Fani I. Gkountakou
| Avrilia Konguetsof | Georgios Souliotis | Basil K. Papadopoulos*
© 2023 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
OPEN ACCESS
Fuzzy rule-based processes have traditionally incorporated a variety of fuzzy implications using modus ponens, modus tollens, and fuzzy negations. This study introduces a novel method of fuzzy implication utilizing Fuzzy Linear Regression (FLR) with triangular fuzzy numbers. This approach was applied to evaluate the relationship between parameters influencing concrete and the compressive strength of sustainable rice husk ash (RHA) concrete. FLR, a technique for modeling relationships between inputs and outputs in a fuzzy environment, was employed to determine a fuzzy output with a specific truth value. This truth value represented the degree of truth of an entire fuzzy implication. The data used in this study were derived from real experimental results. The analysis showed that the FLR method produced accurate outputs, as indicated by a low Theil’s inequality coefficient (Theil’s U=0.1). The results suggest that FLR can effectively manage uncertainties in data and holds potential as an alternative method for fuzzy implication.
Fuzzy Linear Regression (FLR), triangular fuzzy numbers, fuzzy implications, sustainable rice husk ash concrete, mathematical modelling, fuzzy rule-based system, fuzzy modelling
Fuzzy implications are recognized as important factors in fuzzy logic with many applications in approximate reasoning [1], mathematical morphology [2], performing fuzzy conditionals [1, 3] and other fuzzy fields [4-6]. Zadeh [7] proposed that the fuzzy inference rules were applied to evaluate the effectiveness of fuzzy implications based on modus ponens, modus tollens and hypothetical syllogism. Although, there are many definitions of fuzzy implications in the literature, in this study they are expressed as follows [8-10].
Theorem 1. A fuzzy implication is a function defined as $I: [0,1] \mathrm{x}[0,1] \rightarrow[0,1]$ in which the following restrictions are satisfied for every:
$\mathrm{x}_1 \leq \mathrm{x}_2$ then $\mathrm{I}\left(\mathrm{x}_1, \mathrm{y}\right) \geq \mathrm{I}\left(\mathrm{x}_2, \mathrm{y}\right)$, i. e., $\mathrm{I}(\cdot, \mathrm{y})$ (1)
is decreasing
$\mathrm{y}_1 \leq \mathrm{y}_2$ then $\mathrm{I}\left(\mathrm{x}, \mathrm{y}_1\right) \leq \mathrm{I}\left(\mathrm{x}, \mathrm{y}_2\right)$, i. e., $\mathrm{I}(\mathrm{x}, \cdot)$ (2)
is increasing
$I(0,0)=1$ (3)
$I(1,1)=1$ (4)
$I(1,0)=0$ (5)
These conditions demonstrate that the truth values of the initial propositions determine the truth value of the consequence. The following Table 1 categorizes the different implications from the literature which generalize the crisp one to fuzzy logic.
Table 1. Basic fuzzy implications
|
Fuzzy Implications |
Formula |
|
Lukasiewicz [11] |
$\mathrm{I}_{\mathrm{LK}}(\mathrm{x}, \mathrm{y})=\min (1,1-\mathrm{x}+\mathrm{y})$ |
|
Reichenbach [12] |
$I_{R C}(x, y)=1-x+x y$ |
|
Kleene-Dienes [13] |
$I_{K D}(x, y)=\max (1-x, y)$ |
|
Goguen [14] |
$I_{G G}(x, y)=\left\{\begin{array}{l}1, \text { if } x \leq y \\ \frac{y}{x}, \text { if } x>y\end{array}\right.$ |
|
Rescher [15] |
$I_{R S}(x, y)=\left\{\begin{array}{l}1, \text { if } x \leq y \\ 0, \text { if } x>y\end{array}\right.$ |
|
Yager [16] |
$I_{Y G}(x, y)=\left\{\begin{array}{l}1, \text { if } x=0 \text { and } y=0 \\ y^x, \text { if } x>0 \text { or } y>0\end{array}\right.$ |
|
Weber [17] |
$I_{W B}(x, y)=\left\{\begin{array}{l}1, \text { if } x<1 \\ y, \text { if } x=1\end{array}\right.$ |
|
Fodor [18] |
$I_{F D}(x, y)= \begin{cases}1, & \text { if } x \leq y \\ \max (1-x, y), & \text { if } x>y\end{cases}$ |
The use of fuzzy implications had attracted many researchers who were interested in making inferences in a fuzzy environment. Particularly, an algorithmic process was introduced for choosing the most ideal fuzzy implication by using real experimental data [19]. Also, Mylonas and Papadopoulos [20] evaluated the membership function for every variable and defuzzificated them for determining the output y. Then, the deviations of the outputs, that emerged from the implications and the observations, were calculated for considering the best implication process with the smallest deviation values. Moreover, a new fuzzy implication method was proposed [21, 22] which depended on empiristic implication relations.
One step for constructing the parametric implication is to divide the data into language variables in order to be normalized between [0, 1]. This process is necessary for evaluating the membership function of each variable. For example, Makariadis et al. [22] used a new model via fuzzy implications in which the experimental data were classified into linguistic variables through Fuzzy C-means clustering (FCM) algorithm. Also, Botzoris et al. [23] divided the data into low, medium and high according to real published results.
In this study, we used FLR with triangular fuzzy numbers as a fuzzy implication, for evaluating the truth value of every variable. Particularly, a set of 192 experimental observations, derived from previous study [24], was applied for evaluating the relation between the parameters that affect concrete construction and the compressive strength of sustainable RHA concrete. It was demonstrated that [23] the input data were real observations without including the meaning of the fuzziness and consequently the truth value was equal to one. Therefore, by applying FLR method, every experimental output Y belonged to a fuzzy output with a specific truth value, which was the degree of truth of the fuzzy implication. This approach could be verified with the following fuzzy implication axiom, named neutrality of truth [21]:
$I(1, y)=y$ (6)
in which I represented the fuzzy implication.
By evaluating the results, it was concluded that FLR with triangular fuzzy numbers provided accurate inferences between the observations and could be used as a new effective implication method in every case.
The paper is divided into the following sections: The second part refers to the description of sustainable rice husk ash concrete properties and the third part analyzes the FLR method with triangular fuzzy numbers for modelling the relationship among the amount of cement and its compressive strength. The fourth section quotes the model application and the implication process that was followed for evaluating the truth values of every variable. Finally, in the last part of the paper the conclusions of FLR application method are demonstrated.
