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닫기 International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(2): 173-180

Published online June 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.2.173

© The Korean Institute of Intelligent Systems

## Linear Combination-Based Approach for Solving Imprecisely Defined Algebraic Systems of Linear Equations under Fuzzy Uncertainty

Diptiranjan Behera

Department of Mathematics, The University of the West Indies, Mona Campus, Kingston, Jamaica

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

Received: October 6, 2022; Revised: January 10, 2023; Accepted: June 9, 2023

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.

Imprecisely defined systems of linear equations play vital roles in various scientific and engineering applications. In this study, impreciseness is considered in a fuzzy sense. A system of linear equations with crisp coefficients and fuzzy unknowns is established and a new and straightforward method for solving this fuzzy system using the well-known concept of linear combination is proposed. In this approach, the system is first converted into an equivalent interval form, and thereafter, using the linear combination approach, a crisp representation of the system is obtained without loss of generality. Subsequently, by solving the corresponding crisp system using two different sets of linear combination coefficients, a final solution is obtained. To validate the proposed method, various example problems were solved and compared.

Keywords: Fuzzy system of linear equations, Fuzzy number, Linear combination, ξ-cut

In recent years, there has been a growing recognition of the usefulness of the fuzzy set theory in modeling situations where there is limited, incomplete, vague, or imprecise information about the variables or parameters. The presence of fuzzy uncertainty in various physical or real-life problems often results in the creation of a fuzzy system of linear equations (FSLE) or necessities the use of fuzzy technique during the solution process. This type of system has several applications in marketing, transportation, finance, and optimization.

Extensive research has been conducted on fuzzy systems and fuzzy linear equations and documented in the open literature. Notable contributions include the works of Friedman et al. , Behera and Chakraverty [2, 3], and Mikaeilvand et al. . Friedman et al.  proposed an embedding approach considering the existence of a unique solution for a fuzzy linear system. Using the concept of a fuzzy center and radius, Chakraverty and Behera  studied the solution procedure for the FSLE under various types of fuzzy numbers, such as trapezoidal numbers. Ezzati  presented various theories on the existence and uniqueness of solutions for the FSLE. Recently, Mikaeilvand et al.  proposed a novel technique based on an embedding approach for examining the FSLE. They reported that their method used fewer operations than those developed by Friedman et al.  and Ezzati . Behera and Chakraverty  conducted a systematic investigation of the solution procedure for both real and complex fuzzy systems. The applications of FSLE can be explored further in the work of Behera and Chakraverty  in which a static analysis of structural problems under fuzzy and interval loads was conducted.

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

Accordingly, as discussed in this paper, an FSLE with crisp coefficients, a fuzzy unknown, and right-hand side vectors of the form [A]{} = {}is considered, and the 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 describes the general FSLE and the proposed method. Numerical examples are presented in Section 4. Finally, the conclusions are presented in Section 5.

Some notations and definitions are provided in this study .

### 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))xX},

where μΩ̃(x) denotes the membership function or grade of membership of x in Ω̃, which maps X to the membership space ℵ. The range of ℵ is the subset of non-negative real numbers whose supremum is finite. If sup(μΩ̃ (x)) = 1, then Ω̃ is referred to as the 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):[0,1],x,

where μΩ̃ (x) is denoted as the membership function of the fuzzy set and it 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 follows:

μΩ˜(x)={0,xɛ,x-ɛφ-ɛ,ɛxφ,x-λφ-λ,φxλ,0,xλ.

### Definition 4

An arbitrary triangular fuzzy number Ω̃ = (ɛ, φ, λ) can be expressed in an interval form by the ξ − cutapproach as follows:

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 follows:

μΩ˜(x)={0,xɛ,x-ɛφ-ɛ,ɛxφ,1,φxλ,x-ϑλ-ϑ,λxϑ,0,xϑ.

### Definition 6

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

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

where ξ ∈ [0, 1].

### Definition 7

A symmetric Gaussian fuzzy number Ω̃ = (θ, ρ, ρ) is a fuzzy set characterized by a Gaussian distribution with a modal value represented by θ, and the left and right spreads of the distribution represented by ρ. The membership function can be defined as μΩ̃ (x) = exp{−β(x {− θ)2}, where β=12ρ2. Hence the ξ-cut form can be represented as follows:

Ω˜=Ω˜(ξ)=[Ω_(ξ),Ω¯(ξ)]=[θ--logeξβ,θ+-logeξβ],

where ξ ∈ (0, 1].

### Note

In the 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).

However, the parametric form of the Gaussian fuzzy number satisfies the aforementioned requirements, and the left and right bounds are defined as (0, 1].

### Definition 8

Let us consider two fuzzy numbers in the form of ξ-cut for further analysis and comparison.

