# Generator of Fuzzy Implications

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Literature Review-Related Work

_{i}: [0, 1] → [0, ∞), (g(0) = 0) (i = 1, 2, :::, n, n ∈ N) and n + 1 fuzzy negations N

_{i}(i = 1, 2, :::, n + 1, n ∈ N). This method allows authors to use at least two fuzzy negations Ni and one increasing function g in order to generate a new fuzzy implication. Bedregal et al. [7] showed a method of S-implication using two S-implications. The resulting implication S is satisfactory. The new implication is applied to the soundness property and some properties of the known S-implication. Balasubramaniam [8] investigated the conditions under which natural negation is transformed so that implication becomes equal to strong negation. Sufficient conditions are also presented for fuzzy disjunctions to become t-conorms. Jayaram and Mesiar [9] showed that various fuzzy implications are transformable, and gave methods of creating special implications from the given. Wang et al. [10] develop a fast method of intuitive fuzzy clustering analysis. Examples are also given to illustrate and verify their results. Shi et al. [11] showed a fuzzy implication defined as a two-position function on the interval [0, 1]; the authors obtained an extension of the classical binary implication. This paper aimed to highlight the interaction of the eight fuzzy axioms. Fernandez-Peralta et al. [12] presented the family of fuzzy implications in which the central idea is the existence of the completion of a binary function defined on a certain subregion of [0, 1]. Fernandez-Sanchez et al. [13] complemented and generalized some fuzzy implication constructions based on two arbitrary pairs, obtaining new fuzzy implication. Thus, they outlined a general method for constructing fuzzy implications. Madrid and Cornelis [14] refuted the theory of Fodor and Yager that the class of integration measures proposed by Kitainik coincides with that of integration measures based on contrastively fuzzy implications. Pinheiro et al. [15] formulated various distinctive techniques for generating fuzzy implication functions. Zhao and Lu [16] presented a new fuzzy implications construction method that, compared to others, has many advantages. These satisfy the conditions for the resolution of the distribution equations involving fuzzy implications constructed by Drygas and Krol. Massanet et al. [17] presented fuzzy polynomial implications given by a polynomial of two variables. Souliotis and Papadopoulos [18] constructed a new method of generating fuzzy implications based on a given fuzzy negation. So, they made rules aimed at regulation and decision-making, adjusting mathematics to common human logic. Krol [19] dealt with some functions of fuzzy implication within the laws of propositional calculus, leading to new fuzzy implications. Souliotis and Papadopoulos [20] discovered simple mathematical and computational procedures as well as strong fuzzy implications with the help of geometric concepts such as ellipticity and hyperbola.

#### 1.2. Paper Outline

## 2. Theory—New Fuzzy Implication Methods

#### 2.1. Theoretical Framework of Fuzzy Implication

**Definition**

**1.**

- If ${\omega}_{1}\le {\omega}_{2}$ then $f\left({\omega}_{1},y\right)\ge f\left({\omega}_{2},y\right)$ (decreasing as to the first variable);
- If ${\omega}_{1}\le {\omega}_{2}$ then $f\left(x,{\omega}_{1}\right)\le f\left(x,{\omega}_{2}\right)$ (increasing as to the second variable);
- $f\left(0,{\omega}_{1}\right)=1$;
- $f\left(1,{\omega}_{1}\right)=a$;
- $f\left({\omega}_{1},{\omega}_{1}\right)=1$;
- $f\left({\omega}_{1},f\left({\omega}_{2},x\right)\right)=f\left({\omega}_{2},f\left({\omega}_{1},x\right)\right)$;
- If $f\left({\omega}_{1},{\omega}_{2}\right)=1$ then ${\omega}_{1}\le {\omega}_{2};$
- $f\left({\omega}_{1},{\omega}_{2}\right)=f\left(n\left({\omega}_{2}\right),n\left({\omega}_{1}\right)\right)$;
- The function $f$ is continuous.

**Definition**

**2.**

- n(0) = 1 and $n\left(1\right)=0$;
- $n\left(n\left(x\right)\right)=\left(n\circ n\right)\left(x\right)=x,\forall x\in \left[0,1\right]$;
- The n is a genuinely decreasing function.

**Definition**

**3.**

- $x\vee y=y\vee x,\forall x,y\in \left[0,1\right]$ (commutativity property);
- $x\vee (y\vee z)=(x\vee y)\vee z,\forall x,y,z\in \left[0,1\right]$ (associative property);
- $x\vee 0=x,\forall x\in \left[0,1\right]$ (border condition);
- if $\left\{\begin{array}{c}x\le y\\ \omega \le \phi \end{array}\right\}\Rightarrow x\vee \omega \le y\vee \phi ,\forall x,y,\omega ,\phi \in \left[0,1\right]$ (monotonicity);
- Such or satisfying all the above properties is the probor x∨y = x + y − xy.

#### 2.2. The New Proposed Family of Fuzzy Implication

**Theorem**

**1.**

**Proof**

**of Theorem 1.**

- ◾
- For m = 2 the authors have:$$yVy={\widehat{y}}^{2}=y+y-y\xb7y=2y-{y}^{2}$$
- ◾
- For m = 3 the researchers have:$$yVyVy={\widehat{y}}^{3}=2y-{y}^{2}+y-(2y-{y}^{2})\xb7y=3y-3{y}^{2}+{y}^{3}$$
- ◾
- For m = 4 the authors have:$$yVyVyVy={\widehat{y}}^{4}=3y-3{y}^{2}+{y}^{3}+y-(3y-3{y}^{2}+{y}^{3})\xb7y=4y-6{y}^{2}+4{y}^{3}-{y}^{4}$$
- ◾
- For m = 5 the researchers have:$$\mathit{yVyVyVyVy}={\widehat{y}}^{5}=4y-6{y}^{2}+4{y}^{3}-{y}^{4}+y-(4y-6{y}^{2}+4{y}^{3}-{y}^{4})\xb7y=5y-10{y}^{2}+10{y}^{3}-5{y}^{4}+{y}^{5}$$
- ◾
- For m = 6 the authors have:$$\mathit{yVyVyVyVyVy}={\widehat{y}}^{6}=5y-10{y}^{2}+10{y}^{3}-5{y}^{4}+{y}^{5}+y-(5y-10{y}^{2}+10{y}^{3}-5{y}^{4}+{y}^{5})\xb7y=6y-15{y}^{2}+20{y}^{3}-15{y}^{4}+6{y}^{5}-{y}^{6}$$

- The concept of monotonicity is studied with respect to the first variable, and consequently, with respect to x, we consider 0 < x
_{1}< x_{2}so −x_{1}> −x_{2}⇔1 − x_{1}> 1 − x_{2}, that is, n(x_{1}) > n(x_{2}) that is n(x_{1})V${\widehat{y}}^{m}$ > n(x_{2})V${\widehat{y}}^{m}$. Therefore f(x_{1},y) > f(x_{2},y), so the function is decreasing; - Researchers find monotonicity with respect to the second variable, and therefore with respect to y, the authors consider 0 <y
_{1}< y_{2}so ${\widehat{y}}_{\perp}^{m}$ < ${\widehat{y}}_{2}^{m}$. Therefore, n(x)V${\widehat{y}}_{\perp}^{m}$ < n(x)V${\widehat{y}}_{2}^{m}$ so f(x,y_{1}) < f(x,y_{2}), so the function is increasing, and we can thus infer that- ◾
- yVy = ${\widehat{y}}^{2}$ = y + y − y·y = 2y − y
^{2} - ◾
- yVyVy = ${\widehat{y}}^{3}$ = 2y − y
^{2}+ y − (2y − y^{2})·y = 3y − 3y^{2}+ y^{3} - ◾
- yVyVyVy = ${\widehat{y}}^{4}$ = 4y − 6y
^{2}+ 4y^{3}− y^{4} - ◾
- (yVyV…y)′ = (${\widehat{y}}^{m}$)′ = m(1 − y)
^{m}^{−1}≥ 0 namely ${\widehat{y}}^{m}$We assume that (${\widehat{y}}^{m-1}$)′ = (m − 1)(1 − y)^{m}^{−2}. In order to show that (${\widehat{y}}^{m}$)′ = (m)(1 − y)^{m}^{−1},$${\left({\widehat{y}}^{m}\right)}^{\prime}={\left({\widehat{y}}^{m-1}Vy\right)}^{\prime}$$$$={({\widehat{y}}^{m-1}+y-{\widehat{y}}^{m-1}\xb7y)}^{\prime}$$$$=(m-1)\xb7{(1-y)}^{m-2}+1-[{\left({\widehat{y}}^{m-1}\right)}^{\prime}\xb7y+{\widehat{y}}^{m-1}\xb7{\left(y\right)}^{\prime}]$$$$=(m-1)\xb7{(1-y)}^{m-2}+1-(m-1)\xb7{(1-y)}^{m-2}\xb7y-{\widehat{y}}^{m-1}$$$$=(m-1)\xb7{(1-y)}^{m-2}\xb7(1-y)+1-{\widehat{y}}^{m-1}$$$$=(m-1)\xb7{(1-y)}^{m-1}+1-{\widehat{y}}^{m-1}$$$$=(m-1)\xb7{(1-y)}^{m-1}+{(1-y)}^{m-1}$$$$={(1-y)}^{m-1}\xb7(m-1+1)=m{(1-y)}^{m-1};$$

