Inverse Maxwell Distribution and Statistical Process Control: An Efficient Approach for Monitoring Positively Skewed Process
Abstract
:1. Introduction
2. Material and Method
2.1. Inverse Maxwell Distribution and Its Properties
2.2. Derivation of the Control Chart
3. Results
3.1. Performance Evaluation
- The average run length (ARL) is always approximately 370 when the process is in control, (i.e., ) for all considered sample sizes.
- For an in-control process, there is no significant difference between the SDRL values and the corresponding ARL values for any sample size, . For example: The ARL and SDRL values are 370.55 and 369.10, respectively, at . Similarly, when , the ARL and SDRL values are 369.85 and 369.25 respectively.
- The ARL and SDRL values rapidly decrease as the shift increases in the process scale parameter. For example: At 50% increment of shift and for the ARL and SDRL are 28.76 and 28.25, respectively. Under the same condition and at 150% increment of shift, the ARL and SDRL respectively become 3.47 and 2.94. From these values, we can also conclude that the ARL and SDRL are directly proportional to each other.
- In both the in-control and out-of-control situations and for any sample size, the ARL values are greater than the MDRL values. This indicates that the run length distribution of the chart is positively skewed. For example: When and , the ARL and MDRL values are 12.73 and 9, respectively. Additionally, for and , the ARL and MDRL values are 3.36 and 2, correspondingly.
3.2. Simulation Study
- Step 1: Fix the sample size for each random sample.
- Step 2: Generate a random sample of size from.
- Step 3: By taking the square root of , we will get a sample from Maxwell random variable of size .
- Step 4: Obtain a sample from the inverse Maxwell random variable of size by setting .
- Step 5: Estimate the plotting statistics .
- Step 6: Repeat the first five steps until the expected amount of sample subgroups are obtained.
- Step 7: Develop the control limits as proposed in the previous section.
- Step 8: Plot all the values of statistics in contrast to the control limits.
3.3. Comparisons
3.4. Real Life Example
4. Conclusions and Recommendations
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Appendix B
rl.VIM<-function(L1,L2,n,sg,del){ |
sg2 = del*sg^2; lcl = L1*sg^2; ucl = L2*sg^2; |
vim = rl =c() |
for (j in 1:10000) { |
for (i in 1:10000) { |
r = 1/sqrt(rgamma(n, 1.5, scale = 2*sg2)) |
vim[i] = sum(r^-2)/(3*n) |
if (lcl > vim[i] | ucl < vim[i]) |
{ |
rl[j] = i |
break |
} |
else |
{ |
rl[j] = 100000 |
} |
} |
} |
ARL=mean(rl);SDRL=sd(rl);MDRL=median(rl);Q10=quantile(rl,.10);Q25=quantile(rl,.25);Q50=quantile(rl,.50);Q75=quantile(rl,.75);Q90=quantile(rl,.90) |
print(cbind(ARL,SDRL,MDRL,Q10,Q25,Q50,Q75,Q90)) |
} |
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Sample Size (n) | ||||||
