 8.77. 8.36. 3.55. 3. 7.76. 1.81. 1.11. 5.91. 8.26. 7.19. 4. 7. 5. 0.24. 9.57. 0.66. 1.15. 1. 2.34. 0.57. 8.94. 5.54. 11.69. 6.58. 3. 4. 6. 4.21. 8.73. 11.44. 2.89. 2. 19.49. 1.20. 8.01. 6.19. 7.48. 0.07. 4. 6. 7. 15.08. 7.43. 4.31. 6.14. 3. 10.37. 2.33. 1.97. 1.08. 4.27. 14.08. 2. 5. 8. 13.89. 0.30. 3.21. 11.32. 2. 9.90. 4.39. 10.50. 1.70. 10.74. 1.46. 3. 5. 9. 0.03. 12.76. 2.41. 7.41. 2. 1.67. 3.70. 4.31. 2.45. 3.57. 3.33. 0. 2. 10. 12.89. 17.96. 2.78. 3.21. 2. 1.12. 12.61. 4.23. 6.18. 2.33. 6.92. 3. 5. 11. 7.71. 1.05. 1.11. 0.22. 1. 3.53. 0.81. 0.41. 3.73. 0.08. 2.55. 0. 1. 12. 5.81. 6.29. 3.46. 2.66. 2. 4.02. 10.95. 1.59. 5.58. 0.55. 4.10. 1. 3. 13. 2.89. 1.61. 1.30. 2.58. 0. 18.65. 10.77. 18.23. 3.13. 3.38. 6.34. 4. 4. 14. 1.36. 1.92. 0.12. 11.08. 1. 8.85. 3.99. 4.32. 1.71. 1.77. 1.94. 1. 2. 15. 21.52. 0.63. 8.54. 3.37. 2. 6.94. 3.44. 3.37. 6.37. 1.28. 12.83. 3. 5. 學. Nat. n. al. er. io. sit. y. ‧. ‧ 國. 立. 政 治 大. Ch. engchi. 33. i Un. v.
(50) Table 25. The plotting statistics of the SDDS EWMA-AM chart with t=1, 2,. for. , 15.. t. Stage 1. Stage 2 Detect result. 0. 1.520. -1.633. 2. 2. 3.900. -1.291. IC. 2. 1. 1.494. -1.569. IC. -. -. -. -. -. -. 3. 2. 1.519. -0.999. IC. -. -. -. -. -. -. 4. 3. 1.593. -0.073. IC. -. -. -. -. -. -. 5. 1. 1.564. -0.366. IC. -. -. -. -. -. -. 6. 2. 1.585. -0.136. IC. -. -. -. -. -. -. 7. 3. 1.656. 0.501. IC. -. -. -. -. -. -. 8. 2. 1.673. 0.625. IC. -. -. -. -. 9. 2. 1.690. 0.737. -. -. -. -. 10. 2. 1.705. 0.838. -. -. -. -. 11. 1. 1.670. 0.542. IC. -. -. -. -. -. -. 12. 2. 1.686. 0.655. IC. -. -. -. -. -. -. 13. 0. 1.602. 0.016. IC. -. -. -. -. -. -. 14. 1. 1.572. -0.204. IC. -. -. -. -. -. -. 15. 2. 1.593. -0.047. IC. -. -. -. -. -. -. 治 政IC 大 IC -. ‧ 國. 立. Nat. n. al. er. io. sit. y. ‧. 1. 學. 1. Detect result. Ch. engchi. 34. i Un. v.
(51) 政 治 大. 立. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. v ni. Figure 1. The SDDS EWMA-AM chart for t=1, 2,. Ch. Next we plot the 15 statistics EWMA-AM chart. If. engchi U. in the stage 1 of the constructed SDDS. falls outside of control limits of the SDDS EWMA-AM. chart, the process mean is our-of-control. If region (. falls outside within the central. ), we conclude the process mean is in-control. If. warning region (. , 15.. ), then we take the second sample of size. is within the and enter stage 2.. Table 25 shows the results and it can be seen that we need to take the second sample of size. at the first sample since the statistic. the stage2 the statistic. fall in the central region ( 35. is in the WR. However, in ), it indicates that the.
(52) process mean is in-control in the first day. The SDDS EWMA-AM chart shows no signals for t=1, 2,. , 15 (see Figure 1).. To monitor the new service times from the new automatic service system of the bank branch, the new service times and statistic. –. are measured from 4 counters at first. is calculated and plotted in the stage 1 chart, if –. the WR, then the second sample of service times 6 counters and its statistic. is within. are measured from other. is calculated and plotted in the stage 2 chart. Table. 26 shows the detection results and it can be seen that there are 3 samples (t=19, 20, 21) should take the second sample. In stage 2, sample 19 is within the 20 and 21 are outside of the. 立. but samples 治 政 . The mean of the new 大service times is significantly. reduced in Figure 2 and we have 6 out-of-control mean signals after t=20.. ‧ 國. 學. To compare the out-of-control detection performance, we constructed the. chart by applying. y. sit. under. io. al. , and transformed Shewhart. chart with. er. and. transformation. The EWMA-AM chart with. Nat. EWMA-. chart and transformed. ‧. corresponding EWMA-AM chart, transformed Shewhart. n. are constructed as follows:. i C hEWMA-AM ,U n engchi. v. ,. EWMA-AM. ,. (18). , The EWMA-AM chart shows 6 out-of-control mean signals after t=20 (see Figure 3) and the transformed Shewhart signal at t=18 (see Figure 4).. 36. chart shows one out-of-control mean.
