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International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(3): 252-260

Published online September 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.3.252

© The Korean Institute of Intelligent Systems

Computational Technique for Solving Imprecisely Defined Non-negative Fully Fuzzy Algebraic System of Linear Equations

Diptiranjan Behera1 and S. Chakraverty2

1Department of Mathematics, The University of the West Indies, Mona Campus, Kingston, Jamaica
2Department of Mathematics, National Institute of Technology Rourkela, Odisha, India

Correspondence to :
Diptiranjan Behera (diptiranjanb@gmail.com)

Received: July 19, 2022; Revised: August 19, 2022; Accepted: August 22, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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This study proposes a new straightforward approach for solving a fully fuzzy algebraic system of linear equations. The proposed technique was based on a convex combination approach. Here, elements of the fuzzy coefficients matrix, fuzzy unknown vector, and right-hand side fuzzy vector are considered non-negative. Using the fuzzy arithmetic and convex combination concepts, the fuzzy system is converted into an equivalent crisp system. After solving the corresponding system for any two distinct parametric values of the convex combination, the final solution is obtained. Various example problems were solved and compared with existing results for validation.

Keywords: Fully fuzzy system of linear equations, Fuzzy number, Convex combination, ξ-cut

1. Introduction

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Fuzzy set theory is very useful for modelling in instances where there is less, incomplete, vague, or imprecise information about the variables or parameters. Accordingly, many physical or real-life problems, along with fuzzy uncertainty on many occasions, turn to fuzzy systems of linear equations (FSLE) or fully fuzzy systems of linear equations (FFSLE) when determining solutions. These systems have many applications in marketing, transportation, finance, and optimization. The basic difference between FSLE and FFSLE is that the element coefficient matrix, unknown vector, and right-hand side vector of FFSLE are fuzzy, whereas the elements of the coefficient matrix are crisp in FSLE.

Numerous studies have been conducted on FSLE and can be found in the literature [1–4]. Friedman et al. [1] proposed an embedding approach, along with the existence of a unique solution for a fuzzy linear system. Using the concept of fuzzy center and radius, Chakraverty and Behera [5] studied the solution procedure for FSLE under various types of fuzzy numbers, such as trapezoidal numbers. Ezzati [6] presented various theories on the existence and uniqueness of solutions for the FSLE. Recently, Mikaeilvand et al. [4] proposed a novel technique based on an embedding approach to examine FSLE. They mentioned that their method used fewer operations than the methods developed by Friedman et al. [1] and Ezzati [6]. Behera and Chakraverty [2] investigated a systematic solution procedure for real and complex fuzzy systems. Applications of FSLE have been illustrated by Behera and Chakraverty [7]. A static analysis of structural problems under fuzzy and interval loads was conducted.

Various iterative methods were applied by Dehgan and Hashemi [8] to analyze the FSLE. Garg and Singh [9] developed a numerical scheme for the solution of both linear and non-linear fuzzy systems, where uncertainties are modelled using Gaussian fuzzy numbers. Abdullah and Rahman [10] applied four different Jacobi-based iteration methods to solve the same type of system. Inearat and Qatanani [11] also used Jacobi, Gauss-Seidel, and successive over-relaxation iteration schemes, along with convergence analysis for FSLE. Islam et al. [12] used the matrix method to solve the trapezoidal FSLE. Jun [13] used an approximate method with a modification of the crisp Jacobi approach in the solution process to solve the FSLE.

In addition, various studies on FFSLE have recently been reported. Accordingly, Ezzati et al. [14] derived an approach for obtaining a positive solution for FFSLE using a fuzzy arithmetic approach. In their approach, they converted a fuzzy system into a crisp system for the solution. Behera and Chakraverty [15] proposed a double parametric approach for solving the FFSLE, where they considered only non-negative fuzzy numbers for the analysis. Behera et al. [16] proposed a new methodology for finding non-negative and non-positive FFSELE solutions using the core solution and linear programming approach. Otadi and Mosleh [17] investigated a non-negative solution of a fully fuzzy matrix equation using an optimization technique. Jafarian and Jafari [18] presented a new computational method for fully fuzzy non-linear matrix equations. Malkawi et al. [19] discussed the necessary and sufficient conditions to achieve a positive solution for the FFSLE. Recently, Abbasi and Allahviranloo [20] investigated and reported a new concept based on transmission-average-based operations for solving the FFSLE. In addition, the FFSLE with trapezoidal and hexagonal fuzzy numbers was studied by Ziqan et al. [21].

In addition, Akram et al. [22] recently studied new methods for solving the LR-bipolar FSLE. They also investigated the necessary and sufficient conditions for the solution. Najafi and Edalatpanah [23] numerically proposed better iterative algorithms to solve a class of fuzzy linear systems. Moreover, Edalatpanah [24] also solved the extension of a fuzzy set known as a neutrosophic-set-based linear system of equations. Boukezzoula et al. [25] proposed a new concept, known as the thick fuzzy set-based approach, for solving the fuzzy system of equations. A solution algorithm based on finding pseudo-solutions for systems of linear equations and corresponding software was developed by Panteleev and Saveleva [26] to solve a FFSLE.

The remainder of this paper is organized as follows: Section 1 provides an introduction and literature review. The basic definitions and properties related to fuzzy set theory are discussed in Section 2. Section 3 explains the general FFSLE and proposed method. Numerical examples are presented in Section 4. Finally, conclusions are presented in Section 5.

2. Preliminaries

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Here, some notations and definitions are provided concerning the presented study [27–30].

Definition 1

A fuzzy set Ω̃ in X ⊂ ℝ (ℝ represents the set of real numbers) is a set of ordered pairs, such that

Ω˜={(x,μΩ˜(x))∣x∈X},

where μΩ̃(x) is the membership function or grade of membership of x in Ω̃, which maps X to the membership space . The range of is a subset of non-negative real numbers whose supremum is finite. If sup(μΩ̃(x)) = 1, then Ω̃ is called a normalized fuzzy set.

Definition 2

A fuzzy number Ω̃ is a convex normalized fuzzy set Ω̃ of the real line ℝ, such that

μΩ˜(x):ℝ→[0,1],∀x∈ℝ,

where μΩ̃ (x) is called the membership function of the fuzzy set and is piecewise continuous.

Definition 3

A triangular fuzzy number Ω̃ = (ɛ, φ, λ) is a convex, normalized fuzzy set of the real-line ℝ , whose membership function is defined as

μΩ˜(x)={0,x≤ɛ,x-ɛφ-ɛ,ɛ≤x≤φ,x-λφ-λ,φ≤x≤λ,0,x≥λ.

