Review of Type-1 and Type-2 Fuzzy Numbers

Abstract

We review type-1 and type-2 fuzzy numbers in this chapter, and propose one way of perceiving the concept of fuzzy numbers by comparing with that of round numbers. There are some definitions of fuzzy numbers, but we particularly adopt the definition often used in fuzzy analysis. Thereby, we emphasize that fuzzy number theory can be reduced to an argument for interval analysis. Moreover, we explain type-2 fuzzy sets and list two specific type-2 fuzzy numbers, one is a (triangular) perfect quasi-type-2 fuzzy number and the other is a triangular shaped type-2 fuzzy number. Finally, we mention the importance and utility of using type-2 fuzzy numbers.

1. Introduction

A fuzzy number is a special fuzzy set, and the fact that its membership function is unimodal (although “rest stops” along the way up and down the mountain are allowed) and normal is particularly characteristic. The concept of fuzzy sets was introduced by L.A. Zadeh [1] in 1965. There are some definitions of it. We see that in Section 2.3. In each application of fuzzy numbers, you can decide which definition is most useful and choose it on a case-by-case basis. For example, [2] is a well-known reference that defines fuzzy numbers with an eye toward fuzzy analysis (See also [3, 4, 5, 6]).

1.1 Where should the concept of fuzzy numbers be used?

The concept of fuzziness or fuzzy numbers appears when we try to distinguish between two or more things or measure something by sight or feeling. If their specific values were required, there would be no need to force fuzziness into the discussion.

For example, coefficients appearing in differential equations can be treated as fuzzy numbers. Let us consider the radiocarbon dating method. Equation describes how several samples of radioactive material decay

$$\frac{\mathrm{d}N}{\mathrm{d}t}=-\mathit{\lambda N},\lambda >0,$$E1

was presented by E. Rutherford. Here $N=N\left(t\right)$ is the number of atoms in a radioactive material at time $t$. Eq. (1) implies that the larger $\lambda$, the faster the sample decays. $\lambda$ varies with the substance, of course, and is determined experimentally by the observer on a case-by-case basis (see, e.g., [7] for more information). This $\lambda$ is set with some mathematical basis, but in some cases it may be determined empirically, or the accuracy of the observation equipment and the skill of the observer may be highly dependent on it. Therefore, it seems more realistic, appropriate and effective to put fuzziness in $\lambda$ and treat it as a fuzzy number. Differential equations involving fuzzy numbers are generally called fuzzy differential equations. To find the fuzzy solution of a fuzzy differential equation, we can obtain the level-cut sets of the solution by considering its level-cut sets and solving the interval equation, so we can collect them over all levels. However, in order to do so, of course, the fuzzy differential equation must have a (fuzzy) solution. See, e.g., Refs. [3, 6] for the existence and uniqueness of solutions of fuzzy differential equations. Refs. [8, 9, 10, 11] are also helpful in knowing how to solve specific fuzzy differential equations of the type as in Eq. (1). In particular, Ref. [8] covers elementary contents of fuzzy numbers and fuzzy differential equations.

1.2 The aim of this chapter

We treat not only fuzzy sets / numbers, but also “fuzzy-membership-grade fuzzy sets / numbers” in this chapter. These are usually called type-2 fuzzy sets / numbers. In order to distinguish fuzzy sets / numbers from these, they are called type-1 fuzzy sets / numbers with emphasis. These are defined and explained in Section 3. We can generally consider type-$n$ fuzzy sets / numbers, but since type-2 fuzzy sets / numbers are used in practical applications, this chapter also deals exclusively with type-1 and type-2 fuzzy sets/numbers.

More precisely, a type-2 fuzzy set can be said to be a fuzzy set whose membership grades are (type-1) fuzzy numbers. For example, it is a fuzzy set such that the membership grade at which a room feels hot is “about 0.8.” We can thus think that type-2 fuzzy theory is the application of type-1 fuzzy number theory.

Roughly speaking, the membership function of a type-2 fuzzy set is generally in the shape of a mountain standing above the base of a mountain type. This is because the base is the membership function with “width.” The “width” represents the fuzziness of a membership grade (see Figure 1). The red curve is the membership function with zero fuzziness.

Figure 1.

The form of the “bottom” of a type-2 fuzzy set.

Type-2 fuzzy theory is applied mainly to represent linguistic variables in reasoning. In the first place, Zadeh [1] introduced the concept of type-2 fuzzy sets for this purpose in 1975. By introducing it, we can treat truth values such as “approximately true,” “neither true nor false,” “truth unknown,” etc. This has greatly advanced the study of reasoning (see, e.g., [12, 13, 14, 15]). In recent years, type-2 fuzzy number theory has been developing and is being studied in principle as well as in application. Applications to differential equation theory have also been made (see, e.g., [10, 11, 13, 16]. Type-1 fuzzy differential equation theory can be seen in Refs. [3, 6], etc. All of the above studies benefit from the concept of type-1 fuzzy numbers. Indeed, type-2 fuzzy theory is ultimately attributed to type-1 fuzzy theory because any type-2 fuzzy set is represented as a “coupling” of two type-1 fuzzy sets (See Section 3 for details), and hence, concrete computations are also done by level-cutting of type-1 fuzzy sets (see, e.g., [17] for operations for type-2 fuzzy sets). Moreover, the utility of the type-2 fuzzy concept will be explained in Section 3.5.