- It has to be proven that f(0,ω
_{1}) = 1.Actually, f(0,ω_{1}) = n(0)V${\widehat{{\omega}_{1}}}^{m}$= 1 for n(0) = 1, meaning that falsehood implies anything (dominion of falsehood). - It has to be proven that f(1,ω
_{2}) = ω_{2}.Actually, f(1,ω_{2}) = n(1)V${\widehat{{\omega}_{2}}}^{m}$ = ${\widehat{{\omega}_{2}}}^{m}$. This applies to m = 1 and f(1,ω_{2}) = ω_{2}, meaning that truth does not imply anything (truth neutrality). - We must prove that f(ω
_{1},ω_{1}) = 1, that is, n(ω_{1})V${\widehat{{\omega}_{1}}}^{m}$ = 1 and $\{\begin{array}{c}n\left({\omega}_{1}\right)=1therefore{\omega}_{1}=0\\ or\\ {\widehat{{\omega}_{1}}}^{m}=1therefore{\omega}_{1}=1\end{array}$Consequently, f(0,0) = 1 and f(1,1) = 1. - We must prove that f(x,f(y,z)) = f(y,f(x,z)), that is n(x)V f
^{m}(y,z) = n(y)V f^{m}(x,z) andtherefore $\{\begin{array}{c}n\left(x\right)=n\left(y\right)\Rightarrow x=y\mathrm{so}\mathrm{the}\mathrm{original}\mathrm{applies}\\ \mathrm{or}\\ {f}^{m}(y,z)={f}^{m}(x,z)\Rightarrow {\left(n\right(y\left)V{\widehat{z}}^{m}\right)}^{m}={\left(n\right(x\left)V{\widehat{z}}^{m}\right)}^{m}\\ \Rightarrow \left(n\right(y)V{\widehat{z}}^{m}=\left(n\right(x)V{\widehat{z}}^{m}\\ \Rightarrow x=\nu \mathrm{so}\mathrm{the}\mathrm{original}\mathrm{applies}\end{array}$ - If f(x,y) = 1 then x ≤ y.Therefore f(x,y) = 1. Consequently n(x)V${\widehat{y}}^{m}$ = 1, andtherefore $\{\begin{array}{c}n\left(x\right)=1\Rightarrow x=0\mathrm{so}x\le y\\ \mathrm{or}\\ {\widehat{y}}^{m}=1\Rightarrow \nu =1\mathrm{so}x\le \nu \mathrm{which}\mathrm{applies}\end{array}$
- We must find f(x,y) = f(n(y),n(x))so f(x,y) = n(x)V${\widehat{y}}^{m}$f(n(y),n(x)) = n(n(y))V(${\widehat{n\left(x\right))}}^{m}$ = yV(${\widehat{n\left(x\right))}}^{m}$ and these are equal only for m = 1.
- f being producible in both variables means f is continuous.

**Theorem**

**2.**

^{m}where f(x,y) ≥ 0.9, then

**Proof**

**of Theorem 2.**

**Theorem**

**3.**

**Proof**

**of Theorem 3.**

- The concept of monotonicity is studied with respect to the first variable. Therefore, with respect to x, ${N}_{x}^{\prime}\left(x,y\right)=-{\left(1-y\right)}^{m}$ is consequently decreasing .
- The researchers find monotonicity with respect to the second variable, and therefore, with respect to y, ${N}_{y}^{\prime}\left(x,y\right)=-xm{\left(1-y\right)}^{m-1}{\left(1-y\right)}^{\prime}=xm{\left(1-y\right)}^{m-1}$ is consequently increasing .
- It has to be proven that N(0,ω
_{1}) = 1.Actually, N(0,ω_{1}) = N(n(n(0))·(n(ω_{1}))^{m}) = N(n(1)·(n(ω_{1}))^{m}) = N(0·(n(ω_{1}))^{m}) = N(0) = 1. We therefore apply the meaning that falsehood implies anything (dominion of falsehood). - It just has to be proven that N(1,ω
_{2}) = ω_{2}. Actually, N(1,ω_{2}) = N(n(n(1))·(n(ω_{2}))^{m}) = N(n(0)·(n(ω_{2}))^{m}) = N(1·(n(ω_{2}))^{m}) = N(n(ω_{2}))^{m}). This applies to m = 1 and to N(1,ω_{2}) = ω_{2}, meaning that truth does not imply anything (truth neutrality). - We must find that Ν(ω
_{1},ω_{1}) = 1, namely, N(ω_{1},ω_{1}) = N(n(n(ω_{1}))·(n(ω_{1}))^{m}) = N(ω_{1}·(n(ω_{1}))^{m}). For the fifth property to hold, α must be 0 or 1, namely,N(0·(n(0))^{m}) = N(0·1^{m}) = N(0) = 1N(1·(n(1))^{m}) = N(1·0^{m}) = N(0) = 1 - The authors also want to show that N(ω
_{1}, N(ω_{2},x)) = N(ω_{2}, N(ω_{1},x))1 − ω_{1}(1 − N(ω_{2},x))^{m}= 1 − ω_{2}(1 − N(ω_{1},x))^{m}ω_{1}(1 − Ν(ω_{2},x))^{m}= ω_{2}(1 − N(ω_{1},x))^{m}ω_{1}[1 − (1 − ω_{2}(1 − x)^{m})]^{m}= ω_{2}[1 − (1 − ω_{1}(1 − x)^{m})]^{m}ω_{1}[1 − 1 + ω_{2}(1 − x)^{m})]^{m}= ω_{2}[1 − 1 + ω_{1}(1 − x)^{m})]^{m}ω_{1}[ω_{2}(1 − x)^{m}]^{m}= ω_{2}[ω_{1}(1 − x)^{m}]^{m}ω_{1}ω_{2}^{m}= ω_{2}ω_{1}^{m}$$\frac{{{\omega}_{2}}^{m}}{{\omega}_{2}}=\frac{{{\omega}_{1}}^{m}}{{\omega}_{1}}$$ω_{2}^{m}^{−1}= ω_{1}^{m}^{−1}$${\left(\frac{{\omega}_{2}}{{\omega}_{1}}\right)}^{m-1}=1$$⇒ω_{1}= ω_{2} - If N(x,y) = 1, then x ≤ y. Therefore N(x,y) = 1 ⇒ 1 − x(1 − y)
^{m}= 1, andtherefore $\{\begin{array}{c}x=0\mathrm{so}1-y\ne 0\Rightarrow y\ne 1\mathrm{so}x\le y\hfill \\ \mathrm{or}\\ 1-y=0\Rightarrow y=1\mathrm{so}x\le y\mathrm{which}\mathrm{applies}\end{array}$ - N(ω
_{1},ω_{2}) = N(n(ω_{2}),n(ω_{1}))1 − ω_{1}(1 − ω_{2})^{m}= 1 − n(ω_{2})(1 − n(ω_{1}))^{m}ω_{1}(1 − ω_{2})^{m}= n(ω_{2})(1 − n(ω_{1}))^{m}ω_{1}(1 − ω_{2})^{m}= (1 − ω_{2})(1 − (1 − ω_{1}))^{m}$$\frac{{\left(1-{\omega}_{2}\right)}^{m}}{1-{\omega}_{2}}=\frac{{{\omega}_{1}}^{m}}{{\omega}_{1}}$$(1 − ω_{2})^{m}^{−1}= ω_{1}^{m}^{−1}$${\left(\frac{1-{\omega}_{2}}{{\omega}_{1}}\right)}^{m-1}=1$$$$\mathrm{Must}\{\begin{array}{c}\frac{1-{\omega}_{2}}{{\omega}_{1}}=1\u27fa{\omega}_{1}=1-{\omega}_{2}\u27fa{\omega}_{1}+{\omega}_{2}=1\\ \mathrm{or}\\ m-1=0\mathrm{namely}m=1\mathrm{and}\frac{1-{\omega}_{2}}{{\omega}_{1}}\ne 0\mathrm{that}\mathrm{is}{\omega}_{1}+{\omega}_{2}\ne 1\mathrm{which}\mathrm{applies}\end{array}$$ - Ν is producible in both variables, meaning that Ν is continuous.

#### 2.3. A General Framework of the Methodology and an Example for the Implementation of the Fourth Step of the Methodology

#### 2.3.1. Real Data and Area of Study

#### 2.3.2. Implementation of First Step of Methodology Using Matlab Program: The Fuzzification of Real Variables Using Four Membership Degree Functions (Four Cases)

- First Case—Isosceles trapezium (trapezium membership function)

- II.
- Second Case—Random trapezium (trapezoidal membership function)

- III.
- Third Case—Isosceles triangle (triangular membership function)

- IV.
- Fourth Case—Scalene triangle (triangular membership function)

## 3. Results

#### 3.1. General Outcomes of Fuzzy Model—The Results from the First Step of the Methodology

#### 3.2. General Outcomes of Fuzzy Model—The Results from the Second Step of the Methodology

#### 3.3. General Outcomes of Fuzzy Model—The Results from the Third Step of the Methodology

#### 3.4. General Outcomes of Fuzzy Model—The Results from the Fourth Step of the Methodology

## 4. Discussion

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Temperature and humidity values with corresponding membership degrees using isosceles trapezium as the trapezoidal membership function.

