A novel approach for solving fully fuzzy linear systems and their duality

Abstract

This paper deals with uncertain linear systems called fully fuzzy linear systems (FFLSs) and dual FFLSs. The aim is to solve FFLSs and their duality. To get the purpose, a new approach based on the relative-distance-measure fuzzy interval arithmetic (RDM-FIA) is proposed. So far, many approaches based on fuzzy standard interval arithmetic (FSIA) have been suggested for solving FFLSs, and dual FFLSs. However, the suggested approaches suffer from some limitations, e.g. unnatural behavior in modeling (UBM) phenomenon. The limitations are regarded as either the sign of fuzzy numbers or the type of fuzzy numbers considered in systems. However, in this paper, the proposed approach does not have the limitations. Using two theorems, the general form of solutions of FFLSs and dual FFLSs are presented. By a corollary it was demonstrated that a dual FFLS can be regarded as an FFLS. In addition, restrictions associated to the FSIA-based approaches dealing with the FFLSs and dual FFLSs were pointed out. Furthermore, the effectiveness and efficiency of the proposed approach are demonstrated using some comparative examples.

Keywords

Horizontal membership functions RDM arithmetic UBM phenomenon Granular difference Multidimensional RDM fuzzy arithmetic Fuzzy linear systems Fuzzy Arithmetic Granular Computing

1 Introduction

As a mathematical model of linear systems where uncertainty is considered, Fuzzy Linear Systems (FLSs) have attracted considerable attentions in recent years. An FLS is referred to as linear system in which at least one of parameters is a fuzzy number. FLSs are expressed in different forms. The simplest form of an FLS - i.e. ordinary FLS - is as $A \tilde{X} = \tilde{B}$ where A is an (n × n) crisp matrix and $\tilde{X}, \tilde{B}$ are (n × 1) fuzzy matrices. Other form of FLSs is called dual FLSs which is as $A \tilde{X} + \tilde{B} = C \tilde{X} + \tilde{D}$ where A, C are (n × n) crisp matrices, and $\tilde{X}, \tilde{B}, \tilde{D}$ are (n × 1) fuzzy matrices. Two important forms of FLSs are those in which all the matrices are fuzzy matrices. These two forms are expressed as $\tilde{A} \tilde{X} = \tilde{B}$ and $\tilde{A} \tilde{X} + \tilde{B} = \tilde{C} \tilde{X} + \tilde{D}$ . The first form is called Fully Fuzzy Linear Systems (FFLSs) and the latter is called dual FFLSs.

Due to the fact that, in real world applications determining all parameters of a linear system precisely may not be possible, it is immensely important to develop approaches dealing with FLSs. One of the most well-known work done on FLSs proposes that the (n × n) FLS $A \tilde{X} = \tilde{B}$ is replaced by a (2n × 2n) crisp linear system using embedding method introduced by Friedman et al. in [1]. Since the solution obtained from solving (2n × 2n) crisp linear system is not guaranteed to be a fuzzy solution - called also strong fuzzy solution - to the FLS, the so-called weak fuzzy solution was also proposed in [1]. Afterwards, many research works have been done based on the Friedman et al.’s method by this assumption that weak fuzzy solution always exists. However, using a counterexample in [2], it has been shown that the weak fuzzy solution cannot be guaranteed to exist.

In recent years, some considerable advances on FLSs have been achieved such as developing FLSs using type-2 fuzzy sets [3]. As a matter of fact, Najariyan et al. using the triangular perfect quasi type-2 fuzzy numbers [4, 5] introduced type-2 fuzzy linear systems, for the first time in the literature. Moreover, many approaches have been suggested for solving ordinary FLSs, FFLSs, and dual FFLSs. However, the suggested approaches suffer from some limitations. The limitations are regarded as either the sign of fuzzy numbers or the type of fuzzy numbers considered in systems. For example, the studies carried out in [6 –11] have been concentrated on FLSs in which all the elements are non-negative fuzzy numbers. For the second limitation, someone may refer to [12 –15] in which a special class of fuzzy numbers - triangular fuzzy numbers - was only considered, for more details see [16, 17]. A method was proposed in [18] for the algebraic solution of a complex interval linear system in which the coefficient matrix is a complex matrix and the right hand side vector is a complex interval vector. The present study assumed fuzzy numbers as triangular type. The perturbation of an (n × n) fuzzy linear system was analyzed by Liu et al. [19] who focused on fuzzy linear systems with triangular fuzzy numbers. At first, they used the Dehghan’s method to convert the original fuzzy linear systems into three crisp linear systems. Then, the state where the right hand side was somewhat perturbed was taken into consideration while the coefficient matrix was unchanged. Akram et al. [20] described some definite notions such as a bipolar fuzzy number in the parametric form, the interval between two bipolar fuzzy number, and bipolar fuzzy arithmetic. Linear equations and their solution process with right hand side vector were discussed as parametric bipolar fuzzy numbers.A new approach was presented in [21] to solve the static-structural problems based on Artificial Neural Network (ANN). An algorithm was proposed by Guo et al. [22] to evaluate an indefinite static response problem related to structures. A new numerical algorithm named hybrid perturbation Lagrange method (HPLM) was presented in this study through a combination of finite element analysis and a hybrid random convex model for addressing an indefinite static response problem related to structures though mixing random and convex variables.A decomposition technique was presented by Siahlooei et al. [23] to transform a fully fuzzy linear system into two kinds of decomposition in form of interval matrices.Malkawi et al. Introduced a new method to solve the FFLS with hexagonal fuzzy numbers [24]. Hewayda et al. [25] presented an extended embedding solution model called fuzzy compact storage Gauss–Seidel (FCGS) to solve the linear systems of equations with a fuzzy-based right-hand side.

