DUALITY IN MULTIPLE CRITERIA AND MULTIPLE CONSTRAINT LEVELS LINEAR PROGRAMMING WITH FUZZY PARAMETERS

Transcription

1 Ida Joual of Fudameal ad ppled Lfe Sceces ISSN: (Ole) Ope ccess, Ole Ieaoal Joual valable a Vol.5 (S), pp /Noua e al. Reseach cle DULITY IN MULTIPLE CRITERI ND MULTIPLE CONSTRINT LEVELS LINER PROGRMMING WITH FUZZY PRMETERS bbas l Noua, l Paya 2, Elham Poodeh 3 ad Mad Shaee 3 Faculy of Mahemacs, Ssa ad aluchesa Uvesy, Zaheda, Ia 2 Depame of Mahemacs, Zaheda ach, Islamc zad Uvesy, Zaheda, Ia 3 Depame of Mahemacs, Iashah ach, Islamc zad Uvesy, Iashah, Ia uho fo Coespodece STRCT I hs pape, mulple cea ad mulple cosa levels lea poamm wh fuzzy paamees ( F ) s oduced based o fuzzy elaos. The dual of F ha - feasble soluo ad (, ) s cosuced o he bass ca be deceased o a mulple cea lea poamm ( MC ). The coceps of - mamal ad mmal soluos ae defed. Fally, he wea ad so dualy heoems ae obaed. Keywods: Mulple Cea ad Mulple Cosa Levels Lea Poamm, Dualy, Fuzzy Numbes, - Feasble Soluo, (, ) - Mamal ad Mmal Soluos INTRODUCTION I hs pape, mulple cea ad mulple cosa levels lea poamm ( ) ha s oe of he mpoa felds of mulple cea decso ma (MCDM) s cosdeed o sudy. I he lea poamm () ad mulple cea lea poamm ( MC ) models a sle peso s cosdeed chae of decso ma, fo eample he poduco maae. Ths may o be ue assumpo acual pacce. Theefoe, we cosde a oup of decso maes (DMs), clud he pesde, chef facal offce ad coolle shae he decso ma, esposbly. The pefeece values of each volved maae o esouce avalably ca be eaed as a dvdual level of he avalably. The, he obaed model has mulple dscee levels of esouce avalably. poblem wh such sucue s called. The soluo of hs poblem s called a poeal soluo (Sh, 200). Each poeal soluo opmzes he poblem ude a cea ae of decso paamees ha ae he cea ad cosa level weh vecos. poblem, ee poblem. Sh (Sh, 200) suded Sh hs boo (Sh, 200) suded poblem ad aspoao poblems whch obecve fucos ae cosdeed as fuzzy decsos. Isead of fd a se of poeal soluos fo poblem, hey peseed he decso maes oal-see ad compomse behavo o aa a se of sasfc soluos bewee a uppe ad a lowe aspao level. I he classc, all daa ae eacly ow. Howeve, csp daa may o be avalable, because daa may eal applcaos cao be pecsely measued. O he ohe had, dualy cocep povdes he useful fomao abou he pmal poblem. Fuhemoe, solv he dual poblem s ofe ease ha s pmal. Theefoe, hs pape sudes wh fuzzy daa ( FMC 2 ) ad eeds dualy cocep o F. Fo hs, F s fsly oduced ad s coveed o a MC wh fuzzy paamees (FMC) ad he hs poblem s asfom o a lea poamm wh fuzzy paamees (F). I hs suao, he dual of F ca be acued by apply he cocep of possbly ad ecessy Copyh 204 Cee fo Ifo o Techoloy (CITech) 447