Kavala 11:50 O’Clock Measurement | Temperature/ Membership Degrees | Humidity/ Membership Degrees | Kavala 11:50 O’Clock Measurement | Temperature/ Membership Degrees | Humidity/ Membership Degrees |
---|---|---|---|---|---|

1 August 2021 | 34/0.1667 | 0.41/0.4000 | 1 October 2021 | 21/1.0000 | 0.38/0.3000 |

2 August 2021 | 33/0.2500 | 0.46/0.5667 | 2 October 2021 | 21/1.0000 | 0.38/0.3000 |

3 August 2021 | 35/0.0833 | 0.44/0.5000 | 3 October 2021 | 19/1.0000 | 0.52/0.7667 |

4 August 2021 | 33/0.2500 | 0.49/0.6667 | 4 October 2021 | 20/1.0000 | 0.46/0.5667 |

5 August 2021 | 36/0.0000 | 0.55/0.8667 | 5 October 2021 | 20/1.0000 | 0.49/0.6667 |

6 August 2021 | 33/0.2500 | 0.46/0.5667 | 6 October 2021 | 19/1.0000 | 0.46/0.5667 |

7 August 2021 | 29/0.5833 | 0.37/0.2667 | 7 October 2021 | 20/1.0000 | 0.4/0.3667 |

8 August 2021 | 30/0.5000 | 0.52/0.7667 | 8 October 2021 | 15/0.6667 | 0.88/0.2000 |

9 August 2021 | 30/0.5000 | 0.59/1.0000 | 9 October 2021 | 15/0.6667 | 0.82/0.4000 |

10 August 2021 | 32/0.3333 | 0.63/1.0000 | 10 October 2021 | 16/0.7500 | 0.72/0.7333 |

11 August 2021 | 30/0.5000 | 0.59/1.0000 | 11 October 2021 | 16/0.7500 | 0.94/0.0000 |

12 August 2021 | 30/0.5000 | 0.52/0.7667 | 12 October 2021 | 16/0.7500 | 0.94/0.0000 |

13 August 2021 | 30/0.5000 | 0.49/0.6667 | 13 October 2021 | 14/0.5833 | 0.94/0.0000 |

14 August 2021 | 30/0.5000 | 0.38/0.3000 | 14 October 2021 | 13/0.5000 | 0.88/0.2000 |

15 August 2021 | 29/0.5833 | 0.43/0.4667 | 15 October 2021 | 14/0.5833 | 0.94/0.0000 |

16 August 2021 | 30/0.5000 | 0.46/0.5667 | 16 October 2021 | 18/0.9167 | 0.88/0.2000 |

17 August 2021 | 29/0.5833 | 0.58/0.9667 | 17 October 2021 | 18/0.9167 | 0.73/0.7000 |

18 August 2021 | 31/0.4167 | 0.52/0.7667 | 18 October 2021 | 16/0.7500 | 0.88/0.2000 |

19 August 2021 | 25/0.9167 | 0.74/0.6667 | 19 October 2021 | 16/0.7500 | 0.77/0.5667 |

20 August 2021 | 27/0.7500 | 0.45/0.5333 | 20 October 2021 | 18/0.9167 | 0.49/0.6667 |

21 August 2021 | 28/0.6667 | 0.45/0.5333 | 21 October 2021 | 17/0.8333 | 0.73/0.7000 |

22 August 2021 | 29/0.5833 | 0.48/0.6333 | 22 October 2021 | 18/0.9167 | 0.78/0.5333 |

23 August 2021 | 30/0.5000 | 0.29/0.0000 | 23 October 2021 | 19/1.0000 | 0.73/0.7000 |

24 August 2021 | 29/0.5833 | 0.4/0.3667 | 24 October 2021 | 15/0.6667 | 0.77/0.5667 |

25 August 2021 | 28/0.6667 | 0.51/0.7333 | 25 October 2021 | 14/0.5833 | 0.48/0.6333 |

26 August 2021 | 29/0.5833 | 0.58/0.9667 | 26 October 2021 | 15/0.6667 | 0.45/0.5333 |

27 August 2021 | 29/0.5833 | 0.62/1.0000 | 27 October 2021 | 16/0.7500 | 0.52/0.7667 |

28 August 2021 | 29/0.5833 | 0.62/1.0000 | 28 October 2021 | 14/0.5833 | 0.72/0.7333 |

29 August 2021 | 28/0.6667 | 0.66/0.9333 | 29 October 2021 | 16/0.7500 | 0.45/0.5333 |

30 August 2021 | 29/0.5833 | 0.51/0.7333 | 30 October 2021 | 17/0.8333 | 0.45/0.5333 |

31 August 2021 | 28/0.6667 | 0.55/0.8667 | 31 October 2021 | 17/0.8333 | 0.52/0.7667 |

1 September 2021 | 29/0.5833 | 0.48/0.6333 | 1 November 2021 | 17/0.8333 | 0.68/0.8667 |

2 September 2021 | 26/0.8333 | 0.37/0.2667 | 2 November 2021 | 13/0.5000 | 0.94/0.0000 |

3 September 2021 | 25/0.9167 | 0.39/0.3333 | 3 November 2021 | 17/0.8333 | 0.83/0.3667 |

4 September 2021 | 25/0.9167 | 0.44/0.5000 | 4 November 2021 | 21/1.0000 | 0.57/0.9333 |

5 September 2021 | 25/0.9167 | 0.47/0.6000 | 5 November 2021 | 20/1.0000 | 0.73/0.7000 |

6 September 2021 | 25/0.9167 | 0.39/0.3333 | 6 November 2021 | 18/0.9167 | 0.88/0.2000 |

7 September 2021 | 24/1.0000 | 0.41/0.4000 | 7 November 2021 | 18/0.9167 | 0.83/0.3667 |

8 September 2021 | 21/1.0000 | 0.53/0.8000 | 8 November 2021 | 17/0.8333 | 0.83/0.3667 |

9 September 2021 | 22/1.0000 | 0.57/0.9333 | 9 November 2021 | 18/0.9167 | 0.83/0.3667 |

10 September 2021 | 23/1.0000 | 0.65/0.9667 | 10 November 2021 | 14/0.5833 | 0.55/0.8667 |

11 September 2021 | 25/0.9167 | 0.54/0.8333 | 11 November 2021 | 12/0.4167 | 0.51/0.7333 |

12 September 2021 | 27/0.7500 | 0.42/0.4333 | 12 November 2021 | 14/0.5833 | 0.51/0.7333 |

13 September 2021 | 27/0.7500 | 0.42/0.4333 | 13 November 2021 | 13/0.5000 | 0.63/1.0000 |

14 September 2021 | 29/0.5833 | 0.31/0.0667 | 14 November 2021 | 16/0.7500 | 0.68/0.8667 |

15 September 2021 | 27/0.7500 | 0.45/0.5333 | 15 November 2021 | 15/0.6667 | 0.68/0.8667 |

16 September 2021 | 26/0.8333 | 0.58/0.9667 | 16 November 2021 | 15/0.6667 | 0.55/0.8667 |

17 September 2021 | 26/0.8333 | 0.61/1.0000 | 17 November 2021 | 13/0.5000 | 0.51/0.7333 |

18 September 2021 | 27/0.7500 | 0.62/1.0000 | 18 November 2021 | 12/0.4167 | 0.63/1.0000 |

19 September 2021 | 25/0.9167 | 0.65/0.9667 | 19 November 2021 | 11/0.3333 | 0.77/0.5667 |

20 September 2021 | 27/0.7500 | 0.54/0.8333 | 20 November 2021 | 13/0.5000 | 0.72/0.7333 |

21 September 2021 | 24/1.0000 | 0.65/0.9667 | 21 November 2021 | 16/0.7500 | 0.63/1.0000 |

22 September 2021 | 18/0.9167 | 0.73/0.7000 | 22 November 2021 | 11/0.3333 | 0.88/0.2000 |

23 September 2021 | 19/1.0000 | 0.52/0.7667 | 23 November 2021 | 15/0.6667 | 0.88/0.2000 |

24 September 2021 | 20/1.0000 | 0.49/0.6667 | 24 November 2021 | 13/0.5000 | 0.55/0.8667 |

25 September 2021 | 24/1.0000 | 0.5/0.7000 | 25 November 2021 | 11/0.3333 | 0.54/0.8333 |

26 September 2021 | 25/0.9167 | 0.57/0.9333 | 26 November 2021 | 7/0.0000 | 0.93/0.0333 |

27 September 2021 | 25/0.9167 | 0.47/0.6000 | 27 November 2021 | 14/0.5833 | 0.88/0.2000 |

28 September 2021 | 24/1.0000 | 0.47/0.6000 | 28 November 2021 | 16/0.7500 | 0.83/0.3667 |

29 September 2021 | 20/1.0000 | 0.6/1.0000 | 29 November 2021 | 19/1.0000 | 0.68/0.8667 |

30 September 2021 | 21/1.0000 | 0.46/0.5667 | 30 November 2021 | 12/0.4167 | 0.51/0.7333 |

**Table A2.**Temperature and humidity values with corresponding membership degrees using random trapezium as trapezoidal membership function.

