TRIANGULAR MEMBERSHIP FUNCTIONS FOR SOLVING SINGLE AND MULTIOBJECTIVE FUZZY LINEAR PROGRAMMING PROBLEM.

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1 Abbas Iraq Joural of SceceVol 53No Pp TRIANGULAR MEMBERSHIP FUNCTIONS FOR SOLVING SINGLE AND MULTIOBJECTIVE FUZZY LINEAR PROGRAMMING PROBLEM. Iraq Tarq Abbas Departemet of Mathematc College of Scece Uversty of Baghdad.Baghdad-Iraq Abstract I ths paper fuzzy sgle ad multobectve lear programmg models are preseted. Both the obectve fucto ad the costrats are cosdered fuzzyly.the coeffcet of the decso varable the obectve fuctos ad the costrats as well as the rght-had sde of the costrats are assumed to be fuzzy umbers wth tragular membershp fuctos. The possblty programmg approach s utlzed to trasform the fuzzy model to ts crsp equvalet form ad the a sutable method wll be used to solve the crsp problem. Keyword: Fuzzy multobectve lear programmg possblty programmg دالة االنتماء المثلثية لحل مسائل البرمجة الخطية المفردة والمتعددة االهداف الضبابية العساق ع ارق طارق عباس قسن السياضيات كلية العلوم جاهعة بغداد.بغداد الخالصة في هرا البحثقده ا واذج بسهجة خطيةة هرةسدو م هحعةددو ااهةداض اليةبابية.نيث اى كة هةةةي دالةةةة ال ةةةدض مالبيةةةود دبةةةازو دةةةي دمام ضةةةبابية.اى هعةةةاهدت البةةةساز فةةةي دالةةةة ال ةةةدض مالشسمطبااضافة الى الطةسض اايوةي هةي البيةود جةن افحساضة ا ادةداد ضةبابية مبدالةة ا حوةا هثلثيةةة.اى خوازشهيةةة بسهجةةة ااه ةةاى قةةد ادحوةةدت فةةي جحويةة ال وةةوذج اليةةبابي الةةى الغيةةس ضةةبابي الوةةسادض لةة مهةةي ثةةن اخةةح دهث طسيبةةة ددديةةة ه اخةةبة فةةي البسهجةةة ال طيةةة لحةة الوسائ الو اظسو. 1-Itroducto Lear programmg has ts applcatos may felds of operato research whch s cocered wth the optmzatos (mmzato ad/or maxmzato) of a lear fucto whle satsfyg a set of lear equalty ad /or equalty costrats or restrctos. I real stuato the avalable formato the system uder cosderato are ot exact therefore fuzzy lear programmg was troduced ad studed by may authors ad were regularly treated by others.zmmerma [1]proposed the frst formulato of fuzzy ler programmg. Fag ad Hu[2]cosdered lear programmg wth fuzzy costrat coeffcets.vasat et.al[3] appled lear programmg wth fuzzy parameters for decso makg dustral producto plag.malek ad et al [4 5]troduced a le ar programmg problem wth fuzzy varables ad proposed a ew method for solvg these problems usg aauxl aryproblem.mahdav-amr ad Nasser [6] descrbed dualty theory for the fuzzy varable [LP] problem. I ths paper the possblty programmg approach s utlzed to trasform the fuzzy sgle- obectve ad multobectve lear programmg models to ts crsp equvalet problem accordg to ts Iskader ' s modfcato [78].The the crsp sgle ad multobectve lear programmg model are solved usg ay lear programmg method.

