Abstract
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 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 -cut arithmetic on them. -cut method is a method for performing interval operations like addition, multiplication, and subtraction on fuzzy numbers. -cut fuzzy arithmetic is described in detail in [21].
Let us consider an arbitrary trapezoidal fuzzy number , where and . The membership function of is defined as follows:Two trapezoidal fuzzy numbers and are said to be equal, if and only if for .
Given a trapezoidal fuzzy number and membership function and a real number , then the -cut of a fuzzy number is an interval, which is defined aswhere , areNote that The basic arithmetic of two fuzzy numbers and is discussed in [21–23] based on interval arithmeticClearly, for a trapezoidal fuzzy number , we have , .
A matrix (vector) is called a fuzzy matrix (fuzzy vector), if at least one element of matrix (vector) is a fuzzy number. A matrix is called a fully fuzzy matrix, whose elements of are fuzzy numbers. An -cut of an matrix is an interval matrix with the same dimension whose elements are -cut of elements of matrix .
3. Materials and Methods
3.1. Interval Arithmetic
In this section, we briefly explain interval arithmetic and notion. Also, we present a new operation on interval arithmetic.
An interval number is the bounded, closed subset of real numbers, which is indicated by a hat, defined by where and . Let denote the set of such intervals. An interval is a degenerate interval when .
Infimum and supremum of an interval number are and , respectively. Let , say when .
Let denote a binary operation on real space . For Then, for basic arithmetic operations in , we have is the set of all improper intervals which is defined in [24] as follows: The setcontains 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 and let be a matrix with real elements ; we consider if for all and . Basic arithmetic operations on interval matrices are defined similar to real matrices [26].
3.2. New Interval Operations
Let ; we define two new operations and as follows: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 , the interval equation has a unique solution and .
Proof.
Corollary 2. Supposing and and , the equation has no solution in but has exactly one solution in .
Corollary 3. If and , then can be computed by uniquely. On the other hand, the operation is the inverse operation of + in interval arithmetic.
Theorem 4. Considering , the interval equation has a solution , where .
Remark 5. Solution is not usually unique; e.g., the interval equation has solutions . Actually this equation has unlimited solutions. All intervals and where and can be solution of this equation.
In cases that is not unique, we consider as the longest interval as possible. In the above example, will be .
Proof. Let ; Table 1 shows sign of related to signs of and and also shows formulas of multiplication result.
Now, given such that , and . From Table 1, it is deduced that and . Therefore In this case, is calculated uniquely by operation . Other cases, except cases with condition , are provable as above.
In cases and among three possible choices, we could choose the appropriate interval according to the signs of and , while in case all three possible choices are similar in terms of signs of and . To conquer this problem in case , we choose the longest interval as the solution from three possibilities , , and which is
Now, let be an interval where and let be the solution of ; then . It follows thatThen If , then , and . This contradicts with . HenceSimilarly, is obtained asIt is observed simply that is equal to .
In addition, two special cases should be considered separately:(i)If and , there is no such that .(ii)If , then is free and expression is consistent for all , and is considered as unbounded interval that is the longest possible interval satisfying .
Corollary 6. If , , and , then the equation has no solution in but has solution in .
Corollary 7. If and , then can be computed by .
The operation is the inverse operation of in equation where . In case the solution is not unique, but is the longest interval such that . In other words, any that satisfies is a subset of .
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 ,
Lemma 8. For all , .
Proof. All these properties are correct for interval matrices as well.
3.4. A Method for Solving a Fully Fuzzy Linear System
Consider an FFLSwhich is written aswhere the coefficient matrix is the fuzzy matrix of trapezoidal fuzzy numbers, and 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:
(ii) Tolerable trapezoidal fuzzy solution set:
(iii) Controllable trapezoidal fuzzy solution set:Note that and are interval matrices, and , are known right-hand side interval vectors, and and 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 . To find such a solution , it is sufficient to solve the following two interval linear systems:Now, consider the linear system aswhere for all and are unknown intervals. The interval linear system is represented asNote 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 , and there exist two interval matrices such that , where and are lower triangular and upper triangular matrices, respectively. Then using this decomposition method, we can solve the interval linear system (34).
Consider the systemwhere is a known interval matrix, is an unknown lower triangular interval matrix, and is an unknown upper triangular interval matrix. In this system, there exist linear interval equations, and there are unknown variables. This system extends to linear equations with unknowns by adding linear equations .
To solve this linear system, first, we set all diagonal elements of with . Then, using reversing operations, all elements of and can be computed by solving linear interval equations . To solve such linear equations, we need to consider two cases:
Case 1 (). which results in
Case 2 (). which results inUsing 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 or for is , then a dividing-by-zero error occurs, and decomposition is not possible. Otherwise, there exist such that .
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Example 9. Consider the following interval matrix:Initially, assign to , and initialize all other elements of and to . Then, compute the rows of and columns of by annotating Algorithm 1. Compute the rows and columns index of and , following the first loop (lines to ).
In case ,In case ,In case ,It is observed simply that .
Now, to solve for with , we haveTo solve we do the following steps:(1)Decompose into two matrices and .(2)Solve for .(3)Solve for .
Step can be performed by Algorithm 1 and step can be performed as follows:and finally, step can be performed asAlgorithm 2 shows our approach to solve a system of linear interval equation .
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Example 10. Considering the interval matrix in Example 9 and the following , we aim to solve for .In Example 9, ILU decomposition of is obtained. So, we need to solve and . By solving and , and are obtained, respectively, as follows:
4. Numerical Results
Example 11. Consider the following FFLS which is solved in [8, 9, 29]:First, we solve -cut system which is obtained as follows:The -cut crisp solution will be .
Now, we solve -cut system which is obtained as follows:The left-hand side matrix decomposed into the following two matrices:We solve equations and and the solutions areFinally, solution of System (53) is obtained as
Example 12. Consider a FFLS:whereThis system is converted to two interval systems and as follows:Interval matrix is decomposed to :Algorithm 2 solves and obtainsSimilarly, interval matrix is decomposed to as follows: Then, is obtained from by Algorithm 2 asFinally, is found as
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.