Piegat with his co-workers introduced the concept of Horizontal Membership Functions (HMFs) in [26]. The HMFs are a new representation of the ordinary membership functions where some variables called Relative-Distance-Measure (RDM) variables [27] are involved. Based on the HMFs Piegat et al. [28] introduced a new approach in the context of fuzzy arithmetic called RDM Fuzzy Interval Arithmetic (RDM-FIA). On the basis of the results obtained in [29 –34], they showed that RDM-FIA is a more powerful tool than the conventional tool used for fuzzy arithmetic. Afterwards, Mazandarani et al. in [35] introduced the concepts of granular difference, granular metric, granular derivative and four basic operations based on RDM-FIA in a fashion way. They showed that fuzzy differential equations could be revolutionized using the approach proposed in their work. Mazandarani and Pariz also applied this kind of approach for obtaining a sub-optimal control of fuzzy linear dynamical systems [36]. Since then, Najariyan and Zhao in [37] has just extended the fuzzy fractional calculus based on granular differentiability concept and RDM-FIA.The problem of fuzzy time optimal control has been introduced in [38]. With gr-differentiability concept and RDM-FIA approach considered, Mazandarani in [38] proved that there is a unique control input transferring fuzzy states of a fuzzy dynamical system to the origin in a minimum fuzzy time.Recently, Mazandarani in [39], for the first time in the literature, introduced the concept of Z-differential equations which is a more general concept of FDEs. Actually, Pigat’s and Mazandarani’s works can be considered as a turning point in the field of fuzzy mathematics.

In this paper, a new approach based on HMFs and RDM-FIA is put forward to solve FFLSs and dual FFLSs. In the proposed approach, the four basic operations introduced in [35] are utilized. The paper is organized as follows: Section 2 presents some basic concepts concerning with fuzzy arithmetic. The new approach is introduced in Section 3. Some comparative examples are given in Section 4, and then the paper closes by conclusions presented in Section 5.

2 Basic concepts

Throughout this paper, the set of all real numbers is denoted by $ℝ$ , and the set of all the type-1 fuzzy numbers on $ℝ$ by E₁. The left and right end-points of μ-level sets of the fuzzy set $\tilde{A}$ , $[\tilde{A}]^{μ}$ , are denoted by ${\underline{A}}^{μ}$ and ${\bar{A}}^{μ}$ , respectively. The transpose of a matrix Y = [y_ij] _n×n, i, j = 1, . . . , n is denoted by Y^T.

Definition 1. [40]. The fuzzy set $\tilde{u} : ℝ \to [0, 1]$ is called a fuzzy number if it is normal, fuzzy convex, upper semi-continuous and compactly supported fuzzy subsets of the real numbers. The fuzzy number $\tilde{u}$ can be represented in a parametric form by the ordered pair of functions $({\underline{u}}^{μ}, {\bar{u}}^{μ})$ , 0 ≤ μ ≤ 1, satisfying the following properties:

${\underline{u}}^{μ}$ is a bounded non-decreasing left continuous function in (0, 1], and it is right continuous at μ = 0,

${\bar{u}}^{μ}$ is a bounded non-increasing left continuous function in (0, 1], and it is right continuous at μ = 0,

${\underline{u}}^{μ} \leq {\bar{u}}^{μ}$ .

Definition 2. Let $\tilde{u}, \tilde{v} \in E_{1}$ . The addition, subtraction, and multiplication of two fuzzy numbers $\tilde{u}$ and $\tilde{v}$ based on Fuzzy Standard Interval Arithmetic (FSIA) are characterized respectively as:

$[\tilde{u} + \tilde{v}]^{μ} = [{\underline{u}}^{μ} + {\underline{v}}^{μ}, {\bar{u}}^{μ} + {\bar{v}}^{μ}]$

$[\tilde{u} - \tilde{v}]^{μ} = [{\underline{u}}^{μ} - {\bar{v}}^{μ}, {\bar{u}}^{μ} - {\underline{v}}^{μ}]$

$[\tilde{u} \tilde{v}]^{μ} = [min {{\underline{u}}^{μ} {\bar{v}}^{μ}, {\underline{u}}^{μ} {\underline{v}}^{μ}, {\bar{u}}^{μ} {\bar{v}}^{μ}, {\bar{u}}^{μ} {\underline{v}}^{μ}},$

$max {{\underline{u}}^{μ} {\bar{v}}^{μ}, {\underline{u}}^{μ} {\underline{v}}^{μ}, {\bar{u}}^{μ} {\bar{v}}^{μ}, {\bar{u}}^{μ} {\underline{v}}^{μ}}]$ ;

Note 1. [35]. Let $\tilde{u}, \tilde{v}, \tilde{w} \in E_{1}$ . According to Definition 2, it can be proved that, as a whole, $(\tilde{u} + \tilde{v}) \tilde{w} \neq \tilde{u} \tilde{w} + \tilde{v} \tilde{w}$ and $\tilde{u} - \tilde{u} \neq 0$ .