---|---|---|---|---|---|---|
0.005 | 0.0027 | 0.002 | ||||
1 | 0.0150 | 4.7734 | 0.0099 | 5.0294 | 0.0081 | 5.4221 |
2 | 0.0878 | 3.3749 | 0.0706 | 3.6228 | 0.0635 | 3.7430 |
3 | 0.1611 | 2.8292 | 0.1380 | 3.0101 | 0.1280 | 3.0975 |
4 | 0.2218 | 2.5265 | 0.1959 | 2.6722 | 0.1845 | 2.7425 |
5 | 0.2713 | 2.3300 | 0.2442 | 2.4536 | 0.2322 | 2.5132 |
6 | 0.3124 | 2.1901 | 0.2848 | 2.2987 | 0.2725 | 2.3507 |
7 | 0.3471 | 2.0845 | 0.3194 | 2.1819 | 0.3070 | 2.2284 |
8 | 0.3768 | 2.0014 | 0.3493 | 2.0901 | 0.3369 | 2.1324 |
9 | 0.4027 | 1.9338 | 0.3753 | 2.0156 | 0.3630 | 2.0546 |
10 | 0.4254 | 1.8777 | 0.3984 | 1.9538 | 0.3862 | 1.9901 |
Sample Size | |||
---|---|---|---|
0.005 | 0.0027 | 0.002 | |
2 | 0.3379 | 0.2706 | 0.2432 |
3 | 0.8675 | 0.7394 | 0.6851 |
4 | 1.5369 | 1.3517 | 1.2715 |
5 | 2.3004 | 2.0623 | 1.9579 |
6 | 3.1324 | 2.8450 | 2.7180 |
7 | 4.0168 | 3.6833 | 3.5351 |
8 | 4.9431 | 4.5661 | 4.3979 |
9 | 5.9038 | 5.4856 | 5.2984 |
10 | 6.8934 | 6.4360 | 6.2307 |
Sample Size (n) | False Alarm Rate (α) | |||||
---|---|---|---|---|---|---|
0.005 | 0.0027 | 0.002 | ||||
W1 | W2 | W1 | W2 | W1 | W2 | |
2 | 0.8049 | 1.1951 | 0.8438 | 1.1562 | 0.8596 | 1.1404 |
3 | 0.5911 | 1.4089 | 0.6514 | 1.3486 | 0.6770 | 1.3229 |
4 | 0.3726 | 1.6274 | 0.4482 | 1.5518 | 0.4809 | 1.5191 |
5 | 0.1600 | 1.8400 | 0.2470 | 1.7531 | 0.2851 | 1.7149 |
6 | 0.0000 | 2.0441 | 0.0518 | 1.9483 | 0.0939 | 1.9060 |
7 | 0.0000 | 2.2396 | 0.0000 | 2.1367 | 0.0000 | 2.0909 |
8 | 0.0000 | 2.4270 | 0.0000 | 2.3181 | 0.0000 | 2.2696 |
9 | 0.0000 | 2.6068 | 0.0000 | 2.4930 | 0.0000 | 2.4420 |
10 | 0.0000 | 2.7799 | 0.0000 | 2.6618 | 0.0000 | 2.6087 |
1 | 3 | 6 | 10 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
ARL | ||||||||||||
1.00 | 370.14 | 371.53 | 258 | 370.55 | 369.10 | 258 | 369.29 | 368.13 | 257 | 369.85 | 369.25 | 254 |
1.25 | 146.15 | 145.07 | 101 | 96.07 | 93.96 | 67 | 60.40 | 60.34 | 42 | 39.37 | 39.16 | 27 |
1.50 | 62.37 | 61.33 | 44 | 28.76 | 28.25 | 20 | 14.47 | 14.01 | 10 | 8.16 | 7.70 | 6 |
1.75 | 32.32 | 31.69 | 23 | 12.73 | 12.13 | 9 | 6.04 | 5.52 | 4 | 3.31 | 2.69 | 2 |
2.00 | 19.95 | 19.29 | 14 | 7.37 | 6.79 | 5 | 3.36 | 2.83 | 2 | 2.02 | 1.45 | 1 |
2.25 | 13.44 | 13.01 | 9 | 4.71 | 4.12 | 3 | 2.33 | 1.77 | 2 | 1.48 | 0.84 | 1 |
2.50 | 9.96 | 9.41 | 7 | 3.47 | 2.94 | 3 | 1.82 | 1.23 | 1 | 1.26 | 0.58 | 1 |
2.75 | 7.80 | 7.23 | 6 | 2.77 | 2.17 | 2 | 1.52 | 0.90 | 1 | 1.14 | 0.39 | 1 |
3 | 6.39 | 5.99 | 5 | 2.31 | 1.72 | 2 | 1.35 | 0.68 | 1 | 1.08 | 0.29 | 1 |
5 | 2.70 | 2.15 | 2 | 1.26 | 0.57 | 1 | 1.03 | 0.17 | 1 | 1.00 | 0.02 | 1 |
1 | 3 | 6 | 10 | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.00 | 41 | 106 | 258 | 514 | 1089 | 41 | 107 | 258 | 513 | 855 | 39 | 105 | 257 | 517 | 841 | 39 | 106 | 254 | 514 | 853 |
1.25 | 17 | 42 | 101 | 202 | 433 | 11 | 28 | 67 | 134 | 221 | 7 | 18 | 42 | 84 | 138 | 4 | 12 | 27 | 54 | 91 |