Therefore, it can be said to be important to interpret the concept of fuzzy numbers appropriately and consider fuzzy numbers whose level-cut sets are easily computed.

From the above, we see the following in this chapter:

• In Section 2, we review how fuzzy numbers are perceived, comparing them to round numbers. Furthermore, we give a “strict” and “suitable” definition of a (type-1) fuzzy number, which reduces its theory to interval number theory and interval analysis [18].

• In Section 3, we meet the concept and some definitions of type-2 fuzzy sets, and know, via some figures, that type-2 fuzzy sets / numbers are defined by type-1 fuzzy numbers. Type-2 fuzzy sets /numbers are defined by type-1 fuzzy numbers. Furthermore, we give an example of a type-2 fuzzy number whose level-cut sets are easily computed in Section 3.4 and find out why the type-2 fuzzy concept is necessary or what it can do.

2. Fuzzy numbers, their concept, and level-cut sets

We begin by seeing how to perceive fuzziness on numbers, that is, the concept of fuzzy numbers in this section.

2.1 Difference between round numbers and fuzzy numbers

A fuzzy number, e.g., $\tilde{3}$, is often interpreted and called as “about $3$.” However, with this representation, it becomes indistinguishable from round numbers, and there is a risk of confusion. We thus verify the difference between round numbers and fuzzy numbers.

Let us consider the following string of positive and finite length (See Figure 2). And, let us say that we want to know (even roughly) this length. Then, there are two cases:

1. one is measuring, and

2. the other is eyeballing.

Figure 2.

How long is this string?

Suppose we now have a ruler, tape measure, or other object that can measure length. Then, we can measure the length of the string, and suppose the length is 30.2 cm. This is regarded as about 30 cm, and hence this is case a above.

On the other hand, suppose we do not have a ruler, tape measure, or other object that can measure length. Then, we cannot measure the length of the string, so we need to have an approximate idea of the length, for example, by eyeballing it. Let us say the length feels like “30 cm.” Assuming that this visual measurement is perfectly correct, the value $1$ ($100\%$ confidence) is assigned to “30 cm.” The confidence level gradually decreases as “30 cm” is shifted up or down. We thus represent the barometer of confidence by the graph of a function. This function is called a membership function. This is case b above. Like this, the value from $0$ to $1$ assigned to each value of length is called the membership grade, often denoted by $\alpha$. As just explained, the attitude of this chapter is to believe that membership grade represents a degree of confidence.

The concept of round numbers appears in the former case, whereas that of fuzzy numbers appears in the latter case. Indeed, if the length can be measured, there is no need to bring in fuzziness, which complicates the discussion. In other words, the difference between round numbers and fuzzy numbers is whether or not an “exact value” exists. The concept of fuzzy numbers has the advantage that it comes into effect when an “exact value” is not available and can be discussed as if it was values out there. The concrete difference is as follows.

• Round number 3 is like $2\sqrt{2}$, $2.9$, $\pi$, etc. Or, for example, the round number of 10,023 people is $\mathrm{10,000}$ people. We often call this “about 10,000 people.”

• Fuzzy number 3 is the fuzzy set $F$ whose membership function ${\mu }_F:\mathbb{R}\to \left[0,1\right]$ such that

  1. ${\mu }_F\left(3\right)=1$,

  2. ${\mu }_F$ is monotone increasing (resp. deceasing) on $x\in \left(-\infty ,3\right)$ (resp. $x\in \left[3,+\infty \right)$),

  3. for example, ${\mu }_F\left(x\right)=0$ for all $x\le 2$ and $x\ge 4$.

It seems natural to us that ${\mu }_F$ is continuous, but ${\mu }_F$ can be continuous or discontinuous. Moreover, since any fuzzy set is given based on our subjectivity, there are any numbers of membership functions for it. We give an example of a ${\mu }_F$ in Figure 3. It seems more natural that ${\mu }_F$ is smooth, but ${\mu }_F$ can be a broken line as shown in Figure 3. Such a fuzzy number is often called a triangular fuzzy number.

Figure 3.

An example of a membership function of fuzzy number $3$.

The above view of fuzzy numbers is somewhat imprecise and unsuitable for a detailed mathematical discussion. So, a strict definition of fuzzy numbers is given later.

2.2 Notation of fuzzy sets

In what follows, $X$ denotes the universal crisp set, and we write $A\left(x\right)={\mu }_A\left(x\right)$ for the membership grade of a fuzzy set $A$ at $x\in X$. Related to this, following the conventions in fuzzy analysis, we write $A:X\to \left[0,1\right]$ for a fuzzy set on $X$.

It is well known that there exist some representations of a fuzzy set $A$ on $X$:

$$A=\left\{\left(x,{\mu }_A\left(x\right)\right):x\in X\right\},{\int }_{x\in X}{\mu }_A\left(x\right)/x,\mathrm{etc}.$$E2

Remark that “$\int$” in the representation on the right side of Eq. (2) means a continuous union for sets, not an integral. Moreover, “$/$” means a marker, not a division, and “$\int$” is rewritten as “$\sum$” if $A$ is discrete.

2.3 Definitions of fuzzy numbers

There are several definitions for fuzzy numbers. The definition most in line with the senses is as follows.

