Performance Analysis of Interval Type-2 Fuzzy and R Control Charts
Abstract
:1. Introduction
2. Interval Type-2 Fuzzy Sets
2.1. Interval Type-2 Triangular Fuzzy Sets
2.2. Footprint of Uncertainty and Membership Functions
3. Interval Type-2 Fuzzy and R Control Charts
3.1. Fuzzification Method
3.2. Type Reduction and Defuzzification Method
4. Control Chart Performance Measures
5. Results and Discussion
5.1. Average Run Length (ARL) and Standard Deviation Run Length (SDRL) for -R Control Charts
5.2. Run Length Percentile Analysis for -R Control Charts
5.3. Illustrative Example
6. Conclusions
Author Contributions
Funding
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Target | FOU Values (θ) | FOU Relationship |
---|---|---|
Faster response | 0.00 < θ1 = θ2 = θ ≤ 0.50 | They should be equal and small |
Response with lower or no overshoot | 0.50 ≤ θ1 = θ2 = θ < 1.00 | They should be equal and larg |
Smaller overshoot and faster response | 0.30 ≤ θ1 = θ2 = θ < 0.90 | They should be equal and medium sized |
Traditional -R Control Chart ARL | |||||||||
λ | δ (Shift) | ||||||||
# | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 |
1.0 | 370.5 | 228.0 | 84.9 | 30.8 | 12.6 | 6.0 | 3.3 | 2.1 | 1.5 |
1.1 | 119.2 | 88.2 | 44.1 | 20.1 | 9.7 | 5.2 | 3.1 | 2.1 | 1.6 |
1.2 | 48.8 | 40.3 | 25.0 | 13.8 | 7.7 | 4.5 | 2.9 | 2.1 | 1.6 |
1.3 | 24.2 | 21.3 | 15.3 | 9.8 | 6.2 | 4.0 | 2.8 | 2.0 | 1.6 |
1.4 | 13.9 | 12.8 | 10.1 | 7.2 | 5.0 | 3.5 | 2.6 | 2.0 | 1.6 |
1.5 | 8.9 | 8.4 | 7.1 | 5.5 | 4.2 | 3.1 | 2.4 | 1.9 | 1.6 |
2.0 | 2.6 | 2.5 | 2.4 | 2.3 | 2.1 | 1.9 | 1.7 | 1.5 | 1.4 |
2.5 | 1.6 | 1.6 | 1.5 | 1.5 | 1.5 | 1.4 | 1.4 | 1.3 | 1.2 |
Traditional -R Control Chart SDRL | |||||||||
λ | δ (Shift) | ||||||||
# | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 |
1.0 | 370.0 | 227.5 | 84.4 | 30.2 | 12.1 | 5.5 | 2.8 | 1.5 | 0.9 |
1.1 | 118.7 | 87.7 | 43.6 | 19.6 | 9.2 | 4.7 | 2.6 | 1.5 | 1.0 |
1.2 | 48.3 | 39.8 | 24.4 | 13.3 | 7.1 | 4.0 | 2.4 | 1.5 | 1.0 |
1.3 | 23.7 | 20.8 | 14.8 | 9.3 | 5.6 | 3.5 | 2.2 | 1.4 | 1.0 |
1.4 | 13.4 | 12.2 | 9.5 | 6.7 | 4.5 | 3.0 | 2.0 | 1.4 | 1.0 |
1.5 | 8.4 | 7.9 | 6.5 | 5.0 | 3.6 | 2.6 | 1.8 | 1.3 | 1.0 |
2.0 | 2.0 | 2.0 | 1.8 | 1.7 | 1.5 | 1.3 | 1.1 | 0.9 | 0.8 |
2.5 | 0.9 | 0.9 | 0.9 | 0.9 | 0.8 | 0.8 | 0.7 | 0.6 | 0.6 |
IT2TFN -R Control Chart ARL | |||||||||
λ | δ (Shift) | ||||||||
# | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 |
1.0 | 372.45 | 228.54 | 85.32 | 30.43 | 12.55 | 5.93 | 3.35 | 2.11 | 1.56 |