Ω˜=Ω˜(ξ)=[Ω_(ξ),Ω¯(ξ)],

and

I˜=I˜(ξ)=[I_(ξ),I¯(ξ)].

The fuzzy arithmetic operations can be defined as follows:

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

• Multiplication:Ω˜+I˜=Ω˜(ξ)+I˜(ξ)=[min(),max()], 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)],where0I˜, where 0I˜.

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

κΩ˜=κΩ˜(ξ)={[κΩ_(ξ),κΩ¯(ξ)]for κ0,[κΩ¯(ξ),κΩ_(ξ)]for κ<0.

### 3. FSLE and the Proposed Method

Let us consider a n × n fuzzy system of linear equations

[A]{X˜}={b˜},

where [A] = (akj) for 1 ≤ kn and 1 ≤ jn represent an n×n crisp real matrix, { } = {k} is a column vector of fuzzy numbers, and { } = {j} is the vector of fuzzy unknowns.

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

j=1nakjx˜j=b˜k,

for k = 1, ···, n.

Using the parametric or ξ-cut form, Eq. (2) can be also expressed as follows:

j=1nakjx˜j(ξ)=b˜k(ξ),

where j(ξ) = [xj(ξ), χ̄j(ξ)] and k(ξ) = [bk(ξ), k(ξ)].

First, we validate an important result using the concept of a linear combination, as proposed in Theorem 1.

### Theorem 1

If [A]{} = {}, then

ηH_+θH¯=ημ_+θμ¯,

where

H_=akj0akjx_j(ξ)+akj<0akjx¯j(ξ),H¯=akj0akjx¯j(ξ)+akj<0akjx_j(ξ),μ_=b_k(ξ),μ¯=b¯k(ξ),and η,θ+.
Proof

Since [A]{ } = { }, this can be expressed using Eq. (3). Next, by applying fuzzy arithmetic, as defined in Section 2, Eq. (3) can be expressed as follows:

[akj0akjx_j(ξ)+akj<0akjx¯j(ξ),akj0akjx¯j(ξ)+akj<0akjx_j(ξ)]=[b_k(ξ),b¯k(ξ)].

Equating both sides, we obtain

akj0akjx_j(ξ)+akj<0akjx¯j(ξ)=b_k(ξ),akj0akjx¯j(ξ)+akj<0akjx_j(ξ)=b¯k(ξ).

Using the expressions given in the theorem, Eq. (5) can now be represented as follows:

H_=μ_and H¯=μ¯.

The aim is to demonstrate that ηH + θH̄ = ημ + θμ̄. Considering the left-hand side of this expression, i.e., ηH + θH̄, and substituting the expressions obtained in Eq. (6), we obtain

ημ_+θμ¯.

This is represented on the right side of the equation: Hence, the theorem is proven.

### 3.1 Description of the Proposed Method

Since Theorem 1 is true, we can deduce that

ηH_+θH¯=ημ_+θμ¯,

where η, θ ∈ ℝ+. Therefore, for ηi and θi of η and θ, where i = 1 and 2 with the condition η1θ1η2θ2, Eq. (8) can be expressed as follows:

η1H_+θ1H¯=η1μ_+θ1μ¯,η2H_+θ2H¯=η2μ_+θ2μ¯.

Now, solving these crisp systems of equations numerically or analytically provides the values of xj(ξ) and χ̄j(ξ). Accordingly, the solution vector expressed in Eq. (1) can be written in parametric form as [xj(ξ), χ̄j(ξ)] = j(ξ).

Observation 1

For any distinct values of η1, η2, θ1, and θ2 ∈ ℝ+, we obtain the same solution.

Note

Whenever η, θ ∈ ℝ, the proposed method yields the same solution.

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

### Example 1 (Mikaeilvand et al. )

Let us consider a 2 × 2 triangular fuzzy system of linear equations

x˜1=x˜2=(0,1,2),x˜1+3x˜2=(4,5,7).

Next, using the ξ-cut approach, the above system can be expressed as follows:

x˜1(ξ)-x˜2(ξ)=[ξ,2-ξ],x˜1(ξ)+3x˜2(ξ)=[4+ξ,7-2ξ],

where 1(ξ) = [x1(ξ), χ̄1(ξ)] and 2(ξ) = [x2(ξ), χ̄2(ξ)].

Eq. (11) can be expressed using the fuzzy arithmetic as follows:

[x_1(ξ)-x¯2(ξ),x¯1(ξ)-x_2(ξ)]=[ξ,2-ξ],[x_1(ξ)+3x_2(ξ),x¯1(ξ)+3x¯2(ξ)]=[4+ξ,7-2ξ].