Definition 2.1. Let $X=\mathbb{R}$ and $A:\mathbb{R}\to \left[0,1\right]$ be a fuzzy set. $A$ is a fuzzy number on $\mathbb{R}$ if and only if $A$ satisfies that

1. $A$ is normal, that is, there exists $x_0\in \mathbb{R}$ such that $A\left(x_0\right)=1$;

2. $A$ is convex, that is, $A\left(\mathit{tx}+\left(1-t\right)y\right)\ge \mathit{tA}\left(x\right)+\left(1-t\right)A\left(y\right)$ for any $x,y\in \mathbb{R}$ and $t\in \left[0,1\right]$;

3. $A$ is continuous.

An $x_0$ such that condition i is called a core.

Note: Definition 2.1 says that when considering “about $a$,” the confidence level is $1$ (100 $\%$) at $a\in X$ and decreases as the variable $x\in X$ moves away from $a$ to both sides. If $X=\mathbb{Z}$, we should remove condition iii since the membership function is of course discontinuous. In this case, such a fuzzy number is called the discrete fuzzy number.

Recall that any fuzzy set $A$ can be established by its all level-cut sets:

$$A=\bigcup _{\alpha \in \left[0,1\right]}\alpha {\left[A\right]}_{\alpha },$$E3

where ${\left[A\right]}_{\alpha }$ is the $\alpha$-cut set of $A$ defined as

$${\left[A\right]}_{\alpha }=\left\{\begin{array}{ll}\left\{x\in X:A\left(x\right)\ge \alpha \right\}& (\alpha \in \left(0,1]\right),\\ supp\left(A\right)=\mathrm{cl}\left(\left\{x\in \mathbb{R}:A\left(x\right)>0\right\}\right)& \left(\alpha =0\right).\end{array}\right.$$

Here, “$supp$” means “support” and $\mathrm{cl}\left(S\right)$ denotes the closure of a crisp set $S$. $\alpha \left[A_{\alpha }\right]$ is defined as a fuzzy set via the algebraic product operation:

$$\left(\alpha {\left[A\right]}_{\alpha }\right)\left(x\right)=\alpha \cdot \left({\left[A\right]}_{\alpha }\right)\left(x\right),x\in \mathbb{R}.$$

Note: There is another way to establish a fuzzy number by its $\alpha$-cut sets, e.g., Ref. [4]:

$$A=\bigcup _{\alpha \in \left[0,1\right]}\left({\alpha }^{\ast }\cap {\left[A\right]}_{\alpha }\right),$$

where ${\alpha }^{\ast }$ stands for a fuzzy set whose membership function is the constant function, ${\alpha }^{\ast }\left(x\right)\equiv \alpha$.

From this, it is expected that the discussion on fuzzy numbers can be reduced to that on intervals (their level-cut sets). But for that we would need a more rigorous definition of fuzzy numbers. We thus adopt the following definition that is often used in fuzzy analysis, etc.

Definition 2.2. Let $u:\mathbb{R}\to \left[0,1\right]$ be a fuzzy set. $u$ is a fuzzy number on $\mathbb{R}$ if and only if $u$ satisfies

1. $u$ is normal, that is, $u$ has at least one core;

2. $u$ is fuzzy convex, that is, $u\left(\mathit{tx}+\left(1-t\right)y\right)\ge u\left(x\right)\wedge u\left(y\right)$ for any $x,y\in \mathbb{R}$ and $t\in \left[0,1\right]$, where $\wedge$ represents the minimum operation;

3. $u$ is semi-upper-continuous, that is, ${\left[u\right]}_{\alpha }=\left\{x\in \mathbb{R}:u\left(x\right)\ge \alpha \right\}$ is closed for all $\alpha \in \mathbb{R}$;

4. $supp\left(u\right)$ is bounded.

In particular, $u:\mathbb{Z}\to \left[0,1\right]$ is called a discrete fuzzy number (on $\mathbb{Z}$).

Definition 2.2 looses Definition 2.1 by replacing conditions ii and iii with conditions b and c, respectively. In fact,

• condition b of Definition 2.2 does not allow the membership function to be bimodal, but allows it to be non-convex (Figure 4);

• condition c of Definition 2.2 allows the membership function to be a jump function (Figure 5).

Figure 4.

Disconvexity is OK.

Figure 5.

Jump is OK.

On the other hand, however, Definition 2.2 adds a new condition, D, to Definition 2.1. In fact, condition D of Definition 2.2 states that the membership function lands on the $x$-axis on both sides of core $a$. Specifically, think of membership functions like the $C_0^{\infty }$-function

$$u\left(x\right)=\left\{\begin{array}{ll}exp\left(-\frac{1}{1-x^2}\right)& \left(|x|<1\right)\\ 0& \left(|x|\ge 1\right),\end{array}\right.$$

which is often used as an example of a test function in distribution theory; e.g., (Figure 6) [19].

Figure 6.

Definition 2.1 allows membership functions like the left side, but Definition 2.2 requires membership functions like the right side.

Like the above, there are points in Definition 2.2 where the conditions are loosened or strengthened. The reasons for doing so are discussed in the next subsection.

2.4 Correspondence between fuzzy numbers and level-cut sets

Definition 2.2 implies that $u$ is a fuzzy number if and only if ${\left[u\right]}_{\alpha }$ is a bounded closed interval for any $\alpha \in \left[0,1\right]$. In fact,

• condition a says that ${\left[u\right]}_{\alpha }$ is not empty for any $\alpha \in \left[0,1\right]$,

• condition b says that ${\left[u\right]}_{\alpha }$ is an interval on $\mathbb{R}$ for any $\alpha \in \left[0,1\right]$,

• condition c says that ${\left[u\right]}_{\alpha }$ is closed for any $\alpha \in \left[0,1\right]$,

• condition d says that ${\left[u\right]}_{\alpha }$ is bounded for any $\alpha \in \left[0,1\right]$.

