# LU Decomposition Scheme for Solving m-Polar Fuzzy System of Linear Equations

Authors: Ali N. A. Koam; Muhammad Akram; Ghulam Muhammad; Nawab Hussain
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper presents a new scheme for solvingm-polar fuzzy system of linear equations (m-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 m×nm-polar fuzzy matrix equation (m-PFME) and find the least square solution (LSS) of this system by using generalized inverse matrix theory. Moreover, we discuss the strong solution of m-polar fuzzy LSS of the inconsistent m-PFME. In the end, we present a numerical example to illustrate our approach.


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## Body

## 1. Introduction

Certain types of uncertainties arise in several areas of engineering and decision-making. To handle such uncertainties, probability theory, fuzzy set theory [1], 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 [7] 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. [8] described the concept of trapezoidal fuzzy numbers and also other affected investigations were shown in [9, 10]. The notion of m-polar fuzzy set was proposed by Chen et al. [11] as the generalization of bipolar fuzzy set [12]. 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 m-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 generalm-polar fuzzy number rather than a crisp number. A numerical approach that would be suitable to handle and solve a m-polar fuzzy linear system is extremely essential. The notion of m-polar fuzzy linear system with crisp coefficient entries of matrix and the right-hand side is that parametric m-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 m-polar fuzzy environment. It is faster and more numerically stable than computing explicit inverses. LU decomposition method to solve m-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 m-polar fuzzy environment. First we study the basic literature to solve linear system in fuzzy environment. Friedman [18] 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 n×n original system of a fuzzy linear system with the extended 2n×2n system.The iterative scheme to solve the linear system of equation in the formX=AX+U was studied by Wang et al. [19]. Asady et al. [20] developed the general linear system and used an embedding technique to construct the different schemes in a fuzzy environment. Vroman et al. [21] used a parametric technique of fuzzy numbers to find the general solution. Sevastjanov and Dymova [22] introduced a practical approach for interval fuzzy systems. The numerical technique was also studied by Garg and Singh [23] to solve a fuzzy linear system by using the Gaussian fuzzy weight (membership) function. Behera and Chakraverty [24] 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. [40] discussed the solution of linear system in m-polar fuzzy environment. This paper presents a new scheme for solving m-polar fuzzy system of linear equations (m-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 m×nm-polar fuzzy matrix equation (m-PFME) and find the least square solution (LSS) of this system by using generalized inverse matrix theory. Moreover, we discuss the strong solution of m-polar fuzzy LSS of the inconsistent m-PFME. In the end, we present a numerical example to illustrate our approach.The rest of the paper is structured as follows. In Section2, we present the solution of m-polar fuzzy linear system by using LU decomposition method. Section 3 presents the inconsistent m-polar fuzzy matrix equation with some examples and Section 4 develops to obtain the least square solution of m-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.LU Decomposition Method for Solving m-PFSLEs

