The Inverse and General Inverse of Trapezoidal Fuzzy Numbers with Modified Elementary Row Operations

Abstract

Trapezoidal positive/negative fuzzy numbers have no single definition; instead, various authors define them in relation to different concepts. This means that arithmetic operations for trapezoidal fuzzy numbers also differ. For the operations of addition, subtraction, and scalar multiplication, there are not many differences; for multiplication, however, there are many differences. In general, multiplication is divided into various cases. For the inverse operation, there is not much to define; in general, for any trapezoidal fuzzy number $\tilde{u},$ $\tilde{u}\otimes \frac{1}{\tilde{u}}=\tilde{i}=(1,1,0,0)$ does not necessarily apply. As a result of the different arithmetic operations for multiplication and division employed by various authors, several researchers have tackled the same problem and reached different solutions, meaning that the application will also produce different results. To date, many authors have proposed various alternatives for the algebra of the trapezoidal fuzzy number. In this paper, using the parametric form approach to trapezoidal fuzzy numbers, an alternative to multiplication with only one formula is constructed for various cases. Furthermore, based on the definition of multiplication for any trapezoidal fuzzy number, $\tilde{u}$ is constructed $\frac{1}{\tilde{u}}$ so that $\tilde{u}\otimes \frac{1}{\tilde{u}}=\tilde{i}=(1,1,0,0)$. Based on these conditions, we show that various properties that apply to real numbers also apply to any trapezoidal fuzzy number. Furthermore, we modify the elementary row operational steps for the trapezoidal fuzzy number matrix, which can be used to determine the inverse of a trapezoidal fuzzy number matrix with the order $m\times m$. We also give the steps and examples necessary to determine the general inverse for a trapezoidal fuzzy number matrix of the order $m\times n$ with $m \ne n$. This ability to easily determine the inverse and general inverse of a trapezoidal fuzzy number matrix has a number of applications, such as solving fully trapezoidal fuzzy number linear systems and fuzzy transportation problems, especially in applications in fields outside of mathematics; for example, the application of triangular fuzzy numbers in medical problems is a topic currently receiving a significant amount of attention.

1. Introduction

The notational approach to trapezoidal fuzzy numbers has been adopted by various authors with various notations. The interval number approach (parametric form) has facilitated an exciting expansion of operations on trapezoidal fuzzy numbers. This method is an extension of the triangular fuzzy number concept, which is developed into a trapezoidal fuzzy number, as shown in [1,2,3,4,5,6,7,8,9,10,11,12]. For further expansion, many authors also use approaches in the form of intervals, such as in [1,11,13,14,15,16,17,18,19,20,21,22,23,24], but the general notation used for trapezoidal fuzzy numbers is $\tilde{u}=\left(a,b,\alpha ,\beta \right)$. This form can also be changed into the parametric form $\tilde{u}\left(r\right)=\left[\underset{\_}{u}\left(r\right),\overline{u}(r)\right]$. Whatever notation is used, problems always arise regarding the definition of positivity/negativity, as well as the multiplication and inverse forms of a trapezoidal fuzzy number. Most authors only use positive triangular fuzzy numbers [2,3,4,5,9,12,13,14,20,25,26,27,28,29,30,31,32,33]. Several authors have defined negative fuzzy numbers, for example [1,6,11,12,13,20,27,29,32]. Likewise, the multiplication of two trapezoidal fuzzy numbers is explored in [3,4,5,6,10,11,12,25,32] using the concept $\tilde{u}\left(r\right)\otimes \tilde{v}\left(r\right)=\left[\mathit{min} S,\max S\right]$ with $S=\left\{\underset{\_}{u}\left(r\right)\underset{\_}{v}\left(r\right), \underset{\_}{u}\left(r\right)\overline{v}\left(r\right),\overline{u}\left(r\right)\underset{\_}{v}\left(r\right),\overline{u}\left(r\right)\overline{v}\left(r\right)\right\}$ for $\tilde{u}\left(r\right)=\left[\underset{\_}{u}\left(r\right),\overline{u}(r)\right]$ and $\tilde{v}\left(r\right)=\left[\underset{\_}{v}\left(r\right),\overline{v}(r)\right]$.

There are also differences in the definitions of positive numbers. For example, Ref. [14] defines a fuzzy number to be positive if $a-\alpha \ge 0,$ and the number is said to be negative if $b+\beta \le 0$ for $\tilde{a}=\left(a,b,\alpha ,\beta \right)$. Meanwhile, Ref. [26] defines the fuzzy number $\tilde{a}$ to be positive if a > 0 and negative if b < 0. Most writers generally define the fuzzy number $\tilde{u}$ to be positive if ${\mu }_{\tilde{a}}(x)>0,$ and it is said to be negative if ${\mu }_{\tilde{a}}(x)<with x\in \mathbb{R}$. The problem arises for cases outside of these conditions, for example for $\tilde{a}=\left(a,b,\alpha ,\beta \right)$ with $a<0$ but $b>0$, as well as various other cases. Here, can $\tilde{a}$ be said to be positive or negative? Another problem arises from the fact that there is very little that defines the form of $\frac{1}{\tilde{u}}$ for $\tilde{a}=\left(a,b,\alpha ,\beta \right)$ and the form of $\frac{1}{\tilde{a}(r)}$ for $\tilde{a}\left(r\right)=\left[\underset{\_}{a}\left(r\right),\overline{a}(r)\right]$. Some authors, including [20,21,34,35,36], define the form $\frac{1}{\tilde{a}}$, but the results are not necessarily valid for $\tilde{a}\otimes \frac{1}{\tilde{a}}=\tilde{i}=(1,1,0,0)$ for arbitrary $\tilde{a}=\left(a,b,\alpha ,\beta \right)$. As for the parametric form of the triangular fuzzy number $\tilde{a}\left(r\right)=\left[\underset{\_}{a}\left(r\right),\overline{a}(r)\right]$, this also does not necessarily apply for $\tilde{a}\left(\mathrm{r}\right)\otimes \frac{1}{a\left(r\right)}=\tilde{i}=[1,1]$. Every fuzzy number and fuzzy matrix should have an inverse; if it does not have an inverse, this will have an impact on the solving of the trapezoidal fuzzy number linear system. Various authors, such as [1,2,3,4,5,6,7,8,9,10,11,12,13,14,20,22,23,24,25,26,27,28,29,30,34,37,38,39,40,41], have found alternative methods to solve it, which involve modifying various existing concepts in real matrices. This method was not able to show the existence of $\tilde{x}={\tilde{A}}^{-1}\tilde{b}$ for arbitrary trapezoidal fuzzy linear systems in the form $\tilde{A}\otimes \tilde{x}=\tilde{b}$, especially for the full fuzzy linear system problem $\tilde{A}\otimes \tilde{x}\oplus \widetilde{b }=\tilde{C}\otimes \tilde{x}\oplus \tilde{d}$. As a result of the differences in arithmetic operations used by each author, different results are produced; this is shown in [9], which discusses the same problem addressed in [7,8] but obtains different results. The same conditions also occur when different methods and solutions are applied to trapezoidal fuzzy numbers, as discussed in [1,2,3,4,5,6,7,10,11,12,14,22,25,26,28,30,41,42,43]. More specifically, as shown in [41], the standard fuzzy arithmetic operation performed by many authors on the basis of the extension principle does not take into account all of the available information, and the results obtained are inaccurate or, in some cases, wrong. Although the solutions obtained using this method seem reasonable, they sometimes require long procedures and are computationally inefficient. However, the solution offered by [41] is an approach that is essentially no different from the approach given by [22,23]. In applications related to medical diagnoses [44,45,46], the steps commonly used by various authors are as follows. Step 1: Input the fuzzy soft set (F, P) to obtain the patient–disease fuzzy matrix $R_1$. Step 2: Input the fuzzy soft set (G, S) to obtain the symptom–disease fuzzy matrix $R_1$. Step 3: Perform the transformation operation $R_1⨂R_1$ to obtain the patient diagnosis fuzzy matrix $D_1$. In this case, the algebraic multiplication of trapezoidal fuzzy numbers is problematic because it will not ultimately yield good results with which to determine $D_1^{-1}$and $D_1^c$.

