Comparing Fuzzy Numbers Using Defuzzificators on OFN Shapes

Abstract

This chapter concerns an issue of comparing fuzzy numbers. The relationship of similarity is probably the most widely used and most difficult to determine the measure of compliance precisely. Analysis of the similarity between two objects is an essential tool in biology, taxonomy, and psychology, and is the basis for reasoning by analogy. This chapter describes methods for determining the similarity used in fuzzy logic. Many of them were dedicated only to triangular or trapezoidal fuzzy numbers. This was a computing inconvenience and raised the question about the axiological basis for such comparisons. The authors have proposed two new approaches to comparing fuzzy numbers using one of the known fuzzy number extensions that are Ordered Fuzzy Numbers (OFNs). This has allowed us to simplify operations and eliminate said dualism. Two order-sensitive defuzzification methods are presented in the chapter. For OFN numbers with positive order (compliant with the direction of the OX axis increase) the results of defuzzifications are results for numbers of different notations, for example, L-R, whereas for numbers with negative orders, the defuzzification result changes. An important part of the chapter is a catalogue of the shapes of numbers in OFN notation. This is probably the first summary of basic shapes of those numbers with the results of defuzzifications using several methods.

1 Introduction

In all fields of science for a long time it was necessary to compare certain objects. Some branches of science sought to answer the question about the nature of the similarities, whereas others needed precise formal definition. Comparison of two objects or occurrences can be seen as an attempt to determine the relation between them. The most important and most frequently used relations between objects are similarity, difference, and inclusion. In the literature, most attention is dedicated to the issue of the similarity of objects. In recent decades, the theory of fuzzy sets has been used in many areas of science and everyday life. The need to compare fuzzy sets emerged naturally from the very beginning of the theory. There are plenty of methods, often based on those used for conventional sets. Intensive development of fuzzy logic and its applications often need to identify new ways of comparing objects. This issue is particularly important in computer-aided decision support, classification, and processing of natural language. Although the issue of comparison is crucial for many applications of fuzzy set theory, still we failed to formalize clearly basic concepts such as similarity or inclusion. Some researchers concerned with fuzzy logic seek to define the concepts precisely, however, others questioned this approach, saying that imposing a rigid framework limits practical applications. Through years of development of fuzzy logic, many researchers have been developing methods of comparing sets and fuzzy numbers. Among them, it is impossible not to recall that several fuzzy number comparison methods and indices have been researched since 1977 by Zadeh [12], Yager [10, 11], Kaufman [14, 15], Chang [5], and Amado [1]. Bortolan and Degani [5] and Dadgostar [1] reviewed some of the methods for ranking fuzzy sets, including Yager’s first, second, and third indexes, Chang’s algorithm, Adamo’s method, Baas and Kwakernaak’s method [2], Baldwin and Guild’s method [3], Kerre’s method [9], Jain’s method [7, 8], and Dubois and Prade’s four grades [6] of dominance (PD, PSD, ND, NSD). Dadgostar and Kerr [1] proposed a consistent method, called the partial comparison method (PCM). Wang and Kerre [22, 23] proposed several axioms as reasonable properties to determine the rationality of a fuzzy ordering or ranking method and systematically compared a wide array of fuzzy ranking methods.

It appears that although defuzzification in some way deprives a fuzzy number of multidimensionality, this is a natural step preceding the comparison. Subsequent sections present some known defuzzification methods and two new methods proposed specifically for Ordered Fuzzy Numbers (OFNs). It has been proven that both new methods meet properties required for defuzzification operators.

2 Formal Approach to the Problem

The essence of an OFN is discussed in the introduction to this chapter. Redefinition of classic fuzzy sets, where, according to Zadeh, it is an organized pair, has widened the definition by an organized pair of functions. The OFN is defined as follows.

Definition 1

$$\begin{aligned} A=(x_{up},x_{down}) \end{aligned}$$

(6.1)

where $x_{up}, x_{down}: [0,1] \rightarrow R$ are continuous functions.

These functions are called the up-part and down-part, respectively, where both parts are connected by a constant function equal to 1. The order of a fuzzy number is its arrangement so that the up-part is the beginning of the OFN and the down-part is the end of this number.

Fig. 6.1

a OFN example, b OFN presented in relation to a classic fuzzy number, c arrow denotes the orientation and the order of inverted functions: first UP and then DOWN

Interpretation of the Ordered Fuzzy Number is shown in Fig. 6.1, where an example of an OFN is referred to a classic fuzzy number. The defuzzification process, as the last step in the three-step model of fuzzy control, converts a fuzzy set into a single real (defuzzified) value, on which the membership function is defined. The following expression describes defuzzification in a formal way.

Definition 2

$$\begin{aligned} W=\left\{ f:X \rightarrow [0,1] \,\right\} \rightarrow \,X \end{aligned}$$

(6.2)

where W is the defuzzification operator, f is the membership function, and X the universe on which membership functions are defined.

The process can be characterized on the basis of the properties, which are more desirable for a particular system. Considering the type of system, one can distinguish a fuzzy inference system, for which such property as processing power, is less important than for a diffuse control system, for which the processing power is an important parameter. The study [35, 37] introduced criteria of defuzzification operators for classic fuzzy numbers, on the basis of which individual defuzzification methods were assessed. The main conclusion is that there is no all-purpose defuzzification method. Defuzzification methods should be oriented to their field of application. For example, maximization methods, which include LOM (last of maxima) and FOM (first of maxima), are more suitable for inference systems. Research, which has been carried out by the authors of the above-mentioned study, proved that the distribution and field methods are more suitable for applications where control systems are used. Those methods include COG (center of gravity) and COA (center of area).

Upon development of Ordered Fuzzy Numbers, the authors of the paper [7, 20, 26] proposed criteria for defuzzification methods. That gave grounds for guidelines enabling the creation of suitable models of defuzzification operators. The following four conditions should be met for most of the methods.

Definition 3

Each functional $\phi$ is defined on R with the properties:

$$\begin{aligned} \phi (c)=c \end{aligned}$$

(6.3)

$$\begin{aligned} \phi (A+c)=\phi (A)+c \end{aligned}$$

(6.4)

$$\begin{aligned} \phi (cA)=c\phi (A) \end{aligned}$$

(6.5)

$$\begin{aligned} \phi (A)\ge 0\; \; if \; \; A \ge 0 \end{aligned}$$

(6.6)

is called a defuzzification functional,

where $\phi$ is a representation defined on the set of real numbers, and $\phi (c)$ is understood as the defuzzification of the c value on the set of real numbers. In other words, the defuzzification using the singleton method should give a defuzzified number (6.3). Condition (6.4) is related to additiveness, and it requires the defuzzification value for the sum of components to equal the sum of defuzzifications for individual components. Condition (6.5) requires the representation $\phi$ to be homogeneous (first degree); that is, if the argument is multiplied by a factor then the result will also be multiplied by some power of this factor. In this case, that power amounts to one. Condition (6.6) refers to the positive sense of a functional. Detailed interpretation of individual conditions is provided in the study by [5, 30].

3 Defuzzification Methods

It is well known that the defuzzification process reduces the fuzzy set to an individual defuzzified value. The mechanism of that operation consists mainly in the use of an appropriate defuzzification method. Available methods include the following classic solutions.

