Control charts (CCs) are one of the main tools in Statistical Process Control that have been widely adopted in manufacturing sectors as an effective strategy for malfunction detection throughout the previous decades. Measurement errors (M.E’s) are involved in the quality characteristic of interest, which can effect the CC’s performance. The authors explored the impact of a linear model with additive covariate M.E on the multivariate cumulative sum (CUSUM) CC for a specific kind of data known as compositional data (CoDa). The average run length
A widely used method is statistical process control (SPC) to monitor a process. SPC is a decision-making tool used in almost all industrial manufacturing processes to achieve process stability and sustainable quality product. W. A. Shewhart introduced the CCs in the 1920s. The goal of a CC is to find assignable causes of shifts and to minimize the production of a large number of defective items. In practice, determining which CC is appropriate for specific data can be difficult. It is possible to figure out by looking at the distribution of the underlying data process. CCs with compound or simple rules can be computed using the run-length distribution using a Markov chain approach [
Multivariate EWMA CCs are proposed using multiple variables based on an AIC calculation for multivariate SPC [
Compositional data (CoDa) vectors are positive components presented as percentages, ratios, proportions or parts of a whole. Aitchison’s principles and statistical tools from [
The CUSUM charts are well-known for detecting smaller and more frequent changes than Shewhart charts and are among the most common and widely used charts in practice [
CCs are often created with the hypothesis that the factor of the quality feature is assessed accurately. However, M.E’s can occur due to various circumstances. Many researchers examined the impact of M.E’s on CCs. Imprecise measurement instruments or human errors are two main causes contributing to M.E’s. The presence of M.E’s must be taken into account while implementing CCs because it has a huge impact on their performance [
Typically, practitioners monitor processes without considering the possibility of M.E’s. Despite sophisticated measurement techniques and methodologies, M.E’s is commonly caused by operational and environmental factors. As Montgomery et al. [
Although many studies on the assessment of CCs in the presence of M.E’s have been conducted, few have focused on multivariate data, and very few researchers have contributed to CoDa. But as far as the authors are aware, none of them has been dedicated to MCUSUM CC for
A row vector
As CoDa deals with the constraint of constant sum, Euclidean geometry is not supposed to be useful for CoDa. Let us take an example to make it more clear, Suppose there are two CoDa vectors,
• the perturbation operator
• the powering operator
The constant sum constraint makes CoDa difficult to deal with in its original form, hence pre-defined log-ratio transformations (LRT) can be used to transform CoDa from
Another LRT for a CoDa vector
Also, there is
In this paper, the CoDa vectors are denoted without a “*” sign, while the transformed vectors are denoted with a “*” in the superscript.
Assume that there are
One can refer to
Assume we have
The MCUSUM-CoDa CC displays a signal when
To assess the MCUSUM-CoDa CC’s run-length efficiency, the authors implement a Markov chain approximation suggested by Crosier [
Lowry et al. [
When M.E’s is included in the process, the non-centrality parameter is denoted by
Linna et al. [
The
The authors assumed the unit sample size
This section aims to evaluate the performance of the MCUSUM-CoDa CC in the presence of M.E’s. The M.E’s VCM equals Select the most appropriate value for the Select a particular value of Obtain the value of Select the optimal value of
The following six cases for the CoDa VCM
The goal is to investigate the effect of involved parameters
This subsection investigates the effect of the involved parameters The M.E VCM The subgroup size The powering constant
wot M.E’s | ||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.00 | 8.79 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 8.79 | 91.34 | 0.25 | 8.80 | 93.10 | 0.25 | 8.80 | 92.81 | 0.24 | 8.80 | 96.63 | 0.24 | 8.80 | 95.75 | 0.23 | 8.80 | 101.93 | 0.23 | 8.80 | 100.17 |
0.50 | 8.79 | 32.41 | 0.49 | 8.80 | 33.04 | 0.50 | 8.80 | 32.93 | 0.48 | 8.80 | 34.31 | 0.48 | 8.80 | 33.99 | 0.46 | 8.80 | 36.26 | 0.47 | 8.80 | 35.60 |
0.75 | 9.70 | 18.09 | 0.74 | 9.65 | 18.45 | 0.74 | 9.65 | 18.39 | 0.72 | 9.60 | 19.15 | 0.73 | 9.60 | 18.98 | 0.69 | 9.50 | 20.23 | 0.70 | 9.50 | 19.87 |
1.00 | 10.34 | 11.94 | 0.99 | 10.31 | 12.17 | 0.99 | 10.31 | 12.13 | 0.96 | 10.26 | 12.63 | 0.97 | 10.26 | 12.51 | 0.93 | 10.17 | 13.33 | 0.94 | 10.20 | 13.10 |
1.25 | 10.76 | 8.72 | 1.23 | 10.74 | 8.88 | 1.24 | 10.74 | 8.86 | 1.20 | 10.69 | 9.21 | 1.21 | 10.71 | 9.13 | 1.16 | 10.61 | 9.70 | 1.17 | 10.63 | 9.54 |
1.50 | 11.05 | 6.85 | 1.48 | 11.03 | 6.97 | 1.48 | 11.04 | 6.95 | 1.44 | 10.99 | 7.20 | 1.45 | 11.01 | 7.14 | 1.39 | 10.93 | 7.57 | 1.41 | 10.96 | 7.45 |
1.75 | 11.25 | 5.67 | 1.73 | 11.24 | 5.76 | 1.73 | 11.24 | 5.75 | 1.68 | 11.21 | 5.94 | 1.69 | 11.21 | 5.90 | 1.62 | 11.16 | 6.22 | 1.64 | 11.17 | 6.12 |
2.00 | 11.39 | 4.90 | 1.97 | 11.39 | 4.97 | 1.98 | 11.39 | 4.96 | 1.92 | 11.36 | 5.11 | 1.94 | 11.36 | 5.07 | 1.85 | 11.32 | 5.32 | 1.88 | 11.33 | 5.25 |
wot M.E’s | ||||||||||||||||||||
0.00 | 8.79 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 8.79 | 91.34 | 0.22 | 8.80 | 107.24 | 0.23 | 8.80 | 104.59 | 0.24 | 8.80 | 96.63 | 0.24 | 8.80 | 95.75 | 0.25 | 8.80 | 93.98 | 0.25 | 8.80 | 93.54 |
0.50 | 8.79 | 32.41 | 0.45 | 8.80 | 38.26 | 0.46 | 8.80 | 37.25 | 0.48 | 8.80 | 34.31 | 0.48 | 8.80 | 33.99 | 0.49 | 8.80 | 33.35 | 0.49 | 8.80 | 33.20 |
0.75 | 9.70 | 18.09 | 0.67 | 9.38 | 21.32 | 0.68 | 9.44 | 20.78 | 0.72 | 9.60 | 19.15 | 0.73 | 9.60 | 18.98 | 0.74 | 9.65 | 18.62 | 0.74 | 9.65 | 18.53 |
1.00 | 10.34 | 11.94 | 0.89 | 10.10 | 14.05 | 0.91 | 10.13 | 13.69 | 0.96 | 10.26 | 12.63 | 0.97 | 10.26 | 12.51 | 0.98 | 10.28 | 12.28 | 0.98 | 10.28 | 12.22 |
1.25 | 10.76 | 8.72 | 1.12 | 10.55 | 10.21 | 1.14 | 10.59 | 9.96 | 1.20 | 10.69 | 9.21 | 1.21 | 10.71 | 9.13 | 1.23 | 10.72 | 8.96 | 1.23 | 10.72 | 8.92 |
1.50 | 11.05 | 6.85 | 1.34 | 10.88 | 7.94 | 1.36 | 10.91 | 7.76 | 1.44 | 10.99 | 7.20 | 1.45 | 11.01 | 7.14 | 1.47 | 11.02 | 7.02 | 1.48 | 11.03 | 6.99 |
1.75 | 11.25 | 5.67 | 1.56 | 11.11 | 6.50 | 1.59 | 11.13 | 6.36 | 1.68 | 11.21 | 5.94 | 1.69 | 11.21 | 5.90 | 1.72 | 11.23 | 5.81 | 1.72 | 11.24 | 5.78 |
2.00 | 11.39 | 4.90 | 1.79 | 11.28 | 5.54 | 1.82 | 11.30 | 5.43 | 1.92 | 11.36 | 5.11 | 1.94 | 11.36 | 5.07 | 1.96 | 11.37 | 5.00 | 1.97 | 11.38 | 4.99 |
wot M.E’s | ||||||||||||||||||||
0.00 | 8.79 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 8.79 | 91.34 | 0.15 | 8.80 | 171.85 | 0.16 | 8.80 | 159.59 | 0.24 | 8.80 | 96.63 | 0.24 | 8.80 | 95.75 | 0.25 | 8.80 | 91.42 | 0.25 | 8.80 | 91.41 |
0.50 | 8.79 | 32.41 | 0.31 | 8.80 | 68.05 | 0.33 | 8.80 | 61.46 | 0.48 | 8.80 | 34.31 | 0.48 | 8.80 | 33.99 | 0.50 | 8.80 | 32.44 | 0.50 | 8.80 | 32.44 |
0.75 | 9.70 | 18.09 | 0.46 | 8.79 | 36.92 | 0.49 | 8.79 | 33.46 | 0.72 | 9.60 | 19.15 | 0.73 | 9.60 | 18.98 | 0.75 | 9.70 | 18.11 | 0.75 | 9.70 | 18.11 |
1.00 | 10.34 | 11.94 | 0.61 | 9.19 | 24.41 | 0.65 | 9.32 | 22.15 | 0.96 | 10.26 | 12.63 | 0.97 | 10.26 | 12.51 | 1.00 | 10.34 | 11.95 | 1.00 | 10.34 | 11.95 |
