Applied Mathematical Scieces Vl. 7 03. 37 8-87 HIKARI Ltd www.m-hikari.cm Multi-bective Prgrammig Apprach fr Fuzzy Liear Prgrammig Prblems P. Padia Departmet f Mathematics Schl f Advaced Scieces VIT Uiversity Vellre-4 Idia padia6@rediffmail.cm Cpyright 03 P. Padia. This is a pe access article distributed uder the Creative Cmms Attributi Licese which permits urestricted use distributi ad reprducti i ay medium prvided the rigial wrk is prperly cited. Abstract A ew methd amely level-sum methd based the multi-bective liear prgrammig ad the simplex methd is prpsed fr cmputig a ptimal fuzzy sluti t a fuzzy liear prgrammig prblem iwhich fuzzy rakig fuctis are t used. The level-sum methd is illustrated by umerical examples. Keywrds: Fuzzy liear prgrammig prblem Optimal fuzzy sluti Multi-bective liear prgrammig Level-sum methd.. Itrducti Liear prgrammig (LP) is e f the mst applicable ptimizati techiques. It deals with the ptimizatis f a liear fucti while satisfyig a set f liear equality ad/r iequality cstraits r restrictis. I practice a LP mdel ivlves a lt f parameters whse values are assiged by experts / decisi makers. Hwever bth experts ad decisi makers d t precisely kw the value f thse parameters i mst f the cases. Therefre fuzzy liear prgrammig (FLP) prblem [9] was itrduced ad studied. I the literature a variety f algrithms fr slvig FLP have bee studied based fuzzy rakig fucti ad classical liear prgrammig. Taaka et al. [7] Zimmerma [0] Buckley ad Feurig [3] Thakre et al.[8] ad Zhag et al. [] slved FLP prblems usig multi-bective liear prgrammig (MOLP) techique. Padia[6] has prpsed a ew apprach amely sum f bectives (SO) methd fr fidig a prperly efficiet sluti t multi-bective prgrammig prblems.
8 P. Padia I this paper we prpse a ew methd amely level-sum methd fr fidig a ptimal fuzzy sluti t fully FLP prblems which is a crisp LP techique t usig the fuzzy rakig fucti. First we cstruct a crisp MOLP prblem frm the FLP prblem ad the we establish a relati betwee a ptimal fuzzy sluti t the fully FLP prblem ad a efficiet sluti t its related MOLP prblem. Based the relati we develp the prpsed methd. With the help f umerical examples the level-sum methd is illustrated. The advatages f the prpsed methd are that fuzzy rakig fuctis are t used the btaied results exactly satisfy all the cstraits ad the cmputati ca be made by LP slver because it is based ly crisp LP techique.. Prelimiaries We eed the fllwig defiitis f the basic arithmetic peratrs ad partial rderig relatis fuzzy triagular umbers based the fucti priciple which ca be fud i [4 90 ]. Defiiti. A fuzzy umber a is a triagular fuzzy umber deted by ( a where a a ad a3 are real umbers ad its member ship fucti μ ( x) is give belw: a ( x a) /( a a) fr a x a μ a ( x) = ( a3 x) /( a3 a) fr a x a3 0 therwise Let F(R) be the set f all real triagular fuzzy umbers. Defiiti. Let ( a ad ( b b ) be i F (R). The (i) ( a ( b b ) = ( a + b a + b a3 + ). (ii) ( a Θ ( b b ) = ( a a b a3 b ). (iii) k ( a = ( ka k k fr k 0. (iv) k ( a = ( ka 3 k ka ) fr k < 0. ( ab ab a3 ) a 0 (v) ( a ( b b ) = ( a ab a3 ) a < 0 a3 0 ( a ab a3b ) a3 < 0. Defiiti.3 Let A = ( a a a3 ) ad B = ( b b ) be i F (R) the (i) A B iff a i = b i i = 3 ; (ii) A p B iff ai b i i = 3 ad (iii) A f B iff ai b i i = 3.