It is well known that agricultural by-products can be used as supplementary cementitious materials for enhancing the concrete properties and preventing the environment [25, 26]. Rice husk ash is recognized as a sustainable material with distinctive properties, numerous applications in concrete structures and important influence on its constructive performance [27]. Thus, it can be worthily used for replacing the ordinary Portland cement.
The determination of the relation between the parameters that affect concrete and the compressive strength of sustainable RHA concrete is a major process, as it estimates the effect of them on its mechanical properties. Therefore, the utilization of RHA depends on the performance enhancement of concrete mechanical properties avoiding concrete failures. These properties will be described in the following sub-sections.
2.1 Compressive strength
Many researchers studied the compressive strength of sustainable rice husk ash concrete in different proportions. For example, Bheel et al. [28] claimed that the compressive strength of concrete increased by a rate of 2% to 6% with the replacement of 10% RHA instead of conventional concrete. It was also demonstrated that [29] samples containing 10%, 20% and 30% RHA yielded greater compressive strength at the age of 91 days in contrast to the plain concrete. More specifically, for w/b ratio 0.50 the compressive strength was increased by 21%, 22% and 27%, respectively. Another research [30] studied the amount of 0%, 5%, 7.5%, 10%, 12.5%, and 15% RHA by weight and was concluded that the sample with 7.5% RHA revealed higher compressive strength up to 3% increase, than that of the conventional concrete with Portland cement. When the content of RHA increased by more than 7.5% replacement with cement, the compressive strength was reduced, in contrast to the reference specimen. This may due to the fact that higher proportions of RHA contain excessive silica proportions, which cause the reduction in the compressive strength.
2.2 Tensile strength
The determination of tensile strength is a difficult process as it is affected by many factors [31]. Particularly, Ali et al. [32] studied the RHA concrete in the proportions of 0%, 2.5%, 5%, 7.5%, 10%, 12.5%, and 15% by weight and concluded that the tensile strength increased for the proportions between 0% and 10% RHA and decreased for the amount of 12.5%, and 15% RHA. The maximum tensile strength was noticed for the proportion of 10% RHA replacement of cement with 20.41% increase, in contrast to the plain specimen. The increase in the tensile strength was due to inherent correlation between density and strength and the proportion of silica in Rice Husk Ash. However, the reduction in strength caused a decrease in unit weight. Furthermore, [33] the content of 10% RHA increased the tensile strength of sustainable rice husk ash concrete by 3%, 8.8% and 16.5% at 7, 28 and 90 days, respectively, compared to normal concrete. Thus, the combination of 10% RHA and different amounts of nanoparticles in concrete mixtures yielded the best mechanical results.
2.3 Flexural strength
The flexural strength is also an important parameter that must be determined for assessing the constructive behavior of concrete. Depaa et al. [34] compared samples with 15% RHA and normal concrete and cured them for 7, 14, and 28 days. The results revealed that the concrete containing RHA indicated good flexural strength values than plain concrete. At 28 days of hydration, the flexural strength was 4.58MPa for conventional concrete and 4.52MPa for the specimen containing 15% RHA. Also, Meddah et al. [33] found that 10% rice husk ash improved the flexural strength of concrete from 1% to 15% increase, in contrast to the normal mixture. This was probably happening because the addition of 1%-4% content of Al2O3 nanoparticles with pozzolanic behavior enhanced the mechanical properties and durability concrete performance.
Regression analysis is a widely studied method of modelling. It can be applied for evaluating an exact relation between inputs and independent output in many fields of engineering. An extension of this classical analysis is Fuzzy Linear Regression method, in which some variables that compose the model are expressed with fuzzy numbers. More specifically, Fuzzy Linear Regression method is a more advanced form of the classical linear regression that handles uncertainty and vagueness in the data. It applies the principles of fuzzy logic to evaluate a relationship between inputs and outputs, when the data are imprecise. The Fuzzy Linear Regression method, that was proposed by Asai et al. [35], expressed the relation between the dependent output Y and the independent variables X in a fuzzy form, with the following equation [36, 37]:
$Y=A_0+A_1 X_1+\cdots+A_n X_n$ (7)
In this possibilistic model, the uncertainty to determine an accurate relation between the parameters dealt with the triangular fuzzy numbers $A_i=\left(r_i, c_i\right)$, with center $r_i$ that the membership value was equal to one and the half-width $c_i$. Therefore, the membership function of triangular fuzzy numbers expressed as follows [38]:
$\mu_{\mathrm{A}}(\mathrm{x})=\mathrm{L}\left(\frac{\mathrm{x}-\mathrm{r}}{\mathrm{c}}\right)$ (8)
The parameter L(x) was the reference function of the model which obeys the following axioms:
$L(x)=L(-x)$ (9)
$L(0)=1$ (10)
L(x) is decreasing in $[0, \infty)$.
As fuzzy triangular numbers were used in this study, the reference function formed with the following equation, which was decreasing in (0, 1):
$L(x)=\max (0,1-|x|)$ (11)
Then, according to the theorem [39], the membership function of the linear regression model expressed as follows:
$\mu_{Y_j}\left(y_j\right)=L\left[\frac{y_j-\left(r_0+\sum_{i=1}^n r_i x_{i j}\right)}{c_0+\sum_{i=1}^n c_i\left|x_{i j}\right|}\right]$ (12)
in which $i$ represented the number of input data and $j$ expressed the different sets. A degree $h$ was also defined with the following form, which determined that the experimental data should be included in the computed output $Y_j$.
$\mu_{\mathrm{Yj}}\left(\mathrm{y}_{\mathrm{j}}\right) \geq \mathrm{h}$ (13)
In order to define the best fuzzy coefficients for minimizing the whole spread of the outputs, the following objective function was used [40]:
$J=\min \left\{m c_0+\sum_{j=1}^m \sum_{i=1}^n c_i\left|x_{i j}\right|\right\}$ (14)
which was minimized with the Eqs. (15)-(17):
$y_j \geq r_0+\sum_{i=1}^n r_i x_{i j}-(1-h)\left(c_0+\sum_{i=1}^n c_i\left|x_{i j}\right|\right)$ (15)
$y_j \leq r_0+\sum_{i=1}^n r_i x_{i j}+(1-h)\left(c_0+\sum_{i=1}^n c_i\left|x_{i j}\right|\right)$ (16)
$c_i \geq 0, \quad i=1,2, \ldots, n$ (17)
As a result, taking into account the aforementioned theorems, the linear programming problem was formed with the Eqs. (14)-(17) which was solved with simplex method. In this study, Fuzzy Linear Regression was used to determine the relationship between the input parameters and the compressive strength of sustainable rice husk ash concrete, which contains uncertainties due to variability in the recycled materials.
