**3.4 Triangular shaped type-2 fuzzy numbers**

The ð Þ *α*, *β* -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 *α*-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 *A*~ be a type-2 fuzzy set whose core is *a*∈ on . *A*~ is a triangular shaped type-2 fuzzy number (briefly, TST2FN) on if and only if its principal set and secondary membership function at *x* are given by

$$\nu\_A^{\mathfrak{x}}(t) = \begin{cases} \bar{A}\_1(\mathbf{x}) = \max\{\mathbf{1} - |\mathbf{x} - \mathbf{a}|, \ \mathbf{O}\}, \\ 1 - \frac{|t - \bar{A}\_1(\mathbf{x})|}{\min\{\bar{A}\_1(\mathbf{x}), \mathbf{1} - \bar{A}\_1(\mathbf{x})\}} & (\bar{A}\_1(\mathbf{x}) \in (\mathbf{0}, \mathbf{1})), \\ 1 & (t = \bar{A}\_1(\mathbf{x}) \in \{\mathbf{0}, \mathbf{1}\}), \\ \mathbf{0} & (t \neq \bar{A}\_1(\mathbf{x}) \in \{\mathbf{0}, \mathbf{1}\}), \end{cases}$$

respectively (**Figure 10**).

**Figure 10.** *Membership function of a triangular shaped type-2 fuzzy number with core* 2*; Figure 4 [11].*

**Figure 11.**

*Footprint set of "about* 2*" as the meaning of type-2 fuzzy numbers; Figure 5 [11].*

**Note:** For any TST2FN *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 *A*~<sup>1</sup> of *A*~ (see the right sides of **Figures 11** and **12**).

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 *A*~ be a TST2FN with core *a*∈ . For *α*, *β* ∈½ � 0, 1 , the following holds:

$$\left[\hat{A}\right]\_{\beta}^{a} = \begin{cases} \left< \left[a - \frac{2-a-\beta}{2-\beta}, \ a - \frac{\beta-a}{\beta}\right], \ \left[a + \frac{\beta-a}{\beta}, \ a + \frac{2-a-\beta}{2-\beta}\right] \right>, & 0 < a \le \frac{\beta}{2};\\ \left< \left[a - \frac{2-a-\beta}{2-\beta}, \ a - \frac{1-a}{2-\beta}\right], \ \left[a + \frac{1-a}{2-\beta}, \ a + \frac{2-a-\beta}{2-\beta}\right] \right>, & \frac{\beta}{2} < a \le 1 - \frac{\beta}{2};\\ \left< \left[a - \frac{1-a}{\beta}, \ a - \frac{1-a}{2-\beta}\right], \ \left[a + \frac{1-a}{2-\beta}, \ a + \frac{1-a}{\beta}\right] \right>, & 1 - \frac{\beta}{2} < a \le 1. \end{cases} \tag{8}$$

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

**Figure 12.** *Principal set of "about* 2*" as the meaning of type-2 fuzzy numbers; Figure 6 [11].*


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 *λ*). 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* ¼ 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].