FOM, first of maxima: This method is FOM a method concerning the choice of the smallest element of the set core A, where the defuzzification value represents the relationship (6.7).

$$\begin{aligned} FOM(A)= min \, core(A) \end{aligned}$$

(6.7)

LOM, last of maxima: The appropriate choice of the maximum value of an element from the set core A, is the LOM method, the formula of which is presented below:

$$\begin{aligned} LOM(A)= max \, core(A) \end{aligned}$$

(6.8)

MOM, mean of maxima: The formula (6.9) illustrates the use of FOM and LOM as methods, the defuzzification values of which take into account the minimum and maximum elements of the fuzzy set core A. The resulting value is the mean value of those two methods.

$$\begin{aligned} MOM(A)= \frac{min \, core(A)+max \, core(A)}{2} \end{aligned}$$

(6.9)

RCOM, random choice of maxima: The method is also called defuzzification from a core, because the defuzzification value is always included in the core of a fuzzy set. The defuzzification value of this method is a random element $x\in core(A)$ calculated as a probability:

$$\begin{aligned} RCOM(A)=P(x)= \frac{\lambda ({x})}{\lambda ({core(A)})} \end{aligned}$$

(6.10)

where $\lambda$ is the Lebesgue measure in universe X.

MOS, mean of support: Defuzzification method MOS, the defuzzification value of which is the mean value of the A number carrier.

$$\begin{aligned} MOM(A)=\frac{supp(A)}{2} \end{aligned}$$

(6.11)

COG, center of gravity: The most widespread method, which is based on determination of the center of gravity of the analyzed system. In the fuzzy number A defuzzification process, the COG method is expressed as the formula (6.12).

$$\begin{aligned} COG(A)=\frac{\int _{a}^{b}x\mu _A(x)\mathrm {d}x}{\int _{a}^{b}\mu _A(x)\mathrm {d}x} \end{aligned}$$

(6.12)

BADD, basic defuzzification distribution: The defuzzification method proposed [24] as an extension of COG and MOM methods. We obtain the following defuzzification value from the fuzzy set A.

$$\begin{aligned} BADD(A)=\frac{\int _{a}^{b}x\mu _{A}^{\gamma }(x)\mathrm {d}x}{\int _{a}^{b}\mu _{A}^{\gamma }(x)\mathrm {d}x} \end{aligned}$$

(6.13)

Depending on parameter $\gamma \in [0,\infty ]$, BADD may assume the following instances: when $\gamma = 0$, $BADD(A) = MOS(A)$; when $= 1$, $BADD(A) = COG(A)$; and when $\gamma \rightarrow \infty$, $BADD(A) = MOM(A)$.

3.1 Defuzzification Methods for OFN

Classic defuzzification methods presented in the above parts of the chapter are reflected in Ordered Fuzzy Numbers. In the analysis of methods shown below, one of their explanations includes important characteristic elements of OFNs presented in Fig. 6.2.

Fig. 6.2

An OFN number and characteristic elements

In the definition of an Ordered Fuzzy Number expressed by the formula (6.1), an Ordered Fuzzy Number. A can also be defined, according to other approaches to that subject, as an oriented pair of continuous functions:

$$\begin{aligned} A=(f_A,g_A) \end{aligned}$$

(6.14)

where $f_A,g_A: [0,1] \rightarrow R$. The function $f_A$ is called the up-part $UP_A$ (beginning) of an Ordered Fuzzy Number A, and the function $g_A$ is called the-down part $DOWN_A$ (end) of an Ordered Fuzzy Number A.

In the interpretation of OFN defuzzification methods, the value of the $f_A$ function for 0 is $f_A$, for 1 is $f_A(1)$, and of the $g_A$ function it is: for 0 $g_A(0)$ and for 1 is $g_A(1)$.

$$\begin{aligned} \phi _{FOM}(f,g)= f(1) \end{aligned}$$

(6.15)

$$\begin{aligned} \phi _{LOM}(f,g)= g(1) \end{aligned}$$

(6.16)

$$\begin{aligned} \phi _{MOM}(f,g)= \frac{f(1)+g(1)}{2} \end{aligned}$$

(6.17)

$$\begin{aligned} \phi _{ROM}(f,g)=\zeta f(1)+(1-\zeta )g(1),\, \; \; \zeta = [0,1] \end{aligned}$$

(6.18)

$$\begin{aligned} \begin{aligned} \phi _{COG}(f,g)=\\\left\{ \begin{array}{ll} \frac{\int _{0}^{1}\frac{f(s)+g(s)}{2}|f(s)-g(s)|ds}{\int _{0}^{1}|f(s)-g(s)|ds} &{} ,for\, \int _{0}^{1}|f(s)-g(s)|ds\ne 0 \\ \frac{\int _{0}^{1}f(s)ds}{\int _{0}^{1}ds} &{} ,for\, \int _{0}^{1}|f(s)-g(s)|ds= 0 \end{array} \right. \end{aligned} \end{aligned}$$

(6.19)

$$\begin{aligned} \begin{aligned} \phi _{BADD}(A,\lambda )=\\ \begin{array}{ll} \frac{\int _{0}^{1}\frac{f(s)+g(s)}{2}|f(s)-g(s)|\cdot s^{\lambda - 1} ds}{\int _{0}^{1}|f(s)-g(s)|\cdot s^{\lambda - 1} ds}&,for\, \lambda \in [0,1] \end{array} \end{aligned} \end{aligned}$$

(6.20)

$$\begin{aligned} \phi _{GM}(f,g)= \frac{f(1)\cdot g(0)-f(0)\cdot g(1)}{f(1)+g(0)-f(0)-g(1)} \end{aligned}$$

(6.21)

The above formulas (6.15–6.21) are interpretations of classic defuzzification methods. In the discussed OFN theory [30] and in earlier studies, the geometrical mean method is proposed, which was created by D. Wilczyńska-Sztyma [38].

4 Definition of Golden Ratio Defuzzification Operator

At this point we present a proposal for a new method of defuzzification of a fuzzy controller, which is based on the concept of the golden ratio (GR), derived from the Fibonacci series. The origin of the method was the observation of numerous instances of the golden ratio in such diverse fields as biology, architecture, medicine, and painting. A particular area of its occurrence is genetics, where we find the golden ratio in the very structure of the DNA molecule (deoxyribonucleic acid molecules are 21 angstroms wide and 34 angstroms long for each full length of one double helix cycle). This fact makes the ratio in the Fibonacci series in some sense a universal design principle used by man and nature alike.