Fuzzy Linear Regression (FLR) with triangular fuzzy numbers is a widely used method in solving mathematical problems by minimizing the square error among the experimental and predicted outputs. Its application is valid in the estimated value process as it gives an exact relation between the parameters that handles the perceptual uncertainties of linear programming problems [41]. This is achieved by using fuzzy coefficients which express the ambiguity of the relation between the parameters. In this study, FLR method was used as a new method for evaluating the implication between the amount of cement and the compressive strength of sustainable rice husk ash concrete.
The application of ecofriendly cementitious materials has gained the interest of many researchers in the construction industry for investigating more environmentally friendly substitutes [42, 43]. Rice husk ash concrete is a sustainable concrete material that is used instead of traditional concrete for reducing the emission of CO2 in the environment. It consists of recycled materials and cements substitutes without sacrificing the final strength of the material [44]. It is well demonstrated that compressive strength plays a vital role in the concrete quality procedure. Thus, the determination of the parameter contribution of the compressive strength evaluation is an important process. In a previous paper, Iftikhar et al. [24] studied the prediction of the compressive strength of sustainable rice husk ash concrete by using gene expression programming and Random Forest Regression method. The parameters that were used as inputs were the age of concrete, the amount of cement, the rice husk ash, water, super plasticizer and aggregate. The data were selected from published literature. In this study, same data were used for estimating the relation between the inputs and the compressive strength of sustainable rice husk ash concrete with FLR method.
Firstly, FLR method was applied with the following form according to the Eq. (7):
$Y=A_0+A_1 X_1+A_2 X_2+A_3 X_3+A_4 X_4+A_5 X_5+A_6 X_6$ (18)
in which Y was the compressive strength of RHA concrete, X1 was the age of concrete (days), X2 was the amount of cement (Kg/m3), X3 was rice husk ash (Kg/m3), X4 was water (Kg/m3), X5 was super plasticizer (Kg/m3) and X6 was aggregate (Kg/m3). Therefore, the following objective function was emerged:
$J=\min \left\{m c_0+\sum_{i=1}^{192} \sum_{i=1}^6 c_i\left|x_{i j}\right|\right\}$ (19)
By solving the linear programming problem with simplex method from the Eqs. (14)-(17), the estimated output values of Y and the fuzzy parameters were resulted, and the Eq. (7) took the following form:
$\begin{gathered}Y=-0.804+0.339 \cdot X_1+0.063 \cdot X_2+0.050 \cdot X_3-0.050 \cdot X_4+2.938 \cdot X_5 \\ +0.005 \cdot X_6+0.011 \cdot X_3+2.669 \cdot X_5+0.007 \cdot X_6\end{gathered}$ (20)
In Table 2, the calculations of fuzzy parameters were represented.
According to the results of fuzzy triangular numbers, it was worth noting that the variables X3, X5 and X6 contributed to the uncertainty of the Fuzzy Linear Regression ($\left(c_i \neq 0\right)$), while the other variables had $c_i$ values equal to 0. In the rare case where all $c_i$ values turned out to be 0, the model would not contain any fuzziness and would transform into a conventional multiple linear regression model.
Then, the degree of membership function, that specifies the grade to which the experimental output belongs to the fuzzy predicted values between the term [0, 1], was evaluated. Regarding the membership function, the distance of the estimated outputs from the center of the regression plays an important role in the determination of the membership function values. The closer the real demand is to the center of the regression, the higher the degree of participation, taking the value 1 when the real demand coincides with the center of the fuzzy regression and the value 0 when the real demand is on the boundaries of the fuzzy regression. The results of the Fuzzy Linear Regression model were summarized in the Appendix A and the boundaries and the center of the FLR method were shown in Figure 1.
Table 2. The results of the fuzzy triangular numbers
|
Variable |
Estimate Ri |
Estimate Ci |
|
A0 |
-0.804 |
0.000 |
|
A1 |
0.339 |
0.000 |
|
A2 |
0.063 |
0.000 |
|
A3 |
0.050 |
0.011 |
|
A4 |
-0.050 |
0.000 |
|
A5 |
2.938 |
2.669 |
|
A6 |
0.005 |
0.007 |
Figure 1. The parameters of FLR method
For assessing the accuracy of the regression and therefore the fuzzy implication, Theil’s inequality coefficient [45] was evaluated. The range of this coefficient is between 0 to 1. The smaller the coefficient is, the closer the predicted and experimental outputs are. For instance, when the Theil’s inequality coefficient is calculated to be 0, then the Fuzzy Linear Regression exhibits perfect predictive capability. However, in the case where the Theil’s inequality coefficient is estimated to be 1, the fitted Fuzzy Linear Regression model lacks any predictive capability. This coefficient was expressed as follows:
Theil's$U=\frac{\sqrt{\frac{1}{n} \sum_{t=1}^n\left(F_t-A_t\right)^2}}{\sqrt{\frac{1}{n} \sum_{t=1}^n\left(F_t\right)^2}+\sqrt{\frac{1}{n} \sum_{t=1}^n\left(A_t\right)^2}}=0.1$ (21)
in which $A_t$ and $F_t$ were the experimental and predicted variables, respectively, and n was the number of observations. By evaluating the result, it was concluded that the application of Fuzzy Linear Regression method yielded accurate outputs, valid fuzzy parameter and reliable values of membership function, as the value of Theil’s inequality coefficient was low, and hence, the implication provided satisfactory results.
As it proved, Fuzzy Linear Regression can be used not only as prediction method but also as an algorithmic process for evaluating the truth value of every variable. More specifically, the aforementioned method uses a specific equation with fuzzy coefficients for determining a fuzzy output Y for every experimental output, contrary to predictive black box methods based on machine learning [46, 47]. These observed parameters belong to the fuzzy output with a specific degree of truth which represents the truth value of the total fuzzy implication.
