We present a method for solving fully fuzzy linear systems using interval aspects of fuzzy numbers. This new method uses a decomposition technique to convert a fully fuzzy linear system into two types of decomposition in the form of interval matrices. It finds the solution of a fully fuzzy linear system by using interval operations. This new method uses interval arithmetic and two new interval operations ⊖ and ⊘. These new operations, which are inverses of basic interval operations + and ×, will be presented in the middle of this paper. Some numerical examples are given to illustrate the ability of proposed methods.
1. Introduction
In various fields of sciences, in solving real-life problems, where the system of linear equations is noticed, there are situations where the values of the parameters cannot be stated exactly, but their estimation or some bounds on them can be measured. Modeling a lot of these problems leads to a fuzzy linear system because the inexact kind of real numbers can be modeled in fuzzy numbers.
Fuzzy linear systems have been studied by several authors. Friedman et al. [1] proposed a general model for solving an n×n fuzzy linear system whose coefficient matrix is crisp and the right-hand side column is an arbitrary fuzzy vector. Allahviranloo [2] proposed solution of a fuzzy linear system by using an iterative method and later suggested various numerical methods to solve fuzzy linear systems [3]. Some methods for solving these systems can be found in [4–7]. Later, a system, called fully fuzzy linear system (FFLS), is introduced wherein elements in the coefficient matrix, the right-hand side vector, and the vector of unknowns are fuzzy numbers.
This system is solved in [8, 9] using decomposition of the coefficient matrix. Some other methods used iterative methods for solving FFLS [3, 10, 11]. Many researchers studied FFLS and numerical techniques to solve them [12–16]. Kochen et al. proposed a method for solving square and nonsquare FFLS with trapezoidal fuzzy numbers in 2016 [17]. In 2017, Edalatpanah reviewed some iterative methods for solving FFLS [18]. Recently, Allahviranloo and Babakordi introduced a new method for solving an extension of FFLS that has two coefficient matrices [19]. A new method for solving FFLS with hexagonal fuzzy numbers is introduced by Malkawi et al. in 2018 [20].
In Section 2, we present some basics about fuzzy numbers and α-cut arithmetic. Some basics of interval arithmetic, new interval operations and some properties of interval numbers, and also our method for solving FFLS are investigated in Section 3. In Section 4, some examples are solved to show the effectiveness of the proposed method and concluding remarks are contained in Section 5.
2. Preliminaries
In this section we review fuzzy numbers and α-cut arithmetic on them. α-cut method is a method for performing interval operations like addition, multiplication, and subtraction on fuzzy numbers. α-cut fuzzy arithmetic is described in detail in [21].
Let us consider an arbitrary trapezoidal fuzzy number u=(u1,u2,u3,u4), where u1,u2,u3,u4∈R and u1≤u2≤u3≤u4. The membership function μu of u is defined as follows:(1)μux=0u1>xx-u1u2-u1u1≤x≤u21u2≤x≤u3u4-xu4-u3u3≤x≤u40u4<x.Two trapezoidal fuzzy numbers u and v are said to be equal, if and only if ui=vi for 1≤i≤4.
Given a trapezoidal fuzzy number u and membership function μu and a real number α∈{0,1}, then the α-cut of a fuzzy number u is an interval, which is defined as(2)uα=uα̲,uα¯,where uα̲, uα¯ are(3)uα̲=infx∈R∣μux≥α,(4)uα¯=supx∈R∣μux≥α.Note that (5)u=⋃α∈0,1uα.The basic arithmetic of two fuzzy numbers u and v is discussed in [21–23] based on interval arithmetic(6)u+vα=uα+vα=uα̲,uα¯+vα̲,vα¯,(7)u-vα=uα-vα=uα̲,uα¯-vα̲,vα¯,(8)uvα=uα×vα=uα̲,uα¯×vα̲,vα¯.Clearly, for a trapezoidal fuzzy number u=(u1,u2,u3,u4), we have u0=[u1,u4], u1=[u2,u3].