Given this, we denote

$${\left[u\right]}_{\alpha }=\left[u_{-}\left(\alpha \right),u_{+}\left(\alpha \right)\right]$$

for any $\alpha \in \left[0,1\right]$. $u_{\pm }\left(\alpha \right)$ are, of course, crisp numbers (or, the crisp function with respect to $\alpha$, $\left[0,1\right]∍\alpha \mapsto u_{\pm }\left(\alpha \right)\in \mathbb{R}$). Hence, we expect that the discussion of fuzzy number theory can be reduced to that of interval analysis, that is, level-cut theory.

Theorem 2.3 (Representation Theorem for Fuzzy Numbers, e.g. [2, 3, 4, 6]) Let $u:\mathbb{R}\to \left[0,1\right]$ be a fuzzy number. Then, the following holds:

1. ${\left[u\right]}_{\alpha }$ is a non-empty bounded closed interval for any $\alpha \in \left[0,1\right]$.

2. If $0\le {\alpha }_1\le {\alpha }_2\le 1$, then ${\left[u\right]}_{{\alpha }_2}\subset {\left[u\right]}_{{\alpha }_1}$.

3. For the monotone increasing and positive sequence $\left\{{\alpha }_n\right\}$ that converges to $\alpha \in \left(0,1\right]$, one has

$$\bigcap _{n=1}^{\infty }{\left[u\right]}_{{\alpha }_n}={\left[u\right]}_{\alpha }.$$

Conversely, if there is a family of sets ${\left\{P_{\alpha }\right\}}_{\alpha \in \left[0,1\right]}$ satisfying properties 1, 2 and 3 above, then there exists a unique fuzzy number $u$. Moreover, it follows that

$${\left[u\right]}_{\alpha }=P_{\alpha }$$

for any $\alpha \in \left(0,1\right]$ and

$${\left[u\right]}_0\subset P_0.$$E4

Note: As can be seen, Eq. (4) does not guarantee equality. For example, we want to treat fuzzy numbers $u$ in the same way as crisp numbers if possible, so it is necessary to define the four arithmetic operations, etc., for fuzzy numbers. To do so, we only need to well define the operations of level-cut sets (i.e., interval numbers), which corresponds to $P_{\alpha }$ above. It must then be satisfied that $P_{\alpha }={\left[u\right]}_{\alpha }$ for any $\alpha \in \left[0,1\right]$. Representation Theorem and the proof of $P_0\subset {\left[u\right]}_0$ guarantee that the results of the defined operations are fuzzy numbers. Eq. (3) guarantees that ${\cup }_{\alpha \in \left[0,1\right]}\alpha P_{\alpha }$ is a fuzzy set, but does not guarantee that it is a fuzzy number. For this reason, what is needed is the representation theorem. This is detailed in, e.g., Ref. [4].

3. Type-2 fuzzy numbers: Two concrete examples

3.1 Motivation for type-2 fuzzy theory

The key to fuzzy theory is the concept of membership grades, and it is represented by our individual degrees of confidence. However, membership grades we set may be ambiguous. Put another way, determining membership grades means determining the degrees of confidence, but there is also the degree of confidence for the degree of confidence $\alpha$. For example, let us say that we are willing to accept the sentiment that we have determined $\alpha$ ($100\alpha \%$) with a confidence level of 0.8 (80 %). These are called fuzzy membership grades. In summary, type-2 fuzzy theory is a theory of fuzzy sets with fuzzy membership grades.

A type-1 fuzzy set $A$ is characterized by its membership function $A:X\to \left[0,1\right]$ for the universal (crisp) set $X$, whereas a type-2 fuzzy set $\tilde{A}$ is characterized by its membership function $\tilde{A}:X\to {\left[0,1\right]}^{\left[0,1\right]}$. Here, $U^V$ denotes the set of mappings $U\to V$ for crisp sets $U,V$.

There are other advantages of type-2 fuzzy theory. See Section 3.5 for that.

3.2 Type-2 fuzzy sets and those associated with them

We begin with the definition of a type-2 fuzzy set. Type-2 fuzzy theory has many concepts and their terms, but we prepare them required in this chapter. (There is a slight change from the traditional definition and notation.) For example, the membership grade of a type-2 fuzzy set is called the fuzzy membership grade of it. Unlike type-1 fuzzy sets, type-2 fuzzy sets can be three-dimensional figures and are generally difficult to depict.

Definition 3.1. If $\tilde{A}$ is characterized by the membership function

$${\mu }_{\tilde{A}}:I\times J_x∍\left(x,u\right)\mapsto {\mu }_{\tilde{A}}\left(x,u\right)\in \left[0,1\right],$$E5

$\tilde{A}$ is called a type-2 fuzzy set on $X$. Here, $I\subset X$ is the universe for the primary variable $x\in X$, and $J_x\subset \left[0,1\right]$ is the interval determined for each $x\in I$. Then, $I$ and $J_x$ are called the primary and secondary domains of $\tilde{A}$, respectively.

