SOME METHODS FOR SOLVING FULLY FUZZY LINEAR SYSTEM OF EQUATIONS

Transcription

1 SOME METHODS FOR SOLVING FULLY FUZZY LINEAR SYSTEM OF EQUATIONS Thesis submitted i partial fulfillmet of the requiremet for The award of the degree of Masters of Sciece i Mathematics ad Computig Submitted by Gourav Gupta Roll No Uder the guidace of Dr. Amit Kumar JULY 00 School of Mathematics ad Computer Applicatios Thapar Uiversity Patiala (PUNJAB) INDIA

2 DEDICATED TO GOD, MY PARENTS AND MY SUPERVISOR

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5 ABSTRACT System of simultaeous liear equatios plays a major role i various areas such as mathematics, physics, statistics, egieerig ad social scieces. These are importat for studyig ad solvig a large proportio of the problems i may topics i applied mathematics. Usually i may applicatios all the system s parameters are represeted by fuzzy rather tha crisp umbers, ad hece it is importat to develop mathematical models ad umerical procedures that would appropriately treat geeral fully fuzzy liear system ad solve them. This thesis is devoted to some methods for solvig fully fuzzy liear systems (system of liear equatios i which all the parameters are represeted by fuzzy umbers). The chapter-wise summary of the thesis is as follows: Chapter is itroductory i ature. This chapter icludes the basic cocepts used throughout the work. Chapter presets brief review of the work doe i the area of fuzzy liear system of equatios. I Chapter 3, matri iversio method, Cramer s rule ad LU decompositio method to solve the fully fuzzy liear systems are preseted. To illustrate all the preseted methods umerical eamples are solved. I Chapter 4, LU decompositio method, preseted i chapter 3, is modified to show the advatages of the modified method. The umerical eample, solved i chapter 3, is solved by the modified LU decompositio method ad it is show that the results obtaied by both the methods are same while it is easy ad less time iii

6 cosumig to apply the modified LU decompositio method as compared to the LU decompositio method, preseted i chapter 3. I Chapter 5 Jacobi ad Gauss-Seidel methods are used to fid the solutio of fully fuzzy liear system of equatios. To compare the methods a umerical eample i chapter 3 is solved by usig both the methods ad it is show that i umber of iteratios i Gauss-Jacobi method are more tha umbers of iteratios i Gauss-Seidel method. Chapter 6, The methods preseted i the previous chapter ca be applied oly for fidig the solutio of simultaeous fully fuzzy liear system of equatios i.e the fully fuzzy liear system of equatios i which umbers of fuzzy variables equals to umber of equatios. These methods ca t be applied for solvig o-simultaeous fully fuzzy liear system of equatios. To overcome this shortcomig i this chapter a method is preseted which ca be applied for fidig the solutio of simultaeous as well as o-simultaeous fully fuzzy liear system of equatios. To illustrate the preseted method umerical eamples are solved. iv

7 TABLE OF CONTENTS CHAPTER INTRODUCTION -7. Methods to solve the system of liear equatios.. Fuzzy liear system..3.3 Arithmetic operatios o fuzzy umbers. 6 CHAPTER LITERATURE REVIEW 8- CHAPTER 3 SOME DIRECT METHODS FOR SOLVING FULLY FUZZY LINEAR SYSTEM OF EQUATIONS Fully fuzzy liear system ad equatios 3 3. Some direct methods Matri iversio method Cramer s rule LU decompositio method Coclusio..4 CHAPTER 4 MODIFIED LU DECOMPOSITION METHOD FOR SOLVING FULLY FUZZY LINEAR SYSTEM OF EQUATIONS Modified LU decompositio method 5 4. Coclusio....9 v

8 CHAPTER 5 SOME ITERATIVE METHODS FOR SOLVING FULLY FUZZY LINEAR SYSTEM OF EQUATIONS Iterative methods Gauss-Jacobi method Gauss-Seidel method Coclusio..43 CHAPTER 6 LINEAR PROGRAMMING METHOD FOR SOLVING FULLY FUZZY LINEAR SYSTEM OF EQUATIONS Algorithm for solvig liear system of equatios usig liear programmig method Numerical eamples Two-Phase method Big-M method Coclusio.5 BIBLIOGRAPHY vi

9 Chapter INTRODUCTION There are several applicatios i sciece ad egieerig where relevat physical law(s) immediately produces a set of liear algebraic equatios. A system of liear equatios (or liear system) is a collectio of liear equatios ivolvig the same set of variables. A geeral system of m liear equatios with ukows may be writte as: a + a + + a = b a + a + + a = b a + a + + a = b m m m m (.) I matri form this system may be writte as A b where A is a m matri, is a colum vector with etries, ad b is a colum vector with m etries. éa a a ù é ù éb ù ê a a a ú ê ú ê b ú A ê = ú, X = ê ú, b = ê ú ê ú ê ú ê ú ê ú ê ú ê ú ëam am am û ë û ëbm û Defiitio. If all b i are zero the the system of equatios is said to be homogeeous ad if

10 at least oe of b i is ot zero the it is said to be o-homogeeous. Defiitio. The o-homogeeous system has a uique solutio if ad oly if the determiat of A is o-zero. Defiitio.3 The solutio of a liear system is a assigmet of values to the variables,,..., such that each of the equatios is satisfied. The set of all possible solutios is called the solutio set. A liear system of equatios may behave i ay oe of three possible ways: (i) (ii) (iii) The system has ifiitely may solutios. The system has a sigle uique solutio. The system has o solutio. If the system is cosistet the we have either uique solutio or ifiitely may solutios of the o-homogeeous system. (.) has a uique solutio if ad oly if determiat of A is o-zero i.e. A 0 ad the solutio of the system may be writte as A b. The homogeeous system possesses oly a trivial solutio 0 if A 0. But it is iterestig to ote that a homogeeous system has also otrivial solutio provided A 0 i.e. coefficiet matri is sigular.. Methods to solve the system of liear equatios