Next, according to Theorem 1, the above system can be equivalently written in crisp form as follows:

η(x_1(ξ)-x¯2(ξ))+θ(x¯1(ξ)-x_2(ξ))=ηξ+θ(2-ξ),η(x_1(ξ)+3x¯2(ξ))+θ(x¯1(ξ)+3x¯2(ξ))=η(4+ξ)+θ(7-2ξ),

where η, θ ∈ ℝ+. Next, as per the proposed methodology for ηi and θi of η and θ where, i = 1 and 2 with the condition η1θ1η2θ2, we obtain

η1(x_1(ξ)-x¯2(ξ))+θ1(x¯1(ξ)-x_2(ξ))=η1ξ+θ1(2-ξ),η1(x_1(ξ)+3x_2(ξ))+θ1(x¯1(ξ)+3x¯2(ξ))=η1(4+ξ)+θ1(7-2ξ),η2(x_1(ξ)-x¯2(ξ))+θ2(x¯1(ξ)-x_2(ξ))=η2ξ+θ2(2-ξ),η2(x_1(ξ)+3x_2(ξ))+θ2(x¯1(ξ)+3x¯2(ξ))=η2(4+ξ)+θ2(7-2ξ).

Now, particularly for η1 = 1, θ1 = 3, η2 = 2, and θ2 = 4, the above system yields

(x_1(ξ)-x¯2(ξ))+3(x¯1(ξ)-x_2(ξ))=ξ+3(2-ξ),(x_1(ξ)+3x_2(ξ))+3(x¯1(ξ)+3x¯2(ξ))=(4+ξ)+3(7-2ξ),2(x_1(ξ)-x¯2(ξ))+4(x¯1(ξ)-x_2(ξ))=2ξ+4(2-ξ),2(x_1(ξ)+3x_2(ξ))+4(x¯1(ξ)+3x¯2(ξ))=2(4+ξ)+4(7-2ξ).

Finally, solving the crisp system (14) yields

{x_1(ξ)x¯1(ξ)x_2(ξ)x¯2(ξ)}={118+5ξ8238-7ξ878+ξ8118-3ξ8}.

Hence, the solution of Eq. (10) or (11) in the parametric form can be expressed as follows:

x˜1(ξ)=[118+5ξ8,238-7ξ8],

and

x˜2(ξ)=[78+ξ8,118-3ξ8].

These solutions can be equivalently expressed in decimal form as follows:

x˜1(ξ)=[1.375+0.625ξ,2.875-0.875ξ],

and

x˜2(ξ)=[0.875+0.125ξ,1.375-0.375ξ].

We also solved this problem using the proposed method for different sets of η1, θ1, η2, and θ2, as summarized in Table 1. It is evident that for each case, we obtained the same solution. Figure 1 shows the triangular fuzzy solution obtained using the proposed method. Notably, the results obtained using the proposed method are equal to those obtained using other existing methods, such as those reported by Friedman et al. , Mikaeilvand et al. , and Chakraverty and Behera , as summarized in Table 2.

### Example 2 (Chakraverty and Behera )

In this example, let us consider a 3 × 3 trapezoidal fuzzy system of linear equations in ξ-cut form as follows:

x˜1(ξ)+x˜2(ξ)-x˜3(ξ)=[1.5ξ-1,-0.5ξ+2],x˜1(ξ)-2x˜2(ξ)+x˜3(ξ)=[1+ξ,-0.5ξ+3],2x˜1(ξ)+x˜2(ξ)+3x˜3(ξ)=[ξ-2,-ξ+1].

Using the proposed method, we obtained

x˜1(ξ)=[-4113+37ξ13,6313-28ξ13],x˜2(ξ)=[-413-4ξ13,-1713+5ξ26],

and

x˜3(ξ)=[2013-19ξ13,-3213+27ξ26].

This problem was solved using the methods reported by Friedman et al.  and Chakraverty and Behera . It was observed through comparison that the solutions obtained using these methods are exactly same as those of the proposed method. A graphical representation of the trapezoidal fuzzy solutions obtained using the proposed method is shown in Figure 2, which shows that 2(ξ) is a weak trapezoidal fuzzy number. This indicates that a weak fuzzy solution vector exists in this case.

### Example 3 (Garg and Singh )

Here, a 3 × 3 Gaussian FSLE in ξ-cut form is expressed as follows:

4x˜1(ξ)+2x˜2(ξ)-x˜3(ξ)=[-20--50logeξ,-20+-50logeξ],2x˜1(ξ)+7x˜2(ξ)+6x˜3(ξ)=[16--100logeξ,16+-100logeξ],-1x˜1(ξ)+6x˜2(ξ)+10x˜3(ξ)=[44--100logeξ,44+-100logeξ].