1.25 | 10.76 | 8.72 | 0.76 | 9.70 | 17.68 | 0.82 | 9.88 | 16.03 | 1.20 | 10.69 | 9.21 | 1.21 | 10.71 | 9.13 | 1.25 | 10.76 | 8.73 | 1.25 | 10.76 | 8.73 |
1.50 | 11.05 | 6.85 | 0.92 | 10.13 | 13.57 | 0.98 | 10.28 | 12.32 | 1.44 | 10.99 | 7.20 | 1.45 | 11.01 | 7.14 | 1.50 | 11.05 | 6.85 | 1.50 | 11.05 | 6.85 |
1.75 | 11.25 | 5.67 | 1.07 | 10.46 | 10.88 | 1.14 | 10.59 | 9.90 | 1.68 | 11.21 | 5.94 | 1.69 | 11.21 | 5.90 | 1.75 | 11.25 | 5.68 | 1.75 | 11.25 | 5.68 |
2.00 | 11.39 | 4.90 | 1.22 | 10.72 | 9.03 | 1.30 | 10.84 | 8.23 | 1.92 | 11.36 | 5.11 | 1.94 | 11.36 | 5.07 | 2.00 | 11.40 | 4.91 | 2.00 | 11.40 | 4.90 |
This subsection investigates the effect of the involved parameters The M.E VCM The subgroup size The powering constant
wot M.E’s | ||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.00 | 8.79 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 8.79 | 91.34 | 0.25 | 8.80 | 93.10 | 0.25 | 8.80 | 92.75 | 0.24 | 8.80 | 97.38 | 0.24 | 8.80 | 95.33 | 0.23 | 8.80 | 103.43 | 0.24 | 8.80 | 99.34 |
0.50 | 8.79 | 32.41 | 0.49 | 8.80 | 33.13 | 0.50 | 8.80 | 32.88 | 0.48 | 8.80 | 34.58 | 0.49 | 8.80 | 33.84 | 0.46 | 8.80 | 36.82 | 0.47 | 8.80 | 35.30 |
0.75 | 9.70 | 18.09 | 0.74 | 9.65 | 18.50 | 0.74 | 9.65 | 18.36 | 0.72 | 9.55 | 19.30 | 0.73 | 9.60 | 18.89 | 0.69 | 9.44 | 20.54 | 0.71 | 9.55 | 19.70 |
1.00 | 10.34 | 11.94 | 0.99 | 10.31 | 12.20 | 0.99 | 10.31 | 12.11 | 0.96 | 10.23 | 12.73 | 0.97 | 10.28 | 12.46 | 0.92 | 10.13 | 13.53 | 0.94 | 10.20 | 12.99 |
1.25 | 10.76 | 8.72 | 1.23 | 10.74 | 8.91 | 1.24 | 10.74 | 8.85 | 1.20 | 10.67 | 9.28 | 1.21 | 10.71 | 9.09 | 1.15 | 10.59 | 9.85 | 1.18 | 10.65 | 9.46 |
1.50 | 11.05 | 6.85 | 1.48 | 11.03 | 6.98 | 1.49 | 11.04 | 6.94 | 1.44 | 10.98 | 7.25 | 1.46 | 11.01 | 7.12 | 1.38 | 10.92 | 7.67 | 1.42 | 10.96 | 7.39 |
1.75 | 11.25 | 5.67 | 1.72 | 11.24 | 5.77 | 1.73 | 11.24 | 5.74 | 1.67 | 11.20 | 5.98 | 1.70 | 11.22 | 5.87 | 1.60 | 11.15 | 6.30 | 1.65 | 11.18 | 6.08 |
2.00 | 11.39 | 4.90 | 1.97 | 11.38 | 4.98 | 1.98 | 11.39 | 4.95 | 1.91 | 11.35 | 5.14 | 1.94 | 11.36 | 5.06 | 1.83 | 11.31 | 5.38 | 1.89 | 11.34 | 5.22 |
wot M.E’s | ||||||||||||||||||||
0.00 | 8.79 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 8.79 | 91.34 | 0.22 | 8.80 | 109.48 | 0.23 | 8.80 | 103.35 | 0.24 | 8.80 | 97.38 | 0.24 | 8.80 | 95.33 | 0.24 | 8.80 | 94.36 | 0.25 | 8.80 | 93.33 |
0.50 | 8.79 | 32.41 | 0.44 | 8.80 | 39.12 | 0.46 | 8.80 | 36.79 | 0.48 | 8.80 | 34.58 | 0.49 | 8.80 | 33.84 | 0.49 | 8.80 | 33.49 | 0.49 | 8.80 | 33.12 |
0.75 | 9.70 | 18.09 | 0.66 | 8.79 | 21.79 | 0.69 | 9.44 | 20.52 | 0.72 | 9.55 | 19.30 | 0.73 | 9.60 | 18.89 | 0.73 | 9.60 | 18.70 | 0.74 | 9.65 | 18.49 |
1.00 | 10.34 | 11.94 | 0.88 | 10.07 | 14.36 | 0.92 | 10.13 | 13.52 | 0.96 | 10.23 | 12.73 | 0.97 | 10.28 | 12.46 | 0.98 | 10.28 | 12.33 | 0.99 | 10.31 | 12.20 |
1.25 | 10.76 | 8.72 | 1.10 | 10.53 | 10.43 | 1.15 | 10.59 | 9.84 | 1.20 | 10.67 | 9.28 | 1.21 | 10.71 | 9.09 | 1.22 | 10.72 | 9.00 | 1.23 | 10.74 | 8.91 |
1.50 | 11.05 | 6.85 | 1.32 | 10.85 | 8.11 | 1.38 | 10.92 | 7.67 | 1.44 | 10.98 | 7.25 | 1.46 | 11.01 | 7.12 | 1.47 | 11.02 | 7.05 | 1.48 | 11.03 | 6.98 |
1.75 | 11.25 | 5.67 | 1.54 | 11.09 | 6.63 | 1.61 | 11.15 | 6.29 | 1.67 | 11.20 | 5.98 | 1.70 | 11.22 | 5.87 | 1.71 | 11.23 | 5.82 | 1.72 | 11.24 | 5.77 |
2.00 | 11.39 | 4.90 | 1.76 | 11.26 | 5.64 | 1.83 | 11.31 | 5.38 | 1.91 | 11.35 | 5.14 | 1.94 | 11.36 | 5.06 | 1.96 | 11.37 | 5.02 | 1.97 | 11.38 | 4.98 |
wot M.E’s | ||||||||||||||||||||
0.00 | 8.79 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 8.79 | 91.34 | 0.14 | 8.80 | 181.72 | 0.17 | 8.80 | 153.65 | 0.24 | 8.80 | 97.38 | 0.24 | 8.80 | 95.33 | 0.25 | 8.80 | 91.43 | 0.25 | 8.80 | 91.40 |
0.50 | 8.79 | 32.41 | 0.29 | 8.80 | 73.76 | 0.34 | 8.80 | 58.45 | 0.48 | 8.80 | 34.58 | 0.49 | 8.80 | 33.84 | 0.50 | 8.80 | 32.45 | 0.50 | 8.80 | 32.43 |
0.75 | 9.70 | 18.09 | 0.43 | 8.79 | 39.97 | 0.51 | 8.79 | 31.90 | 0.72 | 9.55 | 19.30 | 0.73 | 9.60 | 18.89 | 0.75 | 9.70 | 18.11 | 0.75 | 9.70 | 18.11 |
1.00 | 10.34 | 11.94 | 0.58 | 9.04 | 26.36 | 0.67 | 9.44 | 21.12 | 0.96 | 10.23 | 12.73 | 0.97 | 10.28 | 12.46 | 1.00 | 10.34 | 11.95 | 1.00 | 10.34 | 11.95 |
1.25 | 10.76 | 8.72 | 0.72 | 9.60 | 19.10 | 0.84 | 9.96 | 15.28 | 1.20 | 10.67 | 9.28 | 1.21 | 10.71 | 9.09 | 1.25 | 10.76 | 8.73 | 1.25 | 10.76 | 8.73 |
1.50 | 11.05 | 6.85 | 0.87 | 10.03 | 14.66 | 1.01 | 10.37 | 11.75 | 1.44 | 10.98 | 7.25 | 1.46 | 11.01 | 7.12 | 1.50 | 11.05 | 6.85 | 1.50 | 11.05 | 6.85 |
1.75 | 11.25 | 5.67 | 1.01 | 10.37 | 11.74 | 1.18 | 10.65 | 9.45 | 1.67 | 11.20 | 5.98 | 1.70 | 11.22 | 5.87 | 1.75 | 11.25 | 5.68 | 1.75 | 11.25 | 5.68 |
2.00 | 11.39 | 4.90 | 1.16 | 10.61 | 9.72 | 1.35 | 10.89 | 7.88 | 1.91 | 11.35 | 5.14 | 1.94 | 11.36 | 5.06 | 2.00 | 11.40 | 4.91 | 2.00 | 11.40 | 4.90 |
This subsection investigates the effect of the involved parameters The M.E VCM The subgroup size The powering constant
wot M.E’s | ||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.00 | 8.79 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 8.79 | 91.34 | 0.25 | 8.80 | 92.13 | 0.25 | 8.80 | 93.58 | 0.24 | 8.80 | 95.04 | 0.23 | 8.80 | 99.41 | 0.23 | 8.80 | 99.41 | 0.22 | 8.80 | 108.14 |
0.50 | 8.79 | 32.41 | 0.50 | 8.80 | 32.80 | 0.49 | 8.80 | 33.32 | 0.48 | 8.80 | 33.85 | 0.47 | 8.80 | 35.45 | 0.47 | 8.80 | 35.45 | 0.44 | 8.80 | 38.76 |
0.75 | 9.70 | 18.09 | 0.74 | 9.63 | 18.34 | 0.73 | 9.58 | 18.64 | 0.73 | 9.58 | 18.93 | 0.70 | 9.47 | 19.82 | 0.70 | 9.47 | 19.82 | 0.66 | 9.36 | 21.63 |
1.00 | 10.34 | 11.94 | 0.99 | 10.29 | 12.11 | 0.98 | 10.27 | 12.30 | 0.97 | 10.24 | 12.49 | 0.94 | 10.18 | 13.08 | 0.94 | 10.18 | 13.08 | 0.88 | 10.05 | 14.27 |
1.25 | 10.76 | 8.72 | 1.24 | 10.73 | 8.85 | 1.22 | 10.71 | 8.98 | 1.21 | 10.69 | 9.12 | 1.17 | 10.62 | 9.53 | 1.17 | 10.62 | 9.53 | 1.10 | 10.52 | 10.37 |
1.50 | 11.05 | 6.85 | 1.48 | 11.03 | 6.94 | 1.47 | 11.01 | 7.04 | 1.45 | 11.00 | 7.14 | 1.41 | 10.95 | 7.44 | 1.41 | 10.95 | 7.44 | 1.32 | 10.84 | 8.07 |
1.75 | 11.25 | 5.67 | 1.73 | 11.23 | 5.74 | 1.71 | 11.22 | 5.82 | 1.69 | 11.21 | 5.89 | 1.64 | 11.17 | 6.12 | 1.64 | 11.17 | 6.12 | 1.54 | 11.08 | 6.60 |
2.00 | 11.39 | 4.90 | 1.98 | 11.38 | 4.96 | 1.96 | 11.37 | 5.01 | 1.94 | 11.36 | 5.07 | 1.88 | 11.32 | 5.25 | 1.88 | 11.32 | 5.25 | 1.77 | 11.26 | 5.62 |
wot M.E’s | ||||||||||||||||||||
0.00 | 8.79 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 8.79 | 91.34 | 0.23 | 8.80 | 103.78 | 0.21 | 8.80 | 116.83 | 0.24 | 8.80 | 95.04 | 0.23 | 8.80 | 99.41 | 0.25 | 8.80 | 92.86 | 0.24 | 8.80 | 95.04 |
0.50 | 8.79 | 32.41 | 0.46 | 8.80 | 37.09 | 0.42 | 8.80 | 42.20 | 0.48 | 8.80 | 33.85 | 0.47 | 8.80 | 35.45 | 0.49 | 8.80 | 33.06 | 0.48 | 8.80 | 33.85 |
0.75 | 9.70 | 18.09 | 0.68 | 9.42 | 20.72 | 0.63 | 9.16 | 23.48 | 0.73 | 9.58 | 18.93 | 0.70 | 9.47 | 19.82 | 0.74 | 9.63 | 18.49 | 0.73 | 9.58 | 18.93 |
1.00 | 10.34 | 11.94 | 0.91 | 10.11 | 13.67 | 0.83 | 9.90 | 15.50 | 0.97 | 10.24 | 12.49 | 0.94 | 10.18 | 13.08 | 0.98 | 10.27 | 12.21 | 0.97 | 10.24 | 12.49 |
1.25 | 10.76 | 8.72 | 1.14 | 10.56 | 9.95 | 1.04 | 10.40 | 11.25 | 1.21 | 10.69 | 9.12 | 1.17 | 10.62 | 9.53 | 1.23 | 10.71 | 8.92 | 1.21 | 10.69 | 9.12 |
1.50 | 11.05 | 6.85 | 1.36 | 10.90 | 7.75 | 1.25 | 10.75 | 8.72 | 1.45 | 11.00 | 7.14 | 1.41 | 10.95 | 7.44 | 1.48 | 11.02 | 6.99 | 1.45 | 11.00 | 7.14 |
1.75 | 11.25 | 5.67 | 1.59 | 11.12 | 6.36 | 1.46 | 11.00 | 7.10 | 1.69 | 11.21 | 5.89 | 1.64 | 11.17 | 6.12 | 1.72 | 11.23 | 5.78 | 1.69 | 11.21 | 5.89 |