The Fibonacci series is based on the assumption that it starts with two ones, and each consecutive number is the sum of the previous two. The proposal for the golden ratio method of defuzzification is based on the proportion of the golden ratio. As a result of dividing each of the numbers by its predecessor, we always obtain quotients oscillating around the value of 1.618, the golden ratio number. The exact value of the limit is the golden number itself:

$$\begin{aligned} \lim _{n \rightarrow 0}\frac{k_{n+1}}{k_n}= 1,618033998875\dots = \varPhi \end{aligned}$$

(6.22)

The possibility of using this formula in the process of defuzzification is another example of the universality of the method, as it is applied in the new domain of fuzzy logic theory. Calculation of the classical formula of the golden mean assumes that two values of line segments a and b are in golden ratio $\varPhi$ to each other if:

$$\begin{aligned} \frac{a+b}{a}= \frac{a}{b}= \varPhi \end{aligned}$$

(6.23)

In this case, one method of finding the value of $\varPhi$ is to transform the left-hand fraction of Eq. (6.23) into:

$$\begin{aligned} \frac{a+b}{a}=1+\frac{b}{a}=1+\frac{1}{\varPhi },\, where \, \frac{b}{a}=\frac{1}{\varPhi } \end{aligned}$$

(6.24)

Following subsequent transformations of Eq. (6.24) we obtain the quadratic Eq. (6.25), for which we calculate the roots.

$$\begin{aligned} \varPhi ^2-\varPhi -1=0 \end{aligned}$$

(6.25)

As appropriate, using transformations of the formula in (6.25), we obtain two square roots (6.26).

$$\begin{aligned} \varPhi _1= \frac{1+\sqrt{5}}{2}\; or\; \varPhi _2= \frac{1-\sqrt{5}}{2} \end{aligned}$$

(6.26)

In view of the fact that the value of $\varPhi$ must be positive, in our example we select the positive root, as in Eq. (6.27).

$$\begin{aligned} \varPhi =\varPhi _1= \frac{1+\sqrt{5}}{2}= 1,618033998875\dots \end{aligned}$$

(6.27)

In sum, the ratio between two objects a and b is called the golden ratio when the value of $\varPhi = 1.61803398875\dots$.

Fig. 6.3

Golden ratio defuzzification value

The method of the golden ratio for a fuzzy number is Eq. (6.28):

Definition 4

$$\begin{aligned} \begin{aligned} GR=min(supp(A))+\frac{|supp(A)|}{\varPhi }\;\; \\where\; \varPhi = 1,618033998875\dots \end{aligned} \end{aligned}$$

(6.28)

where GR is the defuzzification operator and supp(A) is the support for fuzzy set A in universe X.

4.1 Golden Ratio for OFN

The mathematical formula (6.28) of the equation as well as the graphic interpretation presented in Fig. 6.3 applies to convex fuzzy numbers. In reference to the OFN, which has orientation, we should use another equation. Therefore, the interpretation of the proposed method is shown in Figs. 6.4 and 6.5.

Fig. 6.4

OFN number A $= [0,2,4,10]$

Fig. 6.5

OFN number B $= [10,4,2,0]$

We note that the individual parts of the two values of line segments a and b take up positions in relation to direction of the OFN. In the first case we have a positive OFN where a larger part of the golden ratio starts from base point f(0). In the second case, which is negative OFN, we have a base point as g(0). The method of the golden ratio for OFN is Eq. (6.29):

Definition 5

$$\begin{aligned} \begin{aligned} GR(A)= \\ \left\{ \begin{array}{ll} \textit{min}(\textit{supp}(A))+\dfrac{|\textit{supp}(A)|}{\varPhi }, &{} \textit{if}\, \textit{order} \,(A)\, \textit{is}\, \textit{positive} \\ &{} \\ max(supp(A))-\dfrac{|supp(A)|}{\varPhi }, &{} \textit{if}\, \textit{order} \,(A)\, \textit{is}\, \textit{negative} \end{array} \right. \end{aligned} \end{aligned}$$

(6.29)

The instrument of the golden ratio, as proposed in this chapter for fuzzy numbers, may serve as another defuzzification method. As a mathematical apparatus that affords wide-ranging possibilities in description and processing of information, it becomes a new solution in constructing models of fuzzy controllers used as tools for inferencing or control.

5 Golden Ratio

Let

$$\begin{aligned} supp(A)=\left\{ \begin{matrix} g_{A}(0)-f_{A}(0) &{} \textit{if} &{} \textit{ordered}\; A\; \textit{is}\; \textit{positive}\\ \\ f_{A}(0)-g_{A}(0) &{} \textit{if} &{} \textit{ordered} \; A\; \textit{is}\; \textit{negative} \end{matrix}\right. \end{aligned}$$

(6.30)

Let

$$\begin{aligned} min(supp(A))=\left\{ \begin{matrix} f_{A}(0)&{} \textit{if} &{} \textit{ordered}\; A\; \textit{is}\; \textit{positive}\\ \\ g_{A}(0)&{} \textit{if} &{} \textit{ordered} \; A\; \textit{is}\; \textit{negative} \end{matrix}\right. \end{aligned}$$

(6.31)

Let

$$\begin{aligned} max(supp(A))=\left\{ \begin{matrix} g_{A}(0)&{} \textit{if} &{} \textit{ordered}\; A\; \textit{is}\; \textit{positive}\\ \\ f_{A}(0)&{} \textit{if} &{} \textit{ordered} \; A\; \textit{is}\; \textit{negative} \end{matrix}\right. \end{aligned}$$

(6.32)

Symbol $\varPhi$ is designated with a golden number, as $\varPhi = 1,680\dots$.

Definition 6

The functional $\varphi _{GR}: R \rightarrow R$, called the golden ratio, is expressed with the formula:

$$\begin{aligned} \varphi _{GR}=\left\{ \begin{matrix} min(supp(A))+\dfrac{supp(A)}{\varPhi } &{} \textit{if} &{} \textit{ordered}\; A\; \textit{is}\; \textit{positive}\\ max(supp(A))-\dfrac{supp(A)}{\varPhi }&{} \textit{if} &{} \textit{ordered} \; A\; \textit{is}\; \textit{negative} \end{matrix}\right. \end{aligned}$$

(6.33)

Theorem 1

Mapping of $\varphi _{GR}: R \rightarrow R$ expressed with the formula (6.33) is a defuzzification functional.

6 Defuzzification Conditions for GR

Recall that, by the definition of adding fuzzy numbers and multiplication of a fuzzy number by a real number, the following equalities apply.

$$\begin{aligned} f_{A+c^+}(s)=f_{A}(s) +c \quad&oraz&\quad g_{A+c^+}(s)=g_{A}(s)+c \end{aligned}$$

(6.34)

dla $s\in R$ oraz $c^+$ -crisp number

$$\begin{aligned} f_{cA}(s)=c\cdot f_{A}(s)\quad&oraz&\quad g_{cA}(s)=c\cdot g_{A}(s) \end{aligned}$$

(6.35)

dla $s\in R$

It is easy to see that

$$\begin{aligned} supp(A+c^+)= supp(A) \end{aligned}$$

(6.36)

$$\begin{aligned} supp(c\cdot A)= c\cdot supp(A) \end{aligned}$$

(6.37)

Namely, using (6.34) we obtain the following for A of positively ordered numbers.

$$\begin{aligned} supp(A+c^+)= g_{A+c^+}(0) - f_{A+c^+}(0)= g_{A}(0)+c -f_{A}(0)-c=supp(A) \end{aligned}$$

(6.38)

By analogy, we obtain (6.36) for A of negatively ordered numbers.