Multi-bective prgrammig apprach 83 Based the tatis f Magasaria [5] we defie the fllwig partial rder relati f as fllws. Defiiti.4 Let A = ( a a a3 ) ad B = ( b b ) be i F (R) the A f B iff ai b i i = 3 ad a r > br fr sme r { 3}. Csider the fllwig multi-bective ptimizati prblem (MP) Miimize f ( x) = ( f( x) f ( x)... fk ( x)) subect t g ( x) 0 x X where f i : X R i =... k ad g : X R where g = ( g... g m ) are differetiable fuctis X a pe cvex subset f R. Nw P = { x X : g ( x) 0 =... m} is the set f all feasible slutis fr the prblem (MP). Defiiti.5: A feasible pit x is said t be efficiet [6] fr (MP) if there exists ther feasible pit x i P such that f i ( x) fi( x ) i =... k ad f x) < f ( x ) fr sme r { Kk}. r ( r m 3. Fully Fuzzy Liear Prgrammig Prblem Csider the fllwig fully FLP with m fuzzy iequality/equality cstraits ad fuzzy variables may be frmulated as fllws: (P) Maximize z c T x subect t A x { p f } b x f 0 where a i c x bi F( R ) fr all ad i m c T = ( c ) A = ( a i ) m x = ( x ) x ad b = ( bi ) mx. Let the parameters z a c x ad b be the triagular fuzzy umbers i ( z z z3) ( p q r ) ( x y t ) ( a i bi ci ) ad ( bi gi hi ) respectively. The the prblem (P) ca be writte as fllws: (P) Maximize ( z z z3) ( p q r ) ( x y t ) = subect t ( ai bi ci ) ( x y t ) { p f } ( bi gi hi ) fr all i =... m = ( x y t ) f 0 = m. i
84 P. Padia Nw usig the arithmetic peratis ad partial rderig relatis we write the give FLPP as a MOLP prblem which is give belw: (M) Maximize Maximize Maximize 3 subect t lwer value f = z = lwer value f ( p q r ) ( x y t )) = middle value f = z = ( p q r ) ( x y t )) z = upper value f ( p q r ) ( x y t )) = middle value f = upper value f = z z ; z3 z ; y x 0 = m. ( a i bi ci ) ( x y t )){ = } bi ( a i bi ci ) ( x y t )){ = } gi ( a i bi ci ) ( x y t )){ = } hi x = m ; y t = m fr all i =... m ; fr all i =... m ; fr all i =... m ; Remark 3.: I the case f a fully FLP prblem ivlvig trapezidal fuzzy umbers ad / r trapezidal fuzzy decisi variables we get a MOLP prblem havig fur bectives. We w prve the fllwig therem which establish a relati betwee a ptimal fuzzy sluti t a fully FLP prblem ad a efficiet sluti t its related MOLP prblems. Therem 3.: Let X = { x y t ; = m} be a efficiet sluti t the prblem (M). The X = { ( x y t ) ; = m} is a ptimal sluti t the prblem (P). Prf: Nw sice X = { x y t ; = m} is a efficiet sluti t the prblem (M) X = { ( x y t ) ; = m} is a feasible sluti t the prblem (P). Assume that X = { ( x y t ) ; = m} is t ptimal t the prblem (P). The there exists a feasible sluti X = { ( x y t ) ; = m} t the prblem (P) such that Z ( X ) f Z( X ) that is zi ( x y t) zi( x y t ) i = 3 ad
Multi-bective prgrammig apprach 85 z ( x y t) z ( x y t r > r ) fr sme r {3 } where x = { x ; = m} y = { y ; = m} t = { t ; = m} x = { x ; = m} y = { y ; = m} ad t = { t ; = m}. This meas that X = { x y t ; = m} is t a efficiet sluti t the prblem (M) which is a ctradicti. Hece the therem is prved. Nw we prpse a ew methd amely level-sum methd fr fidig a ptimal fuzzy sluti t a FLP prblem which is based MOLP ad the simplex methd. The prpsed methd prceeds as fllws: Step : Cstruct a crisp MOLP prblem frm the give FLP prblem. Step : Fid a efficiet sluti t the MOLP prblem btaied i the Step. usig the SO methd [6]. Step 3: The efficiet sluti btaied frm the Step. t the MOLP prblem yields a ptimal fuzzy sluti t the FLP prblem by the Therem 3... Remark 3.: The prpsed methd ca be exteded t fuzzy iteger LP prblems by addig the iteger restrictis ad replacig the simplex methd by a iteger LP techique. The prpsed methd is illustrated by the fllwig examples. Example 3.: Csider the fllwig fully FLP prblem: t Maximize z ( 3 ) x ( 3 4 ) x subect t ( 0 ) x ( 3 ) x ( 0 4 ); ( 3 ) x ( 0 ) x ( 8 ); x ad x f 0. Let x = ( x y t ) x = ( x y t ) ad z = ( z z z3 ). Nw usig the Step. the MOLP prblem related t the give fully FLP prblem is give belw: (M) Maximize z = t + x Maximize z = y + 3y Maximize z 3 = 3t + 4t subect t 0x + x = ; y + y = 0 ; t + 3t = 4 ; x + 0x = ; y + y = 8 ; 3t + t = ; z z ; z3 z ; y x ; y x ; t y ; t y ; x x 0. Nw by the Step. we csider the fllwig LP prblem (S) related t the abve MOLP prblem: (S) Maximize Z = x + y + 3y + t + 4t subect t