11 There are large umbers of methods to solve the system of liear equatios ad systems of three or four equatios ca be easily solved by had usig these methods, but computers are ofte used for larger systems. These methods are geerally divided ito two categories: (i) (ii) Direct method Iterative method There are several direct ad iterative methods i the literature for solvig the system of liear equatios but i this thesis the followig direct methods ad iterative methods are used: Direct methods (i) (ii) (iii) Matri iversio method Cramer s rule LU decompositio method Iterative methods (i) (ii) Gauss-Jacobi method Gauss-Seidel method. Fuzzy liear system For costructig a system of liear equatios it is assumed that there is o ucertaity about ay parameter of the system but i real life there may eist ucertaity about some or all parameters. I such a case it is better to represet all the parameters of system of liear equatios by fuzzy sets (Zadeh, 965). Defiitio.4 3

12 Let X deote a uiversal set the a fuzzy subset A of X is defied by its member- ship fuctio : X [0,] A which assigs a real umber ( ) i the iterval [0, ], to each elemet X, where A the value of ( ) at shows the grade of membership of i A. A fuzzy subset A ca A be characterized as a set of ordered pairs of elemet ad grade ( ) ad is ofte writte as: A, X A A Defiitio.5 A fuzzy set A i X is said to be ormal if there eist X such that ( ). A.Defiitio.6 A fuzzy umber A is called positive (egative), deoted by A 0 ( A 0), if its membership fuctio ( ) 0, 0 ( 0). A Defiitio.7 A fuzzy set with the followig membership fuctio 4

13 m, m m, 0, m ( ), m m, 0, A 0, otherwise. is amed a triagular fuzzy umber ad is symbolically writte as A ( m,, ). Defiitio.8 A triagular fuzzy umber A ( m,, ) is positive if ad oly if m 0 Defiitio.9 Two triagular fuzzy umbers A ( m,, ) ad B (,, ) are said to be equal if ad oly if m,,. Defiitio.0 A matri A ( a ij ) is called a fuzzy matri, if each elemet of A is a fuzzy umber. A fuzzy matri A will be positive ad deoted by A 0 if each elemet of A be positive. We may represet fuzzy matri A ( a ij ) such that (,, ) a a ij ij ij ij with ew otatio A ( A, M, N), where A ( a ) ij, M ( ij ) ad N ( ij ) are three crisp matrices. Defiitio. A square matri A ( a ij ) is called a upper triagular matri if 5

14 a 0 (0,0,0), i j, ij ad a square matri will be lower triagular matri A ( a ij ) if a 0 (0,0,0), i j, ij where 0 = (0,0,0) as a zero triagular fuzzy umber. Defiitio. Cosider the fully fuzzy liear system of equatios: ( a ) ( a ) ( a ) b, ( a ) ( a ) ( a ) b, ( a ) ( a ) ( a ) b. The matri form of the above equatio A b where A ( a ), i, j is a fuzzy matri ad, b F( R), where F ( R ) is the set of all fuzzy umbers. ij j j Defiitio.3 Let A ( a ij ) ad B ( b ) ij be two m ad p fuzzy matrices. We defie A B C ( c ij ) which is the m p matri where ( C ) a b ij ik kj k 6

15 .3 Arithmetic operatios o fuzzy umbers I this sectio arithmetic operatios of triagular fuzzy umbers are preseted. Let A ( m,, ) ad B (,, ) be two triagular fuzzy umbers the () A B ( m,, ) (,, ) ( m,, ). () A ( m,, ) ( m,, ). (3) If A 0 ad B 0 the ( m,, ) (,, ) ( m, m, m ). (4) If is ay scalar the A is defied as ( m,, ), 0 ( m,, ) ( m,, ), 0 7

16 Chapter LITERATURE REVIEW Buckley ad Qu (990) preseted ecessary ad sufficiet coditios for some liear ad quadratic equatios to have a solutio whe the parameters are either real or comple fuzzy umbers. They showed that the solutio will be a real or comple fuzzy umber. Also they preseted applicatios i chemistry, ecoomics, fiace ad physics are preseted for these types of equatios. Buckley ad Qu (99) preseted a method to fid the solutio to the fuzzy matri equatio A b whe the elemets i A ad b are triagular fuzzy umbers. A is square ad o-sigular matri. Zhao ad Govid (99) studied the algebraic equatios ivolvig geeralized fuzzy umbers (which icludes fuzzy umbers, fuzzy itervals, crisp umbers ad iterval umbers) with cotiuous membership fuctios. Buckley (99) proposed ew solutio procedure for solvig fuzzy equatios (liear, o-liear ad differetial) to the followig three problems i ecoomics ad fiace: () Leotief's iput-output model () the iteral rate of retur ad (3) a dyamic supply-demad model where price ad supply are govered by a system of differetial equatios. 8

17 Friedma et al. (998) ivestigated a geeral fuzzy system usig the embeddig approach. They derived the coditios for the eistece of a uique fuzzy solutio to liear system of equatios ad desiged a umerical procedure for calculatig the solutio. They illustrated the applicability of the proposed model with eamples. Rao ad Che (998) proposed the solutio of simultaeous liear equatios that arise i the aalysis of egieerig systems ivolvig fuzzy iput parameters. I additio to a discussio o the eistece of solutio to the problem, they defied ad formalized the umerical solutio to the fuzzy liear system. The proposed methodology cosists of three major aspects ivolvig, computerized selectio of fuzziess, implemetatio of fuzzy operatios ad developmet ad eecutio of a search based algorithm. Feurig et al. (998) proposed a method to fid solutio to the fully fuzzified liear program where all the parameters ad variables are fuzzy umbers. Firstly they chaged the problem of maimizig a fuzzy umber, the value of the objective fuctio, ito a multi-objective fuzzy liear programmig problem. They desiged a evolutioary algorithm to solve the fuzzy fleible program ad applied this program to two applicatios to geerate good solutios. Allahviraloo (004) proposed algorithms for solvig fuzzy system of liear equatios. He discussed the schemes based o the iterative Jacobi ad Gauss-Seidel methods i detail. Allahviraloo (005) trasformed the Gauss-Seidel iterative method for solvig fuzzy system of liear equatios to the successive over relaatio method. 9