2.00 | 11.39 | 4.90 | 1.82 | 11.29 | 5.43 | 1.67 | 11.18 | 6.01 | 1.94 | 11.36 | 5.07 | 1.88 | 11.32 | 5.25 | 1.97 | 11.37 | 4.98 | 1.94 | 11.36 | 5.07 |
wot M.E’s | ||||||||||||||||||||
0.00 | 8.79 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 8.79 | 91.34 | 0.16 | 8.80 | 158.07 | 0.12 | 8.80 | 212.02 | 0.24 | 8.80 | 95.04 | 0.23 | 8.80 | 99.41 | 0.25 | 8.80 | 90.75 | 0.25 | 8.80 | 90.81 |
0.50 | 8.79 | 32.41 | 0.33 | 8.80 | 61.10 | 0.24 | 8.80 | 95.04 | 0.48 | 8.80 | 33.85 | 0.47 | 8.80 | 35.45 | 0.50 | 8.80 | 32.31 | 0.50 | 8.80 | 32.33 |
0.75 | 9.70 | 18.09 | 0.49 | 8.79 | 33.32 | 0.36 | 8.79 | 51.96 | 0.73 | 9.58 | 18.93 | 0.70 | 9.47 | 19.82 | 0.75 | 9.63 | 18.07 | 0.75 | 9.63 | 18.08 |
1.00 | 10.34 | 11.94 | 0.65 | 9.29 | 22.09 | 0.48 | 8.79 | 33.85 | 0.97 | 10.24 | 12.49 | 0.94 | 10.18 | 13.08 | 1.00 | 10.32 | 11.93 | 1.00 | 10.32 | 11.94 |
1.25 | 10.76 | 8.72 | 0.82 | 9.86 | 16.00 | 0.61 | 9.09 | 24.61 | 1.21 | 10.69 | 9.12 | 1.17 | 10.62 | 9.53 | 1.25 | 10.75 | 8.72 | 1.25 | 10.75 | 8.73 |
1.50 | 11.05 | 6.85 | 0.98 | 10.26 | 12.30 | 0.73 | 9.58 | 18.93 | 1.45 | 11.00 | 7.14 | 1.41 | 10.95 | 7.44 | 1.50 | 11.04 | 6.85 | 1.50 | 11.04 | 6.85 |
1.75 | 11.25 | 5.67 | 1.14 | 10.58 | 9.89 | 0.85 | 9.94 | 15.14 | 1.69 | 11.21 | 5.89 | 1.64 | 11.17 | 6.12 | 1.75 | 11.24 | 5.67 | 1.75 | 11.24 | 5.68 |
2.00 | 11.39 | 4.90 | 1.30 | 10.82 | 8.23 | 0.97 | 10.24 | 12.49 | 1.94 | 11.36 | 5.07 | 1.88 | 11.32 | 5.25 | 2.00 | 11.39 | 4.91 | 2.00 | 11.39 | 4.90 |
This subsection investigates the effect of the involved parameters The M.E VCM The subgroup size The powering constant
wot M.E’s | ||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.00 | 8.79 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 8.79 | 91.34 | 0.25 | 8.80 | 92.06 | 0.24 | 8.80 | 93.91 | 0.24 | 8.80 | 94.83 | 0.23 | 8.80 | 100.39 | 0.24 | 8.80 | 98.99 | 0.22 | 8.80 | 110.10 |
0.50 | 8.79 | 32.41 | 0.50 | 8.80 | 32.77 | 0.49 | 8.80 | 33.44 | 0.49 | 8.80 | 33.77 | 0.47 | 8.80 | 35.81 | 0.47 | 8.80 | 35.30 | 0.44 | 8.80 | 39.52 |
0.75 | 9.70 | 18.09 | 0.74 | 9.63 | 18.33 | 0.73 | 9.58 | 18.70 | 0.73 | 9.58 | 18.89 | 0.70 | 9.47 | 20.02 | 0.71 | 9.47 | 19.73 | 0.65 | 9.29 | 22.04 |
1.00 | 10.34 | 11.94 | 0.99 | 10.29 | 12.10 | 0.98 | 10.27 | 12.34 | 0.97 | 10.24 | 12.46 | 0.93 | 10.15 | 13.21 | 0.94 | 10.18 | 13.02 | 0.87 | 10.01 | 14.54 |
1.25 | 10.76 | 8.72 | 1.24 | 10.73 | 8.84 | 1.22 | 10.69 | 9.01 | 1.21 | 10.69 | 9.10 | 1.16 | 10.62 | 9.62 | 1.18 | 10.64 | 9.49 | 1.09 | 10.49 | 10.57 |
1.50 | 11.05 | 6.85 | 1.48 | 11.03 | 6.94 | 1.46 | 11.01 | 7.06 | 1.45 | 11.00 | 7.12 | 1.40 | 10.94 | 7.51 | 1.41 | 10.95 | 7.41 | 1.31 | 10.82 | 8.21 |
1.75 | 11.25 | 5.67 | 1.73 | 11.23 | 5.74 | 1.71 | 11.21 | 5.84 | 1.70 | 11.21 | 5.88 | 1.63 | 11.16 | 6.17 | 1.65 | 11.17 | 6.10 | 1.52 | 11.06 | 6.71 |
2.00 | 11.39 | 4.90 | 1.98 | 11.38 | 4.95 | 1.95 | 11.36 | 5.03 | 1.94 | 11.36 | 5.06 | 1.86 | 11.32 | 5.29 | 1.88 | 11.33 | 5.23 | 1.74 | 1.24 | 5.70 |
wot M.E’s | ||||||||||||||||||||
0.00 | 8.79 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 8.79 | 91.34 | 0.23 | 8.80 | 103.15 | 0.21 | 8.80 | 119.73 | 0.24 | 8.80 | 94.83 | 0.23 | 8.80 | 100.39 | 0.25 | 8.80 | 92.75 | 0.24 | 8.80 | 95.53 |
0.50 | 8.79 | 32.41 | 0.46 | 8.80 | 36.85 | 0.41 | 8.80 | 43.38 | 0.49 | 8.80 | 33.77 | 0.47 | 8.80 | 35.81 | 0.49 | 8.80 | 33.02 | 0.48 | 8.80 | 34.03 |
0.75 | 9.70 | 18.09 | 0.69 | 9.42 | 20.59 | 0.61 | 9.16 | 24.12 | 0.73 | 9.58 | 18.89 | 0.70 | 9.47 | 20.02 | 0.74 | 9.63 | 18.47 | 0.72 | 9.58 | 19.03 |
1.00 | 10.34 | 11.94 | 0.91 | 10.11 | 13.58 | 0.82 | 9.86 | 15.92 | 0.97 | 10.24 | 12.46 | 0.93 | 10.15 | 13.21 | 0.98 | 10.27 | 12.19 | 0.96 | 10.24 | 12.56 |
1.25 | 10.76 | 8.72 | 1.14 | 10.58 | 9.88 | 1.02 | 10.35 | 11.55 | 1.21 | 10.69 | 9.10 | 1.16 | 10.62 | 9.62 | 1.23 | 10.71 | 8.91 | 1.21 | 10.68 | 9.16 |
1.50 | 11.05 | 6.85 | 1.37 | 10.90 | 7.71 | 1.23 | 10.71 | 8.94 | 1.45 | 11.00 | 7.12 | 1.40 | 10.94 | 7.51 | 1.48 | 11.02 | 6.98 | 1.45 | 10.98 | 7.17 |
1.75 | 11.25 | 5.67 | 1.60 | 11.13 | 6.32 | 1.43 | 10.97 | 7.27 | 1.70 | 11.21 | 5.88 | 1.63 | 11.16 | 6.17 | 1.72 | 11.23 | 5.78 | 1.69 | 11.20 | 5.92 |
2.00 | 11.39 | 4.90 | 1.83 | 11.30 | 5.40 | 1.64 | 11.17 | 6.14 | 1.94 | 11.36 | 5.06 | 1.86 | 11.32 | 5.29 | 1.97 | 11.37 | 4.98 | 1.93 | 11.35 | 5.09 |
wot M.E’s | ||||||||||||||||||||
0.00 | 8.79 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 8.79 | 91.34 | 0.17 | 8.80 | 155.08 | 0.11 | 8.80 | 221.93 | 0.24 | 8.80 | 94.83 | 0.23 | 8.80 | 100.39 | 0.25 | 8.80 | 90.74 | 0.25 | 8.80 | 90.83 |
0.50 | 8.79 | 32.41 | 0.33 | 8.80 | 59.56 | 0.23 | 8.80 | 102.89 | 0.49 | 8.80 | 33.77 | 0.47 | 8.80 | 35.81 | 0.50 | 8.80 | 32.30 | 0.50 | 8.80 | 32.33 |
0.75 | 9.70 | 18.09 | 0.50 | 8.79 | 32.52 | 0.34 | 8.79 | 56.59 | 0.73 | 9.58 | 18.89 | 0.70 | 9.47 | 20.02 | 0.75 | 9.63 | 18.07 | 0.75 | 9.63 | 18.08 |
1.00 | 10.34 | 11.94 | 0.66 | 9.36 | 21.56 | 0.46 | 8.79 | 36.75 | 0.97 | 10.24 | 12.46 | 0.93 | 10.15 | 13.21 | 1.00 | 10.32 | 11.93 | 1.00 | 10.32 | 11.94 |
1.25 | 10.76 | 8.72 | 0.83 | 9.90 | 15.61 | 0.57 | 8.93 | 26.68 | 1.21 | 10.69 | 9.10 | 1.16 | 10.62 | 9.62 | 1.25 | 10.75 | 8.72 | 1.25 | 10.75 | 8.73 |
1.50 | 11.05 | 6.85 | 1.00 | 10.29 | 12.01 | 0.69 | 9.42 | 20.53 | 1.45 | 11.00 | 7.12 | 1.40 | 10.94 | 7.51 | 1.50 | 11.04 | 6.85 | 1.50 | 11.04 | 6.85 |
1.75 | 11.25 | 5.67 | 1.16 | 10.60 | 9.66 | 0.80 | 9.81 | 16.43 | 1.70 | 11.21 | 5.88 | 1.63 | 11.16 | 6.17 | 1.75 | 11.24 | 5.67 | 1.75 | 11.24 | 5.68 |
2.00 | 11.39 | 4.90 | 1.33 | 10.85 | 8.04 | 0.92 | 10.11 | 13.55 | 1.94 | 11.36 | 5.06 | 1.86 | 11.32 | 5.29 | 2.00 | 11.39 | 4.90 | 2.00 | 11.39 | 4.91 |
This subsection investigates the effect of the involved parameters The The M.E VCM The subgroup size The powering constant
wot M.E’s | ||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.00 | 8.79 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 8.79 | 91.34 | 0.24 | 10.00 | 122.46 | 0.25 | 10.00 | 119.72 | 0.23 | 10.00 | 132.16 | 0.24 | 10.00 | 123.83 | 0.22 | 10.00 | 146.90 | 0.24 | 10.00 | 130.05 |
0.50 | 8.79 | 32.41 | 0.49 | 10.00 | 38.65 | 0.49 | 10.00 | 37.81 | 0.47 | 10.00 | 41.65 | 0.48 | 10.00 | 39.06 | 0.44 | 10.00 | 46.39 | 0.47 | 10.00 | 40.99 |
0.75 | 9.70 | 18.09 | 0.73 | 10.60 | 20.37 | 0.74 | 10.62 | 19.93 | 0.70 | 10.54 | 21.95 | 0.73 | 10.60 | 20.59 | 0.65 | 10.46 | 24.39 | 0.71 | 10.54 | 21.60 |
1.00 | 10.34 | 11.94 | 0.98 | 10.95 | 13.01 | 0.99 | 10.96 | 12.74 | 0.93 | 10.88 | 13.99 | 0.97 | 10.93 | 13.15 | 0.87 | 10.81 | 15.51 | 0.94 | 10.90 | 13.78 |
1.25 | 10.76 | 8.72 | 1.22 | 11.18 | 9.33 | 1.24 | 11.20 | 9.15 | 1.16 | 11.13 | 9.99 | 1.21 | 11.18 | 9.42 | 1.09 | 11.06 | 11.03 | 1.18 | 11.15 | 9.85 |
1.50 | 11.05 | 6.85 | 1.46 | 11.36 | 7.23 | 1.48 | 11.36 | 7.09 | 1.40 | 11.32 | 7.71 | 1.45 | 11.35 | 7.29 | 1.31 | 11.25 | 8.46 | 1.41 | 11.32 | 7.60 |
1.75 | 11.25 | 5.67 | 1.71 | 11.48 | 5.93 | 1.73 | 11.48 | 5.83 | 1.63 | 11.44 | 6.29 | 1.70 | 11.47 | 5.98 | 1.52 | 11.39 | 6.85 | 1.65 | 11.45 | 6.21 |
2.00 | 11.39 | 4.90 | 1.95 | 11.56 | 5.08 | 1.98 | 11.57 | 5.01 | 1.86 | 11.53 | 5.35 | 1.94 | 11.56 | 5.12 | 1.74 | 11.49 | 5.79 | 1.88 | 11.54 | 5.29 |
wot M.E’s | ||||||||||||||||||||
0.00 | 8.79 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 8.79 | 91.34 | 0.20 | 10.00 | 161.76 | 0.23 | 10.00 | 136.32 | 0.23 | 10.00 | 132.16 | 0.24 | 10.00 | 123.83 | 0.24 | 10.00 | 124.88 | 0.25 | 10.00 | 120.74 |