1.50 | 7 | 19 | 44 | 86 | 184 | 4 | 9 | 20 | 39 | 65 | 2 | 4 | 10 | 20 | 33 | 1 | 3 | 6 | 11 | 18 |
1.75 | 4 | 10 | 23 | 44 | 95 | 2 | 4 | 9 | 17 | 28 | 1 | 2 | 4 | 8 | 13 | 1 | 1 | 2 | 4 | 7 |
2.00 | 2 | 6 | 14 | 28 | 58 | 1 | 2 | 5 | 10 | 16 | 1 | 1 | 2 | 4 | 7 | 1 | 1 | 1 | 3 | 4 |
2.25 | 2 | 4 | 9 | 18 | 39 | 1 | 2 | 3 | 6 | 10 | 1 | 1 | 2 | 3 | 5 | 1 | 1 | 1 | 2 | 3 |
2.50 | 2 | 3 | 7 | 14 | 29 | 1 | 1 | 3 | 5 | 7 | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 2 |
2.75 | 1 | 3 | 6 | 11 | 23 | 1 | 1 | 2 | 4 | 6 | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 2 |
3.00 | 1 | 2 | 5 | 8 | 18 | 1 | 1 | 2 | 3 | 5 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 |
5 | 1 | 1 | 2 | 3 | 7 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Average Run Length | ||
---|---|---|
Lognormal S-Chart | ||
1 | 368.39 | 370.34 |
1.50 | 17.72 | 53.92 |
2.00 | 4.12 | 24.34 |
2.50 | 2.11 | 14.15 |
3.00 | 1.52 | 10.71 |
3.50 | 1.28 | 8.79 |
4.00 | 1.16 | 7.99 |
Sample Number | Observations | ||||||
---|---|---|---|---|---|---|---|
1st | 2nd | 3rd | 4th | 5th | 6th | 7th | |
1 | 22.2 | 23.0 | 24.0 | 28.6 | 21.8 | 17.0 | 26.0 |
2 | 23.2 | 18.9 | 21.9 | 27.3 | 13.8 | 24.0 | 20.1 |
3 | 15.7 | 26.8 | 27.9 | 15.3 | 28.8 | 16.0 | 23.6 |
4 | 53.8 | 21.7 | 28.8 | 17.0 | 16.5 | 15.7 | 28.0 |
5 | 13.3 | 16.5 | 24.2 | 17.6 | 27.8 | 18.3 | 17.7 |
6 | 20.0 | 13.2 | 16.9 | 14.9 | 15.5 | 7.0 | 15.8 |
7 | 15.0 | 38.3 | 11.2 | 38.2 | 26.7 | 17.1 | 29.0 |
8 | 18.3 | 18.4 | 18.2 | 15.9 | 16.4 | 23.6 | 19.2 |
9 | 23.3 | 20.4 | 20.9 | 28.5 | 23.2 | 17.9 | 46.1 |
10 | 39.3 | 11.8 | 17.7 | 30.9 | 22.4 | 45.0 | 18.2 |
11 | 30.2 | 21.8 | 18.2 | 23.0 | 27.2 | 10.9 | 25.5 |
12 | 12.4 | 39.9 | 17.7 | 26.3 | 14.1 | 21.0 | 11.2 |
13 | 10.8 | 25.7 | 32.4 | 13.6 | 19.1 | 16.1 | 53.3 |
14 | 57.3 | 36.5 | 19.7 | 20.8 | 30.8 | 20.0 | 39.6 |
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Omar, M.H.; Arafat, S.Y.; Hossain, M.P.; Riaz, M. Inverse Maxwell Distribution and Statistical Process Control: An Efficient Approach for Monitoring Positively Skewed Process. Symmetry 2021, 13, 189. https://doi.org/10.3390/sym13020189
Omar MH, Arafat SY, Hossain MP, Riaz M. Inverse Maxwell Distribution and Statistical Process Control: An Efficient Approach for Monitoring Positively Skewed Process. Symmetry. 2021; 13(2):189. https://doi.org/10.3390/sym13020189
Chicago/Turabian StyleOmar, M. Hafidz, Sheikh Y. Arafat, M. Pear Hossain, and Muhammad Riaz. 2021. "Inverse Maxwell Distribution and Statistical Process Control: An Efficient Approach for Monitoring Positively Skewed Process" Symmetry 13, no. 2: 189. https://doi.org/10.3390/sym13020189
APA StyleOmar, M. H., Arafat, S. Y., Hossain, M. P., & Riaz, M. (2021). Inverse Maxwell Distribution and Statistical Process Control: An Efficient Approach for Monitoring Positively Skewed Process. Symmetry, 13(2), 189. https://doi.org/10.3390/sym13020189