Advances in Numerical Optimisation: Theory, Models, and ApplicationsView this Special Issue
LU Decomposition Scheme for Solving -Polar Fuzzy System of Linear Equations
This paper presents a new scheme for solving -polar fuzzy system of linear equations (-PFSLEs) by using LU decomposition method. We assume the coefficient matrix of the system is symmetric positive definite, and we discuss this point in detail with some numerical examples. Furthermore, we investigate the inconsistent -polar fuzzy matrix equation (-PFME) and find the least square solution (LSS) of this system by using generalized inverse matrix theory. Moreover, we discuss the strong solution of -polar fuzzy LSS of the inconsistent -PFME. In the end, we present a numerical example to illustrate our approach.
Certain types of uncertainties arise in several areas of engineering and decision-making. To handle such uncertainties, probability theory, fuzzy set theory , and their related models have been proposed as suitable mathematical tools. It helps to define the problem in real form and the solution for these uncertain variables has been obtained. Many researchers [2–6] have studied their basic arithmetical operations and the methods of the fuzzy numbers. Goetschel and Voxman  proposed the concept of fuzzy calculus. They presented the parametric form of fuzzy number by using cut expansion and inserted the class of fuzzy numbers into a topological vector space. Moghadam et al.  described the concept of trapezoidal fuzzy numbers and also other affected investigations were shown in [9, 10]. The notion of -polar fuzzy set was proposed by Chen et al.  as the generalization of bipolar fuzzy set . Nowadays, analysts believe that the world is moving towards multipolarity. Therefore, it comes as no surprise that multipolarity in data and information plays a vital role in various fields of science and technology. The remarkable contribution on applications of -polar fuzzy sets is presented in [13–15].
Linear system plays an important role in many fields of engineering and science. In the wide majority of problems, we often deal with approximate data. Some parameters are represented as a fuzzy number and more general -polar fuzzy number rather than a crisp number. A numerical approach that would be suitable to handle and solve a -polar fuzzy linear system is extremely essential. The notion of -polar fuzzy linear system with crisp coefficient entries of matrix and the right-hand side is that parametric -polar fuzzy number vectors appear in many domains of engineering sciences such as economics, statistics, technology, telecommunications, image processing, social sciences, and physics. Some applications of linear system in fuzzy environment were presented in [16, 17]. LU decomposition method is used to solve many different kinds of systems of linear equations in -polar fuzzy environment. It is faster and more numerically stable than computing explicit inverses. LU decomposition method to solve -polar fuzzy linear system is used in electrical engineering and circuit designing, and this system is used to solve complex circuits. This technique is also used in dynamics to solve Diffusion Load Balancing. However, there is a vast literature in mathematics to solve the fuzzy linear system. But we introduce the new approach to solve linear system in -polar fuzzy environment. First we study the basic literature to solve linear system in fuzzy environment. Friedman  presented the idea of a fuzzy linear system of equations having the crisp coefficient matrix and the right-hand side is parametric fuzzy number vector. Friedman also proposed an embedding scheme and replaced original system of a fuzzy linear system with the extended system.
The iterative scheme to solve the linear system of equation in the form was studied by Wang et al. . Asady et al.  developed the general linear system and used an embedding technique to construct the different schemes in a fuzzy environment. Vroman et al.  used a parametric technique of fuzzy numbers to find the general solution. Sevastjanov and Dymova  introduced a practical approach for interval fuzzy systems. The numerical technique was also studied by Garg and Singh  to solve a fuzzy linear system by using the Gaussian fuzzy weight (membership) function. Behera and Chakraverty  introduced a new scheme for handling the real as well as the complex fuzzy linear system. Moreover, Abbasbandy et al. [25, 26] presented the steepest descent method and the LU decomposition method to solve the fuzzy system of linear equations. Allahviranloo et al. [27–30] developed some important numerical schemes for solving a fuzzy linear system of equations (FLSEs). Moreover, certain methods to solve fuzzy linear systems have been discussed in [31–34]. Akram et al. [35–39] studied certain schemes for solving the bipolar fuzzy linear system of equations. Akram et al.  discussed the solution of linear system in -polar fuzzy environment. This paper presents a new scheme for solving -polar fuzzy system of linear equations (-PFSLEs) by using LU decomposition method. We assume a special case when the coefficient matrix of the system is symmetric and positive definite and then we discuss this point in detail with some numerical examples. Furthermore, we investigate the inconsistent -polar fuzzy matrix equation (-PFME) and find the least square solution (LSS) of this system by using generalized inverse matrix theory. Moreover, we discuss the strong solution of -polar fuzzy LSS of the inconsistent -PFME. In the end, we present a numerical example to illustrate our approach.