A matrix (vector) A is called a fuzzy matrix (fuzzy vector), if at least one element of matrix (vector) A is a fuzzy number. A matrix A is called a fully fuzzy matrix, whose elements of A are fuzzy numbers. An α-cut of an n×m matrix A is an interval matrix with the same dimension whose elements are α-cut of elements of matrix A.
3. Materials and Methods3.1. Interval Arithmetic
In this section, we briefly explain interval arithmetic and notion. Also, we present a new operation on interval arithmetic.
An interval number â is the bounded, closed subset of real numbers, which is indicated by a hat, defined by â=[a̲,a¯], where a̲,a¯∈R and a̲≤a¯. Let IR denote the set of such intervals. An interval â is a degenerate interval when a̲=a¯.
Infimum and supremum of an interval number â are a̲ and a¯, respectively. Let a∈â, say a∈â when a̲≤a≤a¯.
Let ∘ denote a binary operation on real space R. For â,b̂∈IR(9)â∘b̂=a∘b∣a∈â,b∈b̂.Then, for basic arithmetic operations {+,-,×,/} in IR, we have(10)â+b̂=a̲+b̲,a¯+b¯(11)â-b̂=a̲-b¯,a¯-b̲(12)â×b̂=a̲b̲,a¯b¯a̲≥0,b̲≥0a¯b̲,a¯b¯a̲≥0,b̲<0<b¯a¯b̲,a̲b¯a̲≥0,b¯≤0a̲b¯,a¯b¯a̲<0<a¯,b̲≥0a¯b̲,a̲b̲a̲<0<a¯,b¯≤0a̲b¯,a¯b̲a¯≤0,b̲≥0a̲b¯,a̲b̲a¯≥0,b̲<0<b¯a¯b¯,a̲b̲a¯≤0,b¯≤0a¯b¯,a̲b̲a¯≤0,b¯≤0mina¯b̲,a̲b¯,maxa̲b̲,a¯b¯a̲<0<a¯,b̲<0<b¯=mina̲b̲,a̲b¯,a¯b̲,a¯b¯,maxa̲b̲,a̲b¯,a¯b̲,a¯b¯.(13)âb̂=â×1b̂=â×1b¯,1b̲where0∉b̂IR+ is the set of all improper intervals which is defined in [24] as follows: (14)IR+=â=a̲,a¯∣a̲,a¯∈R,a̲≥a¯.The set(15)IR∗=â=a̲,a¯∣a̲,a¯∈Rcontains all interval numbers and improper interval numbers called general interval. The basic arithmetic of general intervals is defined similar to intervals, but the × operation has some differences that are explained in [25].
By an interval matrix, we mean a matrix whose elements are interval numbers. Let  be an interval matrix with elements Âij and let B be a matrix with real elements Bij; we consider B∈ if Bij∈Âij for all i and j. Basic arithmetic operations on interval matrices are defined similar to real matrices [26].
3.2. New Interval Operations
Let â,b̂∈IR∗; we define two new operations ⊖ and ⊘ as follows:(16)â⊖b̂=a̲-b̲,a¯-b¯,(17)â⊘b̂=-∞,+∞a̲=a¯=0,b̲=b¯=00,0a̲=a¯=0,b̲≠0orb¯≠0undefined∃x≠0,x∈â,b̲=b¯=0undefined0∉a,0∈ba̲b̲,a¯b¯a̲≥0,b̲>00,a¯b¯a̲=0,b̲=0a̲b¯,a¯b¯a̲≤0≤a¯,b̲≥0a̲b¯,a¯b̲a¯≤0,b̲>0a̲b¯,0a¯=0,b̲=0â⊘-b̂b¯≤0a̲b¯,a¯b¯a̲≥0,b̲<0<b¯maxa̲b¯,a¯b̲,mina¯b¯,a̲b̲a̲≤0≤a¯,b̲<0<b¯a¯b̲,a̲b̲a¯≤0,b̲<0<b¯The operation ⊖ is defined and used in some articles as an operation in extended interval arithmetic [27]. But, we use ⊖ in a different way. We illustrate that the operation ⊖ can act as an inverse of + in interval arithmetic.
Theorem 1.
