monitoring the coefficient of variation using ewma charts
play

MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHARTS Philippe - PowerPoint PPT Presentation

Sep 09, 2022 •171 likes •1.61k views

MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHARTS Philippe CASTAGLIOLA 1 , Giovanni CELANO 2 , Stelios PSARAKIS 3 1 Universit e de Nantes & IRCCyN UMR CNRS 6597, France 2 Universit` a di Catania, Catania, Italy 3 Athens University

  1. SH- γ chart (Kang et al., JQT 2007) General assumptions subgroups { X k , 1 , X k , 2 , . . . , X k , n } of size n are observed at time k = 1 , 2 , . . . . X k , j ∼ N ( µ k , σ k ). Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  2. SH- γ chart (Kang et al., JQT 2007) General assumptions subgroups { X k , 1 , X k , 2 , . . . , X k , n } of size n are observed at time k = 1 , 2 , . . . . X k , j ∼ N ( µ k , σ k ). from one subgroup to another, µ k and σ k may change, but they are constrained by the relation γ k = σ k µ k = γ 0 when the process is in-control. Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  3. SH- γ chart (Kang et al., JQT 2007) General assumptions subgroups { X k , 1 , X k , 2 , . . . , X k , n } of size n are observed at time k = 1 , 2 , . . . . X k , j ∼ N ( µ k , σ k ). from one subgroup to another, µ k and σ k may change, but they are constrained by the relation γ k = σ k µ k = γ 0 when the process is in-control. Control limits Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  4. SH- γ chart (Kang et al., JQT 2007) General assumptions subgroups { X k , 1 , X k , 2 , . . . , X k , n } of size n are observed at time k = 1 , 2 , . . . . X k , j ∼ N ( µ k , σ k ). from one subgroup to another, µ k and σ k may change, but they are constrained by the relation γ k = σ k µ k = γ 0 when the process is in-control. Control limits � α 0 F − 1 � = 2 | n , γ 0 LCL SH ˆ γ F − 1 � 1 − α 0 � = 2 | n , γ 0 UCL SH γ ˆ Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  5. SH- γ chart (Kang et al., JQT 2007) General assumptions subgroups { X k , 1 , X k , 2 , . . . , X k , n } of size n are observed at time k = 1 , 2 , . . . . X k , j ∼ N ( µ k , σ k ). from one subgroup to another, µ k and σ k may change, but they are constrained by the relation γ k = σ k µ k = γ 0 when the process is in-control. Control limits � α 0 F − 1 � = 2 | n , γ 0 LCL SH ˆ γ F − 1 � 1 − α 0 � = 2 | n , γ 0 UCL SH γ ˆ where α 0 is the type I error rate (ex : α 0 = 0 . 0027). Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  6. SH- γ chart (Kang et al., JQT 2007) General assumptions subgroups { X k , 1 , X k , 2 , . . . , X k , n } of size n are observed at time k = 1 , 2 , . . . . X k , j ∼ N ( µ k , σ k ). from one subgroup to another, µ k and σ k may change, but they are constrained by the relation γ k = σ k µ k = γ 0 when the process is in-control. Control limits � α 0 F − 1 � = 2 | n , γ 0 LCL SH ˆ γ F − 1 � 1 − α 0 � = 2 | n , γ 0 UCL SH γ ˆ where α 0 is the type I error rate (ex : α 0 = 0 . 0027). Comments Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  7. SH- γ chart (Kang et al., JQT 2007) General assumptions subgroups { X k , 1 , X k , 2 , . . . , X k , n } of size n are observed at time k = 1 , 2 , . . . . X k , j ∼ N ( µ k , σ k ). from one subgroup to another, µ k and σ k may change, but they are constrained by the relation γ k = σ k µ k = γ 0 when the process is in-control. Control limits � α 0 F − 1 � = 2 | n , γ 0 LCL SH ˆ γ F − 1 � 1 − α 0 � = 2 | n , γ 0 UCL SH γ ˆ where α 0 is the type I error rate (ex : α 0 = 0 . 0027). Comments Simple two-sided “Shewhart-type” control chart. Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  8. SH- γ chart (Kang et al., JQT 2007) General assumptions subgroups { X k , 1 , X k , 2 , . . . , X k , n } of size n are observed at time k = 1 , 2 , . . . . X k , j ∼ N ( µ k , σ k ). from one subgroup to another, µ k and σ k may change, but they are constrained by the relation γ k = σ k µ k = γ 0 when the process is in-control. Control limits � α 0 F − 1 � = 2 | n , γ 0 LCL SH ˆ γ F − 1 � 1 − α 0 � = 2 | n , γ 0 UCL SH γ ˆ where α 0 is the type I error rate (ex : α 0 = 0 . 0027). Comments Simple two-sided “Shewhart-type” control chart. Unefficient for detecting small change in γ . Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  9. EWMA- γ chart (Hong et al., JSKISE 2008) Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  10. EWMA- γ chart (Hong et al., JSKISE 2008) Monitored statistic Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  11. EWMA- γ chart (Hong et al., JSKISE 2008) Monitored statistic Z k = (1 − λ ) Z k − 1 + λ ˆ γ k Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  12. EWMA- γ chart (Hong et al., JSKISE 2008) Monitored statistic Z k = (1 − λ ) Z k − 1 + λ ˆ γ k Control limits Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  13. EWMA- γ chart (Hong et al., JSKISE 2008) Monitored statistic Z k = (1 − λ ) Z k − 1 + λ ˆ γ k Control limits � λ (1 − (1 − λ ) 2 k ) LCL EWMA − γ = µ 0 (ˆ γ ) − K σ 0 (ˆ γ ) 2 − λ � λ (1 − (1 − λ ) 2 k ) UCL EWMA − γ = µ 0 (ˆ γ ) + K σ 0 (ˆ γ ) 2 − λ Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  14. EWMA- γ chart (Hong et al., JSKISE 2008) Monitored statistic Z k = (1 − λ ) Z k − 1 + λ ˆ γ k Control limits � λ (1 − (1 − λ ) 2 k ) LCL EWMA − γ = µ 0 (ˆ γ ) − K σ 0 (ˆ γ ) 2 − λ � λ (1 − (1 − λ ) 2 k ) UCL EWMA − γ = µ 0 (ˆ γ ) + K σ 0 (ˆ γ ) 2 − λ Approximations for µ 0 (ˆ γ ) and σ 0 (ˆ γ ) � � 0 − γ 2 � � 0 − 3 γ 4 − 7 γ 2 �� 1 + 1 � 0 − 1 � + 1 4 − 7 + 1 32 − 19 γ 2 3 γ 4 0 15 γ 6 0 0 µ 0 (ˆ γ ) γ 0 ≃ 4 n 2 32 n 3 4 128 n � 0 + 7 γ 4 + 3 γ 2 1 � 0 + 1 � + 1 � 0 + 3 � + 1 � + 3 � γ 2 8 γ 4 0 + γ 2 69 γ 6 0 0 σ 0 (ˆ γ ) γ 0 ≃ n 2 n 3 n 2 8 2 4 16 Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  15. EWMA- γ chart (Hong et al., JSKISE 2008) Monitored statistic Z k = (1 − λ ) Z k − 1 + λ ˆ γ k Control limits � λ (1 − (1 − λ ) 2 k ) LCL EWMA − γ = µ 0 (ˆ γ ) − K σ 0 (ˆ γ ) 2 − λ � λ (1 − (1 − λ ) 2 k ) UCL EWMA − γ = µ 0 (ˆ γ ) + K σ 0 (ˆ γ ) 2 − λ Approximations for µ 0 (ˆ γ ) and σ 0 (ˆ γ ) � � 0 − γ 2 � � 0 − 3 γ 4 − 7 γ 2 �� 1 + 1 � 0 − 1 � + 1 4 − 7 + 1 32 − 19 γ 2 3 γ 4 0 15 γ 6 0 0 µ 0 (ˆ γ ) γ 0 ≃ 4 n 2 32 n 3 4 128 n � 0 + 7 γ 4 + 3 γ 2 1 � 0 + 1 � + 1 � 0 + 3 � + 1 � + 3 � γ 2 8 γ 4 0 + γ 2 69 γ 6 0 0 σ 0 (ˆ γ ) γ 0 ≃ n 2 n 3 n 2 8 2 4 16 Comments Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  16. EWMA- γ chart (Hong et al., JSKISE 2008) Monitored statistic Z k = (1 − λ ) Z k − 1 + λ ˆ γ k Control limits � λ (1 − (1 − λ ) 2 k ) LCL EWMA − γ = µ 0 (ˆ γ ) − K σ 0 (ˆ γ ) 2 − λ � λ (1 − (1 − λ ) 2 k ) UCL EWMA − γ = µ 0 (ˆ γ ) + K σ 0 (ˆ γ ) 2 − λ Approximations for µ 0 (ˆ γ ) and σ 0 (ˆ γ ) � � 0 − γ 2 � � 0 − 3 γ 4 − 7 γ 2 �� 1 + 1 � 0 − 1 � + 1 4 − 7 + 1 32 − 19 γ 2 3 γ 4 0 15 γ 6 0 0 µ 0 (ˆ γ ) γ 0 ≃ 4 n 2 32 n 3 4 128 n � 0 + 7 γ 4 + 3 γ 2 1 � 0 + 1 � + 1 � 0 + 3 � + 1 � + 3 � γ 2 8 γ 4 0 + γ 2 69 γ 6 0 0 σ 0 (ˆ γ ) γ 0 ≃ n 2 n 3 n 2 8 2 4 16 Comments More efficient than the SH- γ chart. Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  17. EWMA- γ chart (Hong et al., JSKISE 2008) Monitored statistic Z k = (1 − λ ) Z k − 1 + λ ˆ γ k Control limits � λ (1 − (1 − λ ) 2 k ) LCL EWMA − γ = µ 0 (ˆ γ ) − K σ 0 (ˆ γ ) 2 − λ � λ (1 − (1 − λ ) 2 k ) UCL EWMA − γ = µ 0 (ˆ γ ) + K σ 0 (ˆ γ ) 2 − λ Approximations for µ 0 (ˆ γ ) and σ 0 (ˆ γ ) � � 0 − γ 2 � � 0 − 3 γ 4 − 7 γ 2 �� 1 + 1 � 0 − 1 � + 1 4 − 7 + 1 32 − 19 γ 2 3 γ 4 0 15 γ 6 0 0 µ 0 (ˆ γ ) γ 0 ≃ 4 n 2 32 n 3 4 128 n � 0 + 7 γ 4 + 3 γ 2 1 � 0 + 1 � + 1 � 0 + 3 � + 1 � + 3 � γ 2 8 γ 4 0 + γ 2 69 γ 6 0 0 σ 0 (ˆ γ ) γ 0 ≃ n 2 n 3 n 2 8 2 4 16 Comments More efficient than the SH- γ chart. The paper itself does not provide any thorough investigations (results obtained through simulation only). Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  18. New one-sided EWMA- γ 2 charts Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  19. New one-sided EWMA- γ 2 charts We suggest to ... Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  20. New one-sided EWMA- γ 2 charts We suggest to ... monitor γ 2 instead of γ 1 Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  21. New one-sided EWMA- γ 2 charts We suggest to ... monitor γ 2 instead of γ (more efficient to monitor S 2 than S ). 1 Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  22. New one-sided EWMA- γ 2 charts We suggest to ... monitor γ 2 instead of γ (more efficient to monitor S 2 than S ). 1 define 2 EWMA one-sided charts 2 Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  23. New one-sided EWMA- γ 2 charts We suggest to ... monitor γ 2 instead of γ (more efficient to monitor S 2 than S ). 1 define 2 EWMA one-sided charts (detect shifts more efficiently). 2 Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  24. New one-sided EWMA- γ 2 charts We suggest to ... monitor γ 2 instead of γ (more efficient to monitor S 2 than S ). 1 define 2 EWMA one-sided charts (detect shifts more efficiently). 2 EWMA- γ 2 chart Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  25. New one-sided EWMA- γ 2 charts We suggest to ... monitor γ 2 instead of γ (more efficient to monitor S 2 than S ). 1 define 2 EWMA one-sided charts (detect shifts more efficiently). 2 EWMA- γ 2 chart Upward EWMA- γ 2 chart Z + γ 2 ) , (1 − λ + ) Z + k − 1 + λ + ˆ γ 2 = max( µ 0 (ˆ k ) k Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  26. New one-sided EWMA- γ 2 charts We suggest to ... monitor γ 2 instead of γ (more efficient to monitor S 2 than S ). 1 define 2 EWMA one-sided charts (detect shifts more efficiently). 2 EWMA- γ 2 chart Upward EWMA- γ 2 chart Z + γ 2 ) , (1 − λ + ) Z + k − 1 + λ + ˆ γ 2 k ) , Z + γ 2 ) = max( µ 0 (ˆ 0 = µ 0 (ˆ k Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  27. New one-sided EWMA- γ 2 charts We suggest to ... monitor γ 2 instead of γ (more efficient to monitor S 2 than S ). 1 define 2 EWMA one-sided charts (detect shifts more efficiently). 2 EWMA- γ 2 chart Upward EWMA- γ 2 chart Z + γ 2 ) , (1 − λ + ) Z + k − 1 + λ + ˆ γ 2 k ) , Z + γ 2 ) = max( µ 0 (ˆ 0 = µ 0 (ˆ k � λ + γ 2 ) + K + γ 2 ) UCL EWMA − γ 2 = µ 0 (ˆ 2 − λ + σ 0 (ˆ Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  28. New one-sided EWMA- γ 2 charts We suggest to ... monitor γ 2 instead of γ (more efficient to monitor S 2 than S ). 1 define 2 EWMA one-sided charts (detect shifts more efficiently). 2 EWMA- γ 2 chart Upward EWMA- γ 2 chart Z + γ 2 ) , (1 − λ + ) Z + k − 1 + λ + ˆ γ 2 k ) , Z + γ 2 ) = max( µ 0 (ˆ 0 = µ 0 (ˆ k � λ + γ 2 ) + K + γ 2 ) UCL EWMA − γ 2 = µ 0 (ˆ 2 − λ + σ 0 (ˆ Downward EWMA- γ 2 chart γ 2 ) , (1 − λ − ) Z − γ 2 Z − = min( µ 0 (ˆ k − 1 + λ − ˆ k ) k Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  29. New one-sided EWMA- γ 2 charts We suggest to ... monitor γ 2 instead of γ (more efficient to monitor S 2 than S ). 1 define 2 EWMA one-sided charts (detect shifts more efficiently). 2 EWMA- γ 2 chart Upward EWMA- γ 2 chart Z + γ 2 ) , (1 − λ + ) Z + k − 1 + λ + ˆ γ 2 k ) , Z + γ 2 ) = max( µ 0 (ˆ 0 = µ 0 (ˆ k � λ + γ 2 ) + K + γ 2 ) UCL EWMA − γ 2 = µ 0 (ˆ 2 − λ + σ 0 (ˆ Downward EWMA- γ 2 chart γ 2 ) , (1 − λ − ) Z − γ 2 γ 2 ) Z − = min( µ 0 (ˆ k − 1 + λ − ˆ k ) , Z − = µ 0 (ˆ 0 k Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  30. New one-sided EWMA- γ 2 charts We suggest to ... monitor γ 2 instead of γ (more efficient to monitor S 2 than S ). 1 define 2 EWMA one-sided charts (detect shifts more efficiently). 2 EWMA- γ 2 chart Upward EWMA- γ 2 chart Z + γ 2 ) , (1 − λ + ) Z + k − 1 + λ + ˆ γ 2 k ) , Z + γ 2 ) = max( µ 0 (ˆ 0 = µ 0 (ˆ k � λ + γ 2 ) + K + γ 2 ) UCL EWMA − γ 2 = µ 0 (ˆ 2 − λ + σ 0 (ˆ Downward EWMA- γ 2 chart γ 2 ) , (1 − λ − ) Z − γ 2 γ 2 ) Z − = min( µ 0 (ˆ k − 1 + λ − ˆ k ) , Z − = µ 0 (ˆ 0 k � λ − γ 2 ) − K − γ 2 ) = µ 0 (ˆ 2 − λ − σ 0 (ˆ LCL EWMA − γ 2 Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  31. New one-sided EWMA- γ 2 charts We suggest to ... monitor γ 2 instead of γ (more efficient to monitor S 2 than S ). 1 define 2 EWMA one-sided charts (detect shifts more efficiently). 2 EWMA- γ 2 chart Upward EWMA- γ 2 chart Z + γ 2 ) , (1 − λ + ) Z + k − 1 + λ + ˆ γ 2 k ) , Z + γ 2 ) = max( µ 0 (ˆ 0 = µ 0 (ˆ k � λ + γ 2 ) + K + γ 2 ) UCL EWMA − γ 2 = µ 0 (ˆ 2 − λ + σ 0 (ˆ Downward EWMA- γ 2 chart γ 2 ) , (1 − λ − ) Z − γ 2 γ 2 ) Z − = min( µ 0 (ˆ k − 1 + λ − ˆ k ) , Z − = µ 0 (ˆ 0 k � λ − γ 2 ) − K − γ 2 ) = µ 0 (ˆ 2 − λ − σ 0 (ˆ LCL EWMA − γ 2 γ 2 ) and σ 0 (ˆ γ 2 ) Approximations for µ 0 (ˆ Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  32. New one-sided EWMA- γ 2 charts We suggest to ... monitor γ 2 instead of γ (more efficient to monitor S 2 than S ). 1 define 2 EWMA one-sided charts (detect shifts more efficiently). 