Also, the application of Fuzzy Linear Regression as implication method deals with the uncertainties that are involved in dividing data into linguistic parameters. In this study, the determination of linguistic parameters was unnecessary, as the truth values that were used for the implication process were resulted directly from evaluating the fuzzy outputs. Thus, the evaluation of the membership function emerged from applying FLR method without having to divide data into ascending order and set language variables. Another advantage of this method is that it evaluates the truth value between the output and many different inputs, in contrast to the other implication methods in which the degree of truth is among the output and one input. Thus, FLR method is a valid mathematical process not only for prediction problems, but it can also be used as a successful tool for developing approximate reasoning.
The estimation of an appropriate fuzzy implication is a simplex procedure as it includes many parameters for applying the best mathematical process. In this paper, Fuzzy Linear Regression with triangular fuzzy numbers was used for evaluating the relation between the parameters that affect concrete and the compressive strength of sustainable rice husk ash concrete. Particularly, six inputs such as the age of concrete, the amount of cement, the rice husk ash, water, super plasticizer and aggregate were applied for determining the concrete’s compressive strength. By developing FLR method, each experimental output, that came out from previous publication [24], belonged to a fuzzy output with a specific truth value, which turned out to be the degree of truth of the entire fuzzy implication. This approach was based on the fuzzy implication axiom, named neutrality of truth.
It was also reported that FLR method provides an equation with fuzzy parameters that demonstrates the way that the truth value emerged, which proves the accuracy of the method. Moreover, by applying this method the separation of inputs into language variables is unnecessary which is a useful property in case that there is no information about the division of parameters. Since Fuzzy Linear Regression establishes a direct relationship between inputs and outputs and the membership functions are calculated from the fuzzy outputs, it eliminates the need to partition the inputs into linguistic variables as required in some other fuzzy implication methods. In addition, the use of Fuzzy Linear Regression allows the evaluation of the degree of truth between one output and more than one inputs contrary to the other implication methods that determine the relation among the output and one input.
To sum up, FLR method led to accurate predictions between the coefficients that are involved in concrete construction and the compressive strength of sustainable rice husk ash concrete. According to the results, Fuzzy Linear Regression could effectively handle multiple inputs and determined the degree of truth between the inputs and output. The low value of the Theil’s inequality coefficient (Theil’s U=0.1) proved the capability of this method to be used as fuzzy implication for determining the truth value of the consequence and provided an accurate modelling tool in the approximate reasoning process.
|
I |
fuzzy implication |
|
Y |
dependent output |
|
X |
independent input |
|
Ai ri |
triangular fuzzy numbers center of triangular number |
|
ci |
range of values |
|
L |
reference function |
|
h |
level of confidence |
|
J |
objective function |
|
Theil’s U |