The authors of [16,17,18,19] define the alternative positivity of triangular fuzzy numbers using the concept of area, which is then extended to trapezoidal fuzzy numbers [13,14,15]; these then specify the multiplication form of the trapezoidal fuzzy number in four cases. However, in [13,14,15], for an arbitrary $\tilde{a}=\left(a,b,\alpha ,\beta \right)$, let $\frac{1}{\tilde{a}}=\left(p,q, \gamma ,\delta \right)$ not necessarily apply to $p<q$. Therefore, this form still needs to be perfected. The author of [19,47] proves that interval numbers and triangular fuzzy number fulfill various principles of real numbers. Therefore, the concept of interval numbers presented in [19,47] becomes the basic concept to be developed for trapezoidal fuzzy numbers, namely, by changing each of these trapezoidal fuzzy numbers into interval form; this is also called the trapezoidal fuzzy number parametric form.

For arbitrary trapezoidal fuzzy numbers, $\tilde{a}=\left(a,b,\propto ,\beta \right)$ and $\tilde{b}=\left(c,d,\gamma ,\delta \right)$ can be expressed in parametric form:

$$\tilde{a}=\left(a,b,\propto ,\beta \right)=(a-\left(1-r\right)\alpha ,b+\left(1-r\right)\beta =\left[\underset{\_}{a}\left(r\right),\overline{a}\left(r\right)\right]=\tilde{a}(r)$$

and

$$\tilde{b}=\left(c,d,\gamma ,\delta \right)=(c-\left(1-r\right)\gamma ,d+\left(1-r\right)\delta =\left[\underset{\_}{b}\left(r\right),\overline{b}\left(r\right)\right]=\tilde{b}\left(r\right)$$

By definition, $m(\tilde{a})=\frac{a+b}{2}$ and $m(\tilde{b})=\frac{c+d}{2}$ are constructed of a unique multiplication form of two trapezoidal fuzzy numbers in a standard form and a parametric form. From this multiplication form, $\frac{1}{\tilde{a}}$ or $\frac{1}{\tilde{a}(r)}$ will also be constructed, such that $\tilde{a}\otimes \frac{1}{\tilde{a}}=\tilde{i}=(1,1,0,0)$ or $\tilde{a}\left(\mathrm{r}\right)\otimes \frac{1}{\tilde{a}\left(r\right)}=\tilde{i}=[1,1]$. To date, no author has solved a fuzzy linear equation system in the form of $\tilde{A}\otimes \tilde{x}=\tilde{b}$ into the form of $\tilde{x}={\tilde{A}}^{-1}\tilde{b}$ because it is not possible to determine the value of ${\tilde{A}}^{-1}$ from a fuzzy matrix $\tilde{A}$. Similarly, when using fuzzy matrices to solve a range of problems, authors such as [28,29,30,37] and [31,32,33,38,48,49,50,51,52,53] solve them using various alternatives.

Based on the results of constructing the inverse of multiplication and division described above, it is determined that this method can be used to solve the system of linear equations of trapezoidal fuzzy numbers and fully fuzzy linear systems that produce unique and compatible solutions. The solution step also involves modifying the concept of the reduced row echelon to calculate the inverse of the trapezoidal fuzzy number matrix, so that, to determine the solution, the form $\tilde{x}={\tilde{A}}^{-1}\tilde{b}$ can be used for any trapezoidal fuzzy linear system in the form of $\tilde{A}\otimes \tilde{x}=\tilde{b}$ or in the form of a fully fuzzy linear system. The ability to construct a unique and compatible inverse of a fuzzy matrix can be used to obtain better results in various applications that use the concept of trapezoidal fuzzy number matrices, such as linear programming problems and transportation problems.

2. Preliminaries

The notation for trapezoidal fuzzy numbers is given differently by different authors, for example, [1,2,3,4,5,6,15,16,17,18,19,25,26,32,37,47,51,54,55], but the basic concept is the same and is given below.

Definition 1 

([1,2,3,4,5,6]). A fuzzy set $\tilde{a}$ of $X$ is defined by its membership function ${\mu }_{\tilde{a}}\left(x\right):X\to \left[0,1\right],$ which assigns a real number ${\mu }_{\tilde{u}}\left(x\right)$ in the interval [0, 1] to each element $x \in X$, where the value of ${\mu }_{\tilde{a}}\left(x\right)$ shows the grade membership of x in $\tilde{a}$.

Definition 2 

([15,16,17,18,19,25,26,47,54,55]). A trapezoidal fuzzy number $\tilde{a}=\left(a,b,\alpha ,\beta \right)$ is a fuzzy set on R with a membership function that satisfies the following:

1. 

$\tilde{a}(x)$ is upper semicontinuous;

2. 

$\tilde{a}(x)=0,$outside some interval [0, 1];

3. 

$\tilde{a}(x)$ is a monotonically increasing function on $[a-\alpha ]$;

4. 

$\tilde{a}(x)$ is a monotonically decreasing function on $[b+\beta ]$;

5. 

$\tilde{a}\left(x\right)=1, \mathrm{for} x \in [a,b]$.

The notation of a triangular fuzzy number is $\tilde{a}=\left(a,b,\alpha ,\beta \right)$. This notation is also used by [1,2,3,4,5,6,27,37,54,55]. The membership function is as follows:

$${\mu }_{\tilde{a}}\left(x\right)=\left\{\begin{array}{c}1-\frac{a-x}{\propto }, \mathrm{i}\mathrm{f} a-\propto \le x \le a \\ 1, \mathrm{i}\mathrm{f} a \le x \le b\\ 1-\frac{x-b}{\beta }, \mathrm{i}\mathrm{f} b\le x \le b+\beta \\ 0, Otherwise\end{array}\right.$$

In parametric form, the fuzzy number can be denoted in the form of $\tilde{a}\left(r\right)=\left[\underset{\_}{a}\left(r\right), \overline{a}\left(r\right)\right],$ with $\underset{\_}{a}\left(r\right)=a-(1-r)\alpha$ and $\overline{a}\left(r\right)=b+(1-r)\beta$.

Two fuzzy numbers $\tilde{a}(r)=\left[\underset{\_}{a}\left(r\right), \overline{a}(r)\right]$ and $\tilde{b}(r)=\left[\underset{\_}{b}\left(r\right), \overline{b}(r)\right]$ are said to be equal if $\underset{\_}{a}\left(r\right)=\underset{\_}{b}\left(r\right)$ and $\overline{a}\left(r\right)=\overline{b}(r)$, which, in the form of $\tilde{a}=\left(a,b,\alpha ,\beta \right)$ and $\tilde{b}=\left(c,d,\gamma ,d\right),$ are said to be equal if $a=c, b=d, \alpha =\gamma , \mathrm{and} \beta =d$. The definition of the similarity of two fuzzy sets was agreed upon by all authors. Otherwise, we return the trapezoidal fuzzy numbers in the parametric form $\tilde{a}(r)=\left[\underset{\_}{a}\left(r\right), \overline{a}(r)\right]$ to the form $\tilde{a}=\left(a,b,\alpha ,\beta \right)$ with a = $\underset{\_}{a}\left(1\right)$, b = $\overline{a}\left(1\right)$, $\alpha$ = $a$ − $\underset{\_}{a}\left(r\right),$ and $\beta$ = b − $\overline{a}\left(0\right)$.

Furthermore, the interval algebra in parametric form, as given by [1,11,13,20,56], is as follows:

Definition 3 

([32,37,51]). For two parametric forms of fuzzy numbers $\tilde{a}(r)=\left[\underset{\_}{a}\left(r\right), \overline{a}(r)\right]$ and $\tilde{b}(r)=\left[\underset{\_}{b}\left(r\right), \overline{b}(r)\right]$ and scalar $k\in R$:

(a) 

$\tilde{a}\left(r\right)+\tilde{b}\left(r\right)=\left[\underset{\_}{a}\left(r\right)+\underset{\_}{b}\left(r\right), \overline{a}\left(r\right)+\overline{b}(r)\right]$;

(b) 

$\tilde{a}\left(r\right)-\tilde{b}\left(r\right)=\left[\underset{\_}{a}\left(r\right)-\overline{b}\left(r\right), \overline{a}\left(r\right)-\underset{\_}{b}\left(r\right)\right]$;

(c) 

$\tilde{a}\left(r\right)\otimes \tilde{b}\left(r\right)=\left[\mathit{min}S,\mathit{max}S\right]$,

(d) 

with $S=\left\{\underset{\_}{a}\left(r\right)\underset{\_}{b}\left(r\right),\underset{\_}{a}\left(r\right)\overline{b}\left(r\right),\overline{a}\left(r\right)\underset{\_}{b}\left(r\right),\overline{a}\left(r\right)\overline{b}\left(r\right)\right\}$;

(e) 

$k\tilde{a}\left(r\right)=\left\{\begin{array}{c}\left[k \overline{a}\left(r\right), k\underset{\_}{a}\left(r\right)\right], if k<0\\ \left[k\underset{\_}{a}\left(r\right), k\overline{a}\left(r\right)\right], if k \ge 0\end{array}\right.$.