The rest of the paper is structured as follows. In Section 2, we present the solution of -polar fuzzy linear system by using LU decomposition method. Section 3 presents the inconsistent -polar fuzzy matrix equation with some examples and Section 4 develops to obtain the least square solution of -polar fuzzy matrix equations. Some results are investigated by giving the true reasoning and the conclusion of this research work is in Section 5.
2. Decomposition Method for Solving -PFSLEs
Definition 1 (see ). An -polar fuzzy set on an underlying set is a mapping . The truthness degree of each element is defined aswhere is the -th projection mapping.
Definition 2 (see ). An -polar fuzzy number (-PFN) in parametric form is an -tuple of functions , which satisfy the following properties:(i) is a bounded nondecreasing right continuous function at the point 0 and left continuous over the interval (0, 1](ii) is a bounded nonincreasing right continuous function at the point 0 and left continuous over the interval (0, 1](iii)Throughout the paper, .
Definition 3 (see ). For arbitrary , and , we define , , , and scalar multiplication by as follows:(i), (ii)(iii)(iv), , (v), , (vi), , The family of all -PFNs is denoted by .
Definition 4 (see ). The linear system iswhere the coefficient matrix is a crisp matrix and are known -PFNs and are unknowns which may or may not be -PFNs, which are called -PFSLEs.
Definition 5. The matrix system iswhere the coefficient elements are crisp numbers and in the right-hand matrix are -PFNs which are called a general -PFME. By using matrix equation, we have
Definition 6. An -PFN vector given by , is called a solution of the -PFSLE (2) ifFor a particular p, , we getFrom the expression above, we have the following crisp linear system:orIf any is not specified, it will be perceived as 0.
So, a system in Definition 4 extended to the crisp system (8) where and (8) can be written asOn the base of [18, 41, 42], we investigate the following results.
Theorem 1. The matrix is nonsingular if and only if the matrices and are also nonsingular.
Theorem 2. Suppose that a matrix is nonsingular and a unique solution of always gives -polar fuzzy number for arbitrary vector ; then the necessary and sufficient condition for the inverse of nonnegative matrix exists.
Theorem 3. Let be a matrix that contains all real entries except 0 for leading minors. Then, the matrix has a unique factorization:where and are unit lower-triangular and upper-triangular matrices, respectively.
To eliminate the reduction, the and matrices must be found such that , which iswhere and are lower- and upper-triangular matrices, and are any matrices, and is null matrix, respectively.
Now, we suppose that has decomposition. We haveand then
Therefore, we can write
Theorem 4. Let be a positive definite symmetric matrix; then there is a unique matrix with positive diagonal entries such that
Proof. Suppose that is a positive definite symmetric matrix. We haveand thenTherefore, we can writeBy following Theorem 4 in decomposition method, and should be a positive definite symmetric matrix.
We solve some numerical examples to illustrate our scheme.
Example 1. Consider -polar fuzzy systemThe extended matrix isand henceThe exact solution isThe exact and derived solutions with decomposition of 3-PFN are plotted in Figures 1 and 2.
The Hausdorff norm of errors is 6.3750e−008.
The exact and derived solutions with decomposition of 3-PFN are plotted in Figures 3 and 4.
The Hausdorff norm of errors is 6.3750e−008.
The exact and derived solutions with decomposition of 3-PFN are plotted in Figures 5 and 6.
The Hausdorff norm of errors is 6.3750e−008.
Example 2. Consider the -polar fuzzy system:The extended symmetric positive definite matrix isand hence , whereNow, the exact solution isThe exact and derived solutions with decomposition of 3-PFN are plotted in Figures 7 and 8.
The Hausdorff norm of errors is 3.9705e−004. The exact and derived solutions with decomposition of 3-PFN are plotted in Figures 9 and 10.
The Hausdorff norm of errors is 3.9705e−004.