There are other representations of $\tilde{A}$:

$$\tilde{A}=\left\{\left(x,u;{\mu }_{\tilde{A}}\left(x,u\right)\right):x\in I,u\in J_x\right\},{\int }_{x\in I}{\int }_{u\in J_x}{\mu }_{\tilde{A}}\left(x,u\right)/\left(x,u\right),\mathrm{etc}.$$E6

Remark that “$\int$” in the representation on the right side of Eq. (6) means a continuous union for sets, not an integral. Moreover, “$/$” means a marker, not a division, and “$\int$” is rewritten as “$\sum$” if $\tilde{A}$ is discrete.

Definition 3.2. Let $\tilde{A}$ be a type-2 fuzzy set on $X$. The type-1 fuzzy set for $\tilde{A}$ appears if $x\in I$ is fixed arbitrarily. It is called the vertical slice of $\tilde{A}$, and its membership function

$${\nu }_{\tilde{A}}^x:J_x\to \left[0,1\right].$$

is called the secondary membership function of $\tilde{A}$ at $x$.

$\tilde{A}$ can be said to be characterized by ${\nu }_{\tilde{A}}^x$ as follows:

$$\tilde{A}={\int }_{x\in X}\left({\int }_{u\in J_x}{\nu }_{\tilde{A}}^x\left(u\right)/u\right)/x,$$

where ${\nu }_{\tilde{A}}^x\left(u\right)$ is the value of the secondary membership function at $u$, that is, the secondary membership grade of $\tilde{A}$.

Note: The concepts of vertical slices and secondary membership functions are often treated in the same sense. Because of this, “secondary membership functions” are sometimes also called “vertical slices.”

There are two kinds of cutting with respect to $\beta$ and what is important is what to cut with respect to $\beta$. First, the following is the cutting of vertical slices.

Definition 3.3. Let $\tilde{A}$ be a type-2 fuzzy set on $X$. The crisp set

$$S_{\tilde{A}}\left(x|\beta \right)=\left\{\begin{array}{ll}\left\{u\in J_x:{\nu }_{\tilde{A}}^x\left(u\right)\ge \beta \right\}& (\beta \in \left(0,1]\right),\\ \mathrm{cl}\left(\left\{u\in J_x:{\nu }_{\tilde{A}}^x\left(u\right)>0\right\}\right)& \left(\beta =0\right),\end{array}\right.$$

is called the $\beta$-cut set of the vertical slice of $\tilde{A}$ (Figure 7).

Figure 7.

0-cut set of the vertical slice of “about 1” as the meaning of type-2 fuzzy numbers; Figure 1 [11].

Next, the following is the cutting of type-2 fuzzy sets.

Definition 3.4. Let $\tilde{A}$ be a type-2 fuzzy set on $X$. For $\beta \in \left[0,1\right]$,

$${\tilde{A}}_{\beta }=\bigcup _{x\in I}{S}_{\tilde{A}}\left(x|\beta \right)$$

is called the $\beta$-plane of $\tilde{A}$. In particular, ${\tilde{A}}_1$ and ${\tilde{A}}_0$ are called the principal (or, principle) set and footprint (set) of $\tilde{A}$, respectively.

Roughly speaking, a type-2 fuzzy set can be characterized by a membership function in the form of two mountains. We use the following notation for the $\beta$-plane of $\tilde{A}$ because we want to make it geometrically easy to see what a type-2 fuzzy set looks like. That is, it brings in the notion of “left- and right-sided type-1 fuzzy sets.”

Definition 3.5. If the $\beta$-plane of $\tilde{A}$ is the interval-valued fuzzy set, there exist type-1 fuzzy sets $\underset{¯}{A_{\beta }}$ and $\overline{A_{\beta }}$. Then, we denote

$${\tilde{A}}_{\beta }=⟨\underset{¯}{A_{\beta }},\overline{A_{\beta }}⟩.$$

Here, $\underset{¯}{A_{\beta }}$ and $\overline{A_{\beta }}$ are called the lower membership function (briefly, LMF) and upper membership function (briefly, UMF) on $\tilde{A}$, respectively.

Definition 3.6. Let $\tilde{A}$ be a type-2 fuzzy set on $X$. For each $\beta \in \left[0,1\right]$, the coupling of $\alpha$-cut sets of $\underset{¯}{A_{\beta }}$ and $\overline{A_{\beta }}$ is written as

$${\left[\tilde{A}\right]}_{\beta }^{\alpha }=⟨{\left[\underset{¯}{A_{\beta }}\right]}_{\alpha },{\left[\overline{A_{\beta }}\right]}_{\alpha }⟩,\alpha \in \left[0,1\right],$$E7

and is called the $\left(\alpha ,\beta \right)$-cut set of $\tilde{A}$.

As with type-1, we can discuss type-2 fuzzy sets as their $\beta$-planes or $\left(\alpha ,\beta \right)$-cut sets. Indeed, Hamrawi found the formula that is a type-2 version of Eq. (3) as follows; we leave the details to Ref. [20] for more information on the contents of this neighborhood.

Proposition 3.7. Any type-2 fuzzy set $\tilde{A}$ on $X$ satisfies

$$\tilde{A}=\bigcup _{\beta \in \left[0,1\right]}\beta \bigcup _{\alpha \in \left[0,1\right]}\alpha {\left[\tilde{A}\right]}_{\beta }^{\alpha },$$

where $\alpha {\left[\tilde{A}\right]}_{\beta }^{\alpha }:X\to \left\{0,\alpha \right\}$ is a type-1 fuzzy set.