Definition 3. [35]. Let $\tilde{u} : [a, b] \subseteq ℝ \to [0, 1]$ be a fuzzy number. The horizontal membership function u^gr : [0, 1] × [0, 1] → [a, b] is a representation of $\tilde{u} (x)$ as u^gr (μ, α_u) = x in which “gr” stands for the granule of information included in x ∈ [a, b], μ ∈ [0, 1] is the membership degree of x in $\tilde{u} (x)$ , α_u ∈ [0, 1] is called relative-distance-measure (RDM) variable, and $u^{gr} (μ, α_{u}) = {\underline{u}}^{μ} + ({\bar{u}}^{μ} - {\underline{u}}^{μ}) α_{u}$ .

Note 2. [35, 36]. For the sake of simplicity, the horizontal membership function of $\tilde{u} (x) \in E_{1}$ is also denoted by $H (\tilde{u}) ≜ u^{gr} (μ, α_{u})$ .

According to Definition 3, the horizontal membership function of the triangular fuzzy number $\tilde{u} \in E_{1}$ , denoted as the triple (a, b, c) , a ≤ b ≤ c, can be characterized as $H (\tilde{u}) = [a + (b - a) μ] + [(1 - μ) (c - a)] α_{u}$ . As an example, Fig. 1 shows the triangular fuzzy number $\tilde{u} = (3, 6, 10)$ and its horizontal membership function. In [41] the parametric form of horizontal membership function for an arbitrary trapezoidal fuzzy interval number has been given.

Fig.1

The triangular fuzzy number $\tilde{u} = (3, 6, 10)$ (a), and its horizontal membership function (b)[36].

Note 3. [35]. The μ-level sets of $\tilde{u} \in E_{1}$ which are in fact the span of the information granule can be obtained using $\begin{matrix} H^{- 1} (u^{gr} (μ, α_{u})) = [\tilde{u}]^{μ} \\ = [inf_{β \geq μ} min_{α_{u}} u^{gr} (β, α_{u}), sup_{β \geq μ} max_{α_{u}} u^{gr} (β, α_{u})] \end{matrix}$ (1)

What follows presents the four basic operations defined in RDM fuzzy interval arithmetic.

Definition 4. [35, 36]. Let $\tilde{u}$ and $\tilde{v}$ be two fuzzy numbers whose horizontal membership functions are u^gr (μ, α_u) and v^gr (μ, α_v), respectively, and “⊙” denote one of the four basic operations, i.e. addition, subtraction, multiplication, and division. Then, $\tilde{u} ⊙ \tilde{v}$ is a fuzzy number $\tilde{m}$ such that $H (\tilde{m}) ≜ u^{gr} (μ, α_{u}) ⊙ v^{gr} (μ, α_{v})$ . It should be noted that 0 ∉ v^gr (μ, α_v) when “⊙” denotes the division operator.

It should be noted that the difference between two fuzzy numbers defined in Definition 4 is called granular difference (gr-difference).

Proposition 1. Let $\tilde{m} = \tilde{u} ⊙ \tilde{v}$ . Then, $[\tilde{m}]^{μ} = H^{- 1} (u^{gr} (μ, α_{u}) ⊙ v^{gr} (μ, α_{v}))$ always presents μ-level sets of the fuzzy number $\tilde{m}$ .

Definition 5. [35, 36]. Two fuzzy numbers $\tilde{u}$ and $\tilde{v}$ are said to be equal if and only if $H (\tilde{u}) = H (\tilde{v})$ for all α_u = α_v ∈ [0, 1].

Note 4. [35, 36]. Let $\tilde{u}, \tilde{v}, \tilde{w} \in E_{1}$ . Then, based on Definitions 4 and 5, the following relations hold:

$\tilde{u} - \tilde{v} = - (\tilde{v} - \tilde{u})$ ,

$\tilde{u} - \tilde{u} = 0$ ,

$\tilde{u} \div \tilde{u} = 1$ ,

$(\tilde{u} + \tilde{v}) \tilde{w} = \tilde{u} \tilde{w} + \tilde{v} \tilde{w}$ .

Note 5. According to Note 4 and Definition 4, it is easy to conclude that a u = b v is equivalent to a.

3 Solving a fully fuzzy linear system and its duality

In this section two theorems concerning with solutions of FFLSs and dual FFLSs are presented. The theorems are based on the horizontal membership functions and RDM fuzzy interval arithmetic. Some necessary definitions are given, before introducing the theorems.

Definition 6. The matrix $\tilde{A} = [{\tilde{a}}_{ij}]_{n \times n}, i, j = 1, . . ., n;$ is called a fuzzy matrix if its entries, ${\tilde{a}}_{ij}$ , are fuzzy numbers.