Definition 4

An arbitrary triangular fuzzy number Ω̃ = (ɛ, φ, λ) can be expressed in interval form using the ξ-cut approach

Ω˜=Ω˜(ξ)=[Ω_(ξ),Ω¯(ξ)]=[(φ-ɛ)ξ+ɛ,(φ-λ)ξ+λ],

where ξ ∈ [0, 1].

Definition 5

The membership of a trapezoidal fuzzy number Ω̃ = (ɛ, φ, λ, ϑ) is defined as

μΩ˜(x)={0,x≤ɛ,x-ɛφ-ɛ,ɛ≤x≤φ,1,φ≤x≤λ,x-ϑφ-ϑ,λ≤x≤ϑ,0,x≥ϑ.

Definition 6

An arbitrary trapezoidal fuzzy number Ω̃ = (ɛ, φ, λ, ϑ) can be expressed in interval form using the ξ-cut approach

Ω˜=Ω˜(ξ)=[Ω_(ξ),Ω¯(ξ)]=[(φ-ɛ)ξ+ɛ,(λ-ϑ)ξ+ϑ],

where ξ ∈ [0, 1].

Note

In the above ξ-cut or parametric forms for triangular and trapezoidal fuzzy numbers, the following requirements should be satisfied

  • •Ω(ξ) is a bounded left continuous non-decreasing function over [0, 1].

  • •Ω̄(ξ) is bounded right continuous non-increasing function over [0, 1].

  • •Ω(ξ) ≤ Ω̄(ξ).

Definition 7

Let us consider two fuzzy numbers

Ω˜=Ω˜(ξ)=[Ω_(ξ),Ω¯(ξ)]   and ℑ˜=ℑ˜(ξ)=[ℑ_(ξ),ℑ˜(ξ)],

in the form of ξ – cut, then fuzzy arithmetic operations can be defined as

  • •Addition: .

  • •Subtraction: .

  • •Multiplication: , where

    ℘=[Ω_(ξ)×ℑ_(ξ),Ω¯(ξ)×ℑ¯(ξ),Ω_(ξ)×ℑ¯(ξ),Ω¯(ξ)×ℑ_(ξ)].

  • •Division: Ω˜ℑ˜=Ω˜(ξ)ℑ˜(ξ)=[Ω_(ξ)ℑ¯(ξ),Ω¯(ξ)ℑ_(ξ)], where .

  • •Scalar multiplication: For any scalar κ and fuzzy number Ω̃ = Ω̃ (ξ) = [Ω(ξ), Ω̄(ξ)], we have

    κΩ˜=κΩ˜(ξ)={[κΩ_(ξ),κΩ¯(ξ)],for κ≥0,[κΩ¯(ξ),κΩ_(ξ)],for κ<0.

Definition 8

A triangular fuzzy number Ω̃ = (ɛ, φ, λ) is non-negative if and only if ɛ ≥ 0, φ – ɛ ≥ 0, and λ – φ ≥ 0.

Definition 9

The trapezoidal fuzzy number Ω̃ = (ɛ, φ, λ, ϑ) is said to be non-negative if and only if ɛ ≥ 0, φ – ɛ ≥ 0, λ – φ ≥ 0 and ϑ – λ ≥ 0.

3. Fully Fuzzy System of Linear Equations and the Proposed Method

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Let us consider a n × n FSLE,

[A˜]{X˜}={b˜}.

Here, [Ã] = (ãkj) for 1 ≤ k ≤ n and 1 ≤ j ≤ n is an n × n fuzzy matrix, {b̃} = {b̃k} is a column vector of fuzzy numbers, and {X̃ } = {x̃j} is the vector of fuzzy unknowns. Here, ãkj, x̃j, and b̃k are non-negative ∀k and j.

With these expressions, Eq. (1) can be represented as

∑j=1na˜kjx˜j=b˜k, for k=1,⋯,n.

Using the parametric or ξ-cut form, Eq. (2) can be equivalently written as

∑j=1na˜kj(ξ)x˜j(ξ)=b˜k(ξ), for k=1,⋯,n,

where ãkj(ξ) = [akj(ξ), ākj(ξ)], x̃j(ξ) = [xj(ξ), χ̄j(ξ)] and b̃k(ξ) = [bk(ξ),b̄k(ξ)].

First, some important convex combination results are proposed and proven in terms of Theorems 1 and 2.

Theorem 1

If [Ã] {X̃} = {b̃ }, then

ψ (∑j=1na_kj(ξ)x_j(ξ))+(1-ψ) (∑j=1na¯kj(ξ)x¯j(ξ))=ψ (b_k(ξ))+(1-ψ) (b¯k(ξ)),

where ψ ∈ [0, 1].

Proof

Because [Ã]{X̃} = {b̃ }, this statement can be expressed as Eq. (3). Next, by applying fuzzy arithmetic, as defined in Section 2, Eq. (3) can be equivalently expressed as

[∑j=1na_kj(ξ)x_j(ξ),∑j=1na¯kj(ξ)x¯j(ξ)]=[b_k(ξ),b¯k(ξ)],         for k=1,⋯,n.

Equating the two sides gives

∑j=1na_kj(ξ)x_j(ξ)=b_k(ξ),∑j=1na¯kj(ξ)x¯j(ξ)=b¯k(ξ).

Now, consider the left-hand side of the equation, that is,

ψ (∑j=1na_kj(ξ)x_j(ξ))+(1-ψ) (∑j=1na¯kj(ξ)x¯j(ξ)).

In this expression, substituting the values of Eq. (5) gives

ψ (b_k(ξ))+(1-ψ) (b¯k(ξ)).

This expression represents the right side of the equation. Hence, the theorem is proven.

However, from here, it can be observed that the converse of the theorem is not always true. Therefore, Theorem 2 is proposed accordingly.

Theorem 2

The converse of Theorem 1 is true only when bk(ξ) ≤ b̄k(ξ).

Proof

Assume

ψ (∑j=1na_kj(ξ)x_j(ξ))+(1-ψ) (∑j=1na¯kj(ξ)x¯j(ξ))=ψ (b_k(ξ))+(1-ψ) (b¯k(ξ)).

The aim is to show that [Ã]{X̃} = {b̃}. Since ψ ∈ [0, 1], for ψ = 0 and ψ = 1, from Eq. (7), we obtain

∑j=1na¯kj(ξ)x¯j(ξ)=b¯k(ξ) and ∑j=1na_kj(ξ)x_j(ξ)=b_k(ξ),

respectively.

