Joint Monitoring of the Mean and Variance of Combined Control Charts with Estimated Parameters

Abstract

Joint , two-sided (CUSUM, S²), and (EWMA, S²) control charts are numerically compared when (i) process parameters are known and (ii) process parameters are estimated from retrospective data. In both cases, equations for the conditional and unconditional run length distributions are developed, and expressions for the average run lengths (ARL), the second moment of the run length (SMRL), and the standard deviation of the run lengths (SDRL) are derived for these charts. In-control and out-of-control ARLs and SDRLs are tabulated and compared for a variety of design parameters for each chart. Numerical results and practical recommendations are given.

Keywords:

• Average run length
• Control charts
• CUSUM
• EWMA
• Shewhart

Mathematics Subject Classification:

• Primary 62-XX
• Secondary 62-P30

Notes

Note: 3 sigma limits for the chart and 0.0027 probability limits for the S ² chart.

Note: EWMA parameters (λ = 0.05, h = 0.3963) and 0.0027 probability limits for the S ² chart.

Note: EWMA parameters (λ = 0.25, h = 1.0942) and 0.0027 probability limits for the S ² chart.

Note: EWMA parameters (λ = 0.5, h = 1.7188) and 0.0027 probability limits for the S ² chart.

Note: CUSUM parameters (k = 0.25, h = 7.93) and 0.0027 probability limits for the S ² chart.

Note: CUSUM parameters (k = 0.5, h = 4.759) and 0.0027 probability limits for the S ² chart.

Note: CUSUM parameters (k = 0.75, h = 3.334) and 0.0027 probability limits for the S ² chart.

Note: All S ² charts are constructed with 0.0027 probability limits.