In order to show (6.37) for A of negatively ordered numbers we use (6.35).

$$\begin{aligned} supp(c\cdot A)= g_{c \cdot A}(0) - f_{c \cdot A}(0)= c\cdot g_{A}(0)- c\cdot f_{A}(0)=c\cdot supp(A) \end{aligned}$$

(6.39)

By analogy, we prove (6.37) dla A of negatively ordered numbers. It follows directly from (6.34) and (6.35) that

$$\begin{aligned} min(supp(A+c^+))=min(supp(A))+c\quad&and&\\ \nonumber \quad max(supp(A+c^+))=max(supp(A))+c \end{aligned}$$

(6.40)

and

$$\begin{aligned} min(supp(c\cdot A))=c\cdot min(supp(A))\quad&and&\\ \nonumber \quad max(supp(c\cdot A))=c\cdot max(supp(A)) \end{aligned}$$

(6.41)

6.1 Normalization

Proof

normalized (6.3)

The normalization property results directly from the definition of 6, because

$$\begin{aligned} \begin{aligned} \varphi _{GR}(c^+)=min(supp(c^+))+\frac{supp(c^+)}{\varPhi }=c+\frac{c^+(0)-c^+(0)}{\varPhi }=c \end{aligned} \end{aligned}$$

(6.42)

Therefore, we have shown that $\varphi _{GR}$ fulfills the homogeneity condition; that is:

$$\begin{aligned} \begin{aligned} \varphi _{GR}(c^+)=c \end{aligned} \end{aligned}$$

(6.43)

6.2 Restricted Additivity

Now let us show the restricted additivity property.

Proof

restricted additivity (6.4):

From the Definition 6 of the golden ratio functional, we obtain

$$\begin{aligned} \begin{aligned} \varphi _{GR}(A+c^+)=min(supp(A+c^+))+\frac{supp(A+c^+)}{\varPhi }\\ \quad \textstyle {for\quad positively\quad ordered \quad A} \end{aligned} \end{aligned}$$

(6.44)

and

$$\begin{aligned} \begin{aligned} \varphi _{GR}(A+c^+)=max(supp(A+c^+))+\frac{supp(A+c^+)}{\varPhi }\\ \quad \textstyle {for\quad negatively\quad ordered\quad A} \end{aligned} \end{aligned}$$

(6.45)

It follows directly from (6.40) and (6.36) that

$$\begin{aligned} \begin{aligned} \varphi _{GR}(A+c^+)=c+\left( min(supp(A))+\frac{supp(A)}{\varPhi }\right) =\varphi _{GR}(A)+c\\ \quad \textstyle {for\quad positively\quad ordered \quad A} \end{aligned} \end{aligned}$$

(6.46)

and

$$\begin{aligned} \begin{aligned} \varphi _{GR}(A+c^+)=c+\left( max(supp(A))+\frac{supp(A)}{\varPhi }\right) =\varphi _{GR}(A)+c\\ \quad \textstyle {for\quad negatively\quad ordered\quad A} \end{aligned} \end{aligned}$$

(6.47)

Therefore, we have shown that $\varphi _{GR}$ fulfills the restricted additivity condition; that is:

$$\begin{aligned} \begin{aligned} \varphi _{GR}(A+c^+)=\varphi _{GR}(A)+c \end{aligned} \end{aligned}$$

(6.48)

6.3 Homogeneity

Proof

homogeneity  (6.5):

Based on the Definition 6 we obtain the following.

$$\begin{aligned} \begin{aligned} \varphi _{GR}(c\cdot A)=min(supp(c\cdot A))+\frac{supp(c\cdot A)}{\varPhi }\\ \quad \textstyle {for\quad positively\quad ordered \quad A} \end{aligned} \end{aligned}$$

(6.49)

and

$$\begin{aligned} \begin{aligned} \varphi _{GR}(c\cdot A)=max(supp(c\cdot A))+\frac{supp(c\cdot A)}{\varPhi }\\ \quad \textstyle {for\quad negatively\quad ordered \quad A} \end{aligned} \end{aligned}$$

(6.50)

It follows directly from (6.41) and (6.37) that

$$\begin{aligned} \begin{aligned} \varphi _{GR}(c\cdot A)=c\cdot min(supp(A))+\frac{supp(A)\cdot c}{\varPhi }\\=c\cdot \left( min(supp(A))+\frac{supp(A)}{\varPhi } \right) = c\cdot \varphi \\ \quad \textstyle {for\quad positively\quad ordered \quad A} \end{aligned} \end{aligned}$$

(6.51)

and

$$\begin{aligned} \begin{aligned} \varphi _{GR}(c\cdot A)=c\cdot max(supp(A))+\frac{supp(A)\cdot c}{\varPhi }\\=c\cdot \left( max(supp(A))+\frac{supp(A)}{\varPhi } \right) = c\cdot \varphi _{GR}\\ \quad \textstyle {for\quad negatively\quad ordered \quad A} \end{aligned} \end{aligned}$$

(6.52)

Therefore, we have shown that

$$\begin{aligned} \begin{aligned} \varphi _{GR}(c\cdot A)=c\cdot \varphi _{GR}(A) \end{aligned} \end{aligned}$$

(6.53)

The equalities (6.43), (6.48), and (6.53) imply that $\mathop {\textstyle {\varphi }}\limits _{GR}$ is a defuzzification functional, which was to be proven.

7 Definition of Mandala Factor Defuzzification Operator

Buddhist monks can create amazing pictures with colored sand grains. Those pictures are called mandala. It is difficult to name them paintings because we expect paintings to be rather more lasting. Anyone who has ever seen meditating monks creating, grain after grain, a previously designed picture, remembers such a conclusion for a long time. You can observe the beauty of their art and on the other hand the transitory nature (in the literal sense) of the technique they use is evident. The same reverence is seen in Christianity in the Eastern Orthodox rite when icons are painted, but fortunately for culture and art, the effects of the work can be seen for a long time. The Buddhist mandala is a harmonious combination of a wheel and a square, where the wheel is a symbol of heaven, transcendence, externality, and infinity, and the square depicts the inner sphere, that is, the matters associated with a human and the earth. Both figures are linked by the central point, which is both the start and end of the entire system. The mandala creation process itself, as well as its destruction, is a religious act (Fig. 6.6).

Fig. 6.6

Mandala creation http://wellness.gcublogs.org/tag/sand-mandala/

Fig. 6.7

Mandala factor visualization

The mandala factor defuzzification operator is inspired by mandala. Let A be a given fuzzy number shown in Fig. 6.2. Let it assume the shape of a trapezoid in Fig. 6.7a. A trapezoid can in a particular case come down to ariangle, but we remain at a trapezoid, which makes our analysis more universal. Then one must fill in the outline marked by the sides of the number and the OX axis with virtual grains of sand in Fig. 6.7a. A number of virtual sand grains are collected in this way. Then one must construct a rectangle, the base of which is equal to the support value of the fuzzy number. The rectangle built in such a manner should be filled with virtual sand grains, starting from the outermost left side in Fig. 6.7b. The filling process should be done vertically in columns until all grains are used. A real number obtained as a result of defuzzification is the value above which the last filled column was finished.

Mathematical formalism (6.54) of the above-described mandala factor visualization is shown below. Calculation of the R value uses the mandala factor $\Psi _M$ for the rising edge, falling edge, and core set function integral. Then the obtained value should be scaled from the center of the coordinate system by adding it to the start of the support value of the fuzzy number. When defuzzification is performed in the OFN arithmetic, then in the case of a positive order, one should proceed as described below, whereas in the case of a negative order, one should deduct the calculated value from the first coordinate of the OFN corresponding to the outermost right side of the OFN support.