2 EWMA- γ 2 chart Upward EWMA- γ 2 chart Z + γ 2 ) , (1 − λ + ) Z + k − 1 + λ + ˆ γ 2 k ) , Z + γ 2 ) = max( µ 0 (ˆ 0 = µ 0 (ˆ k � λ + γ 2 ) + K + γ 2 ) UCL EWMA − γ 2 = µ 0 (ˆ 2 − λ + σ 0 (ˆ Downward EWMA- γ 2 chart γ 2 ) , (1 − λ − ) Z − γ 2 γ 2 ) Z − = min( µ 0 (ˆ k − 1 + λ − ˆ k ) , Z − = µ 0 (ˆ 0 k � λ − γ 2 ) − K − γ 2 ) = µ 0 (ˆ 2 − λ − σ 0 (ˆ LCL EWMA − γ 2 γ 2 ) and σ 0 (ˆ γ 2 ) Approximations for µ 0 (ˆ � 3 γ 2 � γ 2 ) ≃ γ 2 0 µ 0 (ˆ 1 − 0 n Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  33. New one-sided EWMA- γ 2 charts We suggest to ... monitor γ 2 instead of γ (more efficient to monitor S 2 than S ). 1 define 2 EWMA one-sided charts (detect shifts more efficiently). 2 EWMA- γ 2 chart Upward EWMA- γ 2 chart Z + γ 2 ) , (1 − λ + ) Z + k − 1 + λ + ˆ γ 2 k ) , Z + γ 2 ) = max( µ 0 (ˆ 0 = µ 0 (ˆ k � λ + γ 2 ) + K + γ 2 ) UCL EWMA − γ 2 = µ 0 (ˆ 2 − λ + σ 0 (ˆ Downward EWMA- γ 2 chart γ 2 ) , (1 − λ − ) Z − γ 2 γ 2 ) Z − = min( µ 0 (ˆ k − 1 + λ − ˆ k ) , Z − = µ 0 (ˆ 0 k � λ − γ 2 ) − K − γ 2 ) = µ 0 (ˆ 2 − λ − σ 0 (ˆ LCL EWMA − γ 2 γ 2 ) and σ 0 (ˆ γ 2 ) Approximations for µ 0 (ˆ � � 3 γ 2 � � � 75 γ 2 �� γ 2 ) ≃ γ 2 γ 2 ) ≃ 0 γ 4 n − 1 + γ 2 2 4 20 0 γ 2 ) − γ 2 0 ) 2 µ 0 (ˆ 1 − , σ 0 (ˆ n + n ( n − 1) + − ( µ 0 (ˆ 0 n 2 n 0 0 Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  34. ARL “local” optimization Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  35. ARL “local” optimization Shift τ Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  36. ARL “local” optimization Shift τ γ 0 = in-control/nominal coefficient of variation. Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  37. ARL “local” optimization Shift τ γ 0 = in-control/nominal coefficient of variation. γ 1 = out-of-control coefficient of variation. Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  38. ARL “local” optimization Shift τ γ 0 = in-control/nominal coefficient of variation. γ 1 = out-of-control coefficient of variation. τ = γ 1 γ 0 denotes the shift size. Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  39. ARL “local” optimization Shift τ γ 0 = in-control/nominal coefficient of variation. γ 1 = out-of-control coefficient of variation. τ = γ 1 γ 0 denotes the shift size. τ ∈ (0 , 1) → decrease of the nominal coefficient of variation. Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  40. ARL “local” optimization Shift τ γ 0 = in-control/nominal coefficient of variation. γ 1 = out-of-control coefficient of variation. τ = γ 1 γ 0 denotes the shift size. τ ∈ (0 , 1) → decrease of the nominal coefficient of variation. τ > 1 → increase of the nominal coefficient of variation. Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  41. ARL “local” optimization Shift τ γ 0 = in-control/nominal coefficient of variation. γ 1 = out-of-control coefficient of variation. τ = γ 1 γ 0 denotes the shift size. τ ∈ (0 , 1) → decrease of the nominal coefficient of variation. τ > 1 → increase of the nominal coefficient of variation. Optimization Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  42. ARL “local” optimization Shift τ γ 0 = in-control/nominal coefficient of variation. γ 1 = out-of-control coefficient of variation. τ = γ 1 γ 0 denotes the shift size. τ ∈ (0 , 1) → decrease of the nominal coefficient of variation. τ > 1 → increase of the nominal coefficient of variation. Optimization ARL = average number of samples before a control chart signals an “ out-of-control ” condition or issues a false alarm. Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  43. ARL “local” optimization Shift τ γ 0 = in-control/nominal coefficient of variation. γ 1 = out-of-control coefficient of variation. τ = γ 1 γ 0 denotes the shift size. τ ∈ (0 , 1) → decrease of the nominal coefficient of variation. τ > 1 → increase of the nominal coefficient of variation. Optimization ARL = average number of samples before a control chart signals an “ out-of-control ” condition or issues a false alarm. Find out the optimal couples ( λ ∗ , K ∗ ) such that : ( λ ∗ , K ∗ ) = argmin ARL ( γ 0 , τγ 0 , λ, K , n ) , ( λ, K ) Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  44. ARL “local” optimization Shift τ γ 0 = in-control/nominal coefficient of variation. γ 1 = out-of-control coefficient of variation. τ = γ 1 γ 0 denotes the shift size. τ ∈ (0 , 1) → decrease of the nominal coefficient of variation. τ > 1 → increase of the nominal coefficient of variation. Optimization ARL = average number of samples before a control chart signals an “ out-of-control ” condition or issues a false alarm. Find out the optimal couples ( λ ∗ , K ∗ ) such that : ( λ ∗ , K ∗ ) = argmin ARL ( γ 0 , τγ 0 , λ, K , n ) , ( λ, K ) subject to the constraint : ARL ( γ 0 , γ 0 , λ ∗ , K ∗ , n ) = ARL 0 = 370 . 4 . Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  45. ARL “local” optimization Shift τ γ 0 = in-control/nominal coefficient of variation. γ 1 = out-of-control coefficient of variation. τ = γ 1 γ 0 denotes the shift size. τ ∈ (0 , 1) → decrease of the nominal coefficient of variation. τ > 1 → increase of the nominal coefficient of variation. Optimization ARL = average number of samples before a control chart signals an “ out-of-control ” condition or issues a false alarm. Find out the optimal couples ( λ ∗ , K ∗ ) such that : ( λ ∗ , K ∗ ) = argmin ARL ( γ 0 , τγ 0 , λ, K , n ) , ( λ, K ) subject to the constraint : ARL ( γ 0 , γ 0 , λ ∗ , K ∗ , n ) = ARL 0 = 370 . 4 . ARL is evaluated using a Brook & Evans’s type Markov chain approach. Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  46. ARL “local” optimization (Markov chain) Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  47. ARL “local” optimization (Markov chain) γ 2 ) and UCL into p Divide the interval between LCL = µ 0 (ˆ γ 2 )) / (2 p ). subintervals of width 2 δ , where δ = ( UCL − µ 0 (ˆ UCL H p H i +1 2 δ H i H i − 1 H 1 µ 0 (ˆ γ 2 ) Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  48. ARL “local” optimization (Markov chain) γ 2 ) and UCL into p Divide the interval between LCL = µ 0 (ˆ γ 2 )) / (2 p ). subintervals of width 2 δ , where δ = ( UCL − µ 0 (ˆ UCL H p H i +1 2 δ H i H i − 1 H 1 µ 0 (ˆ γ 2 ) H j , j = 1 , . . . , p , represents the midpoint of the j th subinterval. Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  49. ARL “local” optimization (Markov chain) γ 2 ) and UCL into p Divide the interval between LCL = µ 0 (ˆ γ 2 )) / (2 p ). subintervals of width 2 δ , where δ = ( UCL − µ 0 (ˆ UCL H p H i +1 2 δ H i H i − 1 H 1 µ 0 (ˆ γ 2 ) H j , j = 1 , . . . , p , represents the midpoint of the j th subinterval. γ 2 ) corresponds to the “restart state” feature. H 0 = µ 0 (ˆ Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  50. ARL “local” optimization (Markov chain) Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  51. ARL “local” optimization (Markov chain) The transition probability matrix Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  52. ARL “local” optimization (Markov chain) The transition probability matrix  · · ·  Q 0 , 0 Q 0 , 1 Q 0 , p r 0 Q 1 , 0 Q 1 , 1 · · · Q 1 , p r 1   Q r    . . . .   = . . . . P =   . . . .    0 T 1   · · · Q p , 0 Q p , 1 Q p , p r p   0 0 · · · 0 1 Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  53. ARL “local” optimization (Markov chain) The transition probability matrix  · · ·  Q 0 , 0 Q 0 , 1 Q 0 , p r 0 Q 1 , 0 Q 1 , 1 · · · Q 1 , p r 1   Q r    . . . .   = . . . . P =   . . . .    0 T 1   · · · Q p , 0 Q p , 1 Q p , p r p   0 0 · · · 0 1 Transient probabilities Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  54. ARL “local” optimization (Markov chain) The transition probability matrix  · · ·  Q 0 , 0 Q 0 , 1 Q 0 , p r 0 Q 1 , 0 Q 1 , 1 · · · Q 1 , p r 1   Q r    . . . .   = . . . . P =   . . . .    0 T 1   · · · Q p , 0 Q p , 1 Q p , p r p   0 0 · · · 0 1 Transient probabilities � µ 0 (ˆ γ 2 ) − (1 − λ + ) H i � � Q + � = F ˆ � n , γ 1 γ 2 i , 0 � λ + Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  55. ARL “local” optimization (Markov chain) The transition probability matrix  · · ·  Q 0 , 0 Q 0 , 1 Q 0 , p r 0 Q 1 , 0 Q 1 , 1 · · · Q 1 , p r 1   Q r    . . . .   = . . . . P =   . . . .    0 T 1   · · · Q p , 0 Q p , 1 Q p , p r p   0 0 · · · 0 1 Transient probabilities � µ 0 (ˆ γ 2 ) − (1 − λ + ) H i � � Q + � = F ˆ � n , γ 1 γ 2 i , 0 � λ + � µ 0 (ˆ γ 2 ) − (1 − λ − ) H i � � � = 1 − F ˆ Q − � n , γ 1 γ 2 � i , 0 λ − Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  56. ARL “local” optimization (Markov chain) The transition probability matrix  · · ·  Q 0 , 0 Q 0 , 1 Q 0 , p r 0 Q 1 , 0 Q 1 , 1 · · · Q 1 , p r 1   Q r    . . . .   = . . . . P =   . . . .    0 T 1   · · · Q p , 0 Q p , 1 Q p , p r p   0 0 · · · 0 1 Transient probabilities � µ 0 (ˆ γ 2 ) − (1 − λ + ) H i � � Q + � = F ˆ � n , γ 1 γ 2 i , 0 � λ + � µ 0 (ˆ γ 2 ) − (1 − λ − ) H i � � � = 1 − F ˆ Q − � n , γ 1 γ 2 � i , 0 λ − � H j + δ − (1 − λ ) H i � H j − δ − (1 − λ ) H i � � � � � � Q i , j = F ˆ � n , γ 1 − F ˆ � n , γ 1 γ 2 γ 2 � � λ λ Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  57. ARL “local” optimization (Markov chain) The transition probability matrix  · · ·  Q 0 , 0 Q 0 , 1 Q 0 , p r 0 Q 1 , 0 Q 1 , 1 · · · Q 1 , p r 1   Q r    . . . .   = . . . . P =   . . . .    0 T 1   · · · Q p , 0 Q p , 1 Q p , p r p   0 0 · · · 0 1 Transient probabilities � µ 0 (ˆ γ 2 ) − (1 − λ + ) H i � � Q + � = F ˆ � n , γ 1 γ 2 i , 0 � λ + � µ 0 (ˆ γ 2 ) − (1 − λ − ) H i � � � = 1 − F ˆ Q − � n , γ 1 γ 2 � i , 0 λ − � H j + δ − (1 − λ ) H i � H j − δ − (1 − λ ) H i � � � � � � Q i , j = F ˆ � n , γ 1 − F ˆ � n , γ 1 γ 2 γ 2 � � λ λ Vector of initial probabilities q = (1 , 0 , . . . , 0) T . Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  58. ARL “local” optimization (Markov chain) Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  59. ARL “local” optimization (Markov chain) Definition The number of steps L until the process reaches the absorbing state is a Discrete PHase-type (or DPH) random variable of parameters ( Q , q ). Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  60. ARL “local” optimization (Markov chain) Definition The number of steps L until the process reaches the absorbing state is a Discrete PHase-type (or DPH) random variable of parameters ( Q , q ). ARL , SRDL Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  61. ARL “local” optimization (Markov chain) Definition The number of steps L until the process reaches the absorbing state is a Discrete PHase-type (or DPH) random variable of parameters ( Q , q ). ARL , SRDL q T ( I − Q ) − 1 1 ν 1 ( L ) = Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  62. ARL “local” optimization (Markov chain) Definition The number of steps L until the process reaches the absorbing state is a Discrete PHase-type (or DPH) random variable of parameters ( Q , q ). ARL , SRDL q T ( I − Q ) − 1 1 ν 1 ( L ) = 2 q T ( I − Q ) − 2 Q1 ν 2 ( L ) = Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  63. ARL “local” optimization (Markov chain) Definition The number of steps L until the process reaches the absorbing state is a Discrete PHase-type (or DPH) random variable of parameters ( Q , q ). ARL , SRDL q T ( I − Q ) − 1 1 ν 1 ( L ) = 2 q T ( I − Q ) − 2 Q1 ν 2 ( L ) = and ARL = ν 1 ( L ) Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  64. ARL “local” optimization (Markov chain) Definition The number of steps L until the process reaches the absorbing state is a Discrete PHase-type (or DPH) random variable of parameters ( Q , q ). ARL , SRDL q T ( I − Q ) − 1 1 ν 1 ( L ) = 2 q T ( I − Q ) − 2 Q1 ν 2 ( L ) = and ARL = ν 1 ( L ) � ν 2 ( L ) − ν 2 SDRL = 1 ( L ) + ν 1 ( L ) Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  65. Optimal ( λ ∗ , K ∗ ) and ARL for EWMA- γ 2 and SH- γ charts n = 7, ARL 0 = 370 . 4 τ γ 0 = 0 . 05 γ 0 = 0 . 1 γ 0 = 0 . 15 γ 0 = 0 . 2 0 . 50 (0 . 5671 , 1 . 8734) (0 . 5637 , 1 . 8480) (0 . 5608 , 1 . 8043) (0 . 5539 , 1 . 7474) (3 . 4 , 18 . 4) (3 . 4 , 18 . 6) (3 . 5 , 18 . 9) (3 . 5 , 19 . 3) 0 . 65 (0 . 2951 , 2 . 1229) (0 . 2902 , 2 . 0932) (0 . 2854 , 2 . 0416) (0 . 2792 , 1 . 9709) (6 . 4 , 69 . 3) (6 . 4 , 69 . 9) (6 . 4 , 70 . 8) (6 . 5 , 72 . 1) 0 . 80 (0 . 1104 , 2 . 2582) (0 . 1088 , 2 . 2142) (0 . 1032 , 2 . 1413) (0 . 0976 , 2 . 0414) (15 . 3 , 212 . 1) (15 . 4 , 213 . 2) (15 . 5 , 215 . 0) (15 . 6 , 217 . 5) 1 . 25 (0 . 1092 , 3 . 0381) (0 . 1101 , 3 . 0831) (0 . 1097 , 3 . 1504) (0 . 1087 , 3 . 2443) (11 . 3 , 32 . 4) (11 . 4 , 32 . 9) (11 . 7 , 33 . 8) (12 . 0 , 35 . 1) 1 . 50 (0 . 2646 , 3 . 5219) (0 . 2603 , 3 . 5538) (0 . 2531 , 3 . 6078) (0 . 2443 , 3 . 6873) (4 . 3 , 7 . 2) (4 . 3 , 7 . 4) (4 . 4 , 7 . 6) (4 . 6 , 8 . 0) 2 . 00 (0 . 5852 , 3 . 