Kavala 11:50 O’Clock Measurement | Temperature/ Membership Degrees | Humidity/ Membership Degrees | Kavala 11:50 O’Clock Measurement | Temperature/ Membership Degrees | Humidity/ Membership Degrees |
---|---|---|---|---|---|

1 August 2021 | 34/0.1538 | 0.41/0.3871 | 1 October 2021 | 21/0.9333 | 0.38/0.2903 |

2 August 2021 | 33/0.2308 | 0.46/0.5484 | 2 October 2021 | 21/0.9333 | 0.38/0.2903 |

3 August 2021 | 35/0.0769 | 0.44/0.4839 | 3 October 2021 | 19/0.8000 | 0.52/0.7419 |

4 August 2021 | 33/0.2308 | 0.49/0.6452 | 4 October 2021 | 20/0.8667 | 0.46/0.5484 |

5 August 2021 | 36/0.0000 | 0.55/0.8387 | 5 October 2021 | 20/0.8667 | 0.49/0.6452 |

6 August 2021 | 33/0.2308 | 0.46/0.5484 | 6 October 2021 | 19/0.8000 | 0.46/0.5484 |

7 August 2021 | 29/0.5385 | 0.37/0.2581 | 7 October 2021 | 20/0.8667 | 0.4/0.3548 |

8 August 2021 | 30/0.4615 | 0.52/0.7419 | 8 October 2021 | 15/0.5333 | 0.88/0.1818 |

9 August 2021 | 30/0.4615 | 0.59/0.9677 | 9 October 2021 | 15/0.5333 | 0.82/0.3636 |

10 August 2021 | 32/0.3077 | 0.63/0.9394 | 10 October 2021 | 16/0.6000 | 0.72/0.6667 |

11 August 2021 | 30/0.4615 | 0.59/0.9677 | 11 October 2021 | 16/0.6000 | 0.94/0.0000 |

12 August 2021 | 30/0.4615 | 0.52/0.7419 | 12 October 2021 | 16/0.6000 | 0.94/0.0000 |

13 August 2021 | 30/0.4615 | 0.49/0.6452 | 13 October 2021 | 14/0.4667 | 0.94/0.0000 |

14 August 2021 | 30/0.4615 | 0.38/0.2903 | 14 October 2021 | 13/0.4000 | 0.88/0.1818 |

15 August 2021 | 29/0.5385 | 0.43/0.4516 | 15 October 2021 | 14/0.4667 | 0.94/0.0000 |

16 August 2021 | 30/0.4615 | 0.46/0.5484 | 16 October 2021 | 18/0.7333 | 0.88/0.1818 |

17 August 2021 | 29/0.5385 | 0.58/0.9355 | 17 October 2021 | 18/0.7333 | 0.73/0.6364 |

18 August 2021 | 31/0.3846 | 0.52/0.7419 | 18 October 2021 | 16/0.6000 | 0.88/0.1818 |

19 August 2021 | 25/0.8462 | 0.74/0.6061 | 19 October 2021 | 16/0.6000 | 0.77/0.5152 |

20 August 2021 | 27/0.6923 | 0.45/0.5161 | 20 October 2021 | 18/0.7333 | 0.49/0.6452 |

21 August 2021 | 28/0.6154 | 0.45/0.5161 | 21 October 2021 | 17/0.6667 | 0.73/0.6364 |

22 August 2021 | 29/0.5385 | 0.48/0.6129 | 22 October 2021 | 18/0.7333 | 0.78/0.4848 |

23 August 2021 | 30/0.4615 | 0.29/0.0000 | 23 October 2021 | 19/0.8000 | 0.73/0.6364 |

24 August 2021 | 29/0.5385 | 0.4/0.3548 | 24 October 2021 | 15/0.5333 | 0.77/0.5152 |

25 August 2021 | 28/0.6154 | 0.51/0.7097 | 25 October 2021 | 14/0.4667 | 0.48/0.6129 |

26 August 2021 | 29/0.5385 | 0.58/0.9355 | 26 October 2021 | 15/0.5333 | 0.45/0.5161 |

27 August 2021 | 29/0.5385 | 0.62/0.9697 | 27 October 2021 | 16/0.6000 | 0.52/0.7419 |

28 August 2021 | 29/0.5385 | 0.62/0.9697 | 28 October 2021 | 14/0.4667 | 0.72/0.6667 |

29 August 2021 | 28/0.6154 | 0.66/0.8485 | 29 October 2021 | 16/0.6000 | 0.45/0.5161 |

30 August 2021 | 29/0.5385 | 0.51/0.7097 | 30 October 2021 | 17/0.6667 | 0.45/0.5161 |

31 August 2021 | 28/0.6154 | 0.55/0.8387 | 31 October 2021 | 17/0.6667 | 0.52/0.7419 |

1 September 2021 | 29/0.5385 | 0.48/0.6129 | 1 November 2021 | 17/0.6667 | 0.68/0.7879 |

2 September 2021 | 26/0.7692 | 0.37/0.2581 | 2 November 2021 | 13/0.4000 | 0.94/0.0000 |

3 September 2021 | 25/0.8462 | 0.39/0.3226 | 3 November 2021 | 17/0.6667 | 0.83/0.3333 |

4 September 2021 | 25/0.8462 | 0.44/0.4839 | 4 November 2021 | 21/0.9333 | 0.57/0.9032 |

5 September 2021 | 25/0.8462 | 0.47/0.5806 | 5 November 2021 | 20/0.8667 | 0.73/0.6364 |

6 September 2021 | 25/0.8462 | 0.39/0.3226 | 6 November 2021 | 18/0.7333 | 0.88/0.1818 |

7 September 2021 | 24/0.9231 | 0.41/0.3871 | 7 November 2021 | 18/0.7333 | 0.83/0.3333 |

8 September 2021 | 21/0.9333 | 0.53/0.7742 | 8 November 2021 | 17/0.6667 | 0.83/0.3333 |

9 September 2021 | 22/1.0000 | 0.57/0.9032 | 9 November 2021 | 18/0.7333 | 0.83/0.3333 |

10 September 2021 | 23/1.0000 | 0.65/0.8788 | 10 November 2021 | 14/0.4667 | 0.55/0.8387 |

11 September 2021 | 25/0.8462 | 0.54/0.8065 | 11 November 2021 | 12/0.3333 | 0.51/0.7097 |

12 September 2021 | 27/0.6923 | 0.42/0.4194 | 12 November 2021 | 14/0.4667 | 0.51/0.7097 |

13 September 2021 | 27/0.6923 | 0.42/0.4194 | 13 November 2021 | 13/0.4000 | 0.63/0.9394 |

14 September 2021 | 29/0.5385 | 0.31/0.0645 | 14 November 2021 | 16/0.6000 | 0.68/0.7879 |

15 September 2021 | 27/0.6923 | 0.45/0.5161 | 15 November 2021 | 15/0.5333 | 0.68/0.7879 |

16 September 2021 | 26/0.7692 | 0.58/0.9355 | 16 November 2021 | 15/0.5333 | 0.55/0.8387 |

17 September 2021 | 26/0.7692 | 0.61/1.0000 | 17 November 2021 | 13/0.4000 | 0.51/0.7097 |

18 September 2021 | 27/0.6923 | 0.62/0.9697 | 18 November 2021 | 12/0.3333 | 0.63/0.9394 |

19 September 2021 | 25/0.8462 | 0.65/0.8788 | 19 November 2021 | 11/0.2667 | 0.77/0.5152 |

20 September 2021 | 27/0.6923 | 0.54/0.8065 | 20 November 2021 | 13/0.4000 | 0.72/0.6667 |

21 September 2021 | 24/0.9231 | 0.65/0.8788 | 21 November 2021 | 16/0.6000 | 0.63/0.9394 |

22 September 2021 | 18/0.7333 | 0.73/0.6364 | 22 November 2021 | 11/0.2667 | 0.88/0.1818 |

23 September 2021 | 19/0.8000 | 0.52/0.7419 | 23 November 2021 | 15/0.5333 | 0.88/0.1818 |

24 September 2021 | 20/0.8667 | 0.49/0.6452 | 24 November 2021 | 13/0.4000 | 0.55/0.8387 |

25 September 2021 | 24/0.9231 | 0.5/0.6774 | 25 November 2021 | 11/0.2667 | 0.54/0.8065 |

26 September 2021 | 25/0.8462 | 0.57/0.9032 | 26 November 2021 | 7/0.0000 | 0.93/0.0303 |

27 September 2021 | 25/0.8462 | 0.47/0.5806 | 27 November 2021 | 14/0.4667 | 0.88/0.1818 |

28 September 2021 | 24/0.9231 | 0.47/0.5806 | 28 November 2021 | 16/0.6000 | 0.83/0.3333 |

29 September 2021 | 20/0.8667 | 0.6/1.0000 | 29 November 2021 | 19/0.8000 | 0.68/0.7879 |

30 September 2021 | 21/0.9333 | 0.46/0.5484 | 30 November 2021 | 12/0.3333 | 0.51/0.7097 |

**Table A3.**Temperature and humidity values with corresponding membership degrees using isosceles triangle as triangular membership function.