inequality coefficient |
|
Ft |
predicted outputs |
|
At |
experimental outputs |
|
Greek Symbols |
|
|
μΑ |
membership function |
|
Subscripts |
|
|
FLR |
Fuzzy Linear Regression |
|
RHA |
Rice Husk Ash |
Appendix A. Compressive strength of sustainable rice husk ash concrete as evaluated by the FLR method across 192 observations
|
Age (days) |
Cement (Kg/m3) |
RHA (Kg/m3) |
Water (Kg/m3) |
Superplasticizer (Kg/m3) |
Aggregate (Kg/m3) |
Compr. Strength (Kg/m3) |
FLR (Left) |
FLR (Center) |
FLR (Right) |
μΑ(yi) |
|
1 |
495 |
55 |
165 |
5.8 |
1819 |
22.7 |
22.5 |
52.0 |
80.2 |
0.0 |
|
1 |
500 |
0 |
160 |
5.5 |
1891 |
20.9 |
20.7 |
49.3 |
76.6 |
0.0 |
|
1 |
400 |
100 |
160 |
6.22 |
1859 |
22 |
18.6 |
49.9 |
80.0 |
0.1 |
|
1 |
425 |
75 |
170 |
5 |
1843 |
19.7 |
18.4 |
46.1 |
72.5 |
0.0 |
|
1 |
495 |
55 |
165 |
6.8 |
1819 |
34.7 |
22.8 |
55.0 |
85.8 |
0.4 |
|
1 |
500 |
0 |
160 |
6.5 |
1891 |
37.8 |
21.0 |
52.2 |
82.2 |
0.5 |
|
1 |
400 |
100 |
160 |
7.36 |
1859 |
34.2 |
18.9 |
53.3 |
86.4 |
0.4 |
|
1 |
425 |
75 |
170 |
6.4 |
1843 |
32.6 |
18.8 |
50.3 |
80.4 |
0.4 |
|
3 |
495 |
55 |
165 |
5.8 |
1819 |
47.9 |
23.2 |
52.7 |
80.9 |
0.8 |
|
3 |
500 |
0 |
160 |
5.5 |
1891 |
41.3 |
21.4 |
50.0 |
77.2 |
0.7 |
|
3 |
400 |
100 |
160 |
6.22 |
1859 |
48.7 |
19.3 |
50.6 |
80.7 |
0.9 |
|
3 |
425 |
75 |
170 |
5 |
1843 |
45.2 |
19.1 |
46.8 |
73.2 |
0.9 |
|
3 |
378 |
42 |
189 |
0 |
1810 |
17.6 |
12.6 |
26.5 |
38.9 |
0.4 |
|
3 |
495 |
55 |
165 |
6.8 |
1819 |
60.8 |
23.5 |
55.6 |
86.5 |
0.8 |
|
3 |
500 |
0 |
160 |
6.5 |
1891 |
63.9 |
21.7 |
52.9 |
82.9 |
0.6 |
|
3 |
400 |
100 |
160 |
7.36 |
1859 |
60.7 |
19.6 |
54.0 |
87.1 |
0.8 |
|
3 |
425 |
75 |
170 |
6.4 |
1843 |
57.3 |
19.4 |
50.9 |
81.1 |
0.8 |
|
7 |
495 |
55 |
165 |
5.8 |
1819 |
60.6 |
24.6 |
54.0 |
82.2 |
0.8 |
|
7 |
500 |
0 |
160 |
5.5 |
1891 |
51 |
22.8 |
51.3 |
78.6 |
1.0 |
|
7 |
400 |
100 |
160 |
6.22 |
1859 |
61.8 |
20.6 |
52.0 |
82.1 |
0.7 |
|
7 |
375 |
0 |
150 |
0 |
1970 |
30 |
13.8 |
28.1 |
41.3 |
0.9 |
|
7 |
356.25 |
18.75 |
142.5 |
0 |
1970 |
31.5 |
13.7 |
28.2 |
41.7 |
0.8 |
|
7 |
337.5 |
37.5 |
135 |
0 |
1970 |
32.5 |
13.6 |
28.3 |
42.0 |
0.7 |
|
7 |
318.75 |
56.25 |
127.5 |
0 |
1970 |
35.5 |
13.5 |
28.4 |
42.3 |
0.5 |
|
7 |
300 |
75 |
120 |
0 |
1970 |
31 |
13.5 |
28.5 |
42.7 |
0.8 |
|
7 |
364 |
19 |
203 |
0 |
1725 |
27.6 |
11.6 |
24.7 |
36.2 |
0.8 |
|
7 |
306 |
77 |
203 |
0 |
1725 |
29.7 |
10.3 |
24.0 |
36.1 |
0.5 |
|
7 |
249 |
134 |
203 |
0 |
1725 |
25.7 |
8.9 |
23.2 |
36.0 |
0.8 |
|
7 |
425 |
75 |
170 |
5 |
1843 |
57.6 |
20.4 |
48.2 |
74.6 |
0.6 |
|
7 |
495 |
55 |
165 |
6.8 |
1819 |
83.6 |
24.8 |
57.0 |
87.8 |
0.1 |
|
7 |
391 |
29 |
189 |
0 |
1810 |
32.4 |
14.3 |
28.0 |
40.2 |
0.6 |
|
7 |
500 |
0 |
160 |
6.5 |
1891 |
76.4 |
23.0 |
54.3 |
84.2 |
0.3 |
|
7 |
400 |
100 |
160 |
7.36 |
1859 |
82.8 |
20.9 |
55.3 |
88.4 |
0.2 |
|
7 |
425 |
75 |
170 |
6.4 |
1843 |
79.2 |
20.8 |
52.3 |
82.4 |
0.1 |
|
14 |
345 |
38 |
203 |
0 |
1725 |
35.3 |
13.6 |
26.9 |
38.5 |
0.3 |
|
14 |
287 |
96 |
203 |
0 |
1725 |
36.1 |
12.2 |
26.1 |
38.4 |
0.2 |
|
28 |
495 |
55 |
165 |
5.8 |
1819 |
72.8 |
31.7 |
61.2 |
89.3 |
0.6 |
|
28 |
500 |
0 |
160 |
5.5 |
1891 |
59.6 |
29.9 |
58.4 |
85.7 |
1.0 |
|
28 |
400 |
100 |
160 |
6.22 |
1859 |
72.7 |
27.7 |
59.1 |
89.2 |
0.5 |
|
28 |
375 |
0 |
150 |
0 |
1970 |
44.5 |
20.9 |
35.3 |
48.5 |
0.3 |
|
28 |
356.25 |
18.75 |
142.5 |
0 |
1970 |
45.5 |
20.8 |
35.4 |
48.8 |
0.2 |
|
28 |
337.5 |
37.5 |
135 |
0 |
1970 |
49.5 |
20.7 |
35.5 |
49.5 |
0.0 |
|
28 |
318.75 |
56.25 |
127.5 |
0 |
1970 |
50 |
20.6 |
35.6 |
50.0 |
0.0 |
|
28 |
300 |
75 |
120 |
0 |