Definition 1 (see [11]).
Anm-polar fuzzy set on an underlying set Z is a mapping M:Z⟶0,1m. The truthness degree of each element z∈Z is defined as(1)Mz=P1∘Mz,P2∘Mz,…,Pm∘Mz,where Pi∘M:0,1m⟶0,1 is the i-th projection mapping.Definition 2 (see [40]).
Anm-polar fuzzy number (m-PFN) in parametric form is an m-tuple K¯1δi,K¯1δi,K¯2δi,K¯2δi,…,K¯mδi,K¯mδi=K¯iδi,K¯iδi of functions K¯iδi,K¯iδi,δi∈0,1m,i=1,2,…,m, which satisfy the following properties:(i)
K¯iδi is a bounded nondecreasing right continuous function at the point 0 and left continuous over the interval (0, 1](ii)
K¯iδi is a bounded nonincreasing right continuous function at the point 0 and left continuous over the interval (0, 1](iii)
K¯iδi≤K¯iδi
Throughout the paper,i=1,2,3,…,m.Definition 3 (see [40]).
For arbitraryK=K¯iδi,K¯iδi, ℐ=ℐ¯iδi,ℐ¯iδi and a>0, we define K+ℐ, K.ℐ, K/ℐ, and scalar multiplication by a as follows:(i)
Ki+ℐi¯δi=K¯iδi+ℐ¯iδi, Ki+ℐi¯δi=K¯iδi+ℐ¯iδi(ii)
Kiℐi¯δi=minK¯iδiℐ¯iδi,K¯iδiℐ¯iδi,K¯iδiℐ¯iδi,K¯iδiℐ¯iδi(iii)
Kiℐi¯δi=maxK¯iδiℐ¯iδi,K¯iδiℐ¯iδi,K¯iδiℐ¯iδi,K¯iδiℐ¯iδi(iv)
Ki/ℐi¯δi=K¯iδi/ℐ¯iδi, Ki/ℐi¯δi=K¯iδi/ℐ¯iδi, K,ℐ>0(v)
aK¯iδi=aK¯iδi, aK¯iδi=aK¯iδi, a≥0(vi)
aK¯iδi=aK¯iδi, aK¯iδi=aK¯iδi, a<0
The family of allm-PFNs is denoted by Ψ.Definition 4 (see [40]).
Then×n linear system is(2)a11K1+a12K2+⋯+a1nKn=m1i,a21K1+a22K2+⋯+a2nKn=m2i,⋮an1K1+an2K2+⋯+annKn=mni,where the coefficient matrix D=apq,1≤p,q≤n is a crisp n×n matrix and mpi,1≤p≤n,1≤i≤m are known m-PFNs and Kq,1≤q≤n are unknowns which may or may not be m-PFNs, which are called m-PFSLEs.Definition 5.
The matrix system is(3)a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮an1an2⋯annK11K12⋯K1nK21K22⋯K2n⋮⋮⋱⋮Kn1Kn2⋯Knn=m11im12i⋯m1nim21im22i⋯m2ni⋮⋮⋱⋮mn1imn2i⋯mnni,where the coefficient elements apq,1≤p,q≤n are crisp numbers and mpqi in the right-hand matrix are m-PFNs which are called a general m-PFME. By using matrix equation, we have(4)DK=Wi.Definition 6.
Anm-PFN vector K1i,K2i,…,KniT given by Kqiδi=Kqi=K¯qiδi,K¯qiδi,1≤q≤n,δi∈0,1m,i=1,2,…,m, is called a solution of the m-PFSLE (2) if(5)∑1≤q≤napqKq¯=∑1≤q≤napqKq¯=m¯pi,∑1≤q≤napqKq¯=∑1≤q≤napqKq¯=m¯pi.
For a particularp, apq>0,1≤p≤n, we get(6)∑1≤q≤napqK¯q=m¯pi,∑1≤q≤napqK¯q=m¯pi.
From the expression above, we have the following2n×2n crisp linear system:(7)NK=Wi,or(8)N1≥0N2≤0N2≤0N1≥0K¯K¯=W¯iW¯i,apq≥0⟹Np,q=apq,Np+n,q+n=apq,Np,q+n=0,Np+n,q=0,apq<0⟹Np,q=0,Np,q+n=−apq,Np+n,q=apq,Np+n,q+n=0,1≤p,q≤n.
If anyNp,q is not specified, it will be perceived as 0.
So, a system in Definition4 extended to the crisp system (8) where A=N1+N2 and (8) can be written as(9)N1K¯+N2K¯=W¯i,N2K¯+N1K¯=W¯i.
On the base of [18, 41, 42], we investigate the following results.Theorem 1.
The matrixN=N1N2N2N1 is nonsingular if and only if the matrices N1−N2 and N1+N2 are also nonsingular.Definition 7.
IfKqiδ=K¯1i,K¯2i,…,K¯ni,K¯1i,K¯2i,…,K¯niT is a solution of system (7) and holds for each 1≤q≤n the inequalities K¯qi≤K¯qi, then the solution is called a strong system solution (7); otherwise it would be a weak solution of system (7).Theorem 2.