9768) (0 . 5725 , 4 . 0146) (0 . 5520 , 4 . 0781) (0 . 5212 , 4 . 1644) (1 . 8 , 2 . 1) (1 . 8 , 2 . 1) (1 . 9 , 2 . 2) (2 . 0 , 2 . 3) Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  66. ( λ ∗ , K ∗ ) nomograms γ 0 = 0 . 05 γ 0 = 0 . 05 1 4.5 0.9 4 0.8 K −∗ K + ∗ λ + ∗ λ −∗ 0.7 3.5 0.6 0.5 3 K λ 0.4 2.5 0.3 n=5 n=5 0.2 2 n=7 n=7 0.1 n=10 n=10 n=15 n=15 0 1.5 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.6 0.8 1 1.2 1.4 1.6 1.8 2 τ τ γ 0 = 0 . 1 γ 0 = 0 . 1 1 4.5 0.9 4 K + ∗ 0.8 K −∗ λ −∗ λ + ∗ 0.7 3.5 0.6 0.5 3 K λ 0.4 2.5 0.3 n=5 n=5 0.2 2 n=7 n=7 0.1 n=10 n=10 n=15 n=15 0 1.5 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.6 0.8 1 1.2 1.4 1.6 1.8 2 τ τ Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  67. ( λ ∗ , K ∗ ) nomograms γ 0 = 0 . 15 γ 0 = 0 . 15 1 4.5 0.9 4 0.8 K −∗ K + ∗ λ + ∗ λ −∗ 0.7 3.5 0.6 0.5 3 K λ 0.4 2.5 0.3 n=5 n=5 0.2 2 n=7 n=7 0.1 n=10 n=10 n=15 n=15 0 1.5 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.6 0.8 1 1.2 1.4 1.6 1.8 2 τ τ γ 0 = 0 . 2 γ 0 = 0 . 2 1 4.5 0.9 4 K + ∗ 0.8 K −∗ λ −∗ λ + ∗ 0.7 3.5 0.6 0.5 3 K λ 0.4 2.5 0.3 n=5 n=5 0.2 2 n=7 n=7 0.1 n=10 n=10 n=15 n=15 0 1.5 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.6 0.8 1 1.2 1.4 1.6 1.8 2 τ τ Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  68. EWMA- γ 2 chart v.s. EWMA- γ (Hong et al., 2008) chart n = 5 τ γ 0 = 0 . 05 γ 0 = 0 . 1 γ 0 = 0 . 15 γ 0 = 0 . 2 0 . 50 (4 . 8 , 4 . 7) (4 . 8 , 4 . 7) (4 . 8 , 4 . 8) (4 . 8 , 4 . 8) 0 . 65 (8 . 7 , 8 . 8) (8 . 8 , 8 . 9) (8 . 8 , 8 . 9) (8 . 8 , 9 . 0) 0 . 80 (20 . 6 , 21 . 1) (20 . 6 , 21 . 2) (20 . 7 , 21 . 3) (20 . 9 , 21 . 5) 0 . 90 (53 . 2 , 56 . 2) (53 . 7 , 56 . 4) (54 . 5 , 56 . 8) (55 . 8 , 57 . 3) 1 . 10 (51 . 0 , 51 . 5) (51 . 2 , 51 . 8) (51 . 7 , 52 . 3) (52 . 4 , 52 . 9) 1 . 25 (15 . 0 , 15 . 5) (15 . 2 , 15 . 6) (15 . 4 , 15 . 8) (15 . 9 , 16 . 0) 1 . 50 (5 . 7 , 5 . 9) (5 . 8 , 5 . 9) (5 . 9 , 6 . 0) (6 . 1 , 6 . 2) 2 . 00 (2 . 4 , 2 . 4) (2 . 4 , 2 . 4) (2 . 5 , 2 . 5) (2 . 6 , 2 . 6) n = 7 τ γ 0 = 0 . 05 γ 0 = 0 . 1 γ 0 = 0 . 15 γ 0 = 0 . 2 0 . 50 (3 . 4 , 3 . 4) (3 . 4 , 3 . 4) (3 . 5 , 3 . 4) (3 . 5 , 3 . 5) 0 . 65 (6 . 4 , 6 . 4) (6 . 4 , 6 . 4) (6 . 4 , 6 . 5) (6 . 5 , 6 . 5) 0 . 80 (15 . 3 , 15 . 6) (15 . 4 , 15 . 6) (15 . 5 , 15 . 8) (15 . 6 , 16 . 0) 0 . 90 (40 . 4 , 41 . 8) (40 . 7 , 42 . 0) (41 . 2 , 42 . 4) (42 . 0 , 42 . 9) 1 . 10 (39 . 2 , 39 . 7) (39 . 5 , 40 . 0) (40 . 1 , 40 . 4) (40 . 9 , 41 . 1) 1 . 25 (11 . 3 , 11 . 5) (11 . 4 , 11 . 6) (11 . 7 , 11 . 8) (12 . 0 , 12 . 1) 1 . 50 (4 . 3 , 4 . 3) (4 . 3 , 4 . 4) (4 . 4 , 4 . 5) (4 . 6 , 4 . 6) 2 . 00 (1 . 8 , 1 . 8) (1 . 8 , 1 . 8) (1 . 9 , 1 . 9) (2 . 0 , 2 . 0) Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  69. ARL “global” optimization Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  70. ARL “global” optimization Drawback of “local” optimization Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  71. ARL “global” optimization Drawback of “local” optimization Usually the quality practitioner does not know in advance the entity of the next shift size because of the lack of related historical data. Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  72. ARL “global” optimization Drawback of “local” optimization Usually the quality practitioner does not know in advance the entity of the next shift size because of the lack of related historical data. The shift size is not deterministic and varies accordingly to some unknown stochastic model. Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  73. ARL “global” optimization Drawback of “local” optimization Usually the quality practitioner does not know in advance the entity of the next shift size because of the lack of related historical data. The shift size is not deterministic and varies accordingly to some unknown stochastic model. New objective function and constraint Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  74. ARL “global” optimization Drawback of “local” optimization Usually the quality practitioner does not know in advance the entity of the next shift size because of the lack of related historical data. The shift size is not deterministic and varies accordingly to some unknown stochastic model. New objective function and constraint Find out the optimal couples ( λ ∗ , K ∗ ) such that : ( λ ∗ , K ∗ ) = argmin EARL ( γ 0 , τγ 0 , λ, K , n ) ( λ, K ) Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  75. ARL “global” optimization Drawback of “local” optimization Usually the quality practitioner does not know in advance the entity of the next shift size because of the lack of related historical data. The shift size is not deterministic and varies accordingly to some unknown stochastic model. New objective function and constraint Find out the optimal couples ( λ ∗ , K ∗ ) such that : ( λ ∗ , K ∗ ) = argmin EARL ( γ 0 , τγ 0 , λ, K , n ) ( λ, K ) with � EARL = f τ ( τ ) ARL ( γ 0 , τγ 0 , λ, K , n ) d τ. Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  76. ARL “global” optimization Drawback of “local” optimization Usually the quality practitioner does not know in advance the entity of the next shift size because of the lack of related historical data. The shift size is not deterministic and varies accordingly to some unknown stochastic model. New objective function and constraint Find out the optimal couples ( λ ∗ , K ∗ ) such that : ( λ ∗ , K ∗ ) = argmin EARL ( γ 0 , τγ 0 , λ, K , n ) ( λ, K ) with � EARL = f τ ( τ ) ARL ( γ 0 , τγ 0 , λ, K , n ) d τ. subject to the constraint EARL ( γ 0 , γ 0 , λ, K , n ) = ARL ( γ 0 , γ 0 , λ, K , n ) = ARL 0 , Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  77. ARL “global” optimization Drawback of “local” optimization Usually the quality practitioner does not know in advance the entity of the next shift size because of the lack of related historical data. The shift size is not deterministic and varies accordingly to some unknown stochastic model. New objective function and constraint Find out the optimal couples ( λ ∗ , K ∗ ) such that : ( λ ∗ , K ∗ ) = argmin EARL ( γ 0 , τγ 0 , λ, K , n ) ( λ, K ) with � EARL = f τ ( τ ) ARL ( γ 0 , τγ 0 , λ, K , n ) d τ. subject to the constraint EARL ( γ 0 , γ 0 , λ, K , n ) = ARL ( γ 0 , γ 0 , λ, K , n ) = ARL 0 , f τ ( τ ) is the p.d.f. of the shift τ Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