0.50 | 8.79 | 32.41 | 0.41 | 10.00 | 51.42 | 0.46 | 10.00 | 42.97 | 0.47 | 10.00 | 41.65 | 0.48 | 10.00 | 39.06 | 0.48 | 10.00 | 39.39 | 0.49 | 10.00 | 38.12 |
0.75 | 9.70 | 18.09 | 0.61 | 10.36 | 26.94 | 0.68 | 10.49 | 22.63 | 0.70 | 10.54 | 21.95 | 0.73 | 10.60 | 20.59 | 0.72 | 10.57 | 20.76 | 0.74 | 10.60 | 20.09 |
1.00 | 10.34 | 11.94 | 0.82 | 10.73 | 17.10 | 0.91 | 10.87 | 14.41 | 0.93 | 10.88 | 13.99 | 0.97 | 10.93 | 13.15 | 0.96 | 10.93 | 13.26 | 0.98 | 10.95 | 12.84 |
1.25 | 10.76 | 8.72 | 1.02 | 10.99 | 12.12 | 1.14 | 11.12 | 10.28 | 1.16 | 11.13 | 9.99 | 1.21 | 11.18 | 9.42 | 1.21 | 11.17 | 9.49 | 1.23 | 11.19 | 9.21 |
1.50 | 11.05 | 6.85 | 1.23 | 11.19 | 9.25 | 1.37 | 11.29 | 7.92 | 1.40 | 11.32 | 7.71 | 1.45 | 11.35 | 7.29 | 1.45 | 11.34 | 7.35 | 1.48 | 11.36 | 7.14 |
1.75 | 11.25 | 5.67 | 1.43 | 11.34 | 7.45 | 1.60 | 11.43 | 6.44 | 1.63 | 11.44 | 6.29 | 1.70 | 11.47 | 5.98 | 1.69 | 11.47 | 6.02 | 1.72 | 11.48 | 5.87 |
2.00 | 11.39 | 4.90 | 1.64 | 11.44 | 6.25 | 1.83 | 11.52 | 5.47 | 1.86 | 11.53 | 5.35 | 1.94 | 11.56 | 5.12 | 1.93 | 11.55 | 5.15 | 1.97 | 11.56 | 5.03 |
wot M.E’s | ||||||||||||||||||||
0.00 | 8.79 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 8.79 | 91.34 | 0.11 | 10.00 | 222.14 | 0.17 | 10.00 | 218.16 | 0.23 | 10.00 | 132.16 | 0.24 | 10.00 | 123.83 | 0.25 | 10.00 | 117.89 | 0.25 | 10.00 | 117.77 |
0.50 | 8.79 | 32.41 | 0.23 | 10.00 | 135.93 | 0.33 | 10.00 | 73.16 | 0.47 | 10.00 | 41.65 | 0.48 | 10.00 | 39.06 | 0.50 | 10.00 | 37.26 | 0.50 | 10.00 | 37.22 |
0.75 | 9.70 | 18.09 | 0.34 | 10.00 | 69.08 | 0.50 | 10.00 | 37.49 | 0.70 | 10.54 | 21.95 | 0.73 | 10.60 | 20.59 | 0.75 | 10.62 | 19.64 | 0.75 | 10.62 | 19.62 |
1.00 | 10.34 | 11.94 | 0.46 | 10.00 | 42.84 | 0.66 | 10.46 | 23.81 | 0.93 | 10.88 | 13.99 | 0.97 | 10.93 | 13.15 | 1.00 | 10.96 | 12.56 | 1.00 | 10.96 | 12.55 |
1.25 | 10.76 | 8.72 | 0.57 | 10.25 | 30.13 | 0.83 | 10.73 | 16.75 | 1.16 | 11.13 | 9.99 | 1.21 | 11.18 | 9.42 | 1.25 | 11.21 | 9.02 | 1.25 | 11.21 | 9.01 |
1.50 | 11.05 | 6.85 | 0.69 | 10.49 | 22.57 | 0.99 | 10.96 | 12.64 | 1.40 | 11.32 | 7.71 | 1.45 | 11.35 | 7.29 | 1.50 | 11.38 | 7.01 | 1.50 | 11.38 | 7.00 |
1.75 | 11.25 | 5.67 | 0.80 | 10.71 | 17.70 | 1.16 | 11.13 | 10.03 | 1.63 | 11.44 | 6.29 | 1.70 | 11.47 | 5.98 | 1.75 | 11.49 | 5.76 | 1.75 | 11.49 | 5.76 |
2.00 | 11.39 | 4.90 | 0.91 | 10.87 | 14.37 | 1.33 | 11.26 | 8.28 | 1.86 | 11.53 | 5.35 | 1.94 | 11.56 | 5.12 | 2.00 | 11.57 | 4.96 | 2.00 | 11.57 | 4.95 |
This subsection investigates the effect of the involved parameters Similar to Case #5, the The M.E VCM The subgroup size The powering constant
wot M.E’s | ||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.00 | 8.79 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 8.79 | 91.34 | 0.25 | 10.00 | 120.25 | 0.25 | 10.00 | 119.51 | 0.24 | 10.00 | 125.45 | 0.24 | 10.00 | 123.22 | 0.23 | 10.00 | 133.32 | 0.24 | 10.00 | 128.81 |
0.50 | 8.79 | 32.41 | 0.49 | 10.00 | 37.97 | 0.50 | 10.00 | 37.75 | 0.48 | 10.00 | 39.56 | 0.49 | 10.00 | 38.88 | 0.46 | 10.00 | 42.01 | 0.47 | 10.00 | 40.60 |
0.75 | 9.70 | 18.09 | 0.74 | 10.60 | 20.01 | 0.74 | 10.62 | 19.90 | 0.72 | 10.57 | 20.85 | 0.73 | 10.60 | 20.49 | 0.69 | 10.54 | 22.14 | 0.71 | 10.54 | 21.40 |
1.00 | 10.34 | 11.94 | 0.99 | 10.96 | 12.79 | 0.99 | 10.96 | 12.72 | 0.96 | 10.93 | 13.31 | 0.97 | 10.95 | 13.09 | 0.93 | 10.88 | 14.11 | 0.95 | 10.92 | 13.65 |
1.25 | 10.76 | 8.72 | 1.23 | 11.19 | 9.18 | 1.24 | 11.20 | 9.13 | 1.20 | 11.17 | 9.53 | 1.22 | 11.18 | 9.38 | 1.16 | 11.13 | 10.07 | 1.18 | 11.15 | 9.76 |
1.50 | 11.05 | 6.85 | 1.48 | 11.36 | 7.12 | 1.49 | 11.37 | 7.08 | 1.44 | 11.34 | 7.37 | 1.46 | 11.35 | 7.26 | 1.39 | 11.31 | 7.76 | 1.42 | 11.32 | 7.54 |
1.75 | 11.25 | 5.67 | 1.73 | 11.48 | 5.85 | 1.73 | 11.48 | 5.82 | 1.68 | 11.47 | 6.04 | 1.70 | 11.47 | 5.95 | 1.62 | 11.44 | 6.33 | 1.66 | 11.45 | 6.16 |
2.00 | 11.39 | 4.90 | 1.97 | 11.56 | 5.02 | 1.98 | 11.57 | 5.00 | 1.92 | 11.55 | 5.16 | 1.94 | 11.56 | 5.10 | 1.85 | 11.53 | 5.39 | 1.89 | 11.54 | 5.26 |
wot M.E’s | ||||||||||||||||||||
0.00 | 8.79 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 8.79 | 91.34 | 0.22 | 10.00 | 141.25 | 0.23 | 10.00 | 134.45 | 0.24 | 10.00 | 125.45 | 0.24 | 10.00 | 123.22 | 0.25 | 10.00 | 121.55 | 0.25 | 10.00 | 120.44 |
0.50 | 8.79 | 32.41 | 0.45 | 10.00 | 44.55 | 0.46 | 10.00 | 42.37 | 0.48 | 10.00 | 39.56 | 0.49 | 10.00 | 38.88 | 0.49 | 10.00 | 38.37 | 0.49 | 10.00 | 38.03 |
0.75 | 9.70 | 18.09 | 0.67 | 10.49 | 23.45 | 0.69 | 10.52 | 22.32 | 0.72 | 10.57 | 20.85 | 0.73 | 10.60 | 20.49 | 0.74 | 10.60 | 20.22 | 0.74 | 10.60 | 20.04 |
1.00 | 10.34 | 11.94 | 0.89 | 10.83 | 14.92 | 0.92 | 10.87 | 14.22 | 0.96 | 10.93 | 13.31 | 0.97 | 10.95 | 13.09 | 0.98 | 10.95 | 12.92 | 0.99 | 10.95 | 12.81 |
1.25 | 10.76 | 8.72 | 1.12 | 11.09 | 10.63 | 1.15 | 11.12 | 10.15 | 1.20 | 11.17 | 9.53 | 1.22 | 11.18 | 9.38 | 1.23 | 11.19 | 9.27 | 1.23 | 11.19 | 9.19 |
1.50 | 11.05 | 6.85 | 1.34 | 11.28 | 8.17 | 1.38 | 11.30 | 7.82 | 1.44 | 11.34 | 7.37 | 1.46 | 11.35 | 7.26 | 1.47 | 11.36 | 7.18 | 1.48 | 11.36 | 7.13 |
1.75 | 11.25 | 5.67 | 1.56 | 11.41 | 6.63 | 1.61 | 11.43 | 6.37 | 1.68 | 11.47 | 6.04 | 1.70 | 11.47 | 5.95 | 1.72 | 11.48 | 5.89 | 1.73 | 11.48 | 5.85 |
2.00 | 11.39 | 4.90 | 1.79 | 11.50 | 5.62 | 1.84 | 11.53 | 5.42 | 1.92 | 11.55 | 5.16 | 1.94 | 11.56 | 5.10 | 1.96 | 11.56 | 5.06 | 1.97 | 11.56 | 5.02 |
wot M.E’s | ||||||||||||||||||||
0.00 | 8.79 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 8.79 | 91.34 | 0.15 | 10.00 | 243.05 | 0.17 | 10.00 | 208.37 | 0.24 | 10.00 | 125.45 | 0.24 | 10.00 | 123.22 | 0.25 | 10.00 | 117.79 | 0.25 | 10.00 | 117.76 |
0.50 | 8.79 | 32.41 | 0.30 | 10.00 | 84.40 | 0.34 | 10.00 | 69.05 | 0.48 | 10.00 | 39.56 | 0.49 | 10.00 | 38.88 | 0.50 | 10.00 | 37.23 | 0.50 | 10.00 | 37.22 |
0.75 | 9.70 | 18.09 | 0.46 | 10.00 | 42.85 | 0.51 | 10.06 | 35.54 | 0.72 | 10.57 | 20.85 | 0.73 | 10.60 | 20.49 | 0.75 | 10.62 | 19.62 | 0.75 | 10.62 | 19.61 |
1.00 | 10.34 | 11.94 | 0.61 | 10.36 | 27.20 | 0.69 | 10.49 | 22.55 | 0.96 | 10.93 | 13.31 | 0.97 | 10.95 | 13.09 | 1.00 | 10.96 | 12.55 | 1.00 | 10.96 | 12.55 |
1.25 | 10.76 | 8.72 | 0.76 | 10.65 | 19.11 | 0.86 | 10.79 | 15.88 | 1.20 | 11.17 | 9.53 | 1.22 | 11.18 | 9.38 | 1.25 | 11.21 | 9.02 | 1.25 | 11.21 | 9.01 |
1.50 | 11.05 | 6.85 | 0.91 | 10.87 | 14.38 | 1.03 | 11.01 | 12.00 | 1.44 | 11.34 | 7.37 | 1.46 | 11.35 | 7.26 | 1.50 | 11.38 | 7.00 | 1.50 | 11.38 | 7.00 |
1.75 | 11.25 | 5.67 | 1.07 | 11.05 | 11.37 | 1.20 | 11.17 | 9.54 | 1.68 | 11.47 | 6.04 | 1.70 | 11.47 | 5.95 | 1.75 | 11.49 | 5.76 | 1.75 | 11.49 | 5.76 |
2.00 | 11.39 | 4.90 | 1.22 | 11.18 | 9.33 | 1.37 | 11.30 | 7.89 | 1.92 | 11.55 | 5.16 | 1.94 | 11.56 | 5.10 | 2.00 | 11.57 | 4.95 | 2.00 | 11.57 | 4.95 |
This subsection investigates the effect of the number of variables involved
Cases | |||
---|---|---|---|
Case #1 | |||
Case #2 | |||
Case #3 | |||
Case #4 | |||
Case #5 | |||
Case #6 |
For shifts For Case #1, when we increase the values of For Case #2, when we increase the values of For Case #3, when we increase the values of For Case #4, when we increase the values of For Case #5, when we increase the values of For Case #6, when we increase the values of
Cases | |||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
no. | |||||||||||||||||||
Case #1 | 0.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 0.24 | 8.80 | 96.63 | 0.24 | 8.80 | 95.75 | 0.24 | 14.67 | 166.41 | 0.24 | 14.67 | 166.41 | 0.24 | 23.34 | 214.71 | 0.24 | 23.34 | 214.71 | |
0.50 | 0.48 | 8.80 | 34.31 | 0.48 | 8.80 | 33.99 | 0.48 | 14.67 | 49.22 | 0.48 | 14.67 | 49.22 | 0.48 | 23.34 | 64.34 | 0.48 | 23.34 | 64.34 | |
