A New Approach to Solve Fuzzy Linear Equation A X + B = C

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1 IOSR Journal of Mathematics (IOSR-JM) e-issn: 8-58 p-issn: 9-65X. Volume Issue 6 Ver. III (Nov. - Dec. ) PP - Md. Farooq Hasan Dr. Abeda Sultana Department of Mathematics Jahangirnagar UniversityDhaka Bangladesh Professor Department of Mathematics Jahangirnagar UniversityDhaka Bangladesh Abstract: Solving fuzzy equations has long been a problem in fuzzy set theory.fuzzy equations were solved by using different standard methods. One of the well-known methods is the method ofα cut. This article proposed a new and simple solution method to solve a fuzzy linear equationa X + B = Cwithout using α cut. The results obtained are also compared with the known solutions and are found to be in good agreement. In this article we also compare the solutions of different methods by plotting them into graphical representation. Keywords: Density Function Fuzzy Number Fuzzy Equation Probability Distribution α cut method Date of Submission: -- Date of acceptance: I. Introduction Fuzzy set theory has been widely acclaimed as offering greater richness in applications than ordinary set theory. Lotfi A. Zadeh[] first introduced the concept of fuzzy set and its applications in 965. One area of fuzzy set theory in which fuzzy numbers and arithmetic operations on fuzzy numbers play a fundamental role is fuzzyequations. Fuzzy equations were investigated by Dubois and Prade []. Sanchez [] put forward a solution of fuzzy equation by using etended operations. Accordingly various researchersbuckley [4] Wasowski [5] Biacino and Lettieri [6] have proposed different methods for solving the fuzzy equations. After this a lot research papers have appeared proposing solutions of various types of fuzzy equations viz. algebraic fuzzy equations a system of fuzzy linear equations simultaneous linear equations with fuzzy coefficients etc. using different methods ([see e.g. Jiang [] Buckley and Qu [8]). Buckley and Qu in [9] [] [] defined the concept of solving fuzzy linear equations A X + B = C where A B C and X are fuzzy numbers by using the method of α cutand their work has been influential in the study of fuzzy linear systems. Mazarbhuiya [] Baruah [] Chou [4] defined the arithmetic operations of fuzzy numbers without using the method of α cuts. In this article we present a new approach of solving fuzzy linear equation A X + B = C without utilizing the standard method. Our procedure is based on fuzzy arithmetic without α cut. We useddistribution function and complementary distribution function to solve the equation with numerical eamples. The paper is arranged as follows. In Section we discuss about the definitions and notations used in this article. In Section we have methodically reviewed the classical methods etension principle α cut and interval arithmetic of solving fuzzy linear equations with numerical eamples and their graphs. In Section 4we introduce a new solution technique of A X + B = Cwithout using α cutwhere we get the solutionx W.C. and compare the new solution with the old solution by graphical representation. All the graphs are plotted by using the computer programming MATLABfunction. In section 5 we give a brief conclusion of our task and lines for future. II. Definitions and Notations A functiona: X []is called a fuzzy seton X wherexis a nonempty set of objects called referential set and (the unit interval) is called valuation set and X; A()represents the grade of membership of. An α cut A α of a fuzzy set A is an ordinary set of elements with membership not less than αfor α. This means A α = X: A α. [5] The height of A is denoted by (A) and is defined as (A) = sup{a : X}.Ais called normal fuzzy set iff A = sup A =. [5] A fuzzy set is said to be conve if all its α cuts are conve sets. [6] A fuzzy number is a conve normal fuzzy set A define on the real line such that A() is piecewise continuous. The support of a fuzzy set Ais denoted by supp(a)and is defined as the set of elements with membership nonzero i.e.supp A = { X: A > }. [5] DOI:.99/ Page