Now, the above expressions can be used to construct the interval form when bk(ξ) ≤ b̄k(ξ).

[∑j=1na_kj(ξ)x_j(ξ),∑j=1na¯kj(ξ)x¯j(ξ)]=[b_k(ξ),b¯k(ξ)].

However, this is equivalent to Eq. (1), that is, [Ã]{X̃} = {b̃}, as discussed above. Hence, the requirement is fulfilled, and the proof is complete.

3.1 Description of the Proposed Method

Since Theorem 1 is true, we have

ψ (∑j=1na_kj(ξ)x_j(ξ))+(1-ψ) (∑j=1na¯kj(ξ)x¯j(ξ))=ψ (b_k(ξ))+(1-ψ) (b¯k(ξ)),

where ψ ∈ [0, 1]. Accordingly, for any two distinct values ψ1, ψ2 of ψ ∈ [0, 1] from Eq. (9), we obtain

ψ1 (∑j=1na_kj(ξ)x_j(ξ))+(1-ψ1) (∑j=1na¯kj(ξ)x¯j(ξ))=ψ1 (b_k(ξ))+(1-ψ1) (b¯k(ξ)),ψ2 (∑j=1na_kj(ξ)x_j(ξ))+(1-ψ2) (∑j=1na¯kj(ξ)x¯j(ξ))=ψ2 (b_k(ξ))+(1-ψ2) (b¯k(ξ)).

Solving these crisp systems of equations numerically or analytically yields the values of xj(ξ) and χ̄j(ξ). Accordingly, the solution vector in Eq. (1) can be written parametrically as

[x_j(ξ),x¯j(ξ)]=x˜j(ξ).
Observation 1

For any two distinct values of ψ ∈ [0, tt1], we obtain the same solution.

In the next section, several examples are solved using the proposed method and compared with the existing results for validation.

4. Numerical Examples

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Example 1 ([31,32])

Let us consider a 2 × 2 triangular FFSLE as

(4,5,6)x˜1+(5,6,8)x˜2=(40,50,67),(6,7,7)x˜1+(4,4,5)x˜2=(43,48,55).

Next, using the ξ-cut approach, the above system can be written as

[4+ξ,6-ξ]x˜1(ξ)+[5+ξ,8-2ξ]x˜2(ξ)=[40+10ξ,67-17ξ],[6+ξ,7]x˜1(ξ)+[4,5-ξ]x˜2(ξ)=[43+5ξ,55-7ξ],

where x̃1(ξ) = [x1(ξ), χ̄1(ξ)] and x̃2(ξ) = [x2(ξ), χ̄2(ξ)].

Accordingly, using the fuzzy arithmetic, Eq. (12) can be expressed as

[(4+ξ)x_1(ξ)+(5+ξ)x_2(ξ),(6-ξ)x¯1(ξ)+(8-2ξ)x¯2(ξ)]=[40+10ξ,67-17ξ],[(6+ξ)x_1(ξ)+4x_2(ξ),7x¯1(ξ)+(5-ξ)x¯2(ξ)]=[43+5ξ,55-7ξ].

Next, by Theorem 1, the above system can be equivalently written in crisp form as

ψ [(4+ξ)x_1(ξ)+(5+ξ)x_2(ξ)]+(1-ψ) [(6-ξ)x¯1(ξ)+(8-2ξ)x¯2(ξ)]=ψ (40+10ξ)+(1-ψ) (67-17ξ),ψ [(6+ξ)x_1(ξ)+4x_2(ξ)]+(1-ψ) [7x¯1(ξ)+(5-ξ)x¯2(ξ)]=ψ(43+5ξ)+(1-ψ)(55-7ξ),

where ψ ∈ [0, 1]. Next, as per the proposed methods, consider any two distinct values, ψ1 and ψ2, of ψ ∈ [0, 1]. Let ψ1 = 0.1 and ψ2 = 0.2 and substitute them in the above system, that is, Eq. (14):

0.1 [(4+ξ)x_1(ξ)+(5+ξ)x_2(ξ)]+0.9 [(6-ξ)x¯1(ξ)+(8-2ξ)x¯2(ξ)]=0.1 (40+10ξ)+0.9 (67-17ξ),0.1 [(6+ξ)x_1(ξ)+4x_2(ξ)]+0.9 [7x¯1(ξ)+(5-ξ)x¯2(ξ)]=0.1(43+5ξ)+0.9(55-7ξ),

and

0.2 [(4+ξ)x_1(ξ)+(5+ξ)x_2(ξ)]+0.8 [(6-ξ)x¯1(ξ)+(8-2ξ)x¯2(ξ)]=0.2 (40+10ξ)+0.8 (67-17ξ),0.2 [(6+ξ)x_1(ξ)+4x_2(ξ)]+0.8 [7x¯1(ξ)+(5-ξ)x¯2(ξ)]=0.2(43+5ξ)+0.8(55-7ξ),

respectively. Finally, by solving these crisp systems simultaneously, we have

{x_1(ξ)x¯1(ξ)x_2(ξ)x¯2(ξ)}={5ξ2+28ξ+55ξ2+7ξ+143ξ2+14ξ-105ξ2+3ξ-265ξ2+37ξ+68ξ2+7ξ+147ξ2+22ξ-139ξ2+3ξ-26}.

Hence, the final solution in parametric form can be expressed as

x˜1(ξ)=[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26],

and

x˜2(ξ)=[5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26].

Next, we solve this problem using the proposed method for different sets of ψ1 and ψ2, as listed in Table 1. For each case, we obtained the same solution, which verifies the statement mentioned in Observation 1. It is worth mentioning that this example was also solved by other existing methods [15,31,32] and was in good agreement with them, as shown in Table 2.

Example 2 ([14,15])

In this example, let us consider a 3×3 triangular FFSLE as

(1,2,5)x˜1+(3,4,4)x˜2+(0,1,2)x˜3=(19,68,115),(2,3,5)x˜1+(0,1,11)x˜2+(4,5,6)x˜3=(30,77,261),(2,5,7)x˜1+(4,6,6)x˜2+(5,7,10)x˜3=(61,167,253).

An equivalent parametric form of the above system can be obtained as

[ξ+1,-3ξ+5]x˜1(ξ)+[ξ+3,4]x˜2(ξ)+[ξ,-ξ+2]x˜3(ξ)=[49ξ+19,-47ξ+115],[ξ+2,-2ξ+5]x˜1(ξ)+[ξ,-10ξ+11]x˜2(ξ)+[ξ+4,-ξ+6]x˜3(ξ)=[47ξ+30,-184ξ+261],[3ξ+2,-2ξ+7]x˜1(ξ)+[2ξ+4,6]x˜2(ξ)+[2ξ+5,-3ξ+10]x˜3(ξ)=[106ξ+61,-86ξ+253].