Considering â,b̂∈IR, the interval equation â+x̂=b̂ has a unique solution x̂=b̂⊖â and x̂∈IR∗.
Supposing x̂=b̂⊖â and x̂∈IR∗ and x̂∉IR, the equation â+x̂=b̂ has no solution in IR but has exactly one solution in IR∗.
Corollary 3.
If â,b̂∈IR and ĉ=â+b̂, then b̂ can be computed by ĉ⊖â uniquely. On the other hand, the operation ⊖ is the inverse operation of + in interval arithmetic.
Theorem 4.
Considering â,b̂∈IR, the interval equation â×x̂=b̂ has a solution x̂=b̂⊘â, where x̂∈IR∗.
Remark 5.
Solution x̂ is not usually unique; e.g., the interval equation [-1,1]×x̂=[-2,2] has solutions [-1,2],[-2,0],[-2,2]. Actually this equation has unlimited solutions. All intervals [u,2] and [-2,v] where -2≤u≤0 and 0≤v≤2 can be solution of this equation.
In cases that x̂ is not unique, we consider x̂ as the longest interval as possible. In the above example, x̂ will be [-2,2].
Proof.
Let â×x̂=b̂; Table 1 shows sign of b̂ related to signs of â and x̂ and also shows formulas of multiplication result.
Now, given â,b̂∈IR such that â×x̂=b̂, a¯≤0 and b̲≤0≤b¯. From Table 1, it is deduced that 0∈x̂ and b̂=[a¯x̲,a¯x¯]. Therefore (20)âx̂=b̂⇔b̂=a̲x¯,a̲x̲⇔b̲=a̲x¯⇔x¯=b̲x̲b¯=a̲x̲⇔x̲=b¯x̲⇔x̂=b̂⊘â.In this case, x̂ is calculated uniquely by operation ⊘. Other cases, except cases with condition 0∈â, are provable as above.
In cases 0≤a̲ and 0≥a¯ among three possible choices, we could choose the appropriate interval according to the signs of b̲ and b¯, while in case 0∈â all three possible choices are similar in terms of signs of b̲ and b¯. To conquer this problem in case 0∈â, we choose the longest interval as the solution from three possibilities 0≤x̲, 0∈x̂, and 0≥x¯ which is 0∈x̂.
Now, let x̂ be an interval where 0∈x̂ and let x̂ be the solution of â×x̂=b̂; then b̂=[min{a¯x̲,a̲x¯},max{a̲x̲,a¯x¯}]. It follows that(21)b̲=mina¯x̲,a̲x¯⇒b̲≤a¯x̲⇒x̲≥b̲a¯,(22)b¯=maxa̲x̲,a¯x¯⇒b¯≥a̲x̲⇒x̲≥b¯a̲.Then x̲≥max{b̲/a¯,b¯/a̲}. If x̲≠max{b̲/a¯,b¯/a̲}, then x̲>max{b̲/a¯,b¯/a̲}, and a¯x̲>max{b̲/,a¯b¯/a̲}≥b̲. This contradicts with b̲=min{a¯x̲,a̲x¯}. Hence(23)x̲=maxb̲a¯,b¯a̲.Similarly, x¯ is obtained as(24)x¯=minb¯a¯,b̲a̲.It is observed simply that x̂ is equal to b̂⊘â.
In addition, two special cases should be considered separately:
If â=[0,0] and b̂≠[0,0], there is no x̂ such that â×x̂=b̂.
If â=b̂=[0,0], then x̂ is free and expression â×x̂=b̂ is consistent for all x̂, and x̂ is considered as unbounded interval [-∞,+∞] that is the longest possible interval satisfying â×x̂=b̂.
Interval multiplication.
â×x̂=b̂
0≤a̲
0∈â
0≥a¯
0≤x̲
[a_x_,a¯x¯]
[a_x¯,a¯x¯]
[a_x¯,a¯x_]
[+,+]
[-,+]
[-,-]
0∈x̂
[a¯x_,a¯x¯]
[mina¯x_,a_x¯,maxa_x_,a¯x¯]
[a_x¯,a_x_]
-,+
[-,+]
[-,+]
0≥x¯
[a¯x_,a_x¯]
[a¯x_,a_x_]
[a¯x¯,a_x_]
[-,-]
[-,+]
[+,+]
Corollary 6.