The authors of [15,16,17] and [54,55] provide a definition of positivity based on the concept of area. Specifically, if the area of the trapezoidal fuzzy number in the first quadrant is greater than the area of the trapezoidal fuzzy number in the second quadrant, it is said to be positive; otherwise, it is said to be negative. However, it is also recognized that this theory will lead to the presence of trapezoidal fuzzy numbers $\tilde{a}=\left(a,b,\alpha ,\beta \right)$ with $a \ne 0 \mathrm{and} b \ne 0$, but the trapezoidal fuzzy number $\tilde{a}=\tilde{0}$. For example, this is the case for a = −b ≠ 0 and $\alpha$ = $\beta$, among other cases. Therefore, based on these conditions, the author feels the need to improve the various concepts that exist for these trapezoidal fuzzy numbers. In particular, the concept of the multiplication and division of two trapezoidal fuzzy numbers and the inverse of a trapezoidal fuzzy number is then used in operations to determine the inverse and general inverse of matrices whose elements are trapezoidal fuzzy numbers.

3. Materials and Methods

3.1. Modification of Elementary Row Operations

The elementary row operations that can be used for the real matrix are as follows:

(a)

Multiplying a row by a non-zero constant;

(b)

Exchanging two rows;

(c)

Adding/subtracting multiples of one row with another row.

The modification of the elementary row operation steps for a matrix whose elements are trapezoidal fuzzy numbers (but which can also be used for triangular fuzzy numbers) is as follows:

(a*)

Multiplying a row by a non-zero trapezoidal fuzzy number;

(b*)

Adding/subtracting the result of multiplying a row with a trapezoidal fuzzy number to another row.

3.2. Algorithm for the Generalized Inverse of a Matrix

Until now, no academic articles have discussed the general inverse of trapezoidal fuzzy matrices. Therefore, the algorithm that can be used to determine the general inverse is an algorithm used for determining the general inverse of the real matrix, as given in [57,58], and it is the same for trapezoidal fuzzy matrices. The general inverse definition for fuzzy matrices is as follows:

Definition 4 

([57,58]). The trapezoidal fuzzy matrix $\tilde{G}(r)$ in parametric form is said to be the general inverse of the matrix $\tilde{A}(r)$ if $\widetilde{A }(r)⨂\widetilde{G }(r)⨂\widetilde{A }(r)$ = $\widetilde{A }(r)$. So, for the arbitrary general inverse $\widetilde{G }(r)$ of the matrix $\widetilde{A }\left(r\right),$ the following will apply:

1. 

$\widetilde{A }(r)⨂\widetilde{G }(r)$=$\widetilde{G }\left(r\right)⨂\widetilde{A }\left(r\right) is symmetrical;$

2. 

$\widetilde{G }(r)⨂\widetilde{A }(r)⨂\widetilde{G }(r)$=$\widetilde{G }(r)$.

Furthermore, the algorithm used to determine the general inverse of a trapezoidal fuzzy matrix $\tilde{A}(r)$ is as follows:

Step 1

If the fuzzy matrix $\widetilde{A }$is not in parametric form, it must first be changed into a matrix whose elements are in parametric form.

Step 2

Any non-singular minor matrix of matrix $\tilde{A}(r)$ of order r is identified where $r(\widetilde{A }(\mathrm{r})=r\le \min \{m,n\}$, denoted by $\tilde{M}(r)$.

$$\tilde{M}\left(r\right)=\left[\begin{array}{cccc}{\tilde{m}}_{11}& {\tilde{m}}_{12\mathrm{l}}& \cdots & {\tilde{r}}_{1r}\\ {\tilde{m}}_{2\mathrm{l}1}& {\tilde{m}}_{2\mathrm{l}2\mathrm{l}}& \cdots & {\tilde{r}}_{2\mathrm{l}r}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{m}}_{r1}& {\tilde{m}}_{r2\mathrm{l}}& \cdots & {\tilde{r}}_{rr}\end{array}\right]$$

Step 3

The inverse matrix $\tilde{M}(r)$ is determined and ${\tilde{M}}^{-1}(r)$ is transposed to obtain ${\left({\tilde{M}}^{-1}(r)\right)}^t$. Next, ${\left({\tilde{M}}^{-1}(r)\right)}^t$ is added to [0, 0] elements for elements outside the minor fuzzy matrix, so that the size is similar to that of matrix $\widetilde{A }(r);\mathrm{t}\mathrm{h}\mathrm{u}\mathrm{s},$ matrix $\widetilde{W }(r)$ is obtained.

Step 4

The matrix $\widetilde{W }(r)$ is transposed and let $\widetilde{G }(r)={\left(\widetilde{W }(r)\right)}^t$. This is the general inverse of the matrix $\tilde{A}(r)$.

To prove that the G-inverse obtained is true, the G-inverse must satisfy the definition of the general inverse, namely, $\widetilde{A }(r)⨂\widetilde{G }(r)⨂\widetilde{A }(r)$ = $\widetilde{A }(r)$.

4. Results and Discussion

The algebraic operations for addition, subtraction, and multiplication devised by various authors satisfy the algebraic properties for addition, subtraction, and multiplication; therefore, the algebraic operations that are used for addition, subtraction, and multiplication remain the same as points (a), (b), and (d) of Definition 3. Furthermore, the multiplication and inverse properties of trapezoidal fuzzy numbers will be modified.

4.1. Modification Concepts

The algebraic operations to be modified are the multiplication, division, and inverse of trapezoidal fuzzy numbers. Algebraic modification is carried out in interval form (parametric form).

Definition 5. 

The trapezoidal fuzzy number $\tilde{p}=\left(a,b,\propto ,\beta \right)$ is said to be positive if $\tilde{m}\left(p\right)>0$ and it is said to be negative if $m(\tilde{p})<0$ with $m(\tilde{p})=\frac{a+b}{2}$.

The concept of the positivity and negativity of fuzzy numbers produces a unique multiplication pattern and a unique inverse for each trapezoidal fuzzy number. Next, let $\tilde{p}$ and $\tilde{q}$ be two trapezoidal fuzzy numbers that, in parametric form, can be written as follows:

$$\tilde{p}=\left(a,b,\propto ,\beta \right)=(a-\left(1-r\right)\alpha , b+\left(1-r\right)\beta =\left[\underset{\_}{p}\left(r\right),\overline{p}\left(r\right)\right]=\tilde{p}\left(r\right)$$

$$\tilde{q}=\left(c,d,\gamma ,\delta \right)=(c-\left(1-r\right)\gamma , d+\left(1-r\right)\delta =\left[\underset{\_}{q}\left(r\right),\overline{q}\left(r\right)\right]=\tilde{q}\left(r\right)$$

where $m(\tilde{p})=\frac{a+b}{2}$ and $\tilde{m}(q)=\frac{c+d}{2}$. The multiplication of two trapezoidal fuzzy numbers is defined as follows:

$$\tilde{p}(r)\otimes \tilde{q}(r)=\left(\underset{\_}{p}\left(r\right)m\left(\tilde{q}\right)+\underset{\_}{q}\left(r\right)m(\tilde{p})-m\left(\tilde{p}\right)m\left(\tilde{q}\right), \overline{p}\left(r\right)m\left(\tilde{q}\right)+\overline{q}\left(r\right)m\left(\tilde{p}\right)-m\left(\tilde{p}\right).m(\tilde{q})\right)$$

$$\begin{gathered} \tilde{p}\left(r\right)\otimes \tilde{q}\left(r\right)=\left[\left(a-\left(1-r\right)\propto \right)m\left(\tilde{q}\right)+\left(c-\left(1-r\right)\gamma \right)m\left(\tilde{p}\right)-m\left(\tilde{p}\right).m\left(\tilde{q}\right), \right. \\ \left(b+\left(1-r\right)\beta \right) m(\tilde{q})+\left(d+\left(1-r\right)\delta \right)m(\tilde{p})-m\left(\tilde{p}\right).m(\tilde{q})] \end{gathered}$$

(1)