Using the proposed method, we obtain

x˜1(ξ)=[-4-170113-2logeξ+90113-logeξ,-4+170113-2logeξ-90113-logeξ],x˜2(ξ)=[70113-2logeξ-170113-logeξ,-70113-2logeξ+170113-logeξ],x˜3(ξ)=[4-25113-2logeξ+20113-logeξ,4+25113-2logeξ-20113-logeξ].

This example problem was also solved using the existing methods reported by Chakraverty and Behera  and Garg and Singh . The results obtained using these methods are listed in Table 3.

The results revealed that the solution obtained using the proposed method and that of Chakraverty and Behera  satisfy Eq. (8), whereas the solution obtained using the method reported by Garg and Singh  approximately satisfied Eq. (8) because they obtained the solution using a numerical approach. This indicated that 2(ξ) and 3(ξ) of the solution vector are weak. Therefore, we generally have a weak fuzzy solution vector. Further details regarding the weak fuzzy solution can be found in the work of Friedman et al. .

A new and straightforward method using the concept of linear combination was successfully proposed for solving a fuzzy system of linear equations with crisp coefficients. To the best of our knowledge, this concept was used for the first time to obtain a solution for the considered fuzzy systems. Various example problems with respect to triangular, trapezoidal, and Gaussian fuzzy uncertainties were solved using the proposed method. In comparison, the results obtained using the proposed method were found to be in good agreement with the results obtained using existing methods. Therefore, the proposed method is an excellent alternative to existing methods. Notably, the proposed method is not limited to producing fuzzy solution vectors. In certain cases, a non-fuzzy solution can also be obtained for fuzzy systems. In future, we aim to further develop the proposed method and apply it in solving a fully fuzzy algebraic system of linear equations.

### Conflict of Interest Fig. 1.

Triangular fuzzy solution obtained using the proposed method for Example 1. Fig. 2.

Trapezoidal fuzzy solution obtained using the proposed method for Example 2.

Table. 1.

Table 1. Solution of Example 1 obtained using the proposed method for different sets of η1, θ1, η2, and θ2.

Solution1(ξ)2(ξ)
η1 = 1, θ1 = 3, η2 = 2, and θ2 = 4[1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]
η1 = 1, θ1 = 4, η2 = 10, and θ2 = 50[1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]
η1 = 1, θ1 = 5, η2 = 6, and θ2 = 10[1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]
η1 = 5, θ1 = 1000, η2 = 6, and θ2 = 50000[1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]
η1 = 0.5, θ1 = 5, η2 = 25, and θ2 = 7.5[1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]
η1 = 85, θ1 = 73, η2 = 91, and θ2 = 23[1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]

Table. 2.

Table 2. Comparison of the results obtained using the proposed method with the results obtained using the methods reported in [1,4,5] for Example 1.

Solution1(ξ)2(ξ)
Proposed method[1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]
Friedman et al. [1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]
Mikaeilvand et al. [1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]
Chakraverty and Behera [1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]

Table. 3.

Table 3. Lower and upper bounds of results obtained by Chakraverty and Behera  and Garg and Singh  for Example 3.

SolutionChakraverty and Behera Garg and Singh 
x1(ξ)-4-170113-2logeξ+90113-logeξ-4-310-50logeξ+788-100logeξ
χ̄1(ξ)-4+170113-2logeξ-90113-logeξ-4+310-50logeξ-788-100logeξ
x2(ξ)70113-2logeξ-170113-logeξ18-50logeξ-320-100logeξ
χ̄2(ξ)-70113-2logeξ+170113-logeξ-18-50logeξ+1493-100logeξ
x3(ξ)4-25113-2logeξ-20113-logeξ4-245-50logeξ-156-100logeξ
χ̄3(ξ)4+25113-2logeξ+20113-logeξ4+245-50logeξ+156-100logeξ

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17. Amirfakhrian, M, Fallah, M, and Rodriguez-Lopez, R (2018). A method for solving fuzzy matrix equations. Soft Computing. 22, 2095-2103. https://doi.org/10.1007/s00500-017-2680-x
18. Behera, D, and Chakraverty, S (2022). Computational technique for solving imprecisely defined non-negative fully fuzzy algebraic system of linear equations. International Journal of Fuzzy Logic and Intelligent Systems. 22, 252-260. http://doi.org/10.5391/IJFIS.2022.22.3.252 Diptiranjan Behera is currently working as a senior lecturer of Mathematics at The University of the West Indies, Mona Campus, Kingston 7, Jamaica, where he 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. He also worked there as a postdoctoral research fellow for three 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 scholarships from China Postdoctoral Science Foundation, Govt. of P. R. China and National Postdoctoral Fellowship by Science and Educational Research Board, Department of Science and Technology, Govt. of India. He has published his research findings in several reputable international journals and has actively engaged in editorial activities, including reviewing papers for those journals. His current research interest is in interval and fuzzy mathematics, computational methods, fractional differential equations, nonlinear differential equations, structural analysis, and optimization problems. E-mail: diptiranjanb@gmail.com