1.1 | 118.56 | 87.10 | 44.23 | 20.32 | 9.56 | 5.17 | 3.11 | 2.10 | 1.59 |
1.2 | 48.54 | 39.54 | 24.89 | 13.77 | 7.67 | 4.54 | 2.95 | 2.08 | 1.58 |
1.3 | 24.10 | 21.09 | 15.45 | 9.71 | 6.14 | 4.03 | 2.76 | 2.00 | 1.60 |
1.4 | 13.91 | 12.61 | 9.77 | 7.27 | 4.99 | 3.50 | 2.56 | 1.97 | 1.60 |
1.5 | 8.70 | 8.33 | 7.06 | 5.52 | 4.10 | 3.08 | 2.41 | 1.92 | 1.59 |
2.0 | 2.56 | 2.50 | 2.42 | 2.29 | 2.05 | 1.87 | 1.70 | 1.54 | 1.42 |
2.5 | 1.57 | 1.55 | 1.54 | 1.50 | 1.45 | 1.40 | 1.36 | 1.29 | 1.26 |
IT2TFN -R Control Chart SDRL | |||||||||
λ | δ (Shift) | ||||||||
# | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 |
1.0 | 371.141 | 226.537 | 84.860 | 29.844 | 12.081 | 5.298 | 2.807 | 1.515 | 0.942 |
1.1 | 119.090 | 85.087 | 43.340 | 19.960 | 9.167 | 4.605 | 2.551 | 1.528 | 0.972 |
1.2 | 47.318 | 39.034 | 24.456 | 13.138 | 7.122 | 3.997 | 2.400 | 1.491 | 0.964 |
1.3 | 23.366 | 20.641 | 14.992 | 9.222 | 5.628 | 3.549 | 2.208 | 1.407 | 0.968 |
1.4 | 13.220 | 12.266 | 9.299 | 6.628 | 4.510 | 3.012 | 1.981 | 1.397 | 0.987 |
1.5 | 8.180 | 7.808 | 6.506 | 4.924 | 3.534 | 2.526 | 1.880 | 1.346 | 0.964 |
2.0 | 2.042 | 1.975 | 1.879 | 1.735 | 1.482 | 1.296 | 1.094 | 0.910 | 0.774 |
2.5 | 0.938 | 0.938 | 0.918 | 0.869 | 0.827 | 0.742 | 0.697 | 0.614 | 0.576 |
5th | λ | δ (Shift) | 25th | λ | δ (Shift) | ||||||||||||||||
# | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 | # | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 | ||
1.0 | 19 | 12 | 5 | 2 | 1 | 1 | 1 | 1 | 1 | 1.0 | 107 | 66 | 25 | 9 | 4 | 2 | 1 | 1 | 1 | ||
1.1 | 7 | 5 | 3 | 2 | 1 | 1 | 1 | 1 | 1 | 1.1 | 35 | 26 | 13 | 6 | 3 | 2 | 1 | 1 | 1 | ||
1.2 | 3 | 3 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1.2 | 14 | 12 | 8 | 4 | 3 | 2 | 1 | 1 | 1 | ||
1.3 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1.3 | 7 | 6 | 5 | 3 | 2 | 2 | 1 | 1 | 1 | ||
1.4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1.4 | 4 | 4 | 3 | 2 | 2 | 1 | 1 | 1 | 1 | ||
1.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1.5 | 3 | 3 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | ||
2.0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2.0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
2.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
50th (MDRL) | λ | δ (Shift) | |||||||||||||||||||
# | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 | ||||||||||||
1.0 | 257 | 158 | 59 | 21 | 9 | 4 | 2 | 2 | 1 | ||||||||||||
1.1 | 83 | 61 | 31 | 14 | 7 | 4 | 2 | 2 | 1 | ||||||||||||
1.2 | 34 | 28 | 17 | 10 | 5 | 3 | 2 | 2 | 1 | ||||||||||||