Let $\tilde{z}\left(r\right)=\left[\underset{\_}{z}\left(r\right),\overline{z}\left(r\right)\right]=\tilde{p}\left(r\right)\otimes \tilde{q}\left(r\right),$ where

$$\underset{\_}{z}\left(r\right)=\left(a-\left(1-r\right)\propto \right)m(\tilde{q})+\left(c-\left(1-r\right)\gamma \right)m(\tilde{p})-m\left(\tilde{p}\right).m\left(\tilde{q}\right)$$

$$\overline{z}\left(r\right)=\left(b+\left(1-r\right)\beta \right)m(\tilde{q})+\left(d+\left(1-r\right)\delta \right)m(\tilde{p})-m\left(\tilde{p}\right).m(\tilde{q})$$

If $\tilde{z}\left(r\right)$ is expressed in the form $\tilde{z}=\left(p,q, \sigma ,\tau \right)$, then

$$\underset{\_}{z}\left(0\right)=\left(a-\propto \right)m(\tilde{q})+\left(c-\gamma \right)m(\tilde{p})-m\left(\tilde{p}\right).m\left(\tilde{q}\right)$$

$$\underset{\_}{z}\left(1\right)=p · 1=am(\tilde{q})+c\tilde{m}\left(p\right)-m(\tilde{p}).\tilde{m}\left(q\right)$$

So

$$\begin{array}{cc}\hfill \sigma & =\underset{\_}{z}\left(1\right)-\underset{\_}{z}\left(0\right)\hfill \\ & =\left(a.m\left(\tilde{q}\right)+c.m(\tilde{p})-m(\tilde{p}).m(\tilde{q})\right)-\left(\left(a-\propto \right)m(\tilde{q})+\left(c-\gamma \right)m(\tilde{p})-m(\tilde{p}).m(\tilde{q})\right)\hfill \\ & =a.m(\tilde{q})+c.m(\tilde{p})-\left(\left(a-\propto \right)m(\tilde{q})+\left(c-\gamma \right)m(\tilde{p})\right)\hfill \\ & =\propto .\tilde{m}\left(q\right)+\gamma .m(\tilde{p})\hfill \end{array}$$

and

$$\overline{z}\left(0\right)=\left(b+\beta \right)m(\tilde{q})+\left(d+\delta \right)m(\tilde{p})-m\left(\tilde{p}\right).m(\tilde{q})$$

$$\overline{z}\left(1\right)=q=b.m\left(\tilde{q}\right)+d.m(\tilde{p})-m\left(\tilde{p}\right).m(\tilde{q})$$

So, in general form, that is

$$\begin{array}{cc}\hfill \tau & =\tilde{z}\left(1\right)-\overline{z}\left(0\right)\hfill \\ & =\left(b+\beta \right)m(\tilde{q})+\left(d+\delta \right)m(\tilde{p})-m(\tilde{p}).m\left(\tilde{q}\right)-\left(b.m\left(\tilde{q}\right)+d.m(\tilde{p})-m\left(\tilde{p}\right).m(\tilde{q})\right)\hfill \\ & =\beta m(\tilde{q})+\delta m(\tilde{p})\hfill \end{array}$$

So that,

$$\tilde{z}\left(r\right)=\tilde{p}\left(r\right)\otimes \tilde{q}\left(r\right)=\left(p,q, \sigma ,\tau \right)$$

where

$$p=a.m\left(\tilde{q}\right)+c.m(\tilde{p})-m(\tilde{p}).m\left(\tilde{q}\right)$$

$$q=b.m\left(\tilde{q}\right)+d.m(\tilde{p})-m(\tilde{p}).m(\tilde{q})$$

$$\sigma =\propto .m\left(\tilde{q}\right)+\gamma .m(\tilde{p})$$

$$\tau =\beta .m\left(\tilde{q}\right)\left(q\right)+\delta .m(\tilde{p})$$

Hence, it can be written as follows:

$$\begin{gathered} \tilde{z}=\left(a.m\left(\tilde{q}\right)+c.m\left(\tilde{p}\right)-m\left(\tilde{p}\right).m\left(\tilde{q}\right), b.\tilde{m}\left(q\right)+d.m\left(\tilde{p}\right)-m\left(\tilde{p}\right).m\left(\tilde{q}\right), \\ \propto m\left(\tilde{q}\right)+\gamma .m\left(\tilde{p}\right),\beta .m\left(\tilde{q}\right)+\delta .m(\tilde{p})\right) \end{gathered}$$

(2)

The result of this multiplication is unique for all cases.

Remark 1. 

Based on Definition 5, Equation (1) or Equation (2) clearly apply:

i. 

If $\tilde{p}\left(r\right)>0$ and $\tilde{q}\left(r\right)>0,$ then $\tilde{p}\left(r\right)\otimes \tilde{q}\left(r\right)>0$;

ii. 

If $\tilde{p}\left(r\right)>0$ and $\tilde{q}\left(r\right)<0,$ then $\tilde{p}\left(r\right)\otimes \tilde{q}\left(r\right)<0$;

iii. 

If $\tilde{p}\left(r\right)<0$ and $\tilde{q}\left(r\right)>0,$ then $\tilde{p}\left(r\right)\otimes \tilde{q}\left(r\right)<0$;

iv. 

If $\tilde{p}\left(r\right)<0$ and $\tilde{q}\left(r\right)<0,$ then $\tilde{p}\left(r\right)\otimes \tilde{q}\left(r\right)>0$.

4.2. The Inverse and Division of Trapezoidal Fuzzy Numbers

Furthermore, because a trapezoidal fuzzy number $\tilde{p}\left(r\right)$ is said to be positive if $m\left(\tilde{p}\right)>0$ and negative if $m\left(\tilde{p}\right)<0,$ the next problem relates to what happens when $\tilde{m}\left(p\right)=0$. This will only happen in two cases: case 1, if $a=-b$ where $\propto \ne 0 \mathit{or} \beta \ne 0$, and case 2, if $a=b=\propto =\beta$. Then, zero trapezoidal fuzzy numbers are also divided into two cases, namely, $\tilde{p}=\left(a,b,\propto ,\beta \right)$ is said to be a zero trapezoidal fuzzy number if $\tilde{m}\left(p\right)=0$; it is denoted by $\widetilde{0_f}=\left(0,0,{\epsilon }_1,{\epsilon }_2\right)$ or, in interval form, it is denoted by $\widetilde{0_f}=[\left(1-r\right){\epsilon }_1,\left(1-r\right){\epsilon }_2]$. The trapezoidal fuzzy number $\tilde{p}$ is said to be pure zero if $m\left(\tilde{p}\right)=(0,0,0,0)$, and it is denoted by $\tilde{0}=\left(0,0,0,0\right),$ which, in interval form, is denoted by $\tilde{0}=[0, 0]$. Then, the identity element is also divided into two cases, namely, the identity element when ${\tilde{i}}_f=[1,1,{\epsilon }_1,{\epsilon }_2$], which, in interval form, is denoted by ${\tilde{i}}_f\left(r\right)=\left[1-\left(1-r\right){\epsilon }_1,1+(1-r){\epsilon }_2\right]$; it is said to be a pure identity element if $\tilde{i}=\left[1,1,0,0\right]$, which, in interval form, is denoted by $\tilde{i}\left(r\right)=[1,1]$. This paper uses pure zero and pure identity fuzzy numbers.

Furthermore, for each trapezoidal fuzzy number $\tilde{p}\left(r\right)$ where $m\left(\tilde{p}\right)\ne 0$, $\tilde{x}(r)$ will be shown to exist, such that $\tilde{p}\otimes \tilde{x}=\tilde{i}=[1,1,0,0]$ where $\tilde{x}=\frac{1}{\tilde{p}}$. Let the following be true:

$$\tilde{p}=\left(a,b,\propto ,\beta \right)=(a-\left(1-r\right)\alpha ,b+\left(1-r\right)\beta =\left[\underset{\_}{p}\left(r\right),\overline{p}\left(r\right)\right]=\tilde{p}\left(r\right);$$

$$\tilde{i}=\left(c,d,\gamma ,\delta \right)=(c-\left(1-r\right)\gamma ,d+\left(1-r\right)\delta =\left[\underset{\_}{I}\left(r\right),\overline{I}\left(r\right)\right]=\tilde{i}\left(r\right).$$

Based on the above concept, for each $m\left(\tilde{p}\right)\ne 0$, whether $m\left(\tilde{p}\right)>0$ or $m\left(\tilde{p}\right)<0$, there will always be $\tilde{x}=\frac{\tilde{i}}{\tilde{p}}$, for which the existence of the inverse of trapezoidal fuzzy numbers can be expressed in the form of the following theorem:

Theorem 1. 

