This paper provides accurate approximate solutions for the symmetric fuzzy linear systems in (Allahviranloo et al.[1])

Ahmad, Nazihah

Ibrahim, Haslinda

Malkawi, Ghassan

UUM Repository

DOI: 10.1515/auom-2015-0034An. S¸t. Univ. Ovidius Constant¸a Vol. 23(2),2015, 173–177A note on “The nearest symmetric fuzzysolution for a symmetric fuzzy linear system”Ghassan Malkawi, Nazihah Ahmad and Haslinda IbrahimAbstractThis paper provides accurate approximate solutions for the symmet-ric fuzzy linear systems in (Allahviranloo et al.[1]).1 IntroductionThe following section reviews basic definitions of fuzzy theory, which will beneeded in the sequel:Definition 1.1. Let X be a universal set. Then, we define the fuzzy subsetA˜ of X by its membership function µA˜ : X → [0, 1] which assigns to eachelement x ∈ X a real number µA˜(x) in the interval [0, 1]; where the valueµA˜(x) represents the grade of membership of x in A˜. A fuzzy set A˜ is writtenas:A˜ = {(x, µA˜(x)), x ∈ X, µA˜(x) ∈ [0, 1]}.Definition 1.2. A fuzzy set A˜ in X = Rn is convex fuzzy set if:Key Words: Symmetric L-L fuzzy number, Symmetric L-L fuzzy linear system, Non-linear programming problem.2010 Mathematics Subject Classification: Primary 46G05, 46L05; Secondary 47A30,47B47.Received: March, 2013.Revised: February, 2014.Accepted: February, 2014.173A NOTE ON THE NEAREST SYMMETRIC FUZZY SOLUTION FOR ASYMMETRIC FUZZY LINEAR SYSTEM 174∀x 1, x 2∈ X ,∀λ ∈ [0,1],µA˜(λx 1+(1 − λ)x 2) ≥min (µA˜(x 1), µA˜(x 2)).Definition 1.3. Let A˜ be a fuzzy set defined on the set of real numbers R.A˜ is called normal fuzzy set if there exist x ∈ R such that µA˜(x) = 1.Definition 1.4. A fuzzy number is a normal and convex fuzzy set, with itsmembership function µA˜(x) defined in real line R and piecewise continuous.Definition 1.5. A fuzzy number A˜ = (a1, a2;α, β)LR is said to be an L-Rfuzzy number, where its membership function satisfyµA˜(x ) =L(a1−xα), x ≤ a1 α > 0,1, a1 ≤ x ≤ a2 ,R(x−a2β ), a2 ≤ x β > 0.Where a1 ≤ a2,and αand β are the left and right spreads, respectively;and the functions L(.),R(.),which are called left and right shape function,satisfying:(1) L(.),R(.) are non–increasing from R+ to [0, 1],(2) L(0) = R(0) = 1, L(1) = R(1) = 0.Also, if α = β and L(x) = R(x) for all x ∈ R we say A˜ is a symmetricL-L fuzzy number.Definition 1.6. (Allahviranloo et al.[1]) Let the shape functions L(.), R(.)are fixed. Consider two L-R fuzzy numbers as A˜ = (a1, a2;α, β), and B˜ =(b1, b2; γ, η). We define the distance between A˜ and B˜ as follows:d(A˜, B˜)=√[(a1−b1)−(α−γ)]2+[(a2−b2)+(β−η)]2+(a1−b1)2+(a2−b2)24 .Definition 1.7. A vector X˜ = (x˜1, x˜2, . . . , x˜n), where x˜i, 1 ≤ i ≤ n areL-R fuzzy numbers, is called an L-R fuzzy vector.Definition 1.8. (Allahviranloo et al.[1]) For two L-R fuzzy vectors X˜ =(x˜1, x˜2, . . . , x˜n) , Y˜ = (y˜1, y˜2, . . . , y˜n) we definedDp(X˜, Y˜)=(n∑i=1dp (x˜i, y˜i)) 1p.as distance between them, where p ≥ 1.A NOTE ON THE NEAREST SYMMETRIC FUZZY SOLUTION FOR ASYMMETRIC FUZZY LINEAR SYSTEM 1752 Numerical examplesIn this section we provide proposed solutions for the examples in [1].Example 2.1. (Allahviranloo et al.