1.3 | 17 | 15 | 11 | 7 | 4 | 3 | 2 | 2 | 1 | ||||||||||||
1.4 | 10 | 9 | 7 | 5 | 4 | 3 | 2 | 1 | 1 | ||||||||||||
1.5 | 6 | 6 | 5 | 4 | 3 | 2 | 2 | 1 | 1 | ||||||||||||
2.0 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | ||||||||||||
2.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||||||
75th | λ | δ (Shift) | 95th | λ | δ (Shift) | ||||||||||||||||
# | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 | # | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 | ||
1.0 | 513 | 316 | 117 | 42 | 17 | 8 | 4 | 3 | 2 | 1.0 | 1109 | 682 | 253 | 91 | 37 | 17 | 9 | 5 | 3 | ||
1.1 | 165 | 122 | 61 | 28 | 13 | 7 | 4 | 3 | 2 | 1.1 | 356 | 263 | 131 | 59 | 28 | 14 | 8 | 5 | 3 | ||
1.2 | 67 | 56 | 34 | 19 | 10 | 6 | 4 | 3 | 2 | 1.2 | 145 | 120 | 74 | 40 | 22 | 13 | 8 | 5 | 4 | ||
1.3 | 33 | 29 | 21 | 13 | 8 | 5 | 4 | 3 | 2 | 1.3 | 72 | 63 | 45 | 28 | 17 | 11 | 7 | 5 | 4 | ||
1.4 | 19 | 17 | 14 | 10 | 7 | 5 | 3 | 2 | 2 | 1.4 | 41 | 37 | 29 | 21 | 14 | 10 | 7 | 5 | 4 | ||
1.5 | 12 | 11 | 10 | 7 | 6 | 4 | 3 | 2 | 2 | 1.5 | 26 | 24 | 20 | 15 | 11 | 8 | 6 | 5 | 4 | ||
2.0 | 3 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 2.0 | 7 | 6 | 6 | 6 | 5 | 4 | 4 | 3 | 3 | ||
2.5 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2.5 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 2 |
5th | λ | δ (Shift) | 25th | λ | δ (Shift) | ||||||||||||||||
# | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 | # | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 | ||
1.0 | 20 | 13 | 5 | 2 | 1 | 1 | 1 | 1 | 1 | 1.0 | 107 | 67 | 25 | 9 | 4 | 2 | 1 | 1 | 1 | ||
1.1 | 7 | 5 | 3 | 2 | 1 | 1 | 1 | 1 | 1 | 1.1 | 34 | 26 | 13 | 6 | 3 | 2 | 1 | 1 | 1 | ||
1.2 | 3 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1.2 | 15 | 12 | 8 | 4 | 3 | 2 | 1 | 1 | 1 | ||
1.3 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1.3 | 7 | 6 | 5 | 3 | 2 | 2 | 1 | 1 | 1 | ||
1.4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1.4 | 4 | 4 | 3 | 3 | 2 | 1 | 1 | 1 | 1 | ||
1.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1.5 | 3 | 3 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | ||
2.0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2.0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
2.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
50th (MDRL) | λ | δ (Shift) | |||||||||||||||||||
# | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 | ||||||||||||
1.0 | 260 | 160 | 59 | 21 | 9 | 4 | 2 | 2 | 1 | ||||||||||||
1.1 | 82 | 61 | 31 | 14 | 7 | 4 | 2 | 2 | 1 | ||||||||||||