Suppose that a matrixN is nonsingular and a unique solution of NKi=Wi always gives m-polar fuzzy number for arbitrary vector Wi ; then the necessary and sufficient condition for the inverse of nonnegative matrix N exists.Theorem 3.
LetPn×n be a matrix that contains all real entries except 0 for leading minors. Then, the matrix Pn×n has a unique factorization:(10)P=LU,where L and U are unit lower-triangular and upper-triangular matrices, respectively.To eliminate theN reduction, the L and U matrices must be found such that N=LU, which is(11)L=L11OL21L22,U=U11U12OU22,where L11,L22 and U11,U22 are lower- and upper-triangular matrices, L21 and U12 are any matrices, and O is null matrix, respectively.Now, we suppose thatP=N1+N2 has LU decomposition. We have(12)N=N1N2N2N1=L11OL21L22U11U12OU22,and then(13)N1=L11U11,N2=L11U12⟹U12=L11−1N2,N2=L21U11⟹L21=N2U11−1,N1=L21U12+L22U22.Therefore, we can write(14)N1−N2N1−1N2=L22U22.Theorem 4.
LetNn×n be a positive definite symmetric matrix; then there is a unique matrix L with positive diagonal entries such that(15)N=LLT.Proof.
Suppose thatNn×n is a positive definite symmetric matrix. We have(16)N=N1N2N2N1=L11OL21L22L11TL21TOL22T,and then(17)N1=L11L11T,N2=L11L21T⟹L21T=L11−1N2,N2=L21L11T⟹L21=N2L11T−1,N1=L21U12+L22U22.
Therefore, we can write(18)N1−N2N1−1N2=L22U22T.
By following Theorem4 in LU decomposition method, N1 and N1−N2N1−1N2 should be a positive definite symmetric matrix.
We solve some numerical examples to illustrate our scheme.Example 1.
Consider2×23-polar fuzzy system(19)3K1−2K2=35+5δ,45−5δ,21+5δ,31−5δ,17+4δ,25−4δ,K1+4K2=9+6δ,21−6δ,13+2δ,17−2δ,16+3δ,22−3δ.
The extended4×4 matrix is(20)N=300−214000−2300014,N1=3014=1.000000.33331.00003004,N1−N2N1−1N2=3.00000.33331.00004.0000=1.000000.33331.00003.00000.333303.8889,and hence(21)N=1.00000000.33331.0000000−0.50001.00000000.33331.00003.000000−2.000004.000000.6666003.00000.33330003.8889.
The exact solution is(22)K1δ=44735+4δ5,50335−4δ5,27935+8δ5,39135−8δ5,547+δ,687−δ,K2δ=−3335+13δ10,5835−13δ10,4435+δ10,5135−δ10,2914+δ2,4314−δ2.
The exact and derived solutions withLU decomposition z11,z21 of 3-PFN are plotted in Figures 1 and 2.
The Hausdorff norm of errors is 6.3750e−008.
The exact and derived solutions withLU decomposition z12,z22 of 3-PFN are plotted in Figures 3 and 4.
The Hausdorff norm of errors is 6.3750e−008.
The exact and derived solutions withLU decomposition z13,z23 of 3-PFN are plotted in Figures 5 and 6.
The Hausdorff norm of errors is 6.3750e−008.Figure 1
Exact solution.Figure 2
The solution withLU decomposition method.Figure 3
Exact solution.Figure 4
The solution withLU decomposition method.Figure 5
Exact solution.Figure 6
The solution withLU decomposition method.Example 2.
Consider the3×33-polar fuzzy system:(23)6K1+2K2−4K3=44+16δ,76−16δ,24+11δ,46−11δ,34+16δ,66−16δ,2K1+16K2+12K3=86+24δ,134−24δ,66+19δ,104−19δ,76+24δ,124−24δ,−4K1+12K2+24K3=62+38δ,138−38δ,42+33δ,108−33δ,52+37δ,126−37δ.
The extended6×6 symmetric positive definite matrix is(24)N=62000−42161200001224−40000−462000021612−40001224,N1=6202161201224=2.4495000.81653.9158003.06453.82212.44950.8165003.91583.0645003.8221,N1−N2N1−1N2=4.90482.0000−0.28572.000016.000012.0000−0.285712.000021.1429=2.2147000.90313.89670−0.12903.10943.38492.21470.9031−0.129003.89673.1094003.3849,and hence N=LLT, where(25)L=2.4495000000.81653.9158000003.06453.822100000−1.04652.2147000000.90313.89670−1.63300.3405−0.2730−0.12903.10943.3849.