0.75 | 0.72 | 9.60 | 19.15 | 0.73 | 9.60 | 18.98 | 0.73 | 15.20 | 25.44 | 0.73 | 15.20 | 25.44 | 0.73 | 23.34 | 32.43 | 0.73 | 23.34 | 32.43 | |
1.00 | 0.96 | 10.26 | 12.63 | 0.97 | 10.26 | 12.51 | 0.97 | 15.60 | 15.70 | 0.97 | 15.60 | 15.70 | 0.97 | 24.71 | 20.91 | 0.97 | 24.71 | 20.91 | |
1.25 | 1.20 | 10.69 | 9.21 | 1.21 | 10.71 | 9.13 | 1.21 | 15.80 | 10.79 | 1.21 | 15.80 | 10.79 | 1.21 | 25.21 | 14.29 | 1.21 | 25.21 | 14.29 | |
1.50 | 1.44 | 10.99 | 7.20 | 1.45 | 11.01 | 7.14 | 1.45 | 15.94 | 7.96 | 1.45 | 15.94 | 7.96 | 1.45 | 25.28 | 10.40 | 1.45 | 25.28 | 10.40 | |
1.75 | 1.68 | 11.21 | 5.94 | 1.69 | 11.21 | 5.90 | 1.69 | 16.04 | 6.17 | 1.69 | 16.04 | 6.17 | 1.69 | 25.31 | 7.97 | 1.69 | 25.31 | 7.97 | |
2.00 | 1.92 | 11.36 | 5.11 | 1.94 | 11.36 | 5.07 | 1.94 | 16.12 | 4.96 | 1.94 | 16.12 | 4.96 | 1.94 | 25.32 | 6.34 | 1.94 | 25.32 | 6.34 | |
Case #2 | 0.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 0.24 | 8.80 | 97.38 | 0.24 | 8.80 | 95.33 | 0.23 | 14.67 | 175.17 | 0.24 | 14.67 | 165.16 | 0.24 | 23.34 | 213.26 | 0.28 | 23.34 | 173.50 | |
0.50 | 0.48 | 8.80 | 34.58 | 0.49 | 8.80 | 33.84 | 0.47 | 14.67 | 51.99 | 0.49 | 14.67 | 48.83 | 0.49 | 23.34 | 63.81 | 0.56 | 23.34 | 50.34 | |
0.75 | 0.72 | 9.55 | 19.30 | 0.73 | 9.60 | 18.89 | 0.70 | 15.12 | 26.80 | 0.73 | 15.20 | 25.24 | 0.73 | 23.34 | 32.20 | 0.83 | 23.34 | 26.37 | |
1.00 | 0.96 | 10.23 | 12.73 | 0.97 | 10.28 | 12.46 | 0.94 | 15.57 | 16.57 | 0.97 | 15.60 | 15.58 | 0.97 | 24.77 | 20.75 | 1.11 | 25.14 | 16.56 | |
1.25 | 1.20 | 10.67 | 9.28 | 1.21 | 10.71 | 9.09 | 1.17 | 15.77 | 11.38 | 1.22 | 15.80 | 10.71 | 1.22 | 25.21 | 14.18 | 1.39 | 25.27 | 11.23 | |
1.50 | 1.44 | 10.98 | 7.25 | 1.46 | 11.01 | 7.12 | 1.41 | 15.91 | 8.39 | 1.46 | 15.94 | 7.90 | 1.46 | 25.28 | 10.32 | 1.67 | 25.30 | 8.19 | |
1.75 | 1.67 | 11.20 | 5.98 | 1.70 | 11.22 | 5.87 | 1.64 | 16.02 | 6.50 | 1.70 | 16.04 | 6.12 | 1.70 | 25.31 | 7.90 | 1.94 | 25.32 | 6.29 | |
2.00 | 1.91 | 11.35 | 5.14 | 1.94 | 11.36 | 5.06 | 1.88 | 16.10 | 5.22 | 1.94 | 16.12 | 4.92 | 1.94 | 25.32 | 6.29 | 2.22 | 25.32 | 5.02 | |
Case #3 | 0.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 0.24 | 8.80 | 95.04 | 0.23 | 8.80 | 99.41 | 0.23 | 14.67 | 175.17 | 0.24 | 14.67 | 170.79 | 0.23 | 23.34 | 234.63 | 0.23 | 23.34 | 228.72 | |
0.50 | 0.48 | 8.80 | 33.85 | 0.47 | 8.80 | 35.45 | 0.47 | 14.67 | 51.99 | 0.48 | 14.67 | 50.60 | 0.45 | 23.34 | 72.11 | 0.46 | 23.34 | 69.73 | |
0.75 | 0.73 | 9.58 | 18.93 | 0.70 | 9.47 | 19.82 | 0.70 | 15.12 | 26.80 | 0.71 | 15.16 | 26.11 | 0.68 | 23.34 | 35.78 | 0.69 | 23.34 | 34.75 | |
1.00 | 0.97 | 10.24 | 12.49 | 0.94 | 10.18 | 13.08 | 0.94 | 15.57 | 16.57 | 0.95 | 15.57 | 16.13 | 0.91 | 24.34 | 23.12 | 0.93 | 24.51 | 22.46 | |
1.25 | 1.21 | 10.69 | 9.12 | 1.17 | 10.62 | 9.53 | 1.17 | 15.77 | 11.38 | 1.19 | 15.79 | 11.09 | 1.14 | 25.14 | 15.93 | 1.16 | 25.16 | 15.43 | |
1.50 | 1.45 | 11.00 | 7.14 | 1.41 | 10.95 | 7.44 | 1.41 | 15.91 | 8.39 | 1.43 | 15.93 | 8.18 | 1.36 | 25.26 | 11.60 | 1.39 | 25.27 | 11.23 | |
1.75 | 1.69 | 11.21 | 5.89 | 1.64 | 11.17 | 6.12 | 1.64 | 16.02 | 6.50 | 1.67 | 16.03 | 6.34 | 1.59 | 25.30 | 8.87 | 1.62 | 25.30 | 8.60 | |
2.00 | 1.94 | 11.36 | 5.07 | 1.88 | 11.32 | 5.25 | 1.71 | 16.05 | 6.05 | 1.91 | 16.11 | 5.05 | 1.82 | 25.31 | 7.05 | 1.85 | 25.32 | 6.84 | |
Case #4 | 0.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 0.24 | 8.80 | 94.83 | 0.23 | 8.80 | 100.39 | 0.23 | 14.67 | 180.54 | 0.24 | 14.67 | 164.39 | 0.21 | 23.34 | 256.62 | 0.22 | 23.34 | 240.51 | |
0.50 | 0.49 | 8.80 | 33.77 | 0.47 | 8.80 | 35.81 | 0.46 | 14.67 | 53.73 | 0.49 | 14.67 | 48.60 | 0.43 | 23.34 | 81.58 | 0.45 | 23.34 | 74.54 | |
0.75 | 0.73 | 9.58 | 18.89 | 0.70 | 9.47 | 20.02 | 0.69 | 15.07 | 27.64 | 0.73 | 15.20 | 25.12 | 0.64 | 23.34 | 39.88 | 0.67 | 23.34 | 36.83 | |
1.00 | 0.97 | 10.24 | 12.46 | 0.93 | 10.15 | 13.21 | 0.92 | 15.54 | 17.10 | 0.98 | 15.60 | 15.51 | 0.85 | 23.53 | 25.62 | 0.89 | 24.24 | 23.78 | |
1.25 | 1.21 | 10.69 | 9.10 | 1.16 | 10.62 | 9.62 | 1.15 | 15.75 | 11.75 | 1.22 | 15.80 | 10.66 | 1.06 | 25.05 | 17.88 | 1.12 | 25.14 | 16.44 | |
1.50 | 1.45 | 11.00 | 7.12 | 1.40 | 10.94 | 7.51 | 1.38 | 15.90 | 8.66 | 1.46 | 15.94 | 7.86 | 1.28 | 25.24 | 13.04 | 1.34 | 25.26 | 11.97 | |
1.75 | 1.70 | 11.21 | 5.88 | 1.63 | 11.16 | 6.17 | 1.61 | 16.01 | 6.71 | 1.71 | 16.05 | 6.10 | 1.49 | 25.28 | 9.97 | 1.56 | 25.29 | 9.16 | |
2.00 | 1.94 | 11.36 | 5.06 | 1.86 | 11.32 | 5.29 | 1.84 | 16.09 | 5.39 | 1.95 | 16.12 | 4.90 | 1.70 | 25.31 | 7.91 | 1.79 | 25.31 | 7.28 | |
Case #5 | 0.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 0.23 | 10.00 | 132.16 | 0.24 | 10.00 | 123.83 | 0.21 | 14.67 | 201.55 | 0.24 | 14.67 | 169.35 | 0.24 | 23.34 | 223.68 | 0.26 | 23.34 | 189.77 | |
0.50 | 0.47 | 10.00 | 41.65 | 0.48 | 10.00 | 39.06 | 0.43 | 14.67 | 60.87 | 0.48 | 14.67 | 50.14 | 0.47 | 23.34 | 67.75 | 0.53 | 23.34 | 55.57 | |
0.75 | 0.70 | 10.54 | 21.95 | 0.73 | 10.60 | 20.59 | 0.64 | 14.76 | 30.98 | 0.72 | 15.16 | 25.89 | 0.71 | 23.34 | 33.90 | 0.79 | 23.34 | 28.65 | |
1.00 | 0.93 | 10.88 | 13.99 | 0.97 | 10.93 | 13.15 | 0.86 | 15.46 | 19.28 | 0.96 | 15.57 | 15.99 | 0.94 | 24.59 | 21.90 | 1.05 | 25.03 | 18.24 | |
1.25 | 1.16 | 11.13 | 9.99 | 1.21 | 11.18 | 9.42 | 1.07 | 15.69 | 13.24 | 1.20 | 15.79 | 10.99 | 1.18 | 25.19 | 15.02 | 1.31 | 25.25 | 12.39 | |
1.50 | 1.40 | 11.32 | 7.71 | 1.45 | 11.35 | 7.29 | 1.29 | 15.85 | 9.75 | 1.44 | 15.94 | 8.11 | 1.41 | 25.27 | 10.93 | 1.58 | 25.29 | 9.03 | |
1.75 | 1.63 | 11.44 | 6.29 | 1.70 | 11.47 | 5.98 | 1.50 | 15.97 | 7.55 | 1.68 | 16.04 | 6.28 | 1.65 | 25.30 | 8.37 | 1.84 | 25.32 | 6.93 | |
2.00 | 1.86 | 11.53 | 5.35 | 1.94 | 11.56 | 5.12 | 1.71 | 16.05 | 6.05 | 1.91 | 16.11 | 5.05 | 1.88 | 25.32 | 6.65 | 2.10 | 25.32 | 5.52 | |
Case #6 | 0.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 | 0.00 | 8.80 | 370.00 |
0.25 | 0.24 | 10.00 | 125.45 | 0.24 | 10.00 | 123.22 | 0.24 | 14.67 | 168.16 | 0.24 | 14.67 | 163.50 | 0.24 | 23.34 | 216.73 | 0.25 | 23.34 | 208.91 | |
0.50 | 0.48 | 10.00 | 39.56 | 0.49 | 10.00 | 38.88 | 0.48 | 14.67 | 49.77 | 0.49 | 14.67 | 48.32 | 0.48 | 23.34 | 65.10 | 0.49 | 23.34 | 62.22 | |
0.75 | 0.72 | 10.57 | 20.85 | 0.73 | 10.60 | 20.49 | 0.72 | 15.16 | 25.71 | 0.73 | 15.24 | 24.99 | 0.72 | 23.34 | 32.76 | 0.74 | 23.34 | 31.52 | |
1.00 | 0.96 | 10.93 | 13.31 | 0.97 | 10.95 | 13.09 | 0.96 | 15.57 | 15.87 | 0.98 | 15.60 | 15.42 | 0.96 | 24.71 | 21.13 | 0.99 | 24.82 | 20.28 | |
1.25 | 1.20 | 11.17 | 9.53 | 1.22 | 11.18 | 9.38 | 1.20 | 15.79 | 10.91 | 1.22 | 15.80 | 10.60 | 1.20 | 25.20 | 14.45 | 1.23 | 25.22 | 13.84 | |
1.50 | 1.44 | 11.34 | 7.37 | 1.46 | 11.35 | 7.26 | 1.44 | 15.94 | 8.05 | 1.47 | 15.95 | 7.82 | 1.44 | 25.28 | 10.52 | 1.48 | 25.28 | 10.07 | |
1.75 | 1.68 | 11.47 | 6.04 | 1.70 | 11.47 | 5.95 | 1.68 | 16.04 | 6.24 | 1.71 | 16.05 | 6.06 | 1.68 | 25.30 | 8.06 | 1.73 | 25.31 | 7.72 | |
2.00 | 1.92 | 11.55 | 5.16 | 1.94 | 11.56 | 5.10 | 1.92 | 16.11 | 5.01 | 1.96 | 16.12 | 4.87 | 1.92 | 25.32 | 6.41 | 1.97 | 25.32 | 6.14 |
For the proposal implementation, two illustrated examples have been incorporated in this study: one involving manufacturing uncoated aspirin tablets with the composition of aspirin, micro-crystalline cellulose, talcum powder and magnesium stearate and the other is the monitoring of machines that are responsible for the percentages of three components (whole-grain cereal, dried fruits, and nuts) in muesli production.