Kavala 11:50 O’Clock Measurement | Temperature/ Membership Degrees | Humidity/ Membership Degrees | Kavala 11:50 O’Clock Measurement | Temperature/ Membership Degrees | Humidity/ Membership Degrees |
---|---|---|---|---|---|

1 August 2021 | 34/0.1379 | 0.41/0.3692 | 1 October 2021 | 21/0.9655 | 0.38/0.2769 |

2 August 2021 | 33/0.2069 | 0.46/0.5231 | 2 October 2021 | 21/0.9655 | 0.38/0.2769 |

3 August 2021 | 35/0.0690 | 0.44/0.4615 | 3 October 2021 | 19/0.8276 | 0.52/0.7077 |

4 August 2021 | 33/0.2069 | 0.49/0.6154 | 4 October 2021 | 20/0.8966 | 0.46/0.5231 |

5 August 2021 | 36/0.0000 | 0.55/0.8000 | 5 October 2021 | 20/0.8966 | 0.49/0.6154 |

6 August 2021 | 33/0.2069 | 0.46/0.5231 | 6 October 2021 | 19/0.8276 | 0.46/0.5231 |

7 August 2021 | 29/0.4828 | 0.37/0.2462 | 7 October 2021 | 20/0.8966 | 0.4/0.3385 |

8 August 2021 | 30/0.4138 | 0.52/0.7077 | 8 October 2021 | 15/0.5517 | 0.88/0.1846 |

9 August 2021 | 30/0.4138 | 0.59/0.9231 | 9 October 2021 | 15/0.5517 | 0.82/0.3692 |

10 August 2021 | 32/0.2759 | 0.63/0.9538 | 10 October 2021 | 16/0.6207 | 0.72/0.6769 |

11 August 2021 | 30/0.4138 | 0.59/0.9231 | 11 October 2021 | 16/0.6207 | 0.94/0.0000 |

12 August 2021 | 30/0.4138 | 0.52/0.7077 | 12 October 2021 | 16/0.6207 | 0.94/0.0000 |

13 August 2021 | 30/0.4138 | 0.49/0.6154 | 13 October 2021 | 14/0.4828 | 0.94/0.0000 |

14 August 2021 | 30/0.4138 | 0.38/0.2769 | 14 October 2021 | 13/0.4138 | 0.88/0.1846 |

15 August 2021 | 29/0.4828 | 0.43/0.4308 | 15 October 2021 | 14/0.4828 | 0.94/0.0000 |

16 August 2021 | 30/0.4138 | 0.46/0.5231 | 16 October 2021 | 18/0.7586 | 0.88/0.1846 |

17 August 2021 | 29/0.4828 | 0.58/0.8923 | 17 October 2021 | 18/0.7586 | 0.73/0.6462 |

18 August 2021 | 31/0.3448 | 0.52/0.7077 | 18 October 2021 | 16/0.6207 | 0.88/0.1846 |

19 August 2021 | 25/0.7586 | 0.74/0.6154 | 19 October 2021 | 16/0.6207 | 0.77/0.5231 |

20 August 2021 | 27/0.6207 | 0.45/0.4923 | 20 October 2021 | 18/0.7586 | 0.49/0.6154 |

21 August 2021 | 28/0.5517 | 0.45/0.4923 | 21 October 2021 | 17/0.6897 | 0.73/0.6462 |

22 August 2021 | 29/0.4828 | 0.48/0.5846 | 22 October 2021 | 18/0.7586 | 0.78/0.4923 |

23 August 2021 | 30/0.4138 | 0.29/0.0000 | 23 October 2021 | 19/0.8276 | 0.73/0.6462 |

24 August 2021 | 29/0.4828 | 0.4/0.3385 | 24 October 2021 | 15/0.5517 | 0.77/0.5231 |

25 August 2021 | 28/0.5517 | 0.51/0.6769 | 25 October 2021 | 14/0.4828 | 0.48/0.5846 |

26 August 2021 | 29/0.4828 | 0.58/0.8923 | 26 October 2021 | 15/0.5517 | 0.45/0.4923 |

27 August 2021 | 29/0.4828 | 0.62/0.9846 | 27 October 2021 | 16/0.6207 | 0.52/0.7077 |

28 August 2021 | 29/0.4828 | 0.62/0.9846 | 28 October 2021 | 14/0.4828 | 0.72/0.6769 |

29 August 2021 | 28/0.5517 | 0.66/0.8615 | 29 October 2021 | 16/0.6207 | 0.45/0.4923 |

30 August 2021 | 29/0.4828 | 0.51/0.6769 | 30 October 2021 | 17/0.6897 | 0.45/0.4923 |

31 August 2021 | 28/0.5517 | 0.55/0.8000 | 31 October 2021 | 17/0.6897 | 0.52/0.7077 |

1 September 2021 | 29/0.4828 | 0.48/0.5846 | 1 November 2021 | 17/0.6897 | 0.68/0.8000 |

2 September 2021 | 26/0.6897 | 0.37/0.2462 | 2 November 2021 | 13/0.4138 | 0.94/0.0000 |

3 September 2021 | 25/0.7586 | 0.39/0.3077 | 3 November 2021 | 17/0.6897 | 0.83/0.3385 |

4 September 2021 | 25/0.7586 | 0.44/0.4615 | 4 November 2021 | 21/0.9655 | 0.57/0.8615 |

5 September 2021 | 25/0.7586 | 0.47/0.5538 | 5 November 2021 | 20/0.8966 | 0.73/0.6462 |

6 September 2021 | 25/0.7586 | 0.39/0.3077 | 6 November 2021 | 18/0.7586 | 0.88/0.1846 |

7 September 2021 | 24/0.8276 | 0.41/0.3692 | 7 November 2021 | 18/0.7586 | 0.83/0.3385 |

8 September 2021 | 21/0.9655 | 0.53/0.7385 | 8 November 2021 | 17/0.6897 | 0.83/0.3385 |

9 September 2021 | 22/0.9655 | 0.57/0.8615 | 9 November 2021 | 18/0.7586 | 0.83/0.3385 |

10 September 2021 | 23/0.8966 | 0.65/0.8923 | 10 November 2021 | 14/0.4828 | 0.55/0.8000 |

11 September 2021 | 25/0.7586 | 0.54/0.7692 | 11 November 2021 | 12/0.3448 | 0.51/0.6769 |

12 September 2021 | 27/0.6207 | 0.42/0.4000 | 12 November 2021 | 14/0.4828 | 0.51/0.6769 |

13 September 2021 | 27/0.6207 | 0.42/0.4000 | 13 November 2021 | 13/0.4138 | 0.63/0.9538 |

14 September 2021 | 29/0.4828 | 0.31/0.0615 | 14 November 2021 | 16/0.6207 | 0.68/0.8000 |

15 September 2021 | 27/0.6207 | 0.45/0.4923 | 15 November 2021 | 15/0.5517 | 0.68/0.8000 |

16 September 2021 | 26/0.6897 | 0.58/0.8923 | 16 November 2021 | 15/0.5517 | 0.55/0.8000 |

17 September 2021 | 26/0.6897 | 0.61/0.9846 | 17 November 2021 | 13/0.4138 | 0.51/0.6769 |

18 September 2021 | 27/0.6207 | 0.62/0.9846 | 18 November 2021 | 12/0.3448 | 0.63/0.9538 |

19 September 2021 | 25/0.7586 | 0.65/0.8923 | 19 November 2021 | 11/0.2759 | 0.77/0.6769 |

20 September 2021 | 27/0.6207 | 0.54/0.7692 | 20 November 2021 | 13/0.4138 | 0.72/0.9538 |

21 September 2021 | 24/0.8276 | 0.65/0.8923 | 21 November 2021 | 16/0.6207 | 0.63/0.9538 |

22 September 2021 | 18/0.7586 | 0.73/0.6462 | 22 November 2021 | 11/0.2759 | 0.88/0.1846 |

23 September 2021 | 19/0.8276 | 0.52/0.7077 | 23 November 2021 | 15/0.5517 | 0.88/0.1846 |

24 September 2021 | 20/0.8966 | 0.49/0.6154 | 24 November 2021 | 13/0.4138 | 0.55/0.8000 |

25 September 2021 | 24/0.8276 | 0.5/0.6462 | 25 November 2021 | 11/0.2759 | 0.54/0.7692 |

26 September 2021 | 25/0.7586 | 0.57/0.8615 | 26 November 2021 | 7/0.0000 | 0.93/0.0308 |

27 September 2021 | 25/0.7586 | 0.47/0.5538 | 27 November 2021 | 14/0.4828 | 0.88/0.1846 |

28 September 2021 | 24/0.8276 | 0.47/0.5538 | 28 November 2021 | 16/0.6207 | 0.83/0.3385 |

29 September 2021 | 20/0.8966 | 0.6/0.9538 | 29 November 2021 | 19/0.8276 | 0.68/0.8000 |

30 September 2021 | 21/0.9655 | 0.46/0.5231 | 30 November 2021 | 12/0.3448 | 0.51/0.6769 |

**Table A4.**Temperature and humidity values with corresponding membership degrees using scalene triangle as triangular membership function.