  78. ARL “global” optimization Drawback of “local” optimization Usually the quality practitioner does not know in advance the entity of the next shift size because of the lack of related historical data. The shift size is not deterministic and varies accordingly to some unknown stochastic model. New objective function and constraint Find out the optimal couples ( λ ∗ , K ∗ ) such that : ( λ ∗ , K ∗ ) = argmin EARL ( γ 0 , τγ 0 , λ, K , n ) ( λ, K ) with � EARL = f τ ( τ ) ARL ( γ 0 , τγ 0 , λ, K , n ) d τ. subject to the constraint EARL ( γ 0 , γ 0 , λ, K , n ) = ARL ( γ 0 , γ 0 , λ, K , n ) = ARL 0 , f τ ( τ ) is the p.d.f. of the shift τ → uniform distribution over [0 . 5 , 1) (decreasing case) Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Multivariate Control Charts Stat 3570 28 Feb, 2013 1 / 13 Multivariate Control Charts In

Multivariate Control Charts Stat 3570 28 Feb, 2013 1 / 13 Multivariate Control Charts In

Multivariate Control Charts Stat 3570 28 Feb, 2013 1 / 13 Multivariate Control Charts In all control charts procedures we discussed so far aims at monitoring one quality characteristic at a time. But in many situations, we are