2 Ida Joual of Fudameal ad ppled Lfe Sceces ISSN: (Ole) Ope ccess, Ole Ieaoal Joual valable a Vol.5 (S), pp /Noua e al. Reseach cle elaos o he obaed F. We cosde he possbly elao fo pmal poblem ad he ecessy elao fo dual poblem. The emde of hs pape s sucued as follows. Seco 2 s abou fuzzy pelmaes. Seco 3 ecalls he fomulao of F ad s popees. I he fouh seco, we cosde fuzzy veso of. Seco 5 s abou mamz obecve fuco ad he e seco, we cosde fuzzy veso of ad eed he dualy cocep fo hs poblem. Coclusos ae ve he las seco. Pelmaes Fuzzy se heoy whch was fsly oduced by Zadeh (Zadeh, 965; Zadeh, 999) has bee eesvely appled much aea of scece. The basc defos of fuzzy se heoy ae as follows: Le X deoes a uvesal se. The a fuzzy subse of X s defed by s membeshp fuco as : X [0,] whch asss o each eleme X a eal umbe ( ) he eval [0,]. The suppo of a fuzzy se o X, deoed by supp( ), s he se of pos X whch ( ) supp ( ) = { X ( ) > 0} The - level se of a fuzzy se s defed as a oday se membeshp fuco eceeds he level : s posve,.e. fo whch he deee of s [ ] = { X ( ) }, [0,]. We deoe he se of all fuzzy subses of X by F (X ). fuzzy subse P of F( X ) F( X ) s called a fuzzy elao o X,.e P F( F( X ) F( X )) fuzzy elao Q o X s called a fuzzy eeso of elao P, f fo each, y X (, y) = (, y) Q P. Le d s a fuzzy uay. Follow oao ae used as: L R d ( ) = f { [ d ] }, d ( ) = sup{ [ d ] } Le ad ae fuzzy ses wh he membeshp fucos : R [0,] ad : R [0,] especvely. We shall cosde, (, ) = sup{ m( ( ), ( y)) y,, yr} ( a) (, ) = f { ma( ( ), ( y)) > y,, yr} ( b) whee euaos (a) ad (b) ae especvely called possbly ad ecessy elaos. sbly ad ecessy elaos (a) ad (b) wee oally oduced as possbly ad ecessy dces (Dubos ad Pade, 983). Ths elaos ca be aleavely we as: (, ) = ( ), (, ) = ( < ) whee ad ae he membeshp fucos of he fuzzy elao o R. y > we mea o <, especvely. Mulple Cea ad Mulple Cosa Lea Poamm I eeal, MC ca be we as: ma C ad Copyh 204 Cee fo Ifo o Techoloy (CITech) 448

3 Ida Joual of Fudameal ad ppled Lfe Sceces ISSN: (Ole) Ope ccess, Ole Ieaoal Joual valable a Vol.5 (S), pp /Noua e al. Reseach cle 0 d () The obecve fuco s o mamze dffee cea we by he ( ) ma C (ceo ma), whle he ( m ) ma s he u cosumpo of esouces ad he ( m ) veco d s he cosa (esouce avalably) level. To cosuco o: ma 0 C D, we eplace d by a ( m ) ma D ad he poblem () s coveed (2) The mples ha s feasble f les a cove se eeaed by he colums of D. Fo solv he poblem (2), Sh (Sh,) oduced wo weh paamees, he cosa paamee (,, ) > 0 ad he cea paamee (,, ) > 0 o he fomulao. Thus, he = ma C D 0 ca be asfomed o: = (3) soluo of called poeal soluo s fomally defed as a eeso of o-domaed soluo fo he mulple cea poam. Coceps of Poeal Soluos (Sefod ad Zhu, 979) bass J s called a poeal bass a poeal soluo fo he 0 0 > 0 ad > 0 such ha J s a opmal bass fo he follow poblem: 0 ma ( ) C 0 D 0 0} Gve a bass J fo a ( ) ( J) = { > 0 D s called pmal paamee se ad ( ) ( J) = { > 0 ( C R C ) 0} s called dual paamee se. R f ad oly f hee ess The maces R ad ae sub-maces of whch R cosss of o-basc vaables ad s a osula ma. Gve a bass J fo a () The bass soluo ( J, ) = D s feasble f ad oly f (J). () The soluo ( J, ) s opmal f ad oly f (J) ad (J). Copyh 204 Cee fo Ifo o Techoloy (CITech) 449

4 Ida Joual of Fudameal ad ppled Lfe Sceces ISSN: (Ole) Ope ccess, Ole Ieaoal Joual valable a Vol.5 (S), pp /Noua e al. Reseach cle () The obecve fuco value of he poblem s deoed by V( J,, ) = C a fuco of (, ). Geeally, () of defo 3 s called he opmaly codo. () J s a pmal poeal bass f ad oly f (J). () J s a dual poeal soluo f ad oly f (J). () J s a poeal bass f ad oly f he caesa poduc ( ) ( J) Specal Poeal Soluo J. If he basc vaable ( J, ) = D, he he codo = ca se as a cosa, ma = = = = C D 2 D, whch s s cosdeed as a paamee fom wh espec he paamee becomes a as 0, 0, =,, (4) C s he h ow of he ( ) ceo ma. o-domaed soluo of (4) Noe ha (4), wh > 0 s called a poeal soluo, ad a o-domaed soluo of (4) wh 0 s called a wea poeal soluo. X = { d, 0 fo some d H( D)} s a wea poeal soluo f ad oly f hee s a, such ha solves (4). ) wh 0, {,2,, } = (, 2,, Fo each ve ( ) = (,,, ) > 0 ad oly f ma ( C 0, wh = s a o-domaed soluo of (5):, C 0 D 2,, C ) 2, = s a poeal soluo of (3) f 0 (5) X s a poeal soluo of (4) f ad oly f hee s a > 0 ad 0, {,2,, } ma = = = = C D, such ha 0, > 0, =,, solves (6): = (, 2,, ) wh (6) Copyh 204 Cee fo Ifo o Techoloy (CITech) 450