2 A fuzzy number A denoted by a triad [a b c] such that A a = = A(c)and A b = wherea() for [a b]is called the left reference function and for [b c]is called right reference function. The left reference function is right continuous monotone and non-decreasing whereasright reference function is left continuous monotone and non-increasing. The above definition of a fuzzy number is called L-R fuzzy number []. If Z is a continuous random variable with density function f(z) then distribution of Z denoted by μ(z) is z defined by μ(z) = Prob Z z = f(z) dz and thecomplementary distribution of Zdenoted by μ (z) is defined by μ z = Prob[Z z] = z f(z) dz. [8] III. Solving Fuzzy Linear Equation A X + B = C by using the Method of α cut LetA B and C be the triangular fuzzy numbers and let= a a a ; B = b b b ; C = [c c c ]. Then A X + B = C. is a fuzzy linear equation. X if it eists will be a triangular shaped fuzzy number. Let X then X = C B is not a solution of A equation (.)...Classical Method [9] We say that this method defines the solutionx c. For anyα Let A α = a α a α ; B α = b α b α ; C α = c α c α andx c α = α α denote respectively the α cuts ofa B C and X in the given equation (.). Substituting these into equation. we find a α a α α α + b α b α = c α c α (..) Eample.: Let A = 8 9 ; B = ; C = 5 ThereforeA α = [8 + α α] ; B α = [ + α α] ; C α = [ + α α] Since A > and C > we must have X c > and equation (..) gives [(8 + α) (α) + ( + α) ( α) (α) + ( α)] = [ + α α] 6+ α 8 α α. Implies that α = and 8+ α α = We see that α is increasing (its derivative is positive) α is decreasing (derivative is negative) and = = 9. The solution X c eists with α cut X α 6+ α 8 α c =. 8+ α α The fuzzy membership function of X c is given by X c = or 4 5 Xc Fig..Solution of classical method X c DOI:.99/ Page

3 ..Etension Principal [] If X e is the value of equation (.) by using etension principal then where X e = ma {π (a b c) c b a = } (..) π a b c = min A a B b C c (..) c b Since the epression a is continuous in a b c we know how to find α cutof X a e. c b e α = min a A α b B α c C α (..) a c b e α = ma a A α b B α c C α (..4) a Therefore X e α = [ e α e α ] (..5) Eample.:Given that A = 8 9 ; B = ; C = 5. C α B α = [4 + α α ]and = A α α 8+ α Now e = 4 + α and α e = α 8 + α Hence X e α = The fuzzy membership function of X e is given by 4 + α α α 8 + α X e = or 5 4 Xe Fig.. Solution X e..interval Arithmetic [] Another way to evaluate equation (.) is to use α cuts and interval arithmetic we get the solution X I where X I α = Cα B α A α (..) to be simplified by the interval arithmetic. Eample.:Given that A = 8 9 ; B = ; C = 5. Cα B α A α = 4 + α α α 8 + α DOI:.99/ Page

4 Here we see that the solutions of equation (.) by etension principle X e and by interval arithmetic X I are same. That is X e = X I. The fuzzy membership function of X I is given by X I = or 5 4 XI Fig.. Solution X I IV. Solving Fuzzy Linear Equation A X + B = C Without Using α cut In this section we are going to solve the simple fuzzy linear equationa X + B = C by the method of without using α cut. A new way to evaluate equation (.) is to use distribution functionand complementary distributionfunction and we get the solution which we denoted by X W.C.. Here we have followed [] [] [4]. Chou [4] has forwarded a method of finding the fuzzy membership function for a triangular fuzzy numbers X. We now define a fuzzy number X = [a b c] with membership function X = L a b R b c oterwise (4.) Where L() being continuous non-decreasing function in the interval [a b] and R() being a continuous non -increasing function in the interval [b c] with L(a) = R(c) = and L(b) = R(b) =. Where L() is left reference function and R() is right reference function of the concerned fuzzy number. In this discussion we are going to demonstrate the easiness of applying this method in evaluating the arithmetic of fuzzy numbers if start from the simple assumption that the left reference function is a distribution function and similarly the right reference function is a complementary distribution function. Accordingly the functions L() and ( R()) would have to be associated with densities d d L in a b and d R in b c. d Now consider X = a b c and Y = [p q r] be two triangular fuzzy numbers. Suppose Z = X Y = a p b q c r be the fuzzy number of X Y where denotes addition(+) and multiplication( ). L a b L y p y q Let X = R b c ; Y y = R y q y r oterwise oterwise Where L() and L(y) are the left reference functions and R() and R(y) are the right reference functions respectively. We assume that L() and L(y) are distribution functions and R() and R(y) are complementary DOI:.99/ Page