459 views • 13 slides
Two Binomial Coefficient Analogues Bruce Sagan Department of Mathematics Michigan State

Two Binomial Coefficient Analogues Bruce Sagan Department of Mathematics Michigan State

Two Binomial Coefficient Analogues Bruce Sagan Department of Mathematics Michigan State University East Lansing, MI 48824-1027 sagan@math.msu.edu www.math.msu.edu/sagan March 4, 2013 The theme Variation 1: binomial coefficients Variation

1.7k views • 140 slides
Business Statistics CONTENTS The correlation coefficient The rank correlation coefficient

Business Statistics CONTENTS The correlation coefficient The rank correlation coefficient

: ESTIMATES AND TESTS Business Statistics CONTENTS The correlation coefficient The rank correlation coefficient Testing the correlation coefficient Non-linear relationships Old exam question Further study THE CORRELATION COEFFICIENT

404 views • 22 slides
1- Fu . t.IE In , f- Flow control of TCP Fairness { over wireless . T cp - estimation ( via

1- Fu . t.IE In , f- Flow control of TCP Fairness { over wireless . T cp - estimation ( via

: finish Up TCP Oct 13 Topics : f- Timeout ( RTO ) Estimation variation RTT , smoothing EWMA . packets to Bytes f- TCP : . I AIM D f- Rough throughput = oasis , 1- Fu . t.IE In , f- Flow control of TCP Fairness { over

466 views • 7 slides
Coefficient of Determination The coefficient of determination, R 2 , is defined as before: y i ) 2

Coefficient of Determination The coefficient of determination, R 2 , is defined as before: y i ) 2

ST 430/514 Introduction to Regression Analysis/Statistics for Management and the Social Sciences II Coefficient of Determination The coefficient of determination, R 2 , is defined as before: y i ) 2 R 2 = 1 SS E ( y i = 1 SS yy

333 views • 15 slides
The Arctic Sea Ice Monitoring and Climate Variation Hiroyuki Enomoto National Institute of Polar

The Arctic Sea Ice Monitoring and Climate Variation Hiroyuki Enomoto National Institute of Polar

Monitoring Climate Change from Space COP21 Japan Pavilion, Dec. 4, 2015 The Arctic Sea Ice Monitoring and Climate Variation Hiroyuki Enomoto National Institute of Polar Research Background Arctic warming rate is twice faster than other

440 views • 19 slides
CS1100: Computer Science and Its Applications Creating Graphs and Charts in Excel Charts

CS1100: Computer Science and Its Applications Creating Graphs and Charts in Excel Charts

CS1100: Computer Science and Its Applications Creating Graphs and Charts in Excel Charts Data is often better explained through visualization as either a graph or a chart. Excel makes creating charts easy: Column Charts Pie

651 views • 37 slides
CS1100: Computer Science and Its Applications Creating Graphs and Charts in Excel Charts

CS1100: Computer Science and Its Applications Creating Graphs and Charts in Excel Charts

CS1100: Computer Science and Its Applications Creating Graphs and Charts in Excel Charts Data is often better explained through visualization as either a graph or a chart. Excel makes creating charts easy: Column Charts Pie

855 views • 41 slides
Control Charts for x and R Subsequent use of the charts The next 20 samples are added to the

Control Charts for x and R Subsequent use of the charts The next 20 samples are added to the

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Control Charts for x and R Subsequent use of the charts The next 20 samples are added to the charts with the control limits derived from

670 views • 14 slides
Monitoring of the W ater Cycle and Clim ate Variation by the Earth Observation Satellites