Now, the exact solution is(26)K1δ=1377112+29δ16,1783112−29δ16,1479224+33δ32,1941224−33δ32,4145224+59δ32,2971224−59δ32,K2δ=−17+δ2,67−δ2,8156+δ4,10956−δ4,121280+11δ20,429280−11δ20,K3δ=1189224+33δ32,1651224−33δ32,1107448+69δ64,2073448−69δ64,93212240+307δ320,136192240−307δ320.
The exact and derived solutions withLLT decomposition z11,z21,z31 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 LLT decomposition z12,z22,z32 of 3-PFN are plotted in Figures 9 and 10.
The Hausdorff norm of errors is 3.9705e−004.
The exact and derived solutions withLLT decomposition z13,z23,z33 of 3-PFN are plotted in Figures 11 and 12.
The Hausdorff norm of errors is 3.9705e−004.Figure 7
Exact solution.Figure 8
The solution withLU decomposition method.Figure 9
Exact solution.Figure 10
The solution withLU decomposition method.Figure 11
Exact solution.Figure 12
The solution withLU decomposition method.Lemma 1 (see [43]).
The solution ofm-PFSLEs exists if and only if the rank of N is equal to that of matrix N,mi; that is,(27)RankN=RankN,mi.WhenRankN=RankN,mi=2n, the system has a unique solution. There are endless solutions to the system if RankN,mi=RankN<2n and there is no solution if RankN,mi>RankN.Theorem 5.
The systemNK=Wiδ has solution if and only if(28)RankN=RankN,miδ,0≤δ≤1.Proof.
LetK=K1,K2,…,Kn, mδi=K1δ,K2δ,…,Kqδ, where(29)Kp=K¯1pδ,K¯2pδ,…,K¯npδ,−K¯1pδ,−K¯2pδ,…,−K¯npδT,mpiδ=m¯1piδ,m¯2piδ,…,m¯npiδ,−m¯1piδ,−m¯2piδ,…,−m¯npiδT,p=1,2,…,q,δ∈0,1.
The matrices below are the same:(30)NK=Wiδ≅NKp=mpiδ,p=1,2,3,…,q.
IfRankN,Wiδ=RankN, then we have RankN,mpiδ=RankN,∀p, since RankN,Wiδ≥RankN,mpiδ≥RankN. By Lemma 1, all linear equations NKp=mpiδ, p=1,2,3,…,q, have solutions. So it makes sense to have the necessary condition.
Conversely, suppose thatNK=Wiδ is solvable; in other words, every linear equation NKp=mpiδ, p=1,2,3,…,q, has solution. Let(31)Kp=K¯1pδ,K¯2pδ,…,K¯npδ,−K¯1pδ,−K¯2pδ,…,−K¯npδT,p=1,2,…,q,δ∈0,1,N=s1,s2,s3,…,s2n,where sp=s1p,s2p,s3p,…,s2n,pT, p=1,2,3,…,2n.
By using equationNKp=mpiδ, p=1,2,3,…,q, we have(32)K¯1pδs1,K¯2pδs2,…,K¯npδsn,−K¯1pδsn+1,−K¯2pδsn+1,…,−K¯npδs2n=mpiδ.
From the above equation, it follows thatmpiδ can be expressed as linear combination of s1,s2,s3,…,s2n; that is, from this equation, the following is that a linear combination of mpiδ can be expressed as s1,s2,s3,…,s2n:(33)RankN=RankN,mpiδ,p=1,2,…,q,RankN=RankN,Wiδ.
From Theorem5, we can deduce the following result about the solvability of (7).Theorem 6.
The equation has an equivalent solution to (7):(34)NKp=mpiδ,∀p=1,2,3,…,q.Theorem 7.
Equation (7) has solution in which the necessary and sufficient conditions for the rows of Wiδ have the same linear relation as the rows of the N matrix.Theorem 8.
If (7) has no solution, then the corresponding m-PFME also has no solution.Corollary 1.
Consider the condition(35)RankN,Wiδ=RankN.IfRankN=2n, 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 m-PFSLE:(36)NKp=mpiδ,∀p=1,2,3,…,q,which has only one solution.