### Article

#### Original Article International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(2): 173-180

Published online June 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.2.173

## Linear Combination-Based Approach for Solving Imprecisely Defined Algebraic Systems of Linear Equations under Fuzzy Uncertainty

Diptiranjan Behera

Department of Mathematics, The University of the West Indies, Mona Campus, Kingston, Jamaica

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

Received: October 6, 2022; Revised: January 10, 2023; Accepted: June 9, 2023

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

Imprecisely defined systems of linear equations play vital roles in various scientific and engineering applications. In this study, impreciseness is considered in a fuzzy sense. A system of linear equations with crisp coefficients and fuzzy unknowns is established and a new and straightforward method for solving this fuzzy system using the well-known concept of linear combination is proposed. In this approach, the system is first converted into an equivalent interval form, and thereafter, using the linear combination approach, a crisp representation of the system is obtained without loss of generality. Subsequently, by solving the corresponding crisp system using two different sets of linear combination coefficients, a final solution is obtained. To validate the proposed method, various example problems were solved and compared.

Keywords: Fuzzy system of linear equations, Fuzzy number, Linear combination, &xi,-cut

### 1. Introduction

In recent years, there has been a growing recognition of the usefulness of the fuzzy set theory in modeling situations where there is limited, incomplete, vague, or imprecise information about the variables or parameters. The presence of fuzzy uncertainty in various physical or real-life problems often results in the creation of a fuzzy system of linear equations (FSLE) or necessities the use of fuzzy technique during the solution process. This type of system has several applications in marketing, transportation, finance, and optimization.

Extensive research has been conducted on fuzzy systems and fuzzy linear equations and documented in the open literature. Notable contributions include the works of Friedman et al. , Behera and Chakraverty [2, 3], and Mikaeilvand et al. . Friedman et al.  proposed an embedding approach considering the existence of a unique solution for a fuzzy linear system. Using the concept of a fuzzy center and radius, Chakraverty and Behera  studied the solution procedure for the FSLE under various types of fuzzy numbers, such as trapezoidal numbers. Ezzati  presented various theories on the existence and uniqueness of solutions for the FSLE. Recently, Mikaeilvand et al.  proposed a novel technique based on an embedding approach for examining the FSLE. They reported that their method used fewer operations than those developed by Friedman et al.  and Ezzati . Behera and Chakraverty  conducted a systematic investigation of the solution procedure for both real and complex fuzzy systems. The applications of FSLE can be explored further in the work of Behera and Chakraverty  in which a static analysis of structural problems under fuzzy and interval loads was conducted.

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

Accordingly, as discussed in this paper, an FSLE with crisp coefficients, a fuzzy unknown, and right-hand side vectors of the form [A]{} = {}is considered, and the 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 describes the general FSLE and the proposed method. Numerical examples are presented in Section 4. Finally, the conclusions are presented in Section 5.

### 2. Preliminaries

Some notations and definitions are provided in this study .

### 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) denotes the membership function or grade of membership of x in Ω̃, which maps X to the membership space ℵ. The range of ℵ is the subset of non-negative real numbers whose supremum is finite. If sup(μΩ̃ (x)) = 1, then Ω̃ is referred to as the 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 denoted as the membership function of the fuzzy set and it 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 follows:

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

### Definition 4

An arbitrary triangular fuzzy number Ω̃ = (ɛ, φ, λ) can be expressed in an interval form by the ξ − cutapproach as follows:

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

where ξ ∈ [0, 1].

### Definition 5

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

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

### Definition 6

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

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

where ξ ∈ [0, 1].

### Definition 7

A symmetric Gaussian fuzzy number Ω̃ = (θ, ρ, ρ) is a fuzzy set characterized by a Gaussian distribution with a modal value represented by θ, and the left and right spreads of the distribution represented by ρ. The membership function can be defined as μΩ̃ (x) = exp{−β(x {− θ)2}, where $β=12ρ2$. Hence the ξ-cut form can be represented as follows:

$Ω˜=Ω˜(ξ)=[Ω_(ξ), Ω¯(ξ)]=[θ--logeξβ,θ+-logeξβ],$

where ξ ∈ (0, 1].

### Note

In the 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).

However, the parametric form of the Gaussian fuzzy number satisfies the aforementioned requirements, and the left and right bounds are defined as (0, 1].

### Definition 8

Let us consider two fuzzy numbers in the form of ξ-cut for further analysis and comparison.