1.2 | 34 | 27 | 17 | 10 | 5 | 3 | 2 | 2 | 1 | ||||||||||||
1.3 | 17 | 15 | 11 | 7 | 4 | 3 | 2 | 2 | 1 | ||||||||||||
1.4 | 10 | 9 | 7 | 5 | 4 | 3 | 2 | 1 | 1 | ||||||||||||
1.5 | 6 | 6 | 5 | 4 | 3 | 2 | 2 | 1 | 1 | ||||||||||||
2.0 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | ||||||||||||
2.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||||||
75th | λ | δ (Shift) | 95th | λ | δ (Shift) | ||||||||||||||||
# | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 | # | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 | ||
1.0 | 517 | 317 | 118 | 42 | 17 | 8 | 4 | 3 | 2 | 1.0 | 1098 | 682 | 258 | 91 | 37 | 17 | 9 | 5 | 3 | ||
1.1 | 164 | 122 | 61 | 28 | 13 | 7 | 4 | 3 | 2 | 1.1 | 358 | 257 | 130 | 60 | 28 | 14 | 8 | 5 | 4 | ||
1.2 | 67 | 55 | 34 | 19 | 10 | 6 | 4 | 3 | 2 | 1.2 | 142 | 118 | 75 | 41 | 22 | 12.1 | 8 | 5 | 4 | ||
1.3 | 33 | 29 | 21 | 13 | 8 | 5 | 4 | 3 | 2 | 1.3 | 71 | 62 | 46 | 28 | 17 | 11 | 7 | 5 | 4 | ||
1.4 | 19 | 17 | 13 | 10 | 7 | 5 | 3 | 2 | 2 | 1.4 | 41 | 37 | 29 | 20 | 14 | 9 | 7 | 5 | 4 | ||
1.5 | 12 | 11 | 10 | 8 | 5 | 4 | 3 | 2 | 2 | 1.5 | 25 | 24 | 20 | 15 | 11 | 8 | 6 | 5 | 3 | ||
2.0 | 3 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 2.0 | 7 | 6 | 6 | 6 | 5 | 4 | 4 | 3 | 3 | ||
2.5 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2.5 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 2 |
Sample Number | Measures (Crisp Values) | Variables | Decision | |||||
---|---|---|---|---|---|---|---|---|
x1 | x2 | x3 | x4 | x5 | R | |||
1 | −0.547 | −0.551 | 1.866 | −0.418 | −0.706 | −0.071 | 2.572 | In control |
2 | −0.382 | 0.075 | −1.504 | 1.395 | −0.152 | −0.114 | 2.899 | In control |
3 | 0.694 | −0.444 | −1.559 | 1.975 | −1.800 | −0.227 | 3.776 | In control |
4 | 0.050 | −0.520 | −1.425 | −0.194 | 0.721 | −0.273 | 2.146 | In control |
5 | 1.368 | 0.014 | −0.207 | 0.518 | −1.400 | 0.059 | 2.769 | In control |
6 | 1.615 | 0.706 | −0.783 | −0.620 | −1.980 | −0.212 | 3.595 | In control |
7 | −2.426 | 1.077 | 0.569 | 1.686 | −0.859 | 0.009 | 4.112 | In control |
8 | 1.824 | 0.195 | −0.659 | −0.415 | −0.581 | 0.073 | 2.483 | In control |
9 | 0.646 | −0.241 | −1.400 | −1.045 | 1.090 | −0.190 | 2.490 | In control |
10 | 1.101 | −0.685 | 1.772 | 0.372 | −0.337 | 0.445 | 2.457 | In control |
11 | 0.259 | −2.005 | 1.247 | 1.204 | 0.223 | 0.186 | 3.252 | In control |
12 | −0.104 | −0.320 | −0.968 | −0.484 | −0.337 | −0.443 | 0.864 | In control |
13 | −2.235 | −0.614 | 0.234 | 0.991 | −0.096 | −0.344 | 3.226 | In control |