Definition 7

$$\begin{aligned} MF(A)=\left\{ \begin{array}{ll} c+r, &{} \textit{if}\, \textit{order} \,(A)\, \textit{is}\, \textit{positive} \\ c-r, &{} \textit{if}\, \textit{order} \,(A)\, \textit{is}\, \textit{negative} \end{array} \right. \end{aligned}$$

(6.54)

where

$$\begin{aligned} r=\frac{1}{d-c}\int _{c}^{d}x\, dx -\frac{c}{d-c}\int _{c}^{d}dx+\frac{f}{f-e}\int _{f}^{e}dx \nonumber \\ -\frac{1}{f-e}\int _{e}^{f}x\, dx + \int _{d}^{e}dx \end{aligned}$$

(6.55)

8 Mandala Factor

Definition 8

The $MF: R \rightarrow R$, called the mandala factor, is expressed by the formula:

$$\begin{aligned} MF(A)=\left\{ \begin{matrix} f_A(0)+\mathop {\textstyle r}\limits _{A} &{} \textit{if} &{} \textit{ordered}\; A\; \textit{is}\; \textit{positive}\\ g_A(0)-\mathop {\textstyle r}\limits _{A} &{} \textit{if} &{} \textit{ordered} \; A\; \textit{is}\; \textit{negative} \end{matrix}\right. \end{aligned}$$

(6.56)

where

$$\begin{aligned} \begin{aligned} \mathop {\textstyle r}\limits _{A}=\frac{1}{f_A(1)-f_A(0)}\int \limits _{f_A(0)}^{f_A(1)}xdx - \frac{f_A(0)}{f_A(1)-f_A(0)}\int \limits _{f_A(0)}^{f_A(1)}dx\\ +\frac{g_A(0)}{g_A(0)-g_A(1)}\int \limits _{g_A(1)}^{g_A(0)}-\frac{1}{g_A(0)-g_A(1)}\int \limits _{g_A(1)}^{g_A(0)}xdx\\ +\int \limits _{f_A(1)}^{g_A(1)}dx \end{aligned} \end{aligned}$$

(6.57)

and $r_A = 0$ for A such that $f_A = const$ lub $g_A = const$ (in particular $r_{c^+} = 0$).

9 Defuzzification Conditions for MF

Proposition

The mandala factor is a defuzzification functional.

9.1 Normalization

Proof

normalized (6.3)

It results directly from the definition that

$$\begin{aligned} MF(c^+)=c+\mathop {\textstyle r}\limits _{c^+}=c \end{aligned}$$

(6.58)

fulfills the normalization condition.

9.2 Restricted Additivity

Proof

restricted additivity (6.4):

Based on the definition of the mandala factor, we obtain the following.

$$\begin{aligned} MF(A+c^+)=\left\{ \begin{matrix} \mathop {\textstyle f}\limits _{A+c^+}(0)+\mathop {\textstyle r}\limits _{A+c^+} &{} \textit{if} &{} \textit{ordered}\; A\; \textit{is}\; \textit{positive}\\ \mathop {\textstyle g}\limits _{A+c^+}(0)-\mathop {\textstyle r}\limits _{A+c^+} &{} \textit{if} &{} \textit{ordered} \; A\; \textit{is}\; \textit{negative} \end{matrix}\right. \end{aligned}$$

(6.59)

whereas

$$\begin{aligned} \begin{aligned} \mathop {\textstyle r}\limits _{A+c^+}=\frac{1}{\mathop {\textstyle f}\limits _{A+c^+}(1)-\mathop {\textstyle f}\limits _{A+c^+}(0)}\int \limits _{\mathop {\textstyle f}\limits _{A+c^+}(0)}^{\mathop {\textstyle f}\limits _{A+c^+}(1)}xdx - \frac{\mathop {\textstyle f}\limits _{A+c^+}(0)}{\mathop {\textstyle f}\limits _{A+c^+}(1)-\mathop {\textstyle f}\limits _{A+c^+}(0)}\int \limits _{\mathop {\textstyle f}\limits _{A+c^+}(0)}^{\mathop {\textstyle f}\limits _{A+c^+}(1)}dx\\ +\frac{\mathop {\textstyle g}\limits _{A+c^+}(0)}{\mathop {\textstyle g}\limits _{A+c^+}(0)-\mathop {\textstyle g}\limits _{A+c^+}(1)}\int \limits _{\mathop {\textstyle g}\limits _{A+c^+}(1)}^{\mathop {\textstyle g}\limits _{A+c^+}(0)}dx-\frac{1}{\mathop {\textstyle g}\limits _{A+c^+}(0)-\mathop {\textstyle g}\limits _{A+c^+}(1)}\int \limits _{\mathop {\textstyle g}\limits _{A+c^+}(1)}^{\mathop {\textstyle g}\limits _{A+c^+}(0)}xdx\\ +\int \limits _{\mathop {\textstyle f}\limits _{A+c^+}(1)}^{\mathop {\textstyle g}\limits _{A+c^+}(1)}dx \end{aligned} \end{aligned}$$

(6.60)

In view of the above Eqs. 6.60 and 6.56 to show the restricted additivity property it is sufficient to demonstrate that

$$\begin{aligned} \mathop {\textstyle r}\limits _A=\mathop {\textstyle r}\limits _{A+c^+} \end{aligned}$$

(6.61)

Operations on OFNs imply that $\mathop {\textstyle \forall }\limits _{s \in R}$

$$\begin{aligned} \left\{ \begin{matrix} \mathop {\textstyle f}\limits _{A+c^+}(s)= \mathop {\textstyle f}\limits _A(s)+c \\ \mathop {\textstyle g}\limits _{A+c^+}(s)= \mathop {\textstyle g}\limits _A(s)+c \end{matrix}\right. \end{aligned}$$

(6.62)

First, we note that

$$\begin{aligned} \int \limits _{\mathop {\textstyle f}\limits _{A}(0)+c}^{\mathop {\textstyle f}\limits _{A}(1)+c}dx=\mathop {\textstyle f}\limits _{A}(1)-\mathop {\textstyle f}\limits _{A}(0)=\int \limits _{\mathop {\textstyle f}\limits _{A}(0)}^{\mathop {\textstyle f}\limits _{A}(1)}dx \end{aligned}$$

(6.63)

and

$$\begin{aligned} \begin{aligned} \int \limits _{\mathop {\textstyle f}\limits _{A}(0)+c}^{\mathop {\textstyle f}\limits _{A}(1)+c}xdx=\\ \frac{1}{2}\left( \left( \mathop {\textstyle f}\limits _{A}(1)\right) ^2 -\left( \mathop {\textstyle f}\limits _{A}(0)\right) ^2 \right) +c\left( \mathop {\textstyle f}\limits _{A}(1) -\mathop {\textstyle f}\limits _{A}(0) \right) \\ =\int \limits _{\mathop {\textstyle f}\limits _{A}(0)}^{\mathop {\textstyle f}\limits _{A}(1)}xdx+c\int \limits _{\mathop {\textstyle f}\limits _{A}(0)}^{\mathop {\textstyle f}\limits _{A}(1)}dx \end{aligned} \end{aligned}$$

(6.64)

Of course, if the function $g_A$ is used in the formulas (6.63) and (6.64), instead of $f_A$ we then get similar equivalences; that is:

$$\begin{aligned} \begin{aligned} \int \limits _{\mathop {\textstyle g}\limits _{A}(1)+c}^{\mathop {\textstyle g}\limits _{A}(0)+c}dx=\int \limits _{\mathop {\textstyle g}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(0)}dx \end{aligned} \end{aligned}$$