The exact and derived solutions with decomposition of 3-PFN are plotted in Figures 11 and 12.
The Hausdorff norm of errors is 3.9705e−004.
Lemma 1 (see ). The solution of -PFSLEs exists if and only if the rank of is equal to that of matrix ; that is,
When , the system has a unique solution. There are endless solutions to the system if and there is no solution if .
Theorem 5. The system has solution if and only if
Proof. Let , , whereThe matrices below are the same:If , then we have , since . By Lemma 1, all linear equations , , have solutions. So it makes sense to have the necessary condition.
Conversely, suppose that is solvable; in other words, every linear equation , , has solution. Letwhere , .
By using equation , , we haveFrom the above equation, it follows that can be expressed as linear combination of ; that is, from this equation, the following is that a linear combination of can be expressed as :From Theorem 5, we can deduce the following result about the solvability of (7).
Theorem 6. The equation has an equivalent solution to (7):
Theorem 7. Equation (7) has solution in which the necessary and sufficient conditions for the rows of have the same linear relation as the rows of the matrix.
Theorem 8. If (7) has no solution, then the corresponding -PFME also has no solution.
Corollary 1. Consider the condition
If , then (7) has unique solution; otherwise an infinite number of solutions exist.
Corollary 2. If there is only one solution in the crisp system (7), then it is equivalent to -PFSLE:which has only one solution.
3. Inconsistent -Polar Fuzzy Matrix Equation
Definition 8. If the crisp matrix equation (7) has no solution, then the associated -PFME iswhere the coefficient matrix , is crisp matrix, the right-hand matrix is -PFN called an inconsistent -PFME.
We consider the following examples.
Example 3. 3-polar fuzzy matrix systemis nonsingular, while the extended matrixis singular. This example shows that even though we represent a nonsingular system, then an extended -polar fuzzy matrix system can have infinite or no solutions.
Example 4. Consider the 3-polar fuzzy matrix systemThe extended matrix isand the augmented matrices areSince, , the original system is therefore inconsistent. Examples 3 and 4 show that -PFME exists without a solution for some time. The approximate solution to this -PFME type is essential. If system (7) is not consistent, then the approximate solution we want can be found by reducing some norm of . We often use the least square solution of (7) for an approximation solution that is described by minimizing Frobenius norms ,This means to minimize the sum of the module squares Now, we define the -polar fuzzy LSS to the inconsistent -PFME by Definition 8.
3.1. -Polar Fuzzy Least Square Solution
We analyze from this investigation that the -PFME is inconsistent if of its extended crisp system (7). When -PFME is inconsistent, then the least square solution may be considered. However, the -PFLSS may not have -PFN matrix. We are limiting our conversation to quadruple -PFNs, that is, , and therefore, are all linear functions of . We can then describe the -polar fuzzy solution to the -polar fuzzy matrix by calculating which are solved by system (7).
Definition 9. Let represent the LSS of system (7). The -PFN matrix defined byis called the -FLSS of . If , are all -PFNs, then , and is called a strong -polar fuzzy LSS. Otherwise, is called a weak -polar fuzzy LSS.
4. Least Square Solution of Fuzzy Matrix Equation in -Polar Fuzzy Environment
We analyze the following Lemma.
Lemma 2. Let . A vector is a -polar fuzzy LSS of the extended crisp function linear equation , that is,which is transformed from inconsistent -PFME (4), if and only if
The LSSs of the abovementioned matrix equation may be expressed in this case bywhere is the least squares generalized inverse of matrix , is unit matrix of order , and are arbitrary vectors with parameter . According to Lemma 2 and the hypothesis of generalized inverse theory, we have the following theorems about the LSS for (7).
Theorem 9. Let . The matrix is the LSS of the matrix system (7), if and only if
The general LSS of system (7) of the crisp matrix equation can be defined by the followingwhere is the least squares generalized inverse of the matrix and are any matrices with the parameters .
Proof. First, we consider the crisp matrix equation (7) in block forms of the matrixwhere , . Let , be the LSS of (51). By following the matrix theory , the matrix equations are inconsistent if and only if at least one of the linear equations , is inconsistent. By following Lemma 2, we havewhere are any matrix with the parameter . Since is the LSSs of the linear equation , we havewhereIn fact,