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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

( 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

( 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

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

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

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

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

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

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

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

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

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

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

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