Pharmaceutical manufacturing is the process of synthesizing pharmaceutical medications on a large scale as part of the pharmaceutical business. The medication production process may be divided into several unit activities, such as milling, granulation, coating, tablet pressing, etc. Here the authors simulated an example of a company producing uncoated aspirin tablets. The base parameters have been taken from [
1 | 1 | 0.33 | 0.33 | 0.33 | 0 | 0 | 0.29 | 0.36 | 0.35 | −0.08 | −0.15 |
2 | 0.37 | 0.32 | 0.32 | 0.06 | 0.11 | ||||||
3 | 0.32 | 0.33 | 0.35 | −0.06 | −0.04 | ||||||
4 | 0.34 | 0.33 | 0.33 | 0.01 | 0.01 | ||||||
5 | 0.37 | 0.30 | 0.34 | −0.02 | 0.15 | ||||||
6 | 0.37 | 0.30 | 0.33 | 0.02 | 0.14 | ||||||
7 | 0.34 | 0.32 | 0.34 | −0.01 | 0.04 | ||||||
8 | 0.38 | 0.28 | 0.34 | −0.03 | 0.23 | ||||||
9 | 0.32 | 0.34 | 0.34 | −0.01 | −0.05 | ||||||
10 | 0.34 | 0.32 | 0.33 | 0.00 | 0.04 | ||||||
2 | 1 | 0.25 | 0.25 | 0.50 | −0.57 | 0.00 | 0.24 | 0.25 | 0.51 | −0.61 | −0.05 |
2 | 0.26 | 0.24 | 0.50 | −0.58 | 0.05 | ||||||
3 | 0.23 | 0.26 | 0.52 | −0.62 | −0.08 | ||||||
4 | 0.24 | 0.25 | 0.51 | −0.61 | −0.03 | ||||||
5 | 0.26 | 0.25 | 0.50 | −0.56 | 0.03 | ||||||
6 | 0.25 | 0.25 | 0.50 | −0.57 | 0.01 | ||||||
7 | 0.23 | 0.26 | 0.51 | −0.61 | −0.09 | ||||||
8 | 0.24 | 0.23 | 0.53 | −0.66 | 0.03 | ||||||
9 | 0.26 | 0.23 | 0.52 | −0.62 | 0.08 | ||||||
10 | 0.24 | 0.26 | 0.50 | −0.56 | −0.04 | ||||||
3 | 1 | 0.25 | 0.50 | 0.25 | 0.28 | −0.49 | 0.26 | 0.50 | 0.24 | 0.33 | −0.46 |
2 | 0.27 | 0.49 | 0.24 | 0.33 | −0.41 | ||||||
3 | 0.26 | 0.50 | 0.24 | 0.33 | −0.47 | ||||||
4 | 0.21 | 0.56 | 0.23 | 0.34 | −0.68 | ||||||
5 | 0.20 | 0.56 | 0.24 | 0.26 | −0.75 | ||||||
6 | 0.26 | 0.48 | 0.26 | 0.26 | −0.44 | ||||||
7 | 0.26 | 0.50 | 0.24 | 0.33 | −0.47 | ||||||
8 | 0.27 | 0.50 | 0.24 | 0.36 | −0.43 | ||||||
9 | 0.25 | 0.51 | 0.24 | 0.30 | −0.52 | ||||||
10 | 0.28 | 0.47 | 0.25 | 0.30 | −0.37 | ||||||
4 | 1 | 0.50 | 0.25 | 0.25 | 0.28 | 0.49 | 0.56 | 0.21 | 0.23 | 0.32 | 0.69 |
2 | 0.52 | 0.23 | 0.25 | 0.25 | 0.58 | ||||||
3 | 0.53 | 0.23 | 0.23 | 0.34 | 0.58 | ||||||
4 | 0.49 | 0.26 | 0.25 | 0.29 | 0.44 | ||||||
5 | 0.52 | 0.23 | 0.25 | 0.25 | 0.57 | ||||||
6 | 0.50 | 0.26 | 0.24 | 0.32 | 0.46 | ||||||
7 | 0.49 | 0.26 | 0.25 | 0.29 | 0.44 | ||||||
8 | 0.51 | 0.23 | 0.25 | 0.26 | 0.55 | ||||||
9 | 0.54 | 0.21 | 0.25 | 0.26 | 0.67 | ||||||
10 | 0.55 | 0.21 | 0.23 | 0.31 | 0.68 |
The estimated values of
1 | 1 | 0.72 | 0.21 | 0.07 | 0.711 | 0.216 | 0.073 | 1.369 | 0.841 | 1.486 |
2 | 0.73 | 0.2 | 0.07 | |||||||
3 | 0.68 | 0.24 | 0.08 | |||||||
2 | 1 | 0.69 | 0.24 | 0.07 | 0.674 | 0.253 | 0.073 | 1.413 | 0.692 | 1.246 |
2 | 0.66 | 0.26 | 0.08 | |||||||
3 | 0.67 | 0.26 | 0.07 | |||||||
3 | 1 | 0.68 | 0.24 | 0.08 | 0.687 | 0.237 | 0.077 | 1.357 | 0.753 | 0.518 |
2 | 0.68 | 0.24 | 0.08 | |||||||
3 | 0.7 | 0.23 | 0.07 | |||||||
4 | 1 | 0.68 | 0.25 | 0.07 | 0.677 | 0.250 | 0.073 | 1.409 | 0.705 | 1.547 |
2 | 0.7 | 0.23 | 0.07 | |||||||
3 | 0.65 | 0.27 | 0.08 | |||||||
5 | 1 | 0.75 | 0.18 | 0.07 | 0.708 | 0.216 | 0.076 | 1.333 | 0.841 | 2.338 |
2 | 0.71 | 0.22 | 0.07 | |||||||
3 | 0.66 | 0.25 | 0.09 | |||||||
6 | 1 | 0.7 | 0.23 | 0.07 | 0.714 | 0.216 | 0.070 | 1.408 | 0.844 | 2.405 |
2 | 0.73 | 0.2 | 0.07 | |||||||
3 | 0.71 | 0.22 | 0.07 | |||||||
7 | 1 | 0.71 | 0.22 | 0.07 | 0.711 | 0.219 | 0.070 | 1.411 | 0.833 | 3.244 |
2 | 0.74 | 0.19 | 0.07 | |||||||
3 | 0.68 | 0.25 | 0.07 | |||||||
8 | 1 | 0.66 | 0.26 | 0.08 | 0.660 | 0.260 | 0.080 | 1.342 | 0.660 | 2.216 |
2 | 0.64 | 0.28 | 0.08 | |||||||
3 | 0.68 | 0.24 | 0.08 | |||||||
9 | 1 | 0.52 | 0.39 | 0.09 | 0.523 | 0.387 | 0.090 | 1.314 | 0.614 | 3.596 |
2 | 0.53 | 0.38 | 0.09 | |||||||
3 | 0.52 | 0.39 | 0.09 | |||||||
10 | 1 | 0.69 | 0.24 | 0.07 | 0.687 | 0.240 | 0.073 | 1.399 | 0.743 | 2.652 |
2 | 0.69 | 0.24 | 0.07 | |||||||
3 | 0.68 | 0.24 | 0.08 | |||||||
11 | 1 | 0.65 | 0.27 | 0.08 | 0.660 | 0.263 | 0.077 | 1.384 | 0.650 | 3.163 |
2 | 0.68 | 0.25 | 0.07 | |||||||
3 | 0.65 | 0.27 | 0.08 | |||||||
12 | 1 | 0.64 | 0.28 | 0.08 | 0.640 | 0.280 | 0.080 | 1.360 | 0.585 | 4.174 |
2 | 0.64 | 0.28 | 0.08 | |||||||
3 | 0.64 | 0.28 | 0.08 | |||||||
13 | 1 | 0.72 | 0.21 | 0.07 | 0.710 | 0.220 | 0.070 | 1.412 | 0.830 | 3.218 |
2 | 0.72 | 0.21 | 0.07 | |||||||
3 | 0.69 | 0.24 | 0.07 | |||||||
14 | 1 | 0.64 | 0.28 | 0.08 | 0.674 | 0.253 | 0.073 | 1.411 | 0.694 | 4.834 |
2 | 0.67 | 0.26 | 0.07 | |||||||
3 | 0.71 | 0.22 | 0.07 | |||||||
15 | 1 | 0.64 | 0.28 | 0.08 | 0.660 | 0.260 | 0.080 | 1.342 | 0.660 | 3.411 |
2 | 0.66 | 0.26 | 0.08 | |||||||
3 | 0.68 | 0.24 | 0.08 | |||||||
16 | 1 | 0.74 | 0.19 | 0.07 | 0.724 | 0.206 | 0.070 | 1.394 | 0.888 | 1.566 |
2 | 0.7 | 0.23 | 0.07 | |||||||
3 | 0.73 | 0.2 | 0.07 | |||||||
17 | 1 | 0.64 | 0.28 | 0.08 | 0.664 | 0.256 | 0.080 | 1.339 | 0.673 | 0.842 |
2 | 0.67 | 0.25 | 0.08 | |||||||
3 | 0.68 | 0.24 | 0.08 | |||||||
18 | 1 | 0.72 | 0.21 | 0.07 | 0.730 | 0.200 | 0.070 | 1.385 | 0.916 | 1.573 |
2 | 0.73 | 0.2 | 0.07 | |||||||
3 | 0.74 | 0.19 | 0.07 | |||||||
19 | 1 | 0.65 | 0.27 | 0.08 | 0.650 | 0.270 | 0.080 | 1.352 | 0.621 | 1.397 |
2 | 0.65 | 0.27 | 0.08 | |||||||
3 | 0.65 | 0.27 | 0.08 | |||||||
20 | 1 | 0.7 | 0.23 | 0.07 | 0.688 | 0.239 | 0.073 | 1.395 | 0.748 | 0.004 |
2 | 0.64 | 0.28 | 0.08 | |||||||
3 | 0.72 | 0.21 | 0.07 |
Using the above values, we can obtain
The results of the MCUSUM-CoDa charts are also presented in
1 | 1 | 0.65 | 0.27 | 0.08 | 0.6535 | 0.2665 | 0.08 | 1.3483 | 0.6343 | 0.5577 |
2 | 0.64 | 0.28 | 0.08 | |||||||
3 | 0.67 | 0.25 | 0.08 | |||||||
2 | 1 | 0.69 | 0.24 | 0.07 | 0.6834 | 0.2434 | 0.0732 | 1.4024 | 0.7302 | 1.9677 |
2 | 0.69 | 0.24 | 0.07 | |||||||
3 | 0.67 | 0.25 | 0.08 | |||||||
3 | 1 | 0.69 | 0.24 | 0.07 | 0.6822 | 0.2462 | 0.0716 | 1.4243 | 0.7206 | 6.8861 |
2 | 0.68 | 0.25 | 0.07 | |||||||
3 | 0.68 | 0.25 | 0.07 | |||||||
4 | 1 | 0.64 | 0.28 | 0.08 | 0.6580 | 0.2723 | 0.0697 | 1.4723 | 0.624 | |
2 | 0.69 | 0.24 | 0.07 | |||||||
3 | 0.65 | 0.29 | 0.07 | |||||||
5 | 1 | 0.64 | 0.28 | 0.08 | 0.6605 | 0.2562 | 0.0833 | 1.3038 | 0.6698 | 7.6805 |
2 | 0.69 | 0.23 | 0.08 | |||||||
3 | 0.65 | 0.26 | 0.09 | |||||||
6 | 1 | 0.64 | 0.28 | 0.08 | 0.6502 | 0.2698 | 0.08 | 1.3512 | 0.6221 | 7.5254 |
2 | 0.67 | 0.25 | 0.08 | |||||||
3 | 0.64 | 0.28 | 0.08 | |||||||
7 | 1 | 0.65 | 0.27 | 0.08 | 0.657 | 0.263 | 0.0801 | 1.3448 | 0.6474 | 6.8381 |
2 | 0.68 | 0.24 | 0.08 | |||||||
3 | 0.64 | 0.28 | 0.08 | |||||||
8 | 1 | 0.73 | 0.2 | 0.07 | 0.7139 | 0.2128 | 0.0733 | 1.3646 | 0.8558 | 4.8713 |
2 | 0.73 | 0.2 | 0.07 | |||||||
3 | 0.68 | 0.24 | 0.08 | |||||||
9 | 1 | 0.66 | 0.26 | 0.08 | 0.6668 | 0.2567 | 0.0765 | 1.3777 | 0.675 | 5.8349 |
2 | 0.68 | 0.25 | 0.07 | |||||||
3 | 0.66 | 0.26 | 0.08 | |||||||
10 | 1 | 0.65 | 0.27 | 0.08 | 0.625 | 0.295 | 0.080 | 1.368 | 0.531 | 7.907 |
2 | 0.56 | 0.36 | 0.08 | |||||||
3 | 0.66 | 0.26 | 0.08 | |||||||
11 | 1 | 0.68 | 0.24 | 0.08 | 0.664 | 0.257 | 0.080 | 1.339 | 0.672 | 6.744 |
2 | 0.66 | 0.26 | 0.08 | |||||||
3 | 0.65 | 0.27 | 0.08 | |||||||
12 | 1 | 0.66 | 0.26 | 0.08 | 0.653 | 0.263 | 0.083 | 1.312 | 0.643 | 5.217 |
2 | 0.66 | 0.26 | 0.08 | |||||||
3 | 0.64 | 0.27 | 0.09 | |||||||
13 | 1 | 0.68 | 0.24 | 0.08 | 0.687 | 0.237 | 0.077 | 1.357 | 0.753 | 4.127 |
2 | 0.7 | 0.23 | 0.07 | |||||||
3 | 0.68 | 0.24 | 0.08 | |||||||
14 | 1 | 0.63 | 0.28 | 0.09 | 0.643 | 0.273 | 0.083 | 1.321 | 0.605 | 4.753 |
2 | 0.65 | 0.27 | 0.08 | |||||||
3 | 0.65 | 0.27 | 0.08 | |||||||
15 | 1 | 0.65 | 0.27 | 0.08 | 0.653 | 0.267 | 0.080 | 1.349 | 0.634 | 5.299 |
2 | 0.66 | 0.26 | 0.08 | |||||||
3 | 0.65 | 0.27 | 0.08 | |||||||
16 | 1 | 0.68 | 0.24 | 0.08 | 0.677 | 0.250 | 0.073 | 1.409 | 0.704 | 5.545 |
2 | 0.68 | 0.25 | 0.07 | |||||||
3 | 0.67 | 0.26 | 0.07 | |||||||
17 | 1 | 0.71 | 0.22 | 0.07 | 0.730 | 0.200 | 0.070 | 1.385 | 0.917 | 3.106 |
2 | 0.74 | 0.19 | 0.07 | |||||||
3 | 0.74 | 0.19 | 0.07 | |||||||
18 | 1 | 0.72 | 0.21 | 0.07 | 0.697 | 0.229 | 0.073 | 1.385 | 0.786 | 2.949 |
2 | 0.71 | 0.22 | 0.07 | |||||||
3 | 0.66 | 0.26 | 0.08 | |||||||
19 | 1 | 0.71 | 0.22 | 0.07 | 0.710 | 0.220 | 0.070 | 1.413 | 0.829 | 4.684 |
2 | 0.71 | 0.22 | 0.07 | |||||||