Using the proposed method, we obtained the solution in parametric form as

x˜1(ξ)=[4ξ+1,-2ξ+7],x˜2(ξ)=[6ξ+6,-2ξ+14], andx˜3(ξ)=[3ξ+7,-2ξ+12].

The results obtained by the present method are compared with those of Ezzati et al. [14] and Behera and Chakraverty [15], as listed in Table 3. The present solution is the same as that of the current results.

Example 3 ([15,33])

Here, a 3 × 3 trapezoidal FSLE in ξ-cut form is considered as

([0.3ξ+0.1,-0.3ξ+0.9][0.4ξ+1,-0.3ξ+1.9][0.19ξ+0.11,-0.4ξ+0.9][0.05ξ+0.1,-0.02ξ+0.2][0.03ξ+0.11,-0.02ξ+0.2][0.1ξ+6,-0.05ξ+6.2][0.1ξ+5,-0.2ξ+5.4][0.2ξ+0.1,-0.05ξ+0.4][0.09ξ+0.11,-0.1ξ+0.4])([x_1(ξ),x¯1(ξ)][x_2(ξ),x¯2(ξ)][x_3(ξ),x¯3(ξ)])=([0.09ξ+0.01,-0.05ξ+0.2][0.03ξ+0.11,-0.04ξ+0.2][0.04ξ+0.1,-0.04ξ+0.2]).

Using the proposed method, we have

{x_1(ξ)x¯1(ξ)x_2(ξ)x¯2(ξ)x_3(ξ)x¯3(ξ)}={125ξ3+7993ξ2-409017ξ+58202110(166ξ3-5386ξ2-1266749ξ-2987081)-(29ξ3+10856ξ2-123600ξ+352000)10(21ξ3-6281ξ2-226540ξ-1208000)17ξ3+482ξ2-478099ξ-363402(166ξ3-5386ξ2-1266749ξ-2987081)-2(94ξ3-4659ξ2-39150ξ+235000)5(21ξ3-6281ξ2-226540ξ-1208000)41ξ3-4387ξ2-31628ξ-53460166ξ3-5386ξ2-1266749ξ-298708133ξ3-1490ξ2+13500ξ-3480021ξ3-6281ξ2-226540ξ-1208000}.

These results are compared with the solutions of Das and Chakraverty [33] and Behera and Chakraverty [15], and the results are identical.

5. Conclusion

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Here, a new method is proposed to solve a non-negative, FFSLE using convex combinations. To the best of our knowledge, this concept has not been previously used to solve these types of fuzzy systems. Various example problems for triangular and trapezoidal fuzzy uncertainties were solved using the proposed method. In comparison, the present results are in good agreement with the results obtained by current methods. Indeed, this method is an excellent alternative to other existing methods. The claim is not that the present method always yields a fuzzy solution vector. However, the future aim is to extend the present idea to solving fuzzy differential equations, non-positive FFSLE, and a more generalized algebraic system of linear equations with fuzzy uncertainty.

Tables

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Table. 1.

Table 1. Solution of Example 1 obtained by the proposed method for different sets of ψ1 and ψ2.

ψ1ψ2Solution
x̃1(ξ)x̃2(ξ)
0.10.2[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]
0.30.9[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]
0.40.1[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]
0.1250.623[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]
01[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]
10[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]
0.50.7[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]
0.20.1[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]

Table. 2.

Table 2. Comparison of solutions obtained by the proposed method along with the methods for Example 1.

Solutionx̃1(ξ)x̃2(ξ)
Behera and Chakraverty [15][5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]
Allahviranloo and Mikaeilvand [31][5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]
Dehghan et al. [32][4311+ξ11,4][5411+ξ11,214-ξ4]
Proposed method[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]

Table. 3.

Table 3. Comparison of the solution obtained by the proposed method along with the methods for Example 2.

Solutionx̃1(ξ)x̃2(ξ)x̃3(ξ)
Ezzati et al. [14][4ξ + 1, −2ξ + 7][6ξ + 6, −2ξ + 14][3ξ + 7, −2ξ + 12]
Behera and Chakraverty [15][4ξ + 1, −2ξ + 7][6ξ + 6, −2ξ + 14][3ξ + 7, −2ξ + 12]
Proposed method[4ξ + 1, −2ξ + 7][6ξ + 6, −2ξ + 14][3ξ + 7, −2ξ + 12]

References

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Biographies

Go to

Diptiranjan Behera is currently working as a senior lecture of Mathematics at The University of the West Indies, Mona Campus, Kingston 7, Jamaica, where he has joined as a lecturer in 2018. Dr. Behera has also worked as a research faculty (assistant researcher) at the Center for System Reliability and Safety, University of Electronic Science and Technology of China, Chengdu, China from March, 2017 to April, 2018. Also he has worked there as a postdoctoral research fellow for 3 years from March, 2015 to April, 2018. Dr. Behera received his M.Sc. and Ph.D. degrees in Mathematics from National Institute of Technology Rourkela, Odisha, India in 2010 and 2015, respectively. He is the recipient of scholarship from China Post-doctoral Science Foundation, Govt. of P. R. China as well as National Postdoctoral Fellowship by Science and Educational Research Board, Department of Science and Technology, Govt. of India. He has published his research findings in many international journal of repute as well as he has involved himself for editorial activities and reviewing papers. His current research interest includes in the areas of interval and fuzzy mathematics, computational methods, fractional differential equations, non-linear differential equations, structural analysis and optimization problems.

