Solution of Fully Fuzzy System of Linear Equations by Linear Programming Approach

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1 Copyright 2015 Tech Sciece Press CMES, vol.108, o.2, pp.67-87, 2015 Solutio of Fully Fuzzy System of Liear Equatios by Liear Programmig Approach Diptiraja Behera 1,2, Hog-Zhog Huag 1 ad S. Charaverty 3 Abstract: Fuzzy systems of liear equatios play a vital role i various applicatios of egieerig, sciece ad fiace problems. This paper proposes a ew method for solvig Fully Fuzzy System of Liear Equatios (FFSLE) usig the liear programmig problem approach. There is o restrictio o the elemets of coefficiet matrix. The proposed method is able to solve the system, whe the elemets of the fuzzy uow vector are both o-egative ad o-positive. Triagular covex ormalized fuzzy sets are cosidered for the preset aalysis. Kow example problems are solved ad compared with the results of existig methods to illustrate the efficacy ad reliability of the proposed method. Keywords: Liear programmig, Triagular fuzzy umber, Fully fuzzy system of liear equatios 1 Itroductio System of liear equatios has great applicatios i various areas such as operatioal research, physics, statistics, egieerig ad social scieces. Equatios of this type are ecessary to solve for the ivolved parameters. A geeral real system of liear equatios may be writte as AX = b, where, A ad b are crisp real matrix ad X is uow real vector. It is simple ad straight forward whe the variables ivolvig the system of equatios are crisp umbers. But i actual case the parameters may be ucertai or a vague estimatio about the variables are ow as those are foud i geeral by some observatio, experimet or experiece. So, to overcome the ucertaity ad vagueess, oe may use the fuzzy umbers i place of the crisp umbers. Thus the crisp system of liear equatios becomes a Fuzzy System 1 Istitute of Reliability Egieerig, School of Mechatroics Egieerig, Uiversity of Electroic Sciece ad Techology of Chia, No. 2006, Xiyua Aveue, West Hi-Tech Zoe, Chegdu, Sichua , P. R. Chia 2 Correspodig author. diptirajab@gmail.com, diptirajab@uestc.edu.c 3 Departmet of Mathematics, Natioal Istitute of Techology Rourela, Odisha , Idia

2 68 Copyright 2015 Tech Sciece Press CMES, vol.108, o.2, pp.67-87, 2015 of Liear Equatios (FSLE) or Fully Fuzzy System of Liear Equatios (FFSLE). There is a differece betwee fuzzy liear system ad fully fuzzy liear system. The coefficiet matrix is treated as crisp i the fuzzy liear system, but i the fully fuzzy liear system all the parameters ad variables are cosidered to be fuzzy umbers. It is a importat issue to develop mathematical models ad umerical techiques that would appropriately treat the geeral fuzzy or fully fuzzy liear systems because subtractio ad divisio of fuzzy umbers are ot the iverse operatios to additio ad multiplicatio respectively. So, this is a importat area of research i the recet years. As such i the followig paragraph few related literatures are reviewed for the sae of completeess of the problem. The cocept of fuzzy set ad fuzzy umber were first itroduced by Zadeh (1965). Related to fuzzy sets several excellet boos have also bee writte by differet authors [Hass (2005); Zimmerma (2001); Ross (2004); Kaufma ad Gupta (1985); Dubois ad Prade (1980)]. We ow that fuzzy umber arithmetic is widely applied i computatio of liear system of equatios, whose parameters are represeted by fuzzy umbers, has a great importace. Solutio of a geeralised FSLE was first proposed by Friedma, Mig, ad Kadel (1998), whose coefficiet matrix ad right-had side colum vector are defied as crisp ad fuzzy respectively. Moreover some methods for solvig this type of system ca be foud i [Charaverty ad Behera (2013); Behera ad Charaverty (2012a); Behera ad Charaverty (2013c); Abbasbady ad Jafaria (2006); Abbasbady, Jafaria, ad Ezzati (2005); Allahviraloo (2004, 2005); Su ad Guo (2009); Yi ad Wag (2009); Gog ad Guo (2011)]. Also these types of system are applied to fid the static resposes of structures usig fuzzy fiite elemet method [Behera ad Charaverty (2013b)]. However FFSLE, was also studied by few authors. As such Behera ad Charaverty (2015); Das ad Charaverty (2012) have studied the solutio procedure for fully fuzzy system of liear equatios, where the authors have cosidered all the ivolved parameters as positive. A umerical approach based o Cholesy decompositio is described by Sethilumar ad Rajedra (2011) to fid the positive solutio of a symmetric fully fuzzy liear system. Recetly, Babbar, Kumar, ad Basal (2013) proposed a ew method to fid the o-egative solutio of a fully fuzzy liear system, where the elemets of the coefficiet matrix are defied as arbitrary triagular fuzzy umbers of the form (m, α, β). Dehgha ad Hashemi (2006); Dehgha, Hashemi, ad Ghatee (2007) have proposed the adomia decompositio method, iterative methods ad some computatioal methods such as Cramer s rule, Gauss elimiatio method, LU decompositio method ad liear programmig approach for fidig the solutios of fully fuzzy system of liear equatios. Muzzioli ad Reyaerts (2007) ivestigated the o-egative solutio procedure of fuzzy system

3 Solutio of Fully Fuzzy System of Liear Equatios 69 by o-liear programmig approach. Otadi ad Mosleh (2012) applied a liear programmig approach to fid the o-egative solutio of a fully fuzzy matrix equatio whose elemets of the coefficiet matrix are cosidered as arbitrary triagular fuzzy umbers. There are o restrictios about the elemets of the coefficiet matrix of the correspodig system. Allahviraloo ad Miaeilvad (2011) discussed fully fuzzy system of liear equatios by usig the embeddig approach. Allahviraloo, Salahshour, ad Khezerloo (2011) proposed the maximal ad miimal symmetric solutios of fully fuzzy liear systems. Recetly Allahviraloo, Hosseizadeh, Ghabari, Haghi, ad Nuraei (2013) also studied a ew approach for fuzzy trapezoidal solutio, amely suitable solutio, for a fully fuzzy liear system (FFLS) based o solvig two fully iterval liear systems (FILSs) that are 1-cut ad 0-cut of the related fuzzy iterval systems. Moreover a approximate solutio of dual fuzzy matrix equatios has bee aalyzed by Gog, Guo, ad Liu (2014) recetly. Also Behera ad Charaverty (2013a) have applied FFSLE for the ucertai static resposes of structures usig fuzzy fiite elemet method. However Yag, Li, ad Cai (2013) have cosidered both radom ad fuzzy variables for the structural reliability. I the followig sectios first basic prelimiaries are give. The a ew method is proposed to solve fully fuzzy system of liear equatios usig the liear programmig approach. Next, umerical examples are solved usig the proposed method. Lastly coclusios are draw. 2 Prelimiaries I this sectio, some otatios, defiitios ad prelimiaries related to the preset wor are give [Kaufma ad Gupta (1985); Zimmerma (2001); Ross (2004); Behera ad Charaverty (2012a, 2014); Otadi ad Mosleh (2012); Fatullayev ad Koroglu (2012)]. Defiitio 2.1 (Fuzzy umber). Fuzzy umber ũ is a covex ormalized fuzzy set ũ of the real lie R such that {µũ(x) : R [0,1], x R} where, µũ is called the membership fuctio of the fuzzy set ad it is piecewise cotiuous. Defiitio 2.2 (Triagular fuzzy umber). Triagular fuzzy umber ũ is a covex ormalized fuzzy set ũ of the real lie R such that i There exists exactly oe x 0 R with µũ(x 0 ) = 1 (x 0 is called the mea value of ũ),where µũ is called the membership fuctio of the fuzzy set. ii µũ(x) is piecewise cotiuous.

4 70 Copyright 2015 Tech Sciece Press CMES, vol.108, o.2, pp.67-87, 2015 Let us cosider a arbitrary triagular fuzzy umber ũ = (a,b,c). The membership fuctio µũ of ũ will be defie as follows 0, x a x a µũ(x) = b a, a x b c x c b, b x c 0, x c The triagular fuzzy umber ũ = (a,b,c) ca be represeted with a ordered pair of fuctios through α cut approach viz. [u(α),ū(α)] = [(b a)α +a, (c b)α + c] where, α [0,1]. Defiitio 2.3. No-egative (No-positive) triagular fuzzy umber A triagular fuzzy umber ũ = (a,b,c) is said to be o-egative (o-positive) if a 0(c 0). Defiitio 2.4. Fuzzy arithmetic Let ũ = (a,b,c) ad ṽ = (e, f,g) be two triagular fuzzy umbers. The fuzzy arithmetic is defied as below i ũ + ṽ = (a + e,b + f,c + g), ii ũ = ( c, b, a), iii ũ ṽ = (a g,b f,c e). Multiplicatio of two arbitrary fuzzy umbers is deoted as [Otadi ad Mosleh (2012)] ũ ṽ = (l,m,r) where, l = mi(ae,ag,ce,cg), m = b f ad r = max(ae,ag,ce,cg). Two triagular fuzzy umbers ũ = (a,b,c) ad ṽ = (e, f,g) are said to be equal if ad oly if a = e, b = f ad c = g. Next let us assume ũ = (a,b,c) is a arbitrary triagular fuzzy umber ad ṽ = (e, f,g) is a o-egative oe, the oe may have (ae,b f,cg) a 0, ũ ṽ = (ag,b f,ce) c 0, (ag,b f,cg) a < 0,c 0.

5 Solutio of Fully Fuzzy System of Liear Equatios 71 3 Fully fuzzy system of liear equatios ad the proposed method The fully fuzzy system of liear equatios may be writte as ã 11 x 1 + ã 12 x ã 1 x = b 1 ã 21 x 1 + ã 22 x ã 2 x = b 2. ã 1 x 1 + ã 2 x ã x = b. (1) I matrix otatio above system may be writte as à X = b, (2) where, the coefficiet matrix à = ã i j,1 i, j is a fuzzy matrix of triagular fuzzy umbers, b = b i,1 i is a colum vector of triagular fuzzy umber ad X = x j, j is the vector of fuzzy uow, where 0 / x j. A fuzzy umber vector X = ( x 1, x 2,, x ) T, give by x j = (y j,x j,z j ),1 j is called the solutio of the fuzzy matrix system (2) if ã i j x j = b i, for 1 i. (3) Let us ow deote the triagular fuzzy umber matrix elemets such as ã i j = (m i j,a i j, i j ), x j = (y j,x j,z j ) ad b i = (g i,b i,h i ). So Eq. (3) may be writte as (m i j,a i j, i j )(y j,x j,z j ) = (g i,b i,h i ). (4) If i the fully fuzzy system of liear equatios (2), each elemet of Ã, X ad b is a o-egative fuzzy umber, the we call the system (2) a o-egative FFSLE. Defiitio 3.1. Cosider a o-egative FFSLE as defied i Eq. (3). We say that x j is a o-egative fuzzy solutio vector if m i j y j = g i a i j x j = b i i j z j = h i (5)

6 72 Copyright 2015 Tech Sciece Press CMES, vol.108, o.2, pp.67-87, 2015 Moreover if y j 0, x j y j 0 ad z j x j 0 the we say that x j is a cosistet solutio of the FFSLE. Let m i j = M,a i j = A, i j = N,g i = g,b i = b,h i = h,y j = Y,x j = X ad z j = Z. Hece system (5) ca be writte i matrix form as MY = g, AX = b, NZ = h. From this oe may get the solutio as Y = M 1 g, X = A 1 b, Z = N 1 h, if M, A, adn are osigular. Next a theorem is stated ad proved as follows for the existece of solutio. This is the special case of the theorem proved by Otadi ad Mosleh (2012). Theorem 3.2. Let à = (M,A,N) 0, b = (g,b,h) 0, ad each of the matrices M,A,N be a product of a permutatio matrix by a diagoal oe. Also let M 1 g A 1 b N 1 h. The the o-egative FFSLE (2) has exist a o-egative cosistet fuzzy solutio. Proof. Hypothesis imply that M 1,A 1,N 1 exists as o-egative matrices (De- Marr 1972). So we have Y = M 1 g 0, X = A 1 b 0, Z = N 1 h 0 with M 1 g A 1 b N 1 h. Hece from this oe may coclude that X is a o-egative solutio of the required system. Next we will proceed for the proposed method where the compoets of the elemets of the coefficiet matrix has o restrictios o their sig. Before this first we will discuss some limitatios of the existig methods to have a better idea about the preset aalysis. 3.1 Limitatios of the existig methods Followig are short comigs of the existig methods for solvig fuzzy ad fully fuzzy system of liear equatios. 1. There exist differet solutio procedures [Charaverty ad Behera (2013); Behera ad Charaverty (2012a); Behera ad Charaverty (2013a); Abbasbady ad Jafaria (2006); Abbasbady, Jafaria, ad Ezzati (2005); Allahviraloo (2004, 2005); Su ad Guo (2009); Yi ad Wag (2009); Friedma, Mig, ad Kadel (1998)] for fuzzy system of liear equatios where (6) (7)

7 Solutio of Fully Fuzzy System of Liear Equatios 73 the coefficiet matrices are cosidered as crisp real matrix. It may be oted that these methods are ot applicable whe system is fully fuzzy. 2. Various methodologies [Das ad Charaverty (2012); Sethilumar ad Rajedra (2011); Dehgha ad Hashemi (2006); Dehgha, Hashemi, ad Ghatee (2007); Muzzioli ad Reyaerts (2007); Otadi ad Mosleh (2012); Allahviraloo ad Miaeilvad (2011)] have bee proposed to solve FFSLE of the form where all the elemets of fuzzy matrices are cosidered as o-egative. These methods are ot able to solve the problem as defie i Example Recetly Otadi ad Mosleh (2012); Babbar, Kumar, ad Basal (2013) proposed solutio techique for FFSLE. The compoets of the elemets of the coefficiet matrix has o restrictios o their sig. But the methods ca oly give the o-egative solutio. These methods are ot applicable whe the uow solutio vector cosists of oly o-positive elemets or both o-egative ad o-positive elemets as cosidered i Examples 2 ad 3. To overcome the above limitatios a ew method has bee proposed i the followig sectio based o liear programmig problem approach. 3.2 Proposed method for solvig FFSLE usig liear programmig Let us first cosider the α cut represetatio of the FFSLE (3), ã i j (α) x j (α) = b i (α). (8) Before proceedig to solve the above system we will first determie the sig of the elemets of the solutio vector of the mai system (1). For this first, we have to fid the core of the solutio vector. That meas we have to solve Eq. (8) for α = 1. Hece correspodig system (8) coverts to a crisp system of liear equatios as ã i j (1) x j (1) = b i (1). (9) Equivaletly the above system ca be writte as a i j x j = b i (10) where ã i j (1) = a i j, x j (1) = x j ad b i (1) = b i. From Eq. (10) oe ca get the core solutio ad may predict the sig of the elemets of solutio vector by the followig propositio.

8 74 Copyright 2015 Tech Sciece Press CMES, vol.108, o.2, pp.67-87, 2015 Propositio 3.3. If ãi j x j = b i, for 1 i where, 0 / x j, the sig of the elemets of the fuzzy solutio vector ca be predicted from the core solutio of the correspodig system by solvig a i jx j = b i where ã i j (1) = a i j, x j (1) = x j ad b i (1) = b i for α = 1. Proof. The core solutio x j ca be obtaied by solvig the crisp system a i jx j = b i. From this we may get the sig of the elemets preset i core solutio. Also we ow that 0 / x j ad core is the ier poit of the fuzzy solutio. Hece oe may predict the sig of the elemets of fuzzy solutio vector accordigly. Next the followig propositios are to be used to fid whether the sig of the elemets of the fuzzy solutio vector are o-egative, o-positive or both oegative ad o-positive. Propositio 3.4. If the elemets of core solutio viz. x j for 1 j, are oegative (o-positive) the the elemets of fuzzy solutio vector x j are o-egative (o-positive). Proof. The proof of the propositio is straight forward. Propositio 3.5. If the core solutio cotais both o-egative ad o-positive elemets, i.e. x j for { j N 1 j } are o-egative ad for { j N + 1 j } are o-positive for all i, where 1 i ad N is the atural umber, the the fuzzy solutio vector x j for { j N 1 j } are o-egative ad for { j N + 1 j } are o-positive for all i. Proof. The proof of the propositio is straight forward. I geeral the obtaied sig of the elemets may be oe of the followig cases: Case 1: All x j are o-egative. Case 2: All x j are o-positive. Case 3: Few x j are o-egative ad few are o-positive. We discuss below the solutio procedure for all the above cases. Case 1: I this case we have cosidered all x j are o-egative. So, Eq. (4) for this case may be coverted to (m i j,a i j, i j )(y j,x j,z j ) + (m i j,a i j, i j )(y j,x j,z j ) m i j 0 i j 0 + (m i j,a i j, i j )(y j,x j,z j ) = (g i,b i,h i ). m i j 0 i j (11)

9 Solutio of Fully Fuzzy System of Liear Equatios 75 Eq. (11) is ow expressed by applyig the geeral rule of fuzzy multiplicatio as (m i j y j,a i j x j, i j z j ) + (m i j z j,a i j x j, i j y j ) m i j 0 i j 0 + (m i j z j,a i j x j, i j z j ) = (g i,b i,h i ) m i j 0 i j (12) Above equatio ca be writte equivaletly m i j y j + m i j 0 i j 0 m i j 0 m i j 0 a i j x j + i j 0 i j z j + i j 0 m i j z j + m i j z j =g i, m i j 0 i j a i j x j + a i j x j =b i, m i j 0 i j i j y j + i j z j =h i. m i j 0 i j (13) Let us deote the above system as w i j = g i q i j = b i u i j = h i where, w i j = m i j y j + m i j 0 i j 0 q i j = a i j x j + m i j 0 i j 0 u i j = i j z j + m i j 0 i j 0 for 1 i, (14) m i j z j + m i j z j, m i j 0 i j a i j x j + a i j x j ad m i j 0 i j i j y j + i j z j. m i j 0 i j Next oe may solve the crisp system (13) directly or may covert Eq. (14) ito the followig Liear Programmig Problem (LPP) to have the solutio. For LPP, the artificial variables r s for s = 1,2,,, + 1, 3 are itroduced. Hece the correspodig LPP ca be defied as

10 76 Copyright 2015 Tech Sciece Press CMES, vol.108, o.2, pp.67-87, 2015 Miimize : r 1 + r r 3 w 1 j + r 1 = g 1, Subjectto : w 2 j + r 2 = g 2,.. w j + r = g, q 1 j + r +1 = b 1, q 2 j + r +2 = b 2, q 1 j + r 2 = b, u 1 j + r 2+1 = h 1, u 2 j + r 2+2 = h 2,. u j + r 3 = h. (15) With the o-egative restrictios, y j,x j,z j ad r s for s = 1,2,,,+1,,3 0. Stadard method may be applied to get the fial solutio of the fully fuzzy system. Case 2: Next i this case let us cosider all x j are o-positive. For this case Eq. (4) ca similarly be writte as (m i j,a i j, i j )(y j,x j,z j ) + (m i j,a i j, i j )(y j,x j,z j ) m i j 0 i j 0 + (m i j,a i j, i j )(y j,x j,z j ) = (g i,b i,h i ). m i j 0 i j (16) The above equatio is ow expressed, by chagig all o-positive variables to o-egative variables as

11 Solutio of Fully Fuzzy System of Liear Equatios 77 ( ˆ i j,â i j, ˆm i j )(ẑ j, ˆx j,ŷ j ) + ( ˆ i j,â i j, ˆm i j )(ẑ j, ˆx j,ŷ j ) ˆm i j 0 ˆ i j 0 + ( ˆ i j,â i j, ˆm i j )(ẑ j, ˆx j,ŷ j ) = (g i,b i,h i ) ˆ i j 0 ˆm i j (17) where, (y j,x j,z j ) = (ẑ j, ˆx j,ŷ j ) ad ( ˆ i j,â i j, ˆm i j ) = (m i j,a i j, i j ). Now applyig the geeral rule of fuzzy multiplicatio we have ( ˆ i j ŷ j,â i j ˆx j, ˆm i j ẑ j ) + ( ˆ i j ẑ j,â i j ˆx j, ˆm i j ŷ j ) ˆm i j 0 ˆ i j 0 + ( ˆ i j ŷ j,â i j ˆx j, ˆm i j ŷ j ) = (g i,b i,h i ) ˆ i j 0 ˆm i j (18) Next we may represet the above system as w i j = g i, q i j = b i, u i j = h i. (19) where, w i j = ˆ i j ŷ j + ˆm i j 0 ˆ i j 0 q i j = â i j ˆx j + ˆm i j 0 ˆ i j 0 u i j = ˆm i j ẑ j + ˆm i j 0 ˆ i j 0 ˆ i j ẑ j + ˆ i j ŷ j, ˆ i j 0 ˆm i j â i j ˆx j + â i j ˆx j ad ˆ i j 0 ˆm i j ˆm i j ŷ j + ˆm i j ŷ j. ˆ i j 0 ˆm i j Similarly for the Case 1 oe ca covert the above system to the followig LPP to fid the optimum solutio

12 78 Copyright 2015 Tech Sciece Press CMES, vol.108, o.2, pp.67-87, 2015 Miimize : r 1 + r r 3 w 1 j + r 1 = g 1, Subjectto : w 2 j + r 2 = g 2,.. w j + r = g, q 1 j + r +1 = b 1, q 2 j + r +2 = b 2, q 1 j + r 2 = b, u 1 j + r 2+1 = h 1, u 2 j + r 2+2 = h 2,. u j + r 3 = h. With the o-egative restrictios ŷ j, ˆx j,ẑ j ad r s for s = 1,2,,,+1,,3 0. Now solvig the above LPP by ay stadard method oe may have the optimum solutio. From which we may obtai the solutio of the FFSLE. Case 3: Fially for this case let us assume the solutio vector x j cotais both o-egative ad o-positive fuzzy umbers. Hece let us cosider that x j for { j N 1 j } are o-egative ad for { j N +1 j } are o-positive for all i, where 1 i ad N is the atural umber. Keepig this i mid oe may ow covert Eq. (4) as (m i j,a i j, i j )(y j,x j,z j ) + (m i j,a i j, i j )(y j,x j,z j ) = (g i,b i,h i ). (21) j=+1 (20) From the discussio of previous two cases, it is possible to write the above expressio as follows

13 Solutio of Fully Fuzzy System of Liear Equatios 79 m i j y j + m i j z j + m i j z j m i j 0 i j 0 m i j 0 i j }{{} for1 j + ˆ i j ŷ j + ˆ i j ẑ j + ˆ i j ŷ j = g i, ˆm i j 0 ˆ i j 0 ˆ i j 0 ˆm i j }{{} for+1 j a i j x j + a i j x j + a i j x j m i j 0 i j 0 m i j 0 i j }{{} for1 j + â i j ˆx j + â i j ˆx j + â i j ˆx j = b i, ˆm i j 0 ˆ i j 0 ˆ i j 0 ˆm i j }{{} for+1 j i j z j + i j y j + i j z j m i j 0 i j 0 m i j 0 i j }{{} for1 j + ˆm i j ẑ j + ˆm i j ŷ j + ˆm i j ŷ j = h i. ˆm i j 0 ˆ i j 0 ˆ i j 0 ˆm i j }{{} for+1 j or w i j + q i j + j=+1 j=+1 w i j = g i, q i j = b i, u i j + u i j = h i. j=+1 (22) (23) Hece correspodig LPP for the above system (23) ca be expressed as

14 80 Copyright 2015 Tech Sciece Press CMES, vol.108, o.2, pp.67-87, 2015 Miimize : r 1 + r r 3 Subject to : w 1 j + w 2 j +.. w j + q 1 j + q 2 j + q j + u 1 j + u 2 j +. u j + j=+1 w 1 j + r 1 = g 1, w 2 j + r 2 = g 2, j=+1 j=+1 j=+1 w j + r = g, q 1 j + r +1 = b 1, q 2 j + r +2 = b 2, j=+1 j=+1 j=+1 q 1 j + r 2 = b, u 1 j + r 2+1 = h 1, u 2 j + r 2+2 = h 2, j=+1 u j + r 3 = h. j=+1 (24) With the o-egative restrictios y j,x j,z j,ŷ j, ˆx j,ẑ j ad r s for s = 1,2,,, + 1, 3 0. Now solvig the correspodig LPP (24) oe may get the solutio accordigly. To illustrate the applicability of the proposed method example problems are solved i the followig sectio. 4 Numerical examples ad discussios Example 1 Let us cosider a 2 2 fully fuzzy system of liear equatios (Otadi ad Mosleh 2012) (1,2,3) x 1 + (2,3,5) x 2 = (4,19,46) ( 2, 1,2) x 1 + (1,2,3) x 2 = ( 13,1,29). (25)

15 Solutio of Fully Fuzzy System of Liear Equatios 81 Suppose x 1 = (y 1,x 1,z 1 ) ad x 2 = (y 2,x 2,z 2 ). The fully fuzzy system of liear e- quatios is ow writte as (1,2,3)(y 1,x 1,z 1 ) + (2,3,5)(y 2,x 2,z 2 ) = (4,19,46) ( 2, 1,2)(y 1,x 1,z 1 ) + (1,2,3)(y 2,x 2,z 2 ) = ( 13,1,29). (26) To fid the core solutio of the above system we have 2x 1 + 3x 2 = 19 1x 1 + 2x 2 = 1. (27) Solvig Eq. (27) we have x 1 = 5 ad x 2 = 3. This meas all the elemets of the solutio vector are o-egative. Hece as per the discussio of Case 1, oe may have the followig system (y 1 + 2y 2,2x 1 + 3x 2,3z 1 + 5z 2 ) = (4,19,46) ( 2z 1 + y 2, x 1 + 2x 2,2z 1 + 3z 2 ) = ( 13,1,29). (28) Eq. (28) is equivaletly writte as y 1 + 2y 2 = 4 2x 1 + 3x 2 = 19 3z 1 + 5z 2 = 46 2z 1 + y 2 = 13 x 1 + 2x 2 = 1 2z 1 + 3z 2 = 29. (29) As such, the correspodig liear programmig of the above system ca be expressed as Miimize : r 1 + r 2 + r 3 + r 4 + r 5 + r 6 1y 1 + 2y 2 + r 1 = 4 2x 1 + 3x 2 + r 2 = 19 3z 1 + 5z 2 + r 3 = 46 Subject to : 2z 1 + 1y 2 + r 4 = 13 1x 1 + 2x 2 + r 5 = 1 2z 1 + 3z 2 + r 6 = 29 (30) where r 1,r 2, r 3,y 1,x 1,z 1,y 2,x 2,z 2 0. Solvig the LPP (30) the optimal solutio may be obtaied as y 1 = 2,x 1 = 5,z 1 = 7,y 2 = 1,x 2 = 3 ad z 2 = 5. Therefore

16 82 Copyright 2015 Tech Sciece Press CMES, vol.108, o.2, pp.67-87, 2015 the required fuzzy solutio is x 1 = (2,5,7) ad x 2 = (1,3,5). Obtaied results are compared with the solutio of Otadi ad Mosleh (2012) ad foud that the results are exactly same. Example 2 Next cosider a 2 2 fully fuzzy system of liear equatios (1,2,3) x 1 + ( 2, 1, 1) x 2 = ( 6, 1,3) ( 3, 2,1) x 1 + (2,4,5) x 2 = ( 12, 2,6). (31) Agai suppose x 1 = (y 1,x 1,z 1 ) ad x 2 = (y 2,x 2,z 2 ), hece Eq. (31) is writte as (1,2,3)(y 1,x 1,z 1 ) + ( 2, 1, 1)(y 2,x 2,z 2 ) = ( 6, 1,3) ( 3, 2,1)(y 1,x 1,z 1 ) + (2,4,5)(y 2,x 2,z 2 ) = ( 12, 2,6). (32) The core solutio of the above system ca be obtaied as x 1 = 1 ad x 2 = 1. From this we may coclude that the elemets of the fuzzy solutio vector are opositive. As per the discussio of Case 2 the above system is ow expressed by covertig the o-positive elemets to o-egative elemets of the solutio as ( 3, 2, 1)(ẑ 1, ˆx 1,ŷ 1 ) + (1,1,2)(ẑ 2, ˆx 2,ŷ 2 ) = ( 6, 1,3) ( 1,2,3)(ẑ 1, ˆx 1,ŷ 1 ) + ( 5, 4, 2)(ẑ 2, ˆx 2,ŷ 2 ) = ( 12, 2,6). (33) where (y 1,x 1,z 1 ) = (ẑ 1, ˆx 1,ŷ 1 ) ad (y 2,x 2,z 2 ) = (ẑ 2, ˆx 2,ŷ 2 ). So Eq. (33) may be writte as ( 3ŷ 1 + ẑ 2, 2 ˆx 1 + ˆx 2, ẑ 1 + 2ŷ 2 ) = ( 6, 1,3) ( ŷ 1 5ŷ 2,2 ˆx 1 4 ˆx 2,3ŷ 1 2ẑ 2 ) = ( 12, 2,6). This is equivalet to 3ŷ 1 + ẑ 2 = 6 2 ˆx 1 + ˆx 2 = 1 ẑ 1 + 2ŷ 2 = 3 ŷ 1 5ŷ 2 = 12 2 ˆx 1 4 ˆx 2 = 2 3ŷ 1 2ẑ 2 = 6. (34) Correspodig liear programmig of the above system ca be expressed as