14 | 2.216 | −0.527 | −3.062 | 0.176 | 0.155 | −0.208 | 5.278 | In control |
15 | 0.960 | 0.729 | 0.772 | 0.304 | 1.699 | 0.893 | 1.395 | In control |
16 | −0.199 | 2.435 | 0.528 | 0.906 | 0.547 | 0.843 | 2.634 | In control |
17 | 1.416 | −0.461 | −0.244 | 0.202 | −0.058 | 0.171 | 1.878 | In control |
18 | 1.633 | −0.748 | −0.642 | −0.832 | 0.872 | 0.056 | 2.465 | In control |
19 | 0.064 | 0.758 | −0.515 | 1.516 | 0.671 | 0.499 | 2.031 | In control |
20 | 0.292 | −0.551 | −0.362 | 0.309 | 0.449 | 0.027 | 1.000 | In control |
Sample Number | x1 | x2 | x3 | x4 | x5 | ||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | −0.587 | −0.575 | −0.547 | −0.526 | −0.517 | −0.566 | −0.561 | −0.551 | −0.535 | −0.528 | 1.838 | 1.846 | 1.866 | 1.890 | 1.900 | −0.461 | −0.448 | −0.418 | −0.391 | −0.380 | −0.736 | −0.727 | −0.706 | −0.696 | −0.692 |
2 | −0.418 | −0.407 | −0.382 | −0.349 | −0.335 | 0.028 | 0.042 | 0.075 | 0.078 | 0.080 | −1.534 | −1.525 | −1.504 | −1.485 | −1.477 | 1.346 | 1.361 | 1.395 | 1.406 | 1.411 | −0.177 | −0.170 | −0.152 | −0.144 | −0.140 |
3 | 0.687 | 0.689 | 0.694 | 0.698 | 0.700 | −0.466 | −0.459 | −0.444 | −0.410 | −0.395 | −1.608 | −1.593 | −1.559 | −1.556 | −1.555 | 1.932 | 1.945 | 1.975 | 1.996 | 2.004 | −1.817 | −1.812 | −1.800 | −1.770 | −1.758 |
4 | 0.008 | 0.021 | 0.050 | 0.079 | 0.091 | −0.538 | −0.532 | −0.520 | −0.488 | −0.474 | −1.428 | −1.427 | −1.425 | −1.402 | −1.392 | −0.209 | −0.204 | −0.194 | −0.167 | −0.155 | 0.690 | 0.700 | 0.721 | 0.743 | 0.752 |
5 | 1.341 | 1.349 | 1.368 | 1.372 | 1.374 | −0.025 | −0.013 | 0.014 | 0.041 | 0.053 | −0.241 | −0.231 | −0.207 | −0.205 | −0.204 | 0.482 | 0.493 | 0.518 | 0.537 | 0.545 | −1.406 | −1.404 | −1.400 | −1.377 | −1.367 |
6 | 1.607 | 1.609 | 1.615 | 1.624 | 1.628 | 0.661 | 0.674 | 0.706 | 0.726 | 0.735 | −0.822 | −0.810 | −0.783 | −0.749 | −0.734 | −0.623 | −0.622 | −0.620 | −0.599 | −0.589 | −1.989 | −1.986 | −1.980 | −1.967 | −1.961 |
7 | −2.475 | −2.460 | −2.426 | −2.409 | −2.401 | 1.067 | 1.070 | 1.077 | 1.112 | 1.127 | 0.537 | 0.547 | 0.569 | 0.576 | 0.579 | 1.664 | 1.671 | 1.686 | 1.701 | 1.708 | −0.866 | −0.864 | −0.859 | −0.849 | −0.845 |
8 | 1.810 | 1.815 | 1.824 | 1.840 | 1.847 | 0.165 | 0.174 | 0.195 | 0.218 | 0.229 | −0.673 | −0.669 | −0.659 | −0.631 | −0.618 | −0.441 | −0.433 | −0.415 | −0.412 | −0.410 | −0.611 | −0.602 | −0.581 | −0.567 | −0.561 |
9 | 0.626 | 0.632 | 0.646 | 0.646 | 0.647 | −0.249 | −0.247 | −0.241 | −0.232 | −0.228 | −1.409 | −1.406 | −1.400 | −1.395 | −1.393 | −1.046 | −1.046 | −1.045 | −1.024 | −1.015 | 1.087 | 1.088 | 1.090 | 1.113 | 1.122 |