E-mail: diptiranjanb@gamil.com


S. Chakraverty has 30 years of experience as a researcher and teacher. Presently he is working in the Department of Mathematics (Applied Mathematics Group), National Institute of Technology Rourkela, Odisha as a senior (higher administrative grade) professor. Prior to this, he was with CSIR-Central Building Research Institute, Roorkee, India. After completing Graduation from St. Columba’s College (Ranchi University), his career started from the University of Roorkee (Now, Indian Institute of Technology Roorkee) and did M.Sc. (Mathematics) and M.Phil. (Computer Applications) from there securing the first positions in the university. Dr. Chakraverty received his Ph.D. from IIT Roorkee in 1993. Thereafter he did his post-doctoral research at the Institute of Sound and Vibration Research (ISVR), University of Southampton, U.K. and at the Faculty of Engineering and Computer Science, Concordia University, Canada. He was also a visiting professor at Concordia and McGill universities, Canada, during 1997–1999 and visiting professor of University of Johannesburg, South Africa during 2011–2014. He has authored/co-authored/Edited 29 books, published 416 research papers (till date) in journals and conferences and two books are ongoing. He is in the Editorial Boards of various international journals, book series, and conferences. He is also the reviewer of around 50 national and international journals of repute. Prof. Chakraverty is a recipient of prestigious awards viz. Indian National Science Academy (INSA) nomination under International Collaboration/ Bilateral Exchange Program (with the Czech Republic), Platinum Jubilee ISCA Lecture Award (2014), CSIR Young Scientist Award (1997), BOYSCAST Fellow. (DST), UCOST Young Scientist Award (2007, 2008), Golden Jubilee Director’s (CBRI) Award (2001), INSA International Bilateral Exchange Award [2010–11 (selected but could not undertake), 2015 (selected)], Roorkee University Gold Medals (1987, 1988) for first positions in M.Sc. and M.Phil. (computer application), etc. He is in the list of 2% world scientists recently (2020, 2021) in artificial intelligence and image processing category based on an independent study done by Stanford University scientists. Also his paper got IOP publishing top cited paper award for one of the most cited articles from India. He has already guided nineteen 19 Ph.D. students and 12 are ongoing. Prof. Chakraverty has undertaken around 16 research projects as principle investigator funded by international and national agencies. His present research area includes differential equations (ordinary, partial, and fractional), numerical analysis and computational methods, structural dynamics (FGM, nano) and fluid dynamics, mathematical and uncertainty modeling, soft computing, and machine intelligence (artificial neural network, fuzzy, interval and affine computations).

E-mail: sne_chak@yahoo.com


Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(3): 252-260

Published online September 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.3.252

Copyright © The Korean Institute of Intelligent Systems.

Computational Technique for Solving Imprecisely Defined Non-negative Fully Fuzzy Algebraic System of Linear Equations

Diptiranjan Behera1 and S. Chakraverty2

1Department of Mathematics, The University of the West Indies, Mona Campus, Kingston, Jamaica
2Department of Mathematics, National Institute of Technology Rourkela, Odisha, India

Correspondence to:Diptiranjan Behera (diptiranjanb@gmail.com)

Received: July 19, 2022; Revised: August 19, 2022; Accepted: August 22, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This study proposes a new straightforward approach for solving a fully fuzzy algebraic system of linear equations. The proposed technique was based on a convex combination approach. Here, elements of the fuzzy coefficients matrix, fuzzy unknown vector, and right-hand side fuzzy vector are considered non-negative. Using the fuzzy arithmetic and convex combination concepts, the fuzzy system is converted into an equivalent crisp system. After solving the corresponding system for any two distinct parametric values of the convex combination, the final solution is obtained. Various example problems were solved and compared with existing results for validation.

Keywords: Fully fuzzy system of linear equations, Fuzzy number, Convex combination, &xi,-cut

1. Introduction

Fuzzy set theory is very useful for modelling in instances where there is less, incomplete, vague, or imprecise information about the variables or parameters. Accordingly, many physical or real-life problems, along with fuzzy uncertainty on many occasions, turn to fuzzy systems of linear equations (FSLE) or fully fuzzy systems of linear equations (FFSLE) when determining solutions. These systems have many applications in marketing, transportation, finance, and optimization. The basic difference between FSLE and FFSLE is that the element coefficient matrix, unknown vector, and right-hand side vector of FFSLE are fuzzy, whereas the elements of the coefficient matrix are crisp in FSLE.

Numerous studies have been conducted on FSLE and can be found in the literature [1–4]. Friedman et al. [1] proposed an embedding approach, along with the existence of a unique solution for a fuzzy linear system. Using the concept of fuzzy center and radius, Chakraverty and Behera [5] studied the solution procedure for FSLE under various types of fuzzy numbers, such as trapezoidal numbers. Ezzati [6] presented various theories on the existence and uniqueness of solutions for the FSLE. Recently, Mikaeilvand et al. [4] proposed a novel technique based on an embedding approach to examine FSLE. They mentioned that their method used fewer operations than the methods developed by Friedman et al. [1] and Ezzati [6]. Behera and Chakraverty [2] investigated a systematic solution procedure for real and complex fuzzy systems. Applications of FSLE have been illustrated by Behera and Chakraverty [7]. A static analysis of structural problems under fuzzy and interval loads was conducted.

Various iterative methods were applied by Dehgan and Hashemi [8] to analyze the FSLE. Garg and Singh [9] developed a numerical scheme for the solution of both linear and non-linear fuzzy systems, where uncertainties are modelled using Gaussian fuzzy numbers. Abdullah and Rahman [10] applied four different Jacobi-based iteration methods to solve the same type of system. Inearat and Qatanani [11] also used Jacobi, Gauss-Seidel, and successive over-relaxation iteration schemes, along with convergence analysis for FSLE. Islam et al. [12] used the matrix method to solve the trapezoidal FSLE. Jun [13] used an approximate method with a modification of the crisp Jacobi approach in the solution process to solve the FSLE.

In addition, various studies on FFSLE have recently been reported. Accordingly, Ezzati et al. [14] derived an approach for obtaining a positive solution for FFSLE using a fuzzy arithmetic approach. In their approach, they converted a fuzzy system into a crisp system for the solution. Behera and Chakraverty [15] proposed a double parametric approach for solving the FFSLE, where they considered only non-negative fuzzy numbers for the analysis. Behera et al. [16] proposed a new methodology for finding non-negative and non-positive FFSELE solutions using the core solution and linear programming approach. Otadi and Mosleh [17] investigated a non-negative solution of a fully fuzzy matrix equation using an optimization technique. Jafarian and Jafari [18] presented a new computational method for fully fuzzy non-linear matrix equations. Malkawi et al. [19] discussed the necessary and sufficient conditions to achieve a positive solution for the FFSLE. Recently, Abbasi and Allahviranloo [20] investigated and reported a new concept based on transmission-average-based operations for solving the FFSLE. In addition, the FFSLE with trapezoidal and hexagonal fuzzy numbers was studied by Ziqan et al. [21].

