Advances in Fuzzy Systems

Volume 2018, Article ID 2104343, 10 pages

https://doi.org/10.1155/2018/2104343

## An Application of Interval Arithmetic for Solving Fully Fuzzy Linear Systems with Trapezoidal Fuzzy Numbers

^{1}Department of Computer Science, Yazd University, Yazd, Iran^{2}Processing Laboratory, Yazd University, Yazd, Iran

Correspondence should be addressed to Seyed Abolfazl Shahzadeh Fazeli; ri.ca.dzay@ilezaf

Received 19 May 2018; Revised 24 June 2018; Accepted 28 June 2018; Published 11 July 2018

Academic Editor: Rustom M. Mamlook

Copyright © 2018 Esmaeil Siahlooei and Seyed Abolfazl Shahzadeh Fazeli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 .