(6.65)

$$\begin{aligned} \begin{aligned} \int \limits _{\mathop {\textstyle g}\limits _{A}(1)+c}^{\mathop {\textstyle g}\limits _{A}(0)+c}dx=\int \limits _{\mathop {\textstyle g}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(0)}xdx + c\int \limits _{\mathop {\textstyle g}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(0)}dx \end{aligned} \end{aligned}$$

(6.66)

$$\begin{aligned} \begin{aligned} \int \limits _{\mathop {\textstyle f}\limits _{A}(1)+c}^{\mathop {\textstyle g}\limits _{A}(1)+c}dx=\int \limits _{\mathop {\textstyle f}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(1)}dx \end{aligned} \end{aligned}$$

(6.67)

Using (6.60) and (6.62) we get

$$\begin{aligned} \begin{aligned} \mathop {\textstyle r}\limits _{A+c^+}=\frac{1}{\mathop {\textstyle f}\limits _{A}(1)-\mathop {\textstyle f}\limits _{A}(0)}\int \limits _{\mathop {\textstyle f}\limits _{A}(0)+c}^{\mathop {\textstyle f}\limits _{A}(1)+c}xdx - \frac{\mathop {\textstyle f}\limits _{A}(0)+c}{\mathop {\textstyle f}\limits _{A}(1)-\mathop {\textstyle f}\limits _{A}(0)}\int \limits _{\mathop {\textstyle f}\limits _{A}(0)+c}^{\mathop {\textstyle f}\limits _{A}(1)+c}dx\\ +\frac{\mathop {\textstyle g}\limits _{A}(0)+c}{\mathop {\textstyle g}\limits _{A}(0)-\mathop {\textstyle g}\limits _{A}(1)}\int \limits _{\mathop {\textstyle g}\limits _{A}(1)+c}^{\mathop {\textstyle g}\limits _{A}(0)+c}dx-\frac{1}{\mathop {\textstyle g}\limits _{A}(0)-\mathop {\textstyle g}\limits _{A}(1)}\int \limits _{\mathop {\textstyle g}\limits _{A}(1)+c}^{\mathop {\textstyle g}\limits _{A}(0)+c}xdx\\ +\int \limits _{\mathop {\textstyle f}\limits _{A}(1)+c}^{\mathop {\textstyle g}\limits _{A}(1)+c}dx \end{aligned} \end{aligned}$$

(6.68)

Now the equalities (6.63), (6.64), (6.65), (6.66), and (6.67) are applied to the above formula (6.68) and we obtain

$$\begin{aligned} \begin{aligned} \mathop {\textstyle r}\limits _{A+c^+}=\frac{1}{\mathop {\textstyle f}\limits _{A}(1)-\mathop {\textstyle f}\limits _{A}(0)} \left( \int \limits _{\mathop {\textstyle f}\limits _{A}(0)}^{\mathop {\textstyle f}\limits _{A}(1)}xdx + c\int \limits _{\mathop {\textstyle f}\limits _{A}(0)}^{\mathop {\textstyle f}\limits _{A}(1)}dx \right) - \frac{\mathop {\textstyle f}\limits _{A}(0)}{\mathop {\textstyle f}\limits _{A}(1)-\mathop {\textstyle f}\limits _{A}(0)}\int \limits _{\mathop {\textstyle f}\limits _{A}(0)}^{\mathop {\textstyle f}\limits _{A}(1)}dx\\ -\frac{c}{\mathop {\textstyle f}\limits _{A}(1)-\mathop {\textstyle f}\limits _{A}(0)}\int \limits _{\mathop {\textstyle f}\limits _{A}(0)}^{\mathop {\textstyle f}\limits _{A}(1)}dx+\frac{\mathop {\textstyle g}\limits _{A}(0)}{\mathop {\textstyle g}\limits _{A}(0)-\mathop {\textstyle g}\limits _{A}(1)}\int \limits _{\mathop {\textstyle g}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(0)}xdx +\frac{c}{\mathop {\textstyle g}\limits _{A}(0)-\mathop {\textstyle g}\limits _{A}(1)}\int \limits _{\mathop {\textstyle g}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(0)}dx\\ -\frac{1}{\mathop {\textstyle g}\limits _{A}(1)-\mathop {\textstyle g}\limits _{A}(0)}\left( \int \limits _{\mathop {\textstyle g}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(0)}xdx + c\int \limits _{\mathop {\textstyle g}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(0)}dx \right) +\int \limits _{\mathop {\textstyle f}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(1)}dx \\ =\frac{1}{\mathop {\textstyle f}\limits _{A}(1)-\mathop {\textstyle f}\limits _{A}(0)}\int \limits _{\mathop {\textstyle f}\limits _{A}(0)}^{\mathop {\textstyle f}\limits _{A}(1)}xdx - \frac{\mathop {\textstyle f}\limits _{A}(0)}{\mathop {\textstyle f}\limits _{A}(1)-\mathop {\textstyle f}\limits _{A}(0)}\int \limits _{\mathop {\textstyle f}\limits _{A}(0)}^{\mathop {\textstyle f}\limits _{A}(1)}dx+ \frac{\mathop {\textstyle g}\limits _{A}(0)}{\mathop {\textstyle g}\limits _{A}(0)-\mathop {\textstyle g}\limits _{A}(1)}\int \limits _{\mathop {\textstyle g}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(0)}dx \\ -\frac{1}{\mathop {\textstyle g}\limits _{A}(0)-\mathop {\textstyle g}\limits _{A}(1)}\int \limits _{\mathop {\textstyle g}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(0)}xdx + \int \limits _{\mathop {\textstyle f}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(0)}dx=\mathop {\textstyle r}\limits _{A} \end{aligned} \end{aligned}$$

(6.69)

It follows directly from the equality (6.61) and the Definition 8 of the mandala factor that

$$\begin{aligned} \begin{aligned} MF(A+c^+)={\mathop {\textstyle f}\limits _{A+c^+}(0)}+\mathop {\textstyle r}\limits _{A+c^+}=\mathop {\textstyle f}\limits _{A}(0)+c+\mathop {\textstyle r}\limits _{A}=MF(A)+c\,\\ \textstyle {if\, order\, A\, is\, positive} \end{aligned} \end{aligned}$$

(6.70)

and

$$\begin{aligned} \begin{aligned} MF(A+c^+)={\mathop {\textstyle g}\limits _{A+c^+}(0)}+\mathop {\textstyle r}\limits _{A+c^+} =\mathop {\textstyle g}\limits _{A}(0)+c+\mathop {\textstyle r}\limits _{A}=MF(A)+c\,\\ \textstyle {if\, order\, A\, is\, negative} \end{aligned} \end{aligned}$$

(6.71)

which proves the restricted additivity property.