$Ω˜=Ω˜(ξ)=[Ω_(ξ), Ω¯(ξ)],$

and

$I˜=I˜(ξ)=[I_(ξ), I¯(ξ)].$

The fuzzy arithmetic operations can be defined as follows:

• $Addition: Ω^+I^=Ω^(ξ)+I^(ξ)=[Ω_(ξ)+I_(ξ),Ω^(ξ)+I^(ξ)]$.

• $Subtraction: Ω˜-I˜=Ω˜(ξ)-I˜(ξ)=[Ω_(ξ)-I¯(ξ), Ω¯(ξ)-I_(ξ)]$.

• $Multiplication: Ω˜+I˜=Ω˜(ξ)+I˜(ξ)=[min(℘),max(℘)]$, where

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

• $Division: Ω˜I˜=Ω˜(ξ)I˜(ξ)=[Ω_(ξ)I¯(ξ),Ω¯(ξ)I_(ξ)], where 0∉I˜$, where $0∉I˜$.

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

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

### 3. FSLE and the Proposed Method

Let us consider a n × n fuzzy system of linear equations

$[A]{X˜}={b˜},$

where [A] = (akj) for 1 ≤ kn and 1 ≤ jn represent an n×n crisp real matrix, { } = {k} is a column vector of fuzzy numbers, and { } = {j} is the vector of fuzzy unknowns.

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

$∑j=1nakjx˜j=b˜k,$

for k = 1, ···, n.

Using the parametric or ξ-cut form, Eq. (2) can be also expressed as follows:

$∑j=1nakjx˜j(ξ)=b˜k(ξ),$

where j(ξ) = [xj(ξ), χ̄j(ξ)] and k(ξ) = [bk(ξ), k(ξ)].

First, we validate an important result using the concept of a linear combination, as proposed in Theorem 1.

### Theorem 1

If [A]{} = {}, then

$ηH_+θH¯=ημ_+θμ¯,$

where

$H_=∑akj≥0akjx_j(ξ)+∑akj<0akjx¯j(ξ),H¯=∑akj≥0akjx¯j(ξ)+∑akj<0akjx_j(ξ),μ_=b_k(ξ), μ¯=b¯k(ξ), and η, θ∈ℝ+.$
Proof

Since [A]{ } = { }, this can be expressed using Eq. (3). Next, by applying fuzzy arithmetic, as defined in Section 2, Eq. (3) can be expressed as follows:

$[∑akj≥0akjx_j(ξ)+∑akj<0akjx¯j(ξ), ∑akj≥0akjx¯j(ξ)+∑akj<0akjx_j(ξ)]=[b_k(ξ), b¯k(ξ)].$

Equating both sides, we obtain

$∑akj≥0akjx_j(ξ)+∑akj<0akjx¯j(ξ)=b_k(ξ),∑akj≥0akjx¯j(ξ)+∑akj<0akjx_j(ξ)=b¯k(ξ).$

Using the expressions given in the theorem, Eq. (5) can now be represented as follows:

$H_=μ_ and H¯=μ¯.$

The aim is to demonstrate that ηH + θH̄ = ημ + θμ̄. Considering the left-hand side of this expression, i.e., ηH + θH̄, and substituting the expressions obtained in Eq. (6), we obtain

$ημ_+θμ¯.$

This is represented on the right side of the equation: Hence, the theorem is proven.

### 3.1 Description of the Proposed Method

Since Theorem 1 is true, we can deduce that

$ηH_+θH¯=ημ_+θμ¯,$

where η, θ ∈ ℝ+. Therefore, for ηi and θi of η and θ, where i = 1 and 2 with the condition η1θ1η2θ2, Eq. (8) can be expressed as follows:

$η1H_+θ1H¯=η1μ_+θ1μ¯,η2H_+θ2H¯=η2μ_+θ2μ¯.$

Now, solving these crisp systems of equations numerically or analytically provides the values of xj(ξ) and χ̄j(ξ). Accordingly, the solution vector expressed in Eq. (1) can be written in parametric form as [xj(ξ), χ̄j(ξ)] = j(ξ).

Observation 1

For any distinct values of η1, η2, θ1, and θ2 ∈ ℝ+, we obtain the same solution.

Note

Whenever η, θ ∈ ℝ, the proposed method yields the same solution.

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

### Example 1 (Mikaeilvand et al. )

Let us consider a 2 × 2 triangular fuzzy system of linear equations

$x˜1=x˜2=(0,1,2),x˜1+3x˜2=(4,5,7).$

Next, using the ξ-cut approach, the above system can be expressed as follows:

$x˜1(ξ)-x˜2(ξ)=[ξ,2-ξ],x˜1(ξ)+3x˜2(ξ)=[4+ξ,7-2ξ],$

where 1(ξ) = [x1(ξ), χ̄1(ξ)] and 2(ξ) = [x2(ξ), χ̄2(ξ)].