## 3. Inconsistentm-Polar Fuzzy Matrix Equation

Definition 8.
If the crisp matrix equation (7) has no solution, then the associated m-PFME is(37)a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮am1m2⋯amnK11K12⋯K1pK21K22⋯K2p⋮⋮⋱⋮Kn1Kn2⋯Knp=m11im12i⋯m1pim21im22i⋯m2pi⋮⋮⋱⋮mm1imm2i⋯mmpi,where the coefficient matrix D=apqi,1≤p≤m,1≤q≤n, is crisp matrix, the right-hand matrix Wi=mpqi is m-PFN called an inconsistent m-PFME.
We consider the following examples.Example 3.
3-polar fuzzy matrix system(38)77777−77−7−7K11K12K13K21K22K23K31K32K33=5+3δ,11−3δ,16+2δ,20−2δ,6+4δ,14−4δ15+4δ,23−4δ,11+3δ,17−3δ,22+6δ,34−6δ17+5δ,27−5δ,13+3δ,19−3δ,7+δ,9−δ4+δ,6−δ,13+4δ,21−4δ,32+6δ,44−6δ25+2δ,29−2δ,35+7δ,49−7δ,14+5δ,24−5δ8+2δ,12−2δ,23+3δ,29−3δ,19+5δ,29−5δ15+6δ,27−6δ,26+3δ,32−3δ,8+δ,10−δ25+3δ,31−3δ,19+4δ,27−4δ,7+3δ,13−3δ18+2δ,22−2δ,28+5δ,38−5δ,9+4δ,17−4δ,is nonsingular, while the extended 6×6 matrix(39)N=777000770007700077000777007770077700,is singular. This example shows that even though we represent a nonsingular system, then an extended m-polar fuzzy matrix system can have infinite or no solutions.Example 4.
Consider the 3-polar fuzzy matrix system(40)77−77−77−777K11K12K13K21K22K23K31K32K33=13+2δ,17−2δ,23+3δ,29−3δ,5+δ,7−δ8+4δ,16−4δ,21+5δ,31−5δ,32+2δ,36−2δ14+5δ,24−5δ,23+4δ,31−4δ,17+δ,19−δ14+3δ,20−3δ,33+4δ,41−4δ,12+6δ,22−6δ5+2δ,9−2δ,15+7δ,29−7δ,24+5δ,34−5δ18+2δ,22−2δ,13+3δ,19−3δ,29+5δ,39−5δ15+6δ,27−6δ,22+3δ,28−3δ,38+δ,40−δ15+4δ,23−4δ,29+4δ,37−4δ,17+3δ,23−3δ28+2δ,32−2δ,8+5δ,18−5δ,19+4δ,27−4δ.
The extended6×6 matrix is(41)N=770007707070077700007770070707700077,and the augmented matrices are(42)N,W1δ=77000713+2δ8+4δ14+5δ70707014+3δ5+2δ18+2δ07770015+6δ15+4δ28+2δ0077702δ−174δ−165δ−240707073δ−202δ−92δ−227000776δ−274δ−232δ−32,N,W2δ=77000723+3δ21+5δ23+4δ70707033+4δ15+7δ13+3δ07770022+3δ29+4δ8+5δ0077703δ−295δ−314δ−310707074δ−417δ−293δ−197000773δ−284δ−375δ−18,N,W3δ=7700075+δ32+2δ17+δ70707012+6δ24+5δ29+5δ07770038+δ17+3δ19+4δ007770δ−72δ−36δ−190707076δ−225δ−345δ−39700077δ−403δ−234δ−27.
Since,RankN≠RankN,Wiδ,i=1,2,3, the original system is therefore inconsistent. Examples 3 and 4 show that m-PFME exists without a solution for some time. The approximate solution to this m-PFME type is essential. If system (7) is not consistent, then the approximate solution we want can be found by reducing some norm of Wiδ−NKδ. We often use the least square solution of (7) for an approximation solution that is described by minimizing Frobenius norms Wiδ−NKδ,(43)Wiδ−NKδF=min,δ∈0,1.
This means to minimize the sum of the module squaresWiδ−NKδ(44)Wiδ−NKδF2=∑q=1q∑p=1um¯p,qiδ−∑t=1nsptK¯t,qδ−sp,n+tK¯t,qδ2+m¯p,qiδ+∑t=1nsp+n,tK¯t,qδ−sp+n,n+tK¯t,qδ2,δ∈0,1.
Now, we define them-polar fuzzy LSS to the inconsistent m-PFME by Definition 8.


### 3.1.m-Polar Fuzzy Least Square Solution

We analyze from this investigation that them-PFME is inconsistent if RankN≠RankN,miδ of its extended crisp system (7). When m-PFME is inconsistent, then the least square solution may be considered. However, the m-PFLSS may not have m-PFN matrix. We are limiting our conversation to quadruple m-PFNs, that is, m¯pqiδ,m¯pqiδ,1≤p≤m,1≤q≤n, and therefore, K¯pqδ,K¯pqδ are all linear functions of r. We can then describe the m-polar fuzzy solution to the m-polar fuzzy matrix by calculating K which are solved by system (7).Definition 9.
Letℛδ=K¯pqδ,−K¯pqδ,1≤p≤m,1≤q≤n represent the LSS of system (7). The m-PFN matrix Uδ=t¯pqδ,t¯pqδ,1≤p≤m,1≤q≤n defined by(45)t¯pqδ=mint¯pqδ,t¯pqδ,t¯pq1,t¯pq1,t¯pqδ=maxt¯pqδ,t¯pqδ,t¯pq1,t¯pq1,is called the m-FLSS of NK=Wiδ. If K¯pqδ,K¯pqδ,1≤p≤m,1≤q≤n, are all m-PFNs, then t¯pqδ=K¯pqδ,t¯pqδ=K¯pqδ,1≤p≤m,1≤q≤n, and Uδ is called a strong m-polar fuzzy LSS. Otherwise, Uδ is called a weak m-polar fuzzy LSS.


## 3.1.m-Polar Fuzzy Least Square Solution

We analyze from this investigation that them-PFME is inconsistent if RankN≠RankN,miδ of its extended crisp system (7). When m-PFME is inconsistent, then the least square solution may be considered. However, the m-PFLSS may not have m-PFN matrix. We are limiting our conversation to quadruple m-PFNs, that is, m¯pqiδ,m¯pqiδ,1≤p≤m,1≤q≤n, and therefore, K¯pqδ,K¯pqδ are all linear functions of r. We can then describe the m-polar fuzzy solution to the m-polar fuzzy matrix by calculating K which are solved by system (7).Definition 9.
Letℛδ=K¯pqδ,−K¯pqδ,1≤p≤m,1≤q≤n represent the LSS of system (7). The m-PFN matrix Uδ=t¯pqδ,t¯pqδ,1≤p≤m,1≤q≤n defined by(45)t¯pqδ=mint¯pqδ,t¯pqδ,t¯pq1,t¯pq1,t¯pqδ=maxt¯pqδ,t¯pqδ,t¯pq1,t¯pq1,is called the m-FLSS of NK=Wiδ. If K¯pqδ,K¯pqδ,1≤p≤m,1≤q≤n, are all m-PFNs, then t¯pqδ=K¯pqδ,t¯pqδ=K¯pqδ,1≤p≤m,1≤q≤n, and Uδ is called a strong m-polar fuzzy LSS. Otherwise, Uδ is called a weak m-polar fuzzy LSS.