2 Abbas Iraq Joural of SceceVol 53No Pp Deftos ad Notatos fuzzy set theory I ths secto some of the fudametal deftos ad basc cocepts of fuzzy sets theory are gve for completeess purpose: Defto (1)[9]. A fuzzy set A X s a set of order pars: A = {(x (x) /xx} (x) A A s called the membershp fucto of x A whch maps X to[01]. If sup x (x) =1 the fuzzy set A s called ormal. Defto (2)[9]. The support of a fuzzy set A o X s the crsp set of all xx such that (x) >0. Defto (3)[9]. The set of elemets that belog to the fuzzy set A o X at least to the degree s called the -cut set: A = {(x (x) [01]}. A Defto (4)[9]. A fuzzy set A o X s sad to be covex f (x+ (1- ) y) m { (x ) (y) } x A A A A A yr ad [0 1]. Note that a fuzzy set s covex f all ts -cuts are covex. Defto (5) [9]. A fuzzy umber a s a covex ormalzed fuzzy set o the real le R such that: 1) There exst at least oe x 0 R wth x ) =1. a ( 0 2) (x) s pecewse cotuous. a A fuzzy umber a s a tragular fuzzy umber see (Fgure 1) f the membershp fucto may be show as: X a a x b ba (x) A = c x b x c cb 0 O.W (x) A Defto (6) [9]. A fuzzy umber M s of LR-type f there exst tow fuctos L (called the Lft fucto) R(called the rght fucto) such that L(x) M 5 0 a b c c c c Fgure 1: The Tragular membershp fucto (x) R(x) xx (Uversal set) ad a scalars a0 b0 wth: m x L( ) for x m (x) M = a x m R( ) for x m b Whe mr whch s called the mea value of M a ad b are called the left ad rght spreads of m respectvely. Symbolcally M s deoted by (m a b) LR. Now applcatos the represetato of a fuzzy umber terms of ts membershp fucto s so dffcult to use therefore a sutable approach for represetg the fuzzy umber terms of ts-level sets s gve as the followg remark: Remark (1): A fuzzy umber M may be uquely represeted terms of ts -level sets as the followg closed tervals of the real le: M = [m- 1 m+ 1 ] or M = [m 1 m] Where m s the mea value of M ad (0 1]. Ths fuzzy umber may be wrtte as M =[ M M ] where M refers to the greatest lower boud of M ad M the least upper x

3 Abbas Iraq Joural of SceceVol 53No Pp boud of M. 3-Possblty Programmg Method Fuzzy Sgle Obectve Fuctos Cosder the formulato of the fuzzy sgle obectve lear programmg model as: Maxmze c x 1 1 a x..(1) b =1 2 m (2) x 0 =1 2.. (3) Where x =12. are o-egatve decso varable c s the fuzzy coeffcet of the th decso varable a represets the fuzzy coeffcet of the th decso varable the th costrat whle b s the fuzzy rght-had sde the th costrat.hece for smplcty c a ad b are cosdered to be of tragular fuzzy umbers.e. usg remark(1)above c = [c - 1 c c + 1 ] a =[a - 1 a a + 1 ] b =[b - 1 b b + 1 ] Thus accordg to the tragular fuzzy umbers the equvalet crsp model for the fuzzy model (1)- (3) s gve below: Maxmze Z= 1 1 [( 1 )c c ] x [( 1 )a a ]x (1 )b b o Where s a predetermed value of the mmum requred possblty (0 1] As a llustrated cosder the followg example: Example (1):- Maxmze Z= 3 x x 2 1 x x x1 + 2 x x1-1 x 2 3 x 1 0 x 2 0 The fuzzy sgle obectve lear programmg tragular form takes the form: Maxmze ( )x ) x 1 + (2-0 ( )x 1 + ( ) x 2 ( ) ( ) x 1 + ( ) x 2 ( ) ( ) x 1 -( ) x 2 ( ) Thus the equvalet crsp sgle obectve lear programmg the case of possblty programmg s stated as: Maxmze [(1- ) (3+ 1 ) +2)] x 2 1 ) +3] x 1 + [(1-)(2- - [(1- ) (1-1 +)] x 1 + [(1- ) (2-1 +2)] x 2 (1- ) ( ) [(1-)(3-1 )+3]x 1 +[(1-)(2-1 +2]x 2 (1-)( ) [(1- ) (1-1 +)] x 1 -[(1- ) (1-1 +] x 2 (1- ) ( ) x 1 x 2 0. Followg Table (1) whch represets the soluto of the above example. Table (1) o Z x x It s remarkable the whe =1 the the soluto of the fuzzy lear programmg s the same soluto of the crsp problem. Example (2):- Maxmze Z = 8 x x 2 4 x1+ 2 x x1 + 4 x x 1 ad x 2 0 The fuzzy sgle obectve lear programmg Tragular form wll be: Maxmze ( ) x 1 + ( ) x 2 ( )x 1 +( )x 2 =( )