Eq. (11) can be expressed using the fuzzy arithmetic as follows:

$[x_1(ξ)-x¯2(ξ), x¯1(ξ)-x_2(ξ)]=[ξ, 2-ξ],[x_1(ξ)+3x_2(ξ), x¯1(ξ)+3x¯2(ξ)]=[4+ξ, 7-2ξ].$

Next, according to Theorem 1, the above system can be equivalently written in crisp form as follows:

$η(x_1(ξ)-x¯2(ξ))+θ(x¯1(ξ)-x_2(ξ))=ηξ+θ(2-ξ),η(x_1(ξ)+3x¯2(ξ))+θ(x¯1(ξ)+3x¯2(ξ))=η(4+ξ)+θ(7-2ξ),$

where η, θ ∈ ℝ+. Next, as per the proposed methodology for ηi and θi of η and θ where, i = 1 and 2 with the condition η1θ1η2θ2, we obtain

$η1(x_1(ξ)-x¯2(ξ))+θ1(x¯1(ξ)-x_2(ξ))=η1ξ+θ1(2-ξ),η1(x_1(ξ)+3x_2(ξ))+θ1(x¯1(ξ)+3x¯2(ξ))=η1(4+ξ)+θ1(7-2ξ),η2(x_1(ξ)-x¯2(ξ))+θ2(x¯1(ξ)-x_2(ξ))=η2ξ+θ2(2-ξ),η2(x_1(ξ)+3x_2(ξ))+θ2(x¯1(ξ)+3x¯2(ξ))=η2(4+ξ)+θ2(7-2ξ).$

Now, particularly for η1 = 1, θ1 = 3, η2 = 2, and θ2 = 4, the above system yields

$(x_1(ξ)-x¯2(ξ))+3(x¯1(ξ)-x_2(ξ))=ξ+3(2-ξ),(x_1(ξ)+3x_2(ξ))+3(x¯1(ξ)+3x¯2(ξ))=(4+ξ)+3(7-2ξ),2(x_1(ξ)-x¯2(ξ))+4(x¯1(ξ)-x_2(ξ))=2ξ+4(2-ξ),2(x_1(ξ)+3x_2(ξ))+4(x¯1(ξ)+3x¯2(ξ))=2(4+ξ)+4(7-2ξ).$

Finally, solving the crisp system (14) yields

${x_1(ξ)x¯1(ξ)x_2(ξ)x¯2(ξ)}={118+5ξ8238-7ξ878+ξ8118-3ξ8}.$

Hence, the solution of Eq. (10) or (11) in the parametric form can be expressed as follows:

$x˜1(ξ)=[118+5ξ8,238-7ξ8],$

and

$x˜2(ξ)=[78+ξ8,118-3ξ8].$

These solutions can be equivalently expressed in decimal form as follows:

$x˜1(ξ)=[1.375+0.625ξ,2.875-0.875ξ],$

and

$x˜2(ξ)=[0.875+0.125ξ,1.375-0.375ξ].$

We also solved this problem using the proposed method for different sets of η1, θ1, η2, and θ2, as summarized in Table 1. It is evident that for each case, we obtained the same solution. Figure 1 shows the triangular fuzzy solution obtained using the proposed method. Notably, the results obtained using the proposed method are equal to those obtained using other existing methods, such as those reported by Friedman et al. , Mikaeilvand et al. , and Chakraverty and Behera , as summarized in Table 2.

### Example 2 (Chakraverty and Behera )

In this example, let us consider a 3 × 3 trapezoidal fuzzy system of linear equations in ξ-cut form as follows:

$x˜1(ξ)+x˜2(ξ)-x˜3(ξ)=[1.5ξ-1, -0.5ξ+2],x˜1(ξ)-2x˜2(ξ)+x˜3(ξ)=[1+ξ, -0.5ξ+3],2x˜1(ξ)+x˜2(ξ)+3x˜3(ξ)=[ξ-2, -ξ+1].$

Using the proposed method, we obtained

$x˜1(ξ)=[-4113+37ξ13,6313-28ξ13],x˜2(ξ)=[-413-4ξ13,-1713+5ξ26],$

and

$x˜3(ξ)=[2013-19ξ13,-3213+27ξ26].$

This problem was solved using the methods reported by Friedman et al.  and Chakraverty and Behera . It was observed through comparison that the solutions obtained using these methods are exactly same as those of the proposed method. A graphical representation of the trapezoidal fuzzy solutions obtained using the proposed method is shown in Figure 2, which shows that 2(ξ) is a weak trapezoidal fuzzy number. This indicates that a weak fuzzy solution vector exists in this case.