## 4. Least Square Solution of Fuzzy Matrix Equation inm-Polar Fuzzy Environment

We analyze the following Lemma.Lemma 2.
LetN∈R2n×2n. A vector Kδ is a m-polar fuzzy LSS of the extended crisp function linear equation NK=Wiδ, that is,(46)s11s12⋯s1,2ns21s22⋯s2,2n⋮⋮⋱⋮s2n,1s2n,2⋯s2n,2nK¯1δ⋮K¯nδ−K¯1δ⋮−K¯nδ=m¯1iδ⋮m¯miδ−m¯1iδ⋮−m¯miδ,which is transformed from inconsistent m-PFME (4), if and only if(47)NK=KN1,3miδ.The LSSs of the abovementioned matrix equation may be expressed in this case by(48)Kδ=N1,3miδ+ℬ2n−N1,3Nziδ,where N1,3 is the least squares generalized inverse of matrix N, ℬ2n is unit matrix of order 2n, and ziδ are arbitrary vectors with parameter δ. According to Lemma 2 and the hypothesis of generalized inverse theory, we have the following theorems about the LSS for (7).Theorem 9.
LetN∈R2n×2n. The matrix Kδ is the LSS of the matrix system (7), if and only if(49)NK=KN1,3miδ.The general LSS of system (7) of the crisp matrix equation can be defined by the following(50)Kδ=N1,3miδ+ℬ2n−N1,3Nziδ,where N1,3 is the least squares generalized inverse of the matrix N and ziδ are 2m×n any matrices with the parameters δ.Proof.
First, we consider the crisp matrix equation (7) in block forms of the matrix(51)NKqδ=Wqiδ,q=1,2,3,…,n,where Kδ=K1δ,…,Knδ, Wiδ=m1iδ,…,mniδ. Let Nq∗δ,q=1,2,3,…,n, be the LSS of (51). By following the matrix theory [44], the matrix equations NKδ=Wiδ are inconsistent if and only if at least one of the linear equations NKqδ=Wqiδ,q=1,2,3,…,n, is inconsistent. By following Lemma 2, we have(52)NKq∗δ=KN1,3Wqiδ,Kq∗δ=N1,3Wqiδ+ℬ2n−N1,3NZiδ,where Ziδ are 2m×n any matrix with the parameter δ. Since Kq∗δ is the LSSs of the linear equation NKqδ=Wqiδ,q=1,2,3,…,n, we have(53)Wqiδ−NKq∗δF2=minWqiδ−NKqδF2,q=1,2,3,…,n,where(54)Wqiδ−NKq∗δF2=∑p=1um¯p,qiδ−∑t=1nsptK¯t,qδ−sp,n+tK¯t,qδ2+m¯p,qiδ+∑t=1nsp+m,tK¯t,qδ−sp+m,n+tK¯t,qδ2,δ∈0,1.
In fact,(55)Wiδ−NKF2=∑q=1nWqiδ−NKqδF2.
The expression(56)Wiδ−NK∗δF2=minWiδ−NKδF2,δ∈0,1,holds which corresponds to the following conditions:(57)NKq∗δ−WqiδF2=minNqKqδ=WiδF2,where(58)K∗δ=K1∗δ,K2∗δ,K3∗δ,…,Kn∗δ,Wiδ−NKF2=∑q=1n∑p=1um¯p,qiδ−∑t=1nsptK¯t,qδ−sp,n+tK¯t,qδ2+m¯p,qiδ+∑t=1nsp+m,tK¯t,qδ−sp+m,n+tK¯t,qδ2,δ∈0,1.
Thus, the matrix(59)K∗δ=K1∗δ,K2∗δ,K3∗δ,…,Kn∗δ,is the LSS of (7). The following results are significant based on the operation of block forms of the matrix(60)NK∗δ=KN1,3Wiδ,where(61)Kq∗δ=N1,3mqiδ+ℬ2n−N1,3NZδ.Remark 1.
It is observed that the LSS is unique only when the full rank isN; i.e., the matrix equation LSS (7) is(62)Kδ=N⊤N−1N⊤Wiδ,RankN=2n,N⊤NN⊤−1Wiδ,RankN=2n.
Otherwise, an infinite set of such solutions can occur in (7).Theorem 10.
Among the general LSSs to system (7),(63)Kδ=N†Wiδ,is one of the minimum norms, where N† is the Moore–Penrose inverse of the matrices N. We know that it is unique N†. System (63) is, therefore, unique. Since the LSS is shown as an m-polar fuzzy matrix, the general inverse of the N matrix is now considered in an exceptional structure. And(64)N=N1N2N2N1.We must follow the following statement.Lemma 3.