1970 |
43 |
20.6 |
35.7 |
49.8 |
0.5 |
|
28 |
383 |
0 |
203 |
0 |
1725 |
37.1 |
19.2 |
32.1 |
43.4 |
0.6 |
|
28 |
326 |
57 |
203 |
0 |
1725 |
41.8 |
17.8 |
31.4 |
43.3 |
0.1 |
|
28 |
268 |
115 |
203 |
0 |
1725 |
37.6 |
16.5 |
30.6 |
43.1 |
0.4 |
|
28 |
425 |
75 |
170 |
5 |
1843 |
67.2 |
27.5 |
55.3 |
81.7 |
0.5 |
|
28 |
495 |
55 |
165 |
6.8 |
1819 |
95.2 |
32.0 |
64.1 |
95.2 |
0.0 |
|
28 |
500 |
0 |
160 |
6.5 |
1891 |
85.7 |
30.2 |
61.4 |
91.3 |
0.2 |
|
28 |
400 |
100 |
160 |
7.36 |
1859 |
94.3 |
28.0 |
62.4 |
95.6 |
0.0 |
|
28 |
420 |
0 |
189 |
0 |
1810 |
40.3 |
22.1 |
35.5 |
47.4 |
0.6 |
|
28 |
357 |
63 |
189 |
0 |
1810 |
46.9 |
20.6 |
34.7 |
47.3 |
0.0 |
|
28 |
425 |
75 |
170 |
6.4 |
1843 |
90.3 |
27.9 |
59.4 |
90.3 |
0.0 |
|
56 |
375 |
0 |
150 |
0 |
1970 |
51.5 |
30.4 |
44.8 |
57.9 |
0.5 |
|
56 |
356.25 |
18.75 |
142.5 |
0 |
1970 |
53.5 |
30.3 |
44.9 |
58.3 |
0.4 |
|
56 |
337.5 |
37.5 |
135 |
0 |
1970 |
56 |
30.2 |
45.0 |
58.6 |
0.2 |
|
56 |
318.75 |
56.25 |
127.5 |
0 |
1970 |
59.5 |
30.1 |
45.1 |
59.5 |
0.0 |
|
56 |
300 |
75 |
120 |
0 |
1970 |
52 |
30.1 |
45.2 |
59.3 |
0.5 |
|
90 |
495 |
55 |
165 |
5.8 |
1819 |
83.2 |
52.7 |
82.2 |
110.3 |
1.0 |
|
90 |
500 |
0 |
160 |
5.5 |
1891 |
66.8 |
50.9 |
79.5 |
106.7 |
0.6 |
|
90 |
400 |
100 |
160 |
6.22 |
1859 |
82.2 |
48.8 |
80.1 |
110.2 |
0.9 |
|
90 |
425 |
75 |
170 |
5 |
1843 |
75.8 |
48.6 |
76.3 |
102.7 |
1.0 |
|
90 |
364 |
19 |
203 |
0 |
1725 |
43.3 |
39.8 |
52.9 |
64.3 |
0.3 |
|
90 |
306 |
77 |
203 |
0 |
1725 |
46 |
38.4 |
52.1 |
64.2 |
0.6 |
|
90 |
249 |
134 |
203 |
0 |
1725 |
37.2 |
37.0 |
51.4 |
64.1 |
0.0 |
|
90 |
375 |
0 |
150 |
0 |
1970 |
55.5 |
41.9 |
56.3 |
69.5 |
0.9 |
|
90 |
356.25 |
18.75 |
142.5 |
0 |
1970 |
56.5 |
41.8 |
56.4 |
69.8 |
1.0 |
|
90 |
337.5 |
37.5 |
135 |
0 |
1970 |
63 |
41.7 |
56.5 |
70.1 |
0.5 |
|
90 |
318.75 |
56.25 |
127.5 |
0 |
1970 |
64 |
41.7 |
56.6 |
70.5 |
0.5 |
|
90 |
300 |
75 |
120 |
0 |
1970 |
61 |
41.6 |
56.7 |
70.8 |
0.7 |
|
90 |
378 |
42 |
189 |
0 |
1810 |
59 |
42.1 |
56.0 |
68.4 |
0.8 |
|
90 |
495 |
55 |
165 |
6.8 |
1819 |
104.1 |
53.0 |
85.1 |
116.0 |
0.4 |
|
90 |
500 |
0 |
160 |
6.5 |
1891 |
94 |
51.2 |
82.4 |
112.3 |
0.6 |
|
90 |
400 |
100 |
160 |
7.36 |
1859 |
103.3 |
49.1 |
83.5 |
116.6 |
0.4 |
|
90 |
425 |
75 |
170 |
6.4 |
1843 |
99.1 |
48.9 |
80.4 |
110.6 |
0.4 |
|
7 |
481 |
48.1 |
169.312 |
3.367 |
1040 |
39.5 |
24.1 |
41.6 |
57.7 |
0.9 |
|
7 |
427 |
85.4 |
163.968 |
3.416 |
1040 |
30.5 |
22.4 |
40.4 |
57.1 |
0.4 |
|
7 |
416 |
41.6 |
183.04 |
1.1232 |
1041 |
29.7 |
18.5 |
29.9 |
40.0 |
1.0 |
|
7 |
370 |
74 |
177.6 |
1.85 |
1041 |
23.6 |
17.3 |
31.0 |
43.4 |
0.5 |
|
7 |
367 |
36.7 |
201.85 |
1.101 |
1041 |
22.7 |
14.2 |
25.7 |
35.5 |
0.7 |
|
7 |
327 |
65.4 |
196.2 |
1.308 |
1041 |
20.8 |
13.2 |
25.5 |
36.2 |
0.6 |
|
28 |
481 |
48.1 |
169.312 |
3.367 |
1040 |
51.4 |
31.2 |
48.7 |
64.8 |
0.8 |
|
28 |
427 |
85.4 |
163.968 |
3.416 |
1040 |
47.4 |
29.6 |
47.6 |
64.2 |
1.0 |
|
28 |
416 |
41.6 |
183.04 |
1.1232 |
1041 |
40.8 |
25.6 |
37.1 |
47.1 |
0.6 |
|
28 |
370 |
74 |
177.6 |
1.85 |
1041 |
39.4 |
24.4 |
38.2 |
50.5 |
0.9 |
|
28 |
367 |
36.7 |
201.85 |
1.101 |
1041 |
34.5 |
21.4 |
32.8 |
42.6 |
0.8 |
|
28 |
327 |
65.4 |
196.2 |
1.308 |
1041 |
35.9 |
20.3 |
32.6 |
43.3 |
0.7 |
|
90 |
481 |
48.1 |
169.312 |
3.367 |
1040 |
64.5 |
52.2 |
69.7 |
85.8 |
0.7 |
|
90 |
427 |
85.4 |
163.968 |
3.416 |
1040 |
68.5 |
50.6 |
68.6 |
85.3 |
1.0 |
|
90 |
416 |
41.6 |
183.04 |
1.1232 |
1041 |
51.5 |
46.6 |
58.1 |
68.1 |
0.4 |
|
90 |
370 |
74 |
177.6 |
1.85 |
1041 |
57.3 |
45.4 |
59.2 |
71.5 |
0.9 |
|
90 |
367 |
36.7 |