17 Solutio of Fully Fuzzy System of Liear Equatios 83 Miimize r 1 + r 2 + r 3 + r 4 + r 5 + r 6 3ŷ 1 + ẑ 2 + r 1 = 6 2 ˆx 1 + ˆx 2 + r 2 = 1 ẑ 1 + 2ŷ 2 + r 3 = 3 Subject to ŷ 1 5ŷ 2 + r 4 = 12 2 ˆx 1 4 ˆx 2 + r 5 = 2 3ŷ 1 2ẑ 2 + r 6 = 6 (35) where r 1,r 2, r 3,ŷ 1, ˆx 1,ẑ 1,ŷ 2, ˆx 2,ẑ 2 0. Solvig the LPP (35) the optimal solutio ca be obtaied as ẑ 1 = 1, ˆx 1 = 1,ŷ 1 = 2,ẑ 2 = 0, ˆx 2 = 1 ad ŷ 2 = 2. Therefore the required fuzzy solutio x 1 = (y 1,x 1,z 1 ) = (ẑ 1, ˆx 1,ŷ 1 ) = (1,1,2) = ( 2, 1, 1) ad x 2 = (y 2,x 2,z 2 ) = (ẑ 2, ˆx 2,ŷ 2 ) = (0,1,2) = ( 2, 1,0). Example 3 I this example agai let us cosider a 2 2 fully fuzzy system of liear equatios ( 2,3,4) x 1 + ( 2,2,3) x 2 = ( 13,2,14) (1,2,2) x 1 + (4,4,5) x 2 = ( 14, 4,0). (36) Agai suppose x 1 = (y 1,x 1,z 1 ) ad x 2 = (y 2,x 2,z 2 ), hece Eq. (36) is ow writte as ( 2,3,4)(y 1,x 1,z 1 ) + ( 2,2,3)(y 2,x 2,z 2 ) = ( 13,2,14) (1,2,2)(y 1,x 1,z 1 ) + (4,4,5)(y 2,x 2,z 2 ) = ( 14, 4,0). (37) Similarly for the above cases we have the core solutio for the above system is x 1 = 2 ad x 2 = 2. So from this we may get that the first ad secod elemets of the fuzzy solutio vector are o-egative ad o-positive respectively. As per the discussio of Case 3 the above system is ow expressed by covertig the o-positive elemet to o-egative as ( 2,3,4)(y 1,x 1,z 1 ) + ( 3, 2,2)(ẑ 2, ˆx 2,ŷ 2 ) = ( 13,2,14) (1,2,2)(y 1,x 1,z 1 ) + ( 5, 4,4,)(ẑ 2, ˆx 2,ŷ 2 ) = ( 14, 4,0) (38) where (ẑ 2, ˆx 2,ŷ 2 ) = (y 2,x 2,z 2 ).

18 84 Copyright 2015 Tech Sciece Press CMES, vol.108, o.2, pp.67-87, 2015 Applyig the geeral rule of fuzzy multiplicatio ad additio we have ( 2z 1 + 3ŷ 2,3x ˆx 2,4z 1 + 2ŷ 2 ) = ( 13,2,14) (y 1 5ŷ 2,2x 1 4 ˆx 2,2z 1 4ẑ 2 ) = ( 14, 4,0). This is equivalet to 2z 1 + 3ŷ 2 = 13 3x ˆx 2 = 2 4z 1 + 2ŷ 2 = 14 y 1 5ŷ 2 = 14 2x 1 4 ˆx 2 = 4 2z 1 4ẑ 2 = 0. (39) (40) The correspodig liear programmig of the above system ca be expressed as Miimize r 1 + r 2 + r 3 + r 4 + r 5 + r 6 2z 1 + 3ŷ 2 + r 1 = 13 3x ˆx 2 + r 2 = 2 4z 1 + 2ŷ 2 + r 3 = 14 Subject to y 1 5ŷ 2 + r 4 = 14 2x 1 4 ˆx 2 + r 5 = 4 2z 1 4ẑ 2 + r 6 = 0 where r 1,r 2, r 3,y 1,x 1,z 1,ŷ 2, ˆx 2,ẑ 2 0. Hece solvig the LPP (41) the optimal solutio ca be obtaied as y 1 = 1,x 1 = 2,z 1 = 2,ẑ 2 = 1, ˆx 2 = 2 ad ŷ 2 = 3. Therefore the required fuzzy solutio x 1 = (y 1,x 1,z 1 ) = (1,2,2) ad x 2 = (y 2,x 2,z 2 ) = (ẑ 2, ˆx 2,ŷ 2 ) = (1,2,3) = ( 3, 2, 1). 5 Coclusios This paper uses arithmetic operatios o fuzzy umbers ad the cocept of liear programmig for the solutio of fully fuzzy system of liear equatios. There is o restrictio o the coefficiet matrix of the correspodig system. The method foud efficiet whe the elemets of the fuzzy solutio vector are oly o-egative, opositive or both. Suitable umerical examples are solved to show the efficiecy of the proposed method. Results obtaied by the proposed method are also compared with the results obtaied by the existig methods ad foud i good agreemet. (41)

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