In addition, Akram et al. [22] recently studied new methods for solving the LR-bipolar FSLE. They also investigated the necessary and sufficient conditions for the solution. Najafi and Edalatpanah [23] numerically proposed better iterative algorithms to solve a class of fuzzy linear systems. Moreover, Edalatpanah [24] also solved the extension of a fuzzy set known as a neutrosophic-set-based linear system of equations. Boukezzoula et al. [25] proposed a new concept, known as the thick fuzzy set-based approach, for solving the fuzzy system of equations. A solution algorithm based on finding pseudo-solutions for systems of linear equations and corresponding software was developed by Panteleev and Saveleva [26] to solve a FFSLE.

The remainder of this paper is organized as follows: Section 1 provides an introduction and literature review. The basic definitions and properties related to fuzzy set theory are discussed in Section 2. Section 3 explains the general FFSLE and proposed method. Numerical examples are presented in Section 4. Finally, conclusions are presented in Section 5.

2. Preliminaries

Here, some notations and definitions are provided concerning the presented study [27–30].

Definition 1

A fuzzy set Ω̃ in X ⊂ ℝ (ℝ represents the set of real numbers) is a set of ordered pairs, such that

Ω˜={(x,μΩ˜(x))∣x∈X},

where μΩ̃(x) is the membership function or grade of membership of x in Ω̃, which maps X to the membership space . The range of is a subset of non-negative real numbers whose supremum is finite. If sup(μΩ̃(x)) = 1, then Ω̃ is called a normalized fuzzy set.

Definition 2

A fuzzy number Ω̃ is a convex normalized fuzzy set Ω̃ of the real line ℝ, such that

μΩ˜(x):ℝ→[0,1],∀x∈ℝ,

where μΩ̃ (x) is called the membership function of the fuzzy set and is piecewise continuous.

Definition 3

A triangular fuzzy number Ω̃ = (ɛ, φ, λ) is a convex, normalized fuzzy set of the real-line ℝ , whose membership function is defined as

μΩ˜(x)={0,x≤ɛ,x-ɛφ-ɛ,ɛ≤x≤φ,x-λφ-λ,φ≤x≤λ,0,x≥λ.

Definition 4

An arbitrary triangular fuzzy number Ω̃ = (ɛ, φ, λ) can be expressed in interval form using the ξ-cut approach

Ω˜=Ω˜(ξ)=[Ω_(ξ),Ω¯(ξ)]=[(φ-ɛ)ξ+ɛ,(φ-λ)ξ+λ],

where ξ ∈ [0, 1].

Definition 5

The membership of a trapezoidal fuzzy number Ω̃ = (ɛ, φ, λ, ϑ) is defined as

μΩ˜(x)={0,x≤ɛ,x-ɛφ-ɛ,ɛ≤x≤φ,1,φ≤x≤λ,x-ϑφ-ϑ,λ≤x≤ϑ,0,x≥ϑ.

Definition 6

An arbitrary trapezoidal fuzzy number Ω̃ = (ɛ, φ, λ, ϑ) can be expressed in interval form using the ξ-cut approach

Ω˜=Ω˜(ξ)=[Ω_(ξ),Ω¯(ξ)]=[(φ-ɛ)ξ+ɛ,(λ-ϑ)ξ+ϑ],

where ξ ∈ [0, 1].

Note

In the above ξ-cut or parametric forms for triangular and trapezoidal fuzzy numbers, the following requirements should be satisfied

  • •Ω(ξ) is a bounded left continuous non-decreasing function over [0, 1].

  • •Ω̄(ξ) is bounded right continuous non-increasing function over [0, 1].

  • •Ω(ξ) ≤ Ω̄(ξ).

Definition 7

Let us consider two fuzzy numbers

Ω˜=Ω˜(ξ)=[Ω_(ξ),Ω¯(ξ)]   and ℑ˜=ℑ˜(ξ)=[ℑ_(ξ),ℑ˜(ξ)],

in the form of ξ – cut, then fuzzy arithmetic operations can be defined as

  • •Addition: .

  • •Subtraction: .

  • •Multiplication: , where

    ℘=[Ω_(ξ)×ℑ_(ξ),Ω¯(ξ)×ℑ¯(ξ),Ω_(ξ)×ℑ¯(ξ),Ω¯(ξ)×ℑ_(ξ)].

  • •Division: Ω˜ℑ˜=Ω˜(ξ)ℑ˜(ξ)=[Ω_(ξ)ℑ¯(ξ),Ω¯(ξ)ℑ_(ξ)], where .

  • •Scalar multiplication: For any scalar κ and fuzzy number Ω̃ = Ω̃ (ξ) = [Ω(ξ), Ω̄(ξ)], we have

    κΩ˜=κΩ˜(ξ)={[κΩ_(ξ),κΩ¯(ξ)],for κ≥0,[κΩ¯(ξ),κΩ_(ξ)],for κ<0.

Definition 8

A triangular fuzzy number Ω̃ = (ɛ, φ, λ) is non-negative if and only if ɛ ≥ 0, φ – ɛ ≥ 0, and λ – φ ≥ 0.

Definition 9

The trapezoidal fuzzy number Ω̃ = (ɛ, φ, λ, ϑ) is said to be non-negative if and only if ɛ ≥ 0, φ – ɛ ≥ 0, λ – φ ≥ 0 and ϑ – λ ≥ 0.

3. Fully Fuzzy System of Linear Equations and the Proposed Method

Let us consider a n × n FSLE,

[A˜]{X˜}={b˜}.

Here, [Ã] = (ãkj) for 1 ≤ k ≤ n and 1 ≤ j ≤ n is an n × n fuzzy matrix, {b̃} = {b̃k} is a column vector of fuzzy numbers, and {X̃ } = {x̃j} is the vector of fuzzy unknowns. Here, ãkj, x̃j, and b̃k are non-negative ∀k and j.

With these expressions, Eq. (1) can be represented as

∑j=1na˜kjx˜j=b˜k, for k=1,⋯,n.

Using the parametric or ξ-cut form, Eq. (2) can be equivalently written as

∑j=1na˜kj(ξ)x˜j(ξ)=b˜k(ξ), for k=1,⋯,n,

where ãkj(ξ) = [akj(ξ), ākj(ξ)], x̃j(ξ) = [xj(ξ), χ̄j(ξ)] and b̃k(ξ) = [bk(ξ),b̄k(ξ)].

First, some important convex combination results are proposed and proven in terms of Theorems 1 and 2.

Theorem 1

If [Ã] {X̃} = {b̃ }, then

ψ (∑j=1na_kj(ξ)x_j(ξ))+(1-ψ) (∑j=1na¯kj(ξ)x¯j(ξ))=ψ (b_k(ξ))+(1-ψ) (b¯k(ξ)),

where ψ ∈ [0, 1].