18 Dehgha ad Hashemi (006) ivestigated a geeral fuzzy liear system of equatios ad the eteded several well-kow umerical algorithms such as Richardso, Jacobi, Jacobi Over Relaatio, Gauss-Seidel, Successive Over Relaatio, Accelerated Over Relaatio, Symmetric ad Usymmetrical Successive Over Relaatio ad Etrapolated Modified Aitke for solvig fuzzy liear system of equatios. Dehgha et al. (006) eteded the Adomai decompositio method for fully fuzzy liear system of equatios. For fid a solutio of the system A b, where A ad b are respectively a fuzzy matri ad fuzzy vector. Mosleh et al. (007) proposed the solutio of the fully fuzzy liear system A b C d, where A ad C are fuzzy matrices, b ad d are fuzzy vectors. Dehgha et al. (007) employed Dubois ad Prade s approimate arithmetic operators o LR fuzzy umbers for fidig a positive fuzzy vector which satisfies A b, where A ad b are a fuzzy matri ad a fuzzy vector, respectively. Fully fuzzy liear system of equatios is trasformed ad proposed iterative techiques such as Richardso, Jacobi, Jacobi over relaatio, Gauss Seidel, Successive Over Relaatio, Accelerated Over Relaatio, Symmetric ad Usymmetrical Successive Over Relaatio ad Etrapolated Modified Aitke are used for solvig fully fuzzy liear system of equatios. I additio, the methods of Newto, Quasi-Newto ad Cojugate Gradiet are proposed from o-liear programmig for solvig a fully fuzzy liear system of equatios. 0

19 Nasseri et al. (008) used a certai decompositio of the coefficiet matri of the fully fuzzy liear system of equatios to obtai a simple algorithm for solvig these systems. The ew algorithm ca solve fully fuzzy liear system of equatios i a smaller computig process. Allahviraloo et al. (008) preseted a method to solve fully fuzzy liear system of equatios. They replaced the fully fuzzy liear system of equatios by three crisp liear system of equatios. They calculated its homomorphic solutio i caoical trapezoidal form based o three crisp liear solutios associated with three parameters, value, ambiguity ad fuzziess. Gao ad Zhag (009) proposed a uified iterative scheme to solve geeral fully fuzzy liear system of equatios A b i which all parameters are LR fuzzy umbers. By this iterative scheme, they preseted Gradiet iterative algorithm ad Leastsquares iterative algorithm for solvig o-square fully fuzzy liear system of equatios. Mikaeilvad ad Allahviraloo (009) foud from the defiitio of eteded operatios o fuzzy umbers, subtractio ad divisio of fuzzy umbers are ot the iverse operatios to additio ad multiplicatio, respectively. Hece for solvig equatios or system of equatios, methods without usig iverse operators must be used. They proposed a ovel method to fid the o-zero solutios of fully fuzzy liear system of equatioss. They splitted system's parameters to two groups of o-positives ad oegatives by solvig oe multi objective liear program ad employig embeddig method to trasform fully fuzzy liear system of equatios to parametric form liear system of equatios ad hece, trasformed operatios o fuzzy umbers to

20 operatios o fuctios. Fially, they used umerical eamples to illustrate their approach. Mosleh et al. (009) discussed a ew decompositio of a o-sigular fuzzy matri, the symmetric times triagular decompositio. By this decompositio, every osigular fuzzy matri ca be represeted as a product of a fuzzy symmetric matri S ad a fuzzy triagular matri T. Rahgooy et al. (009) modeled the fuzzy comple liear equatios i the field of circuit aalysis ad the solved it by a eistig method. Nasseri ad Zahmatkesh (00) cocetrated o solvig fully fuzzy liear system of equatios of equatios ad proposed Huag method for computig a o-egative solutio of the fully fuzzy liear system of equatios. Liu (00) proposed a approach to solve fully fuzzy liear system of equatios that is foud to give more accurate tha the iterative Jacobi ad Gauss-Seidel methods o approimatig the solutio of fully fuzzy liear system of equatios.

21 Chapter 3 SOME DIRECT METHODS FOR SOLVING FULLY FUZZY LINEAR SYSTEM OF EQUATIONS I this chapter, matri iversio method, Cramer s rule ad LU decompositio method to solve A b, where A is a fuzzy matri ad ad b are fuzzy vectors with appropriate sizes are preseted. 3. Fully fuzzy liear system ad equatios Cosider the fully fuzzy liear system of equatios: ( a ) ( a ) ( a ) b, ( a ) ( a ) ( a ) b, ( a ) ( a ) ( a ) b. (3.) The matri form of the above system is A b (3.) where the coefficiet matri A ( a ), i, j is a fuzzy matri ad, b, i are fuzzy vectors. This system is called fully fuzzy liear system. i i ij We wat to fid the o-egative solutio of fully fuzzy liear system A b where A ( A, M, N) 0, b ( b, g, h) 0, ad (, y, z) 0. So we have 3