5 Ida Joual of Fudameal ad ppled Lfe Sceces ISSN: (Ole) Ope ccess, Ole Ieaoal Joual valable a Vol.5 (S), pp /Noua e al. Reseach cle Wh Fuzzy Paamees The eeded fom of ma z = =c =, a P ( d,, d ), =, (2) ca be we as: =, 0, =,, (7) P. Cosde daa as fuzzy umbes, a F ma z = c = =, a P ( d,, d ), =, whee = ( o = o ), = 0, =,, s as: (8) y def he cosa paamee (,, ) > 0 = ad he cea paamee = (,, ) > 0 o he fomulao, he poblem ca be asfomed o: ma c =( = ) a P d, =,, = = = = > 0, =,, 0, =,, (9) whch s euvale o ma c =( = ) a d P 0, =,, = = = = > 0, =,, 0, =,, (0) Now,we cosde ma as follows: a a a d d d a a a d d d a a a d d d = m m m m m m Copyh 204 Cee fo Ifo o Techoloy (CITech) 45

6 Ida Joual of Fudameal ad ppled Lfe Sceces ISSN: (Ole) Ope ccess, Ole Ieaoal Joual valable a Vol.5 (S), pp /Noua e al. Reseach cle c = (,...,,0,0,...,0 such ha c = c ( =,..., ) lso, suppose c c ) X ) 2 c = (,,,,,,, ad 2 2 became as follows: X X P, =,, m = X 0, =,, ma =c m = (0,0,...,0,,,,..., ) = Copyh 204 Cee fo Ifo o Techoloy (CITech) 452, T. Hece, model (0) s () fuzzy se X, whose membeshp fuco X s defed fo all R by m { ( a a d d ( a......,0),...,... P P m a m d m... d m,0), (... P,), ( ) = X (, ),..., (, )} P P 0, ohewse s called he fuzzy se of feasble eo of he F poblem. Fo (0,], a veco [X s called he -feasble soluo of he F (). Noce ha he feasble eo X s a ] fuzzy se. O he ohe had, -feasble soluo s a veco belo o he -cu of he feasble eo X. Mamz he Obecve Fuco Now we loo fo he "bes" fuzzy uaes z wh espec o he ve fuzzy cosas, o, ohe, =,... wods, wh espec o he fuzzy se of feasble eo of (8). Kow he wehs of he obecves we shall deal wh he assocaed poblem (), paculaly, wh he sle obecve fuco z = c... c. Specal elaos ae defed by Ram (Ram, 2006b). Le be a fuzzy elao o R, le a, b be fuzzy ses of R ad le (0,]. I s sad ha a s -less ha b wh espec o ad we a b f ( a, b ) ad, < ( b a ). I s called he - elao o R wh espec o. Noce ha s a bay elao o he se of fuzzy ses F (R) be cosuced fom a fuzzy elao o he level of (0,]. If a ad b ae csp umbes coespod o eal umbes a ad b, especvely, ad s a fuzzy eeso of elao he a b f ad oly f a b. Now, modfy he well ow cocep of poeal soluo opmzao o odomaed soluo opmzao MC, we defe mamzao (o euvalely mmzao) of he obecve fuco of F poblem (). We shall cosde a fuzzy elao o R be a fuzzy eeso of he usual bay elao o R, see also Ram (Ram, 2006a). Hee, we allow fo depede,.e. dffee sasfaco levels:, whee s cosdeed fo he obecve fucos ad fo he cosas (Ram, 2006a). Le c, a ad b,