3.3 Perfect quasi-type-2 fuzzy numbers

We hereafter set $X=\mathbb{R}$.

Hamrawi introduced the following type-2 fuzzy number, which we can call a “triangular type-2 fuzzy number.”

Definition 3.8. ([20], Section 3.4). Let $\mathcal{A}$ be a type-2 fuzzy set on $\mathbb{R}$. $\mathcal{A}$ is a perfect type-2 fuzzy number if and only if

1. UMF and LMF of $\mathrm{FP}\left(\mathcal{A}\right)$ are equal as type-1 fuzzy numbers, and

2. UMF and LMF of $\mathrm{P}\left(\mathcal{A}\right)$ are equal as type-1 fuzzy numbers.

   Moreover, if a perfect type-2 fuzzy number $\mathcal{A}$ satisfies that

3. $\mathcal{A}$ can be completely determined by using its $\mathrm{FP}\left(\mathcal{A}\right)$ and $\mathrm{P}\left(\mathcal{A}\right)$,

such a $\mathcal{A}$ is called the perfect quasi-type-2 fuzzy number (briefly, PQT2FN) on $\mathbb{R}$.

Definition 3.9. A PQT2FN $\mathcal{A}$ is triangular if and only if ${\left[\mathcal{A}\right]}_{\beta }^{\alpha }$ has the $\alpha$-cut set of LMF on $\mathcal{A}$:

$$\begin{array}{lll}{\left[{\underset{¯}{A}}_{\beta }\right]}_{\alpha }& =& \left[L_{{\underset{¯}{A}}_{\beta }}^{\alpha },R_{{\underset{¯}{A}}_{\beta }}^{\alpha }\right];\\ L_{{\underset{¯}{A}}_{\beta }}^{\alpha }& =& X_{A_1}^{\alpha }-\left(1-\beta \right)\left(X_{A_1}^{\alpha }-L_{{\underset{¯}{A}}_0}^{\alpha }\right),\\ R_{{\underset{¯}{A}}_{\beta }}^{\alpha }& =& Y_{A_1}^{\alpha }+\left(1-\beta \right)\left(R_{{\underset{¯}{A}}_0}^{\alpha }-Y_{A_1}^{\alpha }\right),\\ L_{{\underset{¯}{A}}_0}^{\alpha }& =& C_{\mathcal{A}}-\left(1-\alpha \right)\left(C_{\mathcal{A}}-L_{{\underset{¯}{A}}_0}\right),\\ R_{{\underset{¯}{A}}_0}^{\alpha }& =& C_{\mathcal{A}}+\left(1-\alpha \right)\left(R_{{\underset{¯}{A}}_0}-C_{\mathcal{A}}\right)\end{array}$$

and the $\alpha$-cut set of UMF on $\mathcal{A}$:

$$\begin{array}{lll}{\left[{\overline{A}}_{\beta }\right]}_{\alpha }& =& \left[L_{{\overline{A}}_{\beta }}^{\alpha },R_{{\overline{A}}_{\beta }}^{\alpha }\right];\\ L_{{\overline{A}}_{\beta }}^{\alpha }& =& X_{A_1}^{\alpha }-\left(1-\beta \right)\left(X_{A_1}^{\alpha }-L_{{\overline{A}}_0}^{\alpha }\right),\\ R_{{\overline{A}}_{\beta }}^{\alpha }& =& Y_{A_1}^{\alpha }+\left(1-\beta \right)\left(R_{{\overline{A}}_0}^{\alpha }-Y_{A_1}^{\alpha }\right),\\ L_{{\overline{A}}_0}^{\alpha }& =& C_{\mathcal{A}}-\left(1-\alpha \right)\left(C_{\mathcal{A}}-L_{{\overline{A}}_0}\right),\\ R_{{\overline{A}}_0}^{\alpha }& =& C_{\mathcal{A}}+\left(1-\alpha \right)\left(R_{{\overline{A}}_0}-C_{\mathcal{A}}\right),\end{array}$$

where

$$\begin{array}{lll}{X}_{A_1}^{\alpha }& =& C_{\mathcal{A}}-\left(1-\alpha \right)\left(C_{\mathcal{A}}-X_{A_1}\right),\\ Y_{A_1}^{\alpha }& =& C_{\mathcal{A}}+\left(1-\alpha \right)\left(Y_{A_1}-C_{\mathcal{A}}\right).\end{array}$$

They are called the left principle number and right principle number of $\mathcal{A}$, respectively. $C_{\mathcal{A}}$ denotes the core of $\mathcal{A}$, that is, the crisp number ${\left[\mathcal{A}\right]}_1^1$. A triangular perfect quasi-type-2 fuzzy number is abbreviated as TPQT2FN.

Figure 8 shows

Figure 8.