(53) Table 26. The new service times on the SDDS EWMA-AM chart with Stage. t. .. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 3.54. 0.86. 1.45. 1.37. 3.00. 1.59. 5.01. 4.96. 1.08. 4.56. 0.01. 1.61. 0.19. 0.14. 2.46. 3.88. 1.85. 0.55. 0.65. 0.44. 1.33. 1.15. 4.18. 1.54. 0.06. 0.39. 3.10. 1.43. 0.91. 5.61. 7.27. 0.96. 0.18. 1.58. 1.80. 0.54. 1.00. 4.12. 0.88. 2.79. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.564. 1.486. 1.411. 1.341. 1.274. 1.210. 1.150. 1.092. 1.037. 0.986. -0.257. -0.803. -1.311. -1.784. -2.228. -2.644. -3.034. -3.402. -3.749. -4.076. IC. IC. IC. WR. WR. WR. OOC. OOC. OOC. OOC. -. -. -. 2. 3. 4. -. -. -. -. -. -. -. -. -. -. -. -. -. Stage 1. Detect result. 3.25. 1.58. -. -. -. -. 6.01. 2.13. 1.70. -. -. -. -. -. 4.59. 2.22. 0.68. -. -. -. -. -. -. 1.74. 1.37. 1.25. -. -. -. -. -. -. -. 3.92. 2.13. 6.83. -. -. -. -. -. -. -. 4.82. 0.25. 0.31. -. -. -. -. -. -. -. 1. 0. 1. -. -. -. -. -. -. -. 1. 0. 1. -. -. -. -. -. -. 3.755. 3.567. 3.439. -. -. -. -. -. -. -2.293. -3.389. -3.899. -. -. -. -. C -h. IC. OOC. OOC iv n U. -. -. -. -. al. n. Detect result. -. engchi. y. sit. io. -. Nat. -. er. 2. 立. ‧. ‧ 國. 0.45. 學. Stage. 政 治 大. In this case, it can be seen that the new service times data is well detected out by the SDDS EWMA-AM chart and the EWMA-AM chart, they have the same out-of-control mean signals from t=20. However, the transformed Shewhart loses most of the out-of-control mean signals.. 37. chart.
(54) 立. 政 治 大. Figure 2. The SDDS EWMA-AM chart for t=1, 2,. , 25.. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 3. The EWMA-AM chart.. Figure 4. The transformed Shewhart 38. chart..
(55) 3. The SDDS EWMA-AV Chart 3.1.. Construction of the SDDS EWMA-AV Chart We use the double sampling procedure and follow the approach in EWMA-AV. chart (Yang and Arnold (2014)) to construct the SDDS EWMA-AV Chart and monitor the process variance. Denote a critical quality variable observation at time t.. , j=1, 2,. and t=1, 2,. has an in-control variance. . To construct the SDDS. EWMA-AV chart at the stage 1, we take a sample of size. 政 治 大 ,. ‧ 國. )=. , ,. ,. , the statistic. al. ), t=1, 2,. Ch. ), and the value of. Thus, the statistic. .. i Un. engchi. v. ,. , be the mean of. . Hence the mean and variance of. Var(. (20). )=. .. is defined as follows:. ,. E(. (19). will depend on the distribution of. based on the statistic. Let the starting value,. .. follows a binomial distribution with. n ,. , t=1, 2,. sit. io is. ~B(. ,. er. Nat. =. j = 1, 2,. ‧. ), for. 學. E(. P(. parameters (0.5. ,. is even. Define. 立. The mean of. ,. y. from an IC process and. , be the jth. t=1, 2,. .. (21). ; that is,. are ,. ) =. .. (22). Thus, in stage 1 the dynamic double sampling (DDS) EWMA-AV chart constructed based on the mean and variance of the statistic 39. with time.
(56) varying upper control limit ( lower warning limit (. ), upper warning limit (. ) and lower control limit (. ), central line (. ),. ) are , ,. ,. (23) , ,. where. ,. chosen coefficients for the. 立. , 政, and治 , 大 , , and. ,. are appropriately , respectively.. ‧ 國. 學. Let the SDDS EWMA-AV Chart be used by the standardized form. Hence the plotting statistic at stage 1 is defined as follows:. ‧. .. Nat. y. (24). ,. n. er. io. al. sit. The stage 1 standardized control limits, warning limits and central line are given by. Ch. engchi. ,. i Un. v. ,. (25) , .. If. the action region. out of control. If. ,. the central region. process is in control. If ,. ,(. , , [. ,. ], then the. the warning region. ,. ,. , then taking a second sample of size ,. , then the process is. is even and the procedure is entering stage 2. 40. ,.
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