Definition 7. The n × n fuzzy matrix ${\tilde{A}}^{- 1}$ is the inverse of the fuzzy matrix $\tilde{A} = [{\tilde{a}}_{ij}]_{n \times n}$ if and only if $H ({\tilde{A}}^{- 1}) H (\tilde{A}) = I_{n \times n}$ , where I_n×n is the well-known identity matrix.

Definition 8. Determinant of the fuzzy matrix $\tilde{A} = [{\tilde{a}}_{ij}]_{n \times n}$ denoted by $| \tilde{A} |$ is equal to the fuzzy number $\tilde{m}$ such that $H (\tilde{m}) = | H (\tilde{A}) |$ .

Definition 9. The multiplication of the n × n fuzzy matrix $\tilde{A}$ by n × m fuzzy matrix $\tilde{B}$ denoted as $\tilde{A} \tilde{B}$ is equal to the n × m fuzzy matrix $\tilde{M}$ , i.e. $\tilde{A} \tilde{B} = \tilde{M}$ , such that $H (\tilde{M}) = H (\tilde{A}) H (\tilde{B})$ .

Note 6. Since $H (I_{n \times n}) = I_{n \times n}$ , then based on Definitions 7 and 9 we can write ${\tilde{A}}^{- 1} \tilde{A} = \tilde{A} {\tilde{A}}^{- 1} = I_{n \times n}$ .

Definition 10. Consider the n × n linear system of equations: ${\begin{matrix} {\tilde{a}}_{11} {\tilde{x}}_{1} + {\tilde{a}}_{12} {\tilde{x}}_{2} + . . . + {\tilde{a}}_{1 n} {\tilde{x}}_{n} = {\tilde{b}}_{1} \\ {\tilde{a}}_{21} {\tilde{x}}_{1} + {\tilde{a}}_{22} {\tilde{x}}_{2} + . . . + {\tilde{a}}_{2 n} {\tilde{x}}_{n} = {\tilde{b}}_{2} \\ \cdot \\ \cdot \\ \cdot \\ {\tilde{a}}_{n 1} {\tilde{x}}_{1} + {\tilde{a}}_{n 2} {\tilde{x}}_{2} + . . . + {\tilde{a}}_{nn} {\tilde{x}}_{n} = {\tilde{b}}_{n} \end{matrix}$ (2) where ${\tilde{a}}_{ij}, {\tilde{b}}_{j} \in E_{1}, i, j = 1, . . ., n;$ are known, ∃j such that ${\tilde{b}}_{j} \neq 0$ , and the fuzzy matrix $\tilde{X} = [{\tilde{x}}_{j}]_{n \times 1}$ is unknown. The linear system (2) is called an FFLS. By $\tilde{A} = [{\tilde{a}}_{ij}]_{n \times n}$ and $\tilde{B} = [{\tilde{b}}_{j}]_{n \times 1}$ , the FFLS (2) is also can be written in a matrix form as $\tilde{A} \tilde{X} = \tilde{B}$ .

Definition 11. Consider the fuzzy matrices $\tilde{A} = [{\tilde{a}}_{ij}]_{n \times n}, \tilde{C} = [{\tilde{c}}_{ij}]_{n \times n}$ , $\tilde{B} = [{\tilde{b}}_{j}]_{n \times 1}$ and $\tilde{D} = [{\tilde{d}}_{j}]_{n \times 1}$ . A dual FFLS is an n × n linear system of equations as $\tilde{A} \tilde{X} + \tilde{B} = \tilde{C} \tilde{X} + \tilde{D}$ where ∃j such that ${\tilde{d}}_{j} - {\tilde{b}}_{j} \neq 0$ , and the fuzzy matrix $\tilde{X} = [{\tilde{x}}_{j}]_{n \times 1}$ is unknown.

Theorem 1. Consider the fully fuzzy linear system shown in (2). The fuzzy matrix $\tilde{X} = {\tilde{A}}^{- 1} \tilde{B}$ is the fuzzy solution of the FFLS if and only if $\frac{1}{| \tilde{A} |}$ exists.

Proof. Suppose $\tilde{X} = \tilde{C} \tilde{B}$ is the fuzzy solution of the FFLS $\tilde{A} \tilde{X} = \tilde{B}$ meaning $\tilde{A} \tilde{C} \tilde{B} = I_{n \times n} \tilde{B}$ . Based on Definition 5 $H (\tilde{A} \tilde{C} \tilde{B}) = H (I_{n \times n} \tilde{B}) = I_{n \times n} H (\tilde{B})$ . Then, using Definition 9 we have $H (\tilde{A} \tilde{C}) H (\tilde{B}) = I_{n \times n} H (\tilde{B})$ which reads $H (\tilde{A} \tilde{C}) = I_{n \times n}$ . We can also rewrite $H (\tilde{A} \tilde{C}) = I_{n \times n}$ as $H (\tilde{A}) H (\tilde{C}) = I_{n \times n}$ . Thus, based on Definition 7, $H (\tilde{C})$ must be equal to $H ({\tilde{A}}^{- 1})$ meaning $\frac{1}{| \tilde{A} |}$ exists. Now, suppose $\frac{1}{| \tilde{A} |}$ exists. According to Definition 5, $\tilde{A} \tilde{X} = \tilde{B}$ is equivalent to $H (\tilde{A} \tilde{X}) = H (\tilde{B})$ , and based on Definition 9 we have $H (\tilde{A}) H (\tilde{X}) = I_{n \times n} H (\tilde{B})$ . Since, $\frac{1}{| \tilde{A} |}$ exists, the fuzzy matrix $\tilde{A}$ is invertible, then using Note 6 we can write $H (\tilde{A}) H (\tilde{X}) = H (\tilde{A}) H ({\tilde{A}}^{- 1}) H (\tilde{B})$ which reads $H (\tilde{X}) = H ({\tilde{A}}^{- 1}) H (\tilde{B})$ . Therefore, according to proposition 1, $\tilde{X}$ is a fuzzy matrix whose entries are always fuzzy numbers, then $\tilde{X} = {\tilde{A}}^{- 1} \tilde{B}$ is fuzzy solution of fully fuzzy linear system shown in (2).