7 Ida Joual of Fudameal ad ppled Lfe Sceces ISSN: (Ole) Ope ccess, Ole Ieaoal Joual valable a Vol.5 (S), pp /Noua e al. Reseach cle =,..., m, =,..., be fuzzy uaes o R. Le be a fuzzy elao o R, be also a fuzzy eeso of he usual bay elao o R. - feasble soluo of (), [X ] s called he (, ) -mamal soluo of () wh espec o f hee s o [X ] such ha c T T c. Dual Poblems ad Dualy Theoem Model () s a lea poamm ad s dual s as follows: m u = u m =2um m u Q c, =,..., = u 0, =,..., m (2) P =, Q =< o P =<, Q =. I poblem (), mamzao s Hee, we cosde ehe cosdeed wh espec o fuzzy elao P. O he ohe had, mmzao s cosdeed wh espec o fuzzy elao Q, whch ca be fomulaed aalocally. I he follow dualy heoems we pese wo vesos: (I) fo fuzzy elao poblem () ad fuzzy elao < poblem (2), ad (II), fo fuzzy elao < poblem () ad fuzzy poblem (2). I ode o pove dualy esuls we assume = elao. Ohewse, he dualy heoems ou fomulao do o hold, fo moe deals see (Ram, 2006a). Moeove, we assume ha each obecve fuco s assocaed wh a weh > 0, =,...,, such ha = = whee may be epeed as a elave mpoace of he h obecve fuco. The coespod poofs ca be foud he sudy of Ram (Ram, 2006a). Wea Dualy Theoem. Suppose c,,...,, =,..., m, =,..., =. ad ae fuzzy umbes, (I) Le X ad Y ae especvely he feasble eos of poblems () ad (2) whch P = Q =<. If X = (,,...,,,,..., ) [ X ] 2 2 ad u = ( u, u 2,..., u ) [ Y ] m R R m R c c ( ) = ( ) X ( ) u = = = = (II) Le X ad Y ae especvely he feasble eos of poblems () ad (2) whch P =< ad Q = X = (,,...,,,,..., ) [ X u ( u, u,..., u ) [ ] he ] ad, he. If 2 2 ad 2 m L L m L c c ( ) = ( ) X ( ) = = = = So Dualy Theoem. Cosde c, =,...,, =,..., m, =,...,. (I) Suppose [X ] s he feasble eo of poblem () wh poblem (2) wh Q =<. If fo some (0,) u = Y, ad ae fuzzy umbes, P =, ] [Y s he feasble eo of, [X ] ad [Y ] ae oempy, he hee Copyh 204 Cee fo Ifo o Techoloy (CITech) 453

8 Ida Joual of Fudameal ad ppled Lfe Sceces ISSN: (Ole) Ope ccess, Ole Ieaoal Joual valable a Vol.5 (S), pp /Noua e al. Reseach cle ess X espec o = (,,...,,,,..., ) [ X ] whch s (, ) 2 2, ad u = ( u, u 2,..., u ) [ Y ] (2) wh espec o <, such ha R R m R c c ( ) = ( ) X = ( ) = = = = (II) Le [X ] be he feasble eo of poblem () whch Q =< - mamal soluo of () wh whch s (,) - mmal soluo of m u P =< ad ] ] [Y be he feasble eo of poblem (2) whch. If fo some (0,), [X ] ad [Y ae oempy, he hee ess X = (,,...,,,,..., ) [ X ] whch s (,) -mamal soluo 2 2 of () wh espec o < ad u = ( u, u,..., u ) [ Y ] whch s (, ) - mmal soluo of 2 m (2) wh espec o, such ha L L m L c c ( ) = ( ) X = ( ) u = = = = CONCLUSION I hs pape, he dual of whe daa ae fuzzy paamees was peseed, whle he cocep of he possbly ad ecessy elaos was used. I couao, wea ad so dualy heoems coespod o hese models wee saed. s we ow, he possbly ad ecessy elaos do o have self-dualy popey. I he fuue, we wll apply cedbly heoy o dual whch has he self-dualy popey. F o obaed s REFERENCES Dubos D ad Pade H (983). Ra fuzzy umbes he se of possbly heoy. Ifomao Sceces Ram J (2006a). Dualy Fuzzy Lea Poamm wh sbly ad essy Relaos. Fuzzy Ses ad Sysems Ram J (2006b). Dualy fuzzy mulple obecve lea poamm. I: Opeaos Reseach Poceeds (Spe el-hedelbe) Sefod L ad Yu PL (979). Poeal soluos of lea sysems: he Mul-cea mulple cosa levels poam. Joual of Mahemacal alyss ad pplcaos Sh Y (200). Mulple Cea Mulple Cosa Levels Lea Poamm: Coceps, Techues ad pplcao (Wold Scefc Publsh) Zadeh L (965). Fuzzy ses. Ifomao Cool Zadeh L (999). Fuzzy ses as a bass fo a heoy of possbly. Fuzzy Ses ad Sysems Copyh 204 Cee fo Ifo o Techoloy (CITech) 454