LetN be in the form of (64). Then, the matrix(65)N1,3=12N1+N21,3+N1−N21,3N1+N21,3−N1−N21,3N1+N21,3−N1−N21,3N1+N21,3+N1−N21,3,is 1,3-inverse of the matrix N, where N1+N21,3,N1−N21,3 are 1,3-inverse of matrices N1+N2 and N1−N2, respectively. In particular, the Moore–Penrose inverse of the matrix N is(66)N†=12N1+N2†+N1−N2†N1+N2†−N1−N2†N1+N2†−N1−N2†N1+N2†+N1−N2†.The following theorem provided the necessary or sufficient condition for the LSS matrix to them-PFN matrix, given with the arbitrary input m-PFN Wiδ, and the next theorem provides the sufficient condition for one LSS to be the m-PFN matrix.Theorem 11.
For the inconsistentm-PFME (7) and any least squares inverse N1,3 of the coefficient matrix N, the expression Kδ=N1,3Wiδ is the solution of system and, therefore, it admits a strong or weak m-polar fuzzy LSS. Specifically, if N1,3 is nonnegative with the structure (65), the expression Kδ=N1,3Wiδ admits the strong m-polar fuzzy solution for arbitrary m-polar fuzzy matrices Wiδ.Proof.
From Theorem9 and the theory of generalized inverses, the expression is the LSS to the inconsistent m-PFME (7). We used Theorem 9 and one LSS of Definition 5 accompanied by a solution of (7) (from the previous analysis in Theorem 9). Therefore, by Definition 9, it admits a strong or weak m-polar fuzzy LSS. It is sufficient to prove this theorem where the m-PFN definition is set to Kδ. To prove this theorem, it is enough to show that the definition of bipolar fuzzy number holds for Kδ. Let(67)Kδ=K¯iδ,K¯iδT,Wiδ=W¯iδ,−W¯iδT.
We denote(68)N1,3=XYYX,
and we can obtain LSS of system (7), i.e.,(69)K¯iδK¯iδ=N1,3Wiδ=XYYXW¯iδ−W¯iδ=XW¯iδ−YW¯iδXW¯iδ−YW¯iδ.
Since,
X+Y,W¯iδ−W¯iδ,δ∈0,1, are nonnegative, (70)K¯iδ−K¯iδ=X+YW¯iδ−W¯iδ≥0.
SinceW¯iδ is nondecreasing and W¯iδ is nonincreasing, also the bounded left continuity of K¯iδ,K¯iδ is quite simple, and they are in the form of the linear combinations W¯iδ,W¯iδ.Remark 2.
From Theorem11, if N has a least squares generalized inverse N1,3 such as (65) with N1,3≥0, the system Kδ=N1,3Wiδ has strong m-polar fuzzy LSS. Specifically, if N† (Moore–Penrose inverse) such as (66) are nonnegative, the system Kδ=N†Wiδ has also strong m-polar fuzzy LSS. By Theorem 10, it is the minimum norm m-polar fuzzy LSS. Now we are providing a few result for such N1,3 and N† are nonnegative. Usually, ⋅⊤ denotes the transpose of a matrix ⋅.Theorem 12 (see [45]).
The matrixN of rank l except zero columns or rows, which admit the condition N1,3≥0, is necessary and sufficient where there exist certain permutation matrices D,E such that(71)DNE=L,∗,where the direct sum of l positive is L, and matrices are ranked one.Theorem 13 (see [46]).
LetN† be the nonnegative matrix inverse of N, if and only if(72)N†=SX⊤SY⊤SY⊤SX⊤,for some positive diagonal matrix S. In this case,(73)X+Y⊤=SX+Y⊤,X−Y⊤=SX−Y⊤.Example 5.
Consider the following 3-polar fuzzy systems:(74)2K11−2K212K12−2K22−2K11+2K21−2K12+2K22=10+5δ,20−5δ,7+3δ,13−3δ,8+4δ,16−4δ15+3δ,21−3δ,17+6δ,29−6δ,9+5δ,19−5δ6+2δ,10−2δ,3+δ,5−δ,12+4δ,20−4δ8+5δ,18−5δ,11+2δ,15−2δ,6+3δ,12−3δ.
The extended4×4 matrix N is(75)N=2002022002202002.
The augmented matrices form=1,2,3 are(76)N,W1δ=200210+5δ15+3δ02206+2δ8+5δ02205δ−203δ−2120022δ−105δ−18,N,W2δ=20027+3δ17+6δ02203+δ11+2δ02203δ−136δ−292002δ−52δ−15,N,W3δ=20028+4δ9+5δ022012+4δ6+3δ02204δ−165δ−1920024δ−203δ−12.