Theorem 1. Consider the dual fully fuzzy linear system defined in Definition 11. The fuzzy matrix $\tilde{X} = {(\tilde{A} - \tilde{C})}^{- 1} (\tilde{D} - \tilde{B})$ is the fuzzy solution of the dual FFLS if and only if $\frac{1}{| \tilde{A} - \tilde{C} |}$ exists.

Proof. Suppose $\frac{1}{| \tilde{A} - \tilde{C} |}$ exists. Then, based on Definition 4, $\tilde{A} \tilde{X} + \tilde{B} = \tilde{C} \tilde{X} + \tilde{D}$ is equivalent to $H (\tilde{A} \tilde{X} + \tilde{B}) = H (\tilde{C} \tilde{X} + \tilde{D})$ . Using Definition 4 we also can rewrite the system of equations as $H (\tilde{A} \tilde{X}) + H (\tilde{B}) = H (\tilde{C} \tilde{X}) + H (\tilde{D})$ . Thus, we have $H (\tilde{A} \tilde{X}) - H (\tilde{C} \tilde{X}) = H (\tilde{D}) - H (\tilde{B})$ . By considering $\tilde{M} = \tilde{A} \tilde{X}, \tilde{N} = \tilde{C} \tilde{X}$ , according to granular difference definition $H (\tilde{M}) - H (\tilde{N})$ is equal to $H (\tilde{M} - \tilde{N})$ , and $H (\tilde{D}) - H (\tilde{B})$ is also equal to $H (\tilde{D} - \tilde{B})$ . Based on case 4 in Note 4, we can write $\tilde{M} - \tilde{N} = (\tilde{A} - \tilde{C}) \tilde{X}$ which results in the system of equations $H ((\tilde{A} - \tilde{C}) \tilde{X}) = H (\tilde{D} - \tilde{B})$ . By Definition 9, $H ((\tilde{A} - \tilde{C}) \tilde{X}) = H (\tilde{A} - \tilde{C}) H (\tilde{X})$ . Since $\frac{1}{| \tilde{A} - \tilde{C} |}$ exists, then using Note 6 $H (\tilde{A} - \tilde{C}) H (\tilde{X}) = H (\tilde{A} - \tilde{C}) H ({(\tilde{A} - \tilde{C})}^{- 1}) H (\tilde{D} - \tilde{B})$ which reads $H (\tilde{X}) = H ({(\tilde{A} - \tilde{C})}^{- 1}) H (\tilde{D} - \tilde{B})$ , or equivalently - based on Definition 9 - $H (\tilde{X}) = H ({(\tilde{A} - \tilde{C})}^{- 1} (\tilde{D} - \tilde{B}))$ . Therefore, the first part of the proof is completed. The second part of the proof is similar what has been explained in the proof of Theorem 1, then it is omitted.

Corollary 1. Obtaining the solution of the dual FFLS $\tilde{A} \tilde{X} + \tilde{B} = \tilde{C} \tilde{X} + \tilde{D}$ is equivalent to finding the solution of the FFLS $(\tilde{A} - \tilde{C}) \tilde{X} = \tilde{D} - \tilde{B}$ .

Proof. Based on Theorems 1 and 2, the proof is straightforward and hence omitted.

Note 7. It should be noted that Theorems 1 and 2 include general forms of FFLSs and dual FFLSs. In other words, there is no restriction on the sign or type of fuzzy numbers involved in the FFLSs and their duality.

3.1 Restrictions associated to the use of FSIA for solving an FFLS and its duality

Investigating FFLSs and dual FFLSs under FSIA imposes some restrictions which are expressed in this section.