For an arbitrary traezoidal fuzzy number $\tilde{p}=\left[a,b,\propto ,\beta \right]$ where $m\left(\tilde{p}\right)\ne 0$, $\tilde{x}=\frac{\tilde{i}}{\tilde{p}}=\left[\frac{2m\left(\tilde{p}\right)-a}{m^2(\tilde{p})}, \frac{2m\left(\tilde{p}\right)-b}{m^2(\tilde{p})},\frac{-\alpha }{m^2(\tilde{p})}, \frac{-\beta }{m^2(\tilde{p})}\right]$, such that $\tilde{p}\otimes \tilde{x}=\tilde{i}=\left[1,1,0,0\right]=\tilde{x}\otimes \tilde{p}$.

Proof. 

Consider $\tilde{p}=\left[a,b,\propto ,\beta \right],$ where $\tilde{m}\left(p\right)\ne 0$. $\tilde{x}=\left[c,d,\gamma ,\delta \right]$ will be determined such that $\tilde{p}\otimes \tilde{x}=\tilde{i}=\left[1,1,0,0\right]=\tilde{x}\otimes \tilde{p}$. Because it must apply to $\tilde{p}\otimes \tilde{x}=\tilde{i}=\left[1,1,0,0\right]$, $m\left(\tilde{p}\otimes \tilde{x}\right)=m\left(\tilde{p}\right)\otimes m\left(\tilde{x}\right)=1$ or $m\left(\tilde{x}\right)=\frac{1}{m\left(\tilde{p}\right)}.$ Consequently,

$$\tilde{p}\otimes \tilde{x}=\left[a.m\left(\tilde{x}\right)+c.m\left(\tilde{p}\right)-m\left(\tilde{p}\right).m\left(\tilde{x}\right), b.m\left(\tilde{q}\right)+d.m\left(\tilde{p}\right)-m\left(\tilde{p}\right).m\left(\tilde{x}\right),\propto .m\left(\tilde{x}\right)+\gamma .m\left(\tilde{p}\right),\beta .m\left(\tilde{x}\right)+\delta .m\left(\tilde{p}\right)\right]$$

$$\tilde{p}\otimes \tilde{x}=\left[a\frac{1}{m\left(\tilde{p}\right)}+c.m\left(\tilde{p}\right)-1,b\frac{1}{m\left(\tilde{p}\right)}+d.m\left(\tilde{p}\right)-1,\propto \frac{1}{m\left(\tilde{p}\right)}+\gamma m\left(\tilde{p}\right), \beta \frac{1}{m\left(\tilde{p}\right)}+\delta m\left(\tilde{p}\right)\right]=[1,1,0,0]$$

Therefore,

$$a\frac{1}{m\left(\tilde{p}\right)}+cm\left(\tilde{p}\right)-1=1$$

(3)

$$b\frac{1}{m\left(\tilde{p}\right)}+dm\left(\tilde{p}\right)-1=1$$

(4)

$$\propto \frac{1}{m\left(\tilde{p}\right)}+\gamma m\left(\tilde{p}\right)=0$$

(5)

$$\beta \frac{1}{m\left(\tilde{p}\right)}+\delta m\left(\tilde{p}\right)=0$$

(6)

From Equations (3) and (4), we obtain $c=\frac{2m\left(\tilde{p}\right)-a}{m^2(\tilde{p})}$ and $d=\frac{2m\left(\tilde{p}\right)-b}{m^2(\tilde{p})}$, and, from Equations (5) and (6), we obtain $\gamma =\frac{-\alpha }{m^2(\tilde{p})}$ and $\delta =\frac{-\beta }{m^2(\tilde{p})}$.

Thus, for every trapezoidal fuzzy number $\tilde{p}=\left[a,b,\propto ,\beta \right]$where $\tilde{m}\left(p\right)\ne 0$, $\tilde{x}=\frac{1}{\tilde{a}}=\left[\frac{2m\left(\tilde{p}\right)-a}{m^2(\tilde{p})}, \frac{2m\left(\tilde{p}\right)-b}{m^2(\tilde{p})},\frac{-\alpha }{m^2(\tilde{p})}, \frac{-\beta }{m^2(\tilde{p})}\right]$, such that $\tilde{p}\otimes \tilde{x}=\tilde{i}=\left[1,1,0,0\right]=\tilde{x}\otimes \tilde{p}$. □

Based on Theorem 1, the formula for the division of two trapezoidal fuzzy numbers is as follows:

Corollary 1. 

For two arbitrary trapezoidal fuzzy numbers  $\tilde{p}=\left[a,b,\propto ,\beta \right]$ and $\tilde{q}=\left[c,d,\gamma ,\delta \right]$, the division of trapezoidal fuzzy numbers is as follows:

$$\begin{gathered} \frac{\tilde{p}}{\tilde{q}}=\tilde{p}\otimes \frac{1}{\tilde{q}}=\left[\frac{a.m\left(\tilde{q}\right)+2m\left(\tilde{p}\right)m\left(\tilde{q}\right)-cm\left(\tilde{p}\right)-m\left(\tilde{p}\right)m\left(\tilde{q}\right)}{m^2(\tilde{q})}, \right. \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\frac{b.m\left(\tilde{q}\right)+2m\left(\tilde{p}\right)m\left(\tilde{q}\right)-d.m\left(\tilde{p}\right)-m\left(\tilde{p}\right)m\left(\tilde{q}\right)}{m^2(\tilde{q})},\left.\frac{\propto m\left(\tilde{q}\right)-\gamma .m\left(\tilde{p}\right)}{m^2(\tilde{q})},\frac{\beta .m\left(\tilde{q}\right)-\delta m\left(\tilde{p}\right)}{m^2(\tilde{q})}\right] \end{gathered}$$

Proof. 

From Theorem 1, we obtain:

$$\begin{gathered} \frac{\tilde{p}}{\tilde{q}}=\tilde{p}\otimes \frac{1}{\tilde{q}}=\left[a,b,\propto ,\beta \right]\otimes \left[\frac{2m\left(q\right)-c}{m^2(q)}, \frac{2m\left(q\right)-d}{m^2(q)},\frac{-\gamma }{m^2(q)}, \frac{-\delta }{m^2(q)}\right] \\ \tilde{p}\otimes \frac{1}{\tilde{q}}=\left[a\frac{1}{m\left(\tilde{q}\right)}+\left(\frac{2m\left(\tilde{q}\right)-c}{m^2(\tilde{q})}\right)m\left(\tilde{p}\right)-m\left(\tilde{p}\right).\frac{1}{m\left(\tilde{q}\right)},b\frac{1}{m\left(\tilde{q}\right)}+\left(\frac{2m\left(\tilde{q}\right)-d}{m^2(\tilde{q})}\right)m\left(\tilde{p}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-m\left(\tilde{p}\right).\frac{1}{m\left(\tilde{q}\right)},\propto \frac{1}{m\left(\tilde{q}\right)}+\left(\frac{-\gamma }{m^2(\tilde{q})}\right)m\left(\tilde{p}\right),\beta \frac{1}{m\left(\tilde{q}\right)}+\left(\frac{-\delta }{m^2(\tilde{q})}\right)m\left(\tilde{p}\right)\right] \end{gathered}$$

$$\tilde{p}\otimes \frac{1}{\tilde{q}}=\left[\begin{array}{c}\frac{am\left(\tilde{q}\right)+2m\left(\tilde{p}\right)m\left(\tilde{q}\right)-cm\left(\tilde{p}\right)-m\left(\tilde{p}\right)m\left(\tilde{q}\right)}{m^2(\tilde{q})}, \frac{b.m\left(\tilde{q}\right)+2m\left(\tilde{p}\right)m\left(\tilde{q}\right)-d.m\left(\tilde{p}\right)-m\left(\tilde{p}\right)m\left(\tilde{q}\right)}{m^2(\tilde{q})},\\ \frac{\propto m\left(\tilde{q}\right)-\gamma .m\left(\tilde{p}\right)}{m^2(\tilde{q})},\frac{\beta .m\left(\tilde{q}\right)-\delta m\left(\tilde{p}\right)}{m^2(\tilde{q})}\end{array}\right]$$

□

Now, based on Theorem 1 and Corollary 1, it can be shown that, for an arbitrary $\tilde{p}=\left[a,b,\alpha ,\beta \right]$, $\frac{\tilde{p}}{\tilde{p}}=\tilde{i}$will hold.

Corollary 2. 

For an arbitrary trapezoidal fuzzy number $\tilde{p}=\left[a,b,\alpha ,\beta \right]$,  $\frac{\tilde{p}}{\tilde{p}}=\tilde{i}$ .