3 | 0.71 | 0.22 | 0.07 | |||||||
20 | 1 | 0.68 | 0.24 | 0.08 | 0.667 | 0.253 | 0.080 | 1.336 | 0.685 | 2.709 |
2 | 0.65 | 0.27 | 0.08 | |||||||
3 | 0.67 | 0.25 | 0.08 |
For the second example, the authors are using the same example used in [
1 | 1 | 0.33 | 0.33 | 0.33 | 0.34 | 0.33 | 0.33 | 0.0000 | 0.0000 | 0.0122 | 0.0211 |
2 | 0.32 | 0.35 | 0.33 | 0.0115 | −0.0634 | ||||||
3 | 0.33 | 0.35 | 0.32 | 0.0491 | −0.0416 | ||||||
4 | 0.31 | 0.35 | 0.34 | −0.0259 | −0.0858 | ||||||
5 | 0.36 | 0.34 | 0.30 | 0.1255 | 0.0404 | ||||||
6 | 0.34 | 0.32 | 0.34 | −0.0247 | 0.0429 | ||||||
7 | 0.31 | 0.36 | 0.33 | 0.0100 | −0.1057 | ||||||
2 | 1 | 0.60 | 0.20 | 0.20 | 0.62 | 0.19 | 0.19 | 0.4485 | 0.7768 | 0.4828 | 0.8363 |
2 | 0.65 | 0.17 | 0.18 | 0.5009 | 0.9484 | ||||||
3 | 0.59 | 0.22 | 0.19 | 0.5224 | 0.6976 | ||||||
4 | 0.61 | 0.20 | 0.19 | 0.4971 | 0.7885 | ||||||
5 | 0.67 | 0.15 | 0.18 | 0.4621 | 1.0583 | ||||||
6 | 0.65 | 0.17 | 0.18 | 0.5009 | 0.9484 | ||||||
7 | 0.61 | 0.19 | 0.20 | 0.4343 | 0.8248 | ||||||
3 | 1 | 0.20 | 0.60 | 0.20 | 0.17 | 0.65 | 0.18 | 0.4485 | −0.7768 | 0.5009 | −0.9484 |
2 | 0.14 | 0.69 | 0.17 | 0.4926 | −1.1279 | ||||||
3 | 0.21 | 0.60 | 0.19 | 0.5103 | −0.7423 | ||||||
4 | 0.19 | 0.62 | 0.19 | 0.4828 | −0.8363 | ||||||
5 | 0.20 | 0.62 | 0.18 | 0.5479 | −0.8000 | ||||||
6 | 0.16 | 0.66 | 0.18 | 0.4823 | −1.0020 | ||||||
7 | 0.20 | 0.62 | 0.18 | 0.5479 | −0.8000 | ||||||
4 | 1 | 0.20 | 0.20 | 0.60 | 0.18 | 0.20 | 0.62 | −0.8970 | 0.0000 | −0.9668 | −0.0745 |
2 | 0.19 | 0.19 | 0.62 | −0.9657 | 0.0000 | ||||||
3 | 0.21 | 0.19 | 0.60 | −0.8980 | 0.0708 | ||||||
4 | 0.21 | 0.18 | 0.61 | −0.9336 | 0.1090 | ||||||
5 | 0.20 | 0.18 | 0.62 | −0.9668 | 0.0745 | ||||||
6 | 0.19 | 0.19 | 0.62 | −0.9657 | 0.0000 | ||||||
7 | 0.21 | 0.17 | 0.62 | −0.9702 | 0.1494 |
The following estimates are obtained using
To estimate the
1 | 1 | 0.77 | 0.16 | 0.07 | 0.709 | 0.2078 | 0.0832 | 1.2483 | 0.8678 | 0.5965 |
2 | 0.67 | 0.24 | 0.09 | |||||||
3 | 0.68 | 0.23 | 0.09 | |||||||
2 | 1 | 0.64 | 0.27 | 0.09 | 0.6301 | 0.2767 | 0.0932 | 1.2241 | 0.5819 | 0.6735 |
2 | 0.63 | 0.28 | 0.09 | |||||||
3 | 0.62 | 0.28 | 0.1 | |||||||
3 | 1 | 0.76 | 0.16 | 0.08 | 0.6958 | 0.2108 | 0.0934 | 1.1523 | 0.8445 | 2.0531 |
2 | 0.65 | 0.25 | 0.1 | |||||||
3 | 0.67 | 0.23 | 0.1 | |||||||
4 | 1 | 0.65 | 0.27 | 0.08 | 0.6434 | 0.2766 | 0.08 | 1.3575 | 0.5968 | 0.7871 |
2 | 0.64 | 0.28 | 0.08 | |||||||
3 | 0.64 | 0.28 | 0.08 | |||||||
5 | 1 | 0.5 | 0.38 | 0.12 | 0.5268 | 0.3634 | 0.1098 | 1.1289 | 0.2625 | 4.2017 |
2 | 0.54 | 0.36 | 0.1 | |||||||
3 | 0.54 | 0.35 | 0.11 | |||||||
6 | 1 | 0.8 | 0.14 | 0.06 | 0.7677 | 0.1657 | 0.0666 | 1.3699 | 1.0843 | 1.2242 |
2 | 0.74 | 0.19 | 0.07 | |||||||
3 | 0.76 | 0.17 | 0.07 | |||||||
7 | 1 | 0.75 | 0.18 | 0.07 | 0.7304 | 0.1995 | 0.0701 | 1.3843 | 0.9177 | 0.4566 |
2 | 0.71 | 0.22 | 0.07 | |||||||
3 | 0.73 | 0.2 | 0.07 | |||||||
8 | 1 | 0.65 | 0.26 | 0.09 | 0.6534 | 0.2533 | 0.0932 | 1.203 | 0.67 | 0.3416 |
2 | 0.65 | 0.25 | 0.1 | |||||||
3 | 0.66 | 0.25 | 0.09 | |||||||
9 | 1 | 0.65 | 0.27 | 0.08 | 0.6635 | 0.2565 | 0.08 | 1.3389 | 0.6721 | 1.0116 |
2 | 0.66 | 0.26 | 0.08 | |||||||
3 | 0.68 | 0.24 | 0.08 | |||||||
10 | 1 | 0.76 | 0.17 | 0.07 | 0.7534 | 0.1766 | 0.07 | 1.3478 | 1.0257 | 1.1556 |
2 | 0.75 | 0.18 | 0.07 | |||||||
3 | 0.75 | 0.18 | 0.07 | |||||||
11 | 1 | 0.67 | 0.24 | 0.09 | 0.6601 | 0.2499 | 0.09 | 1.2302 | 0.6868 | 0 |
2 | 0.65 | 0.26 | 0.09 | |||||||
3 | 0.66 | 0.25 | 0.09 | |||||||
12 | 1 | 0.53 | 0.36 | 0.11 | 0.5367 | 0.3533 | 0.11 | 1.1234 | 0.2956 | 3.2053 |
2 | 0.54 | 0.35 | 0.11 | |||||||
3 | 0.54 | 0.35 | 0.11 | |||||||
13 | 1 | 0.75 | 0.16 | 0.09 | 0.7251 | 0.1813 | 0.0936 | 1.1058 | 0.9803 | 3.8491 |
2 | 0.67 | 0.23 | 0.1 | |||||||
3 | 0.75 | 0.16 | 0.09 | |||||||
14 | 1 | 0.67 | 0.24 | 0.09 | 0.6701 | 0.2333 | 0.0966 | 1.151 | 0.746 | 5.1974 |
2 | 0.67 | 0.23 | 0.1 | |||||||
3 | 0.67 | 0.23 | 0.1 | |||||||
15 | 1 | 0.64 | 0.27 | 0.09 | 0.6468 | 0.2632 | 0.09 | 1.2431 | 0.6357 | 5.1296 |
2 | 0.64 | 0.27 | 0.09 | |||||||
3 | 0.66 | 0.25 | 0.09 | |||||||
16 | 1 | 0.72 | 0.21 | 0.07 | 0.744 | 0.1928 | 0.0632 | 1.4614 | 0.9548 | 2.0026 |
2 | 0.74 | 0.2 | 0.06 | |||||||
3 | 0.77 | 0.17 | 0.06 | |||||||
17 | 1 | 0.73 | 0.2 | 0.07 | 0.7436 | 0.1899 | 0.0665 | 1.4137 | 0.9652 | 0.59 |
2 | 0.75 | 0.18 | 0.07 | |||||||
3 | 0.75 | 0.19 | 0.06 | |||||||
18 | 1 | 0.65 | 0.28 | 0.07 | 0.6568 | 0.2767 | 0.0665 | 1.5169 | 0.6112 | 3.5386 |
2 | 0.65 | 0.28 | 0.07 | |||||||
3 | 0.67 | 0.27 | 0.06 | |||||||
19 | 1 | 0.76 | 0.17 | 0.07 | 0.7604 | 0.173 | 0.0665 | 1.3846 | 1.0468 | 4.0517 |
2 | 0.74 | 0.19 | 0.07 | |||||||
3 | 0.78 | 0.16 | 0.06 | |||||||
20 | 1 | 0.67 | 0.23 | 0.1 | 0.7013 | 0.2055 | 0.0931 | 1.1475 | 0.8678 | 2.1401 |
2 | 0.68 | 0.22 | 0.1 | |||||||
3 | 0.75 | 0.17 | 0.08 |
Then, using
The authors have computed the MCUSUM-CoDa statistics
Concerning Phase-II,
1 | 1 | 0.68 | 0.23 | 0.09 | 0.6504 | 0.2563 | 0.0933 | 1.205 | 0.6586 | 0.6242 |
2 | 0.63 | 0.27 | 0.1 | |||||||
3 | 0.64 | 0.27 | 0.09 | |||||||
2 | 1 | 0.68 | 0.24 | 0.08 | 0.673 | 0.243 | 0.083 | 1.292 | 0.720 | 0.3082 |
2 | 0.66 | 0.25 | 0.09 | |||||||
3 | 0.68 | 0.24 | 0.08 | |||||||
3 | 1 | 0.62 | 0.31 | 0.07 | 0.654 | 0.276 | 0.070 | 1.471 | 0.610 | 3.3136 |
2 | 0.67 | 0.26 | 0.07 | |||||||
3 | 0.67 | 0.26 | 0.07 | |||||||
4 | 1 | 0.67 | 0.22 | 0.11 | 0.663 | 0.227 | 0.110 | 1.028 | 0.760 | 1.0659 |
2 | 0.65 | 0.24 | 0.11 | |||||||
3 | 0.67 | 0.22 | 0.11 | |||||||
5 | 1 | 0.63 | 0.27 | 0.1 | 0.650 | 0.257 | 0.093 | 1.206 | 0.657 | 1.6727 |
2 | 0.66 | 0.25 | 0.09 | |||||||
3 | 0.66 | 0.25 | 0.09 | |||||||
6 | 1 | 0.77 | 0.15 | 0.08 | 0.770 | 0.150 | 0.080 | 1.180 | 1.158 | 3.5976 |
2 | 0.76 | 0.16 | 0.08 | |||||||
3 | 0.78 | 0.14 | 0.08 | |||||||
7 | 1 | 0.67 | 0.24 | 0.09 | 0.657 | 0.250 | 0.093 | 1.199 | 0.684 | 3.8127 |
2 | 0.66 | 0.24 | 0.1 | |||||||
3 | 0.64 | 0.27 | 0.09 | |||||||
8 | 1 | 0.64 | 0.28 | 0.08 | 0.678 | 0.246 | 0.077 | 1.364 | 0.718 | 1.7546 |
2 | 0.72 | 0.21 | 0.07 | |||||||
3 | 0.67 | 0.25 | 0.08 | |||||||
9 | 1 | 0.72 | 0.21 | 0.07 | 0.700 | 0.227 | 0.073 | 1.383 | 0.798 | 0.0666 |
2 | 0.68 | 0.24 | 0.08 | |||||||
3 | 0.7 | 0.23 | 0.07 | |||||||
10 | 1 | 0.64 | 0.26 | 0.1 | 0.650 | 0.257 | 0.093 | 1.206 | 0.657 | 0.586 |
2 | 0.66 | 0.25 | 0.09 | |||||||
3 | 0.65 | 0.26 | 0.09 | |||||||
11 | 1 | 0.73 | 0.2 | 0.07 | 0.744 | 0.186 | 0.070 | 1.364 | 0.979 | 0.744 |
2 | 0.74 | 0.19 | 0.07 | |||||||
3 | 0.76 | 0.17 | 0.07 | |||||||
12 | 1 | 0.81 | 0.12 | 0.07 | 0.803 | 0.127 | 0.070 | 1.238 | 1.307 | 4.842 |
2 | 0.8 | 0.13 | 0.07 | |||||||
3 | 0.8 | 0.13 | 0.07 | |||||||
13 | 1 | 0.64 | 0.28 | 0.08 | 0.657 | 0.263 | 0.080 | 1.345 | 0.647 | 3.053 |
2 | 0.68 | 0.24 | 0.08 | |||||||
3 | 0.65 | 0.27 | 0.08 | |||||||
14 | 1 | 0.82 | 0.11 | 0.07 | 0.779 | 0.142 | 0.080 | 1.163 | 1.206 | 6.855 |
2 | 0.74 | 0.17 | 0.09 | |||||||
3 | 0.77 | 0.15 | 0.08 | |||||||
15 | 1 | 0.61 | 0.25 | 0.14 | 0.641 | 0.219 | 0.140 | 0.803 | 0.758 | |
2 | 0.66 | 0.2 | 0.14 | |||||||
3 | 0.65 | 0.21 | 0.14 | |||||||
16 | 1 | 0.7 | 0.23 | 0.07 | 0.694 | 0.233 | 0.073 | 1.391 | 0.771 | 9.762 |
2 | 0.71 | 0.22 | 0.07 | |||||||
3 | 0.67 | 0.25 | 0.08 | |||||||
17 | 1 | 0.74 | 0.19 | 0.07 | 0.740 | 0.190 | 0.070 | 1.370 | 0.962 | 9.564 |
2 | 0.73 | 0.2 | 0.07 | |||||||
3 | 0.75 | 0.18 | 0.07 | |||||||
18 | 1 | 0.56 | 0.35 | 0.09 | 0.534 | 0.377 | 0.090 | 1.310 | 0.247 | 4.919 |
2 | 0.51 | 0.4 | 0.09 | |||||||
3 | 0.53 | 0.38 | 0.09 | |||||||
19 | 1 | 0.55 | 0.35 | 0.1 | 0.547 | 0.360 | 0.093 | 1.274 | 0.296 | 2.609 |
2 | 0.55 | 0.36 | 0.09 | |||||||
3 | 0.54 | 0.37 | 0.09 | |||||||
20 | 1 | 0.77 | 0.16 | 0.07 | 0.754 | 0.176 | 0.070 | 1.346 | 1.029 | 1.870 |
2 | 0.76 | 0.17 | 0.07 | |||||||
3 | 0.73 | 0.2 | 0.07 |
The process seems to be
The CCs are an effective statistical process monitoring tool frequently used to assess the stability of manufacturing processes. They monitor the industrial process by giving practitioners an early signal about an
All data, models, and codes supporting the results of this study are accessible on reasonable request from the corresponding author.