1. Multiplicity of the solutions for a dual FFLS

According to the UBM phenomenon explained in [35], as a whole, the solution of the following FFLSs ${\begin{matrix} \tilde{A} \tilde{X} + \tilde{B} = \tilde{C} \tilde{X} + \tilde{D} \\ \tilde{A} \tilde{X} - \tilde{D} = \tilde{C} \tilde{X} - \tilde{B} \\ \tilde{A} \tilde{X} + \tilde{B} - \tilde{D} = \tilde{C} \tilde{X} \\ \tilde{A} \tilde{X} = \tilde{C} \tilde{X} - \tilde{B} + \tilde{D} \\ \tilde{A} \tilde{X} - \tilde{C} \tilde{X} = \tilde{D} - \tilde{B} \\ \tilde{A} \tilde{X} - \tilde{C} \tilde{X} + \tilde{B} = \tilde{D} \\ \tilde{A} \tilde{X} - \tilde{C} \tilde{X} - \tilde{D} = - \tilde{B} \\ - \tilde{C} \tilde{X} + \tilde{B} = - \tilde{A} \tilde{X} + \tilde{D} \end{matrix}$ (3) are not the same, based on FSIA. Therefore, as a whole, obtaining the solution of the FFLSs (3) based on the approaches introduced in [6 –16] does not result in a same fuzzy solution for all the FFLSs (3). In other words, there are multiple solutions for a single uncertain linear system. Nevertheless, based on the proposed approach and RDM fuzzy interval arithmetic, the fuzzy linear systems (3) have the same fuzzy solution which can be characterized according to Theorem 2.

2. There is no fuzzy solution

Consider the following FFLS: $\tilde{A} \tilde{X} + \tilde{B} = 0$ (4) According to Note 1, since, as a whole, $\tilde{u} - \tilde{v} \neq 0$ , then by applying FSIA-based approaches, one cannot obtain the fuzzy solution of FFLS shown in (4). However, based on the note 4, the fuzzy solution of FFLS (4) can be characterized according to Theorem 1.

4 Examples

In this section some comparative examples are given to show the capability and the effectiveness of the proposed approach.

Example 1. Consider the following FFLS: $[\begin{matrix} 1 & \tilde{2} \\ - \tilde{3} & 3 \end{matrix}] [\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \end{matrix}] = [\begin{matrix} - \tilde{2} \\ \tilde{4} \end{matrix}]$ (5) where $\tilde{2} = (1, 2, 4), - \tilde{3} = (- 4, - 3, - 2), - \tilde{2} = (- 3, - 2, - 1)$ , and $\tilde{4} = (1, 4, 5)$ . The FFLS (5) is similar to the FFLS which has been considered in Example 2 presented in [13]. According to Theorem 3.2 in [13], FFLS (5) has no fuzzy solution. However, since $| \begin{matrix} 1 & \tilde{2} \\ - \tilde{3} & 3 \end{matrix} | \neq 0$ then, based on Theorem 1, FFLS (5) has a fuzzy solution which is equal to $\begin{matrix} \tilde{X} = {\tilde{A}}^{- 1} \tilde{B} = [\begin{matrix} - \tilde{1.55} \\ - \tilde{0.22} \end{matrix}] . \end{matrix}$ where $- {\underline{1.55}}^{μ} = \frac{- 23 + 11 μ - 2 μ^{2}}{8 + μ^{2}}$ , $- {\bar{1.55}}^{μ} = \frac{- 5 - 13 μ + 4 μ^{2}}{17 - 10 μ + 2 μ^{2}}$ , $\begin{matrix} - {\underline{0.22}}^{μ} = {\begin{matrix} \frac{- 11 + 10 μ - μ^{2}}{3 + 5 μ + μ^{2}} & 0 \leq μ \leq \frac{- 4 + \sqrt{28}}{2} \\ \frac{- 11 + 10 μ - μ^{2}}{3 + 5 μ + μ^{2}} & \frac{- 4 + \sqrt{28}}{2} \leq μ \leq 1 \end{matrix} \\ - {\bar{0.22}}^{μ} = {\begin{matrix} \frac{3 - 4 μ - μ^{2}}{3 + 5 μ + μ^{2}} & 0 \leq μ \leq \frac{- 4 + \sqrt{28}}{2} \\ \frac{3 - 4 μ - μ^{2}}{22 - 15 μ + 2 μ^{2}} & \frac{- 4 + \sqrt{28}}{2} \leq μ \leq 1 \end{matrix} \end{matrix}$ (6)

In addition, since FFLS (5) includes negative elements, then the approaches introduced in [6 –11] are unable to obtain the fuzzy solution of FFLS (5).

Example 2. Consider the following fuzzy linear system adopted from [2], $[\begin{matrix} 1 & 1 \\ 1 & 2 \end{matrix}] [\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \end{matrix}] = [\begin{matrix} - \tilde{9} \\ - \tilde{15} \end{matrix}]$ (7) where “ $- \tilde{9}$ ” and “ $- \tilde{15}$ ” are fuzzy numbers as follows: $\begin{matrix} - {\underline{9}}^{μ} = {\begin{matrix} 8 μ - 14 & 0 \leq μ \leq \frac{1}{2} \\ 2 μ - 11 & \frac{1}{2} \leq μ \leq 1 \end{matrix} \\ - {\bar{9}}^{μ} = {\begin{matrix} - 1 - 13 μ & 0 \leq μ \leq \frac{1}{2} \\ - 6 - 3 μ & \frac{1}{2} \leq μ \leq 1 \end{matrix} \end{matrix}$ (8)

$\begin{matrix} - {\underline{15}}^{μ} = {\begin{matrix} 12 μ - 24 & 0 \leq μ \leq \frac{1}{2} \\ 6 μ - 21 & \frac{1}{2} \leq μ \leq 1 \end{matrix} \\ - {\bar{15}}^{μ} = {\begin{matrix} - 2 - 18 μ & 0 \leq μ \leq \frac{1}{2} \\ - 7 - 8 μ & \frac{1}{2} \leq μ \leq 1 \end{matrix} \end{matrix}$ (9)