9.3 Homogeneity

Proof

of homogeneity (6.5)

It follows directly from the mandala factor Definition 8 that

$$\begin{aligned} MF(cA)=\left\{ \begin{matrix} \mathop {\textstyle f}\limits _{cA}(0)+\mathop {\textstyle r}\limits _{cA} &{} \textit{if} &{} \textit{ordered}\; A\; \textit{is}\; \textit{positive}\\ \mathop {\textstyle g}\limits _{cA}(0)-\mathop {\textstyle r}\limits _{cA} &{} \textit{if} &{} \textit{ordered} \; A\; is\; \textit{negative} \end{matrix}\right. \end{aligned}$$

(6.72)

whereas

$$\begin{aligned} \begin{aligned} \mathop {\textstyle r}\limits _{cA}=\frac{1}{\mathop {\textstyle f}\limits _{cA}(1)-\mathop {\textstyle f}\limits _{cA}(0)}\int \limits _{\mathop {\textstyle f}\limits _{cA}(0)}^{\mathop {\textstyle f}\limits _{cA}(1)}xdx - \frac{\mathop {\textstyle f}\limits _{cA}(0)}{\mathop {\textstyle f}\limits _{cA}(1)-\mathop {\textstyle f}\limits _{cA}(0)}\int \limits _{\mathop {\textstyle f}\limits _{cA}(0)}^{\mathop {\textstyle f}\limits _{cA}(1)}dx\\ + \frac{\mathop {\textstyle g}\limits _{cA}(0)}{\mathop {\textstyle g}\limits _{cA}(0)-\mathop {\textstyle g}\limits _{cA}(1)}\int \limits _{\mathop {\textstyle g}\limits _{cA}(1)}^{\mathop {\textstyle g}\limits _{cA}(0)}dx - \frac{1}{\mathop {\textstyle g}\limits _{cA}(0)-\mathop {\textstyle g}\limits _{cA}(1)}\int \limits _{\mathop {\textstyle g}\limits _{cA}(1)}^{\mathop {\textstyle g}\limits _{cA}(0)}xdx + \int \limits _{\mathop {\textstyle g}\limits _{cA}(1)}^{\mathop {\textstyle g}\limits _{cA}(1)}dx \end{aligned} \end{aligned}$$

(6.73)

for A such that $\mathop {\textstyle f}\limits _{A}\ne const$ oraz $\mathop {\textstyle g}\limits _{A}\ne const$ and $c\ne 0$ and $\mathop {\textstyle r}\limits _{A} = 0$ dla $c = 0$. It results from operations on OFNs that for $\mathop {\textstyle \forall }\limits _{s \in R}$

$$\begin{aligned} \left\{ \begin{matrix} \mathop {\textstyle f}\limits _{cA}(s)= c*\mathop {\textstyle f}\limits _A(s) \\ \mathop {\textstyle g}\limits _{cA}(s)= c*\mathop {\textstyle g}\limits _A(s) \end{matrix}\right. \end{aligned}$$

(6.74)

Formulas (6.72), (6.74), and (6.56) imply that for any homogeneity it is sufficient to indicate the following equality

$$\begin{aligned} \mathop {\textstyle r}\limits _{cA}=c\mathop {\textstyle r}\limits _{A} \end{aligned}$$

(6.75)

First, we note that

$$\begin{aligned} \int \limits _{\mathop {\textstyle cf}\limits _{A}(0)}^{\mathop {\textstyle cf}\limits _{A}(1)}dx=c\left( \mathop {\textstyle f}\limits _{A}(1)-\mathop {\textstyle f}\limits _{A}(0)\right) =c\int \limits _{\mathop {\textstyle f}\limits _{A}(0)}^{\mathop {\textstyle f}\limits _{A}(1)}dx \end{aligned}$$

(6.76)

and

$$\begin{aligned} \begin{aligned} \int \limits _{\mathop {\textstyle cf}\limits _{A}(0)}^{\mathop {\textstyle cf}\limits _{A}(1)}xdx=\frac{1}{2}c^2\left( \left( \mathop {\textstyle f}\limits _{A}(1)\right) ^2 -\left( \mathop {\textstyle f}\limits _{A}(0)\right) ^2 \right) =c^2\int \limits _{\mathop {\textstyle f}\limits _{A}(0)}^{\mathop {\textstyle f}\limits _{A}(1)}xdx \end{aligned} \end{aligned}$$

(6.77)

Obviously, use of the function $\mathop {\textstyle g}\limits _{A}$ instead of $\mathop {\textstyle f}\limits _{A}$ in the above formulas and results in analogous equalities; that is:

$$\begin{aligned} \begin{aligned} \int \limits _{\mathop {\textstyle g}\limits _{cA}(1)}^{\mathop {\textstyle g}\limits _{cA}(0)}dx=c\int \limits _{\mathop {\textstyle g}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(0)}dx \end{aligned} \end{aligned}$$

(6.78)

$$\begin{aligned} \begin{aligned} \int \limits _{\mathop {\textstyle g}\limits _{cA}(1)}^{\mathop {\textstyle g}\limits _{cA}(0)}dx=c^2\int \limits _{\mathop {\textstyle g}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(0)}xdx \end{aligned} \end{aligned}$$

(6.79)

$$\begin{aligned} \begin{aligned} \int \limits _{\mathop {\textstyle f}\limits _{cA}(1)}^{\mathop {\textstyle g}\limits _{cA}(1)}dx=c\int \limits _{\mathop {\textstyle f}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(1)}dx \end{aligned} \end{aligned}$$

(6.80)

It follows directly from z (6.73) and (6.74) that for dla A such that $\mathop {\textstyle f}\limits _{A}\ne const$ and $\mathop {\textstyle g}\limits _{A}\ne const$ and $c\ne 0$

$$\begin{aligned} \begin{aligned} \mathop {\textstyle r}\limits _{cA}=\frac{1}{c\left( \mathop {\textstyle f}\limits _{A}(1)-\mathop {\textstyle f}\limits _{A}(0)\right) }\int \limits _{\mathop {\textstyle cf}\limits _{A}(0)}^{\mathop {\textstyle cf}\limits _{A}(1)}xdx - \frac{c\mathop {\textstyle f}\limits _{A}(0)}{c\left( \mathop {\textstyle f}\limits _{A}(1)-\mathop {\textstyle f}\limits _{A}(0)\right) }\int \limits _{\mathop {\textstyle cf}\limits _{A}(0)}^{\mathop {\textstyle cf}\limits _{A}(1)}dx\\ + \frac{c\mathop {\textstyle g}\limits _{A}(0)}{c\left( \mathop {\textstyle g}\limits _{A}(0)-\mathop {\textstyle g}\limits _{A}(1) \right) }\int \limits _{\mathop {\textstyle cg}\limits _{A}(1)}^{\mathop {\textstyle cg}\limits _{A}(0)}dx - \frac{1}{c\left( \mathop {\textstyle g}\limits _{A}(0)-\mathop {\textstyle g}\limits _{A}(1)\right) }\int \limits _{\mathop {\textstyle cg}\limits _{A}(1)}^{\mathop {\textstyle cg}\limits _{A}(0)}xdx + \int \limits _{\mathop {\textstyle cf}\limits _{A}(1)}^{\mathop {\textstyle cg}\limits _{A}(1)}dx \end{aligned} \end{aligned}$$

(6.81)

As can be easily seen, using (6.76), (6.77), (6.78), (6.79), and (6.80) we get (6.75) $\mathop {\textstyle r}\limits _{cA}=c\mathop {\textstyle r}\limits _{A}$.

