(5) Specified VP WL Chart

From Figure 80, no samples fall out of the control limits, hence we would use this chart to monitor the mean and variance from now on.

(4) Specified VSI WL Chart

We give the values of the optimal parameters, (

h

₂^*,

h

₁^*)



(0.2,2) and p ^*= 0.5

under

h

₀



1, n₀ 

4

, and



₃ 

3.29

. The control limits and warning limits of the optimal VSI WL chart are UCL = 6.540, LCL = 1.14, UWL = 3.63 and LWL = 2.60.

Then we plot the values of WL on the specified VSI WL chart (Figure 81).

Sample

From Figure 81, all samples fall inside the control limits, hence we would use the control chart to monitor the mean and variance from now on.

(5) Specified VP WL Chart

The control limits and warning limits of the optimal VP WL chart are UCL1 = 8.49, LCL1 = 0.67, UWL1 = 5.71, LWL1 = 1.40, UCL2 = 4.40, LCL2 = 2.16, UWL2 = 3.74, LWL2 = 2.54. Then we plot the values of WL on the specified VP WL chart (Figure 82).

‧

And the sample points are listed in Table 91.

Sample

From Figure 82, all samples fall inside the control limits, hence we would use the control chart to monitor the mean and variance from now on.

Table 91. The Plotting Statistics for Optimal VP WL Chart

No. (

h

_i,

n

_i) WL Which region WL chart

‧

From Figure 83, all samples fall inside the control limits, hence we would use these charts to monitor the mean and variance from now on.

(7)

EWMA YEWMA

ln(

S²

) Chart

‧

We could use the Markov chain (Lucas and Saccucci (1990)) approach to calculate the ARL for the EWMA chart. Under

ARL

₀



740.74 and

 

0.05, using Zero(s) in Fortran IMSL subroutine to obtain 2

L

ln S and

L . So

_Y 2

L

ln S = 2.78 and

L =2.77. The

_Y control limits of these two charts are

EWMA ln S chart EWMA Y chart ²

‧

From Figure 84, all samples fall inside the control limits, hence we would use these charts to monitor the mean and variance from now on.

From the above results, the FP EWMA WL and the VSI EWMA WL charts have three out-of-control samples but the other charts have none. It indicates that the FP EWMA WL and the VSI EWMA WL charts are more effective than other charts.

However, we are concerned with the mean shift and/or the variance shift which is the cause of the three signals. So we could look into the statistic WL. Since WL combines the statistic

S and

²

(

X T

)

², we could compute the deviations of

S from

²

 ˆ

0²

and

(

X T

)

² form

(  ˆ

₀T

)

². Then the bigger one of the two deviations can determine the main cause of the signal.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

CHAPTER 7. CONCLUSION AND FUTURE RESEARCH

The traditional approach to monitor the mean and the variance of a process is using two control charts. But it is laborious and time-consuming. In this project, we proposed a single chart for monitoring the mean and the variance simultaneously. The main advantage of this chart is that the users could control a process by looking at only one chart.

We proposed the FP, VSI, VSSI, VP, optimal FP, optimal VSI, optimal VSSI optimal VP, FP EWMA and VSI EWMA WL charts to monitor the mean and variance simultaneously when the in-control mean may not equal to the target of a process.

Comparing to the X

 S

chart, the WL chart is not only more powerful but also simpler to design and implement. One advantage of the WL chart is that it could vary the weight a to change the weights between the statistics

S and

²

(

X T

)

² such that the performance is more effective under various shifts of the mean and the variance.

However, from the performance comparison, the adaptive WL charts have smaller ATS₁ than the FP WL chart and the optimal WL charts outperform the specified WL charts. Especially the optimal WL charts with the optimal weight a, their ATS1 (ARL1) are smaller than the ATS₁ (ARL₁) of the specified WL charts when the shifts of mean and variance are small. We have shown that the weight a is an important element which could affect the performance of the WL charts. Furthermore, the design is facilitated by the design table provided in this project. The users could find the proper weight a and control limits from table according to the specified



₁,



₂ and



₃. Comparing with the FP WL chart, we found the optimal VSSI and the optimal VP WL charts with the optimal a could save more ATS1. Most of the ATS1 saved % of the two charts are bigger than 90%. And when the shifts of the mean and the variance are

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

small, they also have large ANOS saved % than the other charts. About the specified FP EWMA and VSI EWMA WL charts, they have great performance when



is small for small mean and variance shifts. The ATS₁(ARL₁) saved % of the two EWMA charts is larger than the ATS1(ARL1) saved % of the specified adaptive WL charts and the optimal adaptive WL charts with the optimal a under small



when the mean and the variance shifts are small. As a result, the optimal adaptive WL charts with the optimal weight a have the best performance among these charts proposed in the project.

The design of the WL charts could be easily found. So it makes the WL chart a better choice for the further development of more advanced charts, for example, to construct a chart by CUSUM scheme. And we could also construct the VSSI EWMA or the VP EWMA WL charts. The effectiveness of these charts may be sensitive to small shift of a process.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

REFERENCES

[1] Chan, L. K. and Cheng, S. W. (1996), “Semicircle control chart for variable data,”

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