In [2], it has been proved that there is neither strong nor weak fuzzy solutions for FLS (7) based on Friedman et al.’s method proposed in [1]. Thus, all the approaches introduced in the literature which are on the basis of Friedman et al.’s method are unable to obtain the solution of FLS (7). Nonetheless, the fuzzy solution of FLS (7), based on Theorem 1 is as: $\begin{matrix} {\underline{x}}_{1}^{μ} = {\begin{matrix} 34 μ - 26 & 0 \leq μ \leq \frac{1}{2} \\ 12 μ - 15 & \frac{1}{2} \leq μ \leq 1 \end{matrix} \\ {\bar{x}}_{1}^{μ} = {\begin{matrix} 22 - 38 μ & 0 \leq μ \leq \frac{1}{2} \\ 9 - 12 μ & \frac{1}{2} \leq μ \leq 1 \end{matrix} \end{matrix}$ (10) $\begin{matrix} {\underline{x}}_{2}^{μ} = {\begin{matrix} 21 μ - 23 & 0 \leq μ \leq \frac{1}{2} \\ 5 μ - 15 & \frac{1}{2} \leq μ \leq 1 \end{matrix} \\ {\bar{x}}_{2}^{μ} = {\begin{matrix} 12 - 26 μ & 0 \leq μ \leq \frac{1}{2} \\ 4 - 14 μ & \frac{1}{2} \leq μ \leq 1 \end{matrix} \end{matrix}$ (11) It should be noted that the fuzzy numbers “ $- \tilde{9}$ ” and “ $- \tilde{15}$ ” are not triangular fuzzy numbers and many approaches introduced for solving FLSs – e.g. [12 –15] – are unable to obtain the solution of FLS (7).

Example 3. Consider a dual fully fuzzy linear system as: $[\begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}] [\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \end{matrix}] = [\begin{matrix} 0 & - \tilde{2} \\ \tilde{3} & 0 \end{matrix}] [\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \end{matrix}] + [\begin{matrix} - \tilde{2} \\ \tilde{4} \end{matrix}]$ (12) Based on Corollary 1, and Note 5, dual FFLS 12 can be rewritten as: $([\begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}] - [\begin{matrix} 0 & - \tilde{2} \\ \tilde{3} & 0 \end{matrix}]) [\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \end{matrix}] = [\begin{matrix} - \tilde{2} \\ \tilde{4} \end{matrix}]$ (13) or equivalently $[\begin{matrix} 1 & \tilde{2} \\ - \tilde{3} & 3 \end{matrix}] [\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \end{matrix}] = [\begin{matrix} - \tilde{2} \\ \tilde{4} \end{matrix}]$ (14) which is equal to FFLS shown in (5). Therefore, the solution of dual FFLS (12) is the same as that obtained in Example 1.

Example 4. Regard the electrical circuit indicated in Fig. 2 in which ${\tilde{v}}_{1}$ and ${\tilde{v}}_{2}$ represent the input voltages while ${\tilde{v}}_{3}$ and ${\tilde{v}}_{4}$ represent the output voltages [3]. The circuit is a type of summing amplifier having two inputs and two outputs. The relationship between the input and output voltages is shown below: $[\begin{matrix} 3 & 0.5 \\ - 2 & - 3 \end{matrix}] [\begin{matrix} {\tilde{v}}_{1} \\ {\tilde{v}}_{2} \end{matrix}] = [\begin{matrix} {\tilde{v}}_{3} \\ {\tilde{v}}_{4} \end{matrix}]$ (15)

Fig.2

Electrical circuit considered in Example 4 [3].

The problem that is evaluated here is specifying the input voltages provided that the output voltages are regarded as known but uncertain. That is, e.g., ${\tilde{v}}_{3}$ is “about 16 (volt)”, and ${\tilde{v}}_{4}$ is “about -16 (volt)”. where $\tilde{16} = (14, 16, 18)$ and $- \tilde{16} = (- 18, - 16, - 14)$ . since $\begin{matrix} | \begin{matrix} 3 & 0.5 \\ - 2 & - 3 \end{matrix} | \neq 0 \end{matrix}$ then, based on Theorem 1, FLS (15) has a fuzzy solution which is equal to $\begin{matrix} \tilde{V} = {\tilde{A}}^{- 1} \tilde{B} = [\begin{matrix} \tilde{5} \\ \tilde{2} \end{matrix}] . \end{matrix}$ In which $\tilde{5}$ and $\tilde{2}$ are indicated in Figures 3 and 4.

Fig.3

Input voltage, ${\tilde{v}}_{1}$ ,corresponding to the electrical circuit.

Fig.4

Input voltage, ${\tilde{v}}_{2}$ ,corresponding to the electrical circuit.

Example 5. [21, 42] In this example, a 6-bar truss structure is indicated in Fig. 5. The structure includes six elements. Material, geometric properties, and the applied load parameter $\tilde{P}$ are considered as uncertain. Table 1 indicates the values related to the input variables for the present analysis. The corresponding equilibrium equation for the existing problem is represented as $\tilde{K} \tilde{δ} = \tilde{F}$ (16)

Fig.5

Six-bar truss structure [21, 42].