86 P. Padia 0x + x = ; y + y = 0 ; t + 3t = 4 ; x + 0x = ; y + y = 8 ; 3t + t = ; x + y + 3y + t 0 ; y 3y + 3t + 4t 0 ; y x 0 ; y x 0; t y 0; t y 0 ; x x 0 ad slve it by simplex methd. The ptimal sluti t the prblem (S) is x = ; x = ; y = ; y = 4 ; t = 3 ad t = 6 with Z = 50. Thus ( x = x = y = y = 4 t = 3 t = 6) is a efficiet sluti t the prblem (M). Nw by the Step 3. x ( 3 ) x ( 4 6 ) ad z (6 33 ) is a ptimal fuzzy sluti t the give fully FLP prblem. Remark 3.3 : Fr the fully FLP prblem ( the Example 4.) Amit Kumar et al. [] by the rakig methd btaied the same ptimal fuzzy sluti. Example 3.: Csider the fllwig FLP prblem: t Maximize z ( 704 5) x ( 0 5 35 40 ) x subect t ( 3 4 ) x ( 6 7 ) x p ( 835 ); ( 3 4 6 ) x ( 60 ) x p ( 3 7 9 ); x ad x 0. Let z = ( z z z3 z4 ). Nw usig the Step. the MOLP prblem related t the give FLP prblem is give belw: (M) Maximize z = 7x + 0x Maximize z = 0x + 5x Maximize z 3 = 4x + 35x Maximize z 4 = 5x + 40x subect t x + x 8 ; 3x + 6x 3 ; 4x + 7x 5 ; 3x + x 3 4x + 6x 7 ; 6x + 0x 9 ; z z ; z3 z ; z4 z3 ; x x 0. Nw by the Step. we csider the fllwig LP prblem (S) related t the abve MOLP prblem: (S) Maximize Z = 56x + 0x subect t x + x 8 ; 3x + 6x 3 ; 4x + 7x 5 ; 3x + x 3 4x + 6x 7 ; 6x + 0x 9 ; 3x + 5x 0 ; 4x + 0x 0 ; 9x + 5x 0 ; x x 0 ad slve it by simplex methd. The ptimal sluti t the prblem (S) is x = 0 ad x = 0. 9 with Z = 08. Thus ( x = 0 x = 0.9) is a efficiet sluti t the prblem (M). Nw by the Step 3. x = 0 x = 0. 9 ad z (8 5.5 3.5 36 ) is a ptimal fuzzy sluti t the give FLP prblem. Remark 3.4 : I Thakre et al. [8] by the weighted methd the same ptimal fuzzy sluti t the FLP prblem (the Example 3..) is btaied.
Multi-bective prgrammig apprach 87 4. Cclusi I this paper we prpse the level-sum methd t fid a ptimal fuzzy sluti t a FLP prblem satisfyig all cstraits. The mai advatage f the prpsed methd is that the FLP prblems ca be slved by ay LP slver usig the level-sum methd sice it is based ly simplex methd. The level-sum methd ca serve maagers by prvidig a apprpriate best sluti t a variety f LP mdels havig fuzzy umbers ad variables i a simple ad effective maer. I ear future we exted the level-sum methd t fuzzy MOLP prblems. Refereces [] Amit Kumar Jagdeep Kaur ad Pushpider Sigh Fuzzy ptimal sluti f fully fuzzy liear prgrammig prblems with iequality cstraits Iteratial Jural f Mathematical ad Cmputer Scieces 6 (00) 37-4. [] R. E. Bellma ad L. A. Zadeh Decisi makig i a fuzzy evirmet Maagemet Sciece 7(970) 4-64. [3] J. Buckley ad T. Feurig Evlutiary algrithm sluti t fuzzy prblems: Fuzzy liear prgrammig Fuzzy Sets ad Systems 09(000) 35-53. [4] Gerge J. Klir ad B Yua Fuzzy Sets ad Fuzzy lgic: Thery ad Applicatis Pretice-Hall 008. [5] O.L.Magasaria Nliear prgrammig McGraw Hill New Yrk 969. [6] P. Padia A simple apprach fr fidig a fair sluti t multibective prgrammig prblems Bulleti f Mathematical Scieces & Applicatis (0) 5 30. [7] H. Taaka T Okuda ad K. Asai O fuzzy mathematical prgrammig Jural f Cyberetics ad Systems 3(973) 37-46. [8] P. A. Thakre D. S. Shelar ad S. P. Thakre Slvig fuzzy liear prgrammig prblem as multi bective liear prgrammig prblem Jural f Egieerig ad Techlgy Research (009) 8-85. [9] L. A. Zadeh Fuzzy sets Ifrmati ad Ctrl 8 (965) 338-353. [0] H. J. Zimmerma Fuzzy prgrammig ad liear prgrammig with several bective fuctis Fuzzy Sets ad Systems (978 ) 45-55. [] G. Zhag Y. H. Wu M. Remias ad J. Lu Frmulati f fuzzy liear prgrammig prblems as fur-bective cstraied ptimizati prblems Applied Mathematics ad Cmputati 39( 003) 383-399. Received: Jauary 5 03