201.85 |
1.101 |
1041 |
44.4 |
42.4 |
53.8 |
63.6 |
0.2 |
|
90 |
327 |
65.4 |
196.2 |
1.308 |
1041 |
52.9 |
41.3 |
53.6 |
64.3 |
0.9 |
|
1 |
450 |
0 |
238 |
11.25 |
1405 |
31.5 |
16.2 |
57.0 |
95.9 |
0.4 |
|
1 |
427.5 |
21.375 |
238 |
10.6875 |
1405 |
32.1 |
15.5 |
55.0 |
92.7 |
0.4 |
|
1 |
405 |
40.5 |
238 |
10.125 |
1405 |
33.3 |
14.6 |
52.9 |
89.3 |
0.5 |
|
1 |
382.5 |
57.375 |
238 |
9.5625 |
1405 |
34.5 |
13.7 |
50.7 |
85.7 |
0.6 |
|
1 |
360 |
72 |
238 |
9 |
1405 |
33.6 |
12.7 |
48.3 |
82.0 |
0.6 |
|
1 |
337.5 |
84.375 |
238 |
8.4375 |
1405 |
29.3 |
11.6 |
45.9 |
78.2 |
0.5 |
|
1 |
315 |
94.5 |
238 |
7.875 |
1405 |
29 |
10.5 |
43.3 |
74.3 |
0.6 |
|
28 |
450 |
0 |
238 |
11.25 |
1405 |
41.7 |
25.4 |
66.2 |
105.1 |
0.4 |
|
28 |
427.5 |
21.375 |
238 |
10.6875 |
1405 |
42.7 |
24.6 |
64.2 |
101.8 |
0.5 |
|
28 |
405 |
40.5 |
238 |
10.125 |
1405 |
44.2 |
23.8 |
62.1 |
98.4 |
0.5 |
|
28 |
382.5 |
57.375 |
238 |
9.5625 |
1405 |
46.8 |
22.9 |
59.8 |
94.9 |
0.6 |
|
28 |
360 |
72 |
238 |
9 |
1405 |
43.5 |
21.9 |
57.5 |
91.2 |
0.6 |
|
28 |
337.5 |
84.375 |
238 |
8.4375 |
1405 |
39.5 |
20.8 |
55.0 |
87.4 |
0.5 |
|
28 |
315 |
94.5 |
238 |
7.875 |
1405 |
38.2 |
19.6 |
52.5 |
83.4 |
0.6 |
|
56 |
450 |
0 |
238 |
11.25 |
1405 |
49.1 |
34.8 |
75.7 |
114.6 |
0.3 |
|
56 |
427.5 |
21.375 |
238 |
10.6875 |
1405 |
50.2 |
34.1 |
73.7 |
111.3 |
0.4 |
|
56 |
405 |
40.5 |
238 |
10.125 |
1405 |
52.1 |
33.3 |
71.5 |
107.9 |
0.5 |
|
56 |
382.5 |
57.375 |
238 |
9.5625 |
1405 |
55.3 |
32.4 |
69.3 |
104.4 |
0.6 |
|
56 |
360 |
72 |
238 |
9 |
1405 |
55.2 |
31.4 |
67.0 |
100.7 |
0.7 |
|
56 |
337.5 |
84.375 |
238 |
8.4375 |
1405 |
47 |
30.3 |
64.5 |
96.9 |
0.5 |
|
56 |
315 |
94.5 |
238 |
7.875 |
1405 |
45.9 |
29.1 |
62.0 |
92.9 |
0.5 |
|
90 |
450 |
0 |
238 |
11.25 |
1405 |
52.6 |
46.4 |
87.2 |
126.1 |
0.2 |
|
90 |
427.5 |
21.375 |
238 |
10.6875 |
1405 |
54.9 |
45.6 |
85.2 |
122.8 |
0.2 |
|
90 |
405 |
40.5 |
238 |
10.125 |
1405 |
57.3 |
44.8 |
83.1 |
119.4 |
0.3 |
|
90 |
382.5 |
57.375 |
238 |
9.5625 |
1405 |
61.2 |
43.9 |
80.8 |
115.9 |
0.5 |
|
90 |
360 |
72 |
238 |
9 |
1405 |
55.5 |
42.9 |
78.5 |
112.2 |
0.4 |
|
90 |
337.5 |
84.375 |
238 |
8.4375 |
1405 |
51.9 |
41.8 |
76.1 |
108.4 |
0.3 |
|
90 |
315 |
94.5 |
238 |
7.875 |
1405 |
50.2 |
40.6 |
73.5 |
104.4 |
0.3 |
|
1 |
783 |
87 |
212 |
3.6 |
1277 |
41 |
40.1 |
60.4 |
79.1 |
0.0 |
|
1 |
571 |
0 |
219 |
1 |
1566 |
30 |
21.7 |
36.2 |
49.0 |
0.6 |
|
1 |
514 |
57 |
218 |
1.4 |
1541 |
27 |
20.5 |
36.6 |
50.8 |
0.4 |
|
1 |
457 |
114 |
216 |
2.6 |
1515 |
26 |
19.6 |
39.3 |
57.2 |
0.3 |
|
1 |
400 |
171 |
215 |
3.7 |
1490 |
19 |
18.7 |
41.7 |
63.0 |
0.0 |
|
1 |
383 |
42 |
221 |
0.3 |
1670 |
16 |
11.0 |
24.8 |
36.9 |
0.4 |
|
3 |
783 |
87 |
212 |
3.6 |
1277 |
59 |
40.7 |
61.1 |
79.8 |
0.9 |
|
3 |
571 |
0 |
219 |
1 |
1566 |
46 |
22.4 |
36.9 |
49.6 |
0.3 |
|
3 |
514 |
57 |
218 |
1.4 |
1541 |
41 |
21.2 |
37.2 |
51.5 |
0.7 |
|
3 |
457 |
114 |
216 |
2.6 |
1515 |
38 |
20.3 |
40.0 |
57.9 |
0.9 |
|
3 |
400 |
171 |
215 |
3.7 |
1490 |
32 |
19.3 |
42.4 |
63.7 |
0.5 |
|
3 |
383 |
42 |
221 |
0.3 |
1670 |
26 |
11.7 |
25.5 |
37.6 |
1.0 |
|
7 |
783 |
87 |
212 |
3.6 |
1277 |
62 |
42.1 |
62.5 |
81.1 |
1.0 |
|
7 |
571 |
0 |
219 |
1 |
1566 |
50 |
23.7 |
38.2 |
51.0 |
0.1 |
|
7 |
514 |
57 |
218 |
1.4 |
1541 |
47 |
22.6 |
38.6 |
52.9 |
0.4 |
|
7 |
457 |
114 |
216 |
2.6 |
1515 |
47 |
21.7 |
41.3 |
59.3 |
0.7 |
|
7 |
400 |
171 |
215 |
3.7 |
1490 |
43 |
20.7 |
43.7 |
65.1 |
1.0 |
|
7 |
383 |
42 |
221 |
0.3 |
1670 |
37 |
13.0 |
26.9 |
38.9 |
0.2 |
|
14 |
783 |
87 |
212 |
3.6 |
1277 |
63 |
44.5 |
64.8 |
83.5 |
0.9 |