namely,

$$\begin{aligned} \begin{aligned} \mathop {\textstyle r}\limits _{cA}=\frac{1}{c\left( \mathop {\textstyle f}\limits _{A}(1)-\mathop {\textstyle f}\limits _{A}(0)\right) }\cdot c^2\int \limits _{\mathop {\textstyle f}\limits _{A}(0)}^{\mathop {\textstyle f}\limits _{A}(1)}xdx - \frac{\mathop {\textstyle f}\limits _{A}(0)}{\mathop {\textstyle f}\limits _{A}(1)-\mathop {\textstyle f}\limits _{A}(0)}\cdot c\int \limits _{\mathop {\textstyle f}\limits _{A}(0)}^{\mathop {\textstyle f}\limits _{A}(1)}dx + \frac{\mathop {\textstyle g}\limits _{A}(0)}{\mathop {\textstyle g}\limits _{A}(0) -\mathop {\textstyle g}\limits _{A}(1)}\cdot c\int \limits _{\mathop {\textstyle g}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(0)}dx \\ - \frac{1}{c\left( \mathop {\textstyle g}\limits _{A}(0)-\mathop {\textstyle g}\limits _{A}(1)\right) }\cdot c^2\int \limits _{\mathop {\textstyle g}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(0)}xdx + c\int \limits _{\mathop {\textstyle f}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(1)}dx \\ =c\cdot \left[ \frac{1}{\mathop {\textstyle f}\limits _{A}(1)-\mathop {\textstyle f}\limits _{A}(0)}\int \limits _{\mathop {\textstyle f}\limits _{A}(0)}^{\mathop {\textstyle f}\limits _{A}(1)}xdx-\frac{\mathop {\textstyle f}\limits _{A}(0)}{\mathop {\textstyle f}\limits _{A}(1)-\mathop {\textstyle f}\limits _{A}(0)}\int \limits _{\mathop {\textstyle f}\limits _{A}(0)}^{\mathop {\textstyle f}\limits _{A}(1)}dx+\frac{\mathop {\textstyle g}\limits _{A}(0)}{\mathop {\textstyle g}\limits _{A}(0)-\mathop {\textstyle g}\limits _{A}(1)}\int \limits _{\mathop {\textstyle g}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(0)}dx-\frac{1}{\mathop {\textstyle g}\limits _{A}(0)-\mathop {\textstyle g}\limits _{A}(1)}\int \limits _{\mathop {\textstyle g}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(0)}dx+\int \limits _{\mathop {\textstyle f}\limits _{A}(1)}^{\mathop {\textstyle g}\limits _{A}(1)}dx \right] \\ =c\cdot \mathop {\textstyle r}\limits _{A} \end{aligned} \end{aligned}$$

(6.82)

Therefore

$$\begin{aligned} \begin{aligned} MF(cA)={\mathop {\textstyle f}\limits _{cA}(0)}+\mathop {\textstyle r}\limits _{cA}=c\cdot \mathop {\textstyle f}\limits _{A}(0)+c\cdot \mathop {\textstyle r}\limits _{A}=c\left( \mathop {\textstyle f}\limits _{A}(0)+\mathop {\textstyle r}\limits _{A}\right) =c\cdot MF(A)\,\\ \textstyle {if\, order\, A\, is\, positive} \end{aligned} \end{aligned}$$

(6.83)

and

$$\begin{aligned} \begin{aligned} MF(cA)={\mathop {\textstyle g}\limits _{cA}(0)}+\mathop {\textstyle r}\limits _{cA}=c\cdot \mathop {\textstyle g}\limits _{A}(0)+c\cdot \mathop {\textstyle r}\limits _{A}=c\left( \mathop {\textstyle g}\limits _{A}(0)+\mathop {\textstyle r}\limits _{A}\right) =c\cdot MF(A)\,\\ \text {if}\, \text {order}\, \text {A}\, \text {is}\,\text {negative} \end{aligned} \end{aligned}$$

(6.84)

which ends the proof of the restricted homogeneity property.

10 Catalogue of the Shapes of Numbers in OFN Notation

Figures 6.8 to 6.31 constitute the catalogue of basic shapes of numbers in OFN notation including the results of defuzzifications using several methods (Figs. 6.8, 6.9, 6.10, 6.11, 6.12, 6.13, 6.14, 6.15, 6.16, 6.17, 6.18, 6.19, 6.20, 6.21, 6.22, 6.23, 6.24, 6.25, 6.26, 6.27, 6.28, 6.29, 6.30 and 6.31).

Fig. 6.8

Number A[1, 2, 3, 4] positive oriented

Fig. 6.9

Number A’[4, 3, 2, 1] negative oriented

Fig. 6.10

Number B$[-4,-4,-2,-2]$ positive oriented

Fig. 6.11

Number B’$[-2,-2,-4,-4]$ negative oriented

Fig. 6.12

Number C[1, 2, 2, 3] positive oriented

Fig. 6.13

Number C’[3, 2, 2, 1] negative oriented

Fig. 6.14

Number D[5, 5, 5, 6] positive oriented

Fig. 6.15

Number D’[6, 5, 5, 5] negative oriented

Fig. 6.16

Number E[3, 3, 5, 6] positive oriented

Fig. 6.17

Number E’[6, 5, 3, 3] negative oriented

Fig. 6.18

Number F$[-4,-1,-3,0]$ positive oriented

Fig. 6.19

Number F’$[0,-3,-1,-4]$ negative oriented

Fig. 6.20

Number G[1, 2, 4, 3] positive oriented

Fig. 6.21

Number G’[3, 4, 1, 2] negative oriented

Fig. 6.22

Number H[5, 4, 7, 5] positive oriented

Fig. 6.23

Number H’[5, 7, 4, 5] negative oriented

Fig. 6.24

Number J[2, 3, 7, 4] positive oriented

Fig. 6.25

Number J’[4, 7, 3, 2] negative oriented

Fig. 6.26

Number K[5, 5, 5, 5] singleton

Fig. 6.27

Number L[2, 4, 4, 2] mirror

Fig. 6.28

Number M[1, 4, 4, 3] positive oriented

Fig. 6.29

Number M’[3, 4, 4, 1] negative oriented

Fig. 6.30

Number N[1, 0, 3, 3] positive oriented

Fig. 6.31

Number N’[3, 3, 0, 1] negative oriented

11 Conclusion

The chapter presents two new original defuzzification methods: The golden ratio and mandala factor. Each is characterized by unique properties worthy to be noted. Real number values obtained through the operation of each operator are unique and different from those obtained using known methods. The golden ratio and mandala factor operation can therefore be applied in well-known and widely used arithmetics of fuzzy numbers such as L-R Dubois and Prade notation [20]. As shown in the calculations presented in the previous section the obtained results distinguish new operators from the classic, commonly known solutions. New operators are also characterized by the feature, which is absent in most of the classic operators. This feature is the sensitivity to order (order sensitive). This feature manifests so that different defuzzification values are obtained from one shape of a fuzzy number, depending on the fuzzy number order type (positive or negative) (Ordered Fuzzy Numbers). This is shown in the previous section. Basic shapes of the Ordered Fuzzy Numbers are visualized as shown in this chapter. To sum up, it can be concluded that both defuzzification methods, that is, the Golden ratio and mandala factor, meet all the criteria of defuzzification operators, and are adapted to applications in all fuzzy number arithmetics, including Kosiński’s OFN arithmetic. After defuzzification of OFN numbers, one can trivially use relationship operators in order to make comparisons. This is an easy and intuitive method. It should be added that the defuzzification methods should be selected as a result of empirical research on a given category of data.