SinceRankN≠RankN,Wiδ, the original system is inconsistent. One 1,3-inverse of N is(77)N1,3=122002000002200000,which is nonnegative, and the corresponding solution is given by(78)K1=K111K121K211K221=7δ,14−7δ−3+8δ,13−8δ0,00,0,K2=K112K122K212K222=2+4δ,10−4δ2+8δ,18−8δ0,00,0,K3=K113K123K213K223=−12+8δ,4−8δ−3+8δ,13−8δ0,00,0,and it is a strong 3-polar fuzzy LSS plotted in Figures 13–15.
We havea=K1,b=K2,c=K3. The Moore–Penrose inverse of N is(79)N†=0.5000000.500000.50000.5000000.50000.500000.5000000.5000,which is nonnegative. Therefore the original system has a strong 3-polar fuzzy solution:(80)K1=K¯111δK¯121δK¯211δK¯221δ−K¯111δ−K¯121δ−K¯211δ−K¯221δ=N†W1δ=0.5000000.500000.50000.5000000.50000.500000.5000000.500010+5δ15+3δ6+2δ8+5δ5δ−203δ−212δ−105δ−18,K1=K111K121K211K221=7δ2,7−7δ2−32+4δ,132−4δ−7+7δ2,−7δ2−132+4δ,32−4δ,K2=K¯112sK¯122sK¯212sK¯222s−K¯112s−K¯122s−K¯212s−K¯222s=N†W2δ=0.5000000.500000.50000.5000000.50000.500000.5000000.50007+3δ17+6δ3+δ11+2δ3δ−136δ−29δ−52δ−15,K2=K112K122K212K222=1+2δ,5−2δ1+4δ,9−4δ−5+2δ,−1−2δ−9+4δ,−1−4δ,K3=K¯113δK¯123δK¯213δK¯223δ−K¯113δ−K¯123δ−K¯213δ−K¯223δ=N†W3δ=0.5000000.500000.50000.5000000.50000.500000.5000000.50008+4δ9+5δ12+4δ6+3δ4δ−165δ−194δ−203δ−12,K3=K113K123K213K223=−6+4δ,2−4δ−32+4δ,132−4δ−2+4δ,6−4δ−132+4δ,32−4δ.which leads to the minimum norm 3-polar fuzzy LSS plotted in Figures 16–18. We have d=K1,e=K2,f=K3.Figure 13
3-polar fuzzy LSS (a).Figure 14
3-polar fuzzy LSS (b).Figure 15
3-polar fuzzy LSS (c).Figure 16
Moore–Penrose bipolar fuzzy LSS (d).Figure 17
Moore–Penrose 3-polar fuzzy LSS (e).Figure 18
Moore–Penrose 3-polar fuzzy LSS (f).Remark 3.
Notice that Figures1–3 and 4–6 are plotted to show the differences of the solutions using 1,3-inverse and Moore–Penrose inverse of the matrix N, respectively. Moreover, we obtain the strong m-polar fuzzy least square solution for m=1,2,3 by using 1,3-inverse and Moore–Penrose inverse of the matrix N.


## 5. Conclusion

We have solvedm-polar fuzzy system of linear equations by using LU decomposition method. We have analyzed that if the matrices N1 and N1−N2N1−1N2 have LU or LLT decomposition, then N is also decomposition, and if N is positive definite symmetric matrix, then it has LLT decomposition. The solvability of the LU decomposition method has been discussed in detail and the concept of inconsistent m-PFME was presented. Moreover, we have discussed a class of inconsistent m-PFMEs DK=Wi in which D is an m×n crisp matrix, and the right-hand side vector Wi is m×n arbitrary m-PFN matrix. We also found the m-polar fuzzy least square solution of m-polar fuzzy inconsistent matrix by using the theory of generalized inverse matrix on N. Finally, the strong m-polar fuzzy LSS has been obtained and we illustrated this concept with an example. In the future, this work can help to determine the flow rate of the traffic on the road by using m-polar fuzzy linear system of equations. Moreover, this work can be used in circuit analysis to balance the flow of current in circuit of the system [47–51].



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