4 Abbas Iraq Joural of SceceVol 53No Pp ( )x 1 +( )x 2 =( ) x 1 ad x 2 0 Thus the equvalet crsp sgle obectve lear programmg the case of possblty programmg s stated as: Maxmze [(1- ) (8+ (6+ 1 ) +6] x 2 [(1-)(4- (1-)(60+ [(1-)(2-1 ) +8] x 1 + [(1- ) 1 )+4]x 1 +[(1-)(2-1 )+2]x 2 1 )+60 1 )+2]x 1 [(1-)(4-1 )+4]x 2 (1-)(48+ 1 )+48 x1ad x 2 0. Followg (Table 2) whch represets the soluto of the above example. Table Z x x It s remarkable the whe =1 the the soluto of the fuzzy lear programmg s the same soluto of the crsp problem. 4-Possblty programmg fuzzy multobectve Fuctos Cosder the formulato of the fuzzy multobectve lear programmg: Maxmze c r x r=1 2 p (4) 1 1 a x b =1 2 m (5) x 0 = (6) Where x =1 2. are o-egatve decso varable c r s the fuzzy coeffcet of the th decso varable the rth obectve fucto. a represets the fuzzy coeffcet of the th decso varable the th costrat whle s the fuzzy rght-had sde the th b costrat.hece a ad b are cosder c r to be tragular fuzzy umbers [11].e. c r =[ c r - 1 c r c r + 1 ] a =[ a - 1 a a + 1 ] b =[ b - 1 b + 1 ].Thus accordg to b tragular fuzzy umbers the equvalet crsp model for the fuzzy model (4)-(6) s gve below:- Maxmze Z= 1 1 [( 1 )c c 0 r r 0 ] x [( 1 ) a a ]x (1 ) b b Where s a predetermed value of the mmum requred possblty (0 1]. Now a llustrated example wll be cosdered. Example (3): Maxmze Z 1 (x) = 2 x x 2 Z 2 (x) = 3 x 1-2 x 2 2 x1 + 5 x x1 + 1 x x1 + 1 x x2 1 0 x 1 x 2 0 The fuzzy multobectve lear programmg tragular form wll be: Z 1 = ( ) x 1 + (

5 Abbas Iraq Joural of SceceVol 53No Pp ) x 2 Z 2 = ( ) x )x 1 -(2-1 2 ( ) x 1 + ( ) x 2 ( ) ( )x 1 +( )x 2 ( ) ( ) x 1 ( ) x 2 ( ) ( ) x 2 ( ) Thus the equvalet crsp mult obectve lear programmg the case of possblty programmg s stated as: Z 1 =[(1-)(2+ )(1+ 1 +)]x )]x 1 +[(1- Z 2 = [(1- )( )]x 1 -[(1- ) ] x 2 [(1- ) (2-1 +2)] x 1 [(1- ) (5-1 +5)] x 2 (1- ) ( ) [(1- ) (1-1 +)] x 1 + [(1- ) (1-1 )] x 2 (1- ) (18+ [(1-)( ) 1 +3)]x 1 +[(1-)(1-1 )+]x 2 (1-)( ) [(1- ) (1-1 +)] x 2 (1- ) ( ) x 1 x 2 0. Followg (Table 3) whch represets the soluto of the above example wth dfferet value: Table Z Z X X It s remarkable they whe =1 the the soluto of the fuzzy lear programmg s the same soluto of the crsp problem. Refereces: 1. ZmmermaH. J.1978.Fuzzy program mg ad lear programmg wth seve ral obectve fucto. Fuzzy Sets ad Systems 1: Fag S.C.ad C.F.Hu.1999.Lear prog rammg wth fuzzy coeffcets cos trat.computer Mathematcs Appled. 37: VasatP. R. Nagaraa ad S.Yaacab Decso makg dustral prod ucto plag usg fuzzy lear progr ammg. IMA Joural of Maagemet Mathematcs 15: MalekH.R Rakg fuctos a d ther Applcatos to fuzzy lear prog rammg. Far East Joural. Mathematc s. Scece. 4: MalekH.R. M.Tata ad M. Mashch Lear programmg wth fuzzy v arables.fuzzysets ad Systems 109: MahdavAmr N.ad S.H. Nasser Dualty fuzzy varable lear progra mmg. 4th World Eformatka Coferece WEC'05 Istabul Turkey Jue. 7. Iskader M.G A possblty programmg approaches for stochastc fuzzy multobectve lear fracto programs. Computer ad Mathematcs wth applcatos. 48: Iskader M.G Comparso of fuzzy umbers usg possblty programmg. Computer ad mathematcs wth applcato. 43: Bellma R.E. ad Zadeh L.A Decso makg a fuzzy evromet. Maagemet. Scece. 17: