Appled Mathematcal Sceces, Vol 6, 0, o 4, 63-7 Soluto of Geeral Dual Fuzzy Lear Systems Usg ABS Algorthm M A Farborz Aragh * ad M M ossezadeh Departmet of Mathematcs, Islamc Azad Uversty Cetral ehra Brach, PO Box 385768, ehra, Ira *e-mal: farborzaragh@gmalcom, m_farborz@auctbacr Abstract he ma am of ths paper teds to dscuss the soluto of geeral dual fuzzy lear system (GDFLS AX + F = BX + C where A, B are real m matrx, F ad C are fuzzy vectors, ad the ukow vector X s a vector cosstg of fuzzy umbers, by a specal algorthm based o a class of ABS algorthms called uag algorthm I specal case, we apply the proposed algorthm for the soluto of fuzzy lear system AX = B where elemets of A s crsp ad B ad X are fuzzy vectors Numercal examples show the effcecy of usg ABS algorthm Keywords: Fuzzy lear systems, Geeral dual fuzzy lear system, Fuzzy umber, Fuzzy arthmetc, ABS algorthm, uag algorthm Itroducto he cocept of fuzzy umbers ad fuzzy arthmetc operatos were frst troduced by Zadeh [6] ad Dubos ad Prade [5] Some formato o fuzzy umbers ad fuzzy arthmetc ca be foud Kafma [] Oe of the mportat topcs fuzzy set theory s to solve fuzzy lear systems of equatos Fuzzy Lear systems of equatos play a maor role several applcatos varous areas such as ecoomcs, physcs, statstcs, egeerg, face ad socal sceces Fredma et al [8] proposed a geeral model for solvg fuzzy lear systems by usg the embeddg approach he [7], they used the parametrc form of fuzzy umbers ad replaced the orgal fuzzy lear system by a crsp lear
64 M A Farborz Aragh ad M M ossezadeh system ad studed dualty fuzzy lear systems AX = BX + Y where A, B are real matrx, the ukow vector X s vector cosstg of fuzzy umbers ad the costat Y s vector cosstg of fuzzy umbers Wag et al [5] proposed a teratve algorthm for solvg dual lear system of the form X = AX + U where A s real matrx, the ukow vector X ad the costat U s vectors cosstg of fuzzy umbers Muzzlo ad Reyaerts [3] metoed fuzzy lear systems of the form A x + b = A x + b wth A, A square matrces of fuzzy coeffcets ad b, b fuzzy umber vectors Abbasbady et al [], cosdered the exstece of a mmal soluto of geeral dual fuzzy lear systems of the form AX + F = BX + C where A, B are real m matrx, F ad C are fuzzy vectors, ad the ukow vector X s vector cosstg of fuzzy umbers Xu-dog Su ad S-zog [4], preseted geeral fuzzy lear systems of the form AX = Y ad geeral dual fuzzy lear systems of the form AX + Y = BX + Z wth A, B matrces of crsp coeffcets ad Z, Y fuzzy umber vectors I secto, we troduce some ma deftos ad theorems fuzzy sets theory I secto 3, we recall the ABS algorthms, I secto 4, we propose a algorthm based o the uag algorthm for fdg the soluto of a dualty fuzzy lear systems AX + F = BX + C where A ad B are real m matrces, F ad C are fuzzy vectors, ad the ukow vector X s a vector cosstg of fuzzy umbers he, by usg the ausdorff dstace we compare the umercal soluto obtaed from the proposed algorthm wth aalytcal soluto the sample example Prelmares Defto [7] A fuzzy umber a s of LR-type f there exst shape fuctos L (for left, R (for rght ad scalars α > 0, β > 0 wth m x L, x m α μ a ( x = x m R, x m β a = ( m, α, β LR s a tragular fuzzy umber f L = R = max(0,- x A popular fuzzy umber s a tragular fuzzy umber a = ( m, α, β where m, s a real umber, ad α, β are called the left ad rght spreads, respectvely If α = m α ad β = m + β the we ca also use the otato a = ( m, α, β order to show the fuzzy umber a Defto A fuzzy umber a s called postve (egatve, deoted by m > 0 ( m < 0, f ts membershp fucto μ (x satsfes μ ( x = 0, x 0 ( x 0 a a
Soluto of geeral dual fuzzy lear systems 65 heorem 3 [7] Let x = ( m, α, β, y = (, γ, δ ad λ IR ( λm, λα, λβ λ > 0 (a λ ( m, α, β = ( λm, λβ, λα λ < 0 (b x + y = ( m +, α + γ, β + δ (c x y = ( m, α + δ, β + γ he Defto 4 [0] We represet a arbtrary fuzzy umber by a ordered par of fuctos ( u (, u(, 0 r whch satsfy the followg requremets: u ( s a bouded left cotuous o-decreasg fucto over [ 0, ], u ( s a bouded left cotuous o-creasg fucto over [ 0, ], 3 u( u(, 0 r he set of all these fuzzy umbers s deoted by E whch s a complete metrc space wth ausdorff dstace A crsp umber a s smply represeted by u ( = u( = a, 0 r Defto 5 Let x = ( x(, x( ad ( y(, y( y = be two gve fuzzy umbers space he the ausdorff dstace D ( x, y betwee x ad y s defed as D( x, y = sup max x( y(, x( y(, 0 r ( Defto 6 he m lear system ax + ax + L+ a x = y, ax + ax + L+ a x = y, M amx + amx + L+ am x = ym, ( where the gve matrx of coeffcets A = a, m ad s a real m matrx, the gve y E, m, wth the ukows x E, s called a geeral fuzzy lear system (GFLS For arbtrary fuzzy umbers x = ( x(, x(, y = ( y(, y( ad real umber k, we may defe the addto ad the scalar multplcato of fuzzy umbers by usg the exteso prcple as [] (a x = y f ad oly x ( = y( ad x ( = y ( (b x + y = ( x ( + y (, x ( + y ( (c x y = ( x( - y (, x ( - y ( (
66 M A Farborz Aragh ad M M ossezadeh ( k x, k x, k 0 (d kx = ( k x, k x, k < 0 Defto 7 [8] A fuzzy umber vector, 0 r, s called a soluto of the GFLS ( f ax = ax = y, m = = ax = ax = y = = t ( x, x,, x gve by x ( x (, x ( = ; Defto 8 [] he geeral dual fuzzy lear system (GDFLS s defed as AX + F = BX + C, (3 where A = ( a, B = ( b, m,, are crsp coeffcet matrces wth oegatve elemets, adc ad F are fuzzy umber vectors From (3, we ca wrte ax + f = bx + c = = Sce a 0 ad b 0 for all, the a x + f = b x + c = = It follows that ( a b x = ( a b x = = c = c f f, a x + f = b x + c = = =, K, m (4 herefore, the followg (GFLS s obtaed PX = Q, (5 where the elemets P = p ad Q = q are: p ( = a b ad q = c f, m, ( 3 ABS algorthm for solvg lear system of equatos ABS methods were troduced by Abaffy et al [,3] he ABS algorthm cotas drect teratve methods for computg the geeral soluto of lear systems, lear least
Soluto of geeral dual fuzzy lear systems 67 squares, olear equatos, dophate equatos ad optmzato problems ABS algorthm s used, for solvg a m lear equatos ad ukows wth m [,6,9] Algorthm 3 ABS algorthm Step : Let x IR be arbtrary, ad IR be osgular arbtrary Set =, r = 0 Step : Compute τ = a x b ad s = a Step 3: If s = 0 ad τ = 0 the set x + = x, + =, r + = r ad go to step7 (the -th equato s redudat If s = 0 ad τ 0 the stop (the -th equato ad hece the system s compatble Step 4: { s 0} Compute the search drecto p = z, where z IR s a arbtrary vector satsfyg z a = z s 0 Compute α = τ a p ad set x+ = x α p Step 5: Update to + by aw + =, w a where w IR s a arbtrary vector satsfyg w 0 Step 6: Set r = r + + Step 7: If = m the stop ( xm+ s a soluto else set = + ad go step We ote that after the completo of the algorthm, the geeral soluto of system, f compatble, s wrtte as x = xm+ + m+ q, where q IR s arbtrary Oe of the methods based upo the dea of solvg at the -th step the frst equatos s uag method he uag algorthm s a specal algorthm of the class of ABS algorthms, wth choosg the specal parameters [0] Algorthm 3 uag algorthm Step : Let x IR be arbtrary, ad = I, set =, r = 0 Step : Compute τ = a x b ad s = a Step 3: If s = 0 ad τ = 0 the set x + = x, + =, r + = r ad go to step 7 (the -th equato s redudat If s = 0adτ 0 the stop (the -th equato ad hece the system s compatble Step 4: Compute the search drecto p = z, wth z = a satsfyg z a 0 Compute α = τ a p ad set x+ = x α p Step 5: Update to + by s