5 distribution functions. Accordingly there would eist some density functions for the distribution functions L()and ( R()). Let f = d d L a b and g = d R b c d m t = or t for subtraction and division respectively. We start with equating L() with L(y) andr() with R(y). And so we obtain y = φ and y = φ respectively. Let z = y so we have z = φ and z = φ so that = ψ z and = ψ z (say). Replacing by ψ z in andψ z in g() we get f = η (z)andg = η (z) (say). Now let d dz = d dz ψ d z = m z and dz = d dz ψ z = m (z) Therefore the fuzzy arithmetic without α cut would eists as follows: Addition: Subtraction: Multiplication: Division: (X + Y) = ( Y) y = (X Y) = Y y = a+p η z m z dz a + p b + q η z m z dz y r b+q b + q c + r oterwise η t m t dt r y q y η t m t dt q y p q ap oterwise η z m z dz ap bq η z m z dz bq cr y r bq oterwise η t m t dt r y q y η t m t dt q y p q oterwise Eample 4.:Given that A = 8 9 ; B = ; C = 5. Fuzzy membership function of A B and C are A = or DOI:.99/ Page

6 B = C = + or 5 5 or We have to find the fuzzy membership function of X W.C. that is C B X W.C. = A For fuzzy membership function of(c B)(): (4.) C = 5 5 or (4.) B y = y + y y y y or y (4.4) Suppose Z = C B or Z = C + ( B). Now B = be the fuzzy number of ( B). Let t = y so that m t = dy =. dt Then the density function f(y) and g(y) would be say f (y) = d dy y + = = η (t) y g(y) = d dy ( y) = = η (t) y Then the fuzzy membership function of B is given by μ B y = y y y y y or y (4.5) If L() and L(y) are the left reference function and R() and R(y) are the right reference functions respectively then we assume that L() and L(y) are distribution function and R() and R(y) are complementary distribution function. L = and R = L(y) = y and R(y) = y There would eist some density functions for L and ( R()). Say and Here = d d (L ) = d d g = d d R = d d C + ( B) = [4 ] = 5 = 5 Equating distribution function and complementary distribution function we get L() = L(y) and R() = R(y) = y and = y y = = φ () and y = Let z = + y so we have DOI:.99/ Page = φ ()

7 So that z = + φ () = = z + and z = + φ () = = ψ z and = z + = ψ z respectively. Replacing by ψ z in f andψ z in g we obtain Now let = = η (z) and g() = = η (z) m z = d dz (ψ z ) = and m (z) = d dz (ψ z) = Then fuzzy membership function of C B would be given by (C B) = 4 η z m z dz 4 η z m z dz 4 or (4.6) Hence we get from equation (4.6) (C B) = or (4.) For fuzzy membership function of C B :Again suppose Z = C B = (C B) A A (A ). Then fuzzy membership function of = is given by A 9 8 A y = y y 9 y 8 9 y 8 y or y 8 (4.8) Then by multiplication of fuzzy numbers (C B) = [4 ]and = A 9 8 That is C B Z = A = If L() and L(y) are the left reference function and R() and R(y) are the right reference functions respectively then we assume that L() and L(y) are distribution function and R() and R(y) are complementary distribution function. L = 4 and R() = L y = and R(y) = y y 8 There would eist some density functions of L and ( R()) then = d d (L ) = 4 DOI:.99/ Page

8 g() = d d ( R()) = Equating distribution function and complementary distribution function we obtain y = 4 = φ and y = 4 = φ z = y = φ = and z = y = φ 4 = 4 So that = 4z z + = ψ z and = 4z z + = ψ z Replacing by ψ z in and ψ z in g we obtain respectively. Now let = = η (z) and g() = = η (z) m z = d dz ψ z = z + and m (z) = d dz (ψ z) = (z + ) Then fuzzy membership function of Z = C B C B A = 5 A η z m z dz would be given by η z m z dz (4.9) 5 or 5 4 Hence we get the solution X W.C. from equation 4.9 and the fuzzy membership function of the solution X W.C. is given by X W.C. () = or 5 4 Xw.c Fig 4.. Solution X W.C. The comparison of four solutions in different methods of same problem is shown in the following figure: DOI:.99/ Page

9 Compare of four solutions Xc Xe XI and X w.c..9 Xe XI Xw.c. Xc Fig 4.. Solution X c X e X I X W.C. V. Conclusions The classical methods involving the etension principle and α cuts are too restrictive because very often there is no solution or very strong conditions must be placed on the equations so that there will be a solution. This hampered the application of fuzzy set theory to algebra economics decision theory physics etc. These facts motivated us to develop a new solution technique. In this article we have acquainted with a new technique to solve fuzzy linear equationa X + B = C. We also compared our solution to solutions based onα cuts and the etension principle when they eisted. In future we will try to formulate a full method to solve A X + B = C without using α cut. References []. L. A. Zadeh Fuzzy Sets Information and Control 8 (965) 8-5. []. D. Dubois and H. Prade Fuzzy Set Theoretic Differences and Inclusions and Their Use in The analysis of Fuzzy Equations Control Cybern(Warshaw) Vol. 984 pp []. E. Sanchez Solution of Fuzzy Equations with Etend Operations Fuzzy Sets and Systems Vol. 984 pp doi:.6/65-4(84)9-x. [4]. J. J. Buckley Solving Fuzzy Equations Fuzzy Sets and Systems Vol.5 No. 99 pp. -4. doi:.6/65-4(9)999- E. [5]. J. Wasowski On Solutions to Fuzzy Equations Control and Cybern Vol pp [6]. L. Biacino and A. Lettieri Equation with Fuzzy Numbers Information Sciences Vol. 4 No. 989 pp []. H. Jiang The Approach to Solving Simultaneous Linear Equations That Coefficients Are Fuzzy Numbers Journal of National University of Defence Technology (Chinese) Vol. 986 pp [8]. J. J. Buckley and Y. Qu Solving Linear and Quadratic Equations Fuzzy Sets and Systems Vol. 8 No. 99 pp doi:.6/65-4(9)999-r. [9]. J. J. Buckley and Y. Qu On Using Alpha Cuts to Evaluate Fuzzy Equations Fuzzy Sets and Systems 8 (99) pp. 9. []. J. J. Buckley and Y. Qu Solving Fuzzy Equations: A New Solution Concept Fuzzy Sets and Systems 9 (99) pp. 9. []. J. J. Buckley and Y. Qu Solving Systems of Fuzzy Linear Equations Fuzzy Sets and Systems 4 (99) pp. 4. []. F. A. Mazarbhuiya A. K. Mahanta and H. K. Baruah Fuzzy Arithmetic without Using The method of α cuts Bulletin of Pure and Applied Sciences Vol. E No. pp []. H. K. Baruah K. Hemanta A. Kaufmann A. K. Mahanta International Journal of Latest Trends in Computing(E-ISSN: ) Vol. Issue December. [4]. Chou Chien-Chang The Square Roots of Triangular Fuzzy Number ICIC Epress Letters. Vol. Nos. pp. -. [5]. G. J. Klir and B. Yuan Fuzzy Sets and Fuzzy Logic Theory and Applications Prentice Hall Inc. N. J. U.S.A [6]. G. Q. Chen S. C.Lee and S. H. Yu Eden Application of Fuzzy Set Theory to Economics In: P. P. Wang Ed. Advances in Fuzzy Sets Possibility Theory and Applications Plenum Press New York 98 pp. -5. []. D. Dubois and H. Prade Ranking Fuzzy Numbers in the Setting of Possibility Theory Information Science Vol. No. 98 pp doi:.6/-55(8)95-. [8]. M. Loeve Probability Theory Springer Verlag New York 9. Md. Farooq Hasan "A New Approach to Solve Fuzzy Linear Equation A X+B=C." IOSR Journal of Mathematics (IOSR-JM).6 (): -. DOI:.99/ Page