22 3. Some direct methods A, M, N, y, z b, g, h (3.3) I this sectio some direct methods to solve the equatio (3.) is preseted: 3.. Matri iversio method Usig sectio.3, equatio (3.3) may be writte as ( A, Ay M, Az N) ( b, g, h) (3.4) Usig defiitio.0, we have A b, Ay M g, Az N h. (3.5) i.e A b, Ay g M Az h N. (3.6) Assumig A as a o-sigular matri, equatio (3.6) may be writte as A b, y A g A M z A h A N., (3.7) The fuzzy solutio (, y, z) ca be easily obtaied by usig the values of, y ad z. Eample 3. Cosider the followig fully fuzzy liear system ad solve it by direct method: (5,,) (, y, z ) (6,,) (, y, z ) (50,0,7) (7,,0) (, y, z ) (4,0,) (, y, z ) (48,5,7) So we have, 4

23 A b Similarly by equatios (3.7), 5 6 y 0 4 y 7 4 y y ad Thus the solutio is 3.. Cramer s rule 5 6 z 7 4 z z z (4,,0) (5,, ) For solvig fully fuzzy liear system (3.) with this method, ( i) det( A ),,,...,, i i det( A) where ( i) A deotes the matri obtaied from A by replacig its i th colum by b. The usig solutio, we have yi i det( A) ( i) det( A ),,,...,, 5

24 zi i det( A) ( i) det( A ),,,...,, where ( i) A ad ( i) A deotes matri obtaied from A by replacig its th i colum by h M ad g N, respectively. Eample 3. Cosider the followig fully fuzzy liear system of equatios ad solve it usig Cramer s rule: (4,3, ) (5,,) (3, 0,3) (7,54,76) (7, 4,3) (0, 6,5) (,,) (8,5,9) (6,, ) (7,, ) (5,5, 4) 3 (55,89,5) Here A 7 0 b det( A) (50 4) 5(05 ) 3(49 60) () () (3) A 8 0, A 7 8 A Determiat of each of above matri is 6

25 () det( A ) 84 () det( A ) 368 (3) det( A ) 30 Thus we have, 84 4, ad From the give problem, we have h M g N Now, () () (3) A 46 0 A 7 46 A () det( A ) 9 () det( A ) 38 (3) det( A ) y, y 3 ad y i.e., y 3 7

26 () () (3) Similarly, A 7 0, A 7 7, A () det( A ) 9 () det( A ) 30 (3) det( A ) z, z 5 ad z z 5 4 Thus the solutio of this problem is (4,, ) (8,3,5) (5,, 4) 3..3 LU decompositio method I the LU decompositio method, the coefficiets matri of the liear system of equatios is factored ito the product of two triagular matrices. This method is frequetly used to solve a large system of equatios. Cosider the system of equatios (3.), where A is a o-sigular matri. We first of all write the matri A as the product of a lower triagular matri L ad a upper triagular matri U i the form A L U, where A ( A, M, N), L ( L, L, L ), ad U ( U, U, U ). 3 3 So we have 8

27 ( A, M, N) ( L, L, L ) ( U, U, U ) 3 3 ( A, M, N) ( LU, LU L U, LU L U ) 3 3 A LU, M LU L U, N LU L U. 3 3 (3.8) (3.9) (3.0) I order to fid a uique solutio we either take all the diagoal elemets of L as or all the diagoal elemets of U as. For U, i,,...,, the method is called the Crout s LU decompositio method ad for L, i,,...,, it is called Doolittle s method. Here we will use Doolittle method. ii Firstly, we calculate L ad U such that A LU, where L is a lower triagular crisp matri, havig the diagoal of s ad U is a upper triagular crisp matri with the geeral diagoal. ii l 0 0 l3 l3 0 l l l3 u u u3 u 0 u u3 u 0 0 u33 u 3 = u a a a3 a a a a3 a a3 a3 a33 a 3 a a a3 a which amout to equatios i the ukows l ij ad u ij. The computatios ru as follows: u,,,...,. j a j j (3.) a,,,...,. (3.) i li u ai li i u 9

28 Cotiuig i a recursive way for r,3,...,, we alteratively fid the rows of U ad correspodig colums of L to be r u a l u, j r, r,...,, (3.3) rj rj rk kj k Each row follows by the correspodig colum of L r air likukr lir k u i r, r,...,, rr (3.4) We set the diagoals of L ad L 3 to be cosistig of 0 s ot s. By (3.9), ad l ( l ) with diagoals of 0 s ad U ( u ) we may write ij ij m l u, i, j, l 0. (3.5) ij ik kj ii k With L ad U i had, we ca cotiue our approach to the secod step for L ad U i the followig way: u j mij, j,,...,, (3.6) m l u (3.7) i i li, i,,..., u Cotiuig i a recursive way for r,3,...,, we alteratively fid the rows of U ad correspodig colums of L to be r u m l u, j r, r,...,, rj rj rk kj k r r mir l ikukr liku kr l ir k k u i r, r,...,, rr (3.8) Similarly by (3.0), ad L3 ( l ) ad U3 ( u ) we may write ij ij 0

29 l u l u, i, j. (3.9) ij ik kj ik kj k We ca cotiue our approach to the secod step for fidig L 3 ad U 3 i the followig way: u, j,,...,, j j l u,,,...,. i i li i u (3.0) Fially we alteratively fid the rows of U 3 ad the correspodig colums of L 3 for r,3,..., to be r u ( l u l u ), j r, r,...,, rj rj rk kj rk kj k r r ir l ikukr liku kr l ir k k u i r, r,...,, rr (3.) The solutio to the problem A b could be foud by a two step triagular solve process. A b L U b L b where U (3.) By solvig system (3.), we get the solutio. Eample 3.3 Let A 6,, 4 5,, 3,,,8, 0 4,,5 8,8,0 4,0,34 3,30, 30 0,9, 4

30 be a fuzzy matri Here A 4 8, M 8 8, N From equatios (3.), (3.), (3.3) ad (3.4), we ca fid the elemets of L ad U. u a 6, u a 5, u a a 6 a a 4 l l l 3,, 3 4. u 6 u 6 u 6 u a lu 4 5 4, u3 a3 lu3 8 3 l a l u 4 5, u 4 l a l u u 4 3 Therefore we have, u33 a33 l3u3 l3u , l L 0 U From equatios (3.5), (3.6), (3.7) ad (3.8) we ca obtaied L ad U,

31 L U Similarly, from (3.9), (3.0) & (3.), we get L3 0 0 U Thus LU decompositio method of fuzzy matri A is,0,0 0,0,0 0,0,0 6,,4 5,, 3,, A L U,,,0,0 0,0, 0 0,0,0 4,3,,, 4,,3 3,,,0,0 0,0,0 0,0,0,,3 Now we solve the fully fuzzy liear system of equatios L b i.e,,0,0 0,0,0 0,0,0 58,30,60,,,0,0 0,0,0 y 4,39, 57 4,,3 3,,,0,0 z 36, 97,54 Usig the Cramer s rule, we easily compute i the followig form: 58,30,60 6,, 6, 4, Now we solve the fully fuzzy liear system of equatios U i.e, 6,,4 5,, 3,, 0, 0,0 4,3,,, 6,, 3 58,30,60 0, 0,0 0,0,0,,3 6, 4, Agai usig the Cramer s rule, we may compute the solutio i the followig form: 3

32 (4,,3) (5,0.5, ) (3,0.5,) which is required solutio. 3.3 Coclusio I this chapter the fully fuzzy liear system of equatios, i.e. fuzzy liear system of equatios with fuzzy coefficiets ivolvig fuzzy variables are ivestigated ad some direct methods are applied for solvig these systems. It may be useful i aalysis of physical systems ad other topics i real egieerig problems. The proposed schemes for fidig the positive solutio of the same systems, whe parameters are positive are quite satisfactory. 4

33 Chapter 4 MODIFIED LU DECOMPOSITION METHOD FOR SOLVING FULLY FUZZY LINEAR SYSTEM OF EQUATIONS I this chapter, the LU decompositio method, preseted i chapter 3, is modified. To show the advatages of the modified method, the umerical eample, solved i chapter 3, is solved by the modified LU decompositio method ad it is show that the results obtaied by both the methods are same while it is easy ad less time cosumig to apply the modified LU decompositio method as compared to the LU decompositio method, preseted i chapter Modified LU decompositio method Cosider fully fuzzy liear system of equatios A b, (4.) Where A ( A, M, N ), (, y, z), b ( b, h, g). Here, we cosider L L, 0,0 Let A L U where L ( L,0,0) ad U ( U, U, U ) 3 because our aim is to fid a lower triagular fuzzy matri L istead of L L, L, L 3 with the diagoal of fuzzy idetity umber i LR fuzzy multiplicatio which is,0,0, ot,,. So, A L U ow becomes ( A, M, N) ( L,0,0) ( U, U, U ) 3 5

34 ( A, M, N) ( LU, LU, LU ) (4.) 3 So, from equatios (4.) ad (4.), we have ( LU, LU, LU ) (, y, z) ( b, h, g) 3 By usig sectio.3, we have ( LU, LU LU y, LU LU z) ( b, h, g) 3 LU b, LU LU y h, LU LU z g, 3 ad therefore, U L b, y U L ( h LU ), z U L ( g LU ). 3 (4.3) Eample 4. Cosider the followig fully fuzzy liear system of equatios 6,,4 5,, 3,, 58,30,60,8, 0 4,,5 8,8,0 y 4,39, 57 4,0,34 3,30,30 0,9, 4 z 36, 97, A 4 8, M 8 8, N Firstly, we obtai LU decompositio for matri A as follow: a a a3 0 0 u u u3 a a a l 0 0 u u 3 3 a3 a3 a 33 l3 l3 0 0 u 33 u a 6, u a 5, u a

35 l a a 4 3, l3 4. u 6 u 6 u a l u 4 5 4, u a l u l a l u u 4 3, u a l u l u , Hece we have, L 0, U So we ca obtai U ad U 3 as follows: L , U U L M U L b, U L y U L ( h LU ), 7

36 LU LU h LU y z U L ( g LU ), 3 LU LU g LU z

37 Therefore, (4,,3), y (5,,), z (3,,) 4. Coclusio LU decompositio method, preseted i chapter 3, is modified ad the advatage of modified LU decompositio method, preseted i chapter 3 is show. 9

38 Chapter 5 SOME ITERATIVE METHODS FOR SOLVING FULLY FUZZY LINEAR SYSTEM OF EQUATIONS I the previous chapters, we have preseted some direct methods for solvig fully fuzzy liear system of equatios. I this chapter, two iterative methods (Gauss-Jacobi, Gauss-Seidel) are preseted to fid the solutio of fully fuzzy liear system of equatios. To eplai the method, a umerical eample solved i chapter 3 is solved by usig these two methods ad it is show that the results obtaied by the direct methods ad iterative methods are same. 5. Iterative methods Iterative methods are those which start with a iitial approimatio ad which, by applyig a suitably chose algorithm, lead to successively better approimatios. Eve if the process coverges, we ca oly hope to obtai a approimate solutio by iterative methods. Iterative methods vary with the algorithm chose ad i their rates of covergece. Some iterative method may actually diverge, others may coverge so slowly that they are computatioally useless. Merits of iterative methods are their simplicity ad uiformity of operatios to be performed, which make them well suited for use o computer. 30

39 5.. Gauss-Jacobi method To solve fully fuzzy liear system of equatios A b, we have already discussed a approach i chapter 3. Accordig to which the positive vectors, y ad z ca be foud by solvig followig liear system of equatios sychroously: A b Ay M g Az N h (5.) I this sectio, we employ iterative methods for obtaiig solutio to above problem i.e. we use Gauss-Jacobi or Gauss-Seidel method to solve A b. System (5.) ca be writte as ai ai... ai bi ai y ai y... ai y mi mi... mi gi, i aiz ai z... aiz i i... i hi (5.) Usig above equatios, we ca fid aiii bi aij j j, ji aii yi gi aij y j mij j, i j, ji j aiizi hi aij z j ij j j, ji j (5.3) hece 3

40 i bi aij j aii j, ji yi gi aij y j mij j i a ii j, ji j zi hi aijz j ij j a ii j, ji j (5.4) This ca be re-writte as bi i aij j aii j, ji aii gi yi aij y j mij j i aii j, ji j aii hi zi aij z j ij j aii j, ji j aii (5.5) Equatio (5.5) ca be writte i compact form or matri form as: y J y z z (5.6) where J is called the iteratio matri ad is a vector. To solve system (5.6), we ca make iitial approimatio 0 X of the solutio vector ad substitute ito right had side of equatio (5.6). The solutio of equatio (5.6) will give a vector () X, which is better approimatio to the solutio tha 0 X. We cotiue i this maer util the successive iteratio k X coverges to the solutio up to desired accuracy, which suggests the followig iterative process as the Gauss-Jacobi method for solvig a fully fuzzy liear system of equatios: 3

41 k k i bi aij j (5.7) aii j, ji k k k yi gi aij y j mij j, i 5.8 a ii j, ji j zi hi aijz j ij j a ii j, ji j k k k 5.9 I geeral, we ca write this sequece as ( k) ( k ) ( k) k z z ( k) k y J y k, 0 (5.0) where J is called the iteratio matri of the iterative method ad is a vector. k ad k deote solutio at th k ad k th iteratio respectively. Equatio (5.7) ca be writte i the matri form as: a k k 0 a a3 a b k k a a 0 a3 a b a 3 a3 0 a k 3 b 3 3 a 33 k a 3 0 k a a b a or k k D L U D b A A A A 33

42 Similarly, equatio (5.8) ad (5.9) ca be writte i matri form as: k k k y D L U y D M D g A A A A A z D L U z D N D h k k k A A A A A respectively. Sufficiet coditio: The Gauss-Jacobi iterative method for solvig fully fuzzy liear system of equatios A b coverges if ad oly if the classical Gauss-Jacobi iterative method coverges for solvig the crisp liear system of equatios A correspodig fully fuzzy liear system of equatios. b derived from the If the matri A i the crisp liear system of equatios A b is strictly diagoally domiat i.e., aii aij, i,, 3,... the the iteratios obtaied i j ji classical Gauss-Jacobi iterative method coverges for ay iitial approimatio 0 X. Eample 5. Solve the followig system of equatios usig Gauss-Jacobi method (5,,) (, y, z ) (6,,) (, y, z ) (50,0,7) (7,,0) (, y, z ) (4,0,) (, y, z ) (48,5,7) from the above system we foud that 5 6 A M N ad b g h

43 To solve the above problem by Gauss-Jacobi method, first of all we obtai the followig equatios by the method eplaied i chapter y y 5 (5.) 5 6 z z (5.) ca further be writte as: Sice 5 6 ad 4 7, therefore the above system of equatios is ot diagoally domiat. So writig the above system i diagoally domiat form as: (5.4) Now, the above system of liear equatios is i diagoally domiat form as 7 4 ad 6 5 writte as. Now, to fid the solutio by Gauss-Jacobi method first of all (5.4) ca also be 35

44 Thus, the Gauss-Jacobi s method whe applied to the above system, it gives k k k k, k 0,,,... Now, startig with iitial approimatio vector 0 0,0, we get i.e , hece cotiuig with this we obtai.0953, , , , , , , , , , , , , , , , , ,

45 0 4.97, , , , , , , , , , , , , , , , Sice we have already foud the eact solutio of the above system i chapter 3, Eample 3. ad is foud to be (4,5). It seems that the sequece k, k 0,,,... geerated by the Jacobi method will coverge to the eact solutio. Hece up to two decimal places we obtai, 4.00,5.00 Now, puttig the value of i the equatios (5.) ad (5.3) we obtai, 5 6 y 7 4 y 5 6 z z i.e. 5y 6y 7y 4y ad 5z 6z 3 7z 4z Sice the above equatios are ot i the form of diagoally domiat form. So covertig them to diagoally domiat form as: 37

46 7y 4y 5y 6y (5.5) Now, solvig the above equatios by the same procedure that is used to solve the system (5.), we obtai: y ( 4 y ) 7 y ( 5 ) y 6 Takig the iitial approimatio as 0 0,0 y ad cotiuig with Gauss-Jacobi method we obtai y y y y 0.48, y , , y 0.070, , y.080, , y 0.086,0.086 at 8 th iteratio we obtai 8.086,.086 y which is very close to eact solutio,. Hece the value of the optimal solutio up to two decimal place is: y, y.09,.09 y Similarly solvig 5z 6z 3 7z 4z (5.6) Solvig (5.6) we fid that the value of z coverges at 4 th iteratio as follows: 38

47 z z z z z z 0.857, 0.5 z 0, , z 0, , z 0, , z 0, , z 0, , 0.5 Thus the value of z up to two decimal poits is z, z 0, 0.5 z Hece the solutio of give fully fuzzy liear system of equatios give i eample (5.) is as follows: (4,.09,0) (5,.09, 0.5) which is the required solutio of the give fully fuzzy liear system of equatios. 5.. Gauss-Seidel method displacemets. The Gauss-Seidel method is simple modificatio of the method of simultaeous The system of equatio (5.) ca be writte as: aij j bi aij j, i (5.7) ji ji a y g a y m, i (5.8) ij j i ij j ij j ji ji j 39

48 a z h a z, i (5.9) ij j i ij j ij j ji ji j So, for this the Gauss-Seidel method is defied as: k k k i bi aij j aij j aii ji ji k k k k yi gi aij y j aij y j mij j, i, k 0 aii ji ji j k k k k zi hi aijz j aij z j ij j aii ji ji j System (5.7) ca also be writte as: k k k k k a a a a a b k k k k k a a a a a b k k k k k a a a a a b k k k k k k a a a a a a b I matri form of the system ca be writte as k D L b U k A A A where D, L, U are diagoal, lower triagular ad upper triagular matrices respectively. A A A Similarly (5.8) ad (5.9) ca also be writte i matri form as follows: k k k DA LA y g U Ay M k k k D L z h U z M A A A 40

49 Thus the Gauss-Seidel iterative method for solvig fully fuzzy liear system of equatios is as follows: k k DA LA U A DA LA b k k k k k k y D L U y D L M D L g A A A A A A A A A A A A A A z D L U z D L N D L h I this method also, if A is strictly diagoally domiat the the iteratio always coverges. Gauss-Seidel method will geerally coverge if the Jacobi method coverges ad will coverge at a faster speed. Geerally, for a symmetric matri A, the rate of covergece of the Gauss-Seidel method is twice the covergece rate of the Jacobi s method. This result is true eve if A is ot symmetric. Eample 5. Solve the system of equatios cosidered i Eample 5. usig Gauss Seidel method. Solutio Solvig (5.4) i.e The Gauss-Seidel iterative formula for this system ca be writte as: k k k k 4

50 Takig the, 0, we get hece cotiuig with this, we get , , , , , , , , , Sice we have already foud the eact solutio of the above system i chapter 3, Eample 3. ad is foud to be (4,5). It seems that the sequece k, k 0,,,... geerated by the Gauss-Seidel method will coverge to the eact solutio. So, by the above results it is clear that the value of up to two decimal poits is, 4,5 solvig (5.5) we obtai y y y 0.46, y 0.53, , y , , Hece the value of y up to two decimal places ca be writte as y, y 0.09,0.09 y 4

51 ad solvig (5.6), usig the same method as used for solvig the system (5.4) we obtai z z z z 0.860, 0.60 z 0.36, , z , , z 0.007, , hece from the above results, we fid that the value of z up to two decimal poits is foud to be: z, z 0, 0.5 z hece the solutio of the give fully fuzzy liear system of equatio up to two decimal places is foud to be (4,0.09,0) (5, 0.09, 0.5) 5. Coclusio I this chapter, Gauss-Jacobi ad Gauss-Seidel methods are used to fid the solutio of fully fuzzy liear system of equatios. To compare the methods same umerical eample is solved by usig both the methods ad it is show that i umber of iteratios i Gauss-Jacobi method are more tha umbers of iteratios i Gauss-Seidel method. So, o the basis of these results it ca be cocluded that it is better to use Gauss- Seidel method for solvig fully fuzzy liear system of equatios. 43

52 Chapter 6 LINEAR PROGRAMMING METHOD FOR SOLVING FULLY FUZZY LINEAR SYSTEM OF EQUATIONS The methods, preseted i previous chapters, ca be applied oly for fidig the solutio of simultaeous fully fuzzy liear system of equatios i.e the fully fuzzy liear system of equatios i which umbers of fuzzy variables equals to umber of equatios. These methods ca t be applied for solvig o-simultaeous fully fuzzy liear system of equatios. To overcome this shortcomig i this chapter a method is preseted which ca be applied for fidig the solutio of simultaeous as well as o simultaeous fully fuzzy liear system of equatios. To illustrate the method umerical eamples are solved. 6. Algorithm for solvig liear system of equatios usig liear programmig method To solve liear system of equatios A b, we have already discussed a approach give i chapter 3. Accordig to which the positive vectors, y ad z ca be foud by solvig followig liear system of equatios sychroously: A b, Ay M g, Az N h (6.) A costrait y 0 is employed to create a positive vector solutio. 44

53 I this sectio, we employ a method based o liear programmig for obtaiig solutio to above problem i.e. we use Two Phase or Big-M method to solve A b. The steps of proposed method are as follows: Step : Liear system of equatios defied i (6.) is cosidered as set of costraits Step : Add a artificial vector to each costrait Step 3: Add y 0 as a costrait Step 4: Defie a arbitrary objective fuctio which miimizes the sum of artificial vectors Step 5: Solve the followig liear programmig problem Miimize ' y ' z ' subject to A ' b, M Ay y ' g, N Az z ' h, y 0, ', y ', z ',, y, z 0 (6.) Let * * * * * * ( ', y ', z ',, y, z ) be the optimal solutio of above liear programmig problem. Optimal value of this liear programmig is o-egative. If it is zero, the we have * * * ' y ' z ' 0. Moreover we have * * *, y, z 0. From * 0, the first costrait of (6.) imply that * A b. Similarly for other costraits we have Ay g M ad Az h N. I additio, the fial costrait guaratees positivity property of fuzzy vector. If we have optimal value of (6.) to be positive i.e. if there eists some artificial variables i optimal base, the there does ot eist ay positive fuzzy vector solutio which satisfies A b. Algorithm give i sectio 6. is defied for Two-Phase 45

54 method. To employ Big-M method, algorithm is slightly differet. To illustrate the liear programmig approach by employig Two-Phase method ad Big-M method, umerical eamples are solved. 6. Numerical eamples I this sectio, to illustrate the algorithm a set of simultaeous fully fuzzy liear system of equatios is solved by Two-Phase ad Big-M method ad it is show that by applyig both the methods, same results are obtaied. Also the preseted algorithm is used to a set of o-simultaeous fully fuzzy liear system of equatios. 6.. Two-Phase method I this sectio, Two-Phase method is applied to solve a set of fully fuzzy liear system of equatios. Eample 6. Cosider the followig FFLS (5,,) (, y, z) (6,,) (, y, z) (50,0,7), (7,,0) (, y, z) (4,0,) (, y, z) (48,5,7). Thus we have, 5 6 A M N b g h Usig (6.), we have y z y z y z 46

55 y y z z 0 7 Applyig the algorithm, we get the followig liear programmig problem: Miimize z y y z z subject to , , 5y 6y y 0, 7 y 4y y 5, 5z 6z z 7, 7z 4z z 7, y 0, y 0,,, y, y, z, z,,, y, y, z, z 0. where y z ', y ' ad z ' y z are artificial vectors. Solvig above liear programmig problem usig TORA, we get the optimal solutio as: y y z z 0 ad 4, 5, y 0.09, y 0.09, z 0, z 0.5. Thus the solutio is (4,0.09,0) (5,0.09, 0.5) 47

56 6.. Big-M method If we employ Big-M method to solve the liear system of equatios, the algorithm defied earlier remais same with the differece that ow our objective fuctio becomes Miimize M ' M y ' M z ' 3 where M, M ad M 3 be three positive big umbers. If we select M bigger tha M ad M 3, ca eter to the base faster tha y ad z. Sice is used i the other costraits, it may accelerate the rate of covergece of algorithm. Now we shall solve the same umerical problem with Big-M method. Eample 6. (5,,) (, y, z) (6,,) (, y, z) (50,0,7), (7,,0) (, y, z) (4,0,) (, y, z) (48,5,7). Applyig the algorithm, we get the followig liear programmig problem: Miimize z M M M y M y M z M z subject to , , 5y 6y y 0, 7y 4y y 5, 5z 6z z 7, 7z 4z z 7, y 0, y 0,,, y, y, z, z,,, y, y, z, z 0. M, M ad M are positive big umbers. 3 Solvig above liear programmig problem usig TORA, we get the optimal solutio as: y y z z 0 48

57 ad 4, 5, y 0.09, y 0.09, z 0, z 0.5. Thus the solutio is: (4,0.09,0) (5, 0.09, 0.5) Eample 6.3 A (, 0., 0.) (, 0.4, 0.3) (3, 0.3, 0.4) (4, 0., 0.) (, 0.3, 0.) (3, 0.4, 0.) (, 0., 0.3) (, 0., 0.3) (, 0.3, 0.) (3, 0.4, 0.) (, 0., 0.3) (, 0., 0.3) (, 0.4, 0.5) (4, 0.5, 0.) (, 0.6,.) (3, 0.3, 0.3) (5, 0.5, 0.) (6, 0.4, 0.3) (7, 0.3, 0.5) (5, 0., 0,8) (9,, 0, 9) (, 0.9, 0.) (3, 0.8, 0.3) (4, 0.7, 0.3) (,0.6, 0.3) (3, 0.5, 0.) (, 0.9, 0.) (3, 0.8, 0.3) (4, 0.7, 0.3) (,0.6, 0.3) (3, 0.5, 0.) (, 0., 0.6) (6, 0., 0.3) (8, 0.3, 0.3) (7, 0.4, 0.9) (6, 0.5, 0.) ad b (09.9,3.9,4.) (48,.5, 7.9) (84.9, 7.5,8.3) (03,0.5,3.5) Thus, we have A M N

58 b g h Applyig the liear programmig method, we get the followig liear programmig problem: Miimize y y y y z z z z subject to , , , , y y 3y 4y 5y 6y 7 y 5y 9y y 3.9, y 3y y y y 3y 4y y 3y , y y y y 3y y 9y 5y 4y 6y y 7.5, y 4y y 3y y 6y 8y 7y 6y y 0.5, 50

59 z z 3z 4z 5z 6z 7z 5z 9z z 4., z 3z z z z 3z 4z z 3z z 7.9, z z z 3z z 9z 5z 4z 6z z 8.3, z 4z z 3z z 6z 8z 7z 6z z 3.5, y 0, y 0, y 0, y 0, y 0, y 0, y 0, y 0, y 9 9 0,,,,,,,,,, y, y, y, y, y, y, y, y, y, z, z, z, z , z, z, z, z, z,,,,, y, y, y, y, z, z, z, z Solvig above system of equatios usig some mathematical programmig software package (e.g. TORA), we get the objective value equal to 0. Thus, above fully fuzzy liear system of equatios has positive solutio: 0, 0, 0, 0, y 0, y 0, y 0, y 0,

60 z 0, z 0, z 0, z , 0,.6, 0, 0,.6, 4.47, 3.7, y 0, y 0, y.6, 0, 0, y 0.0, y 0, y 0.07, y z 0, z 0.76, z 0, z 0, z 0, z 0.05, z 0.3, z 0.5, z Thus the solutio is (, y, z) (0,0,0) (, y, z) (0,0,0) ( 3, y3, z3) (.6,.6,0) ( 4, y4, z4) (0,0,0) ( 5, y5, z5) (0,0,0) ( (.6, 0.0, 0.05) 6, y6, z6) ( (4.47,0,0) 7, y6, z6) ( (3.7,0.07, 8, y8, z8) 0.5) ( 9, y9, z9) (5.48, 0.08,0) 6.3 Coclusio I this chapter, the fully fuzzy liear fuzzy systems, i.e. fuzzy liear system of equatios with fuzzy coefficiets ivolvig fuzzy variables are solved usig liear programmig method. This method gives the positive fuzzy vector solutio, if it eists. Moreover, the proposed scheme is more efficiet ad useful i the maer that this ca be applied to solve o-squared fuzzy systems also. 5

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