68 M A Farborz Aragh ad M M ossezadeh where = a w + a w = such that w a = a a + = r + Step 6: Set r Step 7: If = m the stop ( xm+ s a soluto else set = + ad go step, 4 he proposed algorthm ad umercal examples Now, we troduce the followg algorthm order to solve (3 I the algorthm, d s a postve gve umber lke d = 0 ad x = ( m, α, β, are the ukows of the (GFLS (5 Also, q = ( q, q, q, are the elemet of q m α β Algorthm 4 Step : Covert the gve (GDFLS AX + F = BX + C to (GLFS PX = Q accordg to (4 ad (5 Step : For r = 0( d do the followg steps a Solve the crsp m system PM = qm where M = ( m, m, K, m ad q m = ( q m, by usg uag algorthm to obta M b Solve the crsp m lear system PS = where S = ( α, β, ( qα, qβ q = α ( qα, q = q, β β by usg uag algorthm 3 to obta S ( ABS exact Step 3: Fd the ausdorff dstace D ( x, x accordg to ( ABS Step 4: Wrte x = ( m, α β, ad D ( x ABS, x exact he followg example s solved by applyg the algorthm 4 Example 4 [4] Cosder the followg 3 geeral dual fuzzy lear system 4x + x + 3x3 + (r +,4 = x + x + x3 + (3r +,6 x + 3x + x3 + ( r, = x + x + x3 + (r,5 If we mplemet the frst step of the algorthm 4, we have 3x x + x3 = ( r, x + x + x3 = ( r +,3 I ths example, d = 0, m =, = 3 I the table, we cosder the otato a = ( m, α, β, where α = m α ad β = m + β for fuzzy umbers he results of algorthm 4 are demostrated table
Soluto of geeral dual fuzzy lear systems 69 able : Results of example 4 r ABS x ABS x ABS x 3 0 ( 03333,090,04476 ( 06667,095,038 (03333,0905,0476 0 ( 03333,0305,0436 ( 06667,0334,000 (03333,0048,0469 0 ( 03333,049,0448 ( 06667,03695,09638 (03333,090,04476 0 3 ( 03333,0533,0433 ( 06667,04067,0967 ( 03333,0333,04333 0 4 ( 03333,0648,0409 ( 06667,04438,08895 (03333,0476,0490 0 5 ( 03333,076,03905 ( 06667,0480,0854 (03333,069,04048 0 6 ( 03333,0876,03790 ( 06667,058,085 (03333,076,03905 0 7 ( 03333,0990,03676 ( 06667,0555,0778 (03333,0905,0376 0 8 ( 03333,0305,0356 ( 06667,0594,0740 (03333,03048,0369 0 9 ( 03333,039,03448 ( 06667,0695,07038 (03333,0390,03476 ( 03333,03333,03333 ( 06667,06667,06667 ( 03333,03333,03333 I ths example, D ( x ABS, x strog = 0 54 metoed [4], here x strog s the strog soluto of system he followg example at frst was solved by Frdma, et al [8] he, Ghabar ad Mahdav Amr [9] proposed a method based o rakg fucto ad ABS algorthm for solvg ths example I order to fd the soluto of ths system, they preseted a olear programmg whch was solved by applyg the LINGO package order to fd α, he, they foud β from the gve rakg fucto We ca fd the same soluto wth a smpler ad faster method by applyg theorem 3 ad algorthms 3 ad 4
70 M A Farborz Aragh ad M M ossezadeh Example 4 [8,9] Cosder x x = (,, x + 3x = (5,, α + β = m m = α + 3α = Usg theorem 3, we have:, m + 3m = 5 β + α = β + 3β = usg algorthm 3, we obta m =, m = ad α = 0 65, α = 0 5, β = 0875, β = 0 375 herefore, x = ( x, x, wth x = (,065,0875, x = (,05,0375 s the strog or exact soluto obtaed [8,9] 5 Cocluso I ths paper, we foud a umercal fuzzy soluto of a geeral dual fuzzy lear system by troducg a ew algorthm based o the uag algorthm hs ew dea s able to solve a (GDFLS by a less complexty ad smply way Oe ca use the proposed algorthm to solve a olear fuzzy system of equatos by covertg to a (GDFLS or (FLS Refereces [] S Abbasbady, M Otad ad M Mosleh, Mmal soluto of geeral dual fuzzy lear systems, Chaos Soltos & Fractals 37 (008 3-4 [] J Abaffy, CG Broyde, E Spedcato, A class of drect methods for lear systems, Numer Math 45 (984 36-376 [3] J Abaffy ad E Spedcato, ABS Proecto Algorthms: Mathematcal echques for Lear ad olear Equatos, Joh Wley ad Sos, 989 [4] X dog Su ad S zog Guo, Soluto to Geeral Fuzzy Lear System ad Its Necessary ad Suffcet Codto, Fuzzy If Eg 3 (009 37-37 [5] D Dubos, Prade Operatos o fuzzy umbers J Syst Sc 9 (978 63 6 [6] Esmael, N Mahdav-Amr, E Spedcato, A class of ABS algorthms for Dophate lear systems, Numer Math 9 (00 0 5
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