This manuscript has not been simultaneously submitted to more than one journal. So the proposed work is original and has not been published anywhere else in any form or language.
Consent was obtained from all authors for their participation in completing this manuscript and their consent to publish it.
All the authors contributed to this paper.
This work is supported by the
The authors declare that they have no conflicts of interest to report regarding the present study.
Let
Then, an estimation of
Let
As presented in [
Similarly
Following Mardia 1979, the author also used
The Scilab Codes for ARL, when
ARL0 = 370;
d1 = 0.25; d2 = 0.5; d3 = 0.75; d4 = 1; d5 = 1.25; d6 = 1.5; d7 = 1.75; d8 = 2;
b = 1;
mm = 3;
S = [3, 0;0, 3];
SM = 0.3 * eye(p−1, p−1);
b2 = b * b
AA = b2 * inv(b2 * S + SM/mm)
BB = inv(S)
invBB = S [eve, eva] = spec(invBB * AA) if eva(1, 1) < eva(2, 2) then lam1 = eva(1, 1) lam2 = eva(2, 2) v1 = eve(:, 1)
vp = eve(:, 2)
else
lam1 = eva(2, 2)
lam2 = eva(1, 1)
v1 = eve(:, 2)
vp =eve(:, 1)
end
m1 = 30
m2 = 30
dmi1 = d1 * lam1 dma1 = d1 * lam2
dmi2 = d2 * lam1 dma2 = d2 * lam2
dmi3 = d3 * lam1 dma3 = d3 * lam2
dmi4 = d4 * lam1 dma4 = d4 * lam2
dmi5 = d5 * lam1 dma5 = d5 * lam2
dmi6 = d6 * lam1 dma6 = d6 * lam2
dmi7 = d7 * lam1 dma7 = d7 * lam2
dmi8 = d8 * lam1 dma8 = d8 * lam2
[r21,H21,ARL21,SDRL21] = optMEWMAUTM(m1, m2, p−1,ARL0, dmi1)
Z21 = [dmi1;r21;H21;ARL21] [r31,H31,ARL31,SDRL31] = optMEWMAUTM(m1, m2,
p−1,ARL0, dma1)
Z31 = [dma1;r31;H31;ARL31]
disp(Z21) disp(Z31)
[r22,H22,ARL22,SDRL22] = optMEWMAUTM(m1, m2, p−1,ARL0, dmi2)
Z22 = [dmi2;r22;H22;ARL22] [r32,H32,ARL32,SDRL32] = optMEWMAUTM(m1, m2,
p−1,ARL0, dma2)
Z32 = [dma2;r32;H32;ARL32]
disp(Z22) disp(Z32) [r23,H23,ARL23,SDRL23] = optMEWMAUTM(m1, m2, p−1,ARL0, dmi3)
Z23 = [dmi3;r23;H23;ARL23] [r33,H33,ARL33,SDRL33] = optMEWMAUTM(m1, m2,p−1,ARL0, dma3)
Z33 = [dma3;r33;H33;ARL33]
disp(Z23) disp(Z33)
[r24,H24,ARL24,SDRL24] = optMEWMAUTM(m1, m2, p−1,ARL0, dmi4)
Z24 = [dmi4;r24;H24;ARL24] [r34,H34,ARL34,SDRL34] = optMEWMAUTM(m1, m2,
p−1,ARL0, dma4)
Z34 = [dma4;r34;H34;ARL34]
disp(Z24) disp(Z34)
[r25,H25,ARL25,SDRL25] = optMEWMAUTM(m1, m2, p−1,ARL0, dmi5)
Z25 = [dmi5;r25;H25;ARL25] [r35,H35,ARL35,SDRL35] = optMEWMAUTM(m1, m2,
p−1,ARL0, dma5)
Z35 = [dma5;r35;H35;ARL35]
disp(Z25) disp(Z35)
[r26,H26,ARL26,SDRL26] = optMEWMAUT(m1, m2, p−1,ARL0, dmi6)
Z26 = [dmi6;r26;H26;ARL26] [r36,H36,ARL36,SDRL36] = optMEWMAUT(m1, m2, p−1,ARL0, dma6)
Z36 = [dma6;r36;H36;ARL36]
disp(Z26) disp(Z36)
[r27,H27,ARL27,SDRL27] = optMEWMAUTM(m1, m2, p−1,ARL0, dmi7)
Z27 = [dmi7;r27;H27;ARL27] [r37,H37,ARL37,SDRL37] = optMEWMAUTM(m1, m2,
p−1,ARL0, dma7)
Z37 = [dma7;r37;H37;ARL37]
disp(Z27) disp(Z37)
[r28,H28,ARL28,SDRL28] = optMEWMAUTM(m1, m2, p−1,ARL0, dmi8)
Z28 = [dmi8;r28;H28;ARL28] [r38,H38,ARL38,SDRL38] = optMEWMAUTM(m1, m2,
p−1,ARL0, dma8)
Z38 = [dma8;r38;H38;ARL38]
disp(Z28) disp(Z38)
Where the base function “optCUSUMUTM” to find the ARL is given below,
//———————————— function [Q0, q0] = QCUSUM0M(H, k, p) //————————–
// in control [argout, argin] = argn() if (argin<3)|(argin>4) error(“incorrect number of arguments”) end if H<=0 error(“argument “H” must be > 0”)
end if k<=0 error(“argument “k” must be > 0”)
end if p<=0 error(“argument “p” must be > 0”)
end
UCL = H
m = 100
g = 2 * UCL/(2 * m + 1)
Q0 = ones(m + 1, m + 1)
t = [0:1:m]
C = repmat(c, m, 1)
Q02 = cdfchn(“PQ”, up, p * ones(m, m + 1),C)-cdfchn(“PQ”, down, p * ones(m, m + 1),C)
Q0 = [Q01;Q02]’
q0 = zeros(m + 1, 1)
q0(1) = 1
endfunction
//————————— function [ARL,SDRL] = rlCUSUM0M(H, k, p) //—————————
// in-control ARL [argout, argin] = argn() if (argin<3)|(argin>4) error(”incorrect number of arguments”) end if H<=0 error(“argument “H” must be > 0”)
end if k<=0 error(“argument “k” must be > 0”)
end if p<=0 error(“argument “p” must be > 0”)
end [Q0, q0] = QCUSUM0(H, k, p)
W = inv(eye(Q0)-Q0)
Z = q0’ * W
ARL = sum(Z)
if argout == 2
Z = Z * W * Q0
//—————————- function [Q, q] = QCUSUM1M(H, m1, m2, k, p, delta) //——————
//Out-of-control [argout, argin] = argn() if (argin<5)|(argin>6) error(“incorrect number of arguments”) end if H<=0 error(“argument “H” must be > 0”)
end if k<=0 error(“argument “k” must be > 0”)
end if p<=0 error(“argument “p” must be > 0”)
end
UCL = H;
g1 = 2 * UCL/(2 * m1 + 1);
Q1 = ones(2 * m1 + 1, 2 * m1 + 1);
t1 = [1:1:2 * m1 + 1];
c1 = -UCL + (t1–0.5) * g1;
C1 = repmat(c1, 2 * m1 + 1, 1);
tt1 = [1:1:2 * m1 + 1]’;
Tt1 = repmat(tt1, 1, 2 * m1 + 1);
up1 = (-UCL + Tt1 * g1−(1−k) * C1)/k−delta;
down1 = (−UCL + (Tt1–1) * g1−(1−k) * C1)/k-delta;
Q11 = cdfnor(“PQ”, up1, 0 * ones(C1), 1 * ones(C1))-cdfnor(“PQ”, down1, 0 * ones(C1), 1 * ones(C1)); Q1 = Q11’;
g2 = 2 * UCL/(2 * m2 + 1);
Q2 = ones(m2 + 1, m2 + 1);
t2 = [0:1:m2];
Q(:, counter) = 0;
Q(counter,:) = 0;
end
counter = counter + 1;
end
end
q = zeros((2 * m1 + 1) * (m2 + 1), 1)
start = m1 * (m2 + 1) + 1
q(start) = 1
endfunction
//——————— function [ARL,SDRL] = rlCUSUM1M(H, m1, m2, k, p, delta) //—————
//Out-of-Control ARL [argout, argin] = argn() if (argin<5)|(argin>6) error(“incorrect number of arguments”) end if H<=0 error(“argument “H” must be > 0”)
end if k<=0 error(“argument “k” must be > 0”)
end if p<=0 error(“argument “p” must be > 0”)
end [Q, q] = QCUSUM1M(H, m1, m2, k, p, delta)
W = inv(eye(Q)-Q)
Z = q’ * W
ARL = sum(Z)
if argout == 2
Z = Z * W * Q
//——————————- function f = HCUSUMM(H, k, arl0, p) //——————————-
// function of H
if (H<=0)
f = else
ARL = rlCUSUM0M(H, k, p)
f = (arl0-ARL)
end endfunction
//————- function [H,ARL,SDRL] = optMEWMAUTM(m1, m2, p, arl0, delta) //————-
// Final ARL with optimal H [argout, argin] = argn()
if argin = 5 error(”incorrect number of arguments”)
end
lines(0)
k = 0.5
H0 = 7
H = simplexolve(H0,HCUSUMM, list(k, arl0, p), tol = 1e-4); [ARL,SDRL] = rlCUSUM1M(H, m1, m2, k, p, delta)
endfunction