**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., ~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:

a. one is measuring, and

b. 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 *α*. 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.

	- i. *μF*ð Þ¼ 3 1,
	- ii. *μ<sup>F</sup>* is monotone increasing (resp. deceasing) on *x*∈ð Þ �∞, 3 (resp. *x*∈½ Þ 3, þ∞ ),
	- iii. for example, *μF*ð Þ¼ *x* 0 for all *x*≤ 2 and *x*≥4.

It seems natural to us that *μ<sup>F</sup>* is continuous, but *μ<sup>F</sup>* 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 *μ<sup>F</sup>* in **Figure 3**. It seems more natural that *μ<sup>F</sup>* is smooth, but *μ<sup>F</sup>* can be a broken line as shown in **Figure 3**. Such a fuzzy number is often called a triangular fuzzy number.

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.

**Figure 3.** *An example of a membership function of fuzzy number* 3*.*

#### **2.2 Notation of fuzzy sets**

In what follows, *X* denotes the universal crisp set, and we write *A x*ð Þ¼ *μA*ð Þ *x* for the membership grade of a fuzzy set *A* at *x*∈*X*. Related to this, following the conventions in fuzzy analysis, we write *A* : *X* ! ½ � 0, 1 for a fuzzy set on *X*.

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

$$A = \{ (\mathfrak{x}, \mu\_A(\mathfrak{x})) : \mathfrak{x} \in X \}, \quad \int\_{\mathfrak{x} \in X} \mu\_A(\mathfrak{x}) / \mathfrak{x}, \quad \text{etc.} \tag{2}$$

Remark that " Ð " 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 " Ð " is rewritten as " P" 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* ¼ and *A* : ! ½ � 0, 1 be a fuzzy set. *A* is a fuzzy number on if and only if *A* satisfies that


An *x*<sup>0</sup> 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*∈*X* and decreases as the variable *x*∈ *X* moves away from *a* to both sides. If *X* ¼ , 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:

$$A = \bigcup\_{a \in [0,1]} a[A]\_a,\tag{3}$$

where ½ � *A <sup>α</sup>* is the *α*-cut set of *A* defined as

$$[A]\_a = \begin{cases} \{ \mathfrak{x} \in \mathcal{X} : A(\mathfrak{x}) \ge a \} & (a \in (\mathbf{0}, 1]),\\ \operatorname{supp}(A) = \operatorname{cl}(\{ \mathfrak{x} \in \mathbb{R} : A(\mathfrak{x}) > \mathbf{0} \}) & (a = \mathbf{0}). \end{cases}$$

Here, "supp" means "support" and clð Þ*S* denotes the closure of a crisp set *S*. *α*½ � *A<sup>α</sup>* is defined as a fuzzy set *via* the algebraic product operation:

$$(a[A]\_a)(\mathfrak{x}) = a \cdot ([A]\_a)(\mathfrak{x}), \quad \mathfrak{x} \in \mathbb{R}.$$

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

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

where *α*<sup>∗</sup> stands for a fuzzy set whose membership function is the constant function, *<sup>α</sup>*<sup>∗</sup> ð Þ� *<sup>x</sup> <sup>α</sup>*.

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* : ! ½ � 0, 1 be a fuzzy set. *u* is a fuzzy number on if and only if *u* satisfies

a. *u* is normal, that is, *u* has at least one core;

b. *u* is fuzzy convex, that is, *u tx* ð Þ þ ð Þ 1 � *t y* ≥*u x*ð Þ∧*u y*ð Þ for any *x*, *y*∈ and *t*∈ ½ � 0, 1 , where ∧ represents the minimum operation;

c. *u* is semi-upper-continuous, that is, ½ � *u <sup>α</sup>* ¼ f g *x*∈ : *u x*ð Þ≥ *α* is closed for all *α*∈ ;

d. suppð Þ *u* is bounded.

In particular, *u* : ! ½ � 0, 1 is called a discrete fuzzy number (on ).

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**);

**Figure 4.** *Disconvexity is OK.*

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

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*<sup>∞</sup> <sup>0</sup> -function

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

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

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 ½ � *u <sup>α</sup>* is a bounded closed interval for any *α* ∈½ � 0, 1 . In fact,

• condition a says that ½ � *u <sup>α</sup>* is not empty for any *α* ∈½ � 0, 1 ,

**Figure 6.**

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


Given this, we denote

$$[\mathfrak{u}]\_a = [\mathfrak{u}\_-(a), \mathfrak{u}\_+(a)]$$

for any *α*∈ ½ � 0, 1 . *u*�ð Þ *α* are, of course, crisp numbers (or, the crisp function with respect to *α*, 0, 1 ½ �∍*α*↦*u*�ð Þ *α* ∈ ). 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. [2–4, 6]) Let *u* : ! ½ � 0, 1 be a fuzzy number. Then, the following holds:


$$\bigcap\_{n=1}^{\infty} [\mathfrak{u}]\_{a\_n} = [\mathfrak{u}]\_{a}.$$

Conversely, if there is a family of sets f g *P<sup>α</sup> <sup>α</sup>*∈½ � 0,1 satisfying properties 1, 2 and 3 above, then there exists a unique fuzzy number *u*. Moreover, it follows that

$$[u]\_a = P\_a$$

for any *α*∈ ð � 0, 1 and

$$[u]\_0 \subset P\_0. \tag{4}$$

**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<sup>α</sup>* above. It must then be satisfied that *P<sup>α</sup>* ¼ ½ � *u <sup>α</sup>* for any *α* ∈½ � 0, 1 . Representation Theorem and the proof of *P*<sup>0</sup> ⊂½ � *u* <sup>0</sup> guarantee that the results of the defined operations are fuzzy numbers. Eq. (3) guarantees that ∪*<sup>α</sup>*∈½ � 0,1 *αP<sup>α</sup>* 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].