Kavala 11:50 O’Clock Measurement | Temperature/ Membership Degrees | Humidity/ Membership Degrees | Kavala 11:50 O’Clock Measurement | Temperature/ Membership Degrees | Humidity/ Membership Degrees |
---|---|---|---|---|---|

1 August 2021 | 34/0.1481 | 0.41/0.3810 | 1 October 2021 | 21/0.9032 | 0.38/0.2857 |

2 August 2021 | 33/0.2222 | 0.46/0.5397 | 2 October 2021 | 21/0.9032 | 0.38/0.2857 |

3 August 2021 | 35/0.0741 | 0.44/0.4762 | 3 October 2021 | 19/0.7742 | 0.52/0.7302 |

4 August 2021 | 33/0.2222 | 0.49/0.6349 | 4 October 2021 | 20/0.8387 | 0.46/0.5397 |

5 August 2021 | 36/0.0000 | 0.55/0.8254 | 5 October 2021 | 20/0.8387 | 0.49/0.6349 |

6 August 2021 | 33/0.2222 | 0.46/0.5397 | 6 October 2021 | 19/0.7742 | 0.46/0.5397 |

7 August 2021 | 29/0.5185 | 0.37/0.2540 | 7 October 2021 | 20/0.8387 | 0.4/0.3492 |

8 August 2021 | 30/0.4444 | 0.52/0.7302 | 8 October 2021 | 15/0.5161 | 0.88/0.1791 |

9 August 2021 | 30/0.4444 | 0.59/0.9524 | 9 October 2021 | 15/0.5161 | 0.82/0.3582 |

10 August 2021 | 32/0.2963 | 0.63/0.9254 | 10 October 2021 | 16/0.5806 | 0.72/0.6567 |

11 August 2021 | 30/0.4444 | 0.59/0.9524 | 11 October 2021 | 16/0.5806 | 0.94/0.0000 |

12 August 2021 | 30/0.4444 | 0.52/0.7302 | 12 October 2021 | 16/0.5806 | 0.94/0.0000 |

13 August 2021 | 30/0.4444 | 0.49/0.6349 | 13 October 2021 | 14/0.4516 | 0.94/0.0000 |

14 August 2021 | 30/0.4444 | 0.38/0.2857 | 14 October 2021 | 13/0.3871 | 0.88/0.1791 |

15 August 2021 | 29/0.5185 | 0.43/0.4444 | 15 October 2021 | 14/0.4516 | 0.94/0.0000 |

16 August 2021 | 30/0.4444 | 0.46/0.5397 | 16 October 2021 | 18/0.7097 | 0.88/0.1791 |

17 August 2021 | 29/0.5185 | 0.58/0.9206 | 17 October 2021 | 18/0.7097 | 0.73/0.6269 |

18 August 2021 | 31/0.3704 | 0.52/0.7302 | 18 October 2021 | 16/0.5806 | 0.88/0.1791 |

19 August 2021 | 25/0.8148 | 0.74/0.5970 | 19 October 2021 | 16/0.5806 | 0.77/0.5075 |

20 August 2021 | 27/0.6667 | 0.45/0.5079 | 20 October 2021 | 18/0.7097 | 0.49/0.6349 |

21 August 2021 | 28/0.5926 | 0.45/0.5079 | 21 October 2021 | 17/0.6452 | 0.73/0.6269 |

22 August 2021 | 29/0.5185 | 0.48/0.6032 | 22 October 2021 | 18/0.7097 | 0.78/0.4776 |

23 August 2021 | 30/0.4444 | 0.29/0.0000 | 23 October 2021 | 19/0.7742 | 0.73/0.6269 |

24 August 2021 | 29/0.5185 | 0.4/0.3492 | 24 October 2021 | 15/0.5161 | 0.77/0.5075 |

25 August 2021 | 28/0.5926 | 0.51/0.6984 | 25 October 2021 | 14/0.4516 | 0.48/0.6032 |

26 August 2021 | 29/0.5185 | 0.58/0.9206 | 26 October 2021 | 15/0.5161 | 0.45/0.5079 |

27 August 2021 | 29/0.5185 | 0.62/0.9552 | 27 October 2021 | 16/0.5806 | 0.52/0.7302 |

28 August 2021 | 29/0.5185 | 0.62/0.9552 | 28 October 2021 | 14/0.4516 | 0.72/0.6567 |

29 August 2021 | 28/0.5926 | 0.66/0.8358 | 29 October 2021 | 16/0.5806 | 0.45/0.5079 |

30 August 2021 | 29/0.5185 | 0.51/0.6984 | 30 October 2021 | 17/0.6452 | 0.45/0.5079 |

31 August 2021 | 28/0.5926 | 0.55/0.8254 | 31 October 2021 | 17/0.6452 | 0.52/0.7302 |

1 September 2021 | 29/0.5185 | 0.48/0.6032 | 1 November 2021 | 17/0.6452 | 0.68/0.7761 |

2 September 2021 | 26/0.7407 | 0.37/0.2540 | 2 November 2021 | 13/0.3871 | 0.94/0.0000 |

3 September 2021 | 25/0.8148 | 0.39/0.3175 | 3 November 2021 | 17/0.6452 | 0.83/0.3284 |

4 September 2021 | 25/0.8148 | 0.44/0.4762 | 4 November 2021 | 21/0.9032 | 0.57/0.8889 |

5 September 2021 | 25/0.8148 | 0.47/0.5714 | 5 November 2021 | 20/0.8387 | 0.73/0.6269 |

6 September 2021 | 25/0.8148 | 0.39/0.3175 | 6 November 2021 | 18/0.7097 | 0.88/0.1791 |

7 September 2021 | 24/0.8889 | 0.41/0.3810 | 7 November 2021 | 18/0.7097 | 0.83/0.3284 |

8 September 2021 | 21/0.9032 | 0.53/0.7619 | 8 November 2021 | 17/0.6452 | 0.83/0.3284 |

9 September 2021 | 22/0.9677 | 0.57/0.8889 | 9 November 2021 | 18/0.7097 | 0.83/0.3284 |

10 September 2021 | 23/0.9630 | 0.65/0.8657 | 10 November 2021 | 14/0.4516 | 0.55/0.8254 |

11 September 2021 | 25/0.8148 | 0.54/0.7937 | 11 November 2021 | 12/0.3226 | 0.51/0.6984 |

12 September 2021 | 27/0.6667 | 0.42/0.4127 | 12 November 2021 | 14/0.4516 | 0.51/0.6984 |

13 September 2021 | 27/0.6667 | 0.42/0.4127 | 13 November 2021 | 13/0.3871 | 0.63/0.9254 |

14 September 2021 | 29/0.5185 | 0.31/0.0635 | 14 November 2021 | 16/0.5806 | 0.68/0.7761 |

15 September 2021 | 27/0.6667 | 0.45/0.5079 | 15 November 2021 | 15/0.5161 | 0.68/0.7761 |

16 September 2021 | 26/0.7407 | 0.58/0.9206 | 16 November 2021 | 15/0.5161 | 0.55/0.8254 |

17 September 2021 | 26/0.7407 | 0.61/0.9851 | 17 November 2021 | 13/0.3871 | 0.51/0.6984 |

18 September 2021 | 27/0.6667 | 0.62/0.9552 | 18 November 2021 | 12/0.3226 | 0.63/0.9254 |

19 September 2021 | 25/0.8148 | 0.65/0.8657 | 19 November 2021 | 11/0.2581 | 0.77/0.5075 |

20 September 2021 | 27/0.6667 | 0.54/0.7937 | 20 November 2021 | 13/0.3871 | 0.72/0.6567 |

21 September 2021 | 24/0.8889 | 0.65/0.8657 | 21 November 2021 | 16/0.5806 | 0.63/0.9254 |

22 September 2021 | 18/0.7097 | 0.73/0.6269 | 22 November 2021 | 11/0.2581 | 0.88/0.1791 |

23 September 2021 | 19/0.7742 | 0.52/0.7302 | 23 November 2021 | 15/0.5161 | 0.88/0.1791 |

24 September 2021 | 20/0.8387 | 0.49/0.6349 | 24 November 2021 | 13/0.3871 | 0.55/0.8254 |

25 September 2021 | 24/0.8889 | 0.5/0.6667 | 25 November 2021 | 11/0.2581 | 0.54/0.7937 |

26 September 2021 | 25/0.8148 | 0.57/0.8889 | 26 November 2021 | 7/0.0000 | 0.93/0.0299 |

27 September 2021 | 25/0.8148 | 0.47/0.5714 | 27 November 2021 | 14/0.4516 | 0.88/0.1791 |

28 September 2021 | 24/0.8889 | 0.47/0.5714 | 28 November 2021 | 16/0.5806 | 0.83/0.3284 |

29 September 2021 | 20/0.8387 | 0.6/0.9841 | 29 November 2021 | 19/0.7742 | 0.68/0.7761 |

30 September 2021 | 21/0.9032 | 0.46/0.5397 | 30 November 2021 | 12/0.3226 | 0.51/0.6984 |

## References

- Ruan, D.; Kerre, E.E. Fuzzy implication operators and generalized fuzzy method of cases. Fuzzy Sets Syst.
**1993**, 54, 23–37. [Google Scholar] [CrossRef] - Makariadis, S.; Souliotis, G.; Papadopoulos, B. Parametric fuzzy implications produced via fuzzy negations with a case study in environmental variables. Symmetry
**2021**, 13, 509. [Google Scholar] [CrossRef] - Pagouropoulos, P.; Tzimopoulos, C.D.; Papadopoulos, B.K. A method for the detection of the most suitable fuzzy implication for data applications. In Communications in Computer and Information Science, Proceedings of the 18th International Conference on Engineering Applications of Neural Networks (EANN), Athens, Greece, 25–27 August 2017; Iliadis, L., Likas, A., Jayne, C., Boracchi, G., Eds.; Springer: Berlin/Heidelberg, Germany, 2017; Volume 744, pp. 242–255. ISSN 18650929. ISBN 978-331965171-2. [Google Scholar] [CrossRef]
- Pagouropoulos, P.; Tzimopoulos, C.D.; Papadopoulos, B.K. A method for the detection of the most suitable fuzzy implication for data applications. Evol. Syst.
**2020**, 11, 467–477. [Google Scholar] [CrossRef] - Botzoris, G.N.; Papadopoulos, K.; Papadopoulos, B.K. A method for the evaluation and selection of an appropriate fuzzy implication by using statistical data. Fuzzy Econ. Rev.
**2015**, 20, 19–29. [Google Scholar] [CrossRef] - Rapti, M.N.; Papadopoulos, B.K. A method of generating fuzzy implications from n increasing functions and n + 1 negations. Mathematics
**2020**, 8, 886. [Google Scholar] [CrossRef] - Bedregal, B.C.; Dimuro, G.P.; Santiago, R.H.N.; Reiser, R.H.S. On interval fuzzy S-implications. Inf. Sci.
**2010**, 180, 1373–1389. [Google Scholar] [CrossRef] - Balasubramaniam, J. Contrapositive symmetrisation of fuzzy implications-Revisited. Fuzzy Sets Syst.
**2006**, 157, 2291–2310. [Google Scholar] [CrossRef] - Jayaram, B.; Mesiar, R. On special fuzzy implications. Fuzzy Sets Syst.
**2009**, 160, 2063–2085. [Google Scholar] [CrossRef] - Wang, Z.; Xu, Z.; Liu, S.; Yao, Z. Direct clustering analysis based on intuitionistic fuzzy implication. Appl. Soft Comput. J.
**2014**, 23, 1–8. [Google Scholar] [CrossRef] - Shi, Y.; Van Gasse, B.; Ruan, D.; Kerre, E.E. On Dependencies and Independencies of Fuzzy Implication Axioms. Fuzzy Sets Syst.
**2010**, 161, 1388–1405. [Google Scholar] [CrossRef] - Fernandez-Peralta, R.; Massanet, S.; Mesiarová-Zemánková, A.; Mir, A. A general framework for the characterization of (S, N)-implications with a non-continuous negation based on completions of t-conorms. Fuzzy Sets Syst.
**2022**, 441, 1–32. [Google Scholar] [CrossRef] - Fernández-Sánchez, J.; Kolesárová, A.; Mesiar, R.; Quesada-Molina, J.J.; Úbeda-Flores, M. A generalization of a copula-based construction of fuzzy implications. Fuzzy Sets Syst.