Monitoring of the W ater Cycle and Clim ate Variation by the Earth Observation Satellites

Monitoring of the W ater Cycle and Clim ate Variation by the Earth Observation Satellites Misako KACHI Manager for GCOM-W Research/ GPM Applications Earth Observation Research Center (EORC) Japan Aerospace Exploration Agency (JAXA) Dec. 4,

291 views • 14 slides
The application of DNA markers to monitoring adaptation and migration of forest tree species

The application of DNA markers to monitoring adaptation and migration of forest tree species

The application of DNA markers to monitoring adaptation and migration of forest tree species Michele Morgante Rome, 22 February 2008 WHAT IS GENETIC VARIABILITY? GENETIC VARIATION = SEQUENCE VARIATION CHR-A cAAaTTATTA ATAATTTGGT

307 views • 20 slides
Snow spectral albedo variation as a tool for arctic environmental monitoring R. Salvatori

Snow spectral albedo variation as a tool for arctic environmental monitoring R. Salvatori

Final conference Rome October 11 2016 Snow spectral albedo variation as a tool for arctic environmental monitoring R. Salvatori CNR-IIA ARCA project ARCA Task : Continuous Vis-Near Characterization of snow-ice surface in Ny-lesund ( RiS

617 views • 29 slides
Control Charts for x and s Introduction The R chart, based on the sample range , is sensitive

Control Charts for x and s Introduction The R chart, based on the sample range , is sensitive

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Control Charts for x and s Introduction The R chart, based on the sample range , is sensitive in detecting an assignable cause of variation

549 views • 15 slides
Accessible Charts With AChart Reader Project Final G1 Lukas Bodner, Daniel Geiger, Lorenz

Accessible Charts With AChart Reader Project Final G1 Lukas Bodner, Daniel Geiger, Lorenz

Accessible Charts With AChart Reader Project Final G1 Lukas Bodner, Daniel Geiger, Lorenz Leitner 1 of 19 SVG Charts Introduction Accessible Charts ARIA Example Chart 2 of 19 Accessible Charts Scalable Vector Graphics (SVG) charts

461 views • 20 slides
1 Observation error Second finding: Randomness The true daily gains vary at random. = +

1 Observation error Second finding: Randomness The true daily gains vary at random. = +

Outline Filtering techniques applied to monitoring of daily gain in slaughter pigs: The Kalman filter Introduction - and other methods Basic monitoring Anders Ringgaard Kristensen Shewart control charts DLM and the Kalman

115 views • 7 slides
Presenting Data Chapter 2 1 Outline Types of Charts (next) Guidelines for Charts

Presenting Data Chapter 2 1 Outline Types of Charts (next) Guidelines for Charts

3/18/2019 IMGD 2905 Presenting Data Chapter 2 1 Outline Types of Charts (next) Guidelines for Charts Common Mistakes 2 1 3/18/2019 Right Chart Depends on Variable Type Qualitative (Categorical) variables Can have

404 views • 26 slides
Oak Creek Watershed Restoration Plan Summary Chapter 4 2 nd half May 2020 Authors: Laura

Oak Creek Watershed Restoration Plan Summary Chapter 4 2 nd half May 2020 Authors: Laura

Oak Creek Watershed Restoration Plan Summary Chapter 4 2 nd half May 2020 Authors: Laura Herrick, PE, CFM Chief Environmental Engineer Tom Slawski, Ph.D Chief Biologist Joe Boxhorn, Ph.D. Principal Planner Aaron Owens

799 views • 65 slides
COBRA.jl Accelerating Systems Biomedicine JuliaCon 2017 Laurent Heirendt, Ph.D. @laurentheirendt

COBRA.jl Accelerating Systems Biomedicine JuliaCon 2017 Laurent Heirendt, Ph.D. @laurentheirendt

COBRA.jl Accelerating Systems Biomedicine JuliaCon 2017 Laurent Heirendt, Ph.D. @laurentheirendt - June 23 rd , 2017 1 / 25 Outline 1. CO nstraint- b ased R econstruction and A nalysis (COBRA) 2. COBRA & Julia: large- and huge-scale

606 views • 25 slides
Matthieu Da Costa, William Digan, Ccile Jacry Introduction This sphere represents all the water

Matthieu Da Costa, William Digan, Ccile Jacry Introduction This sphere represents all the water

Matthieu Da Costa, William Digan, Ccile Jacry Introduction This sphere represents all the water on Earth Its diameter is about 860 miles Wide range of toxic compounds Introduction PCB Cadmium Lead Nitrite Phenol Introduction PCB

735 views • 26 slides
Spatial and temporal variability of bacterioplankton communities across a river to ocean gradient

Spatial and temporal variability of bacterioplankton communities across a river to ocean gradient

Spatial and temporal variability of bacterioplankton communities across a river to ocean gradient assessed with tag pyrosequencing Caroline Fortunato Bacterioplankton communities vary across a river to ocean gradient with estuaries as an

228 views • 6 slides
and preventing collapse 28 January 2020 Sam Hinton, ADBA ADBA and WBA ADBA has been ADBA

and preventing collapse 28 January 2020 Sam Hinton, ADBA ADBA and WBA ADBA has been ADBA

Feedstock variations, the digestion process and preventing collapse 28 January 2020 Sam Hinton, ADBA ADBA and WBA ADBA has been ADBA representing and 300+ Members 18 Countries supporting the UK AD market since 2009. WBA

533 views • 20 slides
1 Precision medicine: what does it mean? Cardiovascular disease: knowing who to treat

1 Precision medicine: what does it mean? Cardiovascular disease: knowing who to treat

Biomedical Imaging and Genetic (BIG) Data Analytics in Dementia and Oncology Deep Learning for Precision Medicine Wiro Niessen Biomedical Imaging Group Rotterdam Departments of Radiology & Medical Informatics Erasmus MC Imaging

826 views • 10 slides
Broward County, FL RFP No. T2112588P1 Coastal Engineering Consultant Services for Segment III

Broward County, FL RFP No. T2112588P1 Coastal Engineering Consultant Services for Segment III

Broward County, FL RFP No. T2112588P1 Coastal Engineering Consultant Services for Segment III Shore Protection olsen associates, inc. 2618 Herschel Street Jacksonville, FL 32204 904.387.6114 Final Evaluation Committee Meeting

1.01k views • 21 slides
How to Recognize and Avoid False Violations When Using Whole Effluent Toxicity (WET) Test

How to Recognize and Avoid False Violations When Using Whole Effluent Toxicity (WET) Test

tmoore@risk-sciences.com 8/21/2013 How to Recognize and Avoid False Violations When Using Whole Effluent Toxicity (WET) Test Methods to Evaluate Stormwater Samples Timothy F. Moore Risk Sciences Rockvale, TN No Toxics In Toxic

166 views • 16 slides

More recommend