|
14 |
571 |
0 |
219 |
1 |
1566 |
54 |
26.1 |
40.6 |
54.0 |
0.0 |
|
14 |
514 |
57 |
218 |
1.4 |
1541 |
52 |
24.9 |
41.0 |
55.2 |
0.2 |
|
14 |
457 |
114 |
216 |
2.6 |
1515 |
52 |
24.0 |
43.7 |
61.6 |
0.5 |
|
14 |
400 |
171 |
215 |
3.7 |
1490 |
51 |
23.1 |
46.1 |
67.4 |
0.8 |
|
14 |
383 |
42 |
221 |
0.3 |
1670 |
40 |
15.4 |
29.2 |
41.3 |
0.1 |
|
28 |
783 |
87 |
212 |
3.6 |
1277 |
66 |
49.2 |
69.6 |
88.2 |
0.8 |
|
28 |
571 |
0 |
219 |
1 |
1566 |
56 |
30.8 |
45.4 |
58.1 |
0.2 |
|
28 |
514 |
57 |
218 |
1.4 |
1541 |
61 |
29.7 |
45.7 |
61.0 |
0.0 |
|
28 |
457 |
114 |
216 |
2.6 |
1515 |
60 |
28.8 |
48.5 |
66.4 |
0.4 |
|
28 |
400 |
171 |
215 |
3.7 |
1490 |
54 |
27.8 |
50.9 |
72.2 |
0.9 |
|
28 |
383 |
42 |
221 |
0.3 |
1670 |
47 |
20.1 |
34.0 |
47.0 |
0.0 |
|
56 |
783 |
87 |
212 |
3.6 |
1277 |
69 |
58.7 |
79.1 |
97.7 |
0.5 |
|
56 |
571 |
0 |
219 |
1 |
1566 |
60 |
40.3 |
54.8 |
67.6 |
0.6 |
|
56 |
514 |
57 |
218 |
1.4 |
1541 |
62 |
39.2 |
55.2 |
69.5 |
0.5 |
|
56 |
457 |
114 |
216 |
2.6 |
1515 |
61 |
38.3 |
57.9 |
75.9 |
0.8 |
|
56 |
400 |
171 |
215 |
3.7 |
1490 |
60 |
37.3 |
60.4 |
81.7 |
1.0 |
|
56 |
383 |
42 |
221 |
0.3 |
1670 |
51 |
29.6 |
43.5 |
55.5 |
0.4 |
|
90 |
783 |
87 |
212 |
3.6 |
1277 |
74 |
70.2 |
90.6 |
109.3 |
0.2 |
|
90 |
571 |
0 |
219 |
1 |
1566 |
67 |
51.9 |
66.4 |
79.1 |
1.0 |
|
90 |
514 |
57 |
218 |
1.4 |
1541 |
67 |
50.7 |
66.7 |
81.0 |
1.0 |
|
90 |
457 |
114 |
216 |
2.6 |
1515 |
69 |
49.8 |
69.5 |
87.4 |
1.0 |
|
90 |
400 |
171 |
215 |
3.7 |
1490 |
64 |
48.8 |
71.9 |
93.2 |
0.7 |
|
90 |
383 |
42 |
221 |
0.3 |
1670 |
56 |
41.2 |
55.0 |
67.1 |
0.9 |
|
7 |
364 |
19 |
203 |
0 |
1725 |
27.6 |
11.6 |
24.7 |
36.2 |
0.8 |
|
7 |
345 |
38 |
203 |
0 |
1725 |
28 |
11.2 |
24.5 |
36.2 |
0.7 |
|
7 |
326 |
57 |
203 |
0 |
1725 |
29.3 |
10.7 |
24.2 |
36.1 |
0.6 |
|
7 |
306 |
77 |
203 |
0 |
1725 |
29.7 |
10.3 |
24.0 |
36.1 |
0.5 |
|
7 |
287 |
96 |
203 |
0 |
1725 |
28.7 |
9.8 |
23.7 |
36.1 |
0.6 |
|
7 |
268 |
115 |
203 |
0 |
1725 |
27.4 |
9.3 |
23.5 |
36.0 |
0.7 |
|
7 |
249 |
134 |
203 |
0 |
1725 |
25.7 |
8.9 |
23.2 |
36.0 |
0.8 |
|
14 |
364 |
19 |
203 |
0 |
1725 |
34.2 |
14.0 |
27.1 |
38.6 |
0.4 |
|
14 |
345 |
38 |
203 |
0 |
1725 |
35.3 |
13.6 |
26.9 |
38.5 |
0.3 |
|
14 |
326 |
57 |
203 |
0 |
1725 |
36 |
13.1 |
26.6 |
38.5 |
0.2 |
|
14 |
306 |
77 |
203 |
0 |
1725 |
39.3 |
12.6 |
26.4 |
39.3 |
0.0 |
|
14 |
287 |
96 |
203 |
0 |
1725 |
36.1 |
12.2 |
26.1 |
38.4 |
0.2 |
|
14 |
268 |
115 |
203 |
0 |
1725 |
33.5 |
11.7 |
25.9 |
38.4 |
0.4 |
|
14 |
249 |
134 |
203 |
0 |
1725 |
31.1 |
11.3 |
25.6 |
38.4 |
0.6 |
|
28 |
364 |
19 |
203 |
0 |
1725 |
40 |
18.8 |
31.9 |
43.3 |
0.3 |
|
28 |
345 |
38 |
203 |
0 |
1725 |
41.3 |
18.3 |
31.6 |
43.3 |
0.2 |
|
28 |
326 |
57 |
203 |
0 |
1725 |
41.8 |
17.8 |
31.4 |
43.3 |
0.1 |
|
28 |
306 |
77 |
203 |
0 |
1725 |
42.5 |
17.4 |
31.1 |
43.2 |
0.1 |
|
28 |
287 |
96 |
203 |
0 |
1725 |
38.8 |
16.9 |
30.9 |
43.2 |
0.4 |
|
28 |
268 |
115 |
203 |
0 |
1725 |
37.6 |
16.5 |
30.6 |
43.1 |
0.4 |
|
28 |
249 |
134 |
203 |
0 |
1725 |
35.1 |
16.0 |
30.4 |
43.1 |
0.6 |
|
90 |
364 |
19 |
203 |
0 |
1725 |
43.3 |
39.8 |
52.9 |
64.3 |
0.3 |
|
90 |
345 |
38 |
203 |
0 |
1725 |
44.8 |
39.3 |
52.6 |
64.3 |
0.4 |
|
90 |
326 |
57 |
203 |
0 |
1725 |
45.7 |
38.9 |
52.4 |
64.3 |
0.5 |
|
90 |
306 |
77 |
203 |
0 |
1725 |
46 |
38.4 |
52.1 |
64.2 |
0.6 |
|
90 |
287 |
96 |
203 |
0 |
1725 |
43 |
37.9 |
51.9 |
64.2 |
0.4 |
|
90 |
268 |
115 |
203 |
0 |
1725 |
38.7 |
37.5 |
51.6 |
64.2 |
0.1 |
|
90 |
249 |
134 |
203 |
0 |
1725 |
37.2 |
37.2 |
51.4 |
64.1 |
0.0 |
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