**2023**, 456, 197–207. [Google Scholar] [CrossRef] - Madrid, N.; Cornelis, C. Kitainik axioms do not characterize the class of inclusion measures based on contrapositive fuzzy implications. Fuzzy Sets Syst.
**2023**, 456, 208–214. [Google Scholar] [CrossRef] - Pinheiro, J.; Santos, H.; Dimuro, G.P.; Bedregal, B.; Santiago, R.H.N.; Fernandez, J.; Bustince, H. On Fuzzy Implications Derived from General Overlap Functions and Their Relation to Other Classes. Axloms
**2023**, 12, 17. [Google Scholar] [CrossRef] - Zhao, B.; Lu, J. On the distributivity for the ordinal sums of implications over t-norms and t-conorms. Int. J. Approx. Reason.
**2023**, 152, 284–296. [Google Scholar] [CrossRef] - Massanet, S.; Mir, A.; Riera, J.V.; Ruiz-Aguilera, R. Fuzzy implication functions with a specific expression: The polynomial case. Fuzzy Sets Syst.
**2022**, 451, 176–195. [Google Scholar] [CrossRef] - Souliotis, G.; Papadopoulos, B. Fuzzy Implications Generating from Fuzzy Negations. In Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), Proceedings of the 27th International Conference on Artificial Neural Networks (ICANN 2018), Part 1, Artificial Neural Networks and Machine Learning, Rhodes, Greece, 4–7 October 2018; Kurkova, V., Hammer, B., Manolopoulos, Y., Iliadis, L., Maglogiannis, I., Eds.; Springer: Berlin/Heidelberg, Germany, 2018; Volume 11139 LNCS, pp. 736–744. ISSN 03029743. ISBN 978-303001417-9. [Google Scholar] [CrossRef]
- Król, A. Generating of fuzzy implications. In Proceedings of the 8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2013), Milan, Italy, 11–13 September 2013; Pasi, G., Montero, J., Guicci, D., Eds.; Atlantis Press: Amsterdam, The Netherlands, 2013; Volume 32, pp. 758–763, ISBN 978-162993219-4. [Google Scholar] [CrossRef]
- Souliotis, G.; Papadopoulos, B. An algorithm for producing fuzzy negations via conical sections. Algorithms
**2019**, 12, 89. [Google Scholar] [CrossRef] - Giakoumakis, S.; Papadopoulos, B. An algorithm for fuzzy negations based-intuitionistic fuzzy copula aggregation operators in Multiple Attribute Decision Making. Algorithms
**2020**, 13, 154. [Google Scholar] [CrossRef] - Karbassi Yazdi, A.; Hanne, T.; Wang, Y.J.; Wee, H.-M. A Credit Rating Model in a Fuzzy Inference System Environment. Algorithms
**2019**, 12, 139. [Google Scholar] [CrossRef] - Sahin, B.; Yazir, D.; Hamid, A.A.; Abdul Rahman, N.S.F. Maritime Supply Chain Optimization by Using Fuzzy Goal Programming. Algorithms
**2021**, 14, 234. [Google Scholar] [CrossRef] - Koganti, S.; Koganti, K.J.; Salkuti, S.R. Design of Multi-Objective-Based Artificial Intelligence Controller for Wind/Battery-Connected Shunt Active Power Filter. Algorithms
**2022**, 15, 256. [Google Scholar] [CrossRef] - Haghighi, M.H.; Mousavi, S.M. A Mathematical Model and Two Fuzzy Approaches Based on Credibility and Expected Interval for Project Cost-Quality-Risk Trade-Off Problem in Time-Constrained Conditions. Algorithms
**2022**, 15, 226. [Google Scholar] [CrossRef] - Pelusi, D.; Mascella, R.; Tallini, L. Revised Gravitational Search Algorithms Based on Evolutionary-Fuzzy Systems. Algorithms
**2017**, 10, 44. [Google Scholar] [CrossRef] - Miramontes, I.; Guzman, J.C.; Melin, P.; Prado-Arechiga, G. Optimal Design of Interval Type-2 Fuzzy Heart Rate Level Classification Systems Using the Bird Swarm Algorithm. Algorithms
**2018**, 11, 206. [Google Scholar] [CrossRef] - Akisue, R.A.; Harth, M.L.; Horta, A.C.L.; de Sousa Junior, R. Optimized Dissolved Oxygen Fuzzy Control for Recombinant Escherichia coli Cultivations. Algorithms
**2021**, 14, 326. [Google Scholar] [CrossRef] - Fateminia, S.H.; Sumati, V.; Fayek, A.R. An Interval Type-2 Fuzzy Risk Analysis Model (IT2FRAM) for Determining Construction Project Contingency Reserve. Algorithms
**2020**, 13, 163. [Google Scholar] [CrossRef] - Shiau, J.-K.; Wei, Y.-C.; Chen, B.-C. A Study on the Fuzzy-Logic-Based Solar Power MPPT Algorithms Using Different Fuzzy Input Variables. Algorithms
**2015**, 8, 100–127. [Google Scholar] [CrossRef] - Paul, S.; Turnbull, R.; Khodadad, D.; Löfstrand, M. A Vibration Based Automatic Fault Detection Scheme for Drilling Process Using Type-2 Fuzzy Logic. Algorithms
**2022**, 15, 284. [Google Scholar] [CrossRef] - Yang, E. Fixpointed Idempotent Uninorm (Based) Logics. Mathematics
**2019**, 7, 107. [Google Scholar] [CrossRef] - Massanet, S.; Torrens, J.; Shi, Y.; Van Gasse, B.; Kerre, E.E.; Qin, F.; Baczyński, M.; Deschrijver, G.; Bedregal, B.; Beliakov, G.; et al. Advances in Fuzzy Implication Functions, 1st ed.; (Book Series: Studies in Fuzziness and soft Computing Studfuzz, Volume 300 Series Editor Kacprzyk J); Baczynski, M., Beliakov, G., Sola, H.B., Pradera, A., Eds.; Springer: Berlin/Heidelberg, Germany, 2013; Volume VII, p. 209. ISBN 978-3-642-35676-6. e-book ISBN 978-3-642-35677-3; ISSN 1434-9922. E-ISSN 1860-0808. [Google Scholar] [CrossRef]
- Baczynski, M.; Jayaram, B. Fuzzy Implications, 1st ed.; (Book Series: Studies in Fuzziness and Soft Computing STUDFUZZ Volume 231, Series Editor Kacprzyk J.); Springer: Berlin/Heidelberg, Germany, 2008; Volume XVIII, p. 310. ISBN 978-3-540-69080-1. e-book ISBN 978-3-540-69082-5; ISSN 1434-9922. E-ISSN 1860-0808. [Google Scholar] [CrossRef]
- Metcalfe, G.; Montagna, F. Substructural Fuzzy Logics. J. Symb. Log.
**2007**, 72, 834–864. [Google Scholar] [CrossRef] - Ruiz-Aguilera, D.; Torrens, J. Residual implications and co-implications from idempotent uninorms. Kybernetika
**2004**, 40, 21–38. [Google Scholar] - Grammatikopoulos, D.S.; Papadopoulos, B.K. A Method of Generating Fuzzy Implications with Specific Properties. Symmetry
**2020**, 12, 155. [Google Scholar] [CrossRef] - Massanet, S.; Torrens, J. The law of importation versus the exchange principle on fuzzy implications. Fuzzy Sets Syst.
**2011**, 168, 47–69. [Google Scholar] [CrossRef] - Mayor, G. Sugeno’s negations and t-norms. Mathw. Soft Comput.
**1994**, 1, 93–98. [Google Scholar] - Smets, P.; Magrez, P. Implications in fuzzy logic. Int. J. Approx. Reason.
**1987**, 1, 327–347. [Google Scholar] [CrossRef] - Cintula, P. Weakly Implicative (Fuzzy) Logics I: Basic properties. Arch. Math. Log.
**2006**, 45, 673–704. [Google Scholar] [CrossRef] - Klir, G.J.; Yuan, B. Fuzzy Sets and Fuzzy Logic: Theory and Applications, 1st ed.; Prentice Hall Press: UpperSaddle River, NJ, USA, 1995; p. 574, ISBN-10 0131011715; ISBN-13 978-0131011717. [Google Scholar]
- Trillas, E.; Mas, M.; Monserrat, M.; Torrens, J. On the representation of fuzzy rules. Int. J. Approx. Reason.
**2008**, 48, 583–597. [Google Scholar] [CrossRef] - Botzoris, G.; Papadopoulos, B. Fuzzy Sets: Applications in Design-Management of Engineer Projects, 1st ed.; Sofia: Xanthi, Greece, 2015; p. 424, ISBN-13 9789606706868. (In Greek) [Google Scholar]
- Dombi, J.; Jónás, T. On a strong negation-based representation of modalities. Fuzzy Sets Syst.
**2021**, 407, 142–160. [Google Scholar] [CrossRef] - Asiain, M.J.; Bustince, H.R.; Mesiar, R.; Kolesárová, A.; Takác, Z. Negations with respect to admissible orders in the interval-valued fuzzy set theory. IEEE Trans. Fuzzy Syst.
**2018**, 26, 556–568. [Google Scholar] [CrossRef] - Bustince, H.; Burillo, P.; Soria, F. Automorphisms, negations and implication operators. Fuzzy Sets Syst.
**2003**, 134, 209–229. [Google Scholar] [CrossRef] - Drygas, P. Some remarks about idempotent uninorms on complete lattice. In Advances in Intelligent Systems and Computing, Proceedings of the 10th Conference of the European Society for Fuzzy Logic and Technology, Warsaw, Poland, 11–15 September 2017; Kacprzyk, J., Szmidt, E., Zadrozny, S., Atanassov, K.T., Krawczak, M., Eds.; Springer: Cham, Switzerland, 2017; Volume 641, pp. 648–657. ISSN 21945357. ISBN 978-331966829-1. [Google Scholar] [CrossRef]
- Baczynski, M.; Jayaram, B.; Massanet, S.; Torrens, J. Fuzzy Implications: Past, Present, and Future. In Springer Handbook of Computational Intelligence, 1st ed.; Part of the Springer Handbooks Book Series, (SHB); Kacprzyk, J., Pedrycz, W., Eds.; Springer: Berlin/Heidelberg, Germany, 2015; pp. 183–202, ISBN online: 978-366243505-2; ISBN print: 978-366243504-5. [Google Scholar] [CrossRef]
- Available online: https://freemeteo.gr/mobile/kairos/kavala/istoriko/imerisio-istoriko/?gid=735861&station=5222&date=2021-08-01&language=greek&country=greece&fbclid=IwAR3Ph3AbGLWjGn39AWnLMqarYsgjypBRAtAG9gtcEITSWAVkDwEz4Hffn7M (accessed on 12 December 2023).