Proof. 

From Theorem 1, we obtain the following:

$$\begin{gathered} \frac{\tilde{p}}{\tilde{p}}=\tilde{p}\otimes \frac{1}{\tilde{p}}=\left[\left[a,b,\alpha ,\beta \right]\right]\left[\frac{2m\left(\tilde{p}\right)-a}{m^2(\tilde{p})}, \frac{2m\left(\tilde{p}\right)-b}{m^2(\tilde{p})},\frac{-\alpha }{m^2(\tilde{p})}, \frac{-\beta }{m^2(\tilde{p})} \right] \\ \tilde{p}\otimes \frac{1}{\tilde{p}}=\left[a\frac{1}{m\left(\tilde{p}\right)}+\left(\frac{2m\left(\tilde{p}\right)-a}{m^2(\tilde{p})}\right)m\left(\tilde{p}\right)-m\left(\tilde{p}\right).\frac{1}{m\left(\tilde{p}\right)},b\frac{1}{m\left(\tilde{p}\right)}+\left(\frac{2m\left(\tilde{p}\right)-b}{m^2(\tilde{p})}\right)m\left(\tilde{p}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-m\left(\tilde{p}\right).\frac{1}{m\left(\tilde{p}\right)},\propto \frac{1}{m\left(\tilde{p}\right)}+\left(\frac{-\alpha }{m^2(\tilde{p})}\right)m\left(\tilde{p}\right),\beta \frac{1}{m\left(\tilde{p}\right)}+\left(\frac{-\beta }{m^2(\tilde{p})}\right)m\left(\tilde{p}\right)\right] \\ \tilde{p}\otimes \frac{1}{\tilde{p}}=\left[1,1,0,0\right]. \end{gathered}$$

□

Corollary 1 can also be used to prove Corollary 2, and the same result will be obtained. Then, using a similar method to that of the proof above, the following properties can easily be shown.

Theorem 2. 

Let  $\tilde{p}, \tilde{q},\tilde{r}$be trapezoidal fuzzy numbers. Then,

a. 

$\tilde{p}\otimes \tilde{0}=\tilde{0}\otimes \tilde{p}=\tilde{0}$;

b. 

$\tilde{p}\otimes \tilde{i}=\tilde{i}\otimes \tilde{p}=\tilde{p}$;

c. 

$\tilde{p}\otimes \tilde{q}=\tilde{q}\otimes \tilde{p}$;

d. 

$\left(\tilde{p}\otimes \tilde{q}\right)\otimes \tilde{r}=\tilde{p}\otimes \left(\tilde{q}\otimes \tilde{r}\right)$;

e. 

$\left(\tilde{p}\oplus \tilde{q}\right)\otimes \tilde{r}=\left(\tilde{p}\otimes \tilde{r}\right)\oplus \left(\tilde{q}\otimes \tilde{r}\right)$;

f. 

If  $\tilde{p}\otimes \tilde{x}=\tilde{b}$ and $\tilde{p}\ne \tilde{0}$, then $\tilde{x}=\frac{\tilde{b}}{\tilde{p}}$;

g. 

If  $\tilde{p}\otimes \tilde{q}=\tilde{0}$ , then $\tilde{p}=\tilde{0}$ or $\tilde{q}=\tilde{0}$;

h. 

If $\tilde{p}\otimes \tilde{q}=\tilde{p}\otimes \tilde{r}$ and $\tilde{p}\ne \tilde{0}$ , then $\tilde{q}=\tilde{r}$;

i. 

If $\tilde{p}\ne \tilde{0}$, then  $\frac{1}{\tilde{p}} \ne \tilde{0}$ and $\frac{1}{\frac{1}{\tilde{p}}}=\tilde{p}$;

j. 

If $\tilde{p}\ne \tilde{0}$ and $\tilde{q}\ne \tilde{0}$, then $\frac{1}{\tilde{p}\otimes \tilde{q}}=\frac{1}{\tilde{p}}\otimes \frac{1}{\tilde{q}}$.

Proof. 

It is clear. □

Similarly to the identity element and zero element for trapezoidal fuzzy numbers, the zero matrix is divided into two cases. A trapezoidal fuzzy matrix is said to be a trapezoidal fuzzy zero matrix if ${\check{\mathbf{0}}}_f={\left({\tilde{a}}_{ij}\right)}_{nxn}$ with ${\tilde{a}}_{ij}=\left(0,0, {\epsilon }_1, {\epsilon }_2\right)$ for i = j and ${\tilde{a}}_{ij}=\left(0,0, 0, 0\right)$ for $i\ne j$. In interval form, it can be expressed as ${\check{\mathbf{0}}}_f(r)={\left(\left[\underset{\_}{{\tilde{a}}_{ij}}\left(r\right), \overline{{\tilde{a}}_{ij}}(r)\right]\right)}_{mxm}={\left(\left[1-\left(1-r\right){\left({\epsilon }_1\right)}_{ij},1+(1-r){\left({\epsilon }_2\right)}_{ij}\right]\right)}_{mxm}$. Meanwhile, the pure zero fuzzy matrix for trapezoidal fuzzy numbers is defined by $\tilde{\mathit{O}}={\left({\tilde{a}}_{ij}\right)}_{nxn}$ with ${\tilde{a}}_{ij}=\left(0,0, 0, 0\right)$ for all values of i and j, which in interval form can be expressed by $\tilde{\mathit{O}}(r)={\left(\left[\underset{\_}{{\tilde{a}}_{ij}}\left(r\right), \overline{{\tilde{a}}_{ij}}(r)\right]\right)}_{nxn}={\left(\left[0,0\right]\right)}_{nxn}$. This paper also uses $\tilde{\mathit{O}}$ and $\tilde{\mathit{O}}(r)$. When calculating the inverse using the adjoint method, the following form is used: ${\check{\mathbf{0}}}_f$ or ${\check{\mathbf{0}}}_f(r)$.

4.3. Example of the Calculation of the Inverse and General Inverse of a Fuzzy Trapezoidal Matrix

The following is an example of the calculation of the inverse and general inverse of a trapezoidal fuzzy matrix, which is first changed into parametric form. For the inverse of the trapezoidal matrix, an example of a $3\times 3$ matrix is given, and, for the general inverse, an example of a $2\times 3$ matrix is given. For matrices of other sizes, the operation can be conducted in the same way.

Example 1. 

Consider the calculation of the inverse of a $3\times 3$ trapezoidal fuzzy matrix. Let

$$\tilde{A}=\left[\begin{array}{ccc}\left(1,3,1,1\right)& \left(2,4,2,2\right)& \left(0,1,1,3\right)\\ \left(2,6,2,2\right)& \left(1,5,1,1\right)& \left(2,4,1,0\right)\\ \left(-1,2,1,2\right)& \left(-2,1,2,1\right)& \left(-1,3,2,2\right)\end{array}\right]$$

which, in interval form, can be written as follows:

$$\tilde{A}\left(r\right)=\left[\begin{array}{ccc}\left[r,4-r\right]& \left[2r,6-2r\right]& \left[-1+r,4-3r\right]\\ \left[2r,8-2r\right]& \left[r,6-r\right]& \left[1+r,4\right]\\ \left[-2+r,4-2r\right]& \left[-4+2r,2-r\right]& \left[-3+2r,5-2r\right]\end{array}\right]$$

From the form of:

$$\tilde{A}(r)|\tilde{I}(r)=\left[\begin{array}{ccc}\left[r,4-r\right]& \left[2r,6-2r\right]& \left[-1+r,4-3r\right]\\ \left[2r,8-2r\right]& \left[r,6-r\right]& \left[1+r,4\right]\\ \left[-2+r,4-2r\right]& \left[-4+2r,2-r\right]& \left[-3+2r,5-2r\right]\end{array}\left|\begin{array}{ccc}\left[1,1\right]& \left[0,0\right]& \left[0,0\right]\\ \left[0,0\right]& \left[1,1\right]& \left[0,0\right]\\ \left[0,0\right]& \left[0,0\right]& \left[1,1\right]\end{array}\right.\right]$$

Multiply the first row by $\frac{1}{[r,4-r]}$ , so that the following is obtained:

$$\left[\begin{array}{ccc}[1,1]& [\frac{3}{2}+\frac{1}{4}r,\frac{3}{2}-\frac{1}{4}r]& [\frac{-1}{4}+\frac{3}{8}r,\frac{7}{4}-\frac{11}{8}r]\\ [2r,8-2r]& [r,6-r]& [1+r,4]\\ [-2+r,4-2r]& [-4+2r,2-r]& [-3+2r,5-2r]\end{array}\left|\begin{array}{ccc}[1-\frac{1}{4}r ,\frac{1}{4}r]& [0,0]& [0,0]\\ [0,0]& [1,1]& [0,0]\\ [0,0,0]& [0,0]& [1,1]\end{array}\right.\right]$$