Table 1

Input Variables of six bar truss with triangular fuzzy number

Parameters	Values in fuzzified form
Modulus of elasticity ${\tilde{E}}_{i}$ , for i=1 to 6 all elements (N/m²)	(2 × 10¹⁸, 2.1 × 10¹⁸, 2.2 × 10¹⁸)
Cross sectional area i ${\tilde{A}}_{i}$ , for i = 1 to 4 (m²)	(0.9 × 10^-3, 1 × 10^-3, 1.1 × 10^-3)
Cross sectional area of all other elements viz. ${\tilde{A}}_{5}$ and ${\tilde{A}}_{6}$ (m²)	(1 ×10^-3, 10.5 × 10^-3, 1.1 × 10^-3)
$\tilde{P}$ (kN)	(20, 20.5, 21)

In which $\tilde{K}$ , $\tilde{F}$ , and $\tilde{δ}$ respectively refer to the reduced stiffness matrix, load, and displacements vector. Here, $\tilde{K} = [\begin{matrix} \frac{{\tilde{E}}_{1} {\tilde{A}}_{1}}{L_{1}} + 0.36 \frac{{\tilde{E}}_{5} {\tilde{A}}_{5}}{L_{5}} & - 0.48 \frac{{\tilde{E}}_{5} {\tilde{A}}_{5}}{L_{5}} & - \frac{{\tilde{E}}_{1} {\tilde{A}}_{1}}{L_{1}} & 0 \\ - 0.48 \frac{{\tilde{E}}_{5} {\tilde{A}}_{5}}{L_{5}} & \frac{{\tilde{E}}_{3} {\tilde{A}}_{3}}{L_{3}} + 0.64 \frac{{\tilde{E}}_{5} {\tilde{A}}_{5}}{L_{5}} & 0 & 0 \\ - \frac{{\tilde{E}}_{1} {\tilde{A}}_{1}}{L_{1}} & 0 & \frac{{\tilde{E}}_{1} {\tilde{A}}_{1}}{L_{1}} + 0.36 \frac{{\tilde{E}}_{6} {\tilde{A}}_{6}}{L_{6}} & 0.48 \frac{{\tilde{E}}_{6} {\tilde{A}}_{6}}{L_{6}} \\ 0 & 0 & 0.48 \frac{{\tilde{E}}_{6} {\tilde{A}}_{6}}{L_{6}} & \frac{{\tilde{E}}_{4} {\tilde{A}}_{4}}{L_{4}} + 0.64 \frac{{\tilde{E}}_{6} {\tilde{A}}_{6}}{L_{6}} \end{matrix}]$ $\tilde{F} = [\begin{matrix} c \tilde{P} \\ 2 \tilde{P} \\ 2.5 \tilde{P} \\ - 1.5 \tilde{P} \end{matrix}]$ and $\tilde{δ} = [\begin{matrix} c {\tilde{x}}_{2} \\ {\tilde{y}}_{2} \\ {\tilde{x}}_{3} \\ {\tilde{y}}_{3} \end{matrix}]$

The determinant of the matrix $\tilde{K}$ is about 6.3 × 10²¹ with the membership function shown in Fig. 6 then, based on Theorem 1, FFLS (16) has a fuzzy solution which is equal to $\begin{matrix} \tilde{δ} = {\tilde{K}}^{- 1} \tilde{F} = [\begin{matrix} \tilde{0.86} \\ \tilde{0.33} \\ \tilde{0.9} \\ - \tilde{0.31} \end{matrix}] . \end{matrix}$ Where $\tilde{0.86}$ , $\tilde{0.33}$ , $\tilde{0.9}$ and $- \tilde{0.31}$ shown in Figures 7, 8, 9 and 10. The solutions and the results shown in figures 7 to 10 are good agreement with the figures in [21, 42].

Fig.6

The determinant of the matrix $\tilde{K}$ .

Fig.7

Horizontal displacement at node 2 for six bar truss structure.

Fig.8

Vertical displacement at node 2 for six bar truss structure.

Fig.9

Horizontal displacement at node 3 for six bar truss structure.

Fig.10

Vertical displacement at node 3 for six bar truss structure.

5 Conclusions

In this paper, two important forms of uncertain linear systems called fully fuzzy linear systems and their duality was considered to be solved. The new approach for solving the FFLS and dual FFLS was proposed based on RDM fuzzy interval arithmetic and operations put forward by Mazandarani in [35]. Two theorems have been presented based on which we showed the fuzzy solutions of FFLSs and dual FFLSs exist if and only if the coefficients fuzzy matrix is invertible. If the value of the determinant of coefficients matrix fuzzy is zero, this approach will face challenges. In the future scope we aim to investigate fuzzy linear systems, whose coefficients determent value is zero, using multi-granularity idea proposed in [43]. By Corollary 1 it was demonstrated that a dual FFLS can be regarded as an FFLS. In addition, restrictions associated to the FSIA-based approaches dealing with the FFLSs and dual FFLSs were pointed out.

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