If x̂=b̂⊘â, x̂∈IR∗, and x̂∉IR, then the equation â×x̂=b̂ has no solution in IR but has solution in IR∗.
Corollary 7.
If â,b̂∈IR and ĉ=â×b̂, then b̂ can be computed by ĉ⊘â.
The operation ⊘ is the inverse operation of × in equation â×x̂=b̂ where 0∉x̂. In case 0∈x̂ the solution â×x̂=b̂ is not unique, but ẑ=b̂⊘â is the longest interval such that â×ẑ=b̂. In other words, any ŷ∈IR that satisfies â×ŷ=b̂ is a subset of ẑ.
3.3. Some Properties of Interval Numbers
Interval arithmetic and some of their properties over interval numbers and matrices are discussed in [26]. We note some of them here. For any â,b̂,ĉ∈IR,(25)â+b̂=b̂+â,â+b̂±ĉ=â+b̂±ĉ,âb̂=b̂â.
Lemma 8.
For all â,b̂∈IR, (â+b̂)⊖ĉ=â+(b̂⊖ĉ).
Proof.
(26)a^+b^⊖c^=a_+b_-c_,a¯+b¯-c¯=a_+b_-c_,a¯+b¯-c¯=a^+b^⊖c^.All these properties are correct for interval matrices as well.
3.4. A Method for Solving a Fully Fuzzy Linear System
Consider an n×n FFLS(27)a11x1+a12x2+⋯+a1nxn=b1a21x1+a22x2+⋯+a2nxn=b2⋮an1x1+an2x2+⋯+annxn=bn,which is written as(28)Ax=bwhere the coefficient matrix A=(aij)n×n is the fuzzy matrix of trapezoidal fuzzy numbers, and x=(xi)n×1,b=(bi)n×1 are column vectors of trapezoidal fuzzy numbers.
Allahviranloo et al. [14] defined the following solution sets for FFLS (27):
(i) United trapezoidal fuzzy solution set:(29)UTFSS=x∣A1x1=b1,A0x0=b0,
(ii) Tolerable trapezoidal fuzzy solution set:(30)TTFSS=x∣A1x1=b1,A0x0⊆b0,
(iii) Controllable trapezoidal fuzzy solution set:(31)CTFSS=x∣A1x1=b1,A0x0⊇b0.Note that A1=[aij1]n×n and A0=[aij0]n×n are interval matrices, and b1=[bi1]n×1, b0=[bi0]n×1 are known right-hand side interval vectors, and x1=[xi1]n×1 and x0=[xi0]n×1 are unknown interval vectors.
Now, we aim to propose a practical method to obtain the suitable solution of an FFLS. The suitable solution is defined in [14] as solution x∈UTFSS. To find such a solution x∈UTFSS, it is sufficient to solve the following two interval linear systems:(32)A0x0=b0(33)A1x1=b1.Now, consider the n×n linear system as(34)â11x̂1+â12x̂2+⋯+â1nx̂n=b̂1â21x̂1+â22x̂2+⋯+â2nx̂n=b̂2⋮ân1x̂1+ân2x̂2+⋯+ânnx̂n=b̂nwhere âij,b̂i∈IR for all (1≤i,j≤n) and x̂i are unknown intervals. The interval linear system is represented as(35)Âx̂=b̂Note that two systems (32) and (33) are in the form (34). Using operations ⊖ and ⊘, we can propose a new method to decompose an interval matrix  to two interval matrices, similar to LU decomposition method [28]. Then, one can solve systems (32) and (33)) using this decomposition method.
Assume that Â∈IRn×n, and there exist two interval matrices L̂,Û∈IRn×n such that Â=L̂×Û, where L̂ and Û are lower triangular and upper triangular matrices, respectively. Then using this decomposition method, we can solve the interval linear system (34).