10 | 1.100 | 1.100 | 1.101 | 1.120 | 1.128 | −0.724 | −0.712 | −0.685 | −0.681 | −0.679 | 1.737 | 1.748 | 1.772 | 1.796 | 1.806 | 0.323 | 0.338 | 0.372 | 0.381 | 0.385 | −0.350 | −0.346 | −0.337 | −0.321 | −0.315 |
11 | 0.236 | 0.243 | 0.259 | 0.270 | 0.275 | −2.014 | −2.011 | −2.005 | −1.982 | −1.973 | 1.206 | 1.218 | 1.247 | 1.254 | 1.257 | 1.201 | 1.202 | 1.204 | 1.222 | 1.230 | 0.220 | 0.221 | 0.223 | 0.252 | 0.265 |
12 | −0.111 | −0.109 | −0.104 | −0.076 | −0.064 | −0.341 | −0.335 | −0.320 | −0.308 | −0.303 | −1.018 | −1.003 | −0.968 | −0.940 | −0.927 | −0.502 | −0.497 | −0.484 | −0.468 | −0.461 | −0.354 | −0.349 | −0.337 | −0.312 | −0.302 |
13 | −2.281 | −2.267 | −2.235 | −2.203 | −2.189 | −0.662 | −0.648 | −0.614 | −0.609 | −0.607 | 0.197 | 0.208 | 0.234 | 0.265 | 0.279 | 0.973 | 0.979 | 0.991 | 1.006 | 1.012 | −0.112 | −0.107 | −0.096 | −0.088 | −0.085 |
14 | 2.214 | 2.215 | 2.216 | 2.246 | 2.259 | −0.569 | −0.556 | −0.527 | −0.508 | −0.501 | −3.088 | −3.080 | −3.062 | −3.061 | −3.061 | 0.140 | 0.150 | 0.176 | 0.206 | 0.218 | 0.150 | 0.151 | 0.155 | 0.184 | 0.197 |
15 | 0.950 | 0.953 | 0.960 | 0.971 | 0.975 | 0.680 | 0.695 | 0.729 | 0.744 | 0.750 | 0.763 | 0.766 | 0.772 | 0.797 | 0.808 | 0.263 | 0.276 | 0.304 | 0.309 | 0.312 | 1.656 | 1.669 | 1.699 | 1.703 | 1.705 |
16 | −0.244 | −0.230 | −0.199 | −0.176 | −0.166 | 2.389 | 2.403 | 2.435 | 2.444 | 2.447 | 0.499 | 0.508 | 0.528 | 0.538 | 0.543 | 0.878 | 0.886 | 0.906 | 0.913 | 0.917 | 0.524 | 0.531 | 0.547 | 0.576 | 0.589 |
17 | 1.401 | 1.405 | 1.416 | 1.450 | 1.465 | −0.494 | −0.484 | −0.461 | −0.439 | −0.430 | −0.286 | −0.274 | −0.244 | −0.242 | −0.242 | 0.160 | 0.172 | 0.202 | 0.217 | 0.223 | −0.101 | −0.088 | −0.058 | −0.032 | −0.021 |
18 | 1.632 | 1.632 | 1.633 | 1.633 | 1.634 | −0.780 | −0.771 | −0.748 | −0.714 | −0.699 | −0.659 | −0.654 | −0.642 | −0.622 | −0.613 | −0.844 | −0.840 | −0.832 | −0.827 | −0.825 | 0.855 | 0.860 | 0.872 | 0.896 | 0.906 |
19 | 0.015 | 0.030 | 0.064 | 0.098 | 0.113 | 0.740 | 0.745 | 0.758 | 0.774 | 0.781 | −0.521 | −0.519 | −0.515 | −0.490 | −0.480 | 1.491 | 1.498 | 1.516 | 1.546 | 1.559 | 0.650 | 0.656 | 0.671 | 0.703 | 0.717 |
20 | 0.269 | 0.276 | 0.292 | 0.295 | 0.297 | −0.575 | −0.568 | −0.551 | −0.549 | −0.548 | −0.412 | −0.397 | −0.362 | −0.352 | −0.347 | 0.278 | 0.288 | 0.309 | 0.343 | 0.358 | 0.420 | 0.429 | 0.449 | 0.457 | 0.461 |
Sample Number | Defuzzified | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
