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. [1], Behera and Chakraverty [2, 3], and Mikaeilvand et al. [4]. Friedman et al. [1] 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 [5] studied the solution procedure for the 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 for examining the FSLE. They reported that their method used fewer operations than those developed by Friedman et al. [1] and Ezzati [6]. Behera and Chakraverty [2] 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 [7] in which 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 solving both linear and nonlinear fuzzy systems, where uncertainties were modelled using Gaussian fuzzy numbers. Abdullah and Rahman [10] applied four distinct Jacobi-based iteration methods to solve the same type of system. Inearat and Qatanani [11] also used the Jacobi, Gauss-Seidel, and successive over-relaxation iteration schemes, along with a convergence analysis for FSLE. Islam et al. [12] used a matrix-form 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.

Accordingly, as discussed in this paper, an FSLE with crisp coefficients, a fuzzy unknown, and right-hand side vectors of the form [A]{X̃} = {b̃}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 [14–18].

Definition 1

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

where μ_Ω̃(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(μ_Ω̃ (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

where μ_Ω̃ (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:

Definition 4

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

where ξ ∈ [0, 1].

Definition 5

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

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:

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 μ_Ω̃ (x) = exp{−β(x {− θ)²}, where β=12ρ2. Hence the ξ-cut form can be represented as follows:

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.

and

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.

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

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

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

for k = 1, ···, n.

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

where x̃_j(ξ) = [x_j(ξ), χ̄_j(ξ)] and b̃_k(ξ) = [b_k(ξ), b̄_k(ξ)].

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

Theorem 1

If [A]{X̃} = {b̃}, then

where

3.1 Description of the Proposed Method

Since Theorem 1 is true, we can deduce that

where η, θ ∈ ℝ⁺. Therefore, for η_i and θ_i of η and θ, where i = 1 and 2 with the condition η₁ ≠ θ₁ ≠ η₂ ≠ θ₂, Eq. (8) can be expressed as follows:

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

Example 1 (Mikaeilvand et al. [4])

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

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

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

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

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

where η, θ ∈ ℝ⁺. Next, as per the proposed methodology for η_i and θ_i of η and θ where, i = 1 and 2 with the condition η₁ ≠ θ₁ ≠ η₂ ≠ θ₂, we obtain

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

Finally, solving the crisp system (14) yields

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

and

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

and

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. [1], Mikaeilvand et al. [4], and Chakraverty and Behera [5], as summarized in Table 2.

Example 2 (Chakraverty and Behera [5])

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

Using the proposed method, we obtained

and

This problem was solved using the methods reported by Friedman et al. [1] and Chakraverty and Behera [5]. 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 x̃₂(ξ) is a weak trapezoidal fuzzy number. This indicates that a weak fuzzy solution vector exists in this case.

Example 3 (Garg and Singh [9])

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

Using the proposed method, we obtain

This example problem was also solved using the existing methods reported by Chakraverty and Behera [5] and Garg and Singh [9]. 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 [5] satisfy Eq. (8), whereas the solution obtained using the method reported by Garg and Singh [9] approximately satisfied Eq. (8) because they obtained the solution using a numerical approach. This indicated that x̃₂(ξ) and x̃₃(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. [1].

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.

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

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.

Table 3. Lower and upper bounds of results obtained by Chakraverty and Behera [5] and Garg and Singh [9] for Example 3.