Consider the system(36)Â=L̂×Û,where Ân×n is a known interval matrix, L̂n×n is an unknown lower triangular interval matrix, and Ûn×n is an unknown upper triangular interval matrix. In this system, there exist n2 linear interval equations, and there are n2+n unknown variables. This system extends to n2+n linear equations with n2+n unknowns by adding n linear equations liî=1for1≤i≤n.
To solve this linear system, first, we set all diagonal elements of L̂ with [1,1]. Then, using reversing operations, all elements of L̂ and Û can be computed by solving linear interval equations âij=∑k=1nl̂ik×ûkj. To solve such linear equations, we need to consider two cases:
(39)âij=∑k=1j-1l̂ik×ûkj+l̂ij×ûjjwhich results in(40)âij=∑k=1j-1l̂ik×ûkj+l̂ij×ûjj⇒l̂ij×ûjj=âij⊖∑k=1j-1l̂ik×ûkj⇒l̂ij=âij⊖∑k=1j-1l̂ik×ûkj⊘ûjj.Using these results, we can compute interval LU decomposition for an interval matrix. The Algorithm ILU_Decomposition computes such interval LU decomposition and is given in Algorithm 1.
If at least one of u̲ii or u¯ii for 0≤i≤n-1 is [0,0], then a dividing-by-zero error occurs, and decomposition is not possible. Otherwise, there exist L̂,Û∈IR∗n×n such that Â=L̂Û.
Algorithm 1: Computing the interval LU decomposition.
(1) function ILU_DecompositionÂ,L̂,Û,n
(i) Input: Â∈IRn×n
(ii) Output: L̂,Û∈IRn×n such that Â=L̂Û
(2)fori=1 to n do
(3)l̂ii←1,1
(4)for r=1 to n do
(5)i←r
(6)for j=i to n do
(7)sum←∑k=1j-1l̂ik×ûkj
(8)h←âij⊖sum
(9)ûij←h
(10)j←r
(11)for i=j+1 to n do
(12)sum←∑k=1i-1l̂ik×ûkj
(13)ûij←âij⊖sum
(14)l̂ij←sum⊘ûjj
return L̂ and Û
If L∉IRn×n or U∉IRn×n there is no decomposition in
IRn×n but has decomposition in (IR∗)n×n.
Example 9.
Consider the following 3×3 interval matrix:(41)Â=1,40.5,3-3,20.5,30.5,4.5-3,3-3,2-3,3-2,4.25.Initially, assign 1,1 to l11̂,l22̂,l33̂, and initialize all other elements of L̂ and Û to [0,0]. Then, compute the rows of Û and columns of L̂ by annotating Algorithm 1. Compute the rows and columns index of Û and L̂, following the first loop (lines (4) to (14)).
In case r=1,(42)L̂=1,10,00,00.5,0.751,10,0-0.75,0.50,01,1,Û=1,40.5,3-3,20,00,00,00,00,00,0.In case r=2,(43)L̂=1,10,00,00.5,0.751,10,0-0.75,0.5-0.333,0.6661,1,Û=1,40.5,3-3,20,00.25,2.25-0.75,1.50,00,00,0.In case r=3,(44)L̂=1,10,00,00.5,0.751,10,0-0.75,0.5-0.333,0.6661,1,Û=1,40.5,3-3,20,00.25,2.25-0.75,1.50,00,00,1.It is observed simply that Â=L̂×Û.
Now, to solve Âx̂=b̂ for x̂ with Â=L̂Û, we have(45)Âx̂=b̂⇒L̂Ûx̂=b̂⇒L̂ŷ=b̂(46)Âx̂=b̂⇒L̂Ûx̂=b̂⇒Ûx̂=ŷTo solve Âx̂=b̂ we do the following steps:
Decompose  into two matrices L̂ and Û.
Solve L̂ŷ=b̂ for ŷ.
Solve Ûx̂=ŷ for x̂.