Proof

Because [Ã]{X̃} = {b̃ }, this statement can be expressed as Eq. (3). Next, by applying fuzzy arithmetic, as defined in Section 2, Eq. (3) can be equivalently expressed as

[∑j=1na_kj(ξ)x_j(ξ),∑j=1na¯kj(ξ)x¯j(ξ)]=[b_k(ξ),b¯k(ξ)],         for k=1,⋯,n.

Equating the two sides gives

∑j=1na_kj(ξ)x_j(ξ)=b_k(ξ),∑j=1na¯kj(ξ)x¯j(ξ)=b¯k(ξ).

Now, consider the left-hand side of the equation, that is,

ψ (∑j=1na_kj(ξ)x_j(ξ))+(1-ψ) (∑j=1na¯kj(ξ)x¯j(ξ)).

In this expression, substituting the values of Eq. (5) gives

ψ (b_k(ξ))+(1-ψ) (b¯k(ξ)).

This expression represents the right side of the equation. Hence, the theorem is proven.

However, from here, it can be observed that the converse of the theorem is not always true. Therefore, Theorem 2 is proposed accordingly.

Theorem 2

The converse of Theorem 1 is true only when bk(ξ) ≤ b̄k(ξ).

Proof

Assume

ψ (∑j=1na_kj(ξ)x_j(ξ))+(1-ψ) (∑j=1na¯kj(ξ)x¯j(ξ))=ψ (b_k(ξ))+(1-ψ) (b¯k(ξ)).

The aim is to show that [Ã]{X̃} = {b̃}. Since ψ ∈ [0, 1], for ψ = 0 and ψ = 1, from Eq. (7), we obtain

∑j=1na¯kj(ξ)x¯j(ξ)=b¯k(ξ) and ∑j=1na_kj(ξ)x_j(ξ)=b_k(ξ),

respectively.

Now, the above expressions can be used to construct the interval form when bk(ξ) ≤ b̄k(ξ).

[∑j=1na_kj(ξ)x_j(ξ),∑j=1na¯kj(ξ)x¯j(ξ)]=[b_k(ξ),b¯k(ξ)].

However, this is equivalent to Eq. (1), that is, [Ã]{X̃} = {b̃}, as discussed above. Hence, the requirement is fulfilled, and the proof is complete.

3.1 Description of the Proposed Method

Since Theorem 1 is true, we have

ψ (∑j=1na_kj(ξ)x_j(ξ))+(1-ψ) (∑j=1na¯kj(ξ)x¯j(ξ))=ψ (b_k(ξ))+(1-ψ) (b¯k(ξ)),

where ψ ∈ [0, 1]. Accordingly, for any two distinct values ψ1, ψ2 of ψ ∈ [0, 1] from Eq. (9), we obtain

ψ1 (∑j=1na_kj(ξ)x_j(ξ))+(1-ψ1) (∑j=1na¯kj(ξ)x¯j(ξ))=ψ1 (b_k(ξ))+(1-ψ1) (b¯k(ξ)),ψ2 (∑j=1na_kj(ξ)x_j(ξ))+(1-ψ2) (∑j=1na¯kj(ξ)x¯j(ξ))=ψ2 (b_k(ξ))+(1-ψ2) (b¯k(ξ)).

Solving these crisp systems of equations numerically or analytically yields the values of xj(ξ) and χ̄j(ξ). Accordingly, the solution vector in Eq. (1) can be written parametrically as

[x_j(ξ),x¯j(ξ)]=x˜j(ξ).
Observation 1

For any two distinct values of ψ ∈ [0, tt1], we obtain the same solution.

In the next section, several examples are solved using the proposed method and compared with the existing results for validation.

4. Numerical Examples

Example 1 ([31,32])

Let us consider a 2 × 2 triangular FFSLE as

(4,5,6)x˜1+(5,6,8)x˜2=(40,50,67),(6,7,7)x˜1+(4,4,5)x˜2=(43,48,55).

Next, using the ξ-cut approach, the above system can be written as

[4+ξ,6-ξ]x˜1(ξ)+[5+ξ,8-2ξ]x˜2(ξ)=[40+10ξ,67-17ξ],[6+ξ,7]x˜1(ξ)+[4,5-ξ]x˜2(ξ)=[43+5ξ,55-7ξ],

where x̃1(ξ) = [x1(ξ), χ̄1(ξ)] and x̃2(ξ) = [x2(ξ), χ̄2(ξ)].

Accordingly, using the fuzzy arithmetic, Eq. (12) can be expressed as

[(4+ξ)x_1(ξ)+(5+ξ)x_2(ξ),(6-ξ)x¯1(ξ)+(8-2ξ)x¯2(ξ)]=[40+10ξ,67-17ξ],[(6+ξ)x_1(ξ)+4x_2(ξ),7x¯1(ξ)+(5-ξ)x¯2(ξ)]=[43+5ξ,55-7ξ].

Next, by Theorem 1, the above system can be equivalently written in crisp form as

ψ [(4+ξ)x_1(ξ)+(5+ξ)x_2(ξ)]+(1-ψ) [(6-ξ)x¯1(ξ)+(8-2ξ)x¯2(ξ)]=ψ (40+10ξ)+(1-ψ) (67-17ξ),ψ [(6+ξ)x_1(ξ)+4x_2(ξ)]+(1-ψ) [7x¯1(ξ)+(5-ξ)x¯2(ξ)]=ψ(43+5ξ)+(1-ψ)(55-7ξ),

where ψ ∈ [0, 1]. Next, as per the proposed methods, consider any two distinct values, ψ1 and ψ2, of ψ ∈ [0, 1]. Let ψ1 = 0.1 and ψ2 = 0.2 and substitute them in the above system, that is, Eq. (14):

0.1 [(4+ξ)x_1(ξ)+(5+ξ)x_2(ξ)]+0.9 [(6-ξ)x¯1(ξ)+(8-2ξ)x¯2(ξ)]=0.1 (40+10ξ)+0.9 (67-17ξ),0.1 [(6+ξ)x_1(ξ)+4x_2(ξ)]+0.9 [7x¯1(ξ)+(5-ξ)x¯2(ξ)]=0.1(43+5ξ)+0.9(55-7ξ),

and

0.2 [(4+ξ)x_1(ξ)+(5+ξ)x_2(ξ)]+0.8 [(6-ξ)x¯1(ξ)+(8-2ξ)x¯2(ξ)]=0.2 (40+10ξ)+0.8 (67-17ξ),0.2 [(6+ξ)x_1(ξ)+4x_2(ξ)]+0.8 [7x¯1(ξ)+(5-ξ)x¯2(ξ)]=0.2(43+5ξ)+0.8(55-7ξ),

respectively. Finally, by solving these crisp systems simultaneously, we have

{x_1(ξ)x¯1(ξ)x_2(ξ)x¯2(ξ)}={5ξ2+28ξ+55ξ2+7ξ+143ξ2+14ξ-105ξ2+3ξ-265ξ2+37ξ+68ξ2+7ξ+147ξ2+22ξ-139ξ2+3ξ-26}.