A view of a TPQT2FN $\mathcal{A}$ from directly above.

$$L_{{\overline{A}}_0}^{\alpha }\le X_{A_1}^{\alpha }\le L_{{\underset{¯}{A}}_0}^{\alpha }\le C_{\mathcal{A}}\le R_{{\underset{¯}{A}}_0}^{\alpha }\le Y_{A_1}^{\alpha }\le R_{{\overline{A}}_0}^{\alpha }.$$

In particular, the supports of $\mathcal{A}$ are represented by the $\alpha$-cut sets of LMF and UMF of $\mathrm{FP}\left(\mathcal{A}\right)$:

$${\left[{\underset{¯}{A}}_0\right]}_{\alpha }=\left[L_{{\underset{¯}{A}}_0}^{\alpha },R_{{\underset{¯}{A}}_0}^{\alpha }\right],{\left[{\overline{A}}_0\right]}_{\alpha }=\left[L_{{\overline{A}}_0}^{\alpha },R_{{\overline{A}}_0}^{\alpha }\right].$$

Also, the $\alpha$-cut set of $\mathrm{P}\left(\mathcal{A}\right)$ is given as

$${\left[A_1\right]}_{\alpha }=\left[X_{A_1}^{\alpha },Y_{A_1}^{\alpha }\right].$$

Now, recall that the triangular type-1 fuzzy number $u$ is determined by three information, that is, its left end $l$, core $c$ and right end $r$:

$$u=⟨⟨;l;c;r;⟩⟩.$$

In contrast, TPQT2FN $\mathcal{A}$ is determined by seven information, that is, its upper left end $L_{{\overline{A}}_0}$, left principle number $X_{A_1}$, lower left end $L_{{\underset{¯}{A}}_0}$, core $C_{\mathcal{A}}$, lower right end $R_{{\underset{¯}{A}}_0}$, right principle number $Y_{A_1}$ and upper right end $R_{{\overline{A}}_0}$. We then write

$$\mathcal{A}=⟨⟨L_{{\overline{A}}_0},X_{A_1},L_{{\underset{¯}{A}}_0};C_{\mathcal{A}};R_{{\underset{¯}{A}}_0},Y_{A_1},R_{{\overline{A}}_0}⟩⟩.$$

In general, any type-2 fuzzy set/number satisfies that both the principal set and the vertical slice are type-1 fuzzy numbers as shown in Figure 9.

Figure 9.

Membership function of a perfect quasi-type-2 fuzzy number.

3.4 Triangular shaped type-2 fuzzy numbers

The $\left(\alpha ,\beta \right)$-cuts of a perfect quasi-type-2 fuzzy number can be easily obtained, but at cost of its condition being too strict (too ideal). We want to consider a more natural type-2 fuzzy number while still being able to easily compute the $\alpha$-cuts. H. Uesu proposed the following type-2 fuzzy number, and he and the author, et al. [11] introduced in 2022.

Definition 3.10. Let $\tilde{A}$ be a type-2 fuzzy set whose core is $a\in \mathbb{R}$ on $\mathbb{R}$. $\tilde{A}$ is a triangular shaped type-2 fuzzy number (briefly, TST2FN) on $\mathbb{R}$ if and only if its principal set and secondary membership function at $x$ are given by

$$\begin{array}{ll}{\tilde{A}}_1\left(x\right)& =max\left\{1-|x-a|,0\right\},\\ {\nu }_{\tilde{A}}^x\left(t\right)=\left\{\begin{array}{ll}1-\frac{\mid t-{\tilde{A}}_1\left(x\right)\mid }{min\left\{{\tilde{A}}_1\left(x\right),1-{\tilde{A}}_1\left(x\right)\right\}}& \left({\tilde{A}}_1\left(x\right)\in \left(0,1\right)\right),\\ 1& \left(t={\tilde{A}}_1\left(x\right)\in \left\{0,1\right\}\right),\\ 0& \left(t\ne {\tilde{A}}_1\left(x\right)\in \left\{0,1\right\}\right),\end{array}\right.\end{array}$$

respectively (Figure 10).

Figure 10.

Membership function of a triangular shaped type-2 fuzzy number with core $2$; Figure 4 [11].

Note: For any TST2FN $\tilde{A}$, both its left and right footprints are congruent parallelograms of width-length 0.5 (See Figure 11). Moreover, the diagonals of the two parallelograms constitute the principal set ${\tilde{A}}_1$ of $\tilde{A}$ (see the right sides of Figures 11 and 12).

Figure 11.

Footprint set of “about $2$” as the meaning of type-2 fuzzy numbers; Figure 5 [11].

Figure 12.

Principal set of “about $2$” as the meaning of type-2 fuzzy numbers; Figure 6 [11].

Even a fuzzy number by its natural definition is not suitable for application if the computation of its level-cut sets is complex. However, a TST2FN is defined naturally and its level-cut sets can be easily computed.

Theorem 3.11 ([11], Theorem 2.19) Let $\tilde{A}$ be a TST2FN with core $a\in \mathbb{R}$. For $\alpha ,\beta \in \left[0,1\right]$, the following holds:

$${\left[\tilde{A}\right]}_{\beta }^{\alpha }=\left\{\begin{array}{ll}⟨\left[a-\frac{2-\alpha -\beta }{2-\beta },a-\frac{\beta -\alpha }{\beta }\right],\left[a+\frac{\beta -\alpha }{\beta },a+\frac{2-\alpha -\beta }{2-\beta }\right]⟩,& 0<\alpha \le \frac{\beta }{2};\\ ⟨\left[a-\frac{2-\alpha -\beta }{2-\beta },a-\frac{1-\alpha }{2-\beta }\right],\left[a+\frac{1-\alpha }{2-\beta },a+\frac{2-\alpha -\beta }{2-\beta }\right]⟩,& \frac{\beta }{2}<\alpha \le 1-\frac{\beta }{2};\\ ⟨\left[a-\frac{1-\alpha }{\beta },a-\frac{1-\alpha }{2-\beta }\right],\left[a+\frac{1-\alpha }{2-\beta },a+\frac{1-\alpha }{\beta }\right]⟩,& 1-\frac{\beta }{2}<\alpha \le 1.\end{array}\right.$$E8

As we can see from the two type-2 fuzzy numbers above, it may be said that a type-2 fuzzy number is determined by

• what form the FP should take and

• how to connect that FP to the PS.