Step (1) can be performed by Algorithm 1 and step (2) can be performed as follows:(47)1,100…0l̂211,10…0⋮⋮⋱⋮l̂n-11l̂n-12…1,10l̂n1l̂n2……1.1×ŷ1ŷ2⋮ŷn-1ŷn=b̂1b̂2⋮b̂n-1b̂n⇒b̂i=ŷi+∑j=1i-1l̂ij×ŷj⇒ŷi=b̂i⊖∑j=1i-1l̂ij×ŷj,and finally, step (3) can be performed as(48)û11û12û13…ûn10û22û23…û2n⋮⋮⋱⋮00…ûn-1n-1ûn-1n00…0ûnn×x̂1x̂2⋮x̂n-1x̂n=ŷ1ŷ2⋮ŷn-1ŷn⇒x̂i=ûii×x̂i+∑j=i+1nûij×ŷj⇒x̂i=ŷi⊖∑j=i+1nûij×ŷj⊘ûii.Algorithm 2 shows our approach to solve a system of linear interval equation Âx̂=b̂.
Algorithm 2: Solving linear system of intervals using interval LU decomposition.
(1) function ILU_SolveÂ,b̂,n,x̂
(i) Input: Â∈IRn×n,b̂∈IRn
(ii) Output:x̂∈IRn:Âx̂=b̂
(2)call ILU_DecompositionÂ,L̂,Û,n
(3)for i=1 to n do
(4)ŷi=b̂i⊖∑j=1i-1l̂ij×ŷj
(5)for i=n to 1 do
(6)x̂i=ŷi⊖∑j=i+1nûij×ŷj⊘ûii
return x̂
Example 10.
Considering the interval matrix  in Example 9 and the following b̂, we aim to solve Âx̂=b̂ for x̂.(49)b̂=-50.00,47.50-52.50,48.75-47.50,51.25.In Example 9, ILU decomposition of  is obtained. So, we need to solve L̂ŷ=b̂ and Ûx̂=ŷ. By solving L̂ŷ=b̂ and Ûx̂=ŷ, ŷ and x̂ are obtained, respectively, as follows:(50)ŷ=-50,47.5-15,13.1251-1.8750,5andx̂=-5.00,7.50-5.00,2.50-1.875,5.00,
4. Numerical ResultsExample 11.
Consider the following FFLS which is solved in [8, 9, 29]:(51)18,19,2010.5,12,13.55.5,6,6.21.9,2,2.13.9,4,4.41.3,1.5,1.71.9,2,2.21.9,2,2.34.4,4.5,4.6x11,x12,x13x21,x22,x23x31,x32,x33=1469.3,1897,2433.2358,434,543447.2,535.5,667.4First, we solve 1-cut system which is obtained as follows:(52)19126241.5224.5x12x22x32=1897434535.5.The 1-cut crisp solution will be (37,62,75)T.
Now, we solve 0-cut system which is obtained as follows:(53)18,2010.5,13.55.5,6.21.9,2.13.9,4.41.3,1.71.9,2.21.9,2.34.4,4.6x11,x13x21,x23x31,x33=1469.3,2433.2358,543447.,667.4.The left-hand side matrix decomposed into the following two matrices:(54)L̂=1,10,00,00.1050,0.10561,10,00.1055,0.11010.2732,0.28361,1,Û=18,2010.5,13.55.5,6.20,02.7916,2.98260.7194,1.04900,00,03.6154,3.6314We solve equations L̂ŷ=b̂ and Ûx̂=ŷ and the solutions are(55)ŷ=1469.2,2433.3203.2,288.4234.4,321,x̂=29.0993,49.992856.0765,65.581464.8558,88.3868.Finally, solution of System (53) is obtained as(56)x=(29.0993,37,49.9928)(56.0765,62,65.5814)(64.8558,75,88.3868)
Example 12.
