SENSITIVITY ANALYSIS

Hence, the average run length is the expectation ofW₄, that is ( )

(

) (

)

The calculation of ARL in (20)~(27) is complicated. Crowder (1987a, 1987b, 1989) and Lucas

and Saccucci (1990) propose approximated approach and Fortran program to calculate ARL

for a single EWMA control chart. We modify the Fortran program proposed by Crowder (1987b) to obtain the numerical approximated values for α_e1^,βe₁^,αe₂^andβe₂. Consequently,

various ARL of the proposed control charts can be obtained.

6. SENSITIVITY ANALYSIS

The sensitivity analysis illustrates the effects of variation of measurement errors on

the performance of the proposed control charts.

Let ψ₁ and ψ₂ be the ratios of measurement error variation contaminated in the total

variation of the first process and the second process, respectively. That is

2

measurement errors are always less than or equal to all variation from the processes. The values of average run length under various combinations of ψ₁, ψ₂, δ₍₁₀₎ and δ₍₀₁₎ are

shown in Table 1, which is organized into 36 separate panels, corresponding to δ₍₁₀₎ = 0.0,

0.5, 1.0, 1.5, 2.0, 2.5 and 3.0; δ₍₀₁₎= 0.0, 0.5, 1.0, 1.5, 2.0, 2.5 and 3.0; ψ₁= 0.0, 0.1, 0.2, 0.3,

0.4 and 0.5; ψ₂= 0.0, 0.1, 0.2, 0.3, 0.4 and 0.5.

From Table 1, we find that when the first process is out of control but the second process is in control, δ₍₁₀₎≠0 andδ₍₀₁₎ =0, ψ₁ increases, leading to increase in ARL; while ψ₂

increases, ARL is unchanged. When the second process is out of control but the first process is in control, δ₍₁₀₎ =0 andδ₍₀₁₎ ≠0, we found that ψ₁ increases but the ARL is unchanged;

while ψ₂ increases, leading to increase in ARL. When the first and the second processes are

both out of control, δ₍₁₀₎ ≠0andδ₍₀₁₎ ≠0, ψ₁ increases, leading to increase in ARL; while

ψ2 increases, leading to increase in ARL.

Therefore, the imprecision measurement devices seriously affected the performance of the proposed control charts. When ψ₁ and/or ψ₂ increase(s), the ability to detect the

processes shift is poor. The results of sensitivity analysis are summarized in Table 2.

7. CONCLUSIONS

The EWMA control chart and the cause-selecting control chart are proposed to effectively monitor and distinguish the states of two dependent production processes with imprecise measurement devices. The EWMA control chart based on the observed in-coming quality is used to monitor the small mean shifts for the first process, whereas the cause-selecting control chart based on residuals is used to monitor the small mean shifts for the second process.

The performance of the proposed EWMA and cause-selecting control charts under various variations of measurement errors is measured by average run length. It is shown that the presence of measurement error may seriously affect the ability of the proposed control

charts detect processes disturbance quickly. The larger variations of measurement errors lead to small rates of true alarm when at least one of the processes is out of control. If the measurement errors are inherent in the measurement devices and cannot be avoided, then the better way to reduce the effect of measurement errors is to give proper calibration or regular maintenance of measurement devices. The adaptive statistical process control approach may give an improvement for increasing detection ability of the proposed control charts under various variations of measurement errors.

Table 2: Summary of sensitivity analysis

Shift parameter

Ratio of measurement error

variation

ARL

10 0

( )

δ ≠ δ₍₀₁₎ =0

ψ1 q ψ2 q

q -

10 0

( )

δ = δ₍₀₁₎ ≠0 ¹

ψ q

ψ2 q

-

q

10 0

( )

δ ≠ δ₍₀₁₎ ≠0

ψ1 q ψ2 q

q q

ACKNOWLEDGMENTS

Support for this research was provided in part by the National Science Council of the Republic of China, grant No. NSC-91-2118-M-004-008.

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