Then, the second row is subtracted using $[2r,8-2r]\times b_1$ and the third row is subtracted using $[-2+r,4-2r]\times b_1$:

$$\left[\begin{array}{ccc}[1,1]& [\frac{3}{2}+\frac{1}{4}r,\frac{3}{2}-\frac{1}{4}r]& [\frac{-1}{4}+\frac{3}{8}r,\frac{7}{4}-\frac{11}{8}r]\\ [0,0]& [-3r,-6+3r]& [3-r,-4+6r]\\ [0,0]& [-1+\frac{3}{8}r,-4+\frac{17}{8}r]& [\frac{-9}{4}+\frac{25}{16}r,\frac{13}{4}-\frac{13}{16}r]\end{array}\left|\begin{array}{ccc}[1-\frac{1}{4}r ,\frac{1}{4}r]& [0,0]& [0,0]\\ [-2,-2]& [1,1]& \left[0,0\right]\\ [\frac{3}{4}-\frac{3}{8}r,\frac{-7}{4}+\frac{7}{8}r]& [0,0]& [1,1]\end{array}\right.\right]$$

The second row is multiplied by $\frac{1}{[-3r,-6+3r]}$, obtaining the following:

$$\left[\begin{array}{ccc}\left[1,1\right]& \left[\frac{3}{2}+\frac{1}{4}r,\frac{3}{2}-\frac{1}{4}r\right]& \left[\frac{-1}{4}+\frac{3}{8}r,\frac{7}{4}-\frac{11}{8}r\right]\\ \left[0,0\right]& \left[1,1\right]& \left[\frac{-5}{3}+r,2-\frac{8}{3}r\right]\\ \left[0,0\right]& \left[-1+\frac{3}{8}r,-4+\frac{17}{8}r\right]& \left[\frac{-9}{4}+\frac{25}{16}r,\frac{13}{4}-\frac{13}{16}r\right]\end{array}\left|\begin{array}{ccc}\left[1-\frac{1}{4}r ,\frac{1}{4}r\right]& \left[0,0\right]& \left[0,0\right]\\ \left[\frac{4}{3}-\frac{2}{3}r,\frac{2}{3}r\right]& \left[\frac{-2}{3}+\frac{1}{3}r,\frac{-1}{3}r\right]& \left[0,0\right]\\ \left[\frac{3}{4}-\frac{3}{8}r,\frac{-7}{4}+\frac{7}{8}r\right]& \left[0,0\right]& \left[1,1\right]\end{array}\right.\right]$$

The first row is subtracted using $\left[\frac{3}{2}+\frac{1}{4}r,\frac{3}{2}-\frac{1}{4}r\right]\times b_2$ and the third row with $\left[-1+\frac{3}{8}r,-4+\frac{17}{8}r\right]\times b_2$ , obtaining the following:

$$\left[\begin{array}{ccc}\left[1,1\right]& \left[0,0\right]& \left[\frac{9}{4}-\frac{23}{24}r,\frac{-5}{4}+\frac{59}{24}r\right]\\ \left[0,0\right]& \left[1,1\right]& \left[\frac{-5}{3}+r,2-\frac{8}{3}r\right]\\ \left[0,0\right]& \left[0,0\right]& \left[-\frac{25}{6}+\frac{49}{16}r,\frac{47}{12}-\frac{131}{48}r\right]\end{array}\left|\begin{array}{ccc}\left[-1+\frac{7}{12}r,\frac{-7}{12}r\right]& \left[1-\frac{5}{12}r,\frac{5}{12}r\right]& \left[0,0\right]\\ \left[\frac{4}{3}-\frac{2}{3}r,\frac{2}{3}r\right]& \left[\frac{-2}{3}+\frac{1}{3}r,\frac{-1}{3}r\right]& \left[0,0\right]\\ \left[\frac{9}{4}-\frac{35}{24}r,\frac{1}{12}+\frac{7}{24}r\right]& \left[\frac{-3}{4}+\frac{13}{24}r,\frac{-11}{12}+\frac{7}{24}r\right]& \left[1,1\right]\end{array}\right.\right]$$

The third row is multiplied by $\frac{1}{\left[-\frac{25}{6}+\frac{49}{16}r,\frac{47}{12}-\frac{131}{48}r\right]}$ , allowing us to obtain the following:

$$\left[\begin{array}{ccc}\left[1,1\right]& \left[0,0\right]& \left[\frac{9}{4}-\frac{23}{24}r,\frac{-5}{4}+\frac{59}{24}r\right]\\ \left[0,0\right]& \left[1,1\right]& \left[\frac{-5}{3}+r,2-\frac{8}{3}r\right]\\ \left[0,0\right]& \left[0,0\right]& \left[1,1\right]\end{array}\left|\begin{array}{ccc}\left[-1+\frac{7}{12}r,\frac{-7}{12}r\right]& \left[1-\frac{5}{12}r,\frac{5}{12}r\right]& \left[0,0\right]\\ \left[\frac{4}{3}-\frac{2}{3}r,\frac{2}{3}r\right]& \left[\frac{-2}{3}+\frac{1}{3}r,\frac{-1}{3}r\right]& \left[0,0\right]\\ \left[1468-1064r,-1.300+924r\right]& \left[-1028+748r,908-648r\right]& \left[2448-1764r,-2208+1572r\right]\end{array}\right.\right]$$

The first row is subtracted using $\left[\frac{9}{4}-\frac{23}{24}r,\frac{-5}{4}+\frac{59}{24}r\right]\times b_3$ and the second row with $\left[\frac{-5}{3}+r,2-\frac{8}{3}r\right]\times b_3$, obtaining the following:

$$\left[\begin{array}{ccc}\left[1,1\right]& \left[0,0\right]& \left[0,0\right]\\ \left[0,0\right]& \left[1,1\right]& \left[0,0\right]\\ \left[0,0\right]& \left[0,0\right]& \left[1,1\right]\end{array}\left|\begin{array}{ccc}\left[-1850+1344r,1660-1190r\right]& \left[1296-945r,-1160+835r\right]& \left[-3084+2228r,2820-2024r\right]\\ \left[994-724r,-904+654r\right]& \left[-696+509r,632-459r\right]& \left[1656-1200r,-1536+1112r\right]\\ \left[1468-1064r,-1.300+924r\right]& \left[-1028+748r,908-648r\right]& \left[2448-1764r,-2208+1572r\right]\end{array}\right.\right]$$

Then, the inverse is obtained in parametric form, as follows:

$${\tilde{A}(r)}^{-1}=\left[\begin{array}{ccc}[-1850+1344r, 1660-1190r]& [1296-945r,-1160+835r]& [-3084+2228r, 2820-2024r]\\ [994-724r,-904+645r]& [-696+509r, 632-459r]& [1656-1200r,-1536-1112r]\\ [1468-1064r,-1300+924r]& [-1028+748r, 908-648r]& [2448-1764r,-2208+1572r]\end{array}\right]$$

Using Definition 3 (a), (b), and Theorem 1, it can easily be shown that $\tilde{A}(r)⨂{\tilde{A}(r)}^{-1}=\tilde{I}$. In this case, the results obtained are compatible.

Example 2. 

$\tilde{A}$ is considered to be a $2\times 3$ trapezoidal fuzzy matrix. The general inverse of $\tilde{A}$ is then calculated.