Consider a 3×3 FFLS:(57)Ax=bwhere(58)A=-7.2,-1.9,-1.9,5.4-4.4,-2.9,1,5-0.7,0,0.2,1.7-0.2,0,0.5,1-26,6.7,8.4,34.7-23,-3,15.9,24.3-9,-1.1,0.4,3.9-6.7,-3.5,0,1.1-17.4,3.5,4.1,23.2-23,-5.9,5,17.3-9,-1.7,0.7,5.5-7,-2,2,6.2-24,-4.6,0,18-15,-6.8,5.6,25-8,-0.5,1.4,12.7-11,-2,1,8.4,(59)b=-13.9,-2.6,1.3,12.3-63.7,-7.4,13.0,70.6-51,-4.9,7.4,54.8-67.8,-7.9,4.3,58.6.This system is converted to two interval systems A0x0=b0 and A1x1=b1 as follows:(60)-7.2,5.4-4.4,5-0.7,1.7-0.2,1-26,34.7-23,24.3-9,3.9-6.7,1.1-17.4,23.2-23,17.3-9,5.5-7,6.2-24,18-15,25-8,12.7-11,8.4x0=-13.9,12.3-63.7,70.6-51.0,54.8-67.8,58.6.(61)-1.9,-1.9-2.9,10,0.20,0.56.7,8.4-3,15.9-1.1,0.4-3.5,03.5,4.1-5.9,5-1.7,0.7-2,2-4.6,0-6.8,5.6-0.5,1.4-2,1x0=-2.6,1.3-7.4,13.0-4.9,7.4-7.9,4.3.Interval matrix A0 is decomposed to L̂0,Û0:(62)L̂0=1,10,00,00,0-4.8184,-0.64841,10,00,0-3.2230,0.7073-2.2224,1.01021,10,0-1.2122,3.3335-0.1101,2.7049-2.4038,0.89451,1,Û0=-7.2,5.4-4.4,5-0.7,1.7-0.2,10,01.0885,3.0808-0.8088,0.5272-1.8817,0.13640,00,0-2.3497,1.4468-1.8764,1.31120,00,00,0-1.5464,0.1876.Algorithm 2 solves L̂0Û0x0=b0 and obtains(63)x0=-0.4024,1.5463-0.0229,0.0851-0.4682,0.8773-1.5900,2.0357.Similarly, interval matrix A1 is decomposed to L̂1,Û1 as follows:(64)L̂1=1,10,00,00,0-4.4256,-3.52531,10,00,0-2.1571,-1.8455-1.2222,-0.88801,10,00,2.42610.1666,1.0327-0.6347,0.13031,1,Û1=-1.9001,-1.9-2.9,10,0.20010,0.50,01.4142,3.0642-0.1799,0.4037-1.2872,00,00,0-0.7857,0.4715-0.9224,0.43440,00,00,0-0.3958,-0.7927.
Then, x1 is obtained from L̂1Û1x1=b1 by Algorithm 2 as(65)x1=-0.2608,1.0021-0.0145,0.0538-0.3374,0.6323-0.9569,1.2251.Finally, x is found as(66)-0.4024,-0.2608,1.0021,1.5463-0.0229,-0.0145,0.0538,0.0851-0.4682,-0.3374,0.6323,0.8773-1.5900,-0.9569,1.2251,2.0357.
5. Conclusion
In the present paper, we proposed a new method for solving the fully fuzzy linear system with trapezoidal fuzzy numbers. This problem is converted into two interval linear systems; then by introducing a new interval method based on decomposition, we solved these systems. Our method is using two inverse operations ⊖ and ⊘ which are presented in this paper for the first time. Numerical results demonstrated the efficiency of our new method.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
FriedmanM.MingM.KandelA.Fuzzy linear systems199896220120910.1016/S0165-0114(96)00270-9MR1614810Zbl0929.150042-s2.0-0008771415AllahviranlooT.Numerical methods for fuzzy system of linear equations2004155249350210.1016/S0096-3003(03)00793-8MR2077064Zbl1067.650402-s2.0-3242739401AllahviranlooT.Successive over relaxation iterative method for fuzzy system of linear equations2005162118919610.1016/j.amc.2003.12.085MR2107267Zbl1062.650372-s2.0-10444277288AbbasbandyS.EzzatiR.JafarianA.LU decomposition method for solving fuzzy system of linear equations2006172163364310.1016/j.amc.2005.02.018MR2197928DehghanM.HashemiB.GhateeM.Computational methods for solving fully fuzzy linear systems2006179132834310.1016/j.amc.2005.11.124MR2260881Zbl1101.650402-s2.0-33747589674YinJ.