In case of PQT2FNs, the FP is in triangular form, and the FP and the PS are linearly connected. In case of TST2FNs, the FP is in the shape of a parallelogram, and the FP and the PS are curvilinearly connected.

3.5 Utility of the concept of type-2 fuzzy numbers

Type-2 fuzzy theory has an advantage. For example, as discussed in Ref. [9], there are cases where the observer is a veteran or a newcomer to some experiment. In such a case, the coefficients appearing in fuzzy differential equations, etc., may change depending on the former and the latter (in case of Eq. (1), we are talking about the value of $\lambda$). Actually, it can be considered that PS and FP of a type-2 fuzzy set correspond, so to speak, the veteran who make no mistakes at all and the newcomer with no experience at all, respectively. Hence, by discussing type-2 fuzzy theory, we can have the discussion of fuzziness concluded the case of an experiment by any observer (Veteran or not!).

Some readers may think that instead of going to the trouble of discussing type-2 fuzzy numbers, they can simply consider two type-1 fuzzy numbers and compare them. However, doing so would result in obtaining two fuzzy numbers under separate environments (conditions), and it would generally not make sense to compare them, for example. In other words, depending on the nature of the research, one may wish to compare multiple subjects under the same conditions as appropriately as possible. The type-1 fuzzy theory of comparing by each membership function does not, however, seem to be appropriate in general. In fact, we often establish membership functions under unique conditions of the subjects, and hence, the subjects are compared under different conditions. The way of this research will not give appropriate comparison results.

Then, type-$n$ fuzzy theory is useful in overcoming this problem. With type-$n$ fuzzy numbers, membership functions or level-cut sets for all objects under the same conditions can be obtained simultaneously (see Eqs. (7) and (8) as the case $n=2$).

When comparing two objects, the number of times to obtain level-cut sets is the same whether considering two type-1 fuzzy numbers or one type-2 fuzzy number, but basically type-2 fuzzy numbers are more likely to be computationally expensive or unobtainable with respect to the level-cut sets. However, it is not very effective to consider a type-2 fuzzy number that is too convenient for us only because it is easier to calculate. Therefore, when dealing with type-2 fuzzy numbers, we prefer to consider something that is easy to calculate while still being in accordance with our senses. One example of this is TST2FN, Definition 3.10. In addition, one of the applications of this can be seen in Ref. [21].

4. Conclusions

We regarded membership grades as the “degrees of confidence” in this chapter. In particular, Ref. [22] is well known for this same idea of literature.

Although the application aspect is important in fuzzy theory, this chapter focused on how to recognize fuzzy numbers in the first place rather than how to apply them. This is because if the starting point of the discussion is iffy, so will the outcome. Fuzzy theory is not a theory to derive fuzzy results, but the ability to approach fuzziness mathematically (with some rigor) is the real appeal of fuzzy theory. That is, the beginning is crucial, and this mindfulness encourages the proper application of fuzzy numbers.

This chapter reviewed

• how to perceive and find “fuzziness,”

• that to discuss fuzzy numbers, it is sufficient to discuss their level-cut sets,

• that we should define or introduce fuzzy numbers such that their level-cut sets are easily obtained in order to do that.

These things are common to type-$n$ fuzzy numbers for any $n\in \mathrm{\mathbb{N}}$.

We have redefined the concept of fuzzy numbers by comparing them to round numbers. A round number is a number whose exact value is known and whose value is replaced by a tractable number. On the other hand, a fuzzy number is a concept that attempts to estimate its value when the exact value is (forever) unknown and determine its membership function. Even using the same word “about,” they are different in concept itself, let alone approach. In summary, fuzzy numbers are a valid concept for quantities that definitely exist but whose values are difficult to obtain.

Furthermore, the discussion of fuzzy numbers can be reduced to that of interval analysis. Instead of dealing directly with fuzzy numbers, we can discuss them by dealing with the level-cut sets, which are (nonempty bounded closed) intervals. Hence, we want to treat fuzzy numbers whose level-cut sets are easily obtained. Compared to type-1 fuzzy numbers, type-2 fuzzy numbers are generally more complex and difficult to find for their level-cut sets, and in particular, type-2 fuzzy numbers we should be dealing with must be easy to compute. With this in mind, TST2FNs were introduced in Ref. [11]. The application of TST2FNs to type-2 fuzzy differential equation theory can be also seen in [11]. If we want to know about type-2 fuzzy differential equation theory, we can also see in, e.g., [9, 16, 23].

The computation of fuzzy numbers tends to be more tedious and complicated than that of crisp numbers. Therefore, it is not sufficient to make anything a fuzzy number if it is an ambiguous number. It is necessary to appropriately determine what should be regarded as a fuzzy number, even at the expense of calculation tediousness and complexity. One criterion for such a judgment was given in this chapter.

Acknowledgments

I would like to particularly thank Dr. Jiro Inaida. I learned the rigorous formulation of arithmetic, etc. for fuzzy numbers by level-cut sets from him. I also deeply thank my co-researcher, senior and friend, Dr. Hiroaki Uesu. My research on type-2 fuzzy numbers is largely due to him.