### Example 3 (Garg and Singh )

Here, a 3 × 3 Gaussian FSLE in ξ-cut form is expressed as follows:

$4x˜1(ξ)+2x˜2(ξ)-x˜3(ξ)=[-20--50logeξ,-20+-50logeξ],2x˜1(ξ)+7x˜2(ξ)+6x˜3(ξ)=[16--100logeξ, 16+-100logeξ],-1x˜1(ξ)+6x˜2(ξ)+10x˜3(ξ)=[44--100logeξ, 44+-100logeξ].$

Using the proposed method, we obtain

$x˜1(ξ)=[-4-170113-2logeξ+90113-logeξ,-4+170113-2logeξ-90113-logeξ],x˜2(ξ)=[70113-2logeξ-170113-logeξ,-70113-2logeξ+170113-logeξ],x˜3(ξ)=[4-25113-2logeξ+20113-logeξ,4+25113-2logeξ-20113-logeξ].$

This example problem was also solved using the existing methods reported by Chakraverty and Behera  and Garg and Singh . The results obtained using these methods are listed in Table 3.

The results revealed that the solution obtained using the proposed method and that of Chakraverty and Behera  satisfy Eq. (8), whereas the solution obtained using the method reported by Garg and Singh  approximately satisfied Eq. (8) because they obtained the solution using a numerical approach. This indicated that 2(ξ) and 3(ξ) of the solution vector are weak. Therefore, we generally have a weak fuzzy solution vector. Further details regarding the weak fuzzy solution can be found in the work of Friedman et al. .

### 5. Conclusion

A new and straightforward method using the concept of linear combination was successfully proposed for solving a fuzzy system of linear equations with crisp coefficients. To the best of our knowledge, this concept was used for the first time to obtain a solution for the considered fuzzy systems. Various example problems with respect to triangular, trapezoidal, and Gaussian fuzzy uncertainties were solved using the proposed method. In comparison, the results obtained using the proposed method were found to be in good agreement with the results obtained using existing methods. Therefore, the proposed method is an excellent alternative to existing methods. Notably, the proposed method is not limited to producing fuzzy solution vectors. In certain cases, a non-fuzzy solution can also be obtained for fuzzy systems. In future, we aim to further develop the proposed method and apply it in solving a fully fuzzy algebraic system of linear equations.

### Fig 1. Figure 1.

Triangular fuzzy solution obtained using the proposed method for Example 1.

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 173-180https://doi.org/10.5391/IJFIS.2023.23.2.173

### Fig 2. Figure 2.

Trapezoidal fuzzy solution obtained using the proposed method for Example 2.

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 173-180https://doi.org/10.5391/IJFIS.2023.23.2.173

Solution of Example 1 obtained using the proposed method for different sets of η1, θ1, η2, and θ2.

Solution1(ξ)2(ξ)
η1 = 1, θ1 = 3, η2 = 2, and θ2 = 4[1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]
η1 = 1, θ1 = 4, η2 = 10, and θ2 = 50[1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]
η1 = 1, θ1 = 5, η2 = 6, and θ2 = 10[1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]
η1 = 5, θ1 = 1000, η2 = 6, and θ2 = 50000[1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]
η1 = 0.5, θ1 = 5, η2 = 25, and θ2 = 7.5[1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]
η1 = 85, θ1 = 73, η2 = 91, and θ2 = 23[1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]

Comparison of the results obtained using the proposed method with the results obtained using the methods reported in [1,4,5] for Example 1.

Solution1(ξ)2(ξ)
Proposed method[1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]
Friedman et al. [1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]
Mikaeilvand et al. [1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]
Chakraverty and Behera [1.375 + 0.625ξ, 2.875 − 0.875ξ][0.875 + 0.125ξ, 1.375 − 0.375ξ]

Lower and upper bounds of results obtained by Chakraverty and Behera  and Garg and Singh  for Example 3.

SolutionChakraverty and Behera Garg and Singh 
x1(ξ)-4-170113-2logeξ+90113-logeξ-4-310-50logeξ+788-100logeξ
χ̄1(ξ)-4+170113-2logeξ-90113-logeξ-4+310-50logeξ-788-100logeξ
x2(ξ)70113-2logeξ-170113-logeξ18-50logeξ-320-100logeξ
χ̄2(ξ)-70113-2logeξ+170113-logeξ-18-50logeξ+1493-100logeξ
x3(ξ)4-25113-2logeξ-20113-logeξ4-245-50logeξ-156-100logeξ
χ̄3(ξ)4+25113-2logeξ+20113-logeξ4+245-50logeξ+156-100logeξ

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