Hence, the final solution in parametric form can be expressed as

x˜1(ξ)=[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26],

and

x˜2(ξ)=[5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26].

Next, we solve this problem using the proposed method for different sets of ψ1 and ψ2, as listed in Table 1. For each case, we obtained the same solution, which verifies the statement mentioned in Observation 1. It is worth mentioning that this example was also solved by other existing methods [15,31,32] and was in good agreement with them, as shown in Table 2.

Example 2 ([14,15])

In this example, let us consider a 3×3 triangular FFSLE as

(1,2,5)x˜1+(3,4,4)x˜2+(0,1,2)x˜3=(19,68,115),(2,3,5)x˜1+(0,1,11)x˜2+(4,5,6)x˜3=(30,77,261),(2,5,7)x˜1+(4,6,6)x˜2+(5,7,10)x˜3=(61,167,253).

An equivalent parametric form of the above system can be obtained as

[ξ+1,-3ξ+5]x˜1(ξ)+[ξ+3,4]x˜2(ξ)+[ξ,-ξ+2]x˜3(ξ)=[49ξ+19,-47ξ+115],[ξ+2,-2ξ+5]x˜1(ξ)+[ξ,-10ξ+11]x˜2(ξ)+[ξ+4,-ξ+6]x˜3(ξ)=[47ξ+30,-184ξ+261],[3ξ+2,-2ξ+7]x˜1(ξ)+[2ξ+4,6]x˜2(ξ)+[2ξ+5,-3ξ+10]x˜3(ξ)=[106ξ+61,-86ξ+253].

Using the proposed method, we obtained the solution in parametric form as

x˜1(ξ)=[4ξ+1,-2ξ+7],x˜2(ξ)=[6ξ+6,-2ξ+14], andx˜3(ξ)=[3ξ+7,-2ξ+12].

The results obtained by the present method are compared with those of Ezzati et al. [14] and Behera and Chakraverty [15], as listed in Table 3. The present solution is the same as that of the current results.

Example 3 ([15,33])

Here, a 3 × 3 trapezoidal FSLE in ξ-cut form is considered as

([0.3ξ+0.1,-0.3ξ+0.9][0.4ξ+1,-0.3ξ+1.9][0.19ξ+0.11,-0.4ξ+0.9][0.05ξ+0.1,-0.02ξ+0.2][0.03ξ+0.11,-0.02ξ+0.2][0.1ξ+6,-0.05ξ+6.2][0.1ξ+5,-0.2ξ+5.4][0.2ξ+0.1,-0.05ξ+0.4][0.09ξ+0.11,-0.1ξ+0.4])([x_1(ξ),x¯1(ξ)][x_2(ξ),x¯2(ξ)][x_3(ξ),x¯3(ξ)])=([0.09ξ+0.01,-0.05ξ+0.2][0.03ξ+0.11,-0.04ξ+0.2][0.04ξ+0.1,-0.04ξ+0.2]).

Using the proposed method, we have

{x_1(ξ)x¯1(ξ)x_2(ξ)x¯2(ξ)x_3(ξ)x¯3(ξ)}={125ξ3+7993ξ2-409017ξ+58202110(166ξ3-5386ξ2-1266749ξ-2987081)-(29ξ3+10856ξ2-123600ξ+352000)10(21ξ3-6281ξ2-226540ξ-1208000)17ξ3+482ξ2-478099ξ-363402(166ξ3-5386ξ2-1266749ξ-2987081)-2(94ξ3-4659ξ2-39150ξ+235000)5(21ξ3-6281ξ2-226540ξ-1208000)41ξ3-4387ξ2-31628ξ-53460166ξ3-5386ξ2-1266749ξ-298708133ξ3-1490ξ2+13500ξ-3480021ξ3-6281ξ2-226540ξ-1208000}.

These results are compared with the solutions of Das and Chakraverty [33] and Behera and Chakraverty [15], and the results are identical.

5. Conclusion

Here, a new method is proposed to solve a non-negative, FFSLE using convex combinations. To the best of our knowledge, this concept has not been previously used to solve these types of fuzzy systems. Various example problems for triangular and trapezoidal fuzzy uncertainties were solved using the proposed method. In comparison, the present results are in good agreement with the results obtained by current methods. Indeed, this method is an excellent alternative to other existing methods. The claim is not that the present method always yields a fuzzy solution vector. However, the future aim is to extend the present idea to solving fuzzy differential equations, non-positive FFSLE, and a more generalized algebraic system of linear equations with fuzzy uncertainty.

Table 1 . Solution of Example 1 obtained by the proposed method for different sets of ψ1 and ψ2.

ψ1ψ2Solution
x̃1(ξ)x̃2(ξ)
0.10.2[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]
0.30.9[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]
0.40.1[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]
0.1250.623[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]
01[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]
10[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]
0.50.7[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]
0.20.1[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]

Table 2 . Comparison of solutions obtained by the proposed method along with the methods for Example 1.

Solutionx̃1(ξ)x̃2(ξ)
Behera and Chakraverty [15][5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]
Allahviranloo and Mikaeilvand [31][5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]
Dehghan et al. [32][4311+ξ11,4][5411+ξ11,214-ξ4]
Proposed method[5ξ2+28ξ+55ξ2+7ξ+14,3ξ2+14ξ-105ξ2+3ξ-26][5ξ2+37ξ+68ξ2+7ξ+14,118-3ξ87ξ2+22ξ-139ξ2+3ξ-26]

Table 3 . Comparison of the solution obtained by the proposed method along with the methods for Example 2.

Solutionx̃1(ξ)x̃2(ξ)x̃3(ξ)
Ezzati et al. [14][4ξ + 1, −2ξ + 7][6ξ + 6, −2ξ + 14][3ξ + 7, −2ξ + 12]
Behera and Chakraverty [15][4ξ + 1, −2ξ + 7][6ξ + 6, −2ξ + 14][3ξ + 7, −2ξ + 12]
Proposed method[4ξ + 1, −2ξ + 7][6ξ + 6, −2ξ + 14][3ξ + 7, −2ξ + 12]

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