**Figure 1.**The procedure for the fuzzification of temperature values ranging from 7 to 36 based on the vertices of the isosceles trapezium [7, 19, 24, 36].

**Figure 2.**The procedure of fuzzification of humidity values ranging from 0.29 to 0.94 based on the vertices of the isosceles trapezium [0.29, 0.59, 0.64, 0.94].

**Figure 3.**The procedure of fuzzification of temperature values ranging from 7 to 36 based on the vertices of the random trapezium [7, 22, 23, 36].

**Figure 4.**The procedure of fuzzification of humidity values ranging from 0.29 to 0.94 based on the vertices of the random trapezium [0.29, 0.60, 0.61, 0.94].

**Figure 5.**The procedure of fuzzification of temperature values ranging from 7 to 36 based on the vertices of the isosceles triangle [7, 21.5, 36].

**Figure 6.**The procedure of fuzzification of humidity values ranging from 0.29 to 0.94 based on the vertices of the isosceles triangle [0.29, 0.615, 0.94].

**Figure 7.**The procedure of fuzzification of temperature values ranging from 7 to 36 based on the vertices of the scalene triangle [7, 22.5, 36].

**Figure 8.**The procedure of fuzzification of humidity values ranging from 0.29 to 0.94 based on the vertices of the scalene triangle [0.29, 0,605, 0.94].

Value m | Case I ^{1} | Case II ^{2} | |
---|---|---|---|

$\mathit{n}\left(\mathit{x}\right)\mathit{V}{\widehat{\mathbf{y}}}^{\mathit{m}}$ | |||

$\ge 0.9$ | 19 | 20 | |

$=1$ | 239 | 259 |

^{1}Case I: Isosceles trapezium.

^{2}Case II: Random trapezium.

Value m | Case III ^{3} | Case IV ^{4} | |
---|---|---|---|

$\mathit{n}\left(\mathit{x}\right)\mathit{V}{\widehat{\mathbf{y}}}^{\mathit{m}}$ | |||

$\ge 0.9$ | 22 | 21 | |

$=1$ | 289 | 269 |

^{3}Case III: Isosceles triangle.

^{4}Case IV: Scalene triangle.

**Table 3.**The number of temperature–humidity pairs where the new fuzzy implication takes the value $\ge 0.9$ and the optimal value $=1$ in case I—isosceles trapezium.

Value m | Case I with 19 Repetitions | Case I with 239 Repetitions | |
---|---|---|---|

$\mathit{n}\left(\mathit{x}\right)\mathit{V}{\widehat{\mathbf{y}}}^{\mathit{m}}$ | |||

$\ge 0.9$ | 47 | 1 | |

$=1$ | 74 | 120 |

**Table 4.**The number of temperature–humidity pairs wherein the new fuzzy implication takes the value $\ge 0.9$ and the optimal value $=1$ in case II—random trapezium.

Value m | Case II with 20 Repetitions | Case II with 259 Repetitions | |
---|---|---|---|

$\mathit{n}\left(\mathit{x}\right)\mathit{V}{\widehat{\mathbf{y}}}^{\mathit{m}}$ | |||

$\ge 0.9$ | 61 | 1 | |

$=1$ | 60 | 120 |

**Table 5.**The number of temperature–humidity pairs wherein the new fuzzy implication takes the value $\ge 0.9$ and the optimal value $=1$ in case III—isosceles triangle.

Value m | Case III with 22 Repetitions | Case III with 289 Repetitions | |
---|---|---|---|

$\mathit{n}\left(\mathit{x}\right)\mathit{V}{\widehat{\mathbf{y}}}^{\mathit{m}}$ | |||

$\ge 0.9$ | 55 | 1 | |

$=1$ | 66 | 120 |

**Table 6.**The number of temperature–humidity pairs wherein the new fuzzy implication takes the value $\ge 0.9$ and the optimal value =1 in case IV—scalene triangle.

Value m | Case IV with 21 Repetitions | Case IV with 269 Repetitions | |
---|---|---|---|

$\mathit{n}\left(\mathit{x}\right)\mathit{V}{\widehat{\mathbf{y}}}^{\mathit{m}}$ | |||

$\ge 0.9$ | 61 | 1 | |

$=1$ | 60 | 120 |

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## Share and Cite

**MDPI and ACS Style**

Daniilidou, A.; Konguetsof, A.; Souliotis, G.; Papadopoulos, B.
Generator of Fuzzy Implications. *Algorithms* **2023**, *16*, 569.
https://doi.org/10.3390/a16120569

**AMA Style**

Daniilidou A, Konguetsof A, Souliotis G, Papadopoulos B.
Generator of Fuzzy Implications. *Algorithms*. 2023; 16(12):569.
https://doi.org/10.3390/a16120569

**Chicago/Turabian Style**

Daniilidou, Athina, Avrilia Konguetsof, Georgios Souliotis, and Basil Papadopoulos.
2023. "Generator of Fuzzy Implications" *Algorithms* 16, no. 12: 569.
https://doi.org/10.3390/a16120569