[1])According to [1], the symmetric exact solution for S-L-FLS is:X˜v=(x11, x12;α1x, α1x)(x21, x22;α2x, α2x)(x31, x32;α3x, α3x)=(1, 2; 2, 2)(−1, 1; 1, 1)(2, 4; 3, 3) .But X˜v does not correspond to the system, for instance if b11 = 2 isexamined in vector B˜, we get :(−1 )(2 ) + (−1 )(1 ) + (1 )(2 ) = −1 ,By using Definition 1.8. we produce D2(AX˜v, B˜)= D1(AX˜v, B˜)= 3.However, the symmetric exact solution corresponds to the system by solv-ing the associated linear system is as follows:X˜e=(− 54 ,− 14 ; 2, 2)(−7,−5; 1, 1)(− 134 ,− 54 ; 3, 3) ,Dp(AX˜e, B˜)= 0,∀p ≥ 1.Example 2.2 (Allahviranloo et al.[1])According to[1], the nearest symmetric approximate solution is:X˜v =(x11, x12;α1x, α1x)(x21, x22;α2x, α2x)(x31, x32;α3x, α3x) =(2, 2, ; 2, 2)(0.8333, 3.1667; 1, 1)(0.5, 0.5; 1, 1) ,A NOTE ON THE NEAREST SYMMETRIC FUZZY SOLUTION FOR ASYMMETRIC FUZZY LINEAR SYSTEM 176then AX˜v =(b11, b12;α1b , α1b)(b21, b22;α2b , α2b)(b31, b32;α3b , α3b) =(−3.8334, 0.8334; 5, 5)(−4.6667,−2.3333; 4, 4)(−2.6667,−0.3333; 6, 6) ,D1(AX˜v, B) = 1.1666,D2(AX˜v, B) = 0.763763.In fact, there are many nearer (symmetric or non-symmetric) approximatesolutions based on the distance metric function in Definition 1.8.In this note, we illustrate two cases for approximate fuzzy solutions.Case 1: Symmetric approximate solutionThe following L-L fuzzy vector X˜1 is a symmetric approximate solutionfor the system, with distance metric function smaller than distance of solutionX˜v in [1].GivenX˜1=(2, 2; 1, 1)(0.75, 3.25; 0.75, 0.75)(0.5, 0.5; 2.5, 2.5) , then AX˜1 =(−4, 1; 5, 5)(−4.75,−2.25; 4.25, 4.25)(−2.75,−0.25; 5.25, 5.25) ,and the following result is obtained using Definition 1.8.D1(AX˜1, B˜) = 0.707107,D2(AX˜1, B˜) = 0.559017.Case2: Non-symmetric approximate solutionThe following L-R fuzzy number vector X˜2 is a non-symmetric approxi-mate solution for the system, with distance metric function smaller than dis-tance of solution X˜v in [1].GivenX˜2 =(136 ,136 ;76 ,56)(56 ,103 ;56 ,23)(12 ,12 ;52 ,52) , then AX˜2 =(−4, 1; 5, 5)(−5,− 52 ; 4, 92 )(−3,− 12 ; 5, 112 ) ,A NOTE ON THE NEAREST SYMMETRIC FUZZY SOLUTION FOR ASYMMETRIC FUZZY LINEAR SYSTEM 177and we produce the following resultsD1(AX˜2, B˜) = 0.809017,D2(AX˜2, B˜) = 0.612372.Note:Our new solutions are obtained by using distance metric function whichnot only provides L-L fuzzy number vector, but also L-R fuzzy number vector.References[1] T. Allahviranloo, E. Haghi, M. Ghanbari, The nearest symmetric fuzzysolution for a symmetric fuzzy linear system, An. St. Univ. Ovidius Con-stanta, 20( 2012), 151-172.[2] T. Allahviranloo, R. Nuraei, M. Ghanbari, E. Haghi, and A. A. Hossein-zadeh, A new metric for L–R fuzzy numbers and its application in fuzzylinear systems, Soft Computing, 16(2012), 1743–1754.[3] M. Dehghan, B. Hashemi, M. Ghatee, Computational methods for solv-ing fully fuzzy linear systems, Applied Mathematics and Computation,179(2006), 328-343.Ghassan Malkawi,College of Arts & Sciences,University Utara Malaysia,06010 Sintok, Kedah, Malaysia.Email: gassan malkawi@yahoo.comNazihah Ahmad,College of Arts & Sciences,University Utara Malaysia,06010 Sintok, Kedah, Malaysia.Email: nazihah@uum.edu.myHaslinda Ibrahim,College of Arts & Sciences,University Utara Malaysia,06010 Sintok, Kedah, Malaysia.Email: linda@uum.edu.myA NOTE ON THE NEAREST SYMMETRIC FUZZY SOLUTION FOR ASYMMETRIC FUZZY LINEAR SYSTEM 178

A note on “The nearest symmetric fuzzy

solution for a symmetric fuzzy linear system”

http://repo.uum.edu.my/18611/1/An%20st.%20UOC%2023%202%202015%20173-177.pdf

A note on “The nearest symmetric fuzzy solution for a symmetric fuzzy linear system”

Abstract

Similar works

Full text

Available Versions

UUM Repository