DTriT | RDTriT | |||||||||||
1 | −0.102 | −0.093 | −0.071 | −0.052 | −0.043 | 2.529 | 2.542 | 2.572 | 2.617 | 2.636 | −0.072 | 2.578 |
2 | −0.151 | −0.140 | −0.114 | −0.099 | −0.092 | 2.822 | 2.846 | 2.899 | 2.931 | 2.945 | −0.118 | 2.890 |
3 | −0.254 | −0.246 | −0.227 | −0.209 | −0.201 | 3.690 | 3.715 | 3.776 | 3.808 | 3.821 | −0.227 | 3.764 |
4 | −0.295 | −0.289 | −0.273 | −0.247 | −0.235 | 2.082 | 2.101 | 2.146 | 2.170 | 2.180 | −0.269 | 2.138 |
5 | 0.030 | 0.039 | 0.059 | 0.074 | 0.080 | 2.708 | 2.726 | 2.769 | 2.777 | 2.780 | 0.057 | 2.755 |
6 | −0.233 | −0.227 | −0.212 | −0.193 | −0.184 | 3.568 | 3.576 | 3.595 | 3.610 | 3.617 | −0.210 | 3.594 |
7 | −0.014 | −0.007 | 0.009 | 0.026 | 0.033 | 4.065 | 4.079 | 4.112 | 4.161 | 4.183 | 0.009 | 4.119 |
8 | 0.050 | 0.057 | 0.073 | 0.090 | 0.097 | 2.429 | 2.445 | 2.483 | 2.510 | 2.521 | 0.073 | 2.478 |
9 | −0.198 | −0.196 | −0.190 | −0.179 | −0.174 | 2.480 | 2.483 | 2.490 | 2.519 | 2.531 | −0.188 | 2.499 |
10 | 0.417 | 0.426 | 0.445 | 0.459 | 0.465 | 2.417 | 2.429 | 2.457 | 2.508 | 2.530 | 0.443 | 2.466 |
11 | 0.170 | 0.175 | 0.186 | 0.203 | 0.211 | 3.179 | 3.201 | 3.252 | 3.265 | 3.271 | 0.188 | 3.236 |
12 | −0.465 | −0.459 | −0.443 | −0.421 | −0.411 | 0.816 | 0.830 | 0.864 | 0.927 | 0.954 | −0.440 | 0.876 |
13 | −0.377 | −0.367 | −0.344 | −0.326 | −0.318 | 3.162 | 3.181 | 3.226 | 3.273 | 3.293 | −0.346 | 3.227 |
14 | −0.231 | −0.224 | −0.208 | −0.187 | −0.177 | 5.275 | 5.276 | 5.278 | 5.327 | 5.347 | −0.206 | 5.297 |
15 | 0.863 | 0.872 | 0.893 | 0.905 | 0.910 | 1.344 | 1.360 | 1.395 | 1.428 | 1.442 | 0.889 | 1.394 |
16 | 0.809 | 0.820 | 0.843 | 0.859 | 0.866 | 2.555 | 2.579 | 2.634 | 2.674 | 2.691 | 0.840 | 2.628 |
17 | 0.136 | 0.146 | 0.171 | 0.191 | 0.199 | 1.831 | 1.845 | 1.878 | 1.934 | 1.959 | 0.169 | 1.887 |
18 | 0.041 | 0.045 | 0.056 | 0.073 | 0.081 | 2.457 | 2.459 | 2.465 | 2.474 | 2.477 | 0.059 | 2.466 |
19 | 0.475 | 0.482 | 0.499 | 0.526 | 0.538 | 1.970 | 1.988 | 2.031 | 2.065 | 2.080 | 0.503 | 2.028 |
20 | −0.004 | 0.006 | 0.027 | 0.039 | 0.044 | 0.968 | 0.978 | 1.000 | 1.025 | 1.035 | 0.023 | 1.001 |
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Almeida, T.S.; dos Santos Mendes, A.; Rocha Rizol, P.M.S.; Machado, M.A.G.
Performance Analysis of Interval Type-2 Fuzzy
Almeida TS, dos Santos Mendes A, Rocha Rizol PMS, Machado MAG.
Performance Analysis of Interval Type-2 Fuzzy
Almeida, Túlio S., Amanda dos Santos Mendes, Paloma M. S. Rocha Rizol, and Marcela A. G. Machado.
2023. "Performance Analysis of Interval Type-2 Fuzzy