-F.WangK.Splitting iterative methods for fuzzy system of linear equations200920332633510.1007/s10598-009-9039-9MR2655750Zbl1177.650492-s2.0-70349205412SunX.GuoS.Solution to general fuzzy linear system and its necessary and sufficient condition20091331732710.1007/s12543-009-0024-yZbl1275.65019DehghanM.HashemiB.Solution of the fully fuzzy linear systems using the decomposition procedure200618221568158010.1016/j.amc.2006.05.043MR2282597Zbl1111.650402-s2.0-33845225451NasseriS. H.SohrabiM.ArdilE.Solving fully fuzzy linear systems by use of a certain decomposition of the coefficient matrix200823140142MR2461333GaoJ.ZhangQ.A unified iterative scheme for solving fully fuzzy linear systemProceedings of the 2009 WRI Global Congress on Intelligent Systems, GCIS 2009May 2009Xiamen, China4314352-s2.0-70449417599DehghanM.HashemiB.GhateeM.Solution of the fully fuzzy linear systems using iterative techniques200734231633610.1016/j.chaos.2006.03.085MR2327410Zbl1144.650212-s2.0-34147217856AllahviranlooT.SalahshourS.Homayoun-nejadM.BaleanuD.General solutions of fully fuzzy linear systems20132013959327410.1155/2013/593274MoloudzadehS.AllahviranlooT.DarabiP.A new method for solving an arbitrary fully fuzzy linear system2013179172517312-s2.0-8488137547810.1007/s00500-013-0986-xZbl1331.15004AllahviranlooT.HosseinzadehA. A.GhanbariM.HaghiE.NuraeiR.On the new solutions for a fully fuzzy linear system20131819510710.1007/s00500-013-1037-32-s2.0-84891661672RahmanM. M.RahmanG. M. A.Graphical visualization of FFLS to explain the existence of solution and weak solution in circuit analysis20172121639364052-s2.0-8497311577810.1007/s00500-016-2197-8Zbl06847694MalkawiG.AhmadN.IbrahimH.An algorithm for a positive solution of arbitrary fully fuzzy linear system201526343646510.1007/s10598-015-9283-0MR3351798Zbl1343.650292-s2.0-84943348275KockenH. C.AhlatcioglubM.AlbayrakaI.Finding the fuzzy solutions of a general fully fuzzy linear equation system2016302921933EdalatpanahS. A.Modified iterative methods for solving fully fuzzy linear systems20171-355732-s2.0-85021364916AllahviranlooT.BabakordiF.Algebraic solution of fuzzy linear system as: AX + BX = Y201721247463747210.1007/s00500-016-2294-82-s2.0-84982311082MalkawiG.RidaI.AhmadN.An associated linear system approach for solving fully fuzzy linear system with hexagonal fuzzy numberProceedings of the 2018 Advances in Science and Engineering Technology International Conferences (ASET)Feburary 2018Abu Dhabi, United Arab Emirates1710.1109/ICASET.2018.8376919ChangP.-T.HungK.-C.α-cut fuzzy arithmetic: Simplifying rules and a fuzzy function optimization with a decision variable20061444965102-s2.0-3374679646810.1109/TFUZZ.2006.876743DuttaP.BoruahH.AliT.Fuzzy arithmetic with and without using α-cut method: A comparative study20112199107SudhaA.AnithaN.Solving a interval fuzzy linear programming problem using alpha-cut operation2015112101416KaucherE.1980Vienna, AustriaSpringerchapter 4PopovaE. D.Multiplication distributivity of proper and improper intervals20017212914010.1023/A:1011470131086MR1831374Zbl0978.65040MooreR. E.1979Philadelphia, Pa, USASIAMMR551212Zbl0417.65022DimitrovaN. S.MarkovS. M.PopovaE. D.Extended interval arithmetics: new results and applications1992225234RogerA. H.CharlesR. J.20132ndCambridge, UKCambridge University PressMR2978290DaudW.AhmadN.AzizK.2015AIP Publishing LLC