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
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)
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.
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
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.
Here, some notations and definitions are provided concerning the presented study [27–30].
A fuzzy set Ω̃ in
where . The range of
is a subset of non-negative real numbers whose supremum is finite. If sup(
A fuzzy number Ω̃ is a convex normalized fuzzy set Ω̃ of the real line ℝ, such that
where
A triangular fuzzy number Ω̃ = (
An arbitrary triangular fuzzy number Ω̃ = (
where
The membership of a trapezoidal fuzzy number Ω̃ = (
An arbitrary trapezoidal fuzzy number Ω̃ = (
where
In the above
•Ω̄(
Let us consider two fuzzy numbers
in the form of
•Addition: .
•Subtraction: .
•Multiplication: , where
•Division: .
•Scalar multiplication: For any scalar
A triangular fuzzy number Ω̃ = (
The trapezoidal fuzzy number Ω̃ = (
Let us consider a
Here, [
With these expressions,
Using the parametric or
where
First, some important convex combination results are proposed and proven in terms of Theorems 1 and 2.
If [
where
Because [
Equating the two sides gives
Now, consider the left-hand side of the equation, that is,
In this expression, substituting the values of
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.
The converse of Theorem 1 is true only when
Assume
The aim is to show that [
respectively.
Now, the above expressions can be used to construct the interval form when
However, this is equivalent to
Since Theorem 1 is true, we have
where
Solving these crisp systems of equations numerically or analytically yields the values of
For any two distinct values of
In the next section, several examples are solved using the proposed method and compared with the existing results for validation.
Let us consider a 2 × 2 triangular FFSLE as
Next, using the
where
Accordingly, using the fuzzy arithmetic,
Next, by Theorem 1, the above system can be equivalently written in crisp form as
where
and
respectively. Finally, by solving these crisp systems simultaneously, we have
Hence, the final solution in parametric form can be expressed as
and
Next, we solve this problem using the proposed method for different sets of
In this example, let us consider a 3×3 triangular FFSLE as
An equivalent parametric form of the above system can be obtained as
Using the proposed method, we obtained the solution in parametric form as
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.
Here, a 3 × 3 trapezoidal FSLE in
Using the proposed method, we have
These results are compared with the solutions of Das and Chakraverty [33] and Behera and Chakraverty [15], and the results are identical.
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.
No potential conflicts of interest relevant to this study were reported.
Table 1. Solution of Example 1 obtained by the proposed method for different sets of
Solution | |||
---|---|---|---|
0.1 | 0.2 | ||
0.3 | 0.9 | ||
0.4 | 0.1 | ||
0.125 | 0.623 | ||
0 | 1 | ||
1 | 0 | ||
0.5 | 0.7 | ||
0.2 | 0.1 |
Table 2. Comparison of solutions obtained by the proposed method along with the methods for Example 1.
Solution | ||
---|---|---|
Behera and Chakraverty [15] | ||
Allahviranloo and Mikaeilvand [31] | ||
Dehghan et al. [32] | ||
Proposed method |
Table 3. Comparison of the solution obtained by the proposed method along with the methods for Example 2.
Solution | |||
---|---|---|---|
Ezzati et al. [14] | [4 | [6 | [3 |
Behera and Chakraverty [15] | [4 | [6 | [3 |
Proposed method | [4 | [6 | [3 |
E-mail: diptiranjanb@gamil.com
E-mail: sne_chak@yahoo.com
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.
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)
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.
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
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.
Here, some notations and definitions are provided concerning the presented study [27–30].
A fuzzy set Ω̃ in
where . The range of
is a subset of non-negative real numbers whose supremum is finite. If sup(
A fuzzy number Ω̃ is a convex normalized fuzzy set Ω̃ of the real line ℝ, such that
where
A triangular fuzzy number Ω̃ = (
An arbitrary triangular fuzzy number Ω̃ = (
where
The membership of a trapezoidal fuzzy number Ω̃ = (
An arbitrary trapezoidal fuzzy number Ω̃ = (
where
In the above
•Ω̄(
Let us consider two fuzzy numbers
in the form of
•Addition: .
•Subtraction: .
•Multiplication: , where
•Division: .
•Scalar multiplication: For any scalar
A triangular fuzzy number Ω̃ = (
The trapezoidal fuzzy number Ω̃ = (
Let us consider a
Here, [
With these expressions,
Using the parametric or
where
First, some important convex combination results are proposed and proven in terms of Theorems 1 and 2.
If [
where
Because [
Equating the two sides gives
Now, consider the left-hand side of the equation, that is,
In this expression, substituting the values of
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.
The converse of Theorem 1 is true only when
Assume
The aim is to show that [
respectively.
Now, the above expressions can be used to construct the interval form when
However, this is equivalent to
Since Theorem 1 is true, we have
where
Solving these crisp systems of equations numerically or analytically yields the values of
For any two distinct values of
In the next section, several examples are solved using the proposed method and compared with the existing results for validation.
Let us consider a 2 × 2 triangular FFSLE as
Next, using the
where
Accordingly, using the fuzzy arithmetic,
Next, by Theorem 1, the above system can be equivalently written in crisp form as
where
and
respectively. Finally, by solving these crisp systems simultaneously, we have
Hence, the final solution in parametric form can be expressed as
and
Next, we solve this problem using the proposed method for different sets of
In this example, let us consider a 3×3 triangular FFSLE as
An equivalent parametric form of the above system can be obtained as
Using the proposed method, we obtained the solution in parametric form as
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.
Here, a 3 × 3 trapezoidal FSLE in
Using the proposed method, we have
These results are compared with the solutions of Das and Chakraverty [33] and Behera and Chakraverty [15], and the results are identical.
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
Solution | |||
---|---|---|---|
0.1 | 0.2 | ||
0.3 | 0.9 | ||
0.4 | 0.1 | ||
0.125 | 0.623 | ||
0 | 1 | ||
1 | 0 | ||
0.5 | 0.7 | ||
0.2 | 0.1 |
Table 2 . Comparison of solutions obtained by the proposed method along with the methods for Example 1.
Solution | ||
---|---|---|
Behera and Chakraverty [15] | ||
Allahviranloo and Mikaeilvand [31] | ||
Dehghan et al. [32] | ||
Proposed method |
Table 3 . Comparison of the solution obtained by the proposed method along with the methods for Example 2.
Solution | |||
---|---|---|---|
Ezzati et al. [14] | [4 | [6 | [3 |
Behera and Chakraverty [15] | [4 | [6 | [3 |
Proposed method | [4 | [6 | [3 |
Diptiranjan Behera
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