Let

$$\tilde{A}=\left[\begin{array}{ccc}(-3,-2, 1, 1)& (-2, 1, 2, 1)& (-4, 2, 1, 1)\\ (2, 3, 2, 1)& (-1, 3, 2, 2)& (-5, 3, 1, 2)\end{array}\right]$$

which, in interval form, can be written as follows:

$$\tilde{A}\left(r\right)=\left[\begin{array}{ccc}\left[-4+r,-1-r\right]& \left[-4+2r, 2-r\right]& \left[-5+r, 3-r\right]\\ \left[2r, 4-r\right]& \left[-3+2r, 5-2r\right]& \left[-6+r, 5-2r\right]\end{array}\right]$$

because the rank of $\tilde{A}$ = 2, with three minor matrices of $\tilde{A},$ is:

$${ \tilde{M}}_1(r)=\left[\begin{array}{cc}[-4+r,-1-r]& [-4+2r, 2-r]\\ [2r, 4-r]& [-3+2r, 5-2r]\end{array}\right]$$

$${ \tilde{M}}_2(r)=\left[\begin{array}{cc}[-4+2r, 2-r]& [-5+r, 3-r]\\ [-3+2r, 5-2r]& [-6+r, 5-2r]\end{array}\right]$$

$${ \tilde{M}}_3(r)=\left[\begin{array}{cc}[-4+r,-1-r]& [-5+r, 3-r]\\ [2r, 4-r]& [-6+r, 5-2r\end{array}\right]$$

This will result in three general inverses of the matrix of $\tilde{A}$. Using elementary row operations, the inverse of matrix ${ \tilde{M}}_1(r)$ is obtained:

$${\tilde{M}}_1^{-1}\left(r\right)=\left[\begin{array}{cc}\left[\frac{-196}{25}+\frac{88}{25}r, \frac{124}{25}-\frac{56}{25}r\right]& \left[\frac{-208}{25}+\frac{104}{25}r, \frac{152}{25}-\frac{68}{25}r\right]\\ \left[\frac{128}{5}-\frac{56}{5}r, \frac{-96}{5}+\frac{44}{5}r\right]& \left[\frac{144}{5}-\frac{56}{5}r, \frac{-108}{5}+\frac{44}{5}r\right]\end{array}\right]$$

Then, matrix $M_1^{-1}(r)$ is transposed to obtain:

$${\left({\tilde{M}}_1^{-1}(r)\right)}^t=\left[\begin{array}{cc}\left[\frac{-196}{25}+\frac{88}{25}r, \frac{124}{25}-\frac{56}{25}r\right]& \left[\frac{128}{5}-\frac{56}{5}r, \frac{-96}{5}+\frac{44}{5}r\right]\\ \left[\frac{-208}{25}+\frac{104}{25}r, \frac{152}{25}-\frac{68}{25}r\right]& \left[\frac{144}{5}-\frac{68}{5}r, \frac{-108}{5}+\frac{52}{5}r\right]\end{array}\right]$$

Furthermore, element [0, 0] is added to matrix ${\left({\tilde{M}}_1^{-1}(r)\right)}^t$ for anything other than the minor element of the fuzzy matrix, so that the matrix has the same order as $\tilde{A}(r)$. Hence, we obtain a new matrix denoted by $\tilde{W}(r)$:

$$\tilde{W}\left(r\right)=\left[\begin{array}{cc}\left[\frac{-196}{25}+\frac{88}{25}r, \frac{124}{25}-\frac{56}{25}r\right]& \left[\frac{128}{5}-\frac{56}{5}r, \frac{-96}{5}+\frac{49}{5}r\right] \left[0,0\right]\\ \left[\frac{-208}{25}+\frac{104}{25}r, \frac{152}{25}-\frac{68}{25}r\right]& \left[\frac{144}{5}-\frac{68}{5}r, \frac{-108}{5}+\frac{52}{5}r\right]\left[0,0\right]\end{array}\right]$$

Then, the $\tilde{W}$ matrix is transposed so that the ${\tilde{G}}_1({\tilde{W}}^t={\tilde{G}}_1)$ matrix is obtained:

$${\tilde{G}}_1\left(r\right)=\left[\begin{array}{cc}\left[\frac{-196}{25}+\frac{88}{25}r, \frac{124}{25}-\frac{56}{25}r\right]& \left[\frac{-208}{25}+\frac{104}{25}r, \frac{152}{25}-\frac{68}{25}r\right]\\ \left[\frac{128}{5}-\frac{56}{5}r, \frac{-96}{5}+\frac{44}{5}r\right]& \left[\frac{144}{5}-\frac{68}{5}r, \frac{-108}{5}+\frac{52}{5}r\right]\\ \left[0,0\right]& \left[0,0\right]\end{array}\right]$$

The ${\tilde{G}}_1(r)$ matrix is the general inverse of the matrix $\tilde{A}(r)$. Furthermore, using the matrix multiplication formula, it can easily be shown that $\tilde{A}{\left(r\right)\tilde{G}}_1\left(r\right)\tilde{A}\left(r\right)=\tilde{A}\left(r\right),$ which means that $\tilde{A}{\tilde{G}}_1\tilde{A}=\tilde{A}$. In a similar way, we can determine that the general inverses of${ \tilde{M}}_2\left(r\right)$and ${ \tilde{M}}_3(r)$ are ${\tilde{G}}_2(r)$ and ${\tilde{G}}_3\left(r\right),$ as follows:

$${\tilde{G}}_2\left(r\right)=\left[\begin{array}{cc}\left[0,0\right]& \left[0,0\right]\\ \left[\frac{-4}{3}, \frac{8}{9}-\frac{8}{9}r\right]& \left[\frac{2}{3}, \frac{4}{9}+\frac{2}{9}r\right]\\ \left[\frac{14}{3}-2r, \frac{-52}{9}+\frac{16}{9}r\right]& \left[\frac{-4}{3}+r, \frac{1}{9}-\frac{4}{9}r\right]\end{array}\right]$$

and

$${\tilde{G}}_3(r)=\left[\begin{array}{cc}\left[-\frac{17}{50}+\frac{1}{25}r,-\frac{1}{10}r\right]& \left[\frac{7}{50}-\frac{1}{25}r, \frac{2}{5}-\frac{1}{10}r\right]\\ \left[0,0\right]& \left[0,0\right]\\ \left[\frac{43}{20}-\frac{4}{5}r, \frac{-33}{10}+\frac{19}{20}r\right]& \left[\frac{27}{20}-\frac{1}{5}r, \frac{-27}{10}+\frac{11}{20}r\right]\end{array}\right]$$

It can also easily be shown that $\tilde{A}(r){\tilde{G}}_2(r)\tilde{A}(r)=\tilde{A}(r)$ and $\tilde{A}(r){\tilde{G}}_3(r)\tilde{A}(r)=\tilde{A}(r)$, meaning that $\tilde{A}{\tilde{G}}_2\tilde{A}=\tilde{A}$ and $\tilde{A}{\tilde{G}}_3\tilde{A}=\tilde{A}$. Likewise, for the three general inverses, the results obtained are compatible.

5. Conclusions

For two arbitrary trapezoidal fuzzy numbers$\tilde{p}=\left(a,b,\propto ,\beta \right)=(a-\left(1-r\right)\alpha ,b+\left(1-r\right)\beta =\left[\underset{\_}{p}\left(r\right),\overline{p}\left(r\right)=\tilde{p}\left(r\right)\right]$and $\tilde{q}=\left(c,d,\gamma ,\delta \right)=(c-\left(1-r\right)\gamma , d+\left(1-r\right)\delta =\left[\underset{\_}{q}\left(r\right),\overline{q}\left(r\right)\right]=\tilde{q}\left(r\right)$, the multiplication operation is the same as in Equation (1) in interval form; in the standard form, it is the same as in Equation (2). This result proves the existence of the inverse of $\tilde{p}\left(r\right),$ as given in Theorem 1, Corollary 1, and Corollary 2.

Until now, the elementary row operation could not be used for matrices in the form of trapezoidal fuzzy numbers; by using the algebraic operations given in this paper, the elementary row operation can be used to determine the inverse of any square matrix of trapezoidal fuzzy numbers. We found that $\tilde{A}\otimes {\tilde{A}}^{-1}={\tilde{A}}^{-1}\otimes \tilde{A}=\tilde{I}$. The multiplication and inverse formulas of trapezoidal fuzzy numbers given in this paper can also be used to determine the general inverse of an $m\times n$ trapezoidal fuzzy matrix, so that the following is obtained: $\tilde{A}\left(r\right)\otimes \tilde{G}\left(r\right)\otimes \tilde{A}\left(r\right)=\tilde{A}(r)$ or $\tilde{G}\left(r\right)\otimes \tilde{A}\left(r\right)\otimes \tilde{G}\left(r\right)=\tilde{G}\left(r\right).$ However, if we use the determinant method, it will only result in the following: $\tilde{A}(r){\tilde{A}(r)}^{-1}\otimes ={\tilde{A}(r)}^{-1}\otimes \tilde{A}(r)=\widetilde{I_f}(r)$. This result produces a compatible inverse, meaning that it is able to generate more accurate results when applied in various fields that use the inverse matrix of trapezoidal fuzzy numbers. The limitations of this research can be resolved in future research, specifically in studies related to the application of